Faculty of Physics and Astronomy
University of Heidelberg
Diploma thesis
in Physics
submitted by
Anton Piccardo-Selg
born in Stockton, California, USA
2008
Degenerate Quantum Gases:
Towards Bose-Einstein Condensation of Sodium
This diploma thesis has been carried out by Anton Piccardo-Selg at the
Kirchhoff Institut für Physik
under the supervision of
Prof. Dr. M. K. Oberthaler
Auf dem Weg zur Bose-Einstein Kondensation von
Natrium
Gegenstand der vorliegenden Diplomarbeit ist die Inbetriebnahme eines Experiments zur
Erzeugung ultrakalter bosonischer und fermionischer Quantengase. Fermionisches 6 Li
soll mit Hilfe von bosonischem 23 Na sympathetisch gekühlt werden. Es wird die magnetoopische Falle, die Magnetfalle und das evaporative Kühlen von 23 Na untersucht. Eine
anfängliche Thermalisierungszeit von 500 ms wurde in der Magnetfalle gemessen. Mittels
evaporativem Kühlen konnte das runaway Regime erreicht werden. Hierbei wurde die
Phasenraumdichte um zwei Größenordnungen auf 10−4 erhöht. Der Einsatz eines optischen Stöpsels wird die erforderlichen letzen vier Größenordnungen zum Erreichen eines
Bose-Einstein Kondensats ermöglichen.
Towards Bose-Einstein Condensation of Sodium
The topic of this diploma thesis, is the implementation of an experiment for ultracold
bosonic and fermionic quantum gases. Fermionic 6 Li is going to be cooled sympathetically with the help of bosonic 23 Na. The experiment’s magneto-optical trap, magnetic
trap and evaporative cooling are discussed. An initial thermalization time of 500 ms
was measured in the magnetic trap. Using evaporative cooling, the runaway regime was
observed. The phase-space density was raised by two orders of magnitude to 10−4 . An
optical plug will allow for the last four orders of magnitude needed to obtain a BoseEinstein condensate.
I
Contents
1 Introduction
1.1 Degenerate Bose gases . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Degenerate Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The NaLi experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
5
2 The first cooling stages
2.1 Light forces and atomic physics . . . . . . . . . . . . . . .
2.1.1 The scattering force . . . . . . . . . . . . . . . . . .
2.1.2 The dipole force . . . . . . . . . . . . . . . . . . . .
2.1.3 Atoms in static magnetic fields . . . . . . . . . . .
2.1.4 Cold collisions . . . . . . . . . . . . . . . . . . . . .
2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Facts about 23 Na . . . . . . . . . . . . . . . . . . .
2.2.2 Laser setup . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Experimental setup . . . . . . . . . . . . . . . . . .
2.2.4 The Zeeman Slower . . . . . . . . . . . . . . . . . .
2.3 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The magneto-optical trap (MOT) . . . . . . . . . . . . . .
2.4.1 MOT theory . . . . . . . . . . . . . . . . . . . . . .
2.4.2 MOT temperature and Doppler limit . . . . . . . .
2.4.3 Density limitations and loss mechanisms in a MOT
2.4.4 Dark spot MOT . . . . . . . . . . . . . . . . . . . .
2.4.5 Loading of the MOT for different temperatures . .
2.5 Sub-Doppler Cooling . . . . . . . . . . . . . . . . . . . . .
2.5.1 σ + -σ − cooling basics . . . . . . . . . . . . . . . . .
2.5.2 Sub-Doppler cooling in the experiment . . . . . . .
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7
7
7
8
9
10
12
12
14
15
17
17
18
18
20
21
24
24
25
26
28
3 The magnetic trap
3.1 Types of magnetic traps . . . . . . . . . . . . . . . . . .
3.2 Setup of the magnetic trap . . . . . . . . . . . . . . . . .
3.2.1 The Feshbach coils . . . . . . . . . . . . . . . . .
3.2.2 The circuit . . . . . . . . . . . . . . . . . . . . .
3.2.3 The fast turn-off . . . . . . . . . . . . . . . . . .
3.2.4 The field switch . . . . . . . . . . . . . . . . . . .
3.3 Trappable states and the first steps in the magnetic trap
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31
31
33
33
34
37
38
40
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II
Contents
3.3.1 Trappable states of 23 Na . . .
3.3.2 Mode matching . . . . . . . .
3.3.3 Adiabatic compression . . . .
Loss mechanisms and heating . . . .
3.4.1 One-body losses . . . . . . . .
3.4.2 Two-body losses . . . . . . . .
3.4.3 Three-body losses . . . . . . .
3.4.4 Lifetime measurement . . . .
Rethermalization measurement . . .
The plugged quadrupole field . . . .
3.6.1 Plug in the NaLi experiment .
3.6.2 Compensation of astigmatism
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40
40
42
45
45
51
53
53
54
56
56
59
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61
61
66
66
67
5 Conclusion and Outlook
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
71
71
Bibliography
78
3.4
3.5
3.6
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4 Evaporative cooling
4.1 Principle of evaporative cooling . . . . . . . .
4.2 Experimental realization . . . . . . . . . . . .
4.2.1 The microwave setup . . . . . . . . . .
4.2.2 Evaporative cooling in the experiment
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III
List of Figures
1.1
1.2
Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Principle of Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
Breit-Rabi diagram . . . . . . . . . . . . . . . . . . . . . . .
Level scheme of Sodium . . . . . . . . . . . . . . . . . . . .
Setup of the laser table . . . . . . . . . . . . . . . . . . . . .
Setup of the experimental table . . . . . . . . . . . . . . . .
Working principle of the MOT . . . . . . . . . . . . . . . . .
Time-of-flight series of the MOT . . . . . . . . . . . . . . . .
Light-assisted collisions . . . . . . . . . . . . . . . . . . . . .
Loading rate of the Sodium MOT for different temperatures
Clebsch Gordan coefficients . . . . . . . . . . . . . . . . . .
Principle of sub-Doppler cooling . . . . . . . . . . . . . . . .
Measurement of sub-Doppler cooling . . . . . . . . . . . . .
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10
13
14
16
19
21
23
25
26
27
28
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
Magnetic field of the Feshbach coils . . . . . . .
Electric circuit . . . . . . . . . . . . . . . . . .
Current regulator . . . . . . . . . . . . . . . . .
PSpice simulation of the current in the Feshbach
Pre-logic of the IGBT drivers . . . . . . . . . .
Mode matching measurement . . . . . . . . . .
Adiabatic compression of the magnetic trap . .
Majorana losses . . . . . . . . . . . . . . . . . .
Lifetime due to Majorana losses . . . . . . . . .
Lifetime measurements with Majorana losses . .
Fourier spectrum of the current . . . . . . . . .
Lifetime measurement . . . . . . . . . . . . . .
Rethermalization of the magnetic trap . . . . .
Combined magnetic and optical potential . . . .
Plug shot . . . . . . . . . . . . . . . . . . . . .
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34
36
37
38
39
42
44
47
48
49
50
54
55
58
59
4.1
4.2
4.3
4.4
Principle of evaporative cooling . . . . . . . . . . . . . . .
Patch antenna and the S11 parameter . . . . . . . . . . . .
Phase-space density and atom number during evaporation
Phase-space density vs. atom number . . . . . . . . . . . .
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62
67
68
69
. . .
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coils
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2
5
IV
Fundamental Constants
Quantity
Symbol
Value
Unit
Speed of light
c
2.99792458 · 10
ms−1
Boltzmann constant
kB
1.3806503(24) · 10−23
JK−1
Planck constant
h
6.62606876(52) · 10−34
Js
8
−34
h/2π
1.054571596(82) · 10
Js
µB
9.27400949(80) · 10−24
JT−1
Mass
m
3.81754023(30) · 10−26
kg
Saturation intensity
I0
6.2600(51)
mWcm−2
D2-line width
Γ
2π × 9.795(11)
MHz
Doppler temperature
TD
235
µK
Recoil temperature
TRec
2.3998
µK
Bohr magneton
Specific data of Sodium-23
1
Chapter 1
Introduction
In the 5th century B.C. Democritus developed his theory of the “atomos” – the indivisible
entity that builds up the world in limitless varieties of shapes and sizes. Other cultures
had developed this theory even earlier. Modern occidental atomic theory began in the
19th century with John Dalton. His ideas were developed further by Thomas, Rutherford
and many others. With the development of quantum mechanics, it was finally possible
to describe processes in atoms that were previously not intuitively accessible.
Elementary and composed particles are categorized into bosons and fermions, having
symmetric and antisymmetric wave functions, respectively. They are characterized by
their integer and half-integer spin properties. This little difference has great effects on
the behaviour of ensembles of such particles. It is of quantum mechanical nature and
in the classical limit not apparent. For extreme situations as in ultracold gases, where
the particles’ wavepackets start to overlap, this categorization is crucial. This is shown
in figure (1.1). The phase-space density of the system determines whether a gas can be
described in terms of classical physics or has to be treated quantum mechanically.
If the phase-space density of the system is on the order of one, identical bosons
undergo a phase transition called Bose-Einstein-Condensation (BEC) after Satyendra
Nath Bose and Albert Einstein, who first discovered the underlying quantum statistics.
The Bose-Einstein Condensate is described as a macroscopic occupation of the system’s
ground state.
After the publications of Bose [1] and Einstein [2], it would take about 70 years to
realize such a Bose-Einstein condensate experimentally. In 1995 the group of Eric Cornell
and Carl Wiemann at the JILA [3], the group of Wolfgang Ketterle at the MIT [4] and
the group of Randy Hulet [5] at the RICE University accomplished this goal. The alkali
metals 87 Rb, 23 Na and 7 Li were used back then.
Identical fermions, on the other hand, do not show this behaviour. They are governed
by the Pauli-principle that prohibits the occupation of the same quantum state for two
identical fermions. Therefore the condensation process cannot happen. Rather, they
start to fill up the different states, starting from the ground state. This leads to the socalled Fermi sea. The gas does not pass through a phase transition. A degenerate Fermi
gas is obtained when the phase-space density is on the order of one1 . The concept of
the Fermi sea is, for example, used in nuclear physics, the physics of metals and neutron
stars.
1
Degeneracy, in the case of fermions, means that the energy states are filled up with atoms from the
ground state on.
2
1.1 Degenerate Bose gases
Some years after the first production of Bose-Einstein Condensates, there was great
effort to produce degenerate Fermi gases. First success was reported in 1999 [6] using
40
K. Other groups followed using 6 Li [7].
d)
a)
b)
c)
300K
0K
temperature
Figure 1.1: a) Atoms at room temperature behave classically and can be described as “pointlike” particles. b) With decreasing temperature and increasing density the wave nature
of the atoms cannot be disregarded. Collective effects come into play. c) In the case
of fermions the Pauli-principle holds and they fill up the trap energy levels from the
lowest state on. d) At ultra-low temperatures bosons occupy the lowest energy state of
the trap.
1.1
Degenerate Bose gases
In the framework of quantum statistical mechanics the distribution function for Bose
particles is given by
1
n(Eν ) =
.
(1.1)
exp [(Eν − µ)/kB T ] − 1
µ is the chemical potential of the system. It describes the energy needed if a particle
is added to the ensemble. Eν denotes the energy of the particle in state ν. It can be
easily seen that the chemical potential must be less than the energy E0 of the lowest
state. Else unphysical, negative occupation numbers would arise. When µ approches E0
for small decreasing temperatures, the occupation of the ground state enhances. The
critical temperature
2/3
n
h2
Tc =
(1.2)
ζ(3/2)
2πmkB
determines when condensation takes place. Here ζ(3/2) denotes the Riemann zeta function, n the density and m the mass. This equation holds for non-interacting, ideal gases.
The description of a non-ideal Bose gas is immensely more difficult. Mean-field
theories have been developed to take interactions into account. Commonly the model of
1.2 Degenerate Fermi gases
3
Gross [8] and Pitaevskii [9] is used. It holds for T = 0, large atom numbers and weak
interactions. The time evolution is given by the Gross-Pitaevskii equation (GPE)
d
~2 2
2
(1.3)
i~ Ψ(~r, t) = −
∇ + V (~r) + U0 |Ψ(~r, t)| Ψ(~r, t)
dt
2m
with U0 = 4π~2 a0 /m, where the s-wave scattering length a0 gives the scale of the interaction. The last term is the contact potential and describes the short-ranged, weak
interactions of the degenerate Bose gas. Ψ(r, t) is not a single particle wave function but
rather the condensate wave function. The particle density is given by n(~r, t) = |Ψ(~r, t)|2 .
The stationary equation is
~2 2
2
(1.4)
−
∇ + V (~r) + U0 |Ψ(~r)| Ψ(~r) = µΨ(~r)
2m
where µ is the chemical potential. An interesting limit of the GPE is the ThomasFermi approximation. This limit is obtained when the kinetic energy of the particles is
negligible compared to the interaction energy of the ensemble. The kinetic energy term
in the GPE is canceled, resulting in
n(~r) =
1
[µ − V (~r)]
U0
(1.5)
for µ > V (~r) and n(~r) = 0 otherwise. It can be seen that the density distribution of
the BEC is just given by the shape of the confining potential. In a physical perspective
the Thomas-Fermi approximation means that the chemical potential is assumed to be
uniform over the whole sample, such that adding a particle costs the same everywhere
in the trap.
1.2
Degenerate Fermi gases
The distribution function for fermions is given by
n(Eν ) =
1
.
exp [(Eν − µ)/kB T ] + 1
(1.6)
where Eν is defined as above. For T = 0 the distribution function is a Heaviside stepfunction that is unity for Eν < µ and vanishes otherwise. The energy at the edge is
called Fermi energy EF and is equal to the chemical potential at zero temperature. With
this energy the Fermi temperature TF = EF /kB can be defined which gives a measure
of when a fermionic system can be called degenerate. Values for this temperature vary
dramatically in different fields of physics. As described in many textbooks [10, 11], the
Fermi energy of atoms in the homogenous case of a box is
EF =
(6π 2 )2/3 ~2 2/3
n .
2
m
(1.7)
To obtain the density distribution in a trap, the Thomas-Fermi approximation is applied
again. The cost for adding a particle is assumed to be the same all over the trap. Local
4
1.2 Degenerate Fermi gases
Fermi energies EF (~r) can be defined. Setting the sum of the local Fermi energy and the
trapping potential equal to the chemical potential [12],
3/2
1 2m
n(~r) = 2
[µ − V (~r)]
6π
~2
(1.8)
is obtained. The density distribution is different from equation (1.5) and shows an
enhanced profile in the trap center.
The production of a degenerate Fermi gas is much more complicated than the production of a Bose-Einstein Condensate, since collision processes in the ultracold regime
are suppressed between fermionic same-state particles. Therefore several sophisticated
methods have been developed to overcome this difficulty. Jin et al. used several spin
components of 40 K atoms to reach a degeneracy of T /TF = 0.5 [6]. A more recent experiment of Ketterle et al. showed a degeneray of T /TF = 0.09 [13]. Here the cooling scheme
made use of cold bosonic 23 Na atoms. The 23 Na atoms can be cooled efficiently with
evaporative cooling which will be explained in chapter 4. 6 Li atoms are cooled indirectly
via collisions with the cold 23 Na sample. The two-species gas reaches a temperature that
is way below the initial temperature of the lithium sample. The 23 Na atoms are used as
sort of refrigiator. This cooling method is known as sympathetic cooling.
Feshbach resonances
In recent years a main focus has been put on the studies of interactions between particles.
The occurance of Feshbach resonances allows one to tune the s-wave scattering length
and hence gives a method to control the interaction strength of fermion-, fermion-bosonor boson-pairs.
A Feshbach resonance occurs when the kinetic energy of two scattering particles
entering the open channel is close to the total energy of a bound interatomic state in
the closed channel as depicted in figure (1.2). The channel is called closed if the energy
of the particles is below the dissociation threshold. This resonance behaviour can be
realized by tuning an external magnetic field. Different states have different magnetic
moments, hence the field shifts these levels differently. The two channels can be tuned
with respect to each other to have the same energy. The dependence of the scattering
length on the external magnetic field is given by
a(B) = a0 1 −
∆B
B − B0
(1.9)
[14] for small scattering energies, where ∆B is the width of the resonance, B0 the field at
which the resonance occurs and a0 the non-resonant s-wave scattering length. Feshbach
resonances have been found in many alkalis such as 23 Na [15], 6 Li [16] and in interspecies
mixtures [17]. Some resonances are shown in table (1.1). There are many more resonances than the ones listed and research is being done to find new ones at higher fields.
The most important one for 6 Li is the 834 G resonance as it is very broad and therefore
tuning through it can be done easily.
b)
scattering length
5
energy
1.3 The NaLi experiment
repulsive
interactions
c)
d)
a)
0
attractive
interactions
position
magnetic field B
Figure 1.2: left: a) and b) are an open and a closed channel of the interaction potential,
respectively. The particle arriving in the open channel experiences at c) a coupling
between the two channels and can make a transition to the closed channel. This happens
if the particle’s total energy is close to the energy of a bound state in the closed channel.
right: The dependence of the s-wave scattering length on the external magnetic
field in the vincity of a Feshbach resonance is shown. The scattering length diverges
at the resonance. In the case depicted the scattering length changes sign when passing
through the Feshbach resonance. Repulsive interactions are present if the sacttering
length is positive, whereas for negative scattering lengths the interaction is attractive.
Table 1.1: Feshbach resonances of 6 Li and
1.3
23 Na.
The data is from [15, 16, 17].
Species
B0 [G]
∆B[G]
Species
B0 [G]
∆B[G]
6
Li-6 Li
159.14
0.40
6
Li-23 Na
746.0
-
6
6
Li- Li
0.40
6
23
Li- Na
759.6
-
6
6
Li- Li
214.28
0.40
6
23
795.6
-
6
Li-6 Li
543.28
0.40
23
Na-23 Na
907
-
6
6
23
23
855
-
Li- Li
185.09
822 . . . 834
Li- Na
Na- Na
The NaLi experiment
The Sodium-Lithium experiment (NaLi) aims to produce ultracold Fermi gases. Large
samples of fermionic 6 Li are intended to be cooled far into degeneracy. The attained
degeneracies mentioned before certainly determine a goal for us. Models describing
6
1.3 The NaLi experiment
interactions beyond the established mean-field theories need to be investigated. Locally,
the theory department of the university of Heidelberg does research on renormalizationgroups in quantum field theory [18, 19].
But not only pure Fermi gases are of interest in this experiment. Having bosons in
the vacuum chamber makes the study of interspecies effects appealing. With the help of
interspecies Feshbach resonances it is possible to study boson-mediated interactions in a
gas of dilute fermions. Fermions propagating through the bath of bosons are a subject of
interest here. Other fields of interest are gases in lower dimensional traps and possibly
optical lattices. But these projects still lie far in the future.
So far the first steps have been taken. The experimental chamber has been set up
as well as the main laser system. Several more lasers are about to be installed in the
coming months. The cooling steps have been implemented and our first milestone – a
23
Na BEC – seems to be in reach.
In this thesis I will report on the progress made since the experimental framework
of NaLi has been set up. Several cooling steps were introduced during my stay. In
particular I will describe the control of the electrical circuit, the change of configuration
of the current and the fast turn-off of the magnetic trap. The thesis is not structured
as often done in theory and experiment, but rather follows chronologically the different
steps of the experiment.
7
Chapter 2
The first cooling stages
The first part of the atoms’ path to ultracold temperatures is governed by laser cooling.
It seems counterintuitive to cool with light, since light is normally associated with heat.
Still, by almost no other means than laser cooling in combination with evaporative
cooling (see chapter 4) it has been possible to reach temperatures as low as the µK and
nK regime. This makes the lab (on a small scale) one of the coldest places in the entire
universe (to the best of our knowledge).1
2.1
Light forces and atomic physics
The atoms of interest for most groups in the field of ultracold atoms are alkalis, foremost
due to their hydrogen-like electronic structure. These atoms have a pseudo-closed twolevel system which makes the theoretical description of the processes in the experiment
significantly simpler.
The equations governing the temporal evolution of the density matrix ρ of the system
are the optical Bloch equations that can be derived from the von-Neuman equation
i~
dρ
= [H, ρ].
dt
(2.1)
The effect of spontaneous emmision is introduced by hand [20]. H is the system’s Hamiltonian. The Hamiltonian H can be related with help of the Ehrenfest theorem to the
force acting on the atom
D
E
~
F~ = − ∇H
.
(2.2)
~
The expectation value in equation (2.2) is evaluated by taking the trace of ρ∇H.
This
results for the steady-state in two light forces. First, the scattering force which is of
dissipative nature and is used for cooling; second, the conservative dipole force used for
trapping and manipulating atoms.
2.1.1
The scattering force
The scattering force is the underlying force of the magneto-optical trap (MOT) and
the Zeeman slower. It results from the scattering properties of the atom with light.
1
By means of adiabatic demagnetization and the dilution refrigiator comparable temperatures can
be achieved.
8
2.1 Light forces and atomic physics
The atoms can absorb light that is at or close to the transition frequency ω0 . The frequency range around ω0 , at which the light can have an effect on the atom is determined
by the linewidth Γ, ω0 itself as well as the frequency and intensity of the light. The
absorbed photon does not only excite the electron but also transfers the momentum
~~k = (~ω0 /c)~eprop to the atom where ~k is the wave vector of the photon propagating in
the direction of ~eprop . The excited atom decays to the ground state and emits a photon,
causing again a change of momentum. The force depends on how often a photon scatters
from the atom. The scattering force results therefore in
Fscatt = (momentum of a photon) × (scattering rate).
(2.3)
The scattering rate is [20]
Rscatt =
I/I0
Γ
.
2 1 + I/I0 + 4∆2 /Γ2
(2.4)
∆ = ω − ω0 is the detuning of the laser light ω with respect to the transition frequency.
In the case of moving atoms, the Doppler shift has to be taken into account as is done
in subsection 2.4.1. I0 = (hcπΓ)/(3λ3 ) is the saturation intensity of the transition and
is determined by the life time of the excited state. λ is the wavelength corresponding to
the transition frequency ω0 . At I0 one-quarter of the atomic ensemble is in the excited
state if there is no detuning of the light. At very high intensities the scattering rate
does not increase further but causes undesired effects like saturation broadening. The
Lorentzian shape of the scattering rate is just a consequence of the lineshape of the
atomic transition. Finally, the scattering force can be written as
I/I0
Γ
F~ = ~~k
2 1 + I/I0 + 4∆2 /Γ2
(2.5)
with a maximal force of |F~max | = |~~k Γ2 | when the light is in resonance and I/I0 1.
2.1.2
The dipole force
As mentioned before, the dipole force is not dissipative and hence cannot be used for
cooling atoms. It does not rely on spontaneous emission, which causes the irreversibilty
in the case of the scattering force. The force is given by [20]
~∆
Ω
~
F~DIP = −
∇Ω.
2 ∆2 + Ω2 /2 + Γ2 /4
(2.6)
In contrast to the scattering force the dipole force vanishes for resonant light. The
on-resonance Rabi frequency Ω = −(eE0 /~) he| ~r~eprop |gi depends on the electric field
amplitude E0 of the light. e is the electron charge. It is a measure for the electric
dipole transition strength between the ground state |gi and the excited state |ei. The
dipole force is used in experiments with far-off resonant traps. It is useful to deduce the
approximated potential (Γ |∆|, Ω |∆|)
UDIP =
~Γ Γ I
8 ∆ I0
(2.7)
2.1 Light forces and atomic physics
9
for this far-off resonance region. The scattering rate drops as Rscatt ∝ I/∆2 , therefore
the dipole force ceases to be slower than the scattering rate. The force depends on the
sign of the detuning. For blue-detuned light (∆ > 0) the force is repulsive and the atom
will be driven to minima of the light field, whereas in the case of red detuning (∆ < 0)
the atom experiences an attractive force to regions where the light field maximum lies.
This force is commonly used to create optical trapping potentials. In future experiments NaLi will make use of these optical dipole traps. Two different traps will be
installed. A high power laser at 1064 nm and a dye laser, which is only several nm away
from a transition in 6 Li, will be used. The first laser creates an equal potential for both
species, whereas the latter can produce a potential that is significantly stronger for 6 Li
than for 23 Na.
2.1.3
Atoms in static magnetic fields
If atoms are exposed to an external magnetic field, the two will couple. The linear part
of the interaction-Hamiltonian is given by
HJB = −
µB ~ ~
gJ J Bext
~
(2.8)
~ +S
~ of the electron orbiting around the nucleus
The total angular momentum J~ = L
~ ext with µB being Bohr’s magneton. It is
is coupled to the external magnetic field B
~ and the spin S
~ of the electron. The Landé
composed of the angular momentum L
g-factor is given by
gJ = 1 +
J(J + 1) + S(S + 1) − L(L + 1)
.
2J(J + 1)
(2.9)
For most field strengths equation (2.8) is sufficient, but for high magnetic fields also
the quadratic Zeeman effect becomes important. Another relevant part in the total
~ S-coupling
~
Hamiltonian is the L
term resulting from the coupling between the electron’s
spin and the magnetic field produced by the nucleus being a moving charge in the
momentary rest-frame of the electron. The Zeeman effect exists for external magnetic
fields if
~ ext µn gI I~B
~ int
µB gJ J~B
(2.10)
~ int is the internal magnetic field at the nucleus caused by the orbiting
is fulfilled. B
electron. The nuclear spin I~ produced by the nucleons couples to the internal magnetic
field. Analogous to the electron, µn is the nuclear magneton and gI the nuclear g-factor.
If equation (2.10) is fulfilled, then the coupling between I~ and J~ is stronger than the
coupling of each single one of them to the external magnetic field. The total atomic
angular momentum can then be defined as
~
F~ = I~ + J.
(2.11)
The projection states mF of the total atomic angular momentum determine the Zeeman
energy shift
~ ext |.
∆EZeeman = µB gF mF |B
(2.12)
10
2.1 Light forces and atomic physics
in the external magnetic field. If the external field is very strong, not fulfilling equation
(2.10), then I~ and J~ couple separately to the external magnetic field. The energy splitting
in this Paschen-Back regime is
~ ext | + AmI mJ
∆EPaschen-Back = µB gJ mJ |B
(2.13)
with
A=−
~ int |
µI µJ |B
.
IJ
(2.14)
a)
mj=+1/2
F=2
F=1
mj=-1/2
~300
B[G]
energy[a.u.]
energy[a.u.]
~ int | is the in avarage internal magnetic field.
|B
In the intermediate regime the two different forms of coupling need to be treated as
equal. This is described by the Breit-Rabi formula which can be found in textbooks [21].
The result for 23 Na and 6 Li is shown in figure (2.1). Some states bend with increasing
magnetic field strength which is important to keep in mind when thinking about magnetic
trapping.
b)
mj=+1/2
F=3/2
F=1/2
~27
mj=-1/2
B[G]
Figure 2.1: a) shows the Breit-Rabi diagram for sodium.
The bending of the
|F = 1, mF = −1i state occurs at about 300 G which is not relevant for our means. b)
shows the Breit-Rabi diagram of lithium. Here the bending of |F = 1/2, mF = −1/2i
occurs at about 27 G, making this state not efficiently trappable.
When setting up a magnetic trap it is essential to know in which regime the atoms
are captured. In the magnetic trap a gradient of up to 660 Gcm−1 can be achieved.2 If
the cloud of 23 Na atoms is assumed to have a diameter of less than 3 mm, then the field
in which they are captured is at about 100 G, which is safely in the Zeeman regime. 6 Li
atoms, on the other hand, exhibit a change of regimes already at low magnetic fields.
2.1.4
Cold collisions
Cold and ultracold atomic collisions play a crucial role in reaching a BEC. Inelastic,
exoergic collisions in a light field or in the magnetic trap can lead to trap loss. Elastic collisions are an essential part of evaporative cooling, the final cooling stage before
achieving a BEC.
2
The gradient corresponds to the strong axial direction of our quadrupole field. The transversal
gradient is just half the value of the axial gradient.
2.1 Light forces and atomic physics
11
Two atoms passing each other experience an interaction potential that effects their
wave functions. The total wave function of these two atoms is given by the incoming
plane wave and a scattered spherical wave, being modulated by the scattering amplitude
f (E, θ) for a spherical symmetric potential. The scattering amplitude depends on the
relative energy E and the angle of deflection θ with respect to the original relative motion.
The total wave function becomes
exp(ikr)
~
f (E, θ) h~r|ii h~r|ji
(2.15)
Ψ(~r) ∝ exp(i~r · kin ) +
r
with the internal states |ii and |ji of the scattering atoms. This describes elastic scattering where internal changes do not happen and the relative kinetic energy of the two
atoms stays constant. A common method for treating scattering problems is the partial
wave expansion. The incoming plane wave is expanded in terms of spherical waves. The
resulting wave function exhibits for r → ∞ a plane wave behaviour with a phase shift
induced by the interaction potential. An important result is the total cross-section σ(E)
that can be linked to the phase shifts δl induced by each partial wave
σ(E) =
∞
4π X
(2l + 1) sin2 δl .
k 2 l=0
(2.16)
Due to the spherical symmetry of the interaction potential, the angular part of the
wave function is given by spherical harmonics. Therefore, the index of summation in
equation (2.16) corresponds to a relative angular momentum of the two atoms. The
entire information of the potential lies in the phase shifts. The description is general
so far and the advantage for cold collisions arises from the fact that for these collisions
the sum has a cut-off. In the most extreme case only the l=0 component of the sum,
the so-called s-wave, has to be considered. The cut-off is due to the centrifugal barrier
~2 l(l + 1)/(2µr2 ) that arises from the Hamiltonian of this problem. The relative mass µ
of the two colliding atoms is in our case µ = mN a /2. For large relative angular momenta
the centrifugal barrier rises and the classical turning point is shifted far away from the
collision center. The potentials of relevance are often of the form C6 /R6 , created by the
3S − 3S state. This potential vanishes rapidly and therefore higher angular momenta do
not experience the potential. The lower the relative kinetic energy is, the fewer partial
waves take part in the scattering process.
For an entire treatment of the scattering process, the internal degrees of freedom
need to be included as well. The interaction potential depends on the set of quantum
numbers of the colliding atoms. Further, the symmetrization of the two bosons needs to
be taken into account [22].
The s-wave domain
For low relative kinetic energies (k → 0) a scaling law for the phase shift with respect
to the relative kinetic energy can be found [23]. This depends on the power n of the
potential Cn /Rn . For partial waves which obey 2l < (n − 3),
lim k 2l+1 cot δl =
k→0
1
al
(2.17)
12
2.2 Setup
holds. al is the scattering length of the lth partial wave. For the other partial waves the
relation is replaced by
lim k n−2 cot δl = const.
(2.18)
k→0
These partial waves vanish rapidly for the potentials of interest. For 23 Na bosons with
the same quantum number, as in the case in our magnetic trap (|F = 1, mF = −1i),
only even partial waves contribute to the scattering process3 . Therefore it follows from
equations (2.17) and (2.18) that only the s- wave is relevant for the captured 23 Na.
The s-wave scattering length has been measured in [24]. Table (2.1) shows the s-wave
scattering lengths of 23 Na for different internal states of the two colliding atoms. The
scattering lengths are given in units of Bohr radii aBohr .
Table 2.1: Scattering lengths for
23 Na-23 Na.
maF
mbF
Scattering length [aBohr ]
1(−1)
1(-1)
52.98
1(-1)
0(0)
52.98
1
-1
49.23
0
0
51.12
2(-2)
2(-2)
62.51
For k → 0 and with equation (2.16) the elastic cross-section can be approximated by
σ = 8πa20 .
(2.19)
There was a factor of two inserted here, as the treatment above does not account for
the atoms’ boson nature. The s-wave scattering length determines the rethermalization
behaviour of an ultracold gas. By these means, the scattering length in ultracold gases
can be measured.
2.2
Setup
When an experiment on ultracold atoms is to be designed, many things need to be
considered. The choice of atom determines the laser system. The cooling strategies have
to be adapted to the particular species.
2.2.1
Facts about
23
Na
23
Na is an essential part of human life. An adult human being consists of approximately
0.1% sodium, corresponding to 80 g (enough to run the experiment for almost a year).
3
Odd spherical harmonics have negative parity and it is not possible to construct fully symmetric
wave functions under particle exchange.
2.2 Setup
13
But why is 23 Na interesting for our experiment? 23 Na is not the main focus of investigation in NaLi. 6 Li will be cooled to degeneracy with the sympathetic cooling scheme
mentioned before. For this purpose we need large numbers of atoms with similar particle
mass of the colliding species. This ensures an efficient energy transfer between the 23 Na
sample and the 6 Li sample during sympathetic cooling. Since 6 Li is followed by 23 Na in
the alkali group of the periodic table, the prerequisite of similar mass is fulfilled quite
well. Further, it is possible to produce 23 Na BECs with atom numbers exceeding 107
[25].
The level structure of 23 Na consists of the D1 and D2 lines at 589.756 nm and
589.158 nm, respectively, having almost the same linewidths of approximately 2π ×
10 MHz. The wavelength is not easily accessible with common laser technologies and
to date there are only dye lasers commercially available which can be tuned to these
wavelengths. Unfortunately, dye lasers are very expensive and quite difficult to handle.
They need to be adjusted on a daily basis, which is not the case with diode lasers used
in lithium and rubidium experiments.
32P3/2
~14.5 MHz
15.944 MHz 58.326 MHz
34.244 MHz
F’=2 (gF’=2/3)
~92 MHz
F’=1 (gF’=2/3)
15.809 MHz
F’=0 (gF’=2/3)
~350 MHz
589.158 326 4 nm
508.848 716 3 THz
F’=3 (gF’=2/3)
a)
b)
c)
F=2 (gF=1/2)
32S1/2
0.664359 GHz
1.771626GHz
F=1 (gF=-1/2)
Figure 2.2: Level scheme of sodium: The laser frequencies depicted will be further explained
in subsection 2.2.2. a) is the repumper light used in MOT and molasses stages. b) is
the Zeeman slower light that is necessary for the very first cooling stage. c) is the MOT
light.
It is favorable to have a closed two-level system to operate a MOT. In the case of
Na the transition between F = 2 and F 0 = 3 is used. There was also success in the
production of a MOT using the transition between F = 1 and F 0 = 0 [26, 27]. The MOT
will be discussed in more detail in subsection 2.4.1. The level scheme of 23 Na and the
transitions we use in the experiment are depicted in figure (2.2).
23
14
2.2 Setup
2.2.2
Laser setup
The light needed for our experiment is provided by two dye lasers using Rhodamin 6-G
for 23 Na and DCM 4 for 6 Li. The dye lasers are pumped by an Yb:YAG5 laser producing
2 × 10 W. To date, only the light for 23 Na has been used. The dye laser’s output can
range between 1 W and 650 mW depending on how good the laser is adjusted and how
“fresh” the dye is. Details on the lasers can be found in [28]. The light is split into several
paths and manipulated with acousto-optical modulators (AOMs) and an electro-optical
modulator (EOM). Figure (2.3) shows a schematic diagram of the optical setup. There
are several separate sections that are described briefly in the following.
Fiber
Polarizing Mirror λ/2
Coupler Beamsplitter
λ/4
Lens
Glass Photoplate diode
Repumper for
dark spot MOT
MOT
Plug
Repumper
Slower
Imaging
Dye laser
AOM
AOM
AOM
1.72GHz
Yb:YAG
AOM
AOM
spectroscopy
cell
to laser-lock system
Figure 2.3: The fiber couplers correspond to the couplers in figure (2.4). The black dot on the
side of the AOMs depicts the location of their transducers. We have the possibility to
couple repumper light into the MOT fiber hence producing a bright MOT, or sending
everything through the dark spot repumper fiber.
• The laser system is locked onto the crossover peak between F 0 = 2 and F 0 = 3.
We use a 23 Na heat-pipe which is heated to 160◦ C. The crossover peak is located
29 MHz below the cycle transition. The spectroscopy signal is detected by a regular
photodiode and sent to the laser-lock system. The spectroscopic method is based
upon frequency modulation spectroscopy [29].
• The MOT beams need to be red-detuned with respect to the cycle transition as
will be shown later. For other atoms like rubidium the detuning of the MOT is
several linewidths. In our case, this is not possible since the hyperfine splitting
4
5
4-dicyanomethylene-2-methyl-6-p-dimethylaminostyryl-4H-pyran
The active medium is an ytterbium(Yb)-doped yttrium aluminum garnet (YAG) crystal.
2.2 Setup
15
between F 0 = 2 and F 0 = 3 is only 58 MHz and the linewidth is Γ/(2π) = 10 MHz.
Hence the influence of the MOT beam on the F 0 = 2 state would be too strong.
If the atom is in the excited F 0 = 3 state, it can only decay to the F = 2 state
(considering only electric dipole transitions). The frequency detuning of the MOT
beams with respect to the cycle transition is currently set at −14.5 MHz.
• As the excited hyperfine states F 0 = 2 and F 0 = 3 are close to each other and the
linewidth is large compared to the splitting, there can be non-resonant excitations
from the F = 2 state to the F 0 = 2 state. Once in this state, there can be a decay
to the F = 1 hyperfine ground state, producing a dark state. This is the reason a
repumper beam is needed. It drives ground state atoms back to the F 0 = 2/F 0 = 1
state6 , where they can decay to the F = 2 ground state and take part in the
cycle process again. As indicated in figure (2.3) there is the possibility to couple
repumper light into the MOT fiber or into the dark spot MOT fibers when the
experiment is run with a dark spot MOT (see subsection 2.4.4).
• The initial slowing process is done with a Zeeman-Slower. For this apparatus we
need a beam that is red-detuned by several hundert MHz with respect to the cycle
transition. The light is matched to the design of the Zeeman slower. An EOM
modulates repumper sidebands onto the Zeeman slower light. The EOM is located
on the experimental table depicted in figure (2.4).
• Light which is resonant on the cycle transition is used for imaging. When taking
an image the repumper is turned on as well. The atoms are excited several times
during one shot, hence a better signal can be achieved.
• Finally, we have a plug beam for the magnetic trap. This beam has a wavelength of
515 nm and is therefore blue-detuned. The light is taken from the Yb:YAG pump
laser. Due Majorana losses at low magnetic fields in magnetic traps this beam is
essential when producing ultracold samples. The dipole force of the beam pushes
the atoms out of the region of low magnetic fields.
The laser light is prepared on a separate laser table and sent via single-mode fibers to
the experimental table.
2.2.3
Experimental setup
The experiment begins with the atoms being in a steel oven which is heated up to 380◦ C
for 23 Na and about 400◦ C for 6 Li. The situation is depicted in figure (2.4). The mixing
nozzle in between the ovens needs to be heated sufficiently to avoid the deposit of 23 Na.
It is constructed at an angle to enable condensed 23 Na to flow back into its reservoir.
The sodium oven is filled with approximately 25 g of 23 Na which lasts less than half a
year at these temperatures. At the end of this two-species oven, there is an oven nozzle
which creates together with an aperture an atomic beam. In this first section, there is an
ion-getter pump able to pump 55 ls−1 , creating a pressure of nearly 10−8 mbar. Details
on the ovens and the vacuum setup can be found in [30].
Details on why the frequency of the repumper is set between F 0 = 1 and F 0 = 2 and how both
states are used for repumping can be found in [28]
6
16
2.2 Setup
Sodium
Oven
Lithium
Oven
mixing
nozzle
IGP
Valves
Zeeman
Slower
coils
IGP
TSP
Glass window
coils for
magnetic
field
EOM
Imaging MOT Dark-Spot Slower
Plug
CCD-Camera
Figure 2.4: Top view of the experimental table: The beams in the vertical direction are not
sketched here. The different light sources are depicted by the fiber couplers. TSP
denotes the titan-sublimation pump and IGP the ion-getter pumps.
After this section, there is another aperture for beam collimation followed by two differential pumping stages that consist of tubes. They are located between two valves that
can seal ultra-high vacuum to atmospheric pressure and are needed when the samples
in the ovens are replaced. To improve the differential pumping, there is an ion-getter
pump located between the two tubes.
The ultra-high vacuum, starting after the second valve, has a pressure of less than
10−11 mbar. The first cooling stage, the Zeeman slower, has a length of approximately
75 cm. It is a so-called spin-flip Zeeman slower. At the end of the Zeeman slower is
the glass cell where the experiment takes place. The glass cell consists of quarz glass
with a refractive index of n = 1.4585 at a wavelength of λ = 587.6 nm and a Brewster
angle of θB = 55.58◦ . The advantage of this cell is the good optical access in the table
plane. Furthermore, no eddy currents can arise in a glass cell. This is a problem with
octagon-chambers which are made of glass and metal.
Finally, at the end of the vacuum chamber, there is a window where the laser light
for the Zeeman slower enters. For the ultra-high vacuum there is a combination of an
ion-getter pump and a titan-sublimation pump which is able to pump 200 ls−1 . The
window needs to be heated up to 200◦ C to avoid deposit of lithium that could damage
or blur it.
Up to now there are ten beams on the table and the lithium part has not been set
up yet. Two optical dipole traps are yet to come and possibly there will be an extra
imaging direction. This would easily double the amount of beams on the table.
2.3 Imaging
2.2.4
17
The Zeeman Slower
The very first cooling stage of the experiment is the Zeeman slower. Via a magnetic field
the Zeeman sublevels of the incoming atoms are always kept in resonance with the laser
light of the Zeeman slower beam. This works for a wide range of velocities. The main
goal is to compensate the Doppler shift of the moving atoms by the Zeeman shift due
to the magnetic field. For a uniform deceleration and red-detuned light the necessary
magnetic field is [20]
1/2
z
B(z) = B0 1 −
+ BBias
(2.20)
z0
where z0 is the length of the path through the magnetic field. B0 needs to be adapted
to the maximal incoming velocity and BBias can be used to build a Zeeman Slower in
different configurations. We use the spin-flip configuration that has several advantages
over other existing configurations as is explained in [31]. Our spin-flip slower has been
designed to decelerate two species. The average incoming velocity of the 23 Na atoms
is about 800 ms−1 and the slowed beam has an average velocity of 30 ms−1 , enough to
be trapped in our MOT. A disadvantage of the spin-flip configuration is the zero-field
crossing. This causes a flip of the quantization axis. We use an EOM at 1.771 GHz to
pump the atoms again in the right state when they cross the magnetic field zero. The
EOM produces first-order sidebands at about 30% of the carrier wave amplitude.
2.3
Imaging
To obtain knowledge about the captured and cooled atoms, images need to be taken
or fluorescent light has to be collected. There are several standard ways to do this,
e.g. through fluorescence, absorption and phase contrast imaging. Just recently a group
imaged a BEC with the help of a scanning electron microscope [32]. In our case we use
absorption imaging and some measurements are done by collecting the fluorescence light
with a photodiode.
Principally, absorption imaging is counting the photons that never made it to the
camera. All that is needed is a weak collimated laser beam which is resonant for the F =
2 to F 0 = 3 transition. A beam with circular polarization is used and the quantization
axis is provided by turning on a bias magnetic field along the direction of our imaging
system. The imaging pulses have lengths of about 30µ s at imaging intensities that are
around 0.1 mWcm−2 . The beam is sent through an optical system onto a CCD camera.
Each picture consists of three shots with a certain intensity information. The intensities
of shots with atoms Iatoms (y, z), without atoms Iref (y, z) and finally without the laser
beam turned on Iback (y, z) are collected. The background shot is subtracted from the
actual shot and the reference shot to get rid of stray light that might permanently exist.
The optical density of the sample is then
Iatoms (y, z) − Iback (y, z)
.
(2.21)
OD(y, z) = − ln
Iref (y, z) − Iback (y, z)
The change of the laser beam’s intensity through the sample
dI = −hνRscatt ρ(x, y, z)dx
(2.22)
18
2.4 The magneto-optical trap (MOT)
depends on the scattering rate Rscatt and the density ρ(x, y, z) of the sample. As the
imaging intensities used are small, the optical density can be brought into direct relation
with the column density of the sample. This is just the integrated density along the axis
of the laser beam. For this limit the column density [33]
"
2 #
2∆
2I0
1+
× OD(y, z)
(2.23)
n(y, z) = −
Γ~ω
Γ
is directly proportional to the optical density. Equation (2.23) is a form of the LambertBeer law [20]. Taking into account the symmetries of the trap, it is straightforward
to calculate the total atom number from this information. The equations need to be
modified if there is a magnification through the lens system. The relations above hold
for every pixel of the CCD camera chip. Therefore, these images provide information
about the density, the size and the total atom number of the sample.
Taking time-of-flight series [34] gives an insight on the temperature of the trapped
sample. The atoms get released and are imaged at different times. The temporal evolution of the width
r
kB T 2
t
(2.24)
σ(t) = σ0 +
m
of the sample shows for large times a linear behaviour. σ0 is the initial width at t = 0.
2.4
The magneto-optical trap (MOT)
The first success using a MOT was reported in 1987 by Pritchard et al. [35]. Back then
107 atoms were trapped at a temperature of 600 µK with a peak density of approximately
1011 cm−3 . Since then several variations of this technique have been developed. For 23 Na
the so-called dark spot MOT is often used [36].
2.4.1
MOT theory
Before the first MOTs were implemented, there were experiments with optical molasses
[37]. In optical molasses the atoms move in red-detuned light fields produced by three
perpendicular counterpropagating laser beam pairs. This configuration provides a viscous confinement and a cooling of the 23 Na atoms.
For atoms moving along a beam axis with velocity v, the detuning in such an optical
molasses is ∆MOL = ω − ω0 ± kv depending on the motion of the atom with respect
to the propagation direction of the laser beam. The light is more resonant for atoms
moving towards the photons and less for atoms that are moving away from the laser
light. Therefore the atom scatters more light from the direction it is moving to. Since
spontaneous emission is isotropic but absorption is directional, there is a net-force against
the motion of the atom. In the case of optical molasses no magnetic fields are involved,
therefore the polarization of the laser beams does not play a role7 . This net force is
to first order linear in the velocity v. The force does not have a spatial dependence,
therefore trapping is not possible.
7
This is not true in the case of sub-Doppler cooling. There the polarization plays a crucial role.
2.4 The magneto-optical trap (MOT)
19
A spatial dependence of the trapping force can be introduced by applying a special
magnetic field. Two coils in an anti-Helmholtz-like configuration provide a magnetic
quadrupole field which has a linear field gradient in the region of interest. The field is
given by B = B 0 z with the gradient B 0 . The detuning above needs to be modified by
adding a Zeeman shift caused by the magnetic field
∆MOT = ω − ω0 ± kv + ∆Zeeman (z)
(2.25)
with the additional Zeeman shift
µB
(gF’ mF’ − gF mF )B 0 z
(2.26)
~
as in equation (2.12). For the MOT to work, the polarization is crucial. In our multilevel
MOT, the cycle transition is from F = 2 to F 0 = 3 with 5 and 7 Zeeman sublevels,
respectively. Schematically, the situation is depicted in figure (2.5) for the ground state
sublevel |F = 2, mF = 2i. The situation is analogous for the other ground state sublevels.
Atoms at a positive position experience an increasing detuning for the σ + transition and
a decreasing one for the σ − transition as they move further away from the trap center.
Therefore the scattering rate will increase for the later transition and the σ − laser beam
will push the atoms to the trap center. The situation is reversed for atoms on the
negative position side. They are more resonant for the σ + beam and also get pushed
into the center of the trap.
∆Zeeman (z) =
mF’=3
mF’=1
energy[a.u.]
a)
b)
σ+
ω0
ω
σmF=2
0
position[a.u.]
Figure 2.5: Working principle of the MOT: The MOT process is explained in the text. ω0 is
the transition frequency at zero field and ω is the frequency of the red-detuned laser.
The dashed line a) is a measure for the detuning of the σ − transition with respect to
the laser frequency. For atoms on the positive position side this detuning is smaller
than for atoms on the negative position side. b) is the same for the σ + transition.
20
2.4 The magneto-optical trap (MOT)
The force in a MOT [20] in 1D
F = −αv −
αβ
z
k
(2.27)
consists of a term that depends linearly on the velocity and a term that depends linearly
on the position of the atoms. The coeffiecients
α = 4~k 2
I
−2∆/Γ
I0 (1 + I/I0 + (2∆/Γ)2 )2
(2.28)
and
µB
· B0
(2.29)
~
are dependent on the parameters of the MOT. This is for the case of perfect alignment
and beam balancing. The atoms’ motion in the MOT corresponds to an overdamped
oscillation in a harmonic potential. They are accumulated in the center of the trap and
simultaneously cooled.
The MOT is known to be a very robust trap that does not depend sensitively on the
setup. The explanations above suggest that atoms could be arbitrarily cooled down and
densely trapped. But there are limitations to both which makes the use of other cooling
mechanisms necessary.
β = (gF’ mF’ − gF mF )
2.4.2
MOT temperature and Doppler limit
In MOT and optical molasses techniques, photons of a directed beam are redistributed in
random directions. Spontaneous emission imposes a heating on the atoms. The process
is of stochastic nature and can be described by a random walk process8 . A steady state
value between the heating process due to the random walk in momentum space and the
cooling process explained above can be found [20]. The resulting temperature
TD =
~Γ 1 + (2∆/Γ)2
4kB −2∆/Γ
(2.30)
is called Doppler limit and is a theoretical limit for this cooling process. A detuning of
∆ = −Γ/2 results in the minimal temperature, the Doppler temperature. For 23 Na this
is 235 µK. 6 Li has a lower Doppler temperature of 144 µK due to a smaller linewidth of
6
Li compared to 23 Na.
Figure (2.6) shows a time-of-flight series of the MOT atoms. The temperature is
calculated by evaluating the temporal evolution of the cloud’s width as mentioned before. The fitted results give a temperature of 405 µK ± 17 µK. For the detuning of
∆ = −14.5 MHz the theoretical expected temperature is 388 µK. The two temperatures
match within the error, hence the sample is cooled to its theoretical value. More careful
alignment of the beams may cause sub-doppler cooling within the trap as well.
The inital width 1.2 mm of the fit does not correspond to the observed values. This
discrepancy results from the assumption that the initial density distribution is Gaussian
due to the potential ascociated with equation (2.27)9 . The density distribution of the
8
9
This corresponds to a diffusion in the momentum space.
The velocity dependent term is neglected.
2.4 The magneto-optical trap (MOT)
21
trapped sample can only be assumed to be Gaussian for atom numbers smaller than
4 × 104 [38]. Higher atom numbers will cause different density profiles in the MOT due
to effects explained in the next subsection.
4
3.5
T=(405±17)µK
σ [mm]
3
2.5
2
1.5
1
0
2
4
t[ms]
6
8
10
Figure 2.6: Time-of-flight series of the MOT
2.4.3
Density limitations and loss mechanisms in a MOT
Just as the temperature of a MOT is limited, there are processes limiting the achievable
density. The main reason for this is the reabsorption of photons emitted by neighbouring
atoms. Further, light-assisted collisions can lead to trap loss.
Radiation trapping
Radiation trapping [39] produces a repulsive force between the atoms and limits the
achievable atomic density in a MOT by regular means. To fully understand the underlying process, the attractive attenuation force [40] has to be taken into account.
The attenuation force F~att arises from the absorption of the laser beam across an
optically thick gas. The laser intensity diminishes when travelling across the sample.
Therefore, the net-force of the two counterpropagating beams acting on the atom is
always in favour of the beam which pushes the atom to the trap center. This attenuation
gives rise to an overall compression of the MOT. The divergence of the resulting force
can be formulated as
~ · F~att = −6σL2 I∞ n
∇
(2.31)
c
22
2.4 The magneto-optical trap (MOT)
with c being the speed of light [39]. A higher density n will lead to a greater intensity
difference of the counterpropagating beams at positions next to the center of the trap.
I∞ is the initial intensity of a single beam. The cross-section σL for absorption of photons
from the laser beam depends on the detuning and the intensity of the laser beams. The
negative sign implies the compression of the cloud.
In the case of low atomic densities, the radiation trapping force is weak. The emitted
photons pass almost without absorption through the sample. If the atoms have a mean
distance d from each other and are in an light field of intensity I then the force of the
emitting atom on the reabsorbing atom is
|Frad | =
σL I
1
× σR ×
c
4πd2
(2.32)
The last part describes the intensity of the emitted fluorescence light at the position of
the reabsorbing atom. The cross-section σR for reabsorption can be different than the
cross-section σL for absorption of the laser beam. This is due to the shifted frequency of
the emitted photon. The divergence of the radiation trapping force
~ · F~rad = 6σR σL I∞ n
∇
c
(2.33)
has a similar form to the attenuation force. Only the cross-section for reabsorbtion of
the fluorescence light differs. The sign is positive, causing a repulsion of the atoms.
For low temperatures the velocities are small and the first part of equation (2.27)
~
can hbe neglected. Only
i a trapping force of the form Ftrap = −κ~r remains. Setting
~ · F~trap + F~att + Frad = 0 the steady state density distribution of the MOT can be
∇
easily found.
With a laser detuning of ∆0 = −14.5 MHz and a single beam intensity of about
I∞ = 2.5 mWcm−2 , the cross-sections are σL = 2.9 × 10−13 m2 and σR = 4.4 × 10− 13 m2 .
This leads to an achievable density in our MOT of about 3.6×1011 cm3 . This limit has not
been observed in our group so far. The estimation is done with a homogenous intensity
profile. Further we do not have a perfect beam balancing and the counterpropagating
beams are slightly misaligned, since this optimized the total number of atoms in the
MOT. Other groups with a similar setup report on densities of about 1 × 10−11 cm−3 in
a dark spot MOT [25]. Even with this additional technique it was not possible for them
to raise the density in the MOT.
Loss mechanisms
Aside from the principal limit due to the radiation trapping force, there are several loss
processes involved that limit an infinite loading of a MOT.
One-body losses: These losses are due to collisions with background gas and fast
atoms from the atomic beam. Atoms with velocities faster than 689 ms−1 are not decelerated by the Zeeman slower [31]. They fly without hindrance to the experimental cell
and collide with the MOT atoms. There is an atomic beam shutter installed to block
the beam, but of course this stops the loading of the MOT as well. Another part arises
from the limited value of the pressure in the experimental cell. The rate for one-body
losses is independent of the MOT’s density as they are collisions between MOT and
external atoms. They are characterized by a pure exponential decay of the number of
2.4 The magneto-optical trap (MOT)
23
atoms in the MOT when the atomic shutter is closed. A remedy against these losses is
a good vacuum, which is in our case about 1 × 10−11 mbar. Background gas collisions in
magnetic traps will be treated in subsection 3.4.1.
ΔE=ћω1-ћω2
energy
energy
a)
b)
3S1/2+3P3/2
3S1/2+3P3/2
ћω2
ћω1
ΔEFS
A
3S1/2+3P1/2
ћω
B
3S1/2+3S1/2
R̃
R
position[R]
3S1/2+3S1/2
R
position[R]
Figure 2.7: Light-assisted collisions: a) Radiative escape b) Fine-structure-changing collision
Two-body losses: Two atoms inside the MOT take part in these collisions and therefore the rate depends on the density of the MOT. There are several mechanisms and
just two so-called light-assisted collisions are presented briefly. The laser light field of
a MOT can induce inelastic, exoergic collisions between MOT atoms. The energy released is enough for atoms to escape the trap [41]. At the beginning of the processes
two neighbouring MOT atoms are both in the ground state. They experience a ground
state molecular potential corresponding to 3S1/2 + 3S1/2 with a position dependent energy curve E3S1/2 +3S1/2 (R) depicted in figure (2.7). At position R a photon from the
MOT laser beam is absorbed and the molecular potential changes to 3S1/2 + 3P3/2 with
E3S1/2 +3P3/2 (R). The two atoms are accelerated towards each other. During this process
radiative escape [42] and fine-structure-changing collisions can occur as seen in figure
(2.7).
• Radiative escape: As the two atoms are accelerated towards each other, the excited
e
atom can undergo spontaneous emission back to the ground state at position R.
e
The energy difference ∆E = (E3S1/2 +3P1/2 (R)−E3S1/2 +3S1/2 (R))−(E3S1/2 +3P1/2 (R)−
e of the two photons is equally distributed as kinetic energy between
E3S1/2 +3S1/2 (R))
the atoms.
• Fine structure change: In this case the excited atom does not undergo spontaneous
decay to the ground state but turns around at point A in the molecular potential.
At the point of the fine structure crossing it can undergo a change of levels marked
as point B. When the atoms part again, the energy difference ∆EF S of the fine
structure splitting is distributed between the two MOT atoms, causing a trap
escape.
24
2.4 The magneto-optical trap (MOT)
Besides these excited state losses there also exist ground state losses due to hyperfine
changing collisions [22]. These collisions are more important in magnetic traps and in
the process of evaporative cooling. Light-assisted collisions can be suppressed in a dark
spot MOT. The losses in our MOT are thouroughly treated in [30].
2.4.4
Dark spot MOT
To achieve higher densities in MOTs, several techniques have been developed. For 87 Rb a
compressed MOT is often used [43]. In the case of 23 Na the MIT group [36] introduced a
technique of trapping atoms in a dark state – called the dark spot MOT. For this purpose
they simply created a repumper hollow sphere and not a repumper volume. The atoms
in the middle of the hollow sphere decay eventually to the lower ground state and are
transparent for the MOT light since there is no repumper light present inside the hollow
sphere. Therefore, radiation trapping and light assisted collisions are strongly suppressed
in the middle of the MOT. If the atom is moving to the periphery of the MOT, it gets
repumped and pushed back to the middle. The dark spot technique has also been used
with other alkali metals, but often a depumping scheme needs to be applied, since the
decay to the dark state is not strong enough [44]. For lithium this scheme cannot work
since the hyperfine ground state splitting is too small to produce a true dark state.
Experimentally the dark spot is produced by imaging a black dot on a glass plate
onto the middle of the MOT. In our case we just project this black dot onto the MOT,
creating a dark tube. Diffraction effects cause a rather grey dot, however. The intensity
ratio between the bright ring and the dark inside is about 300. A weakly glowing MOT
can be observed. Since we create dark tubes which would lead to a dark loss tunnel, we
need to use two repumper beams that are crossed. This creates the desired repumper
hollow sphere.
The dark spot MOT we setup, seems to be not as efficient as we hoped for. Within
regular measurements there is hardly a difference visible. The rethermalization measurement in subsection 3.5 revealed the same thermalization times for the regular and the
dark spot MOT. If we had a dark spot MOT, the density would rise significantly. This
higher density would cause a higher collision rate, hence the thermalzation time would
decrease10 . This implies that the dark spot is not working as well as it should. At some
point it might be reasonable to go back to this step and improve the density in the dark
spot MOT.
2.4.5
Loading of the MOT for different temperatures
The loading rate of the MOT depends strongly on the flux of atoms coming from the
oven [30]. The flux is about 1017 s−1 for a temperature of 381◦ C and 6 × 1016 s−1 for
a temperature of 364◦ C. This is the flux out of the oven nozzle. The velocity limit of
the Zeeman slower is fixed to a value of about 689 ms−1 for 23 Na atoms. The MaxwellBoltzmann distribution states that only 69.02% and 70.80% of the atoms in the beam are
decelerated. Of the decelerated atoms 60% are captured in the MOT [31]. The resulting
fraction of captured atoms at 381◦ C and 364◦ C are 41.41% and 42.50%, respectively.
10
The collision time τ = 1/(nvσ) is proportional to the thermalization time. n denotes the density, v
the mean velocity and σ the elastic collision cross-section. A gain in density of one order of magnitude
would cause a decrease of one order of magnitude of the thermalization time.
2.5 Sub-Doppler Cooling
25
The ratio of fluxes at these temperatures can be estimated to 1.65. This is equal to the
ratio of the loading rates at these temperatures.
The loading rate of atoms in the MOT is given by the slope of the loading curve at
t = 0. The loading curve of the MOT can be expressed as N (t) = N0 (1 − exp(−t/τ )
with N0 being the final number of atoms and 1/τ the one-body loss rate. The ratio of
the loading rates in figure (2.8) has a value of 2.16 ± 0.06. The measured ratio does not
correspond to the estimated ratio of the loading rates.
16
fluorescence signal[a.u.]
14
12
10
8
6
4
T=381ºC
T=364ºC
2
0
-2
0
0.5
1
1.5
2
2.5
time[s]
3
3.5
4
4.5
Figure 2.8: Loading rate of the Sodium MOT for different temperatures
The used vapor pressure model [45] is only a rough approximation, hence the estimation of the fluxes is only vague. Further the transversal motion of the atoms moving
through the Zeeman slower has not been considered. This leads to a temperature dependent deviation of the calculated fluxes. Transversal motion is treated in detail in [31].
As the temperatures were measured several hours apart from each other the photodiode
might have been accidentally dislocated. Due to the changed effective area of the photodiode with respect to the incoming light a comparison of the two measurements would
not be possible. A temporal changing laser power output, as it is the case sometimes
with our dye laser, would cause a similar effect.
2.5
Sub-Doppler Cooling
The theoretical limit of regular laser cooling is given by the Doppler temperature. With
our detuning this yields a temperature of about 388 µK. Still, it is possible to cool
the atomic sample further down by means of sub-Doppler cooling. This technique was
observed in some of the first MOTs. The measured temperature was below the expected
Doppler temperatures. Many techniques have been developed, some being able to cool
below the recoil limit given by the energy of a single photon. We tried to make use
26
2.5 Sub-Doppler Cooling
of sub-Doppler cooling after the MOT was loaded. The magnetic field was turned off
and homogenous stray fields were cancelled out by making use of bias coils. MOT light,
which was even further red-detuned, was shone in for about 5 ms. As we have circular
polarized laser light, the cooling mechanism is the so-called σ + -σ − cooling [46].
2.5.1
σ + -σ − cooling basics
The σ + -σ − cooling mechanism is based on two counterpropagating, coherent waves with
opposite circular polarization. The combination of the two circular polarized waves
produces a linear polarized wave that rotates around the beam axis, forming a helix.
The length of one turn is just given by the wavelength. Since the polarization always
remains linear, the light shift of the states stays the same in space. Therefore there is
no potential gradient involved like in other sub-Doppler cooling mechanisms.
mF’=3:
1/3
1/2
-1/6
1/5
-4/15
-2
-1
1
-3/10
-4/15
3
-1/6
1/3
1/5
0
2
1/30
1/10
1/10
1/30
mF=2:
0
-1
-2
-3
1
1/2
2
Figure 2.9: Clebsch-Gordan coefficients for the cycle transition [45]
An atom at rest experiences a local polarization of the field along the quantization axis
labeled with y. The wave function is given as eigenfunctions of Fy being the y-component
of the total angular momentum. The only transition driven is the π-tranistion, and the
steady state distribution is N0 = 400/922, N±1 = 225/922 and N±2 = 36/922.11 The
atom’s probability to be in the mF = 0 state is enhanced compared to the other states.
In the case of an atom moving along the z-axis with velocity v, the situation changes.
In the rest frame of the atom the polarization is linear and rotates with an angular
frequency of kv. A transformation into a rotating frame where the polarization is fixed
causes an inertial field along the axis of propagation. The Hamiltonian is then perturbed
by an additional term
Vpert = kvFz
(2.34)
which is proportional to the velocity of the atom12 . This term gives rise to a coupling
11
The populations were normalized to N0 + N1 + N−1 + N2 + N−2 = 1.
The change to a rotating frame can be achieved by a unitary transformation T (t) = exp(−ikvFz /~).
The system’s transformed states are Ψ0 = T (t)Ψ. The transformed Schrödinger equation has the form
d
i~ dt
T (t)Ψ(~r, t) = T (t)H(t)T † (t)T (t)Ψ(~r, t). Evaluating the left hand side of the equation yields the
inertial term.
12
2.5 Sub-Doppler Cooling
27
between the ground states of the atom in the rest frame, since the non-diagonal matrix
elements of the perturbation do not vanish. The result is a velocity-dependent mixing of
the ground states for the moving atom. The steady state expectation value of the total
angular momentum in the propagation direction is given by [46]
~kv
∆0
hFz i ∝
(2.35)
where ∆0 is the light shift of the mFy = 0 state. This means that in terms of the
eigenbasis of Fz there is an imbalance of the states. These populations shall be denoted
by Πi (i ∈ (−2 . . . 2)). For a negative ground state light shift and a positive velocity the
imbalance is in favor of atoms with negative angular projection numbers, meaning
2
X
Π−i >
i=1
2
X
Πi .
(2.36)
i=1
mF=2: -2
population[a.u.]
v<0
population[a.u.]
For a negative velocity the population imbalance is just reversed as depicted in figure
(2.10).
-1
0
1
2
v>0
mF=2: -2
-1
0
1
2
σ-
population[a.u.]
σ+
mF=2: -2
-1
v=0
0
1
2
z
Figure 2.10: σ + -σ − -cooling: The atoms at rest have a steady state population that is equally
populated around mF = 0. Whereas in the moving case, there is a mixing of the states
causing the populations depicted.
This motion-induced population imbalance is the reason for the cooling in this scheme.
For atoms having a positive velocity with the corresponding population probability, the
Clebsch-Gordan coefficients (see figure (2.9)) favour the absorbtion of the σ − -light, which
28
2.5 Sub-Doppler Cooling
they are moving towards (depending, of course, on the projection state). This causes a
radiation pressure, hence damping the atoms. The force on the atoms is
F ∝ ~k 2
hRscatt i
v
∆0
(2.37)
with the mean scattering rate hRscatt i of the ground state atoms. For red-detuned light,
the light shift is negative therefore decelerating the atoms. A full treatment of the
problem can be found in [47] and [48].
2.5.2
Sub-Doppler cooling in the experiment
To use the σ + -σ − cooling scheme, bias fields, resulting from earth’s magnetic field and
stray fields, need to be compensated, since they would create a fixed quantization axis.
The cooling scheme, however, exists because of the rotation of the quantization axis.
We installed rectangular bias coils for each spatial direction (x,y,z) with the possibilty of
sending 1 A with arbitrary polarity through the coils. The field created by the coils is on
the order of 1 G. This is enough to compensate earth’s magnetic field. The currents were
adjusted to have a symmetric, slow expansion of the cloud when the magnetic gradient
field was turned off. We never managed to obtain a very good symmetric expansion,
probably due to imperfect beam alignment. A further reason could be parts close to the
experimental chamber that are magnetizable, producing inhomogenous stray fields which
cannot be compensated by simple bias coils13 . We observed a small MOT produced by
four magnetizable screws in the construction of our bias coils. They were replaced by
teflon screws.
2.5
MOT:
T=(133 ± 12) μK
sub-Doppler: T=( 97 ± 17)μ K
σ [mm]
2
1.5
1
0
2
4
time[ms]
6
8
10
Figure 2.11: Time-of flight series of MOT and sub-Doppler cooled sample.
13
It is though possible to compensate linear magnetic field gradients [49]
2.5 Sub-Doppler Cooling
29
Figure (2.11) shows a time-of-flight series of a bright MOT compared to a time-offlight series of a sub-Doppler cooled atomic sample. The measured temperatures are
TMOT = (133 ± 12) µK and Tsub = (97 ± 17) µK, showing the disturbing fact that our
sub-Doppler cooling works only very inefficiently. The ratio of temperatures was only
0.72 ± 0.14. The limiting temperature for σ + -σ − cooling is on the same order as for lin⊥
lin cooling [46]. With this method a temperature of (40 ± 20) µK was measured [50].
This yields a temperatures ratio of 0.16 ± 0.08 with respect to the Doppler temperature.
The absolute temperature of the MOT is in a range that suggests that its measurement is
not working very well. The measurement was taken after installing a new magnification
of our imaging system. It was changed from a 2 : 1 to a 1 : 1 imaging onto the CCD chip
to have a better resolution when imaging the small dimensions of evaporated clouds.
Compared to the area of the CCD chip, the MOT is a large object. During expansion
atoms move outside of the field of view, obscuring the atom number and limiting the
time an expanded cloud can be observed. Therefore, it seems not surprising that the
absolute values deviate from the measured value of the MOT temperature in subssection
2.4.2. As can be seen in figure (2.11) the initial widths of MOT and sub-Doppler cooling
differ by about 0.5 mm. In the sub-Doppler cooling phase the atoms experience a force
that damps their expansion, hence there cannot be a large deviation of the initial widths.
This gives a hint that our inefficient sub-Doppler cooling scheme might come from bad
beam balancing and misalignment. If the beams passed each other, they would tear the
cloud apart in the most extreme case.
For a dark spot MOT it is questionable if it makes sense to use a molasses sequence.
Due to the higher densities in the dark-spot MOT it would not be possible to use regular
sub-Doppler cooling. Other groups suggested to use a dark form of sub-Doppler cooling,
where the cooling process happens on the shell of the hollow sphere [36]. Nowadays
groups using sodium make use of the dark spot technique but not of sub-Doppler cooling
[25, 51]. We tried to implement a sub-Doppler cooling phase with the dark spot setup
but saw no positive effect. From this data and the resulting phase-space density in the
magnetic trap we decided not to use sub-Doppler cooling.
30
2.5 Sub-Doppler Cooling
31
Chapter 3
The magnetic trap
After having cooled the atoms in a laser light field to or below the Doppler temperature,
they are transferred into a magnetic trap. In the first production of BECs a magnetic
trap was used. In the meanwhile some groups use optical dipole traps instead [52].
3.1
Types of magnetic traps
The first magnetic trapping of neutral atoms was reported with a gas of 23 Na atoms in
1985 [53]. The lifetime of the sample was limited to 0.87 s due to collisions with hot
background gas. This was done with a quadrupole trap. Later, different trap configurations like the Ioffe-Pritchard trap [54] or the TOP trap (time-orbiting potential trap)
[55] were used to capture neutral atoms. The critical temperature TC when condensation
occurs depends on the trapping geometry and the dimensionality [56].
The quadrupole trap
One of the simplest trap designs is the quadrupole trap, consisting of a pair of coils in
an anti-Helmholtz-like configuration. It has the advantage that it can be easily built.
Further, the simple design provides great optical access. The current flows anti-parallel
through the two coaxial coils and produces a cylindrical symmetric field. For analytical calculations of the entire field, elliptical integrals are needed [57] but power series
expansions can be found [58]. Up to third order this is
3zρ2
3
(3.1)
Bz (z, ρ) = b1 z + b3 z −
2
2
ρ
ρz
3ρ2
Bρ (z, ρ) = −b1 − b3
+
2
2
8
Bφ (z, ρ) = 0
for an infinitesimal thin wire in cylindrical coordinates. The coefficients are
b1
b3
3µ0 IdR2
=
(R2 + d2 )5/2
5(4d2 − 3R2 )
=
b1
6(A2 + ρ2 )2
(3.2)
32
3.1 Types of magnetic traps
where R is the loop radius and 2A is the distance between the two coaxial coils that are
supplied by a current I. For small distances from the trapping center the potential can
be approximated in the cylindrical basis as a linear field
 
2z
 
b
1
~ = ρ
B
(3.3)
2  
0
where the field gradient is twice as large in the axial than in the transversal direction of
coil configuration. A disadvantage of this trap is the magnetic zero field at the bottom
of the trap that can lead to Majorana losses due to non-adiabatic transitions. There are
several ways to get rid of the zero field. Our way is to use an optical plug. This scheme
was first demonstrated in [4].
The Ioffe-Pritchard Trap
The Ioffe-Pritchard trap produces a magnetic field that is strong enough to define a
quantization axis everywhere in the trap, hence Majorana losses are not relevant. The
trap consists of two different parts. The first part are the so-called Ioffe-bars that create
a linear quadrupole field. They are formed in a rectangular shape. The second part is
to get rid of the zero magnetic field. So-called pinch coils are added and run with the
same polarity with a distance from each other that is greater than the Helmholtz-mode.
They produce a bias and a curvature field. The bias field is undesirable, as it shifts the
bottom of the trap out of the geometric center. Therefore anti-bias coils are added to
compensate this shift. The trapping field is given by [58]


 
 
−xz
0
x

 
  B 00 
0
~

.



(3.4)
B = B0 0 + B −y  +
−zy


2
z 2 − 21 (x2 + y 2 )
1
0
The Ioffe-Pritchard trap can be approximated by an anisotropic harmonic field. This
is very useful, as the harmonic oscillator is a textbook standard in quantum mechanics
and therefore easy to describe.
This trap is used in an experiment [59] aiming for similar goals as the NaLi experiment. They were able to prepare a BEC of 23 Na in the stretched |F = 2, mF = 2i
state. This state is favorable for sympathetic cooling of 6 Li in the |F = 3/2, mF = 3/2i
state. Spin-exchange collisions are forbidden for these states, since they are both in the
stretched state. The total projection of the spin is conserved in this case. Furthermore
this state can be trapped at high magnetic field values, whereas the lithium state of
|F = 1/2, mF = −1/2i can only be trapped in magnetic fields up to about 27 G (see
figure (2.1)). To prepare the 23 Na BEC in the stretched state it is necessary to get rid of
the |F = 2, mF = 1i and |F = 2, mF = 0i (quadratic Zeeman-shift) states that are also
captured in the magnetic trap. This was done by spectrally separating them with a
homogeneous field of B0 = 80 G using the bias field of the Ioffe-Pritchard trap. The
splitting allowed for selective removal of the undesired states.
This was not considered when designing the magnetic trap of our experiment. We
do not have an extra bias field that could produce 80 G, but in combination with optical
traps, the purification of the upper-hyperfine state could be possible.
3.2 Setup of the magnetic trap
33
The TOP trap
Finally, it should be mentioned that the very first BEC was achieved using a TOP trap
[55]. The essential part is a non-static magnetic bias field that is added transversally to
a quadrupole field. A static addition of a bias field only causes a shift of the magnetic
field zero. Eventually the atoms follow this shift and will spill out of the trap. To avoid
this, the bias field rotates with a speed which the atoms can no longer follow. The
atoms experience a time-averaged, orbiting potential with a minimum having a non-zero
magnetic field. To first order the trapping potential is harmonic. Atoms travelling far
out of the time-avaraged center experience the moving hole, which forms a circle and get
lost. This circle is referred to as the “Circle of Death” and can be used for evaporative
cooling.
3.2
Setup of the magnetic trap
The setup of our magnetic trap is quite simple. The field is produced by quadrupole
coils that can be supplied with 440 A. The coils are not only used for the magnetic trap
but provide also the quadrupole field for the MOT. In future experiments they will be
used to produce homogeneous fields to make use of Feshbach resonances.
3.2.1
The Feshbach coils
The Feshbach coils were custom made by Oswald Elektromotoren GmbH. There are four
coils, each with a double layer of windings, in the double pancake configuration. Four
stacks of windings are above and below the glass cell. The copper wire has an edge length
of 5 mm and a borehole of 1.5 mm radius to allow for water cooling. Each coil has 15
windings with an inner radius of 18 mm and an outer radius of 65 mm. The inductances
of the coils vary between 14 µH and 15 µH. We use a Riedel 25 l chiller to cool the coils.
It can deliver an output pressure of 4 bar, ensuring a flow of 1.5 lmin−1 . The de-ionized
water is stabilized to a temperature of 18 ◦ C with an accuracy of about 1 K. For any
value of the current, no significant heating of the coils was measured. Figure (3.1) shows
the magnetic field produced by the coils in a Helmholtz-like and in an anti-Helmholtz-like
configuration using 5 A current. The field was measured in the axial and in the radial
direction with respect to the symmetry axis.
The measured gradients are (3.60±0.05) Gcm−1 for the radial and (6.86±0.13) Gcm−1
for the axial direction. The expected values were 3.75 Gcm−1 and 7.5 Gcm−1 , respectively.
The deviations result possibly from inaccurate measurements. Further, the calculations
were made for an infinitesimal small wire whereas the extension of the wire is not negligible at these distances.
For the Helmholtz-like mode, polynomial functions were fitted to estimate the region
of linearity. The fit for the axial direction is a simple parabola. The fit parameters are
depicted in figure (3.1). A region of linearity can be defined for which the magnetic field
changes only by 1h of the center value. For the axial direction this yields 1.4 ± 0.1 mm.
This value can be considered only an esimate since the measured points are not very
dense in the region of interest. For the radial direction a polynomial fit to the fourth
order was used, as depicted in figure (3.1). The resulting region of linearity is estimated
to be 314 ± 140 µm. Since the fit coefficients have large errors, the value for the linearity
34
3.2 Setup of the magnetic trap
is only a rough estimate. According to the measurement, the magnetic field at 440 A
would be 1760 G.
b)
a)
20
10
B[G]
0
-10
-20
-5
0
position[cm]
fit: y=ax
a=6.86±0.13
0
-10
-5
5
c)
-20
fit: y=ax
a=3.60±0.05
5
d)
-19.35
0
B[G]
-22
B[G]
-5
-24
5
0
position[cm]
B[G]
B[G]
10
-19.75
-0.6
0
position[cm]
0.6
-10
-26
-28
-3
0
position[cm]
fit: y=a+bx2
a=-20.24±0.40
b=-0.93±0.14
3
-15
-20
-8
-4
0
position[cm]
fit: y=a+bx2+cx4
a=-19.72±0.03
b=-0.22±0.21
c=0.80±0.62
4
8
Figure 3.1: Magnetic field of the Feshbach coils: a) and b) is the magnetic field in axial and
radial direction in the anti-Helmholtz-like mode. c) and d) show the axial and radial
direction for the Helmholtz-like mode, respectively. The inlet in d) is a zoom from
−0.8 cm to 0.8 cm
3.2.2
The circuit
During the work on this thesis, the electrical part of the experiment was installed. The
electrical system can be categorized into two major parts. The first part is responsible
for the current’s turn-off and the change of the coil’s configuration. The second part is
the control of the current by a feedback loop. The circuit is depicted in figure (3.2).
The first part is composed of so-called insulated-gate bipolar transistors (IGBTs)
that are a hybrid of regular bipolar transistors and MOSFETs. They have the capacity
to allow high currents and fast switching. Power-MOSFETs can be used instead. They
are highly sensitive to overvoltage across the emitter-collector (max. 1200 V) and gatecollector (max. 20 V) connections. In the experiment high voltages occur when the
3.2 Setup of the magnetic trap
35
current is turned off. Faraday’s law of induction
Uind = −
d
Φ(t)
dt
(3.5)
states that an induced voltage Uind occurs when the magnetic flux Φ changes. The minus
sign means physically that the system wants to keep up the status quo. When the current
is switched on, the rising current gets damped, whereas for a turn-off of the current, the
induced voltage tries to keep the current alive. The transient voltages can be as high as
several thousand volts. Therefore the systems needs to be protected with varistors and
transient-voltage-suppressor diodes (TVS). The used varistor across emitter-collector of
the IGBT clamps at 1056 V. The TVS diode is bipolar and shows clamping at 15 V.
It is used to clamp the voltage across the gate-emitter connections. The coils must be
protected as well, since they can be damaged when switching off the currents. The
varistors have clamping voltages of 475 V and are accompanied by ring-down resistors
with R = 10 Ω. These resistors are used to damp the induced current. Commercially
available drivers for the IGBTs were used and built into a circuit for remote control.
The IGBT (MBI600U4-120) and the drivers (VLA) are both from FUJI Electric.
The current flows first through a coil-IGBT stretch depicted in the upper part of
figure (3.2). After this stretch, the current passes the high-accuracy current transducer
(LEM IT 600-S). This transducer is specified to follow the current with dtd I = 100 A/µs
and has a linearity of < 1 ppm from 0 A up to 600 A. The transducer works with
fluxgate technology and convertes the measured current down with a ratio of 1500 : 1.
The maximal current of 440A corresponds therefore to an output of 293 mA. This signal
is detected in the Sens-Box that mainly consists of a 10 Ω high-precision burdon resistor,
resulting in a maximal output voltage of 2.93 V. Unfortunately, the signal cannot be
raised by using a higher burdon resistor as the power specifications of the transducer do
not allow higher values. Finally, the voltage is monitored on a groundfree oscilloscope
and sent to the current regulator box.
The next stage after the LEM sensor is the passbank, which is responsible for the
current control. This is an array of NPN-Darlington transitors that are in a parallel
connection in a common-collector follower configuration. The basis resistor values were
adapted to the maximum output power of the current regulator box that sends its setting
signal to the basis of the passbank. For the emitter resistor, paper clips were used
as they have a small resistance and can dissipate a lot of power. Altogether there
are 64 Darlington transistors in parallel. The transistors are water cooled. Although
the maximum current for each transistor lies at 20 A, many are needed, as the power
dissipation is quite bad and overheating can damage them. The worst situation for the
passbank are intermediate currents. This can be understood as follows. The resistance of
each single coil is about 5 mΩ. Including the wiring the total resistance can be estimated
to be between (30 − 40) mΩ. The voltage is constant at 15V. Therefore, the power
dissipated in the passbank is given by
Ppassbank = 15I − Rcoil I 2 W.
(3.6)
Maximizing the power in the passbank yields values between (1.4 − 1.9) kW at currents
between (180 − 250) A. This is only an orientation, since the power dissipated in the
IGBT has not been taken into account.
36
3.2 Setup of the magnetic trap
L1
U
C1
Z2
V2
C1
U
Z1
V1
U
R1
V2
R1
Agilent
6690A
L2
U
V1
earthfree!
Z4
U
C1
V2
C1
LEM-Sensor
...
Q1
R3
R5
Sens-Box
R3
Q1
U
Z3
V2
Current-Regulator
SENS
DRIVE
CNTRL
R5
MON
CPU
Figure 3.2: Electric circuit: Ri and Li correspond to the passive parts in the circuit. The
IGBTs and their drivers are denoted by Zi and Ci , respectively. For overvoltage protection, most components are protected with varistors Vi . The Darlington transistors
Q1 control the current.
The current regulator
The current regulator was built by the electronic division. A schematic diagram is
depicted in figure (3.3). The signal from the sensor box is sent to the SENS input and is
amplified by an instrumentation amplifier. The gain resistor RG is selected to project the
output span of (0 − 2.93) V from the sensor box to (0 − 10) V, hence the sensitivity of the
control voltage is raised. The preamplified signal can be observed at the MON output
where the signal is fed through an isolation amplifier. The amplification ratio is 1 : 1
and it is used only by means of galvanic isolation. The signal is sent to an integrator
3.2 Setup of the magnetic trap
37
circuit, where the capacitor was chosen such that the current showed a fast response
without any oscillations. The CNTRL input provides the setpoint. Finally, the output
stage drives the integrator output. Isolation amplifiers always deliver an offset voltage.
Therefore the control voltage needs to be set slightly negative to actually achieve zero
current.
15V
current
regulator
voltage
in
CNTRL
-
passbank
drive
DRIVE
+
Integrator
Output Stage
Isolation Amplifier
current
monitor
out
MON
+
RG
Isolation Amplifier
sensor
input
SENS
-
Instrumentation Amplifier
Figure 3.3: Current regulator
The amplifier shows a response time of about 7 ms when a current of 440 A is switched
on. To reduce oscillations for such extreme events we use tailored ramp forms for the
CNTRL input.
3.2.3
The fast turn-off
Switching off the magnetic field is essential for our measurements. The high field shifts
the Zeeman levels, hence it is impossible to image all the atoms at once in a gradient field.
Therefore, the magnetic field is supposed to decay as fast a possible. As equation (3.5)
shows, the system wants to keep the current alive when it is turned off, creating high
voltages that may damage the surrounding equipment through sparkovers. A turn off of
440 A in 10 µs would create an induced voltage of more than 2500 V. IGBTs normally
switch in a few ns time, but the induced voltage is not that easy calculable and needs to
be simulated by finite element methods. The estimated voltages above are too high for
the epoxy of the coils which is rated to withstand voltages of approximately 1500 V. The
insulation of the coils could be damaged. Therefore Epcos varistors of type S20K175
with clamping voltages of about 450 V are used over each coil pair.
A direct measurement of the current through the coils was not possible since the
clamp-on ammeters were not specified for this current. Therefore the voltage across the
coils was measured. The voltage does not relate directly to the current as can be seen
38
3.2 Setup of the magnetic trap
in the simulation in figure (3.4). The voltage stays at a level of about 375 V for 25 µs.
This shows that the varistor clamps at a voltage that is below the specified level. The
decay time defined as the time the voltage drops from 90% to 10% of the maximum value
is 90 µs. This is mainly due to the long voltage tail. This tail still is 15 V at 200 µs.
Eventually, the tail extincts at times between (400 − 450) µs. A finite element simulation
using PSpice1 shows that the expected voltage drops in less than 50 µs to zero voltage.
The simulated current through the coils drops linearly in about the same time.
To date it is not quite clear why the measured voltage exhibits this tail. Removing
this tail is crucial to be able to image at very short times. The simulations showed that
the use of higher calmping voltages is the only effective way to reduce the turn-off time.
For too high ring-down resistors the current starts to oscillate and if the resistors are
selected to low the turn-off time increases drastically. The voltage drop accros the whole
system is about 750 V. If a secure shielding around the coils’ ends was installed, the
current varistors could be replaced by varistors with higher clamping voltages, allowing
shorter turn-off times. Nevertheless, this would not solve the problem of the voltage tail.
500
I[A] /U[V]
measured voltage
simulated voltage
simulated current
450
400
90%
350
300
250
200
150
100
50
0
10%
0
~30
100 ~120
time[µs]
200
Figure 3.4: PSpice simulation of the current in the Feshbach coils: The voltage across the
coils was measured during the turn-off of the current. A PSpice simulation shows the
expected transient voltage and the decaying current.
3.2.4
The field switch
As was mentioned previously, the magnetic field is needed in two different configurations:
the gradient field and the homogeneous field. Having only one pair of coils an H-bridge
with four IGBTs is used as depcited in figure (3.2). Two diagonal IGBTs form a pair that
is always switched in the same manner. The two different pairs, however, are switched
in an opposing mode. If one pair is switched on, the other has to be switched off. This
1
by Cadence
3.2 Setup of the magnetic trap
39
causes a change of the current’s polarity through the second coil. The method is also
used in our home-built laser beam shutters [60].
The driver unit has two inputs: one to select the polarity and the other to turn off
all four IGBTs. The pre-logic of the IGBT drivers is depicted in figure (3.5). It can
be catigorized into two main parts. First, the switching part and, second, a frequency
limitation part. The pre-logic elements consist of integrated circuits of the 74LSXXX
series.
b)
a)
ON/OFF
Monoflop
A
to IGBT drivers
Field switch
B
Monoflop
Figure 3.5: a) The first part is for changing the current’s polarity through the second coil
and for the turn-off event. b) The second part is the frequency limitation part.
The first part is constructed only with an inverter and an AND-gate. In the truth
table (3.1) the first two rows represent one polarity mode and the other two rows the
opposing polarity mode. A true signal at the ON/OFF input always sets the outputs to
false. In either polarity mode, one pair is true and the other false, provided the On/OFF
input is false.
Table 3.1: Truth table of the field switch
Field switch
ON/OFF
A
B
1
1
0
0
1
0
1
0
0
1
0
0
0
0
0
1
The frequency limitation part was integrated since an IGBT can only be switched
up to a certain limiting frequency. In case of longer operation times, a high frequency
can lead constantly to a high current density and hence damage the IGBT. In our case,
the limiting frequency is specified to be about 40 kHz. We do not intend to switch that
fast, but as our computer control system has a sampling rate of 1 MHz, there is a chance
that the IGBTs might become accidentally damaged. A combination of a monoflop and
an OR-gate produces pulses that have a length of at least 100 µs. For frequencies higher
40
3.3 Trappable states and the first steps in the magnetic trap
than 12 kHz the output stays constant, therefore protecting the IGBT. Finally, the signal
is sent to an inverter again. This is needed to send the proper signal to the IGBT drivers.
They charge the gate emitter capacity in less than 1 µs.
3.3
Trappable states and the first steps in the magnetic trap
Loading into a magnetic trap is actually a straightforward task. Even so, there were
some difficulties when we first implemented it. Parameters for procedures like mode
matching and adiabatic compression had to be selected in order to improve the loading
and compression of the magnetic trap.
3.3.1
Trappable states of
23
Na
As was seen in subsection 2.1.3, the atoms are in the Zeeman regime and therefore
equation (2.12) holds. The magnetic moment of the atom precesses around the local
magnetic field B(~r) with the Larmor frequency
ωL = mF gF µB
B
.
~
(3.7)
If the precession is rapid compared to the change of the magnetic field, it can follow
adiabatically and the interaction potential is
V (~r) = mF gF µB B(~r)
(3.8)
Earnshaw’s theorem [50] prohibits the existance of a static magnetic feld maximum in
free space, therefore only atoms in low-field seeking states can be magnetically trapped.
Low-field seeking atoms fulfill gF mF > 0.
For 23 Na the possible candidates are |F = 1, mF = −1i with gF = −1/2 for the F = 1
ground state and for the F = 2 ground state |F = 2, mF = 2i, |F = 2, mF = 1i and
very weakly |F = 2, mF = 0i with gF = +1/2. The first 23 Na BEC was realized in the
F = 1 ground state. Just a couple of years ago, a BEC in the F = 2 ground state was
accomplished [61]. In the latter, effects between the different Zeeman sublevels lead to
an extra loss channel.
3.3.2
Mode matching
After the first cooling stages with lasers, the atom cloud needs to be transferred into a
magnetic trap. This transfer is easily a source of heating and loss of phase-space density
if it is not mode matched. It is important that the atoms experience a similar potential
in the magnetic trap to the one they experienced in the MOT. The density is Gaussian
for MOTs with atom numbers below 4 × 104 . Therefore perfect mode matching with a
harmonic trap should be possible. A too steep gradient causes heating whereas a too
shallow trap can cause a loss in density. In the case of large MOTs the density has
different profiles as seen in [38]. In addition, the density distribution depends on the
alignment of the beams. Imperfectly aligned beams can push in a direction of preference
3.3 Trappable states and the first steps in the magnetic trap
41
and deform the MOT. Since all projection states mF are equally present in the MOT,
the maximal transfer of MOT atoms into the magnetic trap can be 33% if atoms are
captured in the F = 1 state. Depending on the condition of the experiment, there are
up to 2 × 108 atoms in the magnetic trap.
The condition for mode-matching can be expressed as
P SDMOT = P SDMAG
(3.9)
where the phase-space density is P SD ∝ T −3/2 N/V . For a transfer that does not change
the temperature, the volume of the magentic trap would have to be three times smaller
than the volume of the MOT. However, normally the volumes are set equal since the
assumption of the same temperature is mostly not fulfilled.
In our quadrupole trap, mode matching is done by selecting a field gradient such
that the trapping volume corresponds more or less to the MOT volume. The density of
atoms in the classical case is given by
2πmkB T
n(~r) =
h2
|
{z
n0
3/2
U (~r)
z exp −
kB T
}
(3.10)
with z thepfugacity and n0 the peak density. The quadrupole potential is U (~r) =
mF gF µB B 0 x2 + y 2 + 4z 2 . An
R 2effectiveRtrapping volume can be defined as Veff = N/neff .
The effective density neff = n (~r)dV / n(~r)dV = n0 /8 accounts for the spatial dependence of n and gives a mean value of the density. Evaluating this with the anisotropic
quadrupole potential leads to
Veff = 32π
kB T
m F gF µB B 0
3
.
(3.11)
A higher temperature means that more atoms are at higher field gradients – the occupied
volume enlargens. An increase in the field gradient is an effective compression which
makes the trapping volume smaller. The relation is Veff ∝ (T /ω)3/2 for the harmonic
trap, with ω being the trapping frequency2 .
The Gaussian density distribution for the MOT atoms can be written as
2
x + y2 + z2
(3.12)
n(~r) = n0 exp −
2σ 2
where σ can be related via equation (2.27) with the temperature and the MOT parameters. The effective volume of the MOT can be equated to
VMOT = (4π)3/2 σ 3
(3.13)
with σ ∝ T 1/2 . Setting equations (3.13) and (3.11) equal, the initial gradient of the
magnetic trap can be easily estimated. For MOTs with a diameter of 6 mm, the strong
gradient would need to be set to 66 Gcm−1 . This seems to be a too small gradient,
2
This shows the advantage of the quadrupole trap. It has a tighter confinement than the other traps
which is preferable for evaporative cooling
42
3.3 Trappable states and the first steps in the magnetic trap
especially when considering the gradients of 130 Gcm−1 used in [4]. The result is not
surprising as the Gaussian approximation fails for large atom number MOTs.
The phase-space density and the atom number in the magnetic trap for different
field gradients were compared in figure (3.6). The phase-space density rises linearly with
decreasing magnetic field gradient. At a certain point the phase-space density should fall
as the density decreases, but these regions were not measured. After the loading process
the magnetic trap was ramped to the same value for all initial field gradients. The
atomnumber in figure (3.6) seems to have a maximum at about 140 Gcm−1 . The change
is only 20% whereas the phase-space density has a change of almost 50%. Currently the
experiment uses an initial magnetic field gradient of 79 Gcm−1 which is only 13 Gcm−1
off of the estimated value.
phase-space density
-6
1.6 x 10
linear fit:
1.5
(16.8-2.6xB’init)x10-7
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
150
100
200
50
initial gradient [Gcm-1]
atom number
7
6.0x 10
5.8
5.6
5.4
5.2
5.0
4.8
4.6
4.4
4.2
4.0
150
100
50
initial gradient [Gcm-1]
200
Figure 3.6: Mode matching measurement: The phase-space density shown in a) increases
when the inital gradient is lowered. The atom number depicted in b) appears to have
a maximum at a gradient of 140 Gcm−1 .
Although there is a dependence of the phase-space density on the initial gradient, it
is not very strong. The initial position of the MOT center and the magnetic trap center
appeared to be far more critical. With bias coils we were able to adjust the position of
the MOT to achieve better mode matching.
3.3.3
Adiabatic compression
Immediately after turning on the magnetic trap, we want to compress the cloud to
increase the density. This results in a higher elastic collision rate which is necessary for
the evaporative cooling stage to be efficient. It is of course not possible to increase the
field gradient instantaneously, as this would give the atoms a kick and cause a loss in
phase-space density. Since phase-space density is the figure of merit in our experiment,
we do not want to lose any of it at any time. In an adiabatic compression where the
atom remains in its trapped state, the phase-space density stays constant. To compress
adiabatically, the trapping length L assocciated with the trapping volume by L ∝ V 1/3
has to change more slowly than the mean velocity of the trapped sample [33]. In this
3.3 Trappable states and the first steps in the magnetic trap
43
case, the atoms do not experience the change of the potential and can adapt to the new
compressed trap. This adiabatic criterion can be expressed as
dL v.
(3.14)
dt or equivalently |dL/dt| = v with the adiabaticity parameter 1. The trapping
volume Veff ∝ T 3 /B 03 from equation (3.11) can be expressed as an effective trapping
length
L ∝ T /B 0 .
(3.15)
This length has to change slowly in order to be adiabatic. If the process is adiabatic,
then the volume and the temperature of an ideal gas fulfill the adiabatic relation [62]
V T 1/(κ−1) = const.
(3.16)
κ is the adiabatic exponent and in the case of an ideal gas it is 5/3. Therefore the
trapping length obays
(3.17)
L ∝ T −1/2 .
With equations (3.17) and (3.15) the adiabatic length scale behaviour can be worked into
our physical situation. The following quantities having a subscript denote the initial
values of these quantities. The quantities without subscript are timedependent, e.g.
L = L(t). The mean velocity of the atoms scales as v ∝ T 1/2 . Using this scaling and
equations (3.17) and (3.15), the following relations
0 −1/3
B
(3.18)
L = L0
B00
0 1/3
B
v = v0
(3.19)
B00
can be obtained. This relates the thermodynamical properties of the gas in the trap to
the field gradient B 0 during an adiabatic compression. The field gradient can be chosen
at will. Equations (3.17) and (3.15) need to be inserted into the adiabatic criterion
(3.14). This yields
0
3v 0
02/3 0−5/3 dB
B0 B
=
.
(3.20)
dt
L0
Solving this equation is straightforward and gives the final time-dependence of the adiabtaic compression
B 0 (t)
1
=
(3.21)
0
B0
(1 − (2v 0 /L0 )t)3/2
At first, the change of compression is slow and accelerates until infinite compression can
be reached in a finite time. By compressing the atoms, they get heated (correspondingly
the density rises such that the phase-space density stays equal), causing a rise in the mean
velocity. Therefore the field gradient can be ramped up faster, causing an even higher
increase in temperature, and so on. The process is self-energizing, hence explaining this
paradox time behaviour.
In our computer control system we have functions to simulate the compression of
the magnetic trap. An exponential ramp of the form A exp(t/τ ) + A0 was chosen
44
3.3 Trappable states and the first steps in the magnetic trap
which is ramped up in 300 ms. Figure (3.7) depicts the adiabtic curve for our trapping parameters and the exponential ramp with time constants τ = 7 ms, τ = 70 ms
and τ = 400 ms. The corresponding adiabaticity parameter of the adiabatic curve is
= 0.01, fulfilling criterion (3.14). This means that when this ramp up process is
driven from the initial value of the trap to the fully compressed value corresponding to
B 0 (t = 300 ms)/B00 = 660 Gcm−1 /79 Gcm−1 = 8.33, the compression can be assumed
adiabatic. The time constant τ = 400 ms used for a long time in the experiment shows
that it fulfills the adiabtic criterion better at the end of the ramp than at the beginning.
This means that the slope of the exponential gradient evolution is larger than the slope
of the adiabatic gradient evolution at the end of the ramp up process, and reversed at
the beginning. Very small time constants, however, fulfill the adiabatic criterion better
at the beginning of the ramp up process than towards the end.
8
7
6
Exponential ramp with
Exponential ramp with
Exponential ramp with
Adiabatic curve with
τ =10 ms
τ =70 ms
τ =400ms
ε =0.011
B(t)/B0
5
4
3
2
1
0
0
0.05
0.1
0.15
t[s]
0.2
0.25
0.3
0.35
Figure 3.7: Adiabatic curve and experimental ramps
The compression parameter τ was scanned from 7 ms to 3 s. There was no significant
difference in the phase-space denstiy or atom number visible. For τ > 400 ms the form
of the curve during the 300 ms ramp time does not change anymore. Small τ are limited
by the speed of the current regulator, hence we could not measure below τ = 7 ms.
To get an idea of why there is no difference visible, the adiabaticity of the different
ramps has to be considered. Using equation (3.15), the change of length is
dL
1 dT
T dB 0
− 02
.
∝ 0
dt
B
dt
B
dt
| {z } | {z }
A
(3.22)
B
Term A causes an expansion of the volume of the cloud whereas term B lets the volume
3.4 Loss mechanisms and heating
45
shrink. For a simple estimation, term A is neglected. Further, it is assumed that the
temperature is the same for the different ramps. Physically this is not true, but the
change in temperature after compression is normally less than 20% whereas the field
gradient changes by more than 800%. Neglecting term A means that the volume shrinks
faster than it would normally do, hence the estimation of the adiabaticity gives an upper
boundary.
The slope of the exponential ramps and the adiabatic curve is calculated and compared at the point t = t̃ of the greatest deviation between the slopes of the two curves.
Dividing the change of lengths of the exponential ramp by that of the adiabatic curve
yields with equation (3.14) the ratio of the adiabaticity parameters
dLexp /dt
=
dLadiab /dt
0
dBexp
(t̃)
0
/Bexp
(t̃)2
dt
0
dBadiab
(t̃)
0
(t̃)2
/Badiab
dt
=
exp
adiab
(3.23)
where the adiabatic curve and the exponential ramps are denoted by subscripts. For a
time constant of 10 ms the ratio is 5.5, resulting in a maximal adiabaticity parameter of
exp = 0.055. This small value explains why there was no change of phase-space density
visible when τ was varied. All the other measured time constants have an even smaller
adiabaticity parameter.
3.4
Loss mechanisms and heating
On the way to a Bose-Einstein condensate there are several processes that work against
us. Loss mechanisms impose a time-scale in which the cooling process and the experiment
has to take place. They can be characterized more or less by the number of particles
inside the trap that are involved in each loss mechanism. Heating, on the other hand,
does not lead directly to loss, but can hinder condensation.
3.4.1
One-body losses
One-body losses are the main loss feature for trapped atoms in dilute gases. Only one
atom in the trap is involed in contrast to two- or three-body losses which result from
interactions of the atoms within the sample. There are several causes for one-body losses.
The most prominent is the collision with background gas, but losses due to stray light
can be significant as well. The one-body loss rate Γ1body is density independent. This is
modeled by
dN
= −Γ1body N
(3.24)
dt
resulting in an exponential decay of the atom number. Majorana losses can be treated
as one-body losses since they do not result from the interaction between two particles in
the trap.
Background gas collisions
Heating and trap loss can be caused by collisions of the trapped sample with background
gas. The background gas is at room temperature, hence having a mean velocity that is
a thousand times larger than that of the trapped sample. The energy transfer is large
46
3.4 Loss mechanisms and heating
enough to expel atoms from the trap. Using the loss rate R = nvσNa-Na [63] a rule of
thumb for the lifetime
6.2 × 10−9
τ=
s
(3.25)
p[mbar]
at a pressure p can be obtained. The lifetime for a pressure of 10−11 mbar is around
10 minutes. The gauge in the experimental section shows a pressure of this magnitude.
Still, the lifetime in our experiment is more than an order of magnitude away from this
value. The measurement of the pressure takes place inside the last pumping section.
The distance between the glass cell and gauge is relatively long. Therefore the walls
have a pumping effect and create a difference between the measured pressure and the
pressure in the glass cell. Even so, this explanation does not account for this big of
a difference. The atomic beam is blocked sufficiently by the mechanical beam shutter.
It was checked if there is a difference in lifetime when the mechanical beam shutter is
opened or closed. The difference was less than a factor of two. To check if the shutter
was not closing properly, the lifetime of the atoms was measured with a closed valve
at the differential pumping stage (see figure (2.4)). This measurement showed that the
atomic beam shutter works just fine.
Majorana losses
As previously mentioned, the magnetic field zero causes Majorana losses [64] in the
quadrupole trap. In a spatially changing magnetic field the orientation of the magnetic
moment needs to be preserved with respect to the local magnetic field while the particle
is moving. The magnetic moment has to reorientate in an adiabatic manner or else a
non-adiabatic transition to untrapped states is possible. If the change in the magnetic
field is too rapid the precessing magnetic moment cannot follow the field. The criterion
for adiabaticity can be formulated as [65]
d B
ωL >> dt .
B (3.26)
For small magnetic fields the Larmor frequency, given by equation (3.7), tends to zero
and the criterion is not fulfilled anymore. Therefore the non-adiabatic region is called
the “hole” of the magnetic trap. For an estimation of this region’s effect on the sample,
the trajectories of the atoms have to be considered. For harmonic traps, the motion
is characterized by the trapping frequencies ωTrapi (i = 1, 2, 3) of the potential. For the
linear potential, trapping frequencies that correspond to circular motion around the trap
center are introduced [66]. A simple classical estimation [55] uses a particle of mass m
being in the z = 0 plane, passing the trap center with velocity v at a distance ρ as
shown in figure (3.8). The atom moves on a circular orbit with ω = v/ρ resulting from
the Lorentz and the centripetal force. In the rest frame of the atom the direction of
the magnetic field is changing in time with v/ρ = | dtd B/B|, hence the Larmor frequency
must exceed the trapping frequencies to avoid losses. An approximated radius
v~
ρ0 ≈
gF mF µB B 0
1/2
.
(3.27)
3.4 Loss mechanisms and heating
47
results if equation (3.7) is set equal to v/ρ. For the Doppler temperature of 23 Na at the
maximal transversal gradient of B 0 = 330 Gcm−1 , the radius is ρ0 = 2.27 µm and for the
lowest gradient used in the magnetic trap the radius becomes ρ0 = 6.52 µm.
b)
a)
d)
c)
Figure 3.8: Majorana losses: The dashed lines depict iso-magnetic fields and the solid line
the limit of the non-adiabatic region. The arrows depict the direction of the field
gradient. a) An atom is orbiting outside of the non-adiabatic region. b) The atom
is still orientated with respect to the local field and remains trapped. c) An atom is
orbiting inside the non-adiabatic region. The change of magnetic field is too rapid for
the magnetic moment to follow. d) Since the atom cannot follow the orientation of the
field the spin is flipped with respect to the local field orientation. The atom is then in
an untrapped state and can escape.
The loss rate can be estimated if the elipticity of the loss region is neglected. The
flux of atoms through the Majorana loss region is given by
FMajorana =
N
AMajorana v
Veff
(3.28)
with N being the total atom number, AMajorana = πρ20 the cross-section of the Majorana
loss volume and v the velocity. The effective volume in equation (3.11) is related to an
effective length of L0 = (3/(4π)Veff )1/3 . Relating p
the velocity and the effective length via
the virial theorem (hU i /2 = hEkin i) yields v = gF mF µB B 0 L0 /m. Combining all this,
the Majorana loss rate can be written as
ΓMajorana
2
~ 241/3 gF mF µB B 0
=
.
m 32
kB T
(3.29)
48
3.4 Loss mechanisms and heating
The dependence of the lifetime on the temperature τ ∝ T 2 can be easily understood. A
decreasing temperature causes the density (∝ T 3 ) to rise. This means that the probability of being in the trap center increases, therefore reducing the lifetime. The radius of
the hole (∝ T 1/2 ) shrinks, since slower atoms see a slower change of the field, resulting
in an increasing effect on the lifetime. Finally, the velocity (∝ T 1/2 ) reduces, therefore
the atoms pass less often through the Majorana loss region. Figure (3.9) depicts the
dependence of the lifetime on the field gradient and the temperature.
10
B’[Gcm-1]
9
8
300
τ[s]
7
250
6
5
200
4
150
3
2
100
1
10
20
40
30
T[µK]
50
60
50
Figure 3.9: The lifetime τ depends quadratically on the temperature and the gradient (measured in the radial direction). The dark blue regions correspond to gradient values that
are not used during the magnetic trap. The temperature dependence is plotted only
up to about 55 µK.
The resulting lifetime at the Doppler temperature with the maximal field gradient
is 183 s. Below a temperature of 20 µK, the lifetime reduces to less than a second with
this model. For a cold gas, Majorana losses can be neglected, whereas in the regime of
ultracold temperatures, they represent a severe loss channel.
The lifetime of the trap is (21.10±0.60) s at a temperature of (402±5) µK as depicted
in figure (3.10). Majorana losses are negligible at this temperature. Making use of
evaporative cooling, explained in chapter 4, the atoms were cooled and the lifetime and
the temperature was measured. Two different sweeps were used, resulting in a lifetime
of (7.15 ± 0.55) s at a temperature of (99 ± 2) µK and a lifetime of (4.72 ± 0.22) s at a
temperature of (63 ± 7) µK. After each sweep, the microwave radiation used for this
cooling sheme, was turned off. Obviously, there is an effect of the hole. To obtain
the Majorana loss rate, the measured rates need to be corrected by the background
loss rate ΓMajorana = Γmeasured − Γbackground . The corresponding lifetimes are τMajorana =
(6.08 ± 0.37) s and τMajorana = (10.81 ± 1.27) s for the two measurements. The theory
states though that the lifetimes should be 11.46 s and 31.20 s at these temperatures.
The simple treatment above accounts only for the region in the trap with a transition
probability to an untrapped state on the order of one. Other low-field regions in the trap
produce losses as well, therefore reducing the lifetime. It is improbable, though, that this
3.4 Loss mechanisms and heating
49
correction could account for the difference between theory and experiment. Finally, it is
interesting that the measurements suggest rather a linear than a quadratic dependence
of the Majorana lifetime on the temperature3 . This is, of course, vague as there are only
two data points.
100
normalized
atom number
τ =(4.720.22)s
T=(637)μK
τ =(7.150.55)s
T=(992)μK
τ =(21.10.6)s
T=(4025)μK
10-1
0
5
10
15
20
t[s]
25
30
35
40
Figure 3.10: Lifetime measurements with Majorana losses: The atom numbers were normalized to the maximum value for comparison reasons. Cooling the atoms reveals the
influence of the hole. The lifetimes are shorter than estimated with the simple model.
Parametric heating
Heating is not directly a source of losses but can prevent the production of a BEC. A
source of heating are fluctuations in the current and therefore also in the magnetic field.
Depending on the frequency of this technical noise and the trap geometry the noise can
lead to a continuous excitation of the atoms from a lower to a higher trap state. This
process is referred to as parametric heating, similar to the parametric oscillator that is
driven by a changing oscillation frequency. For the case of harmonic traps, the heating
rate was calculated in [67]. The following discussion follows this treatment. Our case
is not that easily accessible since coordinates are coupled and the wavefunctions are
not known analytically. Still, a 1D model can give some insight. The following is not
intended to give a stringent introduction and only wants to show the effect of current
fluctuations on the sample. The Hamiltonian for a parametric linear potential is given
as
p2
(3.30)
+ µF gF µB B 0 (1 + (t))|x|
H=
2m | {z }
A
with the relative current fluctuation (t) = (I(t)−I0 )/I0 . I0 is the DC part of the current
and I(t) is the entire time-varying current including AC parts. The fluctuation can be
treated as a perturbation. The averaged transition rate for a transition from |ni to |mi
3 10.81 s
6.08 s
≈ 1.8 and
99 µK
63 µK
≈ 1.6, whereas
(99 µK)2
(63 µK)2
≈ 2.5
50
3.4 Loss mechanisms and heating
is given by
Rn→m
2
Z
2 ∞
A π
2
| hm| |x| |ni | ×
dτ cos ωmn < (t)(t + τ ) >
=
~
4
π 0
|
{z
}
(3.31)
S(ωmn )
with S(ω) being the power spectral density of the perturbation. In the case of our modulus potential, the matrix element is not known. For a harmonic potential, the matrix
elements hm| x2 |ni can be calculated. Due to parity considerations, the transitions exist only from states with quantum number n to states with a quantum number n ± 2i
(i = 1, 2, 3 . . . ). As the wave function overlap of close-by states is greatest, only transitions with n → n ± 2 are relevant. This case is also relevant for the NaLi experiment.
If the quadrupole trap is combined with the optical plug the resulting potential can be
considered harmonic for small temperatures (see subsection (3.6.1)). The heating rate
in the harmonic case is
ΓHeating = π 2 f 2 S(2f ),
(3.32)
giving rise to an exponential growth of the heating in time. f denotes the trapping
frequency of the atoms in the pockets of the plugged quadrupole trap. They are calculated with equation (3.39). The factor 2 in the power spectral density accounts for the
n → n ± 2 condition in the harmonic oscillator case.
10-3
ΔI/I0
10-4
10-5
10-6
102
103
f[Hz]
104
Figure 3.11: Fourier spectrum of the current
With the equipment available, only the spectrum of the current was measured. Still, if
there are spikes in the spectrum plot they are also present in the power spectral density,
hence the spectrum is a good measure whether there are obvious sources of heating
present. Figure (3.11) shows the relative current fluctuations in the frequency domain.
The spectrum was measured from 30 Hz to 90 kHz. The measured values were normalized with respect to the DC current, therefore giving the relative current fluctuation
∆I/I0 . After 10 kHz there is clearly a peak structure visible. The largest resonance is
3.4 Loss mechanisms and heating
51
located at 40 kHz with an amplitude of 2.5×10−4 . Between 100 Hz and 10 kHz the amplitude decays steeply from 2×10−4 to 10−6 . The trapping frequencies in the harmonic plug
potential are calculated with equation (3.39). They are 576 Hz, 993 Hz and 1.743 kHz
for the different directions in the plugged trap. There are no resonances visible at twice
or four times the value of the calculated trapping frequencies. In this region the current
fluctuations are strongly suppressed. Nevertheless, the spikes at higher frequency can
affect the precission of to what value the magnetic field is set to. Most spikes are way
below a value of 10−4 . Still, the source of this noise should be found in order to get rid
of it.
Stray light
Stray light from the laser table can be a severe loss source. To prevent effects from stray
light, we installed a light-proof curtain made of pond foil. It is necessary to seal the
experimental table, as even small non-detectable intensities reflected from somewhere in
the lab can have grave effects. From equation (2.4) the influence of this stray light can
be deduced. The most “dangerous” light is resonant light. For F = 1 state atoms this
would be the repumper light. Whereas for F = 2 the imaging light as well as the MOT
light, due to the small detuning, would cause the greatest loss. For resonant light the
lifetime is
2 I0
+1 .
(3.33)
τ=
Γ I
For an intensity of 200 pWcm−2 the resulting lifetime is already less than one second.
This can hardly be detected, hence we have to rely on thorough light sealing.
3.4.2
Two-body losses
Apart from one-body losses due to stray light and background gas collisions, two-body
losses are an important loss feature for a gas at high densities. We only trap atoms in
one hyperfine ground state. When transferring the atoms from the MOT to the magnetic
trap, the repumper light is turned off several milliseconds before the MOT light. This
ensures that all atoms are depumped in the lower hyperfine ground state F = 1. This can
be verified if the sample is imaged without using the repumper. In case of a mixture of
states there should be a signal of the F = 2 state. Similarly, the upper hyperfine ground
state can be prepared if the repumper is turned off after the MOT beam. The different
trappable states were discussed in subsection 3.3.1. As there are several trappable states
in F = 2 this gives rise to extra loss channels.
Since the interaction occurs between two atoms of the sample, two-body losses are
density dependent. This can be formulated analogous to equation (3.24)
Z
dN
= −Γ1body N − β n2 dV
(3.34)
dt
with the two-body loss coefficient β. The integral can be replaced by making use of the
effective volume Veff and the effective density neff introduced in subsection 3.3.2. The
solution to the quadratic differential equation is [68]
N (t) = N0
Γ1body exp(Γ1body t)
e − exp(−Γ1body t))
Γ1body + N0 β(1
(3.35)
52
3.4 Loss mechanisms and heating
with the initial atom number N0 and βe = β/Veff . In contrast to pure one-body losses
the number of atoms will show an over-exponential decay, provided the density is high
enough. The two-body loss rate is composed of several loss mechanisms explained below.
Spin exchange interaction
During a spin changing collision, the atoms can change their hyperfine ground state
while maintaining the total projection of the spin. For two atoms denoted by |1i and
|2i, the relative angular momentum l, the total spin angular momentum F = F1 + F2
and its projection MF = mF1 + mF2 is conserved. If all the atoms are in the upper hyperfine state, a transition to the lower hyperfine state cannot occur. The only other spin
exchange process is the collision of two atoms in |F = 2, mF = 1i with the exit channel
corresponding to the (almost) untrapped |F = 2, mF = 0i state and the |F = 2, mF = 2i
state [69]. Spin-polarized samples in the stretched |F = 2, mF = 2i state are not affected
by the spin exchange.
Spin dipolar interaction
Apart from spin exchange collisions that conserve the angular momentum and its projection onto the quantization axis, there are dipolar relaxations that conserve the total
angular momentum but not its projection. Therefore the transition to a weaker bound
state or to an unbound state can occur. The spin flip causes a change of the angular momentum lCMS of the center of mass system [70]. The internal spin gets transferred to the
angular momentum of the center of mass system. In subsection 2.1.4 it was shown that
for ultra-cold collisions only s-wave scattering dominates. Howerever, since the dipolar
interaction potential
~
~
~
~
F1 · F2 − 3 F1 · e~r F2 · e~r
(3.36)
Udip-dip = µ0 (gF µB )2
4πr3
is a 1/r3 potential, higher partial waves are allowed as can be seen from equations (2.17)
~i are the total angular
and (2.18). Here |r| e~r is the relative interatomic distance and F
momentum operators of the two colliding atoms. This equation describes the interaction
of the dipole moment of one atom with the magnetic field created by the magnetic dipole
moment of the other atom. In [71] it was shown that in a first-order Born-approximation
of the inelastic cross-section only changes with ∆mF = ±1 of the single atom play a role.
For the two-atom sytem the maximum change is ∆MF = 2. Such a spin flip releases
an energy amount of ∆E = ∆mF µB gF B 0 x. If the exit channel is a d-wave, the relative
energy of the two colliding atoms needs to be large enough to surpass the centrifugal
barrier.
Atoms in the stretched state of F = 2 can be transferred into the |F = 2, mF = 1i
state with additional energy. From there other processes may happen. The atom can
now interact again via spin exchange as explained in the previous subsection. Through
plain elastic collisions it can redistribute its energy, hence heating up the sample. Highenergetic atoms have a higher probability of being far away from the trap bottom than
low-energetic atoms. If evaporative cooling is employed this leads to a loss of atoms, as
atoms in the margins of the trap are removed (see chapter 4). Atoms prepared in the
3.4 Loss mechanisms and heating
53
|F = 1, mF = −1i state are directly transferred to an unbound state. The atom in the
unbound state can only transfer energy on its way out of the trap.
The loss rate of sodium in the stretched state is estimated to be smaller than Γdip =
4 × 10−14 cm3 s−1 [72].
3.4.3
Three-body losses
This loss mechanism gets important just before reaching the BEC. Three trapped atoms
are involved in the process, as the name suggests. Three-body loss is a molecule formation
process in which the extra atom is needed to carry away the released binding energy.
The single atom is heated and eventually leaves the trap. The molecule has a different
magnetic moment than the atom and depending on the configuration will leave the trap
as well. The loss rate is a function of density squared
Γ3body = γn2 .
(3.37)
since three particles are involved. The rate coefficient γ was calculated [73] and measured
[74] for 23 Na and is typically on the order of 10−28 cm6 s−1 . For the initial densities of
(1010 − 1011 ) cm−3 this loss rate is negligible compared to one-body and two-body loss
rates. But for typical BEC densities of the order of 1014 cm−3 , the lifetime is dominated
by three-body losses. Therefore the density is slightly reduced during the process of
evaporative cooling and in the final trap. This is just done by reducing the magnetic
field gradient, therefore decompressing the trap.
3.4.4
Lifetime measurement
The lifetime of the different hyperfine ground states was measured. To directly compare the results between the two hyperfine ground states, the difference of the trapping
strength has to be taken into account. The gradients of the F = 2 measurement were
set to half the value of the F = 1 measurement. The lifetime of the F = 2 state at full
gradient was measured to observe possible effects at increased compression.
The results of the measurement are depicted in figure (3.12). The lifetime of the F = 1
state and the corresponding F = 2 state show an exponential decay curve, suggesting
that the samples were only affected by one-body losses. The lifetimes of (5.8 ± 0.8) s for
the F = 1 state and (5.5±0.5) s for the F = 2 state are extremly short and are due to bad
vacuum. There were severe problems with leaks in our experiment when the lifetimes
were measured. The problems have been fixed and now the lifetime is more than 20 s.
The measurement of the F = 2 state at full gradient shows a different behaviour. The
curve suggests two-body losses, since there appears to be an over-exponential decay at the
beginning. A direct fit with equation (3.35) did not converge properly, hence the problem
had to be treated indirectly by parting the data. For long times the density decreases
and the decay behaviour is dominated by one-body losses. For this reason an exponential
decay curve was fitted to the data values at times larger than 1 s, resulting in Γ1body =
(0.161 ± 0.032) s−1 . This value was set constant in the fit function given by equation
(3.35). The evaluation yields βe = (1.652 ± 0.798) × 109 s−1 . The effective volume for a
typical magnetic trap temperature of 400 µK (as seen in figure (3.10)) is 2.549×10−4 cm3 .
This yields a two-body loss rate coefficient of (6.48 ± 3.13) × 10−14 cm3 s−1 .
54
3.5 Rethermalization measurement
a)
b)
τ = (5.8±0.8)s
atomnumber
0
τ = (5.5±0.5)s
108
atomnumber
108
2
4
t[s]
6
8
10
c)
0
2
4
t[s]
6
8
10
one-body fit
two-body fit
atomnumber
108
107
0
2
4
t[s] 6
8
10
Figure 3.12: Lifetime measurement: a) shows the decay curve of an atomic sample prepared
in the F = 1 state. The empty circles correspond to data that has not been used in
the fit process. It has the same lifetime as the sample prepared in the F = 2 state
with a corresponding gradient, seen in b). In c) the F = 2 sample experiences twice
the compression than the sample in b).
The value corresponds to the theoretical value for dipolar relaxation Γdip = 4 ×
10 14 cm3 s−1 of the stretched state. Figure (3.12) shows that the fit function only
corresponds to the data due to the large errors of the measurement. The prepared
sample consisted not only of the stretched state, since we do not have the possibility to
spin-polarize the sample. Therefore spin-exchange collisions were probably involved as
well.
−
3.5
Rethermalization measurement
When ramping up the magnetic field gradient too quickly, it was noticed that in time-offlight measurements the evolution of the width of the gas behaves differently for the xand y-direction. This means that the different directions had different “temperatures”.
3.5 Rethermalization measurement
55
5
5
4.5
4
3.5
3
2.5
3.5
3
2.5
2
2
1.5
1.5
1
1
0.5
0
2
4
6
t[ms]
a)
8
10
0.5
5
0
4.5
σx:
σy:
4
3
2.5
3.5
6
t[ms]
b)
8
10
3
2.5
2
2
1.5
1.5
1
1
0
2
4
6
t[ms]
c)
8
10
0.5
4.5
0
σx:
σy:
4
0.1
0.2
6
8
10
0.3
0.4
0.5
t[ms]
d)
Tx
500
T[µK]
3.5
3
2.5
2
450
400
350
1.5
Ty
300
1
0.5
2
550
4
σ[mm]
4
σx:
σy:
4
σ [mm]
3.5
0.5
2
5
4.5
σ[mm]
σx:
σy:
4
σ [mm]
σ [mm]
4.5
σx:
σy:
250
0
2
4
6
t[ms]
e)
8
10
0
t[ms]
f)
Figure 3.13: Rethermalization of the magnetic trap: a)-e) are measurements with hold times
of 30 ms, 100 ms, 200 ms, 300 ms and 500 ms, respectively. The evolution of the temperature is depicted in f).
During the ramp process the trapped sample experienced a kick and was not in
thermal equilibrium, hence the difference between the evolution of the widths σx and
σy . By increasing the hold time in the magnetic trap, the sample was given the chance
56
3.6 The plugged quadrupole field
to rethermalize again. The evolution of the widths is compared in figure (3.13). This
measurement was done using the dark spot MOT.
At 500 ms, the difference in temperature of the two directions is less than 10 µK. It
is reasonable to estimate the thermalization time to be close to 500 ms. In [75] it was
shown that each particle needs about 4-5 collisions to reach thermal equilibrium. The
ratio of the thermalization time and the collision rate was estimated to be 2.7 [76]. This
leads in our case to a collision rate of 5.4 s−1 .
The expected value at the densities measured can be calculated with
Rscatt = nvσ
(3.38)
where n is the density, v the mean velocity and σ from equation (2.19). The density
can be extracted from the images. The mean density is estimated only roughly to be
(1.25 − 7.5) × 109 cm−1 . The peak density was divided by a factor of 8 to obtain the
mean density in the trap. For a temperature of 390 µK the mean velocity is 0.6 ms−1 .
The scattering length a0 = 2.8 nm is nowadays well known. This results in a collision
rate of (0.15 − 0.90) s−1 . The value is about one order of magnitude off of the measured
thermalization rate.
With the measured values for the collision rate, the temperature and the density, the
scattering rate yields (5.9 − 14.4) nm. This is on the same order as the acctual scattering
length. First estimations in the 1990’s gave a value of (1.9 − 8.1) nm [77].
The temperature and the thermalization time are measured with an accuracy that
would not correspond to such deviations. An underestimated density is very likly the
reason for the deviation of the collision rate. The density of the sample is extracted from
the time-of-flight series just like the temperature. During the time-of-flight a reduction
of the atom number was observed. Since the cloud was quite large compared to the CCD
chip of the camera, more and more atoms moved out of the field of view. Not having the
total atom number, obscurrs the fit and leads to an underestimation of the density. The
measurement suggests that we had a mean density on the order of (5 × 1010 − 1011 ) cm−3
in the trap.
The same measurement was done with a regular MOT setup. The thermalization
time was similar to the one measured for the dark spot MOT. This implies that there is
no significant increase in density in the dark spot MOT.
3.6
The plugged quadrupole field
Majorana losses can be avoided by using a bias magnetic field as in the case of the IoffePritchard trap or an effective bias field like in the case of the TOP trap. In our case, the
optical plug method is used. This method does not allow the atoms to get in the region
of low magnetic fields. It was applied in the first sodium BEC [4]. The advantage of this
configuration is the tighter confinement of this trap compared to others.
3.6.1
Plug in the NaLi experiment
A blue-detuned, intense laser beam is needed to produce a repulsive potential. The
potential of the beam follows equation (2.7). It depends on the detuning and the intensity
of the laser beam. We use about 350 mW of 515 nm light produced by a frequency
3.6 The plugged quadrupole field
57
doubled Yb:YAG laser. The light is sent via a single-mode fiber to the experimental
table and is focused to a beam waist of about w0 = 10 µm with a corresponding Rayleigh
length of zR = 607 µm. This is realized with a Melles Griot 120 mm achromatic lens. The
laser beam enters the glass cell under Brewster’s angle. This imposes an astigmatism
on the beam. The solution to this is presented in subsection 3.6.2. For conveniance, the
plug direction shall be the y-axis in this section; previously this had been the direction
of the atomic beam.
The combined potential of the magnetic field and the laser beam is depicted in figure
(3.14). The plot shows the potential in the x-z-plane. The potential minimum is located
xmin = 16 µm from the trap center. It has a height of 19.5 µK if the the maximal field
gradient is used. The height of the potential at the Majorana radius with respect to
the minima of the potential is 128.2 µK. An ultracold gas will not be able to penetrate
the region where Majorana losses are relevant. As depicted in figure (3.14) the gas
will eventually be located in two potential pockets, provided it is adequately cold. The
barrier between the two minima has a height of about 17 µK relative to the potential
minima, i.e the pockets. The plug height is approximately 250 µK. In the pockets of the
minima the potential can be approximated by an anisotropic harmonic oscillator with
trapping frequencies [4]
s
ωx =
(4
x2min
− 1)ωy
w02
r
gF mF µB B 0
2mxmin
√
3ωy .
=
ωy =
ωz
(3.39)
In our case, the oscillation frequencies at the maximal field gradient are fx = 1.743 kHz,
fy = 993 Hz and fz = 574 Hz. These frequencies seem quite high, but right before
degeneracy the field gradient will be decreased to avoid three-body collisions. The lower
gradient will reduce the frequencies. The small beam waist is needed since the power in
the plug beam is limited by the single-mode fiber in use. A high power fiber would allow
for more power in the plug, hence larger beam waists could be possible. This would be
advantageous when adjusting the plug, since it is difficult fo find a small hole with a
small plug beam.
The plug is adjusted by evaporatively cooling the atoms, explained in chapter 4. If
the temperature drops, the atom cloud shrinks and atoms pour out through the hole.
The loss rate of the uncooled sample needs to be compared to the loss rate of the cooled
sample to check if the hole has a measureable effect on the atoms. Such a measurement
is shown in figure (3.10).
The outcoupler of the plug-fiber is mounted on a piezo stage and can be controlled
via a high-voltage piezo controller. The effective scanning length of the plug beam at the
location of the atoms is about (30 − 40) µm. First though, the beam has to be adjusted
manually with micrometer screws.
58
3.6 The plugged quadrupole field
a)
U[µK]
70
25
20
15
60
z[µm]
10
5
50
0
−5
40
−10
−15
30
−20
−25
−20
−10
0
x[µm]
10
20
30
20
b)
U[µK]
250
250
200
U[µK]
200
150
150
100
100
50
50
50
0
z[µm]
−50 −40
−20
20
0
x[µm]x[µm]
40
Figure 3.14: Combined magnetic and optical potential: a) shows a contour plot of the plugged
quadrupole trap. The region in the middle is cut off. There are two seperate pockets clearly visible. The pocket structure will determine the BEC shape as given by
equation (1.5). In b) the potential shows a height of about 250 µK in the middle.
The adjustment procedure is done by using resonant light instead of the plug beam
and “shooting away” atoms in the MOT. Once the plug is adjusted, the field gradient
is raised, hence compressing the MOT and repeating the same procedure again. After
several iterations, the procedure is done in the magnetic trap. Once the beam is adjusted
to the scale of the insitu trap size, the piezos are used. Figure (3.15) shows an insitu
3.6 The plugged quadrupole field
59
image of atoms in the magnetic trap, shot away by resonant imaging light. The vertical
position can be adjusted very well, whereas the other directions cannot be positioned
very precisely. After this procedure the resonant light gets switched off and with the
help of the piezo stage a position grid of the blue-detuned plug beam is scanned. At each
grid point the sample is cooled and then imaged. The resulting images are compared to
check if there is an increase in atom number visible. An increase in atom number would
mean that Majorana losses are suppressed. In this case, the plug is at the position of or
at least close to the hole.
130
260
z[µm]
390
540
670
800
930
130 260 390 540 670 800 930 1060 1190 13201450
y[µm]
Figure 3.15: Plug shot: The picture shows that we were using too high intesities and/or had
not positioned the focus of the laser beam onto the cloud. The width of the cut is
52 µm, whereas the beam waist is only 10 µm.
Up to now, we have not found any increase in atom number. There are several reasons
that the mentioned method could not have worked as easily as it sounded. Firstly, the
fibers for the plug beam and the resonant light were two separate fibers. They were
manually placed into the fiber outcoupler. It appeared that the foci of the two beams
were at different positions. The fiber heads were located differently inside the fiber
outcoupler. Therefore the grid scans searched through the wrong region. This was
solved by coupling both wavelenghts in the same fiber. Secondly, there is a displacement
of the resonant beam with respect to the plug beam, when they pass through the lens
and the glass cell. The displacement is about 70 µm, almost twice as large as our scan
width. Considering this displacement it should be possible to plug the hole.
3.6.2
Compensation of astigmatism
Since we want the light of the plug beam to be reflected as little as possible from the
surface of the glass cell, it needs to enter under Brewster’s angle, which is approximately
60
3.6 The plugged quadrupole field
55◦ . The refractive index of the cell wall is n = 1.462 at 515 nm. Using the right polarization yields a maximum of transmission into the glass cell. This advantage imposes an
astigmatism on the beam. Fortuntaly, this astigmatism can be compensated by methods
often used in folded laser resonators, as is explained in the following.
Light travelling under Brewster’s angle through the cell wall experiences an altered
width d, given by [78]
√
d n2 + 1
dsagittal =
(3.40)
2
√n
d n2 + 1
dtangential =
.
(3.41)
n4
In the sagittal plane, the plane perpendicular to the optical table, the beam sees a
larger width than in the tangential plane, the plane of the optical table. Since the beam
experiences different widths, a displacement of the foci of the beam in the sagittal and
the tangential plane will occur.
A second optical device, imposing again an astigmatism, can be used to compensate
the astigmatism produced by the glass cell. If the achromatic lens is tilted by an angle
θ with respect to the propagation direction of the beam, the focus f of the lens changes
to [79]
f
cos θ
= f cos θ.
fsagittal =
ftangential
(3.42)
(3.43)
Tilting the achromatic lens creates an astigmatism. These two effects can now be compensated by chosing the right angle θ. The compensation angle is given by
sin θ tan θ =
2
√
2
Nd
2f
(3.44)
where N = (n −1)n4 n +1 . Our glass cell is 4 mm thick and the focal length of the achromatic lens is 120 mm. Solving equation (3.44) yields an angle of θ = 7.1◦ .
61
Chapter 4
Evaporative cooling
The last cooling step before obtaining a BEC is evaporative cooling. This was first
suggested in 1986 for cooling hydrogen [80]. The ideas were adopted later for alkali
metals [81]. A comprehensive model of evaporative cooling was developed by the MIT
group [82]. A very rigorous and complex treatment is done in [83].
4.1
Principle of evaporative cooling
In the evaporative cooling scheme, energetic atoms are selectively removed from the
atomic trap. After the removal of the energetic atoms, the gas will thermalize again
with a lower temperature than before. If this is done succesively, the process can lead to
ultra-cold temperatures and eventually to a BEC. Evaporative cooling is a lossy method,
hence large numbers of atoms are needed to begin with. The process can be applied to
magnetic [82] as well as to optical traps [52].
For the classical case, the population of energy states follows the Maxwell-Boltzmann
distribution. Depending on the type of trap, there are several methods to remove
the most energetic atoms, thus cutting off the tail of the Maxwell-Boltzmann distribution. After some time this non-equilibrium state will thermalize, resulting in a temperature Tfinal that is smaller than the inital temperature Tinit at the cost of losing
NLoss = Ninit − Nfinal atoms. Normally the process is continuous, hence the atoms only
have the possibility to thermalize incompletely.
In optical traps, the trap depth is lowered, hence energetic atoms are no longer
trapped. The trap depth can be controlled easily by reducing the power of the laser
beam. For magnetic traps, several schemes of evaporative cooling have been developed.
High-energetic atoms have a higher probability of being at high magnetic fields than
low-energetic atoms. In the TOP trap presented in subsection 3.1, the radius of the
“circle of death” is diminished. Only high-energetic atoms can reach this circle and get
lost. To efficiently cool down the sample, the circle is contracted towards the bottom of
the trap.
In our case, the removal of atoms is done by shining in microwave radiation, causing
a transition from the trapped state to the untrapped states. High-energetic atoms at
high magnetic fields have a larger Zeeman shift than low-energetic atoms at the bottom
of the trap. The frequency of the radiation is chosen such that it is only in resonance for
atoms at high magnetic fields. The atoms are removed and the frequency is adapted to
further cool the sample. In our experiment we scan the frequency through approximately
62
4.1 Principle of evaporative cooling
|F=2;mF=2=-2>
energy
energy
150 MHz below the center frequency of 1771 MHz, this being the zero field energy splitting
of the F = 1 and F = 2 hyperfine ground states. This frequency range cools the
|F = 1, mF = −1i state. If we were to cool the |F = 2, mF = 2 > state, the scan would
have to start above the center frequency.
The physical situation is depicted in figure (4.1). Only the transition to one other
trapped state has been taken into account. The whole situation, however, involves more
states. In the dressed-state picture the trapped and untrapped levels approach each
other and produce avoided level crossings.
|F=1;mF=1=-1>
a)
position
b)
position
Figure 4.1: a) The standard way of looking at evaporative cooling is by a simple transition
from a bound state to an unbound state. The frequency is adapted so that only energetic
atoms at high magnetic fields experience the radiation. b) Equivalently, the atom-light
system is treated in the dressed-state picture. There is a level shift leading to avoided
crossings. For small Rabi frequencies the splitting is small, hence the atom has a nonvanishing probability to undergo a Landau-Zener transition to the upper dressed-state.
There it can stay in the trapped state. If the Rabi frequencies are high, though, the
Landau-Zener transition probability vanishes. In this case, the energetic atom follows
adiabatically the dressed-state potential. This eventually leads the atom to a potential
that is no longer trapped. To avoid Landau-Zener transitions, high power rf- and
microwave-amplifiers are used.
After the removal of atoms with energies higher than E, a temperature T arises after
a certain thermalization time. The relation between the cut-off energy and the resulting
temperature is called truncation parameter η = E/(kB T ) [82]. An important quantity
is the efficiency parameter
d ln T
Ṫ /T
γ=
=
.
(4.1)
d ln N
Ṅ /N
4.1 Principle of evaporative cooling
63
It relates the relative change in temperature to the relative change in the number of atoms
during evaporation [83]. Large γ correspond to an efficient cooling process. The relative
change of other quantities scales with Ṅ /N . For the quadrupole trap, the volume scales
as V ∝ T 3 as stated in equation (3.11). The proportionality factor shall be denoted by
α. This can be used to relate the relative change of atom number to the relative change
of quantities like particle density n, phase-space density P SD and so on. For the particle
density this is
d
ṅ
=
ln n
n
dt
N
= d ln
αT 3
= (1 − 3γ)
(4.2)
Ṅ
.
N
It is straightforward to extend this scaling behaviour to other relevant quantities as shown
in table (4.1) [82]. It should be mentioned that the scaling depends on the geometry of
the trap.
Table 4.1: Scaling factors
Quantity
Exponent
Temperature T
γ
Volume V
3γ
Density n
1 − 3γ
Phase-space density PSD
1 − 9/2γ
Elastic collision rate R
1 − 5/2γ
The important quantity in our experiment is the phase-space density. All our cooling
methods aim at increasing the phase-space density, hence the efficiency parameter γ is not
of direct interest. Since we compress and decompress the magnetic trap adiabatically,
the temperature changes, but the phase-space density stays constant. Therefore an
increasing γ can have several origins and should not be confused with an increase in
phase-space density. A better measure is the phase-space efficiency parameter
γ
e=−
˙
P SD/P
SD
.
Ṅ /N
(4.3)
With evaporative cooling, it is possible to decrease the atom’s temperature while
simultaneously increasing the collision rate in the sample. This process is called the
runaway regime. Since the temperature decreases, the effective volume (∝ T 3 ) shrinks.
Depending on how efficiently the atoms are removed, this can give rise to a net gain in
collision rate. This is treated in the next subsection.
64
4.1 Principle of evaporative cooling
Evaporation speed and runaway evaporation
An important parameter of the evaporative cooling scheme is the cooling time. In
the ideal, loss-free situation, a very large cooling time can be chosen. If the MaxwellBoltzman distribution is truncated far into its tail, there is an exponentially vanishing
probability that an atom will have an energy higher than the cut-off energy. This atom
would be removed and would carry away almost the entire energy of the system. The time
scale on which this process happens increases exponentially. In the non-ideal situation
with losses, they impose a timescale on the cooling process. Using forced evaporation,
in which the Maxwell-Boltzmann distribution is cut-off by a temporally changing truncation, it is possible to cool the sample down to degeneracy in less than the loss-induced
lifetime.
The temporal evolution of evaporation and the role of losses are treated in the kinetic
approach to evaporative cooling and can be estimated by the principle of detailed balance
as is done in [83]. The following treatment is a combination of the Amsterdam model [84]
and the MIT model [85]. The treatment starts with the Amsterdam model to motivate
the origin of the atom number’s temporal evolution during evaporation. Scaling laws
and basic insights are gained from the MIT model.
The change of the distribution function f () in time is governed by the Boltzman
transport equation [11] and can be written as
Z
mσ
˙
d1 d2 d3 δ [1 + 2 − 3 − 4 ] ×
(4.4)
ρ(4 )f (4 ) =
π 2 ~3
ρ(min [1 , 2 , 3 , 4 ]) · (f (1 )f (2 ) − f (3 )f (4 )).
| {z } | {z }
A
B
Here ρ() is the density of states and i the energies of the coliding atoms. The deltafunction accounts for energy conservation in the collision process. The physically interesting parts are terms A and B. Term A corresponds to collisions of two atoms with
energies 1 and 2 . This results in the production of two atoms with energies 3 and 4 .
The process has a positive sign since a particle with 4 is produced. Term B describes
the collision of atoms with 3 and 4 . Atoms with 1 and 2 are produced. The negative
sign accounts for the loss of an atom with energy 4 . Up to now this treatment is general.
Evaporative cooling truncates the distribution function
f ()trunc = f ()θ(E − )
(4.5)
with the heaviside function θ(E − ). E is the energy at which the distribution function
is truncated. In the following discussion, the energies are defined as 4 > E > 2 , 1 ,
hence 3 has to be the lowest energy due to energy conservation. Every atom with 4 is
lost immediately from the trap. The rate of evaporation is
Z ∞
Ṅev = −
d4 ρ(4 )f˙(4 ).
(4.6)
E
Equations (4.4) and (4.5) have to be inserted into equation (4.6). But since all atoms
with energies greater than E are removed from the distribution, term B in equation (4.4)
does not contribute to the evaporation rate. This is trivial, since atoms that are lost
from the trap cannot be removed again. Equation (4.6) combined with equations (4.4)
4.1 Principle of evaporative cooling
65
and (4.5) states the following: the rate of producing evaporated atoms is the number of
atoms with energy greater than E (produced in collisions via term A) divided by their
collision time τel . In [83] this is calculated in full detail.
The rate of evaporating atoms can be expressed in terms of the evaporation time τev
Ṅev = −
N
.
τev
(4.7)
The Amsterdam model connects the evaporation time τev with the elastic collision time
τel and the truncation parameter η. For large truncation parameters, this can be written
as
√
2 exp η
τev =
τel .
(4.8)
η
Up to now, losses have been left aside. But one-body losses can be easily incorporated
into the quantity of interest. Firstly, the quantity is calculated in the loss free picture
with the help of table (4.1). Secondly, a loss rate 1/τloss is subtracted from the ideal
calculation [82]. For the elastic collision rate this yields
Ṅ
5
1
Ṙ
(4.9)
=
1− γ −
R |{z}
N
2
τloss
−1
−τel
It should be mentioned that there is a further loss channel. The atoms cannot only
be expelled from the trap by one-body losses and evaporation, but they can also be
spilled from the trap. These atoms do not get lost by means of elastic collisions as in
evaporation but simply have a higher energy than the energy E to which the potential
is truncated. These atoms do not take part in the process of thermalization. Spilling is
treated thoroughly in the Amsterdam model.
To obtain a sustained evaporation it is necessary to get into the runaway regime
where the collision rate increases with time. This is the case if the left side of equation
(4.9) is positive. In terms of the efficiency parameter, this can be stated as
τev
2
1+
.
(4.10)
γ>
5
τloss
As was mentioned before, the efficiency parameter is not of interest, since the temperature can change while the phase-space density remains the same. In the same manner
as equation (4.9) was derived, the phase-space efficiency parameter can be expressed
˙
P SD/P
SD
Ṅ /N
−1
−1
τev 1 − 92 γ + τloss
= −
−1 + τ −1
τev
loss
−1
9
τev
=
γ 1+
− 1.
2
τloss
γ
e = −
(4.11)
This equation relates the two parameters γ and γ
e. The combination of this relation
and the requirement for runaway evaporation given by equation (4.10) yields the simple
condition
4
γ
e>
(4.12)
5
66
4.2 Experimental realization
for our quadrupole trap. This condition states that if four orders of magnitude in phasespace density are gained, maximally five orders of magnitude in atom number may be
lost to have runaway evaporation. This condition is different for other traps, since the
scaling factors in table (4.1) change. The same calculation for the harmonic trap yields
γ
e > 2. The tighter confinement of the quadrupole trap compared to harmonic traps
benefits the conditon of runaway evaporation.
Often, evaporative cooling is mentioned in one word with the ratio of “good” to
“bad” collisions Rgb = τloss /τel . The entire treatment above can be done with this ratio
by making use of the connection between τev and τel in equation (4.8). Elastic collisions
are reffered to as “good” whereas “bad” collisions are simply trap-loss collisions. The
ratio of “good” to “bad” collisions changes during the time of evaporation, and if the
initial ratio is sufficiently large, the ratio will show an accelerated growth. If the initial
ratio is too small, though, the evaporation process ceases. For quadrupole traps, the
ratio of good to bad collisions needs to be on the order of 100 − 200 to have runaway
evaporation [85].
Since the hole is not yet plugged, this ratio will decrease during evaporation due to
Majorana losses. Considering only one-body losses, the results of subsection 3.5 and
3.4.1 can be used to estimate an initial ratio. The results were τel = (1/5.4) s = 0.19 s
and τloss = 21.1 s. The initial ratio of “good” to “bad” collisions is then about Rgb =
21.1/0.19 ≈ 111. This value suggests that runaway evaporation in our experiment is
possible. The loss rate increases, though, once two- and three-body losses come into
play.
4.2
Experimental realization
The driven transition for evaporative cooling in our experiment is from |F = 1, mF = −1i
to |F = 2, mF = −2i. The detuning of these states is in the microwave range. The
first experiment with 23 Na BECs used rf-induced transitions within the lowest hyperfine
ground state. The scan was swept from 30 MHz to a final value of 1 MHz in about 7 s
[4]. This is certainly the easier way to implement evaporative cooling, since the design
of microwave antennas is more complicated, and trial and error methods do not really
work well. Rf-cooling is not used because there is also 6 Li in the experimental chamber.
The upper hyperfine ground state of 6 Li will be trapped since the trappable state of the
lower hyperfine ground state exhibits the bending structure shown in figure (2.1). These
hyperfine states exhibit a splitting similar to the 23 Na atoms and therefore are resonant
for rf-radiation. This leads to transitions to untrapped states. Therefore a frequency
range needs to be used for which the 6 Li atoms are transparent. 6 Li does not have
any energy splitting in the microwave range at the magnetic fields used. This makes
microwave transitions the candidates for our evaporative cooling scheme.
4.2.1
The microwave setup
The microwave antenna was designed and built by Lizzy Brama. The design is similar
to patch antennas used in cellular phones. Unfortunately they emmit radiation several
MHz above our desired frequency range and are not broadband. Therefore they did not
suit our needs. The antenna has an E-shape and uses teflon as dielectric material as can
4.2 Experimental realization
67
be seen in figure (4.2). When the antenna was built, the design focused on evaporating
atoms in the upper hyperfine state of 23 Na. It is impedance-matched for the frequency
range above 1771 MHz. This is the sweep region if the upper hyperfine ground state is
cooled. Currently the lower hyperfine ground state is cooled which needs frequencies
below 1771 MHz.
The antenna’s signal is produced by a high accuracy Rhode&Schwarz signal generator
having a frequency range from 300 kHz − 2000 MHz. This signal is amplified by a 10 W
broadband amplifier. Before reaching the antenna, the signal passes a MECA Electronics
isolator to prevent destruction of the amplifier by reflections. Figure (4.2) shows the S11
parameter of the antenna in use. The S11 two-port parameter measures the reflection of
the microwave sent to the patch antenna. For the frequency range (1751−1866) MHz the
reflection is less than −7 dB. This means that 20% of the power gets reflected and cannot
be used for evaporative cooling. From 1650 MHz to 1720 MHz the reflection is greater
than −3 dB, hence more than 50% of the power gets reflected. This is compensated by
the fact that we use a high-power amplifier. With the corresponding S11 parameters
from figure (4.2), (4 − 8) W of the microwave power can be brought to the atoms. The
antenna is set less than a wavelength of the microwave radiation away from the atoms,
hence the sample is in the near field region of the microwave field.
a)
0
b)
-2
S11 [dB]
-4
-6
-8
-10
-12
-14
-16
1.60
1.70
1.80
f [GHz]
1.90
2.00
Figure 4.2: Patch antenna and the S11 parameter: a) illustrates the S11 parameter of the
antenna shown in b). Originally designed for frequencies above 1771 MHz, the antenna
is fed with frequencies below 1771 MHz. The silvery marks on the antenna in b) were
used to fine-tune its parameters.
4.2.2
Evaporative cooling in the experiment
The temporal evolution of the phase-space density and the atom number of an evaporatively cooled sample is depicted in figure (4.3). The frequency of the microwave radiation
was ramped up within 7.6 s from 1625 MHz to 1745 MHz. The atom number decreases
68
4.2 Experimental realization
in the first 5 s with a time constant τ = (3.22 ± 0.21) s. After this period, the decrease
in atom number is steeper, probably due to Majorana losses. The evaporation time can
be estimated from this data for the first 5 s of evaporation. The measured time constant
needs to be corrected by the loss time τloss = 21.1 ± 0.6 s that was extracted from figure
−1
−1
(3.10). τ −1 = τev
+ τloss
describes the different components of the measured loss rate.
This results in τev = 3.8 ± 0.3 s. It is, however, an estimation that accounts for neither spilling nor two- and three-body losses. The evolution of the phase-space density is
smoother than the evolution of the atom number. The time constant of the phase-space
density decay for the time between 1 s and 5 s is τ = 1.61 ± 0.34. Until 1 s the increase
in phase-space density is smaller.
N
a)
PSD
b)
1010-4
-4
107
10-5
10
-5
1010-6
-6
106
0
N  exp(-t/τ)
τ =(3.22±0.21)s
4
2
t[s]
6
8
10
-7
0
PSD  exp(t/τ)
τ =(1.61±0.34)s
4
2
t[s]
6
8
Figure 4.3: Phase-space density and atom number during evaporation: The atom number in
a) shows a steep decay after about 5 s of evporation. The phase-space density in b)
rises constantly.
Figure (4.4) shows the phase-space density vs. the number of atoms at different times
of the evaporation process. It is the same measurement as depicted in figure (4.3). The
phase-space density rises drastically with decreasing number of atoms. In the double
logarithmic plot, the data points can be fitted with the fit function P SD = αN −β
with the fit parameters α and β. During the first part of the sweep the slope is β =
−2.10±0.26. This means that two orders of magnitude in phase-space density are gained
while losing one order of magnitude in atom number. The fit parameter β is nothing
else than the phase-space density efficiency parameter γ
e defined by equation (4.3). The
phase-space density rises to about 10−4 , which is still more than 4 orders of magnitude
away from condensation. The increase in phase-space density flattens until a constant
level is reached. The flattening is due to Majorana losses. As the sample gets cooler and
cooler Majorana losses become more important. The loss rate due to Majorana losses
needs to be added to the loss rate caused by background gas collisions. Finally, further
cooling of the atoms leads to an entire loss of the sample. At this point the optical plug
needs to be used. This is essential to gain the last four orders of magnitude necessary
for reaching a BEC.
Condition (4.12) is fulfilled by a factor of 2.6 during the first couple of seconds. Even
4.2 Experimental realization
69
10-4
the more stringent requirement for the harmonic potential would probably be fulfilled.
If the slope of β = 2.1 can be maintained during evaporation, we should be able to
produce a BEC with an atom number on the order of 104 . The MIT group produced a
23
Na condensate with 107 atoms to sympathetically cool down the 6 Li atoms [86]. This
suggests that once the BEC is achieved, we will have to go back a few steps and increase
the atom number in the MOT.
10
-6
phase-space density PSD
10-5
PSDN-β
β=2.10±0.26
106
7
atom number N 10
Figure 4.4: Phase-space density vs. atom number: The phase-space density rises untill the
point where Majorana losses dominate the losses.
70
4.2 Experimental realization
71
Chapter 5
Conclusion and Outlook
5.1
Conclusion
Having all necessary components installed and being able to bring the sample into the
runaway regime, a sodium BEC seems to be in close reach. Several weeks after my
arrival at the NaLi experiment, we produced the first sodium MOT in Germany. In the
meantime the magnetic trap as well as evaporative cooling have been implemented.
The installed sub-Doppler cooling scheme for the regular MOT showed moderate
but promising results that give a starting-point for future optimization. In order to
increase the density, a dark spot MOT was installed. A comparison of the thermalization
behaviour of atoms in the magnetic trap revealed that a sample loaded from a dark spot
MOT showed the same thermalization times as observed for an initial cloud produced
by a regular MOT. Therefore it can be concluded that the density gain in the dark
spot MOT configuration is marginal. The dark spot MOT needs to be reconsidered and
eventually remodeled.
The magnetic trap was analyzed and showed a lifetime of more than 20 s. The
different trap steps are examined and compared with theoretical models. A comparison
of different hyperfine ground states in the trap revealed that for the F = 2 state twobody losses are apparent. The setup of the magnetic trap is described in detail. The
source of the current noise, that appears above 10 kHz, needs to be pinned down.
By means of evaporative cooling the cloud was cooled to a temperature of 63 µK,
being the lowest temperature so far in the experiment. The phase-space efficiency parameter, describing the gain of phase-space density vs. the loss of atoms during evaporation, was measured to γ
e = 2.1 ± 0.26. This is significantly higher than the theoretical
minimum requirement for the runaway regime, being γ
e = 4/5. With the achievement
of the runaway regime, one of the last requirements for reaching degeneracy has been
fulfilled; however, Majorana losses hindered a further cooling of the atoms. Therefore
an optical plug beam needs to be installed. If the hole of the magnetic trap is finally
plugged, then, the first condensation of sodium atoms in Germany can take place.
5.2
Outlook
Once a BEC is achieved the next steps seem to be clear. Enhancing the number of atoms
in the BEC will eventually require to revisit all earlier cooling stages, especially the dark
72
5.2 Outlook
spot MOT, since at this cooling stage optimization would be most effective. Alongside
the lithium laser system is installed and will produce a lithium MOT soon. The two
species will be confined in the same trap, allowing for sympathetic cooling of the lithium
atoms.
The first planned experiments at NaLi concern interspecies effects between fermions
and bosons. A Nd:YAG laser will create an optical dipole trap for the captured sodium
and lithium atoms. A second dipole trap, created by a dye laser, is set to a frequency such
that it produces a potential which is significantly stronger for lithium than for sodium.
The lithium atoms inside the tight trap propagate through a bath of bosonic sodium
atoms. Inter-species Feshbach resonances will allow the tuning of the interaction strength
between lithium and sodium atoms. This eventually effects the oscillation frequency of
the lithium sample in the tight dipole trap. In a way, the changing interaction strength
produces a changing effective mass of the lithium atoms inside the trap. This effect is
known in condensed matter physics as superheavy fermions.
73
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81
Danksagung
Zu guter Letzt möchte ich mich bei den Leuten bedanken, die mich während der Zeit
meines Studiums und meiner Diplomarbeit begleitet und unterstützt haben. Mein besonderer Dank gilt hierbei:
• Prof. Markus K. Oberthaler für die Aufnahme in seine Arbeitsgruppe und die
Möglichkeit am neuen NaLi Experiment viele Erfahrungen zu sammeln. Seine
Ideen haben das Experiment immer weitergebracht und in so mancher Sackgasse
wusste er, was zu tun war. Für die erfahrene Unterstützung, sein Vertrauen in
mich und dieses wertvolle Jahr bin ich sehr dankbar.
• Prof. Annemarie Pucci für die Begutachtung meiner Diplomarbeit.
• meinem NaLi-Team:
– den ehemaligen Diplomanden der ersten Generation – Marc Repp, Stefan Weis
und Jan Krieger – die mit vollem Einsatz das Experiment aufgebaut haben.
– der zweiten Generation – Valentin Volchkov, Raphael Scelle und Bernhard
Huber – die mit gleichem Eifer NaLi vorantreiben. Raphael danke ich ausserdem für die Hilfe bei Messungen.
– der neuen Doktorandin Fabienne Haupert, die einen frischen Wind in den
Männerhaufen bringt.
– und zu guter letzt Jens Appmeier, dem alten Platzhirsch am NaLi Experiment.
Er kennt jede Schraube am Experiment und war sich nie zu schade mir die
Dinge, wenn nötig, auch ein zweites Mal zu erklären.
– Lizzy Brama für die raffiniert gebaute Mikrowellenantenne.
• Prof. Peter Krüger, der nicht nur im Labor, sondern vor allem auch ausserhalb
des Labors immer gute Ratschläge wusste und bei kleineren Katastrophen stets
die Ruhe bewahrt hat. Danke und alles Gute für deine Zeit in Nottingham!
• allen Mitgliedern der Arbeitsgruppe für die stets gute Atmosphäre, die Hilfe bei
technischen Problemen und die Grill-, Poker- und Zwitscherstub“-abende.
”
• meinem Elektronikguru und Yogalehrer Sri Maha Sri“Jürgen Schölles. Wenn
”
440 A mal wieder nicht so wollten, wie ich wollte, wusste er mich stets zu beruhigen.
• der mechanischen Werkstatt (insbesondere Herrn Spiegel) und der gesamten Elektronikabteilung.
82
Bibliography
• allen, die sich die Zeit genommen haben, diese Arbeit zu korrigieren.
• meinen Freunden und meiner gesamten Familie, die dafür Verständnis hatten, dass
ich das letze Jahr ein bißchen weniger Zeit für sie hatte als sonst.
• vor allem meinen Eltern, Bonnie Piccardo-Selg1 und Erwin Selg, die stets für mich
da waren und mich immer unterstützt haben, gleichgültig welche Entscheidungen
ich in meinem Leben getroffen habe. DANKE!
1
Illegitimi non carborundum :-)
Erklärung:
Ich versichere, dass ich diese Arbeit selbstständig verfasst und keine anderen als die
angegebenen Quellen und Hilfsmittel benutzt habe.
Heidelberg, den
Unterschrift