Faculty of Physics and Astronomy

Transcription

Faculty of Physics and Astronomy
University of Heidelberg
Diploma thesis
in Physics
submitted by
Christoph Kaup
born in Münster
2011
Single-Atom Detection of
39
Ar
This diploma thesis has been carried out by
Christoph Kaup
at the Kirchhoff Institute for Physics in Heidelberg under the
supervision of
Prof. Dr. M. K. Oberthaler
Einzelatomdetektion von
39 Ar
Diese Diplomarbeit beschreibt und charakterisiert den EinzelatomDetektionsaufbau des Experiments für Atom Trap Trace Analysis (ATTA)
von 39 Ar in Heidelberg. Eine umfangreiche statistische Analyse zeigt, dass mit
dem aktuellen Aufbau, in dem eine magnetooptische Falle (MOT) zwischen einem
Detektions- und einem Einfangmodus umgeschaltet wird, 58% der potentiell fangbaren Atome in natürlichen Wasserproben detektiert werden können. Anschließend
werden die transversale sowie die longitudinale Kühlung des Experiments, welche
beide die Laderate der MOT beeinflussen, charakterisiert und auf potenzielle
Verbesserungsmöglichkeiten hin analysiert. Abschließend werden die zusätzlichen
Verlustkanäle von 39 Ar gegenüber den stabilen Argonisotopen untersucht, die
durch dessen Hyperfeinstruktur bedingt sind.
Single-Atom Detection of
39 Ar
This diploma thesis describes and characterizes the single-atom detection setup
that will eventually be used for Atom Trap Trace Analysis (ATTA) of 39 Ar in
Heidelberg. An extensive statistical investigation of the measured single-atom count
rates finds that the current detection setup, which employs a Magneto-Optical Trap
(MOT) that is toggled between dedicated capture and detection cycles, can detect
58% of the potentially trappable atoms in natural water samples. Subsequently,
the transverse and longitudinal cooling stages of the experimental setup, which
both affect the MOT’s loading rate, are characterized and analyzed for potential
improvements using both experimental data and simulations. Finally, the additional
loss channels that 39 Ar’s hyperfine structure introduces in comparison to those of
the stable Argon isotopes are discussed.
Contents
1. Introduction
1
2. Interaction of Argon With Magnetic Fields
2.1. Electronic Structure of Argon . . . . . .
2.2. Interaction With a Magnetic Field . . . .
2.3. Interaction With Light . . . . . . . . . .
And
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Light
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5
5
7
8
3. Experimental Setup
13
4. Single Atom Detection of Argon 39
4.1. Magneto-Optical Trap . . . . . .
4.1.1. Setup . . . . . . . . . . .
4.1.2. Trapping Time . . . . . .
4.1.3. Capture Velocity . . . . .
4.1.4. Detection vs. Trapping . .
4.2. Single-Atom Detection . . . . . .
4.2.1. Detection Setup . . . . . .
4.2.2. Statistics . . . . . . . . . .
4.2.3. Results . . . . . . . . . . .
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33
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5. Efficiencies and Loss Channels
5.1. Spectroscopy setup . . . . . . . .
5.2. Source . . . . . . . . . . . . . . .
5.3. Collimation . . . . . . . . . . . .
5.3.1. Velocity Profiles . . . . . .
5.3.2. Efficiency for 39 Ar . . . . .
5.4. Zeeman Slower . . . . . . . . . .
5.4.1. Theory . . . . . . . . . . .
5.4.2. Experimental Realization .
5.4.3. Simulation . . . . . . . . .
5.4.4. Characterization . . . . .
5.5. Loss channels of 39 Ar . . . . . . .
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6. Conclusion and Outlook
57
39
6.1. Adequacy of the Detection Setup for ATTA of Ar . . . . . . . . . . . 57
6.2. Limiting Factors and Perspectives for Improvement . . . . . . . . . . . 58
6.3. First Measurements of Natural Samples . . . . . . . . . . . . . . . . . . 59
I
Contents
A. Appendix
A.1. Some Constants . . . . . . . . . . .
A.2. Laser System . . . . . . . . . . . .
A.3. Detailed Transition Scheme of 40 Ar
A.4. Simulation . . . . . . . . . . . . . .
A.4.1. Zeeman Slower . . . . . . .
A.4.2. MOT . . . . . . . . . . . . .
II
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61
61
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64
64
64
1. Introduction
Boon. The counting of single Argon atoms is a cog in the wheel that is the perpetual
pursuit to understand the environment we live in and how it is being affected by human
activities. The environment is a highly complex system; to develop realistic models, a
thorough understanding of the transport mechanisms and mixing processes that take
place throughout the environment is necessary.
Key players in this context are the earth’s miscellaneous bodies of water, e.g. oceans,
lakes, ground water and glaciers: not only do they play a major role in the processes
themselves, but they also contain chemical tracers such as the quantities of particular
isotopes. These can either directly provide information about or allow a reconstruction
of, e.g., the climatic history. An important parameter in the interpretation of these
data is, obviously, the age of the water they were found in.
The paleoclimatologist’s most important tool, when it comes to dating water samples, is isotope analysis. Especially important for hydrology are the long-lived noble
gas radionuclides. Due to their chemical inertness and low solubility in water, they
have ideal geophysical and geochemical properties for dating water samples.
The interpretation of radionuclide data is, in principle, very simple: The number
of radionuclides in the sample decays exponentially with time; the number of stable
nuclides does not change. The ratio of unstable and stable isotopes then contains the
information about how long the sample has been “sealed” (i.e. since the last contact
with atmospheric air).
PRE-BOMB
39
Ar’s half-life makes it particularly inC
Cl
H- He
teresting: The applicable age range is
Kr
Rn
Kr
Ar
roughly 0.1 to 4 times the half-life of an
isotope, depending on the precision of the
10
10
10
10
10
detection method. Most age ranges are
YEARS BEFORE PRESENT
served by a multitude of tracers. However, in the age ranges of all common isotopic tracers, there is a gap between 50 and
1000 years before present. 39 Ar with a half-life of 269 years nicely fills this gap.
39
Ar is almost exclusively produced in the atmosphere by cosmic-ray induced
40
Ar(n, 2n)39 Ar reactions. Short-run fluctuations of that process are very low and
the anthropogenic contribution to the 39 Ar production is less than 5%, so that the
39
Ar/40 Ar ratio has, over the last 1000 years, been constant within 7% [1].
Its distribution is spatially uniform around the globe because 99% of the world’s
Argon resides in the atmosphere and its half-life is much longer than the typical
time scale of global mixing processes [2]. All these properties make 39 Ar an ideal
environmental tracer.
3
222
85
-2
14
3
36
81
39
0
2
4
6
1
CHAPTER 1. INTRODUCTION
Bane. Now here is the catch: Only one in 1.2 · 1015 Argon atoms is 39 Ar. This
is a direfully low abundance: in comparison, the most common isotopic tracer, 14 C,
is about a thousand times more prevalent. One liter of water only contains about
8000 39 Ar atoms. The task of counting single 39 Ar atoms is, figuratively speaking,
comparable to finding a single black sand corn in two thousand five hundred tons of
fine white sand.
The low abundance pushes the limits of current detection methods. Several approaches to tackle this problem have been developed:
Low-Level Counting (LLC) was the first technique available to measure 39 Ar. LLC
directly counts 39 Ar’s β-decay. Its drawbacks are a very long measurement time (' 40
days), a necessity for underground labs to avoid background radiation, and a large
required sample size (' 3000 liters of water).
Other methods are based on isotope selection. Accelerator Mass Spectrometry
(AMS) uses the mass difference of 39 Ar and 40 Ar. It is an effective (short measurement times [' 9h], small required sample size [' 8 liters of water]) but costly method.
Although natural samples have been measured in the past, severe technical difficulties
introduced by a large 39 K background and the fact that Argon does not form negative
ions make substantial progress in the cost effectiveness unlikely.
In laser-based methods, the isotopic selection is based on a variation in the resonance
frequencies of different isotopes. The selectivity, as defined in [2], is s = 4(∆/γ)2 for a
single optical excitation, where ∆ is the frequency difference between the isotopes and
γ is the natural linewidth. One method that uses a single excitation and combines its
selectivity with the mass selectivity of AMS is Resonance Ionization Mass Spectrometry (RIMS), where atoms are selectively ionized by resonant optical excitation and
then passed through a mass spectrometer. RIMS has, however, turned out to be too
difficult to perform reliable analyses on tracers with very low abundances [2].
The selectivity increases exponentially if the method does not rely on only one but
many subsequent optical excitations. This is utilized in the relatively new technique of
Atom Trap Trace Analysis (ATTA). It has first been realized at the Argonne National
Laboratory [3] for 85 Kr and 81 Kr and has since also been applied to rare Calcium
Isotopes [4, 5] and, very recently, to 39 Ar [6]. In ATTA, a magneto-optical trap
(MOT) is used to capture single atoms. Hundreds of thousands of subsequent optical
excitations are necessary to trap an atom, thus this technique takes full advantage of
the optical selectivity. ATTA has already proved to be more efficient than AMS for
81
Kr in terms of the required sample size and on par in terms of the counting time [2].
The detection efficiency necessary for 39 Ar is about a thousand times higher than for
81
Kr, as its isotopic abundance is lower by that factor. To achieve this high detection
efficiency is the key task of this thesis.
Outline This thesis focuses on the single-atom detection of 39 Ar in a magneto-optical
trap and the laser-based manipulations that need to be applied to the atoms in order
to trap them. As mentioned before, the isotope selectivity of ATTA is based on a
difference in the resonance frequency of different isotopes. For the relevant Argon
2
transition, this difference is about 85 times larger than the natural linewidth, thus
providing good selectivity even for single optical excitations. In a MOT, the atoms
are trapped by many subsequent excitations, each of which is followed by the emission
of a photon. These photons can be collected by a single-photon counter focused on
the trap’s center.
The superior selectivity is thus two-fold: First, the stable argon isotopes cannot be
trapped because the laser-based manipulations do not work on them; therefore, they
only exist as a background gas that is not spatially confined to the trap center. Second,
they are much less likely to be excited and emit photons. Thus, in spite of their large
quantity compared to the single trapped 39 Ar atom, the background that limits the
detection is mostly from stray light, not from the background of stable Argon isotopes.
With this inherent selectivity, the limits determining the counting of 39 Ar from natural
samples come down to two parameters: The signal quality (i.e. how many photons
can be collected from one atom and how many background photons there are) and the
loading rate (i.e. how many 39 Ar atoms can be trapped in a given time).
The second chapter gives a review of the properties of Argon and the fundamental
concepts of the interaction of atoms with magnetic fields and light. This section is
deliberately kept very brief as these topics have already been dealt with in more detail
not only in the standard literature, but also in the previous theses documenting the
project at hand [7, 8, 9, 10].
The third chapter describes the experimental setup. The design of most parts of
the machine has been discussed in detail in [8] and [10]; to avoid redundancy, these
topics are, again, dealt with very briefly.
The fourth chapter addresses the magneto-optical trap and the single-atom detection
setup. A substantial part of this chapter is dedicated to the statistics involved in the
detection.
The fifth chapter focuses on the different experimental parts that determine the loading rate and characterize their relevant parameters for this experiment. It furthermore
discusses the question whether the results obtained with stable Argon isotopes are also
valid for 39 Ar.
The last chapter addresses potential areas of improvement and revisits some of the
results to consider them in the broader perspective of analyzing natural water samples.
3
2. Interaction of Argon With
Magnetic Fields And Light
2.1. Electronic Structure of Argon
Argon 40
Isotope
Ratio
Argon’s most abundant isotope (the table on the right
40
Ar
99.6%
shows relative abundances for natural samples) has, in
39
Ar
8.1 · 10−16
its ground state, completely filled electron shells in a
38
2 2
6 2
6
Ar
0.06%
1s 2s 2p 3s 3p configuration [11]. It does not show hy36
Ar
0.33%
perfine splitting because of its even numbers of protons
and neutrons. As for almost all noble gases, ground-state
transition wavelengths are in the vacuum ultraviolet (VUV) region, which is out
of reach for commercial laser systems. As a closed transition for laser cooling, the
3p5 4s(J = 2) → 3p5 4p(J = 3) [1s5 (J = 2) → 2p9 (J = 3) in Paschen’s notation] transition at 811.754nm is used. The 1s5 level is a metastable state that lives 38s—orders
of magnitude longer than the typical timescale of laser cooling processes.
In general, LS-coupling is not suitable for excited states of the Argon atom; the
levels of the 3p5 4p configuration, e.g., are superpositions of different LS terms of the
same J (Racah coupling [12]). The closed transition described before is, however, a
notable exception: The 2p9 level has only one J = 3 configuration; it is therefore
purely 3 D3 . The 1s5 (J = 2) level is purely 3 P2 [13]. Appendix A.3 shows a detailed
transition scheme.
Some of the measurements discussed later on in this thesis were done with 38 Ar,
which is very similar to 40 Ar because it also does not have a nuclear spin.
Argon 39
For 39 Ar, a nuclear spin of I = 7/2 introduces hyperfine splitting into the level scheme.
The interaction Hhf = Ahf I · J shifts the levels in first-order approximation by [14]
∆Ehf =
Ahf
· (F (F + 1) − J(J + 1) − I(I + 1)).
2
(2.1)
The 1s5 (J = 2) and 2p9 (J = 3) levels are split into five and seven hyperfine levels,
respectively. Figure 2.1 shows the relevant level scheme. The F2 = 11/2 → F3 = 13/2
5
CHAPTER 2. INTERACTION OF ARGON WITH MAGNETIC FIELDS AND
LIGHT
REPUMPER 2
REPUMPER 1
COOLER
f40-95MHz
2p9 (J=3)
F
ΔE
MHz
a)
ΔE
MHz
b)
1/2
3/2
5/2
1886
1663
1296
1874
1653
1289
7/2
795
790
9/2
170
168
11/2
-565
-562
13/2 -1391 -1383
3/2
2648
2631
5/2
1877
1865
7/2
827
822
9/2
-471
-468
1s5 (J=2)
11/2 -1980 -1968
Figure 2.1. Relevant HFS levels for 39 Ar. The straight lines indicate transitions that
are driven by lasers, the dotted lines indicate possible deexcitations, and the dashed
lines indicate off-resonant excitations. The values are adapted from [15]a) and [16]b) .
The isotope shift introduces an additional 95MHz deviation from 40 Ar [15].
transition (the subscripts indicate J) is used for cooling because due to the selection
rule ∆F = 0, ±1 this is, in principle, a closed transition. Off-resonant excitation can,
however, induce F2 = 11/2 → F3 = 11/2 transitions. Roughly 2/3 of the atoms in
F3 = 11/2 decay into the F2 = 9/2 level and are effectively lost for the cooling cycle.
The probability is relatively low (about 0.1%, depending on the laser power), but in
the course of tens of thousands of scattering processes, which an atom experiences in
our experiment, nearly all atoms would eventually suffer this fate. Consequently, an
additional repumping laser pumps atoms in the F2 = 9/2 level back into the cycle.
Unfortunately, the same problem comes up for the repumping transition: Off-resonant
excitation can lift the atom to F3 = 9/2, which means it can end up in F2 = 7/2.
For that reason, this experiment uses two repumping sidebands, altogether driving
transitions from F2 = 7/2, 9/2, and 11/2.
6
2.2. INTERACTION WITH A MAGNETIC FIELD
The repumpers have another beneficial
effect: Initially, the atoms with J = 2
F 3/2
are distributed over all hyperfine levels,
deg.
4
weighted with their degeneracy. Without
pop. 10%
sidebands, only 30% of all atoms would
be accessible, compared to 75% with two sidebands.
5/2
6
15%
7/2
8
20%
9/2 11/2
10
12
25% 30%
2.2. Interaction With a Magnetic Field
For atoms without nuclear spin that conform to the LS-coupling scheme, the interaction with a small external magnetic field (Zeeman regime) in z direction is described
by [14]
HZe = −µB = (µB L + gs µB S)B =
hL · J i + gs hS · J i
µB BJz .
J(J + 1)
(2.2)
The corresponding energy shift, assuming gs ' 2, is
EZe = gJ µB BMJ , where gJ =
3 S(S + 1) − L(L + 1)
+
.
2
2J(J + 1)
(2.3)
For Argon isotopes without hyperfine structure, the B-fields in our experiment are
small enough for the Zeeman regime to be a good description of the energy shifts.
For 39 Ar (and other atoms with hyperfine splitting), the total atomic magnetic
moment is
µN µB
(2.4)
µatom = −gJ µB J + gI µN I ' −gJ µB J ,
so that the Hamiltonian for the interaction with an external B-field equals eq. (2.2);
in this approximation it depends only on the electron’s magnetic moment and not on
the nuclear spin because µN = µB · me /Mp ' µB /1836, where me and MP are the
electron and proton mass, respectively. Its expectation value, however, depends on
the hyperfine interaction: In the weak-field case, I and J precess about F , which
itself precesses about B. The magnetic field can be treated as a perturbation to
the hyperfine structure. The projection of the magnetic moments along F results (in
first-order approximation) in
H = gJ µB
hJ · F i
F · B = gF µB F · B = gF µB BFz ,
F (F + 1)
where
gF =
F (F + 1) + J(J + 1) − I(I + 1)
gJ .
2F (F + 1)
The resulting energy shift is
∆E = gF µB BMF .
(2.5)
(2.6)
(2.7)
7
LIGHT
In the strong-field case (Paschen-Back regime), where the interaction with the external field is greater than Ahf I · J , F is not a good quantum number; I and J precess
about B independently. The hyperfine interaction can then be treated as a perturbation to the |IMI JMJ i eigenstates. The resulting energy shift is
EZe = gJ µB BMJ + hIMI JMJ |Ahf I · J |IMI JMJ i = gJ µB BMJ + Ahf MI MJ
(2.8)
In the intermediate-field case, the secular equation H = Hhf + HZe must be solved
numerically. Results for 39 Ar can be found in [10].
2.3. Interaction With Light
Considering a two-level atom, the steady state solutions of the Optical Bloch Equations
provide the scattering rate of light from a laser field [17]
γs =
s0 γ/2
,
1 + s0 + (2δ/γ)2
(2.9)
where γ is the spontaneous decay rate, δ is the overall detuning, and s0 is the onresonance saturation parameter. Because of the fact that an absorbed photon has a
momentum in direction of the laser propagation but the subsequently emitted photon’s
momentum does not have a preferential direction (so that the average momentum of
the emitted photons is zero), an atom experiences a net force in the direction of the
laser propagation
s0
γ
.
(2.10)
F = ~k
2 1 + s0 + (2δ/γ)2
An important parameter is therefore the on-resonance saturation
s0 ≡ 2|Ω2 |/γ 2 = I/Is ,
(2.11)
where Ω = −µeg E0 /~ is the Rabi frequency, µeg is the dipole transition matrix element,
I = 21 0 c|E|2 is the light-field intensity, and Is = ~c0 γ 2 /(4µ2eg ) is the saturation
intensity.
The dipole matrix element µeg can be separated into a radial and an angular part.
The radial part is the same for all relevant transitions and will therefore not be considered. The angular part depends on the fine structure for 40 Ar and on the hyperfine
structure for 39 Ar.
For isotopes without hyperfine structure, the dipole transition matrix element is
given by [17]
µeg = e hα0 J 0 MJ0 |ˆr|αJMJ i .
(2.12)
The optical field only couples to the L component of J , therefore some Clebsch-Gordan
magic is necessary. Expansion in terms of L and S and recoupling of all corresponding
Clebsch-Gordan coefficients leads to [17]
8
2.3. INTERACTION WITH LIGHT
L0 +S−MJ0
µeg ∝ (−1)
0
p
L J0 S
J 1
J0
0
(2J + 1)(2J + 1)
,
J L 1
MJ q −MJ0
(2.13)
where q = ∆MJ , the curly braces represent the 6j symbol, which summarizes the
recoupling of six angular momenta, and the braces represent the 3j symbol, which
summarizes the coupling of three angular momenta. Values for the 1s5 → 2p9 transition are illustrated in fig. 2.2.
Figure 2.2. Magnetic substates of 40 Ar for the levels 2p9 (J = 3) and 1s5 (J = 2) and
the corresponding transition strengths [11].
For atoms with hyperfine structure, e.g.
39
Ar, the result is
p
0
0
0
µeg ∝(−1)1+L +S+J+J +I−MF (2J + 1)(2J 0 + 1)(2F + 1)(2F 0 + 1)
0
0
L J0 S
J F0 I
F 1 F0
.
×
MF q −MF0
J L 1
F J 1
(2.14)
Values for the 1s5 (F2 = 11/2) → 2p9 (F3 = 13/2, 11/2) transition are illustrated in
Figure 2.4. A two-level system is feasible using σ + or σ − light to drive the closed
MF = ±11/2 → MF = ±13/2 transitions. This coupling scheme might raise the
question whether there is actually a necessity to use a repumper in our Zeeman slower
(see ch. 5.4)1 . Perfectly polarized σ − light can only induce ∆MF = −1 transitions.
Experimentally, however, almost all atoms are lost without repumping.
An ensemble of atoms populates all magnetic substates equally. In a magnetic field,
the atoms have a well-defined quantization axis, therefore circularly polarized light
will optically pump them into the outer MF = ±11/2 levels in a small number of
1
The next part will require some basic knowledge about how a Zeeman slower works, which is
explained in detail in ch. 5.4. The remainder of this chapter does not become important before
that chapter, therefore readers without the required knowledge are advised to skip the rest of this
chapter now and return to it later.
9
LIGHT
1
1
0
-9/2
-11/2
0
5
10
15
20
25
-7/2
R e l a t iv e p o p ul a t i o n
R e l a t iv e p o p ul a t i o n
-7/2
30
35
40
45
Nu mb e r o f sc a t t e r i n g p r o ces ses
50
0
-9/2
-11/2
0
5
10
15
20
25
30
35
40
45
N um ber of s cat t eri ng proces ses
50
Figure 2.3. Left: Assuming a uniform distribution over all magnetic substates, after
38 scattering processes with σ − -light 99% of the atoms have MF = −11/2. Right:
If all atoms are initially in MF = +11/2, this number increases only slightly to 42
scattering processes.
absorption-emission cycles, as shown in fig. 2.3. The probability to excite an atom
into the wrong state is very low, so losses during the optical pumping, when atoms
are still in MF > −11/2 and thus could, from angular coupling, be excited to other
states, cannot explain the experimentally observed losses.
The magnetic field does not play a crucial role either: One might assume that atoms
in MF 6= −11/2 with relatively low longitudinal velocities get in resonance with the
slowing light very late in the slower and could be lost due to the different Zeeman shift
of their levels (considering the weak-field case, gF in eq. (2.7) is different for F = 11/2
and 13/2, therefore the Zeeman splitting is not equal for different MF ).
However, all MF > −11/2 actually get in resonance with the slowing light earlier
than MF = −11/2 because of their larger Zeeman shift, so they would be optically
pumped into that state nevertheless. The most probable cause for the high loss without
repumpers is imperfect polarization of the light relative to the B-field.
In the collimator and the MOT, the atoms are in a mixture of orientations and the
light approaches them from various directions with various polarizations (the polarization depends, after all, on the orientation of the quantization axis), therefore the
transitions are not unidirectional and the atoms’ magnetic-sublevel population will
end up in a steady-state distribution. In that case, the coupling is not described by a
single coefficient, but rather by an effective Clebsch-Gordan coefficient calculated from
that distribution [11]. Transitions to MF = −11/2 or − 9/2 are thus not forbidden
from angular coupling.
10
9
72
1
70
1
64
9
54
25
40
49
-11/2
-9/2
-7/2
-5/2
-3/2
-1/2
+1/2
+3/2
25
70
22
81
121
40
54
64
70
72
70
64
54
40
22
1
3
6
10
15
21
28
36
45
55
66
3
78
12
1
-11/2
6
22
-9/2
10
30
-7/2
15
36
-5/2
21
40
-3/2
28
42
-1/2
36
42
+1/2
45
40
+3/2
55
36
+5/2
66
30
+7/2
22
+11/2
+13/2
78
64
22
12
F3=13/2 (2p9)
49
+9/2
+11/2
F2=11/2 (1s5)
81
54
-11/2
121
40
-13/2
22
+5/2
+9/2
+11/2
F3=11/2 (2p9)
+7/2
2.3. INTERACTION WITH LIGHT
Figure 2.4. HFS magnetic substates and transition strengths from eq. (2.14) of
39
Ar for the levels 2p9 (F3 = 13/2, 11/2) and 1s5 (F2 = 11/2). Transition strengths are
normalized to the weakest transition (independently for both sides). The 1s5 state is
placed in the center for reasons of space.
11
3. Experimental Setup
Fig. 3.1 shows a schematic sketch of the experimental setup. In the following, the
different parts will be explained. A photo of the setup is shown in fig. 3.2.
EXHAUST
SAMPLE INTAKE
GETTER
ZSL BEAM
ICP
COLLIMATOR
ZEEMAN
SLOWER
APD
MOT
Figure 3.1. Schematic sketch of the experimental setup (not to scale) including a
simplified scheme of the vacuum setup
Source As mentioned in ch. 2.1, the transition used for the optical manipulation of
the atoms starts in the metastable level 1s5 . To excite the atoms, this experiment
uses an inductively coupled plasma (ICP). In the experiment’s main chamber,
a water-cooled copper coil couples RF power at 13.56MHz from a 300W RF
generator into Argon gas in a glass tube with a diameter of between 1mm and
15mm. Along the glass tube, there is a pressure gradient from ' 10−2 mbar at
the beginning of the tube to ' 10−5 mbar in the main chamber. The source will
be discussed in 5.2 and has been characterized in [8].
Collimator Atoms with large transverse velocities are unlikely to reach the MOT
chamber, therefore a collimator transversely cools the atoms. The design this
experiment uses employs two laser beams, each coupled into a pair of tilted
mirrors, to cool both transverse directions. The collimator is discussed in ch.
5.3 and has been described in [8] and characterized in [10].
Zeeman slower Before the atoms reach the Zeeman slower, they pass a differential
pumping stage that reduces the pressure in the slower by two orders of magnitude
13
CHAPTER 3. EXPERIMENTAL SETUP
Figure 3.2. Photo of the experimental setup. This photo is not up to date; currently
a black box covers the MOT chamber to shield it from ambient light.
compared to the main chamber. The Zeeman slower then longitudinally cools
the atoms down to about 40m/s, which is necessary for the MOT to efficiently
trap them. The change in the Doppler shift that the deceleration introduces
is compensated by a spatially varying magnetic field. The Zeeman slower is
described in detail in ch. 5.4. Along the slower the pressure drops two more
orders of magnitude to ' 5 · 10−9 mbar in the MOT chamber.
MOT The magneto-optical trap (MOT) relies on Doppler cooling to slow the atoms
down and the Zeeman shift to add a central force that confines the atoms in the
trap center. Detailed explanations of magneto-optical traps can be found in the
standard literature as well as in [7]. This experiment’s MOT is characterized in
ch. 4.1.
Single-atom detection The center of the MOT is 1:1 imaged onto a 200µm fiber that
feeds an Avalanche Photo Diode (APD) operated as a single-photon counting
module. The APD feeds TTL pulses to a counting card that is read out by a
computer. The detection setup is described in detail in ch.4.2.1.
Laser system Light with different frequencies is needed for the MOT, the Zeeman
slower and the collimator. Each stage needs a cooling laser and two repumpers.
To obtain this variety of frequencies, an external-cavity diode laser (ECDL) is
locked to a Doppler-free 40 Ar spectroscopy. Two commercial tapered amplifiers
(TAs) are offset-beat-locked to its light. The light from these two commercial
TAs is then sent to a number of acousto-optic modulators (AOMs) that generate
14
the required frequencies. Subsequently, three self-built TAs, one for each stage
of the experiment, amplify the intensity to usable levels. The light is then guided
to the experiment through optical fibers.
A description of the laser system with detailed explanations of all parts involved
is given in [10]. A schematic drawing that lists the generated frequencies can be
found in Appendix A.2.
Vacuum system To obtain appropriate pressures for the different parts of the experiment, 5 turbomolecular pumps (TMPs) are employed. To keep the required
sample size small, the exhaust is closed during measurements and the sample is
recycled. In recycle mode, outgassing from the chambers and small leaks would
drive up the pressure rapidly. To prevent this, a getter pump, which does not
work on noble gases, removes all other gases by adsorption. The limiting factor
for the measurement time is, in the current setup, the outgassing of Argon from
the machine.
15
4. Single Atom Detection of Argon
39
4.1. Magneto-Optical Trap
4.1.1. Setup
The MOT in this experiment works in the common σ + σ − setup that is described in
the standard literature (e.g. [17, 18, 19]) as well as in previous theses about this
experiment [7, 8]. The magnetic-field gradient is produced by a pair of water-cooled
anti-Helmholtz coils located 130mm apart at the top and bottom of the MOT chamber.
Each of the coils has 101 windings and a radius of 54mm. The resulting magnetic field
gradient at 10A is 8.3gauss/cm horizontally and -16.9gauss/cm vertically. Right in
front of the MOT, a compensation coil reduces residual fields from the Zeeman slower.
The combined magnetic field profile is shown in fig. 5.5. The six laser beams have a
diameter of 32mm and a power of about 5mW per beam (the shape of the beams is
Gaussian, therefore their intensities are radially inhomogeneous). The vertical beam
is perpendicular to the atom beam, whereas the two horizontal beams are angled at
45◦ .
4.1.2. Trapping Time
The net loading rate of a MOT is given by [20]
dN
= R − γN − βN 2 ,
dt
(4.1)
where R is the loading rate, γ is the linear loss-rate coefficient1 , and β is the rate
coefficient for two-body collisions. The linear loss-rate coefficient reflects two aspects:
collisions with background particles γc and, for 39 Ar, losses due to leaks in the hyperfine
level structure γL , which are addressed by repumping lasers.
Argon 38 For 38 Ar (and other atoms without hyperfine splitting), γ = γc . For
lifetime measurements, blocking the Zeeman-slower laser rapidly turns off the atom
beam, in which case R ≈ 0. The lifetime is obtained by measuring the fluorescence
with an Avalanche Photo Diode (APD).
1
The linear loss-rate coefficient is not to be confused with the spontaneous decay rate, which is also
denoted γ. The meaning should be clear from the context.
17
CHAPTER 4. SINGLE ATOM DETECTION OF ARGON 39
15
1.1
0.9
10
0.8
−1
γ [s ]
Fluorescence signal [a.u.]
1
0.7
5
0.6
0.5
0.4
0
0.2
0.4
Time [s]
0.6
0
0.8
(a)
0
2
4
6
Pressure [10−9 mbar]
8
(b)
Figure 4.1. (a) MOT Fluorescence signal after switching off the Zeeman slower, fit
with eq. (4.2) (b) Time constant γ vs. MOT chamber pressure (at the pump)
The rate equation’s solution is
N (t) =
N0 e−γt
.
1 + N0 γβ (1 − e−γt )
(4.2)
The linear loss-rate coefficient γ = γc results from scattering processes with the
background gas and with ground-state particles from the atom beam. The collision
rate with the background gas can be approximated by
γbg =
σpv̄bg
v̄bg
= nσv̄bg =
,
λ
kB Tbg
(4.3)
p
where v̄ =
(8kB Tbg /πm) ≈ 400m/s is the mean particle velocity, Tbg ≈ 300K,
λ = (σn)−1 is the mean free path, n = p/kB Tbg is the particle density, and σ is the
total collision cross section of neutral and metastable atoms.
The collision rate with particles from the atom beam is
γbeam =
Ṅ
pS
σ=
σ,
A
kB Tbeam A
(4.4)
where Ṅ = pS/kB Tbeam is the total number of atoms passing an aperture that connects
the MOT chamber with the Zeeman slower tube with an area A = π(38mm/2)2 ,
18
4.1. MAGNETO-OPTICAL TRAP
Tbeam ≈ 350K, and S is the pumping speed of the turbomolecular pump that pumps
the MOT chamber.
A tube with a conductance of L = 24l/s connects the pump, which has a pumping
speed of S = 250l/s, with the vacuum chamber. The vacuum gauge used for the
following measurements is mounted right in front of the pump, therefore the pressure
in the MOT chamber must be corrected by pMOT ≈ (S/L)p. Bringing all the equations
together, the total linear loss-rate coefficient is


 S
Sv̄ 
pMOT v̄bg
pS
.
(4.5)
+
+
= σp 
γ = γbeam + γbg = σ

kB Tbeam A
kB Tbg
kB T A LkB T 
| {z } | {z }
≈6 · 1022
≈1 · 1024
Scattering events with beam particles are about 16 times less likely to occur than
scattering events with background gas.
The loss rate has been measured for various pressures; fig. 4.1b shows the results.
The offset of the linear fit can be explained by a systematic error in the pressures
measured by the vacuum gauge (Varian IMG-100), which is working close to the
lower limit of its specified pressure range. The slope of the linear fit yields the total
2
cross section σ = 1759Å . It is about 40 times higher than the value one would
expect from the ground-state geometric cross section and the van-der-Waals radius
2
σvdW = π(2rvdW )2 = π(2 · 1.88Å)2 = 44Å . This is, in magnitude, comparable to
the results of [21], where the cross section of metastable Neon atoms with Argon was
2
measured to be 556Å , or about 15 times higher than the neutral-neutral geometric
cross section.
For practical purposes, the approximate linear loss-rate coefficient of our setup can
be obtained from the linear fit in fig. 4.1b:
γ = 2 · 109 · p[mbar] − 0.9 · 10−9 s−1 .
(4.6)
Typical values for our MOT are γ = 2 . . . 5/s, which corresponds to lifetimes of τ =
200 . . . 500ms.
Argon 39 Because of 39 Ar’s hyperfine splitting, atoms can also be lost due to leaks
into hyperfine states that are not accessible with our laser system. For clarity, the
notation γ = γc is maintained; the hyperfine leaks γL are added separately.
A system of coupled rate equations describes the time evolution of the atoms in the
MOT. With two repumping lasers, the relevant ground states are F = 11/2, 9/2 and 7/2.
From there, atoms can be excited to F = 13/2, . . . , 5/2. The relevant equations are
thus, for the ground states,
19
Ṅg,11/2 = R − Ng,11/2 γ + W11/2→13/2 + W11/2→11/2 + W11/2→9/2
+ Ne,13/2 Γ13/2→11/2 + Ne,11/2 Γ11/2→11/2 + Ne,9/2 Γ9/2→11/2
..
.
Ṅg,7/2 = Ng,7/2 γ + W7/2→9/2 + W7/2→7/2 + W7/2→5/2
+ Ne,9/2 Γ9/2→7/2 + Ne,7/2 Γ7/2→7/2 + Ne,5/2 Γ5/2→7/2
and for the excited states
Ṅe,13/2 = Ne,13/2 γ + Γ13/2→11/2 + Ng,11/2 W11/2→13/2
..
.
Ṅe,5/2 = Ne,5/2 γ + Γ5/2→7/2 + Γ5/2→5/2 + Γ5/2→3/2 + Ng,7/2 W7/2→5/2 ,
where Γ is the spontaneous decay rate and W is the excitation rate from eq. (2.9).
These equations were solved numerically using MATLAB’s ode15s solver; fig. 4.2
shows the decay times of Ng,11/2 for a set of saturation settings for the cooling beam
and two repumpers, where R = 0 and γc = 0. There are a number of things to learn
from these plots:
1. Without repumpers and for a saturated cooling beam, γL γc , therefore the
lifetime of the MOT is limited by hyperfine leaks.
2. Adding only the first repumper reduces γL noticeably, but still the losses due
to hyperfine leaks happen much faster than the losses due to collisions with
background particles
3. Adding two repumpers, even with very low saturation, drastically reduces hyperfine leaks γL γc . The lifetime is limited by collisions.
4. A high saturation for the repumpers is not beneficial: to leave the accessible levels, the atoms have to populate higher states; a high saturation in the repumpers
speeds up that process. Their intensity should therefore be chosen so that they
pump atoms from otherwise dark states back into the cycle, but do not push
other atoms out.
4.1.3. Capture Velocity
One important parameter that limits the MOT’s loading rate is its capture velocity.
As will be explained in ch. 5.4.4, the Zeeman-slower efficiency decreases with the
final velocity the atoms have when they exit the slower, therefore a high capture
20
4.1. MAGNETO-OPTICAL TRAP
γ vs. Cooler Saturation
L
4
10
3
10
L
−1
γ [s−1]
0
10
−1
10
sRP1=1, sRP2=0
−2
10
γ (typical)
c
−2
10
−1
10
0
scool
10
=1
10
γc (typical)
2
10
1
10
=0
=1
RP1
RP1
γc (typical)
1
0
10
0
10
−1
10
−2
10
10
s
s
2
−1
−2
−3
1
10
=1
cool
3
10
sRP1=0, sRP2=1
−3
=0
RP2
10
sRP2=sRP2=0
L
RP2
−1
1
10
s
s
L
10
γ vs. RP2 Saturation, s
4
10
γ [s ]
2
=1
cool
10
10
γL [s ]
L
3
10
10
γ vs. RP1 Saturation, s
4
10
−3
−2
10
−1
10
0
sRP1
10
1
10
10
−2
10
−1
10
0
sRP2
10
1
10
Figure 4.2. γL for various saturation settings. With two repumpers, hyperfine losses
are much less likely than collision losses. The hyperfine loss rates for the cases in which
only one of the repumpers is active are not perfectly constant but the variations are
too small to be seen in this logarithmic plot.
velocity is desirable. To determine the capture velocity of our MOT, a Monte Carlo
simulation has been set up that is, in detail, explained in Appendix A.4.2. It simulates
the atoms’ trajectories in one dimension2 ; the vertical transverse cooling is neglected.
The simulation only models light forces (further approximations are noted in Appendix
A.4.2). Fig. 4.5 shows simulated trajectories for different detunings. It is obvious that
for higher (negative) detunings the capture velocity increases: The Doppler shift −kv
scales with the velocity, therefore a higher detuning is needed for faster atoms to
compensate it. For very high detunings, the time needed to bring the atoms to rest
is too long; they reach the end of the trapping region and are thus lost. A higher
saturation can shift the onset of this effect to larger detunings, as fig. 4.4 shows.
Fig. 4.3 shows simulated trajectories for two detunings and different B-fields. The
resonant velocity at the beginning of the cooling area increases with a higher gradient.
At some point, there is a maximum in the trappable velocities; if the gradient becomes
even steeper, the maximum deceleration the lasers can apply to the atoms is too small
for the atoms to follow the resonant velocity. This argument is explained in more
detail in ch. 5.4.1 about the Zeeman slower.
First experimental results have shown that an additional longitudinal laser beam
with a relatively high detuning, which is in resonance with relatively fast atoms, can
increase the MOT’s loading rate by a factor of 2-3. The issue that limits the capture
velocity in fig. 4.5 does not play a major role for a truly longitudinal beam, as the
trapping region is limited in longitudinal direction only because of the 45◦ angle of the
horizontal MOT beams (for a truly longitudinal beam the limit would be determined
by the B-field gradient, which is linear in a much larger region).
2
The B-field is only modeled in the in the longitudinal direction, therefore this simulation is, essentially, one-dimensional. The MOT beams, however, are angled at 45◦ and the atoms are allowed
to do a random walk in two dimensions.
21
Figure 4.3. Simulated MOT velocity trajectories for different B-field currents and
two different detunings. The black lines indicate atoms that have been trapped in
MOT’s center at 1.9m. There is an optimal B-field for capturing atoms at ' 10 to 12A.
4.1.4. Detection vs. Trapping
There is a tradeoff between the MOT’s capture velocity and the ability to detect
trapped atoms by fluorescence:
To capture faster atoms, (1) the laser beams should be large to increase the trapping
area, (2) they should be highly saturated to increase the scattering rate and thus the
deceleration, (3) they should have a rather large detuning to compensate the Doppler
shift of fast atoms, and (4) the magnetic field gradient should be at the optimum
setting for capturing atoms.
To efficiently detect already trapped atoms, as discussed in detail in ch. 4.2.1, the
optimal settings are basically reversed: (1) the laser beams should be small to reduce
stray light (the MOT itself is small on the order of hundred µm), (2) they should be
rather weakly saturated ([7] suggests an optimal detection at s ' 1), (3) they should
have a small detuning to maximize the scattering rate, and (4)pthe magnetic field
gradient should be steep because the MOT’s size scales with 1/ dB/dz [7] (which
affects ηimg defined in ch. 4.2.1).
22
4.2. SINGLE-ATOM DETECTION
80
Capture Velocity [m/s]
70
s=0.4
s=0.8
s=1.5
s=4
s=8
60
50
40
30
20
10
0
−50
−40
−30
−20
Detuning [MHz]
−10
0
Figure 4.4. Simulated capture velocities for different detunings and saturations (per
beam) and a B-field current of 10A.
Figure 4.5. Simulated MOT velocity trajectories for different detunings and a Bfield current of 10A. The black lines indicate atoms that have been trapped. For small
detunings, the capture velocity increases with the detuning. For very high detunings,
fast atoms cannot be slowed down before they reach the end of the trapping region.
The elephant in black is a guide to the eye.
Experimental results show that, rather than to find a suitable middle ground, it is
better to toggle between two dedicated sets of parameters, one to capture fast atoms
and one to detect trapped atoms. Chapter 4.2.3 discusses the results for two different
laser detunings; toggling of other parameters could, in the future, increase the singleatom capture and detection efficiencies even further.
4.2. Single-Atom Detection
4.2.1. Detection Setup
Single atoms are detected by analyzing the fluorescence of the MOT for characteristic
signatures: When a single atom enters the trap, the fluorescence increases by a discrete step. The fluorescence signal is detected via an Avalanche Photo Diode (APD)
23
operated as a single photon counting module (PerkinElmer SPCM-AQRH-13-FC) that
feeds TTL pulses to a NI-6601 PCI card that records count rates (usually counts/1ms).
These count rates are then post-processed in MATLAB.
The scattering rate is given in eq. (2.9). With γ = 2π · 5.85MHz, the maximum
scattering rate is γs,max = 18.4MHz. Because of the six MOT beams, the saturation is
6s0 , a typical value is a combined saturation of 2. Typical detunings are in the range
of -4 to -10MHz. The number of photons actually detected by the APD is limited by
a number of factors:
Geometry The objective has a numerical aperture of NA = 0.26 ≈ sinα, where α is
the half-angle of the maximum cone of light that is collected by the objective.
The observable solid angle is therefore
2
NA
πNA2
=
= 0.017 = 1.7%
(4.7)
ηgeo =
4π
2
Transmittance The objective consists of 6 lenses, so there are 12 surfaces with a
specified reflectivity of 1%. The light also passes a vacuum window with about
the same specified reflectivity. The total transmittance is therefore ηtrans =
0.9914 = 0.87.
Imaging The objective projects a 1:1 image of the MOT to a fiber core with a diameter
of 200µm. For a small detuning of the MOT beams, the MOT is smaller than
the fiber head, so that ηimg ≈ 1.
Fiber coupling The 200µm multimode fiber has two uncoated surfaces with a reflectivity of 4%. It is coupled via a GRIN (gradient index) lens with a specified
efficiency of 95% to the APD. As 200µm is larger than the APD’s diameter, only
about 80% (measured) reach the APD. The combined fiber coupling efficiency
is ηfib = 0.962 · 0.95 · 0.8 = 0.70
Detection The PerkinElmer SPCM-AQRH-13-FC has a specified efficiency of 0.45 to
0.6; in the following, the conservative value ηdet = 0.45 will be used.
The combined efficiency is
η = ηgeo · ηtrans · ηimg · ηfib · ηdet = 0.46%,
(4.8)
yielding a theoretical count rate per atom of
γat = γs η = 24kHz
(4.9)
for -6MHz detuning and a combined saturation of 2.
Fig. 4.16a shows a measurement obtained with δ = −4MHz, laser beams with a
diameter of 3.2cm and a power of 5mW each, which yield a combined saturation of
2.6, and a binning time of 40ms. The expected count rate is 1500 counts per 40ms; the
24
APD
Figure 4.6. Single-atom detection setup. From right to left: Center of our MOT
chamber, two lenses inside the chamber, vacuum window, four lenses outside the
chamber, fiber, GRIN lens, APD. The lenses have been dressed to size to fit the
chamber (the figure shows the original size).
actual count rate is about 1000 counts per 40ms. The width of the peaks for few atoms
is an indicator that this is not caused by a lower µimg , as explained in ch. 4.2.3 and
in the following chapter about statistics. The factors for geometry and detection are
relatively certain, so the difference is likely due to either less efficient fiber coupling,
reduced transmittance, or inaccurate measurements of the detuning and (most likely)
the saturation3 .
4.2.2. Statistics
Scattering
In a MOT, atoms scatter photons with a maximum scattering rate of Γrad /2. For
simplicity, the time between two subsequent detected photons will be assumed to be
much larger than the lifetime of the excited state (we only detect a fraction of the
emitted photons); the probability to detect n photons from a single atom in a given
time interval is then described by a Poissonian distribution [23],
Pph (n) =
n̄n −n̄
e .
n!
(4.10)
The average number of detected photons, n̄, depends not only on the scattering rate,
but also on a set of geometric and technical parameters that will be addressed later.
For more than one atom in the trap, it can be shown that the distribution is still
Poissonian with n̄k = kn̄. The number of atoms k in the trap also follows a Poissonian
distribution [24]
k̄ k −k̄
Pat (k) = e ,
(4.11)
k!
where k̄ = R/γ is the mean number of atoms in the MOT defined by the loading rate
and the linear loss-rate coefficient.
3
An accurate measurement of the saturation would furthermore require the knowledge of an exact
value for the saturation intensity. Measurements have, however, yielded varying results [9, 22]
25
The probability to detect n photons from an average of k̄ atoms is
Pph, at (n, k̄) =
∞
X
Pph Pat .
(4.12)
k=0
Including the distribution Pbg (n) of the background noise (which can be assumed to
be Poissonian as well), the total probability to detect n photons from a mean atom
number k̄ is [23]
n
X
P (n, k̄) =
Pbg (m)Pph, at (n − m, k̄),
(4.13)
m=0
i.e. the sum over all background and atom configurations that lead to n photons. For
large count numbers, this equation can√be approximated by a sum of Gaussians with
the Poissonian√
standard deviation σ = n (for the atom peaks, the standard deviation
√
√
is σk = n0 + n1 − n0 + · · · + nk − nk−1 because the sources of the photons are not
correlated). An exemplary plot of this probability distribution is shown in fig. 4.15a.
A useful feature of the Poissonian distribution is the time correlation function
hk(t + τ0 ), k(t)it = hk(t + τ0 ) · k(t)it − hk(t)i2t ,
(4.14)
where τ0 is the parameter of the function. It evolves for the underlying model of eq.
(4.11) (which describes the trapping of few atoms as a Markov process) as [23]
hk(t + τ0 ), k(t)it =
R −γτ0
e
.
γ
(4.15)
and allows a simultaneous measurement of the loading rate R and the linear loss-rate
coefficient γ. The time correlation function will be used in sec. 4.2.3.
Detection
This chapter describes the statistics for the detection of single atoms that evolve from
the considerations above. The first part will focus on an illustrative example; the
second part will focus on the general case.
The 3σ Decision Rule This first part considers a (reasonable but arbitrary) rule to
make the decision if a number of photon counts that has been measured comes from an
atom or is just background noise: If the probability distribution is a sum of Gaussians
with the mean values and standard deviations described in the previous section, the
threshold for a hit (H) must be set to nat − 3σat (where nat is the mean count rate
if one atom is in the MOT and σat is the standard deviation of the one-atom peak
in the histogram) to ensure that 99.87% of the atoms are detected. If the distance
between the atom peaks is large compared to the mean values, this is a reasonable
consideration (if it was not, a better approach would be to set the threshold a couple
of σbg s right of the background peak).
26
The probability P (H|0) for false positives4 , i.e. to falsely identify a backgroundscattering signal as a trapped atom, is the leakage of the probability distribution of
the background over the threshold. For the 3σ decision rule it is [7]
µat − µbg − 3σat
1
√
.
(4.16)
P (H|0) = Erfc
2
2σbg
Probability
This is illustrated in fig. 4.7. The black
area indicates the probability of false positives; its value is obtained with the com!
"
µ at − µ bg − 3σat
1
plementary error function Erfc = 1 −
√
Erfc
2
2σbg
Erf,
is the error function and
h whereErf i
1
√
1 + Erf x−µ
is the cumulative dis2
2σ
tribution function of the normal distribution.
√
With µbg = nbg and σbg = nbg for
800
1000
1200
1400
Counts
the background√and µat = nbg +∆nat and
√
σat = nbg + ∆nat (∆nat = nat − nbg Figure 4.7. Illustrative motivation of eq.
are the additional counts a single atom (4.16). The black area indicates the probproduces on top of the background) for ability for false positives.
a single trapped atom, the probability of
false positives becomes
!
√
√
nat − 3 nbg + ∆nat
1
p
P (H|0) = Erfc
.
(4.17)
2
2nbg
If there are p consecutive positive counts, the probability of a false positive is [P (H|0)]p .
In some cases, e.g. if the MOT is toggled between a detection and a capture cycle, a
single count result (e.g. the number of photons detected in one detection cycle) has
to give information about whether an atom was trapped or not. Fig. 4.8 shows the
minimum number of counts per atom so that P (H|0) for one single incidence is smaller
than a specified value. From eq. (4.17) it follows that, for a given P (H|0), this only
depends on the background scattering.
If, for example, 10 count results are measured per second (which corresponds to a
binning time of 100ms, or, when the MOT is toggled, a cycle time of 100ms), this adds
up to about a million counts per day. If only a couple of atoms per day are expected,
the number of false positives must be very low. To expect only one false positive,
P (H|0) needs to be smaller than 10−6 . Assuming that the background count rate is
known, fig. 4.8 provides information about how good the signal has to be to obtain
usable counting accuracies.
To interpret this information, and use it for very rare events (for old water samples,
the expected count rates are about one atom per hour), another approach, namely
4
Read: the probability that a hit (H) is detected under the condition that 0 atoms are trapped.
27
4
Counts per atom
10
3
10
P(H|0)=1e−2
P(H|0)=1e−3
P(H|0)=1e−4
P(H|0)=1e−5
P(H|0)=1e−6
P(H|0)=1e−7
P(H|0)=1e−8
P(H|0)=1e−9
2
10
0
10
1
10
2
10
3
10
Background counts
4
10
5
10
6
10
Figure 4.8. Minimum counts per atom for a number of background counts and
probabilities for false positives
Bayesian statistics and conditional probabilities, is useful. The probability that a hit
that was measured actually means that an atom was in the trap is [25]
P (1|H) =
P (H|1)Pat (1)
P (H|1)Pat (1)
≈
,
P (H)
P (H|0)Pat (0) + P (H|1)Pat (1)
(4.18)
where
√ 1
P (H|1) = Erfc −3/ 2 = 0.9987
2
Pat (1) = k̄e−k̄
Pat (0) = e−k̄
k̄ = R/γ.
(4.19)
This can be understood as the normalized probability to detect a hit under the condition that an atom was actually trapped; it will be motivated in the discussion of eq.
(4.22). As for very rare events Pat (k) decreases rapidly with k (i.e. it is very unlikely
to have two atoms in the trap), P (H) is approximately P (H|0)Pat (0) + P (H|1)Pat (1).
Bringing it all together, the probability is
P (1|H) =
0.9987 · k̄
.
P (H|0) + 0.9987 · k̄
(4.20)
Fig. 4.9 shows, for different loading rates and background counts, the minimum number of counts per atom to obtain a certainty of 99% that a positive result is a trapped
atom. The lifetime of atoms in the MOT was assumed to be 200ms (γ = 5).
28
4
10
Counts per atom
3
10
2
R=100/h
R=10/h
R=1/h
R=1/10h
R=1/d
R=1/3d
R=1/10d
10
1
10
0
10
1
2
10
10
3
4
10
Background counts
5
10
6
10
10
Figure 4.9. Minimum counts per atom for different background counts and loading
rates to obtain 99% certainty that a positive count is actually a trapped atom.
Optimal Threshold However, the 3σ rule is not necessarily the best decision rule.
To find out the optimal threshold, a more general approach is better: The probability
to have found an atom when n photon counts were measured is
P (1|n) =
P (n|1)Pat (1)
P (n|1)Pat (1)
≈
P (n)
P (n|0)Pat (0) + P (n|1)Pat (1)
2 /2σ 2
at
=
−1 −(n−µat )
σat
e
2
−(n−µbg )2 /2σbg
−1
σbg
e
· k̄
−1 −(n−µat )2 /2σat
· + σat
e
· k̄
2
(4.21)
(4.22)
,
where, again, the probabilities of more than one atom in the trap were neglected.
P(n)
Probability
Fig.
4.10 illustrates this formula:
P (n|1) is normalized (because P (n|1)
evaluates P (n) under the condition that
an atom is trapped), therefore, to obtain
the probability indicated by the gray arrow, P (n|1) needs to be weighted with
Pat (1). The probability now needs to be
set in relation to the total probability to
obtain a count result n, which is indicated by the black arrow in fig. 4.10 and
is accounted for by the denominator in
eq. (4.22).
900
P(n|1)Pat(1)
1000
1100
Counts
1200
1300
This equation can be evaluated for dif- Figure 4.10. Illustrative motivation of eq.
ferent signal qualities and yields the min- (4.22)
imum number of counts per atom to obtain a desired certainty that a hit was actually a trapped atom. This is done for typical
parameters of our MOT in fig. 4.11.
29
1
P(1|n)
0.8
0.6
0.4
R=100/h
R=10/h
R=1/h
R=1/10h
R=1/d
R=1/3d
R=1/10d
0.2
0
5400
5420
5440
5460
5480
5500
Counts n
5520
5540
5560
5580
5600
Figure 4.11. Probability to have found an atom for different count results and typical
count results for 40ms binning time, nbg = 5000 and ∆nat = 1000 and γ = 5. The
vertical lines indicate the count number where P (1|n) = 99%.
Optimal Detection Time for a Toggled MOT
If the MOT is toggled (as motivated in ch. 4.1.4), the loading rate is severely reduced
in the detection cycle. In the following it will be assumed, for simplicity, that an
atom (1) can only be captured during the dedicated capture cycle and (2) can only
be detected if it lives in the MOT until the end of the detection cycle. Then the
probability to capture and detect an atom is the probability that an atom reaches the
MOT in the capture cycle tcap /T times the probability for an atom to arrive at the
MOT in one cycle, to be captured, and to still be trapped at the end of that cycle.
The latter probability is determined by the linear loss rate of the MOT and the time
between the trapping of the atom and the next detection cycle:
Z tcap
e−γT γtcap
tcap 1
·
e−γ(T −t) dt =
e
−1 ,
(4.23)
Pdet =
T tcap 0
γT
where T = tdet + tcap is the length of one cycle and γ is the linear loss rate coefficient
from ch. 4.1.2. This probability can, for either the capture time or the detection
time held fixed, be optimized for different MOT loss rates, as shown in the left part
of fig. 4.12. The right part shows the probability Pdet for the optimized parameters
and different detection times. A quick-and-dirty approach to find the optimal toggling
settings would be to choose a suitable detection time to obtain a clear signal and then
set the capture time according to fig. 4.12.
For the purpose of counting rare single-atom events, a more sturdy analysis of the
statistics involved proves useful. A combination of eq. (4.23) with the results obtained
in the previous chapter yields the optimal settings for the toggling of the MOT. Two
more assumptions are necessary: (1) Atoms that are lost during the detection cycle
are neglected, i.e. the count rate for those atoms is the background count rate. (2)
30
1
10
1
Detection probability with optimal capture time
Optimal capture time [s ]
γ=1
γ=2
γ=5
γ=10
0
10
−1
10
−2
10
−3
10
−2
10
−1
Detection time [s]
10
0
10
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−3
10
γ=1
γ=2
γ=5
γ=10
−2
10
−1
Detection time [s]
10
0
10
Figure 4.12. Left: Capture time that maximizes Pdet for a given detection time and
different linear loss rates. Right: Pdet for the optimal pairs of detection and capture
time.
For simplicity, eq. (4.23) only affects the probability to have one atom in the trap in
(4.22), which originally evolves from (4.11), so that
Ptoggle (1) = Pat (1) · Pdet = k̄e−k̄ · Pdet .
(4.24)
The change in Pat (0) is small and therefore neglected. The optimization procedure
now goes like this:
1. Experimental results yield the values of
• the background count rate Γbg ,
• the additional count rate for one atom in the trap Γat , and
• the MOT’s linear loss-rate coefficient γ.
Further defined as fixed input parameters are
• the loading rate R (which is obtained from loading rates with enriched
samples and the appropriate adjustment for natural abundances or from
the considerations in 6.1), and
• the desired probability P (1|H) that a hit is actually an atom (in the following set to 99%)
and as variables to be evaluated
• the detection time tdet and
• the duration of one cycle T = tdet + tcap .
These values yield
• the mean photon counts for the background nbg = Γbg tdet and for one
trapped atom nat = nbg +Γat tdet and the corresponding standard deviations
under the assumption that the count rates are large so that eq. (4.13) can be
31
Figure 4.13. Ptoggle (H|1) plotted for different tdet and T and three background count
rates.
approximated by a sum of Gaussians with µbg = nbg , µat = nat , σbg =
√
√
and σat = nbg + nat − nbg , and
√
nbg
• the mean number of atoms in the trap (in the capture mode) without the
toggling k̄ = R/γ.
2. The probability
· k̄Pdet
2
−1 −(n−µat )2 /2σat
σat e
2 /2σ 2
at
P (1|n) =
−1 −(n−µat )
σat
e
2
−1 −(n−µbg )2 /2σbg
σbg
e
+
· k̄Pdet
(4.25)
is evaluated for a range of detection times tdet and cycle times T to obtain,
for each combination, the minimal threshold of counts n̂ to ensure that the
probability that a hit actually means that an atom is in the trap P (1|H) > 99%.
3. Each of these sets of tdet , T and n̂ is evaluated for the probability that an atom
that arrives at the MOT during one cycle is counted
n̂ − nat
1
Ptoggle (H|1) = P (H|1) · Pdet = Erfc √
· Pdet .
(4.26)
2
2σat
The combination of tdet and T with the highest Ptoggle (H|1) is the optimal setting.
This probability indicates the detection efficiency of the MOT. However, one
needs to keep in mind that this efficiency is not the total efficiency of the MOT:
It is independent of the capture efficiency, which affects the loading rate. An
improvement in the capture efficiency thus has to be accounted for by adjusting
the input parameters of this optimization procedure.
Applications The next section discusses the implications these considerations have
for our experiment. Fig. 4.13 shows Ptoggle (H|1) for different background count rates.
The values for the optimal settings are Ptoggle (H|1) = 65.8%, 58.7% and 53.5% for
32
Table 4.1. Optimal toggling times for different loading rates and nbg = 125kcps,
∆nat = 25kcps and γ = 5
R=1/min R=1/h R=1/d R=1/10d
tdet [ms]
13
16
18
19
T [ms]
81
91
97
100
Ptoggle (H|1) [ms]
61.2%
58.7%
57.0%
55.9%
Table 4.2. Optimal toggling times for different linear loss-rate coefficients and nbg =
125kcps, ∆nat = 25kcps and R =1/h
γ = 1 γ = 2 γ = 5 γ = 10
tdet [ms]
20
18
16
14
T [ms]
214
146
91
63
Ptoggle (H|1) [ms] 78.7% 71.4% 58.7% 46.9%
nbg = 75, 125 and 175kcps, respectively. The benefit of a lower background is obvious:
less counts are necessary to obtain a high certainty that a hit actually was an atom,
therefore a relatively short detection time can be chosen, which in turn increases Pdet .
Table 4.1 shows the optimal values for different loading rates of the MOT. The
differences are small, therefore settings that work with very old natural samples will
also yield fair results for younger samples. Table 4.2 shows the dependency on the
MOT’s life time. A longer life time (smaller γ) means that the capture cycle can
be longer, which affects Pdet . An improvement in this area can increase Ptoggle (H|1)
noticeably.
4.2.3. Results
Fig. 4.14 shows the fluorescence counts of single-atom signal as a function of time with
a binning time of 40ms. Trapped atoms reveal themselves as discrete peaks. Fig. 4.15a
shows a histogram of this signal. Clearly visible are the peak of the background signal
and, much smaller, the single-atom peak. The dashed line represents the theoretical
Counts per 40ms
1400
1200
1000
800
0
Figure 4.14. Exemplary
10
38
Time [s]
20
30
Ar single-atom signal obtained with a 62.5µm fiber.
33
0.1
Correlation
Occurrence
300
200
0.05
100
0
600
800
(a)
1000
1200
Counts per 40ms
0
1400
0
200
400
τ [ms]
600
800
0
(b)
Figure 4.15. (a) Histogram of the photon counts for a single-atom MOT of 38 Ar
atoms; the dashed line is the theoretical expectation (b) Correlation function of the
atom number using a simple threshold for the detection, exponential fit (solid line)
predictions from eq. (4.13). To make the calculation faster, Pph, at and Pbg were
approximated using a Gaussian distribution instead of the Poisson distribution.
It is evident that the measured peak that corresponds to one trapped atom is wider
than expected. This is due to the fact that, for this measurement, the fiber that feeds
the APD had a diameter of only 62.5µm, so the region that is projected onto the fiber
is smaller than the size of the MOT or, for one single atom, the area about which it
moves. This accounts for a decrease in the imaging efficiency ηimg , but also for the fact
that the assumption underlying eq. (4.10), that the photon count rate only depends
on external factors that change n̄ but not the distribution itself, is not applicable here;
the distribution is much wider than a Poissonian.
For fig. 4.15b, the fluorescence data is separated by a threshold into two states,
with one or zero trapped atoms. On this data, a correlation analysis revealed, using
eq. (4.15), a trapping time of τ = 1/γ = 153ms and a loading rate of R = 0.67/s.
The same analysis for 39 Ar data recorded under the same conditions reveals a trapping time of τ = 142ms. Considering that these results are somewhat dependent on
the chosen threshold, this means that the MOT lifetimes are comparable for both
isotopes.
Toggled MOT
Figures 4.16 and 4.17 show first results for a 38 Ar MOT in two-cycle operation. The
signal quality is better than in fig.4.14 not only because of the toggling but also
because of several signal improvement measures such as a larger fiber and a reduced
pressure in the MOT chamber. In a 120ms long capture cycle, a high laser detuning
of -20MHz implies a high capture velocity; in a subsequent 40ms long detection cycle,
34
a lower laser detuning of -4MHz improves the atoms’ scattering rate. The APD only
records data in the detection cycle, as fig. 4.16b shows. Fig. 4.16a shows the same
signal on a shorter timescale, where the detection gaps were, for visual pleasure, filled
with a signal cloned from the leading detection cycle. The histogram in fig. 4.17a
shows nicely separated peaks for each number of trapped atoms. In fig. 4.17b, the
relative probabilities (which correspond to the area under the histogram peaks and
are calculated from the Gaussian fits) are plotted and compared with a Poissonian fit.
The deviations are probably due to imperfect fits and a noisy signal especially for the
peaks corresponding to more than two trapped atoms.
The standard deviation of the background peak obtained from the fit is about 35%
larger than the expectation value of a Poissonian distribution. A possible reason is
that the MOT is power-stabilized to only one beam. Shifts in the polarization could
change the relative power of the beams and therefore the stray light. The difference
of the standard deviations
√ of the first three single-atom peaks is slightly lower than
the expectation value ∆nat , but lies (for each pair of peaks individually) within the
error of the fit. It is approximately constant for these first three peaks, as expected
from the model. The fourth single-atom peak is an outlier, but its signal-to-noise ratio
is much worse than that of the other peaks, therefore the fit’s error is much higher.
First results show an increase by a factor 4 to 7 in the MOT’s loading rate and a
strong increase in the signal quality (figs. 4.16 and 4.14 are, however, not directly
comparable because the latter was measured with another detection setup). This is
to be expected: from eq. (2.9) it follows that, assuming a saturation of 1, there is a
factor 13 between the scattering rates obtained with -4MHz and those obtained with
-20MHz detuning. This is a huge difference, because this factor only affects the signal,
not the background. The increased loading rate easily compensates the loss from eq.
(4.23), therefore these results make a very strong case for toggling the MOT.
It is also noteworthy that the count rates are sufficient to measure very rare events,
as fig. 4.9 shows (it has to be noted, though, that the larger standard deviation of the
background means that the statistics employed to obtain the figure are not perfectly
applicable).
35
Counts per 40ms (interpolated)
10000
9000
8000
7000
6000
5000
4000
0
10
20
30
40
50
Time [s]
60
70
80
90
100 0
50
100
150
Occurrence
200
250
(a)
Counts per ms
200
150
100
50
0
0
100
200
300
400
Time[ms]
500
600
700
800
900
(b)
Figure 4.16. (a) Interpolated signal with 40ms binning time (the signal is cloned
to fill the capture gaps), histogram of the count rates. The histogram shows clearly
separated peaks, which are evidence for single atoms. (b) Fluorescence signal obtained
in 2-cycle mode with a gated APD. The variation in the height of the peaks indicates
trapped atoms.
250
0.4
0.3
Probability
Occurrence
200
100
0.2
0.1
0
4000
(a)
5000
6000
7000
8000
Counts per 40ms
0
9000
0
1
2
3
Number of trapped atoms
4
(b)
Figure 4.17. (a) Histogram of the photon counts for a few-atom MOT of 38 Ar
atoms with switching between detection and trapping detunings; the dashed line is a
Gaussian fit (b) Normalized histogram of the number of atoms in the trap, Poissonian
fit (black line) for k̄ = 1.26
36
5. Efficiencies and Loss Channels
5.1. Spectroscopy setup
This chapter will widely use spectroscopic measurements done in the MOT chamber.
In the following, the setup is explained briefly.
A spectroscopy setup in the MOT chamber, using the fiber couplers that usually
feed the MOT, provides access to one truly transverse laser beam and two laser beams
angled at 45◦ relative to the atoms’ direction. For v⊥ vk , which holds for most of
our measurements, spectroscopy
√ using one of the tilted beam gives the longitudinal
velocity profile compressed by 2. The Avalanche Photo Diode (APD) feeds a rate
meter set to 1ms integration time. The signal from the rate meter is then recorded
by an oscilloscope. The spectroscopy was performed on 40 Ar, so that no repumping
was necessary; therefore, the beams from the commercial TA that usually feeds the
repumpers could be employed by scanning its beat lock. The results from the scans are
fluorescence signals as a function of time; to obtain velocity profile, the time axis has to
be calibrated. The velocities are directly related to the frequency of the spectroscopy
laser via the Doppler shift (v = δ/k).
To obtain a relative calibration (i.e. frequency/time), the two Zeeman-slower repumpers were used: If they are scanned, they yield a spectroscopic result that shows
two equal but displaced profiles. From the displacement and the difference of their
frequencies (see fig. A.1), the relative calibration can be calculated.
To obtain absolute velocities, the transverse beam was used: The transverse velocity
profile is symmetric, therefore its maximum provides an absolute zero1 .
5.2. Source
ICP The closed transition this experiment uses starts from the metastable level 1s5.
Several excitation techniques have been tested for this experiment; an inductivelycoupled plasma (ICP) has produced the best results. In this setup, RF current running
through a coil wound around a small glass tube attached to the gas outlet heats the
electrons in a plasma. In this plasma, the atoms populate a large variety of energy
levels, most of which decay very fast. Some of these decays end in metastable states;
due to their comparatively long lifetime, they will be substantially populated. A
1
For this calibration, the atom beam is assumed to be perfectly perpendicular to the laser beam.
This is, however, not necessarily the case and that might be a reason for some problems with the
calibration that will arise later in this chapter.
37
CHAPTER 5. EFFICIENCIES AND LOSS CHANNELS
detailed characterization of this experiment’s ICP source can be found in [8], therefore
this section will be kept brief.
Inductive Coupling The initial ignition in ICPs is not induced by inductive coupling
but by capacitive coupling from the potential difference between different parts of the
coil (E-mode). To start inductive coupling (H-mode), a minimum RF power is required
[26]. The transition between these two modes is discontinuous: The coupling efficiency
of the plasma, i.e. the ability to absorb RF power, increases with the electron density
ne . For small ne , this increase is linear. When the RF current reaches a certain level
that is required to maintain an H-mode discharge, a slight increase in the electron
density leads to a better coupling efficiency which in turn increases, again, the electron
density. This process stops when, due to the skin effect, the linearity of coupling
efficiency and electron density breaks down and a new stationary state is reached.
The mode transition is accompanied by a large increase in the plasma’s optical
emission. This effect has been observed with this experiment’s ICP source, fig. 5.1
shows photos of the two modes. The ignition of the H-mode can be introduced by
increasing the RF power or by increasing the pressure, as fig. 5.2 shows.
(a)
(b)
Figure 5.1. (a) ICP source in E-mode. The plasma is confined in the glass tube. (b)
ICP source in H-mode. A bright plasma is visible in front of the glass tube.
Velocity Distributions Measurements show that the velocity distribution of this experiment’s source can be described in longitudinal direction by a Maxwell-Boltzmann
distribution [8]
2
4 vz2 − v̂vz2
z
(5.1)
fz (vz ) = √ 3 e ,
π v̂z
where v̂z is the most probable longitudinal
p velocity, from which the mean longitudinal
velocity can be obtained as v̄z = 2v̂z · 1/π. In transverse direction it is described by
a Gaussian distribution
v2
− r2
1
2σvr
fr (vr ) = √
e
.
(5.2)
2πσvr
Typical parameters of our source are a metastable flow of Ṅ ≈ 1014 atoms
, a metastable
s
14 atoms
flux density in forward direction of q(0) ≈ 10 s · sr [8], a mean longitudinal velocity
of v̄z ≈ 430m/s and transverse width of σvr ≈ 100m/s.
38
5.3. COLLIMATION
0.15
−5
p = 1.10⋅10 mbar
−5
p = 1.80⋅10 mbar
−5
p = 1.90⋅10 mbar
p = 2.50⋅10−5mbar
Fluorescence [a.u.]
p = 4.10⋅10−5mbar
p = 5.50⋅10−5mbar
0.1
−5
p = 5.70⋅10 mbar
p = 8.70⋅10−5mbar
0.05
0
0
500
1000
Longitudinal velocity [m/s]
1500
Figure 5.2. Longitudinal velocity profiles in the MOT chamber for different plasma
chamber pressures. At 5.5 · 10−5 mbar, the plasma starts to burn in H-mode.
Velocity profiles will be an important tool in the following; therefore a word of
caution is appropriate: The Maxwell distribution in longitudinal direction is only valid
in the source. Most of the profiles that will be discussed later are, however, taken in
the MOT chamber, which is about 2.2m behind the source. Not only the height of the
signal changes on the way there, but also the distribution: as longitudinally slower
atoms are more likely to miss the MOT chamber due to their transverse velocity, the
longitudinal velocities will be shifted towards faster velocities, as fig. 5.3 shows. The
resulting profile is not Maxwellian and depends on the width of the transverse velocity
profile.
5.3. Collimation
The MOT chamber is about 2.2m away from the plasma and the trapping region has
a radius of about 16mm. Assuming straight trajectories, only atoms with v⊥ /vk <
16mm/2.2m = 1/138 can reach the trapping region. From these considerations alone—
the situation worsens because the Zeeman slower longitudinally slows the atoms down
before they reach the MOT—it is obvious that, with typical velocity distributions of
our source being v̄k ≈ 430m/s and v̄⊥ ≈ 100m/s, only a small fraction of the atoms can
reach the MOT area. This issue is addressed by a collimator that employs transverse
Doppler cooling to reduce the atoms’ transverse speed. The design with tilted mirrors
that this experiment uses is described in detail in [8]. In short, the Doppler shift’s
dependency on the angle between the laser and the atom’s velocity is used to keep the
laser resonant with a large class of transverse velocities.
39
0
MOT
Source
Occurrence (normalized)
Occurrence (normalized)
MOT
Source
200
400
600
800
1000
1200
−10
−500
0
10
0
Transversal velocity [m/s]
500
Figure 5.3. Simulated velocity profiles directly behind the source and in the MOT
chamber located 2.2m behind the source. Because longitudinally slow atoms have
a longer time of flight, they are more likely to be lost due to transverse drift. The
velocity profile in the MOT chamber is therefore transversely much narrower and
longitudinally faster and non-Maxwellian.
5.3.1. Velocity Profiles
The collimation does not only change the transverse velocity profile, but it also affects
the longitudinal profile: Because of the finite spontaneous decay rate, the atoms can
only be decelerated with a maximum deceleration amax that is defined in eq. (5.6).
For Argon, amax ≈ 2 · 105 sm2 . If the atoms are longitudinally fast, the time they spend
in the collimator decreases and thereby the time they experience the cooling force.
Therefore, the collimation is less efficient for longitudinally fast atoms. As the MOT
is about 2.2 meters away from the collimator, atoms that have not been cooled are
likely to be lost transversely. The consequence is a large increase in the flow of atoms
that have longitudinal velocities that are being affected by the collimator. Fig. 5.4
shows velocity profiles for different detunings of the collimator. The left figure shows
measured results, the right figure shows simulation results obtained with the simulation
described in [8]. The qualitative agreement is obvious (the quantitative differences are
most likely due to different initial velocity distributions and different settings for the
mirror angle).
The resonance condition for the collimator is [8]
!
−k(vz sin β − vr cos β) ≈ −k(vz β − vr ) = −δ,
(5.3)
where β is the laser angle and δ is the detuning. This can be solved for
vz =
δ/k + vr
.
β
(5.4)
Under the condition that vr is limited to transverse velocities that are coolable, the
behavior in fig. 5.4 can be explained: The more positive the detuning, the more is
the resonant class of longitudinal velocities shifted to greater velocities. As can be
40
5.3. COLLIMATION
Table 5.1. Single-atom
mation
Non
ton [s]
38
Ar 675±26 1200
39
Ar 295±17
600
data for consecutive measurements with and without colliRon [s−1 ]
0.56±0.02
0.49±0.03
Noff toff [s]
24±5 1200
20±4 1800
Roff [s−1 ]
η = Ron /Roff
0.02±0.004
28±7
0.011± 0.003
44±12
seen both in the simulation and the experimental data, a very high detuning can
decollimate the atoms of a certain class of longitudinal velocities.
The optimal setting, if only the longitudinal velocity is considered and not the
transverse efficiency of the collimator, would be to push the longitudinal velocities
that are slightly lower than the capture velocity of the Zeeman slower: Those are the
atoms that spend the least amount of time in the slower and are therefore less likely
to be lost to transverse drift.
0.25
Fluorescence [a.u.]
0.2
0.15
−40MHz
−10MHz
−5MHz
0MHz
5MHz
15MHz
25MHz
40MHz
55MHz
Occurrence [a.u.]
δ=44.2MHz
δ=34.2MHz
δ=24.2MHz
δ=14.2MHz
δ=4.2MHz
δ=−5.8MHz
δ=−35.8MHz
0.1
0.05
0
0
500
1000
1500
0
200
400
600
800
1000
Figure 5.4. Velocity profiles for different collimator detunings obtained by fluorescence measurements in the MOT chamber. Left: Measurement. Right: Simulation
result assuming a Maxwell-Boltzmann distribution in the source.
5.3.2. Efficiency for
39
Ar
The collimation efficiency is one of the few accessible parameters to directly compare
the efficiency of laser-cooling for 39 Ar and 38 Ar. It was determined by analyzing singleatom detection rates of two consecutive measurements, where the collimation beam
was blocked in the latter. For 38 Ar, the Zeeman-slower laser intensity was reduced
to obtain MOT loading rates suitable for single-atom analysis2 . Table 5.1 shows
efficiencies that are comparable within their statistical error margins.
2
This could potentially introduce a systematic error because the effect is not the same for all
velocities, as discussed in ch. 5.4.4.
41
5.4. Zeeman Slower
Atoms from an ICP source have a mean longitudinal velocity of several hundred meters per second, while a MOT’s maximum capture velocity is typically about 40 ms .
To effectively trap them, the atoms must therefore be slowed down before they enter the MOT’s trapping region. The Doppler shift for atoms at 600 ms is about
δDoppler ≈ 740MHz; this must be compensated during the deceleration. Two techniques commonly used for neutral atoms are chirped slowing and Zeeman slowing.
A chirped slower sweeps the laser’s frequency to compensate for the Doppler shift.
Consequently, the atoms arrive at the MOT in bunches. The problem with that
approach is this: Whether an atom is resonant with the laser depends on its speed,
but not on its position, so that most atoms will be very slow far away from the detection
region (in our case the MOT). Usually, the detection region is relatively small, so that
a transverse velocity of a few ms can be enough for slow atoms to miss it [27].
It is therefore better to slow the atoms down shortly before they reach the detection
region. A Zeeman slower uses a magnetic field to compensate for the Doppler shift,
therefore the resonance condition depends on the speed and position of the atom.
Losses due to transverse velocity are still prevalent, but strongly reduced compared to
a chirped slower.
5.4.1. Theory
For atoms to be resonant with a counterpropagating laser in a magnetic field, the frequency shifts induced by the Zeeman effect and the Doppler effect have to compensate
for the laser’s detuning δ0 relative to the transition frequency:
!
∆EZe (r)
~
µB
= δ0 − klaser v +
B(r) · (MJ,e gJ,e − MJ,g gJ,g ),
~
0 = δ(r, v) = δ0 − klaser v +
(5.5)
where the last term becomes (MF,e gF,e −MF,g gF,g ) for atoms with hyperfine splitting.
B must therefore be proportional to klaser v − δ0 . An atom that is resonant with the
laser along its way experiences a constant deceleration
a ≤ amax = ~kγ/(2m) ·
s
,
1+s
(5.6)
2
where s = s0 /(1 + (2δ/γ)
p ) is the on-axis saturation. It follows then from basic
kinematics that v(z) = v0 1 − 2az/v02 . The optimal B-field is therefore
s
2a
B(z) = Bb ± B0 1 − 2 · z,
(5.7)
v0
42
5.4. ZEEMAN SLOWER
where the plus and minus signs are valid for σ + light (decreasing-field slower ) and σ −
light (increasing-field slower ), respectively. For σ + light, slowed atoms are resonant
with the laser, which has two severe disadvantages for our experiment: the Zeeman
slower can interfere with the MOT operation, as their lasers’ detuning is similar; furthermore, when the resonance is heavily power-broadened, slow atoms can be further
decelerated or even turned around. Because of their residual transverse velocity, slow
atoms are more likely to leave the trappable region. For σ − light, the laser is resonant with fast atoms in the absence of a magnetic field, therefore it does not interfere
with the MOT operation. The magnetic field, however, has its maximum near the
MOT chamber; a residual magnetic field can leak into the MOT. In our setup, a
compensation coil prevents this leakage.
Length To slow an atom from a velocity v to a final velocity vfinal , the slowing
2
length L is (assuming a constant deceleration a) L = (v 2 − vfinal
)/2a. For a = amax ,
the minimum length to slow Argon atoms from 700m/s to 30m/s is about 1.1m. In
practice, a ≈ 2/3amax [28], therefore this experiment uses a substantially longer slower.
Capture velocity and final velocity The capture velocity for an increasing-field
slower is approximately given by vcapture = δ0 /k. This term does, however, not account
for power broadening, therefore, depending on the saturation, faster atoms can be
captured as well.
For σ − light, there are two ways an atom can go out of resonance: (1) an atom
reaches the end of the Zeeman slower, therefore the resonance velocity increases
strongly as the B-field decreases to zero, or (2) the deceleration needed for the Doppler
shift to cancel out the Zeeman shift is greater than amax (so the resonance velocity
decreases faster than the velocity of the atoms) [29].
In the first case, the final velocity is approximately given by the resonance condition
at the B-field maximum:
δ0
µB (MJ,e gJ,e − MJ,g gJ,g )
Bmax +
~k
k
This approximation does, again, not account for power broadening.
vfinal =
(5.8)
The second case arises from a condition that is obtained by solving eq. (5.5) for v
and differentiating for t ([30]):
µB (MJ,e gJ,e − MJ,g gJ,g )
·
~k
dB dz · v ≤ amax
(5.9)
From this equation it follows that slow atoms can tolerate steeper B-field gradients.
The consequences of this limit are discussed in detail in 5.4.3.
43
5.4.2. Experimental Realization
Bz [gauss]
300
100
0
0
y
400
100
B [gauss]
500
Bz [gauss]
This experiment uses a self-built 1.78m long water-cooled Zeeman slower with a total
of 2544 windings of 4 × 1mm copper wire that is operated with a current of about
12.5A. The resulting B-field is shown in fig. 5.5. This figure shows the total B-field
of our experiment in longitudinal direction, including the fields of the MOT and the
compensation coil, which reduces leakage of the Zeeman slower field into the MOT.
−100
180
−100
185
190
z [cm]
195
200
−10
−5
0
y [cm]
5
10
200
100
0
−0.5
0
0.5
z [m]
1
1.5
2
Figure 5.5. B-field produced by Zeeman-slower coils, compensation coils and MOT
coils with IZSL = 12.6A and IMOT = 10A.
5.4.3. Simulation
A Monte-Carlo simulation, explained in detail in Appendix A.4.1, has been set up to
calculate atom trajectories by adding single scattering events. The following chapter
will use the results of this simulation to explain important processes in the Zeeman
slower; the next chapters will resort to this simulation to illustrate measurement results.
Longitudinal Cooling
The longitudinal deceleration is not constant as was assumed in the theoretical considerations; it rather increases towards the end of the slower. This is beneficial because it
reduces the time the atoms spend in the slower, which in turn reduces the transverse
broadening of the atom beam. The velocity along the slower is plotted for a set of
initial velocities in figure 5.6. Additionally, there is an exemplary velocity distribution
of the flow (atoms/s) for a sample of 500.000 atoms, assuming a Maxwell-Boltzmann
velocity distribution in the source.
In this example, the atoms are slowed to about vfinal = 30m/s. The capture velocity
is about 650m/s. Atoms that are trapped along the way all have the same final
velocity; those that are initially slower than vfinal can be seen to be slowed further
down or even turned around. At the slower’s exit, the light is, again, resonant with
atoms at 650m/s, which is clearly visible in the velocity profile plotted on the left:
44
5.4. ZEEMAN SLOWER
Figure 5.6. Simulated atom trajectories in a Zeeman slower. The dots indicate
scattering processes. Plotted on the same axis is a simulated velocity profile of the flow
of atoms in the MOT in arbitrary units, assuming a Maxwell-Boltzmann distribution
in the source.
there is a dip slightly above and a peak slightly below 650m/s, resonant atoms are
decelerated down to the point where they go out of resonance, again.
For a larger detuning δ0 of the laser, both the capture velocity and the final velocity
increase. Now the considerations about eq. (5.9) become important: The gradient of
B increases towards the slower’s exit. Furthermore, the B-field is not perfectly smooth
but rather stepped. Figure 5.7 illustrates the effect: As they approach the steep field
gradient with greater speed, some atoms cannot follow the B-field and leave the cooling
process early. This results in a final velocity distribution that has two peaks, one at
vfinal and one broader peak at a slightly higher velocity. For high detunings the second
peak can even become greater than the peak at vfinal . The right hand side of fig. 5.7
shows a measured velocity distribution that exhibits just this behavior.
Another parameter that is important in this context is the saturation: In eq. (5.6),
the maximal deceleration amax depends on the laser beam’s saturation. To each longitudinal position can be assigned a maximal velocity that can be slowed, i.e. kept
in resonance with the laser. Fig. 5.9 shows these velocities for a set of saturations.
There are two critical regions:
45
800
600
400
200
0
0
1
2
0
200
400
z [m]
600
800
1000
1200
Figure 5.7. Left: Simulated atom trajectories in a Zeeman slower. Plotted in the
same figure is a simulated velocity profile of the flow of atoms in arbitrary units,
assuming a Maxwell-Boltzmann distribution in the source. Leakage of atoms before
they are slowed down to their final velocity is dominant, as the characteristic double
peak in the velocity profile indicates. Right: Measured velocity profile that shows
similar behavior.
1. At the very beginning of the Zeeman slower, there are rather steep B-field gradients due to the fact that the number of windings increases in integer steps.
This problem can be addressed either by using more windings (and a smaller
current to obtain the same magnetic field) or by using a different design with
a single-layer helix with variable pitch [31]. Both of these approaches are, however, accompanied by a greatly increased mechanical effort and have not been
deemed necessary for this experiment. Fig. 5.8 shows simulated trajectories for
two saturation settings. They behave as expected: there is considerable leakage
at the steeper field gradients; for low saturation, the slowable velocity is slower
than the actual resonant velocity at many gradient maxima, for high saturation,
the resonant velocity is lower than the slowable velocity for most wiggles except
the first few.
2. The very end of the Zeeman slower should have the steepest field gradient according to eq. (5.7). For high detunings, the final velocity of the slowed atoms
is comparable to the slowable velocity; this situation is described above.
Transverse Heating
From the random nature of spontaneous emission it follows that an atom does a
random walk in transverse direction [32].
rms 2
(v⊥
)
46
=
2
σv,⊥
v2
= α rec · N = α
3
~k
m
2
N
,
3
(5.10)
5.4. ZEEMAN SLOWER
s=0.5
s=1
s=2
s=5
s=10
B
900
800
700
500
450
400
350
600
300
500
250
400
200
300
150
200
100
100
50
0
B [gauss]
Maximal slowable longitudinal velocity [m/s]
Figure 5.8. Simulated atom trajectories of two saturation settings for the cooling
laser. Black lines indicate atoms that reach the MOT area, gray lines indicate atoms
that hit a wall.
0
0.2
0.4
0.6
0.8
z [m]
1
1.2
1.4
1.6
1.8
0
Figure 5.9. Maximal slowable velocities for different laser saturations. The bold line
indicates the B-field.
where N is the total number of photons scattered. The factor α = 9/10 corrects the
expression for isotropic scattering for the dipole radiation pattern due to the atoms’
polarization. The factor is, however, small and can usually be neglected [32]. In a
Zeeman slower, the number of scattering processes depends on the atom’s initial velocity (faster atoms are in resonance with the laser and thus being cooled right from
the start whereas slower atoms are far out of resonance at first and only later begin to
scatter substantial numbers of photons). For our slower, the mean number is on the
order of 30000 scattered photons, which leads to σv,⊥ ≈ 1.1m/s. A simulation using a
Maxwellian longitudinal velocity distribution and no initial transverse velocity results
in σv,⊥ = 0.9m/s. However, in our slower, the atoms have a broad initial transverse
velocity distribution (σv,⊥ ' 2.6m/s [10]). The final profile is, depending on the longitudinal velocity profile of the source, relatively narrow due to geometric restrictions,
but it is significantly widened by the random walk, as exemplary simulation results
show in fig. 5.10. The dashed line shows the transverse velocity profile in the MOT
47
number of atoms [a.u.]
trapping area obtained from typical source parameters without random scattering, its
width is σv,⊥ = 0.5m/s. The straight line shows the result for the same initial conditions, but with random scattering. The width is 0.9m/s. This is significant, since
the main loss channel for atoms is transverse loss. It also suggests that the final velocity distribution is determined rather by the randomization on the last centimeters,
when most of the scattering processes happen than by the initial transverse velocity
distribution (which does, however, affect the number of atoms that reach the end of
the slower).
A focused laser beam has been reported to transversely cool the atoms in a Zeeman
slower[32]. The simulation shows a difference in the transverse velocity trajectories
between a focused and an unfocused beam, but the effect is so small that it need not
be considered in the following.
−5
0
transversal velocity [m/s]
5
Figure 5.10. Simulated transverse velocity distributions of longitudinally slow atoms
in the MOT region with (straight line) and without (dashed line) random walk and
their Gaussian fits
5.4.4. Characterization
Laser Saturation
Single-atom count rates of 39 Ar have been measured for various cooling-laser powers
to obtain the slowing efficiency’s dependence on the laser saturation. Fig. 5.11 shows
the results. The power is given in terms of the peak-to-peak voltage of a FabryPerot interferometer signal that scales linearly with the power. The data shows that
there is a maximum in the efficiencies for a distinct—although very broad—region
of saturation intensities and that a higher laser power is not necessarily better. The
simulation could qualitatively reproduce the measured behavior and to illustrate the
dominant processes; they are shown in the small insets of fig. 5.11.
For very low saturations, the atoms are faster than the slowable velocity all the
way along the slower (see sec. 5.4.3) and leak out of the slowing process very early,
48
5.4. ZEEMAN SLOWER
as shown in the left inset. For higher saturations, this leakage decreases, until the
efficiency reaches a maximum. With increasing saturation, the final velocity of the
atoms decreases, and at a certain point, losses due to the atom’s residual transverse
velocity become dominant. This situation is shown in the middle inset, which shows
velocity-trails for the last few centimeters in front of the trapping region of the MOT.
Gray lines indicate atoms that reach the MOT region; black lines are atoms that miss
that region due to transverse drifts. The lower the final velocity, the more dominant
becomes the drift loss. For even higher laser powers, the saturation broadening is so
wide that atoms are stopped and turned around (and then lost transversely) before
they enter the MOT region, as shown in the right inset.
Figure 5.11. 39 Ar count rate for various cooling powers and a fixed detuning. Denoted on the power axis is the peak-to-peak voltage of a Fabry-Perot interferometer
signal that scales with the cooling power. The gray line is a guide to the eye. The
insets are explained in the text.
Detuning
Fig. 5.12 shows measured velocity profiles for different detunings of the Zeeman slower
beam. The behavior can be explained with the knowledge from the previous chapters:
The final velocity increases with the detuning, as discussed in ch. 5.4.1. For large
detunings, the effect explained in 5.4.3 is visible: the atoms are too fast to follow the
steep B-field gradient introduced by the discrete windings and are lost for the slowing
process, this leads to the pronounced side lobes at large detunings.
However, the final velocity for these large detunings is too large for the MOT to
capture the atoms, therefore detunings that lead to lower final velocities are the more
interesting part. For their discussion, however, fig. 5.12 has two major shortcomings:
49
0.4
δ=−742MHz
δ=−772MHz
δ=−792MHz
δ=−812MHz
δ=−822MHz
δ=−832MHz
δ=−842MHz
δ=−852MHz
δ=−862MHz
δ=−872MHz
δ=−892MHz
0.35
100
0.25
0.2
0.15
Final velocity [m/s]
0.3
80
60
40
Meas.
Sim. 40mW (s=4)
Sim. 80mW (s=9)
Sim. 120mW (s=13)
20
0.1
0
−900
−880
0.05
0
−100
−860
−840
−820
ZSL detuning [MHz]
−800
−50
0
50
100
150
Figure 5.12. Zeeman-slowed peaks for different detunings. Saturation in the fluorescence detection caused the equal height of the peaks (see fig. 5.13). Fits to the
peaks gave the final velocities that are shown in the inset. Note that these values have
relatively high errors due to drifts in the calibration of the velocity axis.
a drift of the calibration during the measurement3 and the saturation of the rate
meter. The discussion will therefore be based on simulation results. They have been
qualitatively confirmed by other measurements, but those measurements are lacking
a velocity calibration.
250
90
δ=−812MHz
δ=−822MHz
δ=−832MHz
δ=−842MHz
δ=−862MHz
80
1
150
100
Efficiency
70
Occurrence [a.u.]
Occurrence [a.u.]
200
60
50
0.5
0
40
30
−860 −840 −820 −800
Detuning [MHz]
20
50
10
0
0
10
20
30
40
50
60
70
80
0
1
1.2
1.4
z [m]
1.6
1.8
2
Figure 5.13. Left: Simulated velocity profiles of the slowed atoms. Right: Longitudinal position of atoms that are transversely lost, corresponding efficiency
Fig. 5.13 shows the simulated behavior at low final velocities: The slower the atoms
come out of the Zeeman slower, the more likely they are to be lost transversely. In the
figure on the right, the longitudinal positions of transversely lost atoms are plotted
(i.e. the position where they were lost). It is obvious that for final velocities less than
about 20m/s, the peak shortly before the MOT at 1.9m contains basically all atoms
(there are hardly any slowed atoms in the velocity profile on the left). The inset shows
3
The drift has been assumed to be linear and has been compensated, although with mixed results,
as the leakage of the peaks to negative velocity indicates.
50
5.4. ZEEMAN SLOWER
the efficiency for the different detunings, i.e. the fraction of potentially slowable atoms
that actually reach the MOT chamber (the efficiency will be discussed in more detail
in the next chapter).
Efficiency
Fluorescence [a.u.]
The slower’s efficiency indicates how many of the atoms that can potentially be slowed
actually reach the MOT with the slower’s final longitudinal velocity and have not been
lost to transverse drifts. The flow of slowed atoms can be compared to the reduced flow
of faster atoms to analyze the efficiency. The upper part of fig. 5.14 shows two velocity
profiles for a laser detuning δ = −831MHz; the straight line is a slowed profile, for the
dashed line the slower beam was blocked. It is evident that the slower captures atoms
with longitudinal velocities of up to 700m/s and slows them down to about 50m/s.
The profiles were obtained using fluorescence spectroscopy, which is sensitive to the
Slower on
Slower off
0.2
0.1
0
0
500
1000
Flow [a.u.]
40
1500
Slower on
Slower off
30
20
10
0
0
500
1000
1500
Figure 5.14. To measure the slower’s efficiency, the flow of slow atoms (black area) is
compared to the flow of faster atoms that are missing (gray area). The difference is the
loss during the slowing process. A fluorescence measurement (upper figure) is sensitive
to the atom density; to obtain the flow, the fluorescence signal has to be weighted with
the velocity (lower figure). The vertical line indicates the capture velocity.
atom density ρ = N/V . The important parameter for this experiment, however, is not
the atom density but the flow Ṅ of atoms that reach the MOT’s trapping region in a
given time interval. Therefore, the density for a given velocity class has to be weighted
with its velocity. This is obvious in the one-dimensional case: If atoms reach a target
at a rate N/t, their mean spread is ∆z = vz t/N ; thus, the flow is N/t = vz /∆z = vz ρ,
where ρ is the one-dimensional particle density N/z. If all atoms were slowed down to
the final velocity, the flow would stay the same, but the integral over the fluorescence
signal of the slowed peak would be much larger than the integral over the lacking
faster atoms because the slower atoms are much denser4 .
4
Another way to look at this is to consider the time the atoms spend in the fluorescence volume,
which is inversely proportional to their velocity. Assuming the same flow of atoms at two discrete
velocities, the slower atoms would therefore scatter more photons.
51
The lower part of fig. 5.14 shows the measured distributions weighted with the
velocity. The ratio of the black area, which indicates the flow of slowed atoms5 , and
the gray area, which indicates the initial flow of atoms longitudinally slower than
700m/s minus the flow of atoms with said longitudinal velocity that have not been
slowed, suggests an efficiency of 21%.
This is, however, a flawed argument: Theoretically, all atoms slower than the capture
velocity of the Zeeman slower, which for this detuning is about 700m/s, can theoretically be slowed down. The atoms that have velocities far greater than the slowed
peak but smaller than the capture velocity should therefore also be considered lost, although the reason for their loss was not the transverse drift. The flow that the slowed
peak should actually be compared to is therefore the total flow of atoms with velocities
smaller than the capture velocity (which, in the figure, would resemble the gray area
extended to the base line)6 . The efficiency then becomes only 7%, which is very inefficient.
1
Efficiency
The shape of the velocity profiles suggests that the settings for the slower
might not have been optimal: similar results have been obtained in the simula0.5
tion when the laser saturation was set to
rather low settings. As explained in chapδ=−805MHz
δ=−812MHz
ters 5.4.3 and 5.4.4, the amount of leakδ=−831MHz
age due to steep magnetic field gradients
0
depends heavily on the laser beam’s sat0
2
4
6
8
10
uration. The tail at the right edge of the
Saturation
slowed peak and the high leakage near
the capture velocity are characteristic for Figure 5.15. Simulated slower efficienlow saturations.
cies for different detunings and saturations.
Fig. 5.15 shows simulated efficiencies For even higher saturations, the efficiency
for different detunings and saturations. would start to decrease because the atoms
For low saturations, leakage at steep field would have slower longitudinal velocities
gradients is the dominant loss channel. and thus be more likely to be lost because
For larger detunings, the capture veloci- of transverse drift.
ties and final velocities increase, but also
the resonant velocity at all points in the slower. Fig. 5.9 explains, why the efficiency
for high detunings is lower at low saturations: if the velocity trajectory was plotted in
that figure, it would be higher for high detunings and therefore be above the maximum
slowable velocity more often.
If the saturation is high enough, the effect of the increased longitudinal velocity for
higher detunings reverses: The dominant loss channel is transverse drift. Faster atoms
5
6
A possible error arises from the fact that the profiles are not corrected for power broadening.
This definition of the efficiency was also used in fig. 5.13 and will be used in the following.
52
5.5. LOSS CHANNELS OF
39
AR
Table 5.2. 39 Ar single-atom measurements with different
laser beams blocked. The
√
errors given in the table are purely statistical errors ( N ).
Atoms ∆Atoms Atoms/s ∆Atoms/s
All lasers on (start)
450
21
0.75
0.04
Coll. RP1 blocked
328
18
0.55
0.03
Coll. RP2 blocked
272
16
0.45
0.03
ZSl. RP1 blocked
∼0
ZSl. RP2 blocked
312
18
0.52
0.03
All lasers on (end)
411
20
0.69
0.03
spend less time in the slower and are therefore less likely be lost to transverse drifts.
Thus, the efficiency at high saturations is better for larger detunings.
Other qualitative measurements showed velocity profiles with considerably less leakage, therefore the most likely explanation for the low saturation is that in this measurement, for some reason (human or technical), the Zeeman-slower laser did not operate
at its optimum performance. Combined with the simulation results, these considerations suggest that the achievable efficiency is much higher than 7%. A reasonable
guess would be on the order of 50%.
All these considerations have only discussed the transverse loss in the Zeeman slower;
they have not accounted for the loss that is introduced by the finite capture velocity of
the MOT. The simulations in ch. 4.1.3 predict that the MOT can, if it is operated in
toggled mode, capture atoms with longitudinal velocities of up to 50-60m/s. With a
final velocity of 50m/s, the Zeeman slower can work relatively efficiently, as discussed
in ch. 5.4.4. However, the capture velocity of the MOT has not been experimentally
verified and the simulation neglects some parameters that might change it (as discussed
in A.4.2), therefore the predictions for the efficiency have to be seen as preliminary.
5.5. Loss channels of
39
Ar
The discussion of the loss channels for 39 Ar will be based on table 5.2, which shows
a single-atom measurement where, for each run, one laser beam was blocked. The
measurement time was 600s for each measurement. Two more counting results with
all laser beams on were acquired at the beginning and the end of the measurement
to ensure that the overall efficiency of the machine had not changed during the measurement. The counting results were obtained by applying a peak-detection
algorithm
√
to the measured count rates. The errors are just statistical errors ( N ) and do not
account for possible errors of that algorithm.
Collimator The result for the collimator might, at first sight, seem surprising: The
second repumper alone is more effective than the first repumper alone. This is due to
the fact that the repumpers in the collimator are not necessary to keep the atoms in
53
the cooling cycle, but rather to make more atoms available to the cooling process, as
explained in ch. 2.1.
Assuming that the atoms have a mean longitudinal velocity of about 430m/s, they
stay in the 15cm long collimator for a mere 350µs. The hyperfine loss rate is, according
to fig. 4.2, even for highly saturated lasers, only on the order of 102 /s to 103 /s if no
repumper is used. The collisional loss rate for a pressure of 10−5 mbar is much larger,
according to eq. (4.6). Therefore, even with only the cooling laser active, hyperfine
losses are not the limiting factor in the collimator.
On the other hand, the table on p. 7 shows that the repumpers that make the
substates F = 9/2 and 7/2 available, can increase the number of atoms in the cooling
process by factors of (25+30)/30 = 1.8 and (55+20)/55 = 1.4, respectively. These are,
however, not the factors that were measured, and according to these considerations,
the first repumper should be more effective.
The most probable explanation for the counting results can be found in fig. A.1.
The detuning of the collimator’s first repumper is (relative to the resonance frequency)
about -677MHz. The difference to the Zeeman slower cooling beam, which also passes
the collimator, is just about 110MHz. Considering that this beam does not have to
apply a force on the atoms by inducing many scattering events but just needs to pump
it into the cooling cycle once, it is plausible that, even if the designated repumper
is switched off, some atoms are pumped into the cycle nevertheless by off-resonant
excitation from the Zeeman-slower cooling beam.
The second collimator repumper is about 260Mhz detuned from the first Zeemanslower repumper and might, in part, be replaced by this beam. The larger difference
in frequency and the fact that the Zeeman slower cooling beam is more saturated than
the repumper can provide an explanation for the measurement results: The Zeemanslower cooling beam covers for the first repumper of the collimator more effectively,
therefore the effect of switching the repumper off is reduced. The first Zeeman-slower
repumper covers for the second repumper of the collimator less effectively, therefore
the effect of switching it off is larger.
This measurement makes a strong case for the addition of two more repumpers to
the collimator to make the remaining 25% of the atoms in F = 5/2 and 3/2 available
to the cooling process.
Zeeman Slower In the Zeeman slower, the situation is different. The time atoms
need to pass the Zeeman slower is, depending on their initial velocity, on the order of
10ms. Fig. 4.2 shows that this is a time scale on which hyperfine losses can play a
role, if the considerations in ch. 2.1 about the polarization of the laser are neglected.
Under the assumption that off-resonant excitation by the slower’s cooling beam is
possible, 4.2 explains the measurement results: The pressure at the end of the slower,
where the atoms spend the most time, is about 10−8 mbar, which yields, according to
eq. (4.6), a collisional loss rate of about 20/s. Without repumpers, hyperfine losses
are thus the limiting factor.
54
5.5. LOSS CHANNELS OF
39
AR
The addition of a repumper changes that situation considerably by lowering the
hyperfine loss rate7 . Therefore, without the first repumper, severe hyperfine losses
are expected, which is confirmed by the experiment. The second repumper lowers
the hyperfine loss rate further, but since now collisional losses dominate, the effect of
switching it off is much smaller.
MOT Information about the MOT cannot be obtained by single-atom counting rates
because hyperfine losses in the MOT do not only change the loading rate but also the
lifetime and thus the detection accuracy. This makes a quantification of the hyperfine
losses difficult.
A qualitative analysis of the measurement results yields that switching off the second
repumper has a larger effect than switching off the first repumper. The explanation
comes, again, from fig. A.1. The MOT repumpers have frequencies that are very
similar to the collimator repumpers. The same argument as for the collimator holds
for the MOT: If the first repumper is switched off, it has an effect but it does not
completely spoil the single atom detection as one might expect from fig. 4.2 because
it is covered for by the Zeeman slower cooling laser.
If the second repumper is blocked, the effect is much larger because the replacement
by the first Zeeman-slower repumper works less efficiently. The measured signal showed
fewer single-atom peaks that were too short to accurately decide whether an atom was
trapped or not.
7
A potential loss-channel is introduced by the different Zeeman shifts of the cooling and repumping
lasers. It is discussed in [10].
55
6. Conclusion and Outlook
This conclusion will reconsider the question whether the single-atom detection setup
is sufficient for the needs of ATTA of 39 Ar. It will then give a brief overview of other
limiting factors of this experiment and discuss means to overcome them.
6.1. Adequacy of the Detection Setup for ATTA of
39
Ar
Chapter 4.2.2 has laid out the framework to evaluate whether the single-atom detection
setup is sufficient to be used for the dating of very old water samples. One decisive
parameter in the consideration is the loading rate R of the MOT, i.e. how many atoms
we expect to measure in a given time. The next paragraph will estimate the loading
rate. It can be seen as an update of the considerations in [8] and [10]. To keep a
consistent notation throughout the documentation of this project, in the following the
loading rate will be denoted Ṅ .
Ṅ can be estimated by the multiplication of the total flow with the factors that
affect it:
Ṅ0 = Ṅtotal · A0 · ηHFS · ηvel · ηcoll · ηZsl
(6.1)
Total flow The total flow of metastable Argon atoms arriving at the MOT is the
measured flow density in forward direction (from [33]) times the solid angle of
2
· π 0.016m
= 6.6 · 1010 atoms
the MOT: Ṅtotal ≈ 4 · 1014 atoms
s · sr
2.2m
s
Relative abundance The relative abundance of 39 Ar is, for modern samples, A0 =
8.1 · 10−16 [2]. For older samples, it is A(t) = A0 · e−ln2 · t/269y .
Hyperfine losses The considerations in ch. 5.5 show that no notable losses occur due
to hyperfine leakage once an atom is in the cooling cycle. The hyperfine losses
are therefore limited to those atoms that are initially in hyperfine states that are
inaccessible to our lasers. With two repumpers, according to the table on p. 7,
75% of the atoms are accessible, therefore ηHFS = 0.75.
Losses due to longitudinal velocity This factor addresses the fraction of atoms that
can, from geometrical considerations, reach the MOT, but are longitudinally too
fast to be slowed down by the Zeeman slower. With typical mean velocities
of our source and capture velocities of our Zeeman slower, this factor is about
ηvel ≈ 0.6.
57
CHAPTER 6. CONCLUSION AND OUTLOOK
Collimator gain Chapter 5.3.2 shows that the collimation of
that of 38 Ar. The gain is ηcoll ≈ 40.
39
Ar is comparable to
Zeeman-slower efficiency Ch. 5.4.4 discussed the efficiency of the Zeeman slower.
Although no exact experimental data is available, an efficiency ηZsl ≈ 0.5 is a
reasonable guess under the assumption that the MOT is toggled and thus has a
relatively high capture velocity.
Bringing it all together, the expected loading rate for modern samples is
Ṅ0 ≈ 1.7
atoms
;
h
(6.2)
for a 1000 year old sample, it its Ṅ ≈ 3 atoms
.
day
Table 4.1 shows that, with the current signal quality and a desired detection accuracy
of 99%, detection efficiencies for old samples of about 58% are achievable using a
toggled MOT, yielding a total count rate of 1 atom per hour for modern water samples.
It can therefore be concluded that the detection efficiency is adequate for analyzing
39
Ar in natural water samples. As 58% of the atoms are already a large fraction, room
for improvement in the detection is limited. Figure 4.12 shows that moderate progress
in the signal-to-noise ratio does not change the efficiency substantially. The next
chapter will discuss parts of the experiment that have more potential for significant
improvement.
6.2. Limiting Factors and Perspectives for
Improvement
The statistical
error on the number of 39 Ar counts is, assuming a Poissonian distribu√
tion, 1/ N . To obtain results with a statistical error of less than 10%, 100 atoms have
to be detected. With an estimated count rate of 1 atom per hour, the measurement
time would be about 100 hours for modern samples.
Currently, Argon outgassing in the machine limits the continuous measurement
time to roughly ten hours. Measurements are still possible with consecutive runs and
interjacent flushing, but this procedure significantly increases the required sample size
and the measurement time.
However, according to the estimation, the analysis of younger samples seems within
reach; the necessary increase in the loading rate is not one of orders of magnitude but
rather one by a modest factor. The next segment will briefly discuss potential areas
for improvement.
The ICP source might not run on its full potential. A current excitation efficiency of
' 10−4 [8] suggests that the potential gains are large. Extensive tests have not revealed
clear-cut measures to improve the metastable flow, but they have shown that a fine
tuning of the details, such as the choice of glass tubes and their position, has potential
to enhance the flow. However, an increased metastable flow has qualitatively shown to
58
6.3. FIRST MEASUREMENTS OF NATURAL SAMPLES
often be accompanied by an increase in the longitudinal velocity. To counteract this
effect, a rather extensive measure could be to externally cool the plasma with liquid
nitrogen.
A lower pressure in the MOT area, which is currently sought to be achieved by
the addition of a better differential pumping stage, can further improve the detection
efficiency, as discussed in ch. 4.2.2, and allow a higher pressure in the source to further
increase the flow of metastable atoms.
An additional transverse focusing stage is being used in other ATTA setups [6] to
further decrease transverse losses; first tests in our experiment have shown moderate
improvement.
The laser system has two obvious areas of potential improvement: A gain of 33%
can be achieved by adding two more repumpers, as discussed in ch. 5.5. Furthermore,
first tests with an additional longitudinal cooling beam that is resonant with atoms
with medium longitudinal velocities, as discussed in ch. 4.1.3, have shown promising
results.
6.3. First Measurements of Natural Samples
Once the obstacles described above have been overcome, the machine has to be tested
for contamination from the enriched samples that were used throughout the tests:
If a run with a very old, “dead” sample does not result in zero counts, some parts
of the machine, especially those close to the ICP, may have to be replaced. When
contamination has been ruled out, the machine will have to be carefully calibrated.
Eventually, the ATTA collaboration at the University of Heidelberg, which is formed
by the groups of Markus Oberthaler at the Kirchhoff Institute for Physics and Werner
Aeschbach-Hertig at the Institute of Environmental Physics, will pursue its primary
goal, to date old water samples and find the answers hidden inside them.
59
A. Appendix
A.1. Some Constants
Table A.1. Physical constants and properties of argon [7].
Atomic Mass Unit amu
Planck constant h
Boltzmann constant kB
Bohr magneton µB
Speed of light c
Magnetic permeability µ0
Isotope
Relative abundance
Mass m
Halflife time T1/2
Nuclear spin I
Nuclear magnetic moment µ/µK
Relevant transition
Ground state Landé factor gg
Excited state Landé factor ge
Wavelength λ
Linewidth γ
Lifetime τ
Saturation intensity I0
Doppler temperature TD
1.66053886 · 10−27 kg
2
6.626068 · 10−34 m s· kg
2
1.3806503 · 10−23 ms2 ·· Kkg
9.27400949 · 10−24 TJ
2.99792458 · 108 ms
1.25663706 · 10−6 sm2 ·· Akg2
40
39
Ar
Ar
0.996
8.1 · 10−16
39.96 amu 38.96 amu
269a
0
+ 72
-1.3
1s5 − 2p9
1.506
1.338
811.754 nm
2π · 5.85 MHz
27.21 ns
1.44 mW
cm2
141 µK
61
APPENDIX A. APPENDIX
A.2. Laser System
Ar-789.4
Ar-294.4
Ar-83.8
36
Ar+158.2
39
40
38
AOM
-2*191
Ar-495
Ar+0
38
Ar+210.6
36
Ar+452.6
39
40
AOM
+2*120
Ar-6.4
Ar+488.6
Ar+699.2
36
Ar+941.2
39
39
39
40
40
40
38
38
38
Ar-255
Ar+240
Ar+450.6
36
Ar+692.6 BEATL.1
Ar-407.4
Ar+87.6
Ar+298.2
36
Ar+540.2
-152.4
AOM
+2*200.5
Ar-690.4
Ar-195
Ar-15.2
36
Ar+257.2
ZEEMAN-SLOWER
MOT
39
40
38
AOM
-2*141.5
Ar+6.8
Ar+501.8
Ar+712.4
36
Ar+954.4
MOT
1TER RÜCKPUMPER
39
40
38
AOM
+2*207.1
Ar-677.4
Ar-182.4
Ar+28.2
36
Ar+270.2
KOLLIMATOR
39
40
38
AOM
-2*135
Ar-27.4
Ar+467.6
Ar+678.2
36
Ar+920.2
39
KOLLIMATOR
RÜCKPUMPER
40
38
AOM
+2*190
Ar-2035
Ar-1534
Ar-1323.4
36
Ar-1081.4
ZEEMAN-SLOWER
BOOSTER
39
40
38
AOM
-4*130
BEATL.2
-1254
39
39
40
40
38
38
Ar - 39Ar1R = 683
39
Ar - 39Ar2R = 1247
39
Ar - 39Ar3R = 1672
39
Ar - 39Ar4R = 1942
Figure A.1. Laser system. All values in MHz.
62
Ar-1509
Ar-1014
Ar-803.4
36
Ar-561.4
Ar-1509
Ar-1014
Ar-803.4
36
Ar-561.4
39
Ar-1253.2
Ar-758.2
Ar-547.6
36
Ar-305.6
ZEEMAN SLOWER
2TER RÜCKPUMPER
ZEEMAN-SLOWER
RÜCKPUMPER
39
40
38
AOM
+2*128
MOT/KOLLIMATOR
2TER RÜCKPUMPER
A.3. DETAILED TRANSITION SCHEME OF
A.3. Detailed Transition Scheme of
40
40
AR
Ar
63
APPENDIX A. APPENDIX
A.4. Simulation
A.4.1. Zeeman Slower
The Monte-Carlo simulation used throughout this thesis is inspired by a program developed in [34] to simulate the transverse movement of atoms in a Zeeman slower. It
has been re-written in MATLAB and then extended and modified to fit our experimental setup. An atom’s trajectory is calculated in discrete time steps with a duration
of 1/γ, where γ is the scattering rate. At each step, the B-field and the velocity are
evaluated and an overall detuning is established. The intensity is calculated from the
atom’s distance from the beam, the beam’s size and intensity at the particular position of the atom (which is affected by focusing the beam). From these parameters,
the saturation corresponding to (2.9),
s=
s0
1 + s0 +
2 ,
2δ
γ
(A.1)
is calculated and compared to a random number between 0 and 1. If the saturation is
smaller than the random number, a scattering process is simulated: From the position
of the atom and the laser’s focus parameters, the momentum of the absorbed photon
is calculated and added to the atom’s momentum. The recoil the atom experiences
when the photon is reemitted is then added in a random isotropic direction. The atom
subsequently moves straight for the duration of 1/γ. When an atom is in resonance,
it thus scatters with the maximum scattering rate.
The simulation is then fed with a random sample of atoms, which is obtained using
normally distributed random numbers for the transverse velocity and the transverse
position (the atoms are initially assumed to be at the end of the plasma tube, which
is very small, therefore this distribution does not play a major role) and MaxwellBoltzmann distributed in longitudinal direction. To improve the performance, the
trajectories for all atoms are evaluated simultaneously; atoms that have either reached
the end of the slower or have hit a wall before the slower are dropped out.
A.4.2. MOT
1
s1
s2
s3
s4
64
0
The MOT simulation is different from the Zeeman-slower simulation in that it deals with four laser beams (it neglects the two
truly transverse beams). Therefore, the saturation must be evaluated differently. This simulation follows an approach from [8] and
translates it to discrete steps.
First, the detuning and saturation si of each beam is calculated
individually using the respective k. Then, effective saturations of
each beam
si
P
seff,i =
(A.2)
1 + j sj
A.4. SIMULATION
P
are evaluated. For the cumulative saturation, seff =
i seff,i ∈ [0, 1) holds. The
effective saturations are then stacked as shown on the left. Then a random number
between 0 and 1 is drawn and evaluated: if it is greater than seff , no scattering happens;
otherwise the beam the scattering happens with is determined by the range in the stack
the random number falls into. The scattering process is then modeled as described
above. Atoms are considered trapped when they reach the zero crossing of the B-field
with a negligible longitudinal velocity.
It is important to note that this simulation models only light forces and takes into
account neither the polarization the atoms have when they leave the Zeeman slower,
nor the different absorption characteristics for different light polarizations.
The steps visible in fig. 4.3 near the center of the MOT result from discrete B-field
values.
65
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69
Acknowledgements
Many people have had their share in the infancy and adolescence of this thesis. In
particular I would like to thank:
—Markus Oberthaler for the supervision and grading of my diploma thesis, for his
expertise and his ability to revive his team’s spirit in the course of a lunch break, and,
finally, for the initiation of this project.
—Werner Aeschbach-Hertig for his contribution to the birth and life of this project
and for grading my thesis.
—Joachim Welte and Florian Ritterbusch for letting me join their team and for
their extra-hours to proof-read this thesis. Jo for his level-headed can-do attitude, his
never-ending stock of calculations done in the tram, his great taste in movies, and his
horrible taste in music. Flo for his calmness, his patience to explain things over and
over again, and for steady nutritional support.
—Philippe Bräunig for the hint to knock on the ATTA office’s door and for reliable
companionship.
—Jürgen Schölles for the rate meter and help in all other kinds of questions, be
they about electronics or not.
—Ursula Scheurich for her expertise in glass-blowing all kinds of pipes.
—Knut Azeroth for his great work on the rough parts of the interlock.
—The KIP workshop division for their work on the RF coil and particularly S.
Spiegel for letting me use his machines even after I had pushed the self-destruct switch
on one of them (in my defense—it was not properly labeled).
—Jirka Tomkovic for lots of help with MATLAB and the counting cards.
—The BEClers, BECks, NaLis and AEgISlers for being a nice bunch to have lunch
with. A special thank you to Wolfgang A. M. for comprehensive news analyses and
his prowess in the diverse fields of cartography, Native American tribal ceremonies,
18th century agrarian business principles, and archaeology, and Tilman for his expert
opinions on all kinds of stuff and cameras in particular.
—My parents for everything.
—All the people that helped me keep a clear head during the year of my diploma
thesis and throughout my academic life, especially Bastian, Philipp, and Britta.
Erklärung:
Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als
die angegebenen Quellen und Hilfsmittel benutzt habe.
Heidelberg, den 01.03.2011
.........................................

Faculty of Physics and Astronomy

Transcription

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