Download An Optical Mask for Atomic Interferometry Experiments

An Optical Mask for Atomic
Interferometry Experiments
Simon Coop
a thesis submitted for the degree of
Master of Science
at the University of Otago, Dunedin,
New Zealand.
2013
2
Abstract
This thesis presents work performed to obtain an optical mask for conducting matter-wave interferometry experiments with ultra-cold rubidium-85 atoms. The optical mask is essentially an absorptive
diffraction grating made of light, and it can imprint a periodic density pattern on a cloud of atoms with
a period of half the wavelength of the light used. The mask has an analogous effect to quickly passing
a diffraction grating through the cloud, removing atoms located at the nodes of the grating (though
the optical mask depumps atoms to a different hyperfine ground state rather than actually removes
them). The mask should be useful in performing precision measurements of physical constants such
as the fine-structure constant α, and acceleration due to gravity g. The thesis briefly expounds major
historical developments in atom interferometry and laser cooling. Prerequisites for an optical mask
include a functioning magneto-optical trap, and the ability to perform polarisation-gradient cooling
on trapped atoms. The theory of these two techniques is reviewed and the principle of the the optical
mask is explained. The experimental apparatus that was constructed to realise the optical mask is
described, and technical developments made along the way are presented. Finally, aspects of the experiment relevant to the optical mask are characterised. The experiment can reliably produce samples
of cold atoms at temperatures of around 10 µK, and can use the cold atoms as an optical frequency
reference accurate to ∼ 1 MHz. Clear evidence that the cloud of atoms is being density-modulated by
the optical mask is presented. Improvements required to make the experiment into a fully-functioning
interferometer are also discussed.
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Acknowledgements
There’s a quote attributed to Isaac Newton “If I have seen further it is by standing on ye sholders of
Giants.” I won’t claim to have made the same impact as Newton, but if this thesis is any achievement
at all, it would not have been possible without Mikkel. While I never actually stood on his shoulders,
he is quite tall. Mikkel, your (as far as I can tell) endless patience and knowledge, make you an
inspiration for me, both as a scientist and a person.
I’d probably still be trying to turn on the laser if it wasn’t for Peter, without the benefit of your
experience and humour I’d never have finished this. I wouldn’t have made it much further if it weren’t
for Andrew and Tzahi. Protips from you guys made life a lot easier.
Though much less frequent than it should’ve been, coffee with Fung provided lively discussion of
life, the universe and everything. The excuse for some sunshine and non-HEPA-filtered air was nice
too.
Alicia, it’s not normal to learn Spanish as well as science in a physics lab. From you I learnt large
amounts of both!
Peter and Richard, you made my regular trips to the mechanical workshop an absolute pleasure. In
your professional, efficient approach to your work, your ingenuity, and your lively humour.
Sandy, your remarkable ability to navigate the labyrinthine university bureaucracy with ease, and
your bowl of chocolates both contributed significantly!
All my friends in Dunedin, in and outside of the physics department, you made my time at Otago
pretty awesome. Thanks!
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Contents
1
About this Thesis
1
2
Introduction
3
2.1
Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Laser-Cooled Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3
Background
7
3.1
One-Dimensional Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2.1
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2.2
Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.3
3.4
4
σ+
−
σ−
Polarisation Gradient Cooling . . . . . . . . . . . . . . . . . . . . . . .
11
3.3.1
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.3.2
Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Atomic Interference using a Resonant Optical Standing Wave . . . . . . . . . . . . .
16
The MARIE Experimental Apparatus
21
4.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.2
Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.3
Rubidium Dispenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.4
Lasers and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.4.1
Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.4.2
Tapered Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.4.3
Experiment Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Magnetic Field Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.5.1
Quadrupole Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.5.2
Quenching Field Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.5.3
Compensating Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Making Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.6.1
Photomultiplier Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.6.2
Photodiode for Atom Measurement . . . . . . . . . . . . . . . . . . . . . .
32
4.6.3
PIXIS CCD Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.5
4.6
v
4.6.4
4.7
5
6
Video Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
LabVIEW Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.7.1
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.7.2
Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.7.3
MATLAB Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.7.4
Controlling the Optical Mask
36
. . . . . . . . . . . . . . . . . . . . . . . . .
Characterising MARIE
39
5.1
Counting Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.2
Magneto-Optical Trap and Optical Molasses . . . . . . . . . . . . . . . . . . . . . .
41
5.2.1
Getting Cold Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.2.2
Measuring the Temperature . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.3
Measuring the Laser Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.4
Optical Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.4.1
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.4.2
Saturation with the Optical Mask . . . . . . . . . . . . . . . . . . . . . . .
49
5.4.3
Quenching Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.4.4
Interference in the Optical Mask . . . . . . . . . . . . . . . . . . . . . . . .
49
Summary and Future Work
53
6.1
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
6.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
A Using MARIE’s LabVIEW program
55
A.1 Description of MARIE’s LabVIEW program features . . . . . . . . . . . . . . . . .
55
A.2 List of Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
B Derivation of Time-of-flight Equation
59
References
61
vi
Chapter 1
About this Thesis
This thesis presents work performed between March 2011 and June 2012 related to the construction
and characterisation of an ‘optical mask’ for atom interferometry experiments. The idea for the optical
mask is from [1]. It is hoped that this optical mask will provide New Zealand’s first absolute precision
gravimeter.
The basic principle of the optical mask is that it uses an optical standing wave to act as a diffraction
grating for a cloud of cold atoms. To make one we then need an apparatus that can trap and cool
atoms, the optical mask, and a way of measuring the effect of the optical mask on the cold atoms.
Chapter 2 briefly discusses major historical developments in the fields of matter-wave interferometry and laser cooling.
Chapter 3 provides background information on laser cooling techniques used in this experiment,
and then goes on to discuss the mechanism of the optical mask, and how it could be used to provide
a gravimeter.
Chapter 4 describes the experimental apparatus in detail. The equipment used and a computer
program developed for the experiment are discussed.
Chapter 5 discusses some results in characterising the experiment. Including how the temperature
of the atoms was measured, and evidence the optical mask was working.
The appendices contain information that might prove useful to a future student working on the
experiment.
It should be noted that some of the apparatus discussed in Chapter 4 was not constructed by me
alone, but the apparatus is presented in whole so that this thesis forms a complete description of the
experiment. When I started most of the apparatus used to make the magneto-optical trap was already
installed. Improvement of the system to obtain a functioning magneto-optical trap was performed in
2
About this Thesis
collaboration with another student. Beyond this, all of the content in this thesis is the result of my
own work, except for the compensating coils in Section 4.5.3 which were designed and constructed
by another student.
Chapter 2
Introduction
2.1
Atom Interferometry
Interferometry, the idea of traversing two or more waves along different paths in a device and then
recombining them to observe their interference, has long been a tool for researchers to make precision
measurements of physical phenomena. A famous historical example is the Michelson-Morley experiment of 1887, an optical interferometer designed to measure the effect of the luminiferous ether on
the speed of light [2]. The unambiguous null result of this experiment played a major role in refuting
the theory of the ether and the development of the special theory of relativity. Another example is
Froome’s 1958 measurement of the speed of light using a two-path microwave interferometer. At the
time it was the most accurate measurement of c ever made (his result was c = 299 792 500 ± 100
ms−1 , compared with the modern definition of 299 792 458 ms−1 ) [3].
With the advent of quantum mechanics and the discovery of the wave nature of matter, it was realised particles could be used for interference measurements. As early as 1927, experiments were
conducted that showed electron diffraction [4]. Then in 1936 crude experiments showed neutron
diffraction from crystals. This effect was soon exploited to gain quantitative insights on crystal structures that were unobtainable with conventional x-ray crystallography [5].
Interferometers require some kind of coherence between the waves. The interference signal arises
because of a phase difference between waves that have travelled alternate paths in the interferometer,
so the waves must be initially coherent. With this in mind, any quantum mechanical degree of freedom
can be treated as a wave (e.g. position, momentum, spin, etc), so to interfere quantum particles they
must be first localised in some space to provide the initial coherence. For example, electrons are
prepared in the same spin state in the Stern-Gerlach experiment, or atoms in an atomic beam (i.e.
localised in momentum space) can be made to interfere by diffracting off a grating [4].
Neutral atoms are good candidates for matter-wave interference. They have large optical crosssections compared to neutrons or electrons, sources are cheap to produce, and properties such as
mass and magnetic moment can be selected over a large range. With the invention of laser cooling in
4
Introduction
the 70’s and 80’s, preparing samples of atoms with very narrow momentum distributions has become
commonplace, making them ideal for interference experiments [4].
In a generic interferometer, some mechanism coherently splits the waves to send them down different paths. For example, in a light interferometer this can be done with a diffraction grating or a 50/50
beamsplitter. As mentioned above, atoms can be diffracted off a grating. One very early example of
this is the diffraction of helium atoms from a LiF crystal surface, first done by Estermann and Stern in
1930. The periodicity of the crystal lattice lets it act as a phase grating for the atoms. Modern experiments can make use of nanofabricated structures to make transmission gratings, which can diffract
atomic beams provided the transverse velocity of the beam is low enough. [4]
Another way of making diffraction gratings for atoms is to use optical standing waves, these have
several advantages over gratings made of matter: they can be switched on and off very fast, they have
a tunable period, and the phase can be shifted easily. Far-off resonant light can be used to make phase
gratings for atoms, and on-resonant light can make an absorption grating (such as the one described
in this thesis). [4]
Since the early 90’s, atom interferometry has been instrumental in making many precision measurements. In 1991, Kasevich and Chu made an interferometer that could measure gravity to a precision
of 3 × 10−8 g, [6] and in 1993 Chu made an interferometer that could measure ~/m for cesium atoms
(proportional to the fine-structure constant α) to an accuracy of 10−7 . [7]
More recent experiments include an accurate measurement of G, Newton’s gravitational constant,
using a 500 kg mass moved between two atom interferometers [8], a measurement of the difference in
gravitational acceleration between 85 Rb and 87 Rb atoms as a test of Einstein’s equivalence principle
(they were unable to measure a difference) [9]. An experiment in Paris can continuously measure
gravitational acceleration three times a second to an accuracy of 1 µGal (1 Gal = 1 cm/s2 ) [10].
2.2
Laser-Cooled Atoms
It has been thought for centuries that light exerts a force on matter, as long ago as 1619 Kepler suggested that the shape of comet tails was the result of solar radiation pressure. Radiation pressure was
first quantitatively described with the advent of Maxwell’s equations of electromagnetism in the 19th
century. The first experiments decisively proving that light exerts pressure on matter were performed
independently by Lebedev [11] and Nichols and Hull [12] in 1900 and 1901 respectively. Einstein’s
description of radiation absorption and emission in 1917 explained a mechanism for this pressure
[13]. The next big development was the suggestion (by Townes, [14]) and then implementation (by
Maiman, [15]) of the laser in 1958 and 1960 respectively.
The intense, coherent light source that is the laser catalysed the field of atom manipulation. In the
mid-70’s several researchers suggested the idea of using lasers to reduce the random thermal velocities
Introduction
5
of a sample of atoms - what is now known as laser cooling [16] [17]. Although much earlier, in 1968,
Letokhov suggested using the optical dipole force to trap atoms (see [18] for reference).
The first experiments on laser-cooled atoms were performed in 1978 first by Wineland, Drullinger,
and Walls [19], and then Neuhauser et al [20]. In their experiments both groups laser cooled trapped
ions, rather than neutral atoms.
The first experiments that clearly showed laser-cooled neutral atoms were performed in 1981 at
Moscow’s Institute for Spectroscopy by Andreev et al (see [18] for reference). Over the early-to-mid80’s, Phillips and Metcalf perfected a device they called the Zeeman slower, which uses a spatiallyvarying magnetic field to keep a sample of atoms in resonance with a laser beam as it cools. They
successfully produced the first sample of ‘stopped’ atoms with a temperature of less than 100 mK
using this device [18].
In 1985 Steven Chu and his group laser cooled a sample of sodium atoms to 240µK using what
is now known as Doppler cooling - a standard technique of laser cooling [21]. Shortly after the
demonstration of magnetic traps and optical dipole traps for neutral atoms in 1986 and 1987, Raab,
Prentiss, Cable, Chu and Pritchard successfully used this radiation-pressure cooling technique to
implement a magneto-optical trap, and held atoms in the trap for around two minutes in 1987 [22]
[23]. A magneto-optical trap uses a combination of a magnetic field and radiation pressure to contain
atoms to a small region inside a vacuum chamber, a description is given later in this thesis.
While Chu was working on his magneto-optical trap, Phillips found his laser cooling system was
producing much lower temperatures than what was theoretically predicted to be possible [18]. This
was the first instance of sub-Doppler cooling. It was later realised that all the Doppler cooling models
had assumed simple two-level atoms, but real atomic structure is much more complex than that.
Sub-Doppler cooling that exploited atoms with multi-level structure was theoretically explained by
Dalibard and Cohen-Tannoudji in 1989 [24]. This sub-Doppler cooling regime could produce atomic
samples with temperatures approaching the one photon recoil limit (∼0.4µK for 85 Rb).
Laser cooling has come much further more recently, and has found a myriad of real-world applications. In 1995 laser cooling was instrumental in creating the world’s first Bose-Einstein condensate.
Laser cooling has also been useful in making atomic clocks orders of magnitude more accurate. [25].
6
Introduction
Chapter 3
Background
This chapter reviews the theory of some basic laser cooling techniques that were used in the experiment. The last section is a description of the optical mask, discussing how it works, and how it could
be applied in a gravimetric measurement.
3.1
One-Dimensional Doppler Cooling
The first kind of laser cooling to be suggested - and conceptually the simplest - is Doppler cooling
[16] [17]. Consider an atom whose internal structure can be approximated as consisting of a ground
state and a single excited state. This approximation is valid in the case of an atom in the path of
low-intensity laser light which is tuned close to the frequency difference between the two states. In
such a case an atom is known as a two-level atom.
In the presence of resonant light a two-level atom in the ground state will absorb a photon. This
has two effects: firstly the atom will jump up to the excited state, storing the energy of the photon;
secondly the atom will absorb the momentum of the photon, giving the atom a ‘kick’ in the direction
the photon was travelling. After a short time the atom will spontaneously fall back down to the ground
state, emitting a photon in a random direction and again giving the atom a momentum kick, but in
a direction opposite to the direction of the emitted photon. Over time, these momentum kicks from
photon emission will cancel themselves out but the momentum kicks from absorption are always in
the same direction. This has the net effect of a force on the atom in the direction of the laser beam.
This force is known as the radiation pressure force.
For Doppler cooling, the atom is placed in the path of two co-axial counter-propagating laser beams
tuned slightly below atomic resonance (i.e. red detuned). Experimentally this can be achieved just
by passing the two laser beams through a cloud of gaseous atoms in a vacuum chamber. If the atom
has a velocity towards one of the laser sources, the Doppler effect will cause the laser frequency to
increase in the atom’s rest frame and thus increase the atom’s absorption probability from that laser
beam. The atom will thus feel a net force in the direction opposite to its motion and will slow down.
The force on the atom from either laser beam is given by Eq. 3.1 [25]
8
Background

1 
F ± = ± ~k
2
I
I0 Γ
1+
I
I0
+4
∆∓|ωD |
Γ


2 
(3.1)
where I is the laser intensity, I0 is the saturation intensity for the transition, Γ is the natural
linewidth of the transition ( 2π× 6.1 MHz for the 85 Rb D2 transition), ∆ is the laser detuning from
resonance, k is the laser wavevector, and ωD is the Doppler shift of the laser in the atom’s rest frame
(ωD = 2π × νatom /λ).
The rigorous derivation of Eq. 3.1 is quite long, so it will not be reproduced here. It can, however,
be justified as being reasonable. For a stationary atom in an on-resonance light field (i.e. ∆ = 0 and
νatom = 0), the force becomes F = 12 ~k[ II0 Γ/(1 + II0 )], which is simply the momentum per photon
~k, times the scattering rate. The scattering rate can be interpreted as the rate at which photons are
absorbed and then reemitted by the atom. As the light intensity becomes very large (i.e. I → ∞), the
scattering rate approaches Γ/2. This is exactly the expected result: The excited state decays at rate
Γ, and at high intensity the atom has a 50% probability of being in the excited state. Conversely, at
large detunings the force goes to zero, which is also expected as the atom does not strongly absorb
light that is far from resonance.
In general the total force on the atom is not simply the sum of the forces from each of the laser
beams. There can be sequential effects where one laser beam excites the atom and the other causes
stimulated emission, leading to large velocity-independent changes in the atom’s speed. However in
the case where the light intensity is low enough such that stimulated emission is not important, the
total force on the atom is simply the sum of the contributions from each laser beam, represented by
the solid line in Figure 3.1. [25].
In the absence of any other effects, the temperature of the atom would decrease to zero Kelvin.
However, the effect of spontaneous emission must be included. Every time a photon is absorbed, it is
emitted a random time later in a random direction. The average velocity imparted by these momentum
kicks is zero, but the rms value is finite. The atom can be viewed as undergoing a random walk in
momentum space. The average velocity of the atom is still zero, but the rms velocity slowly grows,
which is the same as heating the atom. Thus the final temperature is determined by an equilibrium
between this heating mechanism and the cooling effect described above. This minimum achievable
temperature is called the Doppler cooling limit, and is roughly given by [18]:
TD ≈
~Γ
2kB
(3.2)
where kB is Boltzmann’s constant. For example, for rubidium this temperature is about 150µK.
This technique can easily be generalised to three dimensions by the use of three orthogonal pairs
of counter-propagating laser beams (i.e. two counter-propagating beams along each of the x-, y-, and
Background
9
0.04
0.03
Force (h̄kΓ)
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−5
−4
−3
−2
−1
0
1
2
3
4
5
Atom Velocity (Γ/k)
Figure 3.1: This figure shows the force on an atom moving in one dimension due to F+ and F−
(dotted lines), and Ftotal , which is the sum of the two contributions (solid line). These plots are
calculated with II0 = 0.1 and ∆ = −Γ.
z-axes), instead of just two as described above. This provides a damping force to the atom’s motion
in all directions. It has been shown that once caught in the intersecting laser beams, the atom has
a diffusive Brownian-like motion. This coupled with the fact Doppler cooling acts very much like
viscous friction has led to use of the term ‘optical molasses’ [18]. It should be noted that optical
molasses does not actually trap neutral atoms, as there is no restoring force for atoms displaced from
the centre of the beams. The atoms are slowed immensely but do eventually escape. A technique for
trapping atoms is discussed in the next section.
3.2
3.2.1
Magneto-Optical Trap
Description
A magneto-optical trap (MOT) uses Doppler cooling and a weak inhomogeneous magnetic field
to trap neutral atoms into a small region. The principle is essentially Doppler cooling as described
above, but the magnetic field Zeeman-splits the energy levels for an atom not at the centre of the trap.
10
Background
The Zeeman splitting causes a spatially-dependent difference in the absorption efficiency of each
laser beam, and a restoring force is introduced. The basic structure of the trap is shown in Fig. 3.2:
There are three orthogonal pairs of counter-propagating laser beams, which provide Doppler cooling
in three dimensions; and two current loops which are arranged to produce a ‘spherical quadrupole’
magnetic field [22]. The magnetic field is zero at the centre of the trap, and increases linearly in every
direction away from the centre (for small distances).
Figure 3.2: A three dimensional magneto-optical trap. The ‘spherical quadrupole’ magnetic field is
generated by two coils with current flowing in opposite directions. Circularly polarised laser light is
indicated by the red arrows. The origin is taken as the centre of the atom cloud.
The trap can be most easily understood in one dimension, with results applicable to the actual
three-dimensional trap. Consider the atom represented in Figure 3.3. It has an angular momentum
J = 0 ground state, and an angular momentum J = 1 (mJ = −1, 0, +1) excited state. In a weak
magnetic field described by B(y) = by the atom’s energy levels are Zeeman-split by an amount
∆E = µB gmJ B = µB gmJ by, where b is the gradient of the magnetic field, µB is the Bohr magneton, and g is the appropriate g-factor for the atomic state [26]. Now introduce circularly-polarised
counter-propagating laser beams as indicated in Fig. 3.2. The laser beams are tuned below the B = 0
resonance frequency, so the atom at y < 0 will be closer to resonance with the σ + laser beam, and
will scatter more photons from this beam. The atom will thus feel a net force towards the origin.
Similarly for an atom at y > 0. This also works in the other two dimensions, so an atom is always
being pushed towards the centre of the trap. If the experimental parameters are chosen appropriately,
the motion of the atoms can also be damped by Doppler cooling as described above. The atoms are
thus cooled and trapped.
Background
11
Figure 3.3: Energy levels of a two-level atom at y < 0 in Figure 3.2. The laser is red-detuned from
resonance with B = 0, Zeeman splitting causes the atom to be resonant with the laser polarisation
that pushes it back to the centre of the trap.
3.2.2
Practical Considerations
In a real-life experiment, there may be many atoms being cooled simultaneously, and there is the
added effect that a photon spontaneously emitted by one atom can be absorbed by another. This
self-heating limits the density of trapped atoms in a MOT to ∼ 1011 cm−3 [25].
In my experiment, the transition used for cooling and trapping 85 Rb was the D2 F = 3 → F 0 = 4
transition (see Figure 3.4). This is a closed transition, but there is still occasionally unavoidable offresonant scattering to the F 0 = 3 or F 0 = 2 excited states, either of which can decay to the F = 2
ground state. This state is dark to the cooling laser and any atom in this state is no longer trapped.
To thwart this effect a weak ‘repump’ laser tuned to the F = 2 → F 0 = 3 transition must be used.
The direction is unimportant as the rate of excitation to the F = 3 excited state is small compared to
the rate of excitation to the F = 4 excited state, consequently the force from the repump laser is also
small.
3.3
σ + − σ − Polarisation Gradient Cooling
This section describes σ + − σ − and not lin⊥lin polarisation gradient cooling (PGC) as this was
the technique used in the experiment. σ + − σ − is practically very easy to implement as it uses the
same laser polarisations as a magneto-optical trap (described in section 3.2), therefore the same optics
and laser source can be used. This section only reviews why σ + − σ − polarisation gradient cooling
works, and does provide a complete derivation. A detailed description can be found in [24].
12
Background
F' = 4
100.205(44) MHz
52P 3/ 2
120.640(68) MHz
20.435(51) MHz
F' = 3
83.835(34) MHz
113.208(84) MHz
63.401(61) MHz
29.372(90) MHz
F' = 2
F' = 1
780.241 368 271(27) nm
384.230 406 373(14) T Hz
12 816.546 784 96(45) cm- 1
1.589 049 139(38) eV
F = 3
1.264 888 516 3(25) GHz
52S1/ 2
3.035 732 439 0(60) GHz
1.770 843 922 8(35) GHz
F = 2
Figure 3.4: Hyperfine structure of the 85 Rb D2 transition. The relative sizes of splittings are indicative
only and should not be compared. Image and data from [26].
3.3.1
Description
σ + − σ − polarisation gradient cooling uses two counter-propagating laser beams - as in Doppler
cooling - to induce motion-dependent population of atomic ground states. Because of this motion-
Background
13
sensitive state population, each laser beam is absorbed with different efficiency giving rise to a
velocity-dependent radiation pressure force which damps an atom’s motion. For 85 Rb, the damping force from PGC is much larger than from Doppler cooling, meaning much lower temperatures
can be acheived. However the capture velocity (the maximum initial velocity an atom can have if it
is to be caught in the laser beams) is much smaller than in Doppler cooling.
The name ‘polarisation gradient cooling’ comes from the properties of the laser beams used. Both
laser beams are circularly polarised, with one beam left-circularly polarised and the other rightcircularly polarised. The beams add to give a light field with linear polarisation at every point, but the
actual direction of polarisation changes as one moves along the propagation axis (see Fig. 3.5).
Figure 3.5: Polarisation gradient in σ + −σ − cooling. The two circularly polarised beams add at every
point along the z-axis to give a linearly polarised light field that rotates in space. Image from [24].
σ + − σ − cooling requires an atom with ground-state angular momentum Jg ≥ 1, so will be
explained using the simplest atom which can undergo this kind of cooling; an atom with Jg = 1 and
excited-state angular momentum Je = 2 (see Fig. 3.6). The description can easily be extended to
atoms with larger ground-state angular momentum.
Say we have a stationary atom subject to a σ + − σ − laser field where both of the constituent laser
beams have a wavevector of magnitude k, in a position where the polarisation is in the ŷ direction.
With the quantisation axis taken to be in the same direction as the local polarisation, it is shown in
[24] that the relative steady state populations of the three ground states |g0 iy , |g−1 iy , and |g+1 iy
- eigenstates of Jy - are 9/17, 4/17, and 4/17 respectively (g indicates an atomic ground state, and
the numerical subscript is the projection of angular momentum along the quantisation axis, the letter
subscript outside the ket indicates the direction of the quantisation axis using the coordinate system
in Figure 3.5).
Now consider an atom moving at velocity v through the laser field in the z−direction. As the atom
moves, the local laser polarisation will change direction. It is convenient to introduce a rotating frame
that moves with the atom such that the polarisation direction is constant. In this frame the atom’s spin
14
Background
Figure 3.6: Atomic level transition scheme and corresponding Clebsch-Gordan coefficients for a
Jg = 1 ↔ Je = 2 atom. Image from [24].
axis will precess about the laser propagation axis, which is the same effect as if the atom were subject
to a magnetic field in that direction (the z-axis in Figure 3.5). This adds a term
Vrot = kvJz
(3.3)
to the Hamiltonian describing atomic evolution in the moving rotating frame.
By making the assumption that the detuning of the laser from resonance is much larger than the
natural linewidth of the transition, and then supposing that the atom is moving slowly through the
light field, then Eq. 3.3 can be treated as a perturbation to the atom’s Hamiltonian [24].
Using first-order perturbation theory, the term in Eq. 3.3 has no first-order effect on the energies of
the different eigenstates of the atom where the quantisation axis is taken in the y−direction. However,
the wavefunction is changed to first-order. The wavefunction of the perturbed |g0 iy state is
|g0 iy = |g0 iy + √
kv
(|g−1 iy + |g+1 iy )
2(∆00 − ∆01 )
(3.4)
where ∆0m is the light shift of the mth magnetic sublevel. The light shift is negative for red-detuned
light, and is approximately proportional to the Rabi frequency squared (so ∆01 = ∆0−1 ). By comparing
at the relative strengths of the π transitions from |g0 i and |g±1 i in Figure 3.6, we can say
4
∆00 = ∆01
3
(3.5)
Similarly to Eq. 3.4:
kv
|g0 iy
2(∆01 − ∆00 )
kv
= |g−1 iy − √
|g0 iy
2(∆01 − ∆00 )
|g+1 iy = |g+1 iy − √
(3.6a)
|g−1 iy
(3.6b)
These equations show that a moving atom initially in the state |g0 iy is ‘contaminated’ by the other
two ground states |g+1 iy and |g−1 iy (and vice versa).
Background
15
The next step is finding the relative steady-state populations of the eigenstates |g0 iz , |g+1 iz and
|g−1 iz . This can be accomplished if we exploit the fact that the expectation value of Jz is proportional
to the relative occupancy of its eigenstates. To this end, we first find the expectation value of Jz with
the quantisation axis in the y−direction (see Appendix A of [24]):
y hg0 |Jz |g0 iy
=
2~kv
∆00 − ∆01
y hg+1 |Jz |g+1 iy
=
y hg−1 |Jz |g−1 iy
(3.7a)
=
~kv
∆01 − ∆00
(3.7b)
These new |gm iy states have the same relative populations as the |gm iy states. Weighting the
expectation values of Jz (Eq. 3.7) by the relative populations of the |gm iy states, one can calculate
the steady-state expectation value of Jz as
2~kv
hJz ist = 0
∆0 − ∆01
9
2
2
−
−
17 17 17
=
40 ~kv
17 ∆00
(3.8)
where Eq. 3.5 has been used. Thus an atom moving in a σ + − σ − laser field has an average Jz
proportional to the velocity of the atom, and hence the two eigenstates |g±1 iz of Jz have different
steady-state populations. Writing Π+1 and Π−1 to represent these populations, from Eq. 3.8 we get
hJz ist = ~(Π+1 − Π−1 ). So:
40 kv
.
(3.9)
Π+1 − Π−1 =
17 ∆00
∆00 is negative for red-detuned light, so it follows that if the laser beams are red-detuned from atomic
resonance, and the atom is moving in the positive z direction, the |J−1 iz state has a larger population
than the |J+1 iz state.
Absorption of a σ + photon increases the eigenvalue of Jz by 1, and absorption of a σ − photon
decreases it by 1. Say our laser beams are configured to match those shown in Figure 3.5. Figure
3.6 shows that an atom in the |g−1 iz state is six times more likely to absorb a σ − photon than a σ +
photon, while the opposite is true for an atom in the |g+1 iz state. Eq. 3.9 shows that if an atom is
moving in the positive z direction with red-detuned light, then the |g−1 iz state is more populated than
the |g+1 iz state. This atom thus scatters more photons travelling in the negative z direction, and the
reverse is true for an atom travelling in the opposite direction. Hence whichever way the atom moves,
its motion is opposed and it slows down.
3.3.2
Practical Considerations
Due to the small capture velocity of PGC, for effective cooling one must start with pre-cooled
atoms. This can be accomplished with a MOT, which Doppler cools the atoms enough for the PGC
to be effective.
16
Background
PGC is assumed to take place in zero external magnetic field. In the presence of an external
magnetic field the atoms will still be cooled, but to some non-zero mean velocity. In other words
the velocity distribution of an ensemble of atoms will narrow, but the mean velocity will not be zero
[27]. An external magnetic field will add to the ficticious magnetic field generated by atomic motion
through the light field: the atom will be cooled to a point where these two magnetic fields add to zero,
which is some non-zero velocity.
Because of the detrimental effect of external magnetic fields, the MOT quadrupole magnetic field
must therefore be turned off for the duration of PGC, and any stray magnetic fields must be eliminated
also (such as Earth’s magnetic field, fields from nearby equipment, etc). The method used in this
experiment for compensating for stray magnetic fields is described in Section 4.5.3.
The optimum parameters for PGC are a compromise. The derivation of PGC assumes the atom is
at steady-state, i.e. the interaction time is long compared to the optical pumping time [25]. However
if there is no MOT quadrupole magnetic field, the atoms are no longer trapped and will eventually
diffuse away. The PGC step of cooling must be long enough such that the atoms have reached the lowest possible temperature, but must be short enough such that the atoms have not moved a significant
distance. Parameters and results from this experiment are described in section 5.2.
3.4
Atomic Interference using a Resonant Optical Standing Wave
The long term goal for this experiment is to accurately measure interference between atomic momentum eigenstates, for the purpose of precision measurements of quantities such as the fine-structure
constant α, and acceleration due to gravity g. This section explains the principle of an atomic interferometer, and how it can be practically realised.
As mentioned in the introduction, atoms can be made to interfere by passing them through a diffraction grating. The kind of diffraction grating used for the experiment described in this thesis is called an
optical mask. The optical mask implemented for this thesis is comprised of two counter-propagating
laser beams resonant with an open atomic transition. The two laser beams interfere such that the light
intensity along the propagation axis is periodic, with the period being half the wavelength of the light
used. Atoms in a particular electronic ground state interact with the mask and are pumped into a
different ground state unless they are close to the nodes of the standing wave where the light intensity
is very low. The atoms that were not at the nodes of the mask are pumped into a far off-resonant
internal state. Once in this state, the atoms are decoupled from the light field and no longer interact
with the laser beam, they have effectively been ‘absorbed’ by the optical mask. The density of atoms
remaining in the original state is now spatially modulated, and looks very similar to how it would if
the atoms had passed through a absorption grating. This is shown in Figure 3.7.
An alternative to the absorptive optical mask considered here is to use a standing wave of far offresonant light. This creates a phase grating for the atoms which can also diffract atoms into different
Background
17
Laser
Laser
85
Rb atoms in the
F=3 ground state
Figure 3.7: Operation of the optical mask. The two lasers form a periodic intensity pattern across the
cloud of atoms. Atoms not at the nodes of the standing wave are depumped to the F = 2 ground
state, atoms at the nodes stay in the F = 3 ground state as they see very little light.
momentum states. In the limit of short pulses, this is known as as Kapitza-Dirac scattering. For more
information see page 1059 of the review [4].
Our experiment uses 85 Rb, which has an energy level structure convenient for interferometry experiments. Figure 3.4 shows the hyperfine structure of the D2 transition utilised in this experiment.
The F = 3 → F 0 = 4 transition is closed: an atom excited to the F 0 = 4 state can only decay back
down to the F = 3 ground state. This means this transition can be used for trapping, cooling and
detecting the atoms. The F = 3 → F 0 = 3 and F = 3 → F 0 = 2 transitions are open and can be
used in the optical mask. The meaning of this is explained in the next paragraph.
Figure 3.8 shows what happens to an atom not at a node of the optical mask. Say the atoms start in
the F = 3 ground state, and the optical mask is resonant with the F = 3 → F 0 = 2 transition. An
atom in the presence of this light will move up to the F 0 = 2 excited state. Provided the light intensity
is low enough such that stimulated emission is not significant, it will fall into the F = 2 ground state
with a probability of 0.79, and into the F = 3 ground state with probability 0.21. Once in the F = 2
ground state it will stay there as the optical mask is about 3 GHz off-resonance from any transition
starting in the F = 2 ground state, and there is no light except that from the optical mask present at
this stage. Now say we have a cloud of atoms subject to an optical mask pulse. If the mask is turned
on for a time equal to several pumping cycles, most of the atoms not at the nodes of the optical mask
will end up in the F = 2 ground state. Atoms at the nodes do not see any light and stay in the F = 3
18
Background
ground state. The atom cloud now has a periodic density pattern of atoms in different states, with the
period being half the wavelength of the light used for the optical mask. We use a laser beam resonant
with the F = 3 → F 0 = 4 transition to detect the atoms, so atoms in the F = 2 ground state can be
considered gone from the system, i.e. absorbed by the optical mask.
F' = 3
F' = 2
0.56
0.79
0.44
0.21
F=3
F=2
Figure 3.8: Relative branching ratios from different 85 Rb hyperfine excited states. Branching to the
F = 2 ground state is much stronger from the F 0 = 2 excited state than from the F 0 = 3 excited state.
Values were calculated from data in [25] by assuming equal population in all magnetic sublevels, and
averaging over all 3 possible light polarisations.
Besides modulating the density pattern, the optical mask has another effect on the atoms known
as the “optical Stern-Gerlach effect” in which the standing wave transfers momentum to the atoms.
For sufficiently short pulses, the process is coherent [4]. The amount of momentum transferred is
proportional to the gradient of the electric field (i.e. the dipole force) [28]. This coherent transfer of
momentum to the atoms is effectively a beamsplitter, diffracting each atom into different momentum
states which evolve coherently, i.e. with a well-defined phase relationship.
After a single optical mask pulse, the density-modulated atoms do not quite constitute an interferometer. While they are very cold, the atoms still have a Maxwell-Boltzmann velocity distribution and
can be thought of as an incoherent mixture of plane waves. While each atom is coherent with itself,
there is no definite phase relationship between different atoms.
To obtain a fully-functioning interferometer, a second optical mask pulse is used. Applied at a time
T after the first pulse, it quenches all atoms except those with an integer multiple of a momentum that
lets them pass through both optical mask pulses without being depumped into the dark state. This
is shown in Figure 3.9. Atoms that survive the first pulse disperse due to their initial momentum
distribution, atoms that survive both pulses must have a momentum such that they moved an integer
number of wavelengths along the optical mask during the time between the pulses. After this second
pulse the remaining atoms are coherent, and form an interference pattern that can be measured (note
that the uncertainty principle does not significantly contribute to broadening of fringes in the interfer-
Background
19
ence pattern [1]). While Figure 3.9 shows a classical picture of the possible interference patterns, one
can measure interference patterns that are classically not possible, as in [29].
An appropriate value for T , the time between optical mask pulses, has been experimentally shown
to be of the order 50 µs [29]. If the pulses are too close together then there will be less selection of
transverse momentum states, if they are too far apart then less atoms will survive overall as they are
being depumped due to stray light fields and they will also eventually move out of the optical mask
beams.
Figure 3.9: Classical trajectories for atoms after the optical mask. The horizontal axis is time, and
the vertical axis is distance along the optical mask. The black lines represent classical trajectories
of atoms. After the first mask atoms are localised in position space but not momentum space. They
diffuse out for time T , then a second mask ‘absorbs’ all atoms except those with an integer multiple
of a particular momentum. Image from [29].
The optical mask could be applied to gravimetry: If the optical mask propagation axis is vertical,
the atomic interference pattern will fall between pulses. A third optical mask can be applied at time
2T , by measuring the number of atoms that survive this third mask as a function of the phase of
the mask, the interference pattern can be ‘mapped’. By mapping the interference pattern around 2T ,
the gravitational acceleration of the atoms can be determined very accurately. This technique can be
made even more accurate by measuring around times that are harmonics of the interference pattern,
i.e. at times t = T (N + 1)/N . The higher-frequency interference pattern means smaller movement
can be resolved. [9]
20
Background
Chapter 4
The MARIE Experimental Apparatus
Figure 4.1: Photo of MARIE’s vacuum chamber and axes definition used in this thesis.
This chapter describes the instruments and equipment used to make the experiment, so that the later
chapter on characterising the experiment can be understood.
22
4.1
The MARIE Experimental Apparatus
Overview
The experiment was named Measuring Atomic Resonances in an Interferometric Experiment,
which coincidentally forms the acronym MARIE. MARIE is a typical cold-atoms experiment: A
vacuum chamber with windows to allow laser light to enter, an internal source of alkali metal atoms,
surrounding optics, cameras/observation instruments, and coils of wire for generating desired magnetic fields.
The entire experiment is controlled at the top level by a LabVIEW program which is basically
a highly configurable arbitrary function generator. Data is collected on an oscilloscope, saved in a
binary file on a computer and then analysed with MATLAB.
4.2
Vacuum Chamber
Figure 4.1 shows the heart of MARIE: a cylindrical vacuum chamber (14 cm radius, 17 cm depth),
with 10 cm radius windows on both of the flat sides, and eight smaller flat faces arranged radially
in an octagonal pattern around the cylinder. Seven of these eight sides have windows in them. Four
of these windows are 2 cm in radius, and the other three are 3.2 cm in radius. The windows are
not anti-reflection coated, and light transmission at normal incidence was measured to be 93% per
window. The entire chamber is oriented such that the cylinder axis is parallel to the ground (see
Figure 4.2 for a schematic). Pressure is kept below the measurement capability of an MKS I-Mag
cold-cathode vacuum gauge (less than 10−11 Torr) with a combination of a Varian StarCell ion pump
running continuously, and a titanium sublimation pump (TSP) which we ran whenever pressure in the
chamber was measurable. The ion pump was installed as far as possible from the main chamber as it
emits quite a strong magnetic field, which can interfere with experiments. Experiments take place in
the centre of the cylindrical region of the system, which is connected to the two pumps by a series of
pipes coming out the bottom of the cylinder.
4.3
Rubidium Dispenser
Rubidium for experiments is sourced from a SAES alkali metal dispenser, commercially produced
for coating applications. It consists of a small amount of rubidium chromate with a reducing agent,
such that when the compound is heated the metal is reduced and pure rubidium is released from the
dispenser as a vapour. The source is mounted on a wire, so that it can be heated by passing a current
through the wire. The dispenser sits behind a 5 cm-long narrow copper tube pointing directly at the
centre of the experiment trapping region, collimating the vapour to increase the trapping efficiency
and decrease the trap load time. Current to heat the dispenser was controlled by MARIE’s LabVIEW
program.
The dispenser is powered by an Agilent 6553A power supply running in voltage-programmed
current mode. The output current is proportional to an input voltage from the computer. Typically
The MARIE Experimental Apparatus
23
Figure 4.2: Schematic of the laser beam and instrument configuration at the vacuum chamber, from
two different perspectives. Axes shown are the same as in Figure 4.1. All of the features are explained
in the text. Each feature is shown in the orientation that is most convenient. The letters label the entry
of different laser beams: O labels the optical mask beams, C the cooling beams, D the detection
beam, and R the repump laser. (a) A side-on view of the vacuum chamber. Two of the cooling beams
and the detection beam are perpendicular to the page. (b) A top-down view.
when ‘on’ a current of 5.3 A was used to heat the dispenser.
To minimise the dead-time waiting for the trap to load each time an experiment was run, the trap
was turned on as quickly as possible after each experiment. This meant most of the atoms from
the previous experiment were re-trapped, minimising the amount of time the dispenser needed to be
turned on.
For a further description and characterisation of the rubidium dispenser see [30].
4.4
Lasers and Optics
Light sources are two frequency-locked and temperature stabilised laser diode systems. One diode
provides repump light, and the other is amplified and split into three beams to provide light for trapping and cooling the atoms, detecting the atoms, and the standing wave required for the optical mask.
24
The MARIE Experimental Apparatus
4.4.1
Diodes
MARIE uses two Sharp Microelectronics GH0781JA2C temperature-stabilised laser diodes. One
diode is locked above the 85 Rb D2 F = 3 → F 0 = 4 transistion. It provides a seed to a tapered
amplifier, the output of which is passed through three acousto-optic modulators (AOMs). This is
shown schematically in Figure 4.3. The locking system used for the two diodes is described elsewhere
[31]. It can provide light with a centre frequency stable to 1 MHz, with a full-width half-maximum
linewidth (FWHM) of 2-4 MHz.
The other diode is locked above the F = 2 → F 0 = 3 transition and is used for the repump
laser. The output of the diode is passed through an Isomet AOM which is used to shift it to down to
resonance, and is also used to switch off the light when it is not needed. The output of the AOM is
delivered directly to the experiment.
4.4.2
Tapered Amplifier
The tapered amplifier (TA) used in the experiment is an Eagleyard Photonics EYP-TPA-078001000-3006-CMT03-0000. This TA was assembled as part of another student’s Honours project, see
[32] for more information on construction and characterisation. The TA is prone to multi-mode when
not aligned properly ∗ .
The TA is seeded with about 40 mW of light from the laser diode. It uses about 2 A of current to
amplify the seed beam up to about 800 mW.
4.4.3
Experiment Laser Beams
The 800 mW output beam of the TA is passed through three AOMs to shift the light to provide
three different laser beams at different frequencies for different purposes (Figure 4.3). After each
AOM the light is coupled into a single-mode polarisation-maintaining optical fibre for delivery to the
experiment. See Figure 4.2 for a diagram of the configuration of the beams at the vacuum chamber.
Due to the poor-quality ouput mode of the TA, coupling efficiency into the fibres is quite low, about
20-30%. Using the fibres is necessary to have Gaussian-shaped beams to use in the experiment. The
laser is above the F = 3 → F 0 = 4 resonance, so AOMs are used to lower the frequency of the laser
to appropriate values.
The AOMs serve a second purpose of providing a mechanism for switching the laser beams on and
off when required. Light deflection could be cut to below detectable levels in less than 200 ns.
Cooling Beam
Light for the MOT and PGC is from the same fibre. The light frequency needs to be about 8-15
∗
A good guide for aligning it can be found at http://www.eagleyard.com/
fileadmin/downloads/app notes/AppNote TPA 2-0.pdf
The MARIE Experimental Apparatus
25
Figure 4.3: Schematic of one of the laser sources for the experiment. Optics such as mirrors, waveplates, optical isolators, etc are omitted for clarity. The names and purposes of the different beams
are explained in the text.
MHz below the F = 3 → F 0 = 4 resonance for the MOT stage of the experiment, and about 18-25
MHz below resonance for the PGC stage. This means the laser frequency needs to be changed during
an experiment. This is accomplished using the locking system (see [31] for an explanation of how
this works). At the end of the MOT stage, the locking system changes the laser frequency by about
-10 MHz within 50 µs. None of the other beams derived from the same diode are being used during
PGC, so this frequency shift does not affect any other part of the experiment.
After the fibre the light is split into six beams, aligned to make the three orthogonal pairs of beams
required for magneto-optical trapping. All are circularly polarised by passing them through polarising
beam splitters followed by quarter-wave plates before entering the vacuum chamber. Two of the MOT
beams enter the vacuum chamber horizontally through the large windows, and the other four beams
enter at 45◦ angles relative to the ground through four of the smaller windows (see Figure 4.2). The
combination of all the beams provides a damping force for the atoms in three dimensions.
The cooling beam fibre outputs light with a 1/e2 diameter of 7.5 mm. Just before entering the
vacuum chamber the beams are expanded by a factor of four, so the 1/e2 beam diameter is 3 cm.
Four of the six cooling beams enter through the smaller windows on the vacuum chamber, which are
4 cm in diameter. Inevitably there is light scattered by clipping the edges of these windows. This
scattered light results in quite a large background on the photomultiplier tube (PMT, described in
Section 4.6.1). With 75 mW coming directly out of the fibre, peak light intensity at the experiment
chamber is about 3 mW/cm2 for each of the six beams, so the peak total light intensity in the trapping
region is about 18 mW/cm2 .
Detection Beam
The purpose of the detection beam is to measure fluorescence from atoms in the F = 3 ground
state. It is on-resonance with the cycling F = 3 → F 0 = 4 transition. We need a separate detection
beam for two reasons: One being that the cooling beams scatter lots of light onto the PMT as mentioned above. The detection beam therefore only passes through a large vertical window in the side
of the vacuum chamber, scattering very little light onto the PMT. The other reason is that the cooling
26
The MARIE Experimental Apparatus
beams are tuned 15 or 25 MHz below resonance, and limitations in the locking system mean it cannot
easily be set to also lock such that the cooling beams are 0 MHz detuned. For the brightest signal
from the atoms we want to detect them with on-resonant light.
The detection beam is coupled in to the same path as the MOT x-beams through a polarisingbeamsplitter cube. At the cube its 1/e2 diameter is 2.1 mm. It is polarised such that it exits through
only one side of the cube, and it is then passed through the same optics as the MOT beams to expand
it by a factor of four and circularly polarise it. The result is that the detection beam forms a circularlypolarised travelling wave. With 30 mW coming out of the fibre, the peak intensity in the vacuum
chamber is 1.5 mW/cm2 .
Figure 4.4: Schematic of the optics used to generate the optical mask.
Optical Mask Beam
The final beam used in the experiment is used to provide the standing wave to induce atomic
interference (described in Section 3.4). This beam is resonant with the F = 3 → F 0 = 2 transition,
and it is switched in the same way as the detection beam.
The transverse mode of this beam was required to be high-quality to ensure optimal interference
between the laser beams. See Figure 4.4 for a diagram of the optics used to generate the optical mask.
At the output of the fibre, there was a Glan-Thompson polariser to provide a very clean and constant
polarisation to a Wollaston prism, which then splits the beam to provide the two counterpropagating
beams for interference. After the Wollaston prism only metal mirrors are used so polarisation is
maintained. Polarising beamsplitters are placed at the windows of the vacuum chamber to correct
for any slight change in polarisation that might have occurred as a result of reflecting off the mirrors,
and in the case of one of the beams, passing through a half-wave plate as the Wollaston prism outputs
beams with orthogonal polarisations while we want both beams of the standing wave to have the same
polarisation. The relative power of the two beams was controlled by changing the angle of the GlanThompson polariser before the Wollaston prism. The standing wave in this experiment is linearly
horizontally polarised.
The MARIE Experimental Apparatus
27
With the experiment in this state, the phase of the optical mask cannot be varied. However this can
be made possible by putting an electro-optic modulator in the path of one of the optical mask beams.
This would be required for gravitometric measurements to be carried out, as described in Section 3.4.
4.5
Magnetic Field Coils
There were three sources of magnetic fields required for the experiment. Quadrupole coils for the
MOT, compensating coils to cancel Earth’s and other stray magnetic fields, and a coil to quench
particular atomic states that are unintentionally dark to the standing wave laser beam.
Figure 4.5: Schematic of the circuits used to switch the (a) quadrupole and (b) quenching magnetic
field coils.
4.5.1
Quadrupole Coils
The coils used to generate the quadrupole magnetic field for the MOT are mounted either side of the
vacuum chamber in a quasi-anti-Helmholtz configuration (the distance between the coils is slightly
more than the radius of each coil). The axes of the coils are parallel with the ground (see Figure
4.2). They have an inner radius of 10.5 cm and an outer radius of 14 cm. Each coil contains 168
turns of wire. They are powered by an Agilent 6543A power supply which provides 5.8 A current.
They generate an estimated magnetic field gradient of 7.3 G/cm in the y−direction, and 3.2 G/cm
in the x− and z−directions near the centre of the trap. PGC requires zero magnetic field, so the
quadrupole coils need to be turned off sufficiently quickly so the atoms do not move significantly
28
The MARIE Experimental Apparatus
from the centre of the beams. Cutting off the power supply suddenly will cause a large voltage
spike across the coils due to their self-inductance. To deal with this, the circuit in Figure 4.5 (a) was
constructed. The coil current is switched off using a IXYS IXFN100N50P N-channel MOSFET, and
an ST Microelectronics transient voltage suppression diode (TVS) absorbs the energy stored in the
coils. The DG642 chip switches the MOSFET gate from the 15V supply (13.6V after the resistor,
12V is needed for saturation) to ground within 500 ns, draining the intrinsic capacitor as quickly as
possible.
Assuming that the moment the MOSFET is switched off, 5.8 A of current is going through the
TVS, then the peak power dissipation is 5.8 A × 300 V = 1740 W. The TVS datasheet says that
for a current dissipation time of 150 µs, the maximum peak power dissipation is about 2500 W. We
therefore operate the TVS within its safe operating range.
Figure 4.6: Quenching coil switching on and off, and quadrupole coils switching off. The switch-off
time of the quenching coil is about 50 µs, and 150 µs for the quadrupole coils. Measured with a LEM
HEME PR200 current probe. The quadrupole coil data is very noisy during the coil switch-off, if
averaged the decay was approximately linear.
4.5.2
Quenching Field Coil
There is an important phenomenon that can adversely affect the performance of the optical mask:
the internal state of the atoms themselves. For the best signal we want all the atoms in the cloud
to react in the same way to the optical mask, however for the case of the D2 F = 3 → F 0 = 3
and F = 3 → F 0 = 2 transistions there are states which cannot be excited despite the presence of
a linearly-polarised resonant laser beam (‘dark states’, see Figure 4.7). For the F = 3 → F 0 = 3
transistion this is the mF = 0 sublevel (where mF is the projection of the total angular momentum on
the quantisation axis) and for the F = 3 → F 0 = 2 transition it is the mF = ±3 magnetic sublevels.
The MARIE Experimental Apparatus
29
This means many atoms in the anti-nodes of the optical mask will not be excited and decay to the
F = 2 ground state, giving an artificially high number of atoms that survive the optical mask. To
mitigate this effect, a magnetic field is applied in a direction perpendicular to the light polarisation.
Figure 4.7: Relative transition strengths between different magnetic sublevels in the 85 Rb D2 transition for linearly-polarised light. Dark states are the mF = 0 state for the F = 3 → F 0 = 3 transition,
and the mF = ±3 states for the F = 3 → F 0 = 2 transition. The F = 3 → F 0 = 4 and F = 1 → F 0
transitions are not shown. Data from [25].
30
The MARIE Experimental Apparatus
The effect can be explained as follows: An atom within the F = 3 ground state manifold |ψi can
be described as being in a superposition of eigenstates of Fz :
|ψi =
X
mF
cmF |mF iz e−iEmF t/~
(4.1)
Where EmF is the energy of the |mF iz eigenstate. In the case of no magnetic field, the |mF iz states
are degenerate and the exponential term can be factored out. In this case the exponential term only
contributes an overall phase which is physically insignificant. This means if we prepare an atom in
a particular |mF iy state (i.e. an eigenstate of Fy ), there is no time dependence and it will stay in its
initial state. However, in the presence of a magnetic field in the z−direction, the |mF iz states are
Zeeman-split, the energies are no longer degenerate and an atom initially in the |mF iy state becomes
a time-dependent superposition of the |mF iz states. This mixing means an atom that initially starts
in a particular magnetic sublevel will change state with time, and will eventually be in a state coupled
to the standing wave laser beam.
The quenching coil in this experiment is a single 7-cm-radius coil mounted above the vacuum
chamber to provide a magnetic field in the z−direction (the standing wave is linearly polarised in the
y−direction, see Figure 4.2 for the position of the quenching coil). The coil is powered by an Agilent
6553A power supply. This power supply is rated to supply a maximum of 15 A, but was measured to
overshoot and supply up to 18 A for a short time when the circuit switch is turned on. The distance
from the centre of the coil to the centre of the MOT is 15 cm. It is switched using the circuit in
Figure 4.5 (b), which functions in a similar way to the quadrupole coil switch. It is dangerous to have
unnecessarily high voltages, and the current was found to decay sufficiently quickly with a 10 V TVS
(timing of the experiment is described in Section 5.4.1). This meant an IRF 3202 MOSFET could
be used rather than the large and expensive IXYS MOSFET used for the quadrupole coils. The IRF
3202 has a lower saturation voltage so the gate can be driven directly from the optocoupler. The most
important characteristic of the quenching coil is that it turns on as quickly as possible after PGC, so
that the experiment can be started while the atoms are still in the centre of the trap. The current was
measured to reach 100% of maximum value 2 ms after being turned on (see Figure 4.6). Atoms at
10µK will move about 500nm in this time due to expansion, and 10µm due to gravity, so they are
still very much in the centre of the experiment region. The coil was estimated to dissipate 110 W of
power when turned on, so the coil will burn if left on indefinitely. For this reason the coil is fused
such that if it is left on, the fuse will blow, keeping the coil and experiment safe (5 A household fuse
wire was found to work best: it was found to not blow after 50 ms but would blow after 1 s, with the
power supply set to 15 A current). When running, the experiment repetition rate was about once every
three seconds. The quenching coil is on for about 3 ms each cycle, so average power consumption
is about 110 mW. The on-axis magnetic field produced 15 cm away from the coil with 3.2 A current
was measured to be 1.0 Gauss, so 18 A is estimated to produce a field of 5.6 G.
The effect of the quenching coil on atoms in the optical mask is characterised in Section 5.4.3.
The MARIE Experimental Apparatus
4.5.3
31
Compensating Coils
As discussed in Section 3.3, in the presence of a magnetic field atoms are cooled about some nonzero velocity i.e. their velocity distribution narrows about some non-zero value. For optimum cooling
we must therefore compensate for any stray magnetic fields - such as Earth’s, or fields from nearby
instruments - in the centre of the vacuum chamber. To do this three orthogonal pairs of large coils
were installed around the experiment. Each pair was placed such that the vacuum chamber centre is
halfway between them, and a perpendicular line drawn from the centre of the coil passes through the
centre of the vacuum chamber. The three pairs of coils produce fields in three orthogonal directions,
such that the magnetic field at the centre of the vacuum chamber can be controlled at will.
Each coil pair is powered by a single Agilent 3615A power supply running in constant current
mode. The x− and z−coil pairs are estimated to produce a field of 1.7 Gauss/A at the trap centre,
and the y−direction coils 1.3 Gauss/A. The optimal currents were found to be Ix = 0.19 A, Iy =
0.67 A, Iz = −0.23 A. This gives the compensating magnetic field as Bc = 0.32x̂ + 0.87ŷ −
0.39ẑ G. The y−axis points 17◦ clockwise from true north. Rotating our axes such that y → y 0 now
points north: Bc 0 = 0.56x̂0 + 0.74ŷ0 − 0.39ẑ0 G. From [33], the magnetic field in Dunedin due to
the Earth is BE 0 = 0.08x̂0 + 0.18ŷ0 + 0.59ẑ0 G. Clearly, in the x0 − and y 0 −directions at least, there
is a major contributor to the local magnetic field besides Earth. This is thought to be the nearby ion
pump, or perhaps current-carrying wires in nearby instruments.
4.6
4.6.1
Making Measurements
Photomultiplier Tube
Quantitative measurements of atomic fluorescence were made using a Hamamatsu H9858-20 photomultiplier tube (PMT). An imaging system above the vacuum chamber centre images an estimated
1.0% of the solid angle of a point source at the chamber centre onto the PMT. The light is passed
through a 780 nm bandpass filter to remove background such as room light. The gain of the PMT
is controlled by a voltage on one of the pins. This gain control voltage should be above 250 mV
and below 900 mV. Note that this is not the actual voltage applied to the PMT, which has an internal
amplifier to generate the high voltages required for charge multiplication. The PMT is affected by
magnetic fields, so it is wrapped in µ-metal as much as possible, and the quadrupole and quenching
magnetic field coils are turned off when taking important measurements.
To calibrate the PMT, a narrow laser beam with known power was shone on to the sensor. The
laser entered the vacuum chamber from the bottom, exited through the top, passed through the lens
and bandpass filter to fall onto the PMT. The laser power was measured before entering the chamber,
and then imperfect transmission through the entry window was corrected for. This determined the
current produced by a known light power in the vacuum chamber, without needing to measure the
absorption of the lens and filter. Data is shown in Table 4.1. Note that there is a systematic error
in this measurement as it assumes transmission through the system is independent of the angle of
32
The MARIE Experimental Apparatus
incidence; the calibration laser had a diameter of about 2 mm and so was transmitted only through
the centre of optics in the system and is incident perpendicular to the surfaces. Light from the MOT
is incident across the entire imaging lens. The lens is 70 mm in diameter and is 152 mm from the
centre of the MOT, so light at the edge of the lens is incident 13◦ from normal. Surfaces reflect more
at higher angles of incidence so this means that the number of atoms is underestimated.
Gain (mV)
A/W
250
300
400
500
600
700
0.818
2.83
63.2
314
1200
3290
Table 4.1: PMT output current to calibration laser power ratio as a function of gain voltage (laser
power in the vacuum chamber, before the chamber window, lens, and filter). Note the PMT collects
1% of the solid angle of the MOT light, so to determine the output light power of atoms in the MOT,
measure PMT current to determine light power before the filter and lens, and then multiply by 100.
The PMT is powered by an Agilent E3615A power supply with 3 V. Noise properties could not
be distinguished whether the PMT was powered by batteries or the power supply, so there was no
detectable mains noise coming through the power supply. The output of the PMT is connected to an
SRS SR570 low-noise current preamplifier. The output of the preamplifier is observed on a Tektronix
TDS 3054B oscilloscope. Data is transferred to the computer the LAN.
4.6.2
Photodiode for Atom Measurement
While the MOT is building we wish to monitor the number of atoms trapped, so the experiment
can be started once a threshold has been reached. The oscilloscope can be used to monitor the PMT
output and trigger the computer to start the experiment once the trap contains the desired number of
atoms. However, data transfer from the oscilloscope takes around 200 ms, so there can be a large
variation in the initial number of atoms due to the slow sampling rate. The solution to this was to
make a dedicated photodiode and amplifier to monitor the trap size and then trigger the computer
through a digital input at exactly the right moment.
A ThorLabs PD100 photodiode was used. It was mounted near the top of one of the large windows
on the vacuum chamber, behind a telescope which collects about 0.8% of the solid angle of the light
emitted by atoms in the MOT and images it onto the diode. The circuit used as the amplifier and
trigger is shown in Figure 4.8. The photodiode does not need to be reverse-biased as high speed
response is not needed in this situation. Biasing also increases noise from the diode † ; given the very
high gain of the amplifier circuit it is critical to eliminate as much signal noise as possible.
†
See,
e.g.
http://sales.hamamatsu.com/assets/applications/SSD/
photodiode technical information.pdf
The MARIE Experimental Apparatus
33
Figure 4.8: Circuit used to amplify photodiode signal and trigger computer. All op-amps are powered
with ±15 V.
Referring to Figure 4.8: the transimpedence amplifier converts the photodiode current to a voltage
and amplifies it, the signal is then amplified again by an inverting amplifier, a 50 Hz notch filter
is needed to remove mains noise from the signal, an RC filter is used to smooth out the signal,
which is then compared to a reference voltage with an analogue comparator. When the amplifier
voltage is greater than the reference voltage, it pulls a digital input on the computer low, triggering
the experiment. The atom number can thus be controlled by the reference voltage. The reference
voltage is proportional to the number of atoms as there is very little background from the lasers (a
typical Vref used was 1-3 V, while the background from the lasers is less than 50 mV).
Rather than calculate the amount of light incident on the photodiode and the gain of the amplifier
circuit, it was easier to calibrate the photodiode by comparing it to the signal from the alreadycalibrated PMT. See Section 5.1.
4.6.3
PIXIS CCD Camera
We used a Princeton Instruments PIXIS 1024 CCD camera to perform time-of-flight (TOF) measurements of the atom cloud. The basic idea is that the atom cloud is left to expand for a brief time,
and then the postion distribution is measured by the camera. By doing this repeatedly for different
expansion times, the velocity distribution of the atoms can be calculated. The cloud temperature can
be calculated from the velocity distribution, as explained in Appendix B.
The camera looked through a mirror up at the MOT from underneath the vacuum chamber. A twolens imaging systems was placed in front of the camera to image the cloud on the CCD detector. A
f = +500 mm lens and a f = +150 mm lens were used, arranged to make the magnification of the
system equal to 0.3.
To calibrate and make sure the camera would focus on the MOT atom cloud, the distance from
the telescope to the vacuum chamber centre was carefully measured. A mirror was placed in front of
the telescope, and a ruler with half-millimetre marks was placed after the mirror. Care was taken to
ensure the ruler was exactly the same distance from the telescope as the MOT centre. The ruler was
34
The MARIE Experimental Apparatus
illuminated with 780 nm light, and the telescope lenses were moved until the ruler was in sharp focus.
By measuring the distance in pixels between marks on the ruler, it was found the conversion factor of
pixels to millimetres is 23.38 pixels/mm. Moving the ruler by ±1 cm changed the conversion factor
by less than 0.5% (1 cm is thought to be larger than possible error in the distance measurement).
Pixel size on the detector is 13 µm × 13 µm. The measured magnification is therefore 23380 px/m
× 13 ×10−6 m/px = 0.3039. This is very close to the calculated magnification of the imaging system.
The camera can be triggered with a digital output on the computer. Once triggered, the shutter
begins opening and will remain open for a pre-programmed exposure time. The time from the trigger
pulse to the shutter being completely open is between 3 and 4 ms. For this reason the camera is
triggered 5 ms before it is needed to photograph something.
The atoms fluoresce in the presence of resonant light, and the camera view is dark otherwise. A
short laser pulse is therefore used as a ‘flash’ so that the atoms are only imaged at exactly the desired
time, and the slow opening/closing of the camera shutter is not a problem.
4.6.4
Video Camera
For qualitative observations of trapped atoms, a CCD video camera was installed in the experiment
to enable real-time viewing of the MOT on a TV. This is useful in optimising various parameters, as
it was found a symmetrical cloud usually corresponded to a symmetrical velocity distribution. Thus
to roughly optimise laser beam powers and compensating magnetic fields, these parameters can be
adjusted until the MOT cloud looks symmetrical. Fine-tuning is done by directly measuring the effect
of changing a parameter on the cloud shape and temperature with the PIXIS camera (temperature
measurements are described in section 5.2).
4.7
LabVIEW Control System
Automatic data collection is a necessary part of any modern atomic physics experiment. Precise
timing requirements and large parameter spaces mean manual control is impossible. For this reason
a LabVIEW program was developed to control MARIE. This program can prepare a sample of cold
atoms and take measurements repetitively without any user input. It is highly customisable, and can
output arbitrary waveforms on 8 digital and 8 analogue channels simultaneously.
4.7.1
Hardware
Data is output using a National Instruments PCI-6733 High-Speed Analogue Output data acquisition card. It has 8 digital I/O ports and 8 analogue output ports with 16-bit resolution. It claimed
to have a maximum output sample rate of 1 MHz, but it was found attempting to run much above
200 kHz would result in a program crash (a possible cause was the computer not being able to load
samples onto the card fast enough, but this was never proven correct). However, this did not prove to
The MARIE Experimental Apparatus
35
be a problem as the fastest output rate needed was 100 kHz. As we will see later, temporal resolution
down to 1 ns was required to control the optical mask, but this was accomplished using an external
arbitrary function generator triggered by the LabVIEW program.
Figure 4.9: Screenshot of MARIE’s LabVIEW interface.
4.7.2
Operation
The LabVIEW program controls the dispenser, magnetic field switches, and AOMs, and triggers
the camera, function generator, and oscilloscope. Optionally, the program can execute MATLAB
scripts before and after running the experiment.
The main feature of the program is the lines of outputs which can be seen in the lower half of Figure
4.9. For each line, the user specifies the output value for each of the 16 output channels and the length
of time the outputs will have that value. The time of any output and the value of an analogue output
can be made to change each run if the user wants to scan over a range of parameters.
A block diagram of operation of the program is shown in Figure 4.10. Once a user clicks ‘Go!’,
the first thing that happens is the program executes a MATLAB script, which is usually used to set
36
The MARIE Experimental Apparatus
instrument parameters for that run. See the next subsection for details. Next, LabVIEW reserves the
hardware for MARIE’s program to stop another program using it, and outputs the values on the first
line. The length of time these values are output for depends on mode selection by the user: In ‘Timed
MOT’ mode the program will simply output the first line for the specified number of seconds, then
continue with the experiment. In ‘Measured MOT’ mode it will output the values on the first line
until it receives a trigger from the photodiode circuit seen in Section 4.6.2. Once the time is up, or the
MOT has reached the desired size, the program outputs the rest of the values at the times specified.
After the outputs are finished, the program can execute another MATLAB script. Once it has finished
the number of runs it was programmed to do, it will go back to an idle state. If told to run repetitively,
it can be made to change the value of an analogue output each run, or change the length of an output
each run.
All the measurements described in this thesis were made possible using this LabVIEW program.
A full description of each feature in Figure 4.9 and a full list of outputs is given in Appendix A.
4.7.3
MATLAB Scripts
One very useful feature of the LabVIEW program is the ability to execute MATLAB scripts. The
program can execute an arbitrary MATLAB script before and after a running an experiment sequence.
Recent versions of MATLAB have support for the VISA communication standard, so it can be used to
program and download data from instruments that support this standard. VISA was used to download
measurements from the oscilloscope over a TCP/IP connection, and as explained in the next section,
it was also used to program the behaviour of the optical mask over a GPIB connection.
4.7.4
Controlling the Optical Mask
Controlling the optical mask required generating pulses much shorter than the sample period of
the LabVIEW program (the authors in [29] use 800 ns pulses). For characterisation it was useful to
generate pulses from 200 ns to 100 µs. An Agilent 33250A Arbitrary Function Generator (AFG) was
used to make short pulses, it can generate pulses with picosecond resolution. It could be programmed
before each experiment run using a MATLAB script and then triggered with an external pulse from
the computer.
The arrangement of the computer, optical mask AOM, and AFG is shown in Figure 4.11. While
a completely arbitrary waveform could be generated in MATLAB and then downloaded to the AFG
over the GPIB connection, this was quite slow (several seconds). For some purposes, as explained
later, it was useful to generate a short optical mask pulse and then a short time later a much longer
pulse. Because of the long download time it was faster to use the AFG’s built-in settings for the first
pulse, and then use the computer to generate the longer pulse.
By measuring the optical mask light intensity with a fast photodiode, it was empirically found that
the optical mask pulses were symmetrical only for particular pulse lengths. For example, a 540 ns
The MARIE Experimental Apparatus
37
Figure 4.10: Block diagram showing conceptual operation of MARIE’s LabVIEW program.
pulse from the AFG produced a 540 ns light pulse followed by a ∼ 1 µs-long tail at about half the
amplitude. A 536 ns pulse produced no such tail. It was never confirmed, but this effect is thought to
be due to an impedence mismatch between the OR gate (shown in Figure 4.11) and the optical mask
AOM driver. ‘Good’ pulse lengths, i.e. pulse lengths that produce symmetrical light pulses are given
in Table 4.2.
38
The MARIE Experimental Apparatus
Figure 4.11: Instruments used to control the optical mask AOM.
Pulse Length (ns)
120
145
265
400
536
670
809
950
Table 4.2: Optical mask pulse lengths that do not produce a tail. This table is mainly included for
reference for a future student. For longer pulses, the tail is short compared to the pulse length so is
not seen as significant. Due to the finite rise and fall time of the AOM, pulses 120 ns and 145 ns long
do not reach maximum amplitude.
Chapter 5
Characterising MARIE
5.1
Counting Atoms
The number of atoms in the MOT can be counted by measuring their fluorescence and calculating
the scattering rate. The scattering rate can be interpreted as the number of photons an atom scatters
from a laser beam every second, and for a two-level atom is given by [34]:
R=
1
21+
I
Isat Γ
I
Isat
+4
∆ 2
Γ
(5.1)
where I is total laser intensity, Isat is the saturation intensity for the transition, Γ is the natural
linewidth of the transition (for the D2 transition in 85 Rb this is 2π× 6.1 MHz), and ∆ is the detuning
of the laser from resonance.
The number of atoms is then:
N=
4π C
Ω GER
(5.2)
where Ω is the solid angle of the MOT light collected by the PMT (for this experiment, Ω/4π =
0.010), C is the PMT current, G is the responsivity of the PMT (see Table 4.1), and E is one photon
energy (E = hc/λ). So the number of atoms is the power incident on the PMT (corrected for
absorption in lens and filter), divided by the fraction of emitted light collected by the PMT, divided
by the power emitted by a single atom.
Figure 5.1 shows the PMT current signal from a cloud of cooled atoms for different initial Vref ’s.
To make this figure the photodiode measured MOT light scattered from the atoms that was 8 MHz
red-detuned, then the atoms were cooled (procedure described in Section 5.2) and all the lasers were
turned off. The detection laser was then turned on for 50 µs to provide the signal to the PMT. In other
words, Figure 5.1 shows the number of atoms trapped for different Vref ’s, with the cooling lasers 8
MHz detuned. The right-hand vertical axis was calculated using Eq. 5.2 and Table 4.1.
Characterising MARIE
4.5
688
4
611
3.5
535
3
458
2.5
382
2
306
1.5
229
1
153
Number of Atoms ( ×103 )
PMT Current ( ×10−8 A)
40
0.5
0
76
0
1
2
3
4
5
6
0
Vref (V)
Figure 5.1: Average PMT current over 50 µs detection pulse as a function of photodiode reference
voltage Vref . Error bars show absolute variation in 5 measurements. Right-hand vertical axis shows
number of atoms calculated corresponding to the fluorescence signal. For these data, the detection
laser was on-resonance with the F = 3 → F = 4 transition and was a circularly polarised travelling
wave. The saturation intensity for this situation is 1.67 mW/cm2 [26]. The detection laser intensity
was about 1.5 mW/cm2 .
In order to measure the light intensity inside the vacuum chamber of the detection beam, the fluorescence of a cloud of cold atoms from a 50 µs detection pulse was measured as a function of light
power at the output of the detection beam fibre. This is shown in Figure 5.2. To work out the light
intensity, Eq. 5.1 was fit to the data with ∆ = 0 (The detuning of the detection beam was measured
to be zero using the technique discussed in Section 5.3), and letting I = αP , where I is the light
intensity, P is the light power coming out of the fibre, and α is the only fitting parameter. α acts as
a scale factor that lets us convert from something we can easily measure (the light power at the fibre
output) to a parameter that we need to know for data analysis (the light intensity experienced by the
atoms). The solid line in Figure 5.2 has α = 0.0472 cm−2 .
Characterising MARIE
41
Fibre output power (mW)
25
0
10.596
21.192
31.788
0
0.5
1
1.5
42.384
Fluorescence (a.u)
20
15
10
5
0
2
2.5
2
Light Intensity (mW/cm )
Figure 5.2: Fluorescence of a cloud of cold atoms subject to a 50 µs detection beam pulse, as a
function of detection beam power/intensity. The top x−axis was measured, and the bottom x−axis
was calculated using the method discussed in the text. Error bars show peak-to-peak variation over
four measurements. The solid line is Eq. 5.1 fitted to the mean of the data with Isat = 1.67 mW/cm2 .
5.2
5.2.1
Magneto-Optical Trap and Optical Molasses
Getting Cold Atoms
Obtaining cold atoms is a two-step process: first a magneto-optical trap (MOT) to collect the atoms
in the centre of the trap, and then polarisation gradient cooling (PGC) to cool the atoms. The optimum
parameters for PGC can be experiment-dependent, as they depend on many variables. A literature
survey found the optimum length of PGC was somewhere between 5 and 10 ms, and optimum laser
detuning during PGC was somewhere greater than 25 MHz below resonance [35] [36]. We found our
PGC worked well even with the cooling lasers only 18 MHz detuned from resonance, but worked best
with about 25 MHz detuning. The procedure we found to be optimum for cooling atoms is shown in
Figure 5.3.
This experiment is able to routinely produce samples of atoms at temperatures of 8-12 µK.
42
Characterising MARIE
Figure 5.3: Optimum parameters for cooling atoms. Frequencies refer to red-detuning of cooling
laser from resonance. Intensities refer to total intensity of cooling laser inside the vacuum chamber
The time taken for the number of atoms in the MOT to reach the desired number depends on how
many atoms are re-trapped from a previous experiment, and how hot the dispenser wire is.
5.2.2
Measuring the Temperature
In this experiment, the temperature of the atom cloud was measured using the time-of-flight technique. Once cooled, the atoms are left to expand freely for a small amount of time and then the cloud
is imaged by flashing the MOT laser beam on for a short time, and the atom cloud can be seen using
the calibrated PIXIS camera. The density of atoms in the cloud is proportional to fluorescence, the
camera can measure fluorescence as a function of position and hence can measure the cloud density
as a function of position. By doing this repeatedly for different times, the speed of expansion can be
measured. Classically, the temperature of a gas can be related to the velocity distribution of its constituent particles. Therefore a measurement of the rate of expansion of the cloud can be interpreted as
a measurement of its temperature.
Assuming a Maxwell-Boltzmann velocity distribution, the density of the cloud along in the x − y
plane at time t should be proportional to (derivation in Appendix B):
√
x2 + y 2
Av03 π
exp − 2
N (x, y) = 2
r0 + t2 v02
r0 + t2 v02
(5.3)
p
where v0 = 2kT /m, T is the temperature of the atoms, m is the mass of an atom, k is Boltzmann’s
3/2
m
constant, r0 is the 1/e width of the cloud at t = 0, and A = 2πkT
. The origin is taken as the
centre of the cloud. Note that this equation describes a Gaussian with a well-defined width.
To obtain a measurement of N , the atoms were prepared as described in Figure 5.3. t is taken as
zero at the moment the laser intensity reaches 0 at the end of PGC. Once the laser was off the camera
shutter was opened, then the atom cloud was left to expand for a short time. At the desired time, the
cooling lasers were briefly (1 ms) turned on to make the atoms fluoresce and provide a ‘flash’ for the
Characterising MARIE
43
camera∗ . Doing this for a range of different expansion times provided a measurement proportional to
N (x, y) as a function of t (the quantum efficiency of the camera is unknown). The cloud is imaged
from directly below, so the temperature in the z−direction cannot be measured using this technique.
Example camera data is shown in Figure 5.4 (a).
Integrating N (x, y) in one direction to get a one-dimensional signal in the other provides the best
signal-to-noise ratio. For example, in the x−direction:
N (x) =
Z
∞
N (x, y)dy
−∞
Av03 π
exp −
2
=p 2
r0 + t2 v0
x2
r02 + t2 v02
(5.4)
So as not to drown the signal with noise, in practice N (x, y) was only integrated over the narrow
slice where the signal is non-zero, as indicated in Figure 5.4 (a). This obtained data that looked like
Figures 5.4 (b) and 5.4 (c). Gaussian functions of the form
(x − c2 )2
y = c1 exp −
c3
+ c4
(5.5)
√
were fit to the data. Comparing Eq. 5.5 to Eq. 5.3, we can see that c1 = αAv03 π/(r02 + t2 v02 ) (where
α is an unknown constant relating the density of the atoms to the magnitude of the camera signal)
and c3 = r02 + t2 v02 . Measuring c1 and c3 for a range of different times gives two ways of measuring
the temperature of the atoms: seeing how the peak height (c1 ) changes with time, and seeing how
the width (c3 ) changes with time. Plotting c1 and c3 against time, and choosing r0 and T to best fit
the data provides a measurement of r0 and T . In practice ten photos of the atom cloud were taken
for expansion times from 5 ms to 23 ms. Each photo is of a different atom cloud. The cooling beam
‘camera flash’ will affect the temperature of the cloud, so each measurement has to be of a different
experiment realisation.
See Figure 5.5 (a) for measurements of c3 , and Figure 5.5 (b) for measurements of c1 .
Some comment should be made on the two different ways of measuring the temperature. Measuring
c3 is more accurate - it is less susceptible to shot-to-shot cooling beam amplitude noise, and atom
number fluctuations. When plotted against t2 as in Figure 5.5 (a), it provides easy interpretation
of the initial size of the cloud (zero crossing of the line), and the temperature (proportional to the
gradient). It does however rely on accurate calibration of the pixel-to-length conversion explained in
Section 4.6.3† . c1 however does not depend at all on the calibration of the camera, only on how the
peak brightness changes with time. The very good agreement between these two methods, as shown
∗
The cooling beams were used for this purpose, rather than the detection beams, simply because the detection beam
was not installed when the temperature measurement technique was developed. It was found to work quite well so there
was no need to change it.
†
Although this was checked in two different ways, which were found to agree, as discussed in Section 4.6.3
44
Characterising MARIE
Figure 5.4: a) Example raw data from the PIXIS camera showing a fluorescence measurement of a
cloud of atoms 5 ms after the trap has been turned off. Dashed white lines show ‘slices’ integrated
to get projections along b) x−axis and c) y−axis. Red lines are Gaussian fits to the data. Axes show
orientation of photo relative to experiment axes.
in Figure 5.5, is a strong indication the temperature is being measured correctly.
5.3
Measuring the Laser Frequency
One thing cold atoms are very useful for is as an absolute frequency reference. There is very
little Doppler broadening of the transition spectrum, so the frequency can be measured to an accuracy
limited by power broadening, the natural linewidth of the transition, and the linewidth of the light used
for the measurement. For an atom at 10 µK, the Doppler broadening should be about 200 kHz, which
is much less than the natural linewidth of the transition (Γ = 6.1 MHz for the 85 Rb D2 transition).
The cooled atoms can be used to measure the frequency of the detection laser. This was practically
accomplished by subjecting a cloud of cold atoms to a 50 µs pulse from the detection laser and
measuring the fluorescence. This was done for a range of detection laser AOM frequencies with the
laser locked, so fluorescence as a function of AOM frequency could be measured. Data is shown in
Figure 5.6. A clear peak can be seen at 140 MHz. The detection beam AOM deflects light into the
-1st order, so Figure 5.6 tells us our laser is locked 140 MHz above the F = 3 → F 0 = 4 transition.
The centre frequency of the AOM used for the measurements was 150 MHz, and light power was
recalibrated after each adjustment of the AOM frequency, so that the same light power was used for
all measurements.
Characterising MARIE
45
−6
x 10
1.2
(a)
r02 + t2 v02 (m2 )
1
0.8
Tx = 7.6µK
0.6
Ty = 7.7µK
r0x = 0.25mm
0.4
r0y = 0.25mm
0.2
0
0
1
2
3
2
4
5
2
6
−4
t (s )
x 10
16
(b)
14
r0x = 0.23mm
r0y = 0.23mm
q
π/ r02 + t2 v02
12
αAv03
√
10
8
Tx = 7.7µK
Ty = 8.1µK
6
4
2
0
0.005
0.01
0.015
0.02
0.025
Time (s)
Figure 5.5: Five measurements of (a) the width of the atom cloud and (b) the ‘peak height’ as given
in Eq. 5.3, as a function of the free-expansion time, with measurements from five separate days.
Errorbars show peak-to-peak variation in measurements. Solid lines are best fits to the mean of all
the measurements. Blue data are measurements in the x−direction and red data are measurements in
the y−direction. Temperatures and radii given correspond to the solid lines. The size of the variation
in different days’ measurements corresponds to a temperature variation of about ±1µK.
46
Characterising MARIE
Fluorescence Signal (a.u.)
25
20
15
10
5
0
110
120
130
140
150
160
170
AOM Frequency (MHz)
Figure 5.6: Fluorescence signal vs. detection laser AOM frequency. The solid blue line is Eq. 5.1
fitted to the mean of the data with I = 1.5 mWcm−2 , it has a FWHM of 8.3 MHz. The peak is
at 140.0 MHz. The red dashed line is a Lorentzian fit to the mean of the data where the width was
chosen by a least-squares algorithm, and has a FWHM of 11.6 MHz. Error bars show peak-to-peak
variation in 10 measurements.
The blue solid line in Figure 5.6 is Eq. 5.1 with I = 1.5 mW/cm2 , and has a FWHM of 8.3 MHz,
the only fitting performed was the amplitude and position of the peak. The red line is a Lorentzian
where the width was chosen by fitting to the data, and has a FWHM of 11.6 MHz. The broader width
of the measured data can be explained by the finite laser linewidth: this measurement was effectively
a convolution of the laser and atomic spectral widths. The laser has a linewidth measured to be 3
MHz, so the red line has a width almost exactly what one would expect. The width of the peak
is also consistent with the atoms being the temperature measured using the time-of-flight technique
described earlier.
Once the laser frequency is known, the cooling laser and optical mask laser frequencies can be
chosen appropriately: The cooling laser should be about 15 MHz red-detuned from the F = 3 →
F 0 = 4 transition, and the optical mask should be resonant with the F = 3 → F 0 = 2 or F 0 = 3
transition.
Characterising MARIE
5.4
47
Optical Mask
As discussed in Section 3.4, the optical mask should act as a comb of transmission slits. Atoms not
at the nodes need to be pumped quickly into the F = 2 ground state, and atoms near the nodes should
stay in the F = 3 ground state. This section describes how the optical mask was characterised and
confirmed to be functioning correctly.
To characterise the optical mask, a mask was applied to a cloud of cold atoms in the F = 3 ground
state. For all cases the optical mask light was resonant with the F = 3 → F 0 = 2 transition. The
fluorescence was then measured after the mask with a pulse from the detection beam. This effectively
performs a measurement of how many atoms were lost to the F = 2 ground state as a result of the
mask (i.e. measures the ‘absorption’ of the mask). This absorption was measured for a range of
different mask parameters to ensure it was working as expected.
5.4.1
Noise
There are two large sources of noise in characterising the optical mask:
1) The initial number of atoms can fluctuate. While the photodiode trigger was made so that the
initial number of atoms was always the same, it was found there are still fluctuations of about ±10%.
The effect of these fluctuations could be minimised by measuring the number of atoms during each
run, and comparing the measured signal to the initial number of atoms.
2) There is background noise on the PMT from scattered detection beam light and fluorescence
from stray atoms (i.e. atoms that fluoresce due to the detection beam but were not in the optical
mask). This background could be minimised by depumping all the atoms with a long optical mask
pulse, and then measuring fluorescence. This measured background can then be subtracted from other
measurements.
To minimise noise, a particular detection pulse sequence was developed. Figure 5.7 shows raw data
from a typical run of the experiment. A typical run of the experiment went as follows (letters indicate
corresponding events in the figure):
• Trap and cool the atoms as in Section 5.2.
• Apply an optical mask pulse (100 ns – 100 µs) (b).
• Measure fluorescence from a detection pulse (50 µs) (d).
• Repump all the atoms to the F = 3 ground state with a 50 µs pulse from the repump laser (50
µs) (e).
• Measure fluorescence from a detection pulse. This is effectively determining the total number
of atoms present (50 µs) (f).
• Subject the atoms to a long (150 µs) optical mask pulse. This depumps them all to the F = 2
ground state so they don’t fluoresce due to the detection laser. (g)
48
Characterising MARIE
• Measure light from a detection pulse. Since all the atoms in the region of the optical mask have
been depumped, this measures background due to stray laser light and fluorescence from stray
atoms (50 µs) (h).
• The cooling laser and quadrupole magnetic field is turned on again to re-trap atoms (i).
0.09
Oscilloscope Voltage (V)
0.08
0.07
0.06
(e)
(i)
0.05
(a)
(f)
0.04
(d)
0.03
0.02
0.01
(c)
(b)
(h)
(g)
0
−0.01
0
0.2
0.4
0.6
Time (s)
0.8
1
−3
x 10
Figure 5.7: Typical PMT signal from oscilloscope. The letters indicate different events: (a) Repump
laser turned off. (b) Short optical mask pulse. (c) Quenching coil turned off. (d) Detection pulse.
(e) Repump pulse. The large signal is due to repump light scattering onto the PMT, not from atomic
fluorescence. (f) Detection pulse. (g) Long optical mask pulse. (h) Detection pulse. (i) Cooling laser
and quadrupole magnetic field turned on.
Notice that there are 50 µs gaps between pulses in Figure 5.7. The PMT amplifier has a low-pass
filtering effect. The cut-off frequency of the low-pass filter can be increased, but at the expense of
noise. Fluorescence was measured by finding the area of each peak, and then dividing by the length
of the laser pulse used to generate that peak. This finds the average oscilloscope voltage over the
length of the pulse, which can then be converted to a light power. The 50 µs gaps enable the signal to
go back to zero between laser pulses, so the area of one pulse is not contaminated by another.
The parameter of interest is how many atoms are transmitted through the optical mask. Having
measured the fluorescence signal at (d) (call it s), the total fluorescence from the atoms at (f) (call it
Characterising MARIE
49
n), and the background at (h) (call it b), the proportion of atoms p transmitted through the mask is
then given by:
p=
s−b
n−b
(5.6)
As confirmation of this technique, p was measured for different optical mask lengths. It was found
to average to ∼ 0.99 for no optical mask, and ∼ 0.01 for long optical mask pulses. See Section 5.4.4
for quantitative data. Ideally, it would average to exactly 1 for no optical mask pulse, and exactly
0 for long pulses. This non-ideality did not affect conclusions made in this thesis, but could have
implications for use of the optical mask in an interferometer. Further improvement is needed with the
detection scheme.
5.4.2
Saturation with the Optical Mask
To minimise the effect of variations in optical mask light intensity, it was kept well above saturation.
Figure 5.8 shows the proportion of atoms remaining in the F = 3 ground state after an optical mask
pulse, as a function of light power coming out of the optical mask fibre. This is proportional to the
optical mask light intensity. There is very little decrease in transmission above around 20 mW. For
all optical mask measurements light power was kept around 30 mW.
5.4.3
Quenching Magnetic Field
As mentioned in Section 4.5.2, there are atomic states dark to the linearly-polarised optical mask
laser. However these states can be quenched using a magnetic field. In order to ensure the quenching
field is working, a cloud of cold atoms was subjected to a optical mask pulse for different currents
through the quenching magnetic field coil. The quenching coil was turned on just after the polarisation
gradient cooling step of the experiment, and was quickly turned off after the optical mask but before
the detection step.
The data in Figure 5.9 shows the proportion of atoms remaining in the F = 3 ground state after a
536 ns optical mask pulse, for different quenching coil currents. In this case one of the beams of the
optical mask were blocked, so that it formed a travelling wave instead of a standing wave. This was
to eliminate any artifacts that might have been due to interference between the laser beams. A clear
increase in depumping efficiency can be seen as the current increases.
5.4.4
Interference in the Optical Mask
The ultimate goal of this project was to make a cloud of cold atoms with a periodic density pattern.
This section presents strong evidence that this goal was successfully achieved, and thus the experiment
is very close to forming a functioning interferometer.
Figure 5.10 shows the proportion of atoms remaining in the F = 3 ground state after an optical
mask pulse, for different pulse lengths. The chance of an atom surviving the optical mask clearly
50
Characterising MARIE
Proportion of atoms remaining
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
Output power of optical mask fibre (mW)
Figure 5.8: Proportion of atoms left in F = 3 ground state after a 536 ns optical mask pulse, for
different optical mask light powers (power is proportional to light intensity). Error bars show peakto-peak variation over four measurements. Crosses show the mean of the data.
decreases as the duration of the pulse increases. To check for interference between the optical mask
beams this measurement was performed for two different cases: the red data in Figure 5.10 survival
for when all of the light power was directed along just one of the optical mask beams, and the blue
data shows survival for when there was equal power in both optical mask beams. Total light power
was the same in both cases.
With all the power in one beam, the light forms a travelling wave with constant intensity. All atoms
in the cloud will experience the same light field. However, with equal light power in both beams the
light will interfere, causing a spatially-varying light intensity along the beams. An atom’s chance of
being depumped to the F = 2 ground state will therefore depend on its position in the beam. Atoms
in the nodes should have a very small chance of being depumped, so survival of atoms should increase
with equal light power in both optical mask beams. This is exactly what is shown in Figure 5.10; for
the same length pulse, more atoms survive the standing wave than the travelling wave, so this must be
due to atoms in the nodes of the standing wave not being depumped, and forming a periodic density
pattern of atoms in the F = 3 ground state. This effect is clearly visible above the noise in the
measurements (the errorbars show absolute variation in 10 measurements). Survival of atoms in the
Characterising MARIE
51
Proportion of atoms after 536 ns TW
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
14
16
18
Current (A)
Figure 5.9: Proportion of atoms left in F = 3 ground state after a 536 ns travelling-wave optical
mask pulse, for different quenching coil currents. Errorbars show peak-to-peak variation over 10
measurements. Crosses show the mean of the data.
standing wave is up to three times the survival of atoms in the travelling wave.
This is the main result of my project, and I want to emphasise that it shows the optical mask is
working correctly. The only difference in the experiments used to obtain the two sets of data in
Figure 5.10 is the composition of the optical mask: the red line shows the survival of atoms after a
travelling wave, and the blue line show atoms after a standing wave. The increased survival shown
by the blue line must be due to atoms at the nodes of the standing wave not being depumped to the
F = 2 ground state, and so the density of atoms in the F = 3 ground state must have the diffraction
grating-like pattern explained in Section 3.4.
Note: As vindication of this conclusion, since I finished working on the experiment another student
has used it to successfully measure atomic interference. The experiment consisted of two standing
wave pulses, followed by a third pulse scanned in time after the second pulse. The survival of atoms
after this third pulse shows clear oscillatory behaviour as a function of the delay between the second
and third standing wave pulses. This shows that the optical mask is definitely working as expected.
Proportion of atoms remaining after pulse
52
Characterising MARIE
1
(a)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
Length of Pulse (µs)
Proportion of atoms remaining after pulse
0.4
(b)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Length of Pulse (µs)
Figure 5.10: Proportion of atoms remaining in the F = 3 ground state after an optical mask pulse.
Red data shows survival for a one-beam optical mask (a travelling wave), blue data a two-beam
optical mask (a standing wave). Error bars show peak-to-peak variation in ten measurements. (b)
simply shows a magnified region of (a).
Chapter 6
Summary and Future Work
6.1
Conclusion
This thesis presented a successful attempt to construct and characterise an optical mask for atomic
interferometry experiments. The work presented here should form a foundation for many interesting interferometry experiments, hopefully culminating in a precision measurement of local g, the
acceleration due to gravity.
The introduction gave a brief history of major developments in the related fields of atom interferometry and laser cooling. This provided a gentle transition into more technical aspects of the background
physics needed in this project: basic theory of laser cooling and trapping. Specifically, I described the
fundamentals of Doppler cooling, magneto-optical traps, and polarisation gradient cooling. Along
the way practical aspects of implementing these techniques were discussed. The last section of the
background chapter explained how to make an optical mask, and most importantly, provided some
motivation for doing so.
Chapter 4 described the anatomy of the experiment in detail. As more detail can already be found
elsewhere, I only briefly described the vacuum chamber and pumps used, how the rubidium dispenser
delivers atoms to the experiment, and the laser sources/tapered amplifier. In more detail I described
the magnetic field coils, the quadrupole coils which were necessary for the magneto-optical trap,
and the quenching coil which was critical to making the optical mask work properly. Both of these
coils have high-performing electronic circuits to switch them on and off quickly and reliably. The
compensating coils were successfully exploited to obtain samples of very cold atoms. Chapter 4 also
described the four instruments for observing trapped atoms: a photomultiplier tube for quantifying
the number of atoms in the experiment, a PIXIS CCD camera that was successfully used to accurate
temperature measurements, a highly linear photodiode trigger to ensure the initial atom number in
each experiment run was repeatable, and a video camera for qualitative observations of the atoms.
The former three are used for quantitative measurements and their characterisation was described.
54
Summary and Future Work
The LabVIEW program developed for this experiment enabled a range of different measurements to
be performed automatically. It can trap and cool atoms and then run a measurement, and it can do this
repetitively for a range of different parameters with no user input. The program was instrumental in
measuring the temperature of cooled atoms, characterising the photodiode trigger module, accurately
determining the laser frequency, and characterising the optical mask. The versatility of this program
should ensure it remains useful well into the future.
The experiment can reliably produce samples of cold atoms with temperatures from 8-12 µK. This
temperature is consistent with literature values, and was measured using two independent parameters,
suggesting that the temperature is indeed being measured correctly.
The main tangible result of this project is the optical mask. There is strong evidence presented here
that the optical mask is writing density patterns on clouds of cooled atoms, and so it should be useful
for interference experiments.
6.2
Future Work
As mentioned previously, the experiment as it was presented here does not constitute a fullyfunctioning interferometer as in [29]. To be able to reproduce their results the experiment needs a
way to control the phase of the optical mask. This could be done by installing an electro-optic modulator in one of the arms of the optical mask. For gravitational measurements the optical mask will
need to propagate vertically - not horizontally as it does currently.
There is a small non-ideality in the detection scheme that might need correcting before interferometry experiments can be performed, as discussed in Section 5.4.1.
Appendix A
Using MARIE’s LabVIEW program
This section is intended as reference for a future student using MARIE’s LabVIEW program.
A.1
Description of MARIE’s LabVIEW program features
Every label appearing in the graphical user interface is listed below with an explanation of its
purpose and how to use it. Text in bold is the label/name of a feature in the main program window.
Counter Parameters
Counter(s) - This tells LabVIEW what clock source to use for timing, you probably don’t need to
change this.
Rate - This is the sampling rate of the digital and analogue outputs. The data sheet of the card says
it should be able to go up to 1 MHz, but running above 200 kHz seems to result in an error. I always
used it at 100 kHz.
Analogue Channel Parameters Features in this box tell LabVIEW the analogue outputs to use and
the maximum and minimum voltages that are allowed to be outputted. ±10 V is the physical limit
for this card. You shouldn’t need to ever change the physical channels option (this card only has 8
analogue outputs).
Digital Channel Parameters Same as the analogue channel parameters except I used the last digital
channel as an input. This card only has 8 digital channels so it shouldn’t be necessary to change these
options either. MOT Selecta tells the program whether to wait on the first line of outputs (the MOT
stage)for the trigger from the photodiode (Measured MOT), or just wait for an alotted amount of
time (Timed MOT). The time to wait is the MOT time (s).
Execute Post-Function? tells LabVIEW whether or not to run a MATLAB function file as soon as
the current experiment run has finished. This is normally the data collection step, i.e. LabVIEW uses
a MATLAB function to transfer data from the oscilloscope. The name and location of the function
is given in the Post-Function and Folder boxes respectively. The text in Datafile name is passed as
an argument to the function, and gives the name of the output data file. Execute Pre-Function? is
basically the same thing, except it runs immediately before the experiment. I normally used this to
change settings on the arbitrary function generator.
56
Using MARIE’s LabVIEW program
Abort? stops program execution. One of the bugs of this program is that it does not do so immediately. As far as I can tell it should stop straight away, but it can take several seconds before the
program will stop running. Check the outputs are what you think they are once it does stop.
Set Output to Zero When Done? Once execution is completed, the program will normally output
whatever the last values were. Toggle this switch to instead only output zeros after the last step.
Run Experiment starts program execution.
# Runs tells the program how many times to run. Loop number tells you what runs the experiment
is up to (note counting starts from zero).
Experiment Running! tells you whether the program is currently running or not, it lights up when it
is running.
The Time column controls how long each row is output for (except the first row, which is controlled
by the MOT Selecta as previously discussed. The value in the boxes is number of samples that row
is ouput for, so the duration of the output is the number of samples divided by the sampling rate.
dt tells the program how much to change the corresponding row in the Time column each run. For
example, if you put ‘10’ in a box in dt, and set the program to run 5 times, it will add 10 to the value
in the Time column each run. This is so you can scan across a range of parameters without needing
to change the time manually.
Ramp to this value? If two adjacent analogue output rows are different values, normally the program
will just jump the output to the new value at the appropriate time. However sometimes it is useful to
linearly ramp to the new value, which is the purpose of this feature. Select a button for the program
to ramp from the previous value to the value in the row corresponding to the button pressed. This will
take place in as many samples as indicated in the same row in the Time column.
Analogue Outputs sets the voltage of each of the eight outputs at the corresponding Time in the
same row. Digital Outputs is the same thing, except ‘on’ sets the output to 5 V and ‘off’ sets it to 0
V.
Some miscellaneous information that might be useful:
• Make sure there are no other LabVIEW programs running simultaneously, if they use the same
hardware there’ll be resource conflicts and neither program will work correctly.
• If the program crashes during an experiment run, the output card will just continue to output the
most recent value, this can cause problems like having the dispenser turned on continuously, be
wary of this!
• A small bug in the program is that all the output vectors (e.g. the time vector, analogue outputs
vectors, etc) need to be the same length.
A.2
List of Outputs
This section lists which instruments were connected to which outputs of the computer, at the time
I finished working on the experiment.
Using MARIE’s LabVIEW program
57
Analogue Outputs
• AO 0 Controls current in the rubidium dispenser
• AO 1 Sets the frequency of the laser through the locking system (changes the voltage on the
FM input of the locking AOM).
• AO 2 Sets the amplitude of the detection laser through the AM input of the AOM, not normally
used.
• AO 3 Controls how big the MOT gets before the program is triggered (Vref in Figure 4.8)
• AO 4 Amplitude of the MOT beams, used in PGC.
• AO 5 Frequency of the MOT beams AOM. This AOM was found to ‘leak’ when switched off,
so the light is deliberately misaligned from the fibre input by changing the AOM frequency
when it is not needed.
• AO 6 Turns the quenching coil on and off, used essentially as a digital switch.
• AO 7 Turns the standing wave on and off, also used as a digital switch.
Digital Outputs
• DO 0 Detection AOM switch
• DO 1 Quadrupole coils switch
• DO 2 PIXIS camera trigger
• DO 3 Repump AOM switch (note this is inverted relative to the rest of the AOM switches: 5 V
is off, 0 V is on)
• DO 4 Arbitrary function generator trigger
• DO 5 MOT AOM switch
• DO 6 Oscilloscope trigger
• DO 7 Is actually a digital input. Program trigger from photodiode circuit.
58
Using MARIE’s LabVIEW program
Appendix B
Derivation of Time-of-flight Equation
Assuming an isotropic Gaussian atom cloud with a Maxwell-Boltzmann velocity distribution, we
can write the initial cloud distribution as the product of its velocity distribution and initial spatial
distribution:
!
νx2 + νy2 + νz2
1
3
3
d3 ν
N (r, v)d νd r = A 3 3/2 exp −
ν02
r0 π
!
rx2 + ry2 + rz2
d3 r
(B.1)
×exp −
r02
where m is atomic mass, νx , νy , and νz are speeds, and
q rx , ry , and rz are distances along the x, y,
3/2
m
and z directions respectively. A = 2πkT
, ν 0 = 2kT
m is the most probable velocity, and r0 is
the 1/e width of the cloud at t = 0. T is temperature.
We want to find what the cloud looks like at time t. Our camera is below the cloud, looking up
along the z-axis. So we can image the cloud projected on to the x−y plane. First we need to transform
from velocity to spatial xyz-coordinates. Using Newtonian mechanics to relate the two coordinate
systems:
x = rx + νx t
(B.2)
y = ry + νy t
(B.3)
1
z = rz + νz t − gt2
2
(B.4)
i.e. an atom’s position at time t is given by its initial position plus its velocity times t. Rearranging:
νx = (x − rx )/t
(B.5)
νy = (y − ry )/t
(B.6)
1
νz = (z − rz + gt2 )/t
2
(B.7)
60
Derivation of Time-of-flight Equation
Therefore the differential d3 ν can be transformed to dxdydz as
d3 ν = dνx dνy dνz
=
(B.8)
dxdydz
t3
(B.9)
We can now rewrite B.1 as:
1
1
A 3 3/2
3
t r0 π
2
2
2
(x − rx ) + (y − ry ) + (z − rz )
dxdydz
×exp −
t2 ν02
!
rx2 + ry2 + rz2
×exp −
d3 r
r02
N (r, x, y, z; t)d3 rdxdydz =
(B.10)
Integrating B.10 over r to get an expression for the density of the cloud as a function of (x, y, z; t)
(working left as exercise for reader. Completing the square is useful here, trust me):
N (x, y, z; t) =
Z
∞
−∞
Z
∞
−∞
Z
∞
N (r, x, y, z; t)d3 r
−∞
x2 + y 2 + z + 12 gt2
Av03
exp
−
= 2
r02 + t2 v02
(r0 + t2 v02 )3/2
2 !
(B.11)
Projecting onto the x − y plane by integrating over z:
N (x, y; t) =
=
Z
∞
N (x, y, z; t)dz
−∞
√
Av03 π
x2 + y 2
exp − 2
r02 + t2 v02
r0 + t2 v02
(B.12)
It is easiest in MATLAB to fit to one-dimensional data, so the cloud is projected onto either the
x− or y−axis then a Gaussian is fitted to the measured data to get the temperature in the respective
direction. The equation to fit the data to is:
N (x; t) =
Z
∞
N (x, y; t)dy
Av03 π
x2
exp − 2
=p 2
r0 + t2 v02
r0 + t2 v02
−∞
(B.13)
One can simply swap y for x to change direction. A series of photos taken of the atom cloud at
different times can be used to find r0 and T .
The above derivation is adapted from a similar calculation in [37]. In this paper the authors calculate the expected fluorescence from a cloud of cold atoms falling through a light sheet.
References
[1] A. Turlapov, A. Tonyushkin, and T. Sleator, “Optical mask for laser-cooled atoms,” Phys. Rev.
A, vol. 68, p. 023408, Aug 2003.
[2] A. A. Michelson and E. W. Morley, “On the relative motion of the earth and the luminiferous
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