Download Bose-Einstein-Condensate Interferometer with

Bose-Einstein-Condensate Interferometer with
Macroscopic Arm Separation
Ofir Garcia-Salazar
San José, Costa Rica
B.S., Eckerd College, 2000
M.A., University of Virginia, 2004
A Dissertation presented to the Graduate Faculty of the
University of Virginia in Candidacy for the Degree of
Doctor of Philosophy
Department of Physics
University of Virginia
January, 2007
Abstract
The basis of our study was to implement an atom interferometer using 87 Rb Bose
Einstein condensates which has advantages in sensitivity over current interferometers
that use cold atoms and light. Interferometers are devices which can accurately
measure phase differences between waves that interfere and originate from a coherent
source (or sources).
We developed a weakly confining waveguide having ωx ≈ 3 Hz, ωz ≈ 3 Hz,
ωy ≈ 1 Hz as characteristic oscillation frequencies. Weak confinement, specially
along the “y” direction, means the condensate can displace along this axis and
interaction energies of the atoms in the condensate are reduced [1].
We have been able to successfully demonstrate condensate interference in our
waveguide using a Mach Zehnder configuration. Coherence times of up to 40 ms
have been observed, and the maximum center to center separation of the condensates
recorded was of 240 µm. At this separation length, the two clouds corresponding
to each of the interferometer’s arms are completely separated. To our knowledge,
this is the first time a picture has been taken of two groups of atoms separated by a
macroscopic distance while in a quantum superposition of being in either cloud. The
coherence time and length measurements presented in our work have been among
the longest ones achieved so far for interferometry using condensed atoms.
Interference visibility of 60% was observed up to 40 ms. We believe technical
limitations in the techniques used to manipulate the atoms are responsible for the
sudden drop in visibility at 44 ms. For example, unwanted laser reflections and
interference patterns in our chamber affect the tecniques used to split and reflect
the atoms. However, we see coherence up to 80 ms from shot to shot, suggesting we
could dramatically improve coherence times. Becasue of the weak confinement of
our trap, we expect to improve coherence times up to an order of magnitude before
running into phase diffusion effects [2].
It is our hope to use our condensate interferometer for future studies in calculating the electric polarizability of 87 Rb. The macroscopic time and length scale
presented are novel in experimental quantum mechanics and of valuable pedagogical
insight.
ii
Acknowledgments
I would like to make an important note of appreciation to all of those who joined me
during these seven years in grad school. In reality many people have been involved
in my life as a student so far. They have, in my opinion, influenced who I am, and
continue to influence me in many ways. The list is very extensive and it would not
be possible to mention every one here, however, I salute and thank all of you (you
know who you are).
I would like to start by expressing my infinite gratitude to my amazing wife!
Karina, I feel you truly deserve much of the credit for the achievements presented
in this work. Thank you for all the tenderness, love, and hard work. They have
provided all the support to fulfill this mission. All my love goes out to you.
My parents, Andres Garcia and Gloria Salazar, there are almost no words to
express how much your influence and guidance have helped me through life. Thank
you for always listening to me under every circumstances and giving me the confidence that I can indefinitely trust you. To my sister Varinia, having you so close
during my graduate studies has been incredible! You have provided me that sense
of family, just like when we were little ones. Thank you for giving me the notion of
remembering where I came from and of the immense value of our immediate family.
You are a huge example to me and I am very proud of you.
Cass Sackett, you have been the best academic mentor and life adviser. It is my
impression that I could have not possibly made a better choice in order to carry out
my graduate studies. I wanted to express to you how meaningful your patience and
efforts towards teaching me physics has been. My aspiration is that I will be able to
pass on that knowledge to many more. Thank you for giving me the motivation and
inspiration to understand things the best I can. Also, thank you for always giving
me confidence in my abilities.
To my lab colleagues (Ben, Jeramy, Jessica, Ken, Au, John), all of you have
been great peers and a true joy to work with. I thank you all for putting up with
my unusual personality. Given that, I do feel all of you got to know me, appreciated
and understood me for who I am! Ben Deissler, I felt impelled to write out your full
name! Thank you for answering all my questions and always considering my point
of view.
As many of you know, In addition to all the experimental work, I have endured
some of the most difficult emotional and psychological times of my life while in
iii
iv
grad school. I would like to thank Jessica Reeves for helping me identify the root
cause of my problem (Obsessive Compulsive Disorder). This can be a very crippling
condition, but thanks to the incredible support network mentioned above, I have
been able to successfully manage it.
Bethany Teachman, thank you for giving me the tools to overcome this condition
and allowing me to re-learn how to ignore thoughts that are not important. I truly
hope your efforts in figuring out techniques in order to help people suffering from
OCD continue to flourish.
I would like to thank my professors Harry Ellis, Anne Cox, Stephen Weppner
and Jerry Junevichus at my undergraduate institution Eckerd College, for giving me
the foundations to pursue this work. Thank you for preparing very well to attend
graduate school.
All my life friends like Kifah Sasa, Carlos Luis Salas, Guillermo Gomez, John
Akl, Amadeo Martinez and many more, thank you for being there constantly even
after many many years of unconditional friendship.
Finally, I would like to end my acknowledgements by expressing my deepest
hopes that the work presented here will have a positive effect on humanity. I have
faith that the more we understand this remarkably complicated universe, the better
we can understand each other and learn how appreciate life for what it is.
Contents
Abstract
ii
Acknowledgments
iii
1 Introduction
1.1 The Nature of BEC . . . . . .
1.2 Interference . . . . . . . . . .
1.3 What is an Interferometer . .
1.4 Coherence . . . . . . . . . . .
1.5 BEC, a Coherent Matter-wave
1.6 Outline of Thesis . . . . . . .
1.7 Experimental conventions . .
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3 Magnetic Waveguide
3.1 Loading a Wave Guide . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Magnetic Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Conventions and Set Up . . . . . . . . . . . . . . . . . . . . .
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Source
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2 Making BEC
2.1 Rubidium Atoms . . . . . . . . . . . . .
2.1.1 External magnetic fields . . . . .
2.2 Creating a MOT . . . . . . . . . . . . .
2.2.1 Supplying 87 Rb . . . . . . . . . .
2.3 Increasing the number density . . . . . .
2.4 Loading a Magnetic Trap . . . . . . . . .
2.4.1 Compressed MOT . . . . . . . .
2.4.2 Optical pumping . . . . . . . . .
2.4.3 Switching The Magnetic Trap On
2.5 Transferring Atoms . . . . . . . . . . . .
2.6 Loading a TOP trap . . . . . . . . . . .
2.7 Evaporative Cooling . . . . . . . . . . .
2.8 Imaging . . . . . . . . . . . . . . . . . .
2.9 Calculating NA . . . . . . . . . . . . . .
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vi
CONTENTS
3.3
3.4
3.5
3.2.2 Superimposing Magnetic Fields .
Generating the Time Averaged Potential
3.3.1 Total Field Approximation . . . .
3.3.2 Calculating the Time Average . .
Design Limitations . . . . . . . . . . . .
3.4.1 Curvature Along “y” . . . . . . .
3.4.2 Trap Characterization . . . . . .
3.4.3 Trap Oscillations . . . . . . . . .
Measuring the Magnetic Field . . . . . .
3.5.1 Connections and Field Directions
3.5.2 Field Gradient & Magnitude . . .
3.5.3 End cap coils . . . . . . . . . . .
3.5.4 Preparing for Interferometry . . .
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62
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4 Interferometry Techniques
4.1 Interferometer Operation . . . . .
4.2 Splitting the Matter Wave . . . .
4.2.1 Two-Level Approximation
4.2.2 The Two Level Solution .
4.2.3 The Light Shift . . . . . .
4.2.4 The Bloch Picture . . . .
4.2.5 Splitting Operation . . . .
4.2.6 Experimental Verification
4.3 Reflecting the Matter Wave . . .
4.3.1 Three Level System . . . .
4.3.2 Experimental Verification
4.4 Recombination . . . . . . . . . .
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78
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5 Experimental Results
5.1 Expected Output State . . . .
5.2 Measurements . . . . . . . . .
5.3 Output Fit Function . . . . .
5.4 Visibility . . . . . . . . . . . .
5.5 Results . . . . . . . . . . . . .
5.6 Decohering Effects . . . . . .
5.7 Intrinsic Limits of the Output
5.8 Number Fluctuation . . . . .
5.9 Atomic Interactions . . . . . .
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113
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6 Conclusion
128
6.1 Achieved Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 Long Arm Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Future Adaptations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS
vii
A Temperature Interlock
132
B Control Panel
134
C Sequences
138
D Image Analysis Program
141
D.1 Fit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
D.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
E Experiment Setup
145
F Mathematical Calculations
149
F.1 Two Level Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
F.2 Matrix Elements of Ĥs . . . . . . . . . . . . . . . . . . . . . . . . . . 152
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
Interfering waves . . . . . . .
Mach-Zehnder interferometer
Double slit experiment . . . .
Coherence examples . . . . . .
Condensate interfering . . . .
Coordinate system . . . . . .
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8
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
Hyperfine structure . . . . . . .
Doppler force . . . . . . . . . .
MOT field configuration . . . .
MOT . . . . . . . . . . . . . . .
MOT 3-D . . . . . . . . . . . .
Atom transfer . . . . . . . . . .
Majorana losses . . . . . . . . .
Circle of death . . . . . . . . .
Evaporative cooling . . . . . . .
Maxwell Boltzmann distribution
Imaging system . . . . . . . . .
Bose-Einstein condensate . . . .
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Coordinates 2 . . .
Trap structure . . .
Trap scale drawing
Trap circuits . . . .
Quadrupole field .
Bias fields . . . . .
Magnetic ramps . .
Pin connections . .
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61
62
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66
67
71
73
4.1
4.2
4.3
4.4
Small arm separation interferometer
Chamber, trap, Bragg beam . . . .
Chamber, trap, Bragg beam 2 . . .
Interferometer configuration . . . .
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79
82
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viii
LIST OF FIGURES
ix
4.5
4.6
4.7
4.8
Michelson interferometer
Interferometer path . . .
Bloch picture . . . . . .
10 ms after split . . . . .
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84
85
104
105
5.1
5.2
5.3
5.4
5.5
Interferometer output . . .
Expected output . . . . .
10-20-10 interferometer . .
Visibility . . . . . . . . . .
Maximum arm separation
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115
116
117
118
122
A.1 Temperature interlock circuit . . . . . . . . . . . . . . . . . . . . . . 133
B.1 Real time control set up . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.2 De-bouncing circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
D.1 Screen shot of AI 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
E.1 Experiment map 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
E.2 Experiment map 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
E.3 Experiment map 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
List of Tables
1.1
List of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1
2.2
2.3
Trapped stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Evaporation sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1
3.2
3.3
Trap field formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Pin polarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Measured trap fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B.1 Control panel channel key . . . . . . . . . . . . . . . . . . . . . . . . 137
C.1
C.2
C.3
C.4
C.5
C.6
Loadcmot sequence . . .
Movetrap sequence . . .
Probesplit sequence . . .
Splitpulse sequence . . .
Reflectpulse sequence . .
Interferometer sequence
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138
139
139
139
139
140
Chapter 1
Introduction
Early in the history of great civilizations, humans developed an inherent desire to
further understand natural phenomena. Cultures like those of the Mayans, Aztecs,
Chinese, Arabs, Babylonians and Greeks have all explored the understanding of
nature through applied mathematics and observation based on the scientific method.
In today’s world, developments in the field of experimental physics seek to add
contributions to our broad understanding of modern science. They fulfill our society’s desire for understanding nature, following the essence of the theoretical and
experimental methods pioneered by these cultures.
The advent of modern physics in the early 1900’s led to the foundations of
quantum mechanics and in particular the understanding of black body radiation
emission. Consolidating this new understanding, in 1916 [3] Einstein established
the theory for stimulated absorption and emission for a material interacting with
electromagnetic radiation [4]. Half a century later, it became the basis for the
development of the laser. As a consequence, techniques like the cooling of atoms
using the coherent nature of laser light emerged, an achievement recognized by the
Nobel prize in 1997 (S. Chu, C. Cohen-Tannoudji and William D. Phillips).
The ability to laser cool atoms at low densities enabled researchers to further
study low temperature quantum mechanical systems. One example is the theoretical
work based on statistical arguments set forth by Bose and Einstein back in 1924, [5].
It proposed the existence of a quantum mechanical phase transition known as BoseEinstein condensation.
To introduce phase transitions, we can think of examples that surround us in
everyday life. For instance, water has three commonly known states, liquid, solid
and gaseous. These physical states occur at differing atmospheric (pressure) and
temperature conditions for water. When enjoying an icy cold drink in a hot summer
day, we are all familiar when small water droplets start forming around the outside of
the container. This happens because the invisible water vapour in the air condenses
as it comes in contact with the container.
Specifically, we can assume the exterior walls of our container are close to 0◦ C.
1
2
CHAPTER 1. INTRODUCTION
Air has a small percentage of water vapour in it. Normally air at sea level will
hold a maximum of ∼ 2.5% of water vapour at 30◦ C. As a result, water vapour
molecules come in contact with the walls of the container, and thermally equilibrate
with the walls. This lowers their temperature well below the dew point of water,
causing the state of the molecule to change from gaseous to liquid. Hence, we see
the droplets forming on our container. This change of physical state is a phase
transition. Here, the physical properties of the system have changed dramatically
given a small change in temperature.
Just as there are phase transitions between the gaseous, liquid and solid states
of matter, there are unique phase transitions for systems described by quantum
statistics. In quantum statistics there are three different types of particles which
determine how states are counted. There are distinguishable particles, fermions and
bosons. For the latter, Bose and Einstein developed a theoretical prediction in which
the particles at a given density will abruptly accumulate in the ground state of the
system if the temperature is lowered below a critical value. This is analogous to the
condensation of vapor to liquid in water. In such a state all the particles would have
the same wave function and exhibit unique wave like properties not seen in classical
systems.
A more precise description of Bose and Einstein’s predicted phase transition [6]
can be derived by thinking of an ensemble of N particles at fixed temperature T .
For this system, the maximum number of particles allowed in the excited states Ne ,
is limited. If the total number of particles N > Ne then the additional particles are
forced to populate the ground state.
From another perspective, for an ensemble with fixed number of particles, there
is a critical temperature Tc . If the temperature of the system drops such that T < Tc ,
then a fraction of the particles will become forced to populate the ground state of
the system. The particles which populate the same single quantum ground state
are said to be in a Bose-Einstein condensate and the discontinuity in the population
growth of the ground state is known as Bose-Einstein condensation. Because all the
particles are in the same quantum state, this new state of matter was predicted to
be coherent (discussed in Sec. 1.4), analogous to photons in a laser.
In 1995, experiments by C. Wieman and E.A. Cornell et al. at the University of
Colorado [7], Hulet et al. at Rice University [8] and W. Ketterle et al. at MIT [9],
verified the quantum mechanical phase transition for bosons. The result’s importance in physics earned it the recognition of the Nobel prize in 2001. Up to date
many experimental groups around the world have successfully achieved BEC. Currently there are various standard procedures in developing a BEC machine. Most
of the experimental techniques used in making BEC are readily understood by the
scientific community and have been documented in detail [10].
The most common elements to use for BEC experiments have been 87 Rb and
23
Na. For other elements, BEC can be hard to achieve or even unstable. For example, experiments [11] have demonstrated that due to lack of ionization suppression,
1.1. THE NATURE OF BEC
3
BEC formation of Ne is not practically possible. In the case of our experiment, we
have opted to use 87 Rb to make our Bose-Einstein condensate.
Our main objective is to create an interferometer with long arm separation using a Bose-Einstein condensate. Interferometers are devices which measure phase
shifts in the interference pattern obtained by combining waves. On many occasions,
interferometers use a single source of waves which is split and later recombined, in
order to measure the interference between them, (see Sec. 1.3). Similarly, we want
to take a wave function and split it, recombine it, and finally measure the output
state which should resemble that of interfering wave functions (a superposition of
wave functions). In light of this, the separation distance of the two waves at any
instant is said to be the arm separation.
To this day, current efforts in implementing BEC interferometers have been successful [12], but not much progress has been done in extending the arm separation.
We pursue a design which will provide large arm separation, allowing for interferometry experiments which require individual access to each arm. A more detailed
account of interferometry and how condensates are suited for them will be given in
the following sections. It turns out that due to their low velocities, a condensate’s
motion is easily manipulated. Granted that, although condensates consist of particles and thus are expected to behave classically, their unique quantum configuration
enables them to exhibit wave-like properties. For this reason, a condensate is said
to behave like a matter-wave. Moreover, we seek to take advantage of certain key
characteristics of our 87 Rb condensate in order to generate large arm separation.
Development of BEC research marked a starting point for experiments like the
one introduced in this thesis which rely on the low velocity of the atoms which
make up a condensate. As will be presented in the following chapters, a low velocity
condensate allows for simple manipulation of the motion of the atoms. In our
experiment, we want to achieve a large arm separation condensate interferometer,
and low velocity atoms enable us to easily achieve this.
As a result, we hope the efforts set forth in the experimental work explained in
this thesis adds a contribution in the applicability of condensates for interferometry.
More generally, we try to provide some pedagogical insight in the behaviour of
macroscopic quantum systems.
1.1
The Nature of BEC
Bose-Einstein condensates will play an essential role in the implementation of our
interferometer. For this reason, we will introduce their physical meaning and then
proceed to outline the key steps needed to create one in the laboratory setting.
Because Bose-Einstein condensation is a state of matter occurring to systems
of many particles, we use statistical mechanics to understand this phase transition.
Furthermore, it is important to comprehend why we must use quantum statistics in
4
CHAPTER 1. INTRODUCTION
order for this unique transition to become evident.
Generally, when studying a dilute system of many particles like that of a monoatomic
gas, the Maxwell-Boltzmann distribution of velocities yields the appropriate thermodynamical properties which describe the system. However, this is a purely classical
approach which does not include any quantum effects. Quantum mechanics asserts
that for a particle of mass m and velocity v, the associated wavelength for any
particle is given by the de Broglie relation using Planck’s constant h, [13].
λd =
h
mv
(1.1)
Similarly, the thermodynamical equivalent of this relationship is the thermal de
Broglie wavelength. It can be thought of as the average de Broglie wavelength for
atoms in a gas at some temperature T :
λt =
h
(2πmkb T )1/2
(1.2)
where kb is the Boltzmann constant. By substituting 13 mhv 2 i for kb T , we obtain:
λt = q
λt
h
2πm2 13 hv 2 i
Ãr !
3
h
=
2π mvrms
(1.3)
(1.4)
which recovers the De Broglie wavelength up to a constant, demonstrating the similarities between λt and λd . The quantity (1.2), in conjunction with the mean
inter-particle spacing (V /N )1/3 , can be used to determine whether or not quantum
effects will become important when modeling the system.
For a gas whose atoms have an inter-particle spacing much greater than their
thermal de Broglie wavelength, the quantum properties of each atom play a negligible role in he macroscopic behavior of the gas. In this situation the gas is said to
be classical. However, if the particle spacing is comparable to or smaller than λt ,
quantum effects become important. This condition can be expressed as:
h
≥ (V /N )1/3
(2πmkb T )1/2
nh3
≥ 1
(2πmkb T )3/2
(1.5)
(1.6)
where the number density n = N/V has been used. One can visualize that each
atom’s associated wavelength is comparable to (or greater than) to the distance of
its nearest neighbour. This means that the atoms’ wave-functions are overlapping
1.1. THE NATURE OF BEC
5
and thus the quantum mechanical effects on the macroscopic system are going to
become important.
Because Bose-Einstein condensation in experiments like ours occurs with n ≈ 1012 cm−3
and temperatures in the order of 35 × 10−9 K, the condition expressed in Eq. (1.6) is
expected to be satisfied. In this way we know that a Bose-condensed gas will exhibit
unique quantum wave like properties that ordinary gases do not have. Additionally,
the condensate’s velocity will be very low, and this will prove to be an advantage
for our interferometer.
The process of Bose-Einstein condensation (as the name suggests) is particular
to bosons and can be fundamentally explained by quantum statistics. Typically,
to derive the condition for the onset of BEC, one must first obtain the formula for
the occupation number as a function of the energy for indistinguishable particles of
integer spin (bosons). This derivation is lengthy and is extensively found in many
books like [14], which is why it will not be covered here.
Nevertheless, with a simple system as an example, we can illustrate why introducing quantum statistical effects to a classical statistical mechanical treatment
changes considerably the macroscopic properties of the system.
In fact, one major characteristic which separates classical systems from quantum
ones is indistinguishability of particles. In principle, for a classical system one could
pinpoint every particle and determine their position and momenta. This means that
one is allowed to label and follow all the dynamics of the system, essentially being
able to distinguish each particle. But according to quantum mechanics, because
we cannot measure something without disturbing it, there is always an uncertainty
which limits the ability to pinpoint every position and momenta. Consequently, it
is not possible to keep track of which particle is which.
For indistinguishable particles, the fundamental difference between fermions and
bosons is their total value of spin angular momentum. Bosons possess an integer
spin value, where as fermions have a half integer spin value. Another characteristic
which separates the two families of particles is that only one fermion can occupy a
single state, this is known as the Pauli exclusion principle. In contrast, bosons do
not have such a restriction.
To place into context our example, it is important to highlight that Bose-Einstein
condensation is a phenomenon which only occurs (as the name suggests) to particles of integer spin, thus excluding fermions. Moreover, as established by the Pauli
exclusion principle [15], the following example will not permit a multiple particle
system1 , making the example inapplicable to fermions. This means that the following illustration need only focus on differentiating between distinguishable bosons
and indistinguishable bosons.
With the following example, one can obtain some basic intuition as to why
quantum statistics differ from classical statistics by considering the differences arising in the counting of states between indistinguishable and distinguishable particles.
1
Given the Pauli condition, no more than one ↓ atom can be in the the system, thus N > 2.
6
CHAPTER 1. INTRODUCTION
Similarly, this difference illustrates why condensation to the ground state requires
quantum statistics.
Consider a case of 10 distinguishable blocks labeled 1 through 10. Each block
has the possibility of being oriented up or down as labeled by an arrow ↑ or ↓. For
instance, let’s give special consideration to the configuration where all the blocks
are pointing downwards like ↓↓↓↓ . . . , and give this state the label N↓ .
Our aim is to figure out what the likelihood to find the system in the configuration
N↓ is if the blocks were to be randomly oriented. In order to obtain the probability
of the state N↓ we must find the total number of states available in our system.
Because each particle only has two possible states, it is not hard to identify that the
total number of states is 2 · 2 · 2 . . . 2 = 210 , which yields a total number of states
Ωs = 1024. Accordingly, the probability to find the system of distinguishable blocks
in state N↓ is given by:
PD (N↓ ) = 1/1024 ∼ 10−3
(1.7)
On the other hand let’s consider the same system, but this time the blocks are
not enumerated making them indistinguishable. Additionally, they are considered
as quantum blocks obeying the symetrization postulate. With this in mind, it is
clear that for any fixed number of ↑ blocks, there is only one arrangement. For
example, there is only one way to get all the blocks like ↓↓↓ . . . ↓, ↑↓↓ . . . ↓ or
↑↑↓ . . . ↓, because for a given configuration, a change in the choice of ↑ blocks can
just be rearranged to give back the same state. For this case the total number
possible states in the system is just 11. Hence the probability to find the system of
indistinguishable blocks in state N↓ is given by:
PI (N↓ ) = 1/11 ∼ 10−1
(1.8)
If we compare PD (N↓ ) and PI (N↓ ), it is clear that PD (N↓ ) < PI (N↓ ), making the
N↓ state for the indistinguishable system much more likely than its distinguishable
counterpart. Extending this example to a larger number of blocks, means the difference in the likelihood for the N↓ state for these systems would just grow further.
On the whole, this example helps illustrate how introducing indistinguishability into
the statistics of blocks makes the N↓ state 2 orders of magnitude more likely in this
particular case. Therefore extending this idea in general we can say that indistinguishable bosons are more likely to be in the same state. In fact, in a realistic
physical model of bosons, we find that for low enough temperatures a macroscopic
fraction of particles in the system will collect in the ground state due to the statistics
shown in the above example.
The example with blocks demonstrates how indistinguishable particles make condensation to the ground state a likely outcome and why quantum statistics are necessary for this process to occur. Specifically, for a system of particles with number
density n, the physical condition for Bose-Einstein condensation to take place is
[14]:
nλ3t ≤ 2.612
(1.9)
1.2. INTERFERENCE
7
Equation 1.9 means that for excited states in a dilute ensemble of integer spin
particles, nλ3t is limited to 2.612. When this number is exceeded, by an increase in
the number of particles or a decrease in the temperature, particles exceeding Ne are
forced to populate the ground state and the quantum mechanical phase transition
known as BEC occurs.
The work presented in this thesis will not focus on BEC creation, but rather,
its application in atomic interferometry. To further motivate this study, it is important to understand what an interferometer consists of and the underlying principles
that explain its functionality. These principles will provide useful insight to the
wave nature of BEC given interferometry requires waves interacting. Similarly, understanding the quantum nature of BEC will explain why they are well suited for
interferometry.
1.2
Interference
In general waves are disturbances in fields which can travel and can transmit information from one point to another. Waves have the fundamental property of
constructively adding or destructively canceling when interacting. The resulting
pattern of the field when the waves add or cancel each other is called an interference pattern. The interference pattern is observed by measuring the intensity of the
resulting waves. It should be noted that the intensity of the wave is the physical
attribute of the field which can be experimentally measured. In the casep
of a elec2
tromagnetic field E, the intensity is defined as I = |E| /2ξ, where ξ = ²0 /µ0 is
the impedance of free space.
A simple example describing this phenomena is seen in Fig. 1.1. In this picture,
two circular waves representing fluctuation in the surface of water create a familiar criss-cross pattern of ripples. The crests are where the waves have added up
(maxima) and the troughs are where they have cancelled each other (minima).
At any rate, interference requires at least two waves interacting in such a way that
the amplitudes of the fields superimpose, resulting in a pattern that can be observed
or measured. It is important to note that there are many physical examples of waves
which can interfere. In the case of light and the probability density in quantum
mechanics, the observable quantity is the modulus squared of the amplitudes of each
field. The corresponding amplitudes are the electric field and the wave function. In
fact, it is crucial to note that the interference pattern which can be observed is the
square of the sum of the amplitudes and not the sum of the squares. This means
that upon squaring, there will be “cross terms” which will either add or detract
from the measurable result.
Equally important to the understanding of interference is the concept of the
wave’s phase. For example, we can think of a two dimensional field like the one
presented in Fig. 1.1 having two point sources (A and B), which oscillate in the
8
CHAPTER 1. INTRODUCTION
Figure 1.1: Mathematical model of two disturbances in a field (e.g. water) lying in
the x, y plane (not labeled), generating concentric waves. The height of the function
represents the amplitude of the wave (along z not labeled). Depending on the type
of field, the amplitude may or may not be a physical observable. For water, it is.
At the points where the waves meet, an interference pattern is created
1.3. WHAT IS AN INTERFEROMETER
9
vertical direction to generate waves. If the two sources began to oscillate with
constant frequency ω0 at the same time and started at the same position, then the
waves created are said to have the same phase. In other words, at some point (x0 , y0 )
equidistant from the two sources, the amplitude of each field will be identical for
all time. When a particular crest is emitted from A and B, it will arrive at exactly
the same time to (x0 , y0 ). The same dynamic would happen for the troughs emitted
from A and B. Hence, at a given time, the fields always add constructively doubling
the amplitude of the total field.
However, if the sources were not synchronized, that is to say, A started to oscillate
before B, then the waves created would have a different phase. Thinking of a snap
shot in time for each wave, if we were to look at a point lying on the axis defined by
the two sources, the phase difference between the waves would be proportional to
the distance between the crests (or troughs) of the waves coming from each source.
Another illustration would include taking pictures of a single wave at time intervals equal to the period of the wave. The resulting images would yield an intensity
pattern that would not vary from shot to shot. If the phase were to change from
one shot to the next, then there would be a forwards or backwards shift in the crests
(or troughs) compared to their original position.
If we take a cross-section somewhere where the two waves overlap and measure
the total field, the resulting field would be the superposition of both waves in this
region. Equivalently, this would be the resulting interference pattern between the
two waves. In the interference pattern itself we would see nodes where the field
remains leveled and sections where the total field oscillates up and down. If we
were to look at the nodes of the interference pattern when the waves from A and
B are in phase, their position would remain fixed over time. However, if we were
to introduce a phase difference between the waves, the nodes of the interference
pattern would shift a distance proportional to the new distance between the crests.
For this reason introducing a relative phase between interfering waves causes a shift
in the interference pattern.
1.3
What is an Interferometer
The word interferometer already conveys substantial meaning. As the word suggests,
it is a “meter” or instrument that measures interference. In general, interferometers
are devices that can accurately measure changes in the interference pattern of waves.
To create an interferometer we must intersect at least two waves. Usually this is
done by interfering two wave sources with constant phase relation.
An illustrative example is the Mach Zehnder interferometer shown in Fig. 1.2.
An incoming laser is split using an optical beam splitter. The two resulting beams
are redirected with mirrors in such a way that an area is enclosed between the arms
of the interferometer. Finally the two beams are redirected to a second beam splitter
10
CHAPTER 1. INTRODUCTION
Figure 1.2: Mach-Zehnder Interferometer. (a)Input Beam. (b)Beam splitter.
(c)Mirrors.(d)Beam splitter for recombination. (e)Intensity detector.
where they are recombined and a photo detector is used to measure the intensity
variation.
Often, interferometers will split the wave emitted by the source and recombine
its parts at a later time to produce an output. The output is the resulting interference pattern of the waves’ intensity. When the arms of the interferometer are
recombined on output, their resulting destructive or constructive interference is seen
as an increase or cancelling of the intensity signal.
When one arm of the interferometer experiences a change in its environment,
the wave travelling through this channel will experience a positive or negative retardation. The retardation in one of the waves will change the relative phase between
the two waves of each arm in the interferometer. A change in phase between the
two arms will be reflected in a change of the interference pattern on the output
which will cause a change in the measured intensity. Consequently, if the shift in
the interference pattern is quantified with accuracy, it can be used to measure with
precision some physical parameter related to the cause of the shift.
So far, there has been no attempt to limit the implementation of interferometers
to any specific kind of wave, but rather to give a general notion of how interferometers operate. Although the example presented above makes use of interfering light
waves, interferometry is not limited to light. In principle, any kind of travelling
wave could be utilized to perform an analogous experiment.
For this reason, it becomes evident that quantum mechanical wave functions can
be used to perform interferometry experiments. Although the wave function itself
is non-observable (i.e a complex quantity), its modulus square corresponds to the
probability of finding a certain state, which is measurable. In this regard, different
wave functions can overlap, meaning that the resulting intensity pattern is just the
resulting probability to find a system in a particular state after the individual wave
functions have interfered.
Atoms can interact with external forces much more so than photons. Usually, the
1.4. COHERENCE
11
interaction between photons and fundamental forces is only observed at extremely
high energies and it is extremely weak [16]. On the other hand, atoms readily
interact with gravity and electromagnetic fields. For this reason, atoms make better
sensory devices to perform experiments with precision measurements involving most
forces. This is one important motivation for the research presented in this thesis.
Because a Bose-Einstein condensate exhibits wave like properties, we are motivated to use its wave function to carry out an interferometer. Although there
are other advantages we will explore when using BEC for interferometry, a condensate also has the added advantage described of increased sensitivity. This makes it
competitive with other types of atom interferometers.
As we will explore in Sec. 1.5, matter can behave like a wave, implying that
potentially atoms could be used for interferometry in a similar way to light. Using
atoms for interferometry does not seem intuitive, but the progress in atomic cooling,
trapping and motion control have made such an experiment accessible.
1.4
Coherence
In many situations on varying applications, waves consist of superpositions of several
different waves. A demonstration of this situation can be explained through Fourier
analysis [17]. By Fourier analyzing any generic wave or pulse, we will decompose
it into its various frequency components. For this reason, the waves we are seeking
to use in an interferometer could consist of many different waves with multiple
frequencies and phases.
The visibility of interference patterns depends on the range of frequencies and
direction of the wave-vectors present in the waves interacting. In the case of light,
the observable is the intensity, proportional to the square of the electric field. When
the range of frequencies present in light is broad, the superposition of the various components makes it harder to detect the interference pattern. Additionally,
if the wave-vectors of the light are randomly pointing in different directions, the
interference pattern can become blurred, making it undetectable [3].
Sunlight is a good example of a wave source that contains multiple frequencies
with randomly pointing wave-vectors. For this reason, observing an interference
pattern with sunlight was not possible for many years. Creating two wave sources
with a stable enough phase relation (see Fig. 1.1) and single-directional wave-vector,
was not experimentally realisable. However, this changed when the wave nature of
light was understood. For example although there are multiple frequencies present
in white light, today we know that in a set up like Fig. 1.2 with a collimated input of
white light, if the path length for each arm is equal, an interference pattern will be
observed. This happens because, even though there are multiple frequencies present,
there are points where the maxima for the different waves always line up.
One of the first experiments to demonstrate that light interfered like water waves
12
CHAPTER 1. INTRODUCTION
do, was performed by Thomas Young in 1805 [3]. He set up a dark room with a
pinhole which let sunlight in. After the first hole, there were two additional holes.
Light emanating out of these two holes was then allowed to overlap over some
distance inside and later imaged at a wall in the back of the closed room. See
Fig. 1.3. It was expected to see interference if light was a wave.
Traditionally, similar experiments were carried out (without the first pinhole),
but no interference would be observed at the back of the room because the light
contained randomly pointing wave-vectors.
However the ingenuity in Young’s experiment was the first pinhole which created
a single point source at normal incidence to the subsequent holes. In this way, the
wavefronts emanating from the first hole are be evenly distributed as a function of
the angle θy Fig. 1.3. This meant that there was one set of wave-vectors coming
from a point source on the subsequent holes and not several sets of vectors from
unrelated random point sources.
Additionally, it created a single source of white light, meaning that the wave
fronts (of each frequency) arriving at the the subsequent holes would have the same
phase relation. Or in other words, the crests (or troughs) would arrive at each hole
at exactly the same time. Hence each hole would see the same field variation for
all time. In this way, the holes would act as two independent wave sources with a
constant phase relation between them, just like the example in Fig. 1.1. Also see
Fig. 1.4 [3].
As a result, the two sources emitted light which interfered, creating a pattern
which was imaged at the back of the room. The pattern resulted in a series of dark
and white fringes indicating the minima and maxima of the interference pattern,
see Fig. 1.3 [3]. This experiment demonstrated the wave nature of light and thus its
ability to interfere. It also provided a useful technique for obtaining wave sources
with a constant phase relation in time and well-defined (non-random) wave-fronts.
Stable interference patterns require the waves interacting to have a well defined
phase relation between them as much as possible. The clarity and thus ability to
measure an interference pattern is best described by the degree of coherence of the
waves superimposing.
There are two distinct types of coherence in waves. There is spatial coherence,
pertaining to the extent the phase of the wave remains constant over some distance.
An example illustrating spatial coherence is seen in Fig. 1.4 where a point source
emits a circular wave, and the wavefronts are depicted by black circles. Here, points
A and B are said to be spatially coherent because the wave will vary identically in
each case.
Equally important, there is temporal coherence between waves. The temporal
coherence is often associated with the coherence length. The degree of temporal
coherence between waves involves how large you can make the arrival time of different wave fronts in a particular region before the interference pattern averages out to
zero, making it undetectable. This is also known as the coherence time of the waves.
1.4. COHERENCE
13
Figure 1.3: Set up for Young’s double slit experiment. 1 A non coherent source of
light is incident on a pinhole. 2. The pinhole outputs spatially coherent light which
in turn is incident on two other pinholes. 3. Because the light is spatially coherent,
the output of the two pinholes acts as two spatially coherent sources generating the
light and dark fringes in 4.
Figure 1.4: Two different types of point sources emitting circular waves. The dark
lines represent maxima and the faint lines minima, 1. Points A and B will experience
the same variation over time, they are said to be completely correlated, the source
is spatially coherent. 2. Having a monochromatic source means that knowing the
wave at C completely determines the wavefront seen at at D, the source is temporally
coherent.
14
CHAPTER 1. INTRODUCTION
An example of the coherence time for waves can be seen in Fig. 1.3. By extending
the distance between the two sources and the imaging screen, the time over which
the waves propagate is increased. In this context, the longer the propagation time is
while still producing the interference pattern shown, the longer the coherence time.
Generally, as long as the waves have a sufficiently constant phase relationship
between them over time and the wave fronts are not random, the interference pattern
will remain stable and clear. If the phase between the waves changes in time, so
will the position of the minima and maxima. This means that the phase between
them can change, but if it happens too quickly (faster than the detector’s sampling
rate), then the interference pattern will become blurred. For this reason, waves
whose phase relation varies slowly over time yield a stable interference pattern.
The constant phase relation will play an important role in many of the interference
examples presented next and most importantly in the creation of our interferometer.
The temporal and spatial coherence mentioned above are relevant to the performance of our interferometer regarding a single run of the experiment. Nevertheless,
it is very important for us to obtain coherence from shot to shot of the experiment.
Initially, we want to to obtain an interference pattern where we have complete control of the relative phase introduced between the arms of the interferometer. To
achieve the desired control in the phase of the output state, it is important that we
eliminate unwanted phase fluctuations which cause decoherence in each run of the
experiment. Unwanted phase shifts from shot to shot will also decrease the contrast
of our interferometer, limiting its sensitivity.
Currently, lasers have tremendously facilitated the study of interference. A version of Young’s double slit experiment can be performed using laser light instead of
white light and the two slits are substituted by a diffraction grating. In a similar
way to the two slits, the diffraction grating will cause an interference pattern that
when imaged reveals a series of light and dark fringes. Nevertheless, contrary to
white light, the imaging plane can be extended for hundreds of meters.
A laser with a 1 MHz line width will maintain its constant phase over a distance
of 300 m (compared to mm for a discharge lamp). For these reasons, laser interferometers have become the popular choice when performing interferometry experiments.
Furthermore, lasers provide a continuous source of monochromatic spatially coherent light, which dramatically improves the visibility of the interference. Having a
collimated beam means the wave fronts are not randomly distributed. Also a single
frequency means no other mode will blur the interference pattern when averaged
over time.
In short, the coherence of waves is an important factor when trying to implement
an interferometer. In order to obtain a functional interference signal, we must make
sure that the type of wave used has long enough coherence times and lengths for
the specific purpose in mind. The next section will motivate why Bose Einstein
condensates which have small energies and small velocity spreads, are well suited
for interferometry. It will also show how these characteristics benefit our aim of
1.5. BEC, A COHERENT MATTER-WAVE SOURCE
15
building an interferometer with individually accessible arms.
1.5
BEC, a Coherent Matter-wave Source
Young’s double slit experiment [3] demonstrated that when light passed through
two slits, its intensity pattern was not consistent with typical particle behavior but
rather established that light exhibited wave characteristics.
Further experiments performed a century and a half later by Jönsson [18] established that fundamental particles such as electrons yielded results identical in
nature to those observed by Young’s famous experiment. The experiment consisted
of passing the electron through a multiple slit configuration like that in Fig. 1.3.
This experiment corroborated the understanding that the electron was not localized
and had gone through multiple slits simultaneously yielding an interference pattern
just like the one obtained for light.
To complicate interpretations further, a variation of the electron experiment was
performed by Tonomura [19]. In this situation a detector was positioned right after
one of the slits to corroborate which path was taken by the electron. This version of
the experiment demonstrated that indeed the electron was localized like a particle
and had gone through one of the slits.
Today, the wavelike behaviour of the electron observed by Jönsson et al. is attributed to the fundamental particle-wave duality of matter. Depending on the experiment, a particle can exhibit wavelike or particle like behaviour as shown above.
Conversely, modern experiments using weak lasers like [20] have shown that if a
single photon is allowed to pass through the two slits, the detector will correspondingly show a pattern of a single photon going through. Moreover, if you allow more
photons to continuously pass through the slits, the wavelike interference pattern will
be reconstructed over time as more photons hit the detector. This corroborates that
in the limit of many particles, wave behaviour can be recovered from particles.
Our current understanding of quantum mechanics has demonstrated that at a
fundamental level, matter exhibits wave-like propperties. It is well understood that
when position and momentum variations approach the quantum limit as described
by Eq. (1.5) and set by the value of h = 6.63 × 10−34 Js, matter is best described
using the dynamics of the Schrödinger wave equation. In this limit, matter can
constructively and destructively interfere.
In our experiment, 87 Rb will transition into BEC at an approximate temperature
of 200 nK. At this temperature, the average kinetic energy of the system is 4.14 ×
10−30 J implying the average velocity of the atoms is approximately 7.6 mm/s.
In this situation, the thermal de Broglie relation (1.2) states that the associated
wavelength of the particles is 0.42 µm [21]. This is much larger than the average
inter-atomic spacing of 87 Rb which is ra = 0.37 µm (at n = 2 × 1013 cm−3 ). For this
reason the physics describing BEC atoms is best described by quantum mechanics,
16
CHAPTER 1. INTRODUCTION
Figure 1.5: Two Bose Einstein condensates, released from a magnetic trap, free
falling and overlapping while expanding. Because a condensate exhibits properties
of a matter wave, as they expand they interfere. The fringes of the interference
pattern are clearly resolved [22].
meaning the wavelike properties of matter will become more prominent. In this
sense BEC makes a good source of atoms that will exhibit wave like properties.
Having a condensate means that all the atoms have the same wave function.
Similarly, because all the atoms are in the same state and time evolve the same, it
is expected for every atom to remain in phase and thus be quantum coherent across
the whole condensate.
In 1997, an atomic wave version of Young’s double slit experiment was conducted
by Andrews et al., at MIT [22]. Two condensates were separated and released
from their magnetic trap and allowed to ballistically expand. As the condensates
expanded, the clouds crossed paths and interfered as matter waves, see Fig. 1.5.
This result proved that the condensed atoms were coherent on the time scale of the
experiment, which imaged the interference pattern after 40 ms of free fall.
Because the condensates are in a minimal energy state (lowest state of the trapping potential ignoring atomic interactions), their kinetic energy is extremely low.
Hence, manipulating the motion of the condensates proves simple and an attractive
proposition for further applications. Our hope is to perform an experiment which
exploits the wave-like behavior of condensates in a controlled way. Instead of using
ballistic expansion we aim to use a waveguide for the atoms.
Currently, interferometry using cold atoms has proven successful [23, 24, 25].
However the condensate atoms have special characteristics which allow them long
temporal and spatial coherence. This gives them some important advantages as
1.6. OUTLINE OF THESIS
17
wave sources for interferometry.
The condensates have long temporal coherence because all the atoms are in the
same state. This means that all the atoms start off at the same velocity and because
each atom’s quantum phase evolves steadily in the same way, their phase relation
does not change over a long period. The result is a condensate that can evolve
for long times without losing coherence. In our case, this means that we could
potentially split the condensate and allow it to travel for long time intervals before
recombining it, and still be able to observe interference.
All the atoms in the condensate are in the same state and each atom’s quantum
phase is identical. Hence the phase relation for each atom is constant, meaning the
condensate is spatially coherent across the whole ensemble. When the condensate
is sitting still in the magnetic trap, all the atoms are in phase. At any instant in
time, the wave function at different points in the condensate is correlated, meaning
the wave function at one point of the condensate can be determined from another
point at any moment.
In the end, long temporal coherence means the frequency variation in the wave
will be small (i.e monochromatic), therefore the energy variation for the wave functions will be minimal such that ~∆ω ∼ ∆E ∼ ∆K.E. for small clouds of atoms.
Additionally, due to spatial coherence, the phase variation across the condensate is
small. Hence the variation in the wavelength of the wave functions will be small
too. For both reasons, it means the spread of velocities ∆v across the condensate is
small too, which is one of the most important characteristics about condensates we
exploit in our experiment. Having a small ∆v at low temperatures will dramatically
improve our ability to control the motion of the atoms, which is what we need to
obtain large arm separation.
We hope to take advantage of the increased coherence length provided by a
condensate to maximize the distance over which we can separate the arms of the
interferometer. In turn this would yield longer propagation times and result in large
coherence times.
1.6
Outline of Thesis
The purpose of this thesis is to concisely describe the steps involved in implementing
a BEC interferometer using a special weak magnetic trap designed to allow long
coherence times and large arm separation.
Chapter 2 gives an introduction to the 87 Rb atom and explains how we produce
BEC. It summarizes the properties of 87 Rb atoms, and explains how we use laser
cooling techniques to obtain an ultra-cold sample of atoms. The atoms are then
loaded into a magnetic trap and evaporatively cooled until BEC is reached. Finally
there is a section on how imaging of the condensate is performed.
Chapter 3 focuses on our novel waveguide design. It starts with an explanation
18
CHAPTER 1. INTRODUCTION
of how the magnetic waveguide is loaded. It continues explaining how the magnetic
fields it uses are achieved. It explains how the trap is used for the evaporative cooling phase and gives a detailed discussion on the role each circuit has in making up
the wave guide. At the end of this section, the total B field is presented, illustrating
what the overall potential the atoms experience is. The measured B field distribution is found to agree well with the observed trap parameters. Finally the chapter
documents in detail the appropriate settings and connections used for the guide.
The main focus of this thesis is to present the implementation of a working one
dimensional analog of the Michelson interferometer in Fig. 4.5. Chapter 4 provides
an in depth explanation of the main experimental techniques that were used to
achieve long coherence times and arm separation. Our implementation requires
operations to both split the condensate into two moving packets, and to reflect
the motion of the packets. These operations are explained and analyzed and the
dependence of the output state on the interferometer phase is determined. We
developed an interferometer scheme which (in sequential order) consisted of four
main operations: a splitting stage, two reflections and a final recombination stage.
In Chapter 5 we study the interference results in detail and obtain the expected
results for the output state of the interferometer. We also discuss these results. The
value of the output state was observed to be consistent with our predicted sinusoidal
function for experiments of up to 44 ms in duration. We were able to identify that
although our interferometer lost visibility after ∼40 ms, coherence was still observed
for longer times up to ∼80 ms from shot to shot. We believe vibrations in our trap
and unwanted fluctuations in laser intensities affecting interferometer operations
(Ch. 4) could be responsible for decoherence effects so we have tried to improve
upon this.
Finally, we attribute the long coherence times to the weak confinement provided
by the trap. We provide some theoretical background explaining the intrinsic quantum limit of the phase contrast. Following this calculation, we analyze the effects
atomic interactions have on the phase contrast observed.
In Chapter 6, the conclusion, we summarize our results and stress their importance as a stepping stone in the applicability of BEC in interferometry. We value the
importance of this experiment as a demonstration of quantum mechanical effects in
a macroscopic scale having important pedagogical value. Future applications of the
trap and adaptations of the current design are discussed.
1.7
Experimental conventions
Through out this thesis there will be several terms that are abbreviated with acronyms,
shorter words or even symbols with the intent of preventing interruptions to the flow
of the topic at hand. Table 1.1 contains the symbol or abbreviation being used along
with a more detailed description of it. Readers can use this table as a quick reference
1.7. EXPERIMENTAL CONVENTIONS
19
Figure 1.6: Two glass chambers are connected via a thin tube 1 cm in diameter
and 28.75 cm in length. The cylindrical chamber on the left is used to create a
MOT and load a quadrupole magnetic trap generated with the two anti-Helmholtz
coils shown on the outside. Atoms are transfered to the second chamber (science
cell) on the right where the interferometer wave guide is located. The center of
the coordinate system used to describe the magnetic fields is located at the center
of the trap structure labeled (0,0,0). The x direction is along the direction of the
mechanical translation of the atoms, z is the vertical direction and y is along the
axis of the waveguide. For chapter 2, the magnetic fields used for the MOT are
referenced to an origin where the x coordinate is offset by D as shown above but is
~ fields.
not reflected in the mathematical expressions of the B
when the context of a particular abbreviation is not sufficient.
20
CHAPTER 1. INTRODUCTION
Symbol or acronym
description
BEC
Bose-Einstein condensate
MOT
Magneto optical trap
TOP
Time-orbiting potential
r.m.s.
root mean squared
σ+
Sigma plus polarized light
σ−
Sigma minus polarized light
~
Planck’s constant / 2π
kB
Boltzmann constant
λt
Thermal de Broglie wavelength
i, j, k
Unit vectors
Table 1.1: List of variables and acronym
Chapter 2
Making BEC
2.1
Rubidium Atoms
Rubidium is an element which is very well suited to make BEC because it has
physical properties that make its cooling and trapping relatively simple. Denoted in
the periodic table by the symbol Rb, rubidium is a metal having a vapor pressure
at 25◦ C of 3.0 × 10−10 Torr. It is found in group 1 of the periodic table of elements,
also known as the alkali metals. Alkali metals all contain a single valence electron.
By containing a single electron in the outer most electron shell, alkali atoms have
the ability to readily ionize, making them highly reactive.
Rubidium has atomic number 37; its most common natural occurrence is 85 Rb
making up ∼ 72.2% of the known isotopes. The second most common ocurrence
is that of 87 Rb with ∼ 27.8% [26]. Currently, this isotope is used extensively in
experiments involving laser cooling which takes advantage of its cycling transition
between hyperfine states, a characteristic shared with all the other alkalis.
The transitions of 87 Rb relevant to laser cooling can be derived from extending
the simple hydrogen electronic shell structure. To an approximation, the quantum
model of the electronic configuration in 87 Rb can be thought of as equivalent to
hydrogen, but with different nuclear mass, higher number of electrons and spin.
Light atoms like hydrogen with low number of electrons demonstrate experimentally a very fine division in their spectral lines, e.g. 4.5 × 10−5 eV [27] for the
H-alpha line (2P1/2 → 2P3/2 ). It is known as the fine structure of the atomic energy
levels. This small effect is due to the interaction between the total orbital angular momentum of the atom and the electron’s spin. It can be thought of as the
interaction between the electron’s magnetic spin and the magnetic field emanating
from the current loop set up by the rotating electron. The energy shift is analogous
to that experienced by a magnet with moment µ surrounded by a loop of current
producing a field B, where ∆E = −µ · B. Accordingly, the interaction is known as
the spin-orbit coupling and depends on the operator L · S.
For atoms that have many electrons and a heavier nucleus (i.e. more neutrons
21
22
CHAPTER 2. MAKING BEC
and protons like 87 Rb), it is important to keep in mind the spin-orbit coupling. For
example, the total angular momentum of an electron Ji = Li + Si can couple to the
total angular momentum of other electrons Jj = Lj + Sj , which means the total
momenta interact with each other (this effect is known as the j-j coupling). It can
be thought of as a repulsive interaction the one active electron feels from the others.
However, a larger effect in the case of 87 Rb is the coulomb interaction between
the nucleus and the electrons. Due to this, different orbitals that would normally be
degenerate are now energetically split. For the case of 87 Rb, the degenerate energy
eigenstates 52 S1/2 , 52 P become split. The resulting split gives rise to transitions
from the 52 S1/2 state to the 52 P states. In turn the 52 P states are split due to the
L · S coupling, wich yields a break in the degeneracy of the 52 P3/2 and 52 P1/2 states.
For this reason, the coulomb interaction mentioned between the 52 S and 52 P
states and the L · S (also known as spin orbit coupling) interaction, give rise to
transitions between the 52 S1/2 and 52 P1/2 states and the 52 S1/2 to 52 P3/2 states.
They are known as the D1 and D2 lines respectively.
Furthermore in 87 Rb, introducing the interaction between the nuclear spin I and
the total angular momentum of the electron J creates another subgroup of split
energy levels known as the hyperfine structure of the atom. The total angular
momentum of the atom is denoted by F which follows the usual rules for addition
of angular momenta:
F=I+J
(2.1)
where the minimum and maximum values for the magnitude of F are given by
|J − I| ≤ F ≤ J + I .
(2.2)
The ground state for 87 Rb is the 52 S1/2 state where L = 0, J = 1/2 and I = 3/2.
This means that the hyperfine structure for this state will consist of two sub-levels
where F = 1 or F = 2. The exited states occur for L = 1, using the addition rules
for angular momenta we obtain that J = 3/2 or J = 1/2 giving us the previously
mentioned excited states. Due to the hyperfine interaction, additional sub levels
appear for each of these excited states. Using equation (2.2), the hyperfine structure
revealed for the two exited states is such that F can take the values of 1 or 2 for the
P1/2 state and 0, 1, 2 or 3 for the P3/2 state. The diagram on Fig. 2.1 represents the
energy levels and splitting due to hyperfine interactions for the ground and exited
states of 87 Rb.
2.1.1
External magnetic fields
For each individual hyperfine state labeled F, there is subset of 2F + 1 states corresponding to a particular projection of the the angular momentum state along the
quantization axis labeled mf . Normally all these states share the same energy, but
in the presence of an external magnetic field their energies are shifted according to
2.1. RUBIDIUM ATOMS
23
Figure 2.1: The atomic states of 87 Rb are split due to the hyperfine interaction. Our
experiment uses the cycling transition found in the D2 line. The magneto-optical
trap uses the transition from F = 2 to F 0 = 3 at 780.246 nm as the main trapping
frequency. A second laser tuned to 780.232 nm drives the F = 1 to F = 2 transition.
The values for the Landé g-factors gf are given for every different F level.
24
CHAPTER 2. MAKING BEC
the Landé g-factor specific to each type of angular momenta S, L, I and the electron
magnetic moment µB . The energy shift due to the external magnetic field for each
mf projection is called the Zeeman effect. Specifically the Hamiltonian describing
this interaction can be expressed as explained in [13, 28]:
HBext =
µB
(gS S + gL L + gI I) · B
~
(2.3)
By choosing the quantization axis to be along the the direction of a magnetic field
in the z direction we obtain
HBext =
µB
(gS Sz + gL Lz + gI Iz ) Bz
~
(2.4)
Moreover, when the energy shift due to the external field is smaller than the
hyperfine shift itself, Eq. (2.4) can be approximated yielding the following energy
shift [26].
µB
∆EBext =
gF mf Bz
(2.5)
~
where mF is the z projection of the total F angular momentum and gf is the Landé
g-factor for each angular momentum configuration. A derivation of the expression
for the Landé g-factor can be found in many texts [29]. Figure 2.1 includes values
for the Landé g-factors in the various | F i sub-levels for the 5S1/2 , 5P1/2 and 5P3/2
states.
If the atom makes a transition from a ground F state to an excited F’ state,
the transition selection rules [28] establish that ∆mF = ±1 or 0. A photon inducing a transition can carry different values of angular momentum depending on its
polarization. If the polarization is linear then ∆mF = 0, and if it is circular then
∆mF = ±1.
For circularly polarized light we use two conventions depending on the helicity of
the photon at hand. If the k vector of the light is aligned with its angular momentum
L, it labeled left hand circularly polarized. For the case of an anti-aligned k and L, it
is labeled right hand circularly polarized. The different cases of circular polarization
are abbreviated as LHC and RHC respectively.
Adittionally, for σ + polarized light (L parallel to an external B), the change in
angular momentum after the transition is ∆mF = 1, on the other hand, σ − light
(L anti-parallel to an external B) induces a ∆mF = −1 change1 . In this way, the
change of angular momentum the atom experiences after a transition will depend
on the angular momenta carried by the photon that caused it.
If resonant light incident on the atom is polarized such that ∆mF = 1, then
after successive transitions, the atom could find itself in the highest | F 0 , mF 0 i state.
At this point it is incapable of transitioning to higher mF 0 states. Its only option is
to decay to the highest | F, mF i state from where it can only go back to up to the
1
more about circular polarization and external B fields is discussed in Sec. 2.2
2.2. CREATING A MOT
25
highest | F 0 , mF 0 i state. Consequently the transition it remains locked to is called a
cycling transition. Rubidium has two cycling transitions according to its frequency
which are useful in optical cooling and magnetic trapping. In particular, as will be
discussed later, cycling transitions are used in order to pump trapped atoms into
the desired hyperfine state.
In order to achieve the desired phase space densities of 87 Rb to obtain BEC, we
use laser cooling as a first step in lowering the the temperature from Tr = 300 K
(room temperature) to approximately T = 100 µK. Understanding the hyperfine
structure of rubidium reveals why this element as well as all the other alkalis, are
suited for laser cooling. As described on Fig. 2.1, the characteristic frequencies that
couple the ground with excited states are found in the near infrared spectrum, this
means we can take advantage of the wide range of commercial diode and Ti:sapphire
continuous wave lasers available in this frequency range to perform laser cooling.
2.2
Creating a MOT
Conceptually, utilizing the force induced by the scattering of laser photons to slow
down the velocity of thermal atoms, was introduced by Letokhov et al. , Wineland
et al. and Hänsch et al. [30, 31, 32, 33] in the mid to late 70’s. Their efforts
culminated 15 years later when S. Chu et al. was awarded the Nobel prize for
experimentally demonstrating laser cooling techniques [34]. A detailed description
of laser cooling can be found in many texts [35]. We give a condensed treatment
here.
Our first step in creating a BEC involves confining atoms which start out as a
dilute gas at room temperature of T = 300 K. This is experimentally achieved by
setting up a magneto-optical trap. It combines viscous forces provided by laser light
tuned to the resonance of the atoms with confining forces resulting from introducing
a magnetic field.
Laser cooling uses light-induced forces to reduce the momentum of thermally
distributed atoms. As the name suggests, the aim is to use resonant light to cool,
or equivalently slow, the atoms, thereby reducing the average kinetic energy of the
sample. Temperature is just a macroscopic thermodynamic property of the system
proportional to the average velocity of a sample of atoms. This can be demonstrated
by the well known statistical result hEi = 3/2kb T , where hEi is the average energy
of a 3 dimensional ideal gas. If one can reduce the average r.m.s velocity of the
atoms in a sample, then the temperature of the system can be reduced.
3
1
mhv 2 i = kb T
(2.6)
2
2
To understand the mechanisms we can use to slow down the atoms, we must
introduce the dynamics of the absorption of photons in a simple two level quantum
system and how the effects of spontaneous emission enter into the two level problem.
K.E. =
26
CHAPTER 2. MAKING BEC
We can include spontaneous emission into the two level problem by using the density
matrix formalism in quantum mechanics. In general, in addition to the wave function
expressed as:
X
| Ψi =
ci | ii,
(2.7)
i
information about what state the system is in, is given by the density operator in
conjunction with the complete set of basis vectors for that system. The density
operator is given by
ρ = | Ψih Ψ|
(2.8)
such that the matrix elements of the density operator are obtained by
ρij = hi|ΨihΨ|ji = ci cj∗
(2.9)
where i and j denote the indexes for two distinct basis vectors in a given space.
Using the example of a simple two level system, the complex coefficients cg (t) and
ce (t) are the time dependent coefficients for the ground and excited states of the
system respectively. For this system, the total wave function can be written as:
| Ψi = cg | gi + ce | ei
(2.10)
Following Eq. (2.9) such a system’s density matrix can be explicitly written as:
·
¸ ·
¸
ce c∗e ce c∗g
ρee ρeg
ρ=
=
(2.11)
ρge ρgg
cg c∗e cg c∗g
Although the two level system will be covered with more detail in Sec. 4, we can
use the results obtained in Metcalf and van der Straten’s Laser Cooling and Trapping
[35] for the purpose of this discussion. It calculates time dependent coefficients cg
and ce regarding a two level system denoted by a ground and an excited state initially
coupled via the Hamiltonian H(t) = −eE(r, t) · r. In this case E is set up as a plane
wave polarized in the z direction. The resulting coefficients are:
¶
µ
Ω0 t i∆t/2
Ω0 t
∆
cg (t) =
cos
− i 0 sin
e
(2.12)
2
Ω
2
Ω0 t −i∆t/2
Ω
e
(2.13)
ce (t) = −i 0 sin
Ω
2
using
Ω0 =
√
Ω 2 + ∆2
(2.14)
0
The parameter Ω ≡ −eE
h e|r| gi is known as the Rabi frequency where E0 is the
~
amplitude of the electric field E and ∆ = ωl − ωa is the detuning of the incident
radiation field from the transition frequency between the two levels. The next step
is to find out the time evolution of the matrix elements ρij . For this purpose, we
2.2. CREATING A MOT
27
take the time derivative of each matrix element ρij as defined by their respective
ci c∗j . Then we substitute in the explicit time dependence of the coefficients ce ,cg and
finally re-write the equations back in terms of the matrix elements ρij . Using the
example of ρgg we obtain:
dc∗g
dρgg
dcg ∗
Ω∗
Ω
=
cg + cg
= i ρeg ei∆t − i ρge e−i∆t
dt
dt
dt
2
2
(2.15)
Additionally we must include in the time evolution of ρij the effects caused by
spontaneous emission. The derivation of how spontaneous emission is obtained will
not be covered here but is discussed in many quantum mechanics texts like [13, 36].
Specifically, it is found that in a system comprised of a two level atom and a photon
field, the state | ei describing the excited atom is short lived. This means the atom
decays to the ground level. The time evolution of the amplitude for this state is
given by the coefficient ce in:
dce (t)
Γ
= − ce (t)
dt
2
(2.16)
which means the excited state amplitude has an exponential spontaneous decay
factor of Γ/2. For 87 Rb the excited state 52 P3/2 will have Γ = 2π × 6.07 MHz.
Consequently the decay rate of the probability is Γ and the lifetime of the excited
state is τ = 1/Γ. We now apply this result to the simple two level problem. If the
excited state probability ρee decays at a rate Γ then the ground state probability ρgg
will grow at a rate Γ. For this reason we introduce this growth factor into Eq. (2.15).
¢
dρgg
i¡ ∗
= Γρee +
Ω ρeg ei∆t − Ωρge e−i∆t
dt
2
(2.17)
The same procedure described above can be performed to the remaining density
matrix elements ρeg , ρge and ρee . This would result in four equations for the time
derivatives of each element of ρ known as the optical Bloch equations. They will
not be presented here, but the result can be found in [35]. Still, we can use two
restrictions imposed by the two level density matrix which allow us to find the steady
state solutions to these matrix elements.
We know that in the two level system, the sum of the population in the two
states must be conserved as ρee + ρgg = 1. Furthermore, the matrix elements ρij are
hermitian as demonstrated by Eq. (2.9). Applying these conditions to the optical
Bloch equations can yield the following:
µ
¶
Γ
iΩ(ρgg − ρee )
dρeg
= −
− i∆ ρeg +
(2.18)
dt
2
2
¢
¡
d(ρgg − ρee )
(2.19)
= −Γ(ρgg − ρee ) − i Ωρ∗eg − Ω∗ ρeg + Γ
dt
28
CHAPTER 2. MAKING BEC
To obtain the steady state solution for the ρij coefficients shown above we can set
the time derivatives to zero and obtain:
ρgg − ρee =
ρeg =
1
1+s
(2.20)
iΩ
2(Γ/2 − i∆)(1 + s)
(2.21)
where we have defined the saturation parameter s = |Ω|2 /2(∆2 + Γ2 /4). Careful
observation reveals that by introducing the value of Γ given in Eq. (2.16), the intensity I = c²0 E02 /2 appears in s. Accordingly we define Is = πhc/3λ3 τ as the
saturation intensity where λ is the wavelength of the radiation field and τ is the life
time defined in Eq. (2.16).
s≡
|Ω|2 /2
I/Isat
=
∆2 + Γ2 /4
1 + (2∆/Γ)2
(2.22)
For the D2 line in 87 Rb λ = 780.24 nm and τ = 26.24 ns, meaning the saturation
intensity Isat = 2.5 mW/cm2 . If the incident light beam is below saturation intensity
then s ¿ 1, meaning ρgg − ρee ≈ 1 and the system is mainly in the ground state.
Otherwise, when s À 1, ρgg − ρee ≈ 0 and the populations for the excited and
ground state are equal. Using the same method utilized to get Eqns. Eq. (2.20) and
Eq. (2.21) we can obtain an expression for ρee :
ρee =
s
I/2Isat
=
2(1 + s)
1 + I/Isat + (2∆/Γ)2
(2.23)
The above expression shows how the probability for the system to be in the excited
state approaches 1/2 as I À Isat . The model depicted above is for a generic two
level system interacting with a an electromagnetic plane wave. Hence, we can apply
it to a real atom like 87 Rb interacting with a laser, in which the D1 or D2 lines act
as a two level system for the valence electron.
Thinking about Eq. (2.23) in the context of an atom in a laser field means that
every time the electron decays to the ground state of the system, there will be a
photon emitted. This allows us to multiply the probability to be in the excited state
ρee by the the decay rate of the excited state (1 photon emission/decay) to obtain
the photon scattering rate Rs as:
µ ¶
I/Isat
Γ
.
(2.24)
Rs = Γρee =
2 1 + I/Isat + (2∆/Γ)2
Now that we have uncovered the mechanism by which a simple two level atom
scatters photons from a laser beam, we can shift towards discussing how to reduce
the thermal energy in a large sample of atoms. Slowing down fast moving atoms
requires an overall momentum exchange. By absorbing a photon that drives the
2.2. CREATING A MOT
29
atom to the excited state, the atom gains an overall momentum of ~k. Here ~ is
related to Plancks constant by ~ = h/2π and k is the wave-vector for the photon.
The atom will eventually decay to the ground state through spontaneous emission.
It will emit a photon in a random direction meaning that on average, over several
emissions, the overall net momentum change due to emission is zero. After the
absorption and emission process, the atom would have an overall momentum change
in the direction of the absorbed photon which results in a “kick” along the direction
of the propagating beam. One can express the net force experienced by the atoms
due to absorption and spontaneous emission as
Fae = ~k Γ ρee .
(2.25)
By writing the force in terms of ρee , it can incorporate the saturation effects that
limit absorption at high intensity light fields. Substituting for ρee in terms of the
detuning ∆ and the ratio of intensity to saturation intensity I/Is of Eq. (2.23),
yields the expression for the force due to absorption and emission.
µ
¶
~kIΓ
1
Fae =
(2.26)
2Is
1 + I/Is + (2∆/Γ)2
The maximum value ρee can take due to saturation effects is 1/2, meaning that
Fae saturates to a value of ~kΓ/2. From the expression above it is clear that a laser
beam tuned near the atomic resonance can effectively be used to impart a force that
would push the atoms in the direction of k, the wave vector of the light. If the atoms
have a component of the velocity v opposite to k then this force can be used to slow
down this particular component of the velocity. One can extend this idea to a pair
of counter propagating beams. The sum of the forces due to both beams results in
a configuration that will slow down atoms from two directions in one dimension.
More generally the Doppler shift ωD = −k · v of the light field experienced by
87
Rb atoms at room temperature is not negligible. As indicated by the dot product,
any component of k which is perpendicular to v will not contribute to the Doppler
shift. In turn, any component of k parallel to v will add or subtract to the Doppler
shift. For example, consider the one dimensional case of an atom whose velocity v
is opposite in direction to the propagation of the light k. The effective field “seen”
by the atom is Doppler shifted to the blue by an amount ωD = kv.
This means that the Doppler-shifted light seen by the atoms can be used advantageously. For laser light slightly tuned red of resonance, an atom moving with
some component of its motion towards the incident beam will experience a light
field closer to resonance. Consequently, it will scatter more photons from that beam
and experience a greater force.
In the one dimensional case of a pair of counter propagating beams, an atom
will preferentially absorb and thus feel a greater force from the beam that has a k
pointing opposite to the velocity of the atom. On the whole the force can be written
as follows:
30
CHAPTER 2. MAKING BEC
Figure 2.2: The two grey curves represent the velocity dependent forces for two
counter propagating laser beams with the same frequency and intensity . The top
curve corresponds to the force field obtained for a beam with k, the bottom curve
represents that of a beam with −k. The black line is the sum of both curves. The
total field is non conservative but damps the motion of the atoms
µ
F=
~s0 Γk
2
¶·
1
1
−
2
1 + s0 + [2(∆ − ωD )/Γ]
1 + s0 + [2(∆ + ωD )/Γ]2
¸
(2.27)
The above equation represents an atom moving with velocity v in a one dimensional force field. There are two main components to the equation. The first term is
due to the force experienced by the atom from the light beam whose wave vector is
pointing in the positive direction, and in the negative direction for the second term.
Both add up to give a total force that is velocity dependent and is illustrated by the
plot shown in Fig. 2.2.
The resultant force field in equation (2.27) can be extended to slow atoms in
three dimensions by adding two additional pairs of counter propagating beams such
that all three pairs are perpendicular to each other. Similar to the one dimensional
case, in the region where the beams overlap a moving atom will always preferentially
absorb a photon from the beams having k vectors anti-parallel to its velocity v.
For atoms moving too fast, ωD is very large and the interaction with any light
beam is negligible. But for slower atoms the net result is a volume where atoms
experience a non-conservative viscous damping force, slowing down their velocity
2.2. CREATING A MOT
31
considerably and consequently their temperature. This configuration of red-detuned
laser beams has the name of optical molasses which describes the dissipative nature
of the force.
A limitation of optical molasses is the lack of confinement provided by the damping forces. As a result atoms are not truly trapped but rather slowed. Typically, an
experimental set up of 87 Rb optical molasses can slow down any one atom to µK
temperatures, in a ms time scale.
Optical molasses are relatively easy technique to set up in the laboratory but
their confinement effect is non-existent. As discussed previously, the net momentum
change per atom due to the absorption and emission process is ~k. Additionally, due
to the discrete nature of photons, the recoil momentum ~k is the minimum obtainable momentum when using a molasses configuration. This intrinsically prevents
Fae from slowing the atoms completely. The recoil limit can be expressed as:
k b Tr =
~2 k 2
MRb
(2.28)
Furthermore, because there are no restoring forces that bring atoms back to the
center of the molasses region, atoms moving in the order of the recoil limit will
normally be lost.
Although optical molasses can cool atoms down to low temperatures (∼ µK),
they can only do so for very dilute samples because of the lack of confinement forces.
We utilize a magneto-optical trap (MOT) in order to provide the necessary confinement for neutral atoms. This technique was first developed by Raab et al. [37]. It
is a very effective way to add a restoring component to the optical molasses viscous
forces by using a magnetic field gradient. The beam configuration is the same to
that of optical molasses, however the polarization of each beam must be chosen
carefully as will be demonstrated later. Additionally, it requires a pair of magnetic
coils to achieve zero field at the center and a linear magnetic gradient in all directions of the volume where the beams overlap. Introducing a magnetic field causes
an energy shift in the atomic states. As a result, an additional shift in the frequency
experienced by the atom (in a light field), is introduced. For this reason the total
detuning experienced by the atoms is now given by:
∆ = ∆0 − k · v + µeg B(r)/~
(2.29)
where ∆0 is now the laser detuning from resonance for a zero velocity atom, −kv
is the Doppler shift, and B(r) is the magnitude of the external magnetic field.
Additionally, µeg = µbe − µbg , where µbe and µbg are the the effective moments
for the excited and ground states, meaning µeg is the difference between magnetic
moments of different states (which depend on gF ). Adding a magnetic field means
that the degenerate states are Zeeman shifted proportionally to the magnitude of
the magnetic field as (for the z direction)
∆E = gF µb mF Bz .
(2.30)
32
CHAPTER 2. MAKING BEC
More detail of how magnetic fields are used to trap 87 Rb atoms will be addressed
in the next section, but it is important to note that the Zeeman energy shifts are also
dependent on the spin quantum number mF and the Landé g-factor gF discussed in
Sec. 2.1.1 given in Fig. 2.1. Depending on the sign of mF , the hyperfine state will
either increase or decrease in energy as the magnitude of the field increases away
from the center.
A key component to creating a MOT is choosing the correct right and left hand
circular polarization of light in order to drive the cycling transitions in 87 Rb. To
figure this out, it is useful to define a convention of polarization relative to the
magnetic field. Circularly polarized photons carrying an angular momentum L
aligned to the magnetic field B, are said to be “sigma plus” or σ + polarized. For
the case of circularly polarized photons carrying an L anti-aligned to the magnetic
field, the light is said to be “sigma minus” polarized or σ − .
Equally important, we must relate the helicity of the light to the σ − , σ + basis.
To do this we use the convention discussed in Sec. 2.1.1 where we define left and
right hand circularly polarized light in terms of angular momentum L carried by
the photon, and its wave vector k.
LHC ⇔ k k L
RHC ⇔ k ¼¹ L
(2.31)
(2.32)
For Zeeman shifted states in 87 Rb, the cycling transitions optimum for laser
cooling occur for σ + and σ − light between the 52 S1/2 , F = 2 and 52 P3/2 , F 0 = 3
transition. To create a MOT we need to use these transitions. Consequently, we
require a magnetic field whose geometry in combination with three pairs of counterpropagating beams will yield these transitions at the right location. See Fig. 2.3.
Consider an atom with the | F, mF i to | F 0 , m0F i transition in a one dimensional
magneto-optical trap. Figure 2.4 shows a schematic for the case of a one dimensional
MOT along the z direction. Atoms positioned away from the center in the +z
direction will experience a higher magnetic field than those in the center. The
further away from the center, the greater the Zeeman energy shift. If the laser is
red detuned the right amount (∼ 15 MHz) from the desired cycling transition, the
Zeeman shift is enough to bring the atom’s energy levels close to resonance with the
light. For example in the case of a | 0, 0i to | 10 , −10 i transition, the shifts lower the
mF = −1 state’s energy enough for a σ − beam to drive the cycling transition.
Similarly, for an atom positioned away from the center in the −z direction, the
mF = 1 state will be shifted such that the σ + transition comes into resonance.
For the case of a one dimensional field, if the magnetic field points away from the
origin along the z axis we get a MOT. Also, the two beams along the axis have
their respective k pointing towards the origin. This means that in order to get a σ −
beam on the +z axis and a σ + beam on the −z axis, the light must be left hand
circularly (LHC) polarized on the beam coming from the +z direction and right
hand circularly polarized (RHC) on the beam coming from the −z direction. In
2.2. CREATING A MOT
33
Figure 2.3: 3 pairs of counter-propagating beams aligned perpendicular to each other
to create a MOT. Each beam should have the right helicity of circular polarization
in order to provide LHC or RHC as shown above. In the x and y axes, to obtain
σ − light, given the direction of B, k will always be anti-parallel to the angular
momentum of the light L. This means the light should be RHC polarized. The
reverse is true in order to obtain σ − light for the z axis. The plane x = 0 shows a
cross section of the quadrupole field relative to the incident beams required to have
a MOT.
34
CHAPTER 2. MAKING BEC
both cases the atoms will be pushed back towards the center of the magnetic field
providing a restoring confinement force which traps the atoms.
In order to extend this restoring mechanism to 3 dimensions, we must keep track
of which transition (whether σ + or σ − ) needs to be driven on each side of every axis
in order for the atoms to receive a momentum transfer towards the center (i.e k is
towards the center). Then, compare this to the direction for the quadrupole field
in that region and establish if LHC or RHC is required. Figure 2.3 illustrates the
required polarizations to obtain a MOT given a particular field.
We use a laser tuned to 780.246 nm to drive the F = 2 → F 0 = 3 transition
required by the six MOT beams. However, there is a small probability for this light
to excite atoms to the F 0 = 2 state. This can become a problem when implementing
a MOT because according to selection rules, atoms in this state can spontaneously
decay to the F = 1 state. The F = 1 ground state is 6.8 GHz detuned from the
F = 2 becoming a dark state to the laser light and disabling the MOT beams from
interacting with the atoms. In order to resolve this problem, we use a re-pump laser
beam tuned to 780.232 nm that drives the F = 1 → F 0 = 2 which gives the atoms
an opportunity to decay back to the F = 2 ground state where they will interact
with the MOT beams (see Fig. 2.1).
A MOT can achieve low temperatures with the added advantage of the confinement forces that trap neutral atoms which is not present in an optical molasses set
up. However, optical molasses can cool further than a MOT configuration. Optical
molasses can reach lower temperatures, but as mentioned earlier their limitation
arises from the single photon recoil momentum present in the absorption emission
process.
Generally MOTs can achieve temperatures in the order of tens to hundreds of µK.
In our experiment we have measured temperatures of ∼ 800 µK for a number density
of 0.8×1010 atoms/cm−3 using ballistic expansion methods (see Sec. 2.8). Compared
to the temperatures and densities required to obtain BEC, which is usually hundreds
of nK at n = 1013 atoms/cm−3 , a MOT does not achieve low enough temperatures
at the given density to achieve the desired phase transition.
Along with the limitations of cooling with lasers addressed briefly in Eq. (2.28),
there are others which prevent us from obtaining the densities required for BEC. In
a MOT, the atoms that are at the core will experience a type of “light shielding”
since atoms near the outer edge will absorb most of the light. This is why atoms
near the core will not get cooled as effectively and high enough densities will not be
achieved. To solve this problem, we perform evaporative cooling. This technique
requires very low background pressures (three orders of magnitude less) in order to
be effective, more details will be explained in section 2.7.
2.2. CREATING A MOT
35
Figure 2.4: The laser frequency ωl , tuned 15 MHz red of the atomic resonance of
Rb, drives the F = 2 → F 0 = 3 transition for atoms whose velocities provide
the adequate Doppler shift. The presence of a magnetic field creates a Zeeman
shift in the hyperfine states, tuning the m = −1 closer to resonance. The chosen
polarization of the beams causes atoms which are positioned away from the center
of the trap to preferentially absorb the light propagating opposite to them, pushing
the atoms back towards the center of the trap. The lower portion of the diagram
shows the magnitude and orientation of the field along z direction for x = 0, y = 0.
87
36
CHAPTER 2. MAKING BEC
Figure 2.5: A 3-D representation of the MOT beams, the coils that make the anti
Helmholtz magnetic field and the glass cell that contains 87 Rb at room temperature.
Typically, a MOT can load 109 atoms in 3 s and achieve a temp of 100 µK. The
small tube extruding from the chamber leads into the UHV region where evaporative
cooling is performed
2.2.1
Supplying
87
Rb
In addition to the light shielding effects mentioned above, the way we supply 87 Rb
atoms to the MOT limits our ability to achieve the necessary phase space densities
to achieve BEC.
We have installed inside the first chamber two pairs of 87 Rb getters that release
Rb vapour. Hence, the first chamber is flooded with 87 Rb vapour in order to
readily capture the atoms using a MOT.
87
In the MOT chamber, a current of approximately 2.2 A is run across the dispensers. This activates a chemical reaction causing the rubidium dispenser to release
hot rubidium gas flooding the chamber with both 87 Rb and 85 Rb. In this chamber
the MOT is created and subsequently a purely magnetic trap is loaded using a pair
of trapping coils which lie outside the cylindrical shaped chamber.
When the getters are on, the recorded pressure is usually P = 7 × 10−10 Torr.
For these kinds of pressures the MOT will suffer a loss rate caused by collisions with
the background gas. With this in mind, background collisions not only limit the
MOT densities, but they also greatly limit the ability to confine atoms in a purely
magnetic trap. We will explore how to solve this problem in Sec. 2.5.
2.3. INCREASING THE NUMBER DENSITY
2.3
37
Increasing the number density
Challenged with the problem of limited phase space density available in a MOT, we
seek to find an alternate method to increase the phase space density of the trapped
87
Rb. One way to achieve this increase in phase space density is to dramatically
lower the temperature of the system. A standard technique used to decrease the
temperature of dilute samples like 87 Rb to produce BEC is evaporative cooling.
Evaporative cooling is an equivalent process as that involved in cooling a cup of
coffee. In coffee, the fastest molecules are the hottest molecules which are already in
a gaseous state. When air is blown over a hot cup of coffee, the fastest molecules are
forced away from the thermal ensemble. The fastest molecules are the most energetic
molecules in the sample, so by removing them, the average energy is decreased and
the system is no longer in thermal equilibrium. After a while, the coffee’s thermal
distribution equilibrates but at a lower average temperature. If we keep blowing
on the cup, this process repeats itself until the coffee is in thermal equilibrium
with the surroundings and in the end we have managed to substantially reduce its
temperature.
An important aspect of evaporative cooling is the ability to hold on to the
atoms while performing the evaporation. For this reason it is important to first
load a purely magnetic trap or “cup” in which to perform evaporation. During
the evaporative cooling of 87 Rb atoms, we use special techniques that remove the
most energetic atoms from our magnetic trap. The average energy of the remaining
atoms is lower, causing the temperature to drop. If we do this enough times, we can
reach the critical temperature, causing the atoms to form a condensate. All of this
assumes that BEC is achieved before the number of atoms in the trap runs out. As
we can see, one of the downfalls of the evaporation is the inherent loss of atoms.
In light of the loss of atoms, we aim to initially load a purely magnetic trap with
as many atoms as possible. That way we will not run out of atoms before reaching
the critical temperature. Because the atoms loaded into the magnetic trap are in
turn loaded from an initial MOT, we seek to have a MOT with large number of
atoms NA . The higher the number of atoms, the higher the initial density right
before evaporating. Above all, since evaporating will always require a loss of atoms,
a MOT with high NA will mean a condensate with a large number of atoms.
Although a MOT configuration will not yield the appropriate phase space density
(see Eq. (1.9)) to achieve BEC, it is a very useful mechanism by which to slow the
atoms enough to load them into a purely magnetic trap. Once loaded into a magnetic
“bowl” evaporative cooling can take place, see Fig. 2.9.
The next section describes the experimental steps taken to load the magnetic trap
and how and why we transfer the atoms to he UHV region to perform evaporative
cooling.
38
2.4
CHAPTER 2. MAKING BEC
Loading a Magnetic Trap
Evaporative cooling is an essential step in the process of making BEC because it
allows us to reach the necessary phase space densities to reach condensation. In
turn, the cooling technique requires a mechanism which will allow us to hold on
to the atoms while we perform evaporation. Normally trying to hold on to neutral
atoms is non-trivial. Because they are not charged (like e.g. ions) it is not possible
to use electric fields and their forces to suspend them. Another possibility is using
laser forces. Nevertheless, as mentioned in the previous section, there are heating
factors which are not trivially overcome.
A solution to this problem can be obtained by suspending neutral atoms via
their spin magnetic moment in a purely magnetic trap. This is our motivation to
load a purely magnetic trap of 87 Rb atoms and physically transfer it to a region
where the pressure is low enough to perform the evaporation. With this in mind, it
becomes important to recall the basic interaction between the magnetic field B and
the atom’s total magnetic moment µm due to the total angular momentum of the
atom. See Sec. 2.1.1.
To motivate how to obtain the energy due to this interaction we can think of
having a magnetic field in the z direction which selects the z component of the
angular momentum operators in equation (2.3). If the energy shift due to the field
is small compared to the hyperfine splitting, then mF becomes a good quantum
number. Also, in many occasions the atomic spin will follow the external magnetic
field, and we choose the quantization axis to be along the direction of the field so
that Bz is in fact just the overall magnitude of the field B(r).
Deriving the final expression for the energy requires subtleties not be addressed
here (due to its length), although a full derivation can be found in [13]. In this way
the energy due to an external magnetic field can be written as:
Umag =
µB
gF mF B(~r)
~
(2.33)
It should be noted that for a potential like this, the normally degenerate mF states
within a particular F manifold will become split. The splitting will occur according
to the Landé g factors for each F level. This splitting has already been discussed
with Eq. (2.5) in Sec. 2.1.1 and Eq. (2.30) in Sec. 2.2.
In order to obtain the force that an external B field exerts on the atoms we take
the gradient of the potential using F = −∇U (r) so that
Fmag = −
µB
gF mF ∇B(r).
~
(2.34)
Equation (2.34) shows that an external magnetic field can be used to trap atoms
if an appropriate B(r) configuration is found. In our experiment we start with
a simple spherical quadrupole field configuration which was initially suggested by
2.4. LOADING A MAGNETIC TRAP
39
Wolfgang Paul [38]. His contributions were later applied to ion traps and successfully
used in 1985 to trap neutral atoms [39] using magnetic fields.
The design of such a configuration can be motivated as follows: for a magnetic
field which is symmetric in x and y, the derivatives along these two directions should
be the same. Above all, the field should also satisfy Maxwell’s equation ∇ · B = 0.
Applying these two constraints we get:
∂B
∂B
1 ∂B
·i=
·j=−
·k
∂x
∂y
2 ∂z
(2.35)
In order to satisfy Maxwell’s condition, the derivative of Bz must be twice the value
and opposite in sign to those of x and y. In this regard, it is not hard to see why
the equation below yields the appropriate symmetry configuration for a spherical
quadrupole field when approximated for small |r|. A cross-section in the x = 0 plane
for the field expressed below can be seen in Fig. 2.3.
x
y
B(r) = B 0 (zk − i − j )
2
2
2.4.1
(2.36)
Compressed MOT
We use a quadrupole field like the one shown in Eq. (2.36) to trap atoms from a
MOT. In order to do so without loosing a lot of phase density we must carry out
an extra step, creating a compressed MOT or CMOT. The aim of this step is to
increase the density of the atoms before they are captured by the purely magnetic
trap.
For this purpose the CMOT stage includes, an increased red detuning for the
main trapping beams and decreased power of the re-pump beams. A higher magnetic field gradient means that the confining forces occur at smaller radius from the
center of the MOT, compressing the atoms more, achieving higher densities. Higher
detuning from resonance decreases the scattering rate, reducing the optical density
of the outer atoms, allowing the trapping light to penetrate deeper into the core of
the MOT, cooling more effectively. Reducing the power of the re-pump means more
atoms will remain in the 52 S1/2 F = 1 ground state making less atoms sensitive to
the trapping light, further increasing the ability of atoms at the core of the MOT
to be compressed by the light. The disadvantage of using a CMOT stage is its
decreased loading rate, so we do not have it on for very long.
2.4.2
Optical pumping
The third stage in creating a BEC is to optically pump the 87 Rb atoms into the
appropriate hyperfine state for loading the magnetic trap. In our case, we use the
atoms in the F = 2 manifold of the ground state 52 S1/2 . Specifically, we want to
have a magnetic trap that is populated by atoms in the | F = 2, mF = 2i hyperfine
40
CHAPTER 2. MAKING BEC
state. Using a decreased power in re-pump frequency during the CMOT stage leaves
most atoms in the dark | F = 1i ground state. In order to get atoms out of this state
we apply a short (∼ 1) ms pulse of re-pump light in addition to circularly polarized
“pumping” light. The re-pump light is to provide a mechanism for atoms in the
| F = 1i state to end up in the | F = 2i ground state (Section 2.2). The circularly
polarized light is tuned to the F = 2 → F 0 = 2 transition.
In addition, we have an appropriately chosen magnetic field which makes the
circularly polarized light σ + . This light not only drives transitions from the | F = 2i
state to the | F 0 = 2i, but changes the m0F value by +1 upon absorption of the photon
to the excited state. When spontaneous emission causes the atom to decay to the
| F = 2i ground state, the atom can change its mF state such that ∆mF = 0, ±1. If
∆mF = 1 upon decay, then the next time it absorbs a σ + photon it will also change
its m0F value by one.
Over many cycles, the atoms will end up in the | F = 2, mF = 2i state, corresponding to the state with maximum mF . The atom’s final state is such that it
cannot absorb any more σ + photons. In other words, the atom is trapped in a
“dark state”, becoming insensitive to the light and unable to transition into any
other state. The final result consists of atoms populating the desired state.
2.4.3
Switching The Magnetic Trap On
In order to finally transfer the atoms into a purely magnetic trap without losing
much phase space density it is important to set the correct trap strength so that
atoms neither expand nor compress non-adiabatically after the optical pumping
stage. To capture the atoms we set the correct initial trap strength by choosing the
appropriate B field gradient. After the atoms are caught, we adiabatically ramp
the field gradient to the desired final strength. By catching the atoms at the correct
B field gradient, sloshing of the atoms in their new trap due to the non-adiabatic
jump in the potential can be avoided.
Finding the correct gradient is done by matching the average cloud size after the
optical pumping stage to the cloud size of the initial magnetic trap. Additionally,
we use the temperature during the optical pumping stage, which is measured to be
200 µK. The B field near the center of the trap (which is lined up with the center of
the MOT) can be approximated as linearly dependant on the coordinates, such that
Umag (r) = µB 0 r. Hence the radius of the atom cloud in the magnetic trap can be
found by equating the thermal energy of the atoms ∼ kb T , to their potential energy
in the magnetic trap and solving for the radius at this energy. Consequently, the
gradient of the field is set by matching the radius of the atoms in the magnetic trap
to the atoms in the optical pumping stage. In our case we have found B 0 = 124
G/cm. Finally, after the magnetic trap has been loaded, the gradient of the field
is ramped up adiabatically to its maximum value of 387 G/cm. This will give us
the necessary densities for evaporative cooling. See Table 2.1 for a list of B field
2.5. TRANSFERRING ATOMS
Stage
41
Detuning
[Mhz]
MOT
CMOT
Optical pumping
Load magnetic trap
Magnetic trap ramp
-15
-28
Magnetic Gradient
Bz0
[G/cm]
Time
10
13.5
0
124
100 → 387
1 × 104
30
1
50
500
[ms]
Table 2.1: Summary of stages leading up to the pureley manetic trap.
settings at different loading stages. Additionally, we measure the 87 Rb atoms in the
compressed magnetic trap to have a temperature of T = 800 µK.
In order to create the quadrupole field shown in Eq. (2.36), two coils made out of
copper tubing were placed outside the glass vacuum chamber. The coils were aligned
on their azimuthal axis as shown on Fig. 2.5. We applied dc currents of opposite
direction through each coil, and an anti-Helmholtz configuration was achieved. The
field gradient used in order to hold the atoms and perform evaporative cooling, was
of 387 G/cm. Each coil was built out of 1/4 inch outer diameter copper and had
a total of 16 turns. The outer radius of the coil structure measured 122 mm and
the inner radius was 45 mm, making the average radius 83.5 mm. To obtain the
maximum field strength desired, the coils had to endure a current of 750 A at a
resistance of 17 mΩ. At these currents the pair of coils dissipated approximately 12
KW of power. In order to dissipate the generated heat, a supply of chilled water
flowed through the copper tubing at a rate of 1.25 l/min [21]. Additionally a water
flow and temperature interlock were set up (appendix A) to prevent water vapour
build up in the coils which could cause hazardous situations involving overheating
explosions.
2.5
Transferring Atoms
When the getters which provide 87 Rb are on, the measured pressure inside the
chamber is Pmot = 3 × 10−10 Torr. At these pressures, the background particles will
be constantly colliding with MOT atoms, ejecting them out of the MOT.
Although atoms trapped in the MOT are lost to background collisions, the MOT
can coexist in the previously mentioned pressures because the lasers continuously
refill it with newly cooled atoms. However, atoms lost due to background collisions
in a pureley magnetc trap will not be replenished, thus greatly reducing the lifetime
42
CHAPTER 2. MAKING BEC
Figure 2.6: (a) The quadrupole coils start at the MOT side of the chamber where
the magnetic trap is loaded. (b) Trapped atoms are moved across a thin tube 1cm in
diameter for a distance of 54.3 cm. (c) The trapped atoms arrive at the science cell
and are positioned on the center of the waveguide structure. We use the waveguide
structure in conjunction with the quadrupole coils to generate the TOP.
of the trapped atoms. The measured life time for the magnetically trapped atoms
in the MOT chamber is τmot = 4 s.
Unfortunately, the lifetime of the magnetically trapped atoms we obtain is not
long enough to perform evaporative cooling. The evaporative cooling technique
relies on atomic collisions which redistribute the energy of the trapped atoms, see
Sec. 2.2. This process of re-thermalization can take several seconds, e.g 10-15 s. If
the evaporation is carried out faster than the atoms’ ability to re-thermalize, then
we would lose all the atoms before ever reaching the desired critical temperature Tc .
For this reason, we first load the magnetic trap from a MOT in the chamber
containing 87 Rb vapour. Afterwards, we transfer the magnetically trapped atoms to
a second ultra high vacuum chamber. In this chamber the pressure is Psci = 4×10−11
Torr, meaning background collisions are much lower and the lifetime of the magnetic
trap is considerably extended. We denote this second chamber as the science cell,
and measure the lifetime of the atoms in the magnetic trap to be τsci = 80 s. As a
result, we can hold on to the atoms long enough to perform evaporative cooling.
Specifically, the chamber used in our experiment consists of two separate glass
2.6. LOADING A TOP TRAP
43
cells joined via a thin tube, see Fig. 1.6. The first cell, where the MOT is created,
is cylindrically shaped, with a diameter of 63.85 mm and a length of 260 mm. The
thin tube connected to one of the ends of the MOT chamber is approximately 1
cm in diameter and 35 cm in length. It separates the chambers enough to allow for
a large pressure differential between the two chambers. Equally important is the
second cell known as the “science cell”. It is box shaped with a length, width and
height of 330 mm, 81.2 mm and 52.2 mm respectively.
Following the scheme by Lewandowski et al. [10], we use a programmable translation stage which supports the coils for the magnetic trap loaded in the first chamber.
After ramping the magnetic trap’s field to its maximum value, we proceed to move
the coils towards the second chamber, moving the atoms via the thin tube into the
UHV region, see Fig. 1.6.
The translation stage has a track allowing a total displacement for the quadcoils of 600 mm. The total distance the atoms must be translated is 543 mm. In
particular, the stage is programed to follow a motion with three stages. First is a
stage accelerating the atoms from rest. Second is a constant velocity period, and
finally a deceleration stage brings the atoms to their final position in the science cell.
More details of how the programmable moving station is operated can be found in
Jessica Reeves’ thesis [21].
Because our ultimate goal is to create a condensate interferometer, we require
a magnetic trap in which we can manipulate the motion of the atoms. For this
purpose, we have created a specially designed magnetic waveguide which traps the
atoms but allows them to easily move in one dimension. This trap is analogous to
a optical fiber, which confines light in all but one direction. The technical details
and operation of the guide will be discussed in the following sections and chapter
3. Because of this requirement, another important function for the science cell is
housing the magnetic waveguide.
We translate the quadrupole coils until the atoms are located in the center of the
guide structure. By performing the evaporation in the center of the guide structure
we take advantage of the bias fields that can be generated from it (see Fig. 3.1) to
create a TOP trap as described in Sec. 2.6.
2.6
Loading a TOP trap
Although evaporative cooling is a very effective technique in reducing the temperature of the trapped atoms and increasing the density, it causes a large loss of atoms
(as will be explained in the next section). Losses also occurr due to stray light and
RF fields which pump the atoms into untrapped states. We can limit the amount of
unwanted stray RF fields and laser light causing loses by shielding the science cell.
Moreover, the quadrupole field configuration described in Eq. (2.36) suffers from a
loss of atoms at the zero of the field.
44
CHAPTER 2. MAKING BEC
When a magnetically trapped atom with velocity ~v finds itself near the zero of
the field, its interaction with the magnetic field is lost momentarily. As the atom
traverses this region, the atom could undergo a non adiabatic transition where its
mF value changes sign, emerging at the non zero region of the field with an opposite
anti-trapping force. This is called a Majorana transition [40, 41]. A change in sign of
the mF quantum number is equivalent to an inversion of the magnetic spin, causing
the atom to be ejected from the trap. The lifetime of the atoms in the magnetic
trap due to Majorana losses is given by [42]:
1
τm = ασF2 W HM
4
(2.37)
in which α = 3.7 × 104 s/cm2 is determined experimentally [42] and σF W HM is the
full width half max of the atom cloud in the trap. We can model the cloud of atoms
as having a Gaussian density distribution of the form:
µ
¶
x2
y2
z2
n(r) = n0 exp −
−
−
(2.38)
wx wy wz
where n0 is the peak number density, and wx = wy = 2wz = w0 is the width of the
symmetric Gaussian distribution. We can relate the width of the Gaussian function
to its full width half max using σF2 W HM = 4w02 ln 2. Therefore, the expression
Eq. (2.37) can be rewritten in terms of w = 1.8w0 , the width of the actual cloud
(which is not non-Gaussian) if modeled by a Gaussian density distribution. This
yields the following expression for the Majorana lifetime:
³ w ´2
= w2 × 80 s/mm2
(2.39)
τm = α ln 2
1.8
Loss of atoms via Majorana spin flips can be greatly reduced by applying a
technique known as the time orbiting potential or TOP [43, 44, 42]. In summary
the technique consists of shifting the zero of the field from the center of the trap
and then rotating it. As a result, the time dependent zero creates a path where the
atoms leave the trap known as the ellipse of death.
Specifically, we can describe the functioning of the TOP trap by calculating the
time dependent field and then averaging it over time. Mathematically the ellipse of
death can be derived as follows. First introduce a bias field at origin (Fig. 1.6) where
the center of the spherical quadrupole is located after the mechanical translation
stage. We denote the oscillation frequency as Ω,
B0 = B0 [cos(Ωt)i + sin(Ωt)k] .
(2.40)
This can be combined with a quadrupole field Bq of the form given in Sec. 2.4 to
yield the following total field:
Bq + B0 = B0 (cos(Ωt)i + sin(Ωt)k) + B 0 (2zk − xi − yj )
BT OP = (B0 cos(Ωt) −B 0 x)i +(B0 sin(Ωt) +2B 0 z)k −(B 0 y)j
(2.41)
2.6. LOADING A TOP TRAP
45
Figure 2.7: Diagram A shows an atom in the mF = 1 state approaching the center
of the field corresponding to a trapping potential given by U (r) = µB gF mF B(~r)/~.
In diagram B the atom goes through the zero, loses its spin information and
transitions to a mF = −1 state where it experiences an anti-trapping potential
U (r) = −µB gF mF B(~r)/~. Undergoing the Majorana transition, the atom is ejected
from the trap.
By using Eq. (2.41) one can obtain expressions for the path where the field is zero,
substituting xe , ye , ze into BT OP and setting it to zero, one can solve for the coordinates where the atoms can escape the magnetic trap via Majorana spin flips. Using
the form of Eq. (2.41) it is simple to obtain independent expressions for x, y and z.
For the y direction the result reveals ye = 0 and the circle or in this case ellipse of
death lies in the x, z plane. For the x and z direction we obtain:
xe =
B0
cos(Ωt)
B0
ze =
B0
sin(Ωt)
2B 0
(2.42)
The set of equations (2.42) describe a point which moves in time along a parametrized ellipse. If the frequency of rotation is faster than the atom’s ability
to follow the zero adiabatically (the oscillation frequency of the atoms in TOP,
ωx = 2π × 78 Hz, ωy = 2π × 110 Hz and ωz = 2π × 156 Hz) they experience a time
averaged potential rather than an instantaneous potential.
It is important not to oscillate the field faster than the Larmor frequency of
the atoms ωl = µb B0 /~ = 14 MHz, which determines how fast their spin precesses
around the magnetic field. Doing so would mean the atom’s spin cannot follow the
field’s direction adiabatically. Extending the size of the ellipse of death larger than
the atom cloud by controlling the currents which yield B 0 and B0 , minimizes the
loss rate from Majorana flips.
46
CHAPTER 2. MAKING BEC
Figure 2.8: Cross section in the z, x plane of the region around the origin defined in
Fig. 1.6. Using superposition, the vector fields of Bq and B0 are added, offsetting the
original position of the zero in Bq denoted by the black arrow. By time varying the
currents in the trap structure that generate B0 , the position of the zero is rotated,
following the ellipse shown above parametrized by the equations for xe and ze
2.6. LOADING A TOP TRAP
47
The actual time averaged field the atoms experience can be calculated by integrating the magnitude of the total field over one full cycle and dividing by the
oscillation period of the field. Denoting θ = Ωt, the time average becomes a function
of θ:
Z 2π
1
h|B|it =
|B(θ)|dθ
(2.43)
2π 0
By squaring each component we can obtain the magnitude of the total TOP field
as:
|BT OP | = [B02 +B 02 (x2 + y 2 + 4z 2 ) − 2B0 B 0 x cos(Ωt) + 4B0 B 0 z sin(Ωt)]1/2
Expanding the square root of the equation above
the time average gives:
µ
B 02 1 2
h|Btop |it = B0 +
x +
B0 4
(2.44)
to second order and calculating
1 2
y + z2
2
¶
(2.45)
which yields a quadratic dependence in the magnitude of BT OP . Plugging in
~ T OP |.
Eq. (2.45) into Eq. (2.33) one can obtain an expression for U in terms of |B
Ignoring any constants that introduce offsets to the energy, the atoms are trapped
on average by a harmonic potential of the form
1
UT OP = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )
2
(2.46)
This allows us to extract by comparison the values for the trap’s oscillation frequencies,
1 µB 02
µB 02
µB 02
ωx2 =
ωy2 =
ωz2 = 2
(2.47)
2 mB0
mB0
mB0
As shown in this section, a magnetic trap consisting of a spherical quadrupole
field and an appropriately chosen bias field can be used to trap atoms minimizing
the Majorana loses.
In the experiment we turn on the TOP trap when the evaporative cooling process
has removed enough atoms that the size of the atomic cloud has shrunk to the point
where it lies inside the ellipse of death. At this stage, T = 85 µK and the density
at the center of the trap is high enough that the Majorana losses are no longer
a negligible loss in the evaporation process. As previously explained (Sec. 2.7),
evaporative cooling relies on the slow re-thermalization of the atoms in the trap,
which in turn relies on the density of the atoms. Minimizing the loss of atoms in
the trap during evaporation is a crucial requisite to achieve BEC.
To help with the focus of this discussion, I summarize the key steps in the process
of loading a condensate into a magnetic waveguide. First we make a MOT where
the measured temperature is at T = 800 µK and NA = 4 × 109 . Second we make
a compressed MOT, reducing the temperature to T = 400 µK and keeping the
48
CHAPTER 2. MAKING BEC
Stage
T [µK]
NA
MOT
CMOT
Optical Pumping
Quadrupole
Quadrupole
Transfer
Quad in Science cell
Evaporation
BEC
800
400
400
800
900
0.2
4 × 109
4 × 109
2 × 109
2 × 109
1.5 × 109
2 × 104
Note
Atoms end in | 2, 2i state
catch at 124 G/cm
ramp 124 V → 387 V
From MOT to Science cell
TOP on
TOP on
Table 2.2: Summary of major steps towards achieving BEC. The voltages shown
reflect the setting on the power supply driving the spherical quadrupole coils.
same number of atoms. Third, we introduce an optical pumping stage where the
atoms appear in the | 2, 2i state. Fourth, the pure magnetic trap is turned on at
the appropriate level to catch NA = 2 × 109 atoms. The coils are then ramped
to provide the maximum gradient and achieve densities necessary for evaporative
cooling giving T = 800 µK. Fifth, the atoms are transferred using the translation
stage to the science cell and positioned inside the waveguide structure. At this stage
there are NA = 1.5 × 109 atoms at T = 900 µK. Sixth, we commence evaporative
cooling. Additionally, during the evaporation, we turn on the TOP field in order to
reduce Majorana losses. We reach critical temperature at Tc = 200 nK and produce
a condensate with NA = 2 × 104 .
2.7
Evaporative Cooling
Evaporative cooling was briefly introduced at the end of Sec. 2.3 but a more detailed
explanation behind the cooling mechanism will be discussed in this section.
Evaporative cooling, as its name suggests, works by evaporating away (thus
removing) hot atoms from the sample you are trying to cool. The principle behind
this technique is analogous to the way a coffee cup is cooled when a current of
air blows over the surface of the coffee. Because temperature is just a measure of
the average energy of the particles, repeatedly removing the most energetic atoms
reduces the temperature of the coffee substantially. See Sec. 2.3.
The same technique can be applied to atoms in a magnetic trap which is analogous to the coffee cup and can be thought of as a “magnetic bowl”. After optically
selecting the Rubidium atoms in the | 2, 2i ground state, loading them into the mag-
2.7. EVAPORATIVE COOLING
49
netic trap and transferring them to the evaporation region, we apply an RF field
on to the atoms using an antenna. The RF is tuned to a range of frequencies that
drive the mF → mF − 1 transition for atoms trapped in the magnetic potential.
Depending on the exact energy of the photons in the RF field, atoms that posses
the appropriate potential energy will absorb a photon. If the atoms absorb a RF
photon, then they transition into a untrapped state and no longer form part of the
original ensemble.
In this context, the appropriate energy corresponds to the atoms that have a
potential Umag which tunes the atom to the transition with frequency difference
∆ν = ∆mF µB gF |B|/~, where ∆mF = 1 (see Fig. 2.9). The RF radiation field can
be tuned so only the atoms with sufficient potential energy are resonant with the
mF → mF − 1 transition.
Because the magnetic field is spatially dependent, it makes the transition spatially dependent as well. If this is the case, then only the atoms which are the
furthest away from the center of the trap will undergo the transition. This is because the more energetic an atom is, the further it can reach out in the magnetic
potential. Therefore, atoms which are distributed spatially on the outer surface of
the cloud will be ejected.
If the RF frequency only removes the outermost atoms from the atomic cloud,
then it acts like the “lip” of a bowl where only the atoms which are at the top can
escape. In fact, the technique involves lowering the “lip” in order to progressively
remove the hottest atoms until Tc is obtained.
For an effective cooling technique the evaporation needs to be done at an appropriate rate. The speed of evaporation will be limited by the the ability of the
sample to equilibrate thermally. Ultimately this depends on how energy is transferred within the atoms by their collisions/interactions [45, 46]. For this reason,
high collision rates between atoms improve energy transfer between them and thus
improve possible evaporation rates.
Sweeping down the RF field is analogous to lowering the lip of the magnetic trap,
allowing only the hottest atoms to escape. It is some times called an“RF knife”
which “chops” off the high velocity atoms from the Maxwell-Boltzmann thermal
distribution. For this process to be effective, the atoms that are left in the trap must
collide with each other and redistribute their kinetic energy among other atoms,
returning close to thermal equilibrium. During the process we do not wait for the
atoms to fully equilibrate as this would take too much time. Instead, we continuously
ramp down the frequency removing at each instant the hottest atoms. In doing so,
the atoms reduce their average velocity and consequently lower their temperature,
see Fig. 2.10. The time it takes for the sample to stay close to thermal equilibrium
limits how fast evaporative cooling can be performed and sets the time scale for the
process.
An important parameter required to perform evaporative cooling is the minimum
energy of the magnetic trapping potential given by gF µB0 . Knowing the minimum
50
CHAPTER 2. MAKING BEC
Figure 2.9: An RF field is incident upon the atoms in the magnetic trap driving
the | 2, 2i → | 2, 1i → | 2, −1i → | 2, −2i transitions which are equivalent to a spin
flip where mF → m0F = −mF . The frequency of the radiation is ramped down
exponentially according to Eq. (2.48), at a rate slower than the re-thermalization
rate, but faster than the loss rate due to background collisions and 3 body loses.
2.7. EVAPORATIVE COOLING
51
Figure 2.10: (a) Maxwell Boltzmann probability distribution for thermal atoms in
terms of the velocity v where hv 2 i ∼ T . The dotted line represents the RF knife,
removing the highest energy atoms down to those having Umag = hν where ν is the
RF frequency. (b) As the distribution approaches thermal equilibrium, T decreases,
and the RF knife is swept lower. (c) The distribution tries to approach equilibrium
once again but the RF knife is lowered more. The result is a substantial drop in the
temperature of the sample.
52
CHAPTER 2. MAKING BEC
energy of the trap will let us know the lowest energy RF field we can apply to the
atoms in order to drive a transition. It is also a parameter used to program the ramp
into our experimental apparatus, in this case, the pulse sequence regarding evaporation, see appendix C. To calculate the bottom frequency of the magnetic trap,
we simply equate the energy of the incoming photon to the energy corresponding to
the minimum value of the magnetic field B0 . Hence ν0 = gF µB B0 /~2 .
The RF sweep is described by the following exponential ramp model which is
used in the evaporation sequence. The final frequency as a function of time is given
in terms of the bottom frequency ν0 , the initial ramp frequency νi and the time
constant τe as:
νf (τ ) = (νi − ν0 )e−t/τe + ν0 .
(2.48)
In order to carefully control the ramp down for the frequency of the RF field
incident on the atoms, we lower the the frequency ν in a series of sub-steps Table 2.3.
Each sub-step consists of smaller frequency changes, covering a particular frequency
range. For our experiment 4 different ramps covered the entire range of the frequency
sweep needed. Also, the sub-steps are useful because we can determine the loss rate
due to evaporation by measuring the number of atoms before and after a sweep
sub-step.
A typical sequence of evaporation steps towards BEC in our experiment is described in Table 2.3. For example, the second line starts the ramp at 30 MHz and
ends it at 10 MHz, uses a time constant of τ = 4 s, a power of -15 dBm and the
bottom frequency of the ramp is set to 2.55 MHz. See Eq. (2.48) for the functional
form of the ramp.
Starting with magnetically trapped atoms at T = 800 µK, we begin the evaporation ramp with ν = 60 MHz and perform several evaporation ramps like those in
Table 2.3. BEC is achieved at νf = 0.18 MHz, using various time constants for each
evaporation sub-step.
Unfortunately, there are atom losses through the zero of the magnetic trap which
complicate the evaporation sequence. Inspecting Eq. (2.36) it is evident that the
quadrupole magnetic field in which the atoms are trapped has a zero. This is a
problem for magnetically trapped atoms because when they cross the B = 0 point,
the spin of the atom is unable to follow the change in B. Consequently the atom
may find itself anti-aligned and thus ejected from the trap because the magnetic
force is reversed, Eq. (2.34). As a result, the trap has an effective leak of atoms
which we need to plug.
To solve this problem we turn on a time varying field resulting in a TOP trap
(Time Orbiting Potential) as explained in Sec. 2.6. Briefly, the TOP trap consists
of a rotating zero for the field, which the atoms cannot catch up to. The path of the
rotating zero is labeled “the circle of death”. This special technique helps out by
2
For the spherical quadrupole field in Eq. (2.36) the minimum is 0. However, this will not be
the case when we use the TOP trap to eliminate Majorana loses, see Sec. 2.6
2.7. EVAPORATIVE COOLING
53
νi [MHz]
νf [MHz]
τ [s]
p [dBm]
ν0 [MHz]
50
30
30
15
12
6
-13
-13
0
0
Table 2.3: A typical evaporation instruction set sent to the RF generator from the
real time controller. For this instruction there are two ramps. See appendix C for
the complete set of evaporation instructions used in the experiment. The power set
here goes to a 50 dBm amplifier before reaching the RF antenna.
effectively plugging the undesired leak, uncompromising the number of atoms left
to make BEC.
The starting size of the atomic cloud in the magnetic trap is larger than the
maximum size for the circle of death. Because of this, the TOP trap is turned on
in the sequence when the size of the cloud drops below that of the circle of death,
Fig. 2.8.
Because the circle of death expels atoms from the trap, we can take advantage
of this effect and use it as another “lip” to expel atoms. For this reason we start RF
evaporating from 50 to 15 MHz, and then stop. Next, we turn the TOP trap on and
gradually reduce its radius, expelling some hot atoms and reducing the temperature.
Finally we resume the RF evaporation from 1.85 MHz to 0.18 Mhz but using the
TOP field as an aid to prevent a loss of density in the trap.
Another complication when performing evaporation is that it cannot be done too
slowly. The upper time limit or slowest rate at which the evaporation can be done
is set by maintaining an adequate balance between the number of atoms lost due to
evaporation and those lost via background collisions and three body collisions.
Three body collisions cause atoms to form molecules which are no longer held
by the trapping potential. Similarly, ejection of atoms from the trap due to stray
RF fields and light are counted as part of the background loses. The total loss rate
not including evaporation is given by:
RL = Rbackg + R3body
(2.49)
where Rbackg ∼ 1/80 s−1 , R3body = G3 n2 depends on the number density n and we
use the measured value of G3 = 1.8 × 10−29 cm6 /s [47].
In the end, using the evaporative cooling technique, we have successfully achieved
BEC’s having Na = 1 × 104 atoms starting with a magnetic trap containing Na =
1 × 109 atoms. The drop in atoms from initial loading of the magnetic trap to the
creation of BEC is quite dramatic but it is the price paid to obtain the critical
density required for condensates. Figure 2.12 shows 3-D absorption images3 in false
3
imaging system will be explained in Sec. 2.8
54
CHAPTER 2. MAKING BEC
color taken after the final RF ramp down for different values of stop frequency νf .
The first image is before critical temperature is achieved, the second at critical
temperature and the final image shows a full BEC.
2.8
Imaging
In general, a large portion of the measurements leading up to and including the
interferometer experiment involve imaging the atoms. Usually the imaging is used to
find out the number of atoms NA and the temperature T , allowing us to derive from
these other physical properties of our trapped atom clouds (e.g. the collision rate).
For example, in the stages prior to the implementation of the BEC interferometer,
measuring T and nλ3t are crucial components to ensure the production of condensed
atoms. These physical properties are calculated from the raw data obtained in the
images of the atom cloud.
The successful operation of the BEC interferometer requires measuring the number of atoms after all its operations are finalized. For this reason, the explanation
presented in this section will focus on the atom clouds obtained after the interferometer operation. However, a detailed discussion of the methods used to obtain
NA , T and the physical properties derived from these prior to the measurement of
interferometer’s output state is given in [21]. It is important to note that all imaging
done in the harmonic trap (in the science cell) is performed the in the same way.
We need to be able to image Bose-Einstein condensates. In our experiment, the
size of these clouds are on the order of tens of microns. For this reason we need an
imaging system that magnifies these small objects and has high resolution, allowing
us to mathematically analyze the signal captured in the image. Additionally in the
condensate, approximately 104 atoms will scatter light from a few 100’s of µW at
780 nm.
We use a technique called absorption imaging to create a picture of the atoms.
This technique consists of shining a near-resonant probe beam in the region where
the atoms are contained. The probe beam should pass through the atoms and
possess a beam waist which is larger than the atom cloud. In our case the waist of
the probe beam needs to cover an area several mm2 in order to probe non-condensed
clouds.
After the light exits the region of the atoms, it is redirected into a camera.
Because the light is near the resonant frequency of the atoms, the light is absorbed
by the atoms, leaving a shadow against a bright background. Mirrors and lenses
are then used to direct and image this light onto the camera. The atoms appear as
dark regions within the area of the probe beam.
To perform temperature measurements we use ballistic expansion. This technique consists of letting the atomic cloud free fall from the magetic trap and taking
pictures of its expansion at several time intervals. We then plot the widths as func-
2.8. IMAGING
55
tions of time and fit the expansion to a quadratic form. Then we extrapolate the
temperature from the fit parameters obtained.
Before flashing the probe beam, the atoms are released from the magnetic potential holding them in place. This means that the atoms drop up to 2 mm before
the image is captured by the camera. Consequently, the imaging system includes a
lens mounted on a micrometer to adjust the focusing if the image plane of the atoms
changes. The magnification of the system is controlled using a 2× or 5× microscope
objective. The camera we use is an Apogee, model AP47 CCD.
Also, the binning of the pixels in the CCD was varied between a 3 × 3 pixel
block and a 1 × 1 pixel block. The 1 × 1 binning permitted a higher resolution but
increased the download time of the images for data acquisition. The download time
of the images varied from 3 s to 26 s (depending on the binning and array size).
When taking pictures of condensates, we mainly used the 5× objective and the 1 × 1
binning, producing images like those in Fig. 4.8.
The absorption image is derived by processing three distinct raw images. The
first image is denoted as the “atom image”, hence it is taken with the probe laser
beam going through the atoms creating the shadow effect described. The second
is a picture without any atoms present or probe light coming into the camera.
This image captures the background light coming into the camera and is therefore
labeled the “background image”. Finally, we take a picture labeled the “no atoms
image” which captures the light with the probe beam going through without any
atoms present. The idea is to simulate the lighting when the probe is on, to later
normalize the atoms image. The resulting processed image is obtained as follows:
S(x, y) =
atoms image − background image
no atoms image − background image
(2.50)
Each image consists of a 2-dimensional array of values representing the intensity of
each pixel on the CCD. We assume the transmission of light through the region of
the atoms is given by e−α (beer’s Law), where α(x, y) is the 2-dimensional absorption
profile of the atom cloud. Using the 3 images described above we write an expression
which relates the processed image to the absorption profile of the atoms:
S(x, y) = e−α(x,y)
(2.51)
In turn the absorption profile α(x, y) is assumed to take form of a Gaussian
function such that:
h
2
2
− x−xa − y−ya
α(x, y) = A + B exp ( wx ) ( wx )
i
(2.52)
where A is an offset allowing for power variation in the probe, B is the peak absorption coefficient, wx , wy are the widths, and xa , ya are the center of the absorption
profile. The image is fit using a special program developed in Matlab discussed in
App[?] called AI 3.
56
CHAPTER 2. MAKING BEC
It is important to mention that due to the expansion, the density of the atoms
is decreased, lowering the absorption coefficient B. In many instances the processed
image is over-saturated, meaning that the absorption signal by the atoms is very high
because the probe is too close to resonance. This means that the true absorption
profile of the image will be hidden, yielding inaccurate widths. Consequently, the
temperature measurement is incorrect.
For this reason, we normally allowed the cloud to expand until the image did
not look saturated having a typical absorption coefficient B between 0.2 and 0.5.
Aditionally, oversaturation has the effect of flattening the shape of the absorption
profile, hiding the true density profile of the atoms.
2.9
Calculating NA
To calculate the number of atoms in the wave packets of the output state, we can use
the absorption profile α(x, y) and the number density function n(x, y, z) of atoms
in the trapping potential in which they are found. A full calculation of how NA is
obtained for the condensate atoms is given in [21]. However an outline of how to
get NA assuming a Gaussian distribution for the number density will be given here.
We start with the expression which gives the the absorption α(x, y) in terms of
the number density of the atoms in the waveguide n(x, y, z).
Z
α(x, y) = σ n(x, y, z) dz
(2.53)
where σ is the scattering cross section for the atoms. The integral over z indicates
summing the absorption over the thickness of the cloud. In our case z represents
the vertical direction. For illustrative purposes we assume that the number density
n(x) has the form of a three dimensional Gaussian.
·
¸
x2
y2
z2
n(x) = n0 exp − 2 − 2 − 2
(2.54)
wx wy wz
R
If the n(x) is normalized such that NA = n d3 r, then we can carry out the
integration and solve for n0 to obtain the peak density for n(x)
n0 =
NA
x wy wz
(2.55)
π 3/2 w
Carrying out the integral of n(x) with respect to the coordinate z we obtain an
expression which can be used to get α.
Z
−
NA
exp
n dz =
πwx wy
y2
x2
2 + w2
wx
y
(2.56)
2.9. CALCULATING NA
57
However, we need an expression for σ. We will not derive σ, nevertheless we will
use the result obtained in [21]
σ0 ΓΓ0
σ = 02
Γ + 4∆2
(2.57)
to get an expression for α. Finally we combine σ with Eq. (2.56) to obtain:
NA
σ0 ΓΓ0
α0 = α(xa , yb ) = 02
Γ + 4∆2 πwx wy
(2.58)
We have chosen the point (xa , ya ) which yields the maximum absorption α0 . The
variable α0 is precisely the fit parameter B in Eq. (2.52), which allows us to introduce
a measured parameter into the result for NA . Finally we solve Eq. (2.58) for NA . In
the end we obtain an expression for NA in terms of the experimental observable α0
NA =
4π 2 × α0 (Γ02 + ∆2 )wx wy
2.8λ2 ΓΓ0
(2.59)
where Γ0 is the observed broadened linewidth, see [21]. In the specific case of the
condensate modeled using the Thomas-Fermi aproximation, from [45, 21] we obtain
the following expression for NA using the method described above.
NA (condensate) =
2πα0
wx wy
5σ
(2.60)
Inspecting the above equation, we use α0 = B, wx , wy and σ to obtain N . Hence
we use the program AI 3 to obtain these fit parameters to get the final result.
The image processing before carrying out the fit was done with a script program
written to obtain Eq. (2.50). In order to facilitate the image processing, we took one
background image per day which was repeatedly used by the script. It is important
that we kept the exposure time of the atoms and no atoms image the same in order
for the processed image to normalize correctly. We used short 30 µs probe flashes at
an intensity of a few µW’s to take the images. As a result, we obtained an imaging
system which performed effectively. The imaging system is illustrated in Fig. 2.11.
58
CHAPTER 2. MAKING BEC
Figure 2.11: Diagram representing the optics used to create the imaging system to
perform absorption imaging. The microscope objective could be swapped, allowing
us to choose between the 5× and 2× objective. Additionally, to account for any
changes in the image plane of the atoms (hence a change in the z position of the
atoms), we have a lens mounted on a micrometer to re-adjust the focus of the image
back onto the plane of the CCD.
2.9. CALCULATING NA
59
Figure 2.12: Three different absorption images of the atoms in the TOP trap in
false color. The vertical direction denotes higher absorption thus higher density
of atoms, the remaining axes are the x and y position in the trap. Each picture
has a corresponding stopping frequency denoting the lowest RF frequency reached
during evaporation. From left to right, stopping evaporation at 2.95 MHz shows a
cold atom cloud on the verge of condensing, atoms start condensing at a stopping
frequency of 2.90 MHz with some atoms still un-condensed, finally stopping at 2.77
MHz most atoms are Bose-condensed in the lowest state possible of the magnetic
trap.
Chapter 3
Magnetic Waveguide
To begin this chapter, it is worth mentioning that much of the motivation, design
considerations and characterization of the interferometer trap for the waveguide are
provided in the thesis project presented by Jessica Reeves in [21].
Our goal is to create a one dimensional BEC interferometer where we can control the relative phase of the arms to control the output state. Keeping in sight
this objective, we need to figure out how to effectively control the motion of the
condensate and perform splitting and recombination operations common to most
interferometers. Moreover, we must optimize the potential in order to reduce the
introduction of unwanted phase shifts and support the atoms against gravity. For
these purposes we have designed a magnetic trap which provides the necessary confinement in the x and z directions (see Fig. 1.6) but permits motion along the y,
allowing us to implement the interferometer.
Given that we must reduce unwanted phase shifts, it is important to understand
the main mechanisms which cause them. One of them is the interactions between
atoms confined in the waveguide. In brief, the interactions between atoms will cause
the arms of the interferometer to develop an extra phase which will modulate the
overall interference pattern of the output state. This modulation will result in a
decrease in contrast of the interference pattern proportional to the number density
of atoms in the trap and the propagation time of the waves. Chapter 5 will explain
this effect in more detail. In this regard, a potential with stronger confinement will
compress the atoms more, resulting in higher number densities. Therefore reducing
the confinement strength of the potential will greatly increase the contrast and
maximum propagation time of our device.
Additionally, it is important to avoid any asymmetries in the trapping potential.
In general, if one arm of the interferometer experiences a different trapping potential,
hence a different potential energy, a relative phase shift will develop between the
arms. We seek to have no development of a relative phase shift as a direct result of
the propagation of the waves. One way to achieve this is to perform the experiment
in a trap that is perfectly symmetric, but this proves unrealistic. Usually there
60
3.1. LOADING A WAVE GUIDE
61
are fringe effects and imperfections in the shapes and materials making up a trap,
limiting their symmetry. For this reason, we can set up an interferometer sequence
where both arms will acquire the same phase shift after all the motion is completed.
In a one dimensional case, this can be done by allowing each wave to travel both
arms of the interferometer, see Ch. 4.
As a result, our magnetic trap design should include minimal confinement along
the x and z directions, along with as little confinement in y as possible. This will
allow for one dimensional propagation of the atoms. These characteristics point
to a very weak trap in general. However, the atoms must be suspended against
gravity. For this reason, we must carefully balance the gravitational potential with
the magnetic potential along the z direction. This ensures the atoms do not fall out
of the trap. This will be addressed in Sec. 3.3.2.
A more detailed account of the experimental procedure utilized to produce and
load the condensate into the waveguide can be found in Reeves et al. [48]. Details
of the Adwin sequences used to make BEC and load the waveguide are given in
appendix C.
3.1
Loading a Wave Guide
We use a 4-rod copper structure located in the science cell (Fig. 1.6) to generate
the rotating bias field for the TOP trap that is used during evaporative cooling.
Additionally, the copper structure is used to generate the magnetic fields used in
the atomic wave guide. A detailed description of how these fields are created using
the trap structure will be discussed in the following sections. In summary, concentric
to the 4 rods are tube-like insulators containing another 4 rods which are used to
generate a linear quadrupole field for the atom waveguide.
To load the guide, we adiabatically ramp down the TOP trap’s quadrupole field
(the field provided by the coils described in Sec. 2.4.3) while the waveguide field is
turned on, allowing the condensate to adiabatically position itself into the atomic
wave guide. After the quadrupole coils are off, the guide remains on.
3.2
3.2.1
Magnetic Trap
Conventions and Set Up
The origin of the coordinate system used to describe the magnetic fields generated
by the trap structure is located at the center of the trap structure. Figure 1.6 shows
the location of the trap relative to the two glass chambers, while Fig. 3.1 zooms
into the science cell providing a more detailed perspective of the trap structure, its
components and the coordinate system used. The trap structure is mounted onto a
set of 33 cm long copper block leads that conduct current to each individual circuit
62
CHAPTER 3. MAGNETIC WAVEGUIDE
Figure 3.1: A different view of the science cell, with the trap structure visible through
the glass chamber. The thin tube connects to the MOT chamber. The center of
trap structure is aligned to the thin tube, which allows the mechanically transferred
atoms to arrive as close as possible to the minimum of the waveguide trap. The
coordinate system used to describe the waveguide fields is centered on the trap as
shown.
in the trap. In turn, these leads are connected to a 4 − 3/4” con-flat feed-through
with 8 pins, which correspond to the 4 independent circuits available in the trap
structure.
One end of the science cell uses several cylindrical sections made of different types
of glass in order to gradually transition into a glass-metal seal. The additional glass
required to make the seal adds an extra 20 cm of length to the chamber. The
chosen length of the trap is set by the necessity to accommodate for the glass metal
transition. A top and side view of the trap in conjunction with the leads is shown
in Fig. 3.2. A pair of y-coils whose axes are aligned to the waveguide axis are also
visible. These are to provide optional confinement along the y direction if necessary.
The geometry of the trap consists of 4 visible rods forming a 5×1.5×1.5 cm3 box
region. When viewed from the side, each rod is centered on and lies perpendicular
to the corner of a 1.5 × 1.5 cm2 square, see Fig. 3.3.
The trap structure consists of four 5-cm copper rods (Fig. 3.3). Each rod is
actually a compound structure consisting of an outer copper tube (5 mm in diameter,
1 mm thick wall) containing an alumina insulating tube, itself containing a copper
wire. The alumina tube has 2.4 mm outer diameter with a 0.8 mm wall making
room for a 1.6 mm inner wire. The outermost tube is made out of oxygen-free, highconductivity copper to minimize its electrical resistance. As shown in Fig. 3.4, the
3.2. MAGNETIC TRAP
63
Figure 3.2: Top image is a side view of the magnetic trap structure including leads
that provide current to generate the fields for the waveguide. Bottom image is a top
view of the structure. Eight copper pins (of which five are visible) are connected to
the leads via a custom made feed through mounted on a 43/4” con-flat flange. The
pins connect to the copper leads corresponding to four independent circuits. At the
right end of the structure, the “y” coils which provide confinement along the “y”
direction are visible.
conductors are connected in three independent circuits. A fourth circuit consisting
of a pair of coils is located perpendicular to and on each side of the rods.
One circuit consists of the four outer copper tubes that create a linear quadrupole
field. Two other circuits are made up of oppositely paired inner rods that generate
the bias field. Finally, the fourth circuit connects the two y-coils visible on Fig. 3.2
in an anti-Helmholtz configuration for confinement along y, but it is omitted for
clarity in the figures. This circuit was not used in the work presented here.
Having a total of four independent circuits means there are eight total connections to be made. These connections correspond to eight pins. Five pins are visible
in Fig. 3.2, extruding from the vacuum chamber through the con-flat feed through.
The assignment of circuits to each pin is summarised in Table 3.2.
3.2.2
Superimposing Magnetic Fields
A motivation for the main considerations of the waveguide design used for our
interferometer were discussed at the beginning of this chapter. Two key factors to
take into account are the need for a weakly confining waveguide in order to decrease
the effects of interactions, and the need for a field strong enough to suspend the
atoms against gravity. A weak field provides weak confinement, but if it its too
weak, the atoms will fall. In light of these contradicting requirements, we developed
64
CHAPTER 3. MAGNETIC WAVEGUIDE
Figure 3.3: Mechanical model of the trap drawn to scale. Four horizontal copper
tubes 50 mm in length and 5 mm in outer diameter with a 1 mm wall provide the
quadrupole field. Inside each tube are alumina insulator tubes which surround an
inner conductor consisting of 1.6 mm diameter copper wire which provide the bias
fields. The tubes, insulator and wire are supported on a pair of boron nitride blocks
which in turn support the copper leads which supply the current to the rods and
wire. The right block is depicted transparently to show the connections.
3.2. MAGNETIC TRAP
65
Figure 3.4: Three of the circuits included in Fig. 3.3 have been separated to demonstrate their spatial configuration and current flow. On each diagram the wires
extending in/out of the page represent the leads which supply the current. Diagrams (a) and (b) represent the thin wires inside the insulators with their respective
currents which provide the bias fields according to Fig. 3.6. Diagram (c) represents
the outer conductor and the corresponding current flow which provides the linear
quadrupole field shown in Fig. 3.5. The loops in the ends help cancel any residual
axial fields.
66
CHAPTER 3. MAGNETIC WAVEGUIDE
a novel solution.
A good candidate for the choice of field to make a waveguide is a quadrupole
field similar to that explained in Sec. 2.4 which possessed symmetry about the axis
in which we seek to guide the atoms (y). This can be achieved by using a linear
quadrupole field having a cross-sectional form just like the one depicted in Fig. 2.3.
However, instead of having cylindrical symmetry about the z axis (like Eq. (2.36)),
it has linear symmetry along y. In other words a (x, z) cross-section of the field at
any point in y looks identical to the field depicted in Fig. 2.3.
Once again we are faced with the loss of condensate atoms via Majorana spin
flips at the zero of the field. But we can implement a rotating bias field to make a
TOP similar to that described in Sec. 2.6. In fact, we use a linear quadrupole field
as described above in conjunction with a bias field that can offset the zero of the
field. We implement a technique similar to the TOP trap but with a modification
to suit our interferometer.
In addition to rotating the bias field, like the case of our TOP to perform evaporative cooling, we vary the linear quadrupole in time. The currents in all the circuits
oscillate with a specific phase relationship (see Eq. (3.4)). This causes the zero of
the field to oscillate in a plane above the atoms. Therefore, they are constantly
attracted to the zero overhead, allowing us to lower the confinement further than
that allowed by a conventional TOP set up.
We use the basic principle of vector superposition to construct a mathematical
model which describes the magnetic waveguide used in the interferometry experiments.
3.3
3.3.1
Generating the Time Averaged Potential
Total Field Approximation
The operation of our trap can be mathematically described by starting with the expression for an oscillating linear quadrupole field. As can be seen from the expression
below, there is no y dependence in the equation. Hence the x, z cross-section of Bq
at every y does not change, a characteristic important in obtaining a waveguide axis
along that direction.
Bq = Bq0 (xi − zk) cos(Ωt)
(3.1)
To create the above field, we use circuit (c) seen in Fig. 3.4, which controls the
current through the four copper tubes of the trap structure generating the field in
Fig. 3.5. Because the tubes which generate the quadrupole are finite in length (i.e.
5 cm), there are fringe effects which introduce y dependence to the field. These
effects, although small, are addressed in Sec. 3.4.1.
In order to offset the zero of the field, we use a bias field similar to that described
in section 2.6. Achieving a bias field at the location of the atoms along the waveguide
3.3. GENERATING THE TIME AVERAGED POTENTIAL
67
Figure 3.5: A cross section looking from the right side of the trap centered on the
waveguide axis, shows the quadrupole field generated from the four copper tubes.
The four copper tubes are shown and highlighted to denote current through them.
The direction of the current flow is shown in Fig. 3.4 (c).
axis requires that we use two of the circuits available in the trap structure. We use
circuit (a) in conjunction with circuit (b) (Fig. 3.4) to generate a field with fixed
magnitude pointing in the −i direction.
Looking from a side perspective (down the y direction from the right in Fig. 3.4)
on circuit (a) and (b), we obtain a view for two pairs of diagonally opposed rods as
seen in Fig. 3.6. We can use the well known result for a concentric magnetic field
produced around a current conducting wire to construct the total bias field [49]. In
Cartesian coordinates, the field due to a single wire carrying current Isw is:
·
¸
µ0 Iw
−(z − z0 )
(x − x0 )
Bsw =
i+
k
(3.2)
2π (x − x0 )2 + (z − z0 )2
(x − x0 )2 + (z − z0 )2
where z0 and x0 establish the position of the wire in the z, x plane and Isw is the
current carried by a single wire and µ0 is the permeability of free space. We can
superpose four fields, each represented by Eq. (3.2), having their respective centers
located at the position of the wires seen in Fig. 3.6. Accordingly, is not hard to see
how the resultant bias field near the origin is produced for circuits (a) and (b).
Using the configuration in Fig. 3.6 for equal magnitude currents through the
wires of circuits (a) and (b), we obtain two perpendicular bias fields. One points in
the k − i direction for circuit (a) and another in the −k − i direction for circuit (b).
By sinusoidally oscillating the current for each bias field and having a π/2 phase
shift between them, the direction of the total bias field will rotate in time causing
the zero of Bq to rotate as depicted on Fig. 2.8.
68
CHAPTER 3. MAGNETIC WAVEGUIDE
Figure 3.6: A cross section looking from the right side of the trap centered on the
waveguide axis, shows the two bias fields generated from the copper wires. For
each field, the corresponding pairs of wires carrying currents are highlighted. The
direction of the current flow is shown in Fig. 3.4 a, b.
In general, with a particular choice of currents, adding the two fields produced
by circuits (a) and (b), yields the bias field1 :
B0 = B0 [sin(Ωt)i + cos(Ωt)k]
(3.3)
As discussed in Sec. 2.6, the rate at which we oscillate the fields must be faster
than the atomic motional frequency (10 Hz) but slower than the Larmor frequency
of the atoms ωl = µb B/~ ∼ 10 MHz. This is to prevent the atoms from catching up
to the field’s zero (avoiding Majorana flips), but still have their spins adiabatically
follow the orientation of the field. It should be noted that noise oscillations much
faster or slower than the rate of rotation of the field tend to cancel out. For these
reasons the 1-100 kHz range is adequate.
Furthermore, as explained in Reeves’ thesis work [21], a noise spectrum of the
laboratory was analyzed in order to choose the most appropriate frequency of oscillation. The noise data revealed that within the 1 − 100 kHz range, the lowest
average noise was around 10 kHz. Taking into account the avoidance of noise peaks
observed in the data, we choose the rotation frequency of the field to be Ω = 11.88
kHz, see [21].
Next, we calculate the total magnitude of the waveguide field Bw = Bq + B0 and
1
In practice the equation for B0 does not correspond to the correct field obtained at t = 0 having
both bias circuits on with the same current. However Fig. 3.6 shows the correct configuration
obtained at t = 0. Choosing B0 in this way makes the calculation of the average potential of the
waveguide simpler.
3.3. GENERATING THE TIME AVERAGED POTENTIAL
69
obtain:
|Bw | = {B02 + Bq02 (x2 + z 2 ) cos2 (Ωt) + Bq0 B0 [cos(Ωt) sin(Ωt) − cos2 (Ωt)]}1/2
Bq0 2
= B0 {1 +
(x + z 2 ) cos2 (Ωt)
B0
Bq0
+ [cos(Ωt) sin(Ωt) − cos2 (Ωt)]}1/2
(3.4)
B0
square root of the above expression using the expansion
√ Approximating the
1 + ² ≈ 1 + ²/2 − ²2 /8 and preserving up to second order terms, we obtain:
Bq02 2
|Bw | = B0 +
(x + z 2 ) cos2 (Ωt) − Bq0 z cos2 (Ωt)
2B0
¤
1 Bq02 £ 2
−
x cos2 (Ωt) sin2 (Ωt) + z 2 cos4 (Ωt)
8 B0
3.3.2
(3.5)
Calculating the Time Average
To calculate the time average as defined by Eq. (2.43) it is useful to know the
following quantities given below. Note that we have used the same definition θ = Ωt:
hcos2 θiθ = hsin2 θiθ = 1/2
hcos4 θiθ = hsin4 θiθ = 3/8
hsin2 θ cos2 θiθ = 1/8
(3.6)
(3.7)
(3.8)
Using the results presented above, we finally obtain an expression for the timeaveraged magnitude for the field used as the interferometer’s waveguide:
Bq02
1
h|Bw |it = B0 − Bq0 z +
(3x2 + z 2 )
2
16B0
(3.9)
The potential energy including gravity for the atoms in the waveguide becomes
Uw = µb h|Bw |i + M gz
³
µb Bq02
µb ´
(3x2 + z 2 )
= µb B0 + M g − Bq0 z +
2
16 B0
(3.10)
(3.11)
where M is the atomic mass, g is the acceleration due to gravity and µb is the
magnetic moment for 87 Rb in a particular hyperfine state. We want to counteract
the effect gravity has on the potential. By setting Bq0 = 2M g/µb we can cancel
out the second term in the above equation and obtain the following 2-D harmonic
potential
1
(3.12)
Uw = µb B0 + M (ωx2 x2 + ωz2 z 2 )
2
70
CHAPTER 3. MAGNETIC WAVEGUIDE
where we identify the trapping frequencies as
µ
¶1
3M g 2 2
ωx
ωx =
(3.13)
and ωz = √ .
2µb B0
3
Using a bias field with a magnitude of 20 G for 87 Rb atoms trapped in the
| 2, 2i state, the waveguide exhibits trapping frequencies ωx = 2π × 5.3 Hz and
ωz = 2π × 3.0 Hz. Because the linear quadrupole and bias fields are independent
of the y direction, the potential calculated does not have any y dependence. This is
a limitation in the model describing the waveguide because there are imperfections
in the trap structure that yield non-zero contributions in the y direction for Bw .
However, the model presented above gives a clear and intuitive picture of a potential
which confines the atoms weakly in the x and z directions. At the same time, the
potential preferentially allows the atoms to move along the y direction, achieving
the function of a channel or guide for the atoms in this direction, hence the name
waveguide.
3.4
Design Limitations
3.4.1
Curvature Along “y”
Because the rods and tubes of the trap structure are finite in length and they are
connected on their side to conducting leads which supply current, the waveguide
field suffers from variations from the mathematical model calculated in section 3.3.
Taking into account every possible correction would be a daunting task, but we improved our basic model with one that incorporates some variations in the quadrupole
and bias fields. For the quadrupole field, there can be an extra term that takes into
account any gradient field in the j direction as follows
Bq = (axi + cyj − bzk) cos(Ωt)
(3.14)
where c = b − a in order for Bq to satisfy Maxwell’s equation ∇ · Bq = 0. Including
also variations in the bias field parametrized by coefficients α, β and γ, one can
approximate the total field by using the procedure described earlier for calculating
the time average and obtain [21]:
µ
¶
1
3
2
h|Bq |it = B0 − bz +
a + αB0 x2
2
16B0
µ
¶
µ
¶
1 2
1 2
2
+
c + γB0 y +
b + βB0 z 2
(3.15)
4B0
16B0
The associated oscillation frequencies for the corrected trapping potential are recognized by comparing them to a harmonic potential form as
¶¸ 21
·
µ 2
2µb
3a
+ αB0
ωx =
M 16B0
3.4. DESIGN LIMITATIONS
71
·
ωy =
·
ωz =
2µb
M
2µb
M
µ
µ
c2
+ γB0
4B0
¶¸ 21
b2
+ βB0
16B0
¶¸ 21
To this end, we made a numerical calculation of the total field using the BiotSavart law. It calculated the field at every point taking into account the leads of
the trap structure and its finite size. We later fit this model to the form expressed
in Eq. (3.15) and obtained values for the parameters a, b, c, α, β and γ. The values
from the fitted model for each parameter (in terms of the respective currents) are
given in [21].
3.4.2
Trap Characterization
After the production of the condensate in the science cell we load it into the magnetic
waveguide. The loading consists of a series of steps which ramp down the quadrupole
magnetic field which makes up the TOP trap in conjunction with steps which ramp
up the fields which make up the waveguide. In general, transferring the condensate
involves reducing the strength of the external quadrupole field, which reduces the
oscillation frequencies of the TOP mentioned in Sec. 2.6. Before the confinement of
the TOP field is totally lost, the linear quadrupole field of the waveguide is turned
up to obtain the weak trapping frequencies in Eq. (3.13). A detailed description of
the the ramps involved in the loading can be seen in Fig. 3.7.
In order to minimize residual oscillations after the loading (which are detrimental
to the interferometer operation (Sec. 5.6)) we try to make the ramps slow so that
the condensate follows the changing magnetic fields adiabatically.
To characterize the waveguide, we measured its frequencies of oscillation along
x y and z. In the x and z directions we observed and measured the oscillation for
the center of mass of the condensate. We used absorption imaging to track the
position of the cloud, and then plotted it in time. In the y direction we observed
and measured oscillations of the semi-major axis of the condensate. In other words
we measured the breathing mode of the condensate. Again, we used absorption
imaging to track the size of the cloud along y as a function of time. We later made
a plot of the position y vs. time.
For all cases x, y and z, we fit the data to a damped sine function and extracted
the oscillation frequencies. More details and the results obtained for the frequencies
are given in [21].
3.4.3
Trap Oscillations
We initially recorded oscillations of up to 200 µm in amplitude of the condensate
upon loading it to the magnetic waveguide. We noted oscillations in the x, y and z
72
CHAPTER 3. MAGNETIC WAVEGUIDE
Figure 3.7: Ramps for different magnetic fields during the loading of the waveguide.
The vertical axis indicates the control voltage for the labeled magnetic field, see
appendix C. (a) B0 the bias field. (b) Bqw the waveguide quadrupole. (c) Bq the
external spherical quadrupole. (d) Initial ramp is not to scale, actual time is 10 s.
The last evaporation occurs after this ramp. Evaporation takes 9 ms and is not
shown. (e) Exponential ramp is not to scale. Ramp takes an approximate of 4s to
complete with an exponential decay constant of 800 ms.
3.5. MEASURING THE MAGNETIC FIELD
73
directions. Initially we thought these were a product of vibrations of the mechanical
trap structure described in Fig. 3.4. To investigate this possibility we watched the
structure with a camera and monitored its motion with an optical interferometer
which used a mirror that was mounted on the structure. The oscillation amounted
to less than 1 µm in amplitude (∼ 10× smaller than the condensate itself), thus
we determined that motion from the trap structure was not responsible for the
condensate oscillation in the waveguide. In the end we fixed the oscillation problem
by optimizing the loading sequence (Fig. 3.7).
It is important to note that we initially also observed a large displacement of
the condensate along the y direction upon its loading to the waveguide [21]. The
condensate moved approximately 2 mm in the process of ramping down the quadrupole coils. This was caused by a misalignment in the minima of the TOP trap and
the waveguide. However we re-adjusted the position of the quadrupole coils at their
final position in the science cell, and realigned the minima. This solved the problem
of the condensate displacement during loading.
Additionally, we observed (by taking pictures of the condensate at rest in the
waveguide) that the minimum of the waveguide varied after some time. Initially we
thought that the shift was due to thermal effects in the waveguide structure. However we tracked down this effect and concluded it was a shift in the tilt of the optical
table which shifted the rest position of the condensate. We realized the pneumatic
system responsible for floating the table was under-pressured, allowing random shifts
in position. We solved this problem by increasing the system’s pressure.
It is very important to consider any motion of the condensate when loading it
to the waveguide. Before beginning the interferometer operations, it is beneficial
for the condensate to be at rest. Having a condensate with v = 0 m/s means the
subsequent splitting reflection and recombining operations explained in Chapter 4,
will work more effectively, making interference measurements possible.
3.5
3.5.1
Measuring the Magnetic Field
Connections and Field Directions
To properly implement the circuits available in the magnetic trap structure shown
in Fig. 3.3, it is important to know which current to apply and how to make the
necessary connections. As shown in Fig. 3.2, the trap structure is supported by
copper leads which in turn are connected to a series of pins. These pins extrude to
the outside of the evacuated chamber using a con-flat feedthrough. The pins are
grouped in four pairs corresponding to each circuit available to the trap structure.
This section documents in detail how the trap structure circuits are configured to
each pin and what fields are obtained given a particular choice of currents. Figure 3.8
maps out the pins to their respective trap circuit. Table 3.1 briefly describes the
direction of characteristic features that each individual field exhibits. Additionally,
74
CHAPTER 3. MAGNETIC WAVEGUIDE
Figure 3.8: A bakeable feed-through is mounted on a 4 − 3/4” con-flat flange connecting eight copper pins to the trap leads. See Table 3.2 to match up corresponding
circuits. The arrow indicates the upright direction for the trap structure. The trap
is mounted with one of the leads (which is neutral) shown in Fig. 3.2 facing up.
to properly connect the magnetic trap and obtain the desired waveguide fields, the
polarity of the pin connectors and the corresponding field direction are summarized
in Table 3.2.
3.5.2
Field Gradient & Magnitude
It is important to document in detail the calibration of the waveguide’s various
magnetic field strengths to the current applied on each of the trap’s four circuits.
This will enable us to properly configure the control voltages, function generators
and amplifiers used to drive current through the connector pins in Fig. 3.8.
To summarize the operation of the trap in dc mode (using dc currents), we describe key parameters of the magnetic fields that are obtained using the settings
described in Table 3.2. We describe in table Table 3.3, the calibration of the gradients and magnitude of the fields with respect to the current applied. It is important
to mention that this calibration was done by running the trap’s circuits at 25 A
dc. A small probe was used to measure the three different directions x, y and z for
four different fields. It was adapted correctly in conjunction with a 3-dimensional
translator to properly measure the magnetic fields.
While the trap structure was not under vacuum, it was held in a fixed position
using an aluminum stand which in turn was bolted to an optics table. The tip of the
3.5. MEASURING THE MAGNETIC FIELD
75
Type of field
Formula
Quadrupole Field
~ q = B10 [−xi + zk]
B
~1 =
B
Bias Field 1
~2 =
B
Bias Field 2
End caps Field
B1
√
[k
2
B2
√
[−i
2
− i]
− k]
~ endc = α(z)k + β(x)i + (α + β)yj
B
Table 3.1: The trap structure contains four independent circuits which provide four
fields described above. This table describes the name of the fields available and their
corresponding mathematical formula
pin
1
8
2
4
6
5
7
3
polarity
+
+
+
+
-
description
k−i
Field
Bias 1
−k − i
Bias 2
−xi + zk
Quadrupole
i+k
Endcap Coils
at y = 0 plane
Table 3.2: Each circuit available in the waveguide has a corresponding pair of pins.
The table above shows the characteristic feature of the field obtained for each circuit
given the polarity applied to each corresponding pair of pins .
76
CHAPTER 3. MAGNETIC WAVEGUIDE
formula
~q =
B
Bi0 (x
− x0q )i +
Bii0 (z
− z0q )k
~ 1 = B1i i + B1ii k
B
~ 2 = B2i i + B2ii k
B
~ endc = α(z − z0 )k + β(x − x0 )i + (α + β)(y − y0 )j
B
parameter
measured
value
Bi0
Bii0
x0q
z0q
B1i
B1ii
B2i
B2ii
α
β
x0
y0
z0
-1.82 G/mm
1.79 G/mm
-2.78 mm
2.14 mm
-6.84 G
7.07 G
-7.08 G
-7.62 G
0.19 G/mm
0.29 G/mm
3.60 mm
2.64 mm
5.55 mm
Table 3.3: The equation for each field available on the trap structure is given corresponding to the polarities in Table 3.2. Additionally, experimentally measured
field parameters are provided. All measurements were taken by applying a 25 A dc
current. The coordinate system used to describe fields is noted in Fig. 3.1.
probe was positioned in the center of the waveguide structure denoting the origin, see
Fig. 3.1. Its starting location (the origin of the coordinate system used to measure
the fields), was carefully recorded by positioning the point of a needle right next to
the tip of the probe. Initially, the probe’s Hall sensor was positioned horizontal to
the table to measure the z component of each field. Then, using the translator, the
probe was scanned throughout the volume enclosed by the rods of the trap. Data
points were taken at 1 mm intervals with the aid of a computer for recording the
data. After this was done, the probe was rotated and re-centered, enabling it to
measure another component of the field. This was done for the remaining i and j
components.
3.5.3
End cap coils
At the moment we do not make use of the end cap coils found on either side of the
waveguide rods which provide axial confinement along the y axis. Originally, we
planned to use them to provide the force required to turn around the condensate
atoms during interferometry, but we have opted to not use a magnetic field to turn
around the condensates.
Instead, we use off-resonant Bragg scattering in a series of pulses to control the
3.5. MEASURING THE MAGNETIC FIELD
77
motion of the condensate in the wave guide. Using the off-resonant beam has proven
to be successful in implementing the interferometer. In the future the end-cap coils
could be used to perform experiments where the atoms turn around due to the
curvature along the guide axis as an alternative to the reflection pulse described in
Sec. 4.3.
As shown in the previous sections, the waveguide potential does posses curvature along y due to the finite size of the trap’s copper rods. This curvature could
be asymmetric in the region where the atoms travel during the interferometer sequence. Therefore, it can add unwanted relative phases between the arms of the
interferometer.
Hypothetically, a set of external coils could be used to create an opposing field
which cancels out unwanted fields. Moreover, if you have independent control of the
coils, there would be flexibility in the type of cancelling field you could generate.
Applying different currents to different coils would yield different strength fields on
each side of the waveguide.
3.5.4
Preparing for Interferometry
At this point it is important to remember that one needs to consider the alignment
between the minimum of the TOP trap provided by the spherical quadrupole (which
determines the atoms’ position after evaporation), and the potential minimum of
the waveguide. A misalignment can cause unwanted oscillations for the condensate
after loading the waveguide.
Having curvature along the y axis means that there could be a misalignment
between the minimum in the TOP and the minimum along y of the waveguide. We
cannot do much about this problem, however we move the final y position in the
science cell of the spherical quadrupole field in order to align minima.
Similarly, there could be a mismatch in the minimum of the TOP and waveguide
along the x direction. But we can move the atoms further (or less) down the track
along x until the minima are matched along x.
Finally, there could be a mismatch in the minima along z. However to our
advantage we can control the minimum of the waveguide along the z by varying the
strength of B0q . In this way, we can try to better match the minima along z and
reduce unwanted oscillations.
In addition to the techniques mentioned above, we developed a slow set of ramps
(including a final exponential ramp) to“ease” the condensate into the waveguide,
reducing oscillations in the waveguide after the loading process (Fig. 3.7).
We will analyze in detail the interferometer operations and their experimental
verification in Ch. 4. But at this stage it is important to mention that loading the
condensate into the waveguide proves to be a crucial step in our experiment.
The effectiveness of subsequent steps like the splitting, reflection and recombination of the condensate heavily depend on any residual motion after the loading of
78
CHAPTER 3. MAGNETIC WAVEGUIDE
the waveguide.
We found out that as we relaxed the TOP trap by reducing the gradient of the
spherical quadrupole field, its trapping frequency crossed the 60 Hz value. There are
many sources of stray noise at this particular frequency in the laboratory, hence they
induced the condensate to oscillate after we crossed this frequency. To solve this
problem, we finished the evaporation sequence such that the bias field for the TOP
trap ended at a higher value. Because ωi2 ∼ Bi02 /B0 (the i denotes the different
spatial coordinates) for the TOP, we chose a final B0 that would make ω lower than
60 Hz. In his way after we performed the final ramp down of the TOP quadrupole
gradient during the loading, the atoms would never experience a potential with a
frequency of 60 Hz. Consequently the atoms’ motion would not resonate with the
background noise at ∼ 60 Hz.
By performing the evaporation as described above, we aim to achieve as little
residual motion as possible after the atoms are first loaded into the waveguide.
Chapter 4
Interferometry Techniques
Atom interferometers are amazingly sensitive devices. In general, atoms have stronger
interactions with forces available in the laboratory setting when compared to photons. For example, atoms readily interact with electromagnetic and gravitational
fields, whereas light does not. For this reason, interferometers using atoms as a wave
source have the potential to substantially outperform those using light. In particular,
atom interferometers used for precision measurements would have a much improved
sensitivity.
For example, interferometers can be configured to perform gyroscopic measurements. Using a Sagnac configuration which encloses an area A and rotates with
angular velocity Ω, the change in phase is given by:
4π
A· Ω
(4.1)
λv
in which λ is the wavelength of the photon or atom and v the velocity of the particle
being used. For an atom travelling at velocity v the wavelength is given by the
de-Broglie relation λa = h/mv. In the case of photons travelling at c = 3 × 108 m/s,
the wavelength is given by λl = 2πc/ω. By substituting the corresponding values of
wavelength for visible light, mass for a 87 Rb atom and the velocity for atoms and
light into Eq. (4.1), the ratio of sensitivities in each case yields:
∆φ =
∆φa
mc2
=
≈ 1011 .
∆φl
~ω
(4.2)
At best, this would represent the largest theoretical gain in sensitivity for an atom
based interferometer. In practice, achieving such an increased gain in sensitivity will
be a difficult task. Among the best examples of experiments which use the increased
sensitivity of atoms is the Sagnac atom interferometer developed by T.L. Gustavson
et al. [50]. It has achieved rotation sensitivities of 2 × 10−8 (rad/s)/Hz, equivalent
to those obtained by the best laser gyroscopes. Therefore improving sensitivities
in atom interferometers promises significant enhancements in these sensory devices.
Other examples of interferometry using thermal atoms are [25, 24, 23].
79
80
CHAPTER 4. INTERFEROMETRY TECHNIQUES
Figure 4.1: Atom beam interferometers cannot achieve large deflection angles due
to the high velocity of the beam, typically atoms move at 290 m/s. This diagram
depicts the small arm separation a = 54 µm obtained. Our design can improve this
limitation increasing a to a = 250 µm.
Nevertheless, interferometers using light do pose certain key advantages over
their atomic counterparts. In general the experimental manipulation of photons is
more simple than that of atoms. For example the deflection and positioning of light
beams is easily achieved by using simple optics, typically mirrors and lenses. In fact,
the use of optics permits light interferometers to achieve arbitrary deflection angles
which permit the enclosure of large areas and thus become useful in cases like that
of Eq. (4.1). Similarly, large deflection angles can allow large arm separation, which
can then allow individual access to the arms.
In contrast the interferometer implemented by Gustavson et al. uses an atom
beam with a velocity of 290 m/s. Due to the high velocity of the beam, it becomes
hard to control the direction and overall motion of the atoms, which makes it hard to
achieve large deflection angles. This limits the area that can be enclosed, hindering
the application of a Sagnac sensing interferometer. Arm separations up to 54 µm
(with 17 µm width beams) have been demonstrated in [51]. Up to date, the largest
packet separations have been achieved in experiments like [52, 53], obtaining up to
13 µm using optical traps and magnetic traps in [54].
Equally important, interferometers using light have a supply of photons which is
orders of magnitude larger than most atomic sources. Hence the rate at which you
can produce an output signal is much higher. This significantly improves the signal
to noise ratio which then increases accuracy in the measurements of the phase shift.
In our experiment we chose to use Bose-Einstein condensates as a coherent source
because it resolved some limitations inherent to atomic beam interferometers, but
still provided the advantage of using highly sensitive atoms. In light of this, it is
important to highlight some of the main advantages BEC interferometers have over
interferometers using thermal atoms.
In the laboratory, condensates are usually produced at temperatures close to
absolute zero. As a result, condensates can be thought of as stationary when com-
81
pared to thermal atoms having velocities in the order of 100 m/s. For this reason,
manipulating the motion of the condensate requires very small forces. Similarly,
this allows for large deflection angles which are useful as mentioned earlier. This is
not the case for thermal beams.
Another key advantage is that condensates are sources of highly coherent matter
waves. As discussed in chapter 1, condensates are spatially coherent across their
entire length, meaning typical coherence lengths are ∼ 100 µm. When performing
an interferometer measurement, an interference pattern will only be seen if the difference in path length is equal or less than the coherence length of the condensate.
On the contrary the coherence length for a beam of atoms is given by the de Broglie
wavelength. For a beam of Cesium atoms travelling at 290 m/s the de Broglie wavelength is tens of pm which is much smaller than the coherence length of condensates.
For this reason, the difference in path lengths allowing measurable interference will
be greater for condensed atoms.
From another perspective, all the atoms in a condensate are in the same state of
the potential trapping them. This makes them analogous to a laser whose photons
are all all in the same state. In contrast, a high velocity beam of atoms has a
wide velocity spread, meaning that the atoms are in multiple translational states,
making them analogous to a beam of white light. Consequently, condensates will
experience longer coherence lengths in comparison to atomic beams, much like the
longer coherence lengths of lasers compared to those of white light.
Furthermore, interferometers using thermal beams of atoms obtain their arm
separation by inducing transitions to the internal states of the atom. Usually when
measuring the output state, the populations of different internal states are recorded.
In turn, these depend on the relative phase of the arms. For this type of splitting, the
relative phase of the arms is sensitive to the phase of the laser. This will introduce
decohering effects to the output state if there is any noise in the laser.
Considering our particular experiment, the low velocities of a condensate greatly
facilitate the implementation of the Michelson type configuration we use. Additionally, low velocities can be exploited to obtain large wave packet separation and
hence a larger area to enclose. The larger area advantage should further motivate
the study of BEC in its application to interferometry. Other examples of BEC in
interferometry are given in [22, 55, 44, 56, 53, 52, 57, 12].
Starting with a condensate at rest enables us to easily split it in two and obtain
clouds moving apart in one dimension. Moreover, we have been able to fully separate
the wave packets from each other such that each can be observed as a separate entity.
With this in mind, we will refer to the distance which separates the center of each
individual atom cloud as the arm separation. In this way the arm separation is
directly related to the actual wave-packet separation. It should be noted that in
most atomic beam experiments the arm separation is attributed to the separation
of the center line of each beam (therefore, the arm separation can be smaller than
the resulting beam widths). However in many cases, the matter waves(beams) never
82
CHAPTER 4. INTERFEROMETRY TECHNIQUES
fully separate from each other; therefore the packets do not fully separate.
As a result of these advantages, we have been able to obtain macroscopic arm
separations which have great potential in many applications where access to a single
arm of the interferometer is essential. Current atom interferometers have limited
individual arm accessibility, hindering experiments where large arm separations are
necessary, see Fig. 4.1. With this purpose in mind, we seek to implement an interferometer that has atomic sensitivity with a motion control that permits large arm
separation.
4.1
Interferometer Operation
Our main objective is to create a one dimensional interferometer where the condensate moves along the axis of the waveguide described in Sec. 3.3. The magnetic
wave guide is an analog of an optical fiber confining photons in two dimension.
The operation of the interferometer is described as follows. First, after the
condensate is loaded into the waveguide at y = 0, we apply an off-resonant standing
light field with vector k to split the atoms into two clouds, giving them an initial
symmetric momentum kick in the y or −y direction of the waveguide. This puts the
condensate in a quantum superposition of two translational states corresponding to
the momenta ±2~k. The atoms propagate for some time, each covering a maximum
spatial amplitude y = ±d. We then apply a second off-resonant standing wave to
reverse the motion of each wave packet. We let the packets propagate crossing each
other at y = 0. When they reach the maximum separation distance y = 2d, the off
resonant pulse is applied a third time to reverse the motion once more. Finally, when
the wave packets overlap for the second time at y = 0, the off resonant standing
wave is used to bring the atoms back to rest. See Fig. 4.4 and 4.6 for illustrations
of these operations. The physics of how all these processes take place is explained
in Sec. 4.2 and Sec. 4.3.
To create the off-resonant standing wave we use a second diode laser beam,
that we refer to as the Bragg beam. The Bragg beam is detuned −7.8 GHz from
resonance and enters the science cell from the −y direction through the magnetic
waveguide structure along the waveguide axis (the y-axis). After the Bragg beam
exits the waveguide and science cell, a mirror retro-reflects it to yield a pair of
counter-propagating beams which generate the standing wave used for splitting and
reflecting the condensate.
Figure (4.2) shows a close up side perspective of the magnetic trap as viewed from
the outside of the science cell. This figure includes the Bragg beam’s orientation
and position relative to the science cell and waveguide structure. The mirror used
to generate the standing wave is also seen. In particular, Fig. E.3 in appendix E,
shows in detail how the Bragg beam is routed on the optics table to finally reach
the configuration shown in Figure 4.2.
4.1. INTERFEROMETER OPERATION
83
Figure 4.2: Side perspective of the waveguide structure as viewed from the outside
of the science cell. (a) The Bragg beam enters the chamber from the y direction
centered on the waveguide axis. (b) The beam traverses the entire waveguide region
and exits the science cell. (c) The Bragg beam is retro-reflected by a mirror outside
the chamber. By having a pair of counter-propagating beams, a standing wave is
created which is used to split and reflect the condensate during the interferometry
experiment.
Figure 4.3: Zoom into the science cell, looking head on to the waveguide region or
“interaction region”. The Bragg beam can be seen traversing along the waveguide
axis (y-axis) setting up a standing wave. The two arrows illustrate the direction
of the Bragg beam and the red sphere represents the condensate atoms suspended
against gravity in the center of the waveguide.
84
CHAPTER 4. INTERFEROMETRY TECHNIQUES
Figure 4.4: Diagram illustrating the wave-packet trajectory along the y-axis. The
solid ovals represent the condensate before the split and reflect operation. The
dashed ovals represent the condensate some time after each of these operations.
The different letters correspond to different times (a) Two red arrows represent the
standing wave used to split the atoms, analogous to an optical beam splitter. A box
at 1 has been drawn to represent the location of the corresponding beam splitter. (b)
The reflection pulse represented by the outermost arrows at 2, is analogous to mirrors
in an optical interferometer. (c) The recombination pulse is identical to the split.
Accordingly, this configuration is a one-dimensional equivalent of the Michelson
interferometer where both packets travel the same path in Fig. 4.5. Imaging camera
at 3 takes pictures of the output, analogous to the photo-detector.
4.1. INTERFEROMETER OPERATION
85
Figure 4.5: Michelson Interferometer. (a) Input beam is divided with beam-splitter
1. (b) First arm of the interferometer is directed to mirror 2 and reflected back to
splitter 1. (c) second arm of the interferometer is directed to mirror 3 and back to
splitter 1. (d) The two arms are recombined at splitter 1 and the output redirected
to the detector 4.
As shown in Fig. 4.4, we implement a one-dimensional interferometer in which
the wave packets travel the same path twice, hence a version of the Michelson
configuration. This particular set up suits the one dimensional magnetic waveguide
obtained from the trap structure and is relatively simple to implement in comparison
to multidimensional configurations. Granted that, having only one dimension means
we are not able to enclose an area with the arms, which makes it impossible for
us to make gyroscopic measurements like [50]. But our first aim is to test the
coherence of our wave packets and achieve an output state which demonstrates
that the condensate is interfering. To this end we will ensure no phase difference
is acquired by the packets during their propagation. However, we will change the
position of the light field when recombining the wave-packets in order to change the
output phase. Additionally we seek to obtain a fully separated pair of wave-packets
at their point of maximum travel (from the center) in the waveguide.
Also, the double reflection technique causes the packets to traverse identical
paths therefore both packets will experience the same potential cancelling any relative phase shift between them. This means our configuration is insensitive to the
gravitational potential, so we cannot make precision measurements of this type.
However, in the future, extending the current trap structure along its axis to form a
loop would yield a 2-D waveguide suitable for Sagnac experiments. The waveguide
potential was initially designed with this conceptual extension in mind.
86
CHAPTER 4. INTERFEROMETRY TECHNIQUES
Figure 4.6: Position vs. time graph summarising path and pulse sequence of the
interferometer. The left black oval represents the condensate starting position and
the black lines its trajectory . First pair of red arrows represents splitting pulse
applied at t = 0, the middle two arrow pairs represent reflection pulses at t = T /4
and t = 3T /4 respectively. A final recombination pulse at t = T is used to produce
the output state. Output states can be: atoms brought to rest, atoms moving at
±2~k or a linear combination of atoms at rest and moving. The latter state is
illustrated by the three black ovals at the end of the condensate trajectory.
4.2. SPLITTING THE MATTER WAVE
4.2
87
Splitting the Matter Wave
The first step in the implementation of the interferometer sequence is splitting the
condensate atoms at rest into two wave packets travelling away from each other.
Next we reflect twice and at the end of the interferometer sequence, the splitting
pulse is used to recombine the atoms and generate the output state. The following
sections will cover the physics required to understand and model the origin of the
splitting, reflection and the dynamics of the wave packets during the interferometer
sequence.
4.2.1
Two-Level Approximation
Applying a pair of counter-propagating beams along the y−axis through the waveguide
results in the condensate atoms experiencing a standing wave pattern that sets up
a periodic potential as will be explained below.
Because the condensate is coherent, we treat it as a quantum wave packet that
follows the Schrödinger equation in the presence of an electric field E(t). In this
system, the Hamiltonian consists of two parts, the time independent part Ĥ0 and
the time dependent component Ĥ 0 such that Ĥ = Ĥ0 + Ĥ 0 . The time evolution of
this problem can be obtained by relating the translational states of the condensate
to a two-level system.
First, it is essential to understand how the laser light interacts with the internal
states of the atom. Understanding this interaction will allow us to find out how the
laser light shifts the energy levels of the atoms. We can then use the generic form
of the two level exact solution to model the translational states of the condensate
in the standing wave.
We begin by considering the internal states of the atoms. In this context, we
demonstrate why it is appropriate to use the two level approach of a ground and
excited state only. Initially we consider a laser beam modeled by using the potential
of a travelling electromagnetic plane wave, and assign it to H 0 . We start by writing
the Schrödinger equation for the valence electron to obtain a general solution.
i~
∂| ψ(t)i
= Ĥ| ψ(t)i
∂t
(4.3)
The wave function can be expanded in terms of the eigen-state vectors with their
corresponding time dependent coefficients and energy phases. In this situation it will
be convenient to explicitly include the energy phases in the expansion of the wave
function. The eigen-states are represented by their respective quantum number n
denoting the energy state
| ψ(t)i =
X
n
cn (t) e−i ωn t | ni
(4.4)
88
CHAPTER 4. INTERFEROMETRY TECHNIQUES
here hr|ni = ψn (r) and coordinate r denotes the electron position. Plugging in the
above equation into (4.3) and multiplying both sides by another wave vector h j|,
we obtain the following expression:
X
X
∂ X
i~
cn hj|nie−i ωn t =
cn hj|Ĥ0 |nie−i ωn t +
cn hj|Ĥ 0 |nie−i ωn t
(4.5)
∂t n
n
n
In order to simplify the calculation we assume that we know the stationary states
of the system such that Ĥ0 | ni = En | ni. Using the ortho-normality relationship
hj|ni = δjn , we can collapse the sums of the first two terms such that:
i~
X
∂
0
cj e−i ωj t = Ej cj e−i ωj t +
Hjn
cn e−i ωn t
∂t
n
(4.6)
0
≡ hj|Ĥ 0 |ni. We carry out the time derivative on the left hand side
Where Hjn
using the product rule. A factor of ωj will be pulled down in one of the resulting
terms, which when multiplied by ~, yields the energy Ej . The term containing Ej ,
then cancels with the first term on the right hand side to give:
X
∂
0
i~ cj =
Hjn
cn ei(ωj −ωn )t
(4.7)
∂t
n
We now define the meaning of the subscript n to consider the internal states
of an atom. Normally Eq. (4.7) would yield an infinite set of coupled differential
equations for the cn ’s, but using an approximation studied by Rabi [58], only two
internal energy levels of the atom are considered (n = 1, 2), truncating the infinite
set. This is done because the probability to populate subsequently higher or lower
energy states is small. Consequently, we only consider the ground and a particular
excited state of the atom coupled via a laser whose frequency ω0 is tuned near the
transition energy between them. In order to justify this approach, we can consider
the rate of driving a transition between the ground state and states separated by
energies much greater or less than ~ω0 .
Specifically, we can consider the coupling during the splitting and reflecting
pulses. We will use a laser frequency detuned 7.8 GHz from the transition between
the 5S1/2 ground, and the 5P3/2 excited state. The closest available transition for a
laser tuned to the D2 line is the D1 line, which we will show has a low transition
rate.
To calculate the transition rate for the D1 line, we can use time dependent perturbation theory on the exact result of Eq. (4.7). The time dependent perturbation
will be given by the following interaction [35, 59, 60]:
H 0 = −eE · r
(4.8)
The above result is not trivial to obtain but can be thought of by considering the
electric potential V (r), such that UE = eV (r). To obtain the potential energy UE ,
4.2. SPLITTING THE MATTER WAVE
89
the electric field E generating the potential must satisfy −∇V (r) = E(r). Noting
that ∇(E · r) = E (for dipole interaction), then we can obtain Eq. (4.8) as our
interaction potential.
Here we assume a simple travelling plane wave form for the electric field and
write it in complex notation as E = k E0 ei(ky−ωl t) . Here, the wavelength of the
light used in the experiment is much larger than that of the wavefunction of the
electron in use. For this reason we can neglect the spatial and directional variation
of the E field relative to the coordinate of the atom (E → k E0 e−iωl t ). This is called
the dipole approximation. As a result, the matrix elements of the time dependent
perturbation can be written as:
0
Hjn
= −~
eE0
hj|z|nie−iωl t
~
(4.9)
We define Ωjn ≡ eE~ 0 hj|z|ni to be the Rabi frequency. Figuring out the integrals in
the Rabi frequency can be a difficult task in many instances, but there are alternate methods to obtain this number. Once we obtain the final expression for the
transition rate, we will plug in values for this important parameter.
0
For now we will focus on the time dependent aspect of Hjn
and relabel it as
0
0 −iωl t
0
Hjn → Hjn e
where Hjn /~ is just the Rabi frequency for the transition between
j and n. This allows us to more clearly obtain the equation for the transition rate.
Equation (4.7) can be expressed in terms of matrix operators and vectors. In it,
0
we include the new labeling of Hjn
which has the oscillation frequency of the laser
ωl and introduce ωjn ≡ ωj − ωn .

 


0 −iωl t
0 i(ω12 −ωl )t
c1
H11
e
H12
e
.
c1
∂
0 i(ω21 −ωl )t
0 −iωl t
(4.10)
i~  c2  =  H21
e
H22
e
.   c2 
∂t
.
.
.
.
.
In principle we would have to solve an infinite set of equations, but a perturbative
approach can be taken. First we use the initial state of the system where c1 = 1,
cj6=1 = 0 at t = 0 and we assume the vector
 
1
 0 
(4.11)
.
to be the zero order solution to the system. Making this assumption means that we
plug in this vector into the right hand side of Eq. (4.10) and solve all the resulting
equations for cj6=1 by integrating. Since j can be any integer up to infinity, and can
be considered the final state where the atoms end up, we can label it by the index
“f” to denote any generic final state. In general we obtain:
Z
−i t 0 i(ωf 1 −ωl )t0 0
dt
H e
cf =
~ 0 f1
90
CHAPTER 4. INTERFEROMETRY TECHNIQUES
−i Hf0 1 i(ωf 1 −ωl )t
(e
− 1)
~ iωf 1
µ
¶
−i i(ωf 1 −ωl )t/2 Hf0 1 sin (t(ωf 1 − ωl )/2)
=
e
~
(ωf 1 − ωl )/2
=
(4.12)
Above, we have rewritten the exponential in terms of a sine function. To obtain
second and higher order corrections, Eq. (4.12) in conjunction with c1 = 1 is plugged
into the right hand side of Eq. (4.10) to generate another infinite set of equations
that can later be integrated.
The first order transition probability is found by taking the magnitude of cf :
1 sin (t(ωf 1 − ωl )/2)2 0 2
|cf | = 2
|Hf 1 |
~ ((ωf 1 − ωl )/2)2
2
(4.13)
For long times t → ∞, the average rate can be calculated by dividing the transition
probability by the time such that:
|cf |2
t→∞
t
R1→f = lim
(4.14)
At this point, we can use a mathematical identity which is an alternate definition
of the delta function
sin [t(ωf 1 − ωl )/2]2
= πδ((ωf 1 − ωl )/2)
t→∞ t[(ωf 1 − ωl )/2]2
lim
(4.15)
and replace it into Eq. (4.14). In the limit where t → ∞, we obtain Fermi’s Golden
rule for the transition rates between states.
π
R1→f = 2 δ [(ωf 1 − ωl )/2] |hf |Ĥ 0 |1i|2
(4.16)
~
We can use this formula to calculate the transition rate of the D1 line of 87 Rb, but
some modifications need to be made. In Eq. (4.16), the delta function ensures energy
conservation, so only the transitions whose energy difference match ~ωf 1 , are driven.
This means that the transition in consideration has an infinitely narrow linewidth.
Experimentally we observe that atomic transitions have a finite line width associated
with them. In particular the D1 line has a linewidth of ΓD1 = 2π × 5.75 MHz, [26].
To incorporate the idea of a finite linewidth, we substitute the delta function for a
Lorentzian in terms of the detuning ∆ ≡ ωf 1 − ωl , which is the difference between
the transition and laser frequencies. Therefore:
δ(∆) →
Γ2
A
+ 4∆2
This function should be normalized over all ∆ such that:
Z ∞
A
d∆ = 1
2
2
−∞ Γ + 4∆
(4.17)
(4.18)
4.2. SPLITTING THE MATTER WAVE
91
which constrains the value of A to 2Γ/π. Finally, we arrive at an expression which
appropriately describes the transition rate for the atomic states of 87 Rb.
µ ¶
2Γ
1
R1→f =
|hf |Ĥ 0 |1i|2 .
(4.19)
2
2
~
Γ + 4∆2
Using a detuning of ∆ = 2π × 6.890 THz (the detuning of the D1 line to the laser
frequency used for the standing wave), an intensity I ∼ 5Isat , ΓD1 = 4.484 mW/cm2
[26] and substituting Hf21 /~2 = Ω2f 1 = Ω2D1 = Γ2D1 I/2Isat [35] into Eq. (4.19), the D1
line will have a transition probability RD1 = 63 × 10−6 s−1 which is small compared
to the lenght of our typical pulses (see Sec. 4.2.5). In other words this transition is
unlikely to occur.
4.2.2
The Two Level Solution
We can proceed to calculate the solution to the two level Rabi problem. To do so,
we will take a slightly different approach in writting the wavefunction as compared
to that in the previous section. Going back to Eq. (4.4), we can expand it with a
different version of ψ,
X
| ψ(t)i =
cn (t) | ni
(4.20)
n
where the time dependent phase has been absorbed into the cn coefficients. When
plugged back into Eq. (4.3) this yields the following expression:
i~
X
∂
0
cj = Ej cj +
Hjn
cn
∂t
n
(4.21)
In the equation above, the two energy levels considered will labeled such that 1 → g
for ground and 2 → e for excited state. To obtain the full Hamiltonian of the system
we use the interaction described by Eq. (4.8). The same dipole approximation is
used, but instead of using the complex representation for the E field we use a real
wave form such that E = k E0 cos (ky − ωt).
0
Hjn
= −~
eE0
hj|z|ni cos (ωt)
~
(4.22)
To simplify the notation, we can introduce the definition of the Rabi frequency,
Ωjn ≡ eE~ 0 hj|z|ni and notice that because of odd parity, the spatial integral hj|z|ji
0
= 0. Because the energy scale
vanishes, making the diagonal matrix elements of Hjn
is defined up to a constant, for the stationary eigenenergies, we choose the value of
Eg = 0 and Ee = ~ω0 . Combining all these ideas together we arrive at the following
set of equations.
¸
¸ ·
¸·
·
∂ cg
0
~Ω cos (ωt)
cg
.
(4.23)
=
i~
ce
~Ω cos (ωt)
~ω0
∂t ce
92
CHAPTER 4. INTERFEROMETRY TECHNIQUES
This translates to the following pair of coupled differential equations for the c0n s
i ċg = Ω cos (ωt) ce
i ċe = Ω cos (ωt) cg + ω0 ce
(4.24)
(4.25)
A mathematical trick can be performed to facilitate solving these equations. One
can map one of the coefficients, namely ce , by performing a unitary transformation
causing the time dependence to be eliminated. We call the newly transformed
coefficient d(t) and define it as follows:
ce (t) ≡ d(t) e−iωt
(4.26)
Using the above definition, we can re-write the differential equations in terms of d
to obtain
ċg = −iΩ cos (ωt)d e−iωt
d˙ = i(ω − ω0 )d − iΩ cos (ωt) cg eiωt
(4.27)
(4.28)
We re-introduce the parameter ∆ ≡ ω −ω0 as the detuning of the laser frequency
from the atomic transition (D2 ) in consideration. At this stage we make use of a
technique known as the rotating wave approximation. This approximation consists
of averaging out the terms whose oscillation frequency is considerably larger than
the detuning. First, we expand the cosine terms in Eqns. (4.27) and (4.28) in terms
of exponentials to obtain:
¤
iΩd £
1 + e−i2ωt
2
¤
iΩcg £ i2ωt
d˙ = i∆d −
e
+1
2
ċg = −
(4.29)
(4.30)
Assuming |∆| ¿ ω we can apply the rotating wave approximation and drop the two
terms which oscillate at a rate 2ω. The above equations yield:
Ω
ċg = −i d
2
Ω
d˙ = i∆d − i cg
2
which written in matrix form gives:
¸ ·
¸
·
¸·
∂ cg
0 Ω2
cg
i
= Ω
d
−∆
∂t d
2
(4.31)
(4.32)
(4.33)
Here, we label the 2 × 2 matrix on the right hand side as Ĥef f . With Eq. (4.33)
we recover the form of the Schrödinger equation which is analogous to Eq. (4.23)
4.2. SPLITTING THE MATTER WAVE
93
but with the advantage that this effective Hamiltonian has no time dependence. We
can check the correspondence of this effective Hamiltonian to the original two level
Hamiltonian by setting the coupling parameter Ω = 0. This means that the effective
Hamiltonian becomes a simple diagonal matrix with two constant energy levels. The
solution in this situation is the plane wave such that d(t) = ei∆t . Carrying out the
transformation back to ce (t), we obtain ce (t) = e−iωt ei∆t . This results in ce (t) =
e−iω0 t which is consistent with the expected result for the two-level Hamiltonian
presented in Eq. (4.23) when Ω = 0.
One approach to solving Eq. (4.33) is to apply one more time derivative to
the equation for d(t), yielding a second order equation that has a term including
ċg (t). This allows us to substitute Eq. (4.31) into Eq. (4.32) to obtain an uncoupled
equation in terms of d(t) and its derivatives. Following the solution of d(t) we can
obtain cg (t) by plugging in the appropriate derivatives of d(t) back into Eq. (4.28).
Appendix F works through the mathematical details of how to obtain the solution
to cg (t), d(t) and introduces the following substitutions which make the solution
more elegant.
√
X =
∆2 + Ω2
(4.34)
c0 = cg (t = 0)
(4.35)
d0 = d(t = 0)
(4.36)
Using the above definitions, we can write solutions to the Schrödinger equation
with the effective Hamiltonian in Eq. (4.33) as an operator Û .
¸
·
¸
·
c0
cg (t)
= Û
(4.37)
d0
d(t)
where,
Û = e
·
i∆t/2
¸
cos(Xt/2) − i∆/X sin(Xt/2)
−iΩ/X sin(Xt/2)
−iΩ/X sin(Xt/2)
cos(Xt/2) + i∆/X sin(Xt/2)
(4.38)
To check, we set Ω = 0 and obtain the following Û operator
· −i∆t/2
¸
e
0
i∆t/2
Û → e
0
ei∆t/2
(4.39)
which yields a result consistent with an effective Hamiltonian that has no laser
coupling. The result becomes cg (t) = c0 and d(t) = ei∆t d0 , as required by a constant
potential Hamiltonian.
Now we can modify the starting Hamiltonian, to one that incorporates a travelling wave with phase φ.
·
¸
0
~Ω cos(ωt + φ)
Ĥ =
(4.40)
~Ω cos(ωt + φ)
~ω0
94
CHAPTER 4. INTERFEROMETRY TECHNIQUES
This means that Eq. (4.31) and (4.32) get modified (as seen in appendix F) to
include an extra phase factor giving a new set of equations
Ω
ċg = −i eiφ d
2
Ω
d˙ = i∆d − i e−iφ cg
2
which means we obtain a new effective Hamiltonian of the form:
·
¸
Ω iφ
e
0
0
2
Ĥ ef f = ~ Ω −iφ
e
−∆
2
(4.41)
(4.42)
(4.43)
We note that the new effective Hamiltonian can be written in terms of the
original effective Hamiltonian in Eq. (4.33) and a transformation matrix Ŝ such
that Ĥ 0 ef f = Ŝ † Ĥef f Ŝ.
· iφ/2
¸·
¸ · −iφ/2
¸
e
0
0 Ω2
e
0
0
Ĥ ef f = ~
(4.44)
Ω
−∆
0
e−iφ/2
0
eiφ/2
2
This means that a new Schrödinger equation in terms of the new effective Hamiltonian can be written in terms of the original effective Hamiltonian as:
i~
∂
| ψi = Ĥ 0 ef f | ψi
∂t
= Ŝ † Hˆef f Ŝ| ψi
(4.45)
If we act on this equation by the Ŝ operator on both sides from the right, we get an
equation equivalent to Eq. (4.33) for the states | ψ 0 i = Ŝ| ψi.
∂
Ŝ| ψi = Ĥ Ŝ| ψi
∂t
∂
i~ | ψ 0 i = Ĥ| ψ 0 i
∂t
i~
(4.46)
As shown, the Ŝ and Ŝ 0 operators let us transform between the | ψi and | ψ 0 i
basis. Similarly, by applying these operators, we can transform the time evolution
of | ψi given by the operator Û . To obtain the solution to Eq. (4.46) we act on the
operator Û from the left and right to obtain the new time evolution Ûφ of the state
| ψ 0 (0)i
Ûφ = Ŝ † Û Ŝ
¸
·
cos(Xt/2) − i∆/2 sin(Xt/2)
−ieiφ Ω2 sin(Xt/2)
i∆t/2
= e
cos(Xt/2) + i∆/2 sin(Xt/2)
−ie−iφ Ω2 sin(Xt/2)
(4.47)
4.2. SPLITTING THE MATTER WAVE
95
The result presented above is the solution to a quantum mechanical two-level
system where the ground and excited state amplitudes cg (t) and ce (t) are coupled via
an electromagnetic plane wave. It is important to note that recovering the amplitude
for the excited state ce (t) can be done by multiplying d(t) by the phase e−iωt . At any
rate, the populations for each state are obtained by taking the modulus squared of
each amplitude so |ce |2 = |d|2 , making the overall phase irrelevant. Above all, this
result will prove useful when we apply the two-level time evolution to the rest and
moving states of the wave-packets in the waveguide with a standing wave potential.
4.2.3
The Light Shift
Equation (4.33) shows the Hamiltonian for a two level system (like an atom), whose
energy levels are coupled via the interaction with the E field. We can use this
Hamiltonian to find out how the light intensity shifts the internal atomic states.
Then we relate the shift to the E field of a standing wave discussed in Sec. 4.1
to obtain the potential generated by the counter propagating beams. The Hamiltonian in Eq. (4.33) can be diagonalized using the requirements on eigenvalue matrix
equations. Specifically, we apply det(H − λ1) = 0, which gives:
λg = ~
Ω2
4∆
λe = −~∆ − ~
(4.48)
Ω2
4∆
(4.49)
For a negative detuning ∆, the light shift reduces the energy of the ground
state by λg . However, the light shift can depend on position. As it was described on
chapter 2, the optical Bloch equations [35] yield the equations defining the saturation
intensity. In turn, these reveal that Ω2 depends on the intensity I of the light seen
by the atoms.
In a 1-D standing wave, like the one we apply to split the atoms, the intensity
of the light will vary as a function of space. For this purpose, we can use the
standard result that the intensity of light I(y) depends on the electric field like
I(y) = |E(y)|2 /2ξ0 , where ξ0 = 1/²0 c = 377 Ω is the impedance of free space [61].
We can then write
Γ2 I(y)
2
(4.50)
Ω (y) =
2Is
where the Is is the saturation intensity for the atoms to scatter photons as discussed
in Sec. 2.2.
4.2.4
The Bloch Picture
In order to obtain an intuitive understanding of the time evolution of the state vector
in the two-level problem, we can make use of the Bloch vector picture developed
96
CHAPTER 4. INTERFEROMETRY TECHNIQUES
by Feynman et al. [62]. In the Bloch representation of a generic two level system
in Eq. (4.33), one constructs a 3-dimensional vector in the {x̂, ŷ, ẑ} basis out of
the time dependent coefficients cg , ce and their complex conjugates [35]. The Bloch
vector R is then made up of three components which evolve in time and represents
the state in which the two-level system can be found.
The Bloch picture also includes the vector Ω, containing information about the
Hamiltonian which couples the two energy levels. Specifically the vector Ω is defined
as:
Ω ≡ Re Ω x̂ + Im Ω ŷ + ∆ ẑ
(4.51)
Where Re Ω = Ω is the Rabi frequency and ∆ the detuning for a two level
problem defined in Sec. 4.2.1. It can be demonstrated, using the equations for the
coefficients in Eq. (4.7), that a vector R constructed from the components:
£
¤
rx = cg c∗e + c∗g cg x̂
(4.52)
£ ∗
¤
∗
ry = i cg ce − cg cg ŷ
(4.53)
£ 2
¤
rz = |ce | − |cg |2 ẑ
(4.54)
obeys the following vector equation known as the Bloch equation of motion
dR
= Ω × R.
dt
(4.55)
The physics that arises from the Bloch equation of motion, describes the dynamics analogous to a classical top whose angular momentum vector Lc precesses
around a constant gravitational field vector g. In the case of a two level quantum
system, the Bloch vector R precesses around the vector Ω. Additionally, according
to equation (4.55), R will evolve in time with constant length. Hence, its motion
is confined to what is known as the Bloch sphere. When R points to the south or
north poles of the sphere, the system is in the ground or excited state respectively.
If the vector lies in the equator, the system is in an equal superposition of ground
and excited states. Because we can experimentally control the direction of Ω, we
can alter the precession angle and obtain different time evolutions for the system.
4.2.5
Splitting Operation
In order to carry out the interferometer sequence we must split the condensate into
two packets moving apart from each other. The atoms interact with a periodic
potential set up by a standing wave from off-resonant beams created as described in
Fig. 4.2 and 4.3. Because a periodic potential is set up, the off-resonant laser beams
interact with the condensates via Bragg scattering [63, 64, 65]. A short time after
the condensate atoms are loaded into the waveguide, we apply the pair of counterpropagating beams along the axis of the waveguide in order to split the condensate.
4.2. SPLITTING THE MATTER WAVE
97
In this section we aim to find a method that models the interaction of the standing
wave created by the counter propagating beams with the condensate.
The counter-propagating beams of a laser can be modeled by two travelling
plane waves with opposite wave vector k. Because we use a mirror to retro-reflect
the incoming beam, the amplitudes and frequencies of each plane wave are very
similar. Adding the two plane waves gives the following field:
Es (y) = k E0 ei(ky−ωt) + k E0 ei(−ky−ωt)
(4.56)
where y is the position of the atom. The magnitude squared of this field yields
|Es (y)|2 = 2E02 + 2E02 cos(2ky)
(4.57)
Using the equation for the intensity in terms of the electric field, we can plug I(y)
into Eq. (4.50) to obtain an expression for Ω which will expose its spatial dependence:
Ω2 (y) =
Γ2
2E 2 [cos(2ky) + 1]
4Is ξ0 0
(4.58)
Next, we can plug the expression for Ω2 (y) into (4.48) to obtain an equation for the
potential experienced by the atoms when a standing wave is applied along the axis
of the waveguide. In the expression below the term corresponding to an over all
constant energy shift has been disregarded.
U (y) =
~Γ2
E02 cos(2ky)
8∆Is ξ0
(4.59)
To obtain the final form for the potential of the standing wave, we introduce the
variable β = Γ2 E02 /8∆ξ0 Is = Γ2 I/4∆Is where I = E02 /2ξ0 in order to make the
above equation more manageable. Eventually, β will become time dependent allowing us to turn on or off the interaction. Additionally we introduce the standing
wave phase φ such that cos(2ky) → cos(2ky + φ). Later, φ will be shown to control
the zero of the standing wave. This gives
U (y) = ~β cos(2ky + φ).
(4.60)
Having derived the above expression for the potential, we can write it as a an
operator Û (y), and proceed to write the 1-D Hamiltonian for atoms that see a
standing wave in the waveguide as
Ĥs =
p̂2
+ ~β cos(2ky + φ)
2m
(4.61)
With p as the atomic momentum. One can write the potential energy term of
this Hamiltonian as a sum of two complex exponentials. Consequently, the Hamiltonian is re-written in terms of two momentum translation operators e±i2~k . These
98
CHAPTER 4. INTERFEROMETRY TECHNIQUES
two momentum transformations will convert stationary states into moving states of
momenta ±2~k.
To get an idea of the coupling mechanism, we can consider a situation where the
interaction times are short thus simplifying the problem. However it should be noted
that this kind of model is not applicable in our case because we will be applying
the standing wave for long times. Similarly we consider a wave-packet confined to
a one-dimensional box. This is also an approximation that will be relaxed later.
For short enough interaction times, we can assume that the kinetic energy term
is negligible since the wavefunction which describes the condensate atoms will not
move while we apply the standing wave.
¤
~β £ i(2ky+φ)
Ĥs '
e
+ e−i(2ky+φ)
(4.62)
2
We start with a stationary wave packet | ψ0 i described by a constant wave function
that is normalized to a 1-D box of length L. Then the evolution of the wave-packet
can be obtained by using the time evolution operator as follows:
| ψ(t)i = e−iĤs t/~ | ψ0 i
(4.63)
To obtain an idea of how the time evolution occurs, we can start with an approximate treatment in which βt ¿ 1. In this way we can approximate the exponential
to get
βt
βt
(4.64)
| ψ(t)i ' 1| ψ0 i − i ei(2ky+φ) | ψ0 i − i e−i(2ky+φ) | ψ0 i
2
2
Which shows the original wave packet | ψ0 i plus two other plane wave states moving
along the y direction. The above example shows how a Hamiltonian of this form can
operate on a stationary wave-packet and generate alternate wave packets travelling
with momentum of 2~k. Other higher momentum states are generated, but their
probability amplitude is small and therefore neglected. This is why we approximated
the exponential to first order only. The result is analogous to an optical grating in
which the zeroth order beam gets split into higher order diffraction beams that have
higher momenta.
It should be noted that this approximation has limitations when comparing it to a
better model (as explained below) of the interaction. If we make the interaction time
too short then we increase the probability to drive atoms into the higher momentum
states ±4~k [64], making this approach invalid.
We could use a long single pulse. However, using a longer pulse will also leave
many atoms at rest. In practice, using a single pulse will at best put 1/3 of the
atoms into the moving states ±2~k, which is not optimum for our interferometer.
Our waveguide is one-dimensional. Implementing the successive operations to
obtain a Michelson interferometer like the one in Fig. 4.4, means the condensate part
| ψ0 i will obstruct the motion of the split condensates. Moreover, due to atomic interactions, the overlap adds phase shifts to the clouds which complicate the operation
of the interferometer.
4.2. SPLITTING THE MATTER WAVE
99
In light of this difficulty, we seek to find an alternative to split the condensate
evenly, without leaving any portion of atoms behind. Consequently we adopt a
technique described by Wu et al. in [66]. In this technique a sequence of two laser
pulses separated by the appropriate time are used to achieve even splitting. The
pulses interact with the atoms through an off-resonant Bragg scattering process.
Subsequently, we calculate the required pulse intensities and times to achieve a
successful splitting technique. This implies that after splitting, 50% of the atoms
move with p = +2~k, the other 50% move at p = −2~k and no atoms are left at
rest. For this purpose, we make use of the Bloch sphere representation of a twolevel system. The Bloch sphere picture will help us visualize the dynamics of the
quantum mechanical states. It will use a time evolving vector which we can track,
helping us choose the right pulse lengths that will yield the correct quantum states.
We start by restricting the Hamiltonian in Eq. (4.61) to a three dimensional
Hilbert space. We no longer assume that βt ¿ 1. This gives us the required
flexibility for the pulse lengths (given a fixed intensity of the laser) needed to split,
reflect and recombine the condensate. Therefore in the following analysis, we choose
to neglect the coupling to momenta states with p = ±4~k or higher [64].
For these reasons, we assume that the time evolution of the condensate’s translational states in the waveguide will be described by the following three states:
| 0i
atoms moving at p = 0~k
| +2i atoms moving at p = +2~k
| −2i atoms moving at p = −2~k
(4.65)
The coordinate representation of the moving states is given by plane waves of the
form ψ0 (y) e±iky and the rest state is ψ0 (y). Here the function ψ0 (y) represents the
condensate wavefunction and it should be normalized.
Using this basis we can compute the matrix elements of Ĥs . Details of how to
obtain each matrix element are given in appendix F, but the results are expressed
by the following matrix using the states in the order of (4.65):

0
β iφ

Ĥs = ~
e
2
β −iφ
e
2
β −iφ
e
2
4ωr
0
β iφ
e
2

0 
4ωr
(4.66)
where ωr = ~k 2 /2m = 2.36 × 104 s−1 is defined to be the recoil frequency of the
atoms.
At this point we make a change of basis that will decouple one of the states in the
three-level system. For now we choose φ = 0 assuming no changes in the phase of
the potential. Nevertheless, φ will become a parameter that has physical significance
in our experiment. The meaning of φ and its relevance will be explained in Sec. 4.4.
We aim to convert this Hamiltonian so that it contains a two-level system to which
we can apply the results obtained in section 4.2.2. This can be achieved using a
100
CHAPTER 4. INTERFEROMETRY TECHNIQUES
change of basis like:
| 0i = | 0i
| +i = √12 (| +2i + | −2i)
| −i = √12 (| +2i − | −2i)
(4.67)
In order to calculate the matrix elements of Ĥs in our new basis, we can use the
results found in Eq. (4.66).
³
´
1
√
h+|Ĥs |0i =
h+2|Ĥs |0i + h−2|Ĥs |0i
2¡
¢
~β
1
(4.68)
= √2 2 + ~β
2
1
= √2 ~β
In a similar way
h−|Ĥs |0i =
√1
2
√1
2
=
= 0
³
´
h+2|Ĥs |0i − h−2|Ĥs |0i
¡ ~β ~β ¢
− 2
2
(4.69)
h
i
h+|Ĥs |−i = 12 h+2|Ĥs |+2i − h+2|Ĥs |−2i + h−2|Ĥs |+2i − h−2|Ĥs |−2i
= 12 [4~ωr − 0 + 0 − 4~ωr ]
= 0
(4.70)
h
i
h+|Ĥs |+i = 21 h+2|Ĥs |+2i + h−2|Ĥs |−2i + h−2|Ĥs |+2i + h+2|Ĥs |−2i
= 12 [4~ωr + 4~ωr + 0 + 0]
= 4~ωr
(4.71)
h
i
h−|Ĥs |−i = 21 h+2|Ĥs |+2i + h−2|Ĥs |−2i − h−2|Ĥs |+2i − h+2|Ĥs |−2i
= 21 [4~ωr + 4~ωr − 0 − 0]
= 4~ωr
(4.72)
h0|Ĥs |0i = 0
(4.73)
Because we know Ĥs is hermitian such that Ĥs† = Ĥs then
†
³
(h+|Ĥs |0i)
=
h0|Ĥs |+i
=
´†
√1 ~β
2
√1 ~β
2
(4.74)
4.2. SPLITTING THE MATTER WAVE
and
101
(h−|Ĥs |0i)† = 0
h0|Ĥs |−i = 0
(4.75)
(h+|Ĥs |−i)† = (4~ωr )†
h−|Ĥs |+i = 4~ωr
(4.76)
Combining all these results we can obtain Ĥs in its new representation using the
| 0i, | +i, | −i basis vectors. The resulting Hamiltonian yields.


0 √β2
0


Ĥs = ~  √β2 4ωr 0 
(4.77)
0
0 4ωr
By inspecting Eq. (4.77), we can identify the {| 0i, | +i} sub-space as being analogous to the effective Hamiltonian found in Eq. (4.33) while the {| −i} state is
decoupled. This means we can treat the {| 0i, | +i} sub-space exactly like the two
level system√
and use the solution found in Eq. (4.38). Making the direct comparison
we get Ω = 2β and ∆ = −4ωr , noting that the detuning is negative.
Accordingly, in this case the ground state represents the rest state | 0i, and the
excited state represents the symmetric superposition √12 (| +2i + | −2i) of the two
travelling plane waves with momentum ±2~k. The latter is the desired output state
after applying the standing wave.
Having a two-level system allows us to take advantage of the Bloch vector model.
We can construct the corresponding Ω and R vectors for our system and use the
time evolution of Eq. (4.38) to get the time evolution for R.
In the case of our standing wave potential, β is the direct analog of Ω in the
generic two level-system. It is real, so we do not worry about the ŷ component. In
particular this is due to Ω not having an imaginary phase. This means the Ω vector
in our case lies in the x̂, ẑ plane.
√
Ω = 2β x̂ − 4ωr ẑ
(4.78)
Experimentally, we can control the angle Ω makes with the x̂ axis. Because it
is not within our control to easily change the recoil frequency of the atoms ωr , the
value of ∆ remains constant. However, the power of the diode laser can be reduced
by controlling the current control of the laser, permitting some intensity control of
the standing wave at the position of the atoms. In this way, we can control the x̂
component of Ω, which in turn changes its angle with the x̂ axis.
The Bloch picture allows us to visually follow the precession of the state vector,
and give us insight as to what Ω should be in order to obtain the desired state. The
atoms start in the ground state so that the Bloch vector points to the south pole of
the Bloch sphere. Our aim is to manipulate Ω so that R precesses in such a way
102
CHAPTER 4. INTERFEROMETRY TECHNIQUES
that its final position after the splitting sequence is done points to the north pole
of the sphere.
We make use of the optimized light pulse sequence technique proposed by Wang
et al. in [12]. They demonstrate that for atoms whose state vector starts in the
south pole of the Bloch sphere, a first pulse with the correct value of Ωx (Ω in the
x̂ direction) causes R to precess around Ω so that at some later time, the Bloch
vector will find itself in the equator of the Bloch sphere. A Ω in the +x̂−ẑ direction,
transfers the state vector to the +x̂ axis at a time corresponding to half the period
of precession
At this point the standing wave is turned off but the recoil frequency −4ωr still
exists. This implies Ω points directly downward and R precess perpendicular to it
along the equator. By waiting a time corresponding to half the period of precession,
the state vector is allowed to rotate 180◦ ending in the −x̂ direction.
Finally, the standing wave is turned back on during a second pulse, causing the
Bloch vector to precess once again about the axis defined by Ω as seen in Fig. 4.7.
Keeping the standing wave on for the appropriate time causes the state vector to
precess until it is pointing directly north.
In terms of our experiment, we can choose the correct value of Ωx by controlling
the intensity of the laser. We can also switch the standing wave interaction on or
off by using an acousto optic modulator which we can switch on and off in ∼ 1 µs.
Using our {| 0i, | +i} basis, for the first pulse we want the following time evolution
at the peak of the precession.
1
| 0i → √ [| 0i + | +i]
2
(4.79)
Inspecting the solution presented in Eqns. (4.38), (4.37) and starting with c0 =
1, d0 = 0 we get the following state vector as a function of time
¸
·
¸
·
cos(Xt/2) − i∆/X sin(Xt/2)
cg (t)
i∆t/2
=e
(4.80)
d(t)
−iΩ/X sin(Xt/2)
From the Bloch picture, given that the vector Ω is in the +x̂ − ẑ direction and
the starting Bloch vector points south, the Bloch vector will precess to the +x̂
position at a the time of half the oscillation cycle. To see this, we can calculate the
components of R(t) with Eqns. (4.52), (4.53), (4.54) and use the initial conditions
set forth by Eq. (4.80).
2Ω∆ 2
sin (Xt/2)
X2
−2Ω
=
sin(Xt/2) cos(Xt/2)
· X2
¸
Ω
∆2
=
−
sin2 (Xt/2) − cos2 (Xt/2)
X2 X2
rx =
(4.81)
ry
(4.82)
rz
(4.83)
4.2. SPLITTING THE MATTER WAVE
103
When the Bloch vector reaches the equator, its rx component is at its maximum.
Inspecting the above equations, specifically the rx component, we can conclude that
this occurs when Xt = (2n + 1)π where n is an integer. Furthermore, by plugging
in t = π/X into Eq. (4.80), we can obtain the following constraints
|∆|
1
= √
X
2
Ω
1
= √
X
2
(4.84)
(4.85)
in order to have a state vector of the form shown in Eq. (4.79), which is the desired
state after the first pulse. This is also consistent with rx = −1 at half the cycle
time, meaning the unit length of R has been preserved. As a result we can obtain
the time needed for Ω to reach the equator as:
τ1 =
(2n + 1)π
√
2|∆|
(4.86)
Immediately after the Bloch vector reaches the equator, we turn off the standing
wave and let R evolve freely. This means Ω is pointing directly to the south pole. In
this case the precession is perpendicular to √
Ω so R rotates in the equatorial plane of
the Bloch
√ sphere. Using√the state vector 1/ 2(| +i + | 0i) as a new initial condition,
c0 = 1/ 2 and d0 = 1/ 2. Then in Eq. (4.38) we obtain the following state vector:
·
cg (t)
d(t)
¸
ei∆t/2
= √
2
·
cos(Xt/2) − i∆/X sin(Xt/2)
cos(Xt/2) + i∆/X sin(Xt/2)
¸
(4.87)
Just as we did earlier, we can find the time evolution for the components of R,
noting that when Ω = 0 means X = |∆|:
rx = cos2 (Xt/2) − sin2 (Xt/2)
ry = 2 sin(Xt/2) cos(Xt/2)
rz = 0
(4.88)
(4.89)
(4.90)
Starting with R(t = 0) = x̂, to achieve a 180◦ rotation, we want R(t) = −x̂. This
implies that Xt = (m + 1)π/2 so that the required time for such a final state is:
τ2 =
(2m + 1)π
.
|∆|
(4.91)
This achieves the desired change in the state vector such that:
1
1
√ [| 0i + | +i] → √ [−| 0i + | +i]
2
2
(4.92)
104
CHAPTER 4. INTERFEROMETRY TECHNIQUES
√
Finally we turn on the second pulse so that Ω = 2βx̂ − 4ωr ẑ, inducing
√ the
Bloch vector
to
precess
once
again.
Using
the
initial
conditions
c
2, and
=
−1/
0
√
d0 = 1/ 2 corresponding to R starting in the −x̂ direction, we obtain the following
time evolution for the state vector.
·
¸
·
¸
ei∆t/2 − cos(Xt/2) + i∆/X sin(Xt/2) − iΩ/X sin(Xt/2)
cg (t)
= √
(4.93)
d(t)
iΩ/X sin(Xt/2) + cos(Xt/2) + i∆/X sin(Xt/2)
2
This in turn yields the following time dependent components of R(t):
· 2
¸
Ω2
∆
2
rx = − cos (Xt/2) − sin(Xt/2)
−
X2 X2
∆
ry = −2 cos(Xt/2) sin(Xt/2)
X
Ω∆ 2
rz = 2 2 sin (Xt/2)
X
(4.94)
(4.95)
(4.96)
The Bloch vector must completely lie along the ẑ direction for the final state vector
to be | +i. Moreover from the Bloch picture, the precession of R must be such that
the maximum rz must occur at half the period of oscillation meaning Xt = (2n+1)π.
Additionally we know that the length of R will not change. According to Eq. (4.80),
we started the sequence with a vector of unit length. Consequently at the end of
the pulse, R should equal ẑ meaning (Ω/X)(∆/X) = 1/2. In order to satisfy the
final condition of t = π/X for Eq. (4.93), we see that Ω/X = |∆|/X, which yields
the constraints
|∆|
1
= √
X
2
Ω
1
= √
X
2
(4.97)
(4.98)
As a result we can see that the time required to reach the north pole of the Bloch
sphere is given by
(2n + 1)π
τ3 = √
(4.99)
2|∆|
The above result demonstrates the motion of R is equivalent to its precession
during the first pulse, but located in the {−x̂, ŷ, ẑ} quadrant of the Bloch space.
This is consistent with the symmetry of the Bloch sphere. Plugging in the constraints
found in Eqns. (4.97), (4.98) and the required time set by (4.99) into Eq. (4.93),
gives the final state of the two level system. The result shows that after the second
pulse is over, the following transformation takes place:
1
√ [−| 0i + | +i] → | +i
2
(4.100)
4.2. SPLITTING THE MATTER WAVE
105
Figure 4.7: Illustration of the Bloch vector precession during the optimized
double
√
pulse sequence. (a) We start with Rx , during the first pulse Ω = 2βx̂ − 4ωr ẑ
so that R precesses along the dotted circle going through the x̂ axis once every
cycle. Waiting t = τ1 places R along the x̂ axis. (b) Turning off the laser means
Ω = −4ωr ẑ, and R rotates along the equator. Waiting t = τ2 leaves R along the
−x̂ axis. (c) A second light pulse is used and R precesses along the dotted line for
t = τ3 corresponding to half the cycle time, giving the final position of R along the
ẑ axis. The time dependence of Ωz is given by the bottom graph showing Ω vs t.
Here τ1 = τ3 .
In short, the dynamics of the Bloch vector for the optimized pulse sequence is
best described and summarized by the three stages shown in Fig. 4.7. According to
the optimized pulse sequence presented, we can summarise the theoretical values for
the parameters of light intensity, pulse time and wait time required to split the rest
condensate into two equal clouds moving at ±2~k. We use the simplest solutions
for τ1√, τ2 and τ3 where n = 0 and m = 0. In our case we identify ∆ = −4ωr and
Ω = 2β from the two level system presented in Eq. (4.77).
√
β1 = 2 2ωr
π
τ1 = √
4 2ωr
π
τ2 =
4ωr
(4.101)
(4.102)
(4.103)
106
CHAPTER 4. INTERFEROMETRY TECHNIQUES
Figure 4.8: Absorption imaging showing two wave packets after applying the double
pulse sequence on the condensate at rest in the waveguide. Here two wave packets
are seen travelling at ±2~k 10 ms after splitting.
4.2.6
Experimental Verification
Experimentally, we have been able to corroborate the efficiency of the double pulse.
Using the parameters calculated in the optimized double pulse sequence we obtained
a ratio 1 : 1 for atoms in the | +2~ki and | −2~ki state respectively. Our absorption
measurement with the camera was only accurate up to 95%. Hence, up to 5% of
the atoms could be left behind by the splitting and we would have not been able
to detect it. This occurs because the absorption coefficients are very small because
there are so litte atoms to absorb the light. For this reason we say that the splitting
using the double pulse technique was 95% efficient. Thus we could see a maximum
of 95% of the atoms separated into two clouds travelling at ±2~k.
The theoretically optimum values of τ1 = τ3 = 24 µs for the pulse durations and
τ2 = 33 µs for the wait time between pulses provided the best results.
To create the Bragg beam we used a laser detuned 7.8 GHz red of the D2 transition. The laser had a power of 0.7 mW and a Gaussian beam waist of approximately 1.5 mm which resulted in a peak intensity of 17.6 mW/cm2 which meant that
β = 52 KHz. This power setting for both pulses corresponded to within 23% with
the expected theoretical intensity given by the relationship in Eq. (4.101), which
relates the recoil frequency ωr to the laser intensity.
The results can be seen in Fig. 4.8.
4.3. REFLECTING THE MATTER WAVE
4.3
4.3.1
107
Reflecting the Matter Wave
Three Level System
Following the splitting of the atoms, we allow the two wave packets to propagate
along the axis of the waveguide until they fully separate from each other. At some
variable time after separation we apply another laser pulse similar to the splitting
in order to reverse the momentum of each wave packet. The objective is to reverse
the momentum of the wave packets twice at t = T /4 and t = 3T /4 as described
by Fig. 4.6 in order for the groups of condensate atoms to travel through identical
paths before becoming recombined. Making each wave packet traverse the same
path means that both packets will experience the same energy phase shift if there
are any asymmetrical imperfections and changes in the potential.
We developed a technique to reflect the atoms via an off-resonant beam, [64].
To understand the reflection sequence we return to using the {| 0i, | +i, | −i} basis.
Using the Hamiltonian containing a potential energy term describing a standing
wave like in Eq. (4.61), we were able to write a representation for φ = 0 where the
{| 0i, | +i} comprised a decoupled subspace from {| −i}.
This meant we could use the two-level time evolution expressed in Eq. (4.38).
If we add an additional decoupled sate with constant energy −4ωr to the two-level
system, the time evolution for the three-level system would just acquire an additional
diagonal term that contained the normal constant energy phase factor.
Û = ei∆t/2 ×

cos(Xt/2) − i∆/X sin(Xt/2)
−iΩ/X sin(Xt/2)
0

−iΩ/X sin(Xt/2)
cos(Xt/2) + i∆/X sin(Xt/2)
0 
0
0
ei∆t/2
(4.104)
where once again ∆ = −4ωr . Our goal is to use this time evolution operator to
transform the states obtained after the splitting sequence, e.g convert | ±2~ki as
follows:
| +2~ki ←→ | −2~ki
1
1
√ [| +i + | −i] ←→ √ [| +i − | −i]
2
2
(4.105)
(4.106)
This means that if our initial state is:


0
| ψ+2k i =  1 
1
(4.107)
then, by applying the time evolution operator Û in Eq. (4.104), the general final
108
CHAPTER 4. INTERFEROMETRY TECHNIQUES
state has the form:


−iΩ/X sin(Xt/2)
1
| ψf i = √ ei∆t/2  cos(Xt/2) + i∆/X sin(Xt/2) 
2
ei∆t/2
To obtain the desired output we would like a final state of the form:


0
| ψf i ∝  1 
−1
(4.108)
(4.109)
which yields a set of constraints that can help us define what the intensity and the
length of the pulse should be in order to achieve reflection. To eliminate sin(Xt/2)
we require that Xt/2 = mπ where m is any integer which in turn means cos(Xt/2) =
(−1)m giving us so far,


0
1
| ψf i = √ ei∆t/2  (−1)m 
(4.110)
2
ei∆t/2
m
Observing the above equation, it is evident we want to have ei∆t/2 = −(−1)
or
√
m±1
2
equivalently (−1)
. It is important to keep in mind that because X = ∆ + Ω2
then ∆ must be smaller than X by definition, thus specifically |∆| < X. The
simplest solution satisfying these conditions is:
∆t
= −π
2
Xt
= 2π
2
(4.111)
(4.112)
This means the time required to reflect the atoms is τref = −2π/∆ = π/2ωr and
X = −2∆. Using the definition of X we obtain:
X 2 = Ω2 + ∆ 2
4∆2 = Ω2 + ∆2
Ω2 = 3∆2
(4.113)
(4.114)
(4.115)
√
In a similar way to the splitting sequence, we identify Ω = 2β and ∆ = −4ωr and
obtain the parameters for intensity and time of the standing wave pulse required to
make the transition in Eq. (4.105).
√
βr = 2 6ωr
(4.116)
π
(4.117)
τr =
2ωr
4.3. REFLECTING THE MATTER WAVE
4.3.2
109
Experimental Verification
Using the above technique to reflect the atoms is not the only way to reflect the
atoms. Previous experiments have used short pulses that couple the | +2~ki to the
| −2~ki through second order Bragg scattering [64]. However this technique is very
sensitive to velocity errors in the wave-packets. Given that, any fluctuation in the
initial velocity of the wave-packets will degrade the performance of the reflection
pulse.
Ordinarily, because we are not interested in populating the | 0i state, a long
weak pulse could be used in order to transfer the atoms from | −2~ki to | +2~ki.
However, the long pulse poses a draw back. The transition less likely to happen if
there are velocity errors present.
A major difficulty arising when performing the interferometer operations is the
residual motion acquired by the atoms after the waveguide is loaded. We have observed that the loading process is very sensitive to fluctuations in external magnetic
fields. This causes the condensates to start with a non-zero velocity when starting
the interferometer sequence. However the initial motion due to loading was resolved
as described in Sec. 3.5.4.
We have observed, using phase contrast imaging, initial condensate velocities of
up to 0.5 mm/s. These offsets in velocity, denoted by δ, can decrease the efficiency
of the reflection pulse. To understand how much these non-zero starting velocities
affect the interferometer, we can analyze how sensitive the reflection pulse is to
velocity variations.
Using Fourier analysis [28], we know that for a light pulse of duration τ , the
width of its frequency distribution is given by the relation:
∆ωτ ≥ 1
(4.118)
If the velocity of a packet is v0 + δ before we apply the reflection pulse, then it
will be −v0 + δ after the transition, giving an energy difference ∆E = 2M v0 δ.
Using the energy-time uncertainty relation we can conclude that for a laser pulse of
duration τr , the width of frequencies available to cause a transition is 1/τr . Hence
the corresponding energy uncertainty width for the laser is ~/τr . As a result, the
energy difference for the reflection transition ∆E, must be small compared to ~/τr ,
giving
1
~
=
(4.119)
|δ| ≤
2M v0 τr
4kτr
in order for the transition to occur. Otherwise it is likely that it will not happen. For
example, Wang et al. used a pulse length of τr = 150 µs. This means the velocity
variation is limited to δ ≤ 0.2 mm/s which is more than 50% less than the variations
we observe. For this reason we opted for the short pulse method described above
and achieved better results.
110
CHAPTER 4. INTERFEROMETRY TECHNIQUES
In the laboratory we tested the method of reflecting the atoms along the waveguide
using the time and intensity parameters calculated in the optimized reflection single pulse sequence. We used an intensity value of 2 times the splitting intensity
having a β within 11% of the theoretical calcalculation. Additionally, we used the
corresponding pulse time of τr = 67 µs calculated above, and obtained reflection efficiencies which varied between 100% and 80% in the waveguide. The shorter pulse
made the reflection technique less sensitive to initial velocity fluctuations. For a
pulse duration of 67 µs, the velocity fluctuations in the waveguide are limited to
δ ≤ 0.5 mm/s. Given our observations for δ, we minimally satisfy this requirement.
4.4
Recombination
Using the optimized splitting and reflecting pulses we can achieve the desired motion of the atoms as described by Fig. 4.6. Up until now, we have assumed that no
external agents change the phase of the condensate wave packets. Applying the optimized splitting double pulse when the atoms regroup at the center of the waveguide
at the end of their oscillation means that all the atoms should come back to rest.
The atoms come back to rest because in quantum mechanical two level systems, like
the one describing {| 0i, | +i}, the operations like splitting are unitary and therefore
reversible. The recombination operation of the condensates will be shown in this
section.
Because the standing wave potential described in Fig. 4.2 is generated by retroreflection from a mirror, we can choose to describe the intensity of the standing
wave as (see Sec. 4.2.5):
I(y) = I0 sin2 (ky − ky0 )
= 1/2I0 [1 − cos(2ky − 2ky0 )]
(4.120)
(4.121)
where y0 is the location of the mirror. In this way, writing the expression for the
potential generated by the standing wave will result in Eq. (4.60). In this case the
additional phase incorporates the location of the retro-reflection mirror at y0 .
In writing the potential for this form, we drop any constant term and obtain:
U (y) = ~β cos(2ky + φ)
(4.122)
where φ = −2ky0 . The above expression is equal to Eq. (4.60). We now see a
physical meaning to φ. A change in φ is caused by a shift in the mirror position, or
more generally any shift in the nodes of the standing wave relative to the atoms.
To observe the functioning of our interferometer, we vary the phase φ. Figure
4.2 shows that the Bragg beam standing wave is generated by using a mirror located
outside the science cell. This mirror is positioned a distance of D = 22.5 cm away
from the atoms along the y axis. At the location of the mirror the there is a node
4.4. RECOMBINATION
111
where the wave vanishes. If the laser frequency is shifted by an amount ∆ν, then
the position of the nodes of the standing wave will change relative to the mirror.
This will be reflected in a change of recombination phase φ.
Specifically, the phase shift φ = 2y0 ∆k can be written in terms of the change in
frequency, yielding φ = 4y0 π∆ν/c . This means that we expect N0 /N = 0 when:
y0 π
π
4
∆ν =
c
2
c
∆ν =
(4.123)
8y0
Using the above equation, for y0 = 22.5 cm the expected change in frequency should
be of 167 MHz. We achieved this change by applying a varying current to the diode
laser. During the experiment, changing the frequency takes approximately 2 ms.
With a nonzero φ, as in Eq. (4.66), the Hamiltonian we obtained using Û (y) in
the {| 0i, | +2i, | −2i} basis is

β −iφ β iφ 
0
e
e
2
2
β iφ

Ĥs = ~
(4.124)
e
4ωr
0 
2
β −iφ
e
0
4ω
r
2
In a similar way to section 4.2.5, we look for a new basis to represent Ĥs such that
one of the levels is decoupled leaving a coupled two level subspace. Likewise, if we
set up a basis{| 0i, | +φ i, | −φ i} to obtain Ĥs | −φ i = 4ωr | −φ i we need


0
1
| −φ i = √  eiφ 
(4.125)
2 −e−iφ
so an orthogonal vector to | −φ i is


0
1
| +φ i = √  eiφ 
2 e−iφ
(4.126)
and the remaining state to complete our basis remains the same, so we keep | 0i. We
proceed to calculate the new representation of Ĥs using the known matrix elements
of Ĥs in the {| 0i, | +2i, | −2i} basis shown in Eq. (4.66). For example,
´
1 ³ iφ
e h0|Ĥs |+2i + e−iφ h0|Ĥs |−2i
h0|Ĥs |+φ i =
2µ
¶
1 β β
β
=
+
(4.127)
=
2 2
2
2
Putting together the new representation of Ĥs we obtain:


0 β2
0
Ĥs = ~  β2 4ωr 0 
0 0 4ωr
(4.128)
112
CHAPTER 4. INTERFEROMETRY TECHNIQUES
which is identical to the result obtained in Eq. (4.77). We note that our approach to
modeling the recombination operation is equivalent to that presented in Sec. 4.2.5
but keeping track of the third state | −i.
Now we can apply the solution to the two level problem to obtain a time evolution
operator for the splitting sequence in the three level system. We start out by using
the general time evolution operator obtained in Eq. (4.104), and plugging in the
parameters for the first pulse in the optimized splitting sequence of section 4.2.5.
This gets the time evolution operator for the first pulse.

−1 1
0

 1 1
0
√ −iπ/(2√2)
2e
0 0

−i
Û1 = √ e
2
√
−iπ/2 2
(4.129)
Next, we use the parameters of the free evolution in the optimized splitting sequence
to get the time evolution operator during the time the laser is turned off.


1 1
0
Ûf ree =  1 −1 0 
0 0 −1
(4.130)
From section 4.2.5 we know that the dynamics of the Bloch vector for the second light
pulse are symmetrical to those of the first pulse. For this reason, the parameters for
the second pulse are identical to that of the first pulse, therefore the time evolution
operator Û2 for the second pulse is just Û2 = Û1 .
To get the total splitting operator for a three level system described by a Hamiltonian of the form in Eq. (4.128) and Eq. (4.77), we multiply the three operators in
the order set by the double pulse sequence so Ûsplit = Û2 Ûf Û1 .

0 1
0
 1 0
0√ 
−iπ/ 2
0 0 e

√
−iπ/ 2
Ûsplit = e
(4.131)
Assuming no relative phase has been introduced between
√ the arms of the interferometer, the state just before recombination is | +i = 1/ 2 (| +2i + | −2i). Here it
has been written in the {| 0i, | +2i, | −2i} basis. It represents the two wave-packets
travelling at ±2~k. The Ûsplit operator used during the recombination pulse is written and valid in the {| 0i, | +φ i, | −φ i} basis. Consequently, we must re-write | +i in
this same basis in order to obtain the output state for the interferometer sequence.
To get the representation of | +i in the desired basis, we project it onto the new
4.4. RECOMBINATION
113
basis.
| +i = h+φ |+i | +φ i + h−φ |+i | −φ i
¢
¢
1 ¡ −iφ
1 ¡ −iφ
=
e + eiφ | +φ i +
e − eiφ | −φ i
2
2
= cos(φ)| +φ i − i sin(φ)| −φ i
(4.132)
Finally we apply Ûsplit to | +i and obtain the output state of the interferometer
as a function of φ.
| ψf inal i = Ûsplit [cos(φ)| +φ i − i sin(φ)| −φ i]
√
= cos(φ)| 0i − i sin(φ)e−iπ/ 2 | −φ i
(4.133)
Inspecting the result above, we can see that there are two states in the output.
One state is the antisymmetric superposition of the plane wave states | ±2i with an
additional phase included, while the the other is the rest state | 0i. In this way the
probability to find the atoms in the rest state is given by:
P (k = 0) = |h0|ψf inal i|2
= cos2 (φ)
(4.134)
In the laboratory we can measure the number of atoms in the rest state by taking a
picture using the absorption imaging technique described in section [?]. Assuming
no loss in the number of atoms loaded into the waveguide N after performing the
interferometer sequence, the fraction of atoms brought back to the rest state N0 is
given by the probability P (k = 0). Conversely, the fraction of atoms left in the
moving state is given by P (k = ±2~k).
For this reason, what we seek to measure during the interferometry experiment
is the phase dependent fraction N0 /N given by:
N0 /N = cos2 (φ)
(4.135)
We notice that if our interferometer is working correctly during the experiment, we
should expect to get N0 = 0 when φ = π/2.
Chapter 5
Experimental Results
Up to now we have discussed the components that to be implemented in order to
create our Bose-Einstein condensate interferometer. These components include the
creation of a 87 Rb condensate, loading the condensate into a magnetic waveguide,
and performing splitting and reflection operations on the condensate.
The coordinate system used to describe the interferometer operations, Bragg
beam and imaging system positioning and alignment is illustrated in Fig. 3.1. In this
reference frame the origin is lined up with the center of the waveguide. To implement
the interferometer we use the sequence of operations described in Chapter 4, and
outlined in Figs. 4.4 and Fig. 4.6. Measurements of the number of atoms in the
initial condensate and subsequent packets produced in the output state are carried
out using absorption imaging. A black and white CCD camera in conjunction with
its imaging lenses are mounted along the z axis directly above the center of the wave
guide. The imaging system is described in Sec. 2.8.
After the splitting and reflecting operations, it is important to ensure that we
can eliminate all uncontrolled phases upon recombination of the wave packets. This
condition guarantees that after having applied and controlled the relative phase
between the packets, no external agents have contributed to the observed phase.
Moreover, we must ensure that the packets remain coherent from shot to shot of
the experiment causing the number fluctuation of the measurement N0 /N to remain
small when taking measurements. This ensures that our number measurements as
a function of the phase (see Sec. 4.4) has a smaller error, causing the contrast of our
measurements to be increased.
Before we took measurements of the output state described in the following section, we started by performing a simpler two-step scheme. The simple interferometer
sequence involved a split, followed by a short propagation time (t1 < 8 ms), then a
reflection followed by another propagation time equivalent to the first one t1 , and
finally a recombination pulse. We label the interferometer sequences by the time
between pulses like t1 − t1 . For example, the simple initial sequence to test the
interferometers’ pulse operations is labeled as 5 - 5. This means 5 ms between the
114
5.1. EXPECTED OUTPUT STATE
115
initial split and the reflection, and 5 ms between the reflection and recombination
pulse.
We performed various t1 − t1 sequences (for t1 < 8 ms) and checked the output
state after a applying no phase change and applying a π phase shift to the recombination pulse. Our results showed that most atoms came back to rest after no phase
was applied in the recombination pulse. For example, in the 2-2 interferometer
∼ 73% of the atoms in the initial condensate returned to rest after applying a φ = 0
phase on the output. This demonstrated that our pulses were working correctly.
The change in phase of the standing wave was achieved by controlling the current
on the diode laser providing the Bragg beam which changed its frequency and in
turn changed the position of nodes in the standing wave potential. The creation of
the standing wave potential is explained in Chapter 4, and the dependence of the
phase φ with the frequency change is given in Sec. 4.4.
5.1
Expected Output State
We use the technique described in Sec. 2.8 to take absorption images of the condensates in the wave guide after they have completed the interferometry sequence
described in Fig. 4.4. The experimental results observed after carrying out the
interferometer sequence were consistent with the output state demonstrated by
Eq. (4.133). Experimentally the results obtained included three basic types of output (see Fig. 5.1). If the phase of the splitting double pulse is kept fixed, then
φ = 0 and all the atoms are brought back to rest. Applying a (2n + 1)π/2 phase
shift (where n is an integer), meant all the atoms are transferred into the ±2~k
states so that no atoms are observed at rest. Finally, any other phase would yield a
linear combination of rest atoms and atoms moving at ±2~k, where the ratio of the
moving atoms is 1 : 1.
Depending on the phase applied upon recombination, the output could include up
to 3 clouds of atoms to image and analyze. For this purpose, we wrote a specialized
2-D fitting program which analyzed the different cloud images.
The fitting program is called AI 3 (Atom interferometer 3). It can simultaneously
fit three wave packets. It also incorporates the ability for the fitter to zoom into
the desired regions where the atoms are, reducing its exposure to background noise
and decreasing the fitting time considerably. An outline of the program and its
operation is given in appendix D.
5.2
Measurements
Raw data is obtained in the form of absorption images which are then fit to a 2-D
Gaussian function to obtain the number of atoms brought to rest N0 . The images
observed match very well the expected output described in Sec. 5.1. We demonstrate
116
CHAPTER 5. EXPERIMENTAL RESULTS
Figure 5.1: Illustration of the expected output upon recombination of the coherent
wave-packets. (a) When φ = 0 the standing wave does not shift so in the output state
all the atoms return to rest. (b) For φ = π/2 all the atoms are evenly distributed
in the p = ±2~k states meaning no atoms will be observed at rest. (c) At an
intermediate phase, a linear combination of p = ±2~k and rest wave packets are
observed.
5.3. OUTPUT FIT FUNCTION
117
Figure 5.2: Absorption images of the output state for different phases of the recombination pulse. The condensate atoms and their position is shown by the dark
clouds. As expected for φ = 0 all the atoms return to the stationary state | 0i. For
φ = π/2 no atoms return to the rest state, implying the atoms must remain in the
| ±2~ki. The phase φ = π/4 is half way between φ = 0 and φ = π/2. Applying it
on recombination yields the expected linear combination of | 0i and | ±2~ki states.
this agreement of the output states for three distinct phases φ of the recombination
pulse in the images shown in Fig. 5.2.
The interferometer sequences used include two reflection pulses between the split
and recombine operations. This is to allow each wave packet to travel an equivalent
path length. Each packet travels through both arms of the interferometer, requiring
two reflection pulses, see Chapter 4. Accordingly, each interferometer sequence
is labeled using the time experienced by the wave packets between splitting and
reflection operations like t1 − 2t1 − t1 . For example, a sequence where the time
between the initial splitting and the first reflection is 5 ms, the first reflection and
the second reflection is 10 ms and the second reflection and the final recombination is
5 ms, is labeled as 5-10-5. The total interaction time of the interferometer operation
is labeled T where T = t1 + 2t1 + t1 .
We obtained data for several different operation times ranging from 3-6-3 ms to a
maximum of 11-22-11 ms. We show the output signal for the 10-20-10 interferometer
obtained from the average of two data sets in Fig. 5.3.
5.3
Output Fit Function
Our interferometer characterization consists of measuring the number of N0 atoms
brought to rest after recombination as compared to the total number of atoms N
prior to splitting. In this respect, following the calculation in Sec. 4.4, we expect that
118
CHAPTER 5. EXPERIMENTAL RESULTS
Figure 5.3: Output signal showing the fraction N0 /N as a function of φ. The data
shown is the average of two runs for the 10-20-10 ms interferometer sequence. The
data is fitted to the function A cos 2φ + y0 showing a visibility of V = 0.45 ± 0.10,
demonstrating a coherence time of 40 ms. The data was obtained in 03/03/06 and
23/02/06.
5.3. OUTPUT FIT FUNCTION
119
Figure 5.4: Visibility of the output function for interferometer sequences of varying
interaction time T . The error ∆V was calculated using the result in Eq. (5.10) and
the visibility for T < 10 ms was obtained using Eq. (5.15).
the fraction N0 /N will depend on the phase φ introduced during the recombination
pulse.
The dependence of the fraction N0 /N in φ is given by the expression in Eq. (4.135).
Consequently, using a squared sinusoidal function with the appropriate phase will
serve as an appropriate fit function. Hence we use the following form:
N0
= A cos2 (φ) + y0
N
1
1
=
A cos (2φ) + + y0
2
2
(5.1)
(5.2)
where A has been selected as a parameter for the amplitude of the probability of
the output state, and y0 is a vertical offset. With this in mind, we can perform the
substitutions 1/2A → A and 1/2 + y0 → y0 , in order to simplify the fit function.
which gives:
N0
Y ≡
= A cos (2φ) + y0 .
(5.3)
N
Using absorption imaging, we expect the output signal of N0 (φ)/N after the
recombination of the wave packets to have the above form.
120
5.4
CHAPTER 5. EXPERIMENTAL RESULTS
Visibility
A technique we can use to characterize the degree of coherence exhibited by the wave
packets used in the interferometer sequence is to measure the visibility of the output
signal over many runs. Before giving the exact equation from which to calculate
the visibility, we must motivate its significance given the mathematical form of our
output state and how it relates to the coherence of the wave packets.
The various values for the fraction N0 (φ)/N will be fitted to the sinusoidal function described by Eq. (5.3). If we do not change the phase of the recombination
pulse as compared to the splitting pulse, we expect all the atoms that were split
into the ±2~k states to return to the rest state. This occurs because as explained
in Sec. 4.4, quantum mechanical operations are reversible, and the recombination
pulse is identical to the split pulse. However, if there are any external agents which
cause the wave packets to acquire a relative phase shift, then on recombination, not
all the atoms in the ±2~k states will come to rest.
Moreover, if we desire to make multiple measurements of N0 /N for a fixed value
of φ in the recombination pulse, any decohering effects will result in the value of
N0 /N shifting from shot to shot. Hence, N0 /N will deviate from the expected value
given by the model in Eq. (5.3), causing a decrease in the contrast of the sinusoidal
signal. This effect has the tendency to “flatten out” the signal observed. In fact,
the decrease in contrast of our signal is analogous to the decohering effects which
cause a blurring in the interference pattern of light discussed in Sec. 1.4.
For an output signal whose maximum and minimum values are given by Imax
and Imin respectively, the visibility function is given by [3]:
V =
Imax − Imin
Imax + Imin
(5.4)
An interpretation for the visibility function V is given by the ratio between the
peak to peak value of the output function and twice its average value. Because the
fraction N0 /N > 1, we multiply the average value by 2 in order to normalize the
visibility such that 0 < V < 1. Applying such normalization means that the visibility value obtained will depend on the overall offset of the output signal. This makes
sense because as the signal’s average value increases, the fluctuations (asuming the
amplitude is fixed) become a smaller fraction of the total signal. This means the
contrast is reduced.
Using the fit function in Eq. (5.3), we obtain that Imax = y0 +A and Imin = y0 −A.
Plugging in these values into Eq. (5.4) we obtain an expression for the visibility in
terms of our fit parameters.
V
V
(y0 + A) − (y0 − A)
(y0 + A) + (y0 − A)
A
=
y0
=
(5.5)
(5.6)
5.5. RESULTS
121
Given that, we can linearize the visibility function with respect to our fit parameters
and obtain an expression for the error in the visibility.
·µ
¶
¸2 ·µ
¶
¸2
∂V
∂V
2
(∆V ) =
∆A +
∆y0
(5.7)
∂A
∂y0
(5.8)
Taking the partial derivatives found in the above expression we find:
∂V
−A
1
∂V
= 2
=
∂A
y0
∂y0
y0
(5.9)
Plugging the above derivatives into Eq. (5.7), we obtain the expression for ∆V
yielding:
sµ
¶2 µ
¶2
∆AV
∆y0 V
∆V =
+
(5.10)
A
y0
sµ
¶2 µ
¶2
∆A
∆y0
∆V = V
+
(5.11)
A
y0
Finally, bringing together the above results, the expression for the visibility in conjunction with its error given our fit function parameter is:
sµ
¶2 µ
¶2
∆y0
∆A
A
±V
+
V =
(5.12)
y0
A
y0
We measure V as a function of the sum of the time between pulses. In this way,
the overall time of the interferometer sequence will reflect the total time the wave
packets have evolved for. We seek to obtain an interferometer sequence in which the
packets are propagating for the longest time, while still yielding a sinusoidal output
signal like Eq. (5.3). In other words, we want to see how large T (the total time
for the interferometer’s operation as explained next) can be while still having the
packets coherent. Therefore, this will achieve the interferometer signal with longest
coherence time.
Applying the expression in Eq. (5.12) to each of the output signals of varying
sequence times, we are able to obtain a quantitative analysis on the coherence of each
sequence. The higher the visibility of each signal, the higher its contrast, therefore
the higher its degree of coherence. Normally, we expect to have a drop in visibility
as the time of the interferometer operation increases.
5.5
Results
For the 10-20-10 interferometer the output signal was fit to the function in Eq. (5.3)
yielding the following fit parameters, A = 0.24 ± 0.05, and y0 = 0.54 ± 0.04. Consequently, the visibility V = A/y0 for this signal yielded a value of V = 0.45 with
122
CHAPTER 5. EXPERIMENTAL RESULTS
an error ∆V = ±0.10. We give the visibility of the output function for interferometer sequences of varying total time T in Fig. 5.4. The range of the total times for
the interferometer sequences varies from 0 to 56 ms. As mentioned in Sec. 4.4, the
rate at which we could change the laser frequency for recombination was limited.
Therefore interferometer sequences with T < 10 ms used a set of points at φ = 0
to estimate the visibility. Assuming that y0 = 1/2 for φ = 0, we can obtain the
minimum and maximum visibilities as:
1
Imax = A +
(5.13)
2
1
Imin = −A +
(5.14)
2
which means that the visibility V is equal to 2A. Using Eq. (5.3), the fit function at
φ = 0 is Y (0) = A+1/2, and can be used to solve and substitute for A. Solving for A
yields A = Y (0)−1/2. Plugging in this result into the visibility yields V = 2Y (0)−1.
We know that Y (0) = N0 /N so we can finally write the average visibility using the
form:
V = 2hN0 /N i − 1.
(5.15)
The result of the output function of the 11-22-11 interferometer with T = 44 ms
shows our longest attainable coherence time to date demonstrated by a visibility of
V = 0.6 ± 0.17. For interferometer sequences with T > 44 ms, the visibility drops
substantially, indicating a loss of coherence for the wave packets (see Sec. 5.6).
Furthermore, the interferometer sequence having T = 44 ms exhibited the
longest matter wave separation while still maintaining coherent wave packets. In
other words, upon applying the recombination pulses using various values for the
phase φ, the wave packets recombined according to the fit function in Eq. (5.3). The
maximum matter wave separation obtained is pictured using absorption imaging in
Fig. 5.5.
We seek to obtain the largest possible arm separation. For this reason, it is
to our advantage to maintain the best visibility for the longest time possible. In
other words, maximizing the coherence time of the wave packets will permit us to
maximize the time between pulses, hence maximizing the path length of the packets
which can allow us to increase the arm separation. As Fig. 5.5 shows, we obtained
two wave packets completely separated by a macroscopic distance of 0.26 mm from
center to center of each packet. It should be noted that the picture shows atoms
which are in a quantum superposition of being in the +2~k and −2~k translational
states [67].
5.6
Decohering Effects
Unfortunately, we observe fluctuations in the measured value of N0 (φ)/N from run
to run. This fluctuation has the effect of lowering the visibility of the interferometer’s
5.6. DECOHERING EFFECTS
123
Figure 5.5: Maximum matter wave separation yielding the largest arm separation
of 260 µm for our Bose-Einstein condensate interferometer. This separation was
obtained from the 11-22-11 interferometer, having imaged the packets 11 ms after
the initial split. Wave packets are pictured using absorption imaging, 1 shows a 3-D
representation of the absorption profile and 2 shows a 2-D image. Throughout this
particular sequence, the wave packets remained coherent as demonstrated by the
non-zero visibility of the T = 44 ms interferometer. Hence the atoms in this picture
were in a quantum superposition of being in both peaks. Note, red color and height
indicate the highest density of atoms.
124
CHAPTER 5. EXPERIMENTAL RESULTS
output signal.
One possible explanation is that fluctuations occur because of an instability in
the frequency of the Bragg beam which creates the standing wave potential. We
checked this supposition by setting up an optical interferometer derived from the
Bragg beam and observed that there were no significant fluctuations in the beam.
We did notice that the Bragg beam contained spatial noise that caused variations
in the Bragg beam intensity of up to 20%. We believe that this fluctuation (from
the expected Gaussian intensity cross-section) in the intensity cross-section of the
beam is caused by imperfections in the glass cell. Additionally, multiple reflections
inside the glass chamber cause random interference patterns see (Fig. 4.2). As a
result the intensity of the Bragg beam varies randomly, causing imperfections in the
standing wave potential.
In turn this causes variations in the coupling strength of the different translational states | 0i, | ±2~ki to the Bragg beam, thus causing variations in splitting and
reflection operations which cause fluctuations in the fraction N0 /N as a function of
φ. The result is a decrease in the contrast of our output signal.
Another decohering effect is the residual velocity experienced by the condensate
after loading it into the magnetic wave guide. As mentioned earlier, this residual
motion causes efficiencies of the reflection pulse between 80 - 100%. Clearly these
ineficiencies will contribute to the lack of conservation in the total number of atoms
in the experiment, consequently casusing the measurements of the fraction N0 /N to
be inaccurate.
Similarly, initial residual motion causes asymmetries in the splitting operation.
This means that after the split, one packet will have a greater number of atoms than
its counterpart. We observe that the difference in the number of atoms from packet
to packet varied as much as 20%. The asymmetry in the number of atoms will cause
one packet to have a larger self interaction energy. As a result, a relative phase shift
between the packets is introduced, reducing the visibility of the output. This phase
shift will fluctuate with velocity and increases with T , which is consistent with the
observed data.
Overall we see that V remains close to 0.5 for interferometer sequences whose
T value ranges from 0 to 44 ms. For longer T ’s the visibility substantially drops to
zero, suggesting a sharp drop of in the form of a step at T = 44 ms. However, the
errors in the measurement are too large to be certain whether this sudden change
is real, or just a reflection of the statistical noise.
5.7
Intrinsic Limits of the Output
Next we discuss and explore the intrinsic limits for the output signal of our interferometer. These limits include fluctuations in the number of atoms as a result of
the splitting ( and recombining) operation and the reduction of the interferometer’s
5.8. NUMBER FLUCTUATION
125
contrast due to the atomic interactions. We refer to the effect of the reduction
in contrast due to interactions as phase diffusion. It will become clear that phase
diffusion is an effect introduced as a direct consequence of the number fluctuation.
5.8
Number Fluctuation
When the atoms are split using the Bragg beam, the number of atoms N+ and N−
in their respective states | +2~ki and | −2~ki is not necessarily even. Due to the
asymmetry in the splitting, the packets will have different self interaction energy
causing a difference in phase between the two packets.
Moreover, if the number of atoms after the split in the | ±2~ki states fluctuates
randomly from run to run, so will the number of atoms N0 brought back to rest
upon recombination. This means that there will be a fluctuation in the fraction
N0 /N for a fixed φ from shot to shot. In the end the decrease in contrast of the
output signal seen in Fig. 5.3 will be reflected in a decrease in visibility given by
Eq. (5.12).
To understand how the number fluctuations come into the variation of the total
number of atoms in the separate clouds with N+ and N− , we make the following
definitions wich represent the translational states after the split (see Sec. 4.2.5):
| ψ+ i = | 2~ki
| ψ− i = | −2~ki
(5.16)
(5.17)
After the split of the condensate with N atoms, we assume the system can be
written in terms of a product of state vectors in different Hilbert spaces, where each
term of the product is a state representing a single atom which has some probability
to be in the +2~k and −2~k state. Consequently there will be N such terms in the
product such that:
·
¸⊗N
| ψ+ i + eiφ | ψ− i
√
| ψi =
(5.18)
2
where ⊗ is the Hilbert space product and φ is once again the relative phase shift
acquired by one of the wave packets. This time the phase shift acquired by the
packets travelling at −2~k is not due to a shift in the mirror position, but rather
acquired through some external agent during its propagation.
We can expand the above equation using the binomial coefficients given by the
choose function:
µ ¶
N
(5.19)
m
This function gives the number of ways you can arrange m things out of N independent of the order. In our case m is an integer for the number of atoms in the −2~k
126
CHAPTER 5. EXPERIMENTAL RESULTS
packet and N is the total number of particles. Writing out the binomial expansion
we get,
·
µ ¶
1
N
N
N −1
iφ
| ψi = N/2 | ψ+ i + N | ψ+ i
| ψ− ie + . . . +
| ψ+ iN −m | ψ− im eimφ +
m
2
i
. . . + N | ψ+ i| ψ− iN −1 ei(N −1)φ + | ψ− iN eiN φ .
(5.20)
To find the expectation value and the variation for the number of atoms in each
packet N+ and N− we define the following operators N̂+ and N̂− .
N̂+ =
N̂− =
N
X
i=1
N
X
| ψ+ ii i h ψ+ |
(5.21)
| ψ− ii i h ψ− |
(5.22)
i=1
which then allows us to obtain the average number of particles in each of the ±2~k
states. We can do this by applying hψ|N̂+ |ψi and hψ|N̂− |ψi giving hN+ i = hN− i =
N/2. Similarly, we can obtain the variance for measuring the number of atoms in
a packet corresponding to the +2~k or −2~k states. This is given by using the
result for the variance of a measurement whose probability is given by the binomial
distribution [17] as
1√
∆N+ = ∆N− =
N.
(5.23)
2
In the end, this result tells us the intrinsic statistical variation in N± from shot to
shot of the experiment and how we can expect the number of atoms in each packet
to vary as a function of N .
5.9
Atomic Interactions
We introduce the effects of atomic interactions by considering the Schrödinger equation using an approximation which models the interaction potential for atoms which
scatter off each other. The function
4π~2 a
U (rij ) =
|ψ(rij )|2
M
(5.24)
gives the interaction between atoms [46, 45]. Above, a is the s−wave scattering
length [15], M is the mass of 87 Rb and rij is the coordinate describing the distance
between two colliding particles. Plugging this interaction into the Schrodinger equation with the relative coordinate rij gives the Gross-Pitaevskii equation [35, 46, 45].
· 2 2
¸
~∇
4π~2 a
∂
2
+ Uext (r) +
|ψ(r, t)| ψ(r, t)
i~ ψ(r, t) = −
(5.25)
∂t
2M
M
5.9. ATOMIC INTERACTIONS
127
Above, Uext is the external potential sensed by the atoms. It should be noted that
the interaction of Eq. (5.24) depends on density of the atoms which is just the
probability |ψ|2 . Consequently, considering atom interactions in our interferometer
would add a phase shift
4π~at N±
φ± =
(5.26)
M V
per atom in the 2~k or −2~k wave-packet. Here we have assumed that for large N± ,
N± − 1 ≈ N± . Setting κ = 4π~at/M V , the total phase shift for N+ and N− atoms
will be given by:
η+ = κN+
η− = κN−
(5.27)
(5.28)
Because atomic interactions will add an additional phase shift to each wave packet,
the observed phase shift in the output state will be the difference in the phase
shifts for each packet η = κ(N+ − N− ). Accordingly, the output function derived in
Eq. (5.3) will have an additional phase induced by interactions such that:
N0
= A cos(2φ + η) + y0
N
(5.29)
As noted, the phase η depends on the difference between the number of atoms in
each packet before recombination. Therefore any uncertainty in the number of atoms
in each packet will be reflected in an uncertainty in the phase due to interactions
∆η.
With this in mind we seek to find how to characterize ∆η. Assuming the total
number of atoms is conserved then N = N+ + N− . This allows us to write N− =
N − N+ . Additionally we can define the quantity q which is proportional to η such
that,
q = N+ − N−
(5.30)
Substituting N− = N − N+ into q yields the expression q = 2N+ − N , in which case
the uncertainty in q is given by the following expression
∆q = 2∆N+
(5.31)
In this way we have found an expression for the uncertainty in the quantity q which
is directly proportional to the extra phase η acquired by the output signal of our
interferometer. We know that η = κq therefore the uncertainty ∆η = κ∆q. From
Eq. (5.23) we can get a result for the uncertainty ∆N+ which we can plug into
Eq. (5.31) to get
√
N
(5.32)
∆(N+ − N− ) =
√
(5.33)
∆η = κ N
128
CHAPTER 5. EXPERIMENTAL RESULTS
This uncertainty in the phase η acquired due to atomic interactions will cause
a reduction in the contrast and therefore the visibility of the output signal. The
effect can be described as follows. We can assume that the probability P (η) to make
a measurement with some value η is described by the Gaussian function which is
normalized over all η:
2
1
− η
√ e (∆η)2
P (η) =
(5.34)
∆η π
Given the above distribution for η, the output signal for a given value of N0 /N
will vary for a fixed value of φ. Hence we take the average of N0 /N over the different
vales of η.
N0 (φ)/N ∼ hcos(2φ + η)iη
Z ∞
2
1
− η
√ e (∆η)2 cos(2φ + η) dη
∼
−∞ ∆η π
2 /4
∼ e(∆η)
cos(φ)
(5.35)
(5.36)
(5.37)
The above equation shows that the signal obtained is equivalent to the original output given in Eq. (5.3) but with a coefficient which reduces its overall amplitude. As
a result, due to the atomic interactions, we observe a signal with reduced visibility.
This example illustrates in a very intuitive way how atomic interactions will cause
a “blurring” effect to the output signal. Moreover, an increase in the interaction
strength κ will increase ∆η hence decrease the visibility exponentially.
Chapter 6
Conclusion
The work presented in this thesis began in the summer of 2002. Specifically, I
and several others had the opportunity to begin the task of setting up most of the
apparatus used to carry out this experiment. Thankfully after four exciting years,
the goal of implementing a functional Bose-Einstein condensate interferometer was
achieved.
6.1
Achieved Interference
Our interferometer has demonstrated that in general we can successfully split the
condensate into two equal parts in the waveguide potential, then let the packets
separate for some distance to later recombine them while maintaining their quantum
coherence, Fig. 4.6. Moreover, upon recombination of the wave packets, the output
state of the interferometer matched the predicted model in Eq. (5.3). Hence, we
were able to achieve interference between matter waves.
As described by Fig. 5.4, we have achieved an output interference signal like
Fig. 5.3 which remains coherent for total interferometer times of T up to 44 ms.
This is demonstrated by the non-zero visibility which remained on average around
0.5, of the output signal for 0 < T < 44 ms. This means that while the wave
packets propagated along the wave guide, they remained in a coherent quantum
superposition for a total time of 44 ms.
Up to date most experiments dealing with coherent matter waves have had coherence times limited to 10 ms [12], however recent experiments ([54]) have reported
coherence times up to 200 ms.
In the latter case, G.-B. Jo et al. have loaded a 23 Na condensate into a single
wire magnetic trap on a chip with radial frequencies such that ωr = 2.1 kHz and the
vertical frequency ωz = 9 Hz. Using a radio frequency field ramp, they separated
the original trap into a double well potential in which the minima of each well were
separated by a distance of 8.7 µm. As a result of the split potential, the condensate
was separated into two packets, located at the minimum of each well. The two
129
130
CHAPTER 6. CONCLUSION
packets were held separated for varying times, then released from the trap causing
them to fall and interfere when their expansion caused them to overlap during the
time of flight. The phase shift between the packets was obtained by controlling the
relative height of the minima in the double well potential. The relative phase of
the packets was obtained by resolving the interference fringes of the overlapping
condensates using absorption imaging.
As their results demonstrate, they observe a dramatic decrease in the number
fluctuations δN for each packet due to inter-atomic interactions. Due to repulsive
interactions when the condensate is split, the atoms preferentially divide evenly
among the wells as this is more energetically favourable. In other words they observe
number squeezed states. In particular they observe squeezing by a factor of ≥ 10.
Consequently for magnetic potentials with trapping frequencies as mentioned above,
the coherence time of the wave-packets is increased by a factor of 100 as compared to
the maximum coherence times limited by phase diffusion effects Sec. 5.6. However,
number squeezing has the effect of increasing the quantum uncertainty in the phase.
Given that, we believe that the leading cause for the long coherence times observed in our interferometer is the weakly confining waveguide potential used for
the interferometer, see Sec. 5.7. Due to lower confinement forces, the density in the
waveguide is reduced, lowering the interaction energies which introduce unwanted
phase shits. These phase shifts cause the wavepackets to decohere, thus eliminating
the contrast observed in the output state. Given the effects described in [2], for a
condensate having a number of atoms NA ≤ 1.5 × 104 , we do not expect the phase
diffusion effects in Sec. 5.7 to reduce the contrast of our interferometer for operation
times of ≤ 1 s.
Olshanii et al. have noted that the interaction between atoms during the initial
splitting can cause a phase gradient in the different packets. As the atoms separate,
one end of a cloud interacts for a time corresponding to a full length travel through
the opposing cloud, while the end that separated first only interacts briefly, corresponding to a minimal travel time through the opposing cloud. If the condensate
has a chemical potential µ, and the separation time is τs then the differential phase
introduced is in the order of µτ /~. Due to this differential phase, when we apply the
recombination pulse using a certain φ, not all the atoms will transfer to the expected
translational state for that φ, therefore decreasing the visibility of the output signal.
For example, applying the Thomas Fermi approximation [45] given the density
in our waveguide, µ for our condensate is ≈ 2π~ × 10 Hz. This gives a differential
phase of 0.2 rad using a separation time τs = 3 ms. We can compare this to the
experiment by Wang et al. [12] which had a differential phase of 3.33 rad. This
demonstrates how a weakly confining trap reduces the differential phase.
To our advantage, the weakly confining trap used in this experiment also reduces
the condensate’s sensitivity to mechanical vibrations of the trap structure. Additionally, it reduces the required precision of alignment of the Bragg beam standing
wave to the guide axis.
6.2. LONG ARM SEPARATION
6.2
131
Long Arm Separation
Another important achievement for this experiment was the long separation between
the arms of the interferometer we had initially aimed for. Up to date, we believe
to have obtained the first literal image of a matter wave that has been split and
separated by a macroscopic distance of 0.26 mm, while still preserving quantum
coherence. In other words, the atoms in Fig. 5.5 are in a quantum superposition
of the translational states | ±2~ki. This is demonstrated by the a visibility of 0.6
Fig. 5.4 for an interferometer whose total interaction time is T = 44 ms.
In this experiment, the coherence demonstrated by the condensates brings more
experimental evidence of the wave-like nature of matter.
6.3
Future Adaptations
The large arm separation shown in this interferometer promises several different
future applications. For example, the long separation between packets may permit
one arm of the interferometer to enter an optical cavity and acquire a phase shift
due to to the electromagnetic field present. This type of experiment could be used
to measure the number of photons in the cavity using non-destructive techniques
similar to experiments by Nogues et al. [68]. In the case of such an interferometer,
it would be more sensitive to smaller phase shifts, permitting the atoms-photon
interaction to be off-resonant, thus making it a simpler technique. This type of
experiment would have applications in quantum communication and QED [69].
Other examples include taking advantage of the arm separation to bounce one of
the packets off a surface. The wave packet would bounce due to quantum reflection
meaning the phase shift due to this effect could be measured. This could be used
to measure effects like the Casimir-Polder force.
Finally, our laboratory plans to set up an experiment where the electric polarizability of 87 Rb can be measured. The long packet separation can be used to position
a set of electrically charged plates at the end of one arm of the interferometer. The
plates will be aligned with the waveguide, permitting one packet to enter it momentarily during the interferometer operation. Then, the electric field of the plates
will induce a phase shift in one of the packets that can be measured to obtain the
polarizability of 87 Rb.
We mentioned our limitations in Sec. 5.6. For this reason we are trying to improve
the efficiency of the interferometer operations with the Bragg beam by improving
the science cells’ windows, thus improving the uniformity of the beam. We also
wish to reduce the residual motion of the condensate after loading the waveguide
by varying the loading sequence. By controlling the magnetic fields we can avoid
unwanted resonant oscillations in the waveguide.
In conclusion, we have achieved and observed an atom interferometer with the
greatest arm spacing to date. We believe our experiment has demonstrated a fun-
132
CHAPTER 6. CONCLUSION
damental principle in quantum mechanics. It has done so by exposing the wave
nature of matter, hence matter’s ability to interfere like a wave. My hope is that
the work presented here will be of pedagogical value to anyone interested in science,
contribute to the understanding of quantum mechanics, science in general and the
human desire in understanding nature.
Appendix A
Temperature Interlock
We use a spherical quadrupole field to generate the magnetic trap to transfer the
atoms to the science cell and to create the TOP trap. In both situations we must
apply 750 A of current (using a custom built high current switch [70]) to obtain
the correct field gradient. This causes a large amount of heating in the coils (see
Sec. 2.4.3). The coils are made of copper tubing through which we can pump chilled
water to prevent overheating. However, the water cooling system requires constant
flow of water at 2.5 L/min. Otherwise, when operating at full current, the water
inside the coils can exceed 100◦ C and cause explosions of steam which cost hours
worth of cleaning optics.
For this reason we configured two interlock systems which were connected in
series. A water flow interlock monitored that the flow remained at the correct level
(see [21]), and a temperature interlock monitored the temperature of the copper
coils themselves. In this way, if any of these two conditions failed, the current to
the quadrupole coils would be turned off. So far the system has worked to great
satisfaction.
The circuit design of the temperature interlock is shown in Fig. A.1. It includes
the usage of a chip specially designed to work in monitoring thermocouple signals,
converting them to a convenient voltage output.
133
134
APPENDIX A. TEMPERATURE INTERLOCK
Figure A.1: Circuit diagram for temperature interlock. Thermocouple leads which
measure temperature are on the upper left corner, solid line represents constantan
(alumel) and dashed represents iron (chromel). The leads are connected to an
amplifier specially designed for thermocouple configurations. The output of the
thermocouple chip is calibrated to 10 mV/◦ C and is fed to a comparator. The
temperature signal is then compared to a set point which can be adjusted as needed.
If the temperature signal exceeds the set point, the relay opens. All the chips on this
circuit are labeled according to their data sheets. Each chip type can be identified
by the part number.
Appendix B
Control Panel
In order to control and synchronize the experimental apparatus used in making
BEC and performing the interferometry experiment, we use the Adwin-Pro real
time controller system.
The current Adwin-Pro controller functions as a stand alone box communicating
to a computer via an Ethernet connection. Adwin-Pro has 24 digital and 8 analog
channels available, doubling the number of apparatus that we could originally control
using the Adwin (computer card). The output bus, having twice as many channels
as the original controller, connects to a new version of the switch panel also made in
the lab see Fig. B.1. A description of the apparatus assigned to each digital/analog
channel is given in Table B.1.
There are two main functions for the switch panel. First, it allows users to
switch the control of a particular device between computer or manual for all digital
channels. In manual operation, the user can choose between a high (5V) or low (0V)
output. Additionally, because the switch component inside the panel introduces
unwanted fluctuations of the output signal, the panel includes a de-bouncing circuit
which filters out any unwanted oscillations that could pass on to the controlled
device. Details of the de-bounce circuit, its components and operation are shown in
Fig. B.2
135
136
APPENDIX B. CONTROL PANEL
Figure B.1: Set up of real time experimental control. Black dots are switches, clear
dots are LED’s, red/black dots with circles are inputs/outputs. (a) Computer is
connected to Adwin Pro via an ethernet connection. (b) Adwin Pro connected to
(c) the output control panel. The left side of the panel contains twenty four threeway switches with LED’s that can be set to manual (on/off) or computer control.
The respective outputs are on the right side in addition to synchronization inputs
and RS 232 control input.(d) Output/monitor panel for the eight analog channels,
user can select to monitor external inputs or internal analog channels. (e) BEC/ALT
output selector panel, user can switch the outputs for different experiments e.g. BEC
or alternative. Eight ethernet analog/digital alternate inputs
137
Figure B.2: De-bounce circuit for a single digital channel in the real-time control
panel. There are twenty-four digital channels, each having a copy of this circuit in
order to eliminate unwanted electrical noise when using the manual on/off setting.
The labeling on each circuit component matches the description found in their data
sheets. Each component is labeled with their respective part number. The Switch
on the top left indicates the pin connections being used (depicted in the 5 V (ON)
manual position).
7
8
1
2
3
4
5
6
Analog Channels
Double pass frequency (−10 V → 10 V)
Bragg beam Amplitude (0 V → 1 V)
Big B field current (0 V → 10 V)
Waveguide Quad Current (−5 V → 5 V)
Wave Guide bias current (−5 V → 5 V)
-EOM bias (0 V → 5 V)
-Waveguide ext.coils amplitude(−5 V → 5 V)
Double pass amplitude (0 V → −1 V)
Bragg beam current control (0 V = φ = 0, 1.1 V φ = π)
Table B.1: Description of the function associated with each digital and analog channel controlled by Adwin-Pro. 15
digital (out of the 24 digital channels) and all 8 analog channels were used. Digital channel “O” was used to trigger a
switcher which selected between two outputs on analog channel 6
G
Big B field on/off
H
Waveguide on/off
I
Pumping bias Field on/off
J
CCD trigger
K
Track move trigger
L
Waveguide Ext
M
Frame Grabber trigger
N
O Analog switcher doubler ch6.
P
Q
R
S
MOT Shutter
T
Bragg beam Shutter (fast)
U
V
W
X
RS232 out
A
B
C
D
E
F
Digital Channels
MOT + Re-pump AOM
Pumping beam AOM
Diode 2 AOM
MBR Probe AOM
138
APPENDIX B. CONTROL PANEL
Appendix C
Sequences
We include below the sequences used to make BEC and perform the interferometer operations using the digital and analog channel convention presented in Sec. B.
Both the BEC sequence and the interferometer sequence have standard subsequences
which are also given. It proved to be very useful and flexible to insert functioning
sub-sequences into the larger more complex sequences. It also gave an object oriented approach to building sequences just like subroutines do in normal coding.
loadcmot
Time [ms]
88
15
0
1
0.2
Active
acgks
agks
aks
ciks
bciks
1
2
0
0
0.5
0.6
0
0.05
3
0.25
0.25
0
3.2
3.2
Analog
4
0
channel
5
6
7
8
0
0
0
0
2
0
5 -0.3
5
0
5
0
Table C.1: Sequence showing steps used to load the CMOT. Blank space denotes
previous value
139
140
APPENDIX C. SEQUENCES
movetrap
Time [ms]
4
1
50
200r
1850
Active
gk
cgk
gk
gk
ga
1
2 3
-1.2 0 3.2
Analog channel
4
5
6
0
0
5
10
10
0
0
7 8
-1 0
0
0
Table C.2: Sequence showing steps used to move the translation stage holding the
spherical quadrupole field. Blank space denotes previous value, r denotes a linear
ramp from the previous value.
probesplit
Time [ms]
0.2
0.05
0.1
30
Active
js
djs
js
a
1
2 3
-0.07 0 0
Analog channel
4
5
6
0
0
5
5
5
7 8
0
0
0
-1
Table C.3: Sequence showing steps used to take absorption images of the condensate
in the waveguide. Blank space denotes previous value
splitpulse
Time [ms]
0.024
0.033
0.024
Active
ht
1
2
3
0.18
-0.016
0.18
Analog channel
4
5
6
7
8
Table C.4: Sequence showing steps used to split the condensate in the waveguide.
Blank space denotes previous value
reflectpulse
Time [ms]
0.067
Active
ht
1
2
3
0.27
Analog channel
4
5
6
7
8
Table C.5: Sequence showing steps used to reflect the condensate in the waveguide.
Blank space denotes previous value
141
interferometer
Time [ms]
loadcmot
movetrap
500
rs232 b ag
500
500
10000r
rs232 d agh
100
1000r
1000r
800e
250r
10
splitpulse
10
reflectpulse
20
reflectpulse
10
splitpulse
40
probesplit
26000
500r
40
probesplit
1850
rs232 a k
100
Active
1
ag
0
ag
agh
agh
0
2
0
3
Analog
4
10
0
agh
ht
2
1
0.007
-0.016
0
-0.016
het
-0.016
ht
-0.016
ht
-0.016
hjs
-0.07 -0.016
a
agh
ghjs
0
0
-0.07
k
acgks
0
0
0
0
0
5
-1
-5
5
-3
3.3
-3
5
8
0
0
0.8
0
0
0
0
0
3.3
0
0
0
0.25
0
7
10
8
agh
agh
channel
5
6
0
5
0
0
5
0
-1
0
0
0
0
0
0
0
Table C.6: Sequence showing 10-20-10 interferometer operation. Blank space denotes previous value and r denotes a linear ramp from previous value. Using e
denotes an exponential ramp. Pre-defined subsequences are used as subroutines.
Rs232 command denotes which evaporation sequence to use with the first letter and
remaining letters are channels that remain constant from the previous step. The RS
232 commands follow the format described in Table 2.3, different letters are used
for different evaporation sequences, see [48]
Appendix D
Image Analysis Program
Two image analysis programs were developed and thus used according to the experimental circumstances. The program Sponge was created to analyze images of
single atom clouds. However, when operating the interferometer, the resulting images contained multiple smaller clouds which became cumbersome to analyse using
Sponge.
For this reason we developed a second image analysis program called by the
acronym AI 3, meaning atom interferometer three. The program’s main feature
is the capability to analyse up to three images in a single run. In summary, the
program uses a two dimensional function described below to fit to each individual
cloud. The fit outputs the absorption and width parameters of the clouds needed
to calculate the number of atoms in each cloud.
Upon start-up, the program consists of a graphical user interface containing a
display for the image being analyzed (main image), 6 smaller displays for a zoom
of the three clouds and their resulting fit, 6 displays for the “x, y” cross-sections of
the fits and boxes for the parameters of the fits.
D.1
Fit Function
AI 3 uses an image S(x, y) which is derived from processing 3 images as described
in Sec. 2.8. A script program is automatically run after taking the last image in
the probing sequence which generates the final image to be analyzed by AI 3. The
function used to fit the two dimensional absorption profile S(x, y) is given by:
·
2
x−x0
F (x, y) = exp A − Be( wx )
2 ¸
y−y
− w 0
y
·
x − x0
×
10
¸4 ·
y − y0
10
¸4
(D.1)
where x0 and y0 are the cloud’s center position, wx and wy are the widths of the
cloud along their respective coordinate, A is an overall offset and B is the absorption
coefficient. AI 3 uses the Matlab built in function “fminsearch” to search for the
142
D.2. OPERATION
143
parameters which minimize the value of a χ2 between the model presented above
and the image values of the array S(x, y).
The last two factors in brackets are there to significantly reduce the χ2 in the
region outside the cloud. Therefore reducing the search performed by Matlab’s
“fminsearch” function.
D.2
Operation
The program is stored in two locations, in the desktop of the computer casslab 6
(set up in case the network drive is not accessible) and the network drive location
R:\\home\casslab\source\AI 3. The program is run by starting up Matlab and
typing AI 3 at the prompt. The directory containing the desired copy of AI 3 to
be run should be set in the path of Matlab. AI 3 has a startup file which sets the
working directory to the previously used directory from where images were analyzed.
Once the program is displayed, the user selects the image to be analyzed by
using the browse button. This also sets the working directory which is stored in the
startup file and displayed in the directory box.
The next step is to provide AI 3 with the centers for each cloud. Depending on
the output described in Sec. 5.1, the user should select the cloud with the highest
“visible” absorption coefficient (we will refer to this as the main cloud). This is
done by clicking the radio button (top, center or bottom) which corresponds to the
relative position of the cloud with strongest absorption, and then clicking the main
image on the center of the main cloud.
AI 3 has a built in function which tries to calculate the center of the two remaining clouds. It does so by using the time in ms set in the “time after rec” box,
which should be the time between the recombination pulse and the time the atoms
image was taken. Additionally, this function requires setting the “x calibration”
and “y calibration” boxes correctly. These boxes contain the number of pixels/mm
along each dimension corresponding to the magnification of each image. Usually
the calibration is done by taking an image of a ruler’s scale at the position of the
atoms.
AI 3 proceeds to draw three sub-images centered and zoomed around the calculated (and user provided) center for each cloud. If the sub-images are not properly
centered on the the clouds, the user can click on the desired new center of each
sub-image to recenter the sub-image. Every time the images are clicked, the corresponding centers are displayed in the “x o pos ” and “y o pos ” boxes (pos
indicates whether it is top center or bottom).
If previously not set (in the startup file), the user sets the “guess” value for each
parameter of the fits by entering the value into their respective boxes. For each
sub-image there are six different parameters. The parameters end with t, c, or b
denoting the top, center and bottom image respectively. The parameters correspond
144
APPENDIX D. IMAGE ANALYSIS PROGRAM
to those described in Eq. (D.1).
The user can also choose to fix the fit parameter by clicking the check box to
the left of each parameter box. In doing so, AI 3 will simply plug this value into
Eq. (D.1) and not vary this parameter during the fminsearch.
To perform the actual fits, the button “Analyze” is pressed. After some time,
the resulting fit parameters overwrite the guess values in their boxes. Additionally,
the fraction N0 /N is displayed in the box “N 0/N”. It calculates this number using
the B coefficients of each cloud.
AI 3 can also export the resulting fit parameters into a running Excel spread
sheet. The user can select the export row by inserting the row number in the
“Export Row” box and the export column by selecting a letter from the scroll menu
next to the parameter to be exported. To export, the user should check the box to
the right of the parameter to be exported.
As a result of the fit, AI 3 will display the results as a two-dimensional absorption
function for each cloud. The absorption for each cloud will be represented by a color
map shown in Fig. D.1 as a function of position. Additionally, a cross section through
the center position of each cloud along “x” and “y” is displayed comparing the fit
function with the measured absorption.
D.2. OPERATION
145
Figure D.1: Screen shot of AI 3. The main image is displayed on the top left.
The upper right boxes include: the working directory, browse button, “x” and “y”
calibration boxes and the “time after recombination” box. Below there are 6 boxes
for each cloud (top, center and bottom). Each box corresponds to the fit parameters
in Eq. (D.1). To the left of each parameter box is a check-box to choose whether the
parameter remains fixed during the fit. To the right there is a check box to export
and a scroll menu to choose the export column. Further below are the “Analyze”
button, “Export Row” box and the “N 0/N” box. The top row of remaining axes
display in adjacent pairs the measured cloud and the fit 2-D function. Similarly, the
bottom row displays adjacent pairs of x y cross-sections comparing the fit function
and the measured values through the centers of each cloud
Appendix E
Experiment Setup
Probably one of the most important experimental tools in our interferometry experiment is the laser. We utilize three different lasers in total. To create a MOT
we use a 10W, V-10 (Verdi) diode laser at 532 nm to pump a Ti:Sapphire crystal,
monolithic ring cavity laser MBR-110 outputting 1.3W of light ranging from 700nm
to 1000nm. The monolithic block design means the entire cavity is machined out
of a single special aluminium alloy, ensuring passive stability. Both units are manufactured by Coherent. The second laser is a diode laser Toptica DL 100 outputting
15-18 mW of light in the 779 nm to 785.2 nm optical range (not mode hop free),
we use this laser to create our standing wave Bragg beam that splits, reflects and
recombines the condensates during the interferometer sequence. We label this laser
Diode I. Finally, a custom laser system using a configuration including a diode in
conjunction with an external cavity was built. It outputs 30 mW - 40 mW of power,
lasing in the range of 780.2 nm to 780.5 nm. We use this system to provide additional re-pump light (780.232nm) to our MOT, increasing the number of atoms
trapped by the MOT and eventually loaded into the magnetic wave-guide. Most
importntly we used it during our optical pumping stage Sec. 2.4.2.
To generate the re-pump laser frequency used in making a MOT we use an electro
optical modulator at 6.8GHz. The output introduces additional frequency sidebands
to the beam offset by ±6.8GHz from the input frequency of 780.246nm. As seen in
Fig. E.1 the EOM does not introduce any spatial shift in the poynting vector of the
laser. The output is used for the vertical beams of the MOT. The sidebands are
created in a crystal inside the unit which interacts with the input light. In turn the
phonons are made and modulated by an RF signal which is controlled and locked
by a custom built driver circuit. Details on the specifics of the driver circuit can be
found in J.M. Reeves’ thesis project [21].
146
147
Figure E.1: A Top view of the layout of the main beam path, not to scale. Beam
starts at the bottom right corner after exiting the Mbr-110 (Ti:Sapphire) leakage
light enclosure. The letters denote beams that are matched up to to the same
lettered beam on Fig. E.2. Dashed lines denote a beam going in or out of the page,
labeled up or down. Although not shown, the glass chamber would be located above
the figure itself. Diode I and Diode II not shown.
148
APPENDIX E. EXPERIMENT SETUP
Figure E.2: Top view of the two glass chambers with MOT and imaging beams
configuration. This diagram continues the layout of the beams labeled by letters on
Fig. E.1. Dashed lines denote a beam going in or out of the page, labeled up or
down. Diode I and Diode II not shown.
149
Figure E.3: A top view of the entire optical table illustrating the beam configuration
for Diode I and Diode II. A flip mirror soon after the beam exits the MBR-110
leakage light enclosure, redirects the normal beam path of the beam to-wards a
fiber coupler which sends the light to a separate table. When the flip mirror is down
the beam path goes to the first λ/2 retarder of the beam path shown in Fig. E.1.
Appendix F
Mathematical Calculations
F.1
Two Level Solution
In order to obtain the time evolution of the state vectors in the two level problem
described in chapter 4, we must find the solution to the Schrödinger equation containing the effective Hamiltonian Ĥef f in Eq. (4.33). Writing out the time dependent
equations we obtain:
ċg = −i
Ω
d
2
Ω
d˙ = i∆d − i cg
2
(F.1)
(F.2)
Taking one more time derivative of Eq. (F.2) yields:
Ω
d¨ = i∆d˙ − i ċg
2
(F.3)
were we can plug in the corresponding expression for ċg found in Eq. (F.1). This
gives a second order equation for d such that:
µ
¶
Ω
Ω
d¨ = i∆d˙ − i
−i d
(F.4)
2
2
Ω2
d
(F.5)
d¨ = i∆d˙ −
4
which can be re-written as a second order homogeneous equation.
Ω2
d¨ − i∆d˙ +
d=0
4
(F.6)
Inspecting Eq. (F.6), we can extract a characteristic quadratic equation corresponding to the orders of the time derivatives of d in terms of the parameter m (which
150
F.1. TWO LEVEL SOLUTION
151
asumes a solution of the form eimt ):
m2 − i∆m +
Ω2
m=0
4
(F.7)
In the equation above, m is no longer used as an integer (as in previous chapters).
Solving the equation for m we obtain two solutions:
m± = i
´
√
1³
∆ ± ∆ 2 + Ω2
2
(F.8)
√
where m± denotes the two distinct solutions. We can define X ≡ ∆2 + Ω2 in order
to make our solution to m more elegant such that m± = i1/2(∆ ± X). Given the
solution to the characteristic equation, we can then write the solution to d(t) as a
sum of exponentials with the powers of m± t.
d(t) = Aeim+ t + Beim− t
¡
¢
d(t) = ei∆t/2 AeiXt/2 + Be−iXt/2
(F.9)
(F.10)
where A and B are constants. By adding the right constants and exponentials, we
can re-write the above equation such that we can make the replacements AeiXt/2 →
A0 cos (Xt/2) and Be−iXt/2 → B 0 sin (Xt/2).
d(t) = ei∆t/2 (A0 cos (iXt/2) + B 0 sin (iXt/2))
(F.11)
We can find A0 by using the initial condition d(0) = d0 , therefore allowing us to
write:
dt = ei∆t/2 (d0 cos (iXt/2) + B 0 sin (iXt/2))
(F.12)
Additionally we can obtain the time derivative of d using the chain rule giving:
µ
¶
−X
X
∆
i∆t/2
0
˙ = i d(t) + e
d0 cos (iXt/2) + B sin (iXt/2)
(F.13)
d(t)
2
2
2
where we can introduce the value for d(t) in Eq. (F.2) giving
µ
¶
∆
−X
X 0
Ω
i∆t/2
d0 sin (iXt/2) + B cos (iXt/2) . (F.14)
i∆d − i cg = i d(t) + e
2
2
2
2
If we apply the initial condition cg (0) = c0 for t = 0 we can obtain a result for B 0
such that:
i∆d0 − i
Ω
∆
X
c0 = i d0 + B 0
2
2
2
∆d
−
Ω
c0
0
B0 = i
X
(F.15)
(F.16)
152
APPENDIX F. MATHEMATICAL CALCULATIONS
Introducing the value of B 0 into Eq. (F.12) we obtain the final expression for d(t):
·
¸
Ω
i∆t/2
d(t) = e
d0 cos (iXt/2) + id0 sin (iXt/2) − i c0 sin (iXt/2) .
(F.17)
X
In order to find the solution for cg (t) we can plug the above result into Eq. (F.2)
and solve for cg (t). First we write:
cg = i
i
2 h˙
d − i∆d
Ω
(F.18)
then the right hand side of the above equation becomes:
·
∆
−X
i∆/2
d˙ − i∆d = −i d + e
−
d0 sin (Xt/2)
2
2
¸
∆
Ω
+i d0 cos (Xt/2) + i c0 cos (Xt/2)
2
2
h
1 i∆t/2
∆2
∆Ω
=
e
− i∆d0 cos (Xt/2) +
d0 sin (Xt/2) −
c0 sin (Xt/2)
2
X
X
i
−Xd0 sin (Xt/2) + i∆d0 cos (Xt/2) − iΩc0 cos (Xt/2)
h
=
∆
Ω i∆t/2
e
− ic0 cos (Xt/2) − c0 sin (Xt/2)
2
X
µ 2
¶
i
∆
X
+
−
d0 sin (Xt/2)
XΩ
Ω
(F.19)
We can simplify the above expression by inspecting the last term and noting
that:
µ
¶
∆2
X
1 ∆2 X 2
−
=
−
(F.20)
XΩ
Ω
X Ω
Ω
µ
¶
1 ∆2 − (Ω2 + ∆2 )
=
(F.21)
X
Ω
Ω
(F.22)
= −
X
so that:
¸
·
Ω
Ω
∆
i∆t/2
d˙ − i∆d = e
−ic0 cos (Xt/2) − c0 sin (Xt/2) − d0 sin (Xt/2) (F.23)
2
X
X
therefore, writing the final form of cg (t) we obtain:
¸
·
Ω
∆
i∆t/2
cg = e
c0 cos (Xt/2) − i c0 sin (Xt/2) − i d0 sin (Xt/2)
X
X
(F.24)
F.2. MATRIX ELEMENTS OF ĤS
153
In the end, the solution for the state vector [cg (t), d(t)] can be written in matrix
form giving:
·
·
¸
¸
cg (t)
c0
i∆t/2
M̂
=e
(F.25)
d(t)
d0
where:
·
M̂ =
Ω
∆
sin (Xt/2)
−i X
sin (Xt/2)
cos (Xt/2) − i X
Ω
∆
−i X sin (Xt/2)
cos (Xt/2) + i X
sin (Xt/2)
¸
(F.26)
Next, we show how to obtain Eqns. 4.41 and 4.42 which use the Hamiltonian
in Eq. (4.33) including an extra phase φ. If we have a Hamiltonian like that in
Eq. (4.40), when we include it in the Schrödinger equation we obtain:
d˙ = i∆d − iΩ cos (ωt + φ) cg
Ω
= i∆d − i (ei2ωt+iφ + e−iφ )cg
2
(F.27)
Similarly for cg we obtain:
ċg = −iΩ cos (ωt + φ)e−iωt d
¢
Ω ¡ iφ
= −i
e + e−i(2ωt+φ) d
2
(F.28)
We apply the rotating wave approximation, which means that the terms oscillating
at a rate ω average out to 0. In this way we get:
Ω
ċg = −i eiφ d
2
Ω
d˙ = i∆d − i e−iφ cg
2
F.2
(F.29)
(F.30)
Matrix Elements of Ĥs
Finally, the following calculation explains how to obtain the matrix elements of Ĥs
presented in Eq. (4.66). We use the coordinate representation of the three states
| ±2~ki and | 0i such that:
ψ0 (y) = hy|0i
ψ0 (y)ei2ky = hy|2~ki
ψ0 (y)e−i2ky = hy|−2~ki
(F.31)
(F.32)
(F.33)
where ±2~k is the recoil momentum given by the standing wave potential in Eq. (4.61).
Here the function ψ0 (y) is normalized over all y space. In this case ψ0 (y) represents
the initial wave-packet loaded into the wave-guide, hence it can be thought as the
154
APPENDIX F. MATHEMATICAL CALCULATIONS
condensate wave-function described by the Thomas Fermi model [45]. First we note
the completeness relation [15] which we will make use of:
Z
| yi ih yi | dyi = 1
(F.34)
where i is an integer representing the different positions in y−space. Additionally
we use the ortho-normality relation of our chosen basis (j is an integer equivalent
to i):
hyi |yj i = δij .
(F.35)
In calculating the matrix elements of Ĥs we note that the kinetic energy term can
be handled as follows:
p̂2
| 0i = 0
2M
p̂2
4~2 k 2
| +2i =
| +2i = 4~ωr | +2i
2M
2M
p̂2
4~2 k 2
| −2i =
| −2i = 4~ωr | −2i
2M
2M
(F.36)
(F.37)
(F.38)
where p̂2 /2M is the kinetic energy operator having p̂ as the momentum and M as
the packet’s mass. In this way we define the recoil frequency ωr = ~k 2 /2M just as
in Eq. (4.66). We use the convention where the rows of Ĥs are indexed according
to the momentum states in the order {0, +2, −2}, representing the order of the row
entry from top to bottom. Similarly the entries for the columns are indexed in the
same way from left to right.
Using the orthogonality relation Eq. (F.35), we can find out that the off-diagonal
kinetic energy terms of Ĥs vanish and the only non-zero terms are those corresponding to Ĥ2,2 and Ĥ−2,−2 . The corresponding kinetic energies for these two terms are
equivalent and equal to 4~ωr .
Similarly we seek to find the matrix representation for the potential shown in
Eq. (4.60) using the above basis. We proceed to obtain the matrix element h2|Ĥs |0i
by calculating the potential energy term using the operator V̂ = ~β cos (2ky + φ).
Therefore by inserting the completeness relation into h+2|V̂ |0i we get:
Z Z
h+2|V̂ |0i =
h+2|yi ihyi |V̂ |yj ihyj |0idyi dyj
(F.39)
yi yj
Z
Z
=
dyi h+2|yi i
dyj V (yj )hyi |yj ihyj |0i
(F.40)
yi
yj
Z
Z
−i2kyi
∗
dyj hyi |yj i V (yj )ψ0 (yj )
=
dyi ψ0 (yi )e
(F.41)
| {z }
yj
yi
δij
Z
=
yi
dyi ψ0∗ (yi )e−i2kyi V (yi )ψ0 (yi )
(F.42)
F.2. MATRIX ELEMENTS OF ĤS
155
Z
dyi |ψ0 (yi )|2 e−i2kyi V (yi )
=
(F.43)
yi
We drop the subscript i because it becomes superfluous, and introduce the
nate representation of the potential V (y) = ~β cos (2ky + φ)
Z
h+2|V̂ |0i =
dy|ψ0 (y)|2 e−i2ky ~β cos (2ky + φ)
Zy
¡
¢
β
=
dy ~ |ψ0 (y)|2 e−i2ky ei(2ky+φ) + e−i(2ky+φ)
2
Zy
¡
¢
β
=
dy ~ |ψ0 (y)|2 eiφ + e−i(4ky+φ)
2
y
Z
Z
iφ β
2
−iφ β
= ~e
dy|ψ0 (y)| + ~e
dy|ψ0 (y)|2 e−i4ky
2 y
2 y
coordi-
(F.44)
(F.45)
(F.46)
(F.47)
As shown above the matrix element yields two terms. Because the wavefunction
ψ0 is normalized, the first term gives ~βeiφ /2. However, the real part of the exponential in the second term oscillates rapidly in the region where |ψ0 |2 is non-zero,
therefore causing it to vanish when performing the integral.
β
h+2|V̂ |0i = ~ eiφ
2
(F.48)
We use the same procedure as illustrated above to obtain the matrix element for
h−2|V̂ |0i. An inspection of Eq. (F.39) reveals that changing the state h +2| to h −2|
yields:
Z
h−2|V̂ |0i =
dy|ψ0 (y)|2 e+i2ky V (y)
(F.49)
y
Z
¡
¢
β
=
dy ~ |ψ0 (y)|2 ei(4ky+φ) + e−iφ
(F.50)
2
y
In this case the first term is oscillating thus causing it to vanish. This gives the
result:
β
(F.51)
h−2|V̂ |0i = ~ e−iφ
2
For the term h−2|V̂ |+2i we can inspect Eq. (F.39) and introduce the states h −2|
and | +2i to obtain:
Z
h−2|V̂ |+2i =
dyi |ψ0 (yi )|2 ei4kyi V (yi )
(F.52)
yi
Z
¢
¡
β
(F.53)
=
dy ~ |ψ0 (y)|2 ei(6ky+φ) + ei(2ky−φ)
2
y
156
APPENDIX F. MATHEMATICAL CALCULATIONS
which means both terms oscillate where ψ0 is non-zero, meaning they average out
to zero when integrated over y yielding:
h−2|V̂ |+2i = 0
(F.54)
Additionally, each diagonal term of V̂ will only contain integrals having oscillating
terms (similar to Eq. (F.47)) that will average out to zero. Hence these diagonal
matrix elements vanish.
We can make use of the hermitian properties of Ĥs in order to calculate the
remaining terms of the potential V . Therefore if Ĥs† = Ĥs then it follows that
V̂ † = V̂ . Consequently we can use the previously found terms in Eqns. F.48, F.51,
and F.54 to find:
µ
¶†
β iφ
†
(h+2|V̂ |0i) =
~ e
(F.55)
2
β
h0|V̂ |+2i = ~ e−iφ
(F.56)
2
and
µ
β
(h−2|V̂ |0i) =
~ e−iφ
2
β
h0|V̂ |−2i = ~ eiφ
2
†
¶†
(F.57)
(F.58)
and
(h−2|V̂ |+2i)† = (0)†
h+2|V̂ |−2i = 0
(F.59)
(F.60)
giving us all the terms of V̂ . Because we know Ĥs = p̂2 /2M + V̂ , we can use the
matrix elements found above to construct the matrix representation of Ĥs according
to our previously defined convention described above. Finally we get:

β −iφ β iφ 
0
e
e
2
2
β
iφ
Ĥs = ~  2 e
(F.61)
4ωr
0 
β −iφ
e
0
4ωr
2
Bibliography
[1] Maxim Olshanii and Vanja Dunjko. Interferometry in dense nonlinear media
and interaction-induced loss of contrast in microfabricated atom interferometers. eprint cond-mat/0505358, 2005.
[2] J Javanainen and M Wilkens. Phase and phase diffusion of a split Bose-Einstein
condensate. Phys. Rev. Lett., 78:4675, 1997.
[3] Eugene Hecht. Optics. Addison Wesley, San Francisco, 4th edition, 2002.
[4] A. Einstein. Über einen die erzeugung und verwandlung des lichtes betreffenden
heuristischen gesichtspunkt. Annalen der Physik, 17:132, 1905.
[5] A Einstein. Quantentheorie des einatomigen idealen gases. Sitzungsber. der
Preuss. Akad. der Wissen., Physik.-Mathem. Klasse, page 3, 1925.
[6] Bose S. N. . Z. Physik, 26:178, 1924.
[7] M H Anderson, J R Ensher, M R Matthews, C E Wieman, and E A Cornell.
Observation of Bose-Einstein condensation in a dilute atomic vapor. Science,
269:198, 1995.
[8] C C Bradley, C A Sackett, J J Tollett, and R G Hulet. Evidence of BoseEinstein condensation in an atomic gas with attractive interactions. Phys. Rev.
Lett., 75:1687, 1995.
[9] K B Davis, M O Mewes, M R Andrews, N J van Druten, D S Durfee, D M
Kurn, and W Ketterle. Bose-Einstein condensation in a gas of sodium atoms.
Phys. Rev. Lett., 75:3969, 1995.
[10] H J Lewandowski, D M Harber, D L Whitaker, and E A Cornell. Simplified
system for creating a Bose-Einstein condensate. J. Low Temp. Phys., 132:309,
2003.
[11] V. P. Mogendorff E. J. D. Vredenbregt and H. C. W. Beijerinck. Coupledchannel analysis of Ne∗ thermalization cross section. Phys. Rev. A, 73:012712,
2006.
157
158
BIBLIOGRAPHY
[12] Y J Wang, D Z Anderson, V M Bright, E A Cornell, Q Diot, T Kishimoto,
M Prentiss, R A Saravanan, S R Segal, and S Wu. Atom Michelson interferometer on a chip using a Bose-Einstein condensate. Phys. Rev. Lett., 94:090405,
2005.
[13] J S Townsend. A Modern Approach to Quantum Mechanics. McGraw-Hill, New
York, 1992.
[14] R. K. Pathria. Statistical Mechanics. Butterworth-Heinemann, Oxford, 2nd
edition, 2000.
[15] Richard L. Liboff. Introductory Quantum Mechanics. Addison-Wesley, San
Francisco, 3rd edition, 1997.
[16] C. Itzykson and Jean-Bernard Zuber. Quantum Field Theory. McGraw-Hill,
New York, 1st edition, 1980.
[17] Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 9th
edition, 1964.
[18] C. Jönsson. Electron diffraction at multiple slits. Am. J. Phys., 42:4, 1974.
[19] A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa. Demonstration of single-electron buildup of an interference pattern. Am. J. Phys., 57:117,
1989.
[20] S.P. Walborn, M. O. Terra Cuhna, S. Pádua, and C. H. Monken. Double-Slit
quantum erraser. Phys. Rev. A, 65:033818, 2002.
[21] J M Reeves. An Atom Waveguide for Interferometry with a Bose-Einstein
Condensate of 87 Rb. PhD thesis, University of Virginia, 2006.
[22] M R Andrews, C G Townsend, H J Miesner, D S Durfee, D M Kurn, and
W Ketterle. Observation of interference between two Bose condensates. Science,
275:637, 1997.
[23] A Peters, K Y Chung, and S Chu. Measurement of gravitational acceleration
by dropping atoms. Nature, 400:849, 1999.
[24] J Schmiedmayer, M S Chapman, C R Ekstrom, T D Hammond, S Wehinger,
and D E Pritchard. Index of refraction of various gases for sodium matter
waves. Phys. Rev. Lett., 74:1043, 1995.
[25] D S Weiss, B C Young, and S Chu. Precision measurement of h/mCs based on
photon recoil using laser-cooled atoms and atomic interferometry. Appl. Phys.
B, 59:217, 1994.
BIBLIOGRAPHY
[26] Daniel A. Steck.
Rubidium
http://steck.us/alkalidata/, 2003.
159
87
D
Line
Data.
available
at
[27] James W. Rohlf. Modern Physics from aα to Z0. Wiley, New York, 1994.
[28] Stephen Gasiorowicz. Quantum Physics. Wiley, New Yok, 3rd edition, 2003.
[29] Hans A. Bethe and Edwin E. Salpeter. Quantum Mechanics of One- and TwoElectron Atoms. Springer-Verlag, Berlin, 1957.
[30] Letokhov and V.G V. S. Minogin. Laser radiation pressure on free atoms. Phys.
Rep, 73:1–65, 1981.
[31] D. Wineland and H. Dehmelt. Proposed 1014 δν/ν laser flourescence spectroscopy on T l+ mono-ion oscillator. Bull. AM. Phys. Soc., 20:637, 1975.
[32] D. Wineland and W. Itano. Laser cooling of atoms. Phys. Rev. A, 20:1521–1540,
1979.
[33] T. Hänsch and A. Schawlow. Cooling of gases by laser radiation. Opt. Commun,
13:68–71, 1975.
[34] S Chu. Nobel lecture: The manipulation of neutral particles. Rev. Mod. Phys.,
70:685, 1998.
[35] H J Metcalf and P van der Straten. Laser Cooling and Trapping. Springer,
New York, 1999.
[36] C. Cohen-Tannoudji. Atoms in strong resonant fields. page 3, Amsterdam,
1977. In Balian et al. , editor.
[37] E. Raab, M Prentiss, A. Cable, S. Chu, and D. Pritchard. Trapping of neutralsodium atoms with radiation pressure. Phys. Rev. Lett., 59:2631, 1987.
[38] W. Paul. Electromagnetic traps for charged and neutral particles. Rev. Mod.
Phys., 62:531540, 1990.
[39] Alan L. Migdall, John V. Prodan, William D. Phillips, Thomas H. Bergeman,
and Harold J. Metcalf. First observation of magnetically trapped neutral atoms.
Phys. Rev. Lett., 54(24):2596–2599, 1985.
[40] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and E. A. Cornell. Observation of anomalous spin-state segregation in a trapped ultracold vapor. Phys.
Rev. Lett., 88(7):070403, 2002.
[41] E. Majorana. Nuovo Cimento, 9(43), 1932.
160
BIBLIOGRAPHY
[42] W Petrich, M H Anderson, J R Ensher, and E A Cornell. Stable, tightly
confining magnetic trap for evaporative cooling of neutral atoms. Phys. Rev.
Lett., 74:3352, 1995.
[43] A S Arnold and E Riis. Bose-einstein condensates in ’giant’ toroidal magnetic
traps. J. Mod. Optics, 49:5861, 1999.
[44] S Gupta, K Dieckmann, Z Hadzibabic, and D E Pritchard. Contrast interferometry using Bose-Einstein condensates to measure h/m and α. Phys. Rev.
Lett., 89:140401, 2002.
[45] F Dalfovo, S Giorgini, L Pitaevskii, and S Stringari. Theory of Bose-Einstein
condensation in trapped gases. Rev. Mod. Phys., 71:463, 1999.
[46] Anthony J. Leggett. Bose-Einstein condensation in the alkali gases: Some
fundamental concepts. Rev. Mod. Phys., 73(2):307–356, 2001.
[47] J Söding, D Guery-Odelin, P Desbiolles, F Chevy, H Inamori, and J Dalibard.
Three-body decay of a rubidium Bose-Einstein condensate. Appl. Phys. B,
69:257, 1999.
[48] J M Reeves, O Garcia, B Deissler, K L Baranowski, K J Hughes, and C A Sackett. A time-orbiting potential trap for Bose-Einstein condensate interferometry.
Phys. Rev. A, 72:051605, 2005.
[49] David J. Griffiths. Introduction to Electrodynamics. Prentice Hall, Upper Saddle
River, New Jersey, 3rd edition, 1999.
[50] T L Gustavson, P Bouyer, and M A Kasevich. Precision rotation measurements
with an atom interferometer gyroscope. Phys. Rev. Lett., 78:2046, 1997.
[51] David M. Giltner, Roger W. McGowan, and Siu Au Lee. Atom Interferometer Based on Bragg Scattering from Standing Light Waves. Phys. Rev. Lett.,
75(14):2638–2641, 1995.
[52] Y Shin, M Saba, T A Pasquini, W Ketterle, D E Pritchard, and A E Leanhardt.
Atom interferometry with Bose-Einstein condensates in a double-well potential.
Phys. Rev. Lett., 92:050405, 2004.
[53] M Saba, T A Pasquini, C Sanner, Y Shin, W Ketterle, and D E Pritchard.
Continuous measurement of the relative phase of two Bose-Einstein condensates
using light scattering. Science, 307:1945, 2005.
[54] G-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, and D. E.
Pritchard. Long Phase Coherence Time and Number Squeezing of Two BoseEinstein Condensates on an Atom Chip. cond-mat/0608585.
BIBLIOGRAPHY
161
[55] K Bongs, S Burger, S Dettmer, D Hellweg, J Arlt, W Ertmer, and K Sengstock.
Waveguide for Bose-Einstein condensates. Phys. Rev. A, 63:031602(R), 2001.
[56] E W Hagley, L Deng, M Kozuma, M Trippenbach, Y B Band, M Edwards,
M Doery, P S Julienne, K Helmerson, S L Rolston, and W D Phillips. Measurement of the coherence of a Bose-Einstein condensate. Phys. Rev. Lett.,
83:3112, 1999.
[57] Y Torii, Y Suzuki, M Kozuma, T Sugiura, T Kuga, L Deng, and E W Hagley.
Mach-Zehnder Bragg interferometer for a Bose-Einstein condensate. Phys. Rev.
A, 61:041602(R), 2000.
[58] I.I. Rabi. Space Quantization in a Gyrating Magnetic Field. Phys. Rep., 51:652,
1937.
[59] M.H. Mittleman. Introduction to the Theory of Laser-Atom Interactions.
Plenum Press, New York, 1982.
[60] D. Suter. The Physics of Laser-Atom Interactions. Cambridge University Press,
Cambridge, 1997.
[61] J.D. Jackson. Classical Electrodynamics. Wiley, New York, 1999.
[62] R. Feynman, F.Vernon, and R. Hellwarth. Geometrical Representation of The
Schrödinger Equation for Solving Maser Problems. J. App. Phys., 28:49, 1957.
[63] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. Pritchard. Bragg scattering of atoms from a standing light wave. Phys. Rev. Lett., 60:515518, 1988.
[64] D M Giltner, R W McGowan, and S A Lee. Theoretical and experimental
study of the Bragg scattering of atoms from a standing light wave. Phys. Rev.
A, 52:3966–3972, 1995.
[65] M Kozuma, L Deng, E W Hagley, J Wen, R Lutwak, K Helmerson, S L Rolson,
and W D Phillips. Coherent splitting of Bose-Einstein condensed atoms with
optically induced Bragg diffraction. Phys. Rev. Lett., 82:871, 1999.
[66] SJ Wu, YJ Wang, Q Diot, and M Prentiss. Splitting matter waves using an
optimized standing-wave light-pulse sequence. Phys. Rev. A, 71:043602, 2005.
[67] O. Garcia B. Deissler K. J. Hughes J. M. Reeves C.A. Sackett. Bose-Einsteincondensate interferometer with macroscopic arm separation. Phys. Rev. A,
74:031601(R), 2006.
[68] G Nogues, A Rauschenbeutel, S Osnaghi, M Brune, J M Raimond, and
S Haroche. Seeing a single photon without destroying it. Nature, 400:239,
1999.
162
BIBLIOGRAPHY
[69] P R Berman, editor. Cavity Quantum Electrodynamics. Academic Press, San
Diego, 1994.
[70] B Deissler. A Magnetic Trap for Evaporative Cooling of Rb Atoms. PhD thesis,
University of Virginia, 2003.