Download MICROWAVE MANIPULATION OF COLD ATOMS

MICROWAVE MANIPULATION OF COLD
ATOMS
Thesis submitted in partial fulfillment of the requirement for the degree of master of science
in the Faculty of Natural Sciences
Submitted by: Ruti Agou
Advisor: Prof. Ron Folman
Department of Physics
Faculty of Natural Sciences
Ben-Gurion University of the Negev
August 15, 2011
Abstract in hebrew
1
Abstract
The coherent control of a two-state quantum system stands at the base of the field of quantum optics. In this work, I have utilized cold rubidium atoms as such a system, and developed
a scheme for imaging multiple Rabi oscillations simultaneously. This may later be used for
fundamental studies as well as metrology.
For this purpose I designed and built, starting with a ’naked’ optical table, a versatile laboratory system capable of trapping 109
87
Rb atoms, cooling them to a temperature
of 60 µK, and manipulating them by direct microwave radiation tuned to the ground-level
hyperfine splitting. In this thesis I describe the experimental setup, including lasers, optics, electro-optics, vacuum, electronics and computers. Following the theoretical review,
I present properties of the cooled atomic cloud, the atomic population oscillations (Rabi
frequency of 10-20 kHz) induced by the microwave (MW) field, and the simultaneous (’one
shot’) observation of multiple spatial Rabi fringes, which ”move” as a function of time as the
populations in the two states evolve. I also present the dependence of the Rabi frequency,
and thereby state populations, on the spatial position of the atoms. I conclude with a summary and brief outlook.
The work described in this thesis was done in parallel to the work of P. Böhi et al.1 ,
who reported a technique that uses ultracold atomic clouds for mapping at a single shot,
microwave near field distribution around a coplanar wave-guide integrated on an atom chip.
The main novelty so far in my results is the ability to map at a single shot, microwave fields
generated from an horn antenna at near and far fields. Since my setup is completely different
and has no atomchip in it I had at my disposal more atoms, enabling the observation of more
fringes. In addition, as I could move and rotate the source of microwave I could also observe
a variety of dependencies of the fringe behavior on parameters such as time, field direction
and polarization.
1
App. Phys. Lett. 97, 051101 (2010)
2
Acknowledgements
During my work on this thesis as part of the AtomChip Group, I had the honor of
meeting and working with brilliant scientists which have made this work possible and to
whom I owe my deepest gratitude.
First of all, I would like to thank my supervisor, Professor Ron Folman, who has given
me the opportunity to learn and apply the knowledge of the quantum atomic physics world.
He guided me through my work with great patience and kindness.
I am grateful to Dr. David Groswasser and Amir Waxman for spending long days, and
even longer nights in the lab with me, offering me assistance whenever necessary and sharing
their rich toolbox of knowledge. It was a great pleasure to work with you, and I thank you
both extensively.
I also want to thank Meny Givon for his professional assistance and his considerable
experience in the Lab, which helped me progress in my studies. Your availability for conversations about physics, politics and religion, during coffee breaks, forged an agreeable and
colourful atmosphere enriching my experience.
I would like to give specials thanks to Shimi Machluf and Ramon Szmuk, the BEC
guys, for always being willing to help. Shimi: thank you for being my teacher, who was
always ready to answer my trivial, and maybe even sometimes annoying, questions. Ramon:
thank you for sharing your software knowledge (especially in 3D). Your love of physics inspired me, and your true friendship means a lot to me.
Many thanks is owed to the PhDs in our group: Yoni Japha, Tal David, Ran Salem,
Julien Chabé and Mark Keil for sharing their great knowledge with me. Also, I would to
thank Jonathan and Asif from Weizmann institute, for lending me the MW amplifier and
therefore allowing me to proceed with my research.
3
My lab work could not have been completed without the supportive, continual help of
our electronics-wizard Zina, who built and fixed most of the electronic devices used in my
work.
None of this would have happened without the endless love, support and backing of
my dear family: my parents, Daniel and Anne Marie Agou, to whom I owe so much, and my
brothers Nathaniel and Jonathan, who were always there for me (Je vous aime énormement).
Most important is the warm presence of my beloved husband Tomer: I thank you for
understanding the endless hours I spent in the lab and the sudden moments of enthusiasm
from physics which occasionally came over me. My appreciation can not be expressed by
words, and for this I love you so much.
And above all, thanks to the infinite kindness and awareness surrounding me, reminding
me on each step of the way just how small I am...
4
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1 Introduction
12
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2
Thesis content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2 Theory of Rabi Oscillations in a Two-state Atom
15
2.1
Bloch sphere representation . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
The Schrödinger equation for a two-state atom . . . . . . . . . . . . . . . . .
16
2.3
Rabi oscillations induced by microwave radiation
18
. . . . . . . . . . . . . . .
3 Theory of Laser Cooling
87
21
3.1
The
Rb alkali metal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2
Theory of light-matter interaction . . . . . . . . . . . . . . . . . . . . . . . .
23
3.2.1
Light force on a two-level atom . . . . . . . . . . . . . . . . . . . . .
23
3.2.2
Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.2.3
Doppler and sub-Doppler limit . . . . . . . . . . . . . . . . . . . . . .
24
Magneto-Optical Trap (MOT) . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.3
4 Experimental Apparatus
29
4.1
Ultra-high vacuum setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.2
Atom source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.3
Lasers and optics system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5
4.3.1
Laser lock and spectroscopy . . . . . . . . . . . . . . . . . . . . . . .
32
4.3.2
Optical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.3.3
Science chamber
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.4
Static and AC fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.5
Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.6
Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.6.1
Temperature and gravitation measurements . . . . . . . . . . . . . .
48
Experimental control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.7
5 Rabi Oscillations in Cold Atoms
53
5.1
Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.2.1
Inducing Rabi oscillations simultaneously over space . . . . . . . . . .
55
5.2.2
Inducing Rabi oscillations over time . . . . . . . . . . . . . . . . . . .
57
5.2.3
The Rabi frequency’s dependence on the position in the cloud in the
5.2.4
far field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
The fringe pattern dependence on the position of the MW antenna .
61
6 Summary and Outlook
63
Bibliography
66
6
List of Figures
2.1
The Bloch Sphere
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Rubidium 87 D1 and D2 transition hyperfine structure.
. . . . . . . . . . . . . . . .
20
3.1
Illustration of a MOT in 1D. The detuning δ is for atoms at rest at the trap’s center. Due to
the Zeeman shift of magnetic sublevels, and the arrangement of laser polarizations, atoms
are driven to the trap’s center. Spatial confinement and cooling are obtained simultaneously
[Met99].
3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MOT pictures from the experiment. (left) Image of the
87
26
Rb cloud as it appears in the
vacuum chamber. (right) Fluorescent image of the MOT using a CCD. The cloud size is
13 mm (2σz ) and contains 9 · 108
3.3
87
Rb atoms, in both images.
. . . . . . . . . . . . .
27
Photograph of optical molasses taken by fluorescent imaging using a CCD. Three orthogonal
. .
28
. . . . . . . . . . . . . . . . . . .
30
MOT beams cross in the center where atoms which are further cooled, glow brightly.
4.1
Image of the vacuum system and science chamber.
4.2
Laser Box. (top) Three home-made lasers: The Cooler and Repumper utilized for the
cooling cycle (the Cooler is also used for imaging and optical pumping). The Tapered
Amplifier (T.A.) amplifies the injected Cooler beam (drawn in red) up to 1.3 W. (bottom)
Polarization spectroscopy scheme. A 3 mW beam is split off from the main laser beam, and
used for spectroscopy. The Rb vapor cell is covered with two layers of µ-metal shielding.
Two photo-diodes are used to produce the error signal. The remaining 30 mW in the main
beam is mode-matched by a telescope to allow for the best injection into the T.A. Then,
the amplified beam exits on the right side of the box, passing through a cylindrical lens to
fix the dispersion of the horizontal axis of the beam with respect to the vertical axis.
7
. . .
34
4.3
Two
87
Rb spectroscopy images used for the laser frequency stabilization: Cooler spec-
troscopy (left), Repumper spectroscopy (right). An absorption spectroscopy signal (top) is
compared to a polarization spectroscopy signal (bottom).
4.4
. . . . . . . . . . . . . . . .
35
Mechanical and thermal housing for the tapered amplifier. (left) Exploded diagram. The
blocks are made of copper (the centring rods are of steel). From [Nym06]. (right) Photograph of our assembled system.
4.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
The frequencies of the four laser beams used for the experimental cycle are superimposed on
the level scheme of the
87
Rb D2 transition hyperfine structure. The ground state levels are
denoted by F, and the excited state levels by F0 . Dashed lines correspond to the cross-over
peaks (1x2, 1x3), to which our lasers are locked. Arrows indicate these lock points, as well
as the frequency of each laser component after passing its AOM.
4.6
. . . . . . . . . . . .
37
Optics behind the laser box. (top) Image of the setup. The Cooler beam is divided into three
paths through three different AOMs, Cooler (blue), Imaging (yellow), Optical Pumping
(red). The Repumper’s AOM is not shown. (bottom) Optical layout from the laser box
into the optical fibers.
4.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
A 3D-image of the science chamber and all relevant optics surrounding it. The red beam
denotes the Cooling and Repumping beams. The gold and purple beams denote the Optical
Pumping and Imaging beams respectively. Both exit from a collimator at the end of the fiber
with a diameter of 1.5 cm. They are circular polarized by a λ/4 wave-plate and combined
through a Beam Splitter (B.S.) cube. The red beam exits from the output collimator of the
fiber with a diameter of 2.5 cm and is split by two passes through a λ/2 wave-plate and a
P.B.S. into three orthogonal beams. The beams are circular polarized by a λ/4 wave-plate
before entering the chamber and are retro-reflected by a mirror and another λ/4 wave-plate.
Also shown in the image are the CCD capturing the Imaging beam, the horn antenna in
grey, and two sets of magnetic coils. The anti-Helmholtz configuration is shown in brown
and the Helmholtz configuration in green. (inset) The chamber with its 3D axis is shown
(Only the Imaging beam is drawn).
. . . . . . . . . . . . . . . . . . . . . . . . . .
8
41
4.8
Optical pumping scheme. Right circularly-polarized light (σ + ) pumps atoms to the highest
mF level of the ground state |F = 2, mF = 2i (yellow arrow) by absorbing and re-emitting
photons until they reach the ”dark state” where they no longer scatter photons (dashed yellow arrow). Atoms absorb ∆mF = +1 but re-emit ∆mF = ±1, 0 (green arrow). Therefore,
on average, ∆mF = 0 for re-emission.
4.9
. . . . . . . . . . . . . . . . . . . . . . . . .
43
Number of atoms in a time-of-flight (T.O.F.) measurement (by absorption imaging) vs.
polarization of the imaging beam determined by the angle of a λ/4 wave-plate. First, the
Optical Pumping and Imaging beams were adjusted to be both σ + polarized. Thus, the
maximum number of atoms are imaged. As the λ/4 wave-plate, which is placed before the
Imaging beam, is rotated, fewer photons are absorbed giving rise to a smaller calculated
value for the number of atoms, until the beams are of opposite polarization, and the minimum number of atoms is counted. The top and bottom insets show the relevant transitions
q
q
1
and their Clebsch-Gordan coefficients, at the maximum ( 12 ) and minimum ( 30
) atom
numbers. The non-smooth curve is assumed to be caused by fluctuations in the MOT atom
number.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.10 Absorption imaging. As the imaging axis is X, what is measured is the column optical
density along the X axis. (left) An image of the laser beam after propagating through
the atomic cloud. The shadow of the cloud is clearly visible. (middle) The corresponding
background image (no atoms). (right) The resulting absorption image. All scales are in [mm].
46
4.11 Imaging analysis interface. The screen is divided into two main parts. On the left, there is
the image with two 1-dimensional cuts along the center of the cloud (absorption imaging
of the initial MOT after 4 ms T.O.F.). The controls of the software and the fit results are
on the right.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.12 Time-Of-Flight measurement after the Molasses stage: Temperature (top) and gravity (bottom). The X and Y-axes in the figure correspond to the original Y and Z-axes respectively,
as they appear in the equations.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.13 Experimental control interface. Analog channels (on the left) and digital channels (on the
right).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
51
5.1
Experimental sequence of the T.O.F. stage from an oscilloscope screen (without the molasses
stage). The Repumper beam (signal from P.D.) is turned off at 0.001 s (which is 1 ms before
the end of the 10 s MOT stage. At 0.002 s, the Cooler beam (signal from P.D.) and the
anti-Helmholtz coils (TTL signal) are turned off. At 0.003 s and 0.00385 s, the Helmholtz
coils and the MW pules (TTL signals) are turned on for 4 ms and 0.4 ms respectively. At
0.00585 the first Imaging pulse (TTL signal) is applied for 0.4 ms. The second Imaging
pulse is not shown since it is activated 30 ms after the first pulse. Due to the limited
number of channels on the oscilloscope, the MW and Imaging pulses are shown on the same
TTL signal. The slow drop times of the light beams (yellow curve) are due to a slow P.D.
(in reality these times are of the order of micro seconds). In the experiment, the Cooler
and the Repumper beams are combined through a P.B.S and then directed to a P.D. The
differences between their polarisations cause differences in the intensities impinging on the
P.D. Therefore, the Cooler and Repumper curves look as if they are of the same power
5.2
although the Repumper is much weaker.
. . . . . . . . . . . . . . . . . . . . . . . .
Absorption image in the YZ plane of the
87
54
Rb cloud in the T.O.F. (free fall) stage for a 2
ms MW pulse. Ten fringes are visible. The MW antenna is positioned 25 cm from the left
side of the fringes, radiating electromagnetic waves in the Z-direction, through the cloud,
and generating multiple Rabi oscillations that can be viewed simultaneously. The atomic
cloud consists of 27 · 106 atoms and it is 7 mm long and 5 mm wide.
5.3
. . . . . . . . . .
56
Graph of optical density vs. the position, for different pulse durations. Each curve is
obtained at a different MW pulse duration starting at a 700 µs pulse with intervals of 50 µs
(from bottom to top). The red linear curve is aligned over the main fringe, which ’slides’
to the right as the pulse duration increases.
5.4
. . . . . . . . . . . . . . . . . . . . . .
58
Absorbtion images of the atomic cloud in the T.O.F. stage for different MW pulse durations.
It can be observed that the fringes are ”moving” - when the first fringe starts to appear
the last one disappears (from left to right).
10
. . . . . . . . . . . . . . . . . . . . . .
58
5.5
Rabi oscillation graphs. Two measurements were taken at two different distances of the
MW antenna: 10 cm (top), 25 cm (bottom). Each data point represents the same pixel
position in the cloud at different MW pulse time durations. The scales in the X-axes are
different for the two fits because they were taken at different times after the MOT stage.
The curves are fits to an exponentially decaying cosine. The Rabi frequency and the decay
constant are: (top) Ω = 21.6 ± 0.03 kHz, T2 = 0.907 ± 0.04 ms, (bottom) Ω = 10 ± 0.5 kHz,
T2 = 0.937 ± 0.025 ms .
5.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A graph of Rabi frequency as a function of position in the cloud. For each pixel position,
the Rabi frequency was calculated and plotted. The curve was fitted to a linear curve.
5.7
59
. .
61
Fringe patterns as a function of the MW antenna’s spatial position (drawn beside the
images). The antenna is positioned at the same YZ-plane, perpendicular to the imaging
beam, but at different distances from the cloud: (a), (b) 10 cm. (c) 25 cm.
6.1
. . . . . . .
62
Images of fringes at different BM W angles. The MW antenna is rotated 360o across its
main axis, which is perpendicular to the quantum axis. (a) 0o (b) 40o (c) 100o (d) 130o
(e) 180o (f) 210o (g) 280o (h) 310o (i) 350o . At 0o the magnetic field’s component of the
electromagnetic wave was parallel to the quantum axis. The number of atoms in images
(c) and (g) is 10 % of the cloud imaged in (a).
11
. . . . . . . . . . . . . . . . . . . . .
65
Chapter 1
Introduction
1.1
Background
Since the quantum phenomenon of Rabi oscillations was first discovered by the Nobel Laureate Isidor Isaac Rabi, who was recognized in 1944 for his discovery of NMR [Rab37], coherent
manipulation has been the subject of many research studies. Two-level quantum systems
and their coherent manipulation are essential to todays atomic clocks and tomorrows quantum computers. The coherence lifetimes in these systems can be measured via the decay of
population oscillations when subjected to a resonant external field.
The field of laser cooling and manipulation of atoms has been one of the most active
fields in physics in recent years. Atoms from room-temperature vapour can be cooled to
temperatures as low as a few nano-Kelvin. At such low temperatures, the wave properties of
the atoms become relevant, giving rise to completely new phenomena such as Bose-Einstein
condensation and allowing experimentation where the matter waves interfere just as usual
waves do. In 1997, the Nobel Prize in physics was awarded to C. Cohen-Tannoudji, S. Chu
and W. D. Phillips for their contribution in this field [Chu98]. In 2001, the Nobel Prize was
presented to E. A. Cornell, W. Ketterle and C. E. Wieman for their observation of BoseEinstein condensation in ultracold dilute atomic gases [Cor02]. In 2005, J. L. Hall and T. W.
Hänsch received the Nobel Prize for the invention of methods for precision spectroscopy that
may lead for example to new clocks based on optical transitions of ultracold atoms [Hal05].
12
The main advantages of neutral alkali atoms as controllable quantum systems are their
weak interactions with the environment and their convenient spectrum of internal energy
levels. The hyperfine interaction between the electron spin and the nuclear spin splits the
neutral alkalis ground state into two levels, whose frequency difference is on the order of
several GHz. The energy difference between these levels and the first excited state corresponds to hundreds of THz. Therefore, alkali atoms such as rubidium are widely used to
realize a two-state system. Moreover, the natural lifetime of their ground state is extremely
long, giving rise to small energy level uncertainties, and therefore, transitions between their
ground-state hyperfine manifold are chosen as a frequency standard in atomic clocks [ACC].
Nowadays, many laboratories and companies are realizing two-state systems by utilizing neutral [Smi02] and charged atoms [Lei03], superconducting circuits [Orl04], nuclear
spins [van01], and also atoms or NV centers inside solid lattices [Bal09]. Technological applications range from clocks [Aka08], magnetometers [Tay08] to quantum communication
[Bet05] and computing [Ben00]. Relevant to this work is a recently reported technique for
mapping microwave field using ultracold atomic clouds on an atom chip [Böh10]. In neutral
atoms, most fundamental studies have transitioned from hot to cold atoms in order to avoid
collisional broadening, Doppler broadening, ”aging” due to chemical processes, and dephasing due to collisions with cell walls.
In our ’Atom chip’ group at Ben-Gurion University, we have previously observed Rabi
oscillations in hot atoms [Wax07, Avi09]. The goal of this work is to build a cold atom
apparatus, to observe Rabi oscillations in cold atoms, and to find new schemes that may
generate new understanding and applications. In this thesis I describe the coherent manipulation of the 5s1/2 ground state manifold in
87
Rb. The driven microwave field was chosen
to be on-resonance with the transition |F = 1, mF = 0i → |F = 2, mF = 0i since, to a first
order approximation, it is not sensitive to magnetic fields. In the presence of an external
magnetic field, the degeneracy of the hyperfine structure is lifted to yield the known Zeeman
splitting, which in
87
Rb, amounts to about 0.7 MHz/G. Using a light field, the atoms are
13
prepared in the |F = 1, mF = 0i state, and with a MW field the atoms are coherently manipulated to yield Rabi oscillations. For this purpose I designed and built an experimental
setup and a corresponding measurement technique, the details of which will be described in
the forthcoming text.
1.2
Thesis content
The structure of the thesis is as follows:
• Chapter 1: An introduction to the field of coherent manipulation in cold atoms is
presented along with the experimental motivation for utilizing the cold atom system.
• Chapter 2: Theoretical concepts are discussed. I review the Bloch sphere representation and present the quantum-mechanical treatment of the two-state atom in the
presence of external interaction. Next, I review the theoretical background necessary
for understanding the physics behind my work i.e Rabi oscillations induced by a direct
microwave radiation.
• Chapter 3: The theory of laser cooling is presented. I begin with the introduction of
the
87
Rb alkali metal atoms, and the light forces interacting with atoms causing them
to slow down, and thereby cooling them. Then, a review of the limits of the cooling
temperature and the concept of the Magneto-Optical Trap, is presented.
• Chapter 4: Experimental concept and results for each step in the sequence of events
used to generate the cold atomic cloud are presented, including the laser and magnetic
fields, imaging and detection setup, the UHV, the atomic sources, and the various
methods of analysis I use in order to quantify the results.
• Chapter 5: The developed scheme of imaging multiple Rabi oscillations simultaneously
and the observed oscillations of the population in the
87
Rb ground state hyperfine
levels induced by a direct microwave radiation, is presented. I describe the results as
a function of different parameters.
14
Chapter 2
Theory of Rabi Oscillations in a
Two-state Atom
Most of the main results discussed in this thesis are explained by the two-level model system,
which is described by the Schrödinger picture. The two-level model system results in the
well-known Optical Bloch equations which are briefly described in this section. This section
also discusses the response to an applied external field and analyzes quantum phenomena
such as Rabi oscillations. This section closely follows [Met99, Sho90], unless otherwise noted.
2.1
Bloch sphere representation
In order to visualize the coherent evolution of the two ground-state hyperfine levels of the
87
Rb atom, which serves as our two-state system, we apply the Bloch sphere (Fig. 2.1)
representation. By definition, the north and the south poles of the Bloch sphere represent the
two pure states |0i and |1i. In our system these states are defined as |0i ≡ |F = 1, mF = 0i
and |1i ≡ |F = 2, mF = 0i. All other points on the sphere can be represented as a
superposition state, namely
θ
θ
|Ψi = cos |0i + eiφ sin |1i,
2
2
(2.1)
where φ and θ are the spherical coordinates. The evolution of the atomic state over time
is expressed in the Bloch sphere by the variation of φ and θ. In this work, manipulation of
15
Figure 2.1: The Bloch Sphere
the atomic state, |Ψi, is achieved via a direct coupling between an atom and a microwave
radiation field.
2.2
The Schrödinger equation for a two-state atom
A two-state atom can be described using a time-dependent Schrödinger representation:
Ψ(t) = e−i
E0 t
~
C0 (t)|ψ0 i + e−i
E1 t
~
C1 (t)|ψ1 i,
(2.2)
where the wavefunction Ψ(t) is expressed as a superposition of the two orthogonal eigenvectors:
hψm |ψn i = δn,m .
(2.3)
The energy En denotes the eigenvalue, and the complex coefficient Cn (t) denotes the amplitude of the eigenvector ψn . These two time-dependent functions are probability amplitudes,
where the probability of finding an atom in state n at time t can be given by:
Pn (t) = |Cn (t)|2 ≡ |hψn |Ψ(t)i|2 ,
(2.4)
following the condition that the probabilities should sum to unity, i.e.
1 = |C0 |2 + |C1 |2 .
(2.5)
16
Hence, under no interactions between the atom and its environment, the system is stationary, and the probability of finding the system in any one of the states is constant.
In order to solve the time-dependent evolution of the system, we introduce the perturbation Hamiltonian (in our experiment, this perturbation is the microwave radiation)
H(t) = H0 + V (t).
(2.6)
To describe the coupling between the two states due to their interaction, we calculate the
matrix elements of V̂ in the basis of H0 , namely
Vnm = hψn |V̂ |ψm i,
(2.7)
where Vnm (t) = Vmn (t)∗ as V̂ is Hermitian. The evolution of the state vector Ψ(t) is described
by the time-dependent Schrödinger equation
~
∂
Ψ(t) = −iH(t)Ψ(t).
∂t
(2.8)
Introducing Eq. 2.6, we get the equation for the amplitudes Cn :

 


C
(t)
E
+
V
(t)
V
(t)
C
(t)
d
0
00
01
= 0
 0
.
i~ 
dt
C1 (t)
V10 (t)
E1 + V11 (t)
C1 (t)
(2.9)
Once V (t) and the initial conditions are specified, Eq. 2.9 provides the evolution of a twostate system subjected to an external interaction.
For a periodically changing interaction, such as the microwave radiation, the matrix
elements of V̂ are given by:

 V =V =0
00
11
 V = V ∗ = 1 |~Ω|e−i(ωt+ϕ)
01
10
(2.10)
2
where Ω, known as the Rabi frequency, is a real-valued quantity that denotes the coupling
strength of the interaction between the atom and the field (for an electric dipole interaction):
Ω=−
E →
−
h0| d |1i,
~
(2.11)
→
−
where h0| d |1i is the dipole matrix element between two non-degenerate states.
Applying the definition from Eq. 2.10 and transforming the reference frame to the rotating
17
wave picture (i.e. to a reference frame that rotates at the field frequency), the time-dependent
Schrödinger equation (2.9) becomes:





−δ |Ω|
C (t)
d  C0 (t) 
i
 0
,
=− 
dt
2
C1 (t)
|Ω| δ
C1 (t)
(2.12)
where δ = ω0 − ω is the detuning between the field frequency and the resonant transition
frequency ω0 . For the initial condition, where only the ground state is populated, C0 (t=0)=1
and C1 (t=0)=0, the differential equation 2.12 can be solved, revealing the probability of
finding the atom in the state |1i, at time t:
1 |Ω|2
P1 (t) = C1 (t) =
[1 − cos(Ω̄t)].
2 Ω̄2
2
(2.13)
This probability oscillates at the flopping frequency (or as it is sometimes referred to ”the
p
generalized Rabi frequency”) Ω̄ = |Ω|2 + δ 2 , which is dependent both on the laser intensity
(expressed by the Rabi frequency) and frequency (expressed by the detuning). The amplitude
of the oscillations, on the other hand, depends solely on the laser frequency (at δ = 0). In
the resonant case (ω = ω0 ), the population alternates between complete concentration in
state |0i and complete inversion to state |1i. As we increase the detuning, the amplitude of
the oscillations is attenuated and the flopping frequency is increased.
2.3
Rabi oscillations induced by microwave radiation
The hyperfine interaction splits the ground state of
87
Rb into two hyperfine levels F = 1
and F = 2. The frequency difference between these two ground states under zero external
magnetic field is stable enough to be chosen as an atomic reference to which some atomic
clocks are locked [Acc]. This frequency is given by ∆HF S = 2π·6.834682610 GHz (see Fig.
2.2). Since both ground states have a spherical orbital symmetry S, the electric dipole
transition is forbidden. Thus, a direct coupling between these states is driven by magnetic
dipole interaction in the microwave regime.
For the 5S1/2
87
Rb, the magnetic dipole potential is given by
−
−
−
V = −(→
µL+→
µS+→
µ I ) · B.
(2.14)
18
The total magnetic moment has three contributions. The orbital magnetic moment results
from the orbit of the valence electron around the nucleus and is given by:
µB
→
−
µL =
gL L,
~
(2.15)
where µB = e~/2me is the Bohr magneton. The spin magnetic moment is given by:
µB
→
−
gS S
µS =
~
(2.16)
and the nucleus magnetic moment is given by:
µN
→
−
µI =
gI I,
~
(2.17)
where L is the orbital angular momentum, S is the spin angular momentum and I is the
nuclear spin angular momentum.
In order to calculate the Rabi frequency resulting from the interaction between the
atom and the magnetic component of the microwave radiation, a quantum axis need to be
define. For this propose we use a weak bias field B0 in the z-direction. For simplicity, let
us assume for now that it is weak enough so that the Zeeman degeneracy is not lifted. The
microwave radiation induces a magnetic field
B = B1 cos(x · k − ωt)ẑ,
(2.18)
which assumes that the magnetic wave is linearly polarized and oscillates in the z-direction.
As we are only interested in the radiation at the location of the atom, we are interested in
one specific value of x for Eq. 2.18, which we may set to zero. Since in the ground state
hyperfine levels L=0, µL = 0, the interaction energy takes the form:
V =
B1
(µB gS Sz + µN gI Iz ) cos(ωt).
~
(2.19)
For the matrix element of the interaction for the ”clock transition” hF = 1, mF = 0| → |F =
2, mF = 0i, we use:
B1
1 1
1 1
V12 = hF = 1, mF = 0|V |F = 2, mF = 0i =
cos(ωt) h , − | + h− , | ×
2~
2 2
2 2
1 1
1 1
−B1
×(µB gS Sz + µN gI Iz ) × | , − i − | − , i =
cos(ωt)(µB gS − µN gI ), (2.20)
2 2
2 2
2
19
Figure 2.2: Rubidium 87 D1 and D2 transition hyperfine structure.
where we have used the |Iz , Sz i basis. Using Eq. 2.20, the solution to the rate equation
(2.12) indicates that atoms, initially prepared in the ground state, will undergo oscillations
between the two states. The rate of the oscillations is determined by the generalized Rabi
frequency:
Ω=
−B1
(µB gS − µN gI ),
2~
(2.21)
where B1 indicates that the frequency is defined as a measure of the interaction strength
√
(i.e. Ω ∝ p), where p is the power of the microwave field.
20
Chapter 3
Theory of Laser Cooling
Our experiment is aimed at studying MW radiation as a tool for the coherent manipulation of
cold atoms. For this purpose
87
Rb atoms were captured, cooled and released under gravity,
forming an isolated sample enabling the investigation of quantum evolution such as Rabi
oscillations. In this section, the theoretical background of cooling and trapping atoms is
briefly introduced.
3.1
The
87
Rb alkali metal
The alkali metals are the elements listed in the first column of the periodic table, and which
have a simple electronic configuration: a closed shell and a single valence electron [Ari77].
In addition, the
87
Rb atoms have two hyperfine ground states, which enables us to treat
the atom as a two level system. There are a few important properties of
87
Rb that make it
especially useful for the cooling and trapping cycle:
• Closed cycling transitions, allowing the atom to scatter many photons, ensuring sufficient momentum transfer for cooling.
• Large magnetic moment, on the order of the Bohr magneton, which is essential for
magnetic trapping.
• The frequencies associated with all the required transitions are very close to the visible
region of the electromagnetic spectrum and are available commercially nowadays via
21
simple diode lasers.
• It is experimentally straightforward to produce alkali metal vapors.
The level structure of the
87
Rb alkali atom, is generated by fundamental interactions.
The fine structure results from the coupling between the orbital angular momentum L of
the valence electron, together with its spin angular momentum S. The total electron angular
momentum is:
J = L + S,
(3.1)
where J is a quantum number which may have the values:
| L − S |≤ J ≤ L + S.
(3.2)
In the ground state of 87 Rb L=0 and S=1/2, hence J=1/2. In the excited state L=1, therefore
J=1/2 or J=3/2. The transition to J=1/2 and J=3/2 are known as the D1 and D2 lines,
where the D2 line is illustrated in Fig. 2.2.
The hyperfine structure F results from the coupling between the total electron angular
momentum J together with the total nuclear angular momentum I and is given by:
F = J + I.
(3.3)
The magnitude of the total atomic spin F can have the values:
| J − I |≤ F ≤ J + I.
In the
87
(3.4)
Rb ground state J = 1/2 and I = 3/2, hence F = 1 or F = 2.
Each of the hyperfine energy levels F, contains 2F + 1 degenerate magnetic sub-levels. In
the presence of an external magnetic field this degeneracy is lifted and split into the Zeeman
sub-levels. The Hamiltonian of this interaction is [Ste01]:
HB =
µB
(gS Sz + gL Lz + gI Iz ) · Bz ,
~
(3.5)
where the magnetic field is parallel to the atomic quantization axis Z. The Landé factors are
gS ∼ 2, gL ∼ 1 and gI ∼ -0.001.
22
3.2
Theory of light-matter interaction
87
Once
Rb atoms are evaporated into the science chamber, they are trapped and cooled
using magnetic and light fields, generating a Magneto-Optical Trap (MOT). Since it was
first demonstrated in 1987 [Raa87] until the present, the MOT has been a fundamental tool
for the study of physics in the field of cold atoms.
3.2.1
Light force on a two-level atom
When a laser frequency is close to an atomic transition, assuming low light intensity and
hence no stimulated emission, the atom spontaneously emits a photon in a random direction.
Because of the spatial symmetry of the emitted fluorescence, there is a net zero average
momentum transfer from a large number of such fluorescence events (the typical cooling
cycle for the
87
Rb excited state is 23ns). On the other hand, since each photon that is
absorbed by the atom comes from the same direction, the atom is ’pushed’ by the beam by
a total force given by [Met99]:
d~p
F~ =
= ~kγabs ,
dt
(3.6)
where
γabs =
s0 Γ/2
,
D 2
)
1 + s0 + ( δ+ω
Γ/2
(3.7)
is the absorption rate of photons by the atoms, Γ is the linewidth of the atomic transition, k
is the wavenumber of the laser light and δ is the detuning of the laser light from the atomic
transition. s0 = I/Isat = I/(πhΓc/3λ3 ) is the saturation parameter and ωD = −~k · ~ν , is the
Doppler shift.
3.2.2
Doppler cooling
Atomic cooling requires that the force exerted by the photons on the atom always be opposed to the atom’s velocity. With two oppositely directed beams and light detuned below
resonance, atoms traveling toward one laser beam see it Doppler shifted upward, closer to
resonance. Since such atoms are traveling away from the other laser beam, they see its light
23
Doppler shifted further downward, hence further out of resonance. Atoms therefore scatter
more light from the beam counter-propagating to their velocity so their velocity is reduced.
This damping mechanism is called optical molasses [Chu85]. The sum of the forces from two
counter propagating beams (F~OM ) is given by:
F~OM = F~+ + F~− ∼
=
4~k 2 s0 δ/(Γ/2)
~υ ≡ −β~υ .
(1 + s0 + [δ/(Γ/2)]2 )2
(3.8)
The optical molasses force is proportional to the velocity and behaves like a friction or
damping force. This force will decrease the atom’s velocity only for a red detuned laser light.
For blue detuning, the light will increase the atom’s velocity. By using three intersecting,
counter-propagating, orthogonal pairs of oppositely directed beams, the movement of atoms
in the intersection region can be suppressed in all three dimensions, and many atoms can
therefore be cooled in a small volume.
3.2.3
Doppler and sub-Doppler limit
It seems from Eq. 3.8 that the damping force could bring the atomic velocities to zero,
giving rise to an absolute temperature of zero. Since this violates thermodynamics, there
must be processes heating the atoms and preventing the cooling from reaching absolute zero
temperature. This heating is a result of the discreteness of the momentum changes 4p = ~k,
for each time that the atom absorbs or emits a photon [Met99].
For two-level atoms, the lowest temperature that can be achieved by the above so-called
Doppler cooling is given by:
TD =
~Γ
,
2kB
(3.9)
where Γ(≈ 2π ·5.9MHz for 87 Rb) is the natural linewidth of the atomic excited state [Koh93].
The cooled atoms reach the Doppler limit TD (≈140 µK for
87
Rb), when the heating due
to emission balances the cooling due to absorption. Another way to determine TD is to recognize that the average momentum transfer of many spontaneous emissions is zero, but the
rms scatter of these about zero is finite. Thus, increasing the number of scattering photon
results in heating.
24
Temperatures below the Doppler limit have been achieved [Let88], and a new theory
containing two models, has been presented [Dal89]. The first is described by a configuration
of two linearly polarized beams, called Lin ⊥ Lin configuration, and the second for a σ + −σ −
configuration. In the σ + −σ − configuration, the polarization is linear and rotates in space. In
this configuration the cooling is due to different probabilities of absorbing a σ + or a σ − photon
due to Clebsch-Gordan coefficients. This new cooling mechanism allows for cooling below
the Doppler limit because it is based on differing absorption probabilities due to different
populations in the ground state, and is not based on on-resonance and off-resonance beams
due to Doppler shifts.
The viscous damping experienced by the atoms in sub-Doppler cooling is much larger than
that in Doppler cooling. However, the cooling capture range is small, so atoms must initially
be Doppler cooled before sub-Doppler cooling can occur.
Traps based on radiation pressure are limited in their ultimate temperature by the recoil
limit Trecoil (≈ 360 nK, for λ = 780 nm). This limit results from the recoil energy the atoms
absorb by spontaneous emission, defined by:
kB Trecoil =
~2 k 2
.
2m
(3.10)
Using the evaporative cooling technique, one can achieve even lower temperatures (several
nK), approaching the Bose-Einstein Condensation (BEC) [Pet95].
3.3
Magneto-Optical Trap (MOT)
In the previous sections, the theory of laser cooling was explained, but in order to trap the
atoms, light alone is not sufficient. An atom can diffuse out of the system, since the force of
light acting on the atoms is strictly velocity dependent. By adding a quadrupole, magnetic
field, and using opposite, circular polarization cooling beams (see Fig. 3.1), one can introduce an additional position-dependent force. This is called a Magneto-Optical-Trap (MOT)
[Raa87, Met99]. The basic concept is that an atom in the ground state (Mg = 0, where M
is the projection of the angular momentum) has three excited states (Me = 0, ±1), and the
probability of being excited to any one of them is not equal. For example, as shown in Fig.
25
Figure 3.1: Illustration of a MOT in 1D. The detuning δ is for atoms at rest at the trap’s center. Due to
the Zeeman shift of magnetic sublevels, and the arrangement of laser polarizations, atoms are driven to the
trap’s center. Spatial confinement and cooling are obtained simultaneously [Met99].
3.1, when the atom is at z’, its Me = -1 state is closer to resonance, hence, it will absorb
more photons from the σ − beam. Consequently, atoms on the right side of the center are
pushed left as they interact only with the beam going from right to left, and vice versa. In
our experiment, although the cooling transition is F = 2 → F’ = 3, the cooling and trapping
scheme is still valid, as for any Zeeman sublevel in the ground state there are three available
sublevels in the excited state. This theory can be applied to two and three dimensions as well.
The common way of generating a MOT is by using two current loops in an antiHelmholtz configuration, together with three pairs of perpendicular counter-propagating
laser beams. In our experiment the three laser beams are reflected from mirrors, overlapping one another at the center of the science chamber (described in section 4.3.3), having
the same polarizations relative to the photon momentum (i.e. same helicity), for any two
counter propagating beams. The polarization in the coils axis is opposite to that in the other
two axes due to the different magnetic field direction. Under those terms,
87
Rb atoms were
trapped, cooled and imaged as shown in Fig. 3.2. In addition, a photograph of the
26
87
Rb
Figure 3.2: MOT pictures from the experiment. (left) Image of the 87 Rb cloud as it appears in the vacuum
chamber. (right) Fluorescent image of the MOT using a CCD. The cloud size is 13 mm (2σz ) and contains
9 · 108
87
Rb atoms, in both images.
cloud in the optical molasses stage is presented in Fig. 3.3.
In order to find the corresponding total force acting on an atom placed in a MOT, we
introduce the magnetic force into Eq. 3.7: the δ + ωD is replaced by δ + ωD + µ0 B~, where
µ0 = µB (ge me − gg mg ) and B = B 0 z. Solving F~+ + F~− for Doppler and Zeeman shifts smaller
than the detuning, results in a similar result to that obtained in Eq. 3.8:
F~ = −β~ν − κ~r,
(3.11)
where β, the momentum damping coefficient, is equivalent to that in Eq. 3.8 and where
κ=
µ0 βB 0
~k
is a position damping coefficient.
27
Figure 3.3: Photograph of optical molasses taken by fluorescent imaging using a CCD. Three orthogonal
MOT beams cross in the center where atoms which are further cooled, glow brightly.
28
Chapter 4
Experimental Apparatus
High precision, coherent manipulation on free falling cold atoms requires high finesse equipment and precise control over the electronics. In this chapter we present in detail the
experimental apparatus we have built for these studies, and describe the main design considerations.
4.1
Ultra-high vacuum setup
At room temperature, background atoms and molecules may heat the sample upon collisions.
In order to maintain isolation of the cold atomic sample, it is essential to keep it under ultrahigh vacuum (UHV) conditions. The main parts of the vacuum system are imaged in Fig.
4.1. All vacuum parts are made of 316L nonmagnetic stainless steel for minimizing stray
magnetic fields. The order of magnitude of our base pressure is 10−10 Torr, below the detection limit of the ion pump’s internal gauge.
The science chamber is an octagon (152 mm wide, 70 mm thick), with two CF100
viewports connected to large windows at both sides, and eight CF40 openings connected to
small windows at both sides around the circumference (the left-most and right-most windows
are indirectly connected via other system parts). All windows are anti-reflection coated and
are of high optical quality. The top window is connected to a T-shaped piece closed by a
window at one end. Connected to the other end of the T-shaped piece is an I-piece contain29
Figure 4.1: Image of the vacuum system and science chamber.
ing two atom dispensers. The top and bottom windows allow optical access for the imaging
beam, optical pumping beam, and bottom video camera. Four of the side ports transfer the
Magneto-Optical Trap (MOT) 45o beams. The microwave horn antenna has a clear line of
sight to the atoms either through the small window on the right (not visible in the image)
or through the large window (as shown in the image). The port on the left connects the
rest of the vacuum setup. The vacuum setup includes a Ti-sublimation pump (TSP), ion
pump, ionization gauge, UHV valve, and turbo pump. All vacuum parts are held on a frame
designed to hold the vacuum system rigidly while allowing optical access from all directions.
To obtain UHV, the first and most important step is to clean all vacuum parts from
any organic substance. Therefore all vacuum parts were cleaned in an ultra-sonic bath with
deionized water, then removed from the bath and subsequently rinsed with acetone, and finally rinsed with methanol or iso-propanol. All parts were dried in an oven, and then quickly
assembled and connected to a 40 l/s ion pump. Then, in order to initiate the pumping, a 70
l/s turbo pump was switched on, backed by a dry scroll pump. After the turbo pump reached
its limit level (10−7 Torr), the ion pump was turned on and worked in parallel with the turbo
30
pump for several hours until it took over the pumping. The ion pump is connected to the
chamber via a 6-way cross and an I-piece nipple. As the ion pump is close to the chamber and
not screened by µ-metal, a residual field of 300 mG exists at the location of the atomic cloud.
While both pumps were working, we started the baking stage to release absorbed
water from the vacuum parts’ inner walls. Heating tape and aluminium foil covered the
entire vacuum system, allowing the temperature, which was monitored by two thermocouples
located on two distant windows, to be gradually increased. Baking was carried out for a
week at a temperature of 120o C. At the end of the baking stage, the turbo pump was
switched off and disconnected from the vacuum system by a CF40 all-metal angle valve.
After cooling to room temperature, the base pressure was below 10−10 Torr. To further
strengthen the vacuum, we operated a TSP, which is also connected to the 6-way cross by a
CF63 nipple. The ideal location of the ionization gauge (as shown in Fig. 4.1) required both
high proximity to the chamber and sufficient distance from the TSP, but due to mechanical
and space limitations, the ionization gauge could not be positioned ideally. After the baking
and pumping stages, one of the two dispensers was initially cleaned by running low current
through it, and the TSP was turned on intermittently.
4.2
Atom source
The Rb source required in the experiment for loading the MOT consists of two Rb dispensers
(SAES Getters Rb/NF/3.4/12), connected in parallel. Each Rb dispenser is connected to
a pair of copper feed-throughs, which is mounted on a CF40 flange and connected by a Tpiece to the top of the octagon. Driving a current of 12 A through the dispensers, initiates
a chemical reaction which releases neutral Rb atoms. The distance between the dispensers
and the location of the MOT is roughly 250 mm - close enough to allow rapid loading of
the MOT (10 sec), but far enough to reduce background collisions during the measurement
time.
31
4.3
Lasers and optics system
The laser setup used in the experimental apparatus is divided into three sections, each of
which is briefly discussed. The first section describes the optical layout in the laser box,
which consists of two lasers, two spectroscopy setups, which are required for each laser
frequency stabilization, and one Tapered Amplifier (T.A.) and its corresponding optics. The
second section details the path outside of the laser box of the two light beams, including
the Acousto-Optical Modulators (AOMs), which shift the frequencies, and the optical fibers,
which guide the light beams into the science chamber. The third section introduces the
science chamber, the lasers which pass through it, and all relevant optics around it.
4.3.1
Laser lock and spectroscopy
The main laser used in the experiment for the Cooling, Optical Pumping and Imaging beams
is a home-made laser generating up to 120 mW of power at 780 nm. It consists of a Toptica
diode laser, a Piezo, a grating for the external cavity, a thermoelectric cooler (TE-cooler),
based on the Peltier effect and a thermocouple. The laser is wired to electronics placed on
a shelf, located above the optical table. The current to the diode is fed by a current supply (LDC202B, Thorlabs). The current to the TE-cooler is fed by a temperature controller
(TED200, Thorlabs). When in scan mode, the piezo is fed by a ’saw’ signal, generated from
a signal generator (gfg-8015g, GW), and amplified by a home-made, high-voltage amplifier.
This allowed us to scan wavelengths in the external cavity, and to mode-match between
the external and internal (the diode) coupled cavities, thus achieving a free spectral range
of several GHz [Azm00]. The coupling between these two cavities can result in collapse of
coherence, which can make the mode-hop-free continuous scanning range much smaller than
the free spectral range of the extended cavity. Therefore, we introduced an improvement
in the cavity’s geometry and in the electronics of the second laser using the feed-forward
technique (built modelling commercial laser type DL100 Toptica). This new design allowed
us to scan the cavity’s frequencies with a free spectral range of 11 GHz without having to
deal with annoying mode-hops.
32
The second laser is used to bring back atoms from the |F = 1i state into the cooling
cycle by re-pumping them to the |F = 2i state. The Repumper is a home-made laser generating up to 80 mW of power at 780 nm, consisting of a Sharp diode laser. This laser was
built and wired in the same way as the Cooler, except for the slight change mentioned above.
Both lasers are built in a metallic box to avoid temperature changes, and to reduce
the influence of air currents and acoustic vibrations. The lasers are located on a breadboard
placed on the optical table, which is flooded with nitrogen to reduce vibrations.
The two lasers beams, as shown in Fig. 4.2, pass through an isolator following which a small
portion is reflected out of the beam by a glass slab to a polarization spectroscopy setup.
The Cooler and Repumper spectroscopes, which are shown in Fig. 4.3, are used to lock the
laser frequency. Fig. 4.3 presents a comparison of saturation and polarization spectroscopy
since both techniques can be utilized, but polarization has a better signal-to-noise ratio and
provides a broader locking range while requiring minimal optical elements. Therefore, polarization is mainly used [Lan99]. The electronics used to lock the laser’s frequency consist of
a home-made proportional-integral-derivative (PID) unit that enables a feedback loop. The
PID subtracts two signals produced by the two photo-diodes in the polarization setup from
each other, and sends the corrected and amplified signal to the high-voltage-amplifier, which
controls the piezo.
Since the Cooler beam is divided and passes through AOMs and fibers, it must be
amplified. For this reason we built the T.A., shown in Fig. 4.4, following the sketches of R.
A. Nyman et. al. [Nym06], introducing slight changes (detailed below). The T.A. is a GaAs
Semiconductor Laser Diode (Eagleyard, EYP-TPA-0780-01000- 3006-CMT03, C-Mount 2.75
mm) with a gain range similar to that of laser diodes. To have enough power, the T.A.
was mounted in a compact self-contained mechanical housing (almost no adjustable parts)
that allows for temperature control of the chip and mechanical stability. The mechanical
supports hold two a-spherical lenses with a focal lens of 4 mm at both sides of the chip to
focus incoming light, to collimate the output light, and to fix the position of the T.A. chip.
This compact design places the chip in the middle of a copper block, which helps control
33
Figure 4.2: Laser Box. (top) Three home-made lasers: The Cooler and Repumper utilized for the cooling
cycle (the Cooler is also used for imaging and optical pumping). The Tapered Amplifier (T.A.) amplifies
the injected Cooler beam (drawn in red) up to 1.3 W. (bottom) Polarization spectroscopy scheme. A 3 mW
beam is split off from the main laser beam, and used for spectroscopy. The Rb vapor cell is covered with
two layers of µ-metal shielding. Two photo-diodes are used to produce the error signal. The remaining 30
mW in the main beam is mode-matched by a telescope to allow for the best injection into the T.A. Then,
the amplified beam exits on the right side of the box, passing through a cylindrical lens to fix the dispersion
of the horizontal axis of the beam with respect to the vertical axis.
34
Figure 4.3: Two 87 Rb spectroscopy images used for the laser frequency stabilization: Cooler spectroscopy
(left), Repumper spectroscopy (right). An absorption spectroscopy signal (top) is compared to a polarization
spectroscopy signal (bottom).
Figure 4.4: Mechanical and thermal housing for the tapered amplifier. (left) Exploded diagram. The blocks
are made of copper (the centring rods are of steel). From [Nym06]. (right) Photograph of our assembled
system.
35
the chip temperature. The copper block also holds the contacts for the amplifier current
supply (LDC340, Thorlabs), and the thermistor, which is used as a temperature monitor
by the temperature-control unit (TED200, Thorlabs). The temperature control is based on
a PID feedback loop, the output of which drives the TE-cooler. If the temperature of the
chip is outside of a defined range, both units shut down, leaving the block to return slowly
to ambient temperature. The current-supply unit has a soft-stop mechanism to protect the
amplifier. In order to improve the mechanical stability, we have introduced a slight change
in the original design. A teflon piece which is a fraction of a mm thicker than the TE-cooler
is placed around it so that the copper structure can be well tightened against it and is still
in good thermal contact with it. In addition, heat conducting paste is also used to ensure
good thermal contact between the TE-cooler and the copper block. The output of the T.A.
is highly divergent and astigmatic. One axis is collimated by the lenses built in the copper
mounting block. The other axis is corrected using a cylindrical lens. The beam is then sent
out of the box through a telescope to produce a collimated beam of approximately 1 mm in
diameter.
4.3.2
Optical layout
After appropriate beam-shaping, the output of the T.A. passes through a metallic slit to
avoid the continued divergence in the horizontal axis, and is then split into different paths.
Each path, which includes a different AOM, is devote to a different functional beam. The
D2 level scheme of the transitions involved in the experiment is shown in Fig. 4.5, where the
lock point of each of the two lasers, and the AOM frequency shifts for each of the beams are
shown. The Zeeman shifts due to stray magnetic fields around the vapor cells are reduced
for locking, by covering them with µ-metal shielding. The main beam is locked between the
52 S1/2 |F = 2i level and the crossover 1x3, which is located between the 52 P3/2 |F 0 = 1i and
52 P3/2 |F 0 = 3i transition. The Repumper beam is locked between the 52 S1/2 |F = 1i level
and the crossover 1x2, which lies between the 52 P3/2 |F 0 = 1i and 52 P3/2 |F 0 = 2i transition.
The T.A. beam is divided into three components as shown in Fig. 4.6: Cooling (blue),
Imaging (yellow), and Optical Pumping (red) beams. Each component goes through an
36
Figure 4.5: The frequencies of the four laser beams used for the experimental cycle are superimposed on
the level scheme of the
87
Rb D2 transition hyperfine structure. The ground state levels are denoted by F,
and the excited state levels by F0 . Dashed lines correspond to the cross-over peaks (1x2, 1x3), to which our
lasers are locked. Arrows indicate these lock points, as well as the frequency of each laser component after
passing its AOM.
37
Figure 4.6: Optics behind the laser box. (top) Image of the setup. The Cooler beam is divided into
three paths through three different AOMs, Cooler (blue), Imaging (yellow), Optical Pumping (red). The
Repumper’s AOM is not shown. (bottom) Optical layout from the laser box into the optical fibers.
38
AOM, which can control the frequency of the locked laser with nanosecond time-resolution.
The Cooling beam is shifted at the beginning of the experimental cycle from the 1x3 crossover peak to which it is locked, by approximately 200 MHz, to be near-resonant with the
cooling transition |F = 2i → |F 0 = 3i. This is done by going through the AOM in a
double-pass configuration. As the AOM leaks some amount of light even when it is off,
mechanical shutters are used to block the light completely. The mechanical shutters have
a slower response time than the AOM. Therefore, shutting the light is performed with the
AOM, and the mechanical shutters (Uniblitz for the Repumper and Thorlabs for Cooler)
are timed to follow up right after the AOM shuts off the light. In order to reduce vibrations
from the operation of the mechanical shutters, they are not connected to the optical table,
but rather hung from the rack above the table. The imaging light is tuned on-resonance
at the cooling transition for regular absorption imaging. This requires a 212 MHz shift,
also done in a double-pass AOM configuration. The optical pumping beam is tuned to the
|F = 2i → |F 0 = 2i transition, which is about 55 MHz red detuned from the 1x3 cross-over.
This is done in a single pass through the AOM. Finally, the Repumper laser is tuned onresonance to the |F = 1i → |F 0 = 2i transition, 78 MHz detuned from the lock point of the
laser. In principle, the Repumper could be locked directly on-resonance, without needing the
AOM. However, because the response time of mechanical shutters (about a few milliseconds)
is much slower than that of the AOMs (a few tens of nanoseconds), the AOM in this case
is used only for the fast switching and control, rather than for frequency shifting from the
resonance. The entire optical layout is presented in Fig. 4.6.
After passing through the AOMs, the Cooler beam and the Repumper are combined
and injected into the same fiber. Each of the three light components is coupled into a
polarization-maintaining single-mode fiber. Each beam is mode-matched by a telescope
such that the light is injected up to 60% efficiency. In order for the polarization of the light
to be stable over time coming out of the fibers, the input polarization has to be set such that
the plane of polarization is parallel to the optical axis of the fiber coupler. For this purpose
a λ/2 wave-plate was inserted before the input coupler, and the beam’s plane of polarization
was aligned using a polarimeter.
39
4.3.3
Science chamber
The setup around the science chamber is shown schematically in Fig. 4.7. A 150 mW
Cooler beam and a 15 mW Repumper beam exit the fiber and are expanded into a 20 mmdiameter beam, which is then split into the three retro-reflected beams required for the MOT
and directed into the chamber. The Horizontal MOT beam is σ + polarized, and the two
45o -angled MOT beams are σ − . After crossing the cloud, reflecting from the mirror, and
passing through a λ/4 wave-plate twice, the beam is moving in the opposite direction with
the opposite polarization relative to the lab frame and equal helicity. The light intensity of
the Imaging beam is 300 µW and only a few µW for the optical pumping beam.
4.4
Static and AC fields
A magnetic trap is required to capture the atoms during the MOT stage. A homogeneous
magnetic field is essential to produce a quantization axis for the optical pumping stage.
Also, an interaction between the microwave field and atoms must occur for the generation of
Rabi oscillations. For all these purposes, two types of magnetic static fields and one electromagnetic AC field are applied during the experiment. One pair of identical round coils are
positioned around the large windows, separated by a distance of 90 mm (see Fig 4.7). A
current of 12 A flows through the coils in opposite directions, yielding the anti-Helmholtz
configuration, which results in a quadrupole with a zero magnetic field at the center of the
chamber. The Helmholtz coils are two identical round coils positioned around the chamber,
centered around the cloud, parallel to table, separated by a distance of 80 mm. A current of
4 A results in a homogeneous magnetic field of 0.4 G at the center of the science chamber,
with good uniformity in the center of the chamber. In addition to the need to generate
the required field values, it is necessary to be able to quickly switch the magnetic coils. For
example, a fast shut down is necessary when we abruptly turn off all fields upon releasing the
atom cloud from the trap before imaging it. A slow switching time, set by the capabilities
of the current source (on the order of tens of milliseconds) as well as by the induction of the
coils, may result in giving the atoms a ’kick’ during the trap release, thereby accelerating
them and hindering the analysis. In our experiment, we accomplish fast shut down of 100
40
Figure 4.7: A 3D-image of the science chamber and all relevant optics surrounding it. The red beam
denotes the Cooling and Repumping beams. The gold and purple beams denote the Optical Pumping and
Imaging beams respectively. Both exit from a collimator at the end of the fiber with a diameter of 1.5 cm.
They are circular polarized by a λ/4 wave-plate and combined through a Beam Splitter (B.S.) cube. The
red beam exits from the output collimator of the fiber with a diameter of 2.5 cm and is split by two passes
through a λ/2 wave-plate and a P.B.S. into three orthogonal beams. The beams are circular polarized by a
λ/4 wave-plate before entering the chamber and are retro-reflected by a mirror and another λ/4 wave-plate.
Also shown in the image are the CCD capturing the Imaging beam, the horn antenna in grey, and two sets
of magnetic coils. The anti-Helmholtz configuration is shown in brown and the Helmholtz configuration in
green. (inset) The chamber with its 3D axis is shown (Only the Imaging beam is drawn).
41
µs by using home-made fast current-shutters.
The AC microwave pulse is transmitted from the signal generator to the rubidium cell
through a horn microwave antenna located about 25 cm from the cloud, as illustrated in Fig.
4.7. The signal generator (SMR 20, Rohde & Schwarz) is synchronized with an atomic clock
(AR40A, Accubeat-Rubidium frequency standard). The 6.8 GHz, microwave pulse passes
through a microwave amplifier (1 W, ZVE-3W-83, Mini-Circuits) before arriving at the antenna. The signal is amplified 20 times. The antenna has a flared rectangular copper portion
(50×100 mm squared), fed by a metallic wave-guide. We use another antenna to analyze
the transmitted signal through the cloud. The second antenna is a flat microwave antenna
made from 50×50 mm squared PCB, with a front (transmitting) 10×10 mm squared copper
printed on one side, while the back (ground) side is all copper.
In the experiment described here, the electromagnetic wave transmitted by the horn
antenna propagates in the direction of the Z-axis (Fig. 4.7). The induced electric and
magnetic fields will thus have perpendicular components in the XY-plane. The polarization
of the microwave and the direction of the components can be controlled by rotating the
antenna around its main axis, on the XY-plane, with respect to the quantized magnetic axis
in the X-direction.
4.5
Optical pumping
After the
87
Rb atoms are released from the MOT stage and fall under gravity, the atoms
must be must be prepared in a specific quantum state before they are manipulated e.g. with
a microwave pulse. At the MOT stage, the atoms occupy all possible Zeeman sublevels of
the |F = 2i hyperfine ground state (|mF = −2, ..., 2i) and the |F = 1i hyperfine ground
state (|mF = −1, 0, 1i). A basic tool for preparing atoms in a well defined quantum state
is optical pumping. We have therefore added this tool to our setup and as a test we have
pumped the atoms to the streched state, |F = 2, mF = 2i. This is done by utilizing a 400
µs light pulse of σ + circularly polarized light (the gold beam shown in Fig. 4.7) propagating
42
Figure 4.8: Optical pumping scheme. Right circularly-polarized light (σ+ ) pumps atoms to the highest
mF level of the ground state |F = 2, mF = 2i (yellow arrow) by absorbing and re-emitting photons until
they reach the ”dark state” where they no longer scatter photons (dashed yellow arrow). Atoms absorb
∆mF = +1 but re-emit ∆mF = ±1, 0 (green arrow). Therefore, on average, ∆mF = 0 for re-emission.
in the X-direction (axes are shown in Fig. 4.7). A magnetic field, using the Helmholtz coils,
is switched on 1 ms before initializing the optical pumping stage to provide a quantization
axis for the atomic spin. The effect of this process can be seen in Fig. 4.8. The optical
pumping frequency is set to resonate with the |F = 2i → |F 0 = 2i transition. Most of the
atoms spontaneously decay to |F = 2i , but about once every 1000 cycles, an atom can be
excited to the |F 0 = 2i state and decay to the |F = 1i ground state [Met99]. Since atoms
in the |F = 1i state can no longer be optically pumped, the Repumper light is added to the
light pulse.
One of the common ways to verify the atoms’ distribution over the mF states, after the
optical pumping stage, is performing a Stern-Gerlach type measurement (as was done previously in our lab [Dav09]). Here, a less accurate but simpler method was implemented, which
can qualitatively indicate the atoms’ final state. We set the Optical Pumping and Imaging
beams, using λ/4 wave-plate, to be σ + and σ − polarized, respectively. As the beams combine
43
Figure 4.9: Number of atoms in a time-of-flight (T.O.F.) measurement (by absorption imaging) vs. polarization of the imaging beam determined by the angle of a λ/4 wave-plate. First, the Optical Pumping and
Imaging beams were adjusted to be both σ + polarized. Thus, the maximum number of atoms are imaged. As
the λ/4 wave-plate, which is placed before the Imaging beam, is rotated, fewer photons are absorbed giving
rise to a smaller calculated value for the number of atoms, until the beams are of opposite polarization, and
the minimum number of atoms is counted. The top and bottom insets show the relevant transitions and their
q
q
1
Clebsch-Gordan coefficients, at the maximum ( 12 ) and minimum ( 30
) atom numbers. The non-smooth
curve is assumed to be caused by fluctuations in the MOT atom number.
44
at the B.S. cube, the Imaging beam (tuned to the atomic transition |F = 2i → |F 0 = 3i),
inverts its polarization when reflected by the B.S. cube (see Fig 4.7), such that both beams
are σ + polarized. Since the Clebsch-Gordan coefficients for the σ + polarized beam for the
Imaging transition, are bigger than for the σ − [Ste01], fewer photons are scattered and hence
a smaller value for the number of atoms is calculated in the latter case. If the assumption
that atoms are pumped to the mF = 2 state is correct, then a decrease in the number of
atoms should be observed as the λ/4 wave-plate of the Imaging beam is rotated by 90o , as
illustrated by the graph in Fig. 4.9.
In order to avoid the effect of magnetic fluctuations, we have used for this experiment
the |F = 1, mF = 0i to |F = 2, mF = 0i transition. Although optical pumping procedures
exist which enable the preparation of the atom in the |F = 1, mF = 0i state, for simplicity
we simply stopped the Repumper before the Cooler, at the end of the MOT stage, thus
populating all three |F = 1i sublevels equally.
4.6
Imaging
By using imaging methods, we can extract information about the atomic cloud and calculate
its properties at any time during the experiment. We use the absorption imaging technique
where two pulses of laser light are imaged on a CCD, after passing through the atomic cloud.
These pulses are on-resonance with the atomic transition |F = 2i → |F 0 = 3i and are applied
successively at intervals of 30 ms, where each lasts 400 µs. The first pulse is absorbed by the
atomic cloud, disperses all the atoms from the trap, and is imaged on the CCD as a shadow.
The second pulse reaches the CCD camera without any losses. It is assumed that the laser
beam did not change its properties (e.g. spatial distribution, intensity, etc.) during the
second pulse. In Fig. 4.10, the image of these two pulses and the resulting absorption image,
are shown. The cloud has to be at the center of the beam where the intensity distribution
is equal for the accuracy of the optical density calculation. Patterns such as unavoidable
fringes, which are imprinted on the beam, or light reflections from metallic parts in the chamber, do not appear in the absorption image (Fig. 4.10 (right)) because they are the same in
45
Figure 4.10: Absorption imaging. As the imaging axis is X, what is measured is the column optical
density along the X axis. (left) An image of the laser beam after propagating through the atomic cloud. The
shadow of the cloud is clearly visible. (middle) The corresponding background image (no atoms). (right)
The resulting absorption image. All scales are in [mm].
both images (Fig. 4.10 (left and middle)), and they are cancelled out when taking the ratio.
Both pulses are de-magnified by two lenses, mounted on a translation stage before the CCD,
allowing accurate focusing such that the de-magnification is 0.3. The pulses are extracted
from the output coupler of the imaging fiber (the purple beam in Fig. 4.7), placed before the
λ/4 and B.S. cube, such that the beam is σ + polarized. This polarization is preferred since
the transition strength from |F = 2i to |F 0 = 3i is higher, and thus the signal is stronger (as
mentioned in Section 4.5). It is a 300 µW beam with a diameter of 1.5 cm. The camera is
an IDS U-eye and has 6×6 µm2 pixels and 8 bits. The CCD is placed at the bottom of the
chamber, where a mirror allows the Imaging beam to be bent over to the YZ-plane (Fig. 4.7).
To form the absorption image and analyze the atomic cloud we use a software, which
allows us to extract the atom number, center of the cloud, optical density, etc. (for further
details see [Mac09]).
In general, the amount of power (Psc ) scattered by a single atom is proportional to the
absorption cross section σ, and is given by:
Psc = σI,
(4.1)
where I is the intensity of the exciting light field.
If the atomic density is denoted by n(z), the intensity of the beam after travelling a small
distance dx through the cloud is attenuated by:
∆I = −σIn(x, y, z)dx.
(4.2)
46
This equation can be integrated to give the Beer-Lambert law [Ket99]:
I(y, z) = I0 (y, z)e−σd(y,z) = I0 (y, z)e−O.D.(y,z) ,
(4.3)
where I denotes the Imaging beam intensity after passing through an atom cloud, I0 is the
second pulse beam intensity (without atoms), O.D. is the optical density and d(y, z) is the
projected particle or column density (integrated over the imaging beam propagation axis),
given by:
Z
d(y, z) =
n(x, y, z)dx.
(4.4)
The Beer-Lambert law (Eq. 4.3) can be used to extract the column density from the intensity
ratio of two successive measurements, one with and one without atoms:
d = −log[
I
N
]/σ = −log[ ]/σ,
I0
N0
(4.5)
where N and N0 denote the counts per pixel acquired by the camera.
In the experiment we use the absorption imaging technique rather than fluorescence
imaging (Fig. 3.3), because it gives a good signal-to-noise ratio even in short pulses, although it is destructive. In fluorescence imaging, only a small fraction of the scattered light
is collected by the camera, while in absorption imaging all of the absorption signal can be
collected, as the shadow acts as a point source. We chose to operate the imaging axis in the
X-direction since the cloud can fall under gravity without deviating from the center of the
Imaging beam, thus allowing longer interrogation times.
When the atomic cloud is thermal (i.e., when T > Tc ), it has a Gaussian spatial
distribution [Ket99]. By taking a cut through the center of the atomic cloud, we can fit the
measured O.D. with a Gaussian function and extract the peak O.D. (A), the dimensions of
the cloud (σy , σz ), and the position of the cloud center (y0 , z0 ), as shown in Fig. 4.11, taken
at the end of the MOT loading event.
47
Figure 4.11: Imaging analysis interface. The screen is divided into two main parts. On the left, there is
the image with two 1-dimensional cuts along the center of the cloud (absorption imaging of the initial MOT
after 4 ms T.O.F.). The controls of the software and the fit results are on the right.
4.6.1
Temperature and gravitation measurements
Once the atomic cloud is cooled, a basic temperature measurement is taken. The measurement is done by shutting down all fields after the MOT stage, and thereby letting the cloud
expand while falling under gravity. This Time-Of-Flight (T.O.F.) measurement includes a
series of images of the cloud, prepared and released each time in the same way, but imaged
at different times. By fitting the cloud images to a 2D Gaussian spatial distribution given
by,
−
P = Ay e
(y−y0 )2
2σy 2
+ Az e
−
(z−z0 )2
2σz 2
,
(4.6)
the amount of the expansion can be determined. By knowing the initial size of the cloud,
before it has been released, its velocity can be found, using the following relationship:
σy (t) =
q
σy (t = 0) + σvy 2 t2 .
(4.7)
48
Figure 4.12: Time-Of-Flight measurement after the Molasses stage: Temperature (top) and gravity (bottom). The X and Y-axes in the figure correspond to the original Y and Z-axes respectively, as they appear
in the equations.
49
Once the velocity of the cloud for each direction is known, its temperature can be estimated
using the following thermodynamic relationship:
mvy 2 = kB T,
(4.8)
where kB and m are the Boltzmann constant and the atomic mass respectively. A temperature measurement taken during the experiment is shown in Fig. 4.12, after the molasses
stage. The two fits represent the two spatial axes in the image. For t = 0 the fits have a
different value due to the different initial size of the cloud and has a vertical differences due
to the initial size of the cloud in the radial and longitudinal directions (the MOT is not perfectly symmetric). The different slope giving rise to a different temperature estimate for the
two axes of the cloud, may point to the fact that the cloud is not in thermal equilibrium. As
a check of the system we used a side imaging to image the fall of the atoms and compared it
to the well known equations of gravity. In Fig. 4.12, a gravity measurement is shown during
the T.O.F., and the corresponding gravitational constant is found to be in good agreement
with the known value.
4.7
Experimental control
The different stages in the experiment require accurate control, both in time and amplitude, of all the electronic devices simultaneously. The sequence must be synchronized by
high-resolution timing (in our experiment about 1 µs) and last for relatively long times (our
cycle is 11 s). The devices listed in Table 4.1 are controlled by analog or digital signals.
We use a dedicated computer, National Instruments PXI, with an independent processor to
manage the entire experimental sequence. It runs in real time mode and sends the signals
that control the experimental sequence.
In addition, we use a LabView program, whose interface is shown in Fig. 4.13, which
loads the experimental sequence to the PXI. For the user’s convenience, we separate the
experimental procedure into several events (according to the experimental stages). For each
of the analog signals, control ramps are defined in the program, with the ability to set delay
times relative to the event start time, the speed of the ramp, and the end value. For the
50
Figure 4.13: Experimental control interface. Analog channels (on the left) and digital channels (on the
right).
digital channels the user defines the TTL trigger timing and logical state (on/off). Once the
user pushes the ”execute” button the data is downloaded to the PXI in the form of a large
matrix of channels, timings and values, and the PXI manages the sequence independently
of any other external clock. The software allows programming automatic loops in which
one or more parameters are changed. We can also save the experimental control data and
load them again later. The analysis program takes the raw image files saved by the camera,
processes them to obtain the absorption image, and executes the fitting algorithm. It can
also operate in a ’loop’ mode, which can be synchronized with the main experimental control
for easy data taking.
The whole experiment is controlled and operated from the control station. Here we
have the PC with the Labview program, the PXI, the BNC cables and one interface box.
The interface box is where some interfaces are located (current shutters for the magnetic
fields, MW current supply). The purpose of the interfaces is to match the signals generated
51
Analog Control
Digital Control
AOM Frequency
AOM Power On/Off
(Cooler, Repumper, Imaging, Opt.Pump.)
(Cooler, Repumper, Imaging, Opt.Pump.)
AOM Amplitude
Mechanical Light Shutter
(Cooler, Repumper, Imaging, Opt.Pump.)
(Cooler, Repumper, Imaging, Opt.Pump.)
Current of Helmholtz and Anti-Helmholtz
Helmholtz and Anti-Helmholtz Coils
Coils
Current Shutter
Dispenser Current
Camera Trigger
MW Amplifier Current
General Trigger
Table 4.1: Analog and digital signals in the experimental control.
by the PXI (only voltages limited in amplitude) to the required signals of each instrument
(e.g. conversion from voltage to currents, amplification). All interfaces are home-made, as
are some of the instruments used in the experiments (AOM drivers, current shutters).
52
Chapter 5
Rabi Oscillations in Cold Atoms
In this chapter we present the sequence applied in the experiment and the results.
5.1
Experimental Sequence
In order to induce and image population oscillations between the F = 1 and the F = 2
hyperfine levels of the
87
Rb ground state, we apply the following sequence (demonstrated in
Fig. 5.1):
• Trapping and cooling. First,
87
Rb atoms are captured, cooled and released in the
vacuum chamber (as described in section 3.3), using the MOT’s beams and the pair
of anti-Helmholtz coils (detailed in sections 4.3.3 and 4.4, respectively). The trapped
cloud is loaded for 10 s after which its properties are measured to be 109 atoms at a
temperature of 150 µK.
• Optical pumping. Then, just before the end of the loading stage the Repumper beam
(the upper yellow curve in Fig. 5.1) is turned off, while the anti-Helmholtz coils and
the Cooler beam (the pink and the bottom yellow curves respectively, in Fig. 5.1) are
kept on. Shutting down the Repumper beam before the Cooler beam pumps all atoms
to the F = 1 ground state (as explained in section 4.5). At the end of the loading the
anti-Helmholtz coils and the Cooler beam are turned off. and the interaction between
the atoms and the applied MW field begins.
53
Figure 5.1: Experimental sequence of the T.O.F. stage from an oscilloscope screen (without the molasses
stage). The Repumper beam (signal from P.D.) is turned off at 0.001 s (which is 1 ms before the end of
the 10 s MOT stage. At 0.002 s, the Cooler beam (signal from P.D.) and the anti-Helmholtz coils (TTL
signal) are turned off. At 0.003 s and 0.00385 s, the Helmholtz coils and the MW pules (TTL signals) are
turned on for 4 ms and 0.4 ms respectively. At 0.00585 the first Imaging pulse (TTL signal) is applied for
0.4 ms. The second Imaging pulse is not shown since it is activated 30 ms after the first pulse. Due to the
limited number of channels on the oscilloscope, the MW and Imaging pulses are shown on the same TTL
signal. The slow drop times of the light beams (yellow curve) are due to a slow P.D. (in reality these times
are of the order of micro seconds). In the experiment, the Cooler and the Repumper beams are combined
through a P.B.S and then directed to a P.D. The differences between their polarisations cause differences in
the intensities impinging on the P.D. Therefore, the Cooler and Repumper curves look as if they are of the
same power although the Repumper is much weaker.
54
• MW pulse. When the Cooler beam is turned off, the atomic cloud falls under gravity.
The Helmholtz coils (green curve in Fig. 5.1) are turned on for 4 ms to generate a
quantum axis. A MW pulse (first blue curve in Fig. 5.1) tuned to the resonance
frequency of ω = 2π · 6.834682610 GHz, is initiated 2 ms after shutting the Cooler
beam. The duration of the pulses must match the expected Rabi frequency in order to
be able to observe Rabi oscillations. In our system we apply a MW pulse which lasts
from 50 µs up to 2 ms.
• Population imaging. 2 ms after initiating the MW pulse, the first Imaging pulse
is applied and after another 30 ms, the second Imaging pulse is turned on. Since the
Imaging beam is ’on-resonance’ with the F = 2 and F 0 = 3 transition, as explained
in section 4.6, the population in F = 2 is detected. The Imaging pulses last 400 µs,
and during each pulse the CCD is opened for 150 µs to capture the light (longer time
saturates the CCD). At the end of the first Imaging pulse the Helmholtz coils are
turned off. The Imaging ends the cycle which lasts about 11 s.
5.2
5.2.1
Results
Inducing Rabi oscillations simultaneously over space
As the sequence detailed above was applied, we observed fringes in the atomic cloud as shown
in Fig. 5.2. The MW antenna transmits spherical waves which are propagating through the
atomic cloud, yielding a change in the population distribution according to Eq. 2.13. Since
the power level decays with distance, and since the cloud is spread over a few mm, the
populations of atoms at each point across the cloud evolve with a different Rabi frequency
according to what MW power they are exposed to. All atoms positioned along each fringe are
excited by the same MW power, thus oscillating at the same Rabi frequency, such that the
population distribution is equal along the Y-axis in each fringe. In other words, the fringes
are iso-Rabi-frequency bands which vary smoothly to yield multiple Rabi oscillations that
can be viewed simultaneously. At maximum intensity points, population inversion occurred
meaning atoms are in the F = 2 state. At minimum intensity points, atoms are in F = 1 state
55
Figure 5.2: Absorption image in the YZ plane of the
87
Rb cloud in the T.O.F. (free fall) stage for a 2
ms MW pulse. Ten fringes are visible. The MW antenna is positioned 25 cm from the left side of the
fringes, radiating electromagnetic waves in the Z-direction, through the cloud, and generating multiple Rabi
oscillations that can be viewed simultaneously. The atomic cloud consists of 27 · 106 atoms and it is 7 mm
long and 5 mm wide.
56
and therefore they are not imaged. All points between the minima and maxima represent
atoms in intermediate superposition states. As our simplified pumping scheme populates
equally all three F = 1 states, and as our MW frequency is tuned to the |F = 1, mF = 0i →
|F = 2, mF = 0i resonance, only 1/3 of the atoms participate in the population oscillation.
It should be noted that the fringes are superposed on the cloud density profile, producing
maximum population at the center of the cloud where the quantity of atoms is highest.
5.2.2
Inducing Rabi oscillations over time
In order to find the evolution of the population over time, we apply a cycle of sequences
as was used in section 5.2.1. This time, however, the MW pulse duration is increased with
each cycle. At the end of each cycle, an image of the fringes is fitted. For each pixel over
the horizontal axis of the cloud (Z-axis), we integrate the observed light beam intensity over
the longitudinal axis of the fringe (Y-axis) and plot the values for each pulse duration as
presented in Fig. 5.3. We start the first cycle with a 700 µs pulse duration and increase the
duration of the pulse in 50 µs steps. The red linear curve presented in Fig. 5.3, connects
the main fringe maxima for varying pulse durations from the shortest pulse duration to the
longest. The positive slope of the red linear curve reflects the rightward ’movement’ of the
main fringe as the pulse duration increases. This ’movement’ can also be observed in the
images of Fig. 5.4 where each image corresponds to a single curve in Fig. 5.3, and where the
pulse duration increases from left to right. The formation of new fringes at the right side
and the disappearance of old fringes at the left side as the pulse duration increases can be
easily observed. The effect arises from the fact that each point across the cloud undergoes a
different Rabi cycle combined with the variation in pulse duration, and therefore each point
experiences a different population evolution of |F = 2, mF = 0i.
In order to quantify the oscillations, we operate the sequence mentioned above, while
taking into account only a narrow band of a few pixels, where we integrate over the Y-axis.
We plot the values for each cycle and fit to an exponential decaying cosine as presented on
the graph in Fig. 5.5. The MW antenna was positioned 10 and 25 cm away from the cloud,
and radiated in the Y and Z-directions, for the top and bottom graphs, respectively. In both
57
Figure 5.3: Graph of optical density vs. the position, for different pulse durations. Each curve is obtained
at a different MW pulse duration starting at a 700 µs pulse with intervals of 50 µs (from bottom to top). The
red linear curve is aligned over the main fringe, which ’slides’ to the right as the pulse duration increases.
Figure 5.4: Absorbtion images of the atomic cloud in the T.O.F. stage for different MW pulse durations. It
can be observed that the fringes are ”moving” - when the first fringe starts to appear the last one disappears
(from left to right).
58
Figure 5.5: Rabi oscillation graphs. Two measurements were taken at two different distances of the MW
antenna: 10 cm (top), 25 cm (bottom). Each data point represents the same pixel position in the cloud at
different MW pulse time durations. The scales in the X-axes are different for the two fits because they were
taken at different times after the MOT stage. The curves are fits to an exponentially decaying cosine. The
Rabi frequency and the decay constant are: (top) Ω = 21.6 ± 0.03 kHz, T2 = 0.907 ± 0.04 ms, (bottom)
Ω = 10 ± 0.5 kHz, T2 = 0.937 ± 0.025 ms .
59
graphs the first pulse duration lasts 400 µs and is increased in 5 µs steps. The data points
are fitted to:
−
f (t) = Ae
t−t0
T2
cos((t − t0 )2πΩ + φ) + c,
(5.1)
where the amplitude of the Rabi oscillations (A), Rabi frequency (Ω), phase (φ) and the
decay constant (T2 ) were found to be for the top fit:
A = 2.289 ± 0.05, Ω = 21.6 ± 0.03 kHz, φ = −1.577 ± 0.001, T2 = 0.907 ± 0.04 ms,
and for the bottom fit:
A = 1.619 ± 0.021, Ω = 10 ± 0.5 kHz, φ = 3.863 ± 0.1, T2 = 0.937 ± 0.025 ms. Since Ω ∝
√
p
(where p is the MW power), the Rabi frequency decreases as the MW antenna is positioned
farther from the cloud.
The very short coherence time (similar to that obtained in vapor cells) requires an
explanation and this will be the subject of further work which is beyond the scope of this
thesis. One possibility is that our bias field is not homogeneous enough in the Y and
X directions. The first is integrated upon in the calculation of every data point, and the
second is perhaps even more crucial as the atoms fall along this axis during the measurement.
Obviously, as the magnetic field changes, the second order Zeeman effect will shift the
transition frequency giving rise to different detunings causing different Rabi frequencies.
A second possibility is that the Y-axis integration those not take account slight curvatures
in the equi-power lines of the MW field.
5.2.3
The Rabi frequency’s dependence on the position in the
cloud in the far field
In order to verify the relationship between the Rabi frequency and the position over the Zaxis in the cloud, we collect the data about the Rabi frequency from the fit of section 5.2.2,
and repeat this measurement for every few pixels over the Z-axis. The data was plotted in
the graph presented in Fig. 5.6. The data is fitted to a linear curve:
f (z) = az + b.
(5.2)
60
Figure 5.6: A graph of Rabi frequency as a function of position in the cloud. For each pixel position, the
Rabi frequency was calculated and plotted. The curve was fitted to a linear curve.
In this measurement the MW antenna was positioned 25 cm away from the cloud. By
definition, the far field is where the distance from the MW antenna is bigger than
2d2
,
λ
with
d being the dimension of the antenna aperture and λ the wavelength [Bal81]. In this region
the power decays with an inverse square law [Bal81]. Since our antenna aperture equals 5 cm
and λ = 4.4 cm, at 25 cm the data can be analyzed using the inverse square law. Therefore,
√
assuming: Ω0 ∼ p ∼ √1z2 ∼ z1 , and introducing this relation into the equation of the Rabi
frequency, for z z0 one can write:
r
1
1
z
ΩR ≈ (
)2 + 42 ∼ (1 − ) ∼ az + b,
z + z0
z0
z0
(5.3)
where z0 is the mean distance of the cloud to the antenna, and where we have assumed zero
detuning. The results are in good agreement with the theory.
5.2.4
The fringe pattern dependence on the position of the MW
antenna
To verify the relationship between the MW radiation and the fringe pattern, we imaged the
cloud for several positions of the MW antenna around the science chamber, as shown in Fig.
5.8. The position of the MW antenna is illustrated in each picture. It can be seen that the
61
(a)
(b)
(c)
Figure 5.7: Fringe patterns as a function of the MW antenna’s spatial position (drawn beside the images).
The antenna is positioned at the same YZ-plane, perpendicular to the imaging beam, but at different
distances from the cloud: (a), (b) 10 cm. (c) 25 cm.
lines of the radiation field are clearly mapped by the cloud. First, the fringe pattern always
follows the position of the antenna. Second, one may clearly observe the lessening curvature
of the radiation as the antenna-cloud distance grows from 10 to 25 cm. We have therefore
found a way to map a MW field in a single shot.
62
Chapter 6
Summary and Outlook
In this work, I experimentally study a method to coherently manipulate a two-state system
with cold atoms. I designed and built, starting with a ’naked’ optical table, a versatile laboratory system capable of trapping 109
87
Rb atoms, cooling them to a temperature of 60 µK,
and permitting them to fall freely under gravity. For these purposes, in chapter 4 I detail
the construction of a vacuum system capable of achieving pressures as low as 10−10 Torr
and two pairs of magnetic coils which facilitate trapping and cooling atoms in a MOT and
which define a quantum axis while at the same time introducing a Zeeman splitting. Also
described is the construction of two lasers which are used for the cooling and repumping
cycle, imaging and optical pumping stages, and a home-made tapered amplifier which yields
a power of up to 1.3 W. The lasers have extended cavities and are stabilized via locking to
an atomic transition by way of Doppler free polarization spectroscopy. I further detail the
path of the laser beams as they are manipulated in frequency and time, split into different
fibers, and impinge on the science chamber from different directions. Then, I describe the
concept of optical pumping and I show a qualitative method which I employed in order to
detect the population distribution over the ’stretched’ state |F 0 = 2, mF = 2i. I also detail
the imaging technique. To test the complete setup I measured the gravitational constant and
obtained a result which is in good agreement with the known value. In chapter 5 I describe
the experimental procedure and discuss relevant results.
Results indicate that the microwave field radiating from the horn antenna induces
63
not only Rabi oscillations over time as the MW pulse duration increases in each cycle, but
also Rabi oscillations simultaneously across the atomic cloud during each individual cycle.
This effect is explained in this work by the spatially varying power of the MW field, which
yields spatially varying Rabi frequencies and therefore spatially varying populations in the
|F = 1, mF = 0i and the |F = 2, mF = 0i states. In quantitative measurements, the Rabi
oscillations were found to be 10 and 20 kHz at different distances from the antenna. I show
that the evolution of the populations across the cloud creates an apparent ’spatial’ movement of the fringes. Using the definition of the far field zone, I could apply a model-based
approach and compare the results to theory. I also present qualitatively the effect of the
MW field spatial distribution on the fringe patterns.
The work described in this thesis was done in parallel to the work of P. Böhi et al.1 ,
who reported a technique that uses ultracold atomic clouds for mapping at a single imaging
shot, microwave near field distribution around a coplanar wave-guide integrated on an atom
chip. The main novelty so far in my results is the ability to map at a single imaging shot,
microwave fields generated from an horn antenna at near and far fields. Since my setup is
completely different and has no atomchip in it I had at my disposal more atoms, enabling the
observation of more fringes. In addition, as I could move and rotate the source of microwave
I could also observe a variety of dependencies of the fringe behavior on parameters such as
time, field direction and polarization.
The results obtained form the base for future work in which we intend to develop
methods for metrology as well as fundamental studies of coherence and decoherence. In the
immediate future I intend to verify and explain preliminary data I collected at the end of
my work and in which I observed an interesting dependence of the fringe spatial periodicity,
spatial angle and visibility on the rotation of the antenna about its longitudinal symmetry
axis, as shown in Fig. 6.1.
1
App. Phys. Lett. 97, 051101 (2010)
64
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 6.1: Images of fringes at different BM W angles. The MW antenna is rotated 360o across its main
axis, which is perpendicular to the quantum axis. (a) 0o (b) 40o (c) 100o (d) 130o (e) 180o (f) 210o (g)
280o (h) 310o (i) 350o . At 0o the magnetic field’s component of the electromagnetic wave was parallel to the
quantum axis. The number of atoms in images (c) and (g) is 10 % of the cloud imaged in (a).
65
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