Download Ultracold Collisions in Atomic Strontium

RICE UNIVERSITY
Ultracold Collisions in Atomic Strontium
by
Sarah B. Nagel
A Thesis Submitted
in Partial Fulfillment of the
Requirements for the Degree
Doctor of Philosophy
Approved, Thesis Committee:
Thomas C. Killian, Chair
Associate Professor of Physics and
Astronomy
Randall G. Hulet
Fayez Sarofim Professor of Physics and
Astronomy
Dan Mittleman
Associate Professor of Electrical and
Computer Engeneering
Houston, Texas
February, 2008
Abstract
Ultracold Collisions in Atomic Strontium
by
Sarah B. Nagel
In this work with atomic Strontium, the atoms are first laser cooled and subsequently trapped, in a MOT operating on the strong E1 allowed transition at 461 nm.
During the operation of this blue MOT, a fraction of the atoms decay into the 3 P2
and 3 P1 states, but can be recovered by applying light from repumper lasers at 679
nm, and 707 nm. Atoms trapped in the blue MOT are subsequently transferred into
a magneto-optical trap operating on the intercombination line at 689 nm, known as
the red MOT. Photoassociation experiments are carried out on these atoms using
an independent tunable source of blue light at 461 nm. These experiments map out
the underlying molecular potentials, and are useful in determining the atom-atom
interaction strength. Additionally, atoms trapped in the red MOT can be transferred
into an optical dipole trap (ODT) operating at 1064 nm, resulting in a cold dense
sample suitable for collision studies, lifetime measurements, and evaporative cooling
towards Bose-Einstein condensation.
Acknowledgments
Many thanks to my committee members, Dr Tom Killian, Dr Randy Hulet, and
Dr Dan Mittleman. Their valuable feedback helped me to improve this thesis in many
ways. Special thanks goes to my thesis advisor Dr. Tom Killian. Without him, this
thesis would not have been possible. I thank him for his patience and encouragement
that carried me on through difficult times, and for his insights and suggestions that
helped to shape my research skills. His valuable feedback contributed greatly to this
thesis.
Thanks also to the post-docs and my fellow graduate students in the Killian lab
for making the experience a memorable one; thanks especially to fellow ”neutrals”
team-members Natali Martinez de Escobar, Pascal Mickelson, and Aaron Saenz,
without whose collaboration this work could not have succeeded.
Thanks to Dwight Dear, for design advice and the fabrication of many custom
components – and for teaching me how to use a lathe. Thanks also to Carter Kittrell,
whose generosity extends almost as far as his collection of spare parts.
For occasionally getting me out of the lab, many thanks to the fellow members
of the Houston Sinfonietta, Crosspoint Church, and my roommates Alexis and Nancy,
whose monthly parties at the house on Drummond will be sorely missed.
Last but not least, I thank my friends and family for their encouragement and
unfailing support; I couldn’t have done it without you.
SDG
Contents
Abstract
i
Acknowledgments
ii
List of Illustrations
v
1 Introduction
1
1.1
Atomic Structure: Past and Present . . . . . . . . . . . . . . . . . . .
1
1.2
Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Optical Dipole Trap
. . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.6
Atomic Structure of strontium . . . . . . . . . . . . . . . . . . . . . .
11
2 The Apparatus
14
2.1
Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
461 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2.1
Seed Light for the Tapered Amplifiers . . . . . . . . . . . . . .
16
2.2.2
Tapered Amplifiers . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2.3
Doubling Cavities . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.4
Frequency Stabilization: Locking to the Atoms . . . . . . . . .
21
Repumper lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.3.1
679 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3.2
688 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3.3
707 nm light . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.4
689 nm Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.5
Photoassociation Laser . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.6
ODT Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.3
3 Typical Performance
35
iv
3.1
Blue MOT Operation and Diagnostics . . . . . . . . . . . . . . . . .
35
3.2
Red MOT Operation and Diagnostics . . . . . . . . . . . . . . . . . .
37
3.3
Dipole Trap Performance . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.4
Measuring The Elastic Cross Section . . . . . . . . . . . . . . . . . .
44
4 Photoassociation at Long Range
53
4.1
Scientific Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.2
Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.3
Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.4
Other Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5 Optical Pumping and Metastable States
5.1
5.2
66
Magnetic Trapping in the 3 P2 state . . . . . . . . . . . . . . . . . . .
67
5.1.1
Scientific Motivation . . . . . . . . . . . . . . . . . . . . . . .
67
5.1.2
Experimental Sequence . . . . . . . . . . . . . . . . . . . . . .
68
5.1.3
Majorana Spin Flips . . . . . . . . . . . . . . . . . . . . . . .
77
5.1.4
Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
ODT Lifetime Measurements of Metastable States . . . . . . . . . . .
80
5.2.1
Scientific Motivation . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.2
Experimental Sequence . . . . . . . . . . . . . . . . . . . . . .
80
5.2.3
Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6 Conclusions
Bibliography
84
88
Illustrations
1.1
Partial strontium Energy Level Diagram . . . . . . . . . . . . . . . .
12
1.2
strontium Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1
Vacuum Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
Mechanical Drawings and Photograph of the Tapered Amplifier Mount 18
2.3
Doubling Cavity Schematic . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4
461 nm Laser System Schematic . . . . . . . . . . . . . . . . . . . . .
23
2.5
Extended Cavity Diode Configurations . . . . . . . . . . . . . . . . .
25
2.6
689 Laser System Schematic . . . . . . . . . . . . . . . . . . . . . . .
29
2.7
689 nm Master . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.8
689 nm Slave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.9
Injection Locking Scope Traces . . . . . . . . . . . . . . . . . . . . .
32
3.1
Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2
Blue MOT Absorption Image . . . . . . . . . . . . . . . . . . . . . .
37
3.3
Blue MOT Temperature Determination . . . . . . . . . . . . . . . . .
38
3.4
Red MOT Temperature Determination . . . . . . . . . . . . . . . . .
40
3.5
Electron Shelving spectroscopy . . . . . . . . . . . . . . . . . . . . .
41
3.6
ODT capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.7
ODT Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.8
Time evolution of Number, Temperature, and η in ODT . . . . . . .
46
3.9
Fitting the curve: one- and two- body loss curves . . . . . . . . . . .
47
3.10 Fitting the curve: one and two body fit parameters . . . . . . . . . .
48
3.11 σel Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.1
Molecular levels and Photoassociative Transitions. . . . . . . . . . . .
55
4.2
Spectra
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.3
Lifetime of1 P1 level. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
vi
4.4
PAS decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
PAS Resonances in
Sr . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.1
Fluorescence Detection of 3 P2 Atoms . . . . . . . . . . . . . . . . . .
71
5.2
Lifetime of atoms in the Magnetic Trap . . . . . . . . . . . . . . . . .
72
5.3
Spatial distribution of 3 P2 atoms . . . . . . . . . . . . . . . . . . . .
74
5.4
3
P2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.5
3
P2 lifetime data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.6
Partial strontium Energy Level Diagram . . . . . . . . . . . . . . . .
80
5.7
Experimental Timing Sequence . . . . . . . . . . . . . . . . . . . . .
82
5.8
Temperatures in the ODT . . . . . . . . . . . . . . . . . . . . . . . .
83
86
61
1
Chapter 1
Introduction
This chapter contains a brief summary of atomic structure, as well as a discussion of optical pumping, a review of forces applied to atoms by light, and forces
applied to a specific atom: strontium. Also included is a discussion of selected applications of basic atomic physics research with this atom.
1.1
Atomic Structure: Past and Present
Theories about the nature of atoms have evolved over time, driven by the desire both
to understand, but also, to control the world around us. The ancient Greek philosopher Democritus, first proposed the idea that matter is made up of tiny particles
that could not be subdivided or made smaller. Our word atom comes from the greek
atomos, meaning ”uncuttable,” but Democritus’s theory quickly fell from popularity,
and was replaced by Aristotle’s ideals for the four humors: earth, fire, air, and water.
This was the prevailing view of the ”elements” that made up all other matter until
about 900 years ago. The next innovations in atomic theory occurred as scientists in
Europe were studying alchemy, in an effort to change ordinary materials into gold.
Their work has developed into modern chemistry. In 1808, John Dalton synthesized
the the ideas of Democritus and those of the alchemists, proposing atoms as tiny
particles that could not be divided, and that each element was made of its own kind
of atom. In 1897, an experimentalist brought empirical evidence to the table. J. J.
Thompson realized that while individual atoms had a net neutral charge, they did,
2
in fact, contain electrons. Thompson proposed that electrons were smaller particles
of an atom, distributed among a positively charged jelly-like substance – rather like
the raisins in plum pudding. In 1911, Earnest Rutherford (a former J.J. Thompson
graduate student) ran an experiment in order to test this plum pudding theory. In
this experiment, a stream of alpha particles were shot at a very thin sheet of gold
foil. Analysis of the scattering pattern revealed that atoms are mostly empty space,
but have a dense central positive core. In 1913, to show why the negatively charged
electrons in an atom are not sucked into this positively charged nucleus, Niels Bohr
proposed that electrons travel in fixed paths around the nucleus. This orbit model
is still often used to gain qualitative understanding of atomic structure, including
quantized energy levels. In his 1922 doctoral thesis, Louis de Broglie generalized this
idea of quantization by expanding it to matter. The de Broglie hypothesis states that
any moving particle or object had an associated wave. The first de Broglie equation
relates the wavelength λ to the particle momentum p as
h
h
h
λ= =
=
p
γmv
mv
r
1−
v2
c2
(1.1)
where h is Planck’s constant, m is the particle’s rest mass, v is the particle’s
velocity, γ is the Lorentz factor, and c is the speed of light in a vacuum. The greater
the energy, the larger the frequency and the shorter (smaller) the wavelength. Given the
relationship between wavelength and frequency, it follows that short wavelengths are more
energetic than long wavelengths. For particles with thermal velocities much smaller than
3
the speed of light in a vacuum, the de Broglie wavelength is given by
λdB = (
, where ~ =
h
2π
2π~2 1/2
)
mkB T
(1.2)
is Planck’s constant over 2π, m is the particle’s rest mass, T is the
temperature, andkB is Boltzmann’s constant.
That same year, a physics student named Werner Hiesenberg met Niels Bohr, and
the two formed a fruitful and lifelong collaboration. In 1927, Heisenberg developed his
uncertainty principle, which states that that the simultaneous determination of two paired
quantities, for example: the position and momentum of a particle, has an unavoidable
uncertainty. Together with Bohr, he formulated the Copenhagen interpretation of quantum
mechanics. After the discovery of the neutron by James Chadwick in 1932, Heisenberg
proposed the proton-neutron model of the atomic nucleus and used it to explain the nuclear
spin of isotopes. Current theories of atomic structure are based on quantum mechanics, the
system of thought that grew from the ideas of de Broglie and Heisenberg. The uncertainty
principle places bounds on how well we can simultaneously know position and momentum,
which leads to a certain amount of ”fuzzyness” associated with a particle.
So, rather than the electron moving in one particular fixed path around the nucleus,
the uncertainty principle allows us to calculate the probability that an electron will be
found in a certain region of space as well as the the shape of of the probability distribution.
1.2
Bose-Einstein Condensation
In 1925, while Heisenberg and de Broglie were working on quantum mechanics, a theorist
named Satyendra Nath Bose predicted a condensation phenomena while working on the
4
statistical mechanics of (massless) photons and (massive) atoms. Bose collaborated with
Einstein, and the result of the efforts of Bose and Einstein is the concept of a Bose gas,
governed by the Bose-Einstein statistics, which describes the statistical distribution of
identical particles with integer spin, now known as bosons. Bosonic particles, which include
the photon as well as atoms such as Helium-4, are allowed to share quantum states with
each other. Einstein demonstrated that cooling bosonic atoms to a very low temperature
would cause them to fall (or ”condense”) into the lowest accessible quantum state, resulting
in a new form of matter. The transition occurs below a critical temperature, which for
a uniform three-dimensional gas consisting of non-interacting particles with no apparent
internal degrees of freedom is given by:
Tc =
n
ζ(3/2)
2/3
h2
,
2πmkB
(1.3)
where Tc is the critical temperature, n is the particle density, m is the mass per boson, h
is Planck’s constant, kB is the Boltzmann constant, and ζ is the Riemann zeta function;
ζ(3/2) ≈ 2.6124.
By rearranging terms, we can see that this transition temperature defines the de
Broglie thermal wavelength for which nλ3dB ∼ 2.62. For temperatures below the critical
temperature, this becomes nλ3dB > 2.62. This gives significant insight: the de Broglie wavelength defines a length scale, and λ3dB defines a three-dimensional volume. By multiplying
the density by this volume, we are actually counting the number of particles present in
that volume. So, at the critical temperature, there are more than 2.6 particles in a volume
defined by the de Broglie wavelength cubed. The spatial extent of the individual atom is
now significantly greater than its de Broglie wavelength. Here, the quantum mechanical
5
nature of the atom dominates the particle picture; the wavefunctions of the atoms overlap and superposition occurs: rather than behaving like a bunch of individual atoms, this
collection of atoms behaves like one coherent quantum object. This parameter nλ3dB is
called the degeneracy parameter, and is a metric by which progress toward Bose-Einstein
condensation is measured.
In 1995, the experimental observation of Bose-Einstein condensation (BEC) in magnetically trapped atomic vapors of rubidium [22], sodium [23], and lithium [24] opened a
new field of study at the intersection of atomic and condensed matter physics, and BECs
have been realized in a number of atomic isotopes. Even now, the atoms that have been
Bose-condensed are those with inherent spin. A BEC of atomic strontium offers a unique
zero-spin groundstate, that would assure a condensate much less susceptible to stray magnetic fields. This sort of spin-zero condensate woudl be especially useful for interferometry
experiments.
1.3
Optical Pumping
Optical pumping is a process in which light is used to promote (or ”pump”) electrons from
a lower energy level in an atom or molecule to a higher one. It is commonly used in laser
construction, to pump the active laser medium so as to achieve population inversion. The
technique was developed by Nobel Prize winner Alfred Kastler in the early 1950’s.
Optical pumping is also used to cyclically pump electrons bound within an atom
or molecule to a well-defined quantum state. For the simplest case of coherent two-level
optical pumping of an atomic species containing a single outer-shell electron, this means
that the electron is coherently pumped to a single hyperfine sublevel (labeled mF ), which
6
is defined by the polarization of the pump laser along with the quantum selection rules.
Upon optical pumping, the atom is said to be oriented in a particular mF sublevel, however
due to the cyclic nature of optical pumping the bound electron will actually be undergoing
repeated excitation and decay between upper and lower state sublevels. The frequency and
polarization of the pump laser determines which mF sublevel the atom is oriented in.[1]
In practice, completely coherent optical pumping may not occur due to powerbroadening of the linewidth of a transition and undesirable effects such as hyperfine structure trapping and radiation trapping. Therefore, the orientation of the atom depends more
generally on the frequency, intensity, polarization, spectral bandwidth of the laser as well
as the linewidth and transition probability of the absorbing transition.
1.4
Laser Cooling and Trapping
Over the last 30 years, laser cooling and trapping techniques have revolutionized experimental atomic physics. Because these techniques allow a significant reduction in translational
energy, these systems approach the ideal ensemble of stationary atoms, allowing us to
probe their interactions among themselves as well as interactions with the environment.
Laser cooling has become a standard laboratory tool for producing cold (< 1 mK), dense
(1010 -1011 cm−3 ) samples of atoms. Magneto-optical traps [7] are now commonplace and
provide a starting point for numerous avenues of research in atomic and quantum physics.
The basic mechanics of laser cooling and trapping are discussed in Metcalf and van der
Straten’s book Laser Cooling and Trapping [9].
Using resonant radiation pressure to cool atoms was proposed independently by Hansch and Schawlow [11] and by Wineland and Dehmelt [12] in 1975. The basic idea, com-
7
monly known as Doppler cooling, is that atoms traveling towards an opposing laser field
will preferentially absorb light that is detuned below (to the red of) the center of the atomic
resonance. This preferential absorption is due to the Doppler effect which causes the light
to be shifted into resonance with the atoms. On average, fluorescence is symmetrically
distributed in space, which leads to a net atomic momentum loss. Momentum transfer
from red-detuned light causes a damping force that opposes the motion of the atom.
Balykin et al. [13] were the first to experimentally observe this effect in 1-D cooling
of a Sodium atomic beam in 1979. Shortly after, Phillips et al. [14, 15] added a magnetic
gradient field to compensate for the changing Doppler shift and to keep the atoms in
resonance with the cooling beam as they slowed down. This cooling scheme is often called
Zeeman slowing. In 1984 Ertmer et al. [16] used a swept-frequency laser chirp to track the
Na atomic resonance as the atoms were slowed from an initial beam velocity to 600 m/s
to a final gas cloud velocity of about 6 m/s (50 mK).
By using three pairs of intersecting, orthogonal, counterpropagating red-detuned
laser beams Chu et al. [17] extended Doppler cooling into three dimensions in 1985.
Atoms with speeds below a certain capture velocity were rapidly cooled to a remarkable
temperature of ∼240 µK. Although there was no position-dependent force, and thus the
sample was not trapped, the overlap region of the laser beams confined the atoms for ∼100
ms. This configuration is known as optical molasses.
Without any position-dependent restoring force, the atoms eventually random walk
out of the cooling beams. The temperature limit of this molasses is found by balancing
the cooling due to the damping force and the heating from the statistical fluctuations of
the force caused by random absorption and emission of photons. Using a Fokker-Planck
8
semiclassical model, several theoretical treatments [18–21] yield the well-known Doppler
limit for laser cooling, given by
kB Td =
~Γcool
,
2
(1.4)
where Td is the Doppler-limited temperature, and Γcool is the linewidth of the atomic
transition used for cooling. This limit says that the minimum kinetic energy of the atoms
is equal to the energy width of the cooling transition. It is interesting to compare this
limit to the recoil-limited temperature, which occurs when the kinetic energy of the atom
is equal to the recoil energy imparted to the atom when it absorbs a single photon:
kB Tr =
~2 k 2
.
M
(1.5)
Here, k is the wavenumber of the light and M is the mass of the atom. For most experiments, this is the hard limit for laser cooling since it involves the minimum interaction
with the laser field. For typical cooling transitions, the Doppler-limited temperature is
100 − 1000 times greater than the recoil-limited temperature.
By adding a magnetic quadrupole gradient field whose zero coincides with the optical
molasses center, Raab et al. [7] utilized the internal structure of the atom to produce a
3-D restoring force that trapped the atoms for long periods of time. By using orthogonal
circular light polarizations to distinguish each counterpropagating beam, spatially-preferred
absorption in three dimensions is caused by Zeeman level shifts as the atoms move away
from the zero of the magnetic field gradient. This gradient field is easily generated by an
anti-Helmholtz coil pair — a Helmholtz configuration with opposite circulating currents
in each coil. The damping force of the optical molasses simultaneously exists with the
restoring force. This apparatus, known commonly as a magneto-optical trap, or MOT,
9
provides a convenient way not only to cool atoms to very low velocities, but also to confine
them to a very small volume (1 mm3 ), producing dense, ultracold atomic samples.
1.5
Optical Dipole Trap
The Optical Dipole Trap (ODT), another confinement technique developed in the 1980s,
is based on an effect quite different from the spontaneous scattering force described above.
While the scattering force acts in the direction of laser propagation, the dipole force acts
in the direction of the gradient of the laser intensity. A good review of the principles
and several different applications of optical dipole traps can be found in Grimm’s 2000
Advances in Atomic, Molecular and Optical Physics paper [6].
The optical dipole force was first observed in 1978 [3], when researchers co-propagated
a tightly- focused laser beam with an atomic beam. They observed that when the laser
was tuned close to the atomic resonance frequency, the laser beam caused deflection and
focusing of the atoms. Thus the light exerted a force on the atoms perpendicular to the
direction of laser propagation.
The energy shift of atoms in a light field is often referred to as the AC Stark shift
or the light shift, and the gradient of this energy shift gives the dipole force. The dipole
force can be understood simply in terms of the atom electric polarizability. In free space,
an atom in its ground state has no electric dipole moment. However, a static electric field
induces a dipole moment in the atom, reducing its potential energy. Therefore a gradient
in electric field draws the atom toward the region of space with higher field. This is simply
the Stark shift.
The AC Stark shift arises in a similar fashion; the laser light is simply an electromag-
10
netic field that reverses direction every few femtoseconds. If the laser frequency is below
the atomic resonance, the dipole induced in the atom can keep up, or stay in phase with
the light, and atoms are drawn to regions of highest intensity. If instead the light frequency
is tuned above the atomic resonance, the dipole moment always opposes the electric field,
and atoms are expelled from regions of highest intensity.
The dipole force was used to demonstrate the first optical trap for atoms in 1986 [4].
In this experiment, about 500 Na atoms were confined at the waist of a focused Gaussian
laser beam. The laser beam contained about 220 mW, detuned about 650 GHz below the
atomic resonance frequency and focused to a 10 µm size. The trap was loaded from an
optical molasses by rapidly alternating between the optical molasses and the optical dipole
trap. The trap laser detuning ∆ was much larger than the width of the transition, but not
large enough to completely suppress the spontaneous scattering force.
Soon after this experiment, the MOT was developed and could trap many more atoms
at much higher densities. Therefore dipole traps were generally abandoned until in 1993
when the far off resonance trap (FORT) was demonstrated [5]. This trap employed more
power and larger detunings, and captured a few thousand atoms. The increased detunings
allowed the much longer trap lifetimes than those of the original dipole traps, in order to
have strong dipole forces and simultaneously have low spontaneous scattering rates that
cause negative effects. The dipole trapping force that actually does the trapping scales as
∆−1 , while the spontaneous photon scattering rate scales as ∆−2 . Spontaneous scattering,
although many orders of magnitude lower in the FORT than the MOT, can still cause
heating, decoherence, and scrambling of the internal states. Additionally, since there is no
dissipation in a FORT optical dipole trap, any cooling must occur via collisions.
11
Michael Chapman[2] in 2001 was the first to use evaporative cooling in a FORT
to cool atoms to quantum degeneracy. Since then, optical dipole traps have become a
standard technology in the world of atomic, molecular, and optical physics.
1.6
Atomic Structure of strontium
The most commonly trapped ultracold atoms are the alkali metals, but there has been
considerable interest in expanding the range of laser-trapped elements. The alkaline earth
metal atoms possess an atomic and nuclear structure which makes them appealing for laser
cooling studies. The most abundant alkaline earth atoms have no nuclear spin (I = 0) and
therefore no complicating hyperfine structure. Furthermore, the two valence electrons can
couple together anti-parallel (S = 0) to produce a singlet state or parallel (S = 1) to
make a triplet state. As a consequence the ground state is a single J = 0 state, and the
alkaline earth atoms approach the ideal J = 0 → J = 1 system which is commonly used
to theoretically describe many laser cooling experiments. The level diagram for strontium
displayed in Figure 1.1 shows this simple atomic structure [25]. Levels for the other alkaline
earth atoms have a similar structure.
All alkaline earth atoms offer strong cycling transitions from the singlet ground state
to the first excited singlet P state. In the case of strontium, the upper state of the 1 S0 → 1 P1
461 nm cycling transition has a 5 ns lifetime [26] and allows rapid laser cooling of the atoms.
If an atom falls into a lower lying D state, it becomes unaccessible for cooling, and lost
from the trap. For strontium, however, the branching ratio to the lower singlet D state is
small enough that atoms can be brought to near zero velocity before decaying out of the
cycling transition.
12
Figure 1.1 : Partial strontium Energy Level Diagram. Decay rates (s−1 ) and selected
excitation wavelengths are given. Taken from Nagel et al. [25]
An important property of the structure of the alkaline earth atoms is the division of all
atomic states into either singlets or triplets. From selection rules, we know that the electric
dipole (E1) operator does not connect ∆S = ±1 transitions, and thus we are justified in
separating the singlet from the triplet states as we have done in Figure 1.1. That is, E1
transitions can occur only between states of the same S. This picture is strictly true for
low mass atoms in which Russell-Saunders LS-coupling holds [27]. However, as an atom’s
mass increases, there is a progressive breakdown of LS-coupling, and spin-orbit effects
will mix singlet and triplet states of the same electronic term. Thus, E1 transitions with
∆S = 1 are possible and become increasingly stronger for heavier atoms. These ∆S = 1
transitions, commonly referred to as intercombination lines, are generally much weaker
than their ∆S = 0 counterparts. Because of their narrow linewidths, intercombination
transitions are useful as potential optical frequency standards. Figure 1.2 compiles the
characteristic numbers for the transitions used in the strontium experiment.
13
Figure 1.2 : This table[8] contains information for each of the transitions used in the
Neutral strontium experiment.
Equipped with the knowledge of capture velocities and Doppler temperatures limits
for the two principle transitions of atomic strontium, the experimental course is clear: the
experiments will start with a MOT operating on the broad transition at 461 nm, cool
the sample in that MOT, then transfer the sample to a MOT operating on the narrow
transition at 689 nm. From there, the sample will be further cooled in an ODT. In the
next chapter, we discuss the apparatus for the experiment, including the laser systems used
to generate light for these experiments.
14
Chapter 2
The Apparatus
Performing experiments on an atomic system requires ultra-high vacuum equipment
and several laser systems. This chapter includes a description of the vacuum system as
well as several laser systems. The subsequent chapters describe each of the experiments in
detail.
2.1
Vacuum System
There are currently two experimental setups in the Killian lab. The first apparatus was used
for early neutral experiments as well as plasma experiments. In late 2005, construction on a
dedicated neutral assembly (called the ”neutrals table”) was completed, and the originally
shared apparatus was designated the ”plasma table.”
In many ways the two assemblies are quite similar. There is a small strontium
reservoir that is heated to 550◦ - 620◦ C. This stream of strontium vapor is transversely
cooled by a 2D-collimator and then travels down a Zeeman slower against a counterpropagating Zeeman beam to the MOT chamber, where three retro-reflected beams and
a pair of anti-Helmholtz coils form the magneto-optical trap. Figure 2.1 shows a sketch.
Notable design differences between the two systems include the orientation of the MOT
chamber to allow more optical access, as well as measures to reduce the pressure in the
system: the addition of a skimmer disk on the new setup, slight modifications to the nozzle
design, and position of the ion pumps. With these improvements, the pressure in the
15
Figure 2.1 : Vacuum Schematic
chamber has been significantly reduced. We determine the background vacuum level by
measuring the lifetime of atoms in the 3 P2 state in the quadrupole magnetic trap formed
by the anti-Helmholz coils during the operation of the blue MOT. The method for this
measurement are detailed in Section 5.1. These lifetimes were measured both in the original
setup ( 500 ms) and in the new neutrals setup (∼7 s) . The 14-fold increase in lifetime
directly maps to a 14-fold reduction in background gas levels. With lower background
gas levels, one body losses from the ODT will be significantly reduced, which allows much
longer interrogation and evaporation times in the ODT.
2.2
461 nm Light
In order to cool on the strong transition in atomic strontium, 461 nm (blue) light is produced via second harmonic generation. While these techniques are well documented in the
literature [118], and in the Killian lab [125], [126], this section documents the implemen-
16
tation of tapered amplifiers (TA), which are used to produce sufficient power at 922 nm.
This infrared light is then coupled into monolithic KN bO3 doubling cavities to produce
blue light via second harmonic generation. The frequency of this blue light is locked to, or
at a certain detuning from, the atomic resonance using saturated absorption spectroscopy.
Figure 2.4 is a schematic of the optics table.
2.2.1
Seed Light for the Tapered Amplifiers
The generation of blue light for cooling on the strong transition in atomic strontium begins
with a source of 922 nm light. There are two such sources in the Killian lab, the Ti:Saph
and a commercially available high-power diode laser from Sacher, and both have been
utilized to seed the tapered amplifiers.
The Ti:Saph has been used as a seed beam in the following manner. 200 mW from
the Ti:Saph is coupled through a polarization-preserving fiber, and then into the tapered
amplifiers. The output mode of this fiber is well known and the frequency of the Ti:Saph is
locked to a strontium absorption cell on the plasma table. Because the Ti:Saph is central
to the plasma setup, using it as a seed beam limits the scope of the neutrals experiments
to one isotope of strontium and a fixed lock frequency.
Using the high powered diode laser from Sacher as a seed beam is also straightforward.
In this case, there is no fiber, just a cylindrical lens to clean up a slight astigmatism.
This laser is coupled directly into the TAs. Here, the frequency is locked to a novel
saturated absorption cell on the neutrals table [123], which gives the experiment much
greater flexibility. This novel cell is discussed in detail in section 2.2.4.
Regardless of which of these sources is used, modulation of the frequency is neces-
17
sary to produce the Pound-Drever-Hall [38] error signal for locking the doubling cavities.
Sidebands written onto the seed light transfer through the TAs without trouble. When
working with the Ti:Saph, an electro-optic modulator (EOM) on the plasma provides 15
MHz sidebands on the seed light, which is written onto the TA output and enables locking
of the doubling cavities. When using the Sacher laser, 15 MHz sidebands are written onto
the seed light via current modulation.
2.2.2
Tapered Amplifiers
Tapered Amplifiers are commercially available solid state devices that function as optical
amplifiers, rather than lasers. We use model EYP-TPA-0915-01500-3006-CNT03 from
Eagleyard Photonics, whose input aperture is 3 µm, rated for 2.5 Amps and 1.5 Watts
of optical power at 922 nm. These high power devices require temperature control with
substantial heat-sinking, a high current source, and very stable input and output alignment.
Simplified mechanical drawings are shown in figure 2.2. The aluminum block serves as a
heat-sink, while three 6 Amp thermoelectric coolers keep the temperature of the copper
block around 20 C. In addition to providing significant heat capacity and physical stability,
the large copper mount acts as the anode for the TA and is electrically isolated from the
aluminum by plastic screws. The other lead cathode of the TA is separated from the
copper by a plastic spacer and secured to an electrical contact with a plastic screw. A
five-axis fiber aligner from New Focus holds the input coupler (Thorlabs C330-TM-B) and
allows precise placement of the input coupling lens. An additional spring-mount aligner
houses the output coupler and allows for similar placement of this identical lens. As a side
note, tapered amplifiers simply work as a gain medium; as such, they can be seeded by
18
Figure 2.2 : Mechanical Drawings and Photograph of the Tapered Amplifier Mount
a collimated 922 nm beam. These devices are are extremely sensitive to optical feedback
and must be protected from reflections by optical isolators.
Maximizing the optical coupling of the seed light into the TA is critical for reliable
performance. In order to do this, it is important that the seed light is collimated. The
first step in aligning the TA is to center the input coupler and collimate the spontaneously
emitted light coming ”in reverse” from the first face of the TA. The flexibility of the New
Focus mount allows the user to move the input coupler lens not only vertically, horizontally,
and longitudinally, but also to adjust tilt and yaw. Once the spontaneously emitted light
19
from the input side of the TA is collimated, this input mount is fixed and is very seldom
adjusted. With the spontaneous emission of the TA collimated, coupling the seed beam into
the device is achieved using the last two mirrors before the input coupler. This alignment
requires daily tweaking to regain the last 10 percent of optical power.
Profiling and collimating the output of the tapered amplifier is a bit tricky. The
output mode of these devices is typically asymmetric and astigmatic. As a first step, it is
useful to roughly collimate the fast-expanding axis using the output coupler. The slowerexpanding axis will still give the beam some astigmatism and requires beam-shaping with
cylindrical lenses. Profiling the beam close to the device, and expecting Gaussian behavior,
is a mistake. Inside 50 cm, non-Gaussian modes are also present and skew the measurements. At distances greater than 2 meters from the TA, however, these non-Gaussian
modes have diffused, the remaining mode fits quite well to a Gaussian and is modeled
quite easily. Profiling in this region while placing the cylindrical lens yields collimated spot
sizes on the order of 1 mm and asymmetries under 5 percent. Once the cylindrical lenses
have been placed, this output mode has been set, and it is seldom revisited.
We use two of these tapered amplifiers to generate light at 922 nm. The frequency
of the seed light for one TA has been shifted relative to the other by an AOM, and the
output of the two TAs go to two separate doubling cavities: one to generate light for the
Zeeman slower, and the other to generate light for the MOT beams, 2D collimator beams,
and the imaging beam.
20
Figure 2.3 : Doubling Cavity Schematic
M=mirror, DC=dichroic, passes 922 nm, reflects 461
2.2.3
Doubling Cavities
Second Harmonic Generation has been used and documented extensively in the Killian lab,
see especially Aaron Saenz’s master’s thesis [126]. Briefly, infrared light is mode-matched
to a cavity formed by a PZT-mounted input coupler (IC) and the back face of a KNbO3
crystal, which has been coated for high reflectivity. By ramping the voltage on the input
coupler PZT, the length of the cavity, and thus its resonance frequency, is changed. When
the mode of the cavity is matched to the mode of the incoming infrared light, the KNbO3
generates blue light. By actively stabilizing the length of the cavity via feedback to the
PZT, blue light is generated. A dichroic mirror (DC) passes the infrared light, but reflects
the blue out of the cavity. With good alignment and temperature stabilization, conversion
efficiencies can be as high a 45%.
21
2.2.4
Frequency Stabilization: Locking to the Atoms
Once blue light has been generated, it is still necessary to stabilize the frequency of this
light to the atomic transition. The locks on the doubling cavities are such that sweeping
the frequency of the seed light results in a sweep of the frequency of blue light. Depending
on which seed light is used, locking to the atomic frequency can be accomplished in either
of the following ways:
If the Sacher laser is functioning as the seed, we use a small reflection of the blue light
generated by the MOT cavity as a saturated absorption beam. This beam goes through an
acousto-optic modulator (AOM) running near 100 MHz. Both the first and zeroth order
beams travel to a novel strontium absorption cell, wound with a variable-current solenoid
[123]. The zeroth order beam functions as the pump, and the first order as the probe. As
the frequency of the 922 nm laser is ramped, the current through the solenoid is dithered,
thus ramping the blue frequency over the atomic resonance and generates an absorption
error signal, which is fed back to the Sacher laser. By changing the offset current in the
solenoid, we shift the atomic resonance, and the laser frequency follows. This capability
allows dynamic ramping of the detuning of the MOT beams, thus giving the experiment
much greater flexibility.
If the Ti:Saph is functioning as the seed, a doubling cavity on the plasma table
generates blue light . A fraction of that light travels through an AOM to a standard heatpipe absorption cell, which generates an error signal that is fed back to the Ti:Saph. Thus,
with the Ti:Saph as seed, the frequency of the blue laser is simply fixed.
Thus, by seeding the TAs and coupling their output into the doubling cavities, we can
22
generate more than the 200 mW of light necessary to run the blue MOT. Figure 2.4 shows
a schematic of the 461 nm laser system. The dependence of the blue MOT performance
on power and detuning are discussed in Chapter 3.
23
Figure 2.4 : 461 nm Laser System Schematic
24
2.3
Repumper lasers
As atoms are loaded into the blue MOT, some fraction of them fall into the D state, and
some fraction fall further into the 3 P2 and 3 P1 states (See Figure 1.1). The atoms that fall
into the 3 P1 state can fall easily back to 1 S0 and are caught up into the blue MOT. But
those that fall into the 3 P2 state are no longer optically accessible. Applying light at 707
nm, however, promotes these 3 P2 atoms to the 3 S1 state, which can then fall back to the
3P
1
and to the ground state. Unfortunately, from the 3 S1 state, atoms can also fall back
into the 3 P0 . Light applied at 679 nm pumps atoms from the 3 P0 back to 3 S1 state, and
now all the atoms are trapped in the blue MOT. Thus, the 707 and 679 lasers are useful
for controlling atoms in the blue MOT and probing the population of the 3 P2 state. An
additional laser at 688 nm, which connects the 3 P1 to the 3 S1 , allows complete control
over all the 3 P levels. Using the 707 nm laser alone yields a factor of three increase in
the number of atoms in the MOT; adding the 679 nm laser as well yields a factor of six
increase in the number of atoms in the MOT. Because there is attrition at each step of the
experiment, and each of the the traps has a finite lifetime, the use of repumpers in this
early stage allows us to extend our experimental capabilities by extending the timescales
on which we have a measurable signal.
Diode lasers are used for generating all of the red light (679-707 nm) in the Killian
lab. The frequency of light produced by a diode laser source depends on temperature and
drive current, as well as optical feedback. As the temperature increases, the frequency
decreases. This is not a smooth tuning; it is subject to mode hops. Likewise, as the drive
current increases, the frequency decreases. This, too, is not a smooth tuning; it is also
subject to mode hops. Nevertheless, these two factors yield very high flexibility, and when
25
Figure 2.5 : Extended Cavity Diode Configurations. a) In the Littrow configuration,
up to 80% of the power is coupled out of the ECDL. b) In the Littman-Metcalf
configuration, the output beam does not wander as the wavelength changes.
working together can yield tuning ranges of hundreds of GHz.
As mentioned earlier, laser diodes are extremely sensitive to optical feedback. We
can use this sensitivity to our advantage by coupling the laser to an external wavelength
selective element such that only a desired frequency is returned to the laser diode. Because
lasers work on amplification principles, the particular frequency that is fed back gains power
and eventually becomes the dominant mode. This reduces the threshold current, narrows
the linewidth, and provides a broad tuning range. Any of several wavelength selective
elements can be used: gratings, etalons, and high finesse cavities [39]. For many of the diode
lasers in the Killian lab, we have chosen to use a grating to form an extended cavity diode
laser (ECDL). Of the many ECDLs configurations, the two common optical configurations
are the the Littrow and Littman-Metcalf, illustrated in Figure 2.5. A description of each
configuration follows.
In the Littrow configuration, the grating is aligned such that the first order diffraction
returns directly to the laser diode. The coarse lasing wavelength is then determined by
the angle of the grating with respect to the laser. Wavelength tunes with this angle. The
output is the zeroth order reflected beam, which can be as high as 80% of the original
26
output power of the diode. The advantages of this configuration are simplicity and output
power. The disadvantage is that the angle of the output beam changes slightly as the angle
of the grating changes.
In the Littman-Metcalf configuration, the output beam from the laser diode is aligned
at grazing incidence with the grating. The first order diffracted beam is then sent to a
mirror which reflects the beam back on itself. This reflected beam then hits the grating
and the first-order diffracted beam couples back into the diode. Here, wavelength tuning
is accomplished by varying the angle of the mirror, which changes the wavelength that
the diode receives as optical feedback. The output is the zeroth order reflected beam off
the grating. Since the grating in this configuration does not move, the output beam angle
does not change as the wavelength is tuned. That is the advantage of the Littman-Metcalf
configuration. However, since the light coupled back into the laser is be diffracted twice by
the grating, a larger fraction of the power must be diffracted, leaving less power available
for the output[40]. The Littman-Metcalf configuration is useful because the beam does not
wander as the angle of the mirror is changed.
In order to manipulate the population of atoms in the 3 P2 and 3 P0 states, we use
a variety of diode-laser technologies to access the different red wavelengths. One of these
lasers uses the very simple Littrow configuration, another uses a Littman-Metcalf, yet
another is an original design, cooled to liquid nitrogen temperatures. Because of pending
patent issues, its design will not be discussed here.
27
2.3.1
679 nm Light
As mentioned above, light at 679 nm connects the 3 P0 state to the 3 S1 . A 30 mW Mitsubishi
ML1012R-12 laser diode is used. The typical wavelength for this laser, free-running, is 685
nm; the maximum current is 120 mA . For wavelength selection, the diode is placed into a
simple Littrow configuration extended cavity. In practice, the diode operates at the target
wavelength at 75 mA current near room temperature. A fraction of the output of the laser
is passed through a strontium discharge cell in order to lock the frequency to the atomic
resonance. To generate an error signal, the grating PZT is dithered at 4 KHz, which
coincides with a mechanical resonance in the PZT mount system. The laser is locked via
PZT feedback. The power available to the experiment is 3 mW, and the linewidth of the
laser is on the order of 2 MHz.
2.3.2
688 nm Light
Light at 688 nm connects the 3 P1 and 3 S1 . A 35 mW Eudyna FLD6A2TK laser diode is
used. The typical free running wavelength for this laser is 685 nm; the maximum current is
60 mA. For wavelength selection (and flexibility) the diode is placed in a Littman-Metcalf
extended cavity. In practice, this diode run at 42 mA current and 22◦ C. The linewidth of
this laser is on the order of 1 MHz and is locked to the wavemeter via mirror PZT feedback.
Optical power available to the system is one the order of 1 mW. This laser serves as a ready
backup to the master laser of the 689 nm system, and holds promise as a tool for future
experiments, including photoassociation on the 689 nm transition.
28
2.3.3
707 nm light
Light at 707 nm connects the 3 P2 and 3 S1 states. A 10 mW Qphotonics QLD-735-10S laser
diode, whose free running wavelength is 735 nm and maximum current is 100 mA, is used.
This novel extended cavity diode laser [124] is cooled to liquid nitrogen temperatures and
locked to an discharge cell via current feedback and affords 0.5 mW of optical power to
the experiment. The linewidth of this laser is on the order of 100 MHz.
2.4
689 nm Light
For cooling on the inter-combination line of atomic strontium, it is necessary to have a
narrow linewidth laser light source. Previously [129], we used a single extended cavity
diode laser in the Littrow configuration, locked to a high finesse cavity, which was, in turn,
locked to a strontium saturated absorption cell. The linewidth of that laser was 100 kHz.
We have been able to further reduce the linewidth of the laser to 50 kHz by making small
changes to the locking electronics, and switching to a Littman-Metcalf configuration. In
this configuration, there is not enough optical power available to run the experiment, so
a master/slave injection locking scheme is used. The locking schemes and example scope
traces are discussed in detail in the following section, with the schematic shown in Figure
2.6.
The master laser in this system (Figure 2.8) is a Littman-Metcalf extended cavity
diode laser locked to a high finesse cavity via the Pound-Drever-Hall (PDH) method, as
detailed in [129]. There are three servo loops: one directly to the diode, the second to the
current modulation input of the current driver, and the third to the laser PZT. An EOM
29
Figure 2.6 : 689 Laser System Schematic
is used to generate 50 MHz sidebands necessary for the PDH error signal.
The slave laser setup is very simple (See Figure 2.8). The diode is temperature
controlled and run at constant current. Of note, in order to eliminate unwanted optical
feedback to the master, a polarizing beam splitter (PBS) cube and an optical isolator
are between the last steering mirror and the slave laser. The polarization of the master
laser is horizontal, and passes through the cube. As it travels through the optical isolator,
the polarization is rotated 45◦ and injected into the slave diode. Light from the slave laser
passes through the optical isolator, which rotates the polarization of the light an additional
45◦ , and at the PBS cube, the polarization of the slave light is vertical and is rejected by
the cube. This polarization concern necessitates physically mounting the diode 45◦ to
horizontal.
In order to injection lock the slave laser to the master, alignment is crucial. It is
necessary to peak up this alignment every few weeks. The steps are as follows: First,
visually inspect the overlap of the seed beam and the slave beam along the optical path.
Determine the threshold current of the slave laser by monitoring the rejected beam from
30
Figure 2.7 : (a) Schematic and Photograph of master Littman-Metcalf ECDL
Figure 2.8 : (b) Schematic and Photograph of Slave Laser
31
the PBS cube in the absence of a seed beam. Then, set the slave laser current just below
the threshold value. While still monitoring the output of the slave laser, use the two
mirrors before the PBS cube to align the seed beam into the slave laser. Injection locking
is assured when the observed output lases in the presence of the seed beam. Next, lower the
current on the slave laser, and realign the seed beam. Do this a few times to minimize the
injection-locked slave laser threshold current. Once the injection-locked current threshold
has been minimized, look at the transmission modes of the diagnostic Fabrey-Perot cavity.
Both the slave and the master have diagnostic beams aligned to this cavity. As the length
of the cavity is changed via a PZT actuator, transmission modes appear (See Figure 1.8).
In the absence of injection-locking, there are two sets of cavity modes: one corresponding
to the master and one corresponding to the slave. If the injection-locking is working well,
there will be one set of modes. Sometimes it is necessary to adjust the current of the slave
laser a bit for the injection locking to work well. The Fabrey-Perot cavity to which the red
laser is locked is an excellent diagnostic as well.
With the master locked to the high finesse cavity, and the slave laser injection locked
to the master, the only detail that remains is locking to the atomic absorption signal. A
fraction of the output of the slave laser is aligned to an optical fiber, which carries about
300 µW of power to a saturated absorption cell [130]. Here, an AOM shifts the frequency
of the pump beam relative to the probe, and by dithering the frequency of the AOM, we
generate an atomic error signal, which is fed back to the high finesse cavity PZTs. Thus,
we lock the 689 nm system to the atomic line.
32
Figure 2.9 : In both cases, the length of the diagnostic Fabrey-Perot cavity is ramped
with both slave and master diagnostic beams present. In the upper trace, The
slave laser is not injection locked, and there are two peaks in the photodiode trace,
corresponding to the two lasers. In the lower trace, the slave has been injection-locked
to the master, and there is a single peak in the photodiode trace, corresponding to
the single frequency operation of the lasers.
33
2.5
Photoassociation Laser
In order to probe the molecular potentials of Sr2 via spectroscopy, an independent source of
high power ( 100 mW) blue light, tunable over several GHz is necessary. For this purpose,
we utilize an additional doubling cavity, and the frequency doubling techniques detailed in
Section 2.2. There are several ways to realize the necessary frequency tunability for this
laser.
In the photoassociation spectroscopy (PAS) experiment detailed in Chapter 4, the
Ti:Saph seeded the Zeeman and MOT doubling cavities, and thus its frequency was locked
to the atomic strontium resonance. A fraction of the Ti:Saph power was diverted by
an AOM and used to seed the additional doubling cavity. In order to achieve frequency
tunability, the output of the PAS doubling cavity was shifted by a series of AOMs. By
varying the AOM drive frequencies, the frequency of the photoassociation beam varied.
Thus, the detuning from atomic resonance here was limited to the frequency ranges of the
AOMs.
In subsequent experiments [122], the Ti:Saph continued to seed the Zeeman and
MOT doubling cavities, and the additional doubling cavity was driven by the 922 nm
Sacher diode laser. Here, the tuning range of the photoassociation light was much larger
and independent of the Ti:Saph.
2.6
ODT Laser
Recently, we have implemented a crossed optical dipole trap by focusing the output of a 20
Watt IPG fiber laser to a beamwaist of 75 µm. This laser is passed through a high-powered
34
AOM, which gives us the ability to vary the power in the ODT beams.
The vacuum and laser systems described in this chapter are tools to perform our
research. In chapter three, I describe some typical performance values for the blue MOT,
the red MOT, and the ODT.
35
Chapter 3
Typical Performance
This chapter includes typical data for operating the experiments on
88 strontium.
Sequentially, the atoms are trapped in the blue MOT, then the red MOT, and then the
ODT. In this chapter, we only used repumpers for data in Section 3.4, but they will feature
heavily in the next two chapters.
I have not included a discussion of repumpers in this chapter, though they will feature
heavily in the next two chapters.
3.1
Blue MOT Operation and Diagnostics
The Magneto-optical trap on the 1 S0 → 1 P1 transition in strontium has been operational in
the Killian lab for several years [25] and is the starting point for all the experiments. In this
setup, atoms are slowed by a Zeeman beam and trapped by three orthogonal retro-reflected
beams. The current in the anti-Helmholtz coils is 35A, corresponding to a field gradient
at the center of 100 G/cm. We measure the number of atoms and sample temperature
using absorption imaging techniques, as illustrated in Figure 3.1. These techniques involve
illuminating the sample with a collimated, near-resonant beam that falls on a Charge
Coupled Device (CCD) camera. The atoms absorb the beam, casting a shadow.
If either the trapping lasers or the magnetic field gradient is not present, the atoms
are not trapped. Starting with a cold trapped sample, we turn off the trapping lasers at a
time t = 0, wait a time t, and then apply the imaging beam. During this time t, the atoms
36
Figure 3.1 : Absorption imaging. A collimated near-resonant beam illuminates the
atomic sample and falls on a ccd camera. The atoms absorb a fraction of the incident
power, thus casting a shadow.
expand ballistically. The CCD camera records a laser intensity pattern with atoms present
I(x, y)atoms , and without I(x, y)back . From Beer’s law, we define optical depth (OD) in the
following manner:
OD(x, y) = ln[
I(x, y)back
] = σabs
I(x, y)atoms
Z
∞
n0 σabs −x2 /2σx2 −y2 /2σy2
dzn(x, y, z) = √
e
,
2πσz
−∞
(3.1)
where n0 is the peak atom density, and σabs is the absorption cross section. We have
inserted a Gaussian density distribution for the atoms in the last line, which leads to the
function used to fit the data. This fitting routine is identical to the one used in [43]. Figure
3.2 shows a typical absorption image, as well as the 2-dimensional gaussian fit and residuals,
taken after a delay time of 3 ms. The number of atoms in the blue MOT depends upon
the amount of power used and the temperature of the oven , but is typically ∼ 100x108 .
This optical density fitting routine yields σx and σy , the RMS widths of the cloud.
These widths expand according to the following equation:
σ 2 = σ02 + v 2 t2 ,
(3.2)
37
Figure 3.2 : Blue MOT Absorption Image. A)The spatial distribution of the atomic
sample after a delay time of 3 ms
where v is given by
p
kb T /M [42].
By fitting our data to this equation, we extract a temperature. Figure 3.3 shows
a typical blue MOT temperature near 1mK. In order to achieve good transfer efficiency
from the blue MOT into the red MOT, we minimize the temperature in the blue MOT by
reducing the intensity in the MOT beams by a factor of 10 for a short hold time before
transferring into the red MOT. This is expected because the temperature in Doppler cooling
depends on the intensity [9]. We have found that for hold times greater than 6 ms, we lose
a significant number of atoms; for hold times under 4 ms, the temperature does not change
much. So, we generally implement a low-intensity hold time of 5 ms before transferring
into the red MOT in order to maximize transfer.
3.2
Red MOT Operation and Diagnostics
The typical blue MOT sample discussed in Section 3.1 has a temperature on the order of
a few mK, which results in a Doppler broadening (3-4 MHz) of the intercombination line
38
Figure 3.3 : Blue MOT Temperature Determination. The fit to the data indicates a
temperature of about 1 mK.
resonance at 689 nm. As mentioned earlier, in order to transfer an appreciable fraction
of the atoms from the blue MOT to the red MOT, it is necessary to artificially broaden
the 689 nm laser. Otherwise, the laser would be on resonance with a very small fraction
of the trapped atoms and the transfer efficiency would be very small (a few percent). We
broaden the laser by modulating the frequency of the red MOT acousto-optic modulator
(AOM) by ∼0.8 MHz at a modulation frequency of 30 kHz.
The experimental setup is very similar to the blue MOT setup; in fact, the red MOT
beams follow the same three retro-reflected paths as the blue MOT beams. This beam
overlap and the use of the same anti-Helmholtz coils ensure that the centers of the two
MOTs overlap. Because the magnetic field gradient required for the red MOT is a factor
of 30 smaller than that for the blue MOT, the red MOT is significantly more susceptible to
39
stray magnetic fields than the blue, and we use trim coils to counteract those stray fields.
Experimentally, the transferring of atoms into the red MOT goes as follows: we start
by trapping atoms in the blue MOT. The field gradient here is 100 G/cm. Then, we switch
off the blue MOT beams, quickly ramp down the magnetic field gradient to 0.2 G/cm,
and switch on the red MOT beams. The detuning of the laser here is -1.2 MHz, with
a frequency modulation of ± 400 kHz. For the first 50 ms of the red MOT operation,
only the magnetic field gradient ramps from 0.2 G/cm to 3 G/cm. Over the next 150 ms,
the field gradient is kept constant at 3 G/cm, while the detuning is ramped to -300 kHz,
and the dither reduced to ± 150 kHz. After 200 ms of red MOT operation, all trapping
parameters are kept constant. To take a temperature at this point, we extinguish the red
MOT beams and turn off the magnets. This is our new t = 0. After waiting a time t, the
same imaging procedure is followed. Figure 3.4 illustrates that the same fitting routine
yields a temperature of 3 µK. At early times, the optical density of the sample is > 2 and
our fitting routines effectively undercount the number of atoms. In order to get a true
value for the density of atoms in the trap, we measure size at early times and number at
late ones. Generally, in the red MOT, we have cooled 30-40 ×106 atoms to the µK regime
with a MOT size of under 1 mm.
As mentioned before, the magnetic field gradient for the red MOT is so small that
zeroing out the earth’s magnetic field is an important part of the experiment. (Even a
stray magnetized allen key on the optics table can move the MOT around by as much as
1 cm.) We use electron shelving techniques in combination with trim coils as a diagnostic
for zeroing out the magnetic fields. The experimental procedure is as follows: atoms are
caught in the blue MOT, transferred to the red MOT, and then released. After about 2
40
Figure 3.4 : Red MOT Temperature Determination. The fit to the data indicates a
temperature of 2.2 µK.
ms, the red MOT beams are pulsed on for ∼ 20 µs, which pumps atoms from the 1 S0 to
the 3 P1 state. The atoms are immediately imaged on the blue transition. By making the
same measurement and varying the frequency of the red MOT beams only during the 20
µs, we perform electron shelving spectroscopy on the cold atomic sample [127]. If there is
a magnetic field at the center of the trap, the degeneracy of the 3 P1 state will have been
lifted, and the three magnetic sublevels are split according to the Zeeman effect. Zeroing
the magnetic field at the center of the trap produces a single Lorentzian lineshape. By
changing the current in three orthogonal sets of Helmholtz coils, the splitting of the peaks
is minimized. Figure 3.5 shows the spectra with and without the field zeroed.
It is also important to note that there is a tradeoff between atom number and temperature in the red MOT. In the setup described above, with 100 million atoms in the blue
41
Figure 3.5 : Upper trace: Line splitting, according to the Zeeman effect, indicates
a very large magnetic field at the center of the trap. Lower trace: A single line
indicates that the magnetic field at the center of the trap is close to zero, and the
trim coils are doing their job.
42
MOT, we can transfer roughly 50 million atoms to the red MOT, with a temperature of 2
µK. By reducing the dither in the latter stages to ∼100 kHz, and reducing the power in
the red MOT beams by a factor of 10, we can reach temperatures down to 800 nK, but
the number of atoms remaining is on the order of 5-6 Million.
3.3
Dipole Trap Performance
Good transfer into the dipole trap depends on the red MOT temperature being small
compared to the trap depth, and on spatial overlap with the two crossed ODT beams. In
this ODT, there is a single beam, which is passed through an AOM to enable switching
and power control. The beam is then focused to a waist at the center of the chamber,
recollimated and recycled through the chamber at 90 degrees to the first beam, with a
second waist overlapping the first. Because the ODT beams are not perfectly horizontal,
the trap is not perfectly symmetric, and atoms will preferentially escape from the trap in
the direction of the downward-tilting beam. We calculate the trap depth based on the
power in the ODT beam, and take into account the gravitational tilt of the potential. So,
trap depth is measured from the minimum energy of the trap to the lower lip of the trap.
In these preliminary ODT studies, the ODT beam is turned on at the beginning of
the red MOT loading time, and held at constant power. After the red MOT beams and
magnets are switched off, the ODT remains on for a given hold time. Then the ODT is
turned off, and the atoms are imaged on the blue transition as before. Figure 3.6 shows
the number of atoms trapped vs trap depth. As the power increases, so does the trap
depth, and so does the number of atoms transferred from the red MOT. Figure 3.7 shows
a temperature measurement for the ODT at its highest power.
43
Figure 3.6 : Number of atoms transferred from the red MOT as a function of power
in the ODT beam.
Figure 3.7 : ODT temperature after 200ms hold time
44
As we increase the hold time in the ODT, the temperature of the sample decreases,
as does the number. This number loss can be attributed to evaporative cooling. Careful
analysis of this data gives us a way to measure the atom-atom elastic cross section.
3.4
Measuring The Elastic Cross Section
The density losses from the ODT are both one-body and two-body:
ṅ = −Γn − g2b n2 ,
(3.3)
where Γ is the one body loss rate, due to collisions with the background gases and g2b is
the two-body loss coefficient. This coefficient takes into account both elastic and inelastic
collisions. This coefficient can be modeled as g2b = (gin +f gel ) VV21 , where gin is the inelastic
scattering coefficient, f is the fraction of elastic collisions that result in one atom being
ejected from the trap, or the evaporation fraction, gel is the elastic scattering coefficient, and
V2 and V2 are effective volumes, given by V1 =
R
e−U (r)/kB T d3 r and V2 =
R
e−2U (r)/kB T d3 r,
where U (r) is the potential. Software has been developed in the Killian research group
to calculate these effective volumes, which depend on the temperature of the sample, the
intensity of the ODT beams, and beam geometry.
Since the ground state of
88 Sr
is spin zero, and inelastic collisions require an addi-
tional degree of freedom for energy balance, the inelastic rates are negligible, so gin ∼ 0,
and the two body loss coefficient is simply g2b = f gel VV21 . In general, η is a dimensionless
parameter that measures the depth of the potential as compared to the temperature of
the sample. That is , η = trapdepth/temperature. For any trap that is sufficiently deep
(where η ≥ 4) the evaporative fraction f → ηe−η [37]. Expressions for f and other evapo-
45
ration parameters have been calculated for η < 4 for a linear quadrupole trap, but are not
available for a crossed beam, optical dipole trap. A logical next step in our analysis would
be to calculate f for the low η situation. That is beyond the scope of this study, so we will
simple restrict our analysis for data for which η > 4.
In the trap, we generally measure atom number, rather than density. Integrating
Equation 3.4, the differential equation becomes
N (t) = −ΓN (t) −
g2b 2
N .
V1
Any equation of this form y 0 = −αy − βy 2 has the solution y(t) = y0
(3.4)
e−αt
,
β
1+y0 α
(1−e−αt )
so the
number of atoms in the trap as a function of time is given by
N (t) = N0
e−Γt
1 + NΓ0 gV2b1 (1 − e−Γt )
(3.5)
.
Thus, by measuring the number of atoms in the ODT as a function of hold time, and
measuring the temperature at each point via ballistic expansion, we can use this function
to fit the data, and extract the two-body loss coefficient.
Figure 3.8 shows that the number of atoms in the ODT decreases with hold time. The
temperatures in each axis are also shown; it is of interest that these temperatures decrease
over the first ∼ 1.3 seconds and then level out. The dimensionless parameter η shows the
relationship of trap depth to temperature, and is also plotted.
Because the two-body loss process is density-dependent, and the density in the trap
changes over time (due to the loss of atoms), it is reasonable to expect that at long times,
the losses will be dominated by one-body processes.
46
Figure 3.8 : We measure the number of atoms and temperature at varying hold
times in the ODT. The dimensionless parameter η is a measurement of trap
depth/temperature.
47
Figure 3.9 : Both fits exclude the first two points, since η is below 4. The one-body
fit is restricted to times > 2 seconds. The two-body fit is restricted to times < 1.3
seconds. At long times, the data fits well with a one-body loss rate; at short times,
a two-body loss comes into play.
Figure 3.9 shows the fits to the data and the residuals. At later times, the data is
well fit by a simple exponential curve N (t) = N0 e−Γt , where Γ is the one body decay rate
is is numerically equal to 0.57 s−1 . This implies a one-body lifetime of ∼1.8 seconds.
For early times (t < 1.3s), the data is not well fit by a single exponential, but requires
the two body loss curve. Here, the fit function is simply N (t) = N0
the fit parameter C containing all the relevant physics.
e−Γt
1+
N0 C
(1−e−Γt )
Γ
, with
48
Figure 3.10 : The values for Γ are in good agreement; values for other parameters
differ slightly
Because the model is based on η > 4 we restrict the fits to timest > 0.4 seconds, and
both the fits are quite good. Notebly, the residuals from the 2-body fit are closer to zero
than those from the 1-body fit at early times, showing that this method is valid. Also, the
value of the one-body loss rate Γ, which is a free parameter in both functions, agrees. The
values and uncertainties are shown in Figure 3.10.
We use the fit parameter C to extract a measurement of the elastic cross-section
49
√
where gel = σel v 2. Solving for σel and
q
8kB T
substituting in the average velocity for a Boltzman distribution v =
πM , the elastic
σel . The fit parameter C =
g2b
V1
=
cross section, can be rewritten σel =
V2
1
V1 f gel V1 ,
2
C V1 √ 1
f V2 4 kB T /πM .
Uncertainties in the fit parameter map directly to uncertainties in σel . And while this
fit parameter C is independent of temperature, the effective trap volume V1 , the effective
two-body volume V2 , and the evaporative fraction f all depend upon the temperature of
the sample. It should be noted that one assumption of this model is that the temperature
of the atoms, and thus η does not change over time. This is not strictly true for the data.
To place bounds on the uncertainty introduced by the changing temperature, we assume a
constant temperature and extract values for σel for the maximum, minimum, and average
temperature observed during the portion of data that is fit to extract the parameter C.
These uncertainties are shown in Figure 3.11A. For comparison purposes, I have chosen the
values that bound the temperature evolution during the time spanned by the data. Thus,
this data reveals an elastic scattering cross section of 1.1 × 10−18 m2 < σel < 1.0 × 10−17 m2 .
In Figure 3.11B, we compare the value of σel determined in this study with other
experimental results. Previous measurements of the scattering length of
88 Sr
via photoas-
sociative spectroscopy performed in the Killian lab [122]Micklson have bounded the elastic
cross section 0 < σel < 1.2 × 10−17 m2 . In that experiment, we directly observed the
position of the node of the
86 Sr
ground state wave function. The location of the node
allowed us to determine the s-wave scattering length for this isotope. Although we did not
directly observe the node in
to use the result from
scattering length of
86 Sr
88 Sr
88 Sr,
a fit-parameter-free mass-scaling relationship allowed us
to fit the88 Sr PAS data taken, and to determine the s-wave
as well. In comparison, the PAS measurement in Katori’s group
50
bounds the elastic cross section 2.0 × 10−17 < σel < 4.9 × 10−17 ( [82] Katori). In this
paper, Katori performed PAS on
88 Sr,
measuring PAS rates on either side of the node,
but excludes the ∼ 15 transitions closest to the node. In the analysis, the author assumes
that the wavefunction is linear as it crosses the node; the validity of that assumption is in
question. Additionally, an ODT thermalization measurement in Tino’s group found that
σel = (3 ± 1) × 10−17 m2 ([83] Tino). In this experiment, atoms are loaded into an optical
trap, which is subsequently perturbed. The thermalization time at a given density and
temperature is then related to the elastic cross section.
It is of interest that the agreement between values taken in the Killian lab is quite
good, and that the agreement between values taken elsewhere is also good. To verify the
value set for σel in the ODT experiment described in this section, it would be useful to
take similar ODT lifetime data under conditions with larger trap depths, so that that the
constraint η > 4 is true for the entire dataset. We might also consider utilizing a technique
identical to Tino’s: perturbing the ODT and measuring thermalization times. Treating
each of these values for σel as valid, we can say is that 0 < σel < 4.9 × 10−17 m2 . This
implies that the absolute value of the s-wave scattering length is less than 26a0 , which is
still rather small, especially when compared to the value for
86 Sr,
which is bounded by
610a0 < a86 < 2300a0 , according to our direct measurement of the position of the node.
Please note: for all other data in this chapter, we have not used repumpers. For the
data presented in this section, where we look at very long hold times in the ODT, we used
repumper lasers to boost our signal.
Thus, in the absence of repumpers, the blue MOT captures about 50 M
88 Sr
atoms
and cools them to about 1 mK. The atoms are then transferred to the red MOT, which
51
Figure 3.11 : Upper plot: Values for σel from this experiment. The temperature
along the x-axis is the temperature assumed in the fit. For comparison I have chosen
values that bound the temperature evolution during the time spanned by the data.
Lower Plot:Comparison of this work with other measurements of σel . Values from this
set compare favorably with those determined via photoassociative spectroscopy [122].
There appears to be a discrepancy with values from an additional ODT evaporation
experiment[83] and PAS experiment [82]
52
captures about 25 M and cools them to about 2 µK. . The atoms are then transferred
to the ODT, which initially captures 2 million atoms at temperatures near 12 µK. Due
to two-body collisions, though, the number of atoms in the ODT decays in a matter of
seconds. Careful study of those decay rates yields a value for the elastic cross section of
88 Sr
on on the order of σel ∼ 2 × 10−17 m2 . This information is useful for planning further
experimental work.
53
Chapter 4
Photoassociation at Long Range
When two atoms collide in the presence of light, there is a probability that they
will form a molecule. The probability of molecule formation depends upon the collision
energy of the two atoms and the frequency of the light. If the collision energy is small,
and the frequency of the light is such that it promotes the two atoms into an excited
molecular state, a molecule is formed. By varying the frequency of the light over these
molecular resonances and counting the number of atoms in the MOT, the frequencies of
these resonances can be observed – thus, photoassociative spectroscopy. By carefully noting
the frequency and spacing of several subsequent levels, we can map out the underlying
atom-atom potential curves, and thus learn about the underlying physics, which may guide
further scientific investigation. The experiment detailed in this chapter was performed in
the shared apparatus, and the discussion here draws heavily on the paper ”Photassociative
Spectroscopy at Long Range in UltraCold strontium” published in 2005, in collaboration
with Robin Côté and Phillipe Pelligrini. [119]
4.1
Scientific Motivation
Photoassociative spectroscopy (PAS) in laser-cooled gases [44] is a powerful probe of molecular potentials and atomic cold collisions. It provides accurate determination of ground
state scattering lengths and excited state lifetimes [115]. Photoassociation occurs naturally
in laser cooling and trapping experiments in which the lasers are red-detuned from atomic
54
resonance, so characterizing the process is also important for understanding and optimizing
the production of ultracold atoms (e.g. [87]).
PAS of alkaline earth atoms differs significantly from more common studies of alkali
metal atoms. The most abundant isotopes of alkaline earth atoms lack hyperfine structure,
making these systems ideal for testing PAS theory. Recent theoretical work [102] emphasized the ability to resolve transitions at extremely large internuclear separation and very
small detuning from the atomic asymptote. The finite speed of light modifies the potential
in this regime through relativistic retardation [48–50].
There is also great interest in alkaline-earth-atom cold collisions because of their importance for optical frequency standards [51–54] and for the creation of quantum degenerate
gases [57, 86, 89]. In addition, collisions involving metastable states [58, 60, 61, 85] display
novel properties that arise from electric quadrupole-quadrupole or magnetic dipole-dipole
interactions.
PAS red-detuned from the principle transitions in calcium [62] and ytterbium [63]
is well characterized, resulting in accurate measurements of the first excited 1 P1 lifetimes
and the ground state s-wave scattering lengths. In spite of its importance for potential
optical frequency standards, before this work, little was known about strontium. The
photoassociative loss rate induced by trap lasers in a 1 S0 → 1 P1 magneto-optical trap
(MOT) has been measured [87], and preliminary results from more extensive spectroscopy
were recently reported [64]. Ab initio strontium potentials have been calculated for small
internuclear separation (R < 9 nm) [65].
In this chapter, we explore PAS of
88 Sr
near the 1 S0 → 1 P1 atomic resonance at
461 nm (Fig. 4.1). The simple spectrum allows us to resolve transitions as little as 600 MHz
55
Figure 4.1 : Resonant excitation occurs near the outer molecular turning point of
states of the 1 Σu potential, and it can lead to trap loss through two processes. The
molecule can radiatively decay at smaller internuclear separation to two ground state
atoms with kinetic energies exceeding the trap depth. This is known as radiative
escape (RE). In a state-changing collision (SC), at small internuclear separation
the molecular state changes to one corresponding at long range to a lower-lying
electronic configuration of free atoms. The atoms exit the collision with greatly
increased kinetic energy and escape the trap.
detuned from the atomic resonance, which produces molecules with greater internuclear
separation than in any previous PAS work. Our determination of the first excited 1 P1 lifetime resolves a discrepancy between experiment and recent theoretical work, and provides
an importance test of atomic structure theory for alkaline earth atoms.
4.2
Experimental Sequence
The most common form of PAS involves resonantly inducing trap loss in a MOT, although
early work was also conducted in optical-dipole traps [115]. Recent experiments have also
studied spectroscopy of atoms in magnetic traps, especially in Bose-Einstein condensates
[66, 67]. The experiments described in this article were performed in a MOT, but the MOT
56
operated on the 1 S0 → 3 P1 intercombination line at 689 nm [90], rather than an electricdipole allowed transition. This results in lower atom temperature, a shallower trap, and
higher atom density than a standard MOT.
Atoms are initially trapped in a MOT operating on the 461 nm 1 S0 → 1 P1 transition,
as described in [69]. During the loading phase, the peak intensity in each MOT beam is 6
mW/cm2 , and the axial magnetic field gradient generated by a pair of anti-Helmholtz coils
is 56 G/cm. The intensity is then reduced by about a factor of 8 for 3.5 ms to reduce the
atom temperature. After this stage the MOT contains about 150 million atoms at 2 mK.
The 461 nm laser cooling light is then extinguished, the field gradient is reduced to 2.1
G/cm, and the 689 nm light for the 1 S0 → 3 P1 intercombination line MOT is switched on.
This MOT also consists of three retro-reflected beams, each with a diameter of 2 cm and
an intensity of 400 µW/cm2 . The frequency of the 689 nm laser is detuned from the atomic
resonance by 0.5 MHz and spectrally broadened with a ±400 kHz sine-wave modulation.
Transfer and equilibration take about 50 ms, after which there are 15 million atoms at a
√
temperature of 5 µK, peak density of about 5 × 1011 cm−3 , and 1/ e density radius of
about 100 µm.
The intercombination-line MOT operates for an adjustable hold time before measuring the remaining number of atoms with absorption imaging using the 1 S0 → 1 P1 transition. The lifetime of atoms in the MOT is approximately 350 ms, limited by background
gas collisions. To detect photoassociative resonances, a PAS laser is applied to the atoms
during hold times of 300 − 400 ms. When the PAS laser is tuned to a molecular resonance,
photoassociation provides another loss mechanism for the MOT, decreasing the number of
atoms.
57
Light for photoassociation is derived from the same laser that produces the 461 nm
light for laser cooling. Several acousto-optic modulators (AOM), detune the light 600 to
2400 MHz to the red of the atomic transition. The laser frequency is locked relative to a
Doppler-free saturated-absorption feature in a vapor cell, with an uncertainty of about 2
MHz. The last AOM in the offset chain, in a double-pass configuration, controls the power
of the PAS laser and scans the frequency up to 200 MHz with minimal beam misalignment.
PAS light is double-passed through the MOT in a standing wave, with a 1/e2 intensity
radius of w = 3 mm.
Thermal broadening (kB T /h ≈ 100 kHz) is negligible. Only s-wave collisions occur,
so only a single rotational level (J = 1) is excited. For higher intensity, the observed line is
broadened slightly because, on resonance, the signal saturates if a large fraction of atoms
is lost.
Figure 4.2B shows a complete spectrum of the 14 different PAS resonances observed
in this study. Center frequencies for each transition are determined by Lorentzian fits.
The typical statistical uncertainty is 2-3 MHz, but there is approximately 2 MHz additional
uncertainty arising from the lock of the laser. The Condon radius for the excitation varies
from 20 nm for the largest detuning to 32 nm for the smallest. These extremely large values
exceed the internuclear spacing of molecules formed in photoassociative spectroscopy to
pure long-range potentials [49, 70].
The region of the attractive 1 Σ+
u molecular potential probed by PAS corresponds to
large internuclear separations, and is typically approximated by
V (R) = D −
C3
~2 [J(J + 1) + 2]
3~λ3
+
,
C
=
,
3
R3
2µR2
16π 3 τ
(4.1)
58
Figure 4.2 : (a) Spectrum for photoassociation to the v = 60 molecular state. Detuning is with respect to the atomic 1 S0 − 1 P1 resonance. The PAS laser intensity
is 1.4 mW/cm2 . The Lorentzian fit yields a FWHM linewidth of 70 ± 7 MHz. (b)
Representative spectra for all transitions observed in this study. Lines connect the
data points to guide the eye. Conditions vary for the individual scans that make up
the full spectrum, so amplitudes and linewidths should not be quantitatively compared. The baseline has been adjusted to match the expected curve for overlapping
Lorentzians. (c) Differences between experimental and calculated positions of the 14
measured levels. Taken from [69]
59
where D is the dissociation energy, µ the reduced mass, and λ = 461 nm. However, at
very large separations, the atom-atom interaction is modified by relativistic corrections.
This retardation effect is well understood [48] and can be included in the analysis of the
spectrum through C3 → C3 [cos(u) + u sin(u)], where u = 2πR/λ. Machholm et al. [102]
discussed this in the context of PAS of alkaline earth atoms.
To extract molecular parameters from the positions of the PAS resonances, we constructed a potential curve by smoothly connecting the long range form (Eq. (4.1)) to a
short range curve at a distance of 1.5 nm. The short range ab initio potential was obtained
using a semi-empirical two-valence-electron pseudopotential method [65]. To account for
uncertainty in the short range potential, the position of the repulsive wall was varied as
a fit parameter. The rovibrational levels in the 1 Σ+
u potential were found by solving the
radial Schrödinger equation using the Mapped Fourier Grid Method [71] with a grid size
typically larger than 500 nm containing about 1000 grid points.
We found that the observed levels range from 48 to 61, starting the numbering from
the dissociation limit. The best fit of the data was achieved with a value of C3 = 18.54 a.u.,
with a reduced chi-squared value of χ2 = 0.79 (See Fig. 4.2C). χ2 varied by less than 10%
for a change of level assignments of ∆v = ±1, so we consider our assignment to be uncertain
by this amount. The value of C3 changed by ±0.5% as the assignments changed, which is
much larger than the statistical uncertainty for a given assignment. We thus take ±0.5%
as our uncertainty in C3 . A fit to the level spacings, as opposed to the absolute positions,
yielded the same value of C3 . From C3 , we derive a natural decay rate of the atomic 5p
1P
1
state of τ = 5.22 ± 0.03 ns. Decay channels other than 1 P1 → 1 S0 can be neglected
at this level of accuracy. The most recent experimental determinations of τ use the Hanle
60
Figure 4.3 : Comparison of experimental (exp.) and theoretical (th.) values for the
lifetime of the 1 P1 level. Adapted from [69]
effect [72, 73]. Our result agrees well with recent theoretical values [75, 76, 107] (Fig. 4.3).
Retardation effects shift the levels by approximately 100 MHz in this regime, which is
similar to the shift seen in a pure long range potential in Na2 [49]. If retardation effects are
neglected and the data is fit using the simple LeRoy semiclassical treatment [77], the level
assignments change significantly, and C3 changes by more than 10%, putting it outside
the range of recent theoretical results. Center frequency shifts due to coupling to the
continuum of the ground state have been ignored as they are less than 50 kHz for typical
experimental temperatures.
61
Figure 4.4 : (a) Number of atoms as a function of time for atoms in the
intercombination-line MOT. The PAS laser is tuned near the v = 59 molecular
resonance. In one data set the PAS laser is on resonance, and in the other it is tuned
to the blue of resonance by 42 MHz, and loss due to photoassociation is small. (b)
β as a function of PAS laser intensity for several molecular resonances. Error bars
denote statistical uncertainties. Taken from [69]
Figure 4.4A shows an example of the number of atoms, N , in the intercombinationline MOT as a function of time. The density in the MOT varies according to ṅ = −Γn−βn2 .
Integrating this equation over volume and solving the differential equation yields
N (t) =
N0 e−Γt
1+
N0 β
√
(1
Γ(2 2πσ)3
− e−Γt )
2 /2σ 2
where we approximate the density as n(r) = n0 e−r
,
(4.2)
, and Γ is the one-body loss due
primarily to background gas collisions. β is the two-body loss rate and it contains important
information about the dynamics of photoassociation and trap loss.
Even with photoassociation, the deviation of N (t) from a single exponential decay is
small, making it difficult to independently extract β and Γ with high accuracy from a single
decay curve. To address this problem, we take advantage of the fact that when the PAS
laser is not on a molecular resonance, the photoassociation rate is small (See Fig. 4.2B), and
within the accuracy of our measurement it can be neglected. All other processes, however,
62
such as off-resonance scattering from the atomic transition, are the same. We thus take
data in pairs of on and off-resonance, and fit off-resonance data to N (t) = N0 e−Γt . The
on-resonance data is then fit with Eq. (4.2) with Γ fixed to the value determined from the
off-resonance data. The resulting two body decay rates are shown in Fig. 4.4B. For the
relatively small intensities used, β is expected to vary linearly with laser power [78].
At the time, we imaged along the direction of gravity and lacked the additional
diagnostic required to obtain information on the third dimension of the atom cloud. Gravity
can distort the equilibrium shape of the intercombination-line MOT because the light
force is so weak. However, we operate the MOT in the regime where the laser detuning
is comparable to the modulation of the laser spectrum, and, as shown in [79], in this
regime the effect is small. Based on the range of distortions we observe in the imaged
dimensions for various misalignments of the MOT beams, we allow for a scale uncertainty
of a factor of two in the volume of the MOT. This contributes an identical uncertainty in
the measurements of β.
Measurements of MOT size suggest that as the PAS laser power increases, the cloud
radius increases slightly (20% for the highest intensities), decreasing the atomic density.
This is expected because off-resonance scattering from the atomic transition should heat
the sample. The data quality does not allow a reliable correction for this effect, however,
and we use the average of all observed σ as a constant σ in Eq. 4.2. This contributes
another approximately 50% uncertainty in our determination of β. Overall, we quote a
factor of three uncertainty due to systematic effects, which dominates the statistical error.
The measured values of β ≈ 1×10−11 cm3 /sec for 1 mW/cm2 in the intercombinationline MOT are comparable to theoretical [102] and experimental [87] values of β for atoms
63
in a MOT operating on the 461 nm transition. This provides some insight into the typical
kinetic energy of the resulting atoms after radiative decay of a molecule to the free-atom
continuum.
A MOT based on an electric-dipole allowed transition would typically have a trap
depth of about 1 K. In our experiment, the transfer efficiency of atoms from the 461 nm
MOT places a lower limit of 0.5 mK on the depth of the intercombination-line MOT. Using
the formalism of [79] and the parameters of the intercombination-line MOT, we calculate
a trap depth of 2 mK. As described in Fig. 4.1, photoassociative loss requires excitation to
a molecular state followed by decay to a configuration of two atoms with enough kinetic
energy to escape from the trap [102]. The fact that the observed linewidths are close to the
theoretical minimum of 1/(πτ ) suggests that radiative escape dominates in this regime, as
predicted in theoretical calculations of PAS rates for magnesium [102]. If β is similar for
trap depths of 1 K and 1 mK, then the energy released during radiative decay must be on
the order of 1 K or greater. Extensive calculations of radiative escape in lithium [80] also
led to an estimate of a few Kelvin for the energy released in this process.
4.3
Implications
This PAS study elucidated dynamics of collisions in an intercombination-line MOT and
provided an accurate measurement of the lifetime of the lowest 1 P1 state of strontium.
This result resolved a previous discrepenancy between experimental and theoretical values
of the lifetime. We have also taken advantage of the simple structure of the spectrum
to measure transitions at very large internuclear separation where the level spacing and
natural linewidth are comparable and retardation effects are large. It should be noted that
64
an additional PAS experiment was performed the following year [82] with an even more
precise lifetime of 5.263(4) ns.
4.4
Other Work
Further experiments along these lines [122], using the Sacher diode laser as the seed for
the additional doubling cavity, allowed tuning of the PAS beam over several additional
PAS resonances in both
88 Sr
and
86 Sr.
Variations in PAS linestrength are indicative of
the shape of the groundstate wavefunction (Figure 4.5), and the experimental location
of the node allows subsequent determination of the s-wave scattering length of the two
most abundant bosonic isotopes of strontium. The analysis for this data was performed by
Pascal Mickelson, in collaboration with Robin Côté and Phillipe Pelligrini, and makes up
the majority of Pascal’s master’s thesis [128].
65
Figure 4.5 : PAS resonances in 86 Sr. An independent seed for the PAS laser allows
variable detuning over several hundred GHz; the linestrength vanishes at -500 GHz,
indicating a node in the wavefunction.
66
Chapter 5
Optical Pumping and Metastable States
Experiments that begin with a stable, cold sample of atoms can illuminate underlying
physics. Two such experiments are detailed in this chapter. The first experiment is magnetic trapping of the 3 P2 atoms in the quadrupole field generated by the MOT magnetic
coils. In addition to providing an alternative cold sample to the blue MOT, measuring
the decay of the number of atoms in the trap over time is a convenient means of testing
the background vacuum level of the apparatus. A second experiment involves transferring
atoms from the blue MOT into the red MOT and then into the ODT. With careful sequencing, we can use the repumper lasers to efficiently transfer atom population in the
ODT from the 1 S0 state to the 3 P2 or 3 P0 states. By measuring the decay of the number
of atoms in the ODT over time, we can probe the elastic scattering cross-sections of these
states, which may be large and therefore favorable for evaporative cooling to quantum
degeneracy.
Discussion of magnetic trapping in the 3 P2 state draws heavily on the paper , Magnetic trapping of 3 P2 strontium atoms, published in 2002 [25]. The data for was taken in
the shared apparatus. To compare the effective background pressure in the new apparatus
to the old one, we compare the liftetimes of the3 P2 atoms in those magnetic traps and find
a marked improvement in the new apparatus.
Discussion of experiments with metastable states in the ODT is fairly preliminary
and illustrates our ability to pump atoms into the the upper 3 P states.
67
5.1
Magnetic Trapping in the 3 P2 state
In the course of operating the blue MOT, some atoms decay into the 3 P2 state, which is
magnetically trappable and has no direct decay channel to the ground state. Atoms are
recovered by applying repumping lasers, and the loading and loss rates of the magnetic trap
are studied. The lifetime of the magnetic trap is a useful diagnostic for determining the
background gas level. Comparison of trap lifetime measurements in the shared apparatus
to those made in the newer neutral apparatus reveal a 14-fold improvement in vacuum
level, which can be attributed to apparatus design improvements.
5.1.1
Scientific Motivation
Laser-cooled alkaline earth atoms offer many possibilities for practical applications and
fundamental studies. The two valence electrons in these systems give rise to triplet and
singlet levels connected by narrow intercombination lines that are utilized for optical frequency standards [105]. Laser cooling on such a transition in strontium may lead to a
fast and efficient route to all-optical quantum degeneracy [89, 90], and there are abundant
bosonic and fermionic isotopes to use in this pursuit. The lack of hyperfine structure in the
bosonic isotopes and the closed electronic shell in the ground states make alkaline-earth
atoms appealing testing grounds for cold-collision theories [87, 102, 117], and collisions between metastable alkaline-earth atoms is a relatively new and unexplored area for research
[85].
In this section, we characterize the continuous loading of metastable 3 P2 atomic
strontium (88 Sr) from a magneto-optical trap (MOT) into a purely magnetic trap. This
idea was discussed in a theoretical study of alkaline-earth atoms and ytterbium [100].
68
Katori et al. [91] and Loftus et al. [101] also reported observing this phenomenon in their
strontium laser-cooling experiments. Continuous loading of a magnetic trap from a MOT
was also described for chromium atoms [113].
This scheme should allow for collection of large numbers of atoms at high density
since atoms are shelved in a dark state and less susceptible to light-assisted collisional
loss mechanisms [87, 92, 102]. It is an ideal starting place for many experiments such as
sub-Doppler laser-cooling on a transition from the metastable state, as has been done with
calcium [88], production of ultracold Rydberg gases [111] or plasmas [94], and evaporative
cooling to quantum degeneracy. Optical frequency standards based on laser-cooled alkalineearth atoms, which are currently limited by high sample temperatures [105], may benefit
from the ability to trap larger numbers of atoms and evaporatively cool them in a magnetic
trap.
5.1.2
Experimental Sequence
Here, we first describe how the operation of the blue MOT loads the magnetic trap with 3 P2
atoms. Then we characterize the loading and decay rates of atoms in the magnetic trap.
This decay rate depends only on the collisions between strontium atoms and background
gases, and so provides a direct measurement of the background pressure in the apparatus.
Finally, we present measurements of the 3 P2 sample temperature.
As detailed in Chapter 3, strontium atoms are loaded from a Zeeman-slowed atomic
beam [108] and cooled and trapped in a standard MOT. [109]. The repumper lasers at 679
nm and 707 nm are not used during the operation of the MOT, but serve to repump atoms
from the 3 P2 level to the ground state via the 3 S1 and 3 P1 levels for imaging diagnostics.
69
In the shared apparatus, a 30 cm long Zeeman slower connects a vacuum chamber
for the Sr oven and nozzle to the MOT chamber. Each chamber is evacuated by a 75 l/s
ion pump. When the Sr oven is operated at its normal temperature of about 550◦ C,
the pressure in the MOT chamber is about 5 × 10−9 torr, and the oven chamber is at
4 × 10−8 torr.
The quadrupole magnetic field for the MOT is produced by flowing up to 80 A of
current in opposite directions through two coils of 36 turns each, with coil diameter of
4.3 cm and separation of 7.7 cm. The maximum current produces a field gradient along
the symmetry axis of the coils of 115 G/cm. Such a large field gradient, about 10 times the
norm for an alkali atom MOT, is required because of the large decay rate of the excited
state (Γ1 P1 = 2π × 32 MHz) and the comparatively large recoil momentum of 461 nm
photons.
Typically 107 −108 atoms are held in the MOT, at a peak density of n ≈ 1×1010 cm−3 ,
with an rms radius of 1.2 mm and temperatures from 2 − 10 mK. The cooling limit for the
MOT is the Doppler limit (TDoppler = 0.77 mK) because the ground state lacks degeneracy
and thus cannot support sub-Doppler cooling. Higher MOT laser power produces higher
MOT temperature, but also a larger number of trapped atoms. These sample parameters
are measured with absorption imaging of a near resonant probe beam. The temperature is
determined by monitoring the velocity of ballistic expansion of the atom cloud [116] after
the trap is extinguished.
Atoms escape from the MOT due to 1 P1 − 1 D2 decay as discussed in [100]. From
the 1 D2 state atoms either decay to the 3 P1 state and then to the ground state and are
recaptured in the MOT, or they decay to the 3 P2 state, which has a lifetime of 17 min [85].
70
The decay rates are given in the energy diagram, Figure 5.6. The resulting MOT lifetime
of 11 − 55 ms was measured by turning off the Zeeman slowing laser beam and monitoring
the decay of the MOT fluorescence. The lifetime is inversely proportional to the fraction
of time atoms spend in the 1 P1 level, which varies with MOT laser power. Light-assisted
collisional losses from the MOT [87] are negligible compared to the rapid 1 P1 − 1 D2 decay.
The mj = 2 and mj = 1 3 P2 states can be trapped in the MOT quadrupole magnetic
field. Such a quadrupole magnetic trap was used for the first demonstration of magnetic
trapping of neutral atoms [103], but in that case atoms were loaded directly from a Zeemanslowed atomic beam.
Near the center of the trap, the magnetic interaction energy for 3 P2 atoms takes the
form
p
Umj = −µmj · B = gµB mj b x2 /4 + y 2 + z 2 /4,
(5.1)
where mj is the angular momentum projection along the local field, g = 3/2 is the g-factor
for the 3 P2 state, µB is the Bohr magneton, and b ≤ 115 G/cm is the gradient of the
magnetic field along the symmetry (y) axis of the quadrupole coil. For the mj = 2 state
and the maximum b, gµB mj /kB = 200 µK/G and the barrier height for escape from the
center of the magnetic trap is 15 mK. Gravity, which is oriented along z, corresponds to
an effective field gradient of only 5 G/cm for Sr and is neglected in our analysis.
Typical data showing the magnetic trapping is shown in Figure 5.1. We are unable to
directly image atoms in the 3 P2 state, so we use the 679 and 707 nm lasers to repump them
to the ground state for fluorescence detection on the 1 S0 −1 P1 transition. The details are
as follows. The MOT is operated for tload ≤ 1300 ms during which time atoms continuously
load the magnetic trap. The MOT and Zeeman slower light is then extinguished, and after
71
Figure 5.1 : Magnetic trapping of 3 P2 atoms. Ground state atoms are detected by
fluorescence from the MOT lasers. If the quadrupole magnetic field is left on during
thold (black trace), large numbers of 3 P2 atoms are magnetically trapped until the
707 nm laser returns them to the ground state. The residual fluorescence after thold
for the gray trace arises from background scatter off atoms in the atomic beam.
Reloading of the MOT from the atomic beam is negligible with the Zeeman slower
light blocked. There is a 1 ms delay between the MOT and 707 nm laser turn on to
allow the MOT light intensity to reach a stable value. Taken from [25]
72
Figure 5.2 : (a) The lifetime of atoms in the magnetic trap is limited by collisions with
background gas molecules. The linear fit extrapolates to zero at zero pressure within
statistical uncertainties. Inset: A typical fit of the decay of the number of trapped
atoms to a single exponential. (b) The magnetic trap loading rate is plotted against
the MOT loss rate. Data correspond to various MOT laser power and slow-atom
fluxes from the atomic beam. Taken from [25]
a time thold , the MOT lasers and repump laser at 707 nm are turned on. The 679 nm laser
is left on the entire time. Any atoms in the 3 P2 state are cycled through the 3 P1 level to
the ground state within 500 µs of repumping, and they fluoresce in the field of the MOT
lasers. If the magnetic field is not left on during thold , Fig. 5.1 shows that a negligible
number of ground state atoms are present in the MOT when the lasers are turned on. If
the magnets are left on, however, the MOT fluorescence shows that 3 P2 atoms were held
in the magnetic trap.
The maximum number of 3 P2 atoms trapped is about 1 × 108 , and the peak density
is about 8 × 109 cm−3 . To determine what limits this number, we varied thold and saw
that the number of 3 P2 atoms varied as N0 e−γthold . The fits were excellent and the decay
rate was proportional to background pressure as shown in Figure 5.2a. This implies that
for our conditions, the trap lifetime is limited by collisions with residual background gas
molecules, and strontium-strontium collisional losses are not a dominant effect.
73
The magnetic trap loading rate was determined by holding thold constant and varying
tload . The loading rate correlates with the atom loss rate from the MOT (Fig. 5.2b). At low
MOT loss rates about 10% of the atoms lost from the MOT are captured in the magnetic
trap. Using the Clebsch-Gordon coefficients to calculate the probability of decay from
the 1 P1 state, and the magnetic sublevel distribution for atoms in the MOT, one expects
that about 40% of the atoms decaying to 3 P2 enter the mj = 1 or mj = 2 states. This
is significantly higher than the largest observed efficiencies, and we may be seeing signs
of other processes, such as losses due to collisions with MOT atoms, which dominated
dynamics during loading of a magnetic trap from a chromium MOT [113].
At larger MOT loss rates (corresponding to higher MOT laser intensities, MOT
temperatures, and trapped 3 P2 atom densities), the efficiency of loading the magnetic trap
decreases by about a factor of two. MOT temperatures approach the trap depth for the
largest loading rates and we attribute the decreasing efficiency to escape of atoms over the
magnetic barrier.
The 500 µs required for repumping is fast compared to the time scale for motion
of the atoms, so absorption images of ground state atoms immediately after repumping
provides a measure of the density distribution of the magnetically trapped sample. For
these measurements, the magnetic trap is loaded for 1.3 s. Then the magnetic field is
turned off and the repump lasers are turned on. After 500 µs, an 80 µs pulse of a weak
461 nm probe beam (I Isat ), 12.5 MHz detuned below resonance, illuminates the atom
cloud and falls on a CCD camera. We record an intensity pattern with atoms present,
I(x, y)atoms , and a background pattern with no atoms present, I(x, y)back . To analyze the
74
Figure 5.3 : Distributions of 3 P2 atoms extracted from absorption images of ground
state atoms shortly after repumping. The fits, which assume thermal equilibrium and
a pure sample of mj = 2 atoms, yield number (1.2×108 ), peak density (8×109 cm−3 ),
and temperature (1.3 mK) of the atoms. Taken from [25]
data, we plot
Z
S(x) =
image
Z
= σabs
dy ln[I(x, y)back /I(x, y)atoms ]
Z ∞
dy
dz n(x, y, z),
image
(5.2)
−∞
and the analogously defined S(y), where σabs is the absorption cross section and n(x, y, z) is
the atom density (Fig. 5.3). Because we do not know the distribution of magnetic sublevels,
we make the simplifying assumption that all atoms are in the mj = 2 state, and the density
is given by
n(x, y, z) = n0 exp[−U2 (x, y, z)/kB T ],
(5.3)
A numerical approximation to Eq. 5.2 fits the data very well, see Figure 5.3.
Our assumption for magnetic sublevel distribution means that the extracted temperatures are upper bounds, but one would expect the mj = 1 level to be less populated. Due
75
Figure 5.4 : The 3 P2 temperature is significantly lower than expected from a simple
model that is described in the text. The inset shows that the temperature tracks
the magnetic trap depth, as expected for evaporative cooling. The scatter of the
temperature measurements is characteristic of our statistical uncertainty, and there
is a scale uncertainty of 25% due to calibration of the imaging system. The magnetic
trap depth is 15 mK for the main figure. Taken from [25].
to the smaller magnetic moment, the trapping efficiency for mj = 1 atoms decreases substantially as the MOT temperature increases, dropping by about a factor of 5 for a MOT
temperature of 12 mK compared to only a factor of two for mj = 2 atoms. Atoms with
mj = 1 can also be lost from the trap through spin-exchange collisions, which are typically
rapid in ultracold gases. Calculated rates for spin exchange collisions for alkali atoms in
magnetic traps are typically 10−11 cm3 /s, although they can approach 10−10 cm3 /s [96].
We have assumed thermal equilibrium in our analysis, but this is reasonable. Thermal
equilibration would need to occur on less than a few hundred ms time scale. Using a recently
calculated s-wave elastic scattering length for 3 P2 atoms of a = 6 nm [86], the collision
rate for identical atoms is nv8πa2 ≈ 9 s−1 for n = 1010 cm−3 and v =
(T = 3 mK).
p
2kB T /M = 1 m/s
76
The most interesting parameter obtained from the fits is the temperature, which is
plotted in Fig. 5.4 as a function of MOT atom temperature. The values are significantly
colder than what one would expect from the simple theory developed in [113] and plotted
in the figure. The expected temperature is determined by assuming the kinetic energy and
density distribution in the MOT are preserved as atoms decay to the metastable state.
The 3 P2 potential energy distribution is then given by the magnetic trap potential energy
corresponding to the density distribution of the MOT. Actual 3 P2 atom temperatures
cluster around 1 mK for a 15 mK trap depth, while the expected temperature approaches
4.5 mK for the hottest MOT conditions. We confirmed these measurements by determining
the 3 P2 atom temperature from ballistic expansion velocities, as is done to measure the
MOT temperature.
As shown in the inset of Fig. 5.4, the temperature decreases with decreasing trap
depth as would be expected for evaporative cooling of the sample [93]. For this data, the
magnetic trap depth is held constant during the entire load and hold time. Confirmation
of this explanation could be achieved with measurement of the collision cross section and
thermalization rate in the trap.
If evaporative cooling is working efficiently, it should be possible to use radio-frequencyinduced forced evaporative cooling to further cool the sample and increase the density.
Majorana spin-flips [103] from trapped to untrapped magnetic sublevels at the zero of the
quadrupole magnetic field will eventually limit the sample lifetime. Even if the sample temperature were 100 µK, this lifetime is still 10 seconds. A short semi-classical calculation of
the Majorana lifetime follows.
77
5.1.3
Majorana Spin Flips
Three important features of the quadrupole magnetic trap are that 1) the magnetic field
direction changes as one crosses the zero point 2) the magnitude of the magnetic field
decreases the closer one gets to the center and 3) the magnetic field direction changes
faster near the origin than far away from it. This is important because atoms have an
intrinsic spin, which, in the presence of an external magnetic field, will precess around this
~
external field at the Larmor frequency: ωLarmor (r) = µ
~ · B(r)/~.
If this external magnetic field changes direction slowly in comparison to the Larmor
frequency, then the interaction of the magnetic moment with the magnetic field will create
a torque that keeps the atoms aligned with this local magnetic field. If the magnetic field
changes faster than the Larmor frequency, the atoms will not be able to keep up and will
lose alignment with the local magnetic field, flip their spins, and be lost from the trap.
As long as an atom in our trap moves in a region where the field magnitude is
sufficiently large, the Larmor precession period will be short compared with the time in
which the atom sees a signicant change of field direction (which is inversely proportional
to the velocity of the atoms). The atom will then remain in the Zeeman sublevel that is
trapped. This is relatively easy for atoms that are far away from the center of the trap
(where the magnetic fields are larger) than it is for atoms near the center. Therefore, higher
energy atoms (which predominantly spend most of their times away from the trap center)
will remain in the trapped state while lower energy atoms will have a higher probability of
undergoing spin flips.[120]
Thus, this loss will occur when the Larmor precession frequency is less than the rate
78
of change of the magnetic field direction, and the loss rate can be easily estimated[121] An
atom with velocity v and mass m, passing within a distance b of the center of the trap (with
radial gradiant dBr /dr = B 0 ), can undergo a nonadiabatic spin flip if the Larmor frequency
∼ µB 0 /~ is smaller than the rate of change of the magnetic field direction v/b. Loss then
occurs within an ellipsoid of radius b0 = (v~/µB 0 )1/2 . The loss rate is given by the flux
through this ellipsoid, that is, the density of atoms n times the area of the ellipsoid (b20 )
~
times the velocity v. Thus the Majorana loss rate can be expressed as ΓM ∼ nb20 v = nv 2 µB
0
We should note here, that when the de Broglie wavelength of the atoms, λdb =
q
2π~2
M kB T ,
is smaller than the size of the Majorana hole, b0 , the atomic motion can be treated
classically (particles falling through a hole). However, when the de Broglie wavelength is
larger than b0 , the entire process must be treated quantum mechanically.
Under the experimental conditions described in this paper, (n ∼ 1010 cm−3 , v =
1m/s (T = 3mK), µ =
3
2 µb ,
B 0 = 115 G/cm where µb is the bohr magneton), the de
Broglie wavelength of the atoms is λdb = 3.39 × 10−9 m and the size of the Majorana
hole, b0 = 2.22 × 10−6 m . Thus it is safe to use a semiclassical picture. The loss rate
~
−4 s−1 , which implies a lifetime of ∼ 2000 seconds, or 33
ΓM ∼ nb2o v = v 2 µB
0 = 5 × 10
minutes. Thus, the losses from the magnetic trap due to Majorana flips are negligible on
the ∼2 second timescale of the experiment.
5.1.4
Implications
The experiment described above was performed in 2002 using the shared apparatus, and
revealed a 3 P2 magnetic trapped lifetime on the order of 500 ms. In 2007, the experiment
was repeated in the new neutrals assembly. Figure 5.5 shows a 3 P2 magnetic trap lifetime
79
Figure 5.5 : 2007 Magnetic Trap lifetime Measurement. The number of atoms in
the magnetic trap decreases as a function of hold time, and is well fit with a single
exponential with a lifetime on the order of 7 seconds.
of 7 seconds in the new apparatus.
This is a factor of 14 increase in trap lifetime, which indicates a proportional reduction
in the background gas level, due to design improvements in the new assembly. This is an
encouraging result for future experiments which require lower vacuum levels, including
evaporative cooling in the ODT.
80
Figure 5.6 : Partial strontium Energy Level Diagram. Decay rates (s−1 ) and selected
excitation wavelengths are given. Taken from Nagel et al. [25]
5.2
5.2.1
ODT Lifetime Measurements of Metastable States
Scientific Motivation
Measuring the lifetime of the atoms in the 3 P0 state trapped in the ODT gives an experimental means of measuring the elastic scattering cross section, which may be useful as a
means to achieving BEC. Currrently, we have demonstrated the ability to promote atoms
in the 1 S0 state trapped in the ODT to the 3 P0 state, trap them in the ODT, and recover
them, using repumper lasers. Because the experimental sequence is a bit complicated, I’ve
re-included the energy level diagram (Figure 5.6).
5.2.2
Experimental Sequence
It should be noted that the 707 nm repumper laser is always on for this experiment. As
detailed in Chapter 3, atoms are first captured in the blue MOT, then transferred to the
red MOT, then transferred to the ODT. Once the atoms have thermalized in the ODT,
81
the trap is released for 2 ms, during which the 689 and 688 nm lasers are pulsed on. This
drives atoms from the 1 S0 state (which was trapped in the ODT) to the 3 P1 state on the
intercombination transition, and then to the 3 S1 via the 688 nm repumper. From the 3 S1
state, atoms can decay to all three 3 P sublevels. Any atoms that fall into the 3 P2 state
are repumped by the always-on 707 laser. Likewise, any atom that fall into the 3 P1 state
are repumped by the 688 laser. Thus, with three repumpers in play, the atoms are forced
into to the 3 P0 state. Then, the ODT is switched back on. After a variable time thold , the
ODT is again released, and this time the 679 nm is pulsed on, which promotes atoms from
the 3 P0 state to the3 S1 state. Again, any atoms that fall into the 3 P2 state are repumped
by the always-on 707 laser, so the combination of these two repumpers forces the atoms
into the 3 P1 state, which decays to the 1 S0 state. The repumpers (679 and 707) remain on
during the following 2 ms delay, and ground state atoms are imaged on the blue transition
at 461 nm. This sequence is illustrated in Figure 5.7.
Figure 5.8 shows the temperature of the atoms in the 3 P0 ODT-trapped state, after
athold time of 200 ms. Comparing this to the ODT temperature of 15 µK , we see that atoms
in the 3 P0 state are at a lower temperature than the atoms in the ODT. This is suggestive
of thermalization of the 3 P0 but more data is necessary for definitive measurements of the
collision cross-section.
5.2.3
Implications
Using repumpers, we have successfully transferred atoms to the 3 P2 , 3 P1 , 3 P0 states. Lifetime studies on each of these states in the ODT may reveal long lifetimes that are would
be useful in evaporative cooling.
82
Figure 5.7 : Experimental Timing sequence for trapping 3 P0 atoms in the ODT
83
Figure 5.8 : (a) 1 S0 ODT Temperature (b) 3 P0 ODT Temperature Note that for the
same amount of hold time, atoms in the 3 P0 state are colder. This is suggestive of
thermalization.
84
Chapter 6
Conclusions
I began graduate school convinced that BECs were terribly interesting and that
working with lasers in the Killian lab would be fun. My plan was to reach quantum
degeneracy in atomic strontium and then graduate.
Experimentally realized BEC’s have been around for over 10 years. So it’s not as
if reaching quantum degeneracy in strontium will garner any sort of high praise for being
”first.” However, all of the atoms that have been bose-condensed to date have intrinsic
spin, and strontium does not. Without intrinsic spin, strontium in its ground state is not
magnetically trappable, so an all-optical route is necessary. This means the combination
of two magneto-optical traps with an optical dipole trap, and then evaporation within the
ODT.
You may well ask, what does a spinless BEC get you? Answer: decreased sensitivity to stray magnetic fields, much simpler collision physics, leading to the possibility of
interfereometers that are an order of magnitude larger.
Also, there are some other interesting things you can do with strontium, again,
because there are two outer electrons, you can take one off and still have one to play
with; so optical imaging on ultracold neutral plasmas has proven to be a fantastic tool for
measuring the plasma’s evolution. I worked on some of the very early plasma experiments
as well.
But back to the neutral atoms. People have been working on strontium, with a view
85
toward quantum degeneracy, for a long time. But they have always stopped just shy of
bose-condensation, unable to get that last factor of 10 decrease in temperature (or increase
in density, either way). What matters for BEC is the phase degeneracy parameter, basically
the number of atoms inside a volume defined by the de Broglie wavelength. If that number
is> 2.62, a transition to BEC occurs. To date, it hasn’t been done.
We began the experiments by implementing a blue MOT. But, instead of looking at
ground state atoms, the first experiment we published was a study of strontium atoms,
in a metastable atomic state, trapped in a magnetic trap. Specifically, we trapped atoms
that were in the 3 P2 state in the quadrupole magnetic trap that was a direct result of
running the anti-Helmholtz coils for the blue MOT. It was interesting to see that these
atoms appeared to thermalize, and we were able to get a rather dense sample in this trap.
About that time, a theoretical paper came out with predictions concerning the 3 P2 state:
at densities necessary for BEC, inelastic loss rates dominated the elastic ones, meaning,
the pursuit of evaporation in the magnetic trap was yet another formula for getting close
to BEC without achieving it. Because of this prediction and some technical issues with
the 707 nm repumper laser, the magnetically trapped 3 P2 sample was dropped as a route
to quantum degeneracy.
We began developing the technology for the next stage in cooling: the red MOT.
The transition that this MOT works on is narrow (∼7 kHz), and requires a laser whose
linewidth is on the same order. My master’s thesis was all about developing this laser
system. Transfer efficiency from the blue MOT to the red can be as high as 70% in the
early stages of red MOT operation. There is a tradeoff, though, between number and
temperature. In general we allow the sample to thermalize in the red MOT, producing
86
colder, more dense samples.
The parameter that really holds all the information about a certain atom (how long
it will take to thermalize in a trap, how atoms in that interact with each other) is the
s-wave scattering length a. The scattering cross section σel is given by σel = 8πa2 . Randy
Hulet’s group at Rice was the first group to use photoassociative spectroscopy to extract
scattering length information on Lithium, and we were happy to follow in his footsteps.
In order to achieve the densities necessary for detectable signals, we performed photoassociative spectroscopy on atoms held in the red MOT. These experiments led to several
results. First, an improved measurement of the lifetime of the 1 P1 excited state. Second,
information about the scattering length.
Until that time, everyone working with strontium was working with the most abundant bosonic isotope,
88 Sr.
After weeks of unsuccessfully searching for the amplitudes of
the PAS resonances to go through a minimum, we decided to trap a less abundant bosonic
isotope,86 Sr. Performing PAS here, we saw the strength of the resonances decrease to zero
and then increase again as we increased the detuning from the atomic reasonance. With
this precise knowledge of the position of the node, we collaborated with Robin Côté and
Phillippe Pelligrini to calculate the s-wave scattering length of 86 Sr. Even better, we could
use a simple mass-scaling relationship to fit the data that we already had for
88 Sr,
and the
fit is very good without any free parameters. So, in these PAS experiments, we found that
the s-wave scattering length for
88 Sr,
the issotope that everyone had been working with, is
somewhere between -1 and 13 bohr. In contrast, the scattering length for
86 Sr
is bounded
by 600 and 2300 bohr. This is a very powerful result: now we know why people have been
having such trouble getting that last factor of 10 – the timescales for thermalization are
87
enormous.
Even in the red MOT, though, the degeneracy parameteter, nλ3dB is still too low to
realize quantum degeneracy. Just for fun, here are the numbers: with the rempumpers on,
in the blue MOT, nλ3dB = 1.6e-8; in the red mot, nλ3dB = 0.01. To increase the degeneracy
parameter even further, we implemented the use of a crossed-dipole trap. The initial value
of the parameter here, without evaporation is nλ3dB = 0.02.
Experiments in the ODT are ongoing: measuring the lifetimes of metastable states,
exploring high-intensity effects in PAS, and trying our hand at evaporative cooling. In
this thesis, we have been able to use the two-body loss rates from the ODT to verify
our value for the
88
Sr elastic cross-section, and the data agree rather well. There is some
discrepancy with values from other experiments, and it is likely that more data is necessary
for a definitive answer.
So, while I did not achieve my original goal (BEC in atomic strontium) during my
graduate career, the work that I have done contributes significantly toward that goal; we
are optimistic that with this knowledge, strontium BEC will be achieved.
88
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