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The Pennsylvania State University
The Graduate School
Eberly College of Science
THE RAPID CONTROL OF INTERACTIONS IN A
TWO-COMPONENT FERMI GAS
A Dissertation in
The Department of Physics
by
Ronald William Donald Stites
c 2012 Ronald William Donald Stites
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2013
The dissertation of Ronald William Donald Stites was reviewed and approved∗ by
the following:
Kenneth M. O’Hara
Associate Professor of Physics
Dissertation Advisor, Chair of Committee
Milton W. Cole
Distinguished Professor of Physics
Kurt Gibble
Professor of Physics
David S. Weiss
Professor of Physics
John V. Badding
Professor of Chemistry
Richard W. Robinett
Professor of Physics
Associate Department Head, Director of Graduate Studies
∗
Signatures are on file in the Graduate School.
Abstract
In this dissertation, we describe a variety of experiments having application to
ultra-cold atomic gases. While the majority of the experimental results focus
on the development of a novel laser source for cooling and manipulating a gas
of fermionic 6 Li atoms, we also report on a preliminary investigation of rapidly
controlling interactions in a two-component Fermi gas.
One of the primary tools for our ultra-cold atomic physics experiments is 671
nm laser light nearly resonant with the D1 and D2 spectroscopic lines of ultracold fermionic 6 Li atoms. Traditionally, this light is generated using dye lasers
or tapered amplifier systems. Here we describe a diode pumped solid state ring
laser system utilizing a Nd:YVO4 gain crystal. Nd:YVO4 has a 4 F3/2 → 4 I13/2
emission line at 1342 nm. This wavelength is double the 671 nm needed for our
experiments. As a part of this investigation, we also measured the Verdet constant
of undoped Y3 Al5 O12 in the near infrared for constructing a Faraday rotator used
to drive unidirectional operation of our ring laser. As an alternative method to
achieve unidirectional, single-frequency operation of the laser, we developed a novel
scheme of “self-injection locking” where a small portion of the output beam is
coupled back into the cavity to break the symmetry. This technique is useful
for high-power, single-frequency operation of a ring laser because lossy elements
needed for frequency selection and unidirectional operation of the laser can be
removed from the internal cavity.
In addition to our laser experiments, we also drive Raman transitions between
different magnetic hyperfine states within 6 Li atoms. For atoms in the two lowest
hyperfine states, there exists a broad Feshbach resonance at 834.1 Gauss whereby
the s-wave scattering length diverges, resulting in strong interactions between the
two species. By using two phase locked lasers to drive a transition from a strongly
interacting state to a weakly interacting state, we can rapidly control the interaction strength of a two component Fermi gas.
iii
Table of Contents
List of Figures
vii
List of Tables
xvi
Acknowledgments
xvii
Chapter 1
Introduction
1.1 Interactions in an Atomic Gas . . . .
1.2 Probing Strongly Correlated Systems
1.3 The Rapid Control of Interactions . .
1.4 A Self-Injected 1342 nm Ring Laser .
1.5 Outline . . . . . . . . . . . . . . . . .
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1
3
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Chapter 2
Fermi Gases
2.1 Properties of 6 Li . . . . . . . . .
2.2 s-Wave Scattering Theory . . . .
2.3 Scattering Resonances . . . . . .
2.4 Interactions in a | 1-| 5 Mixture
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Chapter 3
The Interaction of Atoms with Light
3.1 The Two Level Atom . . . . . . . . .
3.2 The AC Stark Effect . . . . . . . . .
3.3 Magnetic Dipole Transitions . . . . .
3.4 Raman Transitions . . . . . . . . . .
iv
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Chapter 4
The Verdet Constant of Y3 Al5 O12
4.1 Verdet Constant Theory . . . . .
4.2 Y3 Al5 O12 Properties . . . . . . .
4.3 Measurement Setup . . . . . . . .
4.4 Cylindrical Magnet . . . . . . . .
4.5 Verdet Measurement . . . . . . .
4.6 Conclusion . . . . . . . . . . . . .
Chapter 5
Constructing a 1342 nm Ring
5.1 Cavity Layout . . . . . . . .
5.2 Reflectors . . . . . . . . . .
5.3 Nd:YVO4 Crystal . . . . . .
5.4 Pump Laser . . . . . . . . .
5.5 Cooling the Crystal . . . . .
5.6 Additional Components . .
5.7 Cavity Modeling . . . . . .
5.8 Conclusion . . . . . . . . . .
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Laser
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Chapter 6
1342 nm Ring Laser Operation
6.1 Measuring the Nd:YVO4 Gain . . . . . .
6.2 The Free Running Laser . . . . . . . . .
6.3 Modeling Power Output . . . . . . . . .
6.4 Unidirectionality with a Faraday Rotator
6.5 Unidirectionality with Self Injection . . .
6.6 Frequency Tuning . . . . . . . . . . . . .
6.6.1 Intracavity Tuning . . . . . . . .
6.6.2 External Cavity Tuning . . . . .
6.7 Conclusion . . . . . . . . . . . . . . . . .
Chapter 7
Preparing a Two-Component Fermi
7.1 Ultra-High Vacuum System . . . .
7.1.1 The Oven Region . . . . . .
7.1.2 The Zeeman Slower . . . . .
7.1.3 The Experimental Region .
7.1.4 Vacuum Pumps . . . . . . .
7.2 Laser System Overview . . . . . . .
v
Gas
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102
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7.3
7.4
7.5
7.6
Magnetic Field Coils . . .
Magneto-Optical Trapping
Optical Dipole Traps . . .
Atomic Imaging . . . . . .
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Chapter 8
The Rapid Control of Interactions
8.1 Initial Preparation . . . . . . . . . . . . . . . . . .
8.2 Phase Locking of Raman Lasers . . . . . . . . . . .
8.3 Preliminary Investigations of a Non-Interacting Gas
8.4 The Next Steps . . . . . . . . . . . . . . . . . . . .
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Chapter 9
Conclusions and Outlook
131
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Appendix A
Mathematica Code for Calculating Scattering Lengths
137
Bibliography
152
vi
List of Figures
2.1
2.2
2.3
2.4
2.5
The hyperfine energy level structure for 6 Li showing the 22 S ground
state and the two 22 P excited states. . . . . . . . . . . . . . . . .
The magnetic field dependence of the energy for the F = 1/2 and F
= 3/2 ground states of 6 Li. As the applied field becomes stronger,
the atoms enter the Paschen-Back regime, whereby the atom becomes electron spin polarized and the three way splitting arises due
to difference in the nuclear spin of the atoms. . . . . . . . . . . .
A Feshbach resonance occurs when the least bound state of the
close channel molecular potential is tuned via a magnetic field to
be resonant with the collision energy of the particles in the entrance
channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The s-wave scattering length as a function of applied magnetic
field for atoms in the two lowest hyperfine ground states of 6 Li
[1]. Note the broad Feshbach resonance located at 834.1 G. By arbitrarily tuning the applied magnetic field around this resonance,
we can change the strength of the two-body interaction from being
infinitely repulsive to infinitely attractive. . . . . . . . . . . . . .
The result of our coupled channel calculation for s-wave scattering
between atoms in states |1 and |2. To simplify the calculation, we
model the singlet and triplet molecular potentials as finite square
well potentials. Despite this oversimplication, the calculation reproduces the broad Feshbach resonance located at 834.1 G which can
be observed as a divergence in the s-wave scattering length. The
location and width of this resonance is in good agreement with a
coupled channel calculation which uses accurate singlet and triplet
molecular potentials. The good agreement gives us confidence in
our coupled channels calculations using a simple model for the potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
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13
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19
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24
2.6
2.7
3.1
3.2
3.3
4.1
Our simplified coupled-channels model for scattering of atoms in
states |1 and |2 also predict the presence of a narrow Feshbach
resonance. While the position of this resonance is slightly higher
in magnetic field than what has been experimentally observed, this
discrepancy could easily be explained by the different shape of our
simplified potentials. . . . . . . . . . . . . . . . . . . . . . . . . . .
The predicted s-wave scattering lengths as a function of magnetic
field for a |1-|5 mixture. With no Feshbach resonance, the s-wave
scattering length increases to a value of approximately -3 Bohr at
a magnetic field corresponding to 834.1 gauss. In this way, interactions of a two-state |1-|2 mixture at this field could be quickly
suppressed by a simple transfer from state |2 → |5 . . . . . . . . .
Energy level diagram for a simplified atomic system. A two level
atom with ground state | g and excited state | e interacts with
an oscillating electric field having a frequency of ω. This beam is
detuned by an amount δ from the transition frequency ω0 corresponding to the energy difference between the two atomic states. . .
The shift of the energy levels of a two level atomic system due to
the presence of an oscillating electromagnetic field. This AC Stark
effect results in a resonance frequency shift to higher frequencies
when the detuning of the beam is negative and lower frequencies
when the detuning is positive. . . . . . . . . . . . . . . . . . . . . .
The energy level diagram for a two photon Raman transition. Laser
L1 drive the atom from the ground state to a virtually excited
intermediate state i. Laser L2 then drives the atom from state
i back to the desired excited state. It is important to note that
the population in state i is virtual, and losses due to spontaneous
emission are not allowed. By employing this two photon technique,
one can drive state changing transitions where they would otherwise
be forbidden, such as when Δl = 0. . . . . . . . . . . . . . . . . . .
The transmission spectrum of undoped YAG from 200 to 6500 nm.
Note the broad transmission extending from the visible well into
the mid-infrared [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
25
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29
34
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41
4.2
4.3
4.4
5.1
5.2
5.3
5.4
5.5
Light from an external cavity diode laser (ECDL) passes through
a half-wave plate (HWP) and polarizer (P) pair. A double Fresnel
rhomb (FR) is used to adjust the polarization of the light incident
on a cylindrical magnet (M). The light is then split by a polarizing
beam splitter (PBS) before falling on two photodetectors (DET)
which record the power of each beam. . . . . . . . . . . . . . . . . .
(a) Physical layout and dimensions of the right hollow cylindrical
magnet housing the undoped YAG crystal. The crystal, of length
Lc = 18.0mm is located inside the magnet having length L = 19.0
mm and an outer diameter 2b = 22.2 mm. The inner bore of the
magnet has a diameter 2a = 6.3 mm. (b) The measured magnetic
field of the magnet at various distances from the end facets. The
dashed line represents a fit to the measured data for the magnetization M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Verdet constant of undoped YAG in the near infrared. Each data
point represents the average of one hundred measurements. The
inset provides a closer look at the 1300 nm to 1350 nm range. Error
bars are indicative of the standard error of the mean. The dashed
line is a fit to the data as per equation 4.3, demonstrating the
dispersive nature of the Verdet constant. . . . . . . . . . . . . . . .
Energy level diagram of Nd:YVO4 . The crystal has a 4 I9/2 → 4 F5/2
pump absorption line at 808 nm and two emission lines (4 F3/2 →
4
I13/2 and 4 F3/2 → 4 I11/2 ) at 1342 nm and 1064 nm respectively. .
Artistic 3D rendering of the Nd:YVO4 laser cavity. . . . . . . . .
Experimental setup for the laser cavity. A Nd:YVO4 crystal located
inside of a bow-tie ring cavity is double end pumped by two 25 Watt
diode arrays. The cavity consists of a fully reflecting mirror (M1),
an output coupler (M2), and two dichroic mirrors (M3 and M4)
which reflect light at 1342 nm while transmitting light at 808 and
1064 nm. With nothing to break the symmetry of the system, the
laser will lase bidirectionally, resulting in the gain being shared by
the clockwise and counterclockwise modes . . . . . . . . . . . . .
Machine drawing for homemade mirror mount for the dichroic mirror M3 in our 1342 nm laser cavity. . . . . . . . . . . . . . . . . .
Machine drawing for homemade mirror mount for the dichroic mirror M4 in our 1342 nm laser cavity. . . . . . . . . . . . . . . . . .
ix
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5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
The crystal structure of an yttrium ortho-vanadate. Yttrium orthovanadate crystalizes as a zircon tetragonal (tetragonal bipyramidal)
structure, leading to a natural birefringence along its a and c axes.
Shown centered is the yttrium ion surrounded by its vanadium and
oxygen neighbors. This figure is adapted from [3]. . . . . . . . . . .
The absorption spectrum of a 0.27% doped Nd:YVO4 crystal in the
region of the 808 nm band, reproduced from [4]. This plot shows a
peak absorption coefficient of approximately 9.4 cm−1 at 808.7 nm.
Copper mount used to hold the Nd:YVO4 crystal inside the laser
cavity. One end of the mount is connected to a thermo-electric
cooler used to extract heat from the crystal (see section 5.5. The
other end of the mount holds the crystal in a narrow finger-like
region, allowing the crystal to be located in the cavity without
obstructing the beam path. . . . . . . . . . . . . . . . . . . . . . .
Copper cover used to sandwich the gain crystal against the copper
mount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adapter for connecting the sma fiber from the DUO-FAP laser to
the homemade lens mount for the 2:1 pump imaging system. . . . .
Homemade lens mount used to house the imaging optics for focusing
the 808 nm pump laser light onto the Nd:YVO4 gain crystal. Two
lenses, of focal lengths 60 and 30 mm are located so as to provide a
2:1 imaging system, focusing light from the 800 μm diameter pump
laser fibers to a diameter of 400 μm on the gain crystal itself. . . . .
This home built mount is used to hold the lens mount shown in
figure 5.10. By doing so, it allows the lenses used for imaging the
pump laser light onto the gain crystal to be precisely positioned via
a translation stage connected to this mount by the adapter plate
shown in figure 5.13. . . . . . . . . . . . . . . . . . . . . . . . . . .
Machine drawings for an adapter connecting the mount in figure
5.12 to the stainless steel gothc-arch xyz translation stage. . . . . .
Cartoon showing the interface between the copper thermal reservoir
and the water cooled mount [5]. Two heat sinks are located in very
close proximity to one another so as to enable heat transfer without
making a physical connection. This enables heat to be transferred
across the interface without coupling any vibration from the water
cooled mount into the copper thermal reservoir (and subsequently,
the laser gain crystal). . . . . . . . . . . . . . . . . . . . . . . . . .
x
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68
70
5.15 ABCD matrices used for a paraxial resonator analysis of our ring
laser cavity. Included are matrices for propagation of length d in a
uniform medium with index of refraction n as well as a thin lens of
focal length f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 The waist of the gaussian beam for one round trip of propagation
inside the ring laser cavity. The zero position reference is taken to
be the center of the gain crystal. . . . . . . . . . . . . . . . . . . . .
6.1
6.2
6.3
6.4
6.5
Experimental setup for measuring the unsaturated small signal gain
of the Nd:YVO4 crystal when pumped by 2 × 25 Watt pump beams.
The transmitted power of an extended cavity diode laser (ECDL)
whose wavelength is tunable from 1335 to 1350 was measured by a
photodetector (DET) after having passed through the crystal. Not
shown in the setup are the lenses used to mode match the waist of
the probe laser with the pump lenses as they co-propagate through
the crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The unsaturated gain profile of our Nd:YVO4 crystal from 1335
nm to 1350 nm when pumped by 2 × 25 Watt pump beams. The
gain profile is quite broad, spanning several nanometers, and peaks
at approximately 1342 nm. Also of note is a broad excited state
absorption band spanning from 1336 nm to 1341 nm. . . . . . . .
The gain profile of our Nd:YVO4 crystal from 1341 to 1345 nm.
The profile has a peak value of 1.87 at a corresponding wavelength
of 1342.2 nm. A dashed line has been added as a guide to the eye
(see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental setup for measuring the angular dependence of the
unsaturated small signal gain of the Nd:YVO4 crystal. The transmitted power of an extended cavity diode laser (ECDL) whose wavelength is set at 1342 nm was measured by a photodetector (DET)
after having passed through the crystal. A half-wave plate (HWP) is
used to rotate the polarization of light prior to transmission through
the crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Unsaturated small signal gain as a function of polarization angle,
measured with respect to vertical. The dashed line represents a
sinusoidal fit to the data (see text). . . . . . . . . . . . . . . . . .
xi
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6.6
Measurement of the power stability for the free running laser cavity.
For most of the time, the power is split evenly between the two
directions of operation, indicating bidirectionality. However, during
some instances, the laser operates in unidirectional mode, whereby
the recorded power is either twice as high or zero, depending on the
direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Measurement of the frequency characteristics of our free running
ring laser. The graph shows an output of a 300 MHz Fabry-Perot
interferometer. The presence of four distinct peaks is indicative of
the fact that the output laser frequency is multi-longitudinal mode
in structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Results of a beam profile measurement made by the “Mode Master”,
manufactured by Coherent, Inc. The output of our free running
laser is coupled into the Mode Master, which measures values for
the beam radius as the beam propagates over a given distance.
From these measurements, several spatial properties of the laser
beam can be determined and reported. . . . . . . . . . . . . . . .
6.9 Power output from the laser cavity when driven unidirectionally using an intracavity Faraday Rotator for seven different output couplers. The dashed line represents a theoretical fit to the data used
to determine the unsaturated small signal gain and intracavity scattering losses for our home built ring laser. . . . . . . . . . . . . .
6.10 Experimental layout for our novel scheme of self injection. A small
portion of the output light from our laser is picked off from the main
beam using a half-wave plate (HWP) and a polarizing beam splitter
(PBS). This light then travels through a Faraday rotator (FR) and
another half-wave plate to change its polarization back to vertical.
The weak beam is then injected back into the cavity, where it causes
stimulated emission in the gain crystal, breaking the symmetry of
the ring laser and driving unidirectional operation. Also included
in this loop are three lenses (L1, L2, L3) used to shape the injected
beam for mode matching (see text). . . . . . . . . . . . . . . . . .
6.11 Measurement of the power stability for the self-injected laser cavity.
Unlike the free running laser, the unidirectional operation of the
self-injected laser prevents mode competition in the gain crystal
resulting in a drastic reduction of intensity noise on the output of
the laser beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
.
82
.
83
.
84
.
89
.
91
.
92
6.12 The longitudinal mode structure of the self-injected laser as measured by a 300 MHz Fabry-Perot interferometer. Unlike the case of
the free running laser cavity, the lack of mode competition in the
gain crystal enables single frequency operation . . . . . . . . . . .
6.13 The spectral output of a heterodyne measurement of the linewidth
of our self-injected laser. The laser output is beat with the output
of a tunable ECDL and sent to a spectrum analyzer. From the full
width at half maximum of this beat note measurement, it is shown
that the linewidth of our self-injected laser is no larger than 150 kHz.
6.14 The output power of our self-injected laser as a function of output
coupling, δ1 , for seven different output couplers. The dashed line
represents a fit to our data with the intracavity reflectivity being
the only free parameter. Compared to the free running laser, we
observe a 4.5% increase in output power for the optimum output
coupler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.15 The periodic transmission measurement of a thin silicon etalon as
a function of wavelength. The transmission was measured from
1330 nm to 1350 nm using a tunable extended cavity diode laser at
normal incidence to the etalon. The dashed line represents a fit to
the data of equation 6.18, indicating an etalon thickness of 34.2 μm.
6.16 The output power of our laser as a function of wavelength when
tuned via a 34.2 μm intracavity silicon etalon. The dashed line
represents a fit to the data when considering the convolution of the
etalon transmission, the small signal gain of the laser crystal, and
the increase in intracavity scattering losses due to the insertion of
the etalon (see text). . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17 The output power of the laser cavity as a function of wavelength
when tuned via a 250 μm silicon etalon located in the external loop
region of our self-injected laser. By merely rotating the etalon, we
were able to observe a tuning range that exceeded 38.9 GHz for our
laser setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
93
94
95
97
98
99
7.1
7.2
7.3
7.4
7.5
The experimental layout for our UHV system used to cool, trap,
and manipulate a gas of fermionic 6 Li atoms. The apparatus is
divided into three regions, namely an oven used to provide a source
of hot atoms, a Zeeman slower used to reduce the temperature of the
atoms, and an experimental region where the cool atoms are then
trapped and cooled further for use in our experiments. Also shown
are a variety of vacuum pumps used to maintain the low pressure
required for our experiments. This figure has been adapted from
reference [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Layout of the Zeeman slower used in our apparatus [6]. Three electromagnet coils are wired in series to produce a spatially dependent
magnetic field that shifts the energy levels of the atoms to be continuously on resonance with a laser beam. The absorption of photons from this counter propagating laser beam causes a momentum
transfer to the atoms, reducing their velocities and subsequently
cooling the atoms as they enter the experimental chamber. . . . .
Lateral cross section view of the experimental region of our apparatus. From this view, one can see the location of the MOT coils, the
Feshbach coils, and the rf coils used to drive magnetic dipole transitions in our trapped atoms. This figure has been adapted from
reference [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a) The basic layout for the magneto-optical trapping of 6 Li. Three
pair of red detuned orthogonal laser beams overlap the zero magnetic field location between two magnetic coils in anti-Helmholtz
configuration. By choosing the proper polarizations, the combination of magnetic and optical fields provide a restorative cooling
force on the atoms located in this spatially overlapped region. b)
The energy level diagram showing the cooling and repumping laser
transitions for our MOT (solid lines). The dashed lines show the
available channels for spontaneous emission, indicating the necessity of the repumping laser. . . . . . . . . . . . . . . . . . . . . .
Cartoon layout of the three lasers that overlap to form the optical
dipole force trap for our experiments. Two 100 watt 1064 laser
beams intersect at a relative angle of 11◦ with each beam having
a waist of 30μ. A third 1070 nm 100 watt laser overlaps the other
two, forming a deep trap used to capture atoms from our MOT.
Also shown is the side view of one of the anti-Helmholtz coils used
to provide the spherical quadrupole trap for our MOT. . . . . . .
xiv
. 104
. 107
. 111
. 114
. 117
7.6
Absorption image of a cloud of 6 Li atoms trapped in a) two 1064
nm beams forming a crossed dipole trap b) a single 1070 nm beam
c) a combination of the two 1064 nm beams and the 1070 nm beam. 118
8.1
Diagram showing the components making up the phase lock feedback loop. A small amount of power from each of two diode lasers
are combined on a high bandwidth photo detector. The beat note
signal of the two beams is then split where it takes one of two paths.
On the first path, the signal is amplified before being fed into an
optical phase-lock loop circuit where the beat note is compared to
a reference oscillator and a corrective signal is fed back to the piezo
and the FET of the laser diode. The signal is also amplified on the
other path before being mixed with another reference oscillator, filtered, and coupled to the bias-t connector of the same laser. In this
way, we can lock both the frequency and phase of the two diode
lasers to arbitrary values. . . . . . . . . . . . . . . . . . . . . . . .
The beat spectrum of our two phase locked lasers locked at 1.6
GHz. The central peak of 0.35 dBm corresponds to the power
in the carrier while the total power from the occupied bandwidth
measurement (2.2 dBm) can be used to determine the carrier power
fraction, and thus the mean-square phase error of our lock. . . . .
Measurement of the fractional population of atoms in state | 2 as
a function of frequency for a magnetic dipole transition. The frequency of the applied microwave rf field was scanned over a range of
±30 kHz relative to the central transition frequency of 1.493831258
GHz for two different pulse duration times of 100 ms and 200 ms.
Rabi oscillations demonstrating the transfer of atoms from state
| 2 to state | 5 using a two-photon Raman transition. The decay
in coherence is attributed to the small size of the Raman beams
causing a spatially dependent Rabi frequency for atoms located in
different parts of the trap. . . . . . . . . . . . . . . . . . . . . . .
8.2
8.3
8.4
xv
. 124
. 126
. 128
. 129
List of Tables
2.1
5.1
g-factors and Hyperfine constants for the 22 S and 22 P electronic
states of 6 Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Measured reflectivities of available output couplers for our homemade 1342 nm laser. Measurements were made at normal incidence
and at 15◦ for vertically polarized light. The δ value of each output
coupler is another way of representing its reflectivity (also at 15◦ ).
For a further explanation, see the text. . . . . . . . . . . . . . . . .
56
xvi
Acknowledgments
This Dissertation would not have been possible without the endless support of
a number of people who I am honored to consider my mentors, colleagues, and
friends. It has been said that it takes a village to raise a child. In the same vein,
I would argue that it also takes a village to complete a Ph.D.
First and foremost I would like to thank Ken O’Hara for the opportunity to
conduct research in his lab. His positive attitude, keen insight, fervent devotion,
and incessant creativity has served as a constant inspiration, not only to press on
through seemingly impossible obstacles, but to also dream big with lofty goals and
aspirations. I could not have asked for a better advisor.
During my tenure in the O’Hara lab, I have also had the opportunity to rub
elbows with some of the brightest up and coming researchers in the field. The hard
work of Johnny Huckans, Jason Williams, Eric Hazlett, and Yi Zhang have greatly
contributed not only to the success of our research lab, but to my own personal
success as well.
I would also like to thank Kurt Gibble and Dave Weiss for their vast knowledge
and endless patience, both during formal course instruction and during our weekly
AMO journal club meetings. Likewise, I would also like to extend thanks to the
other members of my dissertation committee, Milton Cole and John Badding,
whose willingness to serve in this capacity is truly humbling.
Next I would like to thank my parents, Ron and Connie Stites. Without their
support I could not have achieved success. The completion of this dissertation
should serve as a great example of what can be accomplished through the positive
encouragement and ceaseless dedication of family.
Finally, I would like to thank my new family, starting with my beautiful wife,
Jenna. Her unconditional love and devotion has truly blessed me. Whenever I’m
having a bad day and need someone to listen, I can take comfort in knowing that
she has been and will always continue to be there for me. I only hope that in some
small way I can eventually repay her the endless debt of gratitude that I owe. Last,
I would like to thank my daughter, Hannah. Even though I have only known her
xvii
for a few short months, she has already transformed me— teaching me new lessons
every day about love, patience, and overcoming adversity. There’s no affliction in
the world that one of her smiles cannot cure.
xviii
Chapter
1
Introduction
Ever since Bohr first published his research on the structure of the atom in 1913 [7],
atomic physics, along side the development of quantum mechanics, has been one of
the most studied and fruitful fields of modern physics. However, while the field had
been dominated for many decades by traditional spectroscopy-type experiments,
it has been over the course of the last 30 years that the atomic physics community
has experienced a revolution in research direction, experimental techniques, and
fundamental results. Ushering in this new era of research has been the development
of new techniques for the cooling, trapping, and manipulation of ultra-cold neutral
atomic gasses [8].
One of the reasons that physicists have been so interested in cooling and trapping neutral atomic gases is because these atomic systems provide an ideal test
bed for the simulation of few and many body quantum systems. This is due to the
fact that many of their experimental parameters (such as atomic density, temperature, and scattering length) can easily be tuned by the application of optical and
magnetic fields. Additionally, for the temperature scales involved in these experiments, typically spanning six orders of magnitude from a few hundred nano-Kelvin
to a few milli-Kelvin, the theoretical treatment of interactions in these systems can
be greatly simplified. In this way, fundamental questions regarding the nature of
collisions, chemical reactions, and thermodynamics can be investigated using these
experimental systems.
After it’s initial conception in 1975, the cooling and trapping of ultra-cold gases
using lasers began to achieve success in the early 1980s by Chu at Bell Laboratories
2
in New Jersey [9]. In these experiments, nearly resonant counter propagating
laser light was used to reduce the velocity (and hence the temperature) of sodium
atoms. Though not trapped, it was demonstrated that atoms experiencing this
three dimensional viscous force were cooled to approximately the Doppler limit
of 240 μK. This limit represents the minimum temperature obtainable using this
optical molasses technique [10]. Also in 1985, Phillips, Metcalf, and colleagues at
the National Bureau of Standards (now NIST) first demonstrated the use of field
gradients from a spherical quadrupole magnetic trap to confine these cold atoms
[11]. Repeating the experiments of Chu, they found that the atoms in the trap
had been cooled to a temperature of 40 μK, a factor of six below the theoretical
cooling limit. To explain cooling below this Doppler limit, the idea of polarization
gradient cooling was independently and simultaneously published by Dallibard and
Cohen-Tannoudji [12] as well as Unger, et al. [13]. For the development of methods
to cool and trap atoms with laser light, Chu, Phillips, and Cohen-Tannoudji would
later share the Nobel Prize in Physics in 1997.
Once it had been demonstrated that atoms could be cooled even lower than
the theoretically predicted limit, additional interest was spawned in further cooling
the gas. For an ensemble of ultra-cold atoms, one way of parameterizing the gas
is by it’s phase space density ρ = nλ3dB , where n is the density of the gas and
λdB = 2π2 / (mkB T ) is the thermal de Broglie wavelength for atoms of mass m
and temperature T . When the phase space density equals unity, the de Broglie
wavelength is comparable to the interparticle spacing. It had been predicted that
a system of identical particles with integer spin (bosons) would undergo a phase
transition if the phase space density is further increased to 2.612 whereby the
ground state energy level becomes macroscopically occupied. This so-called “BoseEinstein Condensate” (BEC) was first predicted by Einstein in 1925 [14].
To achieve a BEC, the temperature of the atomic gas had to be cooled even
further. It wasn’t until several years later in 1995 that this hurdle was surpassed
using the idea of forced evaporative cooling [15]. In forced evaporative cooling,
atoms were magnetically trapped in a quadrupole trap in the presence of an rf
magnetic field. This rf field was resonant for atoms with the highest energy and
would cause a spin flip transition to a magnetically untrapped state. By selectively
removing the most energetic atoms from the trap and allowing for rethermalization,
3
the temperature of the trapped gas was lowered until BEC was observed [16, 17, 18].
It was for the observation of BEC that Ketterle, Wieman, and Cornell shared the
Nobel Prize in Physics in 2001. Today, the record for the coldest matter in the
universe at 500 pK is held by an ultracold gas that had been cooled using these
same techniques [19].
For a system of identical particles with half-integer spin (fermions), the experimental story line is fairly similar. In 1957, Bardeen, Cooper, and Shrieffer
published a series of papers attempting to explain superconductivity as a microscopic effect concerning the bose condensation of pairs of fermions [20, 21]. Using
the same techniques of cooling, trapping, and forced evaporative cooling using a
magnetic trap, evidence for the first observation of a non-interacting degenerate
Fermi gas (DFG) of the fermionic isotope
40
K was demonstrated by Jin’s group in
1999 [22].
1.1
Interactions in an Atomic Gas
As mentioned above, the first experimental realization of a DFG in 1999 by Jin
was for
40
K atoms evaporatively cooled in a magnetic trap. In this experiment, it
was noted that when the gas was released from it’s magnetic trapping potential
and allowed to expand before imaging, the time-of-flight absorption image of the
cloud demonstrated isotropic ballistic expansion. This ballistic expansion is the
characteristic signature of a weakly-interacting or non-interacting atomic gas.
For a strongly interacting atomic gas, however, this isotropic expansion is not
expected. As the mean free path becomes shorter than the size of the cloud,
the atoms no longer behave ballistically, but rather hydrodynamically, resulting
in multiple collisions as the cloud expands. This anisotropic or hydrodynamic
expansion was first theorized for a BEC [23], but was later expanded to include
the superfluid nature of a strongly interacting Fermi gas [24, 25]
One tool that atomic physicists have at their disposal for tuning interaction
strengths in ultra-cold atomic samples is the application of a DC magnetic field
in the vicinity of a Feshbach resonance [26] where the s-wave scattering length
diverges. At this resonance for fermions, there exists a universality that connects
the unitary Fermi gas to an ideal Fermi gas [27, 28]. The tuning of the s-wave
4
scattering lengths were first experimentally observed in an atomic gas by Ketterle
in 1998 [29] as demonstrated by an increase in the two-body loss for the system
near this resonance.
For fermionic 6 Li, there was also predicted to be a broad Feshbach resonance
located in the vicinity of 834 Gauss [30]. In 2002, O’Hara et al. performed an
experiment on 6 Li trapped in a conservative optical potential, rather than a magnetic one [31]. This optical potential had the benefit of trapping non-magnetically
trapable states of 6 Li. By releasing the cloud of atoms from the optical trap in
the vicinity of a Feshbach resonance, this strongly interacting Fermi gas expanded
anisotropically in the transverse direction of the cigar shaped optical trap while
remaining nearly stationary in the axial direction [32]. In contrast to ballistic expansion where the column density of the absorption image measurement evolves as
1/t2 , it was found that for anisotropic expansion, the density decreases only as 1/t.
To explain this observed anisotropy, an expansion on the theory of superfluid hydrodynamics was employed [24]. It should be noted that since this first experiment
demonstrating the anisotropic expansion of a highly interactive 6 Li Fermi gas, the
same hydrodynamic expansion has been observed in a
40
K degenerate Fermi gas
[33] as well as for a rotating gas [34, 35].
1.2
Probing Strongly Correlated Systems
Highly correlated atomic systems, such as strongly interacting Bose-Einstein Condensates and Degenerate Fermi Gases, represent a class of materials where a variety
of novel many-body phenomenon can be explored. Such phenomenon include Mottinsulator states, superfluidity, anti-ferromagnetic ordering, frustrated spin systems,
the BEC/BCS crossover, and a variety of other exotic states. To investigate these
systems, however, several innovative techniques have been developed to probe the
atomic systems. One such technique invokes the idea of Bragg diffraction.
Bragg diffraction in an atomic gas is based on similar principles of Bragg diffraction in a crystal system. For an nth order Bragg diffraction, two laser beams can be
used as an optical standing wave to stimulate a 2n photon Raman process where n
photons are absorbed by one of the beams and then emitted into the other. Conservation of energy requires (nPrecoil )2 /2M = nδn where Precoil = 2ksin(θ/2) is
5
the recoil momentum, k = 2π/λ, λ is the wavelength of the light, M is the atomic
mass, and δn is the frequency detuning for the two lasers. When this condition is
met, some of the atoms of the BEC or DFG will diffract from the standing wave
and leave the cloud with a momentum nδn .
The first application of Bragg diffraction to a BEC was by Phillips’ group in
1999 [36]. In this experiment, Bragg diffraction was used as a tool to coherently
split a BEC into two components. Additional use of Bragg diffraction were employed shortly thereafter to measure the excitations spectrum ω(k) as well as the
static structure factor S(k) for these systems [37, 38]. Additionally, it was demonstrated that Bragg scattering could be used to excite phonons in a BEC [39].
Up to this point, all the experiments utilizing Bragg spectroscopy had been
performed on a weakly interacting gas. In 2008, Vale’s group in Australia investigated the crossover from BEC to BCS in a fermionic gas of 6 Li near a Feshbach
resonance using Bragg spectroscopy [40]. In this experiment, it was demonstrated
that using Bragg spectroscopy as a measurement tool in a strongly interacting gas
was non-trivial. As atoms are diffracted they quickly undergo collisions distorting
the diffracted cloud and obfuscating the measurement.
Another tool that has been employed to measure the structure of the highly
correlated atomic systems is the use of spatial quantum noise interferometry [41].
For this technique, the atoms are released from their trapped state and allowed to
expand before an absorption image is taken of the sample. From this image, spatial
correlations are computed to gather information about the initial structure of the
system, such as pair correlations for fermions in momentum space [42]. While this
has been proven to be quite an effective tool for weakly interacting atomic samples,
again this technique has drawbacks in strongly interacting systems. As the gas is
released, collisions quickly destroy the coherence of the cloud and the information
gathered using these spatial correlations is lost.
It should be noted that other experiments have been conducted to measure
the excitation spectrum of these correlated systems using additional techniques.
One example is the use of rf photoemission spectroscopy to measure the excitation
spectrum of a degenerate fermi gas (e.g. see ref. [43]). Additionally, in the group
of Cornell, photon correlation measurements of the laser beams themselves have
recently been demonstrated for Bragg spectroscopy [44].
6
1.3
The Rapid Control of Interactions
With regard to the complicating difficulties that a strongly interacting Fermi gas
imposes on the measurements of Bragg diffraction and spatial correlations in a
highly correlated atomic system, in this dissertation we investigate a method of
rapidly controlling atomic interaction strengths by quickly changing the s-wave
scattering length for a two-component Fermi gas. To do so, we first prepare an
atomic sample of 6 Li atoms in it’s two lowest magnetic hyperfine levels | 1 and | 2.
The s-wave scattering length for atoms in these two states can be widely tuned via
an applied magnetic field in the vicinity of a broad Feshbach resonance located at
834 Gauss. By pulsing on two co-propagating laser beams, a two photon Raman
transition drives the entire population of atoms in the highly interacting state | 2
to a weakly interacting state | 5. The time scale of this Raman transition is on
the order of a few microseconds— much faster than the time scales required to
drive the same transition utilizing an rf field via a magnetic dipole transition. In
this way, the s-wave scattering length can be rapidly reduced by several orders of
magnitude over the course of a few microseconds.
To demonstrate this reduction in interaction strength, we will measure the time
of flight expansion of the two-component cloud of atoms as they are released from
their trapping potential as a function of time. As mentioned above, for atoms in
states | 1 and | 2 near the Feshbach resonance, the s-wave scattering length is
larger than the average interparticle spacing and the sample is in the so called
“hydrodynamic regime”. In this regime, the cloud will expand anisotropically at
a rate of 1/t. This expansion will then be compared to the expansion of atoms in
states | 1 and | 5 at the same magnetic field. Atoms first prepared in states | 1
and | 2 will be release from the trap. Immediately upon their release, the atoms
in state | 2 are transferred to state | 5 using the two-photon Raman pulse. The
time of flight expansion of this cloud is again measured and shown that the cloud
now expands ballistically at a rate of 1/t2 . This technique of rapidly controlling
the interaction strength of a two-component gas has direct application to studies
of Bragg spectroscopy and spatial correlations as outlined above.
7
1.4
A Self-Injected 1342 nm Ring Laser
In addition to our studies of rapidly controlling interactions in a two-component
Fermi gas, a large portion of this dissertation will also be dedicated to describing
the construction of a novel laser source for use in 6 Li atomic experiments. Traditionally, the primary source of the 671 nm laser light used for 6 Li spectroscopy was
generated using dye lasers. While demonstrating a great deal of versatility in the
wavelengths of light able to be produced using these lasers, dye lasers also have
several drawbacks. Since the gain medium of these lasers is a liquid jet stream,
small bubbles in the stream, pressure fluctuations, and even fluctuations in the
stream path can lead to large, noticeable inconsistencies in the repeatability of our
ultra-cold atomic gas experiments. In particular, it was noted that the shot to
shot fluctuations in the number of atoms varied by as much as 50% when using a
dye laser in our experiments [6].
Recently, the development of semiconductor tapered amplifier systems at 671
nm have improved on these instabilities, reducing our shot to shot fluctuations to
less than 10%. However, these tapered amplifiers do not come without problems
of their own. First and foremost, the available power from one of these chips is
currently limited to 500 mW. After spatially filtering the beam and sending it
through a pair of optical isolators to protect the chip from back scattering, the
amount of usable power is already reduced to 300 mW, requiring several amplifier
systems for our experiments. In addition, these tapered amplifiers also have a finite
lifetime, determined experimentally to be on the order of two years. Over time,
the power output from these chips declines even further, making the successful
completion of experiments very challenging.
It was to this end that we developed a solid state laser system utilizing a
Nd:YVO4 gain crystal. The 4 F3/2 →
4
I13/2 transition in Nd:YVO4 has a wave-
length of 1342 nm, double that of the 671 nm light needed for our 6 Li experiments.
Thus, the 671 nm light we require can be generated by frequency doubling the
1342 nm light with non-linear optics. Also, in constructing this laser, we have
developed a novel scheme of “self-injection locking” whereby a small portion of
the output power of the ring laser cavity is re-injected to drive unidirectionality
and single frequency operation. Also during the course of these experiments, we
8
have measured the Verdet constant of undoped YAG in the near infrared. Doing so enabled us to construct a home made Faraday rotator for inclusion in our
laser cavity to compare the output of our laser using this self-injection technique
to a more traditional method for driving unidirectionality. In the end, we have
demonstrated power outputs on the order of 3 Watts at 1342 nm. Even for a modest efficiency of 50%, frequency doubling this light should provide us with a high
powered stable alternative to dye lasers and tapered amplifiers for application to
our 6 Li experiments. Additionally, this power may also enable us to create a deep
optical lattice for Raman cooling of atoms captured directly from a MOT, similar
to the technique described in references [45] and [46].
1.5
Outline
This dissertation is laid out as follows. Chapter 2 will provide the basic background
information regarding the energy level structure of 6 Li and how this structure is
changed by the presence of an applied magnetic field. In addition, a review of
s-wave scattering theory and scattering resonances as they apply to 6 Li will be
provided. Finally, we will report on a simplified calculation for determining the
scattering length of atoms located in states | 1 and | 2 as well as in states | 1 and
| 5.
Chapter 3 will begin with a discussion of atomic transitions while approximating the atom as a two level system. A description of light shifts due to the ac Stark
effect will follow then some background on magnetic dipole transitions. Finally, a
brief discussion of two photon Raman transitions will be presented.
Chapter 4 describes our studies of the Verdet constant of undoped YAG in the
near infrared. By measuring the magnetic field of a right hollow cylindrical magnet
with an axial bore hole, we can determine the total applied magnetic field to an
undoped YAG rod located along its hollow axis. By measuring the rotation of the
plane of polarization for a probe laser beam sent through this magnet housing, we
can extract values of the Verdet constant. We report measurements of the Verdet
constant for wavelengths ranging from 1300 to 1350 nm as well as at 1064 nm.
From these measurements of the dispersive nature of the Verdet constat, we can
extract other properties of the YAG crystal, such as its electron band gap energy,
9
and compare these values to previous measurements.
Chapters 5 and 6 are devoted to describing the construction and testing of our
1342 nm self-injected ring laser cavity. First, the details pertinent to the design,
layout, and physical properties of the ring laser cavity will be described followed by
an extensive investigation into various measurements, including the gain profile, the
power as a function of output coupler, the transverse mode profile, the longitudinal
mode profile, and the spectral linewidth of the beam. In addition, we introduce the
novel idea of self-injection. We then compare our measurements for the self-injected
laser to those using more traditional methods to drive unidirectional operation,
such as the inclusion of a home built Faraday rotator utilizing the Verdet effect
measured in chapter 4. Finally, we demonstrate frequency tuning of this laser
by using an etalon located both inside the laser cavity as well as in the external
self-injection loop region.
In Chapter 7, we describe our experimental system for the creation of a twocomponent Fermi gas. Included in this discussion is an in depth look at our
ultra-high vacuum system used to conduct these investigations. Additional details
pertaining to the tapered amplifier laser systems used to generate the 671 nm
laser light necessary for these experiments will be provided as well as information
about the magnetic field coils used for both trapping our atoms in a magnetooptical trap as well as manipulating the s-wave scattering length via a Feshbach
resonance. Finally, a brief discussion of using absorption imaging to measure
various parameters of our atomic sample will be presented.
In Chapter 8, we will report measurements of the rapid control of interactions
in a two-component Fermi gas. We will begin by describing the process of phase
locking the output of two lasers for use as Raman beams to drive transitions within
our atoms. We also characterize the degree to which these lasers are locked by looking at the phase noise from the beat note signal of these two beams. Additionally,
we will report on our investigations driving | 2 → | 5 state transitions using microwave fields and two-photon Raman pulses for a non-interacting Fermi gas. We
will also provide information about the progress made in rapidly controlling these
interactions near the broad Feshbach resonance in 6 Li.
Next, Chapter 9 will summarize all of the research reported in this dissertation
as well as provide direction for future experiments in our lab group. Finally,
10
Appendix A will provide the Mathematica code that we used to calculate the
s-wave scattering lengths for atoms in different magnetic hyperfine states.
Chapter
2
Fermi Gases
For our experiments with ultra cold Fermi gasses, we are primarily concerned
with two-state mixtures of 6 Li in the two lowest energy hyperfine electronic spin
states. Because we are dealing with atoms whose temperature scales are below
the centrifugal barrier required for higher partial wave contributions to scattering,
the interactions between two fermions can be best described by s-wave collisions.
While the Pauli exclusion principle prevents identical fermions in the same spin
state from interacting, having atoms in two different spin states allows for these
collisions to occur. These s-wave collisions can be parameterized by a single value—
the s-wave scattering length. For certain two-state mixtures, the s-wave scattering
length can be tuned via a broad Feshbach resonance that allows us to tune the
interaction strength of the gas from strongly attractive to strongly repulsive by
simply adjusting an applied magnetic field. In this way, we can prepare a twostate mixture of atoms with arbitrarily strong interactions. On the other hand,
certain two-state mixtures have small s-wave scattering lengths and interact only
weakly. By rapidly changing the internal state for one of our trapped states to
a third non-interacting state, we can rapidly reduce the interaction strength by
several orders of magnitude.
This chapter describes the properties of fermionic 6 Li atoms used in our experiment. Section 2.1 will describe the atomic hyperfine structure of 6 Li and the
Zeeman shift of the energy level of those states in the presence of an applied magnetic field. Section 2.2 will introduce the idea of s-wave scattering for two-atom
collisions in our gas while section 2.3 describes how we can tune this s-wave scatter-
12
ing length using Feshbach resonances. Finally, section 2.4 will describe our method
for estimating the s-wave scattering lengths for other internal spin states.
2.1
Properties of 6Li
The fermionic 6 Li isotope studied in our ultra-cold atomic physics lab shares similar
properties with all other alkali metal atoms in that each of these atoms has a
single unpaired valence electron, greatly simplifying the energy level structure of
the atom. Since the majority of our studies will involve the ground 22 S and excited
22 P electron states, this section will look at the fine and hyperfine splitting of these
states in the presence of a magnetic field.
The fine structure of the atomic energy levels originates from the spin-orbit
coupling of the valence electron and the electric field of the nucleus. The angular
momentum J of the atom is written as the sum of the spin angular momentum of
the electron S and the angular momentum L. For 6 Li, the ground state has values
of S = 1/2 and L = 0, leading to a total angular momentum J = 1/2. For the
excited states, L = 1, yielding two J values (1/2 and 3/2) corresponding to the
fine structure splitting of the D line into the D1 and D2 spectroscopic lines.
Additional splitting of these lines emerges when one considers the interaction
of the total angular momentum J with the angular momentum of the nucleus I.
6
Li has a nuclear angular momentum value I = 1, causing hyperfine splitting in
terms of the total atomic angular momentum F, where F = J + I. The energy
level diagram showing the fine and hyperfine structure of 6 Li is shown in figure
2.1. The values for the ground and excited energy levels were reported in reference
[47]. Additional information about the atomic structure of lithium can be found
in reference [48].
For the majority of our experiments, the energy levels of the atoms will be
shifted due to the Zeeman interaction with a magnetic field. To calculate the effects
of the Zeeman interaction on the energy level structure, we need to first examine
the total Hamiltonian for this system. At small magnetic fields, the Zeeman shift
of the energy levels is no longer small compared to the hyperfine splitting for the
22 S ground state and 22 P excited states of lithium. Because of this, F is no longer
a good quantum number and the magnetic and hyperfine interactions must be
13
F = 1/2
F = 3/2
2 2P3/2
4.4 MHz
F = 5/2
10.056 GHz
D2 = 670.977 nm
F = 3/2
2
2 P1/2
26.1 MHz
F = 1/2
D1 = 670.992 nm
F = 3/2
2 2S1/2
228.2 MHz
F = 1/2
Figure 2.1. The hyperfine energy level structure for 6 Li showing the 22 S ground state
and the two 22 P excited states.
looked at in the |J mJ , I mI basis. The total interaction Hamiltonian is given by
[49]
Hint = Hhf + HZE
(2.1)
where Hhf is the hyperfine interaction Hamiltonian given by
Hhf = Aj I · J +
BJ [3(I · J)2 + 3/2(I · J) − I(I + 1)J(J + 1)]
2I(2I − 1)J(2J − 1)
(2.2)
and Zeeman interaction energy Hamiltonian, HZE is
HZE = −μB (gJ J + gI I) · B
(2.3)
where AJ and BJ are the magnetic dipole and electric quadrupole hyperfine constants for an atom in state J and μB is the Bohr magneton. Values for these
parameters as well as the relevant g-factors for the 22 S ground state and 22 P ex-
14
Property
Total Electronic g-Factor
Nuclear Spin g-Factor
Magnetic Dipole Constant (MHz)
Electric Quadrupole Constant (MHz)
Symbol
gJ (22 S1/2 )
gJ (22 P1/2 )
gJ (22 P3/2 )
gI
A22 S1/2 /h
A22 P1/2 /h
A22 P3/2 /h
B22 P3/2 /h
Value
-2.0023010
-0.6668
-1.335
0.0004476540
152.1368407
17.375
-1.155
-0.10
Table 2.1. g-factors and Hyperfine constants for the 22 S and 22 P electronic states of
6 Li.
cited states of 6 Li are given in table 2.1.
For atoms in the ground state, the angular momentum quantum number L = 0.
As a consequence of this, the angular wavefunction is spherically symmetric and the
electric quadrupole constant is zero, allowing the Hamiltonian to be analytically
solved through diagonalization into six eigenstates labeled |1 through |6 from
least to most energetic [50]. These product states are
|1 = sin Θ+ |1/2 0 − cos Θ+ |−1/2 1
|2 = sin Θ− |1/2 − 1 − cos Θ− |−1/2 0
|3 = |−1/2 − 1
|4 = cos Θ− |1/2 − 1 + sin Θ− |−1/2 0
|5 = cos Θ+ |1/2 0 + sin Θ+ |−1/2 1
|6 = |1/2 1
(2.4)
where the coefficients are given by
1
sin Θ± = 1 + (Z ± + R± )2 /2
cos Θ± = 1 − sin2 Θ±
Z± =
μB
(−gJ (22 S1/2 ) + gI ) ±
A22 S1/2
R± = (z ± )2 + 2
1
2
(2.5)
15
6
5
4
Energy (Joules)
4. μ10 -25
2. μ10 -25
100
200
300
400
500
-2. μ10 -25
-4. μ10 -25
3
2
1
Magnetic Field (Gauss)
Figure 2.2. The magnetic field dependence of the energy for the F = 1/2 and F =
3/2 ground states of 6 Li. As the applied field becomes stronger, the atoms enter the
Paschen-Back regime, whereby the atom becomes electron spin polarized and the three
way splitting arises due to difference in the nuclear spin of the atoms.
The eigenenergies of the ground state levels |1 through |6 as a function of
magnetic field are shown in figure 2.2. As we apply a magnetic field, the degeneracy
of the F = 1/2 and F = 3/2 states are quickly lifted and the six states become
resolved. For field values where the energy μB B is greater than the magnetic
dipole constant A22 S1/2 , the atoms enter the Paschen-Back reginme whereby the
product states become good approximations of the eigenstates of the system. In
this regime, the six states become electron spin polarized with |1 through |3
representing the electron spin down (-1/2) states and |1 through |3 representing
the electron spin up (+1/2) states. The three states in each group thus correspond
to different nuclear spins of the atom.
2.2
s-Wave Scattering Theory
In this section, we take a theoretical look at the scattering of two neutral atoms
interacting via a short ranged molecular potential. The general problem of scattering has been covered in many quantum mechanics textbooks (e.g. [51, 52]) as
well as in previous dissertations within our lab group [6, 53]. The treatment here
will primarily follow that of reference [6].
In the center of mass frame of the two colliding particles, the problem reduces
16
to the scattering of a single particle off the spherically symmetric potential V(r).
For large distances, we can approximate V(r) by the van der Waals potential for
neutral atoms having a J = 1/2 ground state. This van der Waals potential falls
off approximately as
V (r) −C6
r6
(2.6)
where C6 is the van der Waals coefficient. At shorter distances, as the atoms
begin to approach one another, they begin to experience a strong repulsion as
their electrons clouds begin to interact with one another, ultimately becoming
infinitely repulsive as r → 0. The potential well created by these two interacting
neutral particles can support many bound states (a diatomic molecule) and can
be approximated, for states near the potential minimum, by a Morse potential[54].
For low energy collisions, the characteristic length of the interaction potential is
defined by the van der Waals length scale
vdW =
2M C6
2
1/4
(2.7)
where is Planck’s constant divided by 2π and M is the reduced mass of the
two colliding particles. For ultra-cold atoms with high polarizabilities, such as
6
Li, vdW can be much larger than the size of the atom— on the order of several
nanometers.
One way the effect of a scattering event on two particles undergoing a collision
can be interpreted is as a phase shift on the atomic wave function of the particle. To
understand this, we first represent an incident particle as a plane wave traveling
in the +ẑ direction with momentum k. After scattering, the wave function of
the scattered particle in the asymptotic limit will consist of a plane wave plus a
spherical wave, and can be represented as
Ψk = eikz + f (θ, φ)
eikr
r
(2.8)
where the effect of the scattering potential V(r) is contained within the scattering
amplitude f(θ,φ). From this scattering amplitude, the differential scattering cross
17
section can be determined by
dσ
= |f (θ, φ)|2 .
dΩ
(2.9)
Since the scattering potential V(r) is a central potential, we can express the scattering process in terms of an expansion of partial waves. Furthermore, since the
potential has no angular dependence, depending only on r, the scattering amplitude is only a function of θ, where θ is the angle between the incident +ẑ direction
and the direction of the outgoing wave. As a result, we can recast the scattering
amplitude as a series expansion of Legendre polynomials
f (θ) =
∞
(2l + 1)
l=0
e2iδl −1
Pl (cos θ).
2ik
(2.10)
where l represents the value for the orbital angular momentum of the partial waves
in our expansion and δl is the phase shift associated with that partial wave function.
For ultra-cold 6 Li atoms, the long range centrifugal barrier associated with the van
der Waals scattering potential, of the form 2 l(l + 1)/(2mr2 ), has an associated
temperature of 6.5 mK [55]. As the temperature of the 6 Li atoms in our experiment
will almost always be lower than this value, we only need to consider s-wave (l = 0)
collision terms in our theoretical model. With this assumption, equation 2.10
becomes
f = eiδ0
sin(δ0 )
.
k
(2.11)
and the total cross section σ simplifies to
σ=
2
dΩ |f | =
2
sinδ0 2
= 4π sin δ0 .
dΩ k k2
(2.12)
In the zero energy limit, s-wave collisions are characterized by an s-wave scattering length a. It can be shown that tan(δ0 )∝ -ka as k→ 0 [56]. Therefore, for
the low-energy atoms in our experiment, we can define the scattering length a as
tanδ0 (k)
.
k→0
k
a ≡ − lim
(2.13)
18
Using this identity in equation 2.11 for the scattering amplitude yields
sinδ0 (k)
= −a
k→0
k
lim f = lim
k→0
(2.14)
and the total cross section for the collision becomes
σ = 4πa2 .
(2.15)
The physical interpretation of the s-wave scattering length a is simply the
distance between the center of the scattering potential and the location on the
r-axis where the asymptotic wave function crosses zero. It is important to note
that the sign of the s-wave scattering length relates important information about
the nature of the two-particle interaction. For attractive potentials, the scattering
length is negative, whereas for repulsive potentials the scattering length will be
positive.
2.3
Scattering Resonances
In section 2.2 we showed how the s-wave scattering length can be used to describe
the interaction between two scattering particles whose temperatures are colder
than the centrifugal barrier required to freeze out higher partial scattering waves.
Now, we want to take a look at manipulating the value of that s-wave scattering
length by utilizing Feshbach resonances to enhance the value of that scattering
length.
The enhancement of the two-body scattering length was first studied in the
context of nuclear physics of H. Feshbach [57] and then atomic physics by U. Fano
[58]. The basic premise for this resonance can be seen in figure 2.3. Two atoms
approach each other in the triplet molecular potential (entrance channel) that has
a non-zero magnetic moment and can be tuned in a magnetic field. The singlet
state (closed channel) has a zero magnetic moment and is at a higher energy that
exceeds the available kinetic energy of the atoms. The atoms are energetically
forbidden from making the transition to the closed channel. Note that the singlet
state molecular potential becomes energetically accessible for small internuclear
19
Closed Channel (Singlet)
Incident
Energy
Δμ B
0
Entrance Channel (Triplet)
0
Figure 2.3. A Feshbach resonance occurs when the least bound state of the close channel
molecular potential is tuned via a magnetic field to be resonant with the collision energy
of the particles in the entrance channel.
separations. However, because the two scattering states have different magnetic
moments (Δμ), the relative energy of the entrance channel with respect to the
closed channel can be tuned by applying a DC magnetic field. As the least bound
molecular state is tuned such that the energy of the molecular state is just above
the energy of the entrance channel, the s-wave scattering length becomes very large
and negative, diverging when the two states are resonant [59]. Similarly, as the
bound state of the closed channel is just below the energy of the entrance channel,
the s-wave scattering length becomes very large and positive, again diverging when
the two states are resonant. In this way, the scattering length of the two particles
can be tuned by simply applying a DC magnetic field near one of these scattering
resonances (see figure 2.4).
The variation of the s-wave scattering length, a, as a function of magnetic field
B can be described by
a(B) = abg
Δ
1−
B − B0
(2.16)
where abg is the background scattering length for a magnetic field far from the
resonance field B0 and Δ describes the width of the resonance [60]. It is important
to note that Δ can have both positive or negative values.
20
10000
)0a( htgneL gnirettacS
5000
0
-5000
-10000
0
500
1000
1500
2000
Magnetic Field (Gauss)
Figure 2.4. The s-wave scattering length as a function of applied magnetic field for
atoms in the two lowest hyperfine ground states of 6 Li [1]. Note the broad Feshbach
resonance located at 834.1 G. By arbitrarily tuning the applied magnetic field around this
resonance, we can change the strength of the two-body interaction from being infinitely
repulsive to infinitely attractive.
As discussed in section 2.1, the application of a magnetic field to a 6 Li atom
splits the F=1/2 and F=3/2 ground states into six energy levels labeled from
lowest to highest energy. At sufficiently high magnetic fields, greater than 100
Gauss, the atoms enter the Paschen-Back regime, whereby they become electron
spin polarized with states 1-3 having electron spin down and 4-6 having electron
spin up. For the experiments outlined in this dissertation, we want to tune swave interaction strength for atoms in the lowest two spin states. Because of the
atomic properties of 6 Li, the background triplet s-wave scattering length has an
anomalously high value of abg = -2140 a0 where a0 ≈ 0.5Å is the Bohr radius [61].
Figure 2.4 shows the variation of the scattering length as a function of applied
field from a coupled channels calculation based on a precise measurement of the
interaction parameters [1]. As you can see, for states 1-2 there exists a very broad
Feshbach resonance located at a magnetic field B0 of 834.1 G having a width Δ of
-300 G [26]. It is this Feshbach resonance that we will use to tune the interaction
strength of our two-component Fermi gas.
21
2.4
Interactions in a | 1-| 5 Mixture
While the s-wave scattering length for a two-component |1-|5 mixture has not
been explicitly measured or calculated, these scattering lengths becomes important in our experiments as a means to rapidly switch off interactions in a twocomponent Fermi gas. Therefore, we would like to have an estimate of the scale
of the interactions. To do this, we perform a full coupled channels calculation of
s-wave scattering modeling the singlet and triplet scattering potentials as finite
square-well potentials.
The model, with the Mathematica code explicitly reproduced in appendix A,
approximates the singlet and triplet potentials for two body interaction as square
well potentials. The depths and widths of the square well potentials are chosen
to reproduce the known scattering lengths as and at for the singlet and triplet
potentials as well as the binding energy of the most weakly bound molecular state
of the singlet potential. The known values for the background scattering lengths,
as and at , are 45.5 Bohr and -2140 Bohr [61]. There are three parameters we
adjust in this model: the widths of both potentials (R) and the depths (Vt and Vs )
of the triplet and singlet potentials respectively. These parameters are chosen to
reproduce the singlet and triplet scattering lengths and the binding energy of the
ν = 38th vibrational bound state of the singlet molecular potential. In order to
more accurately match the observed location of the Feshbach resonances, we can
adjust the width of the square wells.
The model works as follows. We calculate the s-wave scattering phase shift
(and thereby the scattering length) by solving the coupled channels Schrodinger
equation for all s-wave scattering channels that can be coupled by the spherically
symmetric molecular potentials (i.e. those channels with the same z-projection of
the total spin angular momentum).
We will consider s-wave scattering of atoms in states |1 and |5. The zcomponent of the total spin angular momentum in this case is mF = +1. When
the atoms are asymptotically separated, the two-body spin wave function for the
incident wave is {1, 5} , an anti-symmetrized combination of states |1 and |5
−
defined above. In principle, the molecular potentials could couple this incident
pair of atoms in state {1, 5}− to any channel with a two-body spin wave function
22
having mF = +1. The two possibilities are {2, 6}− and {4, 6}− . However, since
these channels are energetically closed for the energies and fields of interest, the
asymptotic two-body spatial wave function in these channels decay exponentially.
Thus, the asymptotic two body wave function (for r → ∞) is
u(r) = e−ikr + S1,5 eikr {1, 5}− + S2,6 e−κ2,6 r {2, 6}− + S4,6 e−κ4,6 r {4, 6}−
where S1,5 , S2,6 , and S4,6 are the amplitudes for the scattered waves in each respective channel.
The two-body spin wave functions {1, 5}− , {2, 6}− , and {4, 6}− are rele-
vant when the atoms are asymptotically separated as they are eigenstates of the
hyperfine and Zeeman interaction Hamiltonians. However, when the atoms are in
close proximity such that the singlet and triplet potentials dominate over the hyperfine and Zeeman interactions, the relevant two-body spin wave functions should
instead be expressed in terms of the total electron spin S ≡ s1 + s2 and the total
nuclear spin I = i1 + i2 . Again, the two-body spin wave functions must be antisymmetric under exchange of all particle labels. Also, the two-body spin wave
functions of interest must have the same z-projection of the total spin angular momentum as the incoming spin state (i.e. mF = 1 for our example). The relevant
two-body spin wave functions in terms of electron spin singlet and electron spin
triplet states are (written in the |S, mS ; I, mI basis):
|0, 0; 2, 1 ,
|1, 0; 1, 1 ,
|1, 1; 1, 0 .
In the simplified coupled channels model we consider, we assume that the singlet/triplet spin states diagonalize the interaction Hamiltonian for r < R (i.e. the
singlet and triplet molecular potentials dominate) and the spin states {1, 5}− ,
{2, 6} , {4, 6} diagonalize the interaction Hamiltonian for r > R where we
−
−
assume the molecular interactions are zero.
The relative spatial wavefunction u(r) and its derivative must be continuous at
r = R. If we model the singlet and triplet molecular potentials as square wells of
radius R, the value of u and u at r = R are given by
u(R) = AS sin [k(R − as )]
and
u (R) = AS k cos [k(R − as )]
23
for particles in the spin singlet state and
u(R) = AT sin [k(R − at )]
and
u (R) = AT k cos [k(R − at )]
for particles in a spin triplet state. Here, as and at are the singlet and triplet
scattering lengths.
Putting all of this together, continuity of u and u at r = R requires
AS sin [k(R − as )] |0, 0; 2, 1 + AT,0 sin [k(R − at )] |1, 0; 1, 1
+AT,1 sin [k(R − as )] |1, 1; 1, 0
= e−ikR + S1,5 eikR {1, 5}− + S2,6 e−κ2,6 R {2, 6}−
+S4,6 e−κ4,6 R {4, 6}−
and
AS k cos [k(R − as )] |0, 0; 2, 1 + AT,0 k cos [k(R − at )] |1, 0; 1, 1
+AT,1 k cos [k(R − as )] |1, 1; 1, 0
= −ike−ikR + ikS1,5 eikR {1, 5}− − κ2,6 S2,6 e−κ2,6 R {2, 6}−
−κ4,6 S4,6 e−κ4,6 R {4, 6}− .
If we project out the states |0, 0; 2, 1, |1, 0; 1, 1, and |1, 1; 1, 0 in each of the
two equations above we obtain six equations which allow us to determine the six
unknowns: AS , AT,0 , AT,1 , S1,5 , S2,6 , and S4,6 in the limit k → 0. Knowing the
scattering amplitude S1,5 for the {1, 5}− channel allows us to determine the s
wave scattering phase shift and thereby the s-wave scattering length for {1, 5}
−
collisions.
Other than the six unknown quantities AS , AT,0 , AT,1 , S1,5 , S2,6 , and S4,6 , all
of the other variables in the equations above can be determined. The singlet and
triplet scattering lengths for 6 Li are known to be as = +45.5 a0 and at = −2140 a0 .
We choose the radius R of the square well potential to give the correct binding
energy for the most weakly bound vibrational state of the singlet potential. This
bound state in the singlet potential gives rise to the Feshbach resonances in 6 Li.
24
400
)0a( htgneL gnirettacS
200
0
-200
3
-400x10
810
820
830
840
850
860
Magnetic Field (Gauss)
Figure 2.5. The result of our coupled channel calculation for s-wave scattering between
atoms in states |1 and |2. To simplify the calculation, we model the singlet and triplet
molecular potentials as finite square well potentials. Despite this oversimplication, the
calculation reproduces the broad Feshbach resonance located at 834.1 G which can be
observed as a divergence in the s-wave scattering length. The location and width of this
resonance is in good agreement with a coupled channel calculation which uses accurate
singlet and triplet molecular potentials. The good agreement gives us confidence in our
coupled channels calculations using a simple model for the potentials.
The decay constants κ2,6 and κ4,6 are determined from the energies of the two
body spin states {2, 6}− and {4, 6}− relative to the energy of the two-body
spin state for the entrance channel ({1, 5}− ). When projecting out the states
|0, 0; 2, 1, |1, 0; 1, 1, and |1, 1; 1, 0 in the equations above, we compute Clebsch
Gordan coefficients such as 0, 0; 2, 1 {1, 5} .
−
Before looking at the s-wave scattering length as a function of magnetic field
for a |1-|5 mixture, we first want to verify the accuracy of this numerical approximation for the well known values of a |1-|2 interaction. By adjusting R to a
value of 33.2372 Bohr, we can see in figure 2.5 that the calculation predicts a broad
Feshbach resonance centered at a magnetic field of 834.1 gauss. Remarkably, in
addition to the broad Feshbach resonance, the calculation also predicts the existence of a narrow Feshbach resonance at 555.08 gauss (see figure 2.6). While the
location of this resonance is at a slightly higher magnetic field than what has been
25
400
)0a( htgneL gnirettacS
200
0
-200
-400
554.8
555.2
555.6
Magnetic Field (Gauss)
Figure 2.6. Our simplified coupled-channels model for scattering of atoms in states |1
and |2 also predict the presence of a narrow Feshbach resonance. While the position of
this resonance is slightly higher in magnetic field than what has been experimentally observed, this discrepancy could easily be explained by the different shape of our simplified
potentials.
experimentally observed (543.286 G) [53], again, it’s predicted presence provides
confidence in our simplified model.
Looking now at the model for the s-wave scattering length for atoms in a
|1-|5 mixture, the results (using the same parameters as those used to determine the |1-|2 scattering lengths above) are shown in figure 2.7. For values near
zero magnetic field, the scattering length is large and negative, dominated by the
anomalously large background scattering length for the triplet potential. As the
applied magnetic field increases, however, the scattering length increases as well,
approaching zero. For values around the 834.1 gauss Feshbach resonance seen for
a |1-|2 mixture, the |1-|5 scattering length is around -3 Bohr. Therefore, if we
were to create a |1-|2 mixture of atoms at a magnetic field near the broad Feshbach resonance where the scattering length is very large (hundreds of thousands of
Bohr), we could quickly “turn off” interactions between the two atoms by changing
the spin state from |2 → |5, thus reducing the s-wave scattering rate by several
orders of magnitude.
26
0
)0a( htgneL gnirettacS
-10
-20
-30
-40
-50
0
200
400
600
800
1000
Magnetic Field (Gauss)
Figure 2.7. The predicted s-wave scattering lengths as a function of magnetic field for a
|1-|5 mixture. With no Feshbach resonance, the s-wave scattering length increases to a
value of approximately -3 Bohr at a magnetic field corresponding to 834.1 gauss. In this
way, interactions of a two-state |1-|2 mixture at this field could be quickly suppressed
by a simple transfer from state |2 → |5
Chapter
3
The Interaction of Atoms with Light
In this chapter, we consider the effect of electromagnetic radiation on two and
three-level atoms in several different physical contexts relevant to this dissertation
work. We will discuss coherent population transfer and the AC Stark shift in a
two level atom and population transfer via a two-photon Raman transition in a
three-level atom. The Stark shift is relevant for trapping neutral atoms with laser
light and the magnetic dipole and Raman transitions are relevant for transferring
atoms between different hyperfine ground states in 6 Li. The approach taken will
be a semi-classical one, whereby the radiation will be treated using a classical
electric field while the atom itself will be treated quantum mechanically. While
the derivations of the atom-light interactions here will predominantly follow that
of reference [10], the topic itself has been covered many times, with reference [62]
in particular providing a more in depth discussion on the topic.
We will begin our discussion in section 3.1 where we derive from first principles equations describing the coherent evolution of the atom between two internal
quantum states. To do so, we make use of the approximation that the radiation
driving the transition within the atom is far detuned from all other energy levels
in the multi-level atomic system. In this way, we can simplify the problem by
assuming that the atom itself can be approximated as a two level system. As the
presence of electromagnetic radiation can perturb the energy levels of the atomic
system, in section 3.2 we take a theoretical look at the effect of this AC Stark shift,
or light shift, on the energy levels of the atom. The AC Stark shift from a focussed
laser beam can be useful for providing an essentially conservative trapping poten-
28
tial for neutral atoms provided the laser beam is detuned far from resonance. In
section 3.3 we describe the selection rules and dipole matrix elements for using rf
and microwaves to drive magnetic dipole transitions between these states. Finally,
in section 3.4 we look at the coherent evolution of atomic population between two
states using two coherent lasers whose energy difference is equal to that of the
difference between the two energy levels of the atom. This two photon Raman
transition will utilize an intermediate third level that will become virtually populated as the transition occurs. These Raman transitions will become critical to our
experiments involving the rapid control of interactions in a two-component Fermi
gas.
3.1
The Two Level Atom
As mentioned above, in this section we will derive from first principles the coherent
evolution of the internal quantum state of an atom as it interacts with electromagnetic radiation. While the internal structure of the atom can be quite complicated,
we begin our discussion with a simplification. Instead of concerning ourselves with
every possible energy level within the atomic system, we will instead reduce the
atom to a two level system consisting of a ground state | g and an excited state
| e as shown in figure 3.1. While these two energy levels can, in principle, be
any two energy levels within the atomic system where transitions are allowed, we
will primarily be concerned with transitions between different hyperfine Zeeman
sub-levels of our 6 Li gas, for example between | 1 and | 2 (see figure 2.2).
We begin the derivation from the time dependent Schrodinger equation
i
∂Ψ
= HΨ
∂t
(3.1)
where the Hamiltonian, H, consists of two components
H = H0 + HI (t).
(3.2)
The eigenfunctions and eigenvalues of the time independent Schrodinger equation
describe the unperturbed energy levels and wave functions for the atom as discussed
29
δ
e
ω
ω0
g
Figure 3.1. Energy level diagram for a simplified atomic system. A two level atom with
ground state | g and excited state | e interacts with an oscillating electric field having
a frequency of ω. This beam is detuned by an amount δ from the transition frequency
ω0 corresponding to the energy difference between the two atomic states.
in section 2.1. The time dependent interaction Hamiltonian, on the other hand,
will describe the interaction of coherent radiation with an oscillating electric field
which perturbs the energy levels.
To solve the time dependent part of the Schrodinger equation, we assume a
solution of the form
Ψn (r, t) = ψn (r)e−iEn t/
(3.3)
for a wave function with energy En , where we have separated out the time independent wave function ψn . For our two level system, the spatial wave functions
satisfy
H0 ψg (r) = Eg ψg (r)
H0 ψe (r) = Ee ψe (r).
(3.4)
It is important to note that these solutions do not satisfy the time dependent
Schrodinger equation as written in equations 3.1 and 3.2. The complete solution
to the full Hamiltonian can be expressed as
Ψ(r, t) = cg (t)ψg (r)e−iEg t/ + ce (t)ψe (r)e−iEe t/
(3.5)
where cg (t) and cge (t) are the probability amplitudes of the atom being in the
30
ground and excited states respectively. Rewriting equation 3.5 in Dirac notation,
where cg (t) is shortened to cg and ωg =Eg /, yields
Ψ(r, t) = cg | g e−iωg t + ce | e e−iωe t .
(3.6)
For the system to be properly normalized, the two time dependent probability
amplitudes cg and ce must obey the relation
| cg | 2 + | ce |
2
= 1.
(3.7)
We now look at the effect of an oscillating electric field E = E0 cos(ωt) as a
perturbation described by the interaction part of the Hamiltonian
HI = er · E0 cos (ωt) .
(3.8)
This interaction Hamiltonian corresponds to the energy of an electric dipole -er
in the presence of an electric field. Here, r is the position of the electron with
respect to the center of mass of the atom. The effect of this interaction will be to
mix the two states with energies Eg and Ee . In our atomic system, the wavelength
of the radiation will always be much larger than the size of the atom. That is
to say, λ a0 where a0 is the bohr radius. Because of this, we can make the
approximation that the amplitude of the electric field is nearly uniform over the
atomic wave function. This approximation is known as the dipole approximation.
By substituting equation 3.6 into the time dependent Schrodinger equation 3.1,
we find two coupled differential equations
ic˙g = Ωcos (ωt) e−iω0 t ce
(3.9)
ic˙e = Ω∗ cos (ωt) e−iω0 t cg
(3.10)
and
where we have defined ω0 = (Ee − Eg )/ and the Rabi frequency Ω is defined by
g| er · E0 | e
e
Ω=
=
φ∗g (r)r · E0 φe (r)d3 r.
(3.11)
31
We can take the amplitude | E0 | outside the integral in equation 3.11. Therefore,
for linearly polarized light along the x-axis, we obtain
Ω=
eXge |E0 |2
(3.12)
where
Xge = g | x| e .
(3.13)
We now turn our attention back to solving the coupled system of differential
equations 3.9 and 3.10. These two equations can be rewritten as
and
Ω
ic˙g = ce ei(ω−ω0 )t + e−i(ω+ω0 )t
2
(3.14)
Ω∗
ic˙e = cg ei(ω+ω0 )t + e−i(ω−ω0 )t
.
2
(3.15)
It is at this point that we can make another approximation. For our experiments,
the frequency of the radiation is close to the energy level resonance of the atoms.
Because of this, the magnitude of the detuning is small such that | ω0 − ω| ω0 . Therefore, in each of these two equations, the term containing (ω + ω0 )t
oscillates much faster than the term (ω − ω0 )t and subsequently much faster than
any interaction time in the system. As a result, we can make the rotating wave
approximation, whereby each term in the differential equations containing this
summation term goes to zero.
Taking this approximation, our two differential equations can now be written
as
ic˙g = ce eiδt
Ω
2
(3.16)
ic˙e = cg e−iδt
Ω∗
2
(3.17)
and
where we have defined δ = (ω − ω0 ). Combining these two equations yields a
second order differential equation of the form
2
d 2 ce
dce Ω + iδ
+ ce = 0
dt2
dt
2
(3.18)
32
For the initial condition that at time t = 0 the entire population is in the ground
state, the solution of this differential equation gives the time dependent probability
of the atoms being in the excited state as
| ce (t)|
2
Ω2
= 2 sin2
Ω
Ω t
2
(3.19)
where we have defined the generalized Rabi frequency Ω as
Ω2 = Ω2 + δ 2 .
(3.20)
On resonance, when δ = 0 and Ω = Ω, the population evolves as
| ce (t)|
2
= sin
2
Ωt
2
.
(3.21)
The interpretation of this result is relatively straightforward. In the presence of
an oscillating electric field, ignoring spontaneous emission, the atomic population
will coherently oscillate back and forth between the ground and excited states in a
sinusoidal fashion. When the electric field is on resonance, such that the detuning
of the laser light is zero with respect to the difference in energy levels, it becomes
possible to have complete transfer of the atomic population from the ground to
excited state and back. Additionally, this treatment introduces the idea of a π
pulse. On resonance, if the product of the Rabi frequency and pulse length is an
odd-integer multiple of π then the entire population will be transferred from the
ground to the excited state. Similarly, if the product of the Rabi frequency and
pulse length is an even-integer multiple of π then the population will remain in
it’s initial state. In this way, it becomes possible to transfer any arbitrary amount
of population from one state to the other simply by selecting an appropriate pulse
length for the coherent radiation beam.
3.2
The AC Stark Effect
In addition to changing the state populations, the presence of an oscillating electromagnetic field also causes a shift in the energy level eigenvalues of the atom
33
[8]. This shift in energy levels is known as the AC Stark effect. To gain theoretical insight into this effect, we begin by redefining the values for the probability
amplitudes using new variables
c˜g = cg e−iδt/2
(3.22)
c˜e = ce eiδt/2 .
(3.23)
and
Differentiating 3.22 with respect to time yields
iδ
c˜˙g = c˙g e−iδt/2 − cg e−iδt/2 .
2
(3.24)
Now, if we multiply this equation by i and make use of equations 3.16, 3.17, 3.22,
and 3.23, we find the equations reduced a pair of of first order differential equations,
namely
ic˜˙g =
1
(δ c˜g + Ωc˜e )
2
(3.25)
ic˜˙e =
1
(Ωc˜g − δ c˜e ) .
2
(3.26)
and
To solve this system of first order differential equations, we can rewrite them
in in matrix form as
i
d
dt
c˜g
=
c˜e
δ
2
Ω
2
Ω
2
− 2δ
c˜g
c˜e
.
(3.27)
This matrix of equations will have solutions of the form
c˜g
c˜e
=
a
b
e−iλt
(3.28)
where λ is an eigenvalue of the system. To solve for the eigenvalues, we need to
set the determinant of the matrix equal to zero, namely
δ
2
−λ
Ω
2
2 2
δ
Ω
2
=
λ
−
−
= 0.
2
2
− 2δ − λ Ω
2
(3.29)
34
Light
Shift
δ
Figure 3.2. The shift of the energy levels of a two level atomic system due to the presence of an oscillating electromagnetic field. This AC Stark effect results in a resonance
frequency shift to higher frequencies when the detuning of the beam is negative and
lower frequencies when the detuning is positive.
From this equation, one can easily see that the eigenvalues for the system are
λ = ±(δ 2 + Ω2 )1/2 /2. In the absence of the perturbing radiation, that is, when
Ω = 0, the unperturbed eigenvalues are simply λ = ±δ/2, which corresponds to
two energy levels spaced by an energy δ as shown in figure 3.2. These two states
correspond to a ground state having energy Eg and an excited state having energy
Eg + ω, corresponding to the ground state plus a photon of the radiation field.
This picture of the atom plus a photon is referred to as a dressed atom [62].
For the case when the the frequency detuning of the laser light is quite large
with respect to the Rabi frequency, the eigenvalues become
λ±
δ Ω2
+
2 4δ
.
(3.30)
That is to say, in the presence of the perturbing radiation, the energy levels experience a light shift of
Δωlight =
Ω2
.
4δ
(3.31)
It is important to note that this light shift equation is valid for both positive and
negative values of detunings, as this effect will be utilized in our experiments to
trap atoms at the intersection of two far red-detuned laser beams.
35
3.3
Magnetic Dipole Transitions
The experiments described in this dissertation primarily make use of the hyperfine
states within the ground 2 S1/2 manifold of 6 Li (see figure 2.2). Because each of
these six states all lie within the same l = 0 S-level, transitions between them
do not change the orbital angular momentum of the atom’s electrons. Therefore,
in order to conserve angular momentum, transitions between these levels cannot
be driven by an electric dipole transition, as described in section 3.1. To avoid
violating these selection rules and still conserve angular momentum, we instead
employ the use of magnetic dipole transitions.
Because magnetic dipole transitions only change the magnetic quantum number
of the system, they are extremely useful for driving transitions between atomic
energy levels when Δl = ΔL = 0. The transition matrix element between hyperfine
levels is given by
μeg ∝ e| μ · B | g
(3.32)
where μ is the sum of the magnetic dipole moment of the electron and the nucleus,
defined as
μ = gS μB S + gI μB I
(3.33)
where gS and gI are the gyromagnetic ratios of the electron and nucleus, μB is the
Bohr magneton, and S and I are the electron and nuclear spin. Also B = B0 cos ωt
is the oscillating magnetic field. For our 6 Li system in a magnetic field, transitions
between states | 1, | 2, and | 3 or | 4, | 5, and | 6 can be driven using rf fields.
However, transitions requiring an electron spin flip (say from state | 2 to | 5)
require a higher energy and microwave fields must be employed.
Finally, it’s of interest to note that the size of the magnetic dipole matrix
element is much smaller than that of an electric dipole matrix element. This is
due to the fact that the ratio of the magnetic field to the electric field, | B| / | E|,
equals 1/c for an electromagnetic wave. Therefore, magnetic dipole transitions will
be much slower than electric dipole transitions for allowed transitions within an
atom.
36
Δ
i
L2
L1
δ
e
g
Figure 3.3. The energy level diagram for a two photon Raman transition. Laser L1
drive the atom from the ground state to a virtually excited intermediate state i. Laser
L2 then drives the atom from state i back to the desired excited state. It is important
to note that the population in state i is virtual, and losses due to spontaneous emission
are not allowed. By employing this two photon technique, one can drive state changing
transitions where they would otherwise be forbidden, such as when Δl = 0.
3.4
Raman Transitions
In section 3.3 it was noted that electric dipole transitions were forbidden for the
hyperfine state changing transitions in the electronic ground state that we are
primarily interested in for our experiments. Here we examine another possibility
for making these transitions, namely a two photon Raman transition. Two photon
Raman transitions use two laser beams of frequency ωL1 and ωL2 to drive transitions
from state | g to state | e by way of an intermediate state | i as demonstrated
in figure 3.3. While an intermediate level i is used to help aid in the transition,
it’s important to note that at no time is there any real excitation into this virtual
level.
The mathematics behind the two photon Raman process is very similar to that
derived in section 3.1 with the following differences. Since we are now dealing with
a two-component electric field of the form E = EL1 cos (ωL1 t) + EL2 cos (ωL2 t), the
detuning δ of the transition is now defined as
δ = (ωL1 − ωL2 ) − (ωe − ωg ) .
(3.34)
37
Also of importance is the detuning of laser L1 with respect to the intermediate
level i. This detuning Δ is defined as
Δ = ωg + ωL1 − ωi
(3.35)
as shown in figure 3.3.
Also similar to the two level treatment, we can define an effective Rabi frequency for the two photon transition. It should come as no surprise that the
effective Rabi frequency Ωef f has contributing components from the electric fields
of each of the two lasers driving the transition, namely
Ωeff =
Ωei Ωig
e | er · EL2 | i i | er · EL1 | g
=
.
2Δ
2 (ωi − ωg − ωL1 )
(3.36)
In this way, it can be shown that the excited state population also oscillates in time
as it did for electric dipole transitions, except now using Ωef f instead of Ω. On
resonance, this means that the excited state population evolves via the functional
form
2
| ce (t)| = sin
2
Ωeff t
2
.
(3.37)
It’s also important to note that the evolution from state | g to state | e is
coherent in nature and the idea of a π pulse driving all of the population from state
| g to state | e is extended to this treatment as well. In fact, the only requirement
for using two laser frequencies to drive the transition is that the two Raman beams
must be phase coherent with respect to one another. This can be accomplished
experimentally by using the same laser and an acousto-optic modulator to generate
the two beams for difference frequencies in the rf, or by phase locking two different
lasers when the difference frequency is larger.
Finally, because this scheme utilizes electric dipole transitions instead of magnetic dipole transitions, the speed with which one can make state changing transitions is greatly increased. In this way, we can rapidly change the internal state of
the atomic sample, and thus rapidly changing the interaction strength when the
s-wave scattering length has been tuned near a Feshbach resonance.
Chapter
4
The Verdet Constant of Y3Al5O12
The rotation of the polarization of light by a medium in the presence of a magnetic field was first observed by Faraday in 1845 while examining the effect of
magnetic fields on bulk glasses. Using these experiments as a basis for his own
investigation, Verdet conducted a series of experiments to systematically quantify
the dependence of the rotation angle on the various parameters involved. As a
result of these pioneering works on the interplay of light and magnetic fields, the
Faraday effect has since been utilized to construct numerous types of optical devices. At the heart of each of these devices is a crystal characterized by the Verdet
constant— a measure of polarization rotation of transmitted light for an applied
magnetic field over the length of the crystal. Here we describe a measurement
of the Verdet constant of undoped Y3 Al5 O12 (YAG) in the near infra-red. While
previous measurements have been made for the Verdet constant of undoped YAG
in the visible spectrum [63, 64], our measurements represent, to the best of our
knowledge, the first to be made for light at longer wavelengths.
Our motivation for this experiment stemmed from our desire to construct a
home built Faraday rotator which, when inserted into a 1342 nm ring laser cavity,
drives unidirectional operation of the laser. For ring laser cavity configurations,
two stable modes of operation coexist, corresponding to the bidirectional propagation of traveling waves inside the cavity. This bidirectional operation is not
always desirable as the gain from the lasing medium is shared, dividing the total
available power among the two directions. To force unidirectional operation (and
subsequently yield nearly twice the power output for a given beam) one can in-
39
sert an optical isolator into the cavity. However, for some laser wavelengths and
intracavity intensities commercial isolators do not exist. One solution is to use
an optically active crystal combined with a half wave plate to form a homemade
Faraday rotator. For the proper orientation, this system rotates the polarization
of circulating light for only one direction of propagation inside the ring laser cavity. As gain media and intracavity losses can be polarization dependent [65], this
rotation is sufficient to break the symmetry of the system and drive unidirectional
laser operation.
This chapter highlights these measurements of the Verdet constant for undoped
YAG as well as associated parameters that can be extracted from these measurements. Section 4.1 frames the theoretical groundwork for the phenomenon responsible for this effect. Section 4.2 provides background information concerning the
bulk properties of YAG crystals. The experimental setup for measuring the Verdet
constant is laid out in section 4.3 with section 4.4 describing our measurements of
the magnetic field of a hollow right cylindrical magnet. Finally, the results of the
measurements are presented in section 4.5, followed by a concluding section 4.6.
4.1
Verdet Constant Theory
As mentioned above, when linearly polarized light is incident on an optically active
medium in the presence of a magnetic field, the polarization will rotate. This
rotation by an angle dφ can be described per the equation
dφ = V Bz Lc
(4.1)
where Bz is the total applied magnetic field along the propagation axis of a crystal
medium of length Lc , and V is the Verdet constant of the material. The origin of
this rotation stems from the fact that the applied magnetic field causes a Zeeman
splitting of the involved energy levels within the crystal. Linearly polarized light
can be decomposed into components of both right and left circular polarization.
Because of the energy level splitting associated with the application of a magnetic
field, each of these components will experience a slightly different index of refraction within the material. As a result of this difference in index of refraction, the
40
phase relationship between these two components will change as they propagate
through the crystal, corresponding to a rotation of the polarization vector.
The Verdet constant for a single oscillator inside of an optically active medium
is given by
nV = K
ω2
(ω02 − ω 2 )2
(4.2)
where n is the index of refraction of the material, ω is the frequency of the photon
and ω0 is the resonant frequency of the oscillator [66]. It is important to note that
the Verdet constant is dispersive in nature (dependent on the wavelength of the
incident photon) not only for ω, but for the index of refraction n as well. Summing
up the states of all allowed direct transitions in the conduction and valence bands
[67] yields the relation
nV
K
=
x
1
1
√
−√
1−x
1+x
√
√
4
− 2− 1−x− 1+x
x
(4.3)
where we define x ≡ E/Eg with E = ω being the photon energy, Eg is the band
gap energy of the crystal, and K is a dimensionless parameter unique to each
material. It should be noted that for energies larger than the band gap, the index
of refraction becomes imaginary and the material becomes opaque. Therefore, by
simply measuring the rotation angle dφ as a function of wavelength for a crystal of a
given length in a known magnetic field, we can determine Eg and K and extrapolate
the rotation for any desired wavelength, applied field, and crystal length.
4.2
Y3Al5O12 Properties
As the ultimate goal of this investigation was to construct a Faraday rotator for
intracavity insertion into a 1342 nm ring laser, we needed to identify and characterize a material with sufficiently large optical activity to provide enough rotation
when constrained by our cavity size and available magnetic fields. One prime candidate for such a device is undoped Y3 Al5 O12 (YAG). First grown commercially for
scientific use in 1962 [68], YAG has been widely used within the optics community
41
Figure 4.1. The transmission spectrum of undoped YAG from 200 to 6500 nm. Note
the broad transmission extending from the visible well into the mid-infrared [2].
as a host material for various types of rare-earth doped laser gain media, including erbium, ytterbium, thulium, and, most commonly, neodymium [69]. Undoped
YAG, however, is also useful, as some of those same properties that make it an
ideal candidate for active laser gain crystals also make it an ideal candidate for
passive devices as well.
One of the characteristics that makes YAG an ideal material is its broadband
spectral transparency. The transmission spectrum of undoped YAG is shown in
figure 4.1 [2]. From the graph, one can see that YAG has excellent transmission
over the range of 0.2 μm to 5.5 μm, covering the near infra-red spectrum that we
are interested in. In addition, YAG has a high thermal conductivity of 12.9 W m−1
K−1 and a low dn/dT value of +9.1 × 10−6 K−1 at 1064 nm [70]. When subjected
to the high powers circulating inside of a ring laser cavity, these values enable the
crystal to efficiently dissipate any absorbed power with minimal modulation of the
refractive index. As a result, thermal lensing effects are greatly reduced in these
crystals [71].
The crystal structure of YAG is body-centered cubic, having one lattice point
in the center of the cubic unit cell in addition to the eight corner points. Such a
lattice structure exhibits symmetry along the three crystal axes in real space. It is
42
because of this symmetry that undesirable adverse effects, such as birefringence,
are absent within the crystal. Therefore, as linearly polarized light propagates
through the crystal, in the absence of an applied magnetic field, there is no phase
delay in the transverse electric and magnetic field polarizations.
The undoped YAG single crystal rods used in this experiment were grown via
the Czochralski process by United Crystals. A total of six crystals with a diameter
of 5 mm and a length Lc = 18 mm were produced. The crystals were cut along the
111 axis and the two ends of the rods were polished for high optical purity. For
three of these crystals, including the one used for the measurements reported here,
a broadband anti-reflective coating centered at 1342 nm was applied by United
Crystals to the end facets to reduce reflective scattering losses when inserted into
the ring laser cavity.
4.3
Measurement Setup
The setup used in this experiment is similar to that of reference [72] and is shown in
figure 4.2. As mentioned before, to measure the Verdet constant of undoped YAG,
we must first determine the optical activity of the crystal. To do so, we measure
the rotation of the polarization of light as it propagates through the crystal in the
presence of a known magnetic field
One of the sources of laser light used in this experiment is derived from a DL100
External Cavity Diode Laser (ECDL) manufactured by Toptica Photonics. For an
ECDL in the Littrow configuration, the output of the laser diode is collimated
with a short focal length aspheric lens and is incident on a diffraction grating. The
first order of diffraction from this grating is coupled back through the collimation
lens and into the laser diode cavity while the zeroth order continues on for use.
By rotating the angle of the diffraction grating, the wavelength of the laser can be
tuned from approximately 1300 nm to 1350 nm for this laser.
The other source of laser light used in this experiment is derived from a commercial Nd:YAG non-planar ring oscillator (NPRO) laser. This laser was originally
manufactured by Lightwave Electronics (part number 126N-1064-500), now a subsidiary of JDS Uniphase. This single mode laser outputs 500 mW of light at 1064
nm has a linewidth of less than 5 kHz (5 ms integration time).
43
DET
DET
PBS
M
P
FR
HWP
ECDL
Figure 4.2. Light from an external cavity diode laser (ECDL) passes through a halfwave plate (HWP) and polarizer (P) pair. A double Fresnel rhomb (FR) is used to adjust
the polarization of the light incident on a cylindrical magnet (M). The light is then split
by a polarizing beam splitter (PBS) before falling on two photodetectors (DET) which
record the power of each beam.
To make the measurement, radiation from either source was coupled into the
APC end of an APC/FC polished panda style polarization maintaining fiber purchased from Thorlabs (part number PM980-XP). This fiber has a lower wavelength
cutoff of 920 ± 50 nm, well below the desired 1064 and 1342 nm wavelengths that
we are interested in. The fiber has a numerical aperture of 0.12 and a mode field
diameter of 6.6 ± 0.7 μm at a wavelength of 980 nm. An APC/FC fiber was used
to prevent power fluctuations in the transmission of the fiber due to etalon effects.
Additionally, by coupling into the APC end, the laser source (and summarily, the
test wavelength) could be changed without affecting the steering of the beam after
the light is launched out of the fiber.
After the light is launched from the output of the fiber, it first passes through
a half-wave plate and a Glan-Taylor polarizer oriented at a fixed 45◦ with respect
to vertical. By rotating the half wave plate, we ensure a constant 300 μW of
power for each measurement of the polarization rotation. Next, a double Fresnel
rhomb, acting as another half wave plate, is used in conjunction with a polarizing
beam splitter to split the laser light into two beams of equal power. Unlike a
standard zero or multi-order wave plate, the phase delay of the Fresnel rhomb
is wavelength insensitive over several hundreds of nanometers, making it an ideal
44
device when taking measurements over a wide range of frequencies. To measure the
optical activity of the YAG crystal, a magnet with uniform magnetization along its
cylindrical axis and a hole bored down its center containing the crystal is inserted
between the Fresnel rhomb and the polarizing beam splitter. The end facets of the
crystal have a broadband anti-reflective coating to minimize reflective losses of the
incident laser light. After the beam splitter, the beams are then incident upon a
pair of balanced photodetectors and the power of each is simultaneously recorded.
These power measurements are then used to determine the optical activity of the
crystal.
Linearly polarized light incident at 45◦ on a polarizing beam splitter relative to
its s and p axes, in the absence of any rotation, will split into two beams of equal
power. That is to say, I1 = I2 = I0 /2 where I1 and I2 are the intensity of each
beam as measured on detectors 1 and 2 respectively and I0 is the total intensity
incident on the beam splitter. For a small rotation of the polarization of the beam,
a change in intensity equal to
I1 =
Io
Io
(1 − 2dφ) , I2 = (1 + 2dφ)
2
2
(4.4)
will occur [73]. As a result, the difference in the voltage output of the two detectors, proportional to the difference of the incident intensities, assuming identical
transimpedance gains, is
V1 − V2 ∝ 2Io dφ.
(4.5)
Therefore, by measuring the voltages of each detector, the small rotation angle
(optical activity) can be determined by
dφ ≈
1 V1 − V2
.
2 V1 + V 2
(4.6)
The balanced photodetectors used in this experiment are a commercial device
also purchased at Thorlabs (model number PDB-150C). This switchable gain detector is made from InGaAs and has a wavelength response ranging from 800 to
1700 nm. The active area of each detector is 0.3 mm in diameter and has a max
response of 1 Amp/Watt. Each detector also has a fast monitor output that is
proportional to the power incident on the photodetector faces. It is this fast mon-
45
itor output that we record to measure the optical activity (and thus the Verdet
constant) of the undoped YAG.
To record the output from the fast monitors, we employ the use of a U6 Data
Acquisition device manufactured by the LabJack Corporation. This data acquisition card contains a 16 bit analog to digital converter and interfaces with a
computer via a USB connection. To record the data, a simple LabVIEW program
was written to sample the analog outputs of the fast monitors 100 times at a rate
of 10 Hz. The average and standard deviations of these measurements are then
used to calculate the rotation angle of the polarization vector, as per equation.
4.6.
4.4
Cylindrical Magnet
As mentioned before, the magnetic field applied to the undoped YAG crystal is
generated by a right hollow cylindrical magnet of uniform axial magnetization.
The magnet used for this experiment is a rare earth N50 neodymium magnet. As
diagrammed in figure 4.3a, the magnet is of length L = 19.0 mm and has an outer
diameter 2b = 22.2 mm. The inner bore of the magnet has a diameter 2a = 6.3
mm.
To determine the field along the cylindrical axis of the magnet, we begin our
analysis with Maxwell’s equations, notably ∇·B=0 and ∇×H=0, and continue via
the analysis of [74]. Using the relation B=H+4π∇·M, it can be readily shown
that
∇2 Φ = 4π∇ · M
(4.7)
where Φ represents the magnetic scalar potential. For a right-cylindrical permanent magnet with uniform magnetization along its axis, ∇·M vanishes everywhere
except on the pole faces. Solving equation 4.7 using a spherical coordinate system,
the potential at each pole face obeys Laplace’s equation ∇2 Φ = 0 and can be
represented as
Φ(r, θ) =
inf Al rl + Bl r(−l+1) Pl (cos θ)
(4.8)
l=0
where Pl (cos θ) is a Legendre polynomial of order l. The coefficients Al and Bl are
determined by the axial boundary conditions of equation 4.8, which is determined
46
0.03
(a)
Position (meters)
0.02
L
0.01
2a
0.00
2b
Lc
-0.01
-0.02
-0.03
-0.02
0.00
Position (meters)
0.02
0.6
Magnetic Field (Tesla)
(b)
0.4
0.2
0.0
-0.2
-0.4
-0.02
0.00
Position (meters)
0.02
Figure 4.3. (a) Physical layout and dimensions of the right hollow cylindrical magnet
housing the undoped YAG crystal. The crystal, of length Lc = 18.0mm is located inside
the magnet having length L = 19.0 mm and an outer diameter 2b = 22.2 mm. The inner
bore of the magnet has a diameter 2a = 6.3 mm. (b) The measured magnetic field of
the magnet at various distances from the end facets. The dashed line represents a fit to
the measured data for the magnetization M.
by using the Green’s function method to be of the form
Φ(r, θ) = 2πM
√
r 2 + R2 − r
(4.9)
where R is the radius of the pole face. Accounting for the contribution of the
field due to the superposition of each of the four pole faces (two ends of radius a
with two holes of radius b), we find the total magnetic scalar potential along the
47
symmetry axis to be
⎡ Φ(z, 0) = 2πM ⎣−
a2
+
L
+ z−
2
L
a2 + z +
2
2
+
2
−
b2
L
+ z−
2
L
b2 + z +
2
2
2
⎤
⎦
(4.10)
Finally, taking the gradient of the magnetic scalar potential yields the magnetic
field along the symmetry axis of the magnet
⎡
z − L2
z − L2
⎣
+
Bz = 2πM − a2 + (z − L2 )2
b2 + (z − L2 )2
⎤
L
L
z+ 2
z+ 2
⎦
+
−
L
L
a2 + (z + 2 )2
b2 + (z + 2 )2
(4.11)
To determine the magnetization of the magnet, the magnetic field was measured
using a standard gaussmeter at several locations along its symmetry axis extending
out from the magnet faces. The measured magnetic field is shown in figure 4.3b.
We then fit these measurements with equation 4.11, shown by the dashed line in
figure 4.3b. From this fit, we determine the magnet has a magnetization M =0.125
T for our hollow cylindrical magnet.
Given the value for the magnetization, it now becomes possible to determine
the total applied magnetic field along the symmetry axis of the magnet over the
length of the undoped YAG crystal. Assuming the crystal is centrally located
within the bore of the magnet, we can integrate equation 4.11 over the length of
the crystal. Doing so yields a total applied magnetic field of 8.00 × 10−3 T·m
applied along the length of the crystal.
4.5
Verdet Measurement
With the measurements of the rotation angle, dφ, of the polarization vector as a
function of wavelength and knowledge of the integrated applied magnetic field, it
48
Figure 4.4. Verdet constant of undoped YAG in the near infrared. Each data point
represents the average of one hundred measurements. The inset provides a closer look
at the 1300 nm to 1350 nm range. Error bars are indicative of the standard error of
the mean. The dashed line is a fit to the data as per equation 4.3, demonstrating the
dispersive nature of the Verdet constant.
now becomes possible, via equation 4.1 to determine the Verdet constant of the
undoped YAG crystal as a function of wavelength. Figure 4.4 shows the Verdet
constant as a function of wavelength from 1300 to 1350 nm as well as at 1064 nm.
Each data point represents the average of one hundred measurements with error
bars indicating the standard error of the mean. The Verdet constant from 1300 nm
to 1350 nm was found to be of the order 1.38 rad/T·m. Additionally, the optical
activity of the YAG crystal at 1064 nm corresponds to a Verdet constant of 2.12
rad/T·m.
As mentioned in section 4.1, the Verdet constant is dispersive in wavelength.
The dashed line in figure 4.4 is a fit of equation 4.3. From this fit, we find that our
undoped YAG crystal has an energy band gap value of 8.1 eV ± 0.3 eV and a Kparameter of 596 rad/T·m ± 52 rad/T·m, consistent with previous measurements
made with visible light[64].
49
4.6
Conclusion
The optical activity of undoped YAG was measured via a pair of balanced photodetectors and the field of a hollow cylindrical magnet containing the crystal was
determined. From these two results, we report, for the first time, values for the
Verdet constant of undoped YAG in the near-infrared. By fitting this data with
an analytical coupled oscillator theory, we can extract values for the energy band
gap and K-parameter for the undoped YAG crystal and compare them with previous measurements. With this knowledge, it now becomes possible to construct
an appropriate Faraday rotator for insertion into a high power ring laser cavity to
drive unidirectional operation.
Chapter
5
Constructing a 1342 nm Ring Laser
As extensively discussed in chapter 2, many of the experimental investigations undertaken in our atomic physics laboratory revolve around the cooling, trapping,
and manipulation of fermionic 6 Li. As one would expect, one of the vital tools
needed to conduct these experiments is a source of coherent laser radiation whose
wavelength is resonant with the atomic sample. For 6 Li, the D1 and D2 spectroscopic lines lie at a wavelength of 671 nm. Therefore, for experiments involving
ultra-cold 6 Li gases, we require several laser sources near 671 nm with sufficiently
high power and good spectral qualities.
For many ultra-cold atomic gas experiments, with species such as Rubidium and
Cesium, a tunable Ti:Saph laser is sufficient to generate the required wavelengths
of light. However, in the case of 6 Li, the emission spectrum of Ti:Saph lasers are
such that the gain is not high enough at 671 nm to be a viable source of radiation.
Historically, dye lasers have been used to generate the required wavelength. In
fact, the first experiments conducted in our research group were carried out with a
Coherent-899 dye laser pumped by a frequency doubled Nd:YAG solid state laser.
While these lasers are very versatile, producing over 1 Watt of power and having
a tuning range of several hundred nanometers, the spectral linewidth properties
of the emitted light are limited to a few MHz at best. Only by implementing
several “aftermarket” improvements, such as dye dampeners and external cavity
locking schemes, has the linewidth of these lasers been reduced. Additionally,
power stability issues and intensity fluctuations lead to shot-to-shot variation in
the number of atoms trapped in subsequent experimental runs.
51
A more recent solution to this problem arrived with the development of semiconductor chip based tapered amplifier systems. These amplifier chips, when
seeded with tens of milliwatts of power from a seed laser, can produce 500 mW
of 671 laser light. Since these systems are diode based, the spectral output of the
beam is a few hundred kHz. However, while the laser used to seed these amplifiers
has a TEM00 transverse mode profile, often the output of the tapered amplifiers
is a mix of TEM00 and higher order modes. Because of this, the coupling efficiency of light from a tapered amplifier into a fiber optic waveguide is reduced to
50% or less, limiting the available power for the experiment. While two or more
laser-amplifier systems could be used in parallel to generate the required power, a
different solution to this problem may be prudent.
Here we present another solution for generating light at 671 nm— namely, a
solid state laser system. Solid state lasers are much more stable in both frequency
and power than dye lasers and the transverse mode profile of the output laser beams
can be spatially manipulated to be a TEM00 mode. Power outputs of multiple
Watts have been achieved in solid state systems, making them an ideal candidate
for the construction of a 671 nm laser source. The only problem with a solid state
laser solution is the fact that there are currently no known gain crystals that have
a 671 nm lasing transition. However, Nd:YVO4 , whose energy level diagram can
be seen in figure 5.1, may present a solution to that problem. Nd:YVO4 as a lasing
medium is a four level laser system, with a 4 I9/2 →
at 808 nm and two emission lines (4 F3/2 →
4
4
F5/2
pump absorption line
I13/2 and 4 F3/2 →
4
I11/2 ) at 1342
nm and 1064 nm respectively. We have constructed a double end pumped solid
state ring laser using this Nd:YVO4 crystal to produce over 3 Watts of power at
1342 nm. An artistic rendering of this cavity can be seen in figure 5.2. Having
demonstrated this power at 1342 nm, we should then be able to frequency double
the laser output by locking this laser to a frequency doubling cavity containing a
non-linear optical crystal. While we have not demonstrated a very high conversion
efficiency as of yet, with a well-optimized doubling cavity we could expect to have
a 50% conversion efficiency, producing over 1.5 Watts of power at 671 nm.
During the construction of this laser we also developed a novel technique of
“self-injection” [75]. At 1342 nm, high power optical isolators are not commercially
available for insertion into the cavity. Using this novel technique, a small amount
4
F5/2
4
F3/2
4
I15/2
4
I13/2
4
I11/2
4
I9/2
1342 nm
1064 nm
880 nm
808 nm
52
Figure 5.1. Energy level diagram of Nd:YVO4 . The crystal has a 4 I9/2 → 4 F5/2
pump
4
4
4
4
absorption line at 808 nm and two emission lines ( F3/2 → I13/2 and F3/2 → I11/2 )
at 1342 nm and 1064 nm respectively.
of the output power of the laser is mode matched back into the cavity to drive
unidirectional operation. Doing so allows us to remove nearly all the intracavity
elements of our laser. By reducing the number of elements located inside the cavity,
not only will the laser have a higher output power (due to a reduction in intracavity
scattering losses), but the cavity length can be reduced— resulting in a larger free
spectral range. For our laser system having high gain, this additional power due
to the reduction of scattering loss is not significant. However, this technique is
broadly applicable to all ring lasers. Additionally, we demonstrate tunability of
the system via an etalon, both within the cavity itself as well as in the path of the
light injected back into the laser.
Section 5.1 will detail the overall design and layout of the ring laser cavity.
Section 5.2 will discuss the dichroic mirrors used in pumping our laser crystal as
53
Figure 5.2. Artistic 3D rendering of the Nd:YVO4 laser cavity.
well as detailing measurements of the reflectivity of several output couplers used
throughout these investigations. Section 5.3 will provide a discussion about the
physical properties of the Nd:YVO4 gain crystal used in our setup. Section 5.4 will
discuss the methods employed to pump the Nd:YVO4 crystal while section 5.5 will
discuss our setup employed to cool the crystal. Section 5.6 will detail additional
components that could be inserted into the laser cavity for various reasons. Finally,
section 5.7 will present a paraxial resonator analysis of our cavity, followed by a
concluding section 5.8.
5.1
Cavity Layout
Since the goal of this project was to construct a single frequency laser that can
be implemented in an ultra-cold atomic physics experiment, cavity linewidth and
frequency noise considerations played an important role in the design and construction of the laser system. To this end, it was decided that the laser cavity
itself would be mounted within an enclosure that had been machined out of a single block of aluminum. In addition, a top plate aluminum lid was also employed
54
Direction
M1
M2
Pump
Pump
M3
Nd:YVO4
M4
Figure 5.3. Experimental setup for the laser cavity. A Nd:YVO4 crystal located inside
of a bow-tie ring cavity is double end pumped by two 25 Watt diode arrays. The cavity
consists of a fully reflecting mirror (M1), an output coupler (M2), and two dichroic
mirrors (M3 and M4) which reflect light at 1342 nm while transmitting light at 808
and 1064 nm. With nothing to break the symmetry of the system, the laser will lase
bidirectionally, resulting in the gain being shared by the clockwise and counterclockwise
modes
with through holes that enabled the mirrors of the laser cavity to be anchored, not
only to the heavy aluminum base, but to the lid as well. This increased the mirror
stability and vastly improved frequency noise while reducing mode hopping in the
output beam.
The laser cavity layout is shown in figure 5.3. Four reflectors with infinite
radii of curvature form a folded ”bow-tie” ring cavity around the gain medium.
Since we ultimately require single mode operation, a ring cavity configuration was
chosen over a standing wave cavity to prevent gain competition (and thus multimode operation) due to spatial hole burning in the gain medium[76, 77]. The
tradeoff in using a ring cavity geometry is that ring lasers support bidirectional
lasing operation, with the gain being shared between the two symmetric modes
of oscillation, namely clockwise and counterclockwise traveling waves in the laser
cavity. Unless an effort is made to break the symmetry of the laser oscillation,
there will be two output beams emitted from the output coupler and the total
power of the laser will be shared between the two.
In addition to choosing a ring geometry to promote single frequency operation,
we also wanted to minimize the total round trip path length of the cavity as well.
55
For a ring laser, the free spectral range (FSR) of a cavity is given by
FSR =
c
L
(5.1)
where L represents the round trip path length of the cavity and c is the speed of
light. For our bow-tie cavity, L = 0.214 meters, corresponding to a FSR of 1.40
GHz. This means that the stable modes of oscillation for this laser will be spaced,
in frequency, by 1.40 GHz. For the 1342 nm wavelength that we are interested in,
this FSR corresponds to a Δλ value of 0.008 nm. Therefore, if the gain profile is
sharply peaked, or can be made so by the use of an etalon, the laser will be forced
to operate in single mode.
5.2
Reflectors
Here we describe the four reflectors that make up the bow-tie ring cavity (mirrors
M1–M4 in figure 5.3).
Mirror M1 is a highly reflective (taken to have a reflectivity R=1) with infinite
radius of curvature. This mirror has a broadband dielectric coating covering the
1342 nm transition that we are primarily interested in. Mirrors M3 and M4 are
dichroic mirrors with infinite radii of curvature and were manufactured by Advanced Thin Films as well. Each of these mirrors has a dielectric coating that
either transmits or reflects light depending on the wavelength. Because these mirrors need to allow pump laser light to transmit through to the crystal gain medium,
they have an anti-reflective coating centered at 808 nm. Additionally, because we
do not want the cavity to lase on the 4 F3/2 → 4 I11/2 transition of Nd:YVO4 , they
also have an anti-reflective coating centered at 1064 nm as well. Finally, because
we are interested in the 4 F3/2 →
4
I13/2 1342 nm transition, the mirrors have a
highly reflective coating at this wavelength.
Because of space limitations and cavity design, these two mirrors needed to
be mounted as close to the gain crystal as possible while still maintaining an
unobstructed path length. To accomplish this, a pair of custom mirror mounts
were manufactured by the Engineering Machine Shop at Penn State. Although
these mirrors mounts do not allow for the adjustments that a traditional kinematic
56
Label
70%
75%
Laser
#1
#2
93%
98%
R @ 0◦
0.776
0.815
0.900
0.893
0.910
0.950
0.993
R @ 15◦
0.804
0.819
0.892
0.904
0.914
0.956
0.994
δ
0.218
0.200
0.114
0.101
0.0899
0.0450
0.00602
Table 5.1. Measured reflectivities of available output couplers for our homemade 1342
nm laser. Measurements were made at normal incidence and at 15◦ for vertically polarized light. The δ value of each output coupler is another way of representing its
reflectivity (also at 15◦ ). For a further explanation, see the text.
mirror mount would, they were designed to have slight adjustments in their position
via set screws when mounted in the laser cavity. Machine drawings for these two
mirror mounts can be seen in figures 5.4 and 5.5.
To complete the cavity, a partially reflecting mirror M2 is used as an output
coupler for the laser. Seven different output couplers (all flat with infinite radii of
curvature) were available and used in the experiments that follow. Each of these
output couplers were manufactured by CVI Laser. To experimentally determine
the reflectivity, R, of each output coupler, the transmission of each output coupler
was measured using vertically polarized light from an external cavity diode laser
(Toptica DL100) at a wavelength of 1342 nm. By measuring the incident and
transmitted power of this test laser for each output coupler, the transmission, T ,
for each is determined. Using the simple relation R+T =1, we thus determine the
reflectivity. These measurements were made for both normal incidence (θ=0◦ ) as
well as at an angle of 15◦ . The normal incidence measurement was used to verify
the quoted reflectivity of the output coupler while the 15◦ measurement was used
to determine the reflectivity under normal experimental conditions. The results of
these measurements are summarized in table 5.1. Also reported in table 5.1 is the
reflection coefficient of each output coupler using δ notation for mirror reflectivities,
where R = exp[−δ] [78].
0.250
1.625
0.250
0.500
0.750
2-56 TAP THRU
1 HOLE
2.000
0.250
0.313
0.156
1.500
0.125
0.144
0.530
0.757
1/8 DRILL THRU
C'BORE FOR #4 SOC. HD.
(7/32 C'BORE, 3/16 DP.)
2 HOLES
0.026
15.000°
15.000°
0.125
0.480
1.415
SCALE
SIZE
1:1
O'HARA
DWG NO
CONTACT
CAGE CODE
ALUMINUM
DICHROIC
MIRROR MOUNT I
TITLE
1/2 C'BORE WITH
3/8 C'BORE OFFSET BY 0.085
1.500
SHEET
REV
865-7259
1
57
Figure 5.4. Machine drawing for homemade mirror mount for the dichroic mirror M3
in our 1342 nm laser cavity.
A
B
C
D
6
6
A
A
1.500
1.415
5
5
15.000
0.125
0.480
15.000
4
0.026
SECTION A-A
1/2 C’BORE WITH
3/8 C’BORE OFFSET BY 0.085
4
0.757
3
0.144
0.530
1/8 DRILL THRU
C’BORE FOR #4 SOC. HD.
(7/32 C’BORE, 3/16 DP.)
2 HOLES
3
2-56 TAP THRU
1 HOLE
0.750
2.000
SCAL
E
1
0.500
1
0.250
REV
1.625
0.250
1
865-7259
SHEET
QTY.
O’HARA
DWG NO
CONTACT
CAGE CODE
1:1
2
A4
SIZE
ALUMINUM
MATERIAL
DICHROIC
MIRROR MOUNT II
TITLE
0.313
0.156
1.500
0.250
0.125
2
A
B
C
D
58
Figure 5.5. Machine drawing for homemade mirror mount for the dichroic mirror M4
in our 1342 nm laser cavity.
59
5.3
Nd:YVO4 Crystal
The gain crystal used to construct our 1342 nm solid state laser was a 0.27%
neodymium doped yttrium ortho-vanadate (YVO4 ) crystal. First recognized as an
important laser material in 1966 [79], yttrium ortho-vanadate is an excellent host
material for laser crystal construction due to its physical and optical properties.
The structure of the YVO4 crystal is zircon tetragonal (tetragonal bipyramidal)
[3], and shown in figure 5.6. The lattice parameters of the crystal are a = b =
7.119 Å and c = 6.290 Å [80]. Because of this internal structure, YVO4 is naturally
birefringent. The Sellmeier equations [81] are
0.110534
− 0.012267612λ2
− 0.04813
0.069736
n2o = 3.77834 + 2
− 0.0108133λ2
λ − 0.04724
n2e = 4.59905 +
λ2
(5.2)
where λ is the wavelength in units of μm. Because of this natural birefringence
there will be a polarization dependence to the laser gain [65], a fact that we will
exploit to drive unidirectionality in our ring laser. Also, this polarized output
avoids undesirable thermally induced birefringence in our laser.
One potential drawback to YVO4 crystals are their low thermal conductivity.
Parallel to the c-axis, the thermal conductivity coefficient is 5.23 W m−1 K−1 while
perpendicular to the c-axis, the coefficient is 5.10 W m−1 K−1 [80]. These values
are about one third those of YAG and prevent good heat dissipation within the
crystal. As a result, power scaling may be limited in these crystals due to thermal
damage. Also, this lack of dissipation leads to thermal lensing within the crystal
[71, 82].
As shown in the energy level diagram of figure 5.1, Nd:YVO4 has a 4 I9/2 →
4
F5/2
pump absorption line at 808 nm. Nd3+ ions in YVO4 have a large absorption
cross section at this wavelength, with a peak value of about 2.7 × 10−19 cm2 [83]
Figure 5.7 shows the absorption spectrum of a 0.27% doped Nd:YVO4 crystal in
the region of the 808 nm band, reproduced from [4]. At this concentration, this
plot shows a peak absorption coefficient of approximately 9.4 cm−1 at 808.7 nm.
As for stimulated emission from the Nd:YVO4 gain crystal, the 4 F3/2 → 4 I13/2
60
Figure 5.6. The crystal structure of an yttrium ortho-vanadate. Yttrium orthovanadate crystalizes as a zircon tetragonal (tetragonal bipyramidal) structure, leading
to a natural birefringence along its a and c axes. Shown centered is the yttrium ion
surrounded by its vanadium and oxygen neighbors. This figure is adapted from [3].
transition at 1342 nm, though not as strong as the 1064 nm 4 F3/2 →
4
I11/2 line,
still has a stimulated emission cross section value of 6±1.8×10−19 cm2 [84]. Coupled
with a high fluorescent lifetime of 110 μs, such a crystal seems well suited for the
construction of a 1342 nm solid state laser.
The Nd:YVO4 crystals used to construct the laser described in this chapter
were manufactured by Coretech Crystal, a division of Shanda Luneng Information
Technology Co., Ltd. in Shandong China. The crystals were rectangular prisms
in shape, 10 mm in length with ends measuring 3 mm by 3 mm square. The
crystals were cut along their a-axis, with the c-axis being rotated by 45 degrees
relative to the flat 3 x 10 mm faces. The crystal is mounted such that the c-axis
is vertical. Because of the natural birefringence and polarization dependence in
the ortho-vanadate crystal, this particular rotation and orientation of the crystal
establishes the polarization axis of the laser to be in the vertical direction.
For the desired laser output wavelength of 1342 nm, the crystals were doped
with neodymium.
It has been shown that the pump power fracture limit of
Nd:YVO4 is inversely proportional to the doping concentration [85]. Because of
61
Figure 5.7. The absorption spectrum of a 0.27% doped Nd:YVO4 crystal in the region
of the 808 nm band, reproduced from [4]. This plot shows a peak absorption coefficient
of approximately 9.4 cm−1 at 808.7 nm.
this, for higher power operation with pump sources at the tens of Watts level,
as is our case, the optimum concentration for a Nd:YVO4 crystal is in the range
of 0.25% to 0.4% at. [86]. It is for this reason that we chose to use a doping
concentration value of 0.27% at.
The end facets of the gain crystal were also coated with anti-reflection coatings
at 808 nm, 1064 nm, and 1342 nm. The 808 nm coating prevents reflective losses
of the pump light incident upon the crystal, enabling more power to be absorbed
and thus higher output laser powers. The 1064 and 1342 nm coatings are included
to prevent reflections of spontaneously emitted light at the crystal-air interface.
The YVO4 crystal has an index of refraction of 2.18, leading to Fresnel reflections
of approximately 13.7% at the surface. If no measures are taken to prevent these
reflections, the crystal itself can act as a standing wave cavity and begin to lase,
reducing the excited state population of the gain available for the larger ring cavity.
The crystal is mounted in a copper mount shown in figure 5.8 and held in place
by a cover (figure 5.9). Between the crystal and the mount is a layer of indium
foil that provides a compressible barrier to prevent stress fractures of the crystal
while providing a good thermal contact between the mount and the crystal. The
copper mount used to hold the crystal is also part of the cooling system described
in section 5.5.
CONTACT
1:1
COPPER
SCAL
E
MATERIAL
1.583
0.777
1.136
1-80 TAP
1/4 DEEP
2 PLACES
3.250
1.827
1.975
1.275
1.625
1.423
0.117*
90.00
0.957
0.046
* +0.000/-0.002
45.00
0.123
0.094
#41 DRILL
1/4 DP.
2.597
1.447
0.395
1.068
1.742
2.416
0.363
2.456
4-40 TAP 3/8 DP.
4 PLACES
6-32 TAP THRU
4 PLACES
TITLE
GAIN MEDIUM
MOUNT
3.000
QTY.
0.250
2
O’HARA PHONE865-7259
0.250
2.250
62
Figure 5.8. Copper mount used to hold the Nd:YVO4 crystal inside the laser cavity.
One end of the mount is connected to a thermo-electric cooler used to extract heat from
the crystal (see section 5.5. The other end of the mount holds the crystal in a narrow
finger-like region, allowing the crystal to be located in the cavity without obstructing
the beam path.
#48 (0.076) DRILL THRU
C’BORE FOR #1 SOC. HD.
(5/32 C’BORE, 0.085 DEEP)
2 HOLES
30.00
0.096
A
30.00
0.404
SECTION A-A (SCALE 2:1)
0.454
0.187
A
0.374
90.00
45.00
0.117 *
0.188
SCAL
E
1
REV
865-7259
SHEET
QTY.
O’HARA
DWG NO
CONTACT
CAGE CODE
1:1
A4
COPPER
MATERIAL
SIZE
0.550
GAIN CRYSTAL
COVER
TITLE
0.117 *
0.275
* +0.000/-0.002
63
Figure 5.9. Copper cover used to sandwich the gain crystal against the copper mount.
64
5.4
Pump Laser
The pump light used to excite the gain crystal is derived from a DUO-FAP laser
system. This laser system is a commercially available product from Coherent and
can produce 60 total Watts of power (two 30 Watt beams) at a wavelength of
808 nm. This wavelength is resonant with the 4 I9/2 →
4
F5/2
absorption line for
Nd:YVO4 . The spectral width of the laser light is quoted as being less than 3 nm,
with an intensity noise of less than 1% rms. Inside the DUO-FAP laser, light from
two laser diode arrays are coupled into two 800 μm diameter multi-mode fibers
protected by an armored jacket. These fibers are then used to steer the beam to
double end pump the crystal as shown in figure 5.3.
To launch the light from the fibers and onto the crystal, two coupled lens
mounts are used for each fiber. The first mount, shown in figure 5.10, is used
as an adapter and spacer, providing an sma connection to the armored jacket
fiber as well as a connection to the homemade lens mount shown in figure 5.11.
Inside this homemade lens mount are two lenses of focal length 60 mm and 30 mm
respectively. By locating the lenses such that the first is 60 mm from the end of the
fiber sma connector and the second is 30 mm from the center of the gain crystal,
a 2:1 imaging system is used to image the pump light from the DUO-FAP laser
onto the crystal gain medium. Because of the lenses used in this imaging system,
the beam spot size on the crystal is approximately 400 μm.
To properly align the pump light on the center of the gain crystal, each mount,
shown in figure 5.10, was mounted to another homemade laser mount, shown in
figure 5.12. This second mount then connects to an adapter plate, shown in figure
5.13, which is connected to a stainless steel gothic-arch translation stage (model
9061-XYZ) from New Focus. These translation stages have a 25 by 25 mm square
platform that the adapter plate mounts on to. These translation stages allow for
6.25 mm of travel in each of the x, y, and z directions. Used in combination with
Model 9354 actuators (also from New Focus), this allows for precision alignment
of the pump laser beams onto the gain crystal for maximum output laser power.
65
0.875 dia
M20-1.25
SMA Connector
0.734
1.775
2.235
Figure 5.10. Adapter for connecting the sma fiber from the DUO-FAP laser to the
homemade lens mount for the 2:1 pump imaging system.
SHEET
O’HARA
CONTACT
DWG NO
CAGE CODE
1:1
SCAL
E
A4
SIZE
ALUMINUM
MATERIAL
QTY.
PUMP LENS
MOUNT
A
A
1.008 1.200
1.275
0.375
0.787
1.008" DIA.
1.035" - 40 TAP
TITLE
0.694
865-7259
REV
2
0.787" (20 MM) OUTSIDE DIAMETER
M20x1.25 THREAD
SECTION A-A
66
Figure 5.11. Homemade lens mount used to house the imaging optics for focusing the
808 nm pump laser light onto the Nd:YVO4 gain crystal. Two lenses, of focal lengths
60 and 30 mm are located so as to provide a 2:1 imaging system, focusing light from the
800 μm diameter pump laser fibers to a diameter of 400 μm on the gain crystal itself.
O’HARA
SCAL
E
1:1
CONTACT
DWG NO
CAGE CODE
A4
SIZE
ALUMINUM
MATERIAL
OPTICAL DIODE
MOUNT
2-56 TAP THRU
6 HOLES
0.875*
0.250
0.750
1.250
TITLE
1.391
0.109 0.750
0.250
0.656
1.281
1.500
0.750
0.219
0.250
0.750
1.250
1.250
1.188
0.125
1.500
*SLIP FIT NO PLAY
WITH STAINLESS STEEL TUBE PROVIDED
QTY.
SHEET
1
865-7259
REV
67
Figure 5.12. This home built mount is used to hold the lens mount shown in figure
5.10. By doing so, it allows the lenses used for imaging the pump laser light onto the
gain crystal to be precisely positioned via a translation stage connected to this mount
by the adapter plate shown in figure 5.13.
1.500 1.391
0.109
1.000
0.500
0.250
1.250
1.500
0.500
4-40 TAP THRU
4 HOLES
1.000
3/32 DRILL THRU
5/16 C’BORE 0.1 DEEP
4 HOLES
SCAL
E
1
REV
865-7259
SHEET
QTY.
O’HARA
DWG NO
CONTACT
CAGE CODE
1:1
A4
SIZE
ALUMINUM
MATERIAL
ADAPTER PLATE
TITLE
0.188
68
Figure 5.13. Machine drawings for an adapter connecting the mount in figure 5.12 to
the stainless steel gothc-arch xyz translation stage.
69
5.5
Cooling the Crystal
Because Nd:YVO4 is a four level laser with a 4 I9/2 → 4 F5/2 absorption line at 808
nm and a 4 F3/2 → 4 I13/2 emission line at 1342 nm, there exists an energy difference
between the excited pump state and the upper level of the lasing transition (72
nm) as well as a difference between the lower level of the lasing transition and the
ground state of the system (427 nm). These so called “quantum defects” have an
adverse affect on laser operation as these wavelength mismatches are converted to
heat via phonon excitation [87]. It should be noted that the term quantum defect
is ambiguous and is unrelated to Rydberg atom physics. Additionally, while the
YVO4 crystal is highly transmissive at 1342 nm, its transmission is not perfect,
leading to additional heating of the crystal as well. Therefore, because of the power
levels involved with the construction of this laser, actively cooling the laser crystal
is necessary.
As mentioned in section 5.3, the crystal is located at the end of a copper finger
that serves as a mount to hold its position in the laser cavity. In addition to
this function, the mount also plays a role in actively cooling the crystal, as the
other end of the mount is connected to a larger copper block via a thermo-electric
cooler (TEC) that acts as a thermal mass. The TEC used in this experiment is
a UT8,12,F2,2525,TA,W6, UltraTEC manufactured by Melcor Corporation, now
a division of the Laird Group. This TEC is 2.5 cm square and has the capacity
to absorb 69 Watts of power at its cool face. For our laser setup, we operate the
TEC at a constant 4 amps of current.
Simply cooling the laser crystal via a TEC to the copper block reservoir would
not be an effective solution, however, unless heat in that reservoir can be also be
extracted through other means. This extraction occurs by way of an innovative
setup [5] involving two metal heat sinks and a water cooled mount, depicted in
figure 5.14. As can be seen in the picture, the heat sink on the copper reservoir
is located in very close proximity to the heat sink connected to another copper
mount, this one cooled by a circulating water system. The spaces in between
the nearly touching heat sinks are filled with standard ceramic heat sink paste
that acts as a medium for the transfer of heat without creating a solid physical
connection. The reason such lengths are taken to isolate the two heat sinks from
70
Copper Thermal Reservoir
Water Cooled Mount
Figure 5.14. Cartoon showing the interface between the copper thermal reservoir and
the water cooled mount [5]. Two heat sinks are located in very close proximity to one
another so as to enable heat transfer without making a physical connection. This enables
heat to be transferred across the interface without coupling any vibration from the water
cooled mount into the copper thermal reservoir (and subsequently, the laser gain crystal).
direct physical contact is because the flow of water through the copper mount can
cause vibration in the cooling system mounts, which can then be transferred to
the crystal. These crystal vibrations have been observed to be strong enough to
cause frequency noise on the output of the laser beam.
Finally, the water cooled copper block is cooled by a rack mounted solid state
thermoelectric thermal control unit manufactured by ThermoTech. This thermal
control unit circulates an 80/20 mixture of distilled water and corrosion inhibited
Glycol at a temperature of -5◦ C and is itself cooled by the chilled water line of
the building. In all, this cooling system enables to maintain the crystal at a
temperature of approximately 18◦ C when operating at full pump power capacity.
5.6
Additional Components
The only other permanent optical element located inside the cavity besides the
Nd:YVO4 crystal is a 710 μm diameter spatial pinhole filter located halfway between mirror M1 and output coupler M2. At this location in the resonator cavity,
the circulating beam has a waist of 148 μm and an infinite radius of curvature
(see section 5.7). In the absence of this filter, the laser output was observed to
be a mixture of Hermite-Gaussian modes, as determined by the transverse mode
measurement described in section 6.2. This spatial filter ensures a high quality
71
transverse mode structure to the output laser beam by increasing the losses for
those higher order modes.
5.7
Cavity Modeling
To determine the full characteristics of the beam propagating inside of the ring
laser cavity, as well as the thermal lensing of our Nd:YVO4 crystal due to the two
pump beams, we perform a paraxial resonator analysis (ABCD analysis) for the
elements contained within the ring laser cavity assuming a lowest order Hermitegaussian mode [88]. For the laser cavity layout shown in figure 5.3, the two main
components of the laser cavity that need to be modeled in this analysis are the
contributions of free propagation through a uniform medium as well as that of
thin lenses. The corresponding ABCD matrices for each of these two elements are
detailed in figure 5.15.
For this analysis, we determined the ABCD matrix corresponding to the elements inside of our laser cavity for one complete round trip. Choosing the center
of our Nd:YVO4 crystal as our reference plane, a round trip in the cavity consists
of the following elements: 1) 5 mm of free space propagation in a medium with an
index of refraction of 2.18, 2) a thermal lens of unknown focal length, 3) 192 mm
of free space propagation in air (index of refraction of 1.0), 4) A second thermal
lens of unknown focal length, and 5) another 5 mm of free space propagation in a
medium with an index of refraction of 2.18. From this round trip ABCD matrix,
we can define reference values for the radius of curvature, R0 , and spot size, w0
such that
R0 =
and
w0 =
where m ≡
2B
D−A
A+D
2
Aλ
π
1
1 − m2
(5.3)
(5.4)
and λ is the wavelength of the light. From these values, we can
then create a complex q-parameter for our system
q0 = (
1
λ −1
−i
) .
R0
πw0
(5.5)
72
d
1 d/n
0 1
Uniform medium
Index n and length d
Thin lens
Focal length f
1 0
-1/f 1
f
Figure 5.15. ABCD matrices used for a paraxial resonator analysis of our ring laser
cavity. Included are matrices for propagation of length d in a uniform medium with
index of refraction n as well as a thin lens of focal length f .
From the complex q-parameter for the reference plane, we can then determine q(z)
for any arbitrary position inside the cavity
q(z) =
A(z)q0 + B(z)
C(z)q0 + D(z)
(5.6)
where A(z) is the A component of the ABCD matrix for an arbitrary location z
from the reference plane (similarly for B,C, and D). Finally, with this q(z), the
waist and radius of curvature can be determine for that arbitrary position inside
the cavity
w(z) =
and
R(z) =
−λ
1
Im( q(z)π
)
(5.7)
1
1
Re( q(z)
)
(5.8)
To experimentally determine the unknown values of the effective focal lengths
due to thermal lensing at each end of the gain crystal, we perform a measurement
of the beam size and mode properties, as discussed in section 6.2. From these
measurements, for a location halfway between mirror M1 and output coupler M2,
the beam was determined to have a waist of 148 μm and infinite radius of curvature.
Therefore, using this measured value in our complex ABCD calculation above, we
find that the only solution that would yield this waist would be for the thermal
lenses at the ends of the Nd:YVO4 to have an effective focal length of 12.25 cm.
With this knowledge, it now becomes possible to determine the waist and radius
73
0.0005
Waist (meters)
0.0004
0.0003
0.0002
0.0001
0.00
0.05
0.10
0.15
0.20
Position (meters)
Figure 5.16. The waist of the gaussian beam for one round trip of propagation inside
the ring laser cavity. The zero position reference is taken to be the center of the gain
crystal.
of curvature of the beam at any arbitrary location inside of our ring laser cavity.
Figure 5.16 shows the waist of the gaussian beam as it propagates through one
round trip of the cavity, beginning at our reference point located at the center of
the Nd:YVO4 crystal. From this analysis, the waist of the beam at two critical
locations can be determined. First, as was directly measured, at a location halfway
between mirror M1 and output coupler M2, the waist is 148 μm. Second, as the
beam propagates through the active gain crystal, its waist has a value of 316 μm,
an important value in determining the expected power out of the laser cavity.
5.8
Conclusion
In this chapter, we have described the layout and construction of a four mirror
bow-tie ring laser cavity for application to experiments involving the cooling and
trapping of ultra-cold fermionic 6 Li gasses. A 10 mm long a-cut 0.27% at. doped
Nd:YVO4 crystal with its c-axis rotated to the vertical position was used as the gain
medium for this laser. The crystal was double end pumped by two fiber coupled 25
Watt diode arrays resonant with the 808 nm 4 I9/2 →
4
F5/2
absorption line. The
reflectors used to build the cavity were dielectrically coated so as to be transmissive
74
at both 808 and 1064 nm while being highly reflective at 1342 nm. These coatings
suppressed lasing in the cavity of the 4 F3/2 →
4
I11/2 emission line at 1064 nm
while enabling the laser to operate on the F3/2 →
4
4
I13/2 emission line at 1342
nm. An extensive multi-step cooling system was employed to maintain a reasonable
operating temperature of the lasing crystal in order to prevent thermal damage to
the crystal. This system also reduced frequency noise resulting from vibrational
coupling of the water cooling system to the crystal. Also, a theoretical analysis
was done on the cavity to determine the beam properties within the laser cavity,
information that will become pertinent when conducting a theoretical analysis of
the expected power out of the cavity.
In chapter 6 we will continue our study of this 1342 nm ring laser, measuring
its output characteristics while operating in both free running and unidirectional
mode. To drive unidirectionality, two methods will be employed, involving the
use of a home-built Faraday rotator element as well as a novel scheme of “selfinjection.” Various quantitative parameters of these two methods will be compared.
Also, studies involving the tuning of the laser using both intra-cavity and external cavity etalons will be performed. Preliminary results pertaining to frequency
doubling the 1342 nm light to 671 nm will also be reported.
Chapter
6
1342 nm Ring Laser Operation
With construction of our 1342 nm solid state ring laser cavity complete, it now
becomes necessary to measure the output characteristics of our laser for use in
experiments involving ultra-cold 6 Li gases. One important general limitation to
ring laser geometries is that these lasers support bidirectional operation, leading
to instabilities[89]. To combat this, several schemes have been employed to drive
unidirectional operation, such as the inclusion of intracavity optical isolators, the
use of acousto-optic modulators[90], polarization dependent output couplers[91],
and non linear crystals for sum-frequency mixing[92]. The use of such implements,
however, typically involves the insertion of additional elements into the laser cavity, increasing the amount of intracavity scattering loss and reducing the overall
power output of the laser. In this chapter, we report a novel technique of selfinjection locking for forced unidirectional laser operation. Because this technique
does not involve any additional cavity elements, we can achieve stable unidirectional operation with higher output powers than what can otherwise be achieved
using traditional methods.
We begin in section 6.1, where measurements of the small signal gain of the
Nd:YVO4 crystal are reported as a function of wavelength. Also measured is
the angular dependence of that small signal gain at 1342 nm, a consequence of
the natural birefringence in the crystal. Section 6.2 presents our measurements
of the free running output of the laser without making any attempts to either
frequency tune or drive unidirectional operation of the field within the cavity.
The longitudinal mode structure of the laser light is examined using a Fabry-
76
Perot interferometer and the transverse mode structure is measured with an M2
value reported. Also, fluctuations in intensity of the laser output are qualitatively
reported. Section 6.3 provides a theoretical framework for modeling the expected
output power of the laser given values for the unsaturated small signal gain as
well as cavity scattering losses. Next, in section 6.4, we report values for the laser
output when driven unidirectionally by including a home made Faraday rotator
inside the laser cavity. In section 6.5 we introduce a novel scheme of “self-injection”
to drive unidirectional operation and compare this method to that of section 6.5.
Section 6.6 reports results of using etalons to tune the frequency of our lasers and
section 6.7 will conclude the chapter.
6.1
Measuring the Nd:YVO4 Gain
In this section we report measurements of the unsaturated small signal gain for a
single pass as both a function of wavelength and also as a function of polarization
angle. For the wavelength dependent gain measurement, the unsaturated small
signal gain of the Nd:YVO4 crystal was measured when pumped by the 2 × 25
Watt pump beams. The basic layout for this measurement is shown in figure
6.1. To facilitate this measurement, the output coupler M2 of the laser cavity was
removed and an extended cavity diode laser (ECDL) whose wavelength is tunable
from 1335 to 1350 nm was used. Not shown in figure 6.1 but included in the setup
for this measurement were several lenses used to ensure high spatial mode overlap
of the probe laser with the pump beams inside the crystal. Also, as the gain is
polarization dependent, the electric field of the probe beam was rotated so as to
be aligned with the vertical c-axis of the crystal.
The results of our small signal gain measurement are reported in figure 6.2.
From the figure, it can be noted that the gain bandwidth spans several nanometers.
Also significant is a “negative gain” or absorption feature from approximately 1336
to 1341 nm. This feature is the result of excited state absorption of the crystal
and has been previously reported [93]. While this feature is interesting to note,
it’s presence does not prevent laser operation at the desired 1342 nm wavelength
in the Nd:YVO4 crystal.
More important to our discussion of the 1342 nm laser behavior, however, is
77
DE
T
M1
ECDL
Pump
Pump
M3
Nd:YVO4
M4
Figure 6.1. Experimental setup for measuring the unsaturated small signal gain of the
Nd:YVO4 crystal when pumped by 2 × 25 Watt pump beams. The transmitted power of
an extended cavity diode laser (ECDL) whose wavelength is tunable from 1335 to 1350
was measured by a photodetector (DET) after having passed through the crystal. Not
shown in the setup are the lenses used to mode match the waist of the probe laser with
the pump lenses as they co-propagate through the crystal.
the small signal gain peak located at approximately 1342 nm. To get a better and
more insightful understanding of this peak, a zoomed in view of the gain profile
from 1341 nm to 1345 nm is shown in figure 6.3. As a guide to the eye, we have fit
this gain profile over this region to a triple gaussian summed function of the form
Gain(λ) = y + Ae
−(λ−λ1 )2
w1
+ Be
−(λ−λ2 )2
w2
+ Ce
−(λ−λ3 )2
w3
(6.1)
where λ1,2,3 represent the center wavelengths of the three gaussians used for the
fit and w1,2,3 represents the corresponding widths. For the measured values of our
gain profile, A(B, C) = -0.67783(0.48271, 0.38555), λ1 (λ2 , λ3 ) = 1340.1(1342.2,
1342.7), w1 (w2 , w3 )=0.42994(0.30751, 1.4015), and y = 1.0543. Also, from this
data, we observe that the gain profile peaks at approximately 1342.2 nm with a
value of 1.87. This gain value of 1.87 corresponds to a small signal unsaturated
gain coefficient, αm0 , of 0.626 where
α = ln(Gain)
(6.2)
In addition to using values of the small signal gain to determine the expected
power output of the laser, we also have an interest in driving unidirectional operation of the laser within the ring cavity. One way to do this would be to exploit
78
Figure 6.2. The unsaturated gain profile of our Nd:YVO4 crystal from 1335 nm to
1350 nm when pumped by 2 × 25 Watt pump beams. The gain profile is quite broad,
spanning several nanometers, and peaks at approximately 1342 nm. Also of note is a
broad excited state absorption band spanning from 1336 nm to 1341 nm.
the polarization dependence of the gain medium. By inserting a Faraday rotator
coupled with a properly oriented half-wave plate into the laser cavity, light traveling in one direction around the cavity would experience a slight rotation in its
polarization while light from the other direction would not experience any rotation
at all. This small rotation in polarization, coupled with the fact that the Nd:YVO4
crystal is birefringent, should be enough to break the symmetry of the cavity and
force unidirectional operation.
We also wanted to measure how how the small signal gain of the laser changes
as a function polarization. To measure this polarization dependence, we use a
setup similar to the previous gain measurement, portrayed in figure 6.4. Like
the previous measurement, an ECDL is mode matched to the gain crystal and
incident upon a photodetector. However, in this case, a half-wave plate is also
used before the gain crystal to rotate the angle of the polarization of light before
the gain crystal. Also, for this measurement, the wavelength of the ECDL was set
to 1342.2 nm.
Figure 6.5 shows the angular dependence of the unsaturated small signal gain
79
Figure 6.3. The gain profile of our Nd:YVO4 crystal from 1341 to 1345 nm. The profile
has a peak value of 1.87 at a corresponding wavelength of 1342.2 nm. A dashed line has
been added as a guide to the eye (see text).
for our laser crystal from 0 to 90 degrees. As expected, the gain has a peak value
when the polarization of the probe laser light is parallel to the vertically rotated
c-axis of the crystal at 0 degrees. As expected, for this particular polarization, the
gain has a measured peak value of 1.84, consistent with the measurements of gain
as a function of wavelength for this value as reported above.
To further characterize this angular dependence, we fit the measured data
points in figure 6.5 to an offset cosinusoidal function of the form
Gain = y0 + Acos(f θ).
(6.3)
From this fit, we find values of 0.30472 for A, 1.5492 for y0 , and 0.03479 for
f. Therefore, when the polarization is aligned vertically with the c-axis of the
gain crystal, the fitted gain has a value of 1.85 and when the polarization is perpendicular to the c-axis, the gain has a value of 1.55. This fit information about
the angular dependence of the gain will become useful when examining the effects
of inserting a Faraday rotator into the laser cavity to drive unidirectionality, as
described in section 6.4.
80
DE
T
M1
ECDL
HWP
Pump
Pump
M3
Nd:YVO4
M4
Figure 6.4. Experimental setup for measuring the angular dependence of the unsaturated small signal gain of the Nd:YVO4 crystal. The transmitted power of an extended
cavity diode laser (ECDL) whose wavelength is set at 1342 nm was measured by a photodetector (DET) after having passed through the crystal. A half-wave plate (HWP) is
used to rotate the polarization of light prior to transmission through the crystal.
Figure 6.5. Unsaturated small signal gain as a function of polarization angle, measured
with respect to vertical. The dashed line represents a sinusoidal fit to the data (see text).
6.2
The Free Running Laser
As expected, in the absence of any attempts to drive unidirectional operation of
the laser cavity, the 1342 nm ring laser that we have constructed operates in bidirectional mode— the light circulates in both the clockwise and counterclockwise
directions. This can be easily demonstrated by looking at the light emitted from
81
the output coupler M2 (see figure 5.3). For light traveling in the clockwise direction, laser radiation will be emitted from the output coupler traveling in the plane
parallel to the gain crystal. For light traveling in the counter-clockwise direction,
laser radiation will be emitted from the output coupler traveling at an angle of 30
degrees with respect to the clockwise direction output.
For the output coupler with a reflectivity of 0.804 (labeled 70% in table 5.1),
1.62 Watts of laser power were measured in each of two output beams corresponding
to the two directions of bidirectional operation of the laser cavity. To measure the
power, a PM30 thermopile power detector manufactured by Coherent, Inc. was
used. This air convection cooled power meter can measure laser beam powers up
to 30 Watts. Because it is a thermopile detector, there is a 2 second response time
with each measurement. Because of this, fluctuations in the intensity of the laser
light faster that 0.5 Hz will not be discernable with this detector
To qualitatively measure high frequency intensity noise in this free running
laser beam, a small portion (on the order of a few tens of mW) of the output
of one beam was picked off and measured with a model DET10C biased InGaAs
Photodetector from Thorlabs. This photodetector has an active area of only 0.8
mm2 . Because of this small active area and the fact that it is a photodiode rather
than a thermopile detector, the response time is much faster, having a rise time of
only 10 ns. Figure 6.6 shows the intensity of the output beam for a scan of 10 ms.
From the plot, one can get a sense of the competition between the two directions
of operation. For the majority of the time, the power is split evenly between the
two directions, indicated by approximately 20 mW of power. However, at some
times, the power the power measured on the beam is nearly twice this value or
zero, indicating that during brief periods the laser is operating unidirectionally.
Next, we wanted to investigate the frequency mode structure of our free running laser. To measure the mode structure, the laser output was coupled into a
Fabry-Perot interferometer. The Fabry-Perot used in this measurement was also
manufactured by Coherent, Inc. and has a free spectral range of 300 MHz. The
output scan of this Fabry-Perot measurement is recorded in figure 6.7. From the
figure, one can see that the frequency output of the free running laser is operating
in multi-longitudinal mode. In fact, for this particular measurement, four distinct
frequencies are discernable. It is important to note that for our desired applica-
82
Figure 6.6. Measurement of the power stability for the free running laser cavity. For
most of the time, the power is split evenly between the two directions of operation,
indicating bidirectionality. However, during some instances, the laser operates in unidirectional mode, whereby the recorded power is either twice as high or zero, depending
on the direction.
tion of such a laser to experiments involving 6 Li spectroscopy, single frequency
operation is crucial.
Finally, in addition to the longitudinal mode structure of the laser, we also
wanted to measure the transverse mode structure of the beam as well. To do this,
we used a device called a “Mode Master,” also manufactured by Coherent, Inc.
This device samples the beam radius at a variety of locations in order to determine
the divergence of the beam. From this information, a variety of beam parameters
can be computed and reported. The output of the Mode Master scan for this free
running laser is reported in figure 6.8.
One important value measured and reported by the Mode Master is the M2
value of our laser. Also known as the beam quality factor, the M2 value of the
laser refers to the ratio of the beam parameter product (the product of the beams
divergence with its waist) to that of an ideal gaussian beam. For lasers operating
in a pure TEM00 mode, the M2 value of the measurement, by definition, would
be one. As lasers can operate in higher transverse Hermite-Gaussian modes, this
number can be much larger as the beam diverges faster for these higher modes.
83
Figure 6.7. Measurement of the frequency characteristics of our free running ring laser.
The graph shows an output of a 300 MHz Fabry-Perot interferometer. The presence
of four distinct peaks is indicative of the fact that the output laser frequency is multilongitudinal mode in structure.
From the output of the Mode Master, we note that the averaged radial M2 value
for our laser is 1.04, indicative of the fact that, for all intents and purposes, the
laser is operating in the TEM00 mode. This is not surprising as we have included
a pinhole within the laser cavity to prevent lasing in all higher order transverse
modes (see section 5.6).
Another important measurement reported by the Mode Master is the waist of
the laser beam. From this measurement, it is reported that the beam has a waist
of 529.5 μm. From this value, we can then calculate the value for the waist of
the beam inside the ring laser cavity. Not shown in figure 5.3, we use a 17.5 cm
focal length lens located 17.5 cm from the midpoint between mirror M1 and output
coupler M2 in order to collimate the laser beam. By knowing that the beam has
a mode radius of 529.5 μm at the location of this collimating lens, we can use
ABCD matrices to propagate the beam backwards in order to determine that the
beam must have a waist of 147 μm at the halfway position between mirror M1
and output coupler M2. The value for the beam waist at this point is critical for
determining the properties of the propagating beams inside the cavity, as well as
84
Figure 6.8. Results of a beam profile measurement made by the “Mode Master”,
manufactured by Coherent, Inc. The output of our free running laser is coupled into the
Mode Master, which measures values for the beam radius as the beam propagates over
a given distance. From these measurements, several spatial properties of the laser beam
can be determined and reported.
for determining the unknown effective focal lengths for thermal lensing inside the
gain medium (see section 5.7).
It is also interesting to note, from the output of the Mode Master measurement,
that there exists a slight astigmatism to our laser, on the order of 9%. This slight
astigmatism is not surprising, considering the effects of thermal lensing in our
gain crystal. Because the crystal is birefringent, thermal lensing caused by the
pump beams will cause different distortions to the laser wavefront along different
directions, contributing to an astigmatic beam [94].
6.3
Modeling Power Output
It now becomes necessary to develop a theoretical description of the expected output power of our laser cavity, taking into consideration the gain of the laser crystal,
the intracavity reflectivity, and the reflectivity of the output coupler. By then measuring the output power of the laser for a variety of output coupler reflectivities,
85
this theory will enable us to fit the data and perform an independent verification
of our unsaturated small signal laser gain as well as determine the optimum output
coupler for the greatest efficiency of our laser cavity.
The following derivation follows that of reference [78], however with slight modifications for a ring laser cavity instead of a standing wave setup. Also, for this
derivation, it is assumed that the laser is already operating unidirectionally, as
will be the case for the two measurements of laser power vs. output coupler that
follows.
For a laser cavity operating in continuous wave steady state conditions, the
round trip gain for the signal intensity inside the laser cavity must equal the losses
due to intracavity scattering and output coupling. If we assume a traveling ring
laser cavity with homogeneously saturable gain medium, the growth the traveling
wave I(z) inside the cavity will be given by the equation
dI(z)
= [αm − α0 ]I(z)
dz
(6.4)
where α0 is the cavity reflective loss coefficient and αm is the saturated gain coefficient. For a homogeneously saturable gain medium, the saturated gain coefficient
as a function of position along the axis of the laser is
αm =
αm0
1+
I(z)
Isat
(6.5)
where αm0 is the unsaturated small signal gain coefficient and Isat is the saturation
intensity of the transition. As a reminder, the small signal gain coefficient is related
to the small signal gain of the crystal via the relation
Gain = eαm0 .
(6.6)
Also, the saturation intensity, Isat , of the transition is defined as the intensity that
reduces the small signal absorption coefficient down to half its small signal value
and is given by
Isat =
ω
στef f
(6.7)
where is Planck’s constant divided by 2π, ω is the angular frequency of the
86
transition, σ is the stimulated emission cross section for the transition and τef f is
the fluorescent lifetime of the transition. For the 4 F3/2 → 4 I13/2 transition at 1342
nm, σ is reported to have a value of 6±1.8×10−19 cm2 and τef f is 110 μs [84]. This
yields, per equation 6.7, a saturation intensity Isat = 2.24 × 107 Watts/m2 .
If we now make the assumption that the output coupler reflectivity R1 is close
to unity such that the intensity I(z) remains fairly constant during the length of
the cavity, we can then make the approximation that
I(z) ≈ Icirc
(6.8)
where Icirc is the unidirectional circulating intensity inside the laser cavity that
is independent of cavity position. With this approximation, the saturated gain
coefficient also becomes independent of position within the cavity and can be
written as.
αm =
αm0
1 + IIcirc
sat
(6.9)
As mentioned above, for the laser to be operating in steady state, the round
trip gain for a single pass must equal the round trip losses for a single pass as well.
In other words
αm =
αm0
1
= α0 + ln( ) ≡ δ0 + δ1
Icirc
R1
1 + Isat
(6.10)
where e−δ0,1 are the cavity reflectivity (due to scattering losses) and output coupler
reflectivity respectively. Solving equation 6.10 for Icirc we find
Icirc = [
αm0
− 1] × Isat .
δ0 + δ1
(6.11)
For a lightly coupled laser cavity, the output coupler transmissions are given by
T1 = 1 − R1 ≈ δ1 . Therefore, the intensity output from the laser would thus be
δ1 × Icirc or
Icirc = δ1 [
αm0
− 1] × Isat .
δ 0 + δ1
(6.12)
Finally, as intensity is defined as power per unit area, the power output of the laser
cavity can thus be determined by
Power = δ1 [
αm0
− 1] × Isat πw02
δ 0 + δ1
(6.13)
87
where w02 is the waist of the circulating laser beam inside the gain crystal.
As mentioned above, this derivation for the power output from our laser cavity was for one dimensional transport through a homogeneously saturable gain
medium. For our laser system here, this is not the case and equation 6.9 must
be modified appropriately [95]. For situations where the beam converges or diverges in the gain medium, a one dimensional plane wave treatment is insufficient
to describe the incremental gain and higher order terms must be considered to account for this radial radiation transport. A more exact treatment of this additional
consideration would be
I
circ
αm0
γω 2
Isat
αm =
[1
+
]
+
Icirc
4
1 + IIcirc
1
+
I
sat
sat
where
γ=−
2.88
ρ2
(6.14)
(6.15)
is the quadratic term of the zero-order bessel function used to describe the gain
profile of radius ρ. The second term of 6.14 becomes −0.72×ω 2 /ρ2 . In many single
mode oscillators, ω 2 /ρ2 is about 0.5. The last term of 6.14 describes the part of the
radial radiation transport that comes from the change of the saturated gain profile
by the gaussian intensity distribution. For optimized laser systems, Icirc /Isat >> 1
causing the third term to go to one. Therefore, with these corrections, the actual
saturated gain parameter becomes
αm =
1.64αm0
1 + IIcirc
sat
(6.16)
where the 1.64 prefix is the result of radial transport of radiation in the crystal.
Using this value, we can thus rewrite equation 6.13 to reflect this modification,
making the equation for the expected power output of the laser as a function of
output coupler reflectivity
Power = δ1 [
1.64αm0
− 1] × Isat πw02 .
δ0 + δ 1
(6.17)
88
6.4
Unidirectionality with a Faraday Rotator
Traditional techniques to drive unidirectional operation of a ring laser cavity typically involve some form of intracavity element that provides directionally dependent loss to break the symmetry of the system. One such device, a Faraday rotator,
rotates the plane of polarization of the electric field vector in a directionally independent way. This device can be combined with a half-wave plate, oriented
such that the fast axis will rotate the plane of polarization back to its original
orientation for one direction while causing further rotation for the other direction
of transmission. Doing so provides directionally dependent rotation resulting in
directionally dependent loss for our gain crystal. Unfortunately, at the wavelength
and power levels of our 1342 nm laser, such a device does not exist commercially.
Therefore, we have constructed our own unidirectional element consisting of a
home built Faraday rotator and a multi-order half-wave plate for insertion into
our laser cavity
In section 4.4, the applied magnetic field on a 10 mm long undoped YAG crystal from a right hollow cylindrical magnet of uniform magnetization was measured
to be 8.00 × 10−3 T·m along the transmission axis of the crystal contained within
the bore of the magnet. In section 4.5 it was found that the Verdet constant
of undoped YAG was 1.38 rad/T·m. Therefore, our home built Faraday rotator,
when inserted into the ring laser cavity will not rotate the plane of polarization for
the clockwise direction, but will rotate the plane of polarization for the counterclockwise direction by 4.42 × 10−2 radians (2.53 degrees). Since our gain crystal is
polarization dependent, the small signal gain experienced by this rotated polarization is determined to be ∼ 4 × 10−7 less than that for the unrotated polarization
as per equation 6.3. While this difference in gain may seem trivial, it is indeed
enough to drive unidirectional operation of our laser.
Using this intracavity rotation element, we have measured the output power of
the unidirectional laser for the seven different output couplers listed in table 5.1.
The results of this measurement are shown in figure 6.9. For a reflectivity, R1 ,
of 0.804, it was found that the laser had a peak in output power of 3.10 Watts.
Also included in this plot is a theoretical fit of equation 6.17 to our data. From
this fit we can extract values for the unsaturated small signal gain coefficient αm0
89
5
4
)W( rewoP tuptuO
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
Output Coupler (d1)
0.6
0.7
Figure 6.9. Power output from the laser cavity when driven unidirectionally using
an intracavity Faraday Rotator for seven different output couplers. The dashed line
represents a theoretical fit to the data used to determine the unsaturated small signal
gain and intracavity scattering losses for our home built ring laser.
and the cavity reflectivity value R0 . For this measurement, we find αm0 to be
0.637, corresponding to a single pass gain of 1.89 and R0 to be 87.3%. It is very
interesting to note that this value of 1.89 for the unsaturated small signal gain
is very close to the independently measured value of 1.87 reported in section 6.1
using a probe laser beam.
6.5
Unidirectionality with Self Injection
It has been anecdotally reported that in 1865 Christiaan Huygens noticed that
the pendulums of two clocks in his room invariably locked into synchronization
when they were hung close enough to one another yet remained free running when
moved farther apart [96]. While this coupling of the harmonic oscillators of the
pendulums was determined to result from vibrational coupling within the wall
they were hung on, the phenomenon of coupled oscillators still persists through a
variety of systems, including electric circuits [97, 98] and optical cavities [99, 100].
This idea of injection locking, whereby radiation from a master laser is injected
90
into the cavity of a slave laser, controlling its spectral properties, has been often
used to control the output of high power lasers, on the order of several Watts,
using only a weak (tens of milliwatts) master laser beam [101]. In fact, one report
demonstrated injection locking of a 13 Watt cw Nd:YAG ring laser using only a
40 mW laser diode [102].
It is from these studies of using a weak injected laser beam to control the
output characteristics of a more powerful laser cavity that the novel idea of self
injection was derived. Diagramed in figure 6.10, the basic idea of self injection is
as follows [75]. A small amount of output laser light from one output beam (one
direction of intracavity circulation) is picked off using the combination of a halfwave plate and a polarizing beam splitter. This low power beam then propagates
through the external self injected loop region, where it passes through a Faraday
rotator, acting as a unidirectional light diode, allowing light to only travel in the
forward direction. The weak probe beam is then aligned such that the path of its
propagation is aligned perfectly counter to that of the other output direction of the
laser (corresponding to the opposite direction of circulation within the cavity). In
this way, the weak beam is injected back into the laser cavity mode. Also, another
half wave plate is used to change the polarization of injected beam back to vertical.
The addition of this beam to the cavity mode causes stimulated emission in the
gain crystal in the same direction from which the weak beam is sourced, breaking
the symmetry of the ring laser cavity, and driving unidirectional operation of our
laser.
In addition to the half-waveplates, the polarizing beam splitter, and the Faraday rotator, three lenses are also included in the external loop in order to mode
match the weak injected beam to the laser cavity. The first lens, L1, has a focal
length of 17.5 cm and is located 17.5 cm from the midpoint of mirror M1 and
output coupler M2 of the cavity. This lens also acts as a collimation lens for the
laser for the high powered beam that does not get reflected by the polarizing beam
splitter. Also in the loop are two lenses (L2 and L3) having focal lengths of 100 cm
and -10 cm respectively that act as a telescope. From an ABCD calculation of the
beam parameters, at the output coupler M2, the output beam has a waist of 210.8
μm and a radius of curvature of 0.2632 meters while the injected beam has a waist
of 210.3 μm and a radius of curvature of 0.2621 meters. While not perfect, these
91
M
FR
HWP
Direction
M1
M2
L2
L3
HWP
L1
PBS
Pump
Pump
M3
Nd:YVO4
M4
Figure 6.10. Experimental layout for our novel scheme of self injection. A small portion
of the output light from our laser is picked off from the main beam using a half-wave plate
(HWP) and a polarizing beam splitter (PBS). This light then travels through a Faraday
rotator (FR) and another half-wave plate to change its polarization back to vertical. The
weak beam is then injected back into the cavity, where it causes stimulated emission in
the gain crystal, breaking the symmetry of the ring laser and driving unidirectional
operation. Also included in this loop are three lenses (L1, L2, L3) used to shape the
injected beam for mode matching (see text).
two beams are well coupled and the majority of the power injected back into the
laser cavity will be sufficiently mode matched to drive unidirectional operation.
For all the measurements presented in this dissertation used to quantify and
characterize the operation of our self-injected laser, a constant 2% of the output
power of the primary beam is picked off by the polarizing beam splitter to be
injected back into the laser cavity. When this self-injected beam is properly aligned
into the laser cavity, the output power of the laser was measured to be 3.24 Watts
for the output coupler with a reflectivity of 0.804. This power is twice what
was measured for a single output of the free running laser that was reported in
section 6.2, indicating unidirectional operation. Also, as expected, the high power
beam corresponding to the “wrong” direction of circulation within the laser cavity
that emits from the rejection port of the Faraday rotator in the external loop
disappears when the weak beam is injected back into the cavity, further evidence
of unidirectional operation.
As in section 6.2, we again want to qualitatively measure high frequency intensity noise of this self-injected laser. A small portion of the output beam was again
picked off and measured on a fast photodetector. Figure 6.11 shows the intensity
92
Figure 6.11. Measurement of the power stability for the self-injected laser cavity. Unlike
the free running laser, the unidirectional operation of the self-injected laser prevents mode
competition in the gain crystal resulting in a drastic reduction of intensity noise on the
output of the laser beam.
fluctuations of the output beam for a scan of 10 ms. Comparing this plot to figure
6.6, one sees that the intensity noise of the self-injected laser is much quieter than
that of the free running laser. This result is not surprising, given the now lack of
mode competition between the two directions of oscillation within the laser cavity.
In addition to looking at intensity noise fluctuations on the output of this selfinjected laser setup, we also wanted to look at the longitudinal mode structure
corresponding to the frequency components of the output light. To do so, we
again used a 300 MHz Fabry-Perot interferometer, as we did in section 6.2. The
output of the interferometer is reproduced in figure 6.12. From the figure, one can
see that in addition to reducing the intensity noise of the output beam, the lack
of mode competition inside the laser crystal also causes the laser to operate at a
single frequency, a very desirable and necessary quality for application to atomic
physics experiments.
In addition to verifying single mode operation, we also wanted to to check the
spectral linewidth of the output laser beam. To do so, we perform a heterodyne
measurement whereby the output of our self-injected laser is combined with light
93
Figure 6.12. The longitudinal mode structure of the self-injected laser as measured by
a 300 MHz Fabry-Perot interferometer. Unlike the case of the free running laser cavity,
the lack of mode competition in the gain crystal enables single frequency operation
from a tunable ECDL using a 2×2 single mode broadband fiber coupler manufactured by Thorlabs (part number FC1310-70-10-APC). The ECDL is detuned by
∼735 MHz with respect to the ring laser and one of the outputs from the fiber
coupler is sent onto a fast photodetector where the difference frequency of the two
lasers will create a beat note. This beat note signal is then sent from the photodetector into a spectrum analyzer and is reproduced in figure 6.13. The full width
half maximum of the signal measured by the spectrum analyzer was found to be
150 kHz, indicating that the spectral linewidth of the self-injected laser is less than
or equal to 150 kHz. As the reported value for the linewidth of the ECDL ranges
from 100 kHz to 1 MHz, it is impossible to draw a conclusion as to which laser
provides the dominant contribution to our linewidth measurement. Regardless,
150 kHz should be sufficient for our application to lithium experiments.
Finally, we wanted to measure the power output of the self-injected laser as
a function of output coupler, as we did in section 6.4. As before, seven different
output couplers were used and the total output power of our laser was recorded
for each using a thermopile detector. Figure 6.14 shows the result of this measurement. As before, the output coupler having a reflectivity of 0.804 resulted in
94
Figure 6.13. The spectral output of a heterodyne measurement of the linewidth of our
self-injected laser. The laser output is beat with the output of a tunable ECDL and
sent to a spectrum analyzer. From the full width at half maximum of this beat note
measurement, it is shown that the linewidth of our self-injected laser is no larger than
150 kHz.
the highest output power of 3.24 Watts. This corresponds to a 4.5% increase in
total power for the optimum output coupler. We also fit the data in figure 6.14
with equation 6.17 and included the fit as a dashed line in figure 6.14. As the
pump powers and alignments for the Faraday rotator laser measurement and the
self-injected laser were the same, the unsaturated small signal gain coefficient, αm0 ,
was held to be a constant 0.637, leaving the intracavity reflectivity R0 to be the
only free parameter. From this fit, R0 was determined to be 88.8%. This 1.5%
increase in intracavity reflectivity is not surprising, given that, for the self-injected
setup, the Faraday rotating element was removed from the laser cavity, reducing
the intracavity scattering losses. Therefore, by implementing this novel scheme
of injection locking, the output power of a ring laser can be increased due to the
reduction of intracavity elements. As there is nothing unique about the bidirectional operation of our diode pumped solid state ring laser, we would anticipate
this self-injection technique to be broadly applicable to any ring laser cavity.
95
Figure 6.14. The output power of our self-injected laser as a function of output coupling, δ1 , for seven different output couplers. The dashed line represents a fit to our
data with the intracavity reflectivity being the only free parameter. Compared to the
free running laser, we observe a 4.5% increase in output power for the optimum output
coupler.
6.6
Frequency Tuning
If we were only concerned with creating a high powered, single frequency, narrow
linewidth laser our task would be complete at this point. However, as is the case
with most laser applications in atomic physics, the frequency of the laser must be
tunable as well if we are to use such a laser for ultra-cold lithium experiments.
From our measurements of the gain profile in section 6.1, the gain peaks and the
free running laser operates at a wavelength of approximately 1342.6 nm. If we
were to frequency double this light, the resulting wavelength of 671.3 nm would
still be red detuned from the 670.977 nm D2 line of 6 Li. Therefore, we must tune
the wavelength of the solid state ring laser to a value of 1341.954 nm if we wish to
use the frequency doubled light.
To frequency tune the spectral output of our ring laser, we will employ the
use of various optical etalons. When inserted into the laser cavity, the etalon acts
as an additional optical resonator with its transmission periodically varying as a
function of frequency. On resonance, the reflections of the two surfaces destruc-
96
tively interfere and cancel each other out, resulting in full transmission through
the etalon. At the anti-resonances, however, these reflections do not entirely cancel each other out and there will be some reflected loss from the etalon. In this
way, the gain profile of the laser cavity can be affected. Also, as the etalon is
rotated, the resonance frequencies of the etalon are slightly changed. In this way,
the frequency of the laser can be tuned by merely tilting the etalon. For a small
tilt angle θ, the transmission, T , of an etalon is given by
T =
(1 − R)2
1 + R2 − 2Rcos[ 4πnd
(1 +
λ
θ2
)]
n2
(6.18)
where R is the reflectivity of the etalon, n is the index of refraction, and d is the
etalon thickness. For our laser, we will use silicon etalons to tune the frequency.
The index of refraction of silicon at 1342 nm is 3.5, resulting in a reflectivity
of 0.30864. In the subsections that follow, we describe our efforts to frequency
tune our laser using both an intracavity etalon as well as an etalon located in the
external loop for our self-injected laser.
6.6.1
Intracavity Tuning
To tune the laser using an intracavity etalon, a 32 μm silicon etalon was purchased
from Light Machinery for use. Because of the uncertainty in the etalon thickness
as quoted by Light Machinery (±2 μm), the first task was to measure the actual
etalon thickness. To do so, the etalon was oriented normal to a wavelength tunable
extended cavity diode laser (ECDL). The transmission of the etalon as a function
of wavelength was measured from 1330 to 1350 nm and is reproduced in figure
6.15. Also included in this figure is a fit to the etalon transmission using equation
6.18 for an angle θ = 0 and a reflectivity R = 0.30864. From this fit, we find
that the etalon has an actual thickness of 34.2 μm. Knowledge of this thickness is
important in determining the expected power and wavelength output of the laser
cavity when tuned via an intracavity etalon.
To tune the wavelength of our self-injected laser, the 34.2 μm etalon was inserted into the bow tie ring cavity on a rotation mount that allowed for the angular
adjustment of the etalon. Also, for this measurement, the output coupler of the
97
1.0
0.8
noissimsnarT
0.6
0.4
0.2
0.0
1330
1335
1340
1345
Wavelength (nm)
1350
Figure 6.15. The periodic transmission measurement of a thin silicon etalon as a
function of wavelength. The transmission was measured from 1330 nm to 1350 nm using
a tunable extended cavity diode laser at normal incidence to the etalon. The dashed line
represents a fit to the data of equation 6.18, indicating an etalon thickness of 34.2 μm.
laser cavity was changed from one with a reflectivity of 0.804 to one with a reflectivity of 0.956. While this change resulted in a reduction in power for the output of
the laser cavity, it was determined that an output coupler with a higher reflectivity
(lower loss) was necessary in order to tune the laser over a wider range. This can
be understood when considering the fact that in order for the cavity to lase, the
product of the gain and the cavity losses must be greater than one. Since the gain
of the laser is non-uniform (see figure 6.3), the reflective losses in the cavity must
be reduced as the wavelength is tuned away from the peak in the gain profile.
The power output of the laser as a function of wavelength via intracavity etalon
tuning is shown in figure 6.16. The wavelength of the laser light was measured using
an Agilent 86120b wave meter. As the etalon was rotated within the laser cavity,
13 different stable modes were observed. From this measurement, it can be seen
that using an intracavity etalon, we are able to tune the laser over a broad range
of wavelengths from 1341.898 nm to 1343.169 nm, covering the desired 1341.954
nm value for the wavelength doubled D2 line of 6 Li. This range corresponds to a
tunability of over 200 GHz.
98
Figure 6.16. The output power of our laser as a function of wavelength when tuned
via a 34.2 μm intracavity silicon etalon. The dashed line represents a fit to the data
when considering the convolution of the etalon transmission, the small signal gain of the
laser crystal, and the increase in intracavity scattering losses due to the insertion of the
etalon (see text).
It can be also be noted that in figure 6.16 the power output of the laser cavity is
smaller than that reported in figure 6.14 for a self-injected laser having an output
coupler reflectivity of 0.956. To understand this, we convolved the transmission
of the etalon (equation 6.18) with the measured small signal gain (equation 6.3
in order to determine the gain profile of the laser when an intracavity etalon is
inserted in the cavity. Using this convolution as the new small signal gain value,
we fit the data in figure 6.16 with equation 6.17, where the only free parameter is
the intracavity scattering loss parameter, δ0 . From a least squares fit to the data,
it was found that the reflectivity of the cavity R0 was 86.5%, a value lower than
the 88.8% measured before. This discrepancy indicates that the insertion of the
34.2 μm silicon etalon into the laser cavity results in an additional reflective losses
of 3.6%.
99
3.0
2.5
)sttaW( rewoP
2.0
1.5
1.0
0.5
0.0
1342.2
1342.3
Wavelength (nm)
1342.4
Figure 6.17. The output power of the laser cavity as a function of wavelength when
tuned via a 250 μm silicon etalon located in the external loop region of our self-injected
laser. By merely rotating the etalon, we were able to observe a tuning range that exceeded
38.9 GHz for our laser setup.
6.6.2
External Cavity Tuning
Finally, as a proof of principle experiment, we wanted to demonstrate the tunability
of our laser system using an etalon located in the external cavity loop of our selfinjected laser. To do this, we again use a silicon etalon, however this time, the
etalon used has a reported thickness of 250 μm. Also for this measurement, the
output coupler was changed back to having a reflectivity of 0.804 and the polarizing
beam splitter was changed to reflect 30% of the light from the main beam and into
the output coupler. This was done so as to increase the amount of power being
coupled back into the laser cavity via the external loop.
The output power as a function of wavelength measured by tuning the angle
of the external cavity etalon is shown in figure 6.17. As can be noted, 14 stable
modes of operation were observed by rotating the etalon in the external cavity
spanning a range from 1342.169 nm to 1342.403 nm, corresponding to a tuning
range of 38.9 GHz. It is also interesting to note that the data seem to be paired,
with the pairs being separated in wavelength by ∼0.007 nm. We believe this to
be the result of a “mode hop” of the external loop region. The output coupler,
100
beam splitter, and external cavity mirror make up a loop whose path length was
designed to be incommensurate with a multiple of the intracavity path length of
the ring laser cavity. In this way, the loop region can be thought of as an etalon
whereby small changes in the path length could cause mode hops in the output
frequency of the laser cavity.
To test this hypothesis, the path length of the external loop region was reduced
to be an even multiple of four times the path length of the ring laser cavity. The
polarizing beam splitter was located on a translation stage that could be driven by
a piezo-electric transducer (PZT) to finely control its position. In this way, it was
observed that when the path length was as near as possible to an integer multiple
of the cavity length, the wavelength of the laser could be greatly tuned by merely
applying a voltage on the PZT resulting in a small external cavity length change.
Using this technique, a range from 1341.762 nm to 1342.728 nm was observed,
covering the desired 1341.958 nm doubled wavelength for 6 Li. Because the cavity
length of the external loop was extremely sensitive to motion at this even multiple,
it was impossible to record the power output of the laser as a function of wavelength
and further investigation of this phenomenon may be warranted.
6.7
Conclusion
In this chapter, we have described the operation of our home built diode pumped
solid state ring laser system operating at a wavelength of 1342 nm. We began by
measuring the unsaturated small signal gain of our laser as well as the angular dependence of the measurement (stemming from the birefringent nature of the gain
crystal). We’ve also investigated the intensity noise and frequency characteristics
of both the free running laser as well as that for unidirectional operation. We’ve
performed an ABCD matrix calculation to determine the beam properties within
the laser cavity and have used this information to theoretically calculate the expected power output as a function of output coupler, which we have also measured.
Additionally, we constructed a home built Faraday rotator to drive unidirectional
operation of the laser as well as developed a novel technique of “self injection”
laser control as well. Finally, by using etalons both inside the cavity and in the
external loop of this self injected region we have demonstrated tunability of our
101
laser.
The next step necessary for implementing this laser system in ultra-cold atomic
physics experiments involves the efficient frequency doubling the laser light. While
we have observed a modest 10% efficiency using a single pass through periodically poled lithium niobate (PPLN), preliminary investigations of using PPLN
and lithium triborate as non-linear crystals in a four mirror frequency doubling
ring cavity have been met with very limited success. While some red light at 671
nm has been observed (on the order of a few μW), the use of these crystals within
a cavity have been limited due to the fact that as we attempt to actively lock the
cavity length on resonance with the input 1342 nm light, power buildup in the nonlinear crystals cause the crystals to heat and expand, changing the optical path
length and preventing the cavity from locking. Regardless, by overcoming these
technical difficulties, the use of the 1342 nm ring laser as a high power source of
671 nm light will be a welcome tool in the ultra-cold 6 Li community.
Chapter
7
Preparing a Two-Component Fermi
Gas
In order to prepare a two-component gas of ultra-cold 6 Li atoms for our studies of
the rapid suppression of interactions in a fermi gas, many different experimental
subsystems must be constructed and operated in parallel. In this chapter we will
outline the basic components for each of these subsystems. As this dissertation
expands upon the research of previous graduate students in the lab of Dr. Ken
O’Hara, many of the subsystems that follow have been previously described in
great detail. However, for completeness, it is prudent to include an overview for
each of these systems. For additional information and more in-depth discussion,
the reader is directed to references [6] and [103].
The basic outline of this chapter is as follows. In section 7.1, the various components comprising the ultra-high vacuum system required to isolate our experiment
from the surrounding environment are described. This isolation helps ensure that
the trapped atoms will be long lived in the various magnetic and optical traps due
to the reduction in background gas collisions commensurate with the vacuum. In
section 7.2 we discuss the laser systems used to cool and trap 6 Li atoms from a hot
atomic vapor. Section 7.3 describes the various coils of wire used to generate the
magnetic fields and magnetic field gradients for the manipulation of the hyperfine
energy level structure of the 6 Li atoms. In section 7.4, we describe the preliminary
trapping of atoms in a Magneto-Optical Trap (MOT), where the temperature of
the atoms is reduced by over six orders of magnitude.
103
Once the atoms have been sufficiently cooled and trapped in a MOT, section
7.5 describes the process of loading the atoms into an optical dipole trap. Finally,
in section 7.6 we will describe the setup for imaging the cloud of atoms using the
technique of absorption imaging.
7.1
Ultra-High Vacuum System
The basic layout of our experimental apparatus is shown in figure 7.1. This apparatus is composed of three primary regions, namely a lithium oven used to produce
a hot and effusive source of atomic lithium gas, a slower region whereby a narrow
collimated beam of lithium atoms are cooled in preparation to be trapped, and
finally an experimental chamber where the atoms are further cooled and trapped
using a variety of magnetic fields in preparation for further experimentation. To
maintain the ultra-high vacuum required for our experiments, a series of ion pumps,
titanium sublimation pumps, and non-evaporable getters are constantly used to
keep the pressure in the apparatus low. Finally, it should be noted that a gate
valve is also included in the system to isolate the the oven region from the zeeman
slower and experimental region for when we need to open the oven to atmosphere
and replenish the lithium source.
7.1.1
The Oven Region
The oven used as a source of 6 Li atoms for our experiments primarily consists
of a 1-1/3” conflat flange half nipple containing approximately 2 grams of solid
lithium. This nipple is located inside a two inch diameter form fitting aluminum
mount that provides excellent thermal contact with the half nipple ensuring an even
distribution of heat. The aluminum mount itself is wrapped with band heaters that
heat the solid lithium to a temperature of 435 ◦ C, well above the 180.7 ◦ C melting
temperature for lithium. This results in a vapor pressure of 3.6 × 10−4 Torr in the
oven itself [48].
At the top of the oven, the half nipple is connected via a 90◦ right angle elbow
to a blank conflat flange with a 0.25 inch diameter whole drilled in the center. This
hole, approximately 0.15 inches deep, acts as a nozzle for the hot lithium atoms as
104
2
1
1
2
4
5
6
3
1) Ion Pump
2) Ti-Sub Pump
3) Oven Region
4) Gate Valve
5) Zeeman Slower
6) Experimental Chamber
Figure 7.1. The experimental layout for our UHV system used to cool, trap, and
manipulate a gas of fermionic 6 Li atoms. The apparatus is divided into three regions,
namely an oven used to provide a source of hot atoms, a Zeeman slower used to reduce
the temperature of the atoms, and an experimental region where the cool atoms are
then trapped and cooled further for use in our experiments. Also shown are a variety
of vacuum pumps used to maintain the low pressure required for our experiments. This
figure has been adapted from reference [6].
they are emitted from the oven [104]. Because lithium can be very reactive with
copper, a nickel gasket is used at this nozzle in order to prevent the breakdown of
the gasket and the formation of lithium salts. Additionally, to prevent lithium from
condensing at this point, additional band heaters are used at the nozzle location to
raise its temperature to 435 ◦ C. This ensures that the lithium atoms will condense
back into the oven region, which determines the vapor pressure of the atomic gas.
The number density of the atoms inside the oven chamber region can be calculated
as [48]
n0 =
P
= 3.7 × 1010 cm−3 .
kB T
(7.1)
Using this value for the number density, it then becomes possible to determine the
atomic flux Φ as well as the diffusion rate of the atoms through the nozzle Ṅ [105]
105
Φ=
where v =
8kB T
πm
n0v
= 1.48 × 1015 cm−2 s−1
4
Ṅ = ΦAs = 4.7 × 1014 s−1
(7.2)
= 1610 m/s is the velocity of the atoms in the vapor and As is
the area of the nozzle.
Also included in the oven region is a UHV compatible shutter manufactured by
Uniblitz. This shutter is aligned so as to be in the path of the atomic beam. By
activating this shutter, it becomes possible to block the atomic beam and prevent
the atoms from propagating down to the experimental region of our apparatus.
This function is important, as it prevents undesirable collisions between the atomic
beam and the trapped samples of ultra-cold atoms, limiting their lifetime in the
trap.
7.1.2
The Zeeman Slower
To preliminarily cool the atoms emitted from the nozzle of the hot oven region,
we employ the use of a counter propagating resonant laser beam. The basic principle behind this cooling scheme is as follows. As an atom absorbs a photon from
the counter propagating beam, it experiences a change in momentum equal to the
momentum of the absorbed photon. This excited atom will then undergo a spontaneous emission event whereby the photon will then be re-emitted in a random
direction. After the absorption and emission of many photons, the momentum
kicks associated with the spontaneously emitted photon will cancel each other out,
leaving a net change in momentum along the direction of the cooling beam corresponding to the initial directionally dependent absorption of the photon. The
maximum force associated with the absorption of a photon can be calculated as
Fmax =
kΓ
2
(7.3)
where k is the momentum of the absorbed photon and Γ is the spontaneous
emission rate of the atomic transition (the factor of 2 originating from the equal
population of the ground and excited states). For the 2 S1/2 to 2 P3/2 cycling transi-
106
tion used here for our 6 Li atoms, this corresponds to a large maximum acceleration
of 1.8 × 106 m/s2 .
One problem that must be overcome in decelerating atoms using laser light is
the Doppler shift of the light due to the motion of the atoms. As the atoms absorb
photons and begin to slow down, the frequency of the light seen by the atoms is
shifted until the beam is no longer resonant with the cycling transition. Because
of this, only a small velocity class of atoms can absorb photons at any given time.
One way to overcome this limitation is through the implementation of a Zeeman
slower [10]. A Zeeman slower is a device that provides a spatially dependent
magnetic field along the direction of atomic motion. As the atoms travel down
the slower region scattering photons and reducing their velocity, the magnetic field
experienced by those atoms change as well, causing a spatially varying shift of
their energy levels keeping the atoms in resonance with the laser light.
The Zeeman slower that we use in our experiment can be seen in figure 7.2.
Three separate coils of wire are connected in series with one another to provide
a spatially dependent magnetic field. The first two spatially inhomogeneous coils
are wrapped around a long section of vacuum tubing while the third coil is located
at the end of the experimental apparatus and is used to cancel the magnetic field
inside the experimental chamber at the position of the magneto-optical trap. The
wires used in these coils are hollow, enabling them to be water cooled. The magnetic field as a function of location with respect to the apparatus is shown in the
subset of figure 7.2.
The laser light used to drive the cycling transition originates from the spectroscopy laser setup (see section 7.2). Approximately 35 mW of power is coupled
into a polarization maintaining fiber and launched down the slower axis of the
apparatus. The frequency of the light is red detuned by 814 MHz from the D2 line
of 6 Li and is right circularly polarized via a λ/4 wave plate prior to the chamber.
The beam has a diameter of approximately 15 mm at the entrance window of the
experimental chamber and is focused on the nozzle in the oven region. At the zero
crossing inside the slower region, the light is on resonance with the moving atoms.
The intensity of the beam is approximately four times the saturation intensity of
the transition and the velocity of the atoms at this location is approximately 540
m/s. To prevent the atoms from decaying into the 2 S1/2 | F = 1/2 dark state, an
107
Section 3
Section 2
Section 1
λ/4
λ/2
1000
EOM
λ/2
B (Gauss)
Slower Field
Profile
Slower
Launch
-800
Figure 7.2. Layout of the Zeeman slower used in our apparatus [6]. Three electromagnet
coils are wired in series to produce a spatially dependent magnetic field that shifts the
energy levels of the atoms to be continuously on resonance with a laser beam. The
absorption of photons from this counter propagating laser beam causes a momentum
transfer to the atoms, reducing their velocities and subsequently cooling the atoms as
they enter the experimental chamber.
electro-optic modulator is used to provide sidebands at ± 228 MHz for the slower
light. The blue detuned sideband repumps the atoms back into the F=3/2 state
where they can again absorb the carrier photons and continue the cooling process.
As mentioned above, the coils are design to cancel out the magnetic field inside
the experimental chamber after the Zeeman slower. In this region the atoms have
an average velocity of 30 m/s. This velocity is small enough to be captured via a
Magneto-Optical trap as described in section 7.4.
7.1.3
The Experimental Region
At the end of the slower region of our apparatus, an 8 inch multi-conflat spherical
octagon chamber acting as our experimental region is side mounted by one of its
8 2-3/4” conflat flanges. The benefit of using this type of a chamber, as opposed
to a glass cell, is that the 7 remaining side ports as well as the two larger ports
can be sealed with optical windows that have been anti-reflective coated for both
108
671 nm light as well as 1064 nm light. These windows provide excellent optical
access for the variety of lasers required for the additional cooling and trapping of
the pre-cooled atoms in this region. Additionally, the two larger windows are also
recessed, providing nearly 90◦ of optical access to the center of our experimental
region. The recess also allows for the close proximity of additional coils of wire used
to generate the electric field gradients of our MOT as well as the bias magnetic
field of our Feshbach magnets (see section 7.3).
Inside the experimental region, two rf coils are also mounted to the chamber
using groove grabbers. These rf coils are used to drive magnetic dipole transitions
between different magnetic sub levels of our 6 Li atoms. By locating these coils
within the chamber, the power required to drive the transition is greatly reduced,
enabling efficient transfer.
7.1.4
Vacuum Pumps
In order to maintain ultra-high vacuum (UHV) inside the apparatus, several different techniques for pumping out the system are employed. While roughing pumps
and turbo pumps are used to initially reduce the pressure inside the chamber
from atmosphere down to ∼ 1 × 10−6 Torr, these pumps are quite noisy, causing vibrations in the system. Furthermore, these pumps are unnecessary once
we engage additional pumps for continuous use in the system. The three main
types of pumps that are used continuously in the system are ion pumps, titanium
sublimation pumps, and non-evaporable getters.
The principle behind the operation of ion pumps is that background gasses
become ionized and accelerated via a strong dc electric field toward a chemically
active cathode [106]. For the ion pumps used in our setup, this titanium cathode
is specially designed for removing large amounts of noble gases and hydrogen from
the system, operating at pressures ranging from 10−2 to 10−11 Torr.
Titanium sublimation pumps are used to supplement the pumping of the ion
pumps in our system. Since titanium is a highly reactive metal, we can coat the
inside walls of our experimental chamber with a thin layer of titanium to aid in
the removal of unwanted atoms and molecules in our system. To coat the walls, a
high current (on the order of 50 Amps) is sent through a titanium filament located
109
within the vacuum chamber. This current causes the sublimation of a thin layer
of titanium on the walls. This thin layer can pump the chamber at a rate of 10 l
s−1 cm−2 . Over time, the titanium layer becomes saturated and the walls must be
coated with a new layer of titanium to maintain pumping efficiency.
Non-evaporable getters work based on similar principles to titanium sublimation pumps in that they absorb highly reactive molecules from the vacuum chamber. These getters are composed of a TiZrV alloy deposited on an amagnetic strip
that can be cut to the appropriate size for use in our vacuum system. These strips
can pump at rates of 0.1 l s−1 cm−2 , and, much like the titanium, they must be
reactivated from time to time via heating as they surface becomes saturated.
Using a combination of ion pumps, titanium sublimation pumps, and nonevaporable getters, the day to day operating pressure of our apparatus is approximately 7 × 10−9 Torr in our oven region and 4 × 10−10 Torr in our experimental
region. Differential pumping between the two regions is made possible by a four
inch long, quarter inch diameter copper tube that separates these two regions.
This tube restricts the conductance between the two sections of our apparatus and
allows the oven region to operate at a higher pressure at its full operating temperature without increasing the undesirable background gas collisions inside the
experimental region.
7.2
Laser System Overview
In this section, we describe the laser systems that produce the 671 nm light used
in our experiments for cooling, trapping, manipulating, and probing an ultra-cold
6
Li fermi gas. It is intended that this section will provide a brief overview of the
laser systems used for these application without overwhelming the reader with the
numerous details pertaining to the two setups. For a more thorough discussion
of the individual components in each of these systems, please see the excellent
description in reference [6].
Each of the two systems used to generate the required power to complete the
desired tasks originates from a tapered amplifier that is seeded by a grating stabilized, tunable, extended cavity diode laser (ECDL) manufactured by Toptica. The
tapered amplifiers work by taking the relatively low injected power of the seed
110
laser (on the order of 10 mW) and amplifying that power to approximately 500
mW. The first system described, referred to as the spectroscopy laser, generates
the power necessary to operate the Zeeman slower and iodine lock while the second
system, the experimental laser, produces the power for the cooling and repumping
lasers of the MOT as well as the imaging laser for our absorption imaging setup
described in section 7.6.
In order to produce light at the proper frequency for cooling and trapping 6 Li,
the frequency of the amplified laser light from the spectroscopy laser system is
locked using sub-Doppler saturated absorption spectroscopy of molecular iodine
[107]. In particular, the ECDL laser was frequency locked to the a1 0 hyperfine
+
singlet of the R(142)5-6 rovibronic transition of the B0+
u ← X0g electronic system
in 1 27I2 [108]. This singlet was useful as it provides a strong enough line strength
for high resolution feedback to lock the spectroscopy laser. Additionally, this rovibration transition is red-detuned from the 2 S1/2 | F = 3/2 hyperfine transition
by only 931 MHz, making it an ideal frequency reference for our laser system. The
remainder of the light from the spectroscopy laser system is then summarily coupled into a polarization maintaining fiber for use in the Zeeman slower as described
in section 7.1.
As for the experimental laser system, prior to amplification, a small portion
of the output of the ECDL is picked off and coupled into another polarization
maintaining fiber where it is used for imaging the cloud of atoms as described
in section 7.6. The remainder of the seed laser light is then amplified to 500
mW of power using another tapered amplifier setup. After amplification, a small
portion of the beam is picked off and is then combined with a small portion of
the output of the spectroscopy laser onto a high speed photodiode for beat-note
locking the experimental laser to the spectroscopy laser, as described in reference
[109]. In this way, the experimental laser is continuously phase-frequency locked to
the spectroscopy laser (which is in turn frequency locked to the iodine reference)
and can be tuned between over a range of 50 MHz to -2.1 GHz from the D2 line of
lithium. The remainder of the amplified light is then split into two paths for use
as the cooling and repumping beams for our MOT as described in section 7.4.
111
Feshbach Coils
MOT Coils
RF Coils
Zeeman
Slower
Recessed
Viewports
MOT Coil
Housing
Figure 7.3. Lateral cross section view of the experimental region of our apparatus.
From this view, one can see the location of the MOT coils, the Feshbach coils, and the
rf coils used to drive magnetic dipole transitions in our trapped atoms. This figure has
been adapted from reference [6].
7.3
Magnetic Field Coils
In addition to the various optical laser systems necessary to facilitate the cooling
and trapping of our ultra-cold atomic gases, a variety of magnetic field coils are
necessary as well. In this section we will describe three sets of magnetic field
coils used in our apparatus, namely a pair of anti-Helmholtz coils used to create a
magnetic field gradient for the MOT, a pair of Helmholtz coils used to provide the
DC magnetic field to tune the s-wave scattering length and interaction strength
of our two-component Fermi gas near a Feshbach resonance and a pair of rf coils
located inside the experimental region of the vacuum chamber that will be used
to drive magnetic dipole transitions between various hyperfine sub-levels of the
atoms. The location of each of these coils can easily be discerned from the lateral
cross section of the experimental region of our apparatus as shown in Figure 7.3.
We begin our discussion by taking a closer look at the MOT coils. These coils
are actually repurposed coils from previous experiments involving three-component
fermi gasses [6]. In these experiments, the coils were used in Helmholtz configuration to apply a dc magnetic field to boost the old Feshbach coils. Now, these coils
112
are wired in anti-Helmholtz configuration and provide a magnetic field gradient
for the MOT. The coils themselves consist of ten turns of copper wire that fit
within the recessed window housing, as seen in figure 7.3. The are located within
a water cooled housing that keeps their temperature below the 200 ◦ C limit for
our window vacuum seals. They have a radius of 3”, a separation of 2.6”, and
are wired in series. Under normal operating conditions, approximately 25 amps of
current from a Sorenson power supply flow through the coils, providing an on axis
magnetic gradient of 30 gauss/cm for our MOT.
The Feshbach coils used to provide a dc bias field on our trapped atoms consist
of two independent coils of 16 turn shielded copper wire. This wire has a 3/16” ×
3/16” cross sectional area with a 1/16” × 1/16” hollow core that is used to flow
water through for the removal of heat. The coils are located on the outside of the 8”
viewports of our experimental system (see figure 7.3) and are separated by 3.75”,
approximately satisfying the Helmholtz condition. Each coil is also contained
within a plexiglass mount that prevents shifting of the coils as the large currents
are quickly turned off and on.
It should be noted that great lengths have been taken to control and stabilize
the current that flows through each of the Feshbach coils in our system. While an
excellent and thorough description of the control electronics has been described
in the appendix of reference [103], a brief summary is provided here. The current
for each coil is supplied by two current supplies (Agilent 6690A) wired in parallel
for master/slave operation. Together, these supplies are capable of providing up
to 880 amps of current for each coil, well in excess of that needed to generate the
834.1 Gauss for the broad Feshbach resonance as described in section 2.3. The
output current of the two supplies are continuously measured via a Danfysik 867
current transducer. The current then passes through an array of MOSFETs wired
in parallel before continuing on to the Feshbach coils. By applying a gate voltage
to the MOSFET arrays, we can set the current flowing through the coils in our
system.
To apply the appropriate gate voltage for the desired magnetic field, the following feedback loop is employed. First, a commanded current from the timing
computer is converted from an analog signal to a digital signal using a field programable gate array (FPGA). This is done so that the signal, as it travels from
113
the timing computer across the room to the control electronics, does not pick up
any noise on the analog signal. Once across the room, the signal is then sent into
another FPGA that is used as a PID control circuit. The digital output of this
FPGA is then converted to an analog signal again via a digital to analog converter
which then both feeds back to the second FPGA and controls an analog control
circuit. The analog control circuit is used to control the gate voltage of the MOSFET bank which controls the output of the current supplies. As mentioned above,
the current flow is measured via the Danfysik transducers, which then feeds back
to the analog controller. Though seemingly complicated, this setup enables us to
have a magnetic field stability on the order of 250 μG— an impressive feat for an
800 Gauss field.
The final set of coils used in our experiment are a pair of rf coils located within
the vacuum of the experimental region. These coils consist of a single loop of a
copper strip measuring 0.25” tall and 0.01” thick. As seen in figure 7.3, these
loops are shaped into “race-track” shaped structures approximately 3.5” long and
1” wide. The two loops are separated by 1” and centered on the location of the
atomic gas. In this way, the rf current applied to the loops act as a nearly uniform
oscillating magnetic field in the ŷ direction. These loops are connected to bnc
feed-throughs in the walls of our UHV vacuum system. The signal applied to the
rf coils originates from either a home build direct digital synthesizer circuit or an
Agilent waveform generator before being amplified and sent to the coils. In this
way, the frequency of the rf signal can be tuned to be on resonance with the atoms
so as to drive transitions between different magnetic hyperfine levels of the atoms
via magnetic dipole transitions.
7.4
Magneto-Optical Trapping
Even though the velocities of the atoms exiting the Zeeman slower region have
been reduced to approximately 30 m/s, this is still too hot of a temperature for
the atoms to be directly loaded into a crossed dipole trap for our experiments.
Because of this, we first load the atoms into a magneto-optical trap (MOT) to
provide additional cooling.
The MOT is a technique that combines a spherical quadrupole trap with three
114
a)
b)
F = 1/2
F = 3/2
2 2P3/2
4.4 MHz
F = 5/2
σ+
Repumping
Cooling
σ−
σ+
F = 3/2
2 2P1/2
26.1 MHz
F = 1/2
D2 = 670.977 nm
σ
σ
+
−
σ−
F = 3/2
2 2S1/2
228.2 MHz
F = 1/2
Figure 7.4. a) The basic layout for the magneto-optical trapping of 6 Li. Three pair of
red detuned orthogonal laser beams overlap the zero magnetic field location between two
magnetic coils in anti-Helmholtz configuration. By choosing the proper polarizations,
the combination of magnetic and optical fields provide a restorative cooling force on the
atoms located in this spatially overlapped region. b) The energy level diagram showing
the cooling and repumping laser transitions for our MOT (solid lines). The dashed lines
show the available channels for spontaneous emission, indicating the necessity of the
repumping laser.
pairs of counter propagating σ + and σ − polarized laser beams as seen in figure
7.4a. These three pairs of counter propagating beams are orthogonally oriented
and overlap each other in the center of the experimental chamber at the midpoint
between the MOT coils. Each beam is red detuned from the 2 S1/2 | F = 3/2 →2
P3/2 D2 line of 6 Li. This detuning is chosen such that as an atom travels across the
region of space where these beams overlap, there will be a component of the motion
of the atom that travels counter to at least one of the six laser beams. For this
direction, cooling light is doppler shifted into resonance and the atoms will receive
a momentum kick in the opposite direction, similar to the principle behind the
Zeeman slower. This type of cooling has been dubbed “optical molasses” because
of the viscous nature of the cooling beams [10][9].
Because the absorption of a photon from the MOT cooling beams causes a
small amount of heating as well, there is a lower limit to the temperature that can
be obtained using this technique, known as the doppler limit TD . Assuming that
the intensity of the cooling beams I is much less than the saturation intensity, Isat ,
115
the minimum temperature achievable by a MOT is given by [8]
2
Γ 1 + 2δ
Γ
kB T =
−2δ
4
Γ
(7.4)
where Γ is the natural linewidth of the transition and δ = (ω−ω0 ) is the detuning of
the laser. From the equation, one can see that the temperature TD is at a minimum
when δ = Γ/2, yielding a lower limit of 142 μK for 6 Li using this technique.
While these six laser beams can provide additional cooling for the atoms, a
spherical quadrupole magnetic field gradient is required to provide a restorative
confining force for the atoms [10, 8]. The polarization of the six beams and the
orientation of the anti-helmholtz coils are chosen such that as an atom is displaced
from the zero field location between the anti-helmoltz coils, it experiences a net
magnetic field that will cause a Zeeman shift on its energy levels. This shift brings
the detuned laser light from the cooling beams onto resonance with the atom,
which then begins to scatter photons and experience a restoring force pointing in
the direction of the zero field location. In this way, the atoms are both cooled and
trapped using a combination of magnetic fields and optical beams, giving rise to
the name “magneto-optical trap”.
As mentioned previously, the 2 S1/2 | F = 3/2 →2 P3/2 transition is not a closed
transition in 6 Li. Atoms that have been excited to the 2P3/2 state have a finite
probability of spontaneous decay back down to the 2 S1/2 | F = 1/2 state, which
is a dark state to the cooling laser light. If no corrective action were to be taken
to plug this leak, the atoms would quickly be pumped to this lower ground state
and no cooling could effectively take place. To compensate for this, additional
beams resonant with the 2 S1/2 | F = 1/2 →2 P3/2 transition are used to repump
the atoms back into resonance with the cooling laser light, as shown in figure 7.4b.
For our experiment, the cooling and repumping light used for the MOT are
derived from the experimental laser. Light of the two appropriate frequencies are
coupled into a 2 × 6 fiber optic coupler that mixes the two input frequencies and
the six beams used for the MOT. This results in 10 mW of power for the cooling
laser and 2 mW of power for the repumping laser in each of the six beams used
in the MOT. Also, in our experiments, we use a two stage cooling procedure to
optimize the cooling of our trap, as described in reference [6]. In this way, we are
116
able to produce a cloud of ∼ 108 atoms having a temperature of 200 μK with a peak
density of 2×1010 atoms/cm3 for loading into a optical dipole trap for additional
cooling and experimentation.
7.5
Optical Dipole Traps
As mentioned in section 3.2, the presence of an oscillating electric field can shift
the energy levels of an atom via the AC Stark effect. In many atomic physics
experiments, this “light shift” for an atom in the ground state can be exploited to
create an additional trapping potential for the atoms because of their polarizability.
The basic premise for this optical dipole trap is as follows. The dipole force
associated with the gradient of the electric field is given by [10]
Fdipole =
δ
Ω
∂Ω
2 δ 2 + Ω2 /2 + Γ2 /4 ∂z
(7.5)
where δ = ω−ω0 is the detuning of the light from resonance, Ω is the Rabi frequency
of the light, and Γ is the linewidth of the transition. When the detuning | δ| Γ
and for an intensity such that | δ| Ω, the dipole force equals the derivative of
the light shift
Fdipole
∂
−
∂z
Ω2
4δ
.
(7.6)
Therefore, the light shift for an atom in its ground state acts as a potential in
which the atoms can be trapped. This treatment also can be expanded to three
dimensions as
Fdipole = −∇Udipole
where
Udipole Ω2
Γ Γ I
=
4δ
8 δ Isat
(7.7)
(7.8)
where we have defined [10]
I
Isat
≡
2Ω2
.
Γ2
(7.9)
When δ is negative, the light shift provides an attractive trapping potential whereby
the atoms are forced to the region of highest intensity. In this way, a focused laser
beam can be used to create this optical dipole force trap.
117
Y
X
1070 nm
1064 nm
1064 nm
Figure 7.5. Cartoon layout of the three lasers that overlap to form the optical dipole
force trap for our experiments. Two 100 watt 1064 laser beams intersect at a relative
angle of 11◦ with each beam having a waist of 30μ. A third 1070 nm 100 watt laser
overlaps the other two, forming a deep trap used to capture atoms from our MOT. Also
shown is the side view of one of the anti-Helmholtz coils used to provide the spherical
quadrupole trap for our MOT.
It should also be noted that the presence of this trapping potential could also
cause atoms to become ejected from the trap due to the spontaneous emission of
photons from the light. The photon scattering rate [10] is equal to the spontaneous
emission rate times the excited state population, given by
Rscatt =
Γ
Ω/2
.
2
2
2 δ + Ω /2 + Γ2 /4
(7.10)
In the limit that the detuning |δ| Γ, Rscatt becomes
Rscatt Γ Γ2 I
.
8 δ 2 Isat
(7.11)
Comparing this equation to the dipole potential, we see that the scattering rate
goes as I/δ 2 whereas the trapping potential goes as I/δ. Therefore, in order to
provide an effective trapping potential and minimize the effects due to spontaneous
emission, the wavelength of the trap laser must operate at a sufficiently large
detuning.
118
Figure 7.6. Absorption image of a cloud of 6 Li atoms trapped in a) two 1064 nm beams
forming a crossed dipole trap b) a single 1070 nm beam c) a combination of the two 1064
nm beams and the 1070 nm beam.
For our experiments, three intersecting laser beams are used to create our
optical dipole trap as shown in figure 7.5. Two of the laser beams are derived from
a 1064 nm fiber laser while the third beam, along the x-direction, is derived from
a 1070 nm fiber laser. Because these wavelengths are far detuned from the 671 nm
D1 and D2 lines of 6 Li, the effects of spontaneous emission on atoms trapped in
these potentials are minimal. As a result, the intensity gradient of each of these
beams must be very high in order to provide a practical trapping potential. For
our experiment, both the 1064 and 1070 nm laser beams are approximately 100
Watts in power in each beam and have been focused down to a waist of ∼ 30 μm.
This creates a trapping potential that is 400 μK deep and able to trap the 200 μK
atoms from our magneto-optical trap. Additionally, sideband excitations of the
trap by misaligned Raman beams indicate that this crossed dipole potential has
trap frequencies ω = 2π × (7.2, 3.4, 11.2) kHz. In figure 7.6, we show false color
absorption images of our atomic cloud trapped in either the crossed 1064 nm laser
beams, the 1070 nm laser beam, or a combination of all three.
7.6
Atomic Imaging
As seen in the false color image in figure 7.6, experimental information about our
cloud of trapped atoms is measured and recorded via a process known as absorption imaging. The basic premise behind absorption imaging is this. The trapped
atoms are exposed to a laser beam of resonant light that is incident upon a CCD
camera. If there are atoms in the trap, they will absorb photons from the reso-
119
nant light and scatter them in a random direction, leaving a shadow on the CCD
camera corresponding to the location of the atoms. From this absorption image,
information about the total atom number, number of atoms in each spin state,
temperature, density, and quantum degeneracy can all be determined, making it
the primary source of data collection in our experiments.
For a probe beam with initial intensity I0 propagating through a cloud of atoms,
the intensity will be attenuated by the atoms according to
I = I0 e−D
(7.12)
where D is the optical depth of the cloud. For a probe beam that is resonant with
a single transition having saturation intensity Isat , the optical depth is given by
D = D0
1
1+
I
Isat
+
(7.13)
4Δ2
Γ2
where Δ is the detuning of the probe beam from the atomic resonance with natural
linewidth Γ and D0 is given by
D0 = σ0 ñ
(7.14)
where σ0 is the polarization averaged resonant scattering cross section and ñ is
the column density, corresponding to the spatial density integrated over the ẑ
direction.
For our experiments, all images taken of the cloud of atoms occur in the so
called “high field” limit. In this limit, the atoms are imaged in the presence of
a magnetic field causing a Zeeman splitting of the hyperfine energy levels, as described in section 2.1. In this way, the probe light used to image the cloud can
be resonant with only one transition, enabling us to determine the individual populations of the different magnetic hyperfine states. The probe beam is derived
from the experimental laser system (see section 7.2) and is σ − polarized, propagating along the positive ẑ quantization axis of our system. This beam is tuned to
be on resonance with the 1 − 1 1 ± 1, 0 → 3 − 3 1 ± 1, 0 transition in the
2
2
2
2
| J mJ I mI basis. This transition is a closed cycling transition and does not need
repumping light, as would be the case for zero magnetic field imaging.
The actual experimental procedure for taking an absorption image of the cloud
120
of atoms is as follows. First, a short 20 μs resonant imaging pulse Iabs is pulsed on
the atoms and the resulting shadow is recorded by the CCD camera. This pulse
is sufficiently strong so as to rapidly heat all of the atoms out of the trap due to
spontaneous emission. Next, a reference image Iref is taken using an identical pulse
2.159 ms after the first. Finally, a third background image Ibkgrd is taken several
hundred milliseconds later. The transmission of the imaging light as detected by
the camera can then be calculated as
T (x, y) =
Iabs (x, y) − Ibkgrd (x, y)
.
Ires (x, y) − Ibkgrd (x, y)
(7.15)
From this transmission, the density distribution n(x, y) can then be extracted as
n(x, y) = −
1
ln [T (x, y)]
σ0 M 2
(7.16)
where M is the magnification of the lens system in front of the CCD camera. This
density distribution is the primary measurement of our system and is useful for
extracting numerous experimental parameters as mentioned above. For additional
information about the imaging process, see references [6] and [103].
Chapter
8
The Rapid Control of Interactions
In this chapter, we present the development of methods to control interactions in
a two-component Fermi gas. As mentioned before, for atoms in the two lowest
hyperfine magnetic spin states of fermionic 6 Li (| 1 and | 2), there exists a very
broad Feshbach resonance in the vicinity of 834.1 where the s-wave scattering
length diverges. In addition, a much more narrow Feshbach resonance having a
width of 0.1 Gauss located at 543.286 Gauss has also been experimentally verified
[53]. Near this narrow resonance at a field of 527.51 Gauss the scattering length
crosses zero, allowing us to investigate a non-interacting gas. We will first develop
methods to transfer atoms between states | 2 and | 5 near this zero-crossing of
the scattering length and then study Raman transfer near the broad Feshbach
resonance.
To rapidly control interactions in this gas, we will use two phase locked lasers
to quickly drive Raman transitions between different magnetic hyperfine levels.
As calculated in section 2.4, for values around the 834.1 gauss Feshbach resonance
seen for a |1-|2 mixture, the |1-|5 s-wave scattering length is around -3 Bohr.
Therefore, by creating a |1-|2 mixture of atoms at a magnetic field near the broad
Feshbach resonance where the scattering length is very large, we quickly control
the strength of interactions between the two atoms by driving a transition from
|2 → |5, reducing the s-wave scattering rate by several orders of magnitude.
This chapter is divided up as follows. Section 8.1 will describe the basic procedure for preparing an initial two-state Fermi gas starting with loading atoms from
a MOT. Next, we will discuss the various techniques employed to phase lock two
122
lasers in section 8.2. In section 8.3, we demonstrate the rapid transition from state
| 2 to state | 5 for a two-component mixture at the 527.51 Gauss zero-crossing.
Finally, in section 8.4, we will discuss the next steps for our experiment.
8.1
Initial Preparation
To initially prepare a two-component Fermi gas for our experiments, we begin by
loading a magneto optical trap (MOT) with 6 Li atoms that have been pre-cooled
using a Zeeman slower as described in section 7.1. Atoms are then loaded into
the MOT for approximately 1 second. At this point, the detuning and intensity of
the cooling and repumping laser light are changed so as to be closer to resonance
and less intense. This reduces the temperature to approximately 300 μK— cold
enough to then be loaded into the 1064 nm crossed dipole trap.
Once the atoms are loaded into the crossed dipole trap, the repumping light
is turned off and the atoms are optically pumped from the 2 S1/2 F = 3/2 state
into the 2 S1/2 F = 1/2 ground state. After optical pumping, the atoms are in an
incoherent mixture of states | 1 and | 2. Next, a magnetic field of approximately
572 Gauss is applied to the atoms by the Feshbach coils. This causes the atoms in
states | 1 and | 2 to undergo collisions with a positive s-wave scattering length.
It is at this point the atoms are further cooled via forced evaporative cooling by
reducing the intensity of the 1064 nm crossed dipole trapping lasers.
During the evaporative cooling process (and with a final pulse shortly thereafter) we want to prepare the atoms in a mixture of states | 1 and | 2. To do so,
we first transfer the atoms in state | 2 to state | 3 using rapid adiabatic passage.
Light from the imaging laser is then used to clear the atoms from state | 3 and
another rapid adiabatic passage sequence transfers the remaining atoms from state
| 3 to state | 2. For all these experiments, all the atoms are effectively removed
by the first clearing pulse, leaving now only atoms in state | 1. At the end of
evaporation, an additional 1070 nm optical dipole trap is turned on, providing an
additional tight confinement for the atoms along the ŷ direction. Finally, a π/2
pulse is applied via the rf coils located inside the experimental region, creating
a balanced mixture of atoms in states | 1 and | 2. It is at this point that the
applied magnetic field can be changed to the desired field of interest and further
123
experimentation can be undergone.
8.2
Phase Locking of Raman Lasers
To drive a Raman transition between two different hyperfine levels in an atom,
the two lasers must be phase coherent. The simplest way to generate two phase
coherent beams is to use one laser and an acousto-optic modulator (AOM). By
driving the AOM with an rf-signal whose frequency corresponds to the frequency
of the hyperfine transition, light from the 0th and 1st order of diffraction off the
AOM crystal can be used. This technique works well for transitions in 6 Li where
the spin of the electron does not need to be flipped during the transition. That is,
for transitions between states | 1, | 2, and | 3, it is experimentally convenient to
drive the AOM with the ∼80 MHz signal necessary for these transitions.
For transitions involving an electron spin flip, such as | 2 to | 5, the process is
not as simple. As discussed in section 2.1, the difference in energy between states
| 2 and | 5 scale as twice the bohr magneton (2.8 MHz per Gauss) for applied
magnetic fields. Therefore, calculating the difference in energy between the two
states, we find that at the 527.51 Gauss | 1-| 2 zero crossing, the corresponding
frequency for a | 2-| 5 transition is approximately 1.5 GHz. Near the broad Feshbach resonance at 834.1 Gauss, the separation is even larger— on the order of
2.35 GHz. Because of these microwave frequencies required for the transition, it
is challenging to use a single laser source and an AOM to generate the two laser
frequencies as the efficiencies at these frequencies are quite low. Additionally, we
desire to broadly scan the frequency difference to investigate BEC/BCS crossover
physics, but can only generate a difference frequency equal to the resonance of
the AOM. Therefore, we must instead phase lock two lasers with the appropriate
difference frequencies.
The method used to phase lock two lasers is outlined in figure 8.1. This technique is modeled after that used in reference [109]. The output of two DL100 ECDL
from Toptica Photonics operating near 671 nm are incident upon two polarizing
beam splitters where a small portion of each beam is picked off (approximately
1 mW of power in each). The two strong beams are combined and coupled into
a polarization maintaining fiber where they are routed to the main experiment
124
To Experiment
DL100
Bias-T
FET
Piezo
DL100
RF
Signal
Generator
OPLL
DET
+15 dB
Amp
+15 dB
Amp
Split
+15 dB
Amp
Phase
Advance
Filter
Low
Pass
80 MHz
+15 dB
Amp
RF
IF
LO
Mix
RF
Signal
Generator
Figure 8.1. Diagram showing the components making up the phase lock feedback loop.
A small amount of power from each of two diode lasers are combined on a high bandwidth
photo detector. The beat note signal of the two beams is then split where it takes one
of two paths. On the first path, the signal is amplified before being fed into an optical
phase-lock loop circuit where the beat note is compared to a reference oscillator and a
corrective signal is fed back to the piezo and the FET of the laser diode. The signal is
also amplified on the other path before being mixed with another reference oscillator,
filtered, and coupled to the bias-t connector of the same laser. In this way, we can lock
both the frequency and phase of the two diode lasers to arbitrary values.
for use in driving Raman transitions. The weak beams are also combined and
launched into a fiber, where they are incident upon a high speed photo-detector
(ν-focus model 1580). This photo-detector is fast enough (12 GHz bandwidth) to
measure the beat note frequency between the two diode lasers. The output of this
detector is then split using an rf-splitter from Mini-Circuits. For slow feedback to
the lasers, one of the outputs from the splitter is amplified by two +15 dB low
noise amplifiers (Mini-Circuits model ZX60-6013E-S+) and connected to the input
of an optical phase lock loop (OPLL) circuit, as described in reference [109].
The OPLL circuit used in this experiment primarily consists of a digital phasefrequency-discriminator chip (Analog Devices ADF4107). This chip measures the
beat note signal and digitally divides the frequency by a factor of 16. The frequency
and phase of this divided signal is then compared to a reference signal generated by
an 1 GHz MXG Waveform Generator by Agilent Technologies (model N5181A).
This waveform generator allows the user to program the desired difference fre-
125
quency between the two lasers for the OPLL circuit to lock. For small differences
between the beat and reference signals, the phase-frequency-discriminator chip
produces a feedback current proportional to this difference. This feedback current
is then either integrated for feedback to the piezo input of one of the DL100 lasers
(with a bandwidth of approximately 50 kHz), or amplified for direct feedback to
the laser diode current via a faster FET input (with a bandwidth of approximately
5 MHz— limited by the gain bandwidth product of the electronics).
In addition to using the OPLL circuit to lock the laser using a piezo and an
FET, we also apply feedback to the faster bias-t input of the diode laser. The other
output of the rf splitter mentioned above is again amplified by two +15 dB low
noise amplifiers before being mixed with an rf signal generated by another Agilent
N5181A. This Agilent generates frequencies up to 3 GHz and is set such that
the frequency is exactly 16 times that used for the OPLL circuit. Additionally,
the time reference for each of these two generators are synched to a 10 MHz
GPS signal, ensuring a consistent phase difference between their outputs. This
phase difference can then be adjusted by the user on the 3 GHz Agilent waveform
generator. The output of the mixer is then sent through an 80 MHz low pass filter
followed by a phase advance filter [110] before being coupled into the bias-t input
of the same laser. In this way, we can now provide feedback on the locked laser
with a bandwidth of up to 1 GHz for phase locking our two diode lasers.
To characterize how well our lasers are phase locked, we can look at the output
of the beat note signal on a spectrum analyzer (Hewlett Packard HP8595E). From
this signal, we can measure the carrier power fraction by looking at the ratio
between the power in the carrier signal to the total power in the beat note signal.
This carrier power fraction is related to the mean-square phase error Δφ2 by the
relation [111]
Pcarrier
exp − Δφ2 = ∞
P (ν) dν
−∞
(8.1)
where Pcarrier is the power in the carrier and P (ν) is the total power in the beat
spectrum. Figure 8.2 shows the beat spectrum of our laser phase locked to an
arbitrary 1.6 GHz value. From the figure, the total power in the beat spectrum
was measured to be 2.2 dBm with 0.35 dBm power in the carrier. This corresponds
126
Figure 8.2. The beat spectrum of our two phase locked lasers locked at 1.6 GHz. The
central peak of 0.35 dBm corresponds to the power in the carrier while the total power
from the occupied bandwidth measurement (2.2 dBm) can be used to determine the
carrier power fraction, and thus the mean-square phase error of our lock.
to a mean-square phase error of 0.42 rad2 .
127
8.3
Preliminary Investigations of a Non-Interacting
Gas
In this section, we describe the preliminary results of our investigations of using
Raman beams to drive | 2 to | 5 transitions. We began our investigation by
looking at the case of a non-interacting gas. For 6 Li fermions in an equal mixture
of states | 1 and | 2, s-wave scattering length goes to zero for a field value of
527.51 Gauss. From the calculations of the hyperfine splitting in section 2.1, we
find that the energy level splitting between states | 2 and | 5 to be 1.49383 GHz
at this field.
We first wanted to experimentally verify this transition frequency. To do so,
we used a microwave antenna to drive the the magnetic dipole transition from
state | 2 to state | 5 (see section 3.3). A single square loop antenna with 1.5
inch long sides was positioned normal to the 8 inch view port of the experimental
chamber region (see section 7.1). This antenna was powered by an Agilent waveform generator capable of producing frequencies up to 3 GHz. The output of this
function generator was amplified by a 100 Watt rf amplifier and then coupled to
the antenna.
The results of this measurement can be seen in figure 8.3. The frequency of the
waveform generator was scanned over a range of ±30 kHz relative to the central
transition frequency of 1.493831258 GHz for two different pulse duration times of
100 ms and 200 ms. Reported is the number ratio of atoms remaining in state | 2.
This ratio, Nratio is defined as
Nratio =
N2
N2 + N1
(8.2)
where N1 and N2 are the atoms in states | 1 and | 2 respectively. By measuring
Nratio , shot to shot fluctuations in the total atom number can be disregarded,
assuming each measurement begins with an equal distribution of atoms in states | 1
and | 2. From the figure, it can be seen that there is a loss in the number of atoms
in state | 2 for a transition frequency centered at 1.493831258 GHz corresponding
to population transfer from state | 2 to state | 5. While the Rabi spectrum should
resemble a sinc function, magnetic field instabilities on the order of 5 mG over the
128
0.5
0.4
oitaR rebmuN
0.3
0.2
0.1
200ms pulse
100ms pulse
0.0
-30
-20
-10
0
10
Detuning (kHz)
20
30
Figure 8.3. Measurement of the fractional population of atoms in state | 2 as a function
of frequency for a magnetic dipole transition. The frequency of the applied microwave
rf field was scanned over a range of ±30 kHz relative to the central transition frequency
of 1.493831258 GHz for two different pulse duration times of 100 ms and 200 ms.
200 ms time scale of the measurement could explain the broad linewidth. Though
a 5 mG instability on a 527 G magnetic field is pretty good, additional rf power
would enable us to drive faster transitions with a Fourier limited spectral linewidth.
Regardless, it is important to note that the time required to transfer the atoms to
state | 5 using a magnetic dipole transition is still quite long (many milliseconds),
which is much longer than the time-scale for ballistic expansion of the trapped
atoms.
With the transition frequency confirmed for this field, we now wanted to rapidly
transfer population from state | 2 to state | 5 using our phase locked Raman lasers.
To measure the population transfer, we look at the fractional population of atoms
in state | 2 as a function of pulse length. The results of our Rabi oscillation
measurement can be seen in figure 8.4. From the data, it can be seen that the
population of atoms in state | 2 is coherently transferred to state | 5 as described
by a decaying sin2 function of time. Coherent Rabi oscillations with a purely
sin2 dependence are predicted by equation 3.37. Here, the decay in coherence is
observed on a time-scale comparable to that corresponding to a 2 π pulse area.
129
0.5
0.4
oitaR rebmuN
0.3
0.2
0.1
0.0
5
10
15
Hold Time (s)
20
-6
25x10
Figure 8.4. Rabi oscillations demonstrating the transfer of atoms from state | 2 to
state | 5 using a two-photon Raman transition. The decay in coherence is attributed
to the small size of the Raman beams causing a spatially dependent Rabi frequency for
atoms located in different parts of the trap.
This decay in coherence is attributed to the fact that the waist of the Raman beams
used to drive the transition is approximately 25 μm. Such a small waist could lead
to intensity gradients across the sample, resulting in a spatial dependence of the
Rabi frequency [112]. Regardless, from the figure, it can be seen that the time scale
for the π pulse corresponding to complete population transfer will be on the order
of 5-10 μs, many orders of magnitude faster than a magnetic dipole transition.
8.4
The Next Steps
Unfortunately, figure 8.4 represents the extent of the data that has been collected
for our experiments on the rapid control of interactions in a two-component Fermi
gas. We had expanded the size of the Raman beam waist from 25 microns to 75
microns to test our hypothesis that spatially inhomogeneity of the Raman beam
caused the dephasing. However, at this time, our laser system suffered a catastrophic failure derailing our experiments. In switching out the degraded tapered
amplifier chip with both of our two backups, it was discovered that the input coat-
130
ing of both replacement chips had been damaged as well. This damage allowed for
spontaneously emitted photons from the amplifier to reflect back into the amplifier
resulting in amplified spontaneous emission. Because of this, new chips had to be
ordered from Toptica Photonics, an arduous process involving global shipping and
a seemingly endless wait for customs clearance. It is just recently that we have
received the new tapered amplifier chips. Now, the somewhat lengthy process of
installation and realignment of all the beams of our experimental laser system is
currently underway.
Once these issues have been rectified and the system is back online, the completion of this experiment should be relatively straightforward. We have expanded
the Raman beams to a larger beam size, which should allow for the complete population transfer of atoms from state | 2 to state | 5. With this being demonstrated,
we can then change our applied field to the location of the s-wave scattering resonance whereby atoms in states | 1 and | 2 are highly interacting. Repeating our
Rabi spectroscopy measurements at this field will indicate the precise frequency
and duration for a π-pulse to drive the population from state | 2 to state | 5.
Once these values have been determined, we can then create and release a | 1| 2 mixture of atoms from our trap and image the expansion of atoms in state
| 1 as a function of time. Without the application of the Raman π-pulse, the
| 1-| 2 mixture will be highly interacting and hydrodynamic expansion will be
observed. On the other hand, by applying the Raman π-pulse, the | 1-| 5 mixture
will be weakly interacting and will expand ballistically. In this way, we expect to
confirm our claim that through the application of a two-photon Raman pulse, we
can rapidly control interactions in a two-component Fermi gas.
Chapter
9
Conclusions and Outlook
In this dissertation, we have conducted a variety of experiments having application
to ultra-cold atomic gases. While the majority of the experimental results focused
on the development of a novel laser source for cooling and manipulating a gas of
fermionic 6 Li atoms, we have also reported on preliminary investigations of rapidly
controlling interactions in a two-component Fermi gas. The purpose of this chapter
is to summarize these results and to provide an outlook for future experiments in
our research lab.
9.1
Conclusions
In chapter 4, we described an experiment measuring the Verdet constant of undoped Y3 Al5 O12 (YAG) in the near infra-red. While other measurements have
been made of the Verdet constant for visible light, this experiment represents, to
the best of our knowledge, the first reported values for infra-red wavelengths. Linearly polarized light from two different radiation sources (a tunable external cavity
diode laser and a non-planar ring oscillator laser) were incident upon a polarizing
beam splitter that divided the light into two beams with equal powers. The two
beams were then focused onto a pair of balanced photodetectors that measured the
power of each beam. By inserting an undoped YAG rod located in the center bore
hole of a right hollow cylindrical magnet before the polarizing beam splitter, the
plane of polarization of the incident beam was rotated, resulting in a change in the
power measured by the photodetectors. By recording the power of each beam, the
132
small rotation angle of the probe beam polarization was determined as a function
of wavelength from 1300-1350 nm as well as at 1064 nm.
With knowledge of the small rotation angle, it became necessary to know the
integrated magnetic field over the length of the YAG crystal. To determine this,
the on-axis magnetic field for our right hollow cylindrical magnet was measured
using a standard gauss meter for locations extending beyond the end facets of the
magnet. Fitting these measured values, we found a magnetization, M = 0.125
Tesla. Integrating the on-axis magnetic field over the length of the crystal, we
found a total applied magnetic field of 8.00 × 10−3 T·m. Using this value, the
Verdet constant from 1300 nm to 1350 nm was found to be of the order 1.38
rad/T·m. Additionally, the optical activity of the YAG crystal at 1064 nm corresponds to a Verdet constant of 2.12 rad/T·m. As the Verdet constant is dispersive
in wavelength, these values of the Verdet constant were also used to determine
the energy band gap of YAG as well as the dimensionless K-parameter. We found
that our undoped YAG crystal had an energy band gap value of 8.1 eV ± 0.3
eV and a K-parameter of 596 rad/T·m ± 52 rad/T·m, consistent with previous
measurements made with visible light.
In chapters 5 and 6 we described the construction and analysis of a double
end pumped solid state ring laser using a Nd:YVO4 crystal as a gain medium.
Pumped by two 30 Watt fiber coupled diode arrays at 808 nm, this crystal has
a 4 F3/2 →
4
I13/2 transition at 1342 nm, nearly twice the necessary wavelength
for the D1 and D2 lines of 6 Li. The laser consisted of four plane mirrors in a
bow tie configuration stabilized by the thermal lensing of the gain medium. The
unsaturated gain profile of the medium was measured and found to have a peak of
1.87 at a wavelength of 1342.2 nm.
Two different techniques were used to drive unidirectional operation in the ring
laser. The first was a traditional technique of including a Faraday rotator inside
the ring laser cavity. This Faraday rotator was home built, constructed from a
rod of undoped YAG located at the center of a right hollow cylindrical magnetic
in conjunction with a half wave plate. The rotator was inserted into the bow tie
cavity and unidirectional operation was observed. The power output as a function
of output coupler reflectivity was measured using seven different output couplers.
From a fit to this data, the intracavity reflectivity was found to be 87.3% with a
133
single pass gain of 1.89, consistent with the previous independent measurement.
Also, the peak power output of this configuration was 3.10 Watts corresponding
to an output coupler reflectivity of 80.4%.
The other technique used to drive unidirectional operation of our ring laser
was a novel scheme of “self-injection”, whereby a small amount of the output laser
power is picked off from the main beam and injected back into the laser cavity.
This injected beam causes directionally dependent stimulated emission, breaking
the symmetry of the laser cavity and driving unidirectionality. One benefit of
using this technique is that by reducing the number of elements within the laser
cavity, the intracavity scattering losses are reduced as well, leading to higher output
powers. For this technique, 3.24 Watts of power were measured for the output
coupler having a reflectivity of 80.4%.
Next, we wanted to characterize the output quality of our “self-injected” laser.
To do so, the longitudinal mode profile of the laser was measured using a FabryPerot interferometer. From this measurement, the laser was verified as being single
mode. Also, the transverse mode profile was measured and found to have an M2
value of 1.04, indicating a nearly perfect TEM00 mode. Also, we wanted to measure
the linewidth of the laser. To do this, a heterodyne measurement was made by
beating the output of the ring laser with a diode laser on a fast photodetector and
looking at the output on a spectrum analyzer. It was found that the linewidth of
our laser was better than 150 kHz. Finally, we wanted to demonstrate the ability
to tune the wavelength of our laser using both an intracavity etalon and an etalon
located in the external arm of our self-injection loop. For the intracavity etalon,
the wavelength was tuned from from 1341.898 nm to 1343.169 nm, covering the
desired 1341.958 nm value for the wavelength doubled D1 line of 6 Li. For the
external cavity etalon, a range from 1342.169 nm to 1342.403 nm was observed,
corresponding to a tuning range of 38.9 GHz.
Finally, in chapter 8, we describe the experimental progress toward rapidly
controlling the interactions in a two-component Fermi gas. To do so, two external
cavity diode lasers were phase locked using a combination of piezo, FET, and bias-t
feedbacks. The beat note signal of the two lasers was measured on a fast photo
detector and the carrier power fraction was measured via a spectrum analyzer.
From this carrier power fraction measurement, it was determined that the mean-
134
square phase error of our frequency phase lock was 0.42 rad2 .
With the two lasers phase locked, we began our investigations by examining
a two-component non-interacting gas consisting of an equal mixture of the lowest
two magnetic hyperfine states of 6 Li at a magnetic field of 527.51 Gauss. This
field corresponded to the zero crossing of the s-wave scattering length for atoms in
these two spin states. First, we wanted to verify transitions from state | 2 to state
| 5 using a microwave magnetic dipole transition. These transitions were observed
for a time scale exceeding several hundred milliseconds. Next, we wanted to use
the two phase locked lasers to drive a two photon Raman transitions from state
| 2 to state | 5. While transitions were observed for time scales on the order of 10
μs, we were unable to completely transfer the population due to the decoherence
of the Rabi oscillation resulting from the finite 25 μm size of the Raman laser
beams. Unfortunately, this measurement was our last, as the system suffered a
catastrophic tapered amplifier failure while expanding the size of our Raman laser
beams.
9.2
Outlook
As discussed in section 8.4, the repair of the experimental system should be soon
completed and the demonstration of rapidly controlling interactions in a twocomponent Fermi gas should be straightforward. We have expanded the Raman
beams to a larger beam size, which should allow for the complete population transfer of atoms from state | 2 to state | 5. With this being demonstrated, we can
then change our applied field to the location of the s-wave scattering resonance
whereby atoms in states | 1 and | 2 are highly interacting. Repeating our Rabi
spectroscopy measurements at this field will indicate the precise frequency and
duration for a π-pulse to drive the population from state | 2 to state | 5.
Once these values have been determined, we can then create and release a | 1| 2 mixture of atoms from our trap and image the expansion of atoms in state | 1
as a function of time. Without the application of the Raman π-pulse, the | 1-| 2
mixture will be highly interacting and hydrodynamic expansion will be observed.
On the other hand, by applying the Raman π-pulse, the | 1-| 5 mixture will be
weakly interacting and will expand ballistically. In this way, we can make the
135
claim that through the application of a two photon Raman pulse, we can rapidly
control interactions in a two-component Fermi gas.
With the success of this technique, it now becomes possible to conduct an additional series of experiments utilizing the rapid control of interactions. One such
experiment is based on the ideas of reference [41] for quantum noise interferometry
in an expanding cloud. For a strongly interacting two-component Fermi gas on the
BCS side of the Feshbach resonance, Cooper pairs will be formed between atoms
of two different spin states. To identify these pairs, the atoms could be quickly
released from their confining potential and the cloud allowed to expand. Spatial
correlation spectroscopy is then performed to identify those pairs having opposite
momentum states. Traditionally, the highly interacting gases would undergo many
collisions during expansion destroying the correlations. However, by quickly reducing the interaction strength during the release by using this two photon Raman
technique, the cloud of atoms can expand uninhibited. After an amount of time
has passed, an additional Raman pulse could bring the atoms back into the initial state where an absorption imaging measurement could take place and spatial
correlations could be measured.
Another experiment taking advantage of these Raman pulses would be a cooling
experiment using resolved sideband cooling. For our atoms trapped in the 1064
and 1070 nm crossed dipole trap, we have demonstrated that the atoms are in
the resolved sideband regime. By setting the difference frequency between the two
Raman beams to be equal to that of the | 2 − | 5 transition minus the energy of
a few vibrational levels of the trap, the atoms could be transferred to an excited
state | 5 in a lower vibrational energy level of the harmonic trapping potential.
A far detuned pulse operating in the festina-lente limit could then optically pump
the atoms back into state | 2 where the final vibrational level would be lower than
its initial value. In this way, it may be possible to cool the cloud of atoms trapped
in the dipole trap to degeneracy without the use of forced evaporative cooling.
Finally, by changing the directions of the Raman lasers from being co-propagating
to having wavevectors in different directions, it becomes possible to use these lasers
to Bragg scatter atoms from the cloud to different momentum states. By transferring the atoms from a highly interacting state | 2 to a weakly interacting state
| 5, it becomes possible to measure the excitation spectrum and static structure
136
factor of a highly interacting Fermi gas for the BEC to BCS transition across a
Feshbach resonance akin to reference [40].
Appendix
A
Mathematica Code for Calculating
Scattering Lengths
In this appendix, we present the Mathematica code for calculating the s-wave scattering lengths for two atoms as described in section 2.4. The pages that follow show
the code for calculating | 1-| 5 interactions. However, with minor modifications,
the s-wave scattering length for any combination of states | 1 through | 6 can be
calculated.
Initialize Quantum Notation Package
Needs"Quantum`Notation`"
Quantum`Notation` Version 2.3.0. May 2011
A Mathematica package for Quantum calculations in Dirac braket notation
by José Luis GómezMuñoz
This addon does NOT work properly with the debugger turned on. Therefore the
debugger must NOT be checked in the Evaluation menu of Mathematica.
Execute SetQuantumAliases in order to use the keyboard
to enter quantum objects in Dirac's notation
SetQuantumAliases must be executed again in each new
notebook that is created, only one time per notebook.
AsymBoundCond_DiffMixtures_032311.nb
SetQuantumAliases
ALIASES:
ESCketESC
ESCbraESC
ESCbraketESC
ESCopESC
ESC.ESC
ESConESC
ESCtpESC
ESCqpESC
ESCqsESC
ESCsiESC
ESCevESC
ESCeketESC
ESCeeketESC
ESCeeeketESC
ESCebraESC
ESCeebraESC
ESCeeebraESC
ESCebraketESC
ESCeebraketESC
ESCeeebraketESC
ESCketbraESC
ESCeketbraESC
ESCeeketbraESC
ESCeeeketbraESC
ESCherESC
ESCconESC
ESCnormESC
ESCtraceESC
ESCcommESC
ESCantiESC
ESCsuESC
ESCpoESC
ket template
bra template
braket template
operator template
quantum concatenation infix symbol
quantum concatenation infix symbol
tensor product infix symbol
quantum product template
sigma notation for sums template
sigma notation for sums template
eigenvaluelabel template
eigenstate template
twooperatorseigenstate template
threeoperatorseigentstate template
bra of eigenstate template
bra of twooperatorseigenstate template
bra of threeoperatorseigentstate template
braket of eigenstates template
braket of twooperatorseigenstates template
braket of threeoperatorseigentstate template
operator matrix element template
operator matrix element template
operator matrix element template
operator matrix element template
hermitian conjugate template
complex conjugate template
quantum norm template
partial trace template
commutator template
anticommutator template
subscript template
power template
The quantum concatenation infix symbol ESConESC is
used for operator application, inner product and outer product.
SetQuantumAliases must be executed again in each
new notebook that is created, only one time per notebook.
Define Zeeman States
Notation`
AsymBoundCond_DiffMixtures_032311.nb
QuietSymbolize Θ ; QuietSymbolize Θ ;
QuietSymbolize Z ; QuietSymbolize Z ; QuietSymbolize R ;
QuietSymbolize R ; QuietSymbolize aS ; QuietSymbolize aT ;
SetAttributesAhfs, B, , gs, gi, ΜB, Θ , Θ , k, aS , aT , Constant
$Assumptions Ahfs Reals && B Reals && Reals && gs Reals && gi Reals &&
ΜB Reals && Θ Reals && Θ Reals && aS Reals && aT Reals && k Reals &&
Μ Reals && B 0 && 0 && gs 0 && gi 0 && ΜB 0 && k 0 && Μ 0
Ahfs Reals && B Reals && Reals && gs Reals && gi Reals &&
ΜB Reals && Θ Reals && Θ Reals && aS Reals && aT Reals && k Reals &&
Μ Reals && B 0 && 0 && gs 0 && gi 0 && ΜB 0 && k 0 && Μ 0
PhysConsts h 6.62606896 1034 ,
6.62606896 1034 2 Π, gj 2.0023010, gi 0.0004476540,
gs 2.0023193043737, ΜB 1.399624604 106 6.62606896 1034 ,
2
Ahfs 152.1368407 106 6.62606896 1034 6.62606896 1034 2 Π ,
ΜB 927.400915 1030 , aS 45.5 0.529 1010 , aT 2140 0.529 1010 ,
Μ 0.5 1026 , nK 109 , pK 1012 , kB 1.38 1023 h 6.62607 1034 , 1.05457 1034 , gj 2.0023,
gi 0.000447654, gs 2.00232, ΜB 9.27401 1028 , Ahfs 9.06438 1042 ,
ΜB 9.27401 1028 , aS 2.40695 109 , aT 1.13206 107 ,
1
1
Μ 5. 1027 , nK , pK , kB 1.38 1023 1 000 000 000
1 000 000 000 000
Z B_ :
Z B_ :
gi gs ΜB B
2
Ahfs
gi gs ΜB B
2 Ahfs
R B_ :
Z B2 2 ;
R B_ :
Z B2 2 ;
1
;
2
1
;
2
BDepend Θ ArcSin1 Θ ArcSin1 1 Z B R B2 2 ,
1 Z B R B2 2 ;
AsymBoundCond_DiffMixtures_032311.nb
Z1, S_, Sz_, _, z_ :
1
1
1
1
CosΘ ;
,
, 1 , 0z
, , 1 , 1z
SinΘ 2 S
2 Sz
2 S
2 Sz
1
1
Z2, S_, Sz_, _, z_ : SinΘ ,
, 1 , 1z
2 S
2 Sz
1
1
;
CosΘ , , 1 , 0z
2 S
2 Sz
1
1
;
Z3, S_, Sz_, _, z_ :
, , 1 , 1z
2 S
2 Sz
Z4, S_, Sz_, _, z_ :
1
1
1
1
SinΘ ;
,
, 1 , 1z
, , 1 , 0z
CosΘ 2 S
2 Sz
2 S
2 Sz
1
1
Z5, S_, Sz_, _, z_ : CosΘ ,
, 1 , 0z
2 S
2 Sz
1
1
;
SinΘ , , 1 , 1z
2 S
2 Sz
1
1
;
Z6, S_, Sz_, _, z_ :
,
, 1 , 1z
2 S
2 Sz
EZ3 EZ6 EZ1 1
2
1
2
1
8
B 2 gi gj ΜB Ahfs 2 ;
B 2 gi gj ΜB Ahfs 2 ;
4 B gi ΜB 2 Ahfs 2 4 B gi ΜB 2 Ahfs 2 2
16 2 B2 gi gj ΜB2 B2 gj2 ΜB2 Ahfs B gj ΜB 2 2 Ahfs2 4 ;
EZ5 1
8
4 B gi ΜB 2 Ahfs 2 4 B gi ΜB 2 Ahfs 2 2
16 2 B2 gi gj ΜB2 B2 gj2 ΜB2 Ahfs B gj ΜB 2 2 Ahfs2 4 ;
EZ2 1
8
4 B gi ΜB 2 Ahfs 2 4 B gi ΜB 2 Ahfs 2 2
16 2 B2 gi gj ΜB2 B2 gj2 ΜB2 Ahfs B gj ΜB 2 2 Ahfs2 4 ;
EZ4 1
8
4 B gi ΜB 2 Ahfs 2 4 B gi ΜB 2 Ahfs 2 2
16 2 B2 gi gj ΜB2 B2 gj2 ΜB2 Ahfs B gj ΜB 2 2 Ahfs2 4 ;
AsymBoundCond_DiffMixtures_032311.nb
PlotEZ1, EZ2, EZ3, EZ4, EZ5, EZ6 . PhysConsts,
B, 0, 530, PlotRange 5.5 1025 , 5.5 1025 4. 1025
2. 1025
100
200
300
400
500
2. 1025
4. 1025
Define Zeeman States for each Atom
Z11 Z21 Z31 Z41 Z51 Z61 Z1,
Z2,
Z3,
Z4,
Z5,
Z6,
S1,
S1,
S1,
S1,
S1,
S1,
S1z,
S1z,
S1z,
S1z,
S1z,
S1z,
1,
1,
1,
1,
1,
1,
1z;
1z;
1z;
1z;
1z;
1z;
Z12 Z22 Z32 Z42 Z52 Z62 Z1,
Z2,
Z3,
Z4,
Z5,
Z6,
S2,
S2,
S2,
S2,
S2,
S2,
S2z,
S2z,
S2z,
S2z,
S2z,
S2z,
2,
2,
2,
2,
2,
2,
2z;
2z;
2z;
2z;
2z;
2z;
Define Clebsch-Gordan Coefficients
S_S , MS_Sz
, ms1_ , s2_ , ms2_ :
s1_S1
S1z
S2
S2z
Ifms1 ms2 MS && Abss1 s2 S s1 s2,
ClebschGordans1, ms1, s2, ms2, S, MS, 0;
_ , M_z
, mi1_ , i2_ , mi2_ :
i1_1
1z
2
2z
Ifmi1 mi2 M && Absi1 i2 i1 i2,
ClebschGordani1, mi1, i2, mi2, , M, 0;
S_S , MS_Sz
, ms1_ , s2_ , ms2_ , i1_ , mi1_ , i2_ , mi2_ :
s1_S1
S1z
S2
S2z
1
1z
2
2z
Ifms1 ms2 MS && Abss1 s2 S s1 s2,
ClebschGordans1, ms1, s2, ms2, S, MS
, mi1 , i2 , mi2 , 0;
i11
1z
2
2z
AsymBoundCond_DiffMixtures_032311.nb
_ , M_z
, mi1_ , i2_ , mi2_ , s1_ , ms1_ , s2_ , ms2_ :
i1_1
1z
2
2z
S1
S1z
S2
S2z
Ifmi1 mi2 M && Absi1 i2 i1 i2,
ClebschGordani1, mi1, i2, mi2, , M
, _ , M_ S_S , MS_Sz
z
, ms1 , s2 , ms2 , 0;
s1S1
S1z
S2
S2z
,
i1_1
, i2_ , mi2_ , s1_ , ms1_ , s2_ , ms2_ :
mi1_1z
2
2z
S1
S1z
S2
S2z
Ifms1 ms2 MS && Abss1 s2 S s1 s2 && mi1 mi2 M &&
Absi1 i2 i1 i2, ClebschGordans1, ms1, s2, ms2, S, MS
ClebschGordani1, mi1, i2, mi2, , M , 0;
Define Antisymmetrized 2-Body Channel States
For a projected spin angular momentum (total) MT =0
6, 3 Z61 Z32 Z31 Z62 ;
2
QuietSymbolize E63 ; E63 SimplifyEZ6 EZ3 EZ2 EZ1;
5, 4 Z51 Z42 Z41 Z52 ;
2
QuietSymbolize E54 ; E54 SimplifyEZ5 EZ4 EZ2 EZ1;
5, 2 Z51 Z22 Z21 Z52 ;
2
QuietSymbolize E52 ; E52 SimplifyEZ5 EZ1;
4, 1 Z41 Z12 Z11 Z42 ;
2
QuietSymbolize E41 ; E41 SimplifyEZ4 EZ2;
2, 1 Z21 Z12 Z11 Z22 ; QuietSymbolize E21 ; E21 0;
2
ChannelStates 2, 1 ,
4, 1 ,
5, 2 ,
ChannelStateEnergies E21, E41, E52, E54, E63;
5, 4 ,
6, 3 ;
AsymBoundCond_DiffMixtures_032311.nb
Κ Evaluate0,
2 Μ E21 E41 ,
2 Μ E21 E54 ,
2 Μ E21 E52 ,
2 Μ E21 E63 . PhysConsts ;
, 2 , 0 , 0 , 0 SingletStates 0S , 0Sz
, 0S , 0Sz
;
z
z
, 1 , 1 , 1 , 0 , 1 , 1 TripletStates 1S , 1Sz
, 1S , 0Sz
, 1S , 1Sz
;
z
z
z
NumSingletStates LengthSingletStates;
NumTripletStates LengthTripletStates;
NumChannelStates LengthChannelStates;
SingletStateEnergies ConstantArray0, NumSingletStates;
TripletStateEnergies ConstantArray0, NumTripletStates;
Fori 1, i NumSingletStates, i ,
Forj 2, j NumChannelStates, j ,
SingletStateEnergiesi ExpandChannelStateEnergiesj
ExpandSingletStatesi ChannelStatesj2 ;
;
;
Fori 1, i NumTripletStates, i ,
Forj 2, j NumChannelStates, j ,
TripletStateEnergiesi ExpandChannelStateEnergiesj
ExpandTripletStatesi ChannelStatesj2 ;
;
;
SingletStateEnergies SimplifySingletStateEnergies;
TripletStateEnergies SimplifyTripletStateEnergies;
ClearNumSingletStates; ClearNumTripletStates;
ClearNumChannelStates; Cleari; Clearj;
NumSinglets LengthSingletStates;
NumTriplets LengthTripletStates;
SingletTanFncs ConstantArray0, NumSinglets;
TripletTanFncs ConstantArray0, NumTriplets;
Fori 1, i NumSinglets, i ,
SingletTanFncsi EvaluateR aS R R aS 2 Μ SingletStateEnergiesi 2 . BDepend .
PhysConsts;
;
Fori 1, i NumTriplets, i ,
TripletTanFncsi EvaluateR aT . BDepend . PhysConsts;
;
ClearNumSinglets; ClearNumTriplets; Cleari;
AsymBoundCond_DiffMixtures_032311.nb
NumStates LengthChannelStates;
NumSinglets LengthSingletStates;
NumTriplets LengthTripletStates;
SingletCGCoeffs ConstantArray0, NumSinglets, NumStates;
TripletCGCoeffs ConstantArray0, NumTriplets, NumStates;
Fori 1, i NumSinglets, i ,
Forj 1, j NumStates, j ,
SingletCGCoeffsij Evaluate Expand
SingletStatesi ChannelStatesj . BDepend . PhysConsts;
;
;
Fori 1, i NumTriplets, i ,
Forj 1, j NumStates, j ,
TripletCGCoeffsij Evaluate Expand
TripletStatesi ChannelStatesj . BDepend . PhysConsts;
;
;
ClearNumStates; ClearNumSinglets; ClearNumTriplets; Cleari; Clearj;
AsymBoundCond_DiffMixtures_032311.nb
SLengthAndK2BField_, Rad_ : Modulei, j, NumStates,
NumSinglets, NumTriplets, mat, cvect, Svals, K2, Κconst, slen,
NumStates LengthChannelStates;
NumSinglets LengthSingletStates;
NumTriplets LengthTripletStates;
mat ConstantArray0, NumStates, NumStates;
cvect ConstantArray0, NumStates;
Fori 1, i NumSinglets, i ,
cvecti Evaluate SingletCGCoeffsi1 R SingletTanFncsi .
B BField, R Rad;
;
Fori 1, i NumTriplets, i ,
cvecti NumSinglets Evaluate TripletCGCoeffsi1
R TripletTanFncsi . B BField, R Rad;
;
Fori 1, i NumSinglets, i ,
mati1 EvaluateSingletCGCoeffsi1 . B BField;
Forj 2, j NumStates, j ,
matij EvaluateSingletCGCoeffsij
1 Κj SingletTanFncsi . B BField, R Rad;
;
;
Fori 1, i NumTriplets, i ,
mati NumSinglets1 EvaluateTripletCGCoeffsi1 . B BField;
Forj 2, j NumStates, j ,
mati NumSingletsj EvaluateTripletCGCoeffsij
1 Κj TripletTanFncsi . B BField, R Rad;
;
;
Svals ConstantArray0, NumStates;
Svals LinearSolvemat, cvect;
Svals1
a12Tbl TableB, ReSLengthAndK2B, 33.2372 0.529 1010 0.529 1010 ,
B, 554.5, 556, .005;
AsymBoundCond_DiffMixtures_032311.nb
ListPlota12Tbl, Joined True, PlotRange 800, 800
500
554.8
555.0
555.2
555.4
555.6
555.8
556.0
500
a12Tbl TableB, ReSLengthAndK2B, 33.2372 0.529 1010 0.529 1010 ,
B, 800, 860, .1;
ListPlota12Tbl, Joined True, PlotRange 100 000, 100 000
100 000
50 000
810
820
830
840
850
860
50 000
100 000
For a projected spin angular momentum (total) MT =1
6, 4 Z61 Z42 Z41 Z62 ;
2
QuietSymbolize E64 ; E64 SimplifyEZ6 EZ4 EZ5 EZ1;
AsymBoundCond_DiffMixtures_032311.nb
6, 2 Z61 Z22 Z21 Z62 ;
2
QuietSymbolize E62 ; E62 SimplifyEZ6 EZ2 EZ5 EZ1;
5, 1 Z51 Z12 Z11 Z52 ; QuietSymbolize E51 ; E51 0;
2
ChannelStates 5, 1 ,
6, 4 ,
6, 2 ;
ChannelStateEnergies E51, E64, E62;
Κ Evaluate0,
2 Μ E51 E64 ,
2 Μ E51 E62 . PhysConsts;
, 2 , 1 SingletStates 0S , 0Sz
;
z
, 1 , 1 , 1 , 0 TripletStates 1S , 0Sz
, 1S , 1Sz
;
z
z
NumSingletStates LengthSingletStates;
NumTripletStates LengthTripletStates;
NumChannelStates LengthChannelStates;
SingletStateEnergies ConstantArray0, NumSingletStates;
TripletStateEnergies ConstantArray0, NumTripletStates;
Fori 1, i NumSingletStates, i ,
Forj 2, j NumChannelStates, j ,
SingletStateEnergiesi ExpandChannelStateEnergiesj
ExpandSingletStatesi ChannelStatesj2 ;
;
;
Fori 1, i NumTripletStates, i ,
Forj 2, j NumChannelStates, j ,
TripletStateEnergiesi ExpandChannelStateEnergiesj
ExpandTripletStatesi ChannelStatesj2 ;
;
;
SingletStateEnergies SimplifySingletStateEnergies;
TripletStateEnergies SimplifyTripletStateEnergies;
ClearNumSingletStates; ClearNumTripletStates;
ClearNumChannelStates; Cleari; Clearj;
ClearSingletTanFncs; ClearTripletTanFncs;
Clearmat; Clearcvect; ClearSvals
AsymBoundCond_DiffMixtures_032311.nb
NumSinglets LengthSingletStates;
NumTriplets LengthTripletStates;
SingletTanFncs ConstantArray0, NumSinglets;
TripletTanFncs ConstantArray0, NumTriplets;
Fori 1, i NumSinglets, i ,
SingletTanFncsi EvaluateR aS R R aS 2 Μ SingletStateEnergiesi 2 . BDepend .
PhysConsts;
;
Fori 1, i NumTriplets, i ,
TripletTanFncsi EvaluateR aT . BDepend . PhysConsts;
;
ClearNumSinglets; ClearNumTriplets; Cleari;
NumStates LengthChannelStates;
NumSinglets LengthSingletStates;
NumTriplets LengthTripletStates;
SingletCGCoeffs ConstantArray0, NumSinglets, NumStates;
TripletCGCoeffs ConstantArray0, NumTriplets, NumStates;
Fori 1, i NumSinglets, i ,
Forj 1, j NumStates, j ,
SingletCGCoeffsij Evaluate Expand
SingletStatesi ChannelStatesj . BDepend . PhysConsts;
;
;
Fori 1, i NumTriplets, i ,
Forj 1, j NumStates, j ,
TripletCGCoeffsij Evaluate Expand
TripletStatesi ChannelStatesj . BDepend . PhysConsts;
;
;
ClearNumStates; ClearNumSinglets; ClearNumTriplets; Cleari; Clearj;
AsymBoundCond_DiffMixtures_032311.nb
SLengthAndK2BField_, Rad_ : Modulei, j, NumStates,
NumSinglets, NumTriplets, mat, cvect, Svals, K2, Κconst, slen,
NumStates LengthChannelStates;
NumSinglets LengthSingletStates;
NumTriplets LengthTripletStates;
mat ConstantArray0, NumStates, NumStates;
cvect ConstantArray0, NumStates;
Fori 1, i NumSinglets, i ,
cvecti Evaluate SingletCGCoeffsi1 R SingletTanFncsi .
B BField, R Rad;
;
Fori 1, i NumTriplets, i ,
cvecti NumSinglets Evaluate TripletCGCoeffsi1
R TripletTanFncsi . B BField, R Rad;
;
Fori 1, i NumSinglets, i ,
mati1 EvaluateSingletCGCoeffsi1 . B BField;
Forj 2, j NumStates, j ,
matij EvaluateSingletCGCoeffsij
1 Κj SingletTanFncsi . B BField, R Rad;
;
;
Fori 1, i NumTriplets, i ,
mati NumSinglets1 EvaluateTripletCGCoeffsi1 . B BField;
Forj 2, j NumStates, j ,
mati NumSingletsj EvaluateTripletCGCoeffsij
1 Κj TripletTanFncsi . B BField, R Rad;
;
;
Svals ConstantArray0, NumStates;
Svals LinearSolvemat, cvect;
Svals1
aTbl TableB, ReSLengthAndK2B, 33.2372 0.529 1010 0.529 1010 ,
B, 0, 1000, 1;
AsymBoundCond_DiffMixtures_032311.nb
ListPlotaTbl, PlotRange 50, 0, Joined True
200
10
20
30
40
50
400
600
800
1000
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Vita
Ronald William Donald Stites
Education:
Ph.D. in Physics, The Pennsylvania State University,
University Park, PA (2013)
M.S. in Physics, Miami University, Oxford, OH (2005)
B.S. in Physics, Miami University, Oxford, OH (2003)
Employment: Graduate Research Assistant, The Pennsylvania State
University, University Park, PA (2005-2012)
Teaching Assistant and Course Instructor Miami University
Oxford, OH (2003-2005)
Research Assistant Miami University, Oxford, OH (2001-2005)
Awards:
NRC Research Associate Fellowship (2012)
Duncan Graduate Fellowship in Physics (2007, 2009, 2011)
Outstanding Graduate Student Research Award (2004,2005)
John and Genny Snider Award for Promise in Physics (2002-2003)
Selected Publications:
• “The Verdet constant of undoped Y3Al5O12 in the near infrared”
R. W. Stites, and K. M. O’Hara. Opt. Comm. 285, pp. 3997-4000 (2012)
• “Realization of a Resonant Fermi Gas with a Large Effective Range”
E. L. Hazlett, Y. Zhang, R. W. Stites, and K. M. O’Hara. Phys. Rev.
Lett. 108, 045304 (2012)
• “Preparing a highly degenerate Fermi gas in an optical lattice”
J. R. Williams, J. H. Huckans, R. W. Stites, E. L. Hazlett, and K. M.
O’Hara. Phys. Rev. A 82, 011610 (2010)
• “Evidence for an Excited-State Efimov Trimer in a Three-Component
Fermi Gas” J. R. Williams, E. L. Hazlett, J. H. Huckans, R. W. Stites,
Y. Zhang, and K. M. O’Hara. Phys. Rev. Lett. 103, 130404 (2009)
• “Three-Body Recombination in a Three-State Fermi Gas with Widely
Tunable Interactions” J. H. Huckans, J. R. Williams, E. L. Hazlett,
R. W. Stites, and K. M. O’Hara. Phys. Rev. Lett. 102, 165302 (2009)