Building a Better Atomic ClockbyBenjamin J. Bloom
B.S., Massachusetts Institute of Technology, 2008

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2014This thesis entitled:
Building a Better Atomic Clockwritten by Benjamin J. Bloom
has been approved for the Department of Physics

Jun YeJames Thompson

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Bloom, Benjamin J. (Ph.D., Physics)
Building a Better Atomic Clock
Thesis directed by Prof. Jun Ye

Since 1967, the Second has been defined by the CGPM as 9,192,631,770 periods of oscillation
corresponding to a transition between two hyperfine levels in a Cesium atom with zero external
fields. Key to this definition was the fact that as experimentalists found new and more precise ways
to measure this frequency, the definition of the second would become more and more accurate withtime. However, in the last 30 years, new technologies based on tunable lasers addressing opticaltransitions in atoms, ions, and molecules have offered an entirely new approach to defining the Second with significantly higher precision and accuracy. Here, I will show that by trapping thousandsof atoms inside a specially engineered optical lattice one can create an extremely accurate frequency
standard with 2 orders of magnitude improvement over current Cs standards. Furthermore, I will
explain that standards based on this technology are fundamentally more stable than the primary
standard by 3 orders of magnitude. Leapfrogging the currently held accuracy records of ion clocks,
this work documents the first optical lattice clock to best all other atomic clock implementations, amere 8 years after the first proof of principle experiments. In this thesis I will describe how we haveovercome a number of important systematic uncertainties to achieve these results. This revolutionin accuracy and precision opens the door for new experiments utilizing the clock as a probe ofquantum many-body physics. To this end, a new apparatus has been designed that combines theunprecedented precision of clock experiments with the amazing progress attained in quantum gas
experiments.

Dedication

To my family, both old and new.

Acknowledgements

Before I go on to thank all the professors, students, and researchers who helped me achieve
this PhD over the last few years, I first want to thank my family.
To my parents, I cannot imagine a more supportive and loving environment to grow up in.
Whether I was making a mobile of the planets or building an R/C airplane with my dad, they
allowed me to be me and supported me in every way.
To my brother and sister, whether or not you knew it you’ve always acted as a goalpost tome. Pushing me to work harder and achieve more, not just at work but in life too. Some of myhappiest memories are of the three of us on vacation hysterically laughing at god only knows what.
To Ruth, I don’t think I could have done this without you. The strength and love you haveshown me has finally made me realize where my new home is.
To Max, thank you for making me realize how any problem can be overcome with boundlessenthusiasm. To Einstein, thanks for coming back from your walkabout.
When I first showed up to JILA, Jun was a giant. His amazing work and reputation were allthat I knew. But over the last 5 years, I’ve come to learn that exceeding his successes in the lab
is his good nature and enthusiasm for Science! As all the research documented in this thesis andothers will attest to, our projects meander towards their destination tacking left and jibing rightas the grad students and postdocs make daily decisions. But at least once a week, Jun corrects
our course and steers the ship back to our destination. Jun’s mastery of experiment and complete
knowledge of all things ‘A’, ‘M’, and ‘O’ are an inspiration for everybody who works with him.
I also owe a great heartfelt thanks to my undergraduate advisor Vladan Vuletı́c who intro-

vi
duced me to the world of AMO, the most DIY discipline I have ever come across. His hilarious
combination of Comic Sans, Oscar Wilde quotations, and Clifford Algebras enriched everyday inlab as an undergraduate. Over the last 7+ years, since stepping into Vladan’s lab at the CUA,
I’ve come to depend on Jon Simon’s blend of camaraderie and mentorship. My circuitous path to
JILA actually began with a year at Cornell. But after having a change of heart I went back to thefield of AMO. Throughout that year though I got to spend the summer at CERN working on CMS
with Prof. Peter Wittich and volunteer in the new labs of Prof. Mukund Vengalattore. But what
I missed upon leaving Cornell and joining the Ye group were my first year graduate classmates atCornell in the Fall of 2008. Slogging through the problem sets and the grading together, the walks
to CTB, and waiting for what seemed like hours for our pizza to come at The Nines was all madefun by the shared misery of graduate school.
The Sr3 (or publicly the SrII) experiment was comprised of Sebastian Blatt, Travis Nicholson,
and Gretchen Campbell. After a few months Gretchen left for her new job, and except for theoccasional visit from Jan Thomsen, it was just us three. What I remember most from these firsttwo years were the long days and nights of work, ending either with hair brained schemes like baking
vacuum parts in somebody’s apartment at 2 am, or driving out in a snow storm to get a drink from
Conor O’Neill’s. As the Jolly German, Sebastian, was winding down his PhD we were joined by
Jason Williams our new Postdoc. His good spirits and hilarious tales about his endless menagerieof pets helped us push along through the switch to clock mode. Following that Sara Campbell
joined the fun, building the burninator Ti:Sapph which turned out to be so key in achieving ourclock goals. Her exuberance for ‘shredding the gnar’, doing science, and endangering her own life
will no doubt lead the new experiment to greatness :) . And most recently Ross Hutson and Reese
McNally joined the team. The sky’s the limit for the new apparatus and with Sara, Ross, Reese,and Jun leading the charge. I look forward to reading about all their future results.
None of the work in this thesis would be possible without the efforts of Mike Martin, Matt
Swallows, Mike Bishof and Yige Lin. Without the help of these clock experts no SrII would
exist today, especially Matt Swallows, who answered every dumb question I asked him. The new

vii
generation of SrI, Mike Bishof, Xibo Zhang, and Sarah Bromley, pushed to create what was at thetime the second best reported optical lattice clock so that a clock comparison was possible. The
future is bright for the latter two as they bring up their new cavity system while Mike leaves forhis new job. Wei Zhang joined strontium from the combs side of thing and made our comparisonspossible as the new laser guru. His new silicon cavity will one day revolutionize the clock community.
Trying to list all of the Ye labs members that have contributed to my projects by lendinga helping hand or answering questions would be impossible. Needless to say, thank you to all Yelabs members past and present. (Including the honorary ones like Jan Hall!) Special thanks should
also be extended to Judah Levine who helped us get an accurate Cs signal to lock our comb to. Itshould go without saying, this work has benefited in more ways than I can count from the dedicatedprofessionals in the electronics and machine shop. I especially want to thank Hans Green for his
work on the temperature sensors, Blaine Horner for his work on the new chamber, and Carl Sauer
for not only working on all the projects I gave him but also being my goto electronics guy.
Finally, special thanks needs to goto Aviv Keshet, for his chamber pumping designs, thepeople at Sharon Vacuum including Larry Resnick for their excellent machining skills, Tim Darby
at UKAEA, Tom Loftus at AOSense for his help with the new oven and Zeeman slower, and theresearchers at Jefferson Labs who baked and coated our new chamber Philip Adderley, Marcy
Stutzman, and Matt Poelker.

Contents

Chapter
1 Introduction
1.1

1

Optical Lattice Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

The JILA 87 Sr Lattice Clock . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.1

2 Building a Clock
2.1

2.2

The Blue Laser System

12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1

The Old Blue laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2

New Additions to the Blue team . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.3

The Blue Diode System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

The Magic Wavelength Buildup Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1

A Cavity Autolocker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3

Spectrally clean light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4

Easy-to-build DDS Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Running a Clock – A Software Revolution
3.1

The Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1

3.2

42

Synchronizing Two Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

The Slyfox Software Suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1

An Object-Oriented Programming Primer . . . . . . . . . . . . . . . . . . . . 46

3.2.2

An Example: The Slyfox Frequency Synthesizer . . . . . . . . . . . . . . . . . 48

ix
3.2.3

Bringing it Together: The Slyfox Frequency Locker Program . . . . . . . . . 52

4 Where in the world is 1 S0 →3 P0

63

4.1

Statistical Methods for Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2

The Density Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3

The Zeeman Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4

4.5

4.3.1

The First-Order Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.2

The Second-Order Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . . 70

The AC Stark Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.1

The Scalar and Tensor Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.2

The Vector Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.3

The Higher-Order Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

The DC Stark Shift
4.5.1

4.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Active Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Miscellaneous Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6.1

First and Second Order Doppler Shifts . . . . . . . . . . . . . . . . . . . . . . 82

4.6.2

AOM phase chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.6.3

Line Pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6.4

Probe Beam AC Stark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6.5

Background Gas Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Sr – The mini-COBE of Clocks
5.1

86

Understanding the Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.1

Ray Tracing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2

The Temperature Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3

Final SrII Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3.1

SrI–SrII comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

x
6 The New Sr Quantum Gas Extravaganza Machine
6.1

103

Long Vacuum Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1

The Pumpdown Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2

Recessed Viewports for spectroscopic resolution . . . . . . . . . . . . . . . . . . . . . 112

6.3

Optical Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.1

Dipole-Dipole Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.2

Spin-Orbit Physics in Alkaline Earth Atoms . . . . . . . . . . . . . . . . . . . 115

7 Summary & Outlook

121

Bibliography

125

Appendix
A Single Purpose Fast Arduino Example

134

B Technical Drawings for SrIII Apparatus

139

xi

TablesTable
4.1

Second Order Zeeman shift coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1

BBR shift magnitudes for various atom species . . . . . . . . . . . . . . . . . . . . . 87

5.2

SrII Effective Solid Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3

Diode temperature sensor corrections and uncertainties

5.4

SrII Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. . . . . . . . . . . . . . . . 96

Figures

Figure1.1Stability comparison of the primary frequency standards and their competitors . . .

6

1.2

Three leading systematics plaguing optical lattice clocks . . . . . . . . . . . . . . . .

8

1.3

The 87 Sr Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1

Diagram of the 922nm DBR laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

Diagram of Waveguide Doubler System . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3

Blue diode laser layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4

Injection Lock Tracking Servos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5

CAD drawing of Horizontal Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6

ECDL lock to 813nm Buildup Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7

Diagram of injection locked Ti:Sapph lock to 813nm Buildup Cavity . . . . . . . . . 27

2.8

Accelerometer traces of MOT coil shutoff . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9

JILA Slow Loop Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Diagram of the Auto Locker FSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.13 Optical Spectrum Analyzer measurements of Tapered Amplifier systems . . . . . . . 35
2.14 SrI - SrII frequency with and without interference filters . . . . . . . . . . . . . . . . 36
2.15 SrI TA shift drifiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.16 SPI use for controlling DDSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.17 Screenshot of DDS control frontend . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

xiii
3.10 Screenshot of the Frequency Locker program . . . . . . . . . . . . . . . . . . . . . . 56
4.1

Allan deviation of large lever arm density shift measurement. . . . . . . . . . . . . 68

4.2

Field rezeroing using the atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3

Sigma transition minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4

AC Stark measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5

DC Stark shifts in SrII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6

AOM Phase Chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1

Ray tracing model of the UHV vacuum chamber . . . . . . . . . . . . . . . . . . . . 91

5.2

Diode temperature sensor picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3

Zeeman slower heated window’s effect on the temperature sensor . . . . . . . . . . . 98

5.4

SrII - SrI Frequency Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1

AOSense system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2

SrIII Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3

SrIII Recessed Viewports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4

SrIII Multiband Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5

SrIII Broadband Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.6

SrIII coating layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.1

Red MOT absorption image of strontium atoms . . . . . . . . . . . . . . . . . . . . . 124

Chapter 1

Introduction

Measurement is a tricky business. In any quantitative experiment, a researcher tries to gain
understanding of some unknown quantity by performing a differential measurement. Unbeknownst
to most researchers, the differential aspect of their experiments is usually hidden under the monikerof whatever units they quote their result in. For most experiments, this is a trivial, usually forgotten
point, Hertz are Hz, Volts are V, and Henries are H. But for a select few experiments there exist,
at this moment in time, measurements that have no absolute relationship to these units. This has
not always been the case for optical lattice clocks, as even a few years ago frequency measurements,
which were just at the cusp of performing better than the highest performing primary standards,
were still possible via the use of optical frequency combs[1] .
Previous theses from the Ye group have referred to these standards coupled with opticalfrequency combs as rulers with equally spaced tick marks stretching from the radio-frequency (RF)domain all the way to the optical domain [2]. However, it is the fractional frequency accuracy which
is transferred with these ruler ticks, and not their absolute frequency uncertainty. In other words,
looking at an absolute scale from the RF domain to the optical domain, the superbly narrow tickmarks at low frequency become broader and fuzzier at higher frequencies.
While significant progress over the last 40 years has been made on primary frequency standards around the world [3], this technology is by every definition of the word mature. This isperhaps one of the most significant motivations for this work, the fact that new techniques andtechnologies offer a completely different approach to creating a stable frequency source. Essentially2
a top down method, where the standard is created in the optical domain and the stability andaccuracy is transferred for use in the RF domain. The work presented in this thesis is merely an
example of the ease of attaining an order of magnitude improvement to a new technology comparedto squeezing out a factor of 2 from a mature technology. However, the rapid progress exhibitedby the optical clock field comes with its own challenges. Not only do we demand state-of-the-art
performance from both our atomic trapping apparatus and our optical local oscillator, but also
by surpassing the accuracy of conventional frequency measurement techniques we were required toutilize differential measurement techniques, as no standards existed that were accurate enough for
us to compare against. But why is it that Optical clocks have such an advantage over RF clocks?
And what advantage does a neutral lattice clock have over other competing standards?

1.1

Optical Lattice Clocks
A primary standard must excel in three key characteristics – stability, accuracy, and real-

izability. The stability, or precision, of one’s standard is the amount of noise inherent in taking
a measurement with respect to that standard. This greatly affects the utility of the tool you arebuilding. In fact, it has been repeatedly remarked that state-of-the art precision measurement
experiments (eEDM, Atom Interferometry, Atomic Clocks, etc.) routinely attain accuracies near
their quoted 1 day stabilities. Ultimate results will far exceed these numbers, but this is after years
or decades of research [4, 5, 6, 7, 8]. Next, accuracy is the ability of one’s measurement to be
traceable back to an internationally agreed upon standard. In this case, because there is no agreed
upon reference with comparable performance, we usually refer to the total systematic uncertaintyof our clock. This measure is summed up in a list of all the external perturbations we know thatshift our clock’s frequency. The final piece of the puzzle is realizability, the ability to successfully
disseminate this standard to anywhere one might require its accuracy – either via reproduction ofstandards or suitable transfer techniques – and to connect its accuracy to all measurements one
might be interested in. With these three characteristics in mind let’s examine ancient time keeping
solutions, and the three leading contenders for current time keeping solutions.

3
As humanity grew towards technological maturity, constructing time keeping devices andthe definition of what time was became more accurate [9]. Older civilizations based time on
astronomical calendars (including ours pre-1967), measuring the frequency of the rotation of theearth about the sun. Compared to the technology of the time, this oscillator was fairly stable.
However, connecting this orbit to measurements on human time scales like seconds or hours was
not a simple matter. Regardless of this complication, it is easy to argue why this makes a badstandard. Imagine trying to gauge the fractional uncertainty of measuring this frequency. First,
in any measurement, there must be some error or limit to your precision with which you can take themeasurement, δf . And second, one must simply normalize this measurement to get the fractional
1
uncertainty, giving δff or Q where Q is the quality factor. With repeated measurements one might

be able to measure more accurately, but with this figure of merit and an educated guess of δf = 1
day for an astronomical calendar, it is no wonder that error in this method can accrue leading to
the Catholic Church moving the calendar by 10 days in 1582. Looking back it seems obvious thatthe construction of mechanical clocks with significantly higher Qs due to their precise construction
and much higher frequencies was the correct path to take. However, whether these clocks weremechanical pendula in nature, like in the industrial revolution, or electronic feedback from a quartz
oscillator as is prevalent today, there is something unsatisfying with relying on man-made objects
with their sensitivity to construction and external parameters to “keep the time” so to speak.
It was probably with this in mind that I. I. Rabi proposed probing energy levels in atoms
or molecules as a source for a natural frequency (or time) scale. The primary standards we use
today are a direct descendant of the first experiments built by Louis Essen and Jack Parry at the
National Physical Laboratory in the UK and Norman Ramsey and Daniel Kleppner1 at Harvard[10, 11]. To compare the primary Cs standards with optical clocks both ion and neutral, it will be
useful to write down the simplest limit for frequency instability of uncorrelated atoms [12, 13],σy (τ ) ∝
1

1
√
Q τN

I sat behind N. Ramsey at a Colloquium once, and met D. Kleppner a few times as well.

(1.1)

4
where Q is the quality factor of the transition you are probing, τ is the amount of averaging time
related to the number of measurements one has made, and N is the number of atoms. This limit
known as the Quantum Projection Noise (QPN) limit for N atoms can be reached when local
oscillator noise no longer dominates (at some specific line Q given by the local oscillator noise).
Cesium fountains work by probing the energy difference between the 62 S 1 F = 3 and F = 42

hyperfine ground states. By imparting a small upwards velocity to a cold gas of atoms, the atoms
travel through a microwave cavity where they are excited, up into a shielded environment, andback through the same microwave cavity due to the influence of gravity [14, 15]. By double passing
the same cavity many systematic biases associated with the excitation process are suppressed. So
how do cesium fountains rate in each of the key characteristics? First, for stability, the transition
being probed is in the microwave domain and the physical size of the fountain Fourier limits theinterrogated linewidth. The high signal to noise and quiet local oscillators allow operation near the√
QPN Limit for 104 or 105 atoms reaching fractional frequency instabilities of σy = 4 × 10−14 / τ[13]. However, running these clocks accurately with these atom numbers is quite difficult, and lower
atom numbers and correspondingly lower stabilities are normally employed.
New results from a cryogenic Cs fountain clock at the National Institute of Standards andTechnology are showcasing a new way forward for clock accuracy. Systematic error bars, known as
Type B errors, for these systems of 1.1 × 10−16 have been published, limited by technical details
regarding the microwave cavities [8]. However, this does not include collisional shift uncertainties
that are still large at 2.4 × 10−16 , which the authors categorize as statistical errorbars, known
as Type A errorbars. Combining these results a total fractional frequency uncertainty of 4.4 ×
10−16 is reported. Finally, due to the wide use of RF, dissemination of this standard can beaccomplished very easily with off-the-shelf components and contemporary satellite technology [16,17]. Furthermore, the proliferation of Cs fountains in labs and metrology institutes around theworld has proven their reproducibility.
To improve on this seemingly robust standard calls for a fundamental change in clock design.
To significantly lower the instability of the clock it makes sense to dramatically increase the Q of

5the clock by finding narrow transitions to probe with significantly higher frequencies of oscillation,f . Much like the move from an annual frequency to an electronic frequency, a new set of clocks havebeen constructed that interrogate optical frequency transitions rather than RF transitions. The
first of these types of optical clocks to truly challenge the primary standards is the ion clock. The
exquisite control of single quantum particles exhibited in these traps was recently acknowledged by
the Nobel Prize in Physics in 2012 [18]. By confining ions in electrostatic (or effectively electrostatic traps) these ions can be sideband cooled to their motional and electronic ground states [19].
Utilizing an optical local oscillator with good short-term stability, an ion is excited into a superposition of ground and excited states, and a frequency discriminant to steer the laser onto atomic
resonance at long time scales is created via Ramsey or Rabi spectroscopy [20]. As this technique forbuilding a standard is our greatest competitor, due in part to its phenomenal performance, we will
carefully analyze ion clocks in terms of our three figures of merit. Perhaps the biggest limitationof ion clocks is their large QPN as seen in equation 1.1. For a fixed line Q, due to say reaching
the natural transition linewidth, stability is fixed for a single ion system. Regardless, due to the√high Q the instability can be as low as σy = 2.8 × 10−15 / τ [21]. In terms of accuracy, until the
publication of the work documented in this thesis ion clocks held world record total uncertainties
as low as 8.6 × 10−18 [21], a stunning feat given their quoted stabilities. With the advent of thefrequency comb, one could argue that the challenge of propagating this accurate standard to the
RF is a solved problem, however ion clocks are somewhat burdened by the machinery surrounding
their somewhat simple traps. Many of the laser transitions required to trap and cool the ions are
deep within the UV making operation of these clocks difficult, expensive, and hard to miniaturize.
However, as industries need shorter and shorter wavelength lasers and invest in R&D for creating
new classes of lasers, this might become less of an issue.
Turning to the final contender, the optical lattice clock (OLC), we will see that by trying to
combine the many-atom, cold-gas nature of the Cs fountains with the precision afforded by opticalclock techniques of ions might offer the best alternative. Much like a Cs fountain, an OLC starts bycapturing atoms (usually Alkaline-Earth or Alkaline-Earth-like) inside a magneto-optical trap and6

A lla n D e v ia tio n

1 0 -1 3

J IL A O p tic a l L a ttic e C lo c k
N IS T A l+ C lo c k
S Y R T E C e s iu m F o u n ta in

1 0 -1 4
1 0 -1 5
S I S e c o n d

1 0 -1 6
1 0 -1 7
1 0 -1 8
1 0 0

1 0 1

1 0 2

1 0 3

1 0 4

T im e ( s )Figure 1.1: Stability comparison of the primary frequency standards and their competitors. The
green line shows a fit to the demonstrated performance of the JILA SrII apparatus [22]. The blueline shows the performance of the Aluminum ion clock [21]. The solid red line shows the performanceof a Cesium fountain run in high accuracy mode during a Rubidium vs Cesium comparison [23].
The light red arrow shows the current accuracy of many of the world’s best Cesium fountains.

7then transferring them into an optical lattice created by a standing wave of light. While probing
atoms with long wavelength radiation does not require the atoms to be trapped, simple Doppler
shift formulas might fail as the atom traverses the wavefront of a laser. Furthermore, opticallyprobing these atoms along the tightly-confining direction of the lattice allows separation of the
motional and electronic states of the atoms, due to their trapping deep within the Lamb-Dicke andresolved sideband regimes inside the lattice [24]. The large number of trappable atoms, and high Qof the clock transition, create wildly small and presently unattainable instabilities via equation 1.1.
Instead, the stability of these clocks is limited by the state-of-the-art laser local oscillators used
to interrogate them [25]. This technology dependence creates a constant pressure to innovate newadvances in laser frequency control techniques as any progress in the area of frequency stabilization
can immediately be put to use to further improve the performance of the atomic clock. Accuracy
in these systems had until recently lagged behind ion clocks. The added difficulties of controlling
atom-atom interactions, differential light shifts due to the trapping light, and the relatively higher
sensitivity to blackbody radiation shifts on the clock all had to be addressed.
This thesis documents the JILA Sr clock, the first optical lattice clock that bested all otheratomic clocks with a low total systematic uncertainty of 6.4 × 10−18 [22]. A testament to therealizability of this system is the increasing number of OLC experiments popping up around theworld, especially with the Strontium isotope which exhibits laser cooling and trapping wavelengths
all accessible via diode lasers [22, 26, 27, 28, 29, 30]. Perhaps unsurprisingly, the benefits imparted
by using ever more stable laser local oscillators to interrogate the atoms make the state-of-the-art
versions of these clocks very sensitive to thermal noise of both the cavity spacer and mirror coatings,
and presently limits miniaturization of these systems. Future technologies might make these clocks
even more attractive [31].
Examining Fig. 1.1 it is easy to see why optical lattice clocks are considered so promising.
However, moving from a carefully controlled system with one particle to that of many particles wasrife with challenges. As more strontium experiments have been built, an enormous, international
body of work has grown up to defeat the obstacles listed below. First and foremost of these8challenges was the intrinsic deformation of the valence electron cloud of the neutral atom to actuallytrap it for clock interrogation. The so-called “magic” wavelength trap is employed to counteract
these effects, in that the differential light shift between the ground state, 1 S0 , and the excitedclock state, 3 P0 , caused by the trapping light is nearly zero [32]. Higher order corrections might
still come into play, and this shift will need to be measured as precise knowledge about the zerocrossing point is only ever known to the precision with which you measure it. Atom interactions
were thought to be nonexistent in the original proposals, as spin-polarized ultracold fermions shouldhave severely suppressed s-wave collisions and low p-wave collision rates. But as is often the case,at the frontiers of precision measurement, even tiny effects can play a leading role in one’s final
uncertainty [33]. Finally, the largest limitation to accuracy came from the influence of broadband
blackbody radiation (BBR) impinging upon the atoms causing an uncontrolled AC stark shift. A
topic that will be greatly expounded upon in subsequent chapters.
A

B

C

Figure 1.2: The three leading systematics plaguing optical lattice clocks and how they have beenconquered. A) Exhaustive studies of the collision shifts were performed, not just probing theirshift of the clock transition, but their effect on atom shot noise [34]. This image is taken from Ref.
[34] B) The AC stark effect from operating the clock near the magic wavelength has been studied
around the world [22, 26, 35]. This image summarizing AC stark shifts on the clock transition in
the visible and NIR spectrum is taken from Ref. [36]. C) An image from Ref. [37] depicts the DC
polarizability measurement that enabled our use of the temperature probes as a means for inferring
the BBR stark shift. Below the image of the DC electrodes is a CAD file depicting the mountingof our BBR sensors inside the JILA SrII vacuum chamber.

So while the path forward for state-of-the-art clocks seems clear, the inqusitive reader might
wonder what all the fuss is about. While a more mundane answer might center around the advancesin creating UV sources of light and the accuracy of performing spectroscopy with these sources,

9a host of other more exotic uses for these clocks might exist. In fact, comparing optical clocks
made with the same species has already been used to provide more stringent restrictions on the
coupling of fundamental constants to weak gravitational forces [38], and in the future measuring
frequency ratios of highly accurate optical clocks will put stringent limitations on the drift of thefine structure constant with time [38, 39]. As stable long term oscillators, optical lattice clocks
offer a path forward for sub-mm wave very long baseline interferometry (VLBI) [40]. In VLBI one
creates a phase coherent array of antenna that can be used to generate a more detailed pictureof the galactic source one is imaging. The distance between disjoint VLBI stations allows one tocreate planetary scale effective antenna sizes. Current limitations with Hydrogen masers keep thesetechniques well in the RF, whereas a dramatic increase in stability could open the door for future
experiments. The sensitivity of these clocks to gravitational redshifts might also allow these clocks
to be used as tools for relativistic geodesy to map out the earth’s gravitational field [41]. Not
only might these clocks be used to study such fundamental physics like variation of fundamental
constants [42], help in the search for dark matter [43], or the coupling of gravity to other forces innature [38] but also they will continue to allow us to probe complicated artificial quantum manybody systems [34, 44]. Whatever path is taken to create these optical lattice clocks, their futureseems bright.

1.1.1

The JILA 87 Sr Lattice Clock

The results presented in this thesis are based on the second generation JILA Sr lattice clock,referred to throughout this thesis as SrII. The differences between this new apparatus and SrI weremade to fix many of the issues associated with creating a highly accurate lattice clock. But how
does one create a strontium lattice clock? A brief summary is presented below.
An ultracold gas experiment traps incredibly low density gases, at µK temperatures, in
extremely weak traps. This requires an incredibly low background pressure inside the trapping
volume. In the case of strontium cold-atom-machines this poses quite a challenge as the vapor
pressure of strontium, a solid at room temperature, is very low. Heating solid chunks of strontium

10
to 575 K in an effusive oven, the hot vapor can be collimated towards the main trapping chamber.
Heating up the steel of the vacuum chamber creates a large gas load for ultrahigh vacuum pumps
in the region of the oven, so differential pumping between the oven region and the main chamber
is a must. The hot vapor streams down the apparatus, past a mechanical shutter, through adifferential pump hole, and enters a zeeman slower. Strontium has a slightly more complicatedlevel structure compared to an alkali atom as can be seen in Fig. 1.3. However, a mostly closed
dipole-allowed transition exists between 1 S0 and 1 P1 . This fairly broad transition is used first tocool the atoms’ forward-motion in the Zeeman Slower and finally to catch the atoms in the mainchamber with a Magneto-Optical Trap (MOT). This “blue” MOT traps the atoms at a few mK.
The intercombination transition between 1 S0 and 3 P1 is 7.4 kHz broad and allows the atoms to becooled down to less than 1 µK in just a few hundred milliseconds [45]. It is then that the atoms
are loaded into the magic wavelength trap created with a laser at 813 nm forming a standing wave
inside a buildup cavity creating an array of two dimensional isolated atom traps. In this final trap,
the clock transition of the atoms can be probed in the tightly confining direction as a laser tuned
precisely to the transition near 698 nm can coherently Rabi flop the atoms from 1 S0 to 3 P0 . The
ensemble average of the coherent electronic superposition across the cloud can be read out by first
destructively fluorescing atoms in the ground state and then measuring the remaining atoms thatwere shelved in the excited state. This entire experimental sequence can be repeated at a duty cycleof approximately 1 Hz, and the optical local oscillator can be locked to the atoms or used as a stablereference to measure various frequency shifts to the clock transition when external parameters arevaried in a controlled fashion.

113(τ(5=p72.)93ns
P0 ,1), 2 (5(sτ5d
=)16
)
Dns
1, 2 , 3

λ ~ 480 nm(5s 6 s ) 3S1

λ ~ 497 nm)1Pns(τ(5=s55p.22
1 )(5s 4d )1D2

λ = 461 nm
Γ = 32 MHz

λ = 707 nm
λ = 688 nm
λ = 679 nm

λ = 689 nm
Γ = 7.4 kHz(5s5 p )
3
P2
3
P1
3
P0
λ = 698 nm
Γ ~ 1 mHz

1

S0

(5s 2 )1S 0

I = 9/2(τ = 2.9 µs )(53s 4d )D3
3D2
λ = 2.93 μm
3D1
λ = 2.603 μm

87Sr

Figure 1.3: The 87 Sr Level Diagram. Based off of a Figure from Martin Boyd’s thesis [46].

Chapter 2

Building a Clock

When I first arrived as a bright-eyed, happy-go-lucky graduate student the first iterationof many of the major systems on SrII was complete. At the time the majority of our focus was
to explore the optical feshbach resonance technique in 88 Sr. Imaging and final trapping were notquite there, but we had cold atoms and the sky seemed the limit! Unfortunately, it turned out thevacuum lifetime (∼1 s) was the actual limit, and once we were done with the feshbach studies inbosonic strontium, the decision was made to pursue the dream for a far more accurate clock. To
accomplish the feat of switching to the fermionic 87 Sr isotope and creating a really robust systemthat could operate and take data for hours at a time with minimal human intervention a lot ofsystem components needed to be revamped or upgraded. In addition to the system components I
will discuss below there were many components like the MOT coil switching electronics, the bias
field electronics, the atom detection systems, and the red laser systems – to name a few – that
required significant overhauls that are not documented here.

2.1

The Blue Laser System
Until recently, creating a good 461 nm laser system was not something one could take for

granted. The 1 S0 to 1 P1 transition is incredibly important for all laser cooling and trapping ofstrontium. However, due to its relatively high frequency and large linewidth of 32 MHz, comparedto the D1 or D2 lines of alkali elements, its saturation intensity, Isat , is a painfully high 43 mW⁄cm2 [47].
While high intensities are not necessary for optimal first-stage MOT densities, conventional Zeeman

13slower design constraints, and fluorescence detection call for beams approaching the saturation
intensity. Therefore powers approaching or exceeding 50 mW delivered to the atoms at a variety
of detunings are not unheard of requirements for successful fermionic strontium experiments. A
variety of early approaches to creating this light involved either dye lasers or second harmonic
generation of titanium-sapphire (Ti:Sapph) lasers [45].

2.1.1

The Old Blue laser System

The SrI system uses a 922nm external cavity diode laser (ECDL) that seeds a tapered amplifier (TA) that is then coupled into two doubling cavities. These standing-wave second harmonic
generation cavities use KNbO3 with the back flat facet of the potassium niobate crystals having
been HR coated for both the IR light and the blue light. The phase matching temperature of thesecrystals is a toasty 150 ◦ C, however their doubling efficiency exceeded 50%. This system had beenin continuous use for a few years before the first graduate student on SrII, Sebastian Blatt, had
to decide how to build SrII’s blue laser system. Operating temperatures in these systems were
within tens of degrees to the crystals repoling temperature, not only making the use of these cavities cumbersome but also nerve racking. Examining the literature at the time the reports in Ref.
[48] of 75% SHG conversion efficiency were too good to ignore. The system described in Ref. [48]utilized a periodically-poled potasium titanyl phosphate (PPKTP) critically phase-matched crystal
inside a ring resonator and so the decision was made to try a PPKTP system. However, due tothe experience in lab working with standing wave cavities, the PPKTP was built in a standing
wave configuration. Unfortunately, with subsequent years of experience working with this system, it became clear that color center formation severely degraded the performance of this systemthroughout the day, making long experimental runs quite trying. Rejuvenation of the crystals was
possible by baking them overnight, but the constant fluctuations of >60 degrees were an alignmentnightmare. Therefore, the SrII group was always on the lookout for ways to improve the reliabilityof all components in the 461nm setup.
The first major upgrade to the system was an upgrade to the IR laser diode. Over the14course of a few years, we noticed that the Ridge Waveguide lasers that we used from Eagleyard
would work for a few months, and would subsequently become more and more finnicky as time
went on, probably due to decay of the AR coating on the diode’s front facet. Luckily, we foundthat Photodigm, a fairly new company, was trying to expand their offering of distributed bragg
reflector (DBR) lasers. A DBR laser is in some sense a miniaturized ECDL, with the feedback being
provided by a distributed bragg reflector built on the same substrate as the laser gain medium.
This should not be confused with a distributed feedback (DFB) laser where the active gain mediumof the laser is modulated to support lasing on only certain wavelengths. The DBR laser we received
from Photodigm throughout initial testing and day to day use over two years was incredibly stableand very compact. Fig. 2.1 shows the very nice package the laser came in, including both a TEC
and thermistor for good thermal control of the laser diode. Care must be taken with these lasers,
as they are incredibly sensitive to feedback, even allowing for dramatic linewidth reduction given
the correct optical feedback [49]. In fact, when installing this laser into the 461nm SHG system,
additional optical isolation was necessary to allow stable operation. Regardless, I would have to
say that this DBR laser was the best behaved laser I have ever worked with.

2.1.2

New Additions to the Blue team

January 2011 started with a complete blue system realignment. We were switching from themore abundant bosonic isotope of strontium to the less abundant fermionic isotope of strontium.
We needed every blue photon we could muster, and the output power of both SrII blue doublers
had been on a very slow downwards trajectory for almost two years. Three tasks were clear though,more power in the MOT, more Zeeman Slower power, and implement transverse cooling before theatoms entered the Zeeman Slower area.
The blue MOT light was delivered to the atoms via a Schäfter+Kirchoff 1x3 fiber splitter.
This component had been in use for many years, and a careful measurement of the input-ouput
powers of this splitter revealed something quite surprising. Almost 80% of our blue light was being
lost in the fiber couplings into and out of this device. In fact, even at its best, its specified operation

15

A

B

Figure 2.1: A) A diagram of the 922nm DBR laser package including a thermistor and TEC fortemperature control. B) Model of a DBR laser showing both the gain medium and the grating
feedback. Additional SEM images show the construction of a DBR laser at Photodigm. Both
Images are taken from promotional material from the Photodigm website.

16
called for at least 40% of the light to be lost. After contacting the company, purchasing their
specialty tools, and aligning the system from scratch, we were back to operation above specifications.
However, a close eye was kept on the device, and within 6 months a clear degradation of fiber
couplings was seen again. Another solution was needed. Jason Williams, the postdoc on SrII, hadpreviously had a good experience using evanescent wave beamsplitters. We contacted Evanescent
Optics Inc. (formerly Canadian Instrumentation & Research Ltd) and they offered us fiber splittersat a fraction of the cost. These have now been in use for years as well, with no noticeable change
in performance. The MOT problems were solved, and a subsequent realignment of the second
doubling cavity brought the Zeeman Slower powers in line as well.1
However, we were still no closer to garnering enough blue power for transverse cooling. It
should be noted that due to the long distance between the oven and the main chamber, our overall
atom flux was lower than the upgraded SrI system. In the end we utilized a single-pass periodicallypoled lithium niobate (PPLN) waveguide doubler to produce about 60-70 mW of 461 nm light.
Produced by NTT, these waveguide modules can be customized for SHG, SFG, or DFG. These
doublers combine the critical phase matching and high nonlinearity of other periodically poledmaterials built into a waveguide structure to keep the input light at a high intensity throughout
the nonlinear medium. The relative simplicity of using this system can be seen in Fig. 2.2. With
this blue light we were able to construct both a multi-bounce vertical and horizontal transverse
cooling setup that easily increased our final atom number by a factor of 3.

2.1.3

The Blue Diode System

Regardless of the ever-improving performance of SrII system, it was still a rather complicatedsystem for something so basic and intrinsically necessary for a strontium clock. Hopes of portablestrontium lattice clocks hinge on the ease of use of the strontium atom. So it was perhaps with acollective sigh of relief that SrII acquiesced to testing out some new blue diode laser sources for New
Focus/Newport. While it remains unclear exactly the source of the AR coated diodes used in New
1

922 nm powers were measured to be ∼1.1 W, in a poor spatial mode, creating ∼120 mW of blue light.

17xchange waveguides, no mobile protons are formed.

A

B

SinglevMode
922nmvLight

TaperedvAmplifier
EYP-TPA-0915

TovHorizontal
TransversevCooling

NELvWaveguide Doubler

TovVertical
TransversevCooling

Figure 2.2: Diagrams of the waveguide doubler system used for transverse cooling. A) SEM imageof the waveguide structure imprinted into periodically poled Zn:LiNbO3 . Image is from Ref. [50].
B) Schematic diagram of the system used for creating 461nm light for transverse cooling of theatoms. Not all components are pictured for figure clarity.

18
Focus’s laser, the results speak for themselves. The 461nm module started off slightly multimode,
but with an impressive 45 mW of output power. After temperature tuning the diode more carefully
it was back to single mode. Throughout the last 18 months the power has degraded slightly, but
has seemed to level off at about 35 mW. While at first temperature and careful PZT and current
tuning were a necessity, the blue module has seemed to hit its stride and needs much less tuning
than it even did 10 months ago. Luckily, at the same time that New Focus started building these
461nm modules, Nichia started offering 460nm blue diodes. These blue diodes while advertised as
single mode diodes that can put out 120 mW of power, aren’t quite single mode at all at powers
above a few mW. But, they can be used like slave lasers or slave amplifiers. Injection locking ofthese blue diodes is possible with powers as low as 500 µW although stable operation is better at
1 mW injection power. In Fig. 2.3 one can see the new blue laser setup.
Two important design considerations were kept to ensure successful system operation. First,
the master laser is locked directly to the strontium discharge cell. This allows for the master toremain locked for however long is possible given the stability of the New Focus Vortex laser, usuallyan entire day. Second, the vast majority of frequency steering is done before power amplification
to minimize wasting precious blue photons. In periods of quiet operation all lasers, both slave andmaster, remain locked for hours at a time. However, if temperatures are changing quickly in thelab injection lock can be lost. This has spurred an effort into active injection tracking servos thatis still being worked on.

2.1.3.1

Injection Locking Servos

Long term stability of the blue lasers is very important in the operation of a clock. Datasets
can run anywhere from 10 minutes to 10 hours, so having any system component not perform at
100% uptime is unacceptable. Spurred by some curious effects we observed in the photodiode signal
we started looking into active injection locking servos. Before much research had been done into
the subject, Igor Teper and Tom Loftus from AOsense pointed out three very useful papers on
the matter of optimum current set point servos in various diode laser configurations [51, 52, 53].

19

Sr

MasterLaser

Spectrometer
-1st
80

EOM

f0 + 143

f0 + 143

Pre-Spectrometer-1st155(88Sr) or 103(87Sr) MHz

f0 - 64
f0 - 682

TC Tuning-1st Double Pass
103.5 MHz

Inj
Lock

TC to Atoms
f0 - 64

f0 + 23ZS Tuning AOM-1st
825 MHzZS to Atoms
f0 - 682

Inj
Lock

Probe Absorption to Atoms
f0 - 38

MOT/Probe Tuning-1st Double Pass
60 MHz

Probe Fluorescence to Atoms
f0 - 49Inj
Lock

MOT to Atoms
f0 - 91

f0 + 40

Blue Splitter AOM-1st
114 MHz

Fluorescence AOM-1st
72 MHzAbsorption AOM-1st
61 MHzFigure 2.3: Blue diode laser layout showing frequencies and placements of all AOMs. All numbers
are in units of MHz, and f0 represents the Sr 1 S0 -1 P1 transition frequency. All injections locks are
performed by sending in light through an isolator’s rejection port. Not all components are pictured
for figure clarity.

20
For the purpose of explaining how these servos derive their error signal, let us take the example
of an uncoated diode placed inside an external cavity diode laser setup. In this system, there are
two competing modes trying to saturate the laser gain medium, the laser cavity mode associatedwith the laser’s front and back facets, and the external cavity mode between the laser’s back facet
and the feedback grating. When these two cavity modes come close to resonance with each other,
the feedback resonance starts to saturate the laser diode, and the system will exhibit single mode
operation. Bringing these two cavity modes close together can involve walking many parametersthat effect the gain curve of the laser and the length of both cavities. Focusing on the angle of thegrating and current setpoint of the laser diode, an optimum value of the lasing current must exist
where small variations in the resonance position of the external cavity will not push the laser awayfrom single mode operation.

A

B

Slow frequency scan
to see error signal

C
Demodulated Broadband Noise
Demodulated Phase Response

Slow frequency scan
to see error signal

1.2
461
MasterLaser

461
Master
Laser

1.0
0.8

Error Signal (V)0.6

Modulation
Frequency

461
Slave
Laser

SignalModulation
Frequency

0.4
0.2
0.0
-0.2

461
Slave
Laser

-0.4
-0.6

Signal-0.8-1.0
10 MHzHigh Pass

Amp

LIA

RF PowerDetector

-1.2

LIA

-1.4
-1.0 -0.8 -0.6 -0.4 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

Master Laser scanning through optimum lock (arb. units)

Amp

Figure 2.4: A) Diagram showing the procedure of measuring the broadband noise response of aslave diode to determine optimum slave laser locking current using a Lock-In Amplifier (LIA). B)
The signals produced from either the broadband (Green) or phase measurement (Red) of the slave
diode. The Y axis is in volts, however, an arbitrary offset has been subtracted. The position of the
waveforms at x = 0 is where one would lock diode current. Neither seems to occur at the steepest
discriminant in the signal, but a feedback signal seems possible for both methods. C) Diagram
showing the procedure for measuring the phase response of the slave diode.

Finding the optimum setpoint is not difficult, but if parameters change that are out of theexperimentalists control (e.g. temperature or humidity) or even if they are in your control like

21
scanning the frequency of the laser over a large range, it would be nice for the current to track to
the optimum setpoint. In Fig. 2.4 the signals obtained and the layout of two different methods
for obtaining a locking signal are shown. In the first method, the laser’s current is modulated ata frequency fm , and its intensity noise is measured with a bandwidth above fm . The photodiodecurrent signal is then demodulated at fm , and the voltage output from a lock-in amplifier gives thephase relationship between the induced amplitude noise and the current driving signal. The phaserelationship depends on the transfer function of the laser diode, and not only is it stable, but itsmoothly changes with the current of the laser near the optimal setting [51, 53]. In essence, one
is just measuring the frequency modulation to amplitude modulation transfer function of the laser
at one specific frequency, fm . While this method produced a reasonable error signal to lock to andwas less complicated, I experimented with the method in Ref. [52] and found it to be more robust.
In this method we module the laser current at fm but then look at the broadband amplitude noise
at significantly higher frequencies than fm , above 10 MHz. To gain information about the noise,a Minicircuits RF power detector (ZX47-40+) was used. This signal is then demodulated at fm
giving essentially a noise measurement for a sweeping setpoint current. When the laser is mostly
single mode, the amplitude of the high frequency noise goes down, measuring this signal in a phasesensitive manner produces an odd signal that one can feedback to the laser current. Here, one ismeasuring the frequency modulation to amplitude modulation transfer function of the laser at all
frequencies from 10 MHz to the detection bandwidth of our photodiode (180 MHz), and integratingthat response with an RF power detector to arrive at one single voltage for each current setpoint.
While these tests were performed with a commercial lock-in amplifier, which would be a needlessexpense for every injection lock laser, our youngest member on the strontium team is building ahomemade board suitable for creating these error signals.

2.2

The Magic Wavelength Buildup Cavity
Careful comparison of the initial reports of cold collisional shifts in fermionic strontium

clocks [33] and the lack of similar shifts in other clocks, that were still reporting lower measurement

22

A

B

g

Figure 2.5: CAD drawings of the horizontal cavity mounts. A) This CAD drawing shows theplacement of the two horozontal mounts on the Kimball Physics chamber, mounted on two opposing
CF133 viewports. B) A cutaway of the internal structure of the mount that holds the fixed mirror
for the cavity. It is important to note that this mount can both steer the cavity mirror and has anadditional steerable plate for use in retroreflecting a beam for fiber phase noise cancellation.

precision, was for almost a year a point of contention in the optical lattice clock community.
However, key design differences between the JILA and SYRTE atomic clocks were discovered.
While the original JILA clock, SrI, was formed by trapping in the tight focus of a 30 micron
waist sized beam overlapped with the dense red MOT, the SYRTE clock was a drain loaded clock
held in a cavity enhanced lattice eschewing the cold dense red MOT and having very poor atomtransfer between the blue MOT and the lattice. The resulting differences in densities completely
explained the discrepancies [54]. It was with this idea about the fundamental connection between
cold collisional shifts and density (rather than atom number) that we decided to build a nearlyhorizontal build-up cavity on SrII.2 Not only would this setup give us better spatial overlap withthe red MOT, allowing us to transfer a larger number of atoms into the magic wavelength 1D
lattice, but also they would be trapped at a lower density and would induce a lower density shift.
The magic wavelength for the 1 S0 to 3 P0 transition in 87 Sr is at approximately 813nm (SeeFig. 1.2). Luckily this is a wavelength that can be reached with laser diodes and is also accessible
with a cw-Ti:Sapph laser. Here I will describe the use of both versions of an 813nm system suitable
for atom trapping. The build up cavity is external from the vacuum system, which limits the2

The lattice axis is 16◦ above the horizontal, lifting the lattice site degeneracy

23
intracavity power due to the losses in the viewports. However, having a cavity inside the vacuum
is not without its own challenges, especially given the sensitive nature of our spectroscopy. In fact,
placing dielectric surfaces near the atoms caused significant problems for the SYRTE group. Theirmirrors themselves became charged inducing DC stark shifts on their clock transition on the orderof 10−13 [55]. In addition to this concern we have found the ability to move the cavity mode to
the chamber’s magnetic field zero, rather than it’s geometric center, to be very useful. For the
majority of the work presented in this thesis, the cavity mirrors in our system are located on two
opposing CF133 flanges as depicted on the CAD model in Fig. 2.5. The mounts are adjustable,
like a kinematic mirror mount, and allow for small movement of the cavity mode in the system.
They are not identical mounts, as one is built to hold a cavity mirror that has been glued onto astack ring piezo. These piezos were bought from Physik Instrumente and allow beams to enter at
the center of the cavity mirrors.
The cavity supports a TEM00 mode with a beam radius of 160 µm formed by two mirrors
with 20cm radius of curvature. The mirrors are coated for 98% reflectivity at 813nm and 0.04%
reflectivity at 698nm to allow for very little interference, or superlattice effects, in the clock excitation over the atom cloud. Losses through the viewports were measured with an extra viewportfrom the same coating run. A spectrophotometer showed 97.8% transmission through a single AR
coated viewport. The finesse can be calculated in two ways, either with the FSR and the FWHM
of the cavity or with knowledge of the intracavity losses. Both methods agree, showing a finesse of
70. As mirrors and coatings were ordered at the same time as the SrII vacuum chamber, and were
not ordered with a variety of reflectivities the cavity is not impedance matched. This results in a
power build-up factor of about 18. So for 330 mW actually coupled into the cavity, SrII achieves acirculating power of 6 Watts [56, 57]. Impedance matching could in fact bring this factor closer to
37. For that matter better mirrors, that are also impedance matched, could still achieve an overallenhancement factor of 44. This is more than good enough for our purposes, and suggests that thetrouble of building an in vacuum cavity is not worth it if power enhancement is your only goal.
Many options are available to lock the 813nm light to the buildup cavity. Unfortunately

24
the large size of 1 inch mirror glued onto the equally large ring piezo caused fairly low frequency
resonances in the actuation of the piezo. The locking bandwidth of the piezo is purporsefully kept
low to avoid ringing of the cavity. In the case of the diode system, the current of the diode allows
for a large bandwidth lock to a cavity. In Fig. 2.6 one can see how both a fast feedback path tothe diode laser current was created, and a slow feedback path to the cavity piezo was included. A
Pound-Drever-Hall (PDH) lock was constructed for actually keeping the laser on resonance withthe cavity. Excellent writeups abound explaining the PDH locking technique [58]. However, myfavorite and most useful reference for this has been a NIST Time and Frequency writeup that
includes descriptions of the electronic locking circuits involved [59].
Because the atoms react poorly to intensity noise, especially to harmonics of their trap
frequencies, it’s very important that the lock to the cavity is high bandwidth [60]. Any relative
frequency noise or excursion by the laser is in fact transformed into amplitude noise inside thecavity, which can parametrically heat the atoms. Since we are interested in keeping the atoms frombeing heated and holding them in the trap unperturbed, we sample the transmitted light fromthe cavity and add intensity stabilization feedback to further supress any residual amplitude noise.
While SrII used the setup depicted in Fig. 2.6 for many months, one or two oddities of the design
made us rethink our approach in subsequent apparatus upgrades. For example, the large distance
between the EOM which placed sidebands on the laser and the path length of the reflected beam
for the PDH lock were significantly longer than they needed to be. Unfortunately, this turned out
to cause a lot of instability in the system, as residual amplitude modulation (RAM) picked up fromstray reflections throughout the path caused fairly large offset swings.
While this system could be locked to the cavity with a high-bandwidth actuator, the injection
locked Ti:sapph did not naturally have such an actuator. While it is possible with the use of anonlinear transmission line and an EOM to lock light to a cavity, the SrII group decided to try
the more mundane approach of double-passing an acousto-optic modulator (AOM). In Fig. 2.7 one
can see the added complexity of using an injection locked Ti:Sapph. Unfortunately, the locking
bandwidth of this setup to the cavity is only a few hundred kilohertz and the dynamic range is set

AB
To Wavemeter

813
MasterLaser

Tapered Amplifier

PD

Faraday Rotator
PD

Comb

698nm
Sr

PD

Amplitude Information
Goes to filter controlling
AOM Power

PSD of PDH Error Signal (dBm/Hz)

25

-70
-75
-80
-85
-90
-95
-100
-105
-110
-115
-120
103

104

105

106

Frequency (Hz)Figure 2.6: ECDL lock to 813nm Buildup Cavity. A) Diagram of the 813nm ECDL-based system
and its lock to both the frequency comb and the buildup cavity. Not all components are shown indiagram. B) Power spectral density of the in loop locked error signal for the lock to the buildup
cavity. The high bandwidth is achieved due to the fast actuation of the current of the diode andthe large dynamic range of the system, making this very robust against cavity vibrations.

26
by the tuning bandwidth of the AOM and how much steering is caused by changing the frequency.
In fact, this system deals with perturbations to the buildup cavity very poorly.
Jolts or jerks to the system would routinely cause the system to immediately come out oflock. When first set up, the system could not survive the Blue MOT (60 G⁄cm) to red MOT (5⁄cm) MOT coil switch. This fast switch, done in less than 100 µS, caused a sudden jerk to theGsystem as the MOT coils bounced around in their holders. Placing one’s ears near the system, one
could actually hear a metallic ping every time the switch occurred. An accelerometer placed on amezzanine near the cavity clearly saw the effect of the switching (See Fig. 2.8). While a partial
solution to the problem of the cavity coming out of lock every cycle was to slow down the MOT
coil switching, the cavity was still very sensitive and could not be trusted to remain locked overlong timescales. A more permanent solution for all cavity disturbances was needed.

2.2.1

A Cavity Autolocker

A good system knows when it’s beat. The purpose of the cavity autolocker is to gracefully
have the system leave its locking state during strong perturbations and try to re-engage the lockfrom scratch. There are many resources for creating autolocking systems and upon switching to an
AOM cavity lock we tried to copy them [59]. Most center around either a sample and hold technique
for the DC value of the PZT or a breaking of the feedback circuit on the last PI, essentially holding
the feedback capacitor’s charge to keep the output of the circuit the same. And while we had luckwith these systems on the short timescales, disturbances lasting tens to hundreds of milliseconds
would routinely fail at reacquiring lock.
Luckily the “slow” JILA loop filter was built to run in 6 modes, including but not limited to
Acquire, Proportional, and various levels of PI engaged. These settings are implemented througha rotary switch working off a diode-resistor based logic system (DRL) that interfaces with many
DG201 analog switches (See Fig 2.9). Creating a reliable autolocking system really just calls forcreating a Finite State Machine (FSM) that follows the same rules for locking a laser to a cavity
that a human follows. First, with a photodiode measuring transmission through the cavity, see ifCurrent
813
Master
Laser

PZT

PZT

Verdi

27

PD

𝜆

Comb
4

To Wavemeter

698nm

PD

Sr

PZT

PD

Frequency
Amplitude

Faraday Rotator

Frequency Information
Amplitude Information

Figure 2.7: Diagram of injection locked Ti:Sapph lock to 813nm Buildup Cavity and frequency
comb. Note the many frequency locks, including locks for the 2 Cavity PZTs, and the ECDL PZT,
as well as the current of the ECDL and the drive of the double-pass AOM. Not all components are
pictured for figure clarity.

28

B
A

1 0 0

A c c e le r o m e te r R e s p o n s e ( a r b . u n its )A c c e le r o m e te r R e s p o n s e ( a r b . u n its )

0 .2

0 .1

0 .0

-0 .1

F a s t M O T T u rn o ff
S lo w M O T T u r n o f f
5 0

0

-5 0

-0 .2

0

5 0

T im e ( m s )

0

1 0 0

T im e ( m s )Figure 2.8: Accelerometer traces of MOT coil shutoff. A) An accelerometer is placed on a platform
holding MOT optics. The voltage signal from the accelerometer is read out when the MOT coil isswitched from 60 G⁄cm to 5 G⁄cm as fast as possible. A large amount of shaking is seen all around thesystem. B) A different accelerometer is taped onto the horizontal cavity mounts. Here one cansee the difference between shutting off the MOT coils as fast as possible (in Red) or shutting off
the MOT coils over 10 ms (in Blue) in the amount of shaking seen by the accelerometers.

A

B

C

D

2

3

BRN

GRN

BLK

X200

X100

X50

X20

X10

X5

X2

X1

R5A
750

R59
1.5K

R58
3K

R57
7.5K

R56
15K

R55
30K

R54
75K

R53
150K

8

OFF

X500

270

CW

R26
1.00M*

1.00M*
FOR
MOD

R5D
ZERO
OHM

R50
51

R51
160K

C50
5pF

C51
330nF

C52
100nF

C53
33nF

C54
10nF

2

+15V

7
-

-15V

C501
100nF

4

3 +

U1

AD825AR

6

C500
100nF

R24
8.2K

7

R25
82K

C6
100nF

FOR STANDARD UNIT, SHORT R5D, OMIT R5E & C5A.
FOR MODIFIED UNIT, INSTALL VALUES SHOWN FOR MOD.

220nF
FOR
MOD

C5A
OMIT

931K
FOR
MOD

3Hz

10Hz

2

7

+15V

C2A
DNI

C29
47pF

C20
1uF

C21
330nF

R52
51

AS2

AS1
16

1

2

6

J7

3

+

-

3

GRY

1

R30
1K
7

C300
100nF

2
3

1

-15V

J8
+/-RTN

+15V

C301
100nF

AD825AR

6

AGND

FROM J1, PAGE 1

-15V

4

U30
3 +

2

+15V

C3A
DNI

R31
1K

2

-15V

+15V

R3K
620

R3J
200

R3I
300

3

OFF

R3B
430
X1

R3A
430
X3

R39
430
X10

R38
430
X30

R37
430
X100

R36
430
X300

R35
430
X1K

R34
430
X3K

R44
560

CW

R45
5.6k

R46
51k

SM FRONT

R40
51

SM FRONT
R41
51

R42
24k

R43
7.5k

C400
100nF

500Hz

C4A
DNI

-15V

10

7

8
6

11
DG201BDJ
U60C

9DG201BDJ
U60B

C41
0.1uF

C42
33nF

C43
10nF

C44
3.3nF

C45
1nF

5

4

5) INSTALL DIODES D1-D6 W/CATHODES DOWN, BODIES NEXT TO REFERENCE DESIGNATORS.
6) U40 Changed to OPA627 (from OPA637) due to 20 MHz oscillations,13/12/03.

R49
5K

CW

OUTPUT
RANGE
R47
51

QQ030B5

2

DATE: 06/05

1

2

1

J6

SHEET 02 OF 02

1

OUTPUT
J40
RG-174DRAWN BY: M. Whitmore

BUILT FOR:REVISION 10/2011

JILA ELECTRONICS LAB
HIGH SPEED LOOP FILTER

AS4

AS3

C401
100nF

6
3 +U40AD825AR
4

2

7

+15V

CW

1.5kHz

5kHz

15kHz

50KHz

BUILT BY: T. BROWN

AS1

AS2

AS3

AS4

R3H
300

R3G
300

R3F
300

R3E
300

R3D
300

R3C
300

COARSE S30
SERVOGAIN X10K

C46
330pF

4) MOUNT ALL ROTARY SWITCHES AND TRIMPOT R33 ON REAR OF PCB.

+15V

C601
100nF

C600
100nF

2K
R33

CW

(TRIMPOT)

FINE
SERVO
GAIN

-9dB CORNERDESIGN BY: J. Hall/T. Brown

RE
20K

RD
10K

RC
20K

RB
10K

RA
20K

R9
10K

R8
20K

3

HIGH SPEED VERSION 200 kHz PI corner~ 80 nS delay
S40
MOUNT ON REAR
OF PCBR7
10K

43) BNC's & COMPONENTS INSIDE DOTTED LINE ARE PANEL MOUNTED.

THESE SHOULD BE INSTALLED PRIOR TO OTHER COMPONENTS.

2 LOCATED ON REAR OF PCB (R22 & R23).

11 LOCATED ON COMPONENT SIDE (R40 & R41, R71-79).

2) THERE ARE LOCATIONS FOR 13 SURFACE-MOUNT RESISTORS.

D6
1N4153D5
1N4153D4
1N4153D3
1N4153

D2
1N4153

D1
1N4153

MOUNT DIODESCATHODE DOWN

R32
1K<<<NOTES>>>ACQUIRE

2

3

PROPOR

4

5

-6dB+
-6 dB

-9 dB/CPU NC

J9

2

YEL

S2

1

BLU

J5

5

1) * = 1% MF RESISTOR

SERVO
MODE

CW

YEL

GRY

BLU

GAIN
SIGN

R23
51 SM REAR

S21

TOP

S21 WIRING

J50
INPUT MONITOR
RG-17414
DG201BDJ
U60D

1

6

C200
100nF

AD825AR

DG201BDJ
U60A

2

15

-15V

C201
100nF

4

C55
3.3nF

CW

3.3Hz

R12
15.0K*OMIT
R12FOR MOD

CW

60Hz

200Hz

C22
100nF

U20
3 +

R22
51 SM REAR

R21
8.2K

R71
10M

R72
10M

600Hz

C23
33nF

C24
10nF

C25
3.3nF

C26
1nF

C27
270pF

C28
50pF

C57
330pF

300Hz

30Hz

X10

R73
10M

2kHz

6kHz

20kHz

60kHz

6
PI
CORNER

200kHz

INF

S20

C58
82pF

NC
C59
22pF

C56
1nF

100Hz

X100

NO JUMPER
=X1
R10
9.09k

3

2

1

R74
10M

J10

GAIN
JUMPER

R76
10M

R77
10M

R78
10M

R79
10M

R75
10M

R11
820*

1KHz

3KHz

10KHz

30KHz

7

R71-79 ARE SM
ON TOP SIDEINPUT
MONITORBANDWIDTH

100KHz

S51

R5E
OMIT

R20
82k*

10.0K FOR
MOD

INPUT
MONITOR
GAIN
R5B

S50

FROM J3, PAGE 1

1

J4

8

A

B

C

D

29

Figure 2.9: JILA Slow Loop Filter. For autolocker, header J9 is controlled via an arduino microcontroller. Further fine-grained control of the behavior of the loop filter can be obtained via
bypassing the DRL logic and controlling the voltage levels of AS1-4 directly. Early attempts at
sample-and-hold required this extra step.

30
the transmission of light through the cavity is at a level corresponding to the cavity being locked.
If so, do nothing. If not, disengage the loop filter, scan the cavity until a resonance is reached andre-engage the loop filter. A more detailed FSM diagram is shown in Fig. 2.10.
To implement this finite state machine four very simple pieces were assembled, an arduinocontrolled loop filter, a DDS capable of producing triangle waves, a voltage summer for adding
the DDS signal to the output of the loop filter, and a window comparator set to give a logic highwhen the transmission photodiode voltage was considered high. As will be explained in Sec. 2.4
arduinoes are widely used throughout the lab because of their simple programming and ease ofunderstanding. In this project, the arduino controls the setting of the loop filter, reads off thestatus of the photodiode comparator, and can phase coherently switch the DDS from a triangle
wave at a frequency of a few hundred Hz to an effective DC value at 1 µHz via frequency shift
keying (FSK) mode. Much like a human operator would, the system scans through a resonance byengaging the DDS high frequency triangle wave , sees the transmission go up, and stops the scanat the correct DC PZT voltage via the DDS’s FSK mode. The arduino then engages the loop filter
and the entire system is locked.
As this system is so single purpose, many tricks and non-standard uses of the arduino environment were used to speed up the response time of the arduino microcontroller which for the
purposes of this project was an ATMEGA328. Normally, arduinos are used because they hide a lotof complexities of working with microcontrollers. The entirety of any arduino program is encapsulated in two functions; one entitled setup and the other entitled loop. Very simply, the setup
function runs once upon power up, and the loop function continually loops. Arduinos also offer
easy to use functions to read in or write out digital signals onto their ports like digitalRead(pin)or digitalWrite(pin, value). However, a lot of these commands are very slow compared to theclock speed of the ATMEGA chip, and there are a few simple tricks that made the autolocker work
a little faster. These will make your code significantly harder to read, for those unfamiliar with
their use. I’ll highlight a few of the tricks here.
First, entering and exiting the stack of loop causes significant overhead that is just not31

Check Transmission

Previous
Transmission

Set Autolock Mode

Does the Rotaryswitch agree withloop filter mode?

NoNo

Set loop filter to rotaryswitch setting

Yes

Yes

Rotary switch andloop filter set toAutolock/9db?

Rotary Switch

Start over!
What is thecurrent Autolock
Mode is it?
Stop DDS sweeping
Engage Loop Filter

Start over!
No

Set Loop Filter to Acquire
Set Search Mode to true

Yes

SearchMode?

Set DDS to
sweep
Start over!

Figure 2.10: Diagram of the Auto Locker FSM. Design emphasis was placed on minimizing thedelay between scanning past a fringe and trying to engage the loop filter.

32
necessary for this simple a program, especially as we don’t want the autolocker to miss a cavity
fringe because it was busy reinitializing variables that did not need to be reinitialized. To combat
this, one can put the entire program inside a while loop that cannot exit like in Fig. 2.11.
void loop () {while (1)3{... // rest of program
5}}
1

Figure 2.11: Code Snippet 1
Second, controlling or reading digital pins through the arduino commands is also a slow andtime consuming operation. Instead one should directly access the port registers of the microcontroller itself, like PORTB for the output register or PINB for the input register. The use of theseregisters makes the code less portable as the wiring for these port registers is very chip dependent,
but the speed improvements are worth it. For legibility, once the programmer has graduated or
moved on from the lab, these complicated and fast operations should be rewritten with easy tounderstand macros like in Fig. 2.12.
# define setModeProp () PORTD = ( PORTD & ~ modeSetMask ) | (0 b00011000 & modeSetMask )// Set Loop Filter to proportional mode

Figure 2.12: Code Snippet 2
Finally, the interrupt handler in arduino, while convenient, is again a very slow system.
Therefore, in the case of very simple programs, where every cycle counts, one should instead relyon self-programmed poling to check on outside signals changing. In the case of the autolocker, thismeans checking the window comparator digital signal every loop iteration with a minimal amountof overhead both before and after the poll to assure that a cavity fringe isn’t missed while scanning.
While the ability to create fast special purpose projects with the arduino family is nice, I will go
into detail about the versatility of this family of microcontrollers in Sec 2.4. The full source codefor the cavity autolocker is found in Appendix A.

33

2.3

Spectrally clean light
When I arrived in 2009 SrI was in the midst of a Sr-Yb frequency ratio measurement. A

few months into the experiment, however, a great mystery was plaguing the comparisons. SrI,
which had seemed so infalliable only a few months before was showing signs of some uncontrolled
systematic shifting the line day to day on the order of 1 Hz or a few ×10−15 . The most obvious
culprit was the major change that had occurred between the Calcium experimental runs and thosewith Yb, the switching of the injection locked Ti:Sapph laser to the simpler to use tapered amplifiersystem for the 813nm optical lattice trap. Many experiments to determine what was causing thisshift, specifically examining the TA light, were tried, but the slow speed of data taking due toneeding the coordination of two completely separated experimental groups made the results hardto interpret. The comparison was abandoned, but the specter of TA shifts was present in all theminds of the Strontium gang.
The method by which tapered amplifiers could cause frequency shifts was well understood,
even if the results of the first experiments were not. When pumping current through a TA, before
it is seeded with coherent light, some photons can clearly be seen coming out of the chip. This lightis the so-called amplified spontaneous emission (ASE). Spontaneous emission inside the chip, caneasily be amplified as the light passes through the gain medium which has a fairly broad asymmetricgain curve. One can imagine that this is true for all lasing mediums, however one crucial difference
remains. Most lasers operate in a regime where very little light is generated upon each pass throughthe medium and lasing takes place over many passes. In this regime as the seed light passes throughthe medium the gain gets saturated or nearly saturated, it does not matter as the now amplified
seed light will make another pass through the medium. This process repeats according to the finesseof the lasing cavity, but in the end the laser’s gain is saturated and there is very little amplificationof stray spontaneous emission. Operation of a tapered amplifier is in a completely different regime,
where only a single pass through the medium occurs, and the medium is being pumped as hard asheat dissipation can allow. The gain for a single pass remains unsaturated, and an ASE pedestal

34
of light can be seen on an optical spectrum analyzer (see Fig. 2.13).
Even without complete knowledge of the ASE pedastal’s effect on the atoms, two fixes forthis issue were implemented. First, SrII’s large atom build up cavity was colloquially known as the“filter” cavity before it was completed. Second, we purchased transmission filters with very small
bandwidth (FWHM of 0.3 nm) and placed them after the 813nm tapered amplifiers. So in 2011,
when SrII came online as a clock, we immediately tested the effect that the transmission filter hadon the SrI-SrII frequency. In Fig. 2.14 one can see the effect of removing the interference filter from
SrI’s system. Although it is hard to get a number from this rather preliminary test, it was clearthat a large shift occurred. Subsequently, it was no surprise to hear that the SYRTE group also
struggled with ASE issues in their clock comparisons.3

Before accuracy evaluations of SrII were

to commence we decided that this situation of figuring out exactly what shift might be caused by
the TA would be a long and circuitous road that was unnecessary. Therefore SrII rebuilt the oldinjection-locked Ti:Sapph system. SrI, which was going for a much less lofty accuracy evaluation,used two interference filters and a laser filtration cavity to clean up the TA light going towards theatoms. However, during the course of the SrI and SrII comparison we found that there was still a
non-negligible effect being seen on SrI’s AC stark shift (see Fig. 2.15). During the course of the
day the shift would remain stable, but over longer timescales drifts became apparent.
As Sr clocks become more accurate a growing list of seemingly small effects will come into
play, like the ASE pedastal. Reaching 10−19 will surely take considerable effort examining issues
like Raman scattering inside the fiber bringing the light to the atoms or other seemingly improbable effects. It very well might be that reaching low 10−19 fractional uncertainty might require acompletely new single-purpose built machine.

2.4

Easy-to-build DDS Systems
The work presented in this section is a direct descendant from projects done by Sebastian

Blatt and Yige Lin. As the SrI and SrII systems have become more and more complicated the3

As was told to us in private communications at many conferences

35

B
A
B e fo r e B u ild u p C a v ity

2 0
1 5

W ith F ilte r
W ith o u t F ilte r

1 0

A fte r B u ild u p C a v ity

5
0
-5
-1 0
-1 5
-2 0
-2 5

W ith F ilte r
W ith o u t F ilte r-1 0

O p tic a l P o w e r ( d B m / n m )O p tic a l P o w e r ( d B m / n m )

0
-5

-1 5
-2 0
-2 5
-3 0
-3 5
-4 0

-3 0
-4 5

-3 5

-5 0

-4 0
-4 5

-5 5
7 9 0

7 9 5

8 0 0

8 0 5

8 1 0

8 1 5

8 2 0

W a v e le n g th ( n m )

8 2 5

8 3 0

8 3 5

7 9 0

7 9 5

8 0 0

8 0 5

8 1 0

8 1 5

8 2 0

8 2 5

8 3 0

8 3 5

W a v e le n g th ( n m )Figure 2.13: Optical Spectrum Analyzer measurements of tapered amplifier systems with varying
levels of filtration. A) Traces showing the effect of transmissive interference filters before the lightis further filtered by the SrII buildup cavity. B) The same measurements performed with thetransmitted light of the buildup cavity.

36

F r e q u e n c y D iffe r e n c e S r II - S r I (H z )3 .5

N o In te r f e r e n c e F ilte r S r I
In te r f e r e n c e F ilte r S r I

3 .0
2 .5
2 .0
1 .5
1 .0
0 .5
0 .0
-0 .5
-1 .0

3 /1

4 /2

0

0 1
1 2

:4 7
3 /1

4 /2

0

0 1
1 2

:5 0
3 /1

4 /2

0

0 1
1 2

:5 3
3 /1

4 /2

0

0 1
1 2

:5 6
3 /1

4 /2

0

0 1
1 2

:5 9
3 /1

4 /2

0

0 2
1 2

:0 2

Figure 2.14: SrI - SrII frequency with and without interference filters. This test was performed on
the third day of comparing SrI and SrII, and although not all systematics were stabilized at the time,
the quick measurement and huge shift seemed to agree with the idea that an ASE pedastal could
cause a sizable shift and was most likely to blame for issues with the Sr-Yb ratio measurements
three years prior.

37

Figure 2.15: SrI TA shift drifting over the course of comparisons with SrII. Although their AC
stark drifted, the absolute magnitude of the drift was still workable and allowed us to compare ourclocks down to low 10−17 [22].

38need for RF sources around the lab has greatly increased. A majority of these sources need only
be updated on millisecond timescales or need only switch between a few number of frequencies orphases. Rather than purchase many tens of thousands of dollars of RF signal generators we haveinstead created a very simple cost-effective arduino powered DDS system that is easy to replicate.
Systems like these are now present in both strontium experiments, the XUV comb experiments,
the stable laser experiments, and even in the ultracold polar KRb experiment.
My first project on the strontium experiment was programming PIC16f microcontrollers attached to a DAC. While this project wasn’t particularly difficult, I learned a few key lessons fromthis project. First, unless there is a very easy to understand framework for everybody in the lab
to look at and understand, code written for any project will end up being treated like a blackbox. Second, it is much easier to create an extensible platform for use around the lab if there
are many internet resources that other students or postdocs can quickly look up. The arduino
platform satisfied all these conditions and allowed for an inexpensive easy-to-program digital interface. In addition, the arduino platform can very easily be integrated into computer programs or
can be used to control other digital electronics through a variety of industry standardized digitalcommunications protocols.
In fact, the easy-to-use communications library is one of the main selling features for arduino
use in lab. Many digital sensors operate using the Serial Peripheral Interface Bus (SPI) whichallows one master to talk to many slave devices on the same bus. The SPI interface operates with
3 common wires going to every device, and one device-specific wire between the master and slave.4
In Fig. 2.16 one can see the topology of one arduino controlling many DDS slaves and an exampleof how the communications protocol works. The master arduino selects the device it wants to talkto by bringing its slave select pin low, and then sends both clock pulses and data to the slave device.
If data needs to come back, the master continues sending clock pulses pulling data in on its input
line.
So how does one control a direct digital synthesizer (DDS) and how does a DDS create a4

It can also operate with 2 common wires but it is less common to wire devices up like that.

39

SCLK
MOSI
MISO
SS1

SCLK
MOSI
MISO
SS

SS2
SS3

SCLK
MOSI

Timing Diagram

MISO
SS

SCLK
MOSI
MISO
SS

Figure 2.16: Diagram showing the topology of one microcontroller controlling three SPI slaves. The
microcontroller is in charge of the communications clock and 4 different polarities of communication
are possible, as shown in the timing diagram. The timing diagram is taken from Ref. [61].

40sine wave at an arbitrary frequency? A DDS is in fact just a lookup table chip with an integrated
DAC. The rate at which the lookup table is output to the DAC is set by two numbers, the masterclock frequency, and the frequency tuning word (FTW). The FTW is the integer that divides themaster clock which sets the output rate from the lookup table. The higher precision of the FTW
the finer grained steps in frequency are possible. Of course, actually using a DDS requires much
more configuration data being sent to the DDS chip as many chips come with multipliers and phaselocked loops to create high frequency master clocks from stable low frequency inputs.
For high speed communications and changing parameters quickly, some DDSs offer parallel
communication where data is brought in on many digital lines at a time, but programming in thesesituations is much more difficult and is usually reserved for FPGAs or ASICs. However, manyof the DDSs we use offer the ability to program many profiles of frequency, phase, amplitude or
sweeps or any number of specialized routines, making the programming part easier. But even if
the programming and the construction of the hardware is easy, to be a useful tool there must
be a graphical user interface (GUI). Because of the ease of talking to microcontrollers as virtualCOM ports, the GUI frontend for a microcontroller controlling many DDSs can be written in any
language. I chose to write the software in Matlab, a decision that will be fully explained in thenext chapter. To keep the frontend code useful, the code is written to create GUIs for an arbitrary
number of DDSs controlled by one arduino microcontroller. Type 2 into the initialization script
and two totally seperate DDS control boxes are built. The screenshot in Fig. 2.17 shows how one
can control many of the possible parameters of the DDS. This is an incredibly versatile tool that I
hope will continue to be used throughout the Ye labs even after I have graduated.

41

Figure 2.17: Screenshot of DDS control frontend. Interface for controlling the arduino microcontroller and the various parameters of two seperate DDSs are shown. Due to the emphasis on Object
Oriented Programming, controlling an arbitrary number of DDSs does not require any additional
code.

Chapter 3

Running a Clock – A Software Revolution

There are many steps between building a clock and taking frequency comparisons. Getting
all of the hardware in place is only half the battle, actually performing complicated experiments
every second by orchestrating digital, analog, RF channels, and lasers without fail for hours at atime is the other half. During the course of the optical Feshbach resonance experiments we utilized
a hodge podge mix of Labview, Matlab, and even C code. When we decided to switch the focusof the experiment towards precision measurement, the quickest fix would have been to copy SrI’sLabview control code. However, after examining it closely, it seemed that the interim period ofbuilding a clock would be a golden opportunity to try and construct a flexible, extensible code
framework that would allow for significantly more complicated future experiments. Watching the
SrI group struggle to find work-arounds to get their Labview code to do things it was not designed
to do, it seemed like the inflexibility of the system held them back. The issues I saw were notsimple to fix, and would take a concerted effort focusing on modularity and ease of programming
to prepare for the challenges of running a clock that I could think of, and make it easier to expand
the code when new challenges arose.

3.1

The Experimental Sequence
At the end of Section 1.1.1 a brief description of the experimental sequence was explained.

Once the atoms have been cooled and loaded into the lattice, there are a few small details thatneed to be explained to create a usable clock. For the purpose of our accuracy measurements the

43
MOT coils are shut off extremely slowly, and a buffer time of about 250 ms is instituted between
the shutdown of the high current MOT field and the probing of the clock transition. During thistime, the atoms are given their final cooling, as the broadband red MOT is smoothly tuned to asingle-frequency red MOT and the control computer decides if further atom preparation is needed.
After this optional preparation stage of optical pumping, a variable TTL pulse is sent to both the
RF drive electronics for the final clock laser AOM and to the AOM intensity servo built to control
the clock laser intensity. The clock laser interacts with the atoms with a square pulse for a variable
amount of time (usually 160ms). Depending on the frequency of the clock laser and the intensityof the light the atoms will Rabi flop between the ground and excited state, with each atom ending
up in a quantum superposition of these two states. Collecting the information of how the atoms
react to the light frequency relative to the atomic resonance is at the heart of all our experiments.
To read out the coefficients of the quantum superposition the atoms have been placed in, we
first collapse their wavefunction by turning on the blue 1 S0 to 1 P1 light. Leaving this light on, all
the atoms that have collapsed into the ground state scatter photons that we collect on a PMT, andheat out of the trap. Next, we turn on the repumper lasers that connect the 3 P0 and 3 P2 statewith the 3 S1 state allowing the atoms to eventually fall back to the ground state via the 3 P1 state.
Once again the blue light is turned on and the population of atoms in the ground state are counted
by capturing some of the scattered fluorescence. Capturing these PMT signals is an analog signal
digitizer card on our data computer.
Scanning across the clock transition one could see many possible resonances depending on
the nuclear spin state occupation of the atoms and the magnetic bias field vector. One might see
all the π transitions between ground and excited hyperfine states for an unpolarized sample with abias field, a single clean optically pumped π transition, or without a bias field a mess of overlappinglines leading to a broadened unpolarized line. With the excitation signal read from the ensemble ofatoms, which concludes one cycle of a clock measurement, the sequence starts over and a new set ofatoms is loaded and cooled. This repetition of TTL signals is created by a commercial FPGA boardthat does not know what kind of scan has been completed nor does it know what the frequency

44
of any RF oscillators are, it just keeps repeating the same sequence of TTLs over and over again.
To turn this into a smart system something needs to manage all the details, and connect thosedetails with the measurement results. The software that performs all this manipulation and datataking will be explained below. To connect the software to what is happening in the real world,
we look no farther than the digitizer card that reports once a second after its data buffer is full.
This seemingly innocuous “Buffer Full” message is the key to connecting what is going on in thereal world with what is going on in the software’s model world.

3.1.1

Synchronizing Two Clocks

Before I go on to explain the software, another hardware issue remains. The SrI and SrII
experiments were built to be independent, but data taken during comparisons between SrI and
SrII needs to be compared precisely without any timing slop. This is especially true for the case ofsynchronized detection where some amount of common-mode laser noise is removed [28, 62]. While,at first, accomplishing this task seemed easy enough by having one control FPGA trigger the other
FPGA, data analysis showed some very peculiar issues. Each setup would timestamp the data they
had gotten with the time read off by the computer operating system. Even for datasets that were
known to be taken with both experiments synced up to the ns level, the timestamps between thetwo computers showed cyclical drifts on the order of 900 ms. As the duty cycle of the experiment
was very close to this length in time, it became a real problem to try and ‘sync’ up these two data
streams for analysis.
A solution was needed to simplify the timestamping problem and not allow experimenter bias
into the data processing. Our first solution was to use an arduino with an ethernet shield add-on.
One of the FPGAs would trigger a single pulse for each experimental run and the arduino would
count these pulses. Whenever either experiment wanted to add a timestamp to their most recent
data they would ask the arduino over TCP/IP what the current pulse number was. While thisseemed like a simple solution, the inclusion of the ethernet shield to allow communication between
the arduino and both strontium control computers created a strange confluence of blocks for the45
arduino. Issues of communication between the arduino and the ethernet microcontroller, and blocks
when both computers were trying to talk to the arduino caused some pulses to be missed by thearduino resulting in noncontiguous datasets. As most of these issues stemmed from what were
essentially threading problems, it seemed reasonable that a commercial product, like a frequency
counter would have most of these issues worked out. Surprisingly, the task of continually counting
pulses while being asked what the total is is not a common capability of many frequency counters.
Luckily, after searching we found that newer Agilent counters (e.g. the 53220A) could perform
this function. Overall this system worked fairly well and allowed easy frequency comparisons to beanalyzed between the two strontium experiments.

3.2

The Slyfox Software Suite
The Slyfox software suite is a collection of programs I wrote to control many aspects ofthe data taking process. The use of a common framework across all the programs in this suiteof software is to allow maximum code reusability and fast extensibility. Many of the smaller
pieces of the programs are written with object oriented programming (OOP) in mind to allow
for easy access to hardware devices like arduinos, DDSs, analog outputs or software devices likeother computers via TCP/IP communication, PIDs, and data automation programs. The GUIsin all of these programs are separated out from the implementation with the intent to preservea Model-View-Controller (MVC) backend agnostic realization that allows for a clean separation
between coding tasks. The choice to program in Matlab for the control software for SrII was a
conscious decision to minimize the learning curve for future students who might not know how
to program. For the benefit of future generations of strontium students, I’ll include explanations
below of the concepts employed in SlyFox’s various programs and give concrete examples of how
the software uses these techniques. All code is accessible online hosted on its own Google Codepage at https://code.google.com/p/slyfox/.

46
3.2.1

An Object-Oriented Programming Primer

To explain OOP it might be useful to examine an example of how one would describe a realworld setting with OOP principles. Borrowing a page from the Advanced Placement test, let’s
examine the structure of an ocean. Our pretend ocean will have an environment, the sea, andthings that live in this environment like fish and sharks, and we will be charged with showing
the fish moving around and the sharks hunting the fish. The key insight that OOP will bring to
the table is that in this programming scenario there are many pieces of modeling code that are
seemingly redundant. There will be many fish and sharks in the sea, and even if they are not all
exactly the same, they will differ only in small details which can be individually cataloged, andtheir behaviors will in fact be nearly identical. Furthermore, while we might think of sharks and fish
as very different, it’s plain to see that we might model their movement in the exact same manner.
So it seems clear that whatever data structure or form these fish take should somehow make therelationship between fish and sharks evident as well.
While OOP can be implemented in almost any language, I will discuss the framework usedin Matlab and Java as they have many similarities1 . We’ve already discussed that any representation of a fish (or shark) might include details about an individual fish, which will be known as
properties , and also behaviors that the fish has, which will be known as functions . The samecan be said about a shark, that it is somehow a special case or a member of a subset of fish. To
describe our fish we will write a fish Class . This Class file will list all the attributes and behaviors
we could ever want a fish to have. In our example, fish will have three public properties a color,a position and an environment, and will have a few public functions , including swimForward,
swimBackward, swimLeft, swimRight, and a constructor and deconstructor functions. These latter
two functions are included in every Class file.
To fill our pretend sea with fish, one will simply instantiate many objects created from theClass fish. Instantiation is the process of calling the constructor functions, where the programming
1
While I will use the word and explain the concept of an Interface , Matlab does not actually implement thisidea, as Matlab is much less stringently typed than Java.

47
language carves out a piece of computer memory that holds copies of all the data in fish that belong
to each new fish object. Instead of keeping a list of the color of each fish in your main program,when the fish are being constructed, one simply gives each fish a color. If for some reason you wantto know the fish’s color one simply needs to access that chunk of information in each object2 . But
how does this work for a shark? As I mentioned before, in our simulation, a shark will be very
similar to a fish. In fact, it might have all the same public properties and maybe even the same
public functions . The only difference might be in the addition of a few functions or maybe in
the implementation of some of the already present functions. Most object oriented programming
languages have this extension or hierarchy idea built in. What would be created is a Class shark
which “extends” the Class fish. Shark is now a subset of fish, allowing it to act and have the same
memory structure and functions as a fish. In strongly typed languages this hierarchical structure
can incorporate more relationships than just the idea of a subset or special case. By implementing
interfaces one can create certain “types” of objects. For example fish and sharks might bothinclude a function Swim , but so could turtles, and interfaces are used to establish this common
Swimmable behavior.
While this complicated array of relationships and hierarchies between objects might not seemvery important, let’s examine why this might make our implementation of an ocean more efficient.
Imagine that our program has filled its pretend ocean with fish and sharks of various colors andeach object knows its position in our pretend ocean. Running the simulation of our ocean might
be a very simple matter of telling each object to “move”. Rather than have the main program
deal with all the complexities of repositioning each fish, arbitrating whether the fish objects canoverlap at the same position, or causing the sharks to eat the fish, all our main program does is telleach object, maybe in a random order, to move. All behaviors and interactions between objects
will be controlled at the object level. Think of how simple our program’s structure has become, wewould not have to write complicated routines for moving fish and then moving sharks, or carefully
orchestrating each object’s actions, instead it has all been structurally simplified. At the object
2

Some languages might require one to program in “get” functions for retrieving properties about each object.

48
level there is still code to be written, but the behaviors are much simpler to compartmentalize
and write. So not only does OOP allow for easy duplication of common resources, it can also
be a powerful tool for creating structurally simple programs by offloading tasks into well defined
objects.

3.2.2

An Example: The Slyfox Frequency Synthesizer

Lets examine how the slyfox program uses OOP to create a versatile frequency synthesizer
for use in frequency scans for the clock laser, optical pumping light, or sideband cooling. First off,
let’s think about the requirements this interface for hardware instruments is going to need. One, weare going to need to control a lot of different RF synthesizers with this software. This meant thatthe interface needed to agnostically talk to different synthesizers using different commands. Two,
setting up communication between the interface and the instrument should only take a second andshould be very simple. And three, the interface must have a clean and simple user interface (UI)
that allows each instantiation of this object to have very simple to use controls that can be placed
automatically in the main program’s control window. To begin with, I’ll show you a few of thebasics of OOP that are implemented in this frequency synthesizer, and how these requirements are
all addressed.
1classdef FreqSynth < hgsetget% FREQSYNTH Simple interface to control HW instrument

Figure 3.1: Code Snippet 1
The code in Fig. 3.1 begins with the keyword classdef which tells the interpreter that thisfile will be defining a class by the name of FreqSynth which will be a subset of the class hgsetget.
As can be found buried deep within the Matlab documentation, the hgsetget class extends thehandle class. This not only allows the use of set and/or get methods to change the properties ofan object without having to program those all by hand, it also makes sure that this object will
be pass-by-reference versus the default pass-by-value. Making these objects pass-by-reference is anecessary addition to the FreqSynth class because references to these objects will need to be usedby multiple other resources like frequency sweeper or frequency locker programs. Passing these

49
objects by value would create multiple interfaces to the same hardware object which would nothave the same properties or settings.
properties
myFreqCommand = ’ FREQ ’;
myPanel = uiextras . Panel ( ’ Tag ’ , ’ fsynthPan ’) ;myDEBUGmode = 0;
myTopFigure = [];myVISAconstructor = [];myVISAobject = [];myTimeSynth = [];
end

2

4

6

8

Figure 3.2: Code Snippet 2
Following the class declaration are the properties (in Fig. 3.2), data specific to each instantiation of the class3 . The three most interesting properties to point out are myPanel, myVISAobject,and myFreqCommand. MyPanel is a UI panel that will hold all dropdown menus, buttons, and text
inputs. MyVISAobject is a virtual instrument software architecture (VISA) which allows socket-likecommunication between serial, TCP/IP, USB, and GPIB ports. This allows the program communicate with any hardware connected to the control computer regardless of how the connection ismade. The string myFreqCommand will allow us to easily change the behavior of the hardware device, as explained later in this section. With the individual characteristics of each object laid out,
let’s examine the object’s constructor to see how it builds a FreqSynth object.
function obj = FreqSynth ( top , f )obj . myTopFigure = top ;3
obj . myDEBUGmode = getappdata ( obj . myTopFigure , ’ DEBUGMODE ’) ;set ( obj . myPanel , ’ Parent ’ , f ) ;5
h0 = uiextras . VBox ( ’ Parent ’ , obj . myPanel , ’ Tag ’ , ’ freqGrid ’) ;info = instrhwinfo ( ’ visa ’ , ’ ni ’) ;7
visaCMD = uicontrol (...
’ Parent ’ , h0 , ...
9’ Tag ’ , ’ visaCMD ’ , ...
’ Style ’ , ’ popup ’ , ...
11’ String ’ , [{ ’ Use Time Synth Object ’ }; info . Ob ject Const ruct orNa m e ] ,...
’ Callback ’ , @obj . updateFreqSynth ) ;13
h1 = uiextras . HBox ( ’ Parent ’ , h0 , ’ Tag ’ , ’ freqGrid ’) ;...
15
freqComBP = uiextras . BoxPanel (...
’ Parent ’ , h1 ,...
17
’ Tag ’ , ’ freqComBP ’ ,...
1

3

It is possible to make variables that are not instantiation specific, these are denoted by the use of the modifier
static .

50’ Title ’ , ’ Frequency Command ’) ;freqComVB = uiextras . VBox (...
’ Parent ’ , freqComBP ) ;uicontrol (...
’ Parent ’ , freqComVB , ...
’ Style ’ , ’ edit ’ , ...
’ Tag ’ , ’ frequencyCommand ’ , ...
’ String ’ , ’ FREQ ’ ,...
’ BackgroundColor ’ , ’ White ’ , ...
’ Callback ’ , @obj . updateFreqSynth ) ;

19

21

23

25

27

...
myHandles = guihandles ( top ) ;guidata ( top , myHandles ) ;obj . loadState () ;

29

31

end

Figure 3.3: Code Snippet 3
For brevity’s sake, I have removed some of the extraneous UI code (in Fig. 3.3) to allow
us to focus on the important points included in this constructor. Overall, it should be notedthat many of the commands in the FreqSynth constructor are in fact calling other constructors.
For example, all of the function calls starting with uiextras are creating UI objects which getplaced inside the obj.myPanel which in turn gets placed inside of the Panel f. The use of thisuiextras package allows for a dynamically sized and placed grid of UI elements, which makes GUI
programming significantly less time consuming than placing all the elements by hand. Furthermore,this simplifies the difficulty of creating an arbitrary number of FreqSynth objects, as all one needs
to do is carve out a Panel to hold the new object and pass a reference to it in the constructor.
Lines 6 and 11 of the constructor in Fig. 3.3 create a list and dropdown menu for allcomputer connected hardware devices. Choosing which instrument to talk to is accomplished byjust choosing an element from the dropdown list. In addition, line 31 is used in conjuction with asaveState function to create a default selection for the communications dropdown menu and other
UI elements. Finally, to create large hardware and function compatibility, lines 21 to 27 create a text
box allowing for the command that will be sent to the hardware to be entered, the myFreqCommand
string. For example an SRS function generator might expect to receive a command like FREQ
10000000 whereas a dual channel tektronix DDS might expect the command SOUR2:AMPL 0.0 forscanning the second output channels’ amplitude. Rather than focus on the minutiae of initializationand error catching, we’ll instead focus on exactly how this last functionality is in fact implemented.

51

2

4

6

8

10

12

function ret = setFrequency ( obj , newFreqStr )tryfprintf ( obj . myVISAobject , [ obj . myFreqCommand , ’ ’ , newFreqStr ]) ;ret = 1;catch
if obj . myDEBUGmode ~= 1
errordlg ( ’ Abandon hope , could not set frequency . ’)
ret = 0;else
ret = 1;
end
end
end

Figure 3.4: Code Snippet 4
The function setFrequency in Fig. 3.4 is in some ways an all-purpose communications passthrough from the program interacting with the FreqSynth object and the physical hardware. Surrounded by error catching and debugging code, this function writes to the port obj.myVISAobject
via the function fprintf a string of text it creates from the substring obj.myFreqCommand, we
discussed previously, and the substring that is passed into the function call, newFreqStr. Creating
easy access to this function for a variety of programs is in some sense the sole purpose of this entire
object. Furthermore, it allows for easy automated software control of any output parameter of afunction generator that can be characterized by a number.
function quit ( obj )delete ( obj . myPanel ) ;3
obj . myPanel = [];obj . myTopFigure = [];5 end
1Figure 3.5: Code Snippet 5
Finally, to conclude our exposé on all the important features in an object we have a destructorin Fig. 3.5. This piece of code essentially releases all the pieces of memory that the computer might
not realize were associated solely with the object. In the Matlab language, the function delete
knows how to free all base pieces of data, it is up to the programmer to make sure that references
to other objects and instantiations of other objects get correctly freed.
Throughout this examination I’ve tried to highlight 5 mainstays of OOP, constructors, properties, functions, and destructors. With knowledge of how a simple hardware interface can beconstructed, it’s time to see how to orchestrate many objects to create complicated behaviors like

52
locking a clock laser to the atoms and routinely going into a pseudo atom-debug mode all whilestaying synced with another strontium experiment.

3.2.3

Bringing it Together: The Slyfox Frequency Locker Program

My hope for this section is not only to show how one might create a complicated object
oriented program, but also to fully explain how a locking sequence works with the intent thatanybody reading this will be able to come up with their own locking program to accomplish whatevermeasurements they might need. SrII’s control code was built specifically with this objective inmind; a completely new custom locking sequence should take no more than 30 minutes to writeand hopefully a similar amount of time to debug.
To initiate the program one runs one simple function FrequencyControlMain. The functionis actually quite short, and all it does is instantiate a few objects as can be read below.
% Code at the heart of FrequencyControlMain
f1 = FreqSynth (f , tp ) ;3
ardLC = uControlFrontend (f , tp , 1 , ’ LC ’) ;ardCN = uControlFrontend (f , tp , 2 , ’ CN ’) ;5
AVCC = AnalogVoltageCC (f , tp ) ;fs1 = FreqSweeper (f , tp ) ;7
fs1 . setFreqSynth ( f1 ) ;fs1 . setLCuControl ( ardLC ) ;9
fs1 . setCycleNuControl ( ardCN ) ;fL = FreqLocker (f , tp ) ;11
fL . setFreqSynth ( f1 ) ;fL . setFreqSweeper ( fs1 ) ;13
fL . setLCuControl ( ardLC ) ;fL . setCycleNuControl ( ardCN ) ;15
fL . setAnalogCC ( AVCC ) ;gs = G a g e S t r e a m e r C l i e n t F r o n t e n d (f , tp ) ;17
gs . setFreqSweeper ( fs1 ) ;gs . setFreqLocker ( fL ) ;19
pidPan = PID_Panel (f , tp )tp . TabNames = { ’ FreqSynth ’ , ’ LC_Arduino ’ , ’ NC_Arduino ’ , ’AnalogVoltageCC ’ , ’ Sweeper ’ , ’ FreqLocker ’ , ’ GageStreamer ’ , ’ PID Panel ’ };1

Figure 3.6: Code Snippet 6
All of the objects instantiated in Fig. 3.6 are used to interact with hardware surrounding theexperiment or to run programs that control these hardware objects. For example, ardLC and ardCN
are interfaces for talking to an arduino in charge of the liquid crystal waveplate or a cycle number
counter. AVCC is an interface to an NI 6733 analog output card used to control magnetic fields or

53
MOT loading rates. Finally, fs1 and fL (line 6 and 10 respectively) are used to orchestrate all ofthese hardware elements for the purposes of taking sweeps or locking the clock laser, respectively.
Notice how references to all the hardware interfaces are passed to the programs that will utilize
these interfaces as in line 11. This main function just instantiates the objects, and tells them to takeup residence in tp which is a tabbed panel UI object by passing a reference into their constructor.
This is yet again another simplification enabled by using OOP as no effort is wasted on figuring out
how and where to add all the UI elements as the layout manager built into the uiextras package
is algorithmically placing and resizing all the buttons, text fields, and menus for us.
Let’s now focus on the frequency locking program. The full code line count for Freqlocker
is about 4600 lines, so to explain how the majority of the code works I will judiciously be cuttingout a lot of non-important code. Let’s begin with the FreqLocker properties .

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

properties
myPanel = uiextras . Grid () ;myTopFigure = [];myFreqSynth = [];m yGageConfigFrontend = [];myFreqSweeper = [];myLCuControl = [];myCycleNuControl = [];myAnalogStepper = [];myAnalogCC = [];myLockModes = [];mysPID1 = [];myPID1gui = [];mysPID2 = [];myPID2gui = [];mysPID3 = [];myPID3gui = [];mysPID4 = [];myPID4gui = [];myDataToOutput = [];myDriftPlotHandle = [];myDriftFitHandle = [];myTempSensor = [];endproperties ( Constant )bufferSize = 128;
plotSize = 40;
refTime = 1361903296619;
end

Figure 3.7: Code Snippet 7

54
Hopefully many of the handles in Fig. 3.7 are self explanatory. The Constant or static
properties are ones that do not need to change between instantiations of the object, and are mostly
aesthetic information used to make the plots look nice. There are two new kinds of objects thatare very important to the locking process that I have not shown the reader yet, the seriesPID and
the PID gui objects. These objects are exactly what they sound like, a seriesPID is a very simple
object used to calculate the response of two standard form PIDs in series. Where of course, thestandard form is written as in equation 3.1. And the PID gui object was written to keep the PID
and seriesPID objects to be as simple and versatile as possible without getting caught up in UI
code.


Z
d
1 t
e(t) dt + Td e(t)C = Kp e(t) +
Ti 0
dt

(3.1)

The instantiation of 4 of these completely independent objects will be the backbone of some of themost complicated locking sequences, and in the future with more experimental conditions under
digital feedback instantiating more digital locks is just a few simple lines of code. The constructor
for FreqLocker places and/or creates many of the objects listed in the properties leaving a tab
panel like that shown in Fig. 3.10. To allow FreqLocker access to already created objects like a
FreqSynth, handles are passed to the FreqLocker via helper functions after its construction likein Fig. 3.8.

2function setFreqSynth ( obj , fs )obj . myFreqSynth = fs ;end

Figure 3.8: Code Snippet 8
With all the pieces set, let’s examine how the program initiates a lock and runs the system.
After taking a Rabi scan, guesses for where to start each of the four PIDs are loaded into the text
boxes on the right of Fig. 3.10 by the user. The user chooses which locking protocol to use from a
dropdown list and hits “Start Acquire”. This button’s callback below is fairly simple as it offloads
a lot of work to a different function.
1function sta rtAcq uire _Call back ( obj , src , eventData )myHandles = guidata ( obj . myTopFigure ) ;55
set ( myHandles . stopAcquire , ’ Enable ’ , ’ on ’) ;set ( myHandles . startAcquire , ’ Enable ’ , ’ off ’) ;5
setappdata ( obj . myTopFigure , ’ run ’ , 1) ;obj . s t a rtLockingProtocol () ;7 end
3Figure 3.9: Code Snippet 9
The startLockingProtocol reads off which protocol has been selected from the dropdown
menu and launches that protocol’s initialization function. For our example using the locking protocol “4 PID Locks with Rezero” it calls the function obj.start4PeakLockRezero initialize().
In the future when new protocols have been written, not only will they need to be added to the
dropdown list in the FreqLocker constructor, they must be added to this launching function as
well.
function s t a r t 4 P e a k L o c k R e z e r o _ i n i t i a l i z e ( obj )% % Initialize a bunch of fields for later use
3
seqPlace = 0;
% 0 = left side of PID1 - low Freq5
% 1 = right side of PID1 - low Freq
% 2 = left side of PID2 - low Freq7
% 3 = right side of PID2 - low Freq
% 4 = left side of PID3 - high Freq9
% 5 = right side of PID3 - high Freq
% 6 = left side of PID4 - high Freq11
% 7 = right side of PID4 - high Freq
rezeroSeqPlace = 9;13
% 0 = Unpol , Ix = Ix0 - Delta% 1 = Unpol , Ix = Ix015% 2 = Unpol , Ix = Ix0 + Delta% 3 = Unpol , Iy = Iy0 - Delta
17% 4 = Unpol , Iy = Iy0% 5 = Unpol , Iy = Iy0 + Delta
19% 6 = Unpol , Iz = Iz0 - Delta% 7 = Unpol , Iz = Iz021% 8 = Unpol , Iz = Iz0 + Delta% 9 and on ..... do whatever seqPlace says
23
% Initialize Liquid Crystal Waveplate% Initialize Frequency Synthesizer
25
obj . myFreqSynth . initialize () ;% Start Frequency Loop / Check ’Run ’
27% Try writing column headers for data
try
29
...
fid = fopen ([ path filesep fileName ] , ’a ’) ;31...
colNames = { ’ Frequency ’ , ’ Norm ’ , ’ GndState ’ , ...
33’ ExcState ’ , ’ Background ’ , ’ TStamp ’ , ’ BLUEGndState ’ , ...
’ BLUEBackground ’ , ’ BLUEExcState ’ , ...
35’ err1 ’ , ’ cor1 ’ , ’ servoVal1 ’ , ...
’ err2 ’ , ’ cor2 ’ , ’ servoVal2 ’ , ...
37’ err3 ’ , ’ cor3 ’ , ’ servoVal3 ’ , ...
1

56

Figure 3.10: Screenshot of the Frequency Locker program. Note the many independent PIDs on
the left, and the controls on the right for choosing the locking protocol.

57

39

41

43

45

47

49

51

53

’ err4 ’ , ’ cor4 ’ , ’ servoVal4 ’ , ...
’ Vx ’ , ’ Vy ’ , ’ Vz ’ , ...
’ freqSr1 ’ , ’ delta1 ’ , ’ freqSr2 ’ , ’ delta2 ’ , ’ cycleNum ’ , ’ badData ’ };n = length ( colNames ) ;for i =1: nfprintf ( fid , ’% s \ t ’ , colNames { i }) ;end
fprintf ( fid , ’\ r \ n ’) ;catch
disp ( ’ Could not open file to write to . ’) ;end
% Set to first frequency point
curFrequency = newCenterFreqL1 - linewidth /2;
ret = obj . myFreqSynth . setFrequency ( num2str ( curFrequency ) ) ;if ~ ret
setappdata ( obj . myTopFigure , ’ run ’ , 0) ;
end

55

57

59

61

% % save various fields to be accessed by subsequent function calls including
most importantly
setappdata ( obj . myTopFigure , ’ nextStep ’ , @obj . s t a r t 4 P e a k L o c k R e z e r o _ t a k e N e x t P o i n t );pause (0.5)
guidata ( obj . myTopFigure , myHandles ) ;setappdata ( obj . myTopFigure , ’ readyForData ’ , 1) ;end

Figure 3.11: Code Snippet 10
As can be seen in this greatly simplified code above, only a few major pieces of code actually
get executed when the user hits the start button. After initializing a few fields, two very importantvariables are declared, seqPlace and rezeroSeqPlace. As was mentioned in Sec. 3.1.1 the best
method we found for synchronizing two clocks was having them share a “Cycle Number” thatgave each individual experimental run a numerical ID. This ID not only allowed for easy datatabulation, but it also opened up the door to programmatically cycle through the behaviors of thelocking protocol. In these programs this is accomplished by reading off the numerical ID (known
as the cycle number) and finding its modulo with the number of different locking behaviors or how
often some rare behavior is supposed to occur. For example, if a field rezeroing is set to occur
every 81 cycles the program will simplify find the answer to cycle number % 81 and compare theresult with the recipe in lines 12 through 22 in Fig. 3.11. As can be read in those lines, as longas the answer is 9 or higher the program is told to go on about its business. In that case, tofind where the program is in the sequence, it takes the cycle number and finds its modulo with 8.
As will be seen soon, the action taken by the program will follow the protocol laid out in lines 4

58
through 11. While much of the rest of this code is boilerplate for beginning a locking sequence,
the most important line is 57 in Fig. 3.11. This places a hook in the program that will launch
the correct function when data has been collected from the atoms. Designing a new sequence canbe accomplished very simply then by clearly laying out what the order of the sequence is and any
strange cyclical behaviors like rezeroing, making sure all the data one is interested will be correctly
written down like in lines 28-48, and setting the correct hook for your new program. All the realmeat of interfacing with hardware and taking and analyzing the data is encapsulated in each of the
protocol’s takeNextPoint functions.
function s t a r t 4 P e a k L o c k R e z e r o _ t a k e N e x t P o i n t ( obj , data )% % Access and save fields for later use
3...
if get ( myHandles . cycleNumOnCN , ’ Value ’) % using cycle number ?
5cycleNum = str2double ( obj . myCycleNuControl . getCycleNum () ) ;if runNum ~= 0
7
prevCycleNum = getappdata ( obj . myTopFigure , ’ prevCycleNum ’) ;end
9rezeroSeqPlace = mod ( cycleNum -2 , rezeroSequenceNumber ) ; % 1 for previous
measurement and 1 for mike bishof ’ s convention
NE XT rezeroSeqPlace = mod ( rezeroSeqPlace + 1 , rezeroSequenceNumber ) ;11
seqPlace = mod ( cycleNum -2 ,8) ; % 1 for previous measurement and 1 for mikebishof ’ s convention
fprintf (1 ,[ ’ Cycle Number for data just taken : ’ num2str ( cycleNum -1) ’\rTherfore we just took seqPlace : ’ num2str ( mod ( cycleNum -2 ,8) ) ’\ n ’ ]) ;13
if runNum ~= 0 && prevCycleNum ~= ( cycleNum -1)
fprintf (1 , ’ Cycle Slipped \ n ’) ;15
badData = 1;
pause (0.1) ;17
end
setappdata ( obj . myTopFigure , ’ prevCycleNum ’ , cycleNum ) ;19
end
pointDone = 0;21while ( getappdata ( obj . myTopFigure , ’ run ’) && ~ pointDone )% IMMEDIATELY READJUST LC WAVEPLATE and set frequency for23% next point . Also decide if next time is a rezeroing% unpolarized sequence , and set comp coil voltages
25% accordingly .
if ( N EXTrezeroSeqPlace > 8) % run frequency locks
27
switch mod ( seqPlace +1 ,8)
case 0 % left side of line 1
29
curFrequency = newCenterFreqL1 - linewidth /2;
case 1 % right side of line 1
31
curFrequency = newCenterFreqL1 + linewidth /2;
case 2 % left side of line 2
33
curFrequency = newCenterFreqL2 - linewidth2 /2;
case 3 % right side of line 2
35
curFrequency = newCenterFreqL2 + linewidth2 /2;
case 4 % left side of line 3
37
curFrequency = newCenterFreqH1 - linewidth /2;
1

59

39

41

43

45

47

49

51

53

55

57

59

61

63

65

67

69

71

73

75

77

79

81

83

85

87

case 5 % right side of line 3
curFrequency = newCenterFreqH1 + linewidth /2;
case 6 % left side of line 4
curFrequency = newCenterFreqH2 - linewidth2 /2;
case 7 % right side of line 4
curFrequency = newCenterFreqH2 + linewidth2 /2;
end
if get ( myHandles . bounceLCwaveplate , ’ Value ’) && strcmp ( get ( myHandles .
openSerialLC , ’ Enable ’) , ’ off ’)disp ([ ’ :2; c ’ int2str ( mod ( seqPlace +1 ,8) ) ’; d0 ; t80000 ’ ]) ;fprintf ( obj . myLCuControl . mySerial , [ ’ :2; c ’ int2str ( mod ( seqPlace +1 ,8) ) ’; d0 ;t80000 ’ ]) ;end
else if ( runNum > NEXTrezeroSeqPlace ) % i think this helps with bootstrapping .
switch mod ( rezeroSeqPlace +1 , rezeroSequenceNumber )case 0 % Unpol , Ix = Ix0 - Delta
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (0 , previousVoltages (1) - voltageStepSize ) ;case 1 % Unpol , Ix = Ix0
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (0 , previousVoltages (1) + voltageStepSize ) ; % needto add because of previous step in sequence .
case 2 % Unpol , Ix = Ix0 + Delta
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (0 , previousVoltages (1) + voltageStepSize ) ;case 3 % Unpol , Iy = Iy0 - Delta
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (1 , previousVoltages (2) - voltageStepSize ) ;case 4 % Unpol , Iy = Iy0
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (1 , previousVoltages (2) + voltageStepSize ) ; % needto add because of previous step in sequence .
case 5 % Unpol , Iy = Iy0 + Delta
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (1 , previousVoltages (2) + voltageStepSize ) ;case 6 % Unpol , Iz = Iz0 - Delta
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (2 , previousVoltages (3) - voltageStepSize ) ;case 7 % Unpol , Iz = Iz0
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (2 , previousVoltages (3) + voltageStepSize ) ; % needto add because of previous step in sequence .
case 8 % Unpol , Iz = Iz0 + Delta
curFrequency = ( newCenterFreqL1 + newCenterFreqH1 ) /2;
obj . changeAnalogVoltage (2 , previousVoltages (3) + voltageStepSize ) ;end
if get ( myHandles . bounceLCwaveplate , ’ Value ’) && strcmp ( get ( myHandles .
openSerialLC , ’ Enable ’) , ’ off ’)disp ( ’ :4; c0 ; d0 ; t80000 ’) ;fprintf ( obj . myLCuControl . mySerial , ’ :4; c0 ; d0 ; t80000 ’) ;end
else
curFrequency = newCenterFreqL1 - linewidth /2;
end
end
ret = obj . myFreqSynth . setFrequency ( num2str ( curFrequency ) ) ; % temporarily setsclock laser frequency
...

60
89

end

Figure 3.12: Code Snippet 11
The function in Fig. 3.12 has a few main objectives: first, it must setup the experiment
for the next experimental run, and second, it must analyze and save the previously taken data.
To accomplish this the program first queries an outside object for the cycle number. Following
this, a series of switch and if statements steer the behavior of the program to correctly prepare
the atoms and RF synthesizers. Above one can see two examples of this. The first works off ofthe seqPlace variable setting the right RF frequency for the next experimental run. The next
bit of code similarly works off of the rezeroSeqPlace to correctly take the data for rezeroing themagnetic fields. Although I have removed the rest of the code, many similar decisions are made in
the rest of the function call.
function s t a r t 4 P e a k L o c k R e z e r o _ t a k e N e x t P o i n t ( obj , data )...
3% 9. Write Data to file .
...
5
if ( getappdata ( obj . myTopFigure , ’ run ’) ) % Prepare for a new data point
runNum = runNum + 1; % update runNum
7
seqPlace = mod ( seqPlace + 1 ,4) ; % update seqPlace% save various data19else % close everything done
% % Delete and clear all elements
end

11

13

end

Figure 3.13: Code Snippet 12
As a reminder for future graduate students, the last bit of code to remember is to correctly
deal with all variables and data that were created and plotted throughout running of the frequency
locker. In fact, subsequent calls to start4PeakLockRezero takeNextPoint only communicate toeach other through storing variables in the appdata structure provided by Matlab. So all future
sequences need to correctly remove all data and objects foisted onto this structure in place of line
11 of the code above or else running the locker might result in Matlab taking more and more of
the control computer’s RAM until the program crashes.

61
3.2.3.1

Arduinos: Simple, flexible Timing for Complicated Sequences

To accomplish many of the tasks required to change the optical pumping polarization, turnon and off bias fields, and switch analong switches the FreqLocker program needs to be able to sendout digital TTL signals on demand. The use of an arduino or any microcontroller makes this very
simple. All that is needed is to create a common language for the arduino and the Matlab programto speak to each other. A careful reading of some of the code above will show that the program
routinely writes data like :4;c0 to the object myLCuControl. This is a two part command thatfirst tells the arduino which ‘Mode’ it is in (4) and second what cycle number is coming up (0).4
With this tool it’s very easy to create constant TTLs that can be changed via the FreqLocker
program.
But because this is a microcontroller, one is not limited to only creating DC values. In fact,
one can feed TTL signals into the microcontroller and depending on the cycle number or ‘Mode’change the behavior of the arduino’s output to be an AND, or an OR, or any logical calculation of
the TTL signals. As seen below this is easily done with macros.
# define pinTTL_IN1 7 // Reads this TTL In# define pinTTL_OUT1 5 // Depending on state either mirrors TTL_IN1 or is held low .
3 # define s etPinTTL_OUT1_HIGH () PORTD |= 0 b00100000 ;# define setPinTTL_OUT1_LOW () PORTD &= 0 b11011111 ;5 ...
void loop () {7 if ( mirrorTTL1 ) {if ( digitalRead ( pinTTL_IN1 ) ) {9
setPinTTL_OUT1_HIGH () ;}11else {
se tP inTTL_OUT1_LOW () ;13}}
15 else {se tP in TTL_OUT1_LOW () ;
17
}
...
19 }
1

Figure 3.14: Code Snippet 13
4
The rest of the message was built to house instructions for Ramsey TTL signals, however arduino timing jitterfor slow clock speed arduinos may introduce noise into the atom signal.

62
Inside loop, the behavior is governed by state variables that are set by parsing the messages
from FreqLocker. Although this means that complicated frequency locker programs are essentiallyspread out over both Matlab and arduino files, the array of what is possible to do in these sequences
is wildly expanded.

Chapter 4
Where in the world is 1 S0 →3 P0

2008 was a tumultuous year for America. With the subprime mortgage crisis leading thecountry into a recession, a pretty divisive presidential election, and ultracold fermion optical latticeclocks seeing a very clear atom-dependent collisional frequency shift, people were unsure about what
the future would hold. Many obstacles stood in the way of optical lattice clocks. In some ways,
these systems brought with their claims of high stability an unexplored territory of systematics
that needed to be understood and characterized to incredible levels of precision. The first roundof strontium systematic uncertainties just barely edged by the state-of-the-art cesium fountains,reaching total systematic uncertainties of 1.5 × 10−16 [1, 63]. To do better than this level of
performance two things needed to occur. First, either considerable measurement time or the useof a much more stable local oscillator was necessary. And second, measurement techniques orfundamentally different clock designs were needed to address the three largest systematics: densitydependent shifts, AC Stark shifts, and BBR external temperature shifts.
A major obstacle to this work was the inability to take absolute measurements, as we wereno longer looking for a factor of two improvement over current standards. This required us toinnovate where the additional relative measurement precision, afforded to us by the use of the morestable laser, was not enough to achieve our goals [62]. The following two chapters are expanded
versions of Ref. [22] made to fully explain all measurements, arguments, and statistical methods
employed to reach our final systematic uncertainty of 6.4 × 10−18 . Where applicable, I will mention
additional measurements or ideas for improvements to each of these systematic effects for future64
clock evaluations.

4.1

Statistical Methods for Data Analysis
For all systematic measurements, residual non-white noise introduces over scatter in the data.

Thanks to the influence of precision measurement experiments to the statistical techniques employedin our strontium group at JILA, I believe we have taken a very conservative approach to quoting
our results. Much of the data taken in this paper consists of a sequence of difference measurements
taken between two oscillating experimental conditions. Of course there are many ways to find thedifference between subsequent measurements. Imagine a sequence with frequency measurements aand b taken like so: (a1 , b1 , a2 , b2 , a3 , b3 , ....). We could perform “forward differences” like a2 − b1 ,or we could perform “backward differences” like a1 − b1 , or we could create a weighted difference
between the two. In fact, there are a seemingly infinite number of choices. The creation of these
correctly weighted sums of differences is known, at least in our group, as point-string analysis. In
equation 4.1 one can read off exactly how to weight each of these frequency measurements to create
ever longer strings [64].
1
1
δν =
n−2
M −n+12MX−n+1 Xnl=1

m=1(−1)l+m−1(n − 1)!
νl+m−1(n − m)! (m − 1)!

(4.1)

Where M is the total number of frequency points, n is the point-string length, and ν are thefrequency measurements. For example for the third point string in the series above with n = 3,δν(l = 3) = −ν3 + 2ν4 − ν5 .
It should be noted that due to the overlapping nature of this data analysis method, eachpoint string is not statistically independent. A correction factor according to equation 4.2 must
be included when looking at the statistical significance of results produced via the point-string
method. Any standard deviations calculated using point strings should be multiplied by the factor
noted in equation 4.2. An added benefit for this method is that for increasingly larger n, the data
becomes immune to polynomial-like correlations between points. For example, starting with n = 3

65
the data is immune to linear drifts in the data [64].
fcorrection to std err = s
n 
P
m=1

2n−1
(n−1)!
(n−m)!(m−1)!

(4.2)
2

Data analysis on many of the experiments begins by examining the simple means, standard errors,and Allan deviation of these point-string quantities.
To quote a final number for a measurement, however, is a little bit more involved. This
process begins by binning the data into small chunks. The bin size can and should vary greatly,
again to examine if the data is behaving in a consistent manner for quoting a statistically significant
result. For all bins, a mean and standard error, including corrections for point string and student-t
effects (undersampling of the standard deviation), are calculated. This allows for the calculationof the reduced chi-squared, χ2ν . As is the case for almost all our data, residual 1/f noise pushes thevalue of χ2ν above 1. In instances where this occurs a simple weighted mean of the bins would seemto underestimate the error with which the final result is known. Instead, the bins’ errors are inflatedpby the calculated χ2ν , thereby making the error on the final quantity larger and more statistically
relevant. However, minor changes in bin size, although they should not effect the mean, can cause
fairly large fluctuations in the final quoted error. To combat this, the analysis is performed forall bin sizes ranging from 35 to 75 points per bin where undersampling is less of an issue and the
distribution of deviations is flat, and the error quoted is the simple mean of all bin sizes’ calculated
error. This conservative approach is applied to all measurements in this thesis unless otherwise
noted.

4.2

The Density Shift
The initial observations of cold collisional shifts in a Sr OLC showed only a very small

signature of shifts at a few parts in 10−16 [1]. Reaching accuracies below 10−17 would require precise
knowledge and understanding of these shifts. While work proceeded on finding novel suppression
techniques of these shifts, progress on a complete understanding of these shifts was slowed due

66
to the complexity of modeling Rabi shifts [64]. Subsequent work based on Ramsey spectroscopy
for both Sr and Yb, however, finally explained a completely self-consistent picture of 2D p-wave
collisions occurring inside the lattice pancakes [65, 34, 66]. Regardless of the s- or p-wave natureof the shift1 , as a clockmaker chasing the always on the move accuracy target, we needed to findan easy way to suppress collisions.
While significant advances had been made by working at the so-called “zero crossing” of thedensity shift in Ramsey clock operation, the complexity of Ramsey operation and the stability ofthe zero-crossing to other parameters had not yet been explored [65, 34]. Instead, SrII focused on
the reason why no other strontium groups had yet to report statistically significant density shifts.
Regardless of the lower precision of some other competing experiments, the explanation for thedisagreement was the much more mundane fact that referring to the shift as being caused by somenumber of atoms is incomplete [54]. The number of atoms is not as important as the number ofcollisions that the atoms can undergo, which is of course more of a function of density and latticeoccupation. Not only do the loading dynamics into one’s trap matter but also the trap frequencies
and the atoms’ temperature. The SrI system was seeing large density shifts because they wereloading a dense sample into a very tight trap with densities at the mid 1011 cm−3 .
The simplest way for SrII to decrease the effect of the density shift was to operate at lower
densities around a few 108 cm−3 . This lowers the magnitude of the density shift by almost a factor of
30 even at SrI’s lowest attainable densities. This was accomplished via the use of the build up cavity
mentioned in Chapter 2 which traps the atoms in a significantly wider beam waist for moderate
input trapping powers. A much higher power laser could also accomplish this task without a cavity.
An added benefit of the larger trap volume was of course the ability to better overlap the latticewith the cold red MOT, loading more atoms. As has been seen both in explanations of the density
shift and been borne out of experimental data the density shift is, at our current precision, linear
in atom number. Therefore, the ability to run the system with upwards of 100,000 spin-polarizedatoms allowed us to measure a significantly larger shift and scale down to the atom number for our1

In 2D, p-wave collisions are temperature independent which is plus for making 1D lattice clocks.

67
clock operation with exceedingly high precision. Measurements like this are showcased in Figure
4.1 where the lever arm has allowed us to measure the shift per atom very precisely very quickly.
However, the measurements reported in Ref. [62] are taken at deeper lattice depths thanwe can accurately run the clock at. Running the clock at lower lattice depths reduces the atom
density further, decreasing the shift, in a very nonlinear manner. In fact, if we were somehow to
run the lattice at 5000 Er , even an incredibly modest number of atoms could cause frequency shifts
on the order of 10−14 . So as not to rely on pure density scaling arguments we proceeded to takedensity shift measurements as close to our operating lattice depth as possible. Extrapolation of thedensity shift due to changes in the lattice trapping potential was used only when trap frequencies
had changed by less than 10%. Additional error from this extrapolation is included in the final
quoted uncertainty, although it did not significantly change the final error. For a SrII latticewith depth of 87 Er , the final shift for ∼2000 atoms was 4.7(6) × 10−18 . This corresponds toapproximately 1.8 atoms per pancake, with axial temperatures near 2 µK and radial temperaturesnear 3 µK.2 Without careful measurements of the cloud temperatures at these low density limitsof SrII, uncertainty in the temperatures restricts the accuracy of our average density in an occupied
lattice site to 1.0(0.7) × 109 cm−3 [54].
Future experiments should find a healthy balance between SrI and SrII’s density shifts. Whilethe use of a buildup cavity to ensure low densities is cumbersome, using higher power lasers willallow one to forgo the many complications of using a buildup cavity. Furthermore, the use of colder
atomic samples and vertical lattices will allow atoms to be loaded into even less dense lattices
further reducing the density shift. Active cancellation via feedback operation at the zero-crossingof the Ramsey density shift for low density lattices is another avenue worth exploring in the future
as well [65].
2Recall that in 2D p-wave collisions are to first-order temperature insensitive.

68

Figure 4.1: An Allan deviation of a large lever arm density shift measurement, reproduced fromref. [62]. Disregarding the modulation of the atom number, this data was taken at a higher thannormal clock operating density. The shift per count has been scaled to reflect clock operation
uncertainty for 2000 atoms. Other measurements have shown even higher stability, and make it
clear that the density shift, although an annoyance, should not be considered a problem for future
higher accuracy clocks.

69

4.3

The Zeeman Shifts
Alkaline earth elements’ paired electrons have an inherent immunity towards magnetic field

related shifts. For instance, the boson, 88 Sr, has no nuclear spin and does not exhibit any magnetic
field tuning of the ground state scattering length. The fermion’s tiny nuclear moment couples very
weakly with the field, and the differential g-factor between the 1 S0 and the 3 P0 leads to a small shiftcoefficient of 108mF Hz/Gauss between π transitions. Of course, even this small shift for 1 Gauss ofmagnetic field is almost six orders of magnitude larger than we are trying to control our clock to.
Luckily, how to control the first and second order shifts Zeeman effects is well understood.

4.3.1

The First-Order Zeeman Shift

The first-order Zeeman shift is canceled by alternately interrogating opposite nuclear spin
stretched states. By sending the 1 S0 to 3 P1 (F = 9/2) optical pumping laser through a liquid
crystal waveplate, one can easily switch between polarizing the atoms in the mF = +9/2 or the
mF = −9/2. After each line center acquisition of the clock laser, the atom polarization is swapped.
The quoted Sr frequency is the running average between the two previous measurements, cancelingout any linear magnetic field effects in mF . Key to the efficacy of this scheme is that changes in
the magnetic field over two line center acquisitions are either negligible or averaged away.
The servo controlling the bias field splitting between the nuclear spin states was unfortunately
not perfect, and would slowly drift over long timescales. Not only did this cause some extra
uncorrelated noise between SrI and SrII in comparisons, but it also made us worried that the firstorder Zeeman shift was not being perfectly canceled by the alternating interrogation scheme. If
an overall breathing mode of the field persisted throughout the measurements one could imagine
this causing a shift. A residual first-order Zeeman shift is calculated by examining the overall driftof the magnetic field splitting between the two stretched nuclear spin states, and extrapolating
the inherent shift this drift would induce. If a biased derivative existed in the splitting data it
could be extrapolated to cause a shift. Luckily, no statistically significant drift existed in the data.

70
Table 4.1: Second order Zeeman shift coefficients and their excellent agreement around the world.
Location
JILA SrIPTB
LNS-SYRTEJILA SrII

Year
2008 [63]2011 [67]2011 [35]2014 [22]Coefficient (Hz−1 )
2.58
2.43
2.46
2.48

Uncertainty (Hz−1 )
0.35
0.03
0.02
0.02

Furthermore, a recent update to this coil servo has suppressed the field drift.

4.3.2

The Second-Order Zeeman Shift

The second-order Zeeman shift, caused by the presence of an applied bias and a residual
magnetic field during the clock interrogation sequence, is subtracted off point by point. The
determination of the second-order Zeeman shift coefficient is made via a fast modulation experiment
whereby four digital locks are modulated in sequence: two locks at a high magnetic field, one fora measurement of each stretched spin component, and two locks at a low magnetic field. This
experiment was performed for stretched states splitting ranging from 300 Hz to 1,200 Hz. Because
the shift is quadratic in nature and we end up running the clock at the low end of this measurement
we gain a large lever arm for determining the second-order shift coefficient.
To fit this data, a quadratic function, cS 2 , was used. Here c was found to be −0.248(2)×
10−6 Hz−1 , where S is the measured stretched state splitting in Hz. Extra care was taken to ensurethat modulation of the applied bias field did not cause any rotation of the polarization axis, which
would induce an unwanted differential lattice tensor shift. Table 4.1 compares our measurementof this coefficient with previously reported measurements, showing excellent agreement. By fitting
this coefficient in terms of the frequency splitting measured rather than in SI units of field, we
bypass our imperfect knowledge of the δg factor3 . The second-order Zeeman shift uncertainty of
1.2 × 10−18 is quoted for a 500-Hz splitting between the mF = ±9/2 stretched states, which allows
for a relatively strong bias magnetic field of about half a gauss (50 µT) to be applied to the atoms
3

Although, we introduce other errors as described in Section 4.4.2

71
during clock operation. Future measurements with the new coil servo have already been performed
at higher field splittings further reducing the uncertainty in the second order Zeeman coefficientand its shift of the clock transition [68].

4.4

The AC Stark Shifts
The basis of many of the state-of-the-art experiments dealing with ultracold atoms includes

their trapping or storage inside a far detuned optical trap [69]. Whether this trap is formed at
the intersection between different beams that are either phase coherent or not simply changes thegeometry of the trap. Fundamental to the idea of trapping neutral atoms is that the electron
cloud around the atom becomes polarized, intrinsically deforming the various wavefunctions in theatoms. This changes the energy splittings between various levels, and would on the surface at least
discount the very possibility of using neutral atoms as atomic clocks. The saving grace of using atransition inside a multi-level atom as an atomic clock is of course the fact that the polarizabilities
as a function of frequency for both the ground and excited states goes through many resonances
and there are gentle crossings where these polarizabilities are equal [32]. These “magic wavelength”
trapping points ideally leave no differential trapping shift during clock operation.
In a real system we can only hope to asymptotically approach this ideal limit. The ability
to which we can measure where the magic wavelength trap is located is limited by the precisionlevel of the measured residual trapping shift. Furthermore, higher order shifts related to either
the hyperpolarizability (essentially two-photon coupling) [70] or non-E1 couplings [71] are hard tomeasure due to their weak nature. In this section we will examine our most recent exploration
into all of these shifts, and cross check with work performed around the world also regarding theseshifts in strontium OLCs.

4.4.1

The Scalar and Tensor Shifts

To ease comparison with other published papers on lattice shifts I will use the notation foundin Ref. [35]. In this section we will discuss the shift coming from scalar and tensor components of

72
the polarizability,∆ν E1 = ∆κs + ∆κt β U02where β = 3 |~ε · ~eB | − 1



(4.3)

3m2F − F (F + 1) ; the trap depth, U0 , is expressed in units of latticephoton recoil, Er ; F refers to the total spin of the atom, mF , refers to the projection of this spin
onto the quantization axis; and ~ε and ~eB are the unit vectors of the lattice polarization and themagnetic field quantization axis, respectively. Varying the angle between the quantization axis
and the lattice polarization allows one to separate out the different contributions of the scalar andtensor shifts. The SYRTE group did just that, and reported ∆κs = 0 when trapping with a laser
tuned to νtrap = 368554693(5) MHz and ∆κt = −0.0577(23)mHz/Er [35]. While it is useful to know
these numbers, the actual operation of a clock does not really require one to separate these two
effects which are linear in lattice depth.
Instead, we treat both effects as if they came from the same source. We are able to do
this because we make the magnetic bias field direction to be the same as the lattice polarization,
and always run our clock on the mF = ±F stretched states. However, due to the unfortunateproximity of our ion pump magnet to the MOT coils, a slow DC magnetic field wander is observed
as elements of the apparatus slowly change their magnetization on the tens of minutes time scale.
Active control of the magnetic field was implemented to combat this. We use the atoms themselves
as a collocated magnetometer for the clock. Every two minutes during the clock operation, thecomputer-based frequency locking program pauses to interrogate unpolarized atomic samples undera zero applied magnetic field. A drift in the background magnetic field away from zero results ina reduced excitation signal for the peak of an unpolarized excitation, because all ten nuclear spinstates will experience different Zeeman shifts. As can be seen in figure 4.2, the magnetometer-based
feedback loop not only keeps the field direction constant throughout the clock operation, but also
automatically nulls the background field without an operators intervention.
Once the clock has entered into the servo routine, it takes three peak excitation measurements
for each pair of magnetic field compensation coils (along three orthogonal spatial directions) at

73

Excitation Fraction

0.5
0.4
0.3
0.2
0.1
0
1 3 5 7 9 11
Servo Iteration

-20

-10

0
10
Detuning (Hz)20

Figure 4.2: Using the atomic cloud as a collocated magnetometer, a residual non-zero magnetic
field is inferred via the peak excitation of an unpolarized Rabi lineshape. The left panel shows theservo action of zeroing the residual magnetic field. The slow action is due to the implementationof a simple FIR mean filter on the servo’s output, trading off stability for speed. The right panel
shows a clock transition lineshape for an unstabilized magnetic field (red circles) and an improvedlineshape under the stabilized magnetic field (blue filled circles). Reproduced from ref. [22].

74
different currents (I0 − ∆I, I0 , I0 + ∆I) where I0 is the current of the previous iteration and ∆I
is a small trial step. In situations where the residual field is far from zero, the software steps thecurrent by a fixed amount in the direction indicated by the increasing excitation. When the field
is near zero, a parabola is fit between the three measurements and the current is stepped to thefitted value filtered via a low pass finite impulse response filter. On stopping the servo routine andmeasuring the unpolarized line, the residual magnetic field wander was measured to be less than
0.3 µT.
I would like to note that the choice of the parallel orientation between the quantization axis
and the lattice polarization is very beneficial for clock operation. While the β term in equation 4.3
can be set to zero at the “magic” angle, this is not the optimal orientation to run a clock! By setting
the angle between the two axes to be 0, our clock is to first order linearly insensitive to variationsof this angle. While this cannot be said for the magic angle. Precision alignment of these two axes
is performed via iterative minimization of the appearance of residual σ +(−) transitions from thestretched 1 S0 mF = −(+)9/2 states to the 3 P0 mF = −(+)7/2. Figure 4.3 shows an iteration of theminimization process where the signal for setting each coil’s current is clearly visible. Coupled withthe zero field holding magnetic field servo, this greatly relieves the burden on how exactly to set
your quantization axis parallel to the lattice polarization. This was especially necessary in our casedue to the somewhat random birefringence in our vacuum viewports which caused us to choose apolarization mode set by the optics themselves.
To successfully measure and extrapolate the shift with respect to trap depth, a methodfor determining the trap depth at each measurement point was necessary. Assuming that thepower detecting photodiode had a perfect relationship to the trap frequency would ignore issues
like gravitational sag thereby introducing needless systematics in the measurement. Instead, trap
frequency measurements were performed at each measured AC Stark point via a high-resolution
sideband scan. As explained in ref. [72], radial motion only brings the sidebands closer to thecarrier, so the true longitudinal sideband frequency is found by looking at the farthest edge of theblue sideband. This edge corresponds to the contributions of atoms that are distributed at the75

Figure 4.3: To more precisely align the field of the clock laser, lattice polarization, and quantization
axis we iteratively minimize the peak of any visible transitions driven by circularly polarized light.
Depicted above is one of the minimization iterations of the Y-coil. The clock laser intensity is run
4 or so orders of magnitude higher than during clock operation. Further iterations lose track of thesigma transition as it cannot be seen below the noise. The blue line is a fit to the data used to picka value for the Y-coil before moving on to other coils.

76center of the Gaussian profile for the lattice beam. A scan of the carrier was taken simultaneouslywith these high-resolution sideband scans, and a fit to the carrier was used to determine the exact
fourier-limited linewidth. The hard, outer, vertical edge of the sideband is fit with a straight line
and its frequency-intercept (x-intercept) is noted. This intercept can then be related to the interceptof a Lorentzian’s tangent line at its inflection point. Using both the intercept and linewidth, anapproximate position for the center of the lorentzian can be found. Once the longitudinal trap
frequency was determined, the trap depth could be extracted using the procedures outlined in ref.
[72].
Alternating between four atomic servos, with the magnetic field control activated, we measured differential shifts between a variety of high and low lattice depths from 87Er to 300Er for the
SrII apparatus. To minimize systematic uncertainties caused by differential atomic density shifts,
both high and low lattice depths were operated with an absolute density shift below 1 × 10−17 . As
was mentioned previously, atomic interaction shifts follow a power-law behavior in trap frequency,
and must be taken into account especially at very high lattice depths. Atom numbers were chosen
to provide similar density shifts for both high and low lattice depths allowing for common-mode
cancellation. Furthermore, individual measurements for particular values of lattice depth difference
were performed at 1 × 10−17 statistical uncertainty. Many measurements at different lattice depths
were needed to achieve the low uncertainty reported here (see Fig. 4.4). Combined with otheranalysis to determine higher order shifts, discussed in a subsequent section, this data represents
one of the most precise measurements of the AC stark shift yet.
Although many of these measurements took place many MHz away from the scalar+tensor
magic wavelength, the system was briefly used to pin-point where this zero crossing wavelength
existed. Even though this was not a full systematic study, I believe this number should be used
to at least guide further study of this colloquial magic wavelength, ν ∗ = 368.554485 THz. Future
experiments should focus on a few different systemwide optimizations that would allow for a morerobust AC stark shift and a lower uncertainty. To begin with, our highest lattice depths werelimited by laser power due to a faulty 10 W Verdi pumping the Ti:Sapphire laser used to generate

770Frequency Difference (Hz)

-0.05
-0.10
-0.15
-0.20
-0.25
-0.30
-0.35
-0.40
-0.45
0

50

100

150

200

Differential Lattice Depth (Er)Figure 4.4: To determine the lattice AC Stark effect accurately, a variety of lattice depths wereused. This effect is depicted as a function of the differential lattice depth with binning chosen for
figure clarity (average bin size 68 min, corresponding to an average of 1,600 points). Given the
results of the F-test, the best fit is a linear model. Reproduced from ref. [22].

78
the lattice light. A better pump laser and impedance matching of the cavity mirrors would greatly
raise the attainable power. Second, our low lattice depths were limited by the need to trap radiallyagainst gravity. Vertical lattices are much better suited to creating more accurate clocks. Finally,
if magnetic field direction sensitivity ever becomes a problem again the use of the mF = ±5/2 states
will decrease the tensor shift magnitude by a factor of 6. I believe that many of these suggestions
will become necessities as clocks push below 10−18 .

4.4.2

The Vector Shift

The vector Stark shift manifests itself as a fictitious magnetic field. Borrowing notation from
Ref. [35] again,∆ν E1 = (∆κv mF ξ~ek · ~eB ) U0

(4.4)

where ξ~ek is the degree of circularity in the lattice. Because the shift is odd under the projectionof the nuclear spin, the shift is cancelled by alternately measuring opposite nuclear spin stretchedstates just like the first order Zeeman effect. Previous measurements with the lattice, before theprecision alignment of both the lattice polarization and the magnetic field direction, were shownto have a lattice vector shift on each state of a few tens of mHz. With the addition of an extra
polarizer that both the clock laser and lattice copropogated through, a spectroscopic alignmentof the quantizaiton axis was performed following this vector shift discovery. A worst case extra
splitting of 100 mHz was included in the error budget. This extra splitting would introduce asystematic error to the second order Zeeman shift correction. Calculations, however, show that
this residual vector shift is far below 10−19 .

4.4.3

The Higher-Order Shifts

Higher order shifts, if they were statistically significant, should have been seen in the datain figure 4.4. Model fits for both a hyperpolarizability, quadratic in lattice depth, and an E2/M1
contribution, proportional to the square root of the lattice depth, revealed no statistically significant
contribution, based on a Fisher test with a 1σ threshold. To be clear, without precise knowledge

79of the magnitude of either of these two terms, a Fisher test for the inclusion of an additional termin the fitting equation was used. This test, as outlined in ref. [73] section Test of Additional Term,compares whether the decrease in the χ2ν of the fit to the data is consistent with white noise or not.
Even at this measurement precision, it’s possible that the small magnitude of these shifts
allowed some measure of their contributions to be included in the linear fit. Our fitted hyperpolarizability, 0.48(47) µHz/Er2 , is not inconsistent with previously reported coefficients of 0.45(10) µHz/Er2[27]. However, our data, as shown by the Fisher test, supports a linear fit only. We thus list onlya single overall systematic uncertainty for the lattice AC Stark shift, treating the entire data set
in the most statistically consistent manner. We note that using a prior reported hyperpolarizability coefficient, our overall AC Stark uncertainty would change only slightly from 3.7 × 10−18 to
4.1 × 10−18 . In the previous measurement by the SYRTE group, in order to gain a very large leverarm for the measurement of the hyperpolarizability, a very high (5000 Er ) lattice depth was used,
but the atomic density effects in such tight traps were seemingly not considered. Simple scaling of
the density shift would imply enormous frequency shifts even for small numbers of atoms. In summary, although this data is not inconsistent with the hyperpolarizability measurements previously
reported, this work is not an independent verification of their measurement. Future work, involving
higher stability clock lasers, will no doubt observe and catalogue these more exotic shifts.

4.5

The DC Stark Shift
Strontium is somewhat cursed with a fairly large differential polarizability between the 1 S0

and the 3 P0 states. So the presence of any external DC field, or low frequency radiation thatgets rectified by the atoms, will cause a significant shift.

This is what leads to strontium’s

higher sensitivity to blackbody radiation than other alkaline earth atoms. In the past, incomplete knowledge of the differential polarizability was also a leading contributer to the systematic
uncertainty. However research done at PTB, where precise knowledge of an applied electric field
was correlated with frequency shifts, allowed full determination of the DC differential polarizability,∆α = 4.07873(11)×10−39 Cm2/V [74]. This measurement had an enormous influence on the outcome

80of this thesis, and its importance to the Sr OLC community cannot be overstated.
0

DC Stark Shift (Hz)

-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-250 -200 -150 -100

-50

0

50

100

150

200

250

Electrode Voltage (V)Figure 4.5: Measurements of DC electric-field-induced Stark shift show a quadratic behaviour. The
red circles show that a residual shift due to the stray DC field was -1.3×10−16 . The blue squares
show a greatly reduced shift after purging the vacuum chamber with N2 gas. Dashed lines show aquadratic fit to the data. Solid vertical lines show the locations of a zero net electric field. Tracingout the quadratic form of the shift is not strictly necessary for measuring the shift. Reproduced
from ref. [22]Although we expected no shift, we decided to see if there were any stray DC electric fields
in our system. We were worried that patch charges could be immobilized in the vacuum chambers
fused silica viewports [55]. A pair of disk-shaped quadrant electrodes was placed near the twolargest viewports in the system (separated along the vertical direction) and shifts were recorded as
the electrode polarity was switched or grounded. Differences between the frequency shifts inducedby oppositely charged electrodes indicate the presence of stray background electric fields, as shownin figure 4.5. To be clear, to successfully observe a residual DC shift with only one clock, without
tracing out the quadratic behavior of the shift, three repeated measurements need to be performed.
2~ bg + E~ applied +First, one must apply a field in the (arbitrary) positive direction: ν↑ = − 12 α E
νlaser . Second, one must measure the frequency of the laser with the electrodes grounded: νgnd =2~ bg + νlaser . Third, one measures the effect of the applied field in the (arbitrary) negative− 21 α E2~ bg − E~ applied +νlaser . Combining and manipulating these equations results
direction: ν↓ = − 12 α E

81
in a measured shift as seen below.
(ν↑ − ν↓ )2δνbg =
8 (ν↑ − νgnd + ν↓ − νgnd )(4.5)

On first measuring the DC Stark effect on SrII, a residual, stable −1.3 × 10−16 shift was
discovered. However, when the vacuum chamber was filled with clean nitrogen and then reevacuated while installing temperature probes, we reduced the measurable DC Stark effect to−1.6(1.0) × 10−18 . Further, we found that by cleaning a large viewport we could induce anotherDC charge, and although we tried high power UV LEDs shining into the chamber to dissipate thecharge, this method seemed much less effective than purging the system with gas. The exact mechanism for removing the charges was theorized to be water molecules contaminating the nitrogengas. To complete the full evaluation of the DC Stark effect, we performed similar measurements
along the horizontal directions and determined its effect at −1.9(1.9) × 10−18 .

4.5.1

Active Control

This feed forward method of measuring the DC stark shift and then running the clock issomewhat nerve racking as compared to many of the other systematics that are either actively
measured or cancelled. Recent work has shown that due to the nature of the statistics involvedin equation 4.5, a lower residual field will average down faster. So by intelligently biasing theelectric field plates, one can not only actively cancel the field at the atoms but also have real time
information on whether this field has drifted. Recent unpublished results show that this active
cancellation can indeed measure an actively cancelled shift to an uncertainty of 1 × 10−19 in only
15 minutes of running the clock. Because of optical lattice clocks high stability, active systems likethese will be the future of OLCs.

4.6

Miscellaneous Shifts
While we have discussed many of the major shifts of the clock transition, save the BBR shift

which will be discussed in the next chapter, a great many more shifts still need to be examined. In

82
any new experiment hoping to push the frontiers of frequency metrology to even higher levels of
precision all of these shifts must be considered, and perhaps some that have yet to be discovered.

4.6.1

First and Second Order Doppler Shifts

By trapping the strontium atoms deep within the Lamb-Dicke regime, the motional stateof the atoms becomes separated from the internal electronic state of the atoms. Assuming one
is probing along the tightly confined direction, the Lamb-Dicke parameter can be calculated asclockη 2 = ωωRecoil, which for our case is far below 1. However, even with this condition met, firstsidebandorder Doppler shifts can still occur due to relative motion of the atoms with respect to the wavefrontof the clock laser. For small velocities, the fractional frequency change is approximately equal toµm/s would cause a few 10−15 shift.
β = vrelc , meaning even a persistent velocity of a few

Luckily, the atoms position is fixed by the lattice, and a lattice mirror can be used as a fixedreference point to the atoms. We physically connect a mirror to the same mount that holds the
fixed cavity lattice mirror and use a reflection off that system as our fiber phase noise reference
plane [75]. By having the wavefront phase of the clock laser referenced to the same point as thelattice via a high bandwidth transducer we make the clock laser phase follow the motion of theatoms’ trap precisely. Thus canceling residual first order Doppler shifts.
Using the measured temperature of around 3 µK, we calculate an average total velocity of
3 cm/s. A Taylor expansion of the full relativistic Doppler shift has second-order terms from both
longitudinal and transverse motion. Overall, the second-order Doppler effect results in a fractional
shift less than 10−20 .

4.6.2

AOM phase chirp

Turning on and off the clock laser light for performing detuned π pulses can cause unexpectedconsequences. In the SrII experiment we decided to perform this procedure using an AOM. However,
the heating, cooling, and initial formation of the phonon waves inside the AOM can cause distortionof the clean phase of the clock light [76]. To be more specific, as different parts of the AOM heat up

83
in a non-uniform manner one can imagine very strange pathlength modulations, resulting in opticalphase shifts. This effect was measured in SrII by the creation of a Mach-Zehnder interferometer
examining the phase of the beated light on a digital phase detector [77]. To more easily measure anappreciable signal the effect of higher RF input powers into the AOM were examined. Convolving
the response function of the atoms for Rabi spectroscopy (whose sensitivity peaks in the middle ofthe pulse) with the raw measured phase of the light shows the shift magnitude in Fig. 4.6A. Properchoice of neutral density filters allow one to operate with the intensity servo set to give the lowest
shift measured 6(4) × 10−19 . I have no doubt that this effect can easily be lowered in the future byrunning the clock and measuring the shift at even lower RF powers.
A(radians)
Phase(milliradians)
Phase

B

-10

-20

-30

-40

-50
0

1

2

Time (s)Time (milliseconds)

3

Figure 4.6: AOM Phase Chirps present in the SrII system. A) This panel shows the calculated
clock shift due to the tuning AOM’s phase chirp when the light is pulsed on and off. Longer
interrogation times will decrease this shift even further in the future. B) The in-loop signal seenfrom the fiber phase noise canceller when its signal was amplitude modulated. Even with thispronounced ringing this does not produce a particularly large shift. The analysis of these effects
was included in an internal Ye labs writeup written by Jason Williams.

A secondary concern was the response of the fiber phase noise canceller to pulsing the firstorder light off and on. This pulsing amplitude-modulates the 0th order light and subsequently
the fiber phase noise signal [75]. The servo can be seen to ring at the beginning of each pulse
due to AM-PM conversion in the RF mixer. Luckily, due to the insensitivity of the atoms to thebeginning to the Rabi pulse, this ringing observed in figure 4.6B, would cause a frequency shift ofonly 8 × 10−20 .

84
4.6.3

Line PullingLine pulling can be caused by a variety of effects. These effects include a slight ellipticity inthe clock laser polarization driving detuned sigma transitions, imperfect optical pumping combinedwith the effect of driving a detuned mF = ±7/2 transition, and clock laser-induced tunnelingbetween different pancakes. Each of these effects is modeled as a deformation of the perfect Rabilineshape. The spectral narrowness of the 160 ms (∼5-Hz) Fourier-limited linewidth with which alldata was taken in this work greatly reduced the effect of any possible line pulling. It has been noted
in recent work that line pulling effects should be modeled more formally as coherently driving a Λsystem, which similarly gives incredibly low shifts of 2×10−19 for extremely conservative parameter
estimates [26].

4.6.4Probe Beam AC Stark

The only exception to taking data with spectrally narrow features was the investigation of
the AC Stark shift induced by the clock laser itself. This systematic was evaluated by measuring
the frequency difference between our clock transition interrogated with 50-ms and 200-ms π pulses.
For this measurement, the clock was run with our largest possible bias magnetic field to avoid any
residual line-pulling effects during the 50-ms clock interrogation. Including the errors in determining
our π pulse exactly, we estimate a 1.3 × 10−18 uncertainty in the probe beam AC stark shift for
SrII’s normal clock operation. More recent experiments to pin this systematic down even further
have found a host of systematics related to taking this measurement. However, recent work witha HyperRamsey interrogation scheme with the octupole transition in Yb ions has shown that evenstrong probe beam AC stark shifts can be cancelled to incredibly high precision [78, 79]. This
technique involves a detuned and time-imbalanced spin echo interrogation sequence to actively
cancel the probe beam AC stark induced by driving the weak transition. Future experiments
should take this last information to heart and consider revisiting bosonic strontium clocks should
the line Q or atom number ever turn out to be a limitation. A main detraction for 88 Sr clocks is

85
the need to greatly power broaden the incredibly weak synthetic clock state created by applying astate mixing magnetic field.

4.6.5

Background Gas Collisions

Two different methods were examined to quote an uncertainty for background gas collisions.
The first method was to scale down the shifts measured in Ref. [80]. However, these measurements
were not performed using hydrogen gas. By far the largest residual gas in our ultrahigh-vacuum,oven-loaded, vacuum system is hydrogen. The second method followed the procedure outlined inRef. [81] and was what was included in Ref. [22]. Differential C6 coefficients for the Sr 1 S0 and
3P

0 states for their resonant dipole interactions were scaled to the Cs ground state – ground-state

C6 coefficients [82]. Both the Sr 1 S0 -H2 C6 coefficient and the Sr 3 P0 -H2 C6 coefficient were thenestimated by scaling with respect to the non-resonant dipole CsH2 C6 coefficient. Atomic trapping
lifetimes of about 1 s, average excitations during the detuned Rabi pulse, and interrogation timesof 160 ms were combined to estimate the background gas collisional shift uncertainty. In our final
uncertainty budget, no shift correction is quoted, and we provide only an upper bound of thisuncertainty.

Chapter 5

Sr – The mini-COBE of Clocks

The Nobel prize in physics in 2006 was given to the Cosmic Background Explorer (COBE).
One of the key features of this experiment was the measurement of the background blackbody
radiation left over from the early stages of the universe. Interestingly, the last systematic to
be discussed in a neutral atom clock has a lot in common with the COBE satellite. While wenormally consider our atoms to be trapped inside a nearly perfect vacuum, with nothing but a few
stray hydrogen molecules bouncing off the walls, the truth is all the components of our vacuum
chamber are sitting at room temperature spewing out blackbody photons throughout the entire
EM spectrum. Luckily, the spectrum peaks at about 10 µm for room temperature, 298 K, whichis incredibly far detuned from many of the atoms’ electronic transitions.
While there is very little resonant light contained in the BBR spectrum of 298 K, it behooves
us to examine how poorly all leading atomic clocks are shifted by the broadband radiation. As canbe seen in table 5.1, strontium is seemingly the most difficult neutral atom to create an opticallattice clock with. Of course this is a trade off, because strontium has more transitions in thevisible and NIR it is more susceptible to BBR radiation but easier to work with than say neutral
mercury. Compared to the alkali atoms used for RF clocks, strontium just barely has the upper
hand on BBR-related shifts. However lost in the numbers is the necessarily simpler system thata strontium OLC apparatus has compared to a fountain clock. The atoms in an OLC stay in one
place during the interrogation, experiencing only one temperature environment whereas the cesium
atoms in a primary standard are thrown upwards into an elongated chamber being exposed to a87
Table 5.1: BBR shift magnitudes for room temperature radiation for a variety of atomic species.
Reproduced from ref. [83] unless otherwise cited.
Atom
87 Rb
133 Cs
Ca+
Sr+
Al+
Yb+
Hg
Mg
Ca
Yb
Sr

Clock Transition
5s(F = 2 – F = 3)6s(F = 4 – F = 3)
2S2
1/2 – D5/2
2S
2
1/2 – D5/2
1S – 3P
0
0
2S
2
1/2 – F7/2
1S – 3P
0
0
1S – 3P
0
0
1S – 3P
0
0
1S – 3P
0
0
1S – 3P
00δν/ν0−1.3 × 10−14−1.7 × 10−14
9.2 × 10−16
5.6 × 10−16−9 × 10−18 [21]−1.1 × 10−16 [37]−1.6 × 10−16−3.9 × 10−16−2.6 × 10−15−2.5 × 10−15 [84]−5.3 × 10−15 [22]range of viewing angles from many surfaces.
To conclude our investigation of the strontium optical lattice clock, I will discuss our recent
efforts to solve the BBR shift uncertainty and present the final uncertainty budget for SrII, ending
with the results of a SrI–SrII comparison.

5.1

Understanding the ShiftThis shift is simply the integrated spectral response of the atoms to the blackbody radiation

field
2π
δνi = −hZ ∞u(ω)αi (ω)dω

(5.1)

0

where δνi is the frequency shift of state i, u(ω) is the spectral energy density of the BBR, andαi (ω) is the polarizability of the state. For a perfect blackbody spectra, after much manipulation
this integral can be replaced with an asymptotic expansion in terms of the transition frequency
between resonant states i and l, ωil , and the temperature T in terms of the quantity ωTil [85, 86].
Summing over all the resonances in the atom, the final result is a decreasing series of contributions
that go like T 4 , T 6 , T 8 and so on. Not much intuition can be gained by this however, and a closer
examination of the first two terms in this series is warranted as they are the only terms that are

88statistically relevant at our targeted accuracy level.
The blackbody radiation shift, given by the T 4 and T 6 terms, is most easily understood as two
distinct effects acting upon the atom. The first effect, known as the BBR static shift, encompasses
the rectification of the radiation to an effective DC electric field experienced by the atoms. Over
90% of the shift is encompassed in this term alone, and this is why it was so instrumental for
the PTB group to measure the DC atomic polarizability of both the 1 S0 and the 3 P0 states [74].
The second effect, known as the BBR dynamic shift, is the frequency-weighted spectrum of theradiation and accounts for photon occupancy at and near specific transitions that occur between
the two aforementioned states and various other higher excited states in the atom. Assuming aperfect BBR environment, a very convenient closed-form simplification of the BBR shift is written
below from Ref. [74].
δνBBR = νstat

T
300 K

4


+ νdyn

T
300 K

6
(5.2)

Throughout the course of this work we will take νstat = −2.13023(6) Hz and νdyn = 0.1484(16) Hz,
which for the latter is the mean between the two most recent theory derivations of this shift coefficient1 [74, 87]. The shift coefficient is also lent more credence due to the recent work measuring the
dynamic shift as the atoms’ surrounding temperature was varied from 95K to room temperature,
where a fitted shift coefficient of νdyn = 0.1480(26) Hz was obtained [88]. Disregarding the uncertainty in the coefficients themselves, blindly achieving 4 × 10−18 uncertainty in either of these shifts
would be possible with 60 mK temperature uncertainty for the static shift and 580 mK temperature
uncertainty for the dynamic shift. While these numbers might at first seem intimidating, research
will quickly show that even commercial off-the-shelf temperature sensors are capable of beatingthese limits, precisely because of the compact nature of the Sr vacuum chamber.
Before we embark on how we use equation 5.2 to get to our low BBR uncertainty, it’s
important to better understand the relationship between these coefficients and their underlying
physics. Examining the static shift, it should be apparent from the T 4 scaling of the shift that
there is a direct scaling relationship between the static shift and the Stefan-Boltzmann law for
1

The error is the difference between these two most recent values.

89
measuring power emitted from a blackbody. For consistency, I will use the notation from Ref. [89].
νstat = −4∆α4π 5 kB∆αT4
E 2 = (For a Perfect BB Spectrum)2h15c30 0 h4(5.3)

Realizing the connection between these two ideas, that (1) the cause of the static shift is proportional
to E 2 and that (2) this quantity is proportional to the heat exchanged between radiatively coupledobjects, was a huge leap forward in our understanding of the BBR shift. Fundamentally, this allowed
us to make the cognitive leap that a successful measurement of an object in equilibrium with its
surroundings would shed light on the actual shift experienced by the atoms. But how does thisrelate to the frequency-weighted spectrum, namely the dynamic shift?
While the vast majority of this dynamic shift comes solely from the state with the lowest
transition frequency, 3 P0 – 3 D1 at 2.6 microns, I will nonetheless write the more general expressionthat takes into account all of the couplings [89].




1
~ωik
kB T 3 X 2Jk + 1 Aki
νi, dyn = −(5.4)
3 G k T~
2Ji + 1 ωik
(2π)2
B
k

R ∞ x3  1
1
2
where G(y) = 0 ex −1 y−x + y+x − y dx, Aki refers to specific transition’s Einstein A coeffecients, and νdyn = νe, dyn − νg, dyn . This equation can of course be transformed back into an
equation based solely on the electric field of the radiation, allowing easy use of frequency dependent emissivities and custom spectra. However, many people like the use of the function G as it
is very similar to the function F1 introduced by Farley and Wing, which allows for an asymptotic
expansion of these integrals for a perfect BBR spectrum [90, 89, 87, 86]. As an aside for future BBR
modelers, while these substitutions might be convenient, Mathematica, with a little bit of handholding, can perform these integrals regardless of the singularities present, and Matlab’s quadgk
function works on the first try. Understanding that an arbitrary electric field’s effect on δνBBR canbe calculated, it remains for us to understand exactly what kind of spectra the atoms see, how boththe static shift and the dynamic shift diverge from the perfect BBR shift, and most importantly
how their divergences are correlated.
Taking inspiration from an older paper, not about BBR shifts, but about metastable state
lifetimes, we found references to simple techniques for evaluating what sample BBR spectra could

90look like [91, 92]. The methods presented and used in these two papers can be distilled into an
exercise in photon counting. By tabulating all the surrounding surfaces where photons can comefrom and keeping track of those surfaces’ temperatures and emissivities one can write down theiremissivity weighted solid angles and an approximation of the BBR spectrum experienced at any
point inside a closed object. However, absolute knowledge of an effective temperature requires
extremely detailed knowledge of emissivities of surfaces at a multitude of angles and frequencies.
Playing around with these emissivity weighted solid angles one should quickly see that without thisknowledge it would be fruitless to try and claim knowledge of an absolute effective temperature
known from even these simple models.
While previous experiments have sufficed with placing temperature sensors surrounding theirchambers, additional knowledge seems to be required for achieving lower BBR uncertainties. In fact,
while it has been recently argued that these temperature sensors create a boxcar like uncertainty
allowing for the use of a special rule in the BIPM’s ‘GUM: Guide to the Expression of Uncertainty
in Measurement’ which allows for a rather low quoted uncertainty, I do not agree with its use inthis case [26, 93]. Furthermore, the GUM’s use of this rule comes with two caveats found in Section
4.3.7 [93]. First, one must have a uniform probability of finding the value in the boxcar distribution,
which I argue is incorrect as components with high emissivity will dominate the temperature felt
by the atoms. Second, it is noted that this method should not be used if the results of thisuncertainty are considered particularly important to the overall result, which for the case of theleading systematic in an OLC it is. Measurements only on the outside of the chamber make it
incredibly difficult, even with aggressive temperature modulations and state of the art lasers toestimate the spectrum seen by the atoms. Instead, we would need to combine our understandingof the spectrum with a knowledge about what is going on internally in the vacuum chamber.

5.1.1

Ray Tracing Model

Creating an emissivity weighted solid angle distribution from the very center of the chamber
is nothing more than a complicated geometry problem. Our extended octagon from Kimball Physics

91(Part No. MCF800-ExtOct-G2C8A16) has very detailed dimensions. Sitting down with a notebook
and calculating what each and every components solid angle is from the very center of the chamber
is easily accomplished. But what about slightly off-center? Or what if one would like to see themodeled spectrum for a variety of positions around the chamber? Indeed, as we will see, knowledgeof temperature gradients throughout the system will shine light on the non-perfectness of blackbody
radiation spectrum.

Figure 5.1: Ray tracing model of the UHV vacuum chamber. The point of origin for these rays is
the exact center of the chamber. Each colored dot represents a ray hitting a surface of the chamber.
Different surfaces have been color coded.

To model the chamber, I decided to use a custom ray tracing program. While many options
exist for writing a ray tracing algorithm, I decided to write what I thought was the most conceptuallysimple ray tracing algorithm. Rays would be cast out of a point in (x,y,z) space inside of the chamber
at a specified direction. The chamber’s boundaries would be specified inside an auxiliary function,and binary search would be performed on the point of origin and some maximum distance awayalong the ray to find the intersection between the chamber boundaries and the light ray. Theintersection is then noted and the surface that is struck by the ray is written down. Although thisprocedure turned out to be very inefficient, reasonable results could be obtained on a few year old
laptop in about 20 minutes per origin point2 . Even though one’s first instinct might be to cast the2

Hindsight being 20/20, a quick web search on how to perform ray tracing analysis will reveal a myriad number

92
Table 5.2: Effective solid angles when sitting at the exact center of the chamber. Although thesenumber do not include first-order reflections, these reflections will just reemphasize the importanceof high emissivity temperature components like the top and bottom viewports.
Name of component
Top Viewport
Bottom Viewport
Electropolished Metal Chamber
2 3/4” Viewport
1 1/3” Viewport
Zeeman Slower Window

Number of components visible
1
1
1
8
15
1

Total Effective Solid Angle
37.4 %
37.4 %
0.3 %
17.9 %
6.7 %0.5 %rays using a Monte Carlo algorithm, studying the error associated with estimating solid angles via
this method quickly showed that this method required significantly more rays. Instead, an iterative
method known as the Hammersley point sets were used [94]. The results can be seen in figure 5.1
where a color coded representation of the main vacuum chamber shows where all the rays from theexact center have struck.
Using emissivities from Ref. [91] and the viewport manufacturers specifications combinedwith temperatures measured on the outside of our chamber we can start to get an idea about
the kinds of spectra inside. Looking at table 5.2 recounting the emissivity-weighted solid angles
as seen from the very center of the chamber a few things become clear. First, because of thehigh emissivity of top and bottom viewports (∼ 0.92) and how close those two viewports are to
the atoms they play a huge role in determining the radiation spectra seen by the atoms. Second,
the low effective solid angle of the metal chamber due to its lower modeled emissivity (∼ 0.003)
means the temperature will be dominated solely by the temperature of the various viewports on
the system. One might worry that because these emissivity values are soft that we might not begaining any useful information from this exercise.
To see why we can gain an upper hand on this problem, let us instead focus on what an
ideal radiatively coupled temperature probe might read out. This ideal temperature probe wouldof more efficient algorithms for encoding and traversing environments to perform ray tracing.

93
exchange radiative power with its environment, and its response to any arbitrary spectra can becalculated. In fact, as the temperature probe reads out a specific number, we have fundamentally
gained some understanding of the intravacuum spectra. Although, as an experimentalist given only
this one number, it would be impossible to figure out how poorly the spectra matches a perfect
Planck spectrum. However, by varying parameters in the model and keeping track of both theassumed perfect BBR temperature read out by the ideal sensor and the actual shift caused by themodelled spectra we can examine how useful a perfect BBR sensor would be. Surprisingly, becauseof small temperature differentials (3 K) across our chamber, the emissivities of the metal or the
viewports would need to change by a factor of 20 to introduce a 1×10−18 error into the temperature
sensor’s shift errors. This means that even with very little knowledge of component emissivities,
an ideal temperature probe, under the right conditions, would measure the temperature accurately
enough for our purposes. Furthermore, by taking temperature measurements outside of the system
on viewports and moving the intravacuum sensor around to measure temperature gradients themodel can be validated3 .

5.2

The Temperature Sensors
The gauntlet had been thrown, keep temperature differentials small and try to create an ideal

and accurate set of temperature sensors in situ. In a wholly uncharacteristic show of cooperationfrom the experimental apparatus, we were lucky to find that our carefully built stray-light enclosure
acted as a strong temperature homogeneity enforcer for the entire system. Fueled mostly by warm
exhaust coming out of a CCD camera cooling unit, after closing the enclosure in the morning,temperature probes all around the system and hanging in the enclosure slowly came to equilibrium
at about 301 K with at the very most 3 K temperature differential between the minimum and the
maximum registered temperatures. As a further benefit, long comparison measurements over days
clearly showed the lab temperature swinging by many degrees, effecting the SrI apparatus, while
SrII only registered a drift of ∼200 mK in its enclosure.
3

Similarly, one could just have many spatially separated temperature probes.

94
Many obstacles exist in taking a good measurement of temperature, and I would be remiss to
not give special thanks to Hans Green for designing and building the mounts for our temperature
sensors, as it was those mounts that allowed us to accurately measure the temperature as well as we
did. While we ended up using temperature sensors that had previously been purchased for anotherproject, I will nonetheless make the case for their use. Future theses will go significantly more in
depth on this issue. Many different kinds of sensors exist to measure temperature: thermistors,
platinum resistance temperature detectors (RTDs), crystal oscillators, and silicon diodes. While it
might be that platinum RTDs are the more conservative choice, silicon diodes offer a few advantages
which seem to minimize their systematics. First, silicon diode sensors have a very linear response
in voltage vs temperature making their calibration easier [95]. Second, the signal size of ∼0.5 V
for a very tiny current make their readout much easier, and more immune to contact resistances as
the effective resistance of the diode is quite high. Third, compared to RTDs which can dissipate
100 µWatts during operation with self-heating on the order of ∼0.4 mK/µW, diodes dissipate 5µWatts with self-heating coefficients close to ∼0.2 mK/µW making their calibrations more immune
to differences in mounting4 . With these considerations in mind we used a DT-471 calibrated sensor
from Lakeshore Cryotronics.
To decrease heat conduction from the chamber to the temperature sensor, two diodes were
mounted at the end of two elongated glass tubes. These glass proboscises also allow the tube itself
to radiatively couple to the environment. Figure 1.2c shows a CAD drawing of the two sensor
mounts, both a retractable movable mount and a shorter fixed mount, while figure 5.2 shows aphotograph of a sensor. The metal conflat flange to glass tip distance on the movable sensor is150.6mm while the fixed sensor is 85.2mm5 . The pyrex glass that the mount is made of has a
7mm diameter with 1mm thick walls. The diode sensor is epoxied onto a short glass platform at
the end of the tube using Epotek 353ND. The leads of the sensor are then silver soldered with
4

The SD package for our diodes also places the diode junction inside a relatively gigantic thermal load further
reducing systematic errors in the calibration.
5
While this was the design goal, full extension of the movable sensor seemed to place the sensor slightly past thecenter of the chamber by ∼8 mm.

95

Figure 5.2: A picture of the diode temperature sensor mounted on the pyrex glass proboscis. The
top facet of the diode has been painted with high emissivity vacuum compatible paint. One cansee the silver soldered phosphor bronze wires leading away from the diode strain relieved on special
glass tie downs. A CAD model of the two sensors is depicted in figure 1.2C.

96
Table 5.3: Diode temperature sensor corrections and uncertainties.
Corrections
Calibration (including self-heating)Residual conduction
Temperature gradientLead resistance
Lattice light heating
Totals∆T (mK)
0
0
40
7.7
-15
32.7

σT (mK)
16
0.7
20
1.5
7.5
26.7

Fusion solder paste WCC-430-801 made up of 96.5% Sn and 3.5% Ag to phosphor bronze leads.
The lead material is thermally insulating and fairly electrically conductive. To increase radiative
coupling, facets of the sensor that could be painted with high emissivity vacuum-compatible paintwere coated with Ball VacKote 48816. Because of the high-emissivity of the glass, this is less of arequirement, however it very likely increases the sensor’s radiative coupling. The phosphor bronze
leads are strain relieved around notches in the glass, and wind around the proboscises towards themetal flange where they are connected to vacuum compatible electrical feedthroughs.
Vacuum baking of the sensor was performed before installation of the sensor into the main
UHV trapping chamber. Unfortunately, the baking temperature was limited to 200 ◦ C by thesilver solder, not the sensor itself. Although care was taken to not stress the sensor leads, themovable sensor was sadly struck off its holder causing a severe change in its calibration. Later tests
comparing the sensors showed that relative temperature measurements below a few tens of degrees
were still possible with the broken sensor, but absolute measurements would require a complete
sensor recalibration. Upon installation into the chamber re-baking of the sensor to 100 ◦ C was
required to pump down to 5 × 10−10 Torr at the system’s ion gauge. While originally theorized to
be an issue with the porousness of the high-emissivity paint, newer work has suggested that thesensor packaging itself, or at least the damaged sensor, might create a virtual leak by trapping asmall amount of water vapor or gas.
Many sources of error for extrapolating the temperature at the atoms needed to be addressed97
to take the measurement from the fixed sensor and extrapolate the temperature at the atoms.
Reading table 5.3 one will understand some of the largest systematics we investigated. Sensor calibration which was done by LakeShore, claimed an accuracy of 16 mK at 300 K. Although this seems
small, this is actually less than a 1 ppm measurement accuracy, state-of-the-art temperature measurements near the triple-point of water can reach as low an uncertainty as a few µK [96]. Residual
conduction was measured by wrapping the conflat flange of the fixed sensor in nichrome wire andheating up the flange by ten degrees. Although previous experiments showed rapid response of thesensor to changes in high-emissivity objects’ temperature, no clear signal due to heat conduction
was seen after taking data for many minutes. An uncertainty was made by taking the peak topeak amplitude measured throughout the experiment and scaling that number down linearly withnormal temperature differentials between the flange and the sensor. A temperature gradient fromthe retracted movable sensor’s position to its extended position where the atoms would normally
sit was measured with the system running, but the oven blocked. Auxilary temperature sensors
confirmed that the bottom viewport was slightly warmer than the top viewport, most likely due toinsufficient airflow inside the enclosure. Although the model and the measurement agreed on themagnitude of the gradient, a conservative uncertainty of 50% was added to this systematic. Leadresistance and its effect on the measured voltage was estimated from the length of the phosphor
bronze wires. Finally, stray lattice light scatter caused a very clear 15 mK change in the measured
sensor’s temperature, again a conservative uncertainty of 50% was added. Other systematics likemagnetic field effects or thermocouple effects in the vacuum feedthroughs were tested or researched,
but were found not to be an issue.
A few simple tests of the emissivity-weighted solid angle model revealed surprisingly goodresults. In fact, measurements around the chamber and of the top and bottom viewport predicted
an absolute temperature within 100 mK of the measured temperature. However, these results were
too dependent on the emissivities to make this a viable temperature measurement technique without
careful study of the material properties. Agreement was also found when the movable sensor was
moved into view of the Zeeman slower window with the heater still on (see Fig. 5.3). Not only

98
did the broad shape of this Zeeman slower window gradient match what the model predicted, thepeak height of 30 mK temperature increase also matched predictions. Keeping the Zeeman slower
window at 150 ◦ C however, would create a > 1 × 10−18 spectrum shift due to the very different
temperature of the window with respect to the rest of the chamber. Lowering the temperature to
12 ◦ 5C would make the spectrum shift lower than 1 × 10−18 , however this would place a fairly largeemphasis on modeling and its associated uncertainties. Instead, one can just run the clock for shortperiods of time without the window heated, or obscure the atoms’ view of this radiation.

R e la tiv e T e m p e r a tu r e (m K )

2 0

1 0

0

-1 0

-2 0

-3 0
-1 0

-5

D is ta n c e fr o m

0

5

1 0

1 5

2 0

C e n te r o f C h a m b e r (m m )Figure 5.3: A measurement of the influence of the Zeeman slower heated window on the temperature
sensor. Although this measurement was not taken under running conditions for the clock, a broad
spatial gradient can be seen of at least 30 mK, in addition to the peaked response of the sensor
when placed in view of the Zeeman slower window.

Putting all this information together we get an approximately ∼27 mK temperature uncertainty in our measurement. While this is actually better than we initially hoped, the uncertainty inthe BBR dynamic coefficient far exceeds the residual uncertainty in the temperature measurement.
To decrease the coefficient uncertainty future experiments will need to measure the Einstein A
coefficient for the 3 P0 - 3 D1 transition at 2.6 microns. While work can be done to further decrease
the uncertainty, via better sensor calibration techniques, this method will seem to bottom out after
only a factor of a few improvement. Future clock work to reach uncertainties in the 10−19 will most
likely be accomplished by trapping the atoms in cryogenic environments specifically designed to99
occlude temperature inhomogeneities or through the use of other atomic species.

5.3

Final SrII Error Budget
Tabulating all the systematic uncertainties discussed in the previous two chapters we can

combine them to get a rather astonishing result. As seen in table 5.4 the total systematic uncertainty
for SrII is 6.4 × 10−18 , this represented the first time an OLC held the mantle of the lowest total
systematic uncertainty. While this result is a welcome sight for the OLC community, it’s interesting
to reexamine the measurement with some hindsight. The two leading systematics that make up thevast majority of this number come from the dynamic BBR coefficient and the AC stark effect. Bothof these are solvable problems that do not interfere with this method of temperature readout. As
such, a renewed measurement of the AC stark effect with a larger lever arm (more powerful pumplaser) and a lower trap operating depth (vertical lattice) will allow SrII to squash that systematic
down. Secondly, a lifetime measurement of the 3 D1 line will help pin down the dynamic BBR
coefficient. In the end, following these two simple prescriptions should cut the total SrII frequency
uncertainty down by a factor of 2 at least. Work along this line is ongoing in the lab.

5.3.1

SrI–SrII comparison

As the systematic evaluation of SrII was coming to a close, SrI, led by Mike Bishof and
Xibo Zhang, took up the challenge of a systematic evaluation allowing us to perform clock-clock
comparisons. Since this research is meant to highlight the utility of high stability clocks, it wasparticularly important for us to perform a clock comparison between two OLCs. In fact, the low
instability of these Sr clocks (3 × 10−18 at about 10,000 s), displayed as the Allan deviation of theirfrequency comparison in Fig. 5.4a, is perhaps their greatest selling point.
Figure 5.4b documents a comparison of the SrI and SrII clocks over a period of one month.
After the data from the comparisons had been post-processed by each individual strontium team,time-stamped and corrected frequencies were shared. Although the overall systematic uncertaintyof the comparison is 5.4 × 10−17 , as a consistency check for the comparison a variety of methods

100

Table 5.4: SrII Systematic Uncertainties. Shifts and uncertainties are given in fractional frequency
units multiplied by 10−18 . Uncertainties are quoted as 1σ standard errors. They are determinedwith the square root of thepquadrature sum of the systematic error and statistical error, with thelatter quantity inflated by χ2ν . Reproduced from ref. [22].
ShiftsBBR Static
BBR Dynamic
Density Shift
Lattice Stark
Probe Beam AC starkFirst-order Zeeman
Second-order Zeeman
Residual Lattice Vector
Line Pulling and Tunneling
DC Stark
Background Gas Collisions
AOM Phase Chirp
Second-order Doppler
Servo Error
Totals∆ν-4962.9-345.7-4.7-461.50.8-0.2-144.5
0
0
-3.5
0
0.6
0
0.4
-5921.2

σν
1.8
3.7
0.6
3.7
1.3
1.1
1.2
< 0.1
< 0.1
2.1
0.6
0.4
< 0.1
0.6
6.4

101
10

a

-15

−16
3.4×10
√
τ

-16

10

-17

10

-18

Allan Deviation

10

10

0

10

1

10

2

10

3

10

4

Averaging Time (s)3×10-160ν -ν (Fractional Frequency)
SrII SrIb-3×10-16
1×10-16±5.4x10-170-1×10-16

00

2:

0
ul
-J

17

-J

17

00

6:

0
ul

1:

18

-J

2
ul

00
-J

19

00

1:

0
ul

19

-J

00

5:

0
ul

g

Au

17

0
:0

01

Au

17

g

0

:0

05

g

Au

20

0

:0

03

20

g

Au

07

0
:0

Coordinated Universal Time (UTC)

Figure 5.4: pSrII - SrI Frequency Comparison. a) Allan deviation of the SrI and SrII comparison
divided by (2) to reflect the performance of a single clock. The red solid line is the calculated
quantum projection noise for this comparison. The green dashed line is a fit to the data, showing
the worst case scenario for the averaging of a single clock of 3.4 × 10−16 at one second. b) The
absolute agreement between SrI and SrII recorded at the indicated Coordinated Universal Time.
The light-green region denotes the 1σ combined systematic uncertainty for the two clocks under
the running conditions at that time. The top panel shows the frequency record binned at 60 s;
in the bottom panel each solid circle represents 30 min of averaged data. The greenp
dashed lines
represent the 1σ standard error inflated by the square root of the reduced chi-squared, χ2ν , for theweighted mean of these binned comparison
data. The final comparison over 52,000 s of data showedp−17 ( χ2 =10.5) for the 30-min averaging time and −2.8(2) × 10−17 (agreement
at−2.7(5)×10νpχ2ν =3.5) for the 60-s averaging time. Reproduced from ref. [22].

102
were used to show the agreement of the two clocks within this confidence interval. A simple meanof all the data gives the difference between the two clocks to be −2.4 × 10−17 . Binning the data
in small chunks, of approximately one minute per data point (as in the top panel of Fig. 5.4b)p
gives agreement of −2.8(2) × 10−17 . The uncertainty on this number has been inflated by χ2ν
p, as χ2ν = 3.5 denoted overscatter in the data. Binning the data in 30-min chunks (as in thebottom panel of Fig. 5.4b) clearly shows that there are systematic fluctuations still present in thepcomparison, with χ2ν = 10.5 and an agreement of −2.7(5) × 10−17 . Again, this uncertainty ispinflated by χ2ν . The greater overscatter in the data at longer timescales is probably caused byimprecise knowledge of the BBR environment for SrI, which allows for fluctuations within the 1σcomparison uncertainty.
With the successful completion of the SrII systematic evaluation and frequency comparison,
it was time to look forward to new experiments with strontium.

Chapter 6

The New Sr Quantum Gas Extravaganza Machine

The study of quantum gases is a topic riddled with misnomers. While people in the fieldcontinually refer to various systems as having a high-phase-space density, it’s easy to forget thatdeeply degenerate samples of alkali’s or alkaline earth atoms are actually at an exceedingly low
density. In fact, the nitrogen molecules in air are at significantly higher spatial densities [97]. It isonly in comparison with atoms’ de Broglie wavelength at ultracold temperatures that the densities
seem “high”. It is the exquisite control and isolation of the particles in an ultracold gas that makethis field so special, and allow interactions to be studied in attainable timescales. But of course,
there are many experiments all around the world focusing on quantum gases of alkali atoms, sowhat does an alkaline earth atom have to offer to this already rich field?
In the hunt for studying and understanding strongly correlated quantum systems, alkaline
earth atoms have shown a surprising path towards achieving these delicate, correlated, many-bodystates. Due to the separation of energy scales between quantized motional levels and interactionenergies, decoupling between the internal spins and external degrees of freedom of the atoms caneasily be obtained. Although this restricts our atoms to probe weak interactions, the evolution of
the system can be dominated by these weak scale interactions as other single-particle dynamics
can be frozen out even at thermal (µK) temperatures. Given a sufficiently long probe time we canthus achieve strongly correlated dynamics. Compared to research with alkali atoms, fundamentallyquantum many-body processes [44, 34] are accessible in this regime that would otherwise be hidden
behind entropy scales that have yet to be experimentally obtained. Further studies to better control

104
and engineer these many-body states in the presence of large thermal energies might lead to morerobust quantum simulation protocols and push the frontiers of quantum science towards a seemingly
more classical regime.
While alkali atoms excel in their sensitivity to magnetic fields, alkaline earth atoms excel attheir sensitivity to optical fields. It is true that there are trade offs in pursuing the study of an
atom lacking magnetic feshbach resonances or a convenient RF transition, however strontium offers
narrow linewidths for spin manipulation [98], a host of nuclear spin states [44], a closed narrow
optical transition with no scatter [99], IR transitions for low recoil cooling, and the possibilityof anisotropic inter-atomic interaction feshbach resonances between metastable and ground states[100]. Given the opportunities to bring forth a new tool for the study of exotic quantum phenomena,
multiple groups have achieved degenerate quantum gases of strontium [101, 102]. It is my hope,that due to our group’s extensive background and expertise with spectroscopy that we might offer
an interesting perspective on ultracold quantum gas experiments with strontium that utilize theunique energy scales accessible in these systems.
While the task of creating a quantum gas has become somewhat routine, the level structure
in strontium offers a few different paths to degeneracy. Because of strontium’s rich level structure,laser cooling to quantum degeneracy is indeed an option [101]. However, the most deeply degenerate samples of strontium have continued to be created via the use of evaporation [102]. While
future experiments might focus on improving these laser cooling techniques, having an apparatus
that can produce quantum gases using more tried and true methods will only make things easier.
Therefore, significant emphasis on vacuum lifetime was placed when designing the new SrIII apparatus. Furthermore, future experiments that seemed promising required the use of optical access at
approximately 30 degree intervals (30, 60, 90, 120), putting stringent restrictions on what our mainchamber would need to look like. This following chapter will shed light on the design decisions
made for the SrIII apparatus, will document its construction, and will offer a glimpse at some of
the many experiments possible with this new apparatus.

105

6.1

Long Vacuum Lifetime
To ensure that quantum degeneracy could be reached with the SrIII apparatus a vacuum

lifetime in the 10s of seconds would be needed. The design goals of SrII had also been for a similar
vacuum lifetime, but as is usually the case first tries are for ironing out the bugs in the system. Theoven region in SrII was designed with capillary tubes to collimate the hot strontium vapor, whilethese tubes have performed admirably (having never run out of strontium) their large surface area
at an extremely high temperature presents a huge gas load for the pumps in that region [103]. And
while the oven was placed farther away from the main chamber, for differential pumping concerns,
atom flux at the main chamber suffered. The second bug we battled with was the low conductancepath to the main chamber ion pump, this reduced the pumps ability to take care of the extra gasload from the oven/Zeeman slower region. Adding insult to injury, even though there was low
conductance to the pump, the pump magnet was much too close to the atoms and MOT coils andcaused a drifting magnetic field. Finally, without the inclusion of a titanium sublimation pump,
and with a mostly steel chamber, there was simply not enough pumping of Hydrogen. This resulted
in vacuum lifetimes of ∼1 s for base pressures of high 10−10 Torr. So with these lessons in mind,
significant thought was placed into the ultimate level of vacuum that could conceivably be reachedby SrIII.
The first decision to be made for SrIII was whether or not to construct a double chamberedsystem1 . Multi-chamber systems are very popular in vapor-gas loaded systems, because they allow
one to create a very small clean science chamber distinct from a dirtier atom loading area. While
many schemes for atom transfer between chambers exist like pusher beams, magnetic guiding, andoptical trapping, only the last scheme would work for strontium because of the lack of appreciablemagnetic moment in the ground state. Optical dipole trapping would require that the first chamber
have good enough vacuum to allow for µK temperatures requiring a setup as complicated as SrII
was already, in addition to a new science chamber reached via translating a lens on an air-bearing
1
Not counting our oven region, which is necessary due to the low vapor pressure of strontium metal, and the lackof demonstration of a chemical dispenser.

106stage. This level of complexity seemed not only slow, given our concern about duty cycles inclocks, but also unnecessarily complicated. Instead, we decided to focus on single-chambered UHV
systems, where collection of atoms and final interrogation occurred at the same place.
A survey of single chamber designs, loaded by a Zeeman slower, came up with all steel
chambers and two-point-contacted glass chambers [104]. While glass chambers were a good option,
the fragility of the glass and the difficulty and expense of a nicely AR coated glass chamber werelarge negatives. Focusing on steel chambers, we first needed to decide how to feed atoms into oursystem. While our current system was performing adequately enough, we nonetheless wanted tomake improvements rather than keep the status quo. Our old Zeeman slower had never really had
a true bake, as the epoxy used when wrapping it had a fairly low melting point, and many groups
had come up with interesting new permanent magnet slower designs that get around this problem[105, 106]. However, during the course of researching options for our new system, a company offeredto build an oven, Zeeman slower, and 2D MOT system for us2 .

AOSense Atomic Beam Chambers

Figure 6.1: Picture of the AOSense oven, Zeeman slower, transverse cooling, and 2D-MOT system. Many of the components are intravacuum pieces like the racetrack transverse cooling system.
Picture is taken from promotional material given to us by AOSense.

As can be seen in figure 6.1 the entire system has an incredibly small footprint. While detailed
2

AOSense Inc.

107
specifications can be obtained from AOSense, I would like to highlight some of the reasons why
this system was attractive. First, the tiny footprint and in-vacuum oven are almost an order ofmagnitude smaller than the SrII oven region, leaving lots of table space for other optics and lasers.
Second, the oven successfully uses capillary tubes made out of a non-stainless-steel material that
outgasses significantly less. Third, not only is there a permanent magnet Zeeman slower with aheated window, but due to the inclusion of a 2D-MOT which deflects the atomic beam, the heatedwindow and oven will not be visible to atoms in the main chamber. Finally, the specified pressure
after the oven of < 5 × 10−9 Torr is significantly better than SrII’s, and the pressure farther awayfrom the oven after the 2D-MOT and before the differential pump tube should be even better.
Needless to say, a ready built system like this was hard to pass up.
For the main chamber, we examined purchasing another Kimball physics chamber or designing
our own custom system. However, as our main goal was to achieve long vacuum lifetimes, stockparts did not seem like an option. Getting enough pumping into the main chamber meant either
sacrificing one of the large viewports or adding a custom piece onto one of the large viewports.
As we did not want to lose any of our large viewports, a fully custom solution seemed the smart
choice. After examining many groups chambers we found a design that was both practical andproven. Aviv Keshet’s thesis from Wolfgang Ketterle’s group wonderfully outlined the constructionof a chamber with a large (many 100s of L/s) conductance path to vacuum pumps, and was niceenough to offer CAD files so large modifications could be included in his chamber’s design [107].
As can be seen in figure 6.2 the main attraction to this chamber design is the large rectangularpath intersecting both the main chamber and auxiliary pumping chambers. Furthermore, viewports
surrounding the chamber could be placed in approximately 30 degree intervals meeting our opticalaccess requirements as well.
The main chamber construction was a fairly complicated job that required professional vacuum machinists. After talking with a few companies, we proceeded with Sharon Vacuum who
had previously built Aviv Keshet’s chamber as well. Appendix B includes the CAD drawings ofthe chamber. The final chamber was built with 304 stainless steel and electropolished by Sharon

108

Figure 6.2: The SrIII Main Chamber. The vacuum system was built with most ports giving a 30
degree angle between neighboring ports, allowing for 30, 60, 90, or 120 degree angles all accessible.
The chamber was constructed by Sharon Vacuum where it was electropolished and leak tested.
After verifying the leak test at JILA, it was shipped to Jefferson Lab where it was vacuum baked
at 400 C for 100 hours. Following this, it was sputter coated for 100 Hours with 40 mA going
through the TiZrV wires, and 30 Hours at 150 mA. All blanks used in this process were speciallymachined with jack screws for prying off stuck blanks, furthermore belleville washers were usedwith all the screws to help alleviate metal fusing at these high temperatures. More details about
the main chamber can be found in the engineering diagrams in Appendix B.

109
Vacuum. The bottom of the L-shaped auxiliary pumping chamber in other designs had held a
75 L/s ion pump while the top of the L held a titanium sublimation pump, final quoted pressure
values for the main chamber seemed to put the main chamber pressure at low 10−11 Torr. Rather
than just strictly adapt this chamber design, it seemed worthwhile to examine where it might be
improved. Although equations for gas conductance in the molecular flow regime are valid only
under large lengths, L, compared to tube diameters, D. We can nonetheless use the equation as aq
3
T
guide, C = 3.64 π4 DL M
, where D and L are in cm, T is the temperature of the gas in K, M is therelative atomic mass of the gas, and C is computed in units of L/s. Using this equation, the pathway
to the pump would seem to have many hundreds of L/s conductance of hydrogen, so increasing thepump speed would directly benefit the ultimate pressure in the main chamber.
However, purchasing larger and larger ion pumps requires both heavier pumps, putting physical strain on the new apparatus, and larger pump magnets possibly interfering with the magnetic
field at the atoms. Furthermore, pumping speeds of ion pumps decrease at lower vacuum levels causing diminishing gains. Instead, we purchased a pump from SAES Getters based on Non-Evaporable
Getter (NEG) technology where residual gases are sorbed onto porous sintered vanadium zirconium
iron (VZrFe) discs. This technology has the benefits of being compact and requiring no power after
activation, however it is completely incapable of pumping away certain gas species and the inclusionof a small ion pump is usually required. SrIII includes a 300 L/s combination NEG-ion pump. All
the NEG material is hidden by the elbow of the L-shape from a titanium sublimation pump (TSP),
which also has no line of sight to any vacuum viewports. Both of these pumps excel at pumpinghydrogen gas, which is the predominant residual gas in clean XHV systems.
An additional pumping mechanism has been included in the main chamber with the helpof staff members at Jefferson Labs. Thanks to the efforts of Philip Adderley, Marcy Stutzman,
and Matt Poelker our chamber walls were coated with additional NEG material. This material,titanium zirconium vanadium (TiZrV), although a less effective pump, has a significantly reduced
activation temperature of ∼180 ◦ C. This allows us to activate the NEG material on surfaces very
close to coated vacuum viewports that are specified to only go up to 200 ◦ C. Although commercially110
manufactured coatings of this type by SAES quote pumping speeds fairly comparable to sputteredtitanium from a TSP, our coating is expected to perform not as well. However, the coating itself
also acts as a buffer between the steel and the vacuum lowering the outgassing rate of the steel
chamber. All of these pumps combined with their different activation temperatures and properties
make for a fairly complicated pumping procedure from atmospheric pressure all the way down to
the system’s ultimate base pressure.

6.1.1

The Pumpdown Procedure

Many NEG materials can be overwhelmed by large gas loads, and might require reactivationof the NEG material to achieve high pumping speeds. In many instances reactivation does not justmore deeply sorb the material but rather pushes it back out into the vacuum, so reactivation should
nominally be done in the presence of a turbo-molecular pump3 . However, since reactivation or
sputtering of titanium temporarily increases the gas load, these procedures should be performed ina manner where they do not interfere greatly with each other, neither procedure overwhelming thepumping benefits of the other. In fact, careful operation and preparation of a TSP pump should
allow short, quick sputtering that should only raise the overall pressure to low 10−8 or high 10−9Torr as read by a nude ion gauge.
Given the lowest sorption capacities listed for our NEG pump of ∼1 Torr·Liter of carbon
monoxide, it would take approximately 1 million seconds, or 11.5 days, of pumping a 10−8 vacuum
to sorb these capacities of CO. Therefore, the goal is to correctly prime the TSP to allow for clean
sputtering, that will not overwhelm the NEG pump, nor will it overwhelm the NEG coating. Manyof the ideas presented in this list are pulled together from the manuals for the NEG ion pump, theTSP, and the turbo-molecular pump. In addition to the instructions in this list, it is important that
all maximum temperature requirements for the viewports, pump magnet, electrical cords, turbo
pump flange, and any other components be followed exactly.
3

Some quantities are sorbed deep into the material, so reactivation still does lessen the overall gas load in thelong term.

111(1) After engaging the roughing pump, turn on the turbo pump while the system is still atroom temperature.
(2) Make sure the cables, metal container, and pump magnets are removed from the ion pump.
Heat the ion pump to a safe baking temperature (max 350 ◦ C), and bake out for at leasta day or two.
(3) Cool the ion pump to room temperature, temporarily reassemble the ion pump magnets
and prime the ion pump at least once. This needs to be done before ever activating anyof the NEG elements of the chamber. Remove pump magnets, etc. and prepare for a fullsystem bake.
(4) Slowly heat the entire vacuum system to a temperature well below the TiZrV activation
temperature, say ∼150 ◦ C, place the NEG pump into bakeout/conditioning mode whichheats the pump without activating it. Additionally, run ∼25 A through each TSP filament
to keep them hot.
(5) After the background pressure is well below 10−8 Torr, begin to clean each TSP filament
according the the instructions in the manual. Run 37 to 42 A through each filament
for approximately 1 minute each. Between filament outgassings continually run 25 Amps
through each filament to keep them hot.
(6) Continue baking the chamber at a temperature well below the TiZrV activation temperature, keep performing the TSP filament outgassing until the pressure rise from outgassing
any filament peaks in the high 10−9 . Once this is accomplished, the filaments should beclean enough for use post-bake4 .
(7) Once pumping down at this temperature starts to slow down, raise the temperature to
above the activation temperature of TiZrV (180 ◦ C). This temperature must be kept for
Other groups in JILA have recorded 49 A for 7 minutes raising pressure to 3 × 10−8 , and pressure falling oncethe current has been shut off. Depending on the performance of the TSP filaments, this could also be a goal during
the bake.
4

112
at least 24 hours or longer to ensure activation of the TiZrV.
(8) If at the end of this high temperature activation phase, the pressure is mid or low 10−9Torr proceed with activation of the VZrFe NEG pump. This will greatly increase the gas
load.
(9) After the activation of the VZrFe NEG pump is over, begin cooling the system down. If
the pressure was too high before to activate the NEG pump, activate the NEG pump oncethe system has cooled. Remember to cool all components of the system including the TSP
filaments.
(10) After the system is room temperature, reassemble the ion pump magnets and turn the ionpump back on.
(11) Close the all-metal gate valve, and turn off the turbo. If after the gate valve has beenclosed for a few hours the pressure is not low enough, fire the now clean TSP pump.

6.2

Recessed Viewports for spectroscopic resolution
In the spirit of adding additional capabilities to SrIII that neither SrI nor SrII had, the

new system will include recessed top and bottom viewports. Not only do these recessed viewports
allow imaging lenses to be placed significantly closer to the atoms, they also will allow us toattain incredibly large magnetic field gradients up to 1000 Gauss/cm. This will allow spectroscopic
resolution of single pancakes of atoms in our optical lattice, a technique that could be especiallyinteresting for looking at atomic noise correlations [34]. Construction of these recessed viewports
was accomplished by the United Kingdom Atomic Energy Authority (UKAEA)5 . In addition totheir expertise in constructing these specialized UHV windows, they have extensive experiencebuilding these windows in a piecewise manner. They first braze the glass onto the bottom disc,before connecting the disk to a CF flange with a recessed wall. This allows us to have the front
5

These viewports were designed to be similar to those found in Wolfgang Ketterle’s apparatus, with some minorchanges for creating an intra-vacuum mounting system for future experiments.

113
and back of the glass viewports AR coated before the completed “bucket” makes coating them
impossible due to shadowing. Figure 6.3 shows what these recessed viewports will look like, andAppendix B holds mechanical drawings of the viewport. Special thanks should be given to Tim
Darby, an engineer at UKAEA.
The ability to perform experiments on individual pancakes, or to address individual atoms
in a 3D lattice, will further the development of precision optical control of strontium atoms. It
is my hope that with a combination of magnetic field gradients and easily attainable resolution
requirements for the clock laser, SrIII could one day be considered a quantum gas microscope in
its own manner.

6.3

Optical Access
While in experiments with alkali atoms a great emphasis is placed on both magnetic field

performance and the system’s ability to deliver RF to the atoms, for alkaline earth atoms given
the myriad number of ways light and matter interactions are introduced to the atoms the system’s ability to deliver clean light to the atomic sample is incredibly important. While the use ofcommercial-off-the-shelf (COTS) chambers allows quick and easy design choices, not only have webeen disappointed with the viewport angular placement, we have also found that not all viewports
are usuable once mounting structures and magnetic coils have been attached. By going custom
with sufficient planning of viewport placement we have higher hopes for the future of the SrIII
main chamber. Anti-reflection (AR) coating was also of great concern to us, especially as SrII’s
successful demonstration of an extra-vacuum cavity hinged upon our viewports’ AR coating.
Previous work in SrII, had in fact suffered from insufficient AR coating planning. While
multiband coatings were implemented that covered the 1 S0 to 1 P1 , 3 P1 , 3 P0 transitions, and various
magic wavelengths, no attention had been paid to widely available dipole trapping wavelengths in
the NIR. In fact, our current viewports reflect 15% of 1064 nm light per surface, reducing the single
pass power delivered to the atoms by at least 30%. Not only would SrIII require multiband coatings
for ports that would deliver clock light, but broadband coatings would need to be found that would

114
not limit any future laser light choices. While many companies were contacted, American Photonics
offered the best multiband coating given our requirements. Their quoted specifications are given
in Figure 6.4. While many companies competed for our multiband coatings, the search for a truly
broadband coating that could stretch to include 461 nm and 1064 nm, was fairly difficult. In theend, TAKOS came through for us offering a fairly impressive broadband coating (see Figure 6.5).
Careful selection of which viewports to coat with which coating was performed with an eye
towards future endeavors. Figure 6.6 shows a top-down view of the SrIII apparatus, with colorcoding indicating the coating for each viewport. The top, bottom, and large side viewports (ports
4a and 4b) were chosen to have the multiband coating. This would allow for superior absorption
imaging, fluorescence detection, the possibility of magic wavelength enhancement cavities, and clock
laser launching. Port pairs 6 and 2 were also chosen because their 60 degree angle would possibly be
useful in studying dipole-dipole interactions. All other ports are coated with broadband coatings,
such that the use of 688nm, anti-magic wavelength lasers, or any other visible or NIR wavelength
would never be limited by the viewports themselves.
Having briefly discussed some of the features of SrIII, we will now examine how SrIII will
support investigations into two different experiments.

6.3.1

Dipole-Dipole Interactions

The study of long-range interactions is a goal for much of atomic physics, whether the interaction is mediated through optical cavities, dipolar interactions, or even coulomb interactions.
Precision control of long-range interactions might lead to a better understanding of strongly correlated materials, or might be used to entangle spatially separated qubits. While proposals exist forachieving dipole interaction energies in strontium gases comparable to polar molecule interactionenergies, these proposals have not yet been implemented [108]. However, our work with atomic
clocks has taught us a great deal about the power of using clock shifts themselves to measure
incredibly small interaction energies [109], which could unexpectedly turn into a tool for creating
quantum correlated states when other single particle dephasing has been suppressed.

115
The atoms themselves when placed in a coherent superposition between the ground state 1 S0
and the electronically excited 3 P0 state do have a tiny dipole moment of approximately 40 µDebye.
By changing the geometric position of an array of atoms in a 3D lattice, one can create spatiallycoherent arrays of dipoles that will resonantly shift the clock transition [110]. Observing theseshifts will require high phase space density samples of atoms, loaded into lattices with very specificintersection angles. The SrIII apparatus has been designed with a majority of viewports being
separated by 30 degrees so that achieving the ∼64.8 degree resonant angle is possible. Furthermore,creating cubic lattices which will suppress this shift will also be possible with viewports set 90
degrees from each other, making a differential measurement possible. The final smoking gun willbe the atomic number scaling of this shift which should go like N 2/3 . After many years of work,
these shifts are finally within our reach of observing as they should be high 10−18 in magnitude forreasonable atom numbers.
Not only will this experiment need to be performed with deeply degenerate samples of atoms,
necessitating long vacuum lifetimes, the optical access requirements were not met with SrII.

6.3.2

Spin-Orbit Physics in Alkaline Earth Atoms

The study of artificial gauge fields in cold atoms to simulate the behavior of charged particles
moving around in a magnetic field is a fast growing topic around the world [111, 112, 113, 114]. By
coherently addressing atoms and imparting phase shifts as they traverse closed paths in either thebulk of the trap or around specific lattice sites, an analog to the Aharnov-Bohm phase of enclosed
magnetic flux is created [115]. Creating these effects on the level of single particles has beenachieved, however heating either via scattering field-dressing lasers or other less well understoodmechanisms has slowed the progress of this field. To study interesting effects, like the Hofstedter
Butterfly transition or to create topologically non-trivial states, requires not only engineering single
particle dispersion curves that are flat and highly degenerate, but creating a system where theinterparticle interactions become the dominant energy scale while still obeying the dressed statesthat have been imposed on the atoms.

116
The connection between dressed-state interactions and some of the first proposed density shiftmechanisms for OLCs (based on laser inhomogeneity induced interactions) was even commented
upon by Prof. Jing Zhang on a visit to our group [33]. In fact, the creation of velocity and position
dependent Rabi coupling to our clock state has in the past been used for mundane diagnostics
such as Doppler temperature scans. While some of the original proposals for spin-orbit coupling
utilized the narrowness of our clock state due to its lack of scattering, excited state loss due toinelastic collisions might put a damper on these proposals. However, multi-clock-photon coupling
to different ground state nuclear spin states might remain a saving grace for implementing manyof these proposals.
While options for spin-orbit coupling for atoms in shallow lattices abound, focusing on theexperiments in the bulk also show off the benefits of working with alkaline earth atoms. Creating a
dressing field that increases the flux of a fictitious field that scales favorably is not particularly easy.
The first implementations of spin orbit coupling could only scale the flux with a single lengthscaleof the cloud. Achieving high fictitious flux increases the number of vortices decreasing the filling
factor (N/Nv ) and bringing the sample closer to the behavior of a superfluid electron gas, where theparticles find it energetically favorable to form composite particles with the vortices present in thesystem [116]. By intersecting multiple dressing beams with specific detunings and angles one canhowever create an optical flux lattice [117]. These lattices will present an ever increasing amountof flux for closed paths in 2D, scaling with area and not just length or width.
Creating these optical flux lattices again requires significant optical flexibility that SrII did
not have. For example, the proposal in Ref. [118] required multiple beams to intersect a 2D
pancake of atoms at 120 degrees from each other in the pancake plane. Furthermore, achieving
meaningful correlated atom states requires an incredibly cold and dense atomic sample. While
many of the proposals might not work out due to inelastic loss, or low interparticle interactionenergies in strontium, I believe SrIII will have the necessary tools and the experience of using clock
lasers to coherently address atoms and push the field of spin-orbit physics in an as of yet unexplored
direction.

117

Figure 6.3: SrIII Recessed Viewports. The top viewport is depicted as a transparent wireframe
for clarity. UKAEA will construct these viewports, first delivering the bottom of the bucket withthe brazed window so it can be coated and then reattached to the rest of the bucket. More details
about the bucket window can be found in the engineering diagrams in Appendix B.

118

Figure 6.4: SrIII Multiband Coating from American Photonics. Special attention was paid toperformance at 689 nm and 813 nm. Coating performance at both 461 nm and 1064 nm wasconsidered a secondary goal.

119

Figure 6.5: SrIII Broadband Coating from TAKOS. The solid line is the expected coating performance, while the dashed line is uncoated BK-7 glass. We wanted a coating that could be used formost of the visible and NIR, including more exotic wavelengths like the anti-magic wavelengths in
the 650 nm region.

120

Figure 6.6: SrIII coating layout. The blue colored viewports (including the top and bottom
bucket windows) are using the multiband coating. The green shaded viewports are coated with thebroadband antireflection coating.

Chapter 7

Summary & Outlook

SrIII will no doubt push the boundaries of what is possible in a cold atomic gas machine.
By combining the techniques the Ye group has honed in the frequency metrology business with asuccessful quantum gas apparatus, the next few years should prove very interesting [119]. But what
does the future hold for atomic clocks? Let me first state unequivocally, I believe there is no end
to accuracy or stability of clocks in sight. Whether or not we call future super-precise frequency
metrology devices ‘clocks’ or ‘quantum sensors’, I believe that there are still decades of advances
to be made.
In the field of ion clocks, at least three new contenders will emerge to battle the now triumphant optical lattice clocks. The first contender will be the Yb ion octupole transition clock.
This extremely narrow transition offers the ability to push ion clock stability to the absolute limitof the stability of the optical local oscillator’s stability. Because these clocks are still quantum
projection noise limited with one ion, interrogation times will need to be incredibly long to achievethis. The second contender is the nuclear transition 229 Th ion clock. This transition, once it isfound, is immune to many of the systematics associated with clocks as its nuclear nature shields
it considerably from external perturbations [120]. Furthermore, it’s nuclear nature allows for the
possibility of embedding this isotope directly into a solid creating a crystal clock [121]. However,finding the exact frequency of this transition might require the use of novel technologies like an
XUV comb or the Advanced Light Source at Lawrence Berkeley National Lab. [122, 123]. And
finally, other more accessible transitions in highly charged ions are also better shielded from ex-

122
ternal perturbations and offer promising routes to systematic uncertainties of 10−19 [124]. While
OLCs might have got the upper hand for now, it would be a mistake to count out ion clocks in therunning for redefinition of the second.
The high stability of neutral lattice clocks will continue to increase, as new materials andprocesses are tried to obtain better and better laser local oscillators. While our most recent results
were driven by our new 40 cm long ULE cavity [62], a new round of results will appear resulting
from cryogenically cooled silicon cavities [125] and custom crystalline Bragg mirrors [126]. Alongthe way more active laser technologies have been theorized and at least principally demonstrated
that replace the phase memory of cavity mirrors with that of atomic dipoles creating incredibly
narrow frequency sources [127, 128]. It has been noted that frequency metrology improves by afactor of 10 with every decade [129] and so far I would say we are more than well on track.
But what of strontium? The blackbody radiation shift will remain its biggest obstacle. Whilemodulation of the environment temperature coupled with clock comparisons might be a long term
viable option, single system performance will continue to be limited. Neither temperature sensors
nor small, cold enclosures will allow strontium to dip to far below 1 × 10−18 [22, 88]. It may be thata combination of the two technologies will suffice. Instead, I suspect that for strontium to reachdeep into fractional uncertainties of 10−19 or below, significant work will be needed to completely
enclose the strontium cloud in a cold environment. Whether via a specially built cryostat chamberwith light delivered by fiber or, as I have joked many times, an industrial freezer keeping theentire chamber at a chilly 187 K, the next factor of 10 of systematic uncertainty will need a lowtemperature trapping environment.
The field of optical lattice clocks is rich though, and the methods pioneered in strontium by
us and other groups around the world will one day be employed in a variety of species. A Yb clockusing our temperature sensor procedure would have a 2 times smaller BBR uncertainty than ours,a Hg clock 50 times smaller. The high stability that these clocks can offer will for the forseeable
future continue to be a selling point for their use, even if in the end it is as a flywheel technology
used much like a hydrogen maser. This first triumph after more than a decade of research does not123mark the beginning of the end of optical lattice clocks, but maybe the end of the beginning.

124

Figure 7.1: Red MOT absorption image of strontium atoms incorrectly aligned through the imaging
optics. Boldly going where no optical lattice clock has gone before.

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Appendix A

Single Purpose Fast Arduino Example

1/*
Arduino Code to Control Slow Loop Filter

3

Language : Arduino language .
5

Environment : Arduino 1.0

7

Copyright ( C ) 2013 by Ben Bloom
MIT License

9

Ver : 0.1
11*/
# define pinLED 13 //13# define isTransHigh () PINC & 0 b00000001

15# define setPinSWEEP_HIGH () PORTC |= 0 b00000010# define setPinSWEEP_LOW () PORTC &= 0 b11111101

17// pins for reading out Mode dial PORTB# define modeReadMask 0 b00011111

19# define readCurrentMode () PINB &= modeReadMask

21# define modeSetMask 0 b11111000# define setModeAcq () PORTD = ( PORTD & ~ modeSetMask ) | (0 b00001000 & modeSetMask )23# define setModeProp () PORTD = ( PORTD & ~ modeSetMask ) | (0 b00011000 & modeSetMask )135# define setMode6dB () PORTD = ( PORTD & ~ modeSetMask ) | (0 b00111000 & modeSetMask )25# define setMode6dBP () PORTD = ( PORTD & ~ modeSetMask ) | (0 b01111000 & modeSetMask )# define setMode9dB () PORTD = ( PORTD & ~ modeSetMask ) | (0 b11111000 & modeSetMask )27# define setModeSearch () PORTD = ( PORTD & ~ modeSetMask ) | (0 b11100000 & modeSetMask) // for sweeping to find a fringe# define setMode3dB () PORTD = ( PORTD & ~ modeSetMask ) | (0 b11101000 & modeSetMask )// for craziness29

31

byte isLocked ;byte previousIsLocked ;33

byte isAcquiring ;byte cur rentModeSwitch ;35

byte setMode ;byte currentALmode ;37

byte transmission ;byte prevTransmission ;39

byte searching ;unsigned long timeOut ;41

unsigned long currentTime ;43

void setup () {45

pinMode ( pinLED , OUTPUT ) ;pinMode ( A0 , INPUT ) ;47

pinMode ( A1 , OUTPUT ) ;digitalWrite ( A1 , LOW ) ;49

DDRB &= 0 b11100000 ; // Sets the pins for reading current mode
PORTB |= 0 b00011111 ; // Sets pullup resistors for current mode rotary switch

51

DDRD &= 0 b11111000 ; // Sets the pins for WRITING current mode
53

setMode = 0;
searching = 0;

136
setPinSWEEP_HIGH () ;

55

}
57

void loop () {59

while (1){61transmission = isTransHigh () ;currentALmode = ( prevTransmission << 1) | transmission ;63

if ( setMode == 15){

65

if ( currentALmode == 1) // Just passed a fringe !
{setPinSWEEP_HIGH () ;67

delayMicroseconds (8000) ;setModeProp () ;69

delayMicroseconds (1000) ;setMode6dB () ;71

delayMicroseconds (1000) ;setMode6dBP () ;73

delayMicroseconds (1000) ;setMode9dB () ;75

setMode = 15;
delayMicroseconds (16383) ;

77

}
79

else if ( currentALmode == 2) // Just fell out of lock{searching = 1;

81

setModeAcq () ;setPinSWEEP_LOW () ;

83

}
85

else if ( currentALmode == 0) // Out of lock{87

if ( searching == 0)

137{searching = 1;

89

setModeAcq () ;}91

setPinSWEEP_LOW () ;}93else { // lock is working !
setPinSWEEP_HIGH () ;95

searching = 0;
}

97

}
99

cur rentModeSwitch = readCurrentMode () ;if ( setMode != currentModeSwitch )101{switch ( currentModeSwitch )103{
case 30:

105

setModeAcq () ;break ;

107

case 29:
setModeProp () ;109

break ;
case 27:

111

setMode6dB () ;break ;

113

case 23:
setMode6dBP () ;115

break ;
case 15:

117

setMode9dB () ;break ;

119

default :break ;

138
}

121

setMode = currentModeSwitch ;setPinSWEEP_HIGH () ;

123

}
125

PORTB |= 0 b00011111 ; // Necessary , seems related to endless loop

127

prevTransmission = transmission ;
}

129

}

code/LoopFilterAutoLock.ino

Appendix B

Technical Drawings for SrIII Apparatus

What follows are a collection of technical CAD drawings of the SrIII main chamber and therecessed viewports.

9.00

11.00

11.62

7.00

Thet

Y

a

P hi

SolidWorks Student License
Academic Use Only

6.75

Z

P1, X = 0, Y = -9, Z = -7

P0, X = 0, Y = 0, Z = 0

X

P1, X = 0, Y = -9, Z = -7

X

P0, X = 0, Y = 0, Z = 0

APPV'D

Q.A

MFG

NAME

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESSURFACE FINISH:TOLERANCES:
LINEAR:ANGULAR:

CHK'D

DRAWN

FINISH:

SIGNATUREDATEDWG NO.

SCALE 1:2

WEIGHT:

TITLE:

MATERIAL:

DEBUR AND
BREAK SHARPEDGESREVISIONSHEET 1 OF 7

Sr3_MC_v5

JILA Sr Main Chamber

DO NOT SCALE DRAWING

A1

140

?

1 1/3

2 3/4

2 1/8

4 5/8

2 1/8

2 3/4

2 3/4

2 3/4

10

8

8

1(a/b)

2(a/b)

3(a/b)

4(a/b)

5(a/b)

6(a/b)

7(a/b)

8(a/b)

9(t/b)

10

11

P1

P1

P0

P0

P0

P0

P0

P0

P0

P0

Focal Point

SolidWorks Student License
Academic Use Only

Flange OD (in)Port Number

90

90

90

90

a: 90
b: -90a: 120b: -120
a: 150b: -150
a: 172b: -172

18.62

6.75

3.47

Approximate
FlangeCoordiantes
a:X=7,Y=-11,Z=-7
b:X=-7,Y=-11,Z=-7

14.50

10.00

8.00

6.00

180

90

a: 60
b: -60

8.00

0

90

t: 0
b: 18090

a: 30
b: -30

8.00

90

a: 8
b: -8

a: 6.30
b: 5.00

Phi

Theta

Flange Distance(in)sniffer port alignedwith z axis

Sniffer Port alignedwith y axis

Centerline of the
Horizontal Tube fromPart 10, Exact
Dimensions Not asimportanta: Clearence Holes
must be aligned
vertically. Port 1a is
atom beam entrance
and needs to befocused on Focal
Point P0 to < 0.5
degrees accuracy
b:tapped CF133
Flange

Notes

10

7b

7a

3a

6a
8a

6b
8b

5a

2a

5b

1a

4a

1b

9b

9t

4b

3b

2b

11

141

0.25

3.25

0.92

0.73

6.35

4.73

10.00

4.00

1in spacing between holes

0.27 for all mounting holes

7.00
3.00
4.00

8.00

SolidWorks Student License
Academic Use Only

3.00

1.00

Q.A

MFG

APPV'DNAME

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESSURFACE FINISH:TOLERANCES:
LINEAR:ANGULAR:

CHK'D

DRAWN

FINISH:

SIGNATURE

DATEDO NOT SCALE DRAWINGREVISIONDWG NO.

SCALE:1:2

WEIGHT:SHEET 3 OF 7

A1
main_chamber_sr3_v5

TITLE:

MATERIAL:

DEBUR AND
BREAK SHARP
EDGES

142

15.50

0.75

SolidWorks Student License
Academic Use Only

0.50

3.00

1.69

1.00

3.25

1.00

0.50
2.04

All Mounting Holes are 0.27 in diameter 1/4-20 clearance holes with 1 inch spacings

APPV'D

Q.A

MFG

NAME

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESSURFACE FINISH:TOLERANCES:
LINEAR:ANGULAR:

CHK'D

DRAWN

FINISH:

SIGNATURE

DATEDO NOT SCALE DRAWINGREVISIONDWG NO.

SCALE: 2:1

WEIGHT:SHEET 4 OF 7

A1
main_chamber_sr3_v5

TITLE:

MATERIAL:

DEBUR AND
BREAK SHARP
EDGES

143

R4
.0
0

SolidWorks Student License
Academic Use Only

1.00

0.50

7.00

1.00

0.50

All Mounting Holes are 0.27 in diameter 1/4-20 clearance holes with 1 inch spacings

APPV'D

Q.A

MFG

NAME

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESSURFACE FINISH:TOLERANCES:
LINEAR:ANGULAR:

CHK'D

DRAWN

FINISH:

SIGNATURE

DATEDO NOT SCALE DRAWINGREVISIONDWG NO.

SCALE: 2:1

WEIGHT:SHEET 5 OF 7

A1
main_chamber_sr3_v5

TITLE:

MATERIAL:

DEBUR AND
BREAK SHARP
EDGES

144

SolidWorks Student License
Academic Use Only

0.50

1.00

3.00
1.00

0.50

All Mounting Holes are 0.27 in diameter 1/4-20 clearance holes with 1 inch spacings

APPV'D

Q.A

MFG

NAME

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESSURFACE FINISH:TOLERANCES:
LINEAR:ANGULAR:

CHK'D

DRAWN

FINISH:

SIGNATURE

DATEDO NOT SCALE DRAWINGREVISIONDWG NO.

SCALE: 5:1

WEIGHT:SHEET 6 OF 7

A1
main_chamber_sr3_v5

TITLE:

MATERIAL:

DEBUR AND
BREAK SHARP
EDGES

145

1.00
0.85

4.00

SolidWorks Student License
Academic Use Only

1.00

6.00

0.42

0.50
R3
.0
0

R4
.00

All Mounting Holes are 0.27 in diameter 1/4-20 clearance holes with 1 inch spacings

APPV'D

Q.A

MFG

NAME

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESSURFACE FINISH:TOLERANCES:
LINEAR:ANGULAR:

CHK'D

DRAWN

FINISH:

SIGNATURE

DATEDO NOT SCALE DRAWINGREVISIONDWG NO.

SCALE: 2:1

WEIGHT:SHEET 7 OF 7

A1
main_chamber_sr3_v5

TITLE:

MATERIAL:

DEBUR AND
BREAK SHARP
EDGES

146

147