Thank you to the organizers, Zhirong Huang, Tor
Raubenheimer, Yunhai Cai, and Naomi Nagahashi, for inviting me to participate
in this symposium honoring the career of Alex Chao.

Alex was my PhD supervisor from 2000 to 2006. When I first joined the PhD program in
physics at Stanford in 1999, I had never heard of accelerator physics as a
discipline and had not imagined that this would be my path. I had studied math and physics at Reed
College (a small liberal arts college in Portland, Oregon), and, in starting
the PhD program at Stanford, thought that I might pursue condensed matter
physics or possibly particle physics.

I had taken a year off after graduating from Reed, because
initially, I didn’t get into the physics programs I applied to: Princeton, Harvard, and Cal Tech, if I recall
correctly…. I spent a year in Portland, working for Dr. Richard Crandall in a
basement of a small house above Reed College called “The Center for Advanced
Computation.”

This year, I learned how to program computers more
rigorously for the first time, worked on some interesting algorithmic problems
relating to (among other things), data compression for Pixar movies, and
studied and improved my physics GRE scores, such that I now had a more
respectable application, and was accepted to Stanford, along with Columbia
University, and UC Santa Cruz.

To say a little more about my undergraduate experience, I
initially heard about Reed because my father, Peter Nash, went there in the
1950’s planning to study medicine. Like
Steve Jobs, however, he didn’t finish at Reed. Instead, he took a year
traveling around Europe, trying to be a writer and find his direction in life. He finally returned to the US, finishing
undergraduate education at San Francisco State University and going on to earn
his MD at University of Southern California.

I was at Reed from 1994 to 1998, and I found it highly
stimulating intellectually, although a bit unbalanced socially. I enjoyed the universal humanities
requirements, covering Greek and Roman history, literature, and
philosophy. And I took some more modern
philosophy and literature courses as well.

I started at Reed as a biology major, thinking I would
ultimately study mathematical biology, but the physics department was more
flexible, and allowed me to take advanced courses earlier on. I took electrodynamics, quantum mechanics,
and particle physics from David Griffiths, a wonderful teacher and expositor.

As a senior, all Reed students are required to conduct
original research in terms of a senior thesis.
All of the theses are then collected and displayed in a “thesis tower”
in a actual tower atop the Reed College library. I worked with Drs. Nicholas Wheeler of the
physics department and Thomas Wieting of the math department. It was Nick Wheeler who first gave me my
thesis problem, asking the rather general question of what it means to move a
physics system around in the world (a problem that would have many echoes with
my later career in accelerator physics).
As a kid who grew up with divorced parents moving between houses twice a
week, this problem had a certain emotional and psychological appeal to me. I simplified the problem to be a single
particle moving in a potential energy function and asked what happened to that
particle as the system was moved from point A to point B. This problem led me to issues such as
adiabatic invariance and pushed me to read VI Arnold’s classical mechanics text
that defined action-angle variables for a variety of physical systems.

In my Reed thesis, I treated the Foucault pendulum, a clock
on a rocket ship, and transport of a central Coulomb potential such as a
hydrogen atom.

Although I was satisfied with what I accomplished in this
thesis, I was so focused on this work that I failed to get into graduate
school. As mentioned, however, after a
year working with Dr Crandall, I was ready to start at Stanford.

At Stanford, one has a full year to find a research group
and I started in my first quarter in experimental particle physics. I was not very satisfied with this work and
continued to explore my options.
Sometime during that quarter, there was an event where all the different
research groups were represented, and one could ask questions and find an appropriate
group. I spoke with someone at this
event and described my undergraduate research.
They suggested that I talk to Ron Ruth at SLAC and consider accelerator
physics. I had little idea what that
might involve, but I followed up on this.
I talked to Ron and he arranged that I give a talk on my undergraduate
work in the ARDA BIG (Beam Instability Group) meeting. Alex was there at this meeting and we started
to talk about working together after that.

The first problem that Alex gave me was a two macroparticle
model for the head-tail instability. I
like that it was something I could get to work on immediately and feel like I
might make a contribution to the field.
He showed me data about the so-called “sawtooth instability” and
suggested a simple model may be able to explain it.

I worked on a number of such “small” problems with Alex
before finding a larger thesis project to develop. I worked out analytic expressions for fringe
fields for a solenoid, and though about whether one could observe quantum
effects in beam dynamics by analyzing the evolution of the Wigner
function. These problems all planted
seeds that I would later return to, with other instability problems, and use of
the Wigner function to describe partially coherent synchrotron radiation.

I remember the day that Alex had an idea that was to form
the foundation for my thesis work. He
drew a simple diagram on his white board with two particles performing Coulomb
scattering. Given the initial positions
and momenta of the particles, one could compute the change in momentum
resulting from the scattering. Now apply
this to al the particle pairs in the beam distribution, and one has a new way
to analyze intrabeam scattering. By
forgoing the use of cross sections, and just computing classical orbits
directly, there was a hope that we might be able to formulate the IBS growth
rates in a way that avoided the logarithmic divergences of the usual
approaches.

By considering the time to the distance of closest approach
t_min and limiting the scatterers in a time Delta t to t_min < Delta t, I
was able to derive expressions for IBS diffusion and damping coefficients that
did not have the logarithmic divergence.
I was even able to reduce the 12-D integral to a 2-D bounded integral in
the case of a Gaussian distribution. So
the result could be computed without too much difficulty! I showed how this formulation could be
reduced to all other IBS formulations that I knew of. By comparing to results such as those of
Piwinski and Bjorken-Mtingwa, one could “derive” the Coulomb logarithm rather
than have to put it in by hand with an ill-defined b_min and b_max cut-off.

Through the whole process, Alex was always available and
always able to give just enough feedback to encourage me to continue and
occasionally to discourage me from following unpromising lines of
research. Piece by piece, and conversation
by conversation with Alex, the work came together. [PAC PAPER]

Now that we had a general approach for deriving the
diffusion and damping in IBS, I needed to better understand the Gaussian beam
distributions in electron storage rings.
These distributions arise by equilibrium between diffusion and damping
from synchrotron radiation.

Alex proposed that I look at what happens near a
synchro-betatron coupling resonance, particularly when the coupling term came
from dispersion at an RF cavity, or due to a crab cavity, which is a topic of
interest in colliders, creating so called “crabbed collisions “.

I tried to formulate the problem using perturbation theory,
considering the coupling term as small.
It toom me some time to realize that working near a resonance means that
one needs to do degenerate perturbation theory.
On resonance, two eigenvalues are equal, and on needs to find the right
way to break this degeneracy and find the “good” linear combinations of
eigenvectors. Using the tools of
degenerate perturbation theory in quantum mechanics that I had learned from
David Griffiths at Reed but applied to symplectic matrices instead of Hermitian
operators from QM, I could solve this problem.
After finding the expressions for the eigenvalues an eigenvectors near
resonance, I included the damping and diffusion from synchrotron radiation to
find equilibrium eigen-emmitance and beam second moments.

Of course, everyone has heard of Alex’s so-called SLIM
formalism. My contribution was to find
analytic expressions near resonances.

The last piece was to add on the effect of IBS to get a
theory that included coupling and IBS together.
I wrote all this up in my thesis.

I want to take a moment to acknowledge some of the other people
besides Alex who helped me with my PhD thesis.
First, and foremost, I’d like to acknowledge Juhao Wu. It was really a pleasure to work with him,
and he helped a lot both with talking through the concepts, and with
implementation of the equations to get concrete numerical results out of
it.

Overall, I enjoyed interacting with all the members of the
Beam Instability group in ARDA at SLAC.
Notably, Karl Bane also helped with the work on IBS and I had many
helpful discussions with Sam Heiffets, Ron Ruth, and Gennady Stupakov.

Another memory I have of this time is a period of at least 6
weeks when my main activity was trying to track down a factor of 2 in the
overall normalization of the IBS formulas.
I’m appreciative to Alex for letting me move along at my own slow pace,
with confidence that I would finally solve the given issue in a finite amount
of time.

I finally finished my thesis in 2006 and defended it to
achieve my doctorate degree!

I applied for post-doctoral positions at this time and I recall
that I was debating between working with Alex Dragt at University of Maryland
on abstract beam dynamics and other mathematical topics, and a position in the
design team for the NSLS-II at Brookhaven National Lab. Although Alex Chao has the highest respect
for Alex Dragt, he encouraged me to get involved with the more practical beam
dynamics work at NSLS-II, and after some reflection, I followed this
advice.

At NSLS-II, I worked under the direction of Johan Bengtsson,
moving from linear dynamics to non-linear dynamics, computing dynamics aperture
and Touschek lifetime with errors for the NSLS-II lattice. Although I ended up disagreeing with Johan on
many things, I’m very thankful for how he pushed me to include so many
realistic effects in my beam dynamics calculations.

In a way, this approach I learned from Johan was the
opposite approach I had used during my PhD work with Alex, in which one seeks
simple mechanisms underlying complex dynamics.
I came to appreciate that for machine design work, both approaches have
their place. I later worked under Sam
Krinsky, doing both practical simulations and some theoretical work as well.

At the end of my post-doc position at NSLS-II, I found two
options for my next position. The first
was in Campinas, Brazil, working on the design for the new Sirius light
source. The other position resulted from
a visit from Pascal Elleaume from ESRF in Grenoble, France. I emailed him asking about a position, and
they opened one up, and I had an offer to work there with him and Laurent
Farvacque. I decided to take the
position in France.

I spent 8 years in Grenoble at the ESRF first as a post-doc,
and then as a scientist. I had an
opportunity to work on many electron beam dynamics topics, from collective
effects to non-linear dynamics, and even spin dynamics via resonant
depolarization. In addition, one of my
main interests while at ESRF was working with beamline scientists to understand
their experiments and the relation to the electron beam dynamics. In addition, I met experts on x-ray optics
and learned about ray tracing and wave front propagation through x-ray
beamlines.

I really liked the broad range of science being done on the
x-ray beamlines and enjoyed explaining how the electron beam dynamics
worked. I was amazed to find out that very
few beamline scientists knew even basics about accelerator physics and beam
dynamics. I prepared a series of
lectures at ESRF with the help and encouragement of Luigi Paolasini, and
explained to beamline scientists and other technicians at ESRF how radiation
damping and diffusion works, about betatron and synchrotron motion and about
beam growth and loss mechanisms such as IBS, Touschek scattering and collective
instabilities driven by impedance.

I felt I was carrying on the tradition I had learned from
Alex in which accelerator physics and beam dynamics is seen as a topic of great
interest in its own right, with all the needs for development and academic
rigor of any other physics domain.

After 8 years at ESRF, my time in France came to an end, and
I decided to return to the US. I found a
position at RadiaSoft in Boulder, Colorado and have been working there for the
past 2 years. My work at RadiaSoft
continues several threads that I have followed in my career in accelerator
science. One strong interest I developed
throughout my several different positions was in ease of use issues with beam
dynamics and x-ray optics codes. At
NSLS-II, I worked with the code Tracy, managed by Dr. Bengtsson. At ESRF, I worked with Accelerator Toolbox
(AT), first developed here at SLAC by Andrei Terebilo. In both cases, I sought to develop an open
source collaboration for the software.
There are so many codes, and each is very challenging in its own
way. Finding ways to develop
collaboration and improve documentation and ease of use of these codes is
something I believe I can continue with through my work at RadiaSoft. At the same time, I continue with some of the
challenging accelerator science, studying topics such as polarization evolution and
preservation in electron ion colliders and magnetic undulator design and x-ray transport in synchrotron beamlines.

I would never have followed this journey through high
energy, short wavelengths, and fantastic complexity manifesting out of simple mechanisms
if I hadn’t had the good fortune to encounter and work with Alex Chao in my
career. And thus, I add my own story to
all the others we hear today to get a glimpse of the impressive legacy that
Alex has given us and how his may contributions and academic approach to
accelerator science and beam dynamics will continue to bear fruit in future
generations. Thanks for all this,
Alex. And thanks to all of you today for
listening to me tell my story!