Category Archives: Real science

Spotlight on research performed by teachers and students as part of our outreach efforts.

Holography and the MERA

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The AdS/MERA correspondence has been making the rounds of the blogosphere with nice posts by Scott Aaronson and Sean Carroll, so let’s take a look at the topic here at Quantum Frontiers.

The question of how to formulate a quantum theory of gravity is a long-standing open problem in theoretical physics. Somewhat recently, an idea that has gained a lot of traction (and that Spiros has blogged about before) is emergence. This is the idea that space and time may emerge from some more fine-grained quantum objects and their interactions. If we could understand how classical spacetime emerges from an underlying quantum system, then it’s not too much of a stretch to hope that this understanding would give us insight into the full quantum nature of spacetime.

One type of emergence is exhibited in holography, which is the idea that certain (D+1)-dimensional systems with gravity are exactly equivalent to D-dimensional quantum theories without gravity. (Note that we’re calling time a dimension here. For example, you would say that on a day-to-day basis we experience D = 4 dimensions.) In this case, that extra +1 dimension and the concomitant gravitational dynamics are emergent phenomena.

A nice aspect of holography is that it is explicitly realized by the AdS/CFT correspondence. This correspondence proposes that a particular class of spacetimes—ones that asymptotically look like anti-de Sitter space, or AdS—are equivalent to states of a particular type of quantum system—a conformal field theory, or CFT. A convenient visualization is to draw the AdS spacetime as a cylinder, where time marches forward as you move up the cylinder and different slices of the cylinder correspond to snapshots of space at different instants of time. Conveniently, in this picture you can think of the corresponding CFT as living on the boundary of the cylinder, which, you should note, has one less dimension than the “bulk” inside the cylinder.

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Even within this nice picture of holography that we get from the AdS/CFT correspondence, there is a question of how exactly do CFT, or boundary quantities map onto quantities in the AdS bulk. This is where a certain tool from quantum information theory called tensor networks has recently shown a lot of promise.

A tensor network is a way to efficiently represent certain states of a quantum system. Moreover, they have nice graphical representations which look something like this:

 mera

Beni discussed one type of tensor network in his post on holographic codes. In this post, let’s discuss the tensor network shown above, which is known as the Multiscale Entanglement Renormalization Ansatz, or MERA.

The MERA was initially developed by Guifre Vidal and Glen Evenbly as an efficient approximation to the ground state of a CFT. Roughly speaking, in the picture of a MERA above, one starts with a simple state at the centre, and as you move outward through the network, the MERA tells you how to build up a CFT state which lives on the legs at the boundary. The MERA caught the eye of Brian Swingle, who noticed that it looks an awfully lot like a discretization of a slice of the AdS cylinder shown above. As such, it wasn’t a preposterously big leap to suggest a possible “AdS/MERA correspondence.” Namely, perhaps it’s more than a simple coincidence that a MERA both encodes a CFT state and resembles a slice of AdS. Perhaps the MERA gives us the tools that are required to construct a map between the boundary and the bulk!

So, how seriously should one take the possibility of an AdS/MERA correspondence? That’s the question that my colleagues and I addressed in a recent paper. Essentially, there are several properties that a consistent holographic theory should satisfy in both the bulk and the boundary. We asked whether these properties are still simultaneously satisfied in a correspondence where the bulk and boundary are related by a MERA.

What we found was that you invariably run into inconsistencies between bulk and boundary physics, at least in the simplest construals of what an AdS/MERA correspondence might be. This doesn’t mean that there is no hope for an AdS/MERA correspondence. Rather, it says that the simplest approach will not work. For a good correspondence, you would need to augment the MERA with some additional structure, or perhaps consider different tensor networks altogether. For instance, the holographic code features a tensor network which hints at a possible bulk/boundary correspondence, and the consistency conditions that we proposed are a good list of checks for Beni and company as they work out the extent to which the code can describe holographic CFTs. Indeed, a good way to summarize how our work fits into the picture of quantum gravity alongside holography and tensors networks is by saying that it’s nice to have good signposts on the road when you don’t have a map.

Mingling stat mech with quantum info in Maryland

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I felt like a yoyo.

I was standing in a hallway at the University of Maryland. On one side stood quantum-information theorists. On the other side stood statistical-mechanics scientists.* The groups eyed each other, like Jets and Sharks in West Side Story, except without fighting or dancing.

This March, the groups were generous enough to host me for a visit. I parked first at QuICS, the Joint Center for Quantum Information and Computer Science. Established in October 2014, QuICS had moved into renovated offices the previous month. QuICSland boasts bright colors, sprawling armchairs, and the scent of novelty. So recently had QuICS arrived that the restroom had not acquired toilet paper (as I learned later than I’d have preferred).

Interaction space

Photo credit: QuICS

From QuICS, I yoyo-ed to the chemistry building, where Chris Jarzynski’s group studies fluctuation relations. Fluctuation relations, introduced elsewhere on this blog, describe out-of-equilibrium systems. A system is out of equilibrium if large-scale properties of it change. Many systems operate out of equilibrium—boiling soup, combustion engines, hurricanes, and living creatures, for instance. Physicists want to describe nonequilibrium processes but have trouble: Living creatures are complicated. Hence the buzz about fluctuation relations.

My first Friday in Maryland, I presented a seminar about quantum voting for QuICS. The next Tuesday, I was to present about one-shot information theory for stat-mech enthusiasts. Each week, the stat-mech crowd invites its speaker to lunch. Chris Jarzynski recommended I invite QuICS. Hence the Jets-and-Sharks tableau.

“Have you interacted before?” I asked the hallway.

“No,” said a voice. QuICS hadn’t existed till last fall, and some QuICSers hadn’t had offices till the previous month.**

Silence.

“We’re QuICS,” volunteered Stephen Jordan, a quantum-computation theorist, “the Joint Center for Quantum Information and Computer Science.”

So began the mingling. It continued at lunch, which we shared at three circular tables we’d dragged into a chain. The mingling continued during the seminar, as QuICSers sat with chemists, materials scientists, and control theorists. The mingling continued the next day, when QuICSer Alexey Gorshkov joined my discussion with the Jarzynski group. Back and forth we yoyo-ed, between buildings and topics.

“Mingled,” said Yigit Subasi. Yigit, a postdoc of Chris’s, specialized in quantum physics as a PhD student. I’d asked how he thinks about quantum fluctuation relations. Since Chris and colleagues ignited fluctuation-relation research, theorems have proliferated like vines in a jungle. Everyone and his aunty seems to have invented a fluctuation theorem. I canvassed Marylanders for bushwhacking tips.

Imagine, said Yigit, a system whose state you know. Imagine a gas, whose temperature you’ve measured, at equilibrium in a box. Or imagine a trapped ion. Begin with a state about which you have information.

Imagine performing work on the system “violently.” Compress the gas quickly, so the particles roil. Shine light on the ion. The system will leave equilibrium. “The information,” said Yigit, “gets mingled.”

Imagine halting the compression. Imagine switching off the light. Combine your information about the initial state with assumptions and physical laws.*** Manipulate equations in the right way, and the information might “unmingle.” You might capture properties of the violence in a fluctuation relation.

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With Zhiyue Lu and Andrew Maven Smith of Chris Jarzynski’s group (left) and with QuICSers (right)

I’m grateful to have exchanged information in Maryland, to have yoyo-ed between groups. We have work to perform together. I have transformations to undergo.**** Let the unmingling begin.

With gratitude to Alexey Gorshkov and QuICS, and to Chris Jarzynski and the University of Maryland Department of Chemistry, for their hospitality, conversation, and camaraderie.

*Statistical mechanics is the study of systems that contain vast numbers of particles, like the air we breathe and white dwarf stars. I harp on about statistical mechanics often.

**Before QuICS’s birth, a future QuICSer had collaborated with a postdoc of Chris’s on combining quantum information with fluctuation relations.

***Yes, physical laws are assumptions. But they’re glorified assumptions.

****Hopefully nonviolent transformations.

Generally speaking

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My high-school calculus teacher had a mustache like a walrus’s and shoulders like a rower’s. At 8:05 AM, he would demand my class’s questions about our homework. Students would yawn, and someone’s hand would drift into the air.

“I have a general question,” the hand’s owner would begin.

“Only private questions from you,” my teacher would snap. “You’ll be a general someday, but you’re not a colonel, or even a captain, yet.”

Then his eyes would twinkle; his voice would soften; and, after the student asked the question, his answer would epitomize why I’ve chosen a life in which I use calculus more often than laundry detergent.

http://www.sell-buy-machines.com/2013/02/why-prefer-second-hand-equipment-over-new.html

Many times though I witnessed the “general” trap, I fell into it once. Little wonder: I relish generalization as other people relish hiking or painting or Michelin-worthy relish. When inferring general principles from examples, I abstract away details as though they’re tomato stains. My veneration of generalization led me to quantum information (QI) theory. One abstract theory can model many physical systems: electrons, superconductors, ion traps, etc.

Little wonder that generalizing a QI model swallowed my summer.

QI has shed light on statistical mechanics and thermodynamics, which describe energy, information, and efficiency. Models called resource theories describe small systems’ energies, information, and efficiencies. Resource theories help us calculate a quantum system’s value—what you can and can’t create from a quantum system—if you can manipulate systems in only certain ways.

Suppose you can perform only operations that preserve energy. According to the Second Law of Thermodynamics, systems evolve toward equilibrium. Equilibrium amounts roughly to stasis: Averages of properties like energy remain constant.

Out-of-equilibrium systems have value because you can suck energy from them to power laundry machines. How much energy can you draw, on average, from a system in a constant-temperature environment? Technically: How much “work” can you draw? We denote this average work by < W >. According to thermodynamics, < W > equals the change ∆F in the system’s Helmholtz free energy. The Helmholtz free energy is a thermodynamic property similar to the energy stored in a coiled spring.

http://www.telegraph.co.uk/property/propertyadvice/jeffhowell/8013593/Home-improvements-Slime-does-come-out-in-the-wash.html

One reason to study thermodynamics?

Suppose you want to calculate more than the average extractable work. How much work will you probably extract during some particular trial? Though statistical physics offers no answer, resource theories do. One answer derived from resource theories resembles ∆F mathematically but involves one-shot information theory, which I’ve discussed elsewhere.

If you average this one-shot extractable work, you recover < W > = ∆F. “Helmholtz” resource theories recapitulate statistical-physics results while offering new insights about single trials.

Helmholtz resource theories sit atop a silver-tasseled pillow in my heart. Why not, I thought, spread the joy to the rest of statistical physics? Why not generalize thermodynamic resource theories?

The average work <W > extractable equals ∆F if heat can leak into your system. If heat and particles can leak, <W > equals the change in your system’s grand potential. The grand potential, like the Helmholtz free energy, is a free energy that resembles the energy in a coiled spring. The grand potential characterizes Bose-Einstein condensates, low-energy quantum systems that may have applications to metrology and quantum computation. If your system responds to a magnetic field, or has mass and occupies a gravitational field, or has other properties, <W > equals the change in another free energy.

A collaborator and I designed resource theories that describe heat-and-particle exchanges. In our paper “Beyond heat baths: Generalized resource theories for small-scale thermodynamics,” we propose that different thermodynamic resource theories correspond to different interactions, environments, and free energies. I detailed the proposal in “Beyond heat baths II: Framework for generalized thermodynamic resource theories.”

“II” generalizes enough to satisfy my craving for patterns and universals. “II” generalizes enough to merit a hand-slap of a pun from my calculus teacher. We can test abstract theories only by applying them to specific systems. If thermodynamic resource theories describe situations as diverse as heat-and-particle exchanges, magnetic fields, and polymers, some specific system should shed light on resource theories’ accuracy.

If you find such a system, let me know. Much as generalization pleases aesthetically, the detergent is in the details.

The dance of the electrons

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On the day I returned to the lab, Marcus Teague, a post-doc leader in the Yeh Group, was orienting two summer undergraduate interns. As he asked the students questions regarding superconductors and scanning tunneling microscopes (STM) I was happy with the amount of information I readily recalled from previous years. It was a good sign that I was ready to build on the content and skills I had already mastered. This may be expected of a graduate student at Caltech, but I am not a graduate student at Caltech. I am a High School physics teacher.

One of the first tasks assigned to me this summer, as I returned for my third year as an IQIM research intern, was to build and solder a connecting cord for the cryocooler in which we stored superconductors. That’s right – I have been working on cutting edge research involving high temperature superconductivity. But back to the soldering. This cord would communicate information, such as temperature and voltage, from the sensor. It had been years since my undergraduate soldering days at UCLA, but it turned out I was not so rusty. I was able to solder the tiny connecting pieces and show the undergraduate research students how to solder.

This summer I made additional tips for STM. Again, we chemically etched the tips using a Calcium Chloride solution in order to reach an ideal thickness of 1 atom across. We also employed the physical stretch and clip approach to make tips. Each person in the lab has their own philosophy on the best method to make tips. It’s amusing and interesting to compare the different techniques for attempting to make sharp, symmetric tips. Whether chemical or physical, the process to make sharp tips is tedious and time consuming, but imperative for a scan with good resolution.

I also worked with Kyle Chen again for certain projects, one of which involved making a low pass filter which would cut out signals above 2 GHz. We wrapped copper wire around a skewer in order to make a tiny solenoid. Care was taken to ensure the number of loops in one direction was matched in the other to avoid creating a net magnetic field. My first attempt was pathetic, but with each trial and error, I was able to construct decent solenoids. The completed solenoids were soldered to a SMA connector (like the one used for coaxial cable) which became one end of the RF filter. In order to connect the copper tube and SMA electrically, a silver epoxy is used. The solenoids were slid carefully into a small copper tube and then filled with the epoxy, a mixture of copper powder and Stycast A and B at a ratio of 100:28. Silver apoxy needs high temperature baking, above 100 degrees Celsius, in order to set, so the system was then baked in order to solidify our new low pass filter to be placed alongside the sample for STM testing.

This summer we also continued testing the YBCO superconducting samples I helped to make two summers ago with Professor Feng. After etching, the procedure to load the sample into the long, cylindrical, central tube required detailed planning. As with last year, the hood was flushed with Argon gas and then vacuum pumped in order to reduce contaminants. Using gloves in the hood, the sample is daintily set into place. Lifting the giant cylinder with the sample took four of us: two for lifting, one for holding it from the base within the hood and one person adjusting the gas levels.

Next we took the cylinder with the sample downstairs to the subbasement to be vacuum pumped and baked in order to expel all gaseous particles. Finally the central tube was loaded into the STM central chamber and the cooling process began, first with liquid nitrogen and then with liquid helium. Finally, it was time to scan the superconducting samples.

Last summer’s scans of the superconductor were in the 500 mV range. However, this summer’s scans are within 150 mV which will lead to much better resolution of the image and possibly valuable enough data to publish.

Each Friday Nai-Chung Yeh has a group meeting with everyone working in the lab. She thoughtfully discusses each persons’ progress, contributions, and questions from the past week. Her genuine curiosity and passion to discuss the best methods for experimentation are inspirational. She is expressive with her hands as she explains a concept, such as how she thinks the polymer doping is affecting the graphene samples. Each week she inevitably goes to the whiteboard and draws a picture of the hypothesized phenomenon observed in the data for that week. She gives insights from her vast wealth of background knowledge and suggests applicable equations, troubleshooting techniques, and information found in the current literature. It is fascinating to watch her warmly lead this group to a deeper understanding of the research topics at hand. I am moved by her work ethic and ability to balance oversight of the graphene projects, topological insulators, superconductors, and possible new solar cell technology. She supervises each sub-projects’ progress while writing papers, traveling the world over to present, as well as secure funding for research.

It is difficult to believe yet another year has passed. Again it is time to return to my classroom to meet my new 150 students, to get them fired up for learning about the exciting world of Physics. I am eager to share my learning experiences from the last three summers at Caltech.

The Graphene Effect

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Spyridon Michalakis, Eryn Walsh, Benjamin Fackrell, Jackie O'Sullivan

Lunch with Spiros, Eryn, and Jackie at the Athenaeum (left to right).

Sitting and eating lunch in the room where Einstein and many others of turbo charged, ultra-powered acumen sat and ate lunch excites me. So, I was thrilled when lunch was arranged for the teachers participating in IQIM’s Summer Research Internship at the famed Athenaeum on Caltech’s campus. Spyridon Michalakis (Spiros), Jackie O’Sullivan, Eryn Walsh and I were having lunch when I asked Spiros about one of the renowned “Millennium” problems in Mathematical Physics I heard he had solved. He told me about his 18 month epic journey (surely an extremely condensed version) to solve a problem pertaining to the Quantum Hall effect. Understandably, within this journey lied many trials and tribulations ranging from feelings of self loathing and pessimistic resignation to dealing with tragic disappointment that comes from the realization that a victory celebration was much ado about nothing because the solution wasn’t correct. An unveiling of your true humanity and the lengths one can push themselves to find a solution. Three points struck me from this conversation. First, there’s a necessity for a love of the pain that tends to accompany a dogged determinism for a solution. Secondly, the idea that a person’s humanity is exposed, at least to some degree, when accepting a challenge of this caliber and then refusing to accept failure with an almost supernatural steadfastness towards a solution. Lastly, the Quantum Hall effect. The first two on the list are ideas I often ponder as a teacher and student, and probably lends itself to more of a philosophical discussion, which I do find very interesting, however, will not be the focus of this posting.

The Yeh research group, which I gratefully have been allowed to join the last three summers, researches (among other things) different applications of graphene encompassing the growth of graphene, high efficiency graphene solar cells, graphene component fabrication and strain engineering of graphene where, coincidentally for the latter, the quantum Hall effect takes center stage. The quantum Hall effect now had my attention and I felt it necessary to learn something, anything, about this recently recurring topic. The quantum Hall effect is something I had put very little thought into and if you are like I was, you’ve heard about it, but surely couldn’t explain even the basics to someone. I now know something on the subject and, hopefully, after reading this post you too will know something about the very basics of both the classical and the quantum Hall effect, and maybe experience a spark of interest regarding graphene’s fascinating ability to display the quantum Hall effect in a magnetic field-free environment.

Let’s start at the beginning with the Hall effect. Edwin Herbert Hall discovered the appropriately named effect in 1879. The Hall element in the diagram is a flat piece of conducting metal with a longitudinal current running through. When a magnetic field is introduced normal to the Hall element the charge carriers moving through the Hall element experience a Lorentz force. If we think of the current as being conventionHallEffectal (direction flow of positively charged ions), then the electrons (negative charge carriers) are traveling in the opposite direction of the green arrow shown in the diagram. Referring to the diagram and using the right hand rule you can conclude a buildup of electrons at the long bottom edge of the Hall element running parallel to the longitudinal current, and an opposing positively charged edge at the long top edge of the Hall element. This separation of charge will produce a transverse potential difference and is labeled on the diagram as Hall voltage (VH). Once the electric force (acting towards the positively charged edge perpendicular to both current and magnetic field) from the charge build up balances with the Lorentz force (opposing the electric force), the result is a negative charge carrier with a straight line trajectory in the opposite direction of the green arrow. Essentially, Hall conductance is the longitudinal current divided by the Hall voltage.

Now, let’s take a look at the quantum Hall effect. On February 5th, 1980 Klaus von Klitzing was investigating the Hall effect, in particular, the Hall conductance of a two-dimensional electron gas plane (2DEG) at very low temperatures around 4 Kelvin (- 4520 Fahrenheit). von Klitzing found when a magnetic field is applied normal to the 2DEG, and Hall conductance is graphed as a function of magnetic field strength, a staircase looking graph emerges. The discovery that earned von Klitzing’s Nobel Prize in 1985 was as unexpected as it is intriguing. For each step in the staircase the value of the function was an integer multiple of e2/h, where e is the elementary charge and h is Planck’s constant. Since conductance is the reciprocal of resistance we can view this data as h/ie2. When i (integer that describes each plateau) equals one, h/ie2 is approximately 26,000 ohms and serves as a superior standard of electrical resistance used worldwide to maintain and compare the unit of resistance.

Before discussing where graphene and the quantum Hall effect cross paths, let’s examine some extraordinary characteristics of graphene. Graphene is truly an amazing material for many reasons. We’ll look at size and scale things up a bit for fun. Graphene is one carbon atom thick, that’s 0.345 nanometers (0.000000000345 meters). Envision a one square centimeter sized graphene sheet, which is now regularly grown. Imagine, somehow, we could thicken the monolayer graphene sheet equal to that of a piece of printer paper (≈ 0.1 mm) while appropriately scaling up the area coverage. The graphene sheet that originally covered only one square centimeter would now cover an area of about 2900 meters by 2900 meters or roughly 1.8 miles by 1.8 miles. A paper thin sheet covering about 4 square miles. The Royal Swedish Academy of Sciences at nobelprize.org has an interesting way of scaling the tiny up to every day experience. They want you to picture a one square meter hammock made of graphene suspending a 4 kg cat, which represents the maximum weight such a sheet of graphene could support. The hammock would be nearly invisible, would weigh as much as one of the cat’s whiskers, and incredibly, would possess the strength to keep the cat suspended. If it were possible to make the exact hammock out of steel, its maximum load would be less than 1/100 the weight of the cat. Graphene is more than 100 times stronger than the strongest steel!

Graphene sheets possess many fascinating characteristics certainly not limited to mere size and strength. Experiments are being conducted at Caltech to study the electrical properties of graphene when draped over a field of gold nanoparticles; a discipline appropriately termed “strain engineering.” The peaks and valleys that form create strain in the graphene sheet, changing its electrical properties. The greater the curvature of the graphene over the peaks, the greater the strain. The electrons in graphene in regions experiencing strain behave as if they are in a magnetic field despite the fact that they are not. The electrons in regions experiencing the greatest strain behave as they would in extremely strong magnetic fields exceeding 300 tesla. For some perspective, the largest magnetic field ever created has been near 100 tesla and it only lasted for a few milliseconds. Additionally, graphene sheets under strain experience conductance plateaus very similar to those observed in the quantum Hall effect. This allows for great control of electrical properties by simply deforming the graphene sheet, effectively changing the amount of strain. The pseudo-magnetic field generated at room temperature by mere deformation of graphene is an extremely promising and exotic property that is bound to make graphene a key component in a plethora of future technologies.

Graphene and its incredibly fascinating properties make it very difficult to think of an area of technology where it won’t have a huge impact once incorporated. Caltech is at the forefront in research and development for graphene component fabrication, as well as the many aspects involved in the growth of high quality graphene. This summer I was involved in the latter and contributed a bit in setting up an experimenKodak_Camera 1326t that will attempt to grow graphene in a unique way. My contribution included the set-up of the stepper motor (pictured to the right) and its controls, so that it would very slowly travel down the tube in an attempt to grow a long strip of graphene. If Caltech scientist David Boyd and graduate student Chen-Chih Hsu are able to grow the long strips of graphene, this will mark yet another landmark achievement for them and Caltech in graphene research, bringing all of us closer to technologies such as flexible electronics, synthetic nerve cells, 500-mile range Tesla cars and batteries that allow us to stream Netflix on smartphones for weeks on end.

Caltech InnoWorks 2014, More Than Just a Summer Camp

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“More, we need more!”

Adding more fuel to “Red October”, I presented the final product to my teammates. With a communal nod of approval, we rushed over to the crowd.

“1, 2, 3, GO!”

It was the semi-finals. Teams Heil Hydra! and The Archimedean Hawks ignited their engines and set their vehicles onto the starting line. Nascar? F1? Nope, even better. Homemade steamboat races! Throughout the cheers and yelling, we discovered that more isn’t better. Flames were devouring Team Heil Hydra!’s Red October. Down went the ship. Despite the loss, the kids learned about steam as a source of energy, experimentation, and teamwork. Although it may have been hard to tell the first day, by the end of this fourth day of the camp, all students were visibly excited for another day of the InnoWorks summer program at Caltech.

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What is InnoWorks? An engaging summer program aimed for middle school students with disadvantaged backgrounds, InnoWorks offers a free of charge opportunity to dig into the worlds of science, technology, engineering, mathematics, and medicine (STEM^2). In his own life experience, William Hwang (founder of InnoWorks) was blessed with the opportunities to attend several summer camps throughout his childhood, but he had a friend who did not share the same opportunities. Sparked with the desire to start something, Hwang founded the non-profit organization, United InnoWorks Academy. With the first program to begin in 2004, the InnoWorks Academy developed these summer programs to help provide underprivileged kids with hands-on activities, team-building activities, and fast-paced competitive missions. Starting with just 34 students and 17 volunteers in a single chapter, InnoWorks has now grown to more than a dozen university chapters that have hosted above 60 summer programs for 2,200 middle school students, all done with the help of over 1000 volunteers.

Monday, August 11th, 2014 marked the first day of Caltech’s 3rd annual summer InnoWorks program. Last year, my younger brother participated in the program and had such a great experience that he wanted to become a junior mentor this year. After researching the program and listening to my brother’s past experiences, I was ecstatic to accept this journey as a mentor for Caltech’s InnoWorks program. Allow me to take you on a ride through my team’s and my own experience of InnoWorks.

First Day of Caltech InnoWorks 2014. My first team member that I checked in was Elliot. “Are you ready for InnoWorks!?” Perhaps I was a little overly excited. I received a shrug and “What’re we eating for breakfast?” Not the response I was hoping for, but that was going to change. As the rest of my team, which included Frank, Megan, Ethan, and my junior mentor, Elan, arrived, I began peppering them with icebreakers left and right. Soon enough, we dubbed ourselves Heil Hydra! and by the end of the second day, I couldn’t get them to be quiet.

“What are we doing next?”

“Guys, GUYS! Let’s use the green pom-poms as chloroplasts.”

“Hey, the soap actually smells good.”

“Hm. If you add another rubber band, the cup won’t vibrate as much, and it makes a lower sound.”

Sometimes they would have endless questions, which was great! Isn’t that what science is all about?

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Most of the days during camp were themed with a specific subject, including biology, chemistry, physics, and engineering. Before each activity, both mentors and junior mentors gave a brief, prepared introduction to the science used during the experimentation. Here’s a quick synopsis of some of the activities and the students’ experiences:

Camera Obscura. After a short explanation of light, and how a lens works, we split the room up into 3 groups to build their very own camera obscura, which is an optical device that projects an image of its surroundings onto a screen (or in our case, the ground). Using a mirror, a magnifying glass, some PVC piping, and a black tarp, the kids constructed a camera obscura. I was impressed by how many students encumbered the heat of the black tarp and concrete all in the name of science.

Build Your Own Instrument. The title says all. I let my junior mentor, Elan, lead the group in this activity. Tasked with creating an instrument based on accurate pitch of 3 whole note tones, creativity, efficiency, and performance, the students went straight to work. Children have endless imaginations. Give kids PVC pipes, rubber bands, balloons, cups, and paper clips, and they’ll make everything! Working together, the groups created an instrument (often more than one) to present in front of everyone. Teams were required to explain how their instrument created sound (vibrations), and attempt to play “Mary Had a Little Lamb” (which most succeeded). I came across paperclip rain sticks, PVC didgeridoos, test tube pan flutes, red solo cup drums, and even PVC balloon catapults and rubber band ballistas!

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Liquid Nitrogen. One of the highlights of the camp was liquid nitrogen! We were very honored to have Glen Evenbly and Olivier Landon-Cardinal, IQIM postdocs, join us. After pouring the liquid into a bowl, Glen showed the kids how nitrogen gas enveloped the area. Liquid nitrogen’s efficiency as a coolant is limited by the fact that it boils immediately upon contact with a warmer object, surrounding the object with nitrogen gas on which the liquid surfs. This effect is known as the Leidenfrost effect, which applies to any liquid in contact with an object significantly hotter than its boiling point. 

InnoWorks-38InnoWorks-40 

However, liquid nitrogen is still extremely cold, and when roses were placed into the bowl with liquid nitrogen, the pedals froze right before everyone’s eyes.

Lego Mindstorms. The last activity of the camp was building a lego robot and programming it to track and follow a black tape trail using its light sensor. Since each of my team members had experience with these lego kits, they went to work right away. Two of my students worked on building the robot, while the other two retrieved the pieces. After awhile, they prompted each other to switch roles.

InnoWorks-3

Programming the robot was a struggle, but manipulating the code and watching the aftermath was all part of the experiment. After many attempted tries, the group was unable to accurately get the robot to follow the black line (some groups were successful!). However, without any outside help (including myself), Team Heil Hydra! programmed the robot to move and sing (can you guess?) “Mary Had a Little Lamb”. Teamwork for the win! Team spirit bloomed in my group – each day of camp my InnoWorkers agreed on a matching t-shirt color. As a mentor, I could not have been more proud.

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I know that I am not only speaking for myself when I say that the InnoWorks family, the students, and the program itself has burrowed its way into my heart. I have watched these students develop teamwork skills, enthusiasm for learning new things, and friendships. I have heard these students speak the minimal amount on their first day, only to find that their chatterboxes won’t stop the last day. To overlook InnoWorks as just a science camp where students come to learn about science is an understatement. InnoWorks is where students experience, engage, and conduct science, where they learn not just about science, but also about collaboration, leadership, and innovation. 

I must end on this last note: Heil Hydra! 

Editor’s Note: Ms. Rebekah Zhou is majoring in mathematics at CSU Fresno. In her spare time, she enjoys teaching piano and tutoring.

Reading the sub(linear) text

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Physicists are not known for finesse. “Even if it cost us our funding,” I’ve heard a physicist declare, “we’d tell you what we think.” Little wonder I irked the porter who directed me toward central Cambridge.

The University of Cambridge consists of colleges as the US consists of states. Each college has a porter’s lodge, where visitors check in and students beg for help after locking their keys in their rooms. And where physicists ask for directions.

Last March, I ducked inside a porter’s lodge that bustled with deliveries. The woman behind the high wooden desk volunteered to help me, but I asked too many questions. By my fifth, her pointing at a map had devolved to jabbing.

Read the subtext, I told myself. Leave.

Or so I would have told myself, if not for that afternoon.

That afternoon, I’d visited Cambridge’s CMS, which merits every letter in “Centre for Mathematical Sciences.” Home to Isaac Newton’s intellectual offspring, the CMS consists of eight soaring, glass-walled, blue-topped pavilions. Their majesty walloped me as I turned off the road toward the gatehouse. So did the congratulatory letter from Queen Elizabeth II that decorated the route to the restroom.

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I visited Nilanjana Datta, an affiliated lecturer of Cambridge’s Faculty of Mathematics, and her student, Felix Leditzky. Nilanjana and Felix specialize in entropies and one-shot information theory. Entropies quantify uncertainties and efficiencies. Imagine compressing many copies of a message into the smallest possible number of bits (units of memory). How few bits can you use per copy? That number, we call the optimal compression rate. It shrinks as the number of copies compressed grows. As the number of copies approaches infinity, that compression rate drops toward a number called the message’s Shannon entropy. If the message is quantum, the compression rate approaches the von Neumann entropy.

Good luck squeezing infinitely many copies of a message onto a hard drive. How efficiently can we compress fewer copies? According to one-shot information theory, the answer involves entropies other than Shannon’s and von Neumann’s. In addition to describing data compression, entropies describe the charging of batteries, the concentration of entanglement, the encrypting of messages, and other information-processing tasks.

Speaking of compressing messages: Suppose one-shot information theory posted status updates on Facebook. Suppose that that panel on your Facebook page’s right-hand side showed news weightier than celebrity marriages. The news feed might read, “TRENDING: One-shot information theory: Second-order asymptotics.”

Second-order asymptotics, I learned at the CMS, concerns how the optimal compression rate decays as the number of copies compressed grows. Imagine compressing a billion copies of a quantum message ρ. The number of bits needed about equals a billion times the von Neumann entropy HvN(ρ). Since a billion is less than infinity, 1,000,000,000 HvN(ρ) bits won’t suffice. Can we estimate the compression rate more precisely?

The question reminds me of gas stations’ hidden pennies. The last time I passed a station’s billboard, some number like $3.65 caught my eye. Each gallon cost about $3.65, just as each copy of ρ costs about HvN(ρ) bits. But a 9/10, writ small, followed the $3.65. If I’d budgeted $3.65 per gallon, I couldn’t have filled my tank. If you budget HvN(ρ) bits per copy of ρ, you can’t compress all your copies.

Suppose some station’s owner hatches a plan to promote business. If you buy one gallon, you pay $3.654. The more you purchase, the more the final digit drops from four. By cataloguing receipts, you calculate how a tank’s cost varies with the number of gallons, n. The cost equals $3.65 × n to a first approximation. To a second approximation, the cost might equal $3.65 × n + a√n, wherein a represents some number of cents. Compute a, and you’ll have computed the gas’s second-order asymptotics.

Nilanjana and Felix computed a’s associated with data compression and other quantum tasks. Second-order asymptotics met information theory when Strassen combined them in nonquantum problems. These problems developed under attention from Hayashi, Han, Polyanski, Poor, Verdu, and others. Tomamichel and Hayashi, as well as Li, introduced quantumness.

In the total-cost expression, $3.65 × n depends on n directly, or “linearly.” The second term depends on √n. As the number of gallons grows, so does √n, but √n grows more slowly than n. The second term is called “sublinear.”

Which is the word that rose to mind in the porter’s lodge. I told myself, Read the sublinear text.

Little wonder I irked the porter. At least—thanks to quantum information, my mistake, and facial expressions’ contagiousness—she smiled.

 

 

With thanks to Nilanjana Datta and Felix Leditzky for explanations and references; to Nilanjana, Felix, and Cambridge’s Centre for Mathematical Sciences for their hospitality; and to porters everywhere for providing directions.

“Feveral kinds of hairy mouldy fpots”

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The book had a sheepskin cover, and mold was growing on the sheepskin. Robert Hooke, a pioneering microbiologist, slid the cover under one of the world’s first microscopes. Mold, he discovered, consists of “nothing elfe but feveral kinds of fmall and varioufly figur’d Mufhroms.” He described the Mufhroms in his treatise Micrographia, a 1665 copy of which I found in “Beautiful Science.” An exhibition at San Marino’s Huntington Library, “Beautiful Science” showcases the physics of rainbows, the stars that enthralled Galileo, and the world visible through microscopes.

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Beautiful science of yesterday: An illustration, from Hooke’s Micrographia, of the mold.

“[T]hrough a good Microfcope,” Hooke wrote, the sheepskin’s spots appeared “to be a very pretty fhap’d Vegetative body.”

How like a scientist, to think mold pretty. How like quantum noise, I thought, Hooke’s mold sounds.

Quantum noise hampers systems that transmit and detect light. To phone a friend or send an email—“Happy birthday, Sarah!” or “Quantum Frontiers has released an article”—we encode our message in light. The light traverses a fiber, buried in the ground, then hits a detector. The detector channels the light’s energy into a current, a stream of electrons that flows down a wire. The variations in the current’s strength is translated into Sarah’s birthday wish.

If noise doesn’t corrupt the signal. From encoding “Happy birthday,” the light and electrons might come to encode “Hsappi birthdeay.” Quantum noise arises because light consists of packets of energy, called “photons.” The sender can’t control how many photons hit the detector.

To send the letter H, we send about 108 photons.* Imagine sending fifty H’s. When we send the first, our signal might contain 108– 153 photons; when we send the second, 108 + 2,083; when we send the third, 108 – 6; and so on. Receiving different numbers of photons, the detector generates different amounts of current. Different amounts of current can translate into different symbols. From H, our message can morph into G.

This spring, I studied quantum noise under the guidance of IQIM faculty member Kerry Vahala. I learned to model quantum noise, to quantify it, when to worry about it, and when not. From quantum noise, we branched into Johnson noise (caused by interactions between the wire and its hot environment); amplified-spontaneous-emission, or ASE, noise (caused by photons belched by ions in the fiber); beat noise (ASE noise breeds with the light we sent, spawning new noise); and excess noise (the “miscellaneous” folder in the filing cabinet of noise types).

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Beautiful science of today: A microreso-nator—a tiny pendulum-like device— studied by the Vahala group.

Noise, I learned, has structure. It exhibits patterns. It has personalities. I relished studying those patterns as I relish sending birthday greetings while battling noise. Noise types, I see as a string of pearls unearthed in a junkyard. I see them as “pretty fhap[es]” in Hooke’s treatise. I see them—to pay a greater compliment—as “hairy mouldy fpots.”

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*Optical-communications ballpark estimates:

  • Optical power: 1 mW = 10-3 J/s
  • Photon frequency: 200 THz = 2 × 1014 Hz
  • Photon energy: h𝜈 = (6.626 × 10-34 J . s)(2 × 1014 Hz) = 10-19 J
  • Bit rate: 1 GB = 109 bits/s
  • Number of bits per H: 10
  • Number of photons per H: (1 photon / 10-19 J) (10-3 J/s)(1 s / 109 bits)(10 bits / 1 H) = 108

 

An excerpt from this post was published today on Verso, the blog of the Huntington Library, Art Collection, and Botanical Gardens.

With thanks to Bassam Helou, Dan Lewis, Matt Stevens, and Kerry Vahala for feedback. With thanks to the Huntington Library (including Catherine Wehrey) and the Vahala group for the Micrographia image and the microresonator image, respectively.

The return of the superconducting high school teacher

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Last summer, I was blessed with the opportunity to learn about the basics of high temperature superconductors in the Yeh Group under the tutelage of visiting Professor Feng. We formed superconducting samples using a process known as Pulse Laser Deposition. We began testing the properties of the samples using X-Ray Diffraction, AC Susceptibility, and SQUIDs (superconducting quantum interference devices). I brought my new-found knowledge of these laboratory techniques and processes back into the classroom during this past school year. I was able to answer questions about the formation, research, and applications of superconductors that I had been unable to address prior to this valuable experience.

This summer I returned to the IQIM Summer Research Institute to continue my exploration of superconductors and gain even deeper research experience. This time around I have accompanied Caltech second year graduate student Kyle Chen in testing samples using the Scanning Tunneling Microscope (STM), some of which I helped form using Pulse Laser Deposition with Professor Feng last summer. I have always been curious about how we can have atomic resolution. This has been my big chance to have hands-on experience working with STM that makes it possible!

The Scanning Tunneling Microscope was invented by the late Heinrich Rohrer and Gerd Binnig at IBM Research in Zurich, Switzerland in 1981. STM is able to scan the surface contours of substances using a sharp conductive tip. The electron tunneling current through the tip of the microscope is exponentially dependent on the distance (few Angstroms) to the substance surface. The changing currents at different locations can then be compiled to produce three dimensional images of the topography of the surface on the nano-scale. Or conversely the distance can be measured while the current is held constant. STM has a much higher resolution of images and avoids the problems of diffraction and spherical aberration from lenses. This level of control and precision through STM has enabled scientists to use tools with nanometer precision, allowing scientists even to manipulate atoms and their bonds. STM has been instrumental in forming the field of nanotechnology and the modern study of DNA, semiconductors, graphene, topological insulators, and much more! Just five years after building their first STM, Rohrer and Binning’s work rightfully earned them the 1986 Nobel Prize in Physics.

Descending into the Sloan basement, Kyle and I work to prepare and scan several high temperature superconducting (HTSC) Calcium Doped YBCO (\rm Y_{1-x} Ca_x Ba_2 Cu_3 O_{7-\delta}) samples in order better to understand the pairing mechanism that causes Cooper Pairs for superconductivity. In regular metals, the pairing mechanism via phonon lattice vibrations is fairly well understood by physicists. Meanwhile, the pairing mechanism for HTSC is still a mystery. We are also investigating how this pairing changes with doping, as well as how the magnetic field is channeled up vortices within HTSC.

One of our first tasks is to make probe tips for STM. Adding Calcium Chloride to de-ionized water, we are preparing a liquid conductive path to begin the chemical etching of the probe tip. Using a 10V battery, a wire bent into a ring is connected to the battery and placed in the Calcium Chloride solution. Then a thin platinum iridium wire, also connected to the voltage source, is placed at the center of the conductive ring. The circuit is complete and a current of about half an Ampere is used to erode uniformly the outer surface of the platinum iridium wire, forming a sharp tip. We examine the tip under a traditional microscope to scrutinize our work. Ideally, the tip is only one atom thick! If not, we are charged with re-etching until we reach a more suitable straight, uniform, sharp tip.   As we work to prepare the platinum iridium tips, a stoic picture of Neils Bohr looks down at our work with the appropriate adjacent quotation, ” When it comes to atoms, language can be used only as in poetry.  The poet, too, is not nearly so concerned with describing facts as with creating images. ”  After making two or three nearly perfect tips, we clean and store them in the tip case and proceed to the next step of preparation.

We are now ready to clean the sample to be tested. Bromine etching removes any oxidation or impurities that have formed on our sample, leaving a top bromine film layer. We remove the bromine-residue layer with ethanol and then plunge further into the (sub)basement to load the sample into the STM casing before oxidation begins again. The STM in the Yeh Lab was built by Professor Nai-Chang Yeh and her students eleven years ago. There are multiple layers of vacuum chambers and separate dewars, each with its own meticulous series of steps to prepare for STM testing. At the center is a long, central STM tube. Surrounding this is a large cylindrical dewar. On the perimeter is an exterior large vacuum chamber.

First we must load the newly etched YBCO sample and tip into the central STM tube. The inner tube currently lays across a work bench beneath desk lamps. We must transfer the tiny tip from the tip case to just above the sample. While loading the tip with an equally minuscule flathead screwdriver, it became quite clear to me that I could never be a surgeon! The superconducting sample is secured in place with a small cover plate and screw. A series of electronic tests for resistance and capacitance must be conducted to confirm that there are no shorts in the numerous circuits. Next we must vacuum pump the inner cylindrical tube holding the sample, tip, and circuitry until the pressure is 10^{-4} Bar. Then we “bake” the inner chamber, using a heater to expel any other gas, while the vacuum pump continues until we reach approximately 10^{-5} Bar. The heater is turned off and the vacuum continues to pump until we reach 10^{-6} Bar. This entire vacuum process takes approximately 15 hours…

During this span of time, I have the opportunity to observe the dark, cold STM room. The door, walls and ceiling are covered with black rubber and spongy padding to absorb vibration. The STM room is in the lowest level basement for the same reason. The vibration from human steps near the testing generates noise in the data, so every precaution is made to minimize noise. Giant cement blocks lay across the STM metal box to increase inertia and decrease noise. I ask Kyle what he usually does with this “down” time. We discuss the importance of reading equipment manuals to grasp a better understanding of the myriad of tools in the lab. He says he needs to continue reading the papers published by the Yeh Lab Group. In knowing what questions your research group has previously answered, one has a better understanding of the history and the direction of current work.

The next day, the vacuum-pumped inner chamber is loaded to the center of the STM dewar. We flush the surrounding chambers with nitrogen gas to extricate any moisture or impurities that may have entered since our last testing. Next we can set up the equipment for a liquid nitrogen transfer which lasts approximately 2 hours, depending on the transfer rate. As the liquid nitrogen is added to the system, we meticulously monitor the temperature of the STM system. It must reach 80 Kelvin before we again test the electronics. Eventually it is time to add the liquid helium. Since liquid helium is quite expensive, additional precautions are taken to ensure maximum efficiency for helium use. It is beautiful to watch the moisture in the air deposit in frost along the tubing connecting the nitrogen and helium tanks to the STM dewar. The stillness of the quiet basement as we wait for the transfer is calming. Again, we carefully monitor the temperature drop as it eventually reaches 4.2 Kelvin. For this research, STM must be cooled to this temperature because we must drop below the critical temperature of the sample in order to observe superconductivity. The lower the temperature, the more of the superconducting component manifests itself. Hence the spectrum will have higher resolution. Liquid nitrogen is first added because it can carry over 90% of the heat away due to its higher mass. Nitrogen is also significantly cheaper than liquid helium. The liquid helium is added later, because it is even cooler than liquid nitrogen.

After adding additional layers of rubber padding on top of the closed STM, we can move over to the computer that controls the STM tip. It takes approximately one hour for the tip to be slowly lowered within range for a tunneling current. Kyle examines the data from the approach to the surface. If all seems normal, we can begin the actual scan of the sample!
An important part of the lab work is trouble shooting. I have listed the ideal order of steps, but as with life, things do not always proceed as expected. I have grown in awe of the perseverance and ingenuity required for daily troubleshooting. The need to be meticulous in order to avoid error is astonishing. I love that some common household items can be a valuable tool in the lab. For example, copper scrubbers used in the kitchen serve as a simple conducting path around the inner STM chamber. Floss can be used to tie down the most delicate thin wires. I certainly have grown in my immense respect for the patience and brilliance required in real research.

I find irony in the quiet simplicity of recording and analyzing data, the stillness of carefully transferring liquid helium juxtaposed to the immense complexity and importance of this groundbreaking research. I appreciate the moments of simple quiet in the STM room, the fast paced group meetings where everyone chimes in on their progress, or the boisterous collaborative brainstorming to troubleshoot a new problem. The summer weeks in the Sloan basement have been a welcome retreat from the exciting, transformative, and exhausting year in the classroom. I am grateful for the opportunity to learn more about superconductors, quantum tunneling, vacuum pumps, sonicators, lab safety, and more. While I will not be bromine etching, chemically forming STM tips, or doing liquid helium transfers come September, I have a new-found love for the process of research that I will radiate to my students.

High School Physics Teacher Embedded on A Quest to Squash Quantum Noise

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Date: 8/22/2013

Location: Caltech Cryo Lab, West Bridge:

Hello: I am Steve Maloney, a Physics and Chemistry teacher intern from Duarte High School, sponsored by IQIM (Institute for Quantum Information and Matter), doing whatever I can to be of assistance to Dr. Nicolas Smith-Lefebvre. Upon meeting him in mid-June I soon learned that our mission for the length of my visit was to assist him in determining with a greater degree of certainty the linear expansion coefficient of silicon in and around 125 K. (See below)

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Fig. 1: Silicon cavity.

The temperature of 125 K is of special interest to operators of LIGO (Laser Interferometer Gravitational Wave Observatory), because that is one of two temperatures where the thermal expansion coefficient, \alpha, of silicon is equal to zero. A zero linear expansion coefficient is of special interest to LIGO researchers because a small change in temperature inside the cryostat, (see Fig. 2, below) will not result in a significant change in length for the silicon cavity shown in Fig. 1.

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Fig. 2: Inside the Cryostat

Scientists working at LIGO need to know with great precision the length of the resonance cavity, because as gravitational waves pass through the cavity, they simultaneously compress the length of the cavity and stretch in a direction perpendicular to the shortening (warp). The arrival of a gravity wave produces a signal in the Fabry-Perot Interferometer, shown in Fig. 3, below.

Fig. 3: LIGO set-up with Fabry-Perot cavity.

Fig. 3: LIGO set-up with Fabry-Perot cavity.

Because the interferometer is sensitive to changes in length of as little as 1X10-15 m, sources of noise must be reduced to an absolute minimum. This brings us back to establishing the Thermal Coefficient of Linear Expansion, \alpha. Knowing the value of \alpha to a greater certainty will provide LIGO researchers the mathematical tools to better correct for small changes in temperature for the cavity, thus reducing the noise, therefore increasing the sensitivity of the Gravity Wave Detector.

So Where Do I Fit In?

In the Cryo-Lab on Thursday, July 11 2013, Nicolas Smith-Lefebvre, with my assistance, fed a radio frequency of 160.13 Mhz by means of a frequency-to-voltage transducer. The frequency fed into the transducer was changed by a fixed amount, and the change in voltage was noted. The hz / volt constant obtained was 253.9 hz / mvolt.

Nicolas then locked the east-west cryo-cavities so that the beat signal was approximately 0 (zero) volts.  See the plot, below:

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We then sent a 3.16 second pulse (approximated) of a 3.6 mWatt, 532 nm laser, (green) onto the surface of a mirror that reflects in the infrared, but absorbs in visible wavelengths. (Note top graph) I manufactured the electrical power interface for the laser by modifying the casing of a BIC disposable pen. The mirror was situated at the aperture of a silicon spacer.

The goal of the experiment was to determine the absorbance of the silicon mirror of the 532 nm laser.

Assuming we know the quantity of energy pulsed into the mirror:

3.6 mWatt * 3.16 s = .011376 joules,

the change in length of the cavity was determined by: Change in f/f1550nm = change in L/ L0

The change in volts (.025) gave us a change in f of 6.3475 x 103 hz.

With L0 having a value of 10 cm, that means change in L was 3.2794 x 10-10cm.

The specific heat capacity of Si is 700j/k*kg.

Coefficient of linear expansion for Si = 2.6 x 10 -6/K.

To calculate increase in temperature we need to obtain the change in L.

If 2.6 x 10 -6/C  x 10 cm x DT = DL = 3.2794 x 10-10 cm, then DT must be 1.2613 x 10 -5 K.

If DT = 1.2613 x 10-5 K , then Q must equal .41 kg x 700 j/kg Kx1.2613 x 10-5 K = 3.6 x10-3 j,

Q/Epulse=  3.6 x 10-3j / 1.1376 x 10-2j = .316 absorbance.

In other words, the silicon cavity reflected about 68% of the light that struck it.

What have I come away with from this experience?

What struck me first and foremost during this summer internship in the Cryo-Lab was the importance of future knowledge workers of having certain key skills. Among them:

Proficiency in Language

Proficiency in Math

Proficiency in Science

Proficiency in Coding

I will share my insights with my local school district and I intend to capitalize on the connections I made during my experience at Caltech.

Acknowledgements:

I would like to thank the Duarte Unified School District for giving me a leave of absence, Rana Adhikari for, yet again, finding space for me in spite of my general ineptitude regarding General Relativity, Spyridon Michalakis (Spiros) for inviting me back and letting me participate in cutting-edge science, and most of all I would like to thank Nicolas Smith-Lefebvre (softball savant),  and David Yeaton-Massey (D-Mass), for their patience, generosity, and mentoring.

 

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