# Breaking up the band structure

Note from the editor: During the Summer of 2019, a group of thirteen undergraduate students from Caltech and universities around the world, spent 10 weeks on campus performing research in experimental quantum physics. Below, Aiden Cullo, a student from Binghampton University in New York, shares his experience working in Professor Yeh’s lab. The program, termed QuantumSURF, will run again during the Summer of 2020.

This summer, I worked in Nai-Chang Yeh’s experimental condensed matter lab. The aim of my project was to observe the effects of a magnetic field on our topological insulator (TI) sample, ${(BiSb)}_2{Te}_3$. The motivation behind this project was to examine more closely the transformation between a topological insulator and a state exhibiting the anomalous hall effect (AHE).

Both states of matter have garnered a good deal of interest in condensed matter research because of their interesting transport properties, among other things. TIs have gained popularity due to their applications in electronics (spintronics), superconductivity, and quantum computation. TIs are peculiar in that they simultaneously have insulating bulk states and conducting surface state. Due to time-reversal symmetry (TRS) and spin-momentum locking, these surface states have a very symmetric hourglass-like gapless energy band structure (Dirac cone).

The focus of our particular study was the effects of “c-plane” magnetization of our TI’s surface state. Theory predicts TRS and spin-momentum locking will be broken, resulting in a gapped spectrum with a single connection between the valence and conduction bands. This gapping has been theorized and shown experimentally in Chromium (Cr)-doped ${(BiSb)}_2{Te}_3$ and numerous other TIs with similar make-up.

In 2014, Nai-Chang Yeh’s group showed that Cr-doped ${Bi}_2{Se}_3$ exhibit this gap opening due to the surface state of ${Bi}_2{Se}_3$ interacting via the proximity effect with a ferromagnet. Our contention is that a similar material, Cr-doped ${(BiSb)}_2{Te}_3$, exhibits a similar effect, but more homogeneously because of reduced structural strain between atoms. Specifically, at temperatures below the Curie temperature (Tc), we expect to see a gap in the energy band and an overall increase in the gap magnitude. In short, the main goal of my summer project was to observe the gapping of our TI’s energy band.

Overall, my summer project entailed a combination of reading papers/textbooks and hands-on experimental work. It was difficult to understand fully the theory behind my project in such a short amount of time, but even with a cursory knowledge of topological insulators, I was able to provide a meaningful analysis/interpretation of our data.

Additionally, my experiment relied heavily on external factors such as our supplier for liquid helium, argon gas, etc. As a result, our progress was slowed if an order was delayed or not placed far enough in advance. Most of the issues we encountered were not related to the abstract theory of the materials/machinery, but rather problems with less complex mechanisms such as wiring, insulation, and temperature regulation.

While I expected to spend a good deal of time troubleshooting, I severely underestimated the amount of time that would be spent dealing with quotidian problems such as configuring software or etching STM tips. Working on a machine as powerful as an STM was frustrating at times, but also very rewarding as eventually we were able to collect a large amount of data on our samples.

An important (and extremely difficult) part of our analysis of STM data was whether patterns/features in our data set were artifacts or genuine phenomena, or a combination. I was fortunate enough to be surrounded by other researchers that helped me sift through the volumes of data and identify traits of our samples. Reflecting on my SURF, I believe it was a positive experience as it not only taught me a great deal about research, but also, more importantly, closely mimicked the experience of graduate school.

# “A theorist I can actually talk with”

Haunted mansions have ghosts, football teams have mascots, and labs have in-house theorists. I found myself posing as a lab’s theorist at Caltech. The gig began when Oskar Painter, a Caltech experimentalist, emailed that he’d read my first paper about quantum chaos. Would I discuss the paper with the group?

Oskar’s lab was building superconducting qubits, tiny circuits in which charge can flow forever. The lab aimed to control scores of qubits, to develop a quantum many-body system. Entanglement—strong correlations that quantum systems can sustain and everyday systems can’t—would spread throughout the qubits. The system could realize phases of matter—like many-particle quantum chaos—off-limits to most materials.

How could Oskar’s lab characterize the entanglement, the entanglement’s spread, and the phases? Expert readers will suggest measuring an entropy, a gauge of how much information this part of the system holds about that part. But experimentalists have had trouble measuring entropies. Besides, one measurement can’t capture many-body entanglement; such entanglement involves too many intricacies. Oskar was searching for arrows to add to his lab’s measurement quiver.

In-house theorist?

I’d proposed a protocol for measuring a characterization of many-body entanglement, quantum chaos, and thermalization—a property called “the out-of-time-ordered correlator.” The protocol appealed to Oskar. But practicalities limit quantum many-body experiments: The more qubits your system contains, the more the system can contact its environment, like stray particles. The stronger the interactions, the more the environment entangles with the qubits, and the less the qubits entangle with each other. Quantum information leaks from the qubits into their surroundings; what happens in Vegas doesn’t stay in Vegas. Would imperfections mar my protocol?

I didn’t know. But I knew someone who could help us find out.

Justin Dressel works at Chapman University as a physics professor. He’s received the highest praise that I’ve heard any experimentalist give a theorist: “He’s a theorist I can actually talk to.” With other collaborators, Justin and I simplified my scheme for measuring out-of-time-ordered correlators. Justin knew what superconducting-qubit experimentalists could achieve, and he’d been helping them reach for more.

How about, I asked Justin, we simulate our protocol on a computer? We’d code up virtual superconducting qubits, program in interactions with the environment, run our measurement scheme, and assess the results’ noisiness. Justin had the tools to simulate the qubits, but he lacked the time.

Know any postdocs or students who’d take an interest? I asked.

Chapman University’s former science center. Don’t you wish you spent winters in California?

José Raúl González Alonso has a smile like a welcome sign and a coffee cup glued to one hand. He was moving to Chapman University to work as a Grand Challenges Postdoctoral Fellow. José had built simulations, and he jumped at the chance to study quantum chaos.

José confirmed Oskar’s fear and other simulators’ findings: The environment threatens measurements of the out-of-time-ordered correlator. Suppose that you measure this correlator at each of many instants, you plot the correlator against time, and you see the correlator drop. If you’ve isolated your qubits from their environment, we can expect them to carry many-body entanglement. Golden. But the correlator can drop if, instead, the environment is harassing your qubits. You can misdiagnose leaking as many-body entanglement.

Our triumvirate identified a solution. Justin and I had discovered another characterization of quantum chaos and many-body entanglement: a quasiprobability, a quantum generalization of a probability.

The quasiprobability contains more information about the entanglement than the out-of-time-ordered-correlator does. José simulated measurements of the quasiprobability. The quasiprobability, he found, behaves one way when the qubits entangle independently of their environment and behaves another way when the qubits leak. You can measure the quasiprobability to decide whether to trust your out-of-time-ordered-correlator measurement or to isolate your qubits better. The quasiprobability enables us to avoid false positives.

Physical Review Letters published our paper last month. Working with Justin and José deepened my appetite for translating between the abstract and the concrete, for proving abstractions as a theorist’s theorist and realizing them experimentally as a lab’s theorist. Maybe, someday, I’ll earn the tag “a theorist I can actually talk with” from an experimentalist. For now, at least I serve better than a football-team mascot.

# Humans can intuit quantum physics.

One evening this January, audience members packed into a lecture hall in MIT’s physics building. Undergraduates, members of the public, faculty members, and other scholars came to watch a film premiere and a panel discussion. NOVA had produced the film, “Einstein’s Quantum Riddle,” which stars entanglement. Entanglement is a relationship between quantum systems such as electrons. Measuring two entangled electrons yields two outcomes, analogous to the numbers that face upward after you roll two dice. The quantum measurements’ outcomes can exhibit correlations stronger than any measurements of any classical, or nonquantum, systems can. Which die faces point upward can share only so much correlation, even if the dice hit each other.

Dice feature in the film’s explanations of entanglement. So does a variation on the shell game, in which one hides a ball under one of three cups, shuffles the cups, and challenges viewers to guess which cup is hiding the ball. The film derives its drama from the Cosmic Bell test. Bell tests are experiments crafted to show that classical physics can’t describe entanglement. Scientists recently enhanced Bell tests using light from quasars—ancient, bright, faraway galaxies. Mix astrophysics with quantum physics, and an edgy, pulsing soundtrack follows.

The Cosmic Bell test grew from a proposal by physicists at MIT and the University of Chicago. The coauthors include David Kaiser, a historian of science and a physicist on MIT’s faculty. Dave co-organized the premiere and the panel discussion that followed. The panel featured Dave; Paola Cappellaro, an MIT quantum experimentalist; Alan Guth, an MIT cosmologist who contributed to the Bell test; Calvin Leung, an MIT PhD student who contributed; Chris Schmidt, the film’s producer; and me. Brindha Muniappan, the Director of Education and Public Programs at the MIT Museum, moderated the discussion.

think that the other panelists were laughing with me.

Brindha asked what challenges I face when explaining quantum physics, such as on this blog. Quantum theory wears the labels “weird,” “counterintuitive,” and “bizarre” in journalism, interviews, blogs, and films. But the thorn in my communicational side reflects quantum “weirdness” less than it reflects humanity’s self-limitation: Many people believe that we can’t grasp quantum physics. They shut down before asking me to explain.

Examples include a friend and Quantum Frontiers follower who asks, year after year, for books about quantum physics. I suggest literature—much by Dave Kaiser—he reads some, and we discuss his impressions. He’s learning, he harbors enough curiosity to have maintained this routine for years, and he has technical experience as a programmer. But he’s demurred, several times, along the lines of “But…I don’t know. I don’t think I’ll ever understand it. Humans can’t understand quantum physics, can we? It’s too weird.”

Quantum physics defies many expectations sourced from classical physics. Classical physics governs how basketballs arch, how paint dries, how sunlight slants through your window, and other everyday experiences. Yet we can gain intuition about quantum physics. If we couldn’t, how could we solve problems and accomplish research? Physicists often begin solving problems by trying to guess the answer from intuition. We reason our way toward a guess by stripping away complications, constructing toy models, and telling stories. We tell stories about particles hopping from site to site on lattices, particles trapped in wells, and arrows flipping upward and downward. These stories don’t capture all of quantum physics, but they capture the essentials. After grasping the essentials, we translate them into math, check how far our guesses lie from truth, and correct our understanding. Intuition about quantum physics forms the compass that guides problem solving.

Growing able to construct, use, and mathematize such stories requires work. You won’t come to understand quantum theory by watching NOVA films, though films can prime you for study. You can gain a facility with quantum theory through classes, problem sets, testing, research, seminars, and further processing. You might not have the time or inclination to. Even if you have, you might not come to understand why quantum theory describes our universe: Science can’t necessarily answer all “why” questions. But you can grasp what quantum theory implies about our universe.

People grasp physics arguably more exotic than quantum theory, without exciting the disbelief excited by a grasp of quantum theory. Consider the Voyager spacecraft launched in 1977. Voyager has survived solar winds and -452º F weather, imaged planets, and entered interstellar space. Classical physics—the physics of how basketballs arch—describes much of Voyager’s experience. But even if you’ve shot baskets, how much intuition do you have about interstellar space? I know physicists who claim to have more intuition about quantum physics than about much classical. When astrophysicists discuss Voyager and interstellar space, moreover, listeners don’t fret that comprehension lies beyond them. No one need fret when quantum physicists discuss the electrons in us.

Fretting might not occur to future generations: Outreach teams are introducing kids to quantum physics through games and videos. Caltech’s Institute for Quantum Information and Matter has partnered with Google to produce QCraft, a quantum variation on Minecraft, and with the University of Southern California on quantum chess. In 2017, the American Physical Society’s largest annual conference featured a session called “Gamification and other Novel Approaches in Quantum Physics Outreach.” Such outreach exposes kids to quantum terminology and concepts early. Quantum theory becomes a playground to explore, rather than a source of intimidation. Players will grow up primed to think about quantum-mechanics courses not “Will my grade-point average survive this semester?” but “Ah, so this is the math under the hood of entanglement.”

Sociology restricts people to thinking quantum physics weird. But quantum theory defies classical expectations less than it could. Measurement outcomes could share correlations stronger than the correlations sourced by entanglement. How strong could the correlations grow? How else could physics depart farther from classical physics than quantum physics does? Imagine the worlds governed by all possible types of physics, called “generalized probabilistic theories” (GPTs). GPTs form a landscape in which quantum theory constitutes an island, on which classical physics constitutes a hill. Compared with the landscape’s outskirts, our quantum world looks tame.

GPTs fall under the research category of quantum foundations. Quantum foundations concerns why the math that describes quantum systems describes quantum systems, reformulations of quantum theory, how quantum theory differs from classical mechanics, how quantum theory could deviate but doesn’t, and what happens during measurements of quantum systems. Though questions about quantum foundations remain, they don’t block us from intuiting about quantum theory. A stable owner can sense when a horse has colic despite lacking a veterinary degree.

Moreover, quantum-foundations research has advanced over the past few decades. Collaborations and tools have helped: Theorists have been partnering with experimentalists, such as on the Cosmic Bell test and on studies of measurement. Information theory has engendered mathematical tools for quantifying entanglement and other quantum phenomena. Information theory has also firmed up an approach called “operationalism.” Operationalists emphasize preparation procedures, evolutions, and measurements. Focusing on actions and data concretizes arguments and facilitates comparisons with experiments. As quantum-foundations research has advanced, so have quantum information theory, quantum experiments, quantum technologies, and interdisciplinary cross-pollination. Twentieth-century quantum physicists didn’t imagine the community, perspectives, and knowledge that we’ve accrued. So don’t adopt 20th-century pessimism about understanding quantum theory. Einstein grasped much, but today’s scientific community grasps more. Richard Feynman said, “I think I can safely say that nobody understands quantum mechanics.” Feynman helped spur the quantum-information revolution; he died before its adolescence. Besides, Feynman understood plenty about quantum theory. Intuition jumps off the pages of his lecture notes and speeches.

Landscape beyond quantum theory

I’ve swum in oceans and lakes, studied how the moon generates tides, and canoed. But piloting a steamboat along the Mississippi would baffle me. I could learn, given time, instruction, and practice; so can you learn quantum theory. Don’t let “weirdness,” “bizarreness,” or “counterintuitiveness” intimidate you. Humans can intuit quantum physics.

# Doctrine of the (measurement) mean

Don’t invite me to dinner the night before an academic year begins.

You’ll find me in an armchair or sitting on my bed, laptop on my lap, journaling. I initiated the tradition the night before beginning college. I take stock of the past year, my present state, and hopes for the coming year.

Much of the exercise fosters what my high-school physics teacher called “an attitude of gratitude”: I reflect on cities I’ve visited, projects firing me up, family events attended, and subfields sampled. Other paragraphs, I want off my chest: Have I pushed this collaborator too hard or that project too little? Miscommunicated or misunderstood? Strayed too far into heuristics or into mathematical formalisms?

If only the “too much” errors, I end up thinking, could cancel the “too little.”

In one quantum-information context, they can.

Imagine that you’ve fabricated the material that will topple steel and graphene; let’s call it a supermetatopoconsulator. How, you wonder, do charge, energy, and particles move through this material? You’ll learn by measuring correlators.

A correlator signals how much, if you poke this piece here, that piece there responds. At least, a two-point correlator does: $\langle A(0) B(\tau) \rangle$. $A(0)$ represents the poke, which occurs at time $t = 0$. $B(\tau)$ represents the observable measured there at $t = \tau$. The $\langle . \rangle$ encapsulates which state $\rho$ the system started in.

Condensed-matter, quantum-optics, and particle experimentalists have measured two-point correlators for years. But consider the three-point correlator $\langle A(0) B(\tau) C (\tau' ) \rangle$, or a $k$-point $\langle \underbrace{ A(0) \ldots M (\tau^{(k)}) }_k \rangle$, for any $k \geq 2$. Higher-point correlators relate more-complicated relationships amongst events. Four-pointcorrelators associated with multiple times signal quantum chaos and information scrambling. Quantum information scrambles upon spreading across a system through many-body entanglement. Could you measure arbitrary-point, arbitrary-time correlators?

Supermetatopoconsulator (artist’s conception)

Yes, collaborators and I have written, using weak measurements. Weak measurements barely disturb the system being measured. But they extract little information about the measured system. So, to measure a correlator, you’d have to perform many trials. Moreover, your postdocs and students might have little experience with weak measurements. They might not want to learn the techniques required, to recalibrate their detectors, etc. Could you measure these correlators easily?

Yes, if the material consists of qubits,2 according to a paper I published with Justin Dressel, José Raúl González Alsonso, and Mordecai Waegell this summer. You could build such a system from, e.g., superconducting circuits, trapped ions, or quantum dots.

You can measure $\langle \underbrace{ A(0) B (\tau') C (\tau'') \ldots M (\tau^{(k)}) }_k \rangle$, we show, by measuring $A$ at $t = 0$, waiting until $t = \tau'$, measuring $B$, and so on until measuring $M$ at $t = \tau^{(k)}$. The $t$-values needn’t increase sequentially: $\tau''$ could be less than $\tau'$, for instance. You’d have to effectively reverse the flow of time experienced by the qubits. Experimentalists can do so by, for example, flipping magnetic fields upside-down.

Each measurement requires an ancilla, or helper qubit. The ancilla acts as a detector that records the measurement’s outcome. Suppose that $A$ is an observable of qubit #1 of the system of interest. You bring an ancilla to qubit 1, entangle the qubits (force them to interact), and look at the ancilla. (Experts: You perform a controlled rotation on the ancilla, conditioning on the system qubit.)

Each trial yields $k$ measurement outcomes. They form a sequence $S$, such as $(1, 1, 1, -1, -1, \ldots)$. You should compute a number $\alpha$, according to a formula we provide, from each measurement outcome and from the measurement’s settings. These numbers form a new sequence $S' = \mathbf{(} \alpha_S(1), \alpha_S(1), \ldots \mathbf{)}$. Why bother? So that you can force errors to cancel.

Multiply the $\alpha$’s together, $\alpha_S(1) \times \alpha_S(1) \times \ldots$, and average the product over the possible sequences $S$. This average equals the correlator $\langle \underbrace{ A(0) \ldots M (\tau^{(k)}) }_k \rangle$. Congratulations; you’ve characterized transport in your supermetatopoconsulator.

When measuring, you can couple the ancillas to the system weakly or strongly, disturbing the system a little or a lot. Wouldn’t strong measurements perturb the state $\rho$ whose properties you hope to measure? Wouldn’t the perturbations by measurements one through $\ell$ throw off measurement $\ell + 1$?

Yes. But the errors introduced by those perturbations cancel in the average. The reason stems from how we construct $\alpha$’s: Our formula makes some products positive and some negative. The positive and negative terms sum to zero.

The cancellation offers hope for my journal assessment: Errors can come out in the wash. Not of their own accord, not without forethought. But errors can cancel out in the wash—if you soap your $\alpha$’s with care.

1and six-point, eight-point, etc.

2Rather, each measured observable must square to the identity, e.g., $A^2 = 1$. Qubit Pauli operators satisfy this requirement.

With apologies to Aristotle.

# I get knocked down…

“You’ll have to have a thick skin.”

Marcelo Gleiser, a college mentor of mine, emailed the warning. I’d sent a list of physics PhD programs and requested advice about which to attend. Marcelo’s and my department had fostered encouragement and consideration.

Suit up, Marcelo was saying.

Criticism fuels science, as Oxford physicist David Deutsch has written. We have choices about how we criticize. Some criticism styles reflect consideration for the criticized work’s creator. Tufts University philosopher Daniel Dennett has devised guidelines for “criticizing with kindness”:1

1. You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way.

2. You should list any points of agreement (especially if they are not matters of general or widespread agreement).

3. You should mention anything you have learned from your target.

4. Only then are you permitted to say so much as a word of rebuttal or criticism.

Scientists skip to step four often—when refereeing papers submitted to journals, when posing questions during seminars, when emailing collaborators, when colleagues sketch ideas at a blackboard. Why? Listening and criticizing require time, thought, and effort—three of a scientist’s most valuable resources. Should any scientist spend those resources on an idea of mine, s/he deserves my gratitude. Spending empathy atop time, thought, and effort can feel supererogatory. Nor do all scientists prioritize empathy and kindness. Others of us prioritize empathy but—as I have over the past five years—grown so used to its latency, I forget to demonstrate it.

Doing science requires facing not only criticism, but also “That doesn’t make sense,” “Who cares?” “Of course not,” and other morale boosters.

Doing science requires resilience.

So do measurements of quantum information (QI) scrambling. Scrambling is a subtle, late, quantum stage of equilibration2 in many-body systems. Example systems include chains of spins,3 such as in ultracold atoms, that interact with each other strongly. Exotic examples include black holes in anti-de Sitter space.4

Imagine whacking one side of a chain of interacting spins. Information about the whack will disseminate throughout the chain via entanglement.5 After a long interval (the scrambling time, $t_*$), spins across the systems will share many-body entanglement. No measurement of any few, close-together spins can disclose much about the whack. Information will have scrambled across the system.

QI scrambling has the subtlety of an assassin treading a Persian carpet at midnight. Can we observe scrambling?

A Stanford team proposed a scheme for detecting scrambling using interferometry.6 Justin Dressel, Brian Swingle, and I proposed a scheme based on weak measurements, which refrain from disturbing the measured system much. Other teams have proposed alternatives.

Many schemes rely on effective time reversal: The experimentalist must perform the quantum analog of inverting particles’ momenta. One must negate the Hamiltonian $\hat{H}$, the observable that governs how the system evolves: $\hat{H} \mapsto - \hat{H}$.

At least, the experimentalist must try. The experimentalist will likely map $\hat{H}$ to $- \hat{H} + \varepsilon$. The small error $\varepsilon$ could wreak havoc: QI scrambling relates to chaos, exemplified by the butterfly effect. Tiny perturbations, such as the flap of a butterfly’s wings, can snowball in chaotic systems, as by generating tornadoes. Will the $\varepsilon$ snowball, obscuring observations of scrambling?

It needn’t, Brian and I wrote in a recent paper. You can divide out much of the error until $t_*$.

You can detect scrambling by measuring an out-of-time-ordered correlator (OTOC), an object I’ve effused about elsewhere. Let’s denote the time-$t$ correlator by $F(t)$. You can infer an approximation $\tilde{F}(t)$ to $F(t)$ upon implementing an $\varepsilon$-ridden interferometry or weak-measurement protocol. Remove some steps from that protocol, Brian and I say. Infer a simpler, easier-to-measure object $\tilde{F}_{\rm simple}(t)$. Divide the two measurement outcomes to approximate the OTOC:

$F(t) \approx \frac{ \tilde{F}(t) }{ \tilde{F}_{\rm simple}(t) }$.

OTOC measurements exhibit resilience to error.

Physicists need resilience. Brian criticizes with such grace, he could serve as the poster child for Daniel Dennett’s guidelines. But not every scientist could. How can we withstand kindness-lite criticism?

By drawing confidence from what we’ve achieved, with help from mentors like Marcelo. I couldn’t tell what about me—if anything—could serve as a rock on which to plant a foot, as an undergrad. Mentors identified what I had too little experience to appreciate. You question what you don’t understand, they said. You assimilate perspectives from textbooks, lectures, practice problems, and past experiences. You scrutinize details while keeping an eye on the big picture. So don’t let so-and-so intimidate you.

I still lack my mentors’ experience, but I’ve imbibed a drop of their insight. I savor calculations that I nail, congratulate myself upon nullifying referees’ concerns, and celebrate the theorems I prove.

I’ve also created an email folder entitled “Nice messages.” In go “I loved your new paper; combining those topics was creative,” “Well done on the seminar; I’m now thinking of exploring that field,” and other rarities. The folder affords an umbrella when physics clouds gather.

Finally, I try to express appreciation of others’ work.7 Science thrives on criticism, but scientists do science. And scientists are human—undergrads, postdocs, senior researchers, and everyone else.

Doing science—and attempting to negate Hamiltonians—we get knocked down. But we can get up again.

Around the time Brian and I released “Resilience” two other groups proposed related renormalizations. Check out their schemes here and here.

1Thanks to Sean Carroll for alerting me to this gem of Dennett’s.

2A system equilibrates as its large-scale properties, like energy, flatline.

3Angular-momentum-like quantum properties

4Certain space-times different from ours

5Correlations, shareable by quantum systems, stronger than any achievable by classical systems

6The cancellation (as by a crest of one wave and a trough of another) of components of a quantum state, or the addition of components (as two waves’ crests)

7Appreciation of specific qualities. “Nice job” can reflect a speaker’s belief but often reflects a desire to buoy a receiver whose work has few merits to elaborate on. I applaud that desire and recommend reinvesting it. “Nice job” carries little content, which evaporates under repetition. Specificity provides content: “Your idea is alluringly simple but could reverberate across multiple fields” has gristle.

# The Curious Behavior of Topological Insulators

IQIM hosts a Summer Research Institute that invites high school Physics teachers to work directly with staff, students, and researchers in the lab.  Last summer I worked with Marcus Teague, a highly intelligent and very patient Caltech Staff Scientist in the Yeh Group, to help set up an experiment for studying exotic material samples under circularly polarized light.  I had researched, ordered, and assembled parts for the optics and vacuum chamber.  As I returned to Caltech this summer, I was eager to learn how the Yeh Group had proceeded with the study.

Yeh group (2017): I am the one on the front-left of the picture, next to Dr. Yeh and in front of Kyle Chen. Benjamin Fackrell, another physics teacher interning at the Yeh lab, is all the way to the right.

The optics equipment I had researched, ordered, and helped to set up last summer is being used currently to study topological insulator (TI) samples that Kyle Chien-Chang Chen, a doctoral candidate, has worked on in the Yeh Lab.  Yes, a high school Physics teacher played a small role in their real research! It is exciting and humbling to have a connection to real-time research.

Quartz quarter-wave plates are important elements in many experiments involving light. They convert linearly polarized light to circularly polarized light.

Kyle receives a variety of TI samples from UCLA; the current sample up for review is Bismuth Antimony Telluride $\mathrm{(BiSb)}_2\mathrm{Te}_3$.  Depending on the particular sample and the type of testing, Kyle has a variety of procedures to prep the samples for study.  And this summer, Kyle has help from visiting Canadian student Adrian Llanos. Below are figures of some of the monolayer and bilayer structures for topological insulators studied in the lab.

Pictures of samples from UCLA

Under normal conditions, a topological insulator (TI) is only conductive on the surface. The center of a TI sample is an insulator. But when the surface states open an energy gap, the surface of the TI becomes insulating. The energy gap is the amount of energy necessary to remove an electron from the top valence band to become free to move about.  This gap is the result of the interaction between the conduction band and valence band surface states from the opposing surfaces of a thin film. The resistance of the conducting surface actually increases. The Yeh group is hoping that the circularly polarized light can help align the spin of the Chromium electrons, part of the bilayer of the TI.  At the same time, light has other effects, like photo-doping, which excites more electrons into the conduction bands and thus reduces the resistance. The conductivity of the surface of the TI changes as the preferentially chosen spin up or spin down is manipulated by the circularly polarized light or by the changing magnetic field.

A physical property measurement system.

This interesting experiment on TI samples is taking place within a device called a Physical Property Measurement System (PPMS).  The PPMS is able to house the TI sample and the optics equipment to generate circularly polarized light, while allowing the researchers to vary the temperature and magnetic field.  The Yeh Group is able to artificially turn up the magnetic field or the circularly polarized light in order to control the resistance and current signal within the sample.  The properties of surface conductivity are studied up to 8 Tesla (over one-hundred thousand times the Earth’s magnetic field), and from room temperature (just under 300 Kelvin) to just below 2 Kelvin (colder than outer space).

Right-Hand-Rule used to determine the direction of the magnetic (Lorentz) force.

In the presence of a magnetic field, when a current is applied to a conductor, the electrons will experience a force at a right angle to the magnetic field, following the right-hand rule (or the Physics gang sign, as we affectionately call it in my classroom).  This causes the electrons to curve perpendicular to their original path and perpendicular to the magnetic field. The build up of electrons on one end of the conductor creates a potential difference. This potential difference perpendicular to the original current is known as the ordinary Hall Effect.  The ratio of the induced voltage to the applied current is known as the Hall Resistance.

Under very low temperatures, the Quantum Hall Effect is observed. As the magnetic field is changed, the Hall Voltage increases in set quantum amounts, as opposed to gradually. Likewise, the Hall Resistance is quantized.  It is a such an interesting phenomenon!

For a transport measurement of the TI samples, Kyle usually uses a Hall Bar Geometry in order to measure the Hall Effect accurately. Since the sample is sufficiently large, he can simply solder it for measurement.

Transport Measurements of TI Samples follow the same setup as Quantum Hall measurements on graphene: Current runs through electrodes attached to the North/South ends of the sample, while electron flow is measured longitudinally, as well as along the East/West ends (Hall conductance).

What is really curious is that the Bismuth Antimony Telluride samples are exhibiting the Hall Effect even when no external magnetic field is applied!  When the sample is measured, there is a Hall Resistance despite no external magnetic field. Hence the sample itself must be magnetic.  This phenomenon is called the Anomalous Hall Effect.

According to Kyle, there is no fancy way to measure the magnetization directly; it is only a matter of measuring a sample’s Hall Resistance. The Hall Resistance should be zero when there is no Anomalous Hall Effect, and when there is ferromagnetism (spins want to align in the direction of their neighbors), you see a non-zero value.  What is really interesting is that they assume ferromagnetism would break the time-reversal symmetry and thus open a gap at the surface states.  A very strange behavior that is also observed is that the longitudinal resistance increases gradually.

Running PPMS

Typically the quantum Hall Resistance increases in quantum increments.  Even if the surface gap is open, the sample is not insulating because the gap is small (<0.3 eV); hence, under these conditions this TI is behaving much more like a semiconductor!

Next, the group will examine these samples using the Scanning Tunneling Microscope (STM).  The STM will be able to provide local topological information by examining 1 micron by 1 micron areas.  In comparison, the PPMS research with these samples is telling the story of the global behavior of the sample.  The combination of information from the PPMS and STM research will provide a more holistic story of the behavior of these unique samples.

I am thrilled to see how the group has used what we started with last summer to find interesting new results.  I am fascinated to see what they learn in the coming months with the different samples and STM testing. And I am quite excited to share these applications with my students in the upcoming new school year.  Another summer packed with learning!

# The light show

A strontium magneto-optical trap.

How did a quantum physics experiment end up looking like a night club? Due to a fortunate coincidence of nature, my lab mates and I at Endres Lab get to use three primary colors of laser light – red, blue, and green – to trap strontium atoms.  Let’s take a closer look at the physics behind this visually entrancing combination.

The spectrum

The electronic spectrum of strontium near the ground state.

The trick to research is finding a problem that is challenging enough to be interesting, but accessible enough to not be impossible.  Strontium embodies this maxim in its electronic spectrum.  While at first glance it may seem daunting, it’s not too bad once you get to know each other.  Two valence electrons divide the spectrum into a spin-singlet sector and a spin-triplet sector – a designation that roughly defines whether the electron spins point in the opposite or in the same direction.  Certain transitions between these sectors are extremely precisely defined, and currently offer the best clock standards in the world.  Although navigating this spectrum requires more lasers, it offers opportunities for quantum physics that singly-valent spectra do not.  In the end, the experimental complexity is still very much manageable, and produces some great visuals to boot.  Here are some of the lasers we use in our lab:

The blue

At the center of the .gif above is a pulsating cloud of strontium atoms, shining brightly blue.  This is a magneto-optical trap, produced chiefly by strontium’s blue transition at 461nm.

461nm blue laser light being routed through various paths.

The blue transition is exceptionally strong, scattering about 100 million photons per atom per second.  It is the transition we use to slow strontium atoms from a hot thermal beam traveling at hundreds of meters per second down to a cold cloud at about 1 milliKelvin.  In less than a second, this procedure gives us a couple hundred million atoms to work with.  As the experiment repeats, we get to watch this cloud pulse in and out of existence.

The red(s)

689nm red light.  Bonus: Fabry-Perot interference fringes on my camera!

While the blue transition is a strong workhorse, the red transition at 689nm trades off strength for precision.  It couples strontium’s spin-singlet ground state to an excited spin-triplet state, a much weaker but more precisely defined transition.  While it does not scatter as fast as the blue (only about 23,000 photons per atom per second), it allows us to cool our atoms to much colder temperatures, on the order of 1 microKelvin.

In addition to our red laser at 689nm, we have two other reds at 679nm and 707nm.  These are necessary to essentially plug “holes” in the blue transition, which eventually cause an atom to fall into long-lived states other than the ground state.  It is generally true that the more complicated an atomic spectrum gets, the more “holes” there are to plug, and this is many times the reason why certain atoms and molecules are harder to trap than others.

The green

After we have established a cold magneto-optical trap, it is time to pick out individual atoms from this cloud and load them into very tightly focused optical traps that we call tweezers.  Here, our green laser comes into play.  This laser’s wavelength is far away from any particular transition, as we do not want it to scatter any photons at all.  However, its large intensity creates a conservative trapping potential for the atom, allowing us to hold onto it and even move it around.  Furthermore, its wavelength is what we call “magic”, which means it is chosen such that the ground and excited state experience the same trapping potential.

The quite powerful green laser.  So powerful that you can see the beam in the air, like in the movies.

The invisible

Yet to be implemented are two more lasers slightly off the visible spectrum at both the ultraviolet and infrared sides.  Our ultraviolet laser will be crucial to elevating our experiment from single-body to many-body quantum physics, as it will allow us to drive our atoms to very highly excited Rydberg states which interact with long range.  Our infrared laser will allow us to trap atoms in the extremely precise clock state under “magic” conditions.

The combination of strontium’s various optical pathways allows for a lot of new tricks beyond just cooling and trapping.  Having Rydberg states alongside narrow-line transitions, for example, has yet unexplored potential for quantum simulation.  It is a playground that is very exciting without being utterly overwhelming.  Stay tuned as we continue our exploration – maybe we’ll have a yellow laser next time too.