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.

Seesaw

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?

New material

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.

Success

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.

Balance 2

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.

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?

Carpet

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?

Snowball

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.

Arm 2

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

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.

7234_ZOQuartWavplatMount_1

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.

2016 0808 sample from UCLA

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.

PPMS

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

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.

hall_resitance_featured

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

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

Atoms 2

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

Sr level structure

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.

IMG_3379

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)

IMG_3380

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.

IMG_3369

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.

 

Gently yoking yin to yang

The architecture at the University of California, Berkeley mystified me. California Hall evokes a Spanish mission. The main library consists of white stone pillared by ionic columns. A sea-green building scintillates in the sunlight like a scarab. The buildings straddle the map of styles.

Architecture.001

So do Berkeley’s quantum scientists, information-theory users, and statistical mechanics.

The chemists rove from abstract quantum information (QI) theory to experiments. Physicists experiment with superconducting qubits, trapped ions, and numerical simulations. Computer scientists invent algorithms for quantum computers to perform.

Few activities light me up more than bouncing from quantum group to info-theory group to stat-mech group, hunting commonalities. I was honored to bounce from group to group at Berkeley this September.

What a trampoline Berkeley has.

The groups fan out across campus and science, but I found compatibility. Including a collaboration that illuminated quantum incompatibility.

Quantum incompatibility originated in studies by Werner Heisenberg. He and colleagues cofounded quantum mechanics during the early 20th century. Measuring one property of a quantum system, Heisenberg intuited, can affect another property.

The most famous example involves position and momentum. Say that I hand you an electron. The electron occupies some quantum state represented by | \Psi \rangle. Suppose that you measure the electron’s position. The measurement outputs one of many possible values x (unless | \Psi \rangle has an unusual form, the form a Dirac delta function).

But we can’t say that the electron occupies any particular point x = x_0 in space. Measurement devices have limited precision. You can measure the position only to within some error \varepsilon: x = x_0 \pm \varepsilon.

Suppose that, immediately afterward, you measure the electron’s momentum. This measurement, too, outputs one of many possible values. What probability q(p) dp does the measurement have of outputting some value p? We can calculate q(p) dp, knowing the mathematical form of | \Psi \rangle and knowing the values of x_0 and \varepsilon.

q(p) is a probability density, which you can think of as a set of probabilities. The density can vary with p. Suppose that q(p) varies little: The probabilities spread evenly across the possible p values. You have no idea which value your momentum measurement will output. Suppose, instead, that q(p) peaks sharply at some value p = p_0. You can likely predict the momentum measurement’s outcome.

The certainty about the momentum measurement trades off with the precision \varepsilon of the position measurement. The smaller the \varepsilon (the more precisely you measured the position), the greater the momentum’s unpredictability. We call position and momentum complementary, or incompatible.

You can’t measure incompatible properties, with high precision, simultaneously. Imagine trying to do so. Upon measuring the momentum, you ascribe a tiny range of momentum values p to the electron. If you measured the momentum again, an instant later, you could likely predict that measurement’s outcome: The second measurement’s q(p) would peak sharply (encode high predictability). But, in the first instant, you measure also the position. Hence, by the discussion above, q(p) would spread out widely. But we just concluded that q(p) would peak sharply. This contradiction illustrates that you can’t measure position and momentum, precisely, at the same time.

But you can simultaneously measure incompatible properties weakly. A weak measurement has an enormous \varepsilon. A weak position measurement barely spreads out q(p). If you want more details, ask a Quantum Frontiers regular; I’ve been harping on weak measurements for months.

Blame Berkeley for my harping this month. Irfan Siddiqi’s and Birgitta Whaley’s groups collaborated on weak measurements of incompatible observables. They tracked how the measured quantum state | \Psi (t) \rangle evolved in time (represented by t).

Irfan’s group manipulates superconducting qubits.1 The qubits sit in the physics building, a white-stone specimen stamped with an egg-and-dart motif. Across the street sit chemists, including members of Birgitta’s group. The experimental physicists and theoretical chemists teamed up to study a quantum lack of teaming up.

Phys. &amp; chem. bldgs

The experiment involved one superconducting qubit. The qubit has properties analogous to position and momentum: A ball, called the Bloch ball, represents the set of states that the qubit can occupy. Imagine an arrow pointing from the sphere’s center to any point in the ball. This Bloch vector represents the qubit’s state. Consider an arrow that points upward from the center to the surface. This arrow represents the qubit state | 0 \rangle. | 0 \rangle is the quantum analog of the possible value 0 of a bit, or unit of information. The analogous downward-pointing arrow represents the qubit state | 1 \rangle, analogous to 1.

Infinitely many axes intersect the sphere. Different axes represent different observables that Irfan’s group can measure. Nonparallel axes represent incompatible observables. For example, the x-axis represents an observable \sigma_x analogous to position. The y-axis represents an observable \sigma_y analogous to momentum.

Tug-of-war

Siddiqi lab, decorated with the trademark for the paper’s tug-of-war between incompatible observables. Photo credit: Leigh Martin, one of the paper’s leading authors.

Irfan’s group stuck their superconducting qubit in a cavity, or box. The cavity contained light that interacted with the qubit. The interactions transferred information from the qubit to the light: The light measured the qubit’s state. The experimentalists controlled the interactions, controlling the axes “along which” the light was measured. The experimentalists weakly measured along two axes simultaneously.

Suppose that the axes coincided—say, at the x-axis \hat{x}. The qubit would collapse to the state | \Psi \rangle = \frac{1}{ \sqrt{2} } ( | 0 \rangle + | 1 \rangle ), represented by the arrow that points along \hat{x} to the sphere’s surface, or to the state | \Psi \rangle = \frac{1}{ \sqrt{2} } ( | 0 \rangle - | 1 \rangle ), represented by the opposite arrow.

0 deg

(Projection of) the Bloch Ball after the measurement. The system can access the colored points. The lighter a point, the greater the late-time state’s weight on the point.

Let \hat{x}' denote an axis near \hat{x}—say, 18° away. Suppose that the group weakly measured along \hat{x} and \hat{x}'. The state would partially collapse. The system would access points in the region straddled by \hat{x} and \hat{x}', as well as points straddled by - \hat{x} and - \hat{x}'.

18 deg

Finally, suppose that the group weakly measured along \hat{x} and \hat{y}. These axes stand in for position and momentum. The state would, loosely speaking, swirl around the Bloch ball.

90 deg

The Berkeley experiment illuminates foundations of quantum theory. Incompatible observables, physics students learn, can’t be measured simultaneously. This experiment blasts our expectations, using weak measurements. But the experiment doesn’t just destroy. It rebuilds the blast zone, by showing how | \Psi (t) \rangle evolves.

“Position” and “momentum” can hang together. So can experimentalists and theorists, physicists and chemists. So, somehow, can the California mission and the ionic columns. Maybe I’ll understand the scarab building when we understand quantum theory.2

With thanks to Birgitta’s group, Irfan’s group, and the rest of Berkeley’s quantum/stat-mech/info-theory community for its hospitality. The Bloch-sphere figures come from http://www.nature.com/articles/nature19762.

1The qubit is the quantum analog of a bit. The bit is the basic unit of information. A bit can be in one of two possible states, which we can label as 0 and 1. Qubits can manifest in many physical systems, including superconducting circuits. Such circuits are tiny quantum circuits through which current can flow, without resistance, forever.

2Soda Hall dazzled but startled me.

Standing back at Stanford

T-shirt 1

This T-shirt came to mind last September. I was standing in front of a large silver-colored table littered with wires, cylinders, and tubes. Greg Bentsen was pointing at components and explaining their functions. He works in Monika Schleier-Smith’s lab, as a PhD student, at Stanford.

Monika’s group manipulates rubidium atoms. A few thousand atoms sit in one of the cylinders. That cylinder contains another cylinder, an optical cavity, that contains the atoms. A mirror caps each of the cavity’s ends. Light in the cavity bounces off the mirrors.

Light bounces off your bathroom mirror similarly. But we can describe your bathroom’s light accurately with Maxwellian electrodynamics, a theory developed during the 1800s. We describe the cavity’s light with quantum electrodynamics (QED). Hence we call the lab’s set-up cavity QED.

The light interacts with the atoms, entangling with them. The entanglement imprints information about the atoms on the light. Suppose that light escaped from the cavity. Greg and friends could measure the light, then infer about the atoms’ quantum state.

A little light leaks through the mirrors, though most light bounces off. From leaked light, you can infer about the ensemble of atoms. You can’t infer about individual atoms. For example, consider an atom’s electrons. Each electron has a quantum property called a spin. We sometimes imagine the spin as an arrow that points upward or downward. Together, the electrons’ spins form the atom’s joint spin. You can tell, from leaked light, whether one atom’s spin points upward. But you can’t tell which atom’s spin points upward. You can’t see the atoms for the ensemble.

Monika’s team can. They’ve cut a hole in their cylinder. Light escapes the cavity through the hole. The light from the hole’s left-hand edge carries information about the leftmost atom, and so on. The team develops a photograph of the line of atoms. Imagine holding a photograph of a line of people. You can point to one person, and say, “Aha! She’s the xkcd fan.” Similarly, Greg and friends can point to one atom in their photograph and say, “Aha! That atom has an upward-pointing spin.” Monika’s team is developing single-site imaging.

Solvay

Aha! She’s the xkcd fan.

Monika’s team plans to image atoms in such detail, they won’t need for light to leak through the mirrors. Light leakage creates problems, including by entangling the atoms with the world outside the cavity. Suppose you had to diminish the amount of light that leaks from a rubidium cavity. How should you proceed?

Tell the mirrors,

T-shirt 2

You should lengthen the cavity. Why? Imagine a photon, a particle of light, in the cavity. It zooms down the cavity’s length, hits a mirror, bounces off, retreats up the cavity’s length, hits the other mirror, and bounces off. The photon repeats this process until a mirror hit fails to generate a bounce. The mirror transmits the photon to the exterior; the photon leaks out. How can you reduce leaks? By preventing photons from hitting mirrors so often, by forcing the photons to zoom longer, by lengthening the cavity, by shifting the mirrors outward.

So Greg hinted, beside that silver-colored table in Monika’s lab. The hint struck a chord: I recognized the impulse to

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The impulse had led me to Stanford.

Weeks earlier, I’d written my first paper about quantum chaos and information scrambling. I’d sat and read and calculated and read and sat and emailed and written. I needed to stand up, leave my cavity, and image my work from other perspectives.

Stanford physicists had written quantum-chaos papers I admired. So I visited, presented about my work, and talked. Patrick Hayden introduced me to a result that might help me apply my result to another problem. His group helped me simplify a mathematical expression. Monika reflected that a measurement scheme I’d proposed sounded not unreasonable for cavity QED.

And Greg led me to recognize the principle behind my visit: Sometimes, you have to

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to move forward.

With gratitude to Greg, Monika, Patrick, and the rest of Monika’s and Patrick’s groups for their time, consideration, explanations, and feedback. With thanks to Patrick and Stanford’s Institute for Theoretical Physics for their hospitality.

The power of information

Sara Imari Walker studies ants. Her entomologist colleague Gabriele Valentini cultivates ant swarms. Gabriele coaxes a swarm from its nest, hides the nest, and offers two alternative nests. Gabriele observe the ants’ responses, then analyzes their data with Sara.

Sara doesn’t usually study ants. She trained in physics, information theory, and astrobiology. (Astrobiology is the study of life; life’s origins; and conditions amenable to life, on Earth and anywhere else life may exist.) Sara analyzes how information reaches, propagates through, and manifests in the swarm.

Some ants inspect one nest; some, the other. Few ants encounter both choices. Yet most of the ants choose simultaneously. (How does Gabriele know when an ant chooses? Decided ants carry other ants toward the chosen nest. Undecided ants don’t.)

Gabriele and Sara plotted each ant’s status (decided or undecided) at each instant. All the ants’ lines start in the “undecided” region, high up in the graph. Most lines drop to the “decided” region together. Physicists call such dramatic, large-scale changes in many-particle systems “phase transitions.” The swarm transitions from the “undecided” phase to the “decided,” as moisture transitions from vapor to downpour.

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Sara versus the ants

Look from afar, and you’ll see evidence of a hive mind: The lines clump and slump together. Look more closely, and you’ll find lags between ants’ decisions. Gabriele and Sara grouped the ants according to their behaviors. Sara explained the grouping at a workshop this spring.

The green lines, she said, are undecided ants.

My stomach dropped like Gabriele and Sara’s ant lines.

People call data “cold” and “hard.” Critics lambast scientists for not appealing to emotions. Politicians weave anecdotes into their numbers, to convince audiences to care.

But when Sara spoke, I looked at her green lines and thought, “That’s me.”

I’ve blogged about my indecisiveness. Postdoc Ning Bao and I formulated a quantum voting scheme in which voters can superpose—form quantum combinations of—options. Usually, when John Preskill polls our research group, I abstain from voting. Politics, and questions like “Does building a quantum computer require only engineering or also science?”,1 have many facets. I want to view such questions from many angles, to pace around the questions as around a sculpture, to hear other onlookers, to test my impressions on them, and to cogitate before choosing.2 However many perspectives I’ve gathered, I’m missing others worth seeing. I commiserated with the green-line ants.

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I first met Sara in the building behind the statue. Sara earned her PhD in Dartmouth College’s physics department, with Professor Marcelo Gleiser.

Sara presented about ants at a workshop hosted by the Beyond Center for Fundamental Concepts in Science at Arizona State University (ASU). The organizers, Paul Davies of Beyond and Andrew Briggs of Oxford, entitled the workshop “The Power of Information.” Participants represented information theory, thermodynamics and statistical mechanics, biology, and philosophy.

Paul and Andrew posed questions to guide us: What status does information have? Is information “a real thing” “out there in the world”? Or is information only a mental construct? What roles can information play in causation?

We paced around these questions as around a Chinese viewing stone. We sat on a bench in front of those questions, stared, debated, and cogitated. We taught each other about ants, artificial atoms, nanoscale machines, and models for information processing.

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Chinese viewing stone in Yuyuan Garden in Shanghai

I wonder if I’ll acquire opinions about Paul and Andrew’s questions. Maybe I’ll meander from “undecided” to “decided” over a career. Maybe I’ll phase-transition like Sara’s ants. Maybe I’ll remain near the top of her diagram, a green holdout.

I know little about information’s power. But Sara’s plot revealed one power of information: Information can move us—from homeless to belonging, from ambivalent to decided, from a plot’s top to its bottom, from passive listener to finding yourself in a green curve.

 

With thanks to Sara Imari Walker, Paul Davies, Andrew Briggs, Katherine Smith, and the Beyond Center for their hospitality and thoughts.

 

1By “only engineering,” I mean not “merely engineering” pejoratively, but “engineering and no other discipline.”

2I feel compelled to perform these activities before choosing. I try to. Psychological experiments, however, suggest that I might decide before realizing that I’ve decided.