May | 2020 | Quantum Frontiers

Mr. Mole is trapped indoors, alone. Spring is awakening outside, but he’s confined to his burrow. Birds are twittering, and rabbits are chattering, but he has only himself for company.

Sound familiar?

Spring—crocuses, daffodils, and hyacinths budding; leaves unfurling; and birds warbling—burst upon Cambridge, Massachusetts last month. The city’s shutdown vied with the season’s vivaciousness. I relieved the tension by rereading The Wind in the Willows, which I’ve read every spring since 2017.

Project Gutenberg offers free access to Kenneth Grahame’s 1908 novel. He wrote the book for children, but never mind that. Many masterpieces of literature happen to have been written for children.

One line in the novel demanded, last year, that I memorize it. On page one, Mole is cleaning his house beneath the Earth’s surface. He’s been dusting and whitewashing for hours when the spring calls to him. Life is pulsating on the ground and in the air above him, and he can’t resist joining the party. Mole throws down his cleaning supplies and tunnels upward through the soil: “he scraped and scratched and scrabbled and scrooged, and then he scrooged again and scrabbled and scratched and scraped.”

The quotation appealed to me not only because of its alliteration and chiasmus. Mole’s journey reminded me of research.

Take a paper that I published last month with Michael Beverland of Microsoft Research and Amir Kalev of the Joint Center for Quantum Information and Computer Science (now of the Information Sciences Institute at the University of Southern California). We translated a discovery from the abstract, mathematical language of quantum-information-theoretic thermodynamics into an experimental proposal. We had to scrabble, but we kept on scrooging.

Over four years ago, other collaborators and I uncovered a thermodynamics problem, as did two other groups at the same time. Thermodynamicists often consider small systems that interact with large environments, like a magnolia flower releasing its perfume into the air. The two systems—magnolia flower and air—exchange things, such as energy and scent particles. The total amount of energy in the flower and the air remains constant, as does the total number of perfume particles. So we call the energy and the perfume-particle number conserved quantities.

We represent quantum conserved quantities with matrices $Q_1$ and $Q_2$ . We nearly always assume that, in this thermodynamic problem, those matrices commute with each other: $Q_1 Q_2 = Q_2 Q_1$ . Almost no one mentions this assumption; we make it without realizing. Eliminating this assumption invalidates a derivation of the state reached by the small system after a long time. But why assume that the matrices commute? Noncommutation typifies quantum physics and underlies quantum error correction and quantum cryptography.

What if the little system exchanges with the large system thermodynamic quantities represented by matrices that don’t commute with each other?

Colleagues and I began answering this question, four years ago. The small system, we argued, thermalizes to near a quantum state that contains noncommuting matrices. We termed that state, $e^{ - \sum_\alpha \beta_\alpha Q_\alpha } / Z$ , the non-Abelian thermal state. The $Q_\alpha$ ’s represent conserved quantities, and the $\beta_\alpha$ ’s resemble temperatures. The real number $Z$ ensures that, if you measure any property of the state, you’ll obtain some outcome. Our arguments relied on abstract mathematics, resource theories, and more quantum information theory.

Over the past four years, noncommuting conserved quantities have propagated across quantum-information-theoretic thermodynamics.¹ Watching the idea take root has been exhilarating, but the quantum information theory didn’t satisfy me. I wanted to see a real physical system thermalize to near the non-Abelian thermal state.

Michael and Amir joined the mission to propose an experiment. We kept nosing toward a solution, then dislodging a rock that would shower dirt on us and block our path. But we scrabbled onward.

Imagine a line of ions trapped by lasers. Each ion contains the physical manifestation of a qubit—a quantum two-level system, the basic unit of quantum information. You can think of a qubit as having a quantum analogue of angular momentum, called spin. The spin has three components, one per direction of space. These spin components are represented by matrices $Q_x = S_x$ , $Q_y = S_y$ , and $Q_z = S_z$ that don’t commute with each other.

A couple of qubits can form the small system, analogous to the magnolia flower. The rest of the qubits form the large system, analogous to the air. I constructed a Hamiltonian—a matrix that dictates how the qubits evolve—that transfers quanta of all the spin’s components between the small system and the large. (Experts: The Heisenberg Hamiltonian transfers quanta of all the spin components between two qubits while conserving $S_{x, y, z}^{\rm tot}$ .)

The Hamiltonian led to our first scrape: I constructed an integrable Hamiltonian, by accident. Integrable Hamiltonians can’t thermalize systems. A system thermalizes by losing information about its initial conditions, evolving to a state with an exponential form, such as $e^{ - \sum_\alpha \beta_\alpha Q_\alpha } / Z$ . We clawed at the dirt and uncovered a solution: My Hamiltonian coupled together nearest-neighbor qubits. If the Hamiltonian coupled also next-nearest-neighbor qubits, or if the ions formed a 2D or 3D array, the Hamiltonian would be nonintegrable.

We had to scratch at every stage—while formulating the setup, preparation procedure, evolution, measurement, and prediction. But we managed; Physical Review E published our paper last month. We showed how a quantum system can evolve to the non-Abelian thermal state. Trapped ions, ultracold atoms, and quantum dots can realize our experimental proposal. We imported noncommuting conserved quantities in thermodynamics from quantum information theory to condensed matter and atomic, molecular, and optical physics.

As Grahame wrote, the Mole kept “working busily with his little paws and muttering to himself, ‘Up we go! Up we go!’ till at last, pop! his snout came out into the sunlight and he found himself rolling in the warm grass of a great meadow.”