Quantum dynamics
The Schrödinger equation underpins all of quantum mechanics, and yet we know very little about the behavior that can arise from it in systems of many quantum particles. Much of what we know about quantum many-body systems has to do with their stationary or low-energy states: solutions of the time-independent Schrödinger equation. In large part, this has been due to practical limitations: solid-state experiments natural prepare states at thermal equilibrium. Meanwhile, numerical simulations of evolving systems are challenging due to the rapid buildup of entanglement. Fortunately, non-equilibrium dynamics is the natural mode of operation of modern quantum devices. This presents the exciting opportunity to explore uncharted territory and map out what is possible in many-body dynamics.
Yet all things have precedent and we have certain expectations for how an out-of-equilibrium quantum system should behave:
1. Quantum chaos and thermalization… – Quantum chaos originated in the 90’s from attempts to extend then-new results in chaotic classical dynamical systems. Yet foundational concepts such as the Lyapunov exponent are difficult to generalize and quantum chaos is arguably, still in search of a basic definition. The situation is further complicated with many-body quantum systems, which do not have classical limits (unlike, say, an electron in a stadium-shaped potential, which has a classical limit of a particle in a stadium-shaped box).
Despite this, most interacting quantum systems are expected to be chaotic and a set of phenomenology has emerged, with close ties to quantum thermalization: the process in which a system reaches thermal equilibrium.
Broadly speaking, modern notions of quantum chaos study how perturbations (“information”) spread in the exponentially large Hilbert space of many-body quantum systems, in particular over space and time. There are certain universal aspects of this behavior which are independent of a system’s microscopic details. Such universality is not only fascinating, it can also be harnessed for practical applications.
Exciting broad questions about quantum chaos that motivate me include:
To what extent do chaotic systems behave randomly, and how do they interact with structures like conservation laws? This is a long-standing direction in the community, broadly under the banner of the Eigenstate Thermalization Hypothesis. In part, this was answered by our maximum entropy principle, but I believe the question runs deeper…
How can we efficiently describe thermal structures? Much work has gone into describe thermal correlation lengths of few-body observables, but modern quantum devices beget a broader class of observables which may also have tantalizing structure: the 
How can we predict thermal properties? To me, the holy grail of predictive power would be: You hand me a many-body Hamiltonian that describes a system, tell me its temperature, and I predict its thermal properties. A simple question, yet to my knowledge no general recipe exists.
Of course, phenomena such a thermal phase transitions mean that the partition function, which determines many thermal properties, are non-analytic and hence hard to predict. Yet, local quantities in quantum many-body systems also seem to be roughly (but not quite!) described by a thermal state of a local Hamiltonian. Therein lies the challenge…
2. Hydrodynamics – Despite the “chaotic” nature of quantum dynamics*, its local quantities are expected to exhibit simple behavior: at sufficiently large length- and time-scales, non-conserved quantities die out (more correctly, they spread into large operators and are never seen again), and only conserved quantities remain. Local densities (e.g. energy or charge densities) of these quantities take time to redistribute themselves to steady state, and they do so in accordance to hydrodynamic equations, typically diffusion but sometimes more exotic equations.
*Think about this like hydrodynamics emerging from many interacting classical particles, whose microscopic dynamics can be classically chaotic.
3. Exceptions to the rule. – Reality often exceeds the capacity for human imagination, and quantum dynamics is no exception: quantum simulation experiments have begun to yield surprises that defy theoretical expectations. As an example, many-body scars were an unexpected discovery in a quantum simulation experiment: of a system which exhibited simple, non-thermalizing behavior when initialized in certain special states. This set off a flurry of theoretical activity, discovering scars in a panoply of systems beyond the initial experiment.
Part of this flurry was my undergraduate work: we developed a common framework [Phys. Rev. B 101, 195131] to explain a collection of results in the literature. Using this framework, we identified Hubbard-like models that feature exact many-body scar states [Phys. Rev. B 102, 075132].
Stay tuned for more on quantum dynamics…