Friday Quantum Seminar
The fundamental assumption of statistical mechanics is that the long-time average of any observable is equal to its average over the microcanonical ensemble. In classical mechanics, this stems from Boltzmann’s ergodic hypothesis, by which a generic initial state in an ergodic system visits the neighborhood of all states in phase space with the same energy. However, wavelike effects in quantum mechanics have made it difficult to identify what it even means for a quantum system to be ergodic, except on a case-by-case basis for individual observables.
In parallel, a famous conjecture states that “quantum chaos” should be associated with the eigenvalue statistics of Haar random unitary matrices. While this has been a useful observable-independent guiding principle (and is sometimes adopted as a definition of quantum chaos), several puzzles remain, including exactly what kind of “quantum chaos” the conjecture refers to.
In this talk, we will seek to address these two issues, providing a qualitative overview of Ref. . We will show that a discretized version of ergodicity can be quantized, without reference to specific observables, in terms of an analogous property for orthonormal bases in the Hilbert space (typically involving a discrete Fourier transform of the energy eigenstates). We will also show that it is this form of ergodicity that is quantitatively determined by the system’s eigenvalue statistics, with random matrix statistics emerging as a special case of a more general set of constraints. A key implication is that the kind of “quantum chaos” associated with eigenvalue statistics has little to do with the popular understanding of chaos in terms of exponential instabilities or unpredictability.
 A. Vikram and V. Galitski. “Dynamical quantum ergodicity from energy level statistics.” Phys. Rev. Res. 5, 033126 (2023). arxiv:2205.05704 [quant-ph].
Pizza and drinks will be served after the seminar in ATL 2117.