Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

Cornelia Vogel (University of Tübingen)

To study the long-time behaviour of the unitary evolution of an isolated macroscopic quantum system, we assume, following von Neumann, that different macro states correspond to mutually orthogonal high-dimensional subspaces $\mathcal{H}_\nu$ (macro spaces) of the Hilbert space $\mathcal{H}$. Let $P_\nu$ denote the projection to $\mathcal{H}_\nu$. A pure state $\psi_t \in \mathbb{S}(\mathcal{H})$ will generally be a superposition of contributions $P_\nu\psi_t$ from different $\mathcal{H}_\nu$'s. We ask how the superposition weights $\|P_\nu\psi_t\|^2$ of these contributions evolve for a typical initial state $\psi_0$ starting in some (possibly non-equilibrium) initial macro space $\mathcal{H}_\mu$. We show that for large dimension $d_\mu := \dim \mathcal{H}_\mu$ the weights $\|P_\nu\psi_t\|^2$ evolve nearly deterministically for most $\psi_0\in\mathbb{S}(\mathcal{H}_\mu)$ (w.r.t. Haar measure) and approach certain time- and $\psi_0$-independent values $M_{\mu\nu}$ for large $t$. These results modify, extend, and in a way simplify the concept of dynamical typicality introduced by Bartsch, Gemmer, and Reimann, as well as the concept, introduced by von Neumann, now known as normal typicality. Lower bounds on the values $M_{\mu\nu}$ can be obtained for certain random band matrices.
This is joint work with Stefan Teufel and Roderich Tumulka.