Abstract
We address the problem of understanding, from first principles, the conditions under which a quantum system equilibrates rapidly with respect to a concrete observable. On the one hand, previously known general upper bounds on the time scales of equilibration were unrealistically long, with times scaling linearly with the dimension of the Hilbert space. These bounds proved to be tight since particular constructions of observables scaling in this way were found. On the other hand, the computed equilibration time scales for certain classes of typical measurements, or under the evolution of typical Hamiltonians, are unrealistically short. However, most physically relevant situations fall outside these two classes. In this paper, we provide a new upper bound on the equilibration time scales which, under some physically reasonable conditions, give much more realistic results than previously known. In particular, we apply this result to the paradigmatic case of a system interacting with a thermal bath, where we obtain an upper bound for the equilibration time scale independent of the size of the bath. In this way, we find general conditions that single out observables with realistic equilibration times within a physically relevant setup.
- Received 12 December 2015
DOI:https://doi.org/10.1103/PhysRevX.7.031027
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Many macroscopic systems naturally evolve towards equilibrium, one example being a cup of hot coffee cooling down to room temperature. It has been shown that this macroscopic behavior can emerge from the underlying quantum dynamics, but demonstrating that it happens in a reasonable time has proven to be a challenge. In this article, we focus on a class of physically relevant measurements and give, for the first time, general conditions necessary for quantum systems to equilibrate in physically realistic times.
Although the conditions under which quantum systems equilibrate are very general, little was known about the times involved in this process before this work. In fact, it has been shown that (unphysical) observables exist with equilibration times proportional to the dimension of the Hilbert space, which surpasses the age of the Universe even for small systems. We focus on classes of observables with physical significance, and we identify conditions on the initial state and Hamiltonian that ensure equilibration occurs in much more reasonable times. Furthermore, these conditions match what one would expect of practical systems. In particular, we show that a system interacting with a thermal bath equilibrates in a time that is independent of the bath size.
Our findings constitute a major step forward in the topic of equilibration time scales of quantum systems, providing tight and calculable upper bounds on how long we can maintain practical systems out of equilibrium.