Macroscopic Irreversibility in Quantum Systems: Free Expansion in a Fermion Chain

TL;DR

This paper proves that a quantum fermion chain exhibits macroscopic irreversibility, specifically ballistic diffusion, at large times without relying on randomness in initial states or Hamiltonian, using large deviation bounds akin to ETH.

Contribution

It demonstrates the emergence of irreversibility in a quantum system from arbitrary initial states without randomness, using a novel large deviation approach.

Findings

Measured density distribution becomes almost uniform at large times

Irreversible behavior emerges in quantum unitary evolution without randomness

Large deviation bounds for energy eigenstates are key to the proof

Abstract

We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).