Quantum scar

Perturbation-induced quantum skipping scar in a disordered quantum well with an external magnetic field.[1]

In quantum mechanics, quantum scarring is a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits.[2][3] The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.

A classically chaotic system is also ergodic, and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform manner up to random fluctuations in the semiclassical limit. However, scars are a significant correction to this assumption. Scars can therefore be considered as an eigenstate counterpart of how short periodic orbits provide corrections to the universal spectral statistics of the random matrix theory. There are rigorous mathematical theorems on quantum nature of ergodicity,[4][5][6] proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. Nonetheless, the quantum ergodicity theorems do not exclude scarring if the quantum phase space volume of the scars gradually vanishes in the semiclassical limit.

On the classical side, there is no direct analogue of scars. On the quantum side, they can be interpreted as an eigenstate analogy to how short periodic orbits correct the universal random matrix theory eigenvalue statistics. Scars correspond to nonergodic states which are permitted by the quantum ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure. In addition to conventional quantum scars, the field of quantum scarring has undergone its renaissance period, sparked by the discoveries of perturbation-induced scars and many-body scars (see below).

  1. ^ Cite error: The named reference :9 was invoked but never defined (see the help page).
  2. ^ Heller, Eric J. (1984-10-15). "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits". Physical Review Letters. 53 (16): 1515–1518. Bibcode:1984PhRvL..53.1515H. doi:10.1103/PhysRevLett.53.1515.
  3. ^ Heller, Eric Johnson (2018). The semiclassical way to dynamics and spectroscopy. Princeton: Princeton University Press. ISBN 978-1-4008-9029-3. OCLC 1034625177.
  4. ^ Zelditch, Steven (1987-12-01). "Uniform distribution of eigenfunctions on compact hyperbolic surfaces". Duke Mathematical Journal. 55 (4). doi:10.1215/S0012-7094-87-05546-3. ISSN 0012-7094.
  5. ^ Shnirelman, Alexander (1974). "Ergodic properties of eigenfunctions". Uspekhi Matematicheskikh Nauk. 29: 181–182.
  6. ^ Colin de Verdière, Yves (1985). "Ergodicité et fonctions propres du laplacien". Communications in Mathematical Physics. 102 (3): 497–502. Bibcode:1985CMaPh.102..497D. doi:10.1007/BF01209296. S2CID 189832724.