Editing Quantum chaos (section)

== Recent directions ==
One open question remains understanding quantum chaos in systems that have finite-dimensional local [[Hilbert space]]s for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such [[Many-body system|quantum many-body systems]].<ref>{{Cite journal|last1=Kos|first1=Pavel|last2=Ljubotina|first2=Marko|last3=Prosen|first3=Tomaž|date=2018-06-08|title=Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory|journal=Physical Review X|volume=8|issue=2|pages=021062|doi=10.1103/PhysRevX.8.021062|arxiv=1712.02665 |bibcode=2018PhRvX...8b1062K |doi-access=free}}</ref><ref>{{Cite journal|last1=Chan|first1=Amos|last2=De Luca|first2=Andrea|last3=Chalker|first3=J. T.|date=2018-11-08|title=Solution of a Minimal Model for Many-Body Quantum Chaos|journal=Physical Review X|language=en|volume=8|issue=4|pages=041019|doi=10.1103/PhysRevX.8.041019|arxiv=1712.06836 |bibcode=2018PhRvX...8d1019C |issn=2160-3308|doi-access=free}}</ref>

The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions of various chaotic Hamiltonian. For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation ([[Quantum ergodicity]]). However, a quantum eigenstate of a classically chaotic system can be scarred:<ref name=":0">{{Cite journal |last=Heller |first=Eric J. |date=1984-10-15 |title=Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits |url=https://link.aps.org/doi/10.1103/PhysRevLett.53.1515 |journal=Physical Review Letters |volume=53 |issue=16 |pages=1515–1518 |doi=10.1103/PhysRevLett.53.1515|bibcode=1984PhRvL..53.1515H |url-access=subscription }}</ref> the probability density of the eigenstate is enhanced in the neighborhood of a periodic orbit, above the classical, statistically expected density along the orbit ([[Scar (physics)|scar]]s).  In particular, scars are both a striking visual example of classical-quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos.  For example, this is evident in the perturbation-induced quantum scarring:<ref>{{Cite journal |last1=Keski-Rahkonen |first1=J. |last2=Ruhanen |first2=A. |last3=Heller |first3=E. J. |last4=Räsänen |first4=E. |date=2019-11-21 |title=Quantum Lissajous Scars |url=https://link.aps.org/doi/10.1103/PhysRevLett.123.214101 |journal=Physical Review Letters |volume=123 |issue=21 |pages=214101 |doi=10.1103/PhysRevLett.123.214101|pmid=31809168 |arxiv=1911.09729 |bibcode=2019PhRvL.123u4101K |s2cid=208248295 }}</ref><ref>{{Cite journal |last1=Luukko |first1=Perttu J. J. |last2=Drury |first2=Byron |last3=Klales |first3=Anna |last4=Kaplan |first4=Lev |last5=Heller |first5=Eric J. |last6=Räsänen |first6=Esa |date=2016-11-28 |title=Strong quantum scarring by local impurities |journal=Scientific Reports |language=en |volume=6 |issue=1 |pages=37656 |doi=10.1038/srep37656 |issn=2045-2322 |pmc=5124902 |pmid=27892510|arxiv=1511.04198 |bibcode=2016NatSR...637656L }}</ref><ref>{{Cite journal |last1=Keski-Rahkonen |first1=J. |last2=Luukko |first2=P. J. J. |last3=Kaplan |first3=L. |last4=Heller |first4=E. J. |last5=Räsänen |first5=E. |date=2017-09-20 |title=Controllable quantum scars in semiconductor quantum dots |url=https://link.aps.org/doi/10.1103/PhysRevB.96.094204 |journal=Physical Review B |volume=96 |issue=9 |pages=094204 |doi=10.1103/PhysRevB.96.094204|arxiv=1710.00585 |bibcode=2017PhRvB..96i4204K |s2cid=119083672 }}</ref><ref>{{Cite journal |last1=Keski-Rahkonen |first1=J |last2=Luukko |first2=P J J |last3=Åberg |first3=S |last4=Räsänen |first4=E |date=2019-01-21 |title=Effects of scarring on quantum chaos in disordered quantum wells |url=https://doi.org/10.1088/1361-648X/aaf9fb |journal=Journal of Physics: Condensed Matter |language=en |volume=31 |issue=10 |pages=105301 |doi=10.1088/1361-648x/aaf9fb |pmid=30566927 |arxiv=1806.02598 |bibcode=2019JPCM...31j5301K |s2cid=51693305 |issn=0953-8984}}</ref><ref>{{Cite book |last=Keski-Rahkonen |first=Joonas |url=https://trepo.tuni.fi/handle/10024/123296 |title=Quantum Chaos in Disordered Two-Dimensional Nanostructures |date=2020 |publisher=Tampere University |isbn=978-952-03-1699-0 |language=en}}</ref>  More specifically, in quantum dots perturbed by local potential bumps (impurities), some of the eigenstates are strongly scarred along periodic orbits of unperturbed classical counterpart.

Further studies concern the parametric (<math>R</math>) dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on  wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the [[standard map]] and the [[kicked rotator]] are considered to be prototype problems.

Works are also focused in the study of driven chaotic systems,<ref>{{Cite arXiv |eprint=quant-ph/0403061 |first=Cohen |last=Doron |title=Driven chaotic mesoscopic systems, dissipation and decoherence|date=2004 }}</ref> where the Hamiltonian  <math>H(x,p;R(t))</math> is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting ''many-body'' quantum systems far from semi-classical regimes as well as a large effort in quantum chaotic scattering.<ref>{{Cite journal|last=Gaspard|first=Pierre|date=2014|title=Quantum chaotic scattering|journal=Scholarpedia|language=en|volume=9|issue=6|pages=9806|doi=10.4249/scholarpedia.9806|bibcode=2014SchpJ...9.9806G |issn=1941-6016|doi-access=free }}</ref>