Editing Quantum tunnelling (section)

== Dynamical tunneling ==
[[File:Quantum tunneling in phase space.gif|left|thumb|upright=1|Quantum tunneling oscillations of probability in an integrable double well of potential, seen in phase space]]

The concept of quantum tunneling can be extended to situations where there exists a quantum transport between regions that are classically not connected even if there is no associated potential barrier. This phenomenon is known as dynamical tunnelling.<ref>{{Cite journal|last1=Davis|first1=Michael J.|last2=Heller|first2=Eric J.|date=1981-07-01|title=Quantum dynamical tunneling in bound states|url=https://aip.scitation.org/doi/10.1063/1.441832|journal=The Journal of Chemical Physics |volume=75 |issue=1 |pages=246–254 |doi=10.1063/1.441832 |bibcode=1981JChPh..75..246D |issn=0021-9606}}</ref><ref>{{Cite book|last1=Keshavamurthy|first1=Srihari|url=https://books.google.com/books?id=c6HMBQAAQBAJ&q=dynamical+tunneling+theory+and+experiment|title=Dynamical Tunneling: Theory and Experiment |last2=Schlagheck |first2=Peter |date=2011-03-09 |publisher=CRC Press|isbn=978-1-4398-1666-0|language=en}}</ref>

=== Tunnelling in phase space ===
The concept of dynamical tunnelling is particularly suited to address the problem of quantum tunnelling in high dimensions (d>1). In the case of an [[integrable system]], where bounded classical trajectories are confined onto [[Torus|tori]] in [[phase space]], tunnelling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori.<ref>{{Cite journal|last=Wilkinson|first=Michael|date=1986-09-01|title=Tunnelling between tori in phase space|url=https://dx.doi.org/10.1016%2F0167-2789%2886%2990009-6|journal=Physica D: Nonlinear Phenomena|language=en|volume=21|issue=2|pages=341–354|doi=10.1016/0167-2789(86)90009-6|bibcode=1986PhyD...21..341W|issn=0167-2789}}</ref>

=== Chaos-assisted tunnelling ===
[[File:Chaos-assisted tunneling in phase space.gif|left|thumb|upright=1|Chaos-assisted tunnelling oscillations between two regular tori embedded in a chaotic sea, seen in phase space]]

In real life, most systems are not integrable and display various degrees of chaos. Classical dynamics is then said to be mixed and the system phase space is typically composed of islands of regular orbits surrounded by a large sea of chaotic orbits. The existence of the chaotic sea, where transport is classically allowed, between the two symmetric tori then assists the quantum tunnelling between them. This phenomenon is referred as chaos-assisted tunnelling.<ref>{{Cite journal|last1=Tomsovic|first1=Steven|last2=Ullmo|first2=Denis|date=1994-07-01|title=Chaos-assisted tunneling|url=https://link.aps.org/doi/10.1103/PhysRevE.50.145|journal=Physical Review E|volume=50|issue=1|pages=145–162|doi=10.1103/PhysRevE.50.145|pmid=9961952|bibcode=1994PhRvE..50..145T}}</ref> and is characterized by sharp resonances of the tunnelling rate when varying any system parameter.

=== Resonance-assisted tunnelling ===
When <math>\hbar</math> is small in front of the size of the regular islands, the fine structure of the classical phase space plays a key role in tunnelling. In particular the two symmetric tori are coupled "via a succession of classically forbidden transitions across nonlinear resonances" surrounding the two islands.<ref>{{Cite journal|last1=Brodier|first1=Olivier|last2=Schlagheck|first2=Peter|last3=Ullmo|first3=Denis|date=2002-08-25|title=Resonance-Assisted Tunneling|url=http://www.sciencedirect.com/science/article/pii/S0003491602962810|journal=Annals of Physics|language=en|volume=300|issue=1|pages=88–136|doi=10.1006/aphy.2002.6281|arxiv=nlin/0205054|bibcode=2002AnPhy.300...88B|s2cid=51895893|issn=0003-4916}}</ref>