Exploiting Chaos for Quantum Control
The field of Quantum Chaos is referred to as the study of quantum behaviors of systems whose corresponding classical dynamics are chaotic, or study of quantum manifestations of classical chaos. Equivalently, it means that quantum behaviors depend on the nature of the classical dynamics, implying that classical chaos can be used to control or manipulate quantum behaviors. We discuss two examples here: using transient chaos to control quantum transport in nanoscale systems and exploiting chaos to regularize relativistic quantum tunneling dynamics in Dirac fermion and graphene systems.
KeywordsQuantum Tunneling Classical Dynamic Quantum Transport Tunneling Rate Unstable Periodic Orbit
The main idea of using chaos to manipulate quantum behaviors was generated through extensive discussions with Dr. L. Pecora from Naval Research Laboratory in January 2011 at Dr. M. Shlesinger’s ONR Program Review Meeting at UCSD. The computations and theoretical analyses reported in the references [4, 5, 30, 31] on which this Review is based were mainly carried out by Dr. R. Yang, Dr. X. Ni, and Dr. L. Huang, all formerly affiliated with ASU.
- 7.See, for example, R. A. Jalabert, H. U. Baranger, and A. D. Stone, Phys. Rev. Lett. 65, 2442 (1990)Google Scholar
- 9.R.P. Taylor, R. Newbury, A.S. Sachrajda, Y. Feng, P.T. Coleridge, C. Dettmann, N. Zhu, H. Guo, A. Delage, P. J. Kelly, Z. Wasilewski. Phys. Rev. Lett. 78, 1952 (1997)Google Scholar
- 19.Given a closed Hamiltonian system that exhibits fully developed chaos in the classical limit, one might expect the quantum wavefunctions associated with various eigenstates to be more or less uniform in the physical space. However, in the seminal work of McDonald and Kaufman [Phys. Rev. Lett. 42, 1189 (1979) and Phys. Rev. A 37, 3067 (1988)], it was observed that quantum eigen-wavefunctions can be highly non-uniform in the chaotic stadium billiard. A systematic study was subsequently carried out by Heller [Phys. Rev. Lett. 53, 1515 (1984)], who established the striking tendency for wavefunctions to concentrate about classical unstable periodic orbits, which he named quantum scars. Semiclassical theory was then developed by Bogomolny [Physica D 31, 169 (1988)] and Berry [Proc. Roy. Soc. (London) A 423, 219 (1989)], providing a general understanding of the physical mechanism of quantum scars. The phenomenon of quantum scarring was deemed counterintuitive and surprising but only for chaotic systems, as the phase space of an integrable system is not ergodic so that the quantum wavefunctions are generally not expected to be uniform. Relativistic quantum scars in chaotic graphene systems have also been reported [L. Huang, Y.-C. Lai, D. K. Ferry, S. M. Goodnick, and R. Akis, Phys. Rev. Lett. 103, 054101 (2009)].Google Scholar
- 22.See, for example, Chapter 18 in J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge 1999)Google Scholar
- 30.R. Yang, L. Huang, Y.-C. Lai, C. Grebogi, and L. M. Pecora, Chaos 23, 013125 (2013)Google Scholar
- 31.X. Ni, L. Huang, Y.-C. Lai, C. Grebogi, Phys. Rev. E 86, 015702 (2012)Google Scholar
- 38.G.-L. Wang, L. Ying, Y.-C. Lai, and C. Grebogi, Phys. Rev. E 87, 052908 (2013)Google Scholar