J P Keating et al 2008 Nonlinearity 21 2591 doi:10.1088/0951-7715/21/11/007
On the resonance eigenstates of an open quantum baker map
J P Keating1, S Nonnenmacher2,3, M Novaes1 and M Sieber1
Show affiliationsRecommended by B Eckhardt
We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, |zmin| ≤ |z| ≤ |zmax|. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius r. We prove that if the moduli converge to r = |zmax| then the sequence of eigenstates is associated with a fixed phase space measure ρmax. The same holds for sequences with eigenvalue moduli converging to |zmin|, with a different limit measure ρmin. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius |zmin| < r < |zmax| there is no unique limit measure, and we identify some families of eigenstates with precise self-similar properties.
- PACS
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05.45.Mt Quantum chaos; semiclassical methods
03.65.Sq Semiclassical theories and applications
03.65.Yz Decoherence; open systems; quantum statistical methods
- MSC
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81Q50 Quantum chaos (See also 37Dxx)
37A25 Ergodicity, mixing, rates of mixing
70K55 Transition to stochasticity (chaotic behavior) (See also 37D45)
81Rxx Groups and algebras in quantum theory
81Q20 Semiclassical techniques including WKB and Maslov methods
- Subjects
- Dates
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Issue 11 (November 2008)
Received 4 June 2008, in final form 29 August 2008
Published 29 September 2008
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On the resonance eigenstates of an open quantum baker map
J P Keating et al 2008 Nonlinearity 21 2591
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