Semi-classical Scar functions in phase space

Abstract

We develop a semi-classical approximation for the scar function in the Weyl-Wigner representation in the neighborhood of a classically unstable periodic orbit of chaotic two dimensional systems. The prediction of hyperbolic fringes, asymptotic to the stable and unstable manifolds, is verified computationally for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus. Characteristic fringe patterns can be distinguished even for quasi-energies where the fixed point is not Bohr-quantized. Also the patterns are highly localized in the neighborhood of the periodic orbit and along its stable and unstable manifolds without any long distance patterns that appears for the case of the spectral Wigner function.

19 pages, 2 figures. Submited to Journal Of Physics A

References in corpus (14)

• Sub-Planck spots of Schroedinger cats and quantum decoherence
• Scarred eigenstates for quantum cat maps of minimal periods
• Measuring Scars of Periodic Orbits
• The Weyl representation on the torus
• Semiclassical quantization with short periodic orbits
• Semiclassical Theory of Short Periodic Orbits in Quantum Chaos
• Eigenstate structures around a hyperbolic point
• Signatures of homoclinic motion in quantum chaos
• General Quantum Surface of Section Method
• Quantization of multidimensional cat maps
• Localization properties of groups of eigenstates in chaotic systems
• Crystal properties of eigenstates for quantum cat maps
• Scar functions in the Bunimovich Stadium billiard
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Cited by in corpus (8)

• Scarring in open quantum systems
• Short periodic orbit approach to resonances and the fractal Weyl law
• Wigner distributions for finite state systems without redundant phase point operators
• Relevant OTOC operators: footprints of the classical dynamics
• Stable classical structures in dissipative quantum chaotic systems
• The classical skeleton of open quantum chaotic maps
• Semiclassical matrix elements for a chaotic propagator in the Scar functions basis
• On Bargmann Representations of Wigner Function