Quantized baker map

LEONARDO ERMANN; MARCOS SARACENO

Scholarpedia

a

Lugar: San Diego; Año: 2012 vol. 7 p. 9860 - 9860

1941-6016

The baker´s transformation is one of the simplest models where all features of chaos are present. It was devised by E. Hopf in 1937 in the context of ergodic theory [1]. It is defined geometrically, just like the horseshoe map and the cat map. Unlike other simple chaotic models coming from Hamiltonian systems as the Chirikov Standard map and Kicked Harper model all its classical structures can be described analytically. The quantum version of the map is used to explore the relationship between classical and quantum mechanics in the semiclassical limit, when the dynamics is chaotic (i.e. quantum chaos). The relative simplicity of the classical description - provided by symbolic dynamics - and the elegant and flexible quantization scheme devised by Balazs and Voros allow very detailed analytical and/or numerical studies of central open questions in the semiclassical description of chaotic systems, like the commutativity of quantization with time propagation and the comparison of classical and quantum invariant structures [2],[3],[4]. This scheme is sufficiently general to be applied to most piecewise linear maps like the tri-baker and the non-dissipative horseshoe map. When implemented on tensor product Hilbert spaces (in particular on a system of qubits) it provides a simple implementable model of unitary propagation in the circuit model of quantum computation. It has also been used to study the emergence of fractal Weyl laws in the quantum theory of open systems.