Spectral properties of noisy classical and quantum propagators

Abstract

We study classical and quantum maps on the torus phase space, in the presence of noise. We focus on the spectral properties of the noisy evolution operator, and prove that for any amount of noise, the quantum spectrum converges to the classical one in the semiclassical limit. The small-noise behaviour of the classical spectrum highly depends on the dynamics generated by the map. For a chaotic dynamics, the outer spectrum consists in isolated eigenvalues (``resonances'') inside the unit circle, leading to an exponential damping of correlations. On the opposite, in the case of a regular map, part of the spectrum accumulates along a one-dimensional ``string'' connecting the origin with unity, yielding a diffusive behaviour. We finally study the non-commutativity between the semiclassical and small-noise limits, and illustrate this phenomenon by computing (analytically and numerically) the classical and quantum spectra for some maps.

35 pages, 6 .eps figures, to be published in Nonlinearity. I added some references and comments

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• Realization of quantum process tomography in NMR
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Cited by in corpus (22)

• Chaos signatures in the short and long time behavior of the out-of-time ordered correlator
• Random Lindblad Dynamics
• Ruelle-Pollicott resonances for real analytic hyperbolic map
• Classical decays in decoherent quantum maps
• Dissipation time and decay of correlations
• Spectral properties and classical decays in quantum open systems
• Noise models for superoperators in the chord representation
• Momentum dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems
• Entanglement-screening by nonlinear resonances
• Classical approach to equilibrium of out-of-time ordered correlators in mixed systems
• Classical nonlinear response of a chaotic system: Langevin dynamics and spectral decomposition
• Theory of Irreversibility in Quantum Many-Body Systems
• Classical to quantum correspondence in dissipative directed transport
• Phase space contraction and quantum operations
• Spectral Approach to Chaos and Quantum-Classical Correspondence in Quantum Mapas
• Classical projected phase space density of billiards and its relation to the quantum Neumann spectrum
• Ruelle-Pollicott Decay of Out-of-Time-Order Correlators in Many-Body Systems
• Irreversibility in quantum maps with decoherence
• Symmetry Properties of Quantum Dynamical Entropy
• Coarse Grained Liouville Dynamics of piecewise linear discontinuous maps
• Time Glasses: Symmetry Broken Chaotic Phase with a Finite Gap
• Relaxation Time of Quantized Toral Maps