This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo- absorption spectrum. Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1 The number of periodic orbits proliferates exponentially as a function of action. This difficulty is also present when applying periodic-orbit theory to regular systems.

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The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states energy levels , one can use standard quantum mechanical perturbation theory to compute eigenvalues energy levels and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits.

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Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform. Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books. Closed-orbit theory was developed by J.

Delos, M. Du, J. Gao, and J. It is similar to periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density observable photo-absorption spectrum from a specified initial state whereas periodic-orbit theory yields the density of states. Only orbits that begin and end at the nucleus are important in closed-orbit theory.

Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state.

For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time. It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave.

There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the standard map and the kicked rotator are considered to be prototype problems. Recent [ when? There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting many-body quantum systems far from semiclassical regimes.

In , Berry and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable. From Wikipedia, the free encyclopedia. Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory. This article may be too technical for most readers to understand.

Please help improve it to make it understandable to non-experts , without removing the technical details.

## Instabilities and Chaos in Optics

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Berry and M. Tabor, Proc. Altland, P. Braun, and F. Haake, , Phys. Physical Review B. Bibcode : PhRvB.. Garbaczewski and R.

Olkiewicz Springer, Bibcode : EL Jan Notices of the AMS. Chaos theory. Yorke Lai-Sang Young.

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Views Read Edit View history. Together they will serve as a comprehensive and tutorial pair of companion volumes. This school was directed by Prof.

Massimo Inguscio, of the Department of Physics, University of Naples, Naples, Italy to whom we express our gratitude on behalf of all lecturers and students. The Scientific Advisory Committee consisted of N. Abraham of Bryn Mawr College; F. Lugiato of the Politechnic Institute of Torino.

The school continues the long tradition of Europhysics Summer Schools in Quantum Electronics which have provided instruction and training for young researchers and advanced students working in this field for almost twenty years.

## Instabilities and Chaos in Quantum Optics | SpringerLink

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