| What is quantum chaos? |
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| Quantum chaos |
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Our group has been studying "quantum chaos" since 1998. In these pages, you can find a breef summary of the main ideas concerning quantum chaos and of our activity in the field.
Quantum chaos is the study of quantum systems which are chaotic in the classical limit. In general, quantum chaos is very different from classical chaos, for a number of rather fundamental reasons:
| The Schrödinger
equation is linear |
In classical physics (Newton's law of motion), the spatial cordinate x is both the dynamical variable and the parameter of the force. This means that by taking a nonlinear force on easily generates nonlinear dynamics. Nonlinearity is responsible for the sensitivity to initial conditions which is at the heart of chaotic behavior. 
In Schrödinger's equation, the dynamical variable is the wave function, whereas x is a parameter of the potential. Thus, taking a nonlinear potential does not produce a nonlinear dynamics. It is only by consider interparticle interactions in the quantum regime (e. g. in a Bose-Einstein condensate) that one can have a nonlinear quantum equation. For more on that, see our pages on quasi-classical chaos with Bose-Einstein condensates. |
| Trajectories
are not well defined in quantum mechanics |
Many concepts of classical
chaos (periodic orbits, atracttors, phase space maps) rely on the classical
notion of trajectory. In general, due to Heisenberg's uncertainty principle,these
notions does not translate well into the quantum world. |
| However,
there are "signatures of quantum chaos" |
Fortunately, one founds large
classes of universality in quantum chaos. These universal behaviors are
named "signatures" of the quantum chaos. For example, the statistics
of level-spacing of quantum-chaotic systems are rather different from
that of non-chaotic systems. In time-periodic quantum-chaotic systems,
one observes another signature, the dynamical localization, that is one
of the main themes of our studies. |