The physics and mathematics of the theory of quantum systems, of their manipulations and of their applications – for instance for the technology of information – is a fast evolving and important field. Researcher at IPhT are involved in many directions of research: quantum non-demolition measurements and the properties of "quantum jumps" and "quantum spikes" ; quantum chaos, escape rates and hyperbolicity; number theory, the Riemann hypothesis, and its relationship with quantum chaos.
These themes overlap with other themes pursued at IPhT, in mathematical physics (from integrable systems to string theory and black holes) and in condensed matter and statistical mechanics.
When subjected to indirect non-demolition measurements, the evolution of a quantum system under a deterministic conservative (Hamiltonian) or dissipative (Lindbladian) dynamics becomes stochastic. A parameter controls the interpolation between the deterministic regime when measurements have a very small effect, and a stochastic regime dominated by quantum jumps triggered by the measurements. Results by IPhT and ENS give a complete picture of the highly stochastic regime. The correlation function for a finite number of times can be computed in terms of some explicit effective Markov process and our predictions are consistent with actual experiments exploring this fascinating realm. Surprisingly, correlations involving an infinite number of times may exhibit anomalies, involving aborted quantum jumps that we call spikes. The mathematical description of spikes forced us to conjecture and prove new limit theorems for stochastic differential equations, actively studied by mathematicians. There is some hope that these anomalies can be seen in real experiments in a foreseeable future. Our techniques opens also the way to original analysis methods for more general intermittent signals, as in turbulence and seismology.
In the presence of a scattering potential, an initially localized quantum wave function will "escape" towards infinity in the long time limit. In the semiclassical regime, the associated escape rate is highly influenced by the classically trapped set, i.e. Hamiltonian trajectories which do not escape to infinity. Reseracher from IPhT/IMO consider the case where this set forms a normally hyperbolic symplectic submanifold of the phase space, a situation relevant e.g. for the dynamics of chemical reactions or when studying wave propagation on a Kerr black hole in general relativity. They show that the decay rate of the quantum correlations is then related to the transverse hyperbolicity of the Hamiltonian flow. Paradoxically, this "quantum" result allows to also recover properties of the correlation decay for strongly chaotic (Anosov) classical flows.
Progress was done around the celebrated Riemann Hypothesis (RH), a major conjecture of number theory about the Zeta function. The RH admits an apparently simple criterion, given by X.-J. Li in 1997, based on the behavior of a specific (Keiper-Li) sequence, whose terms are however very hard to calculate. A totally explicit and elementary variant of this sequence is obtained, leading by semiclassical analysis to a new criterion: if the RH is true, the new sequence grows logarithmically; if it is false, the sequence eventually oscillates with amplitude increasing as a power law. A first multi-precision parallel programming allowed to reach the index n = 500000 (compared to 100000 for Keiper-Li) but a real test of the RH would require reaching n = 1013. For a more general Zeta-type function (Davenport-Heilbronn) which clearly violates the RH, the increasing oscillation of the new associated sequence is visible from n = 100, confirming the proposed new criterion.
|Stéphane NONNENMACHER *
* present address, IMO, Université Paris-sud
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