4th Meeting of the GDR Quantum Dynamics

We consider the problem of propagation of photons in the quantum theory of non-relativistic matter coupled to electromagnetic radiation. Our aim will be to justify that photons do not propagate faster than the speed of light. More precisely, using the method of propagation observables, we will show that the probability to find photons at time t at a distance from the matter greater than ct, where c is the speed of light, vanishes as t goes to infinity as an inverse power of t. If time permits, we will also discuss minimal velocity estimates for photons. This is joint work with J.-F. Bony and I.M. Sigal.

Frederic Faure : A natural quantization for chaotic maps. Slides

An old standing question in mathematics of quantum mechanics is about the existence of a quantization procedure such that the semiclassical formula are exact. We consider any Anosov map on a compact symplectic manifold which is a standard model for chaotic dynamics. We will show that there exists a "quantization" namely a family of finite rank operators which depend on h (small), such that the Gutzwiller trace formula is exact for large time and Egorov theorem is also exact. Such a quantization is unique. We will explain its construction from a classical transfer operator acting on the prequantum bundle. We will also discuss the relation with geometric quantization and perspectives for geodesic flows. This is a joint work with Masato Tsujii.

Giulia Ferrini : Macroscopic superpositions in the presence of phase noise in a Bose Joseph junction Slides

A Bose Josephson Junction (BJJ) [1] is a system composed of bosons which can occupy two modes, which can be coupled. This system models either an ultracold bosonic gas trapped in a double-well potential, either a gas of ultracold bosons in two different hyperfine states, trapped in the same harmonic potential. Both congurations have been experimentally realized, and find interesting applications, e.g. in high-precision interferometry [2, 3]. If the system is initially prepared in a coherent state and then is let evolve after suppressing the coupling of the two modes, at short time the state undergoes squeezing [3, 4], and at larger time macroscopic superpositions of coherent states are formed [5]. One of the most relevant sources of noise in such a system, together with particle losses, is phase noise, resulting from stochastic fluctuations of the energies of the two modes [3]. We address the question how the presence of phase noise affects the formation of squeezed states and macroscopic superpositions, showing in particular how these latter display an unexpected robustness against decoherence induced by the noise considered [6]. (Joint work with D. Spehner, A. Minguzzi and F.W.J. Hekking)
[1] A. Smerzi et al, Phys. Rev. Lett. 79, 357 (1997).
[2] M. Albiez et al, Phys. Rev. Lett. 95, 010402 (2005); J. Esteve et al, Nature 455, 1216 (2008).
[3] C. Gross et al, Nature 464, 1165 (2010).
[4] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
[5] F. Piazza, L. Pezze and A. Smerzi, Phys. Rev. A 78, 051601 (2008).; G. Ferrini, A. Minguzzi and F.W.J. Hekking, Phys. Rev. A 78, 023606 (2008).
[6] G. Ferrini et al, Phys. Rev. A 82, 033621 (2010); G. Ferrini et al, Phys. Rev. A 84, 043628 (2011).

Jean-Claude Garreau : Sensitivity to initial conditions in a disordered lattice with interactions Slides

We study the quantum dynamics of a wavepacket evolving in a disordered lattice, which realizes the well-known "Anderson model", leading to an exponential spatial localization of the wavepacket, in sharp contrast with the delocalized Bloch functions observed in a regular lattice. We added to such a system atom-atom interactions modeled (in the mean field approximation) by an additional quadratic nonlinear term in the corresponding Schrodinger equation. Dynamics in such a system is very complex, because the nonlinearity brings into the problem sensitivity to the initial conditions (SIC), which is absent of "standard" (linear) quantum mechanics. Three main dynamical regimes emerge: For weak nonlinearity, the localization is preserved at least for short enough times. For intermediate nonlinearity, the SIC induces a chaotic (in the classical sense) diffusive dynamics that destroys the localization. For strong nonlinearity, the diffusion is again suppressed by a nonlinear effect known as "self-trapping" (which is independent of the disorder). The characterization of such behaviors is made difficult by the SIC, which in particular prevents one from drawing a "phase diagram" in the parameter plane nonlinearity vs. disorder. Using a few reasonable assumptions, however, we demonstrate that this system obey scaling laws with respect to the size of the initial state, which allows one to characterize the dynamics independently of the initial state. This is a first step in providing this important problem with a language able to take into account the effects of the nonlinearity. More generally, it puts into evidence the necessity of adapt the methods of quantum mechanics to nonlinear systems whose development will undoubtedly be driven, in the next years, by the fast experimental progress in the physics of quantum degenerate systems of ultracold atoms. (Joint work with Benoit Vermersch.)

Francois Germinet : Eigenvalues statistics of random Schrodinger operators in the localized regime.

David Guery-Odelin : The first steps of guided atom optics Slides

L'expose retrace quelques etapes de nos travaux de recherche qui vise au developpement de l'optique atomique guidee. Dans un premier temps, je propose d'expliquer la demarche experimentale qui a permis la realisation des premiers laser à atomes guides monomode transverse. Ils constituent pour les ondes de matiere l'equivalent des fibres monomodes pour l'optique. Dans un deuxieme temps, je detaillerai plusieurs experiences menees dans mon groupe où de telles ondes de matiere guidees sont exploites comme des sondes de potentiels complexes. Nous avons ainsi recemment demontre la mise au point d'un reflecteur de Bragg sur un potentiel structure dont les couches successives sont fournis par des murs de lumiere de dimension sub-micrometrique. Si ces "murs" de lumiere ont une intensite qui depend du temps, il est de plus possible de realiser des filtres de vitesse tres selectifs. Ces travaux ouvrent en particulier une nouvelle voie pour la realisation de cavite à ondes de matiere ou encore pour le developpement de composants d'atomtronique. Enfin, nous discuterons d'autres resultats experimentaux où l'onde de matiere guidee explore des potentiels classiquement chaotiques. Plusieurs geometries on ete etudiees, certaines d'entre elles donnent des pistes pour la realisation de separatrices à ondes de matiere guidees. Ces systemes semblent egalement ouvrir de nouvelles possibilites pour l'exploration du chaos classique et du chaos quantique.

Rafik Imekraz : Almost global existence for Klein-Gordon equations with small Cauchy data on a Toeplitz structure Slides

We prove a long time existence for hamiltonian nonlinear equations with small Cauchy data in Sobolev spaces. Manifolds are endowed with Toeplitz structures in the sense of Boutet de Monvel and Guillemin. This allows us to consider Szego projectors in nonlinearities. Our work uses a method developed by Delort-Szeftel and Bambusi-Delort-Grebert-Szeftel on spheres and Zoll manifolds.

Jeremy Joudioux : Decay of Dirac fields of arbitrary spin using Hertz potentials Slides

The purpose of the talk is to recover the result of decay of zero rest mass fields on flat space-time using two methods introduced by Penrose, known as spin raising and spin lowering processes. The spin raising process consists in representing the fields with potentials satisfying a scalar wave equation. The spin lowering process consists in a contraction with a spinor satisfying the twistor equation, so that the spin lowered field also satisfies a zero rest mass field equation (or a wave equation). The decay properties of these fields can then be inferred by using the standard decay result for solutions of the scalar wave equation. The reference decay result of Christodoulou & Klainerman for spin 1 and spin 2 is recovered.

Fabricio Macia : Semiclassical measures for the Schrodinger flow

We address the problem of describing semiclassical measures for sequences of solutions to the Schrodinger equation on a compact Riemannian manifold. The scaling we are interested in corresponds to performing the semiclassical limit at time scales that tend to infinity as the characteristic length scale of the oscillations of the initial data tends to zero. We present a complete description in the case of the flat torus, as part of a joint work with Nalini Anantharaman. The techniques involved include a precise description of energy concentration on resonant linear manifolds by means of two-microlocal semiclassical measures.

Kirone Mallick : Nonequilibrium Fluctuation Relations for Quantum Dynamical Systems Slides

Fluctuations in non-equilibrium systems have been shown to satisfy various remarkable relations, such as the Gallavotti-Cohen theorem and the Jarzynski-Crooks identity, that were discovered during the last twenty years. These results have lead to fierce discussions concerning the nature of heat, work and entropy, raising the fundamental issue of understanding the interactions between a given system and its environment. For a classical system, these questions have been clarified for various types of dynamics (Hamiltonian, Markovian, Langevin...). In this talk, we shall review the state of the art for quantum dynamical systems far from thermal equilibrium. In particular, we shall present an extension of the fluctuation relations to an open quantum system modeled by a Lindblad master equation that takes into account the interactions with the environment as well as measurement processes.

Phan Thanh Nam : Bogoliubov correction to the ground state energy of a large bosonic atom.

We consider the ground state energy of an atom with a large nuclear charge $Z$ and $N$ "bosonic" electrons, where the ratio $N/Z$ is fixed. It is well known that the leading term of the energy is of order $Z^3$ and it is determined by the Hartree theory. We shall discuss about the next term, which is of order $Z^2$ and comes from the Bogoliubov theory. This is joint work with Jan Philip Solovej.

Gianluca Panati : Maximally localized Wannier functions for periodic Schroedinger operators: the Marzari-Vanderbilt functional and the geometry of the Bloch bundle Slides

We consider a periodic Schrodinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum.
A crucial problem in solid state physics is the construction of a basis of Wannier functions which are exponentially localized in space, since such a basis allows to develop computational methods which scale linearly with the system size, makes possible the description of the dynamics by tight-binding effective Hamiltonians, and plays a prominent role in the modern theories of macroscopic polarization and magnetization.
The problem of proving the existence of exponentially localized Wannier functions was raised in 1964 by W. Kohn, who solved it in dimension $d=1$ in the case of a single isolated Bloch band.
We show that the obstruction to the existence of exponentially localized Wannier functions is equivalent to the triviality of a Hermitian vector bundle, canonically associated to the Schrodinger operator and the family of Bloch bands. Exploiting this insight, we are able to show that exponentially localized Wannier functions exist for any periodic Schrodinger operator which is time-reversal symmetric.
Moreover, we show that such exponentially localized Wannier functions can be obtained by minimizing the localization functional introduced by Marzari and Vanderbilt, proving existence and exponential localization of its minimizers, in dimension $d \leq 3$. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds. The latter result is a joint work with Adriano Pisante.

Quang Sang Phan : Spectral monodromy for nonselfadjoint operators. Slides

The inspiration of the quantum monodromy which is defined for the joint spectrum of a system of n commuting operators (given by Vu Ngoc San, [CMP-1999]), we ask a question mysterious :Can we define (and detect) the monodromy for the spectrum of single pseudo-differential operator?
The question is not known in the selfadjoint setting. However it turns out to work well for 2-D non-selfadjoint operators. We treat here two cases:
1. A normal operator C: = A + iB with A, B commuting.
2. A small perturbation of a self-adjoint operator of the form A + iB with principals symbols a, b commute and h converges to 0.
We also relate this monodromy, the quantum monodromy and the classical monodromy (given by Duistermaat).

Constanza Rojas Molina :Dynamical localization for Delone-Anderson operators. Slides

We review recent results on dynamical localization for Delone-Anderson models in the context of non-ergodic random Schrodinger operators. We consider both a randomly perturbed Laplacian operator and Landau Hamiltonian, where the random perturbation is of Anderson-type with the impurities lying in an aperiodic Delone set, which yields a break of ergodicity. To prove dynamical localization we use the Bootstrap Multiscale Analysis adapted to the non-ergodic setting, for which we prove uniform Wegner estimates and uniform initial length scale estimates at the bottom of the spectrum and near band edges, for the perturbed Laplacian and the perturbed Landau Hamiltonian, respectively. For the former, we get a description of the dynamical localization region with explicit dependence on the parameters of the underlying Delone set. Furthermore, for this model we prove the almost-sure existence of the Integrated Density of States (IDS) under some assumptions on the extent of aperiodicity of the Delone set. The IDS is proven to be Lipschitz continuous and to exhibit Lifshitz tails at the bottom of the spectrum.

Jacques Smulevici : Wave equations and asymptotically anti-de-Sitter spacetimes. Slides

The aim of this talk is to present recent results, obtained in collaboration with Gustav Holzegel, concerning the behaviour of waves propagating in the Schwarzschild-anti-de-Sitter and Kerr-anti-de-Sitter Lorentzian manifolds. Our main motivation is to try to understand the mechanisms for linear and non-linear stability or instability of some of the simplest solutions of the equations of general relativity, the Einstein equations. After a crash course in relativity and an brief introduction to stability problems in this context, I will focus on our two main results, a linear decay estimate for Klein Gordon fields in Kerr-anti-de-Sitter and a proof of asymptotic stability for Schwarzschild-anti-de-Sitter for the spherically-symmetric Einstein-Klein-Gordon system.

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