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Beyond Pilot Wave Dynamics: Non-Linearity and Non-Equilibrium in Quantum Mechanics

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Beyond Pilot Wave Dynamics: Non-Linearity and Non-Equilibrium in Quantum Mechanics ECOLE CENTRALE MARSEILLE -INSTITUT FRESNEL THÈSE PRÉSENTÉE À L’ECOLE CENTRALE DE MARSEILLE POUR OBTENIR LE GRADE DE DOCTEUR Beyond pilot wave dynamics: non-linearity and non-equilibrium in quantum mechanics Autheur: Directeur de thèse: Mohamed HATIFI Pr. Thomas DURT Spécialité: Physique Théorique Ecole Doctorale 352 - Physique et Sciences de la Matière Soutenue le 24 septembre 2019 devant le jury composé de : M. Christos EFTHYMIOPOULOS Research Center for Astronomy and Rapporteur Applied Mathematics M. Jacques ROBERT Laboratoire de Physique des Gaz Rapporteur et des Plasmas (LPGP) M. Alexandre MATZKIN Laboratoire de Physique Théorique Examinateur et Modélisation (LPTM) Mme Fabienne MICHELINI Institut Matériaux Microélectronique Examinatrice Nanosciences de Provence (IM2NP) M. Thomas DURT Institut Fresnel Directeur de thèse iii “L’existence du quantum d’action (. ) implique une sorte d’incompatibilité entre le point de vue de la localisation dans l’espace et dans le temps et le point de vue de l’évolution dynamique (. ) La localisation exacte dans l’espace et le temps est une sorte d’idéalisation statique qui exclut toute évolution et toute dynamique. (. ) Dans la mécanique classique, il était permis d’étudier pour eux-mêmes les déplacements dans l’espace et de définir ainsi les vitesses, les accélérations sans s’occuper de la façon dont sont matériellement réalisés ces déplacements : de cette étude abstraite des mouvements, on s’élevait ensuite à la dynamique en introduisant quelques principes physiques nouveaux. Dans la mécanique quantique, une semblable division de l’exposé n’est plus en principe admissible puisque la localisation spacio-temporelle qui est à la base de la cinématique est acceptable seulement dans une mesure qui dépend des conditions dynamiques du mouvement...” Louis de Broglie - « La physique nouvelle et les quanta » - (1937) v ECOLE CENTRALE MARSEILLE - INSTITUT FRESNEL Abstract (French) Institut Fresnel Ecole Doctorale 352 - Physique et Sciences de la Matière Doctor of Philosophy Beyond pilot wave dynamics: non-linearity and non-equilibrium in quantum mechanics by Mohamed HATIFI La mécanique quantique a modifié notre façon d’interpréter ce que jadis l’on appelait communément "réalité physique". A titre d’exemple, selon l’interprétation standard de la mécanique quantique (dite interprétation probabiliste de Copenhague) les propriétés d’un objet quantique n’ont pas de réalité physique, du moins, pas avant que l’observateur ne les mesure. De plus, tout semble se passer comme s’il y avait un indéterminisme in- trinsèque à la dynamique quantique qui ne permettrait pas de prédire avec certitude le résultat d’une mesure. Dès lors, plusieurs interprétations physiques et philosophiques ont vu naissance afin de décrire (notre con- naissance de) cette réalité. C’est au cours de la conférence de Solvay en 1927 que Louis de Broglie, un opposant à l’interprétation prob- abiliste, proposa une solution alternative qui permettait d’une part de restaurer le déterminisme (ainsi que le réalisme) et d’autre part de remettre au premier plan la notion de trajectoire. Par la suite cette théorie fut redécouverte et complétée par David Bohm pour donner naissance à la théorie connue aujourd’hui sous l’appellation de théorie de l’onde pilote. John Bell a dit1 à propos de cette interprétation : " En 1952, l’impossible a été rendu possible. C’était dans l’article de David Bohm. Bohm a montré explicitement comment une de- scription indéterministe pouvait être transformée en théorie déterministe." Les travaux présentés dans ce manuscrit de thèse s’inscrivent dans la continuité de la vision de de Broglie et consistent en deux parties, chacune d’elles ayant pour but de répondre à une problématique particulière. Dans la première, on considère deux formalismes du type onde pilote, une version déterministe (dynamique de de Broglie-Bohm chapitre2) ainsi qu’une de ses extensions stochastiques (dynamique de Bohm-Hiley- Nelson chapitre3). On s’attardera notamment sur l’émergence de la probabilité quantique jY(x, t)j2 à partir de ces dynamiques dans l’ approche dite du "Quantum Non-Equilibrium". Cette approche permet entre autres de s’affranchir du statut axiomatique de la distribution de probabilité jY(x, t)j2 mais aussi de la justifier par des arguments similaires à ceux que l’on retrouve en mécanique statistique. Parmi ces arguments on retrou- vera à titre d’exemple la notion d’ergodicité, de chaos, de mixing ainsi que d’autres propriétés qui feront l’objet d’une étude approfondie (chapitre4). En particulier, l’ émergence de l’ équilibre s’accompagne d’un processus de relaxation que nous allons caractériser dans chacune de ces dynamiques (dans le chapitre3 nous dériverons un théorème H qui décrit quantitativement ce processus dans le cas stochastique). Par ailleurs, nous nous efforcerons, dans une approche phénoménologique, d’appliquer ces théories quantiques d’onde pilote à la dynamique macroscopique des gouttes d’huile rebondissantes dans un bain (chapitre5). La deuxième problématique quant à elle, repose sur une hypothétique généralisation non-linéaire de la mé- canique quantique. En particulier, nous considérerons l’équation de Schrodinger Newton comme une pre- mière proposition a cette généralisation. Cette équation non-linéaire découle d’une approximation semi- classique de la gravité et a été entre autres proposée par Roger Penrose pour expliquer le collapse de la fonction d’onde. Nous montrerons dans un premier temps comment le programme de la double solution de Louis de Broglie se développe dans ce contexte (chapitre6). Par la suite nous verrons comment tester cette généralisation non-linéaire par deux propositions expérimentales (chapitre7). En particulier, l’une de ces propositions nous conduira à étudier des effets de décohérence lors du refroidissement laser (Doppler cooling, chapitre8). Pour cela on utilisera le modèle de Ghirardi–Rimini–Weber (GRW) comme modèle de décohérence. Ce qui nous permettra par la suite de généraliser les résultats obtenus auparavant par GRW dans leur modèle. 1“On the impossible pilot wave” -1982 vii ECOLE CENTRALE MARSEILLE - INSTITUT FRESNEL Abstract (English) Institut Fresnel Ecole Doctorale 352 - Physique et Sciences de la Matière Doctor of Philosophy Beyond pilot wave dynamics: non-linearity and non-equilibrium in quantum mechanics by Mohamed HATIFI The quantum theory has modified the way we interpret what in the past was commonly called "physical reality". As an example, according to the standard interpretation of quantum mechanics (the so-called prob- abilistic interpretation of Copenhagen), the properties of a quantum object have no physical reality, at least not before the observer measures them. Moreover, everything seems to happen as if there was an intrinsic indeterminism in the quantum dynamics that forbids to predict with certainty the result of a measurement. From then, several physical and philosophical interpretations were born to describe (our knowledge of) this reality. It is in 1927, during the Solvay conference, that Louis de Broglie, an opponent of the probabilistic interpreta- tion, proposed an alternative solution to that problem. He proposed on the one hand to restore determinism (as well as realism) and on the other hand to bring back the notion of trajectory to the foreground. Subse- quently this theory was rediscovered and supplemented by David Bohm to give birth to the theory known today as pilot wave theory. John Bell said2 about this interpretation: " In 1952, I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how .... the indeterministic description could be transformed into a deterministic one." The works carried out in this manuscript are in continuity with de Broglie’s view and can be summed up in two main parts, each of them having the aim of answering a particular problem. In the first part, we con- sider two versions of the pilot wave theory: a deterministic version (de Broglie-Bohm dynamics in chapter2) as well as one of its stochastic extensions (Bohm-Hiley-Nelson dynamics in chapter3). In the framework of what is called the "Quantum non-equilibrium" approach we shall see how the quantum probability jY(x, t)j2 emerges from those dynamics. This approach makes it possible to get rid of the axiomatic status of the proba- bility distribution jY(x, t)j2 but also to justify it by arguments similar to those found in statistical mechanics. Among these arguments we shall for instance find ergodicity, chaos, mixing and other properties that will be studied in depth (chapter4). In particular, the emergence of the quantum probability is accompanied by a relaxation process that will be characterized for both dynamics (in chapter3 we derive a strong H-theorem for the stochastic dynamics which quantitatively describes how this process occurs). In addition, we will try in a phenomenological approach to apply these quantum pilot wave theories to the macroscopic dynamics of bouncing oil droplets (chapter5). The second problem is linked to a hypothetical nonlinear generalization of the quantum theory. In partic- ular, we considered the Schrodinger Newton equation as a first proposal to this generalization. In a nutshell, this non-linear equation derives from a semi-classical approximation of gravity and has been proposed by Roger Penrose among others to explain the collapse of the wave function. We shall first show how it is related to the double solution program of Louis de Broglie (chapter6). Subsequently we will see how to test this nonlinear generalization by considering two experimental proposals (chapter7). In particular, one of these proposals will lead us to study the interplay between decoherence and Doppler cooling (chapter8). To do this we shall use the model of Ghirardi-Rimini and Weber (GRW) as a decoherence model, which will allow us to generalize their original results. 2“On the impossible pilot wave” -1982 ix Acknowledgements Au nom de Dieu. Papa, je me souviens encore de ce jour, ce jour où tes dernières paroles ont résonné à travers mon cœur et mon âme. Ce jour-là, je t’ai fait entre autres la promesse de faire des études.
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