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Quantum Reflection from the Casimir-Polder Potential

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Quantum Reflection from the Casimir-Polder Potential THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité : Physique École doctorale : “Physique en Île-de-France” réalisée au Laboratoire Kastler-Brossel présentée par Gabriel DUFOUR pour obtenir le grade de DOCTEUR DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Sujet de la thèse : Réflexion quantique sur le potentiel de Casimir-Polder – Quantum reflection from the Casimir-Polder potential soutenue le 20 novembre 2015 devant le jury composé de M Andreas Buchleitner Rapporteur M David Guéry-Odelin Rapporteur Mme Marie-Christine Angonin Examinatrice M Olivier Dulieu Examinateur M Valery Nesvizhevsky Examinateur Mme Astrid Lambrecht Directrice de thèse i À la mémoire de mes grands-pères, Jean Dufour et Michel Durin ii Quand j’ai poussé la porte du Laboratoire Kastler Brossel en novembre 2011, je re- venais d’un mois de voyage à vélo sur les routes et chemins d’Italie. Astrid Lambrecht et Serge Reynaud m’ont convaincu de me lancer dans une nouvelle aventure, avec son lot d’ascensions ardues, de petites victoires, de problèmes techniques et de découvertes grisantes. Astrid et Serge m’ont accompagné sans relâche au cours du stage et de la thèse qui ont suivi. Je les remercie chaleureusement pour leur gentillesse et leur patience, leurs mots d’encouragement et les nombreuses discussions passionnantes que nous avons eues. Marie-Pascale Gorza, Romain Guérout, Manuel Donaire et Axel Maury ont été mes compagnons de route au sein de l’équipe “fluctuations quantiques et relativité”. Leurs conversations stimulantes et animées ont égayé bien des austères journées de travail. J’ai aussi partagé de très bons moments avec mes collègues et amis du laboratoire et d’au delà, que ce soit dans les courants d’air de la cantine, sous le soleil des arènes ou dans l’ambiance chaleureuse d’une cafétéria. En plus de cette atmosphère amicale, j’ai bénéficié au Laboratoire Kastler Brossel et à l’Université Pierre et Marie Curie de conditions de travail exceptionnelles, je voudrais en remercier les équipes techniques et administratives. J’ai eu la chance de participer à la collaboration GBAR, un projet ambitieux qui rassemble des chercheurs de tous les horizons géographiques et scientifiques, de Tokyo à Swansea et de la physique des accélérateurs à celle des particules ultrafroides. J’ai hâte que cette belle expérience voie le jour et livre ses premiers résultats. J’ai notamment col- laboré étroitement avec Alexei Voronin et Valery Nesvizhevsky que je remercie vivement pour leurs idées fructueuses et leur enthousiasme communicatif. Merci aussi à Valery pour son accueil à l’Institut Laue-Langevin et l’inoubliable visite du réacteur. Valery me fait également l’honneur de participer à mon jury de thèse, aux côtés de Marie-Christine Angonin, Olivier Dulieu, David Guéry-Odelin et Andreas Buchleitner que je remercie sincèrement de l’intérêt dont ils témoignent ainsi pour mon travail. Je suis particulièrement reconnaissant envers David Guéry-Odelin et Andreas Buchleitner qui ont accepté d’être rapporteurs de ma thèse et je me réjouis d’aller travailler à Freiburg avec Andreas l’année prochaine. Un grand merci enfin à ma famille et à mes amis, qui m’ont encouragé, questionné, accompagné, consolé, soutenu et supporté, de près ou de loin, tout au long de ce périple. Gabriel Dufour Contents Notation vi Introduction1 The GBAR experiment . .2 Quantum mechanics in a gravitational field . .4 Casimir forces . .5 Quantum reflection . .6 Outline of the thesis . .8 I Quantum reflection9 I.1 Classical and quantum mechanics in one dimension . .9 I.1.a Classical mechanics in one dimension . .9 I.1.b Schrödinger equation in one dimension . 11 I.1.c Transmission and reflection on a step . 14 I.2 Quantum reflection and the WKB approximation . 17 I.2.a The WKB approximation . 17 I.2.b The badlands function . 20 I.2.c Coupled WKB waves . 21 I.2.d Scattering matrix . 24 I.3 Liouville transformations of the Schrödinger equation . 25 I.3.a Liouville transformations . 25 I.3.b Trivial coordinates . 27 I.3.c Dimensionless coordinate . 28 I.3.d WKB coordinate . 28 I.3.e Langer coordinate . 30 I.3.f Extension to higher dimensions . 32 I.4 Semiclassical and quantum regimes . 34 II Reflection from the Casimir-Polder potential 38 II.1 Casimir-Polder interaction . 39 II.1.a Casimir free energy for two objects in vacuum . 39 II.1.b Scattering on a plane and an atom . 40 II.1.c Expression of the Casimir-Polder potential . 43 iii CONTENTS iv II.2 Casimir-Polder potential near various surfaces . 44 II.2.a Thick and thin slabs . 44 II.2.b Potential near an undoped graphene sheet . 50 II.2.c Effective medium treatment of porous materials . 51 II.3 One-way quantum reflection . 56 II.3.a Behavior of the badlands function . 56 II.3.b Boundary condition on the surface . 58 II.3.c Using the coupled equations for WKB waves . 60 II.3.d Liouville transformation from a well to a wall . 61 II.3.e Reflection probabilities . 66 II.3.f Scattering lengths . 68 II.3.g Comparison with model potentials . 69 II.3.h Consequences for GBAR . 76 II.4 Two-way quantum reflection . 77 II.4.a Changing the boundary condition on the surface . 77 II.4.b Successive scattering processes . 78 IIIQuantum free fall 83 III.1 Free fall of a matter wave . 83 III.1.a Schrödinger equation in an accelerated frame . 83 III.1.b Particle in a uniform gravity field . 86 III.1.c Wigner function . 88 III.1.d Free falling wavefunctions . 91 III.2 Propagation of errors for the fall of a wavepacket . 94 III.2.a A first, simple calculation . 94 III.2.b Quantum probability current through a surface . 95 III.2.c Squeezed arrival time distribution . 98 III.3 Stationary states . 98 III.3.a Airy wavefunctions . 98 III.3.b Decompositions on the stationary state basis . 101 III.3.c WKB approximation . 102 IV Quantum bouncers 104 IV.1 Toy models . 104 IV.1.a Infinite step . 105 IV.1.b Waveguide . 108 IV.2 Atom bouncing above a real surface . 110 IV.2.a Quasi-stationary states . 110 IV.2.b Resonances in Langer coordinate . 112 IV.3 Shaping the distribution of vertical velocities . 116 IV.3.a Limits on the precision of GBAR . 116 IV.3.b Velocity shaper for GBAR . 121 IV.3.c Classical trajectories in the shaper . 121 IV.3.d Estimation of the statistical uncertainty . 125 CONTENTS v Conclusions and perspectives 129 Appendices 131 A The Schrödinger equation in 1D . 131 A.1 Wronskians . 131 A.2 Transfer and scattering matrices . 134 A.3 Unitarity and reciprocity . 136 A.4 Sturm-Liouville equations and their transformations . 138 B Reflection from homogeneous potentials . 140 B.1 Scattering lengths . 140 B.2 Liouville transformation to the WKB coordinate . 142 B.3 Exact solution for the C4 potential . 144 C Airy functions . 146 C.1 Definition . 147 C.2 Asymptotic behavior and zeros . 148 C.3 Integrals . 148 C.4 Airy wavepackets . 149 D Time dependent quantum harmonic oscillator . 150 D.1 Solution in terms of classical trajectories . 151 D.2 Parametric creation of quanta . 152 D.3 Liouville transformations . 155 D.4 Shortcuts to adiabaticity . 157 List of publications 159 Bibliography 161 Notation General • ~r = x~ex + y~ey + z~ez is a position vector written in the usual Cartesian basis. • We often single out one direction and denote z or ζ the corresponding coordinate. • The time variable is denoted t or τ. df • f(x) denotes a function of the variable x and f 0(x) = (x) is its derivative; a dx ∂ partial derivative with respect to x is denoted . ∂x • The Schwarzian derivative {f, x} is defined by f 000(x) 3 f 00(x)2 {f, x} = − . f 0(x) 2 f 0(x) • The Wronskian of two functions f(x) and g(x) is W(f, g) = f(x)g0(x) − f 0(x)g(x) . • Stars and daggers denote complex and hermitian conjugation, respectively. Constants −34 • Reduced Plank constant ~ ≈ 1.05457 × 10 J.s • Speed of light c ≈ 2.99792458 × 108 m.s−1 −12 −1 • Vacuum permittivity ε0 ≈ 8.85419 × 10 F.m −23 −1 • Boltzmann constant kB ≈ 1.38065 × 10 J.K • Fine structure constant α ≈ 1/137.036 −27 • Mass of the hydrogen atom mH ≈ 1.67353 × 10 kg vi Notation vii Units All equations are written in the International System of units but numerical values are sometimes expressed in more convenient units. The energies involved in quantum reflection processes are of the order of the nanoelectronvolt: neV ≈ 1.60217646 × 10−28 J Atomic units (a.u.) are well adapted to express scattering lengths and Casimir-Polder potentials, the atomic length and energy units are respectively −11 • Bohr radius a0 ≈ 5.29177 × 10 m −18 • Hartree Eh ≈ 4.35974 × 10 J Quantum effects in a gravitational field involve the following scales: 2 !1/3 • Length z = ~ ≈ 5.87 × 10−6 m g 2m2g !1/3 2mg2 • Energy E = ~ ≈ 9.64 × 10−32 J ≈ 6.02 × 10−4 neV g 2 2 1/3 −29 −1 • Momentum pg = (2~m g) ≈ 1.80 × 10 kg.m.s 2 1/3 • Time t = ~ ≈ 1.09 × 10−3 s g mg2 where the numerical values are given assuming that m is the mass of the hydrogen atom and g = g. Acronyms and symbols • WKB Wentzel, Kramers and Brillouin semiclassical approximation • GBAR Gravitational Behavior of Antihydrogen at Rest • H hydrogen atom H antihydrogen atom • p proton p antiproton • e− electron e+ positron • Ps positronium Notation viii Special functions We follow the definitions and notations of the NIST Handbook of Mathematical Func- tions [1]: • Ai(x), Bi(x) Airy functions • Ai(−an) = 0 zeros of the Airy function • F (a, b, c; x) Gauss hypergeometric function (1,2) • Hn (x) Hankel functions • Ln(x) Laguerre polynomials • M(a, b, x), U(a, b, x) Kummer confluent hypergeometric functions Introduction In 1924, de Broglie made the groundbreaking hypothesis that every microscopic particle of momentum p is associated with a wave of wavelength h λ = , (1) dB p where h = 2π~ is Planck’s constant [2].
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