Quantum interferences of gravitational quantum states 

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Testing the equivalence principle for antimatter

The recognition that the motion of objects in a gravitational field is independent of their mass and composition was central to the birth of modern science in the 17th century. The universality of free fall or weak equivalence principle is a cornerstone of Einstein’s General Relativity. Today, the fact that all bodies undergo the same acceleration g at a given point on the surface of the Earth is verified with ever increasing precision both for macroscopic test masses [40] and atoms [41, 42]. The space mission MICROSCOPE tests the weak equivalence principle with a relative precision of 10−15 by comparing the free fall of a macroscopic platinum and titanium masses. A preliminary result confirms the weak equivalence principle [43] up to a relative precision of 2 • 10−14.
However gray areas remain. The mysteries surrounding dark matter and energy re-mind us that our knowledge of gravitational forces is still incomplete. Our ignorance of how gravity and other interactions are articulated at a fundamental level leaves some room for alternative proposals which include violations of the weak equivalence principle. In particular, the possibility of an asymmetry in the gravitational behavior of matter and antimatter has been raised [4, 44–47]. Although theoretical arguments and exper-imental observations have been put forward against « antigravity” [5, 7, 48], a direct, model-independent test of the universality of free fall for antimatter is still lacking. A direct measurement of the acceleration g of an antimatter particle in the Earth’s gravity field is a longtime objective of physicists. Early experiments with charged antiparticles were thwarted by the preponderance of electromagnetic forces over gravity [49]. Current experimental endeavors are thus concentrating on neutral particles, especially the an-tihydrogen atom. The antihydrogen atom (H) is the bound state of an antiproton (p¯) and a positron (e+); it was first produced at high energies in CERN in 1995 [50]. Since then, much progress has been made towards lower temperatures and longer lifetimes in several experiments based around CERN’s Antiproton Decelerator [51, 52].
At CERN, the new deceleration ring ELENA [53] provides cooler antiprotons to a new generation of antimatter experiments. Antiprotons are produced by collisions of 26 GeV protons with a target and cooled down to approximately 100 keV by the AD and ELENA rings. Antiprotons are then used in different experiments. AEGIS aims to measure the deflection of a produced beam of antihydrogen atoms using a Moiré deflectometer [54]. There is also a proposal to build an interferometric gravimeter in the ALPHA experiment [55].
In particular, the GBAR experiment will consist in cooling antihydrogen to the ground state of a harmonic ion trap before releasing it in the Earth’s gravity field and timing its free fall [56, 57]. For doing that, antiprotons coming from the ELENA ring are slowed down electrostatically to approximately 1 keV before reaching the reaction chamber. As explained in [36], the specificity of GBAR is that it will produce the antihydrogen ion H+, two positrons orbiting an antiproton, in order to take advantage of ion trapping and cooling techniques [13]. Once the ion is cold, a laser will be used to photodetach the excess positron, letting the neutral antihydrogen atom fall freely towards a detection plate.
The H+ ion will be produced by the successive reaction of an antiproton with two positroniums (the bound state of electron and positron):
Producing these reactants and bringing them together in the right conditions will con-stitute an impressive experimental feat.
Positronium is formed by implanting positrons in a porous silica sample. The positron captures an electron and the resulting positronium diffuses in the network of nanometric pores until it is expelled back into the vacuum with a well defined energy [58]. The positrons themselves are obtained in the collision of 10 MeV electrons from a linear accelerator (LINAC) with a target. They must then be moderated, accumulated in a Penning trap and sympathetically cooled by a cloud of electrons before being sent on the porous silica sample. The positronium will be excited by laser to a higher energy state in order to maximize the cross-section of the reactions [59].
At the output of the reaction chamber, the H+ ions are separated from the neutral H and the negatively charged p¯ and collide with a Coulomb crystal of laser-cooled Be+ and HD+ ions. Their energy is efficiently transferred to the ions of the crystal thanks to the small mass ratios between these three species. One H+-Be+ pair is then trans-fered to a Paul trap. Raman sideband cooling is performed on the beryllium ion, which has the effect of sympathetically cooling the antihydrogen ion to the ground state of the harmonic trap [60]. A laser pulse is then used to photodetach the extra positron, defining the starting time of the free fall of the neutral antihydrogen atom. The fall ends by the annihilation of the anti-atom on a detection plate some 10 cm below. The annihilation products (pions and gamma photons) are detected by Micromegas detectors [61] and scintillation counters placed outside the vacuum vessel. The acceleration g of antihydrogen is deduced from the free fall time. A 1% precision on g is expected in this scheme.

Quantum reflection

The classical description of an antihydrogen atom as a point particle is not sufficient at low temperature. In the formalism of the quantum mechanics, cold antihydrogen atom is described by a wavefunction obeying the Schrödinger equation. The atoms live in a gravitational potential described classically (relativistic effects are negligible), and the Casimir-Polder potential.
The confinement of a cold atom trapped above a perfectly reflecting surface in the gravity field leads to gravitationally bound states. These gravitationally bound states have first been observed with ultracold neutrons at the Institut Laue-Langevin [62–64]. Though atomic mirrors have been realized using inhomogeneous electric or magnetic fields [65–72], gravitationally bound quantum states of atoms remain to be observed.
The presence of the attractive Casimir-Polder potential induces quantum reflection phenomena: antihydrogen atoms have a probability to bounce above the surface, pre-venting their annihilation. Observation of the phenomenon has only been achieved rela-tively recently for atoms, because very low temperatures are required to reach sufficiently large wavelengths. The first experiments where carried out in the 1980s with helium and hydrogen atoms scattering off a liquid helium surface [23, 73, 74]. It took several more years before reflection from solid surfaces was observed, first with beams of atoms inci-dent on a surface at grazing incidence [26, 27] and later with Bose-Einstein condensates falling towards the surface at normal incidence [28]. Since, a number of experiments have been carried out with rough or nanostructured surfaces [31, 32, 75–80].
In this thesis, we compute precisely the Casimir-Polder potential, taking also into account the heterogeneities in the particular case of metallic surfaces and see how it modifies the potential and eventually the quantum reflection. Despite the progresses in the understanding of the Casimir effect, there still exists a disagreement between exper-iment and theory in the Casimir community, called the « Drude vs plasma puzzle ». Some experiments [81–83] fit better with a theoretical model where the finite conductivity of the metallic medium (Drude model) is supposed infinite (plasma model). With the present work, we thus give now a perspective to probe the model of dissipation in the Casimir-Polder formula, testing the « Drude vs plasma puzzle ».
The name « quantum reflection” emphasizes the contrast between the classical and quantum dynamics. In fact, this phenomenon is a general feature of wave propagation in inhomogeneous media [84]. It is met for atmospheric and oceanic waves for example, or electromagnetic waves in dielectrics and transmission lines. Roughly speaking, quantum reflection occurs in regions where the wavelength varies rapidly. In this thesis, we use frequently the Liouville transformations that give an intuitive picture of the quantum reflection by transforming the reflection on a well into a reflection on a wall.
If we consider the gravity acting identically for atoms and antiatoms, the antihydrogen atoms are trapped between gravity pulling them downwards and quantum reflection balancing their free fall [37, 38]. It leads to the existence of quantum levitation states for ultracold matter waves. We perform for the first time a full quantum treatment of gravity and Casimir-Polder interaction. We compute precisely the Casimir-Polder shifts on gravitationally bound quantum states induced by the Casimir-Polder interaction. A precise knowledge of this phenomenon leads to a new proposition to measure the free fall acceleration for antihydrogen by quantum measurement techniques [85–87].
In this thesis, we use the phenomenon of quantum reflection to propose a new exper-imental setup keeping almost all antihydrogen atoms, consisting of adding a reflecting surface to generate interferences between quantum states, before the free fall of the an-tihydrogen on the detector, thus producing interference pattern containing much more information on g that the classical free fall time. The quantum nature of the quan-tum levitation states are thus fully involved, producing beautiful interference figures. Doing that, we prove that the accuracy of the initial GBAR design by three orders of magnitude.

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Outline of the thesis

Chapter I of this thesis is devoted to the theoretical study of quantum reflection on a Casimir-Polder potential. We precisely calculate the quantum reflection on a liquid helium bulk, that leads to the highest reflectivity of an antihydrogen atom above a surface and to a surprising manifestation of a shape resonance when varying the thickness of liquid helium film. We also introduce the Liouville transformations, an important tool to understand the quantum reflection and make quantitative calculations.
In chapter II we perform a full quantum treatment of gravity and Casimir-Polder potential. We introduce the Liouville-Langer coordinates to study the quantum levitation states, in a new physical picture corresponding to a cavity built up with two mirrors, a partly reflective one associated with quantum reflection and a perfectly reflecting one due to gravity. We develop also a new effective range theory improving the expansion of the scattering length at low energy. We finally calculate the properties of the quasi-stationary states of the quantum bouncer with a high accuracy.
In chapter III we propose a new measurement technique of the antihydrogen free fall acceleration making interfering quantum levitation states. We describe the experimental setup, that could be implemented in the GBAR experiment, and derive the evolution of the atomic wave packet from the photodetachment to the detection. We also present statistical methods to extract an estimation of g and give the standard deviation that is much smaller than the one achieved with the free fall timing experiment.
Chapter IV is focused on the details of the Casimir-Polder interaction. We describe the fluctuations of the Casimir-Polder potential due to the heterogeneities of the medium and see if it could affect the quantum reflection. We also make a more general discussion on the model used for describing the dissipation in metal in the « Drude vs plasma » puzzle, by finding a different asymptotic behavior for the two models.
The first chapter introduces the phenomenon of quantum reflection. Quantum reflec-tion has been studied theoretically for the attractive van der Waals potential since the early days of quantum mechanics [20, 21]. Theoretical treatments of the effect are re-viewed for example in [22, 88]. It was first observed experimentally for H and He atoms [23, 73, 74] and then for ultracold atoms or molecules on solid surfaces [26–28, 31].
The interest of studying quantum reflection also for antimatter has been noticed more than ten years ago [85, 89, 90] and it should play a key role in experiments with anti-hydrogen atoms [85, 90–93]. The precise knowledge of this phenomenon is in particular crucial for spectroscopic studies of the quantum levitation states [62, 94] of antihydrogen atoms trapped by quantum reflection and gravity [37, 38].
I.1. We also present the scattering length, an important quantity in the scattering theory that encodes the reflection. We introduce the Liouville transformation of the Schrödinger equation in section I.2, an elegant tool that helps us to intuitively and quantitatively appreciate the counter-intuitive phenomenon of quantum reflection by changing the po-tential landscape while preserving the reflection amplitudes. Then, we discuss the case of an atom interacting with a surface, through the attractive Casimir-Polder potential. We compute the quantum reflection for a liquid helium bulk in section I.3, that offers a very high reflectivity. We finish by taking into account the finite thickness of the liquid helium film in section I.4. We also compute the Casimir-Polder potential and the quan-tum reflection for different film thicknesses and highlight scattering length oscillations that are explained by an adapted Liouville transformation.

Table of contents :

I Quantum reflection on the Casimir-Polder potential 
I.1 Quantum reflection
I.1.a Definition
I.1.b Helmholtz and Schrödinger equations
I.1.c The WKB approximation
I.1.d Badlands
I.1.e Reflection amplitude
I.1.f Scattering matrix
I.1.g Reciprocity theorem
I.1.h Scattering length
I.2 Liouville transformation of the Schrödinger equation
I.2.a Liouville transformation group
I.2.b Liouville transformations for the V4 potential
I.2.c Analytical solution with Mathieu coordinates
I.2.d WKB phase: badlands
I.3 Quantum reflection on liquid helium bulk
I.3.a Casimir interaction
I.3.b Casimir-Polder interaction
I.3.c Casimir-Polder potential for a liquid helium bulk
I.3.d Quantum reflection on liquid helium bulk
I.4 Quantum reflection on liquid helium film
I.4.a Casimir-Polder potential dependence on film thicknesses
I.4.b Scattering length oscillations
I.4.c Shape resonance
II Casimir-Polder shifts on quantum levitation states 
II.1 Gravitational quantum states
II.1.a Quantum bouncers
II.1.b Airy functions
II.1.c Quantization of gravitational bound states
II.1.d Scattering length approximation
II.2 Improved effective range theory
II.2.a Effective range theory for the V4 potential
II.2.b Effective range theory for a potential with a V4 tail
II.2.c Necessity of a new effective range theory
II.2.d V3 tail
II.2.e Scattering matrix composition
II.2.f Derivation of A˜
II.2.g Advantages of the new effective range theory
II.3 Liouville-Langer transformation
II.3.a Turning point
II.3.b Langer coordinates
II.3.c Transformed potential landscape
II.4 Quantum levitation states
II.4.a Fabry-Perot cavity
II.4.b Round-trip factor: numerical analysis
II.4.c Round-trip factor: analytical expression
II.4.d Complex Casimir-Polder shifts
III Quantum interferences of gravitational quantum states 
III.1 Free fall of a matter wave
III.1.a Description of the free fall timing experiment
III.1.b Time evolution of the wavefunction
III.1.c Wigner function
III.1.d Current on the detector
III.2 Estimation of the uncertainty
III.2.a Classical time uncertainty
III.2.b Uncertainty estimation from the probability current distribution
III.2.c Monte-Carlo simulation
III.2.d Cramer-Rao lower bound
III.3 Interferences of gravitational quantum states
III.3.a Experimental setup
III.3.b Interferences above mirror
III.3.c Interference pattern on the detector
III.3.d Uncertainty estimation
IV Casimir-Polder fluctuations 
IV.1 Mean Casimir-Polder potential
IV.1.a Qualitative description of the interaction between an atom and a metallic medium
IV.1.b Green tensor of the Helmholtz equation
IV.1.c Fresnel coefficients
IV.1.d Drude vs plasma puzzle
IV.2 Fluctuations of the Casimir-Polder potential
IV.2.a General expression of fluctuations
IV.2.b Correlations in metal
IV.2.c Magnitude of the fluctuations
IV.2.d Asymptotic behavior of fluctuations
IV.2.e Possibility of an experimental test
IV.3 Effect of the temperature
IV.3.a Casimir-Polder potential at finite temperature
IV.3.b Fluctuations of the potential at finite temperature
Conclusion and perspectives

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