# Ebook: Atom Optics and Space Physics

**Description**

The goal of this volume is to discuss the rapidly moving field of atom optics and interferometry with all its intricate aspects ranging from fundamental physics to applications and the theory of relativity. The breathtaking success in manipulating atoms using lasers has encouraged these two so far disjunct communities to move closer together and begin collaborations. After an introduction to atom optics and Bose-Einstein condensation, the theoretical foundations of cold atom interferometers, their use to test gravity, and their implementation in laboratory measurements of the Earth rotation and of Newton's gravitational constant are discussed. Several papers discuss the characteristics of gyroscopes and interferometers as sensors for inertial forces, starting from gyroscopes based on light waves and comparing their sensitivity to those based on matter waves. The final topic is the variation of fundamental constants, a subject that during the last years has attracted a lot of attention from different communities of physics.

**Preface**

The International School of Physics “Enrico Fermi” on Atom Optics and Space Physics was organized by the Italian Physical Society in Villa Monastero, Varenna, Italy, during July 2-13, 2007, in collaboration with the Wilhelm und Else Heraeus-Stiftung. In the tradition of the Fermi Schools the main goal was to highlight within the physics community an advanced topic in a rapidly expanding area of research.

Wave optics with cold and ultracold atoms has experienced an enormous progress during the last few years, as recognized by the Nobel prizes for laser cooling in 1997, and for Bose-Einstein condensation in 2001. The high control in the production of ultracold atoms and molecules, and in the quantum engineering of the internal and external degrees of freedom of those systems has provided us with new tools for detailed investigations. Moreover, quantum optics of atom waves has reached a new level of refinement with the production of squeezed atomic samples and the non-linear generation of matter waves. Finally major breakthroughs with interferometers based on cold atoms demonstrated that atom optics has become a mature technology which produces matter wave sensors with unprecedented sensitivity for metrology and fundamental physics.

So far most atom optics experiments have been performed on Earth. Only a few have been made in free-fall, either in airplanes or in a drop tower, as to eliminate the influence of gravitation on the motion of the atoms. However, ultra-high precision experiments with atoms ask for ultra-low velocities to surpass the present limitations. In this regime experiments on Earth are limited by gravitation which is one of the main motivations for placing a cold atom interferometer in Space. This approach has to be viewed in the context of ongoing activities using state-of-the art cryogenic, superconducting or mechanical sensors to measure predictions of general relativity such as the Lense-Thirring effect or to test the Equivalence Principle. The Lense-Thirring effect is a manifestation of a “force” produced by the rotation of a massive object. It can also be thought of as the result of the “dragging of inertial frames”, a name proposed by Albert Einstein.

The goal of our School was to teach the rapidly moving field of atom optics and interferometry with all its intricate aspects ranging from fundamental physics to applications together with the theory of relativity. The breathtaking success in manipulating atoms using lasers has encouraged these two so far disjunct communities to move closer together and begin collaborations. However, further progress in this highly advanced branch of physics requires the training of young researchers with a combined expertise.

The general outline of the School can be characterized by a few topics. Starting points were the basic concepts of special and general relativity, including the formulations of rotation in relativity, the Sagnac effect and the basic elements of the Lense-Thirring effect and gravito-magnetism. Seminars on theoretical as well as experimental aspects of gravito-magnetism including lectures on the Gravity Probe B experiment completed this theme. The following lectures covered dark matter and dark energy, tests of general relativity in the Solar System including the Pioneer anomaly, and the gravitational decoherence modifying the response of a matter wave interferometer. Then detection of gravitational waves with resonant bar detectors, and with light interferometers on Earth and in Space was discussed.

A major issue of the School was the interface between cold atoms and relativity. After an introduction to atom optics and Bose-Einstein condensation, the theoretical foundations of cold atom interferometers, their use to test gravity, and their implementation in laboratory measurements of the rotation of the Earth and of Newton's gravitational constant were presented. Several lecturers reviewed the characteristics of gyroscopes and interferometers as sensors for inertial forces, starting from gyroscopes based on light waves and comparing their sensitivity to those based on matter waves. The scientific objectives and the technological developments of the atom interferometers associated with the space programs and ground projects were discussed. The presentation of a unifying theoretical approach towards space-time sensors based on optical or matter wave interferometers using optics in five dimensions and of new free-fall absolute gravimeters completed the theme of cold atoms and relativity.

The final topic of our School was the variation of fundamental constants, a subject that during the last years has attracted a lot of attention from different communities of physics. The underlying theories and the state of the art of light-based precision measurements of, for example, the time variation of the electron-proton mass ratio to put constraints on the variation of the fine structure constant were presented.

Unfortunately, a few of our lecturers are not represented in these Proceedings. For example, William D. Phillips (Joint Quantum Institute, National Institute of Standards and Technology, Gaithersburg, Maryland, and University of Maryland, College Park, Maryland, USA) introduced the School to atom optics and Bose-Einstein condensation. He started with an overview of some laboratory experiments for fundamental physics in Space and convinced the audience that in the near future they will lead to real experiments in Space. In this way we can address some of the most penetrating questions of the physics of the 21st century. The principles of laser cooling and trapping of atoms, atom optics, Bose-Einstein condensation, manipulation of ultracold atoms, and atom interferometry were explained with his typical excitement and followed his famous intuitive approach. Emphasis was placed on modern tools of experimental quantum optics such as clocks using cold atoms or interferometers with ultracold coherent matter waves. They are instrumental in the context of fundamental physics as well as in tracking and navigation of satellites in deep-space exploration. The lectures by Phillips were based on his notes prepared for the 2006 Summer School and published in: K. Helmerson and W. D. Phillips: Cooling, trapping and manipulation of atoms and Bose-Einstein condensates: Applications to metrology, in Proceedings of the International School of Physics “Enrico Fermi”, Course CLXVI: Metrology and Fundamental Constants edited by T. W. Hänsch, S. Leschiutta and A.J. Wallard (IOS Press, Amsterdam and SIF, Bologna) 2007, pp. 211-262.

In his lectures Christof Wetterich (Institut für Theoretische Physik, Universität Heidelberg, Germany) covered quintessence, dark matter and dark energy. He focused on the evolution of the universe on the basis of a cosmological constant or quintessence, that is the presence of dynamical dark energy generated by a scalar field called cosmon. The principles of cosmodynamics and the consequence that “fundamental constants” are not really constant anymore were presented at an elementary level. These dramatic predictions require tests based on quantum metrology.

Ignazio Ciufolini (Dipartimento d'Ingegneria dell'Innovazione, Università di Lecce, Italy) reported on a more than twenty years lasting research effort to measure the Lense-Thirring effect through the modifications of the orbits of the LAGEOS and GRACE satellites induced by gravito-magnetism.

Christophe Salomon (Laboratoire Kastler Brossel, Ecole Normale Supérieure, Paris, France) summarized the subject of cold atom clocks on Earth and in Space to test the foundations of physics. Here he put special emphasis on the performance and limits of atomic fountains. Moreover, he presented the Ultra-stable Clocks in Space (ACES) Project approved by the European Space Agency to place ultrastable clocks inside the International Space Station. In this way a cold atom clock and a maser clock will be orbiting around the Earth and their time will be compared to clocks on the ground. The final goal of ACES is to test general relativity. The timetable of this project as well as the effort required to produce a reliable experimental apparatus was presented. The application of atomic clocks to prove or disprove that fundamental physical constants indeed vary with time was also discussed by Salomon.

The School was actively attended by more than 60 participants including students, lecturers and seminar speakers from all over the world. Several students presented their own research and a number of informal sessions with extremely lively discussions took place. An evening Round Table examined the future plans of the European and USA Space Agencies aimed at exploring the limits of fundamental physics in Space.

All activities were inspired by the breathtaking beauty of Lake Como, the Villa Monastero and its gardens, and by the rich scientific heritage of the Enrico Fermi International School. The success of the School is also due to the excellent organizational and administrative support provided by the staff of the Italian Physical Society and the generous monetary support of the Wilhelm und Else Heraeus-Stiftung.

During the completion of these proceedings we have been shocked and deeply saddened by the untimely death of Jürgen Ehlers one of the main lecturers of our School. Jürgen was a highly gifted teacher, an outstanding scientist, and a great friend to many of us. His lectures included in this volume are a living testimony to his deep insight into and love for relativity. Unfortunately his premature departure did not allow him to complete the proof reading of his article. We are most grateful to Endre Kajari for taking this task upon himself. He has done a wonderful job. Many thanks Endre!

Jürgen's participation through his crystal clear lectures, his penetrating questions and the numerous illuminating discussions was vital to the success of our School. For this reason we have decided to dedicate this volume to the memory of this great pioneer of relativity. A more detailed obituary of Jürgen Ehlers may be found in these proceedings. We are grateful to his widow Anita Ehlers and to his longterm collaborator Bernd Schmidt for their help and to Graham Allen as well as Elke Müller for providing us with the pictures of Jürgen shown in these Proceedings.

E. Arimondo, W. Ertmer, E. M. Rasel and W. P. Schleich

**Abstract**

These lectures outline the basic concepts of general relativity, which unifies Newtonian gravitational mechanics and Maxwellian electrodynamics by generalizing special relativity. The metric structures underlying both relativity theories, which are pre-requisites for the physical laws governing matter and fields, are explained. Special attention is paid to local, inertial and non-intertial reference frames which relate the theories to each other and to laboratory experiments.

**Abstract**

We compare and contrast the different points of view of rotation in general relativity, put forward by Mach, Thirring and Lense, and Gödel. Our analysis relies on two tools: i) the Sagnac effect which allows us to measure rotations of a coordinate system or induced by the curvature of spacetime, and ii) computer visualizations which bring out the alien features of the Gödel Universe. In order to keep the paper self-contained, we summarize in several appendices crucial ingredients of the mathematical tools used in general relativity. In this way, our lecture notes should be accessible to researchers familiar with the basic elements of tensor calculus and general relativity.

**Abstract**

We survey theoretical and experimental/observational results on general-relativistic spin (rotation) effects in binary systems. A detailed discussion is given of the two-body Kepler problem and its first post-Newtonian generalization, including spin effects. Spin effects result from gravitational spin-orbit and spin-spin interactions (analogous to the corresponding case in quantum electrodynamics) and these effects are shown to manifest themselves in two ways: a) precession of the spinning bodies per se and b) precession of the orbit (which is further broken down into precessions of the argument of the periastron, the longitude of the ascending node and the inclination of the orbit). We also note that ambiguity that arises from use of the terminology frame-dragging, de Sitter precession and Lense-Thirring precession, in contrast to the unambiguous reference to spin-orbit and spin-spin precessions. Turning to one-body experiments, we discuss the recent results of the GP-B experiment, the Ciufolini-Pavlis Lageos experiment and lunar-laser ranging measurements (which actually involves three bodies). Two-body systems inevitably involve astronomical observations and we survey results obtained from the first binary pulsar system, a more recently discovered binary system and, finally, the highly significant discovery of a double-pulsar binary system.

**Abstract**

This paper describes the flight hardware, on-orbit operations, and preliminary data analysis for the Gravity Probe B satellite.

**Abstract**

Tests of gravity performed in the Solar System show a good agreement with general relativity. The latter is however challenged by observations at larger, galactic and cosmic scales which are presently cured by introducing “dark matter” or “dark energy”. A few measurements in the Solar System, particularly the so-called “Pioneer anomaly”, might also be pointing at a modification of gravity law at ranges of the order of the size of the Solar System. The present lecture notes discuss the current status of tests of general relativity in the Solar System. They describe metric extensions of general relativity which have the capability to preserve compatibility with existing gravity tests while opening free space for new phenomena. They present arguments for new mission designs and new space technologies as well as for having a new look on data of existing or future experiments.

**Abstract**

The quite different behaviors exhibited by microscopic and macroscopic systems with respect to quantum interferences suggest that there may exist a naturally frontier between quantum and classical worlds. The value of the Planck mass (22μg) may lead to the idea of a connection between this borderline and intrinsic fluctuations of space-time. We show that it is possible to obtain quantitative answers to these questions by studying the diffusion and decoherence mechanisms induced on quantum systems by gravitational waves generated at the galactic or cosmic scales. We prove that this universal fluctuating environment strongly affects quantum interferences on macroscopic systems, while leaving essentially untouched those on microscopic systems. We obtain the relevant parameters which, besides the ratio of the system's mass to Planck mass, characterize the diffusion constant and decoherence time. We discuss the feasibility of experiments aiming at observing these effects in the context of ongoing progress towards more and more sensitive matter-wave interferometry.

**Abstract**

This paper is a transcript of the lecture of the same name given in Summer 2007 in Varenna. It summarises the technologies needed for the direct detection of gravitational waves and their current status. Despite their different heritage and different measurement principle, resonant bar detectors and interferometric detectors share common problems and solutions to technological challenges. The technological challenges faced in a space mission to detect gravitational waves are briefly reviewed.

**Abstract**

The resolution of km-length detectors is limited by the thermal noise of mirror suspensions and mirror bulks, as well as by the shot noise. We give the status of actual interferometers. Advanced detectors, in operation in the next decade, open new technical challenges.

**Abstract**

Review of recent works devoted to the variation of the fine-structure constant α=e^{2}/ħc, strong interaction and fundamental masses (Higgs vacuum) is presented. Theories unifying gravity with other interactions suggest temporal and spatial variation of the fundamental “constants” in expanding Universe. The spatial variation can explain fine tuning of the fundamental constants which allows humans (and any life) to appear. We appeared in the area of the Universe where the values of the fundamental constants are consistent with our existence. There are some hints for the variation in quasar absorption spectra and Big-Bang nucleosynthesis data. However, a majority of publications report only limits on possible variation. A stringent limit on the variation of the strong interaction and fundamental masses follows from the Oklo natural nuclear reactor data. A very promising method to search for the variation consists in comparison of different atomic clocks. Huge enhancement of the relative variation effects happens in transitions between close atomic and molecular energy levels. We suggest several new cases where the levels are very narrow. A new idea is to build a “nuclear” clock based on UV (7.6 eV) transition in thorium nucleus. This may allow to improve sensitivity to the variation by several orders of magnitude. Large enhancement of the variation effects is also possible in cold atomic and molecular collisions near Feshbach resonance. How changing physical constants and violation of local position invariance may occur? The fundamental constants may depend on scalar fields which very naturally appear in modern cosmological models. Space-time evolution of these fields in expanding Universe may lead to the variation of the fundamental constants. Massive bodies (galaxies, stars, planets) can also affect physical constants. They have large scalar charge S proportional to number of particles which produces a Coulomb-like scalar field U=S/r. This leads to a variation of the fundamental constants proportional to the gravitational potential, e.g., δα/α=k_{α}δ(GM/rc^{2}). We compare different manifestations of this effect. The strongest limits are obtained from the measurements of dependence of atomic frequencies on the distance from Sun (the distance varies due to the ellipticity of the Earth's orbit).

**Abstract**

Modern capabilities in quantum state engineering of ultracold matter and precise control of light fields now permit accurate control of interactions between matter and field to suit applications for precision tests of fundamental physics. We report on our recent development of a highly stable and accurate optical atomic clock based on ultracold neutral Sr atoms confined in an optical lattice. We discuss precision tools for the lattice clock, including stabilized lasers with sub-Hz linewidth, femtosecond-comb based technology allowing accurate clock comparison in both microwave and optical domains, and clock transfer over optical fibers. With microkelvin Sr atoms confined in an optical lattice that provide a zero differential a.c. Stark shift between two clock states, we achieve a resonance quality factor >2×10^{14} on the ^{1}S_{0}−^{3}P_{0} doubly forbidden ^{87}Sr clock transition at 698 nm. High-resolution spectroscopy of spin-polarized atoms is used for both high-performance clock operations and accurate atomic structure measurement. The overall systematic uncertainty of the clock has been evaluated at the 10^{−16} level, while the stability approaches the 10^{−15} level at 1 s. These developments in precise engineering of light-atom interactions can be extended to the field of ultracold molecules, bringing new prospects for precision measurements, quantum control, and determinations of the constancy of the fundamental constants.

**Abstract**

Today, atom interferometers are used for a variety of experiments to determine fundamental constants and to investigate fundamental forces, such as demonstrated in the measurement of the photon recoil for the determination of the fine-structure constant, the measurement of the gravitational constant G, as well as gravimeters and gravity-gradiometers or atomic gyroscopes. Tests of the principle of equivalence, the measurement of relativistic effects or the realization of precise inertial references will challenge the full potential of atom interferometry. This lecture reports on our work towards a high-resolution Sagnac atom interferometer for measuring tiny rotations and accelerations. Beside the development and test of new atom-optical methods, future fields of applications of this apparatus are the monitoring of variations of the Earth's rotation rate, as well as the exploration of fundamental physical effects. The set-up is designed as a transportable device which permits comparisons with the largest stationary existing ring laser gyroscopes which are to date one of the most sensitive rotation sensors. The device is based on a dual Mach-Zehnder–type interferometer using laser-cooled rubidium atoms in a differential measurement scheme. Starting from the interferometers basic concepts, the lecture discusses key elements of the interferometer including the atomic sources, the coherent manipulation of atoms and methods for in situ diagnostics for the optimisation and characterisation of the devices. In future, major improvements of experimental tests of relativity and gravity are expected to be achieved in space. Space permits operation of experimental platforms which provide an environment with reduced inertial noise even down to very low Fourier frequencies corresponding to minutes and hours of measurement times. They also allow for an extension of the free fall, which lasts for seconds in drop towers, to months and years. The second part of the lecture presents research directed towards experiments with atom sensors in space. The lecture focuses on atomic interferometers, as proposed for the HYPER mission, which aims to monitor the spatial structure of the Lense-Thirring effect close to the Earth, as well as on the measurement of the gravitational acceleration including tests of the universality of the free fall of matter waves. In space, the duration of the experiments are eventually limited by the temperature of the atoms. Reduction of systematic effects (such as tidal effects) has to be achieved by the perfect control of the atomic wave function. Dilute Bose-Einstein condensates and ultra-cold Fermi gases are in this respect the ideal source for these experiments. Therefore, the last part of the lecture presents the project QUANTUS (Quanten Gase unter Schwerelosigkeit), which aims at a compact, robust and mobile experiment for the creation of a BEC to be operated in the drop tower facility at the ZARM in Bremen. The QUANTUS apparatus serves as a platform for studying the manipulation of dilute quantum gases at lowest energy scales, probing their coherent evolution over seconds and exploring their potential for precision measurements.

**Abstract**

The techniques of laser cooling combined with atom interferometry make possible the realization of very sensitive and accurate inertial sensors like gyroscopes or accelerometers. Besides earth-based developments, the use of these techniques in space should provide extremely high sensitivity for research in fundamental physics, Earth's observation and exploration of the solar system.

**Abstract**

Accurate optical frequency standards employ single trapped and laser cooled ions, we review recent experiments. Optical frequencies can be measured with fractional uncertainties of only a few parts in 10^{17} demonstrating the most accurate clocks available today. Demonstrated quantum information techniques help to further improve on determining residual systematic errors. Additionally, entanglement is also demonstrated to increase the signal-to-noise ratio. For generating the entanglement of a large number of ions, novel ions traps are currently under development which will allow for a coherent manipulation of the order of 10 to 50 ions. As one application of this maturing technology, we outline a space mission investigating all aspects of gravity in the outer Solar System, using a single-ion clock as one of the main onboard instruments. We describe the mission and its major scientific goals centered on tests of fundamental physics and exploration of outer Solar System objects. It will allow tests of special and general relativity with unprecedented sensitivity and will yield detailed information on the Kuiper belt circumsolar disk.

**Abstract**

The Plebanski-Demianski solution is a very general axially symmetric analytical black hole solution of Einstein's field equations generalizing the Kerr solution. This solution depends on 6 parameters which include the mass and cosmological constant. In this paper we present a general description of matter wave interferometry in this general space-time. We show that it is possible to have access to all parameters separately except a combination of electric and magnetic charge.

**Abstract**

A new framework is proposed to compare and unify photon and atom optics, which rests on the quantization of proper time. A common wave equation written in five dimensions reduces both cases to 5D-optics of massless particles. The ordinary methods of optics (eikonal equation, Kirchhoff integral, Lagrange invariant, Fermat principle, symplectic algebra and ABCD matrices, …) are used to solve this equation in practical cases. The various phase shift cancellations, which occur in atom interferometers, and the quantum Langevin twin paradox for atoms, are then easily explained. A general phase-shift formula for interferometers is derived in five dimensions, which applies to clocks as well as to gravito-inertial sensors.

**Abstract**

The light-pulse atom interferometry method is reviewed. Applications of the method to inertial navigation and tests of the Equivalence Principle are discussed.

**Abstract**

Experiments we are performing using atom interferometry to determine the gravitational constant G and test the Newtonian gravitational law at micrometric distances are discussed. Ongoing projects to develop transportable atom interferometers for applications in geophysics and in space are also presented.

**Abstract**

In this paper, I summarize the main applications and history of precision measurement of Earth gravity (g). Two common types of gravimeters are discussed: the relative, static, spring type and the dynamic, free-fall, absolute gravimeter. In particular, the design, construction, and performance of our latest free-fall gravimeters (MPG-2) are presented in detail. Various sources of error have to be carefully considered and removed to reach a precision at the micro-Gal (10^{−9} g) level. The paper is a summary of the talk given at the Enrico Fermi School during the summer of 2007.