Ebook: Atom Interferometry

Matter wave interference is at the basis of quantum mechanics. After early experiments with electrons and neutrons, about 20 years ago atom interferometers were first realized. Since then, atom interferometry led to growing applications in basic and applied science. It has been used, for example, to measure gravity acceleration, rotations and fundamental physical quantities with unprecedented precision. Future applications range from tests of general relativity to the development of next generation inertial navigation systems. Several laboratories in the world are actively working in this field. The improving performances of atom interferometric apparatus also stimulated a deep understanding of the theory required to connect the measured interferometric phase shift to the physically relevant quantities.

The goal of the 2013 Varenna School on “Atom Interferometry” and of this volume, which is published about one year later, has been to cover the basic experimental and theoretical aspects and to provide an updated review of the current activities in the field, the main achievements, open issues and future prospects. Main topics include theoretical background and experimental schemes for atom interferometry, ultracold atoms and atom optics, comparison of atom, light, electron, neutron interferometers and applications, high precision measurements with atom interferometry and applications to tests of fundamental physics, gravitation, intertial measurements and geophysics, measurement of fundamental constants, interferometry with quantum degenerate gases, matter wave interferometry beyond classical limits, large area interferometers, atom interferometry on chips, interferometry with molecules. The School was organized as a series of minicourses, some seminar lectures on specific topics and poster presentations by the participants.

The large number of students attending the lectures, a demonstration of the interest and vitality of the field, was at the limit of the School capacity; everything ran smoothly thanks to the organization by Barbara Alzani and her team. Special thanks are due to the Scientific Secretary of the School, Dr. Fiodor Sorrentino. The inspiring atmosphere of Villa Monastero stimulated scientific discussions amongst the students and with the lecturers and likely new ideas that will lead to further advances of atom interferometry in the future.

This volume collects the notes carefully prepared by all the lecturers after the School with the assistance of Monica Bonetti (editorial office) and Marcella Missiroli (production office). We are confident that it will be a reference for students, newcomers and experts in the field of atom interferometry.

G.M. Tino and M.A. Kasevich

Optics and interferometry with matter waves is the art of coherently manipulating the translational motion of particles like neutrons, atoms and molecules. Coherent atom optics is an extension of techniques that were developed for manipulating internal quantum states. Applying these ideas to translational motion required the development of techniques to localize atoms and transfer population coherently between distant localities. In this view position and momentum are (continuous) quantum mechanical degrees of freedom analogous to discrete internal quantum states. In our contribution we start with an introduction into matter wave optics in sect. 1, discuss coherent atom optics and atom interferometry techniques for molecular beams in sect. 2 and for trapped atoms in sect. 3. In sect. 4 we then describe tools and experiments that allow us to probe the evolution of quantum states of many-body systems by atom interference.

We discuss modern developments in quantum optics with organic molecules, clusters and nanoparticles — in particular recent realizations of near-field matter wave interferometry. A unified theoretical description in phase space allows us to describe quantum interferometry in position space and in the time domain on an equal footing. In order to establish matter wave interferometers as a universal tool, which can accept and address a variety of nanoparticles, we elaborate on new quantum optical elements, such as diffraction gratings made of matter and light, as well as their absorptive and dispersive interaction with complex materials. We present Talbot-Lau interferometry (TLI), the Kapitza-Dirac-Talbot-Lau interferometer (KDTLI) and interferometry with pulsed optical ionization gratings (OTIMA) as the most advanced devices to study the quantum wave nature of composite matter. These experiments define the current mass and complexity record in interferometric explorations of quantum macroscopicity and they open new avenues to quantum-assisted metrology with applications in physical chemistry and biomolecular physics.

We first give a short historical introduction to the fundamentals of atom interferometry with internal excitation, based on density matrix diagrams and a Liouville space approach. Then, 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, 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. This contribution is an update of a previous presentation of 5D matter-wave optics and interferometry. Electromagnetic interactions are explicitly added in the 5D metric tensor in complete analogy with Kaluza's work. The 5D Lagrangian is rederived and an expression for the Hamiltonian suitable for the parabolic approximation is presented. The corresponding equations of motion are also given. The 5D action is shown to cancel for the actual trajectory which is a null geodesics of the 5D metric. This presentation is mainly devoted to the classical aspects of the theory and only general consequences for the quantum phase of matter-waves are outlined. The application to Borde-Ramsey interferometers is given as an illustration.

We provide an introduction into the formulation of non-relativistic quantum mechanics using the Wigner phase-space distribution function and apply this concept to two physical situations at the interface of quantum theory and general relativity: i) the motion of an ensemble of cold atoms relevant to tests of the weak equivalence principle, and ii) the Kasevich-Chu interferometer. In order to lay the foundations for this analysis we first present a representation-free description of the Kasevich-Chu interferometer based on unitary operators.

These lectures were given in the Varenna summer school on atom interferometry. They discuss a general relativistic formulation of atom interferometry and the use of this formulation to understand gravitational wave detection with atom interferometry. These notes were compiled from our previous papers on these subjects, which contain more detail. Significant thanks are owed to all the coauthors on these papers.

This paper is divided into three parts. In the first, we demonstrate that all of quantum mechanics can be derived from the fundamental property that the propagation of a matter-wave packet is described by the same gravitational and kinematic time dilation that applies to a clock. We will do so in several steps, first deriving the Schroedinger equation for a non-relativistic particle without spin in a weak gravitational potential, and eventually the Dirac equation in curved space-time describing the propagation of a relativistic particle with spin in strong gravity. In the second part, we present interesting consequences of the above quantum mechanics: that it is possible to use wave packets as a reference for a clock, to test general relativity, and to realize a mass standard based on a proposed redefinition of the international system of units, wherein the Planck constant would be assigned a fixed value. The clock achieved an absolute accuracy of 4 parts per billion (ppb). The experiment yields the fine structure constant α=7.297 352 589(15)×10⁻³ with 2.0 ppb accuracy. We present improvements that have reduced the leading systematic error about 8-fold and improved the statistical uncertainty to 0.33 ppb in 6 hours of integration time, referred to α. In the third part, we present possible future experiments with atom interferometry: A gravitational Aharonov-Bohm experiment and its application as a measurement of Newton's gravitational constant, antimatter interferometry, interferometry with charged particles, and interferometry in space. We will give a review of previously published material when appropriate, but will focus on new aspects that have not been published before.

In Paris, we are using an atom interferometer to precisely measure the recoil velocity of an atom that absorbs a photon. In order to reach a high sensitivity, many recoils are transferred to atoms using the Bloch oscillations technique. In this paper, I will present in details this technique and its application to high precision measurement. I will especially describe in details how this method allows us to perform an atom recoil measurement at the level of 1.3×10⁻⁹. This measurement is used in the most precise determination of the fine structure constant that is independent of quantum electrodynamics.

We developed two types of atom interferometer for gravitational physics experiments: The first is a double Raman interferometer with Rb atoms that we operate as a gravity gradiometer. The second is based on Bloch oscillations of Sr atoms in optical lattices. As in my lectures at the 2013 “E. Fermi” School in Varenna, the aim of these notes is to describe the characteristics of the two interferometers taking as examples the experiments that we performed. I present the schemes of the interferometers, the key steps that allowed us to optimize the setups, their operation and performances. I discuss experiments that we carried out, namely, a precision measurement of the value of the gravitational constant G using the Rb Raman interferometer and the measurement of gravity at small spatial scale using the Sr apparatus, with implications in the search for deviations from Newtonian gravity and as tests of general relativity. I also discuss prospects and possible ideas for future experiments in laboratories on ground and in space.

The past three decades have shown dramatic progress in the ability to manipulate and coherently control the motion of atoms. This exquisite control offers the prospect of a new generation of inertial sensors with unprecedented sensitivity and accuracy, which will be important for both fundamental and applied science. In this article, we review some of our recent results regarding the application of atom interferometry to inertial measurements using compact mobile sensors. This includes some of the first interferometer measurements with cold ³⁹K atoms, which is a major step toward achieving a transportable, dual-species interferometer with rubidium and potassium for equivalence principle tests. We also discuss future applications of this technology, such as remote sensing of geophysical effects, gravitational wave detection, and precise tests of the weak equivalence principle in space.

Atom interferometers have demonstrated high sensitivities in measurements of accelerations, rotations and different physical constants. In this article, we present the mobile instrument GAIN (Gravimetric Atom Interferometer) which allows to utilize this potential for high-precision on-site measurements of the local gravitational acceleration. Important influences and systematics arising from both the experimental realization as well as the environment and possibilities to distinguish between them are reviewed. In addition, we give a compact overview of other established gravimeters and discuss the performance of GAIN in the context of several comparison campaigns.

Prospects for gravitational wave detection using light-pulse atom interferometry are reviewed. We describe recent proposals for a satellite detector and discuss how the top-level system architecture is impacted by the details of the atomic physics processes used to implement the atom interferometry. Much of the required atom technology development for such a detector can be done in ground-based experiments. In this context, we present recent results from long-time atom interferometry performed in a 10-meter atomic fountain.

After a short review of the properties of gravitational waves as described in Einstein general relativity, the problem of characterizing suitable detectors is addressed in order to define suitable parameters for comparison. The concept of sensitivity as a combination of frequency response function and noise budget at fixed signal to noise ratio is used, looking at the performances of existing optical gravitational waves antennas and of a symmetric Ramsey-Borde (Mach-Zehnder geometry) atom interferometer studied as a prototype of new detectors for gravitational radiation.

Nowadays inertial sensing with atom interferometers has reached a great level of maturity. Several limits on a wider application of cold-atom–based inertial sensors are imposed by the size of the devices and the short coherence times of the sources. A new generation of matter-wave interferometers utilizing Bose-Einstein condensates (BEC) is continuously making the latter statement less severe. Autonomous, compact and high-flux BEC machines are realized and push forward the use of atomic sensors thanks to the source intrinsic coherence. A spectacular boost in the performance of high-precision tests of fundamental theories is expected when such sensors are considered in the ideal conditions of micro-gravity platforms and satellite missions.

Advancements in physics are often motivated/accompanied by advancements in our precision measurements abilities. The current generation of atomic and optical interferometers is limited by shot noise, a fundamental limit when estimating a phase shift with classical light or uncorrelated atoms. In the last years, it has been clarified that the creation of special quantum correlations among particles, which will be called here “useful entanglement”, can strongly enhance the interferometric sensitivity. Pioneer experiments have already demonstrated the basic principles. We are probably at the verge of a second quantum revolution where quantum mechanics of many-body systems is exploited to overcome the limitations of classical technologies. This review illustrates the deep connection between entanglement and sub shot noise sensitivity.

The precision of classical interferometers is limited by the quantum mechanics of single particles. The so-called projection noise limit fundamentally appears in the measurement process when the individual particles, being in a superposition of the two interferometer arms, are projected to one of the two output modes. However, entanglement among the particles can be used as a resource that allows for measurements beyond this limit. In this lecture note we give an introduction to the concept of spin squeezing to achieve quantum enhanced precision. We summarize the Heidelberg experiments that realized spin-squeezed states and interferometric measurements beyond the standard quantum limit using few mode Bose-Einstein condensates. These experimental results showed that direct interatomic interactions in a Bose-Einstein condensate (next to cavity-mediated interactions, quantum non-demolition measurements or state transfer from non-classical light) can be employed to achieve spin squeezing