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New Journal of Physics

Representation-free description of atom interferometers in time-dependent linear potentials

Matthias Zimmermann, Maxim A. Efremov, W Zeller, Wolfgang P. Schleich, J P Davis, Frank A. Narducci

Published Jul 1, 2019• 14 citations
Publisher

Abstract

Abstract In this article we present a new representation-free formalism, which can significantly simplify the analysis of interferometers comprised of atoms moving in time-dependent linear potentials. We present a methodology for the construction of two pairs of time-dependent functions that, once determined, lead to two conditions for the closing of the interferometer, and determine the phase and the contrast of the resultant interference. Using this new formalism, we explore the dependency of the interferometer phase on the interferometer time T for different atom interferometers. By now, it is well established that light pulse atom interferometers of the type first demonstrated by Kasevich and Chu (1991 Phys. Rev. Lett. 67, 181–4; 1992 Appl. Phys. B 54, 321–32), henceforth referred to as Mach–Zehnder (MZ) atom interferometers, have a phase scaling as T 2. A few years ago, McDonald et al (2014 Europhys. Lett. 105, 63001) have experimentally demonstrated a novel type of atom interferometer, referred to as the continuous-acceleration bloch (CAB) interferometer, where the phase reveals a mixed scaling which is governed by a combination of T 2 and T 3. Moreover, we have recently proposed a different type of atom interferometer (Zimmermann et al 2017 Appl. Phys. B 123, 102), referred to as the T 3-interferometer, which has a pure T 3 scaling, as demonstrated theoretically. Finally, we conclude that the CAB interferometer can be shown to be a hybrid of the standard MZ interferometer and the T 3-interferometer.

DOI
10.1088/1367-2630/ab2e8c

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Abstract In this article we present a new representation-free formalism, which can significantly simplify the analysis of interferometers comprised of atoms moving in time-dependent linear potentials. We present a methodology for the construction of two pairs of time-dependent functions that, once determined, lead to two conditions for the closing of the interferometer, and determine the phase and the contrast of the resultant interference. Using this new formalism, we explore the dependency of the interferometer phase on the interferometer time T for different atom interferometers. By now, it is well established that light pulse atom interferometers of the type first demonstrated by Kasevich and Chu (1991 Phys. Rev. Lett. 67, 181–4; 1992 Appl. Phys. B 54, 321–32), henceforth referred to as Mach–Zehnder (MZ) atom interferometers, have a phase scaling as T 2. A few years ago, McDonald et al (2014 Europhys. Lett. 105, 63001) have experimentally demonstrated a novel type of atom interferometer, referred to as the continuous-acceleration bloch (CAB) interferometer, where the phase reveals a mixed scaling which is governed by a combination of T 2 and T 3. Moreover, we have recently proposed a different type of atom interferometer (Zimmermann et al 2017 Appl. Phys. B 123, 102), referred to as the T 3-interferometer, which has a pure T 3 scaling, as demonstrated theoretically. Finally, we conclude that the CAB interferometer can be shown to be a hybrid of the standard MZ interferometer and the T 3-interferometer.
Motivated by a recent claim by Müller et al (2010 Nature 463 926-9) that an atom interferometer can serve as an atom clock to measure the gravitational redshift with an unprecedented accuracy, we provide a representation-free description of the Kasevich-Chu interferometer based on operator algebra.We use this framework to show that the operator product determining the number of atoms at the exit ports of the interferometer is a c-number phase factor whose phase is the sum of only two phases: one is due to the acceleration of the phases of the laser pulses and the other one is due to the acceleration of the atom.This formulation brings out most clearly that this interferometer is an accelerometer or a gravimeter.Moreover, we point out that in different representations of quantum mechanics such as the position or the momentum representation the phase shift appears as though it originates from different physical phenomena.Due to this representation dependence conclusions concerning an enhanced accuracy derived in a specific representation are unfounded.
A substantial part of this thesis consolidates, summarizes, and deepens the theoretical work that has led to the recent observations of the Kennard phase and the construction of a novel T³ Stern-Gerlach interferometer for matter waves. Our novel device is sensitive to a linear potential as originating from the gravitational field. In comparison to existing atom interferometers, it displays a faster scaling of the interferometer phase with regard to the interferometer time T. The resulting cubic dependence originates from the Kennard phase and benefits from employing state-dependent forces that are induced by magnetic field gradients. In addition, several parts of this thesis exceed this particular topic. By developing a general and easily applicable formalism based on branch-dependent dynamics, we simplify, unify, and generalize the description of matter-wave interferometers. Within our approach, we compare and contrast classical and quantum concepts from a fundamental perspective while concentrating on the contrast and phase shift of matter-wave interferometers. Based on our formalism, we propose schematic rules which enable a straightforward analysis of a given interferometer geometry in the presence of state- and time-dependent forces. With the help of these rules, we pursue the classification of existing and the construction of novel interferometer geometries that display a sensitivity to particular aspects of spatially and temporally varying external fields. Our method is supported by several examples of novel interferometers which still await an experimental implementation.
Motivated by the recent debate on whether the Kasevich-Chu atom interferometer can measure the gravitational redshift, we show that in different representations of quantum mechanics chosen for the calculation, the observed phase shift appears as though it originates from different physical phenomena. In particular, we demonstrate that the decomposition of the total phase shift into three dynamical phases, which emerges in a semiclassical approach and is at the very heart of the redshift controversy, does not appear in an exact treatment based on a representation-free analysis. Here only two phenomena determine the phase shift: the difference of the laser phases and the acceleration of the atom. Hence, the Kasevich-Chu interferometer is an accelerometer or gravimeter.
Matter - wave interferometry began with the development of electron interferometers and neutron interferometers in the 1950s and 1960s, but it wasn't until 1991 that the first atom matter - wave interferometer was realized. This was followed by a surge of activity sparked by the advent of sources of ultracold laser-cooled atoms. Some of the potential advantages of atom interferometry over neutron interferometry include: much higher particle fluxes, lower particle velocities, larger particle masses, particles with internal states, and laboratory-based sources. This volume is the first text on atom interferometry. It consists of ten comprehensive review articles by leading researchers in the field, many of whom pioneered the first atom interferometers in 1991. The articles describe the different types of atom interferometers developed to date, ranging from the three-grating Mach - Zehnder interferometer based on nanofabricated grating structures (MIT) or standing-wave light-field gratings (Innsbruck), to the Talbot - Lau atom interferometer (Berkeley and Michigan), to atom interferometers based on internal-state labelling such as separated light-field interferometers (PTB, Hannover and Paris-Nord) and light-pulse interferometers (Stanford). Some of the recent applications of atom interferometry described in the articles include high precision measurements of rotations, the gravitational acceleration, and the fine structure constant (Stanford); measurements of topological phases (PTB and Hannover); correlation experiments with atoms and photons (Konstanz) and with pairs of atoms (Tokyo); holographic manipulation of atoms (Tokyo); and experiments on the quantum theory of measurement (Garching, Harvard and Texas A&M). In summary, this volume provides an excellent up-to-date account of atom interferometry and is recommended as essential reading for all researchers and would-be researchers in the field. At the same time it contains sufficient background material and theory to serve as an excellent introduction for the non-specialist. The field is presently moving at such a rapid pace, however, that many of the latest results reported in this volume will already have been superseded by the time it reaches the library shelves.
In an recent work with the title ``Asking Photons Where They Have Been,'' Danan et al. experimentally demonstrate an intriguing behavior of photons in an interferometer [Phys. Rev. Lett. 111, 240402 (2013)]. In their words: ``The photons tell us that they have been in the parts of the interferometer through which they could not pass.'' They interpret the results using the two-state vector formalism of quantum theory and say that, although an explanation of the experimental results in terms of classical electromagnetic waves in the interferometer is possible (and they provide a partial description), it is not so intuitive. Here we present a more detailed classical description of their experimental results, showing that it is actually intuitive. The same description is valid for the quantum wave function of the photons propagating in the interferometer. In particular, we show that it is essential that the wave propagates through all parts of the interferometer to describe the experimental results. We hope that our work helps to give a deeper understanding of these interesting experimental results.
More than two decades ago a new state of matter called Bose-Einstein Condensation was proved experimentally although predicted long before in 1924-25.At extremely low temperatures and controlled densities, this exotic state can be achieved where all atoms share the same energy and linear momentum similarly as the light behaves in a laser.Therefore, it is interesting to analyze the matterwave counterparts for the nowadays well-known optical devices.Here, we present a numerical implementation of a Mach-Zehnder interferometer which we engineered by means of time-dependent external trapping potentials.We use an appropriate implementation of the finite-difference method for the spatial discretization and the Runge-Kutta technique for the time evolution.The dynamics of both conjugate variables, namely density and phase of the matter-wave functions are shown.Our aim is also to show that this kind of problems and techniques can be addressed in standard courses of Quantum Mechanics and Computational Physics.