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Physical Review A

Sagnac experiment with electrons: Reanalysis of a rotationally induced phase shift for charged particles

Richard Neutze, FRANZ HASSELBACH

Published Jul 1, 1998• 16 citations
Publisher

Abstract

Using a path-integral formalism, we present a reanalysis of the Sagnac experiment with electrons of Hasselbach and Nicklaus [Phys. Rev. A 48, 143 (1993)]. Our analysis, from an inertial frame of reference, makes explicit the central role of the electrostatic biprism in generating interference fringes and recovers the Sagnac formula for the phase shift induced when rotating the interferometer. This phase shift, however, is shown to differ significantly in physical origin from the similar experiment with neutrons. In the electron Sagnac experiment, rotation displaces all classical electron paths closer to (or further from, depending upon direction) the electrostatic biprisms, thereby perturbing the electrostatic potential along these paths, and a phase shift of an electrostatic nature arises. Our analysis yields a corrected expression for the projected area of the interferometer and small numerical corrections to previous work are consequently given.

DOI
10.1103/physreva.58.557

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Using a path-integral formalism, we present a reanalysis of the Sagnac experiment with electrons of Hasselbach and Nicklaus [Phys. Rev. A 48, 143 (1993)]. Our analysis, from an inertial frame of reference, makes explicit the central role of the electrostatic biprism in generating interference fringes and recovers the Sagnac formula for the phase shift induced when rotating the interferometer. This phase shift, however, is shown to differ significantly in physical origin from the similar experiment with neutrons. In the electron Sagnac experiment, rotation displaces all classical electron paths closer to (or further from, depending upon direction) the electrostatic biprisms, thereby perturbing the electrostatic potential along these paths, and a phase shift of an electrostatic nature arises. Our analysis yields a corrected expression for the projected area of the interferometer and small numerical corrections to previous work are consequently given.
A Sagnac experiment with electron waves in vacuum is reported. The phase shift caused by rotation of an electron biprism interferometer placed on a turntable has been measured. It was found to agree with prediction within error margins of about 30%. A compact ruggedized electron interferometer was used. It is based on a high-precision optical bench of 36-cm length. This interferometer is less sensitive by orders of magnitude to mechanical vibrations and electromagnetic stray fields than conventional electron interferometers. A beam of low-energy electrons (150--3000 eV) emitted by a field-emission electron source was used. For the most part, electrostatic electron optical components were employed. The magnified interference fringe pattern was intensified by a dual-stage multichannel-plate intensifier, recorded by a charge-coupled-device video camera, transmitted from the turntable to the laboratory system via a slip ring, and evaluated by an image-processing system. Both the rotation rate and the area enclosed between the two partial waves were varied (up to values of 0.5 ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ and 3.9 mm2{\mathrm{mm}}^{2}mm2, respectively). Fringe shifts on the order of 5% of a fringe period were attained. Some historical aspects of the Sagnac effect as well as some aspects of its interpretation are mentioned. A brief informal discussion is included of the interpretation of the Sagnac phase shift as a geometric phase (``Berry phase'') caused by the global anholonomy of the local phase factor that is produced by the gauge field induced by rotation.
The Sagnac effect is an important phase coherent effect in optical and atom interferometers where rotations of the interferometer with respect to an inertial reference frame result in a shift in the interference pattern proportional to the rotation rate. Here, we analyze the Sagnac effect in a mesoscopic semiconductor electron interferometer. We include in our analysis the Rashba spin-orbit interactions in the ring. Our results indicate that spin-orbit interactions increase the rotation-induced phase shift. We discuss the potential experimental observability of the Sagnac phase shift in such mesoscopic systems.
After the first matter wave version of Sagnac’s classical light optical experiment of 1913, performed by Mercereau and Zimmermann with electron Cooper pairs in 1965, and the Sagnac experiment realized with neutrons by Werner et al. in 1979 , we report here on the first observation of the rotational phase shift of electron waves in vacuum. Theory. The Sagnac effect links classical physics, quantum physics and relativity. Using the special theory of relativity it can be derived that coherent waves, e.g. of light, neutrons or electrons, travelling around a finite area A experience a relative phaseshift
A rotating interferometer with paths that enclose a physical area exhibits a phase shift proportional to this area and to the rotation rate of the frame. Understanding the origin of this so-called Sagnac effect has played a key role in the establishment of the theory of relativity and has pushed for the development of precision optical interferometers. The fundamental importance of the Sagnac effect motivated the realization of experiments to test its validity for waves beyond optical, but precision measurements remained a challenge. Here, we report the accurate test of the Sagnac effect for matter waves, by using a Cesium atom interferometer featuring a geometrical area of 11 cm 2 and two sensitive axes of measurements. We measure the phase shift induced by Earth’s rotation and find agreement with the theoretical prediction at an accuracy level of 25 parts per million. Beyond the importance for fundamental physics, our work opens practical applications in seismology and geodesy.
We carry out a complete examination of the effects of rotation on the physics of spin-\textonehalf{} particles. The origin of these effects is connected to the fact that the frame of the rotating experiment is Fermi-Walker transported, and is related to the inertia rest frame (with respect to which the apparatus rotates with angular velocity $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}$) by an instantaneous Lorentz boost plus a time-dependent spatial rotation, in that order. Two distinct sets of effects are obtained. The first depends on the spin-rotation interaction, and consists of the Mashhoon effect and a mass split effect due to the Lorentz boost mentioned above. In a neutron interferometer the latter effect produces a phase shift smaller than the Mashhoon phase shift by a factor of order O(v2c2)O(\frac{{v}^{2}}{{c}^{2}})O(c2v2​), where vvv is the velocity of the particles in the beams of the interferometer. The detection of the Mashhoon effect is crucial to decide if free spin-\textonehalf{} particles actually behave as gyroscopes. The second set appears already in the eikonal approximation and is due to the active boost of the particles by the rotating apparatus; it results in the Sagnac effect. We also discuss a criterion to fix the frame of the experiment among all mathematically admissible frames in the Brill-Wheeler formulation, and show that frames used in the literature to derive the Sagnac-Mashhoon effect cannot be physically realized.
The Sagnac effect, manifest in the phase shift of a rotating interferometer proportional to its physical area and to the rotation rate of the frame, plays a crucial role in modern physics being generalized to any kind of interferometer. However, as compared to light, the accurate experimental validation of the Sagnac effect for matter-waves remains challenging. We report the accurate measurement of the Sagnac phase shift, induced by the Earth’s rotation, using a large-area cold atom interferometer. We probe the atomic Sagnac phase shift along the two orthogonal horizontal axes in various orientation of the sensor with respect to geographic north and obtain an agreement with the theoretical prediction at the 20 ppm level, improving by twenty five-fold over the previous achievements for matter waves. Our results underline the universality of the Sagnac effect and open new horizons for testing other fundamental theories, as well as for a number of technological applications in seismology, geodesy and inertial navigation systems.