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The Sagnac Phase Shift suggested by the Aharonov-Bohm effect for relativistic matter beams

Guido Rizzi, Matteo Luca Ruggiero

Published Jan 1, 2003• 15 citations
OpenAlex

Abstract

The phase shift due to the Sagnac Effect, for relativistic matter beams counter-propagating in a rotating interferometer, is deduced on the bases of a formal analogy with the Aharonov-Bohm effect. A procedure outlined by Sakurai, in which non relativistic quantum mechanics and Newtonian physics appear together with some intrinsically relativistic elements, is generalized to a fully relativistic context, using the Cattaneo's splitting technique. This approach leads to an exact derivation, in a self-consistently relativistic way, of the Sagnac effect. Sakurai's result is recovered in the first order approximation.

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arXiv:1806.02710•Jun 7, 2018•0 citations
The phase shift due to the Sagnac Effect, for relativistic matter beams counter-propagating in a rotating interferometer, is deduced on the bases of a formal analogy with the Aharonov-Bohm effect. A procedure outlined by Sakurai, in which non relativistic quantum mechanics and Newtonian physics appear together with some intrinsically relativistic elements, is generalized to a fully relativistic context, using the Cattaneo's splitting technique. This approach leads to an exact derivation, in a self-consistently relativistic way, of the Sagnac effect. Sakurai's result is recovered in the first order approximation.
A gyroscope based on Sagnac interferometer measures the rotation rate relative to an inertial frame of reference. Sagnac effect originally has been derived and experimentally demonstrated with optical waves. Later, matter wave based Sagnac interferometers were developed due to inherent sensitivity over a photon based system. However in any interferometer whether it is photon or matter wave based the resultant phase shift due to counter-rotating waves is independent of the wave velocity. Here we show that one can have a larger phase shift with slower matter waves using Aharonov-Bohm effect: the phase difference of the counter propagating waves is proportional to the inverse square of the particle velocity.
This chapter presents the derivation of Berry's phase. Many classical phenomena have corresponding quantum mechanical analogue. Therefore, it is natural to identify a classical analogue of Berry's phase. The concept of Berry's phase has been used in the study of quantized Hall Effect, the spin statistics properties of quasiparticle excitations, the rotation of photon polarization in helical optical fibers, etc. The chapter discusses various applications of Berry's phase in solid state physics. The Berry's phase in one-dimension is usually called Zak phase and is used for a better understanding of electric polarization in dielectrics, protected edge states and Wannier–Stark ladders and integer quantum Hall effect. The chapter discusses the Aharonov–Bohm and Sagnac effects. The Aharonov–Bohm experiment consists of splitting an electron beam, passing around both sides of a solenoid and then recombining the beam. The Sagnac effect has been studied in resonant microcavities and light propagation in a wave guide also.
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.
We canonically quantize a spin-less non-relativistic point particle in a rigidly rotating cylinder symmetric reference system. The resulting quantum mechanics is investigated in the case of both a two-dimensional cylindrical shell and in the case of the full rotating cylinder; in both cases energy eigenstates exist which exhibit rotation induced negative energies. Based on a reanalysis of the classical Sagnac effect a novel way to compute the quantum Sagnac phase shift is deduced. It is pointed out that certain states on both the cylindrical shell and in the bulk give rise to a novel anomalous quantum Sagnac effect.