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arXiv:0810.4111 · Physical Review A

Sagnac effect in a chain of mesoscopic quantum rings

Christopher P. Search, John R. E. Toland, Marko Zivkovic

Published May 6, 2009• 14 citations
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Abstract

The ability to interferometrically detect inertial rotations via the Sagnac effect has been a strong stimulus for the development of atom interferometry because of the potential ${10}^{10}$ enhancement of the rotational phase shift in comparison to optical Sagnac gyroscopes. Here we analyze ballistic transport of matter waves in a one-dimensional chain of $N$ coherently coupled quantum rings in the presence of a rotation of angular frequency $\ensuremath{\Omega}$. We show that the transmission probability, $T$, exhibits zero transmission stop gaps as a function of the rotation rate interspersed with regions of rapidly oscillating finite transmission. With increasing $N$, the transition from zero transmission to the oscillatory regime becomes an increasingly sharp function of $\ensuremath{\Omega}$ with a slope $\ensuremath{\partial}T/\ensuremath{\partial}\ensuremath{\Omega}\ensuremath{\sim}{N}^{2}$. The steepness of this slope dramatically enhances the response to rotations in comparison to conventional single ring interferometers such as the Mach-Zehnder interferometer and leads to a phase sensitivity well below the quantum shot-noise limit typical of atom interferometers.

DOI
10.1103/physreva.79.053607

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The ability to interferometrically detect inertial rotations via the Sagnac effect has been a strong stimulus for the development of atom interferometry because of the potential 1010{10}^{10}1010 enhancement of the rotational phase shift in comparison to optical Sagnac gyroscopes. Here we analyze ballistic transport of matter waves in a one-dimensional chain of NNN coherently coupled quantum rings in the presence of a rotation of angular frequency \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ. We show that the transmission probability, TTT, exhibits zero transmission stop gaps as a function of the rotation rate interspersed with regions of rapidly oscillating finite transmission. With increasing NNN, the transition from zero transmission to the oscillatory regime becomes an increasingly sharp function of \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ with a slope \ensuremath∂T/\ensuremath∂\ensuremathΩ\ensuremath∼N2\ensuremath{\partial}T/\ensuremath{\partial}\ensuremath{\Omega}\ensuremath{\sim}{N}^{2}\ensuremath∂T/\ensuremath∂\ensuremathΩ\ensuremath∼N2. The steepness of this slope dramatically enhances the response to rotations in comparison to conventional single ring interferometers such as the Mach-Zehnder interferometer and leads to a phase sensitivity well below the quantum shot-noise limit typical of atom interferometers.
Atom interferometry is of considerable interest in part because of the ability to interferometrically detect inertial rotations via the Sagnac effect with a potential sensitivity 1010 greater than optical gyroscopes. It has been shown recently that a coherently coupled array of identical interferometers can significantly enhance the sensitivity to rotations due to the appearance of transmission bands as a function of the inertial rotation rate Ω. Here we consider phase coherent transport of atomic matter waves in a chain of ring interferometers with a single occupied transverse mode in the presence of a rotation, Ω, and study the effect of variations in the size of the rings. We show that for randomly sized rings, the entire array functions as a highly sensitive Sagnac interferometer provided the level of random size fluctuations does not exceed a few per cent of the mean size. We also analyse how the use of individual defect states and controlled variations of the sizes in the array can be used to further enhance the sensitivity by creating narrow transmission resonances inside of a zero transmission gap.
The utilization of atom interferometry for gyroscopes is of considerable interest because the detection of inertial rotations via the Sagnac effect has a potential 1010{10}^{10}1010 greater sensitivity than optical gyroscopes. Here we analyze, in the presence of a rotation $\ensuremath{\Omega}$, phase-coherent transport of atomic matter waves in a one-dimensional chain of NNN ring interferometers in which the sizes of the rings are chirped symmetrically about the center ring. The gyroscope scale factor of the area-chirped array for $N\ensuremath{\lesssim}10$ is several orders of magnitude larger than that for the unchirped array, which itself is larger than the scale factor of a Mach-Zehnder interferometer with the same enclosed area. This enhancement is achieved with total enclosed areas that are larger or smaller than that of the unchirped array by less than 25%. As such, the area chirp achieves rotation sensitivities with $N\ensuremath{\lesssim}10$ interferometers that would normally require $N\ensuremath{\gg}100$ rings of identical sizes.
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.
The light-matter-wave Sagnac interferometer based on ultraslow light proposed recently in Zimmer and Fleischhauer, Phys. Rev. Lett. 92, 253201 (2004), is analyzed in detail. In particular the effect of confining potentials is examined and it is shown that the ultraslow light attains a rotational phase shift equivalent to that of a matter wave, if and only if the coherence transfer from light to atoms associated with slow light is associated with a momentum transfer and if an ultracold gas in a ring trap is used. The quantum sensitivity limit of the Sagnac interferometer is determined and the minimum detectable rotation rate calculated. It is shown that the slow-light interferometer allows for a significantly higher signal-to-noise ratio as possible in current matter-wave gyroscopes.
Matter-wave interferometry with atoms propagating in a guiding potential is expected to provide compact, scalable and precise inertial sensing. However, a rotation sensing device based on the Sagnac effect with atoms guided in a ring has not yet been implemented despite continuous efforts during the last two decades. Here we discuss some intrinsic effects that limit the coherence in such a device and propose a scheme that overcomes these limitations and enables a multi-pass guiding Sagnac interferometer with a Bose-Einstein condensate (BEC) on a chip in a ring potential. We analyze crucial dephasing effects: potential roughness, phase diffusion due to atom-atom interactions and number uncertainty, and phase fluctuations. Owing to the recent progress in achieving high momentum beam splitting, creating smooth guides, and manipulating the matter-wavepacket propagation, guided interferometry can be implemented within the coherence time allowed by phase diffusion. Despite the lower particle flux in a guided Sagnac ring and the miniaturization of the interferometer, the estimated sensitivity, for reasonable and practical realizations of an atom chip-based gyroscope, is comparable to that of free-space interferometers, reaching 45 nrads^{-1}Hz^{-1/2}. A significant improvement over state-of-the-art free-space gyrocope sensitivities can be envisioned by using thermal atoms instead of a BEC, whereby the interferometer can be operated in a continuous fashion with the coherence limited by the scattering rate of the atoms with the background gas. Taking into account the sensitivity times length of the interferometer as the figure of merit which takes into account compactness, our configuration is expected to deliver a potential improvement of 2-4 orders of magnitude over state-of-the-art free-space gyroscopes for a BEC, and 4-6 orders of magnitude for thermal atoms.
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.