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arXiv:1409.2137 · Physical Review A

Mesoscopic mechanical resonators as quantum noninertial reference frames

B. Katz, M. P. Blencowe, Keith Schwab

Published Oct 8, 2015• 12 citations
arXivHTMLPDF

Abstract

An atom attached to a micrometer-scale wire that is vibrating at a frequency $\ensuremath{\sim}100\phantom{\rule{0.28em}{0ex}}\mathrm{MHz}$ and with displacement amplitude $\ensuremath{\sim}1\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$ experiences an acceleration magnitude $\ensuremath{\sim}{10}^{9}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}\phantom{\rule{0.28em}{0ex}}\mathrm{s}}^{\ensuremath{-}2}$, approaching the surface gravity of a neutron star. As one application of such extreme noninertial forces in a mesoscopic setting, we consider a model two-path atom interferometer with one path consisting of the $100\phantom{\rule{0.28em}{0ex}}\mathrm{MHz}$ vibrating wire atom guide. The vibrating wire guide serves as a noninertial reference frame and induces an in principle measurable phase shift in the wave function of an atom traversing the wire frame. We furthermore consider the effect on the two-path atom wave interference when the vibrating wire is modeled as a quantum object, hence functioning as a quantum noninertial reference frame. We outline a possible realization of the vibrating wire, atom interferometer using a superfluid helium quantum interference setup.

DOI
10.1103/physreva.92.042104

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An atom attached to a micrometer-scale wire that is vibrating at a frequency \ensuremath∼100MHz\ensuremath{\sim}100\phantom{\rule{0.28em}{0ex}}\mathrm{MHz}\ensuremath∼100MHz and with displacement amplitude \ensuremath∼1nm\ensuremath{\sim}1\phantom{\rule{0.28em}{0ex}}\mathrm{nm}\ensuremath∼1nm experiences an acceleration magnitude \ensuremath∼109ms\ensuremath−2\ensuremath{\sim}{10}^{9}\phantom{\rule{0.28em}{0ex}}{\mathrm{m}\phantom{\rule{0.28em}{0ex}}\mathrm{s}}^{\ensuremath{-}2}\ensuremath∼109ms\ensuremath−2, approaching the surface gravity of a neutron star. As one application of such extreme noninertial forces in a mesoscopic setting, we consider a model two-path atom interferometer with one path consisting of the 100MHz100\phantom{\rule{0.28em}{0ex}}\mathrm{MHz}100MHz vibrating wire atom guide. The vibrating wire guide serves as a noninertial reference frame and induces an in principle measurable phase shift in the wave function of an atom traversing the wire frame. We furthermore consider the effect on the two-path atom wave interference when the vibrating wire is modeled as a quantum object, hence functioning as a quantum noninertial reference frame. We outline a possible realization of the vibrating wire, atom interferometer using a superfluid helium quantum interference setup.
Neutron interferometry has been performed on an accelerated interferometer under conditions where gravitational effects on the interfering beams very nearly compensate each other. The observed neutron phase shift is found to be in excellent agreement with that calculated from Schr\"odinger's equation transformed to the accelerated frame where it is found essential to include the spherical wave aspect. The present experiment thus verifies the validity of the classical transformation laws for noninertial frames also in the quantum limit.
Atom interferometers are among the most sensitive inertial sensors, yet deployment outside the laboratory is limited by vibration noise, conventionally mitigated by external sensors or bulky isolation. Here we demonstrate a hybrid inertial sensor which fuses an optomechanical resonator and an atom interferometer by exploiting the resonator's test mass as the interferometer's reference mirror. This allows for better correlation than with two separate sensors, whose unknown transfer function is replaced by the static response of one mechanical element. The resonator achieves a displacement sensitivity of 8.6⋅10−158.6\cdot 10^{-15}8.6⋅10−15 m/Hz\sqrt{\mathrm{Hz}}Hz​ with suppressed 1/f noise and yields a minimum acceleration sensitivity of 1.1⋅10−61.1\cdot10^{-6}1.1⋅10−6 m/s2^22/Hz\sqrt{\mathrm{Hz}}Hz​ over a bandwidth extending from sub-Hertz to 2.5 kHz. Under ambient laboratory conditions the integrated system removes vibration-induced phase ambiguity for accelerations up to 50⋅10−350\cdot 10^{-3}50⋅10−3 m/s2^22 and reaches the interferometer's technical noise limit, which a commercial force-balance accelerometer does not. Because the resonance-tracking readout is largely independent of the mechanical design, the architecture transfers directly to other precision sensing platforms.
Neutron interferometry performed in a noninertial frame is essentially influenced by dynamical diffraction effects occurring in the interferometer crystal which hitherto have not been taken into account. We have developed a corresponding theory from which a corrected phase shift is obtained which applies likewise to interferometry performed under the influence of gravity or on an accelerated neutron interferometer. The theory also predicts fading of interference contrast with increasing acceleration as was observed in the Colella-Overhauser-Werner gravity experiments but could not be explained unambiguously.
Quantum sensing using atom interferometry possesses immense potential for application in gravimetry, inertial navigation and for gravitational wave detection, apart from carrying out tests of fundamental physics. Atom interferometry can be used to perform absolute inertial measurements with unprecedented long term stability, unlike classical sensors which are prone to drift and incur calibration errors. However, atom interferometers seldom achieve their fundamental sensitivity due to the coupling of ambient vibrations. It is the dominant noise source that not only impairs performance but also affects its operability. Ideal compensation of the effects due to vibration noise using a versatile accelerometer, while concurrently enabling miniaturized atom interferometer setups has not been pioneered. Such a compact hybrid quantum sensor will enable reliable operation with an enhanced performance in field applications, paving the way for robust atom interferometry beyond laboratory conditions. A novel optomechanical resonator has been designed, developed and successfully hybridized with an atom interferometer to suppress the effects of vibration noise, and the results are presented in this thesis. The optomechanical resonator’s inertial test mass is simultaneously used to create a continuous intracavity optical lattice to sense its displacement, and a pulsed optical lattice to manipulate the atoms. For the first time, a broadband accelerometer is seamlessly hybridized with an atom interferometer via mechanical resonator mediated coupling of the optical lattices to measure acceleration. This enables elimination of vibration induced measurement ambiguity in our atom interferometer with a well defined mechanical transfer function. Vibration noise correlation is demonstrated up to our prototype atom interferometer’s most sensitive operating mode of T = 6 ms using double Bragg diffraction, ultimately limited by its technical noise. The optomechanical resonator measures acceleration with a sensitivity down to 1.1×10−6 ms−2/ √Hz corresponding to a displacement sensitivity of 8.6 × 10−15 m/ √Hz. Substantial 1/f noise suppression due to the developed optical readout scheme enables a measurement bandwidth spanning from sub-hertz frequencies up to several kilohertz. Multiple versions of the optomechanical resonator with different characteristics were developed and hybridized with our atom interferometer. This paves the way for developing tailored optomechanical resonators for specific inertial sensing applications.
Abstract The fundamental dual nature of neutrons –sometimes a particle (when detected) and sometimes a wave (when traversing the interferometer) – is wonderfully mani-fested by the non-local nature of the neutron interferometry experiments discussed in this chapter. We begin with a discussion of the series of experiments, called gravitationally induced quantum interference, in Section 7.1. Then in Section 7.2, the effect of the Earth’s rotation on the phase of a neutron de Broglie wave is described. This is the quantum mechanical analogue of the Michelson et al. (1925) experiment with light. An experiment in which this Sagnac phase shift was observed due to rotation of the interferometer on a turntable phase within the laboratory frame is also discussed (Atwood et al. 1984). An experiment related to the small effective mass of the neutron propagating in a crystal under Bragg reflecting conditions and its deflection due to the Coriolis force is discussed in Section 7.2.3 (Raum et al. 1997). Acceleration-induced interference, which is related to the gravity experiments by the principle of equivalence, is discussed in Section 7.3 (Bonse and Wroblewski 1983). Phase shifts caused by the motion of matter within the interferometer are related to the optical Fizeau effect. Several neutron Fizeau-type experiments are discussed in Section 7.4.
In the presence of Earth gravity and gravity-gradient forces, centrifugal and Coriolis forces caused by the Earth rotation, the phase of the time-domain atom interferometers is calculated with accuracy up to the terms proportional to the fourth degree of the time separation between pulses. We considered double-loop atom interferometers and found appropriate condition to eliminate their sensitivity to acceleration to get atomic gyroscope, or to eliminate the sensitivity to rotation to increase accuracy of the atomic gravimeter. Consequent use of these interferometers allows one to measure all components of the acceleration and rotation frequency projection on the plane perpendicular to gravity acceleration. Atom interference on the Raman transition driving by noncounterpropagating optical fields is proposed to exclude stimulated echo processes which can affect the accuracy of the atomic gyroscopes. Using noncounterpropagating optical fields allows one to get a new type of the Ramsey fringes arising in the unidirectional Raman pulses and therefore centered at the two-quantum line center. Density matrix in the Wigner representation is used to perform calculations. It is shown that in the time between pulses, in the noninertial frame, for atoms with fully quantized spatial degrees of freedom, this density matrix obeys classical Liouville equations.