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Publication# Foucault pendulum properties of spherical oscillators

Abstract

The Foucault pendulum provides a demonstration of the turning of the Earth. The principle at work is that linear oscillations of a two-degree-of-freedom isotropic harmonic oscillator remain unchanged in an inertial frame of reference, so appear to precess in a rotating frame of reference. In recent work, we applied two-degree-of-freedom isotropic oscillators to mechanical timekeeping. In this paper, we note that the spherical oscillators we considered have qualitatively different behavior in a non-inertial frame. We show that when in a rotating frame, linear oscillations precess at one half the rotational speed of the rotating frame. We validate this result experimentally by designing and constructing a proof of concept demonstrator placed on a motorized rotating table. The demonstrator consists of a spherical isotropic oscillator, a launcher to place the oscillator on planar orbits, a motorized rotating table, video recording for qualitative observation, and a laser measurement setup for quantitative results. The experimental data recorded by the lasers strongly validate the physical phenomenon.

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Why are classical theories often sufficient to describe the physics of our world even though everything around us is entirely composed of microscopic quantum systems? The boundary between these two fundamentally dissimilar theories remains an unsolved problem in modern physics. Position measurements of small objects allow us to probe the area where the classical approximation breaks down. In quantum mechanics, Heisenbergâs uncertainty principle dictates that any measurement of the position must be accompanied by measurement induced back-action---in this case manifested as an uncertainty in the momentum. In recent years, cavity optomechanics has become a powerful tool to perform precise position measurements and investigate their fundamental limitations. The utilization of optical micro-cavities greatly enhances the interaction between light and state-of-the-art nanomechanical oscillators. Therefore, quantum mechanical phenomena have been successfully observed in systems far beyond the microscopic world. In such a cavity optomechanical system, the fluctuations in the position of the oscillator are transduced onto the phase of the light, while fluctuations in the amplitude of the light disturb the momentum of the oscillator during the measurement. As a consequence, correlations are established between the amplitude and phase quadrature of the probe light. However, so far, observation of quantum effects has been limited exclusively to cryogenic experiments, and access to the quantum regime at room temperature has remained an elusive goal because the overwhelming amount of thermal motion masks the weak quantum effects. This thesis describes the engineering of a high-performance cavity optomechanical device and presents experimental results showing, for the first time, the broadband effects of quantum back-action at room temperature. The device strongly couples mechanical and optical modes of exceptionally high quality factors to provide a measurement sensitivity $\sim\!10^4$ times below the requirement to resolve the zero-point fluctuations of the mechanical oscillator. The quantum back-action is then observed through the correlations created between the probe light and the motion of the nanomechanical oscillator. A so-called âvariational measurementâ, which detects the transmitted light in a homodyne detector tuned close to the amplitude quadrature, resolves the quantum noise due to these correlations at the level of 10% of the thermal noise over more than an octave of Fourier frequencies around mechanical resonance. Moreover, building on this result, an additional experiment demonstrates the ability to achieve quantum enhanced metrology. In this case, the generated quantum correlations are used to cancel quantum noise in the measurement record, which then leads to an improved relative signal-to-noise ratio in measurements of an external force. In conclusion, the successful observation of broadband quantum behavior on a macroscopic object at room temperature is an important milestone in the field of cavity optomechanics. Specifically, this result heralds the rise of optomechanical systems as a platform for quantum physics at room temperature and shows promise for generation of ponderomotive squeezing in room-temperature interferometers.

Patrick Robert Flückiger, Simon Nessim Henein, Ilan Vardi

In 1851 Léon Foucault created a sensation with his pendulum providing a direct demonstration of the turning of the Earth. This simple device consists of a pendulum which is launched in a purely planar orbit. Following Mach's principle of inertia, the mass will continue to oscillate in the same planar orbit with respect to absolute space. For an observer on Earth, however, the plane of oscillation will turn. Conceptually speaking, Foucault constructed a very precise demonstrator showing that, when put on a rotating table, planar oscillations of an isotropic two degree of freedom oscillator remain planar with respect to an inertial frame of reference. These oscillators have currently been under study in order to construct new horological time bases. A novel concept was a spherical isotropic two degree of freedom oscillator. Theoretical computations indicate that when put on a rotating table, planar oscillations of the spherical oscillator neither remain planar in the inertial frame nor in the rotating frame of reference, but in a frame of reference rotating at exactly half the rotational speed of the rotating table. This intriguing result led to the design, construction and experimental validation of a proof of concept demonstrator placed on a motorized rotating table. The demonstrator consists of a spherical isotropic oscillator, a launcher to place the oscillator on planar orbits, a motorized rotating table and a measurement setup. The experimental data recorded by the lasers validates the physical phenomenon.

Patrick Robert Flückiger, Simon Nessim Henein, Ilan Vardi

The Foucault pendulum is a well-known mechanism used to demonstrate the rotation of the Earth. It consists in a pendulum launched on linear orbits and, following Mach’s Principle, this line of oscillation will remain fixed with respect to absolute space but appear to slowly precess for a terrestrial observer due to the turning of the Earth. The theoretical proof of this phenomenon uses the fact that, to first approximation, the Foucault pendulum is a harmonic isotropic two degree of freedom (2-DOF) oscillator. Our interest in this mechanism follows from our research on flexure-based implementations of 2-DOF oscillators for their application as time bases for mechanical timekeeping. The concept of the Foucault pendulum therefore applies directly to 2-DOF flexure based harmonic oscillators. In the Foucault pendulum experiment, the rotation of the Earth is not the only source of precession. The unavoidable defects in the isotropy of the pendulum along with its well-known intrinsic isochronism defect induce additional precession which can easily mask the precession due to Earth rotation. These effects become more prominent as the frequency increases, that is, when the length of the pendulum decreases. For this reason, short Foucault pendulums are difficult to implement, museum Foucault pendulum are typically at least 7 meters long. These effects are also present in our flexure based oscillators and reducing these parasitic effects, requires decreasing their frequency. This paper discusses the design and dimensioning of a new flexure based 2-DOF oscillator which can reach low frequencies of the order of 0.1[Hz]. The motion of this oscillator is approximatelyplanar, like the classical Foucault pendulum, and will have the same Foucault precession rate. The construction of a low frequency demonstrator is underway and will be followed by quantitative measurements which will examine both the Foucault effect as well as parasitic precession.

2020