Thermal noise limitations to force measurements with torsion pendulums: Applications to the measurement of the Casimir force and its thermal correction
S. K. Lamoreaux, W. T. Buttler
备注：
one figure, five pages, to be submitted to Phys Rev E
A general analysis of thermal noise in torsion pendulums is presented. The specific case where the torsion angle is kept fixed by electronic feedback is analyzed. This analysis is applied to a recent experiment that employed a torsion pendulum to measure the Casimir force. The ultimate limit to the distance at which the Casimir force can be measured to high accuracy is discussed, and in particular the prospects for measuring the thermal correction are elaborated upon.
We want to introduce an atomic pendulum whose driving force (torque) is due to the quantum vacuum fluctuations. Applying the well-known Casimir-Polder effect to a special configuration (a combined structure of an atomic nanostring and a conducting plate), an atomic pendulum (Casimir atomic pendulum) is designed. Using practically acceptable data corresponding to the already known world of nanotechnology and based on reasonable/reliable numerical estimates, the period of oscillation for the pendulum is computed. This pendulum can be considered as both a new Micro (Nano)-Eelectromechanical syste…
We discuss a proposal to measure the Casimir force in the parallel plate configuration in the $1-10μ$m range via a high-sensitivity torsional balance. This will allow to measure the thermal contribution to the Casimir force therefore discriminating between the various approaches discussed so far. The accurate control of the Casimir force in this range of distances is also required to improve the limits to the existence of non-Newtonian forces in the micrometer range predicted by unification models of fundamental interactions.
Precise measurements of the displacement of, and force acting on, a mechanical oscillator can be performed by coupling the oscillator to an optical cavity. Brownian thermal forces represent a fundamental limit to measurement sensitivity which impedes the ability to use precise force measurements as a tool of fundamental enquiry, particularly in the context of macroscopic quantum measurements and table-top gravitational experiments. A torsion pendulum with a low mechanical resonant frequency can be limited by very small thermal forces - from its suspensions - at frequencies above resonance. Her…
We characterized a double torsion pendulum system, including measurements of the photon-spin-induced torque. Our experimental strategy was similar to that used in Beth's experiment, which was performed in 1936 to measure photon-spin-induced torque using forced oscillation caused by polarization modulation of light incident on a suspended object. Through simple passive isolation of the suspended object from external vibration noise, the achieved torque sensitivity was 2 x 10^{-17} N m in a measurement time of 10^4 s, which is close to the thermal noise limit and one order smaller than the minim…
We report development of a high-sensitivity torsion balance to measure the thermal Casimir force. Special emphasis is placed on experimental investigations of a possible surface electric force originating from surface patch potentials that have been recently noticed by several experimental groups. By gaining a proper understanding of the actual contribution of the surface electric force in real materials, we aim to undertake precision force measurements to resolve the Casimir force at finite temperature in real metals, as well as in other semiconducting materials, such as graphene.
We have measured the short-range attractive force between crystalline Ge plates, and found contributions from both the Casimir force and an electrical force possibly generated by surface patch potentials. Using a model of surface patch effects that generates an additional force due to a distance dependence of the apparent contact potential, the electrical force was parameterized using data at distances where the Casimir force is relatively small. Extrapolating this model, to provide a correction to the measured force at distances less than 5 $μ$m, shows a residual force that is in agreement, w…
Repulsion, induced from quantum vacuum fluctuations, for an anisotropically polarizable atom on the symmetry axis of an anisotropically polarizable annular disc is studied. There exists two torsion free points on each side of the annular disc, where the interaction energy is orientation independent. The position of second of the two torsion free points, on either side of the disc, is shown to determine the orientation dependence of the atom at distances far from the plate, revealing a geometrical dependence. In the ring limit of the annular disc, new repulsion emerges.
The presence of finite energy in quantum vacuum has profound implications to physics at the microscopic and macroscopic levels. One of the direct consequences of vacuum energy is the Casimir Force, which is a force of attraction experienced by a pair of infinite planar conductive plates placed in vacuum. Experiments to measure Casimir force have been carried out since the 1950s. This thesis presents a measurement of Casimir force between a flat and a spherical conducting surface using a torsion balance for separations in the range of 1 micron to 8 micron. The details of the construction of the…
We address several different Casimir experiments where theory and experiment disagree. First out is the classical Casimir force measurement between two metal half spaces; here both in the form of the torsion pendulum experiment by Lamoreaux and in the form of the Casimir pressure measurement between a gold sphere and a gold plate as performed by Decca et al.; theory predicts a large negative thermal correction, absent in the high precision experiments. The third experiment is the measurement of the Casimir force between a metal plate and a laser irradiated semiconductor membrane as performed b…
We recently proposed a new approach to the Casimir effect based on classical ray optics (the "optical approximation"). In this paper we show how to use it to calculate the local observables of the field theory. In particular we study the energy-momentum tensor and the Casimir pressure. We work three examples in detail: parallel plates, the Casimir pendulum and a sphere opposite a plate. We also show how to calculate thermal corrections, proving that the high temperature `classical limit' is indeed valid for any smooth geometry.
Probing the boundary between classical and quantum mechanics has been one of the central themes in modern physics. Recently, experiments to precisely measure the force acting on milligram scale oscillators with optical cavities are attracting interest as promising tools to test quantum mechanics, decoherence mechanisms, and gravitational physics. In this paper, we review the present status of experiments using milligram scale optomechanical systems. We compare the feasibility of reaching the quantum regime with a pendulum, torsion pendulum, and optically levitated mirror. Considerations for de…
Gravity generated by large masses has been observed using a variety of probes from atomic interferometers to torsional balances. However, gravitational coupling between small masses has never been observed so far. Here, we demonstrate sensitive displacement sensing of the Brownian motion of an optically trapped 7-mg pendulum motion whose natural quality factor is increased to $10^8$ through dissipation dilution. The sensitivity for an integration time of one second corresponds to the displacement generated by the gravitational coupling between the probe and a mm separated 100 mg mass, whose po…
The Casimir force between uncharged metallic surfaces originates from quantum mechanical zero point fluctuations of the electromagnetic field. We demonstrate that this quantum electrodynamical effect has a profound influence on the oscillatory behavior of microstructures when surfaces are in close proximity (<= 100 nm). Frequency shifts, hysteretic behavior and bistability caused by the Casimir force are observed in the frequency response of a periodically driven micromachined torsional oscillator.
We compare theoretical expectations for the Casimir force with the results of precise measurements. The force is calculated at finite temperature for multilayered covering of the bodies using the Lifshitz theory. We argue that the dielectric function of the metallization has to be directly measured to reach the necessary precision in the force calculation. Without knowledge of this function one can establish a well defined upper limit on the force using parameters of perfect single-crystal materials. The force measured in the torsion pendulum experiment does not contradict to the upper limit. …
The shift of nanosized metal configurations relative to one another can lead to an increase in the range of optimal lengths of wings and angles of the opening of cavities, at which noncompensated Casimir forces in them are maximal. The possibility of the existence of the effect of Casimir expulsion is demonstrated with a trapezoid cavity with shifted wings which are opened at different angles. It is also shown that shift in parallel structures leads to the appearance of forces tending to bring the configuration back to the situation before the shift. In some conditions, in systems with shifted…
We report measurements of the Casimir force between a gold sphere and a silicon surface with an array of nanoscale, rectangular corrugations using a micromechanical torsional oscillator. At distance between 150 nm and 500 nm, the measured force shows significant deviations from the pairwise additive formulism, demonstrating the strong dependence of the Casimir force on the shape of the interacting bodies. The observed deviation, however, is smaller than the calculated values for perfectly conducting surfaces, possibly due to the interplay between finite conductivity and geometry effects.
The first precise measurement of the Casimir force between dissimilar metals is reported. The attractive force, between a Cu layer evaporated on a microelectromechanical torsional oscillator, and an Au layer deposited on an Al$_2$O$_3$ sphere, was measured dynamically with a noise level of 6 fN/$\sqrt{\rm{Hz}}$. Measurements were performed for separations in the 0.2-2 $μ$m range. The results agree to better than 1% in the 0.2-0.5 $μ$m range with a theoretical model that takes into account the finite conductivity and roughness of the two metals. The observed discrepancies, which are much larger…
Casimir effect is the attractive force which acts between two plane parallel, closely spaced, uncharged, metallic plates in vacuum. This phenomenon was predicted theoretically in 1948 and reliably investigated experimentally only in recent years. In fact, the Casimir force is similar to the familiar van der Waals force in the case of relatively large separations when the relativistic effects come into play. We review the most important experiments on measuring the Casimir force by means of torsion pendulum, atomic force microscope and micromechanical torsional oscillator. Special attention is …
We compare the results of capacitance measurements in the lens-plane and sphere-plane configurations with theoretical predictions from various models of a spherical surface. It is shown that capacitance measurements are incapable of discriminating between models of perfect and modified spherical surfaces in an experiment demonstrating the anomalous scaling law for the electric force. Claims to the contrary in the recent literature are explained by the use of irregular comparison. The data from capacitance measurements in an experiment measuring the Casimir force using a micromechanical torsion…
We review new constraints on the Yukawa-type corrections to Newtonian gravity obtained recently from gravitational experiments and from the measurements of the Casimir force. Special attention is paid to the constraints following from the most precise dynamic determination of the Casimir pressure between the two parallel plates by means of a micromechanical torsional oscillator. The possibility of setting limits on the predictions of chameleon field theories using the results of gravitational experiments and Casimir force measurements is discussed.
We report on current efforts to detect the thermal and dissipative contributions to the Casimir force. For the thermal component, two experiments are in progress at Dartmouth and at the Institute Laue Langevin in Grenoble. The first experiment will seek to detect the Casimir force at the largest explorable distance using a cylinder-plane geometry which offers various advantages with respect to both sphere-plane and parallel-plane geometries. In the second experiment, the Casimir force in the parallel-plane configuration is measured with a dedicated torsional balance, up to 10 micrometers. Para…
A micromechanical torsion oscillator has been used to strengthen the limits on new Yukawa forces by determining the Casimir pressure between two gold-coated plates. By significantly reducing the random errors and obtaining the electronic parameters of the gold coatings, we were able to conclusively exclude the predictions of large thermal effects below 1 $μ$m and strengthen the constraints on Yukawa corrections to Newtonian gravity in the interaction range from 29.5 nm to 86 nm.
We report measurements of the short-range forces between two macroscopic gold-coated plates using a torsion pendulum. The force is measured for separations between 0.7 $μ$m and 7 $μ$m, and is well described by a combination of the Casimir force, including the finite-temperature correction, and an electrostatic force due to patch potentials on the plate surfaces. We use our data to place constraints on the Yukawa-type "new" forces predicted by theories with extra dimensions. We establish a new best bound for force ranges 0.4 $μ$m to 4 $μ$m, and, for forces mediated by gauge bosons propagating i…
We report new constraints on extra-dimensional models and other physics beyond the Standard Model based on measurements of the Casimir force between two dissimilar metals for separations in the range 0.2--1.2 $μ$m. The Casimir force between an Au-coated sphere and a Cu-coated plate of a microelectromechanical torsional oscillator was measured statically with an absolute error of 0.3 pN. In addition, the Casimir pressure between two parallel plates was determined dynamically with an absolute error of $\approx 0.6$ mPa. Within the limits of experimental and theoretical errors, the results are in…
It is well known that, beginning in 2000, the behavior of the thermal correction to the Casimir force between real metals has been hotly debated. As was shown by several research groups, the Lifshitz theory, which provides the theoretical foundation for the calculation of both the van der Waals and Casimir forces, leads to different results depending on the model of metal conductivity used. To resolve these controversies, the theoretical considerations based on the principles of thermodynamics and new experimental tests were invoked. We analyze the present status of the problem (in particular,…
We present results on the determination of the differential Casimir force between an Au-coated sapphire sphere and the top and bottom of Au-coated deep silicon trenches performed by means of the micromechanical torsional oscillator in the range of separations from 0.2 to 8 $\upmu$m. The random and systematic errors in the measured force signal are determined at the 95\% confidence level and combined into the total experimental error. The role of surface roughness and edge effects is investigated and shown to be negligibly small. The distribution of patch potentials is characterized by Kelvin p…
We report an improved dynamic determination of the Casimir pressure between two plane plates obtained using a micromachined torsional oscillator. The main improvements in the current experiment are a significant suppression of the surface roughness of the Au layers deposited on the interacting surfaces, and a decrease in the experimental error in the measurement of the absolute separation. A metrological analysis of all data permitted us to determine both the random and systematic errors, and to find the total experimental error as a function of separation at the 95% confidence level. In contr…
