We point out a general framework that encompasses most cases in which quantum effects enable an increase in precision when estimating a parameter (quantum metrology) The typical quantum precision-enhancement is of the order of the square root of the number of times the system is sampled We prove that this is optimal and we point out the different strategies (classical and quantum) that permit to attain this bound

Quantum metrology

Nanomechanical resonators can now be realized that achieve fundamental resonance frequencies exceeding 1 GHz, with quality factors (Q) in the range 10^3<=Q<=10^5. The minuscule active masses of these devices, in conjunction with their high Qs, translate into unprecedented inertial mass sensitivities. This makes them natural candidates for a variety of mass sensing applications. Here we evaluate the ultimate mass sensitivity limits for nanomechanical resonators operating in vacuo that are imposed by a number of fundamental physical noise processes. Our analyses indicate that nanomechanical resonators offer immense potential for mass sensing—ultimately with resolution at the level of individual molecules.

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Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems

This review provides a general introduction into the theory of cantilever sensors, including practical design concepts, examples for liquid-phase sensing with particular emphasis on biological applications, and a discussion of future prospects.

Cantilever-based biosensors

The damping characteristics of beam-shaped microactuators that oscillate in the transverse direction are analytically evaluated to establish a design method that minimizes energy consumption and increases dynamic performance. The damping force due to airflow is calculated using the Navier-Stokes equation, the accuracy of which is verified by comparing the calculated damping with experimental results. The contributions to the damping due to squeeze force, internal friction, and support loss are calculated by using the Reynolds equation, structural damping theory, and a two-dimensional theory of elasticity, respectively. The final formulae, obtained in simple and closed forms for easy use in the actual design process, are then used to evaluate the relationships between the beam size and the damping ratio of silicon cantilevers, Permalloy cantilevers, and Permalloy spiral springs. Finally, the step response of a Permalloy cantilever is calculated and the relationship between beam size and settling time is determined.

Damping characteristics of beam-shaped micro-oscillators

Dissipation in nanoelectromechanical systems

A bending and stretching mode crystal oscillator as a friction vacuum gauge

The impedance dependence of a tuning‐fork‐shaped quartz oscillator on gas pressure is theoretically analyzed on the basis of a string‐of‐beads model of the quartz oscillator. The theoretical results are numerically compared with experimental data. The analysis has been already carried out in molecular flow and viscous flow region. In this paper, the impedance behavior in the intermediate flow region is formulated by making use of the theory of slip effect. Furthermore, by the formal extension of this formula to the molecular flow region, a single formula describing the impedance dependence on gas pressure over the whole pressure region is obtained. This formula is found to be in good agreement with experimental data. However, there is still a slight discrepancy (<20%) between the two in the intermediate flow region. This shows the limit of the slip theory. For this reason, the Millikan’s empirical formula is adopted in the intermediate flow region. The unified formula obtained thusly is in numerical agreement with experimental data for various kinds of gases.

Unified formula describing the impedance dependence of a quartz oscillator on gas pressure

The frequency shifts of a tuning‐fork‐shaped quartz oscillator with gas pressure were studied experimentally and theoretically. The experimental measurements showed that the frequency shift of the oscillator was very small in the pressure region below 1 Torr and much larger in the higher pressure region above 1 Torr. The experimental results were theoretically analyzed on the basis of a string‐of‐beads model for a quartz oscillator. The theoretical analysis showed that the resonance frequency shift of the quartz oscillator was small in the molecular flow region. In the viscous flow region the frequency shift was found to be proportional to the pressure. These theoretical results are in good agreement with the experimental data. The frequency shift of the oscillator in the higher pressure region was also found to be proportional to the molecular weight of the gas species in both theory and experiment.

Frequency dependence of a quartz oscillator on gas pressure

Measurement of outgassing characteristics from a vacuum chamber fabricated for pressure calibration in UHV/XHV region

A quartz friction vacuum gauge (QF gauge) is based on the fact that the change in resonance impedance of a quartz oscillator is proportional to the frictional force of ambient gas. The QF gauge adopting a self‐oscillating circuit has been developed in order to obtain quick response and high resolution. The sensor head is the tuning fork shaped (33 kHz) quartz oscillator, which is kept at about 50 °C by a temperature controller. The driving circuit consists of a phase shifter, a zero‐cross detector, and analog switches connected to constant voltage suppliers, which send the quartz oscillator 200 mV peak‐to‐peak square‐wave voltage. The I/V converter transforms the current through the quartz oscillator into the voltage. And the voltage is sent to the phase shifter and a meter driver. The testing results of the QF gauge are as follows. Measurable pressure range 10−1–105 Pa, pressure resolution 5×10−3 Pa at 10−1 Pa, stability 1.3×10−3 Pa, response time 0.2 s at the transition from 10−4 to 105 Pa.

Kiyohide Kokubun

Papers

A bending and stretching mode crystal oscillator as a friction vacuum gauge

Unified formula describing the impedance dependence of a quartz oscillator on gas pressure

Frequency dependence of a quartz oscillator on gas pressure

Measurement of outgassing characteristics from a vacuum chamber fabricated for pressure calibration in UHV/XHV region

Design and testing of a quartz friction vacuum gauge using a self‐oscillating circuit