Nanomechanical resonators (3)

Abstract

What is the ultimate resolution of mirror position measurement using an interferometric setup ? This question was the subject of the third lesson, where we assumed perfect homodyne detection of the light signal passing through the Fabry-Pérot cavity with moving mirror that had been introduced in the first two lessons. we first recalled the quadrature formalism of the harmonic oscillator. These dynamic variables, well suited to discussing the oscillator in the quantum regime, behave like the oscillator's position and impulse, but are normalized so that a unit amplitude corresponds to an excitation photon. With this language, it is possible to represent the coherent states of the harmonic oscillator by a fresnel " lollipop ", rather than by a simple vector. The lollipop stick is the square root of the average number of vibrational quanta, and the disk radius is the standard deviation of the quantum fluctuations, which is 1/2 in the quadrature system.

Let's consider a propagative time mode of the incident field in the Fabry-Pérot cavity. This mode is itself an oscillator. After passing through the cavity, the mode's Fresnel lollipop will have rotated by a small angle, proportional to the displacement of the mirror, which is assumed to be stationary as the light passes through (adiabatic approximation). By interfering the propagative mode with a reference signal on a splitter plate leading to two hotodetectors, and subtracting their two signals, we obtain the homodyne signal of the mirror position measurement. This signal generally contains three noises. The first is the intrinsic noise of the system being measured, corresponding to fluctuations in the mirror's position, ideally zero-point if the thermal fluctuations of mechanical motion have been eliminated. The second is shot noise, corresponding to the particulate nature of the photons used in the measurement. The third is the most subtle : it corresponds to the quantum feedback introduced into the mirror pulse by the measurement of its position. Although the device only measures the instantaneous position of the mirror, it is sensitive to this feedback action. In fact, for an oscillator, a perturbation of the pulse is converted, a quarter of a period later, into a perturbation of the position. thus, the spectral density of fluctuations in the homodyne signal for interferometric position detection of a mechanical oscillator will present three components at the oscillator's natural frequency. Optimal conditions for this type of measurement are achieved when the photon shot noise component, which governs the measurement inaccuracy, is equal to the component arising from the feedback action. In recent years, a number of research groups have achieved the experimental feat of placing themselves in this regime.