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Journal of Modern Optics

Quantum Theory of High-resolution Length Measurement with a Fabry-Perot Interferometer

M. Ley, R. Loudon

Published Feb 1, 1987• 99 citations
Publisher

Abstract

The quantum limits on measurements of small changes in the length of a Fabry-Perot cavity are calculated. The cavity is modelled by a pair of dissimilar mirrors oriented perpendicular to a one-dimensional axis of infinite extent. The continuous spectrum of spatial modes of the system is derived, and the electromagnetic field is quantized in terms of a continuous set of mode creation and destruction operators. Coherent state and squeezed vacuum-state excitations of the field are characterized by energy flow, or intensity, variables. The determination of small changes in the cavity length by observations of fringe intensity is considered for schemes in which the cavity is simultaneously excited by coherent and squeezed vacuum-state inputs. The contributions to the limiting resolution from photocount and radiation-pressure length uncertainties are evaluated. These properties of the Fabry-Perot cavity are compared with the corresponding results for the Michelson interferometer.

DOI
10.1080/09500348714550251

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The quantum limits on measurements of small changes in the length of a Fabry-Perot cavity are calculated. The cavity is modelled by a pair of dissimilar mirrors oriented perpendicular to a one-dimensional axis of infinite extent. The continuous spectrum of spatial modes of the system is derived, and the electromagnetic field is quantized in terms of a continuous set of mode creation and destruction operators. Coherent state and squeezed vacuum-state excitations of the field are characterized by energy flow, or intensity, variables. The determination of small changes in the cavity length by observations of fringe intensity is considered for schemes in which the cavity is simultaneously excited by coherent and squeezed vacuum-state inputs. The contributions to the limiting resolution from photocount and radiation-pressure length uncertainties are evaluated. These properties of the Fabry-Perot cavity are compared with the corresponding results for the Michelson interferometer.
A full quantum theory of the Fabry-Perot interferometer is presented. The quantum limits on the determination of small changes in the cavity length are considered, studying the case in which the interferometer is simultaneously excited by coherent and squeezed-vacuum inputs. The contributions to the limiting resolution from photon noise and radiation pressure uncertainty are evaluated in a unified calculation by means of a unitary transformation which relates the input modes to the output modes. These results are compared with the corresponding results for the Michelson interferometer.
This communication has considered the case of a Fabry-Perot cavity with light impinging on one end while the field distribution on the other end is observed. It has been shown that if one starts with a wave that can be expanded into the modes of the integral operator, and if the medium in the cavity is passive, more than one mode would be observed at the other end although the amplitude of the dominant mode can be much greater than that of the other modes. It was further shown that if there is present an active medium with sufficient gain there will be only a single mode, and the single mode is necessary and sufficient for complete spatial coherency.
We find an upper bound for the sensitivity of Fabry-P\'erot interferometric length measurements using inefficient photodetectors. This bound applies to all possible quantum-mechanical states of the light entering the interferometer. It applies to standard photodetection, differenced photodetection, and homodyne detection. In the limit of large photon number we identify the optimal detection scheme and input states. They are homodyne detection with a coherent state in one input port, and a nonvacuum quadrature squeezed state in the other. No other quantum states of the input light give better sensitivity, provided that the interferometer output is measured by either photodetection or homodyning, and that the photodetectors are inefficient.
We have developed an experiment of ultrasensitive interferometric measurement of small displacements based on a high-finesse Fabry-Perot cavity. We have observed the internal thermal noise of mirrors and fully characterized their acoustics modes with a sensitivity of 3 × 10 −20 m/Hz 1/2 . This unique sensitivity is a step towards the first observation of the radiation pressure effects and the resulting standard quantum limit in interferometric measurements. Our experiment may become a powerful facility to test quantum noise reduction schemes such as the use of squeezed light or quantum locking of mirrors. As a first result, we present the observation of a cancellation of radiation pressure effects in our cavity. This back-action noise cancellation was first proposed within the framework of gravitational-wave detection with dual resonators, and may drastically improve the sensitivity of such measurements.
A description is given of several novel experimental methods for quantum-electrodynamic vacuum confinement of emission modes close to a single electromagnetic boundary and in a microscopic Fabry–Perot optical cavity of length of the same order of magnitude as the wavelength of the recorded radiation. Strong anomalies (Purcell effect) of the spontaneous emission process and of a phase transition with spontaneous–stimulated decay of states in the stimulated emission process are demonstrated experimentally. It is shown that in the context of the phase transition theory an active microscopic cavity behaves as a superconductor with an extremely high critical transition temperature. This unusual behavior can be interpreted as a manifestation of a perturbation induced in a quantum system by "measurement apparatus" of dimensions comparable with the de Broglie wavelength of a test particle.