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Journal of the Optical Society of America B

Quantum mechanically formulated fluctuation-characterization method using symmetric and asymmetric interferometers

Tatsuya Tomaru

Published May 20, 2011• 2 citations
Publisher

Abstract

A fluctuation-characterization method using an interferometer is ordinarily classically formulated by considering frequency fluctuations. However, it is preferable to treat it quantum mechanically because the method measures q-number quadrature fluctuations at the same time. We formulated an interferometric fluctuation-characterization method by treating quadrature fluctuations as q-number variables and frequency fluctuations as a c-number stochastic variable, and we gave the power spectral density of the fluctuations for pulse light. This method measures fluctuations that are calibrated with shot noise level, and it makes it possible to estimate the fluctuation angle in phase space, which is the quantity that determines transmission characteristics in optical communications. Fluctuations in a laser diode were characterized as an example of measurement.

DOI
10.1364/josab.28.001502

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A fluctuation-characterization method using an interferometer is ordinarily classically formulated by considering frequency fluctuations. However, it is preferable to treat it quantum mechanically because the method measures q-number quadrature fluctuations at the same time. We formulated an interferometric fluctuation-characterization method by treating quadrature fluctuations as q-number variables and frequency fluctuations as a c-number stochastic variable, and we gave the power spectral density of the fluctuations for pulse light. This method measures fluctuations that are calibrated with shot noise level, and it makes it possible to estimate the fluctuation angle in phase space, which is the quantity that determines transmission characteristics in optical communications. Fluctuations in a laser diode were characterized as an example of measurement.
A Michelson interferometer with noise reduction to sub-shot noise levels is proposed and realized. Multiple measurements of a single signal beam are taken and the quantum property of light plays an essential role in the principle underlying this interferometry. The method makes use of the coherent state of light and requires only a simple modification to the standard Michelson interferometer. The surface fluctuation spectra of liquids are measured using this method down to a few orders of magnitude below the shot noise level. The spectrum derived from hydrodynamical considerations agrees well with the observed results for water. However, for oil, slight deviations are seen at high frequencies (≳1 \gtrsim1\,≳1MHz), perhaps indicating its more complex underlying physics. The measurement requires a relatively low light power and a short time, so that it has a wide range of applicability.
Quantum fluctuations in the phase and amplitude quadratures of light set limitations on the sensitivity of modern optical instruments. The sensitivity of the interferometric gravitational-wave detectors, such as the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO), is limited by quantum shot noise, quantum radiation pressure noise, and a set of classical noises. We show how the quantum properties of light can be used to distinguish these noises using correlation techniques. Particularly, in the first part of the paper we show estimations of the coating thermal noise and gas phase noise, hidden below the quantum shot noise in the Advanced LIGO sensitivity curve. We also make projections on the observatory sensitivity during the next science runs. In the second part of the paper we discuss the correlation technique that reveals the quantum radiation pressure noise from the background of classical noises and shot noise. We apply this technique to the Advanced LIGO data, collected during the first science run, and experimentally estimate the quantum correlations and quantum radiation pressure noise in the interferometer.
According to quantum mechanics, if we keep observing a continuous variable we generally disturb its evolution. For a class of observables, however, it is possible to implement a so-called quantum nondemolition measurement: by confining the perturbation to the conjugate variable, the observable is estimated with arbitrary accuracy, or prepared in a well-known state. For instance, when the light bounces on a movable mirror, its intensity is not perturbed (the effect is just seen on the phase of the radiation), but the radiation pressure allows one to trace back its fluctuations by observing the mirror motion. In this work, we implement a cavity optomechanical experiment based on an oscillating micromirror, and we measure correlations between the output light intensity fluctuations and the mirror motion. We demonstrate that the uncertainty of the former is reduced below the shot-noise level determined by the corpuscular nature of light.
The interferometers now being developed to detect gravitational waves work by measuring the relative positions of widely separated masses. Two fundamental sources of quantum-mechanical noise determine the sensitivity of such an interferometer: (i) fluctuations in number of output photons (photon-counting error) and (ii) fluctuations in radiation pressure on the masses (radiation-pressure error). Because of the low power of available continuous-wave lasers, the sensitivity of currently planned interferometers will be limited by photon-counting error. This paper presents an analysis of the two types of quantum-mechanical noise, and it proposes a new technique---the "squeezed-state" technique---that allows one to decrease the photon-counting error while increasing the radiation-pressure error, or vice versa. The key requirement of the squeezed-state technique is that the state of the light entering the interferometer's normally unused input port must be not the vacuum, as in a standard interferometer, but rather a "squeezed state"---a state whose uncertainties in the two quadrature phases are unequal. Squeezed states can be generated by a variety of nonlinear optical processes, including degenerate parametric amplification.
Central topic of this thesis is the investigation of the quantum nature of light. This investigation is carried out in two separate experiments which are described in part I and part II respectively. In part I, classical and non-classical laser radiation is characterized at the quantum mechanical level with respect to its amplitude and phase uctuations, its photon number distribution and other observable quantities. This is done by employing recently developed methods of quantum state reconstruction. Such a complete characterization is of fundamental interest, since it can provide a much more detailed experimental description of light than previously known. Furthermore, since many experimental systems are analyzed by optical means, these methods may in future and important applications in the characterization of such systems in full quantum mechanical detail, by determining the state of the light held used as a probe before and after the interaction with the system. In part II, high precision position measurements via laser interferometry are investigated. Such measurements play an important role in the microscopic domain (optomechanical sensors, modern microscopy techniques) as well as in the macroscopic domain (development of large scale interferometers for the detection of gravitational waves). The goal of the second experiment is to explore the quantum mechanical limits in the precision with which the position of a macroscopic body can be determined. One common conceptual aspect of both experiments, besides the similar optical techniques employed, is that both attempt a high precision characterization of a harmonic oscillator system disturbed by stochastic noise. In part I, this oscillator is the light held, subject to quantum noise, in part II, it is a mechanical harmonic oscillator excited by thermal noise.
An operator is constructed for the spectral density of a fluctuating dynamical variable. The formulation is based on an analysis of method of measurement. An essential feature of the operator is the replacement of classical dynamical variables by time-dependent Heisenberg operators. The expectation value of the spectral density includes fluctuations of statistical origin, as well as fluctuations of quantum mechanical origin. It can be computed for any physical system when the initial state has been specified. Two applications are given: Nyquist's law is rigorously deduced for a physical system that is similar to the one discussed by Callen and Welton, and the shot effect is calculated for free uncorrelated electrons. A quantum correction is obtained whose origin lies in the wavelike character of the electron.