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Cambridge University Press eBooks

Linear Detectors

Andrew N. Jordan, Irfan A. Siddiqi

Published Feb 15, 2024• 0 citations
Publisher

Abstract

In Chapter 7, we discuss the fundamental limits of quantum amplification. When quantum effects are amplified to classical signals, noise is added to the signal (there are exceptions, but other trade-offs come into play). A detailed discussion of linear response theory is given, which is applicable to many kinds of quantum-limited measurements. This theory is applied to mesoscopic charge detectors and resonant optical cavities. While fundamental bounds quantum mechanics gives to amplification are important, they do not tell you how to invent a quantum-limited amplifier.

DOI
10.1017/9781009103909.008

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In Chapter 7, we discuss the fundamental limits of quantum amplification. When quantum effects are amplified to classical signals, noise is added to the signal (there are exceptions, but other trade-offs come into play). A detailed discussion of linear response theory is given, which is applicable to many kinds of quantum-limited measurements. This theory is applied to mesoscopic charge detectors and resonant optical cavities. While fundamental bounds quantum mechanics gives to amplification are important, they do not tell you how to invent a quantum-limited amplifier.
The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and atomic, molecular, optical--quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, the basics of weak continuous measurements are described. Particular attention is given to the treatment of the standard quantum limit on linear amplifiers and position detectors within a general linear-response framework. This approach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quantum optics and its application to the case of electrical circuits is illustrated, including mesoscopic detectors and resonant cavity detectors.
Using standard linear response relations, we derive the quantum limit on the sensitivity of a generic linear-response position detector, and the noise temperature of a generic linear amplifier. Particular emphasis is placed on the detector's effective temperature and damping effects; the former quantity directly determines the dimensionless power gain of the detector. Unlike the approach used in the seminal work of Caves [Phys. Rev. D 26 1817 (1982)], the linear-response approach directly involves the noise properties of the detector, and allows one to derive simple necessary and sufficient conditions for reaching the quantum limit. Our results have direct relevance to recent experiments on nanoelectromechanical systems, and complement recent theoretical studies of particular mesoscopic position detectors.
Quantum optics is the field of physics describing the interaction of individual photons with matter, but in recent years it has expanded beyond pure physics to become an important driving force for technological innovation. This book starts with an elementary description of the underlying physics and then builds up a more advanced treatment. The theory begins with the quantum description of the simple harmonic oscillator, and is subsequently extended to provide the tools required to discuss coherent states, the interaction of light with atoms, entangled states, quantum noise and dissipation, linear optical amplifiers, and the fundamental issues associated with Bell's theorem. There is an equally strong emphasis on experimental methods. A quantum description of lenses, mirrors, beam splitters, Y-junctions, circulators, and stops is applied to a collection of important experiments in linear optics. A description of the most important methods of primary photon detection is followed by an explanation of heterodyne and homodyne techniques. Spontaneous down conversion and quantum tomography are discussed, together with important experimental applications. These experimental and theoretical considerations come together in a chapter briefly discussing quantum noise and its suppression in telecommunications; the limitations and possibilities for quantum cloning; the principles and techniques of quantum cryptography; and the physical basis for quantum computing.
If the uncertainty principle of quantum mechanics is applied to the process of signal measurement, two theorems relating to amplifier noise performance can be deduced. First, it can be shown that it is impossible to construct a linear noiseless amplifier. Second, if the amplifier is characterized as having additive white Gaussian noise, it can be shown that the minimum possible noise temperature of any linear amplifier is T.= In 2 -IIG -hv 1-1/ IG k In the limit of high gain G this expression reduces to that previously derived for the ideal maser and parametric amplifier. It is shown that the minimum noise amplifier does not degrade the signal but rather allows the use of an inaccurate detector to make measurements on an incoming signal to the greatest accuracy consistent with the uncertainty principle.
Nonlinear amplifiers, such as the transistor, are ubiquitous in classical technology. Little is understood about the noise properties and applications of quantum nonlinear amplifiers. We introduce a class of nonlinear amplifiers that allow one to measure any normal operator with a linear detector while adding a half-quantum of vacuum fluctuations as noise at the output. When these nonlinear amplifiers are used in conjunction with noisy linear detectors, the resulting measurement in the large gain limit becomes equivalent to ideal projective measurement of the normal operator.
After a quick review of the basic theory of quantum optomechanical systems, based largely on linearized Heisenberg–Langevin equations, this chapter focuses on selected topics related to quantum measurement and quantum optomechanics. Included are: a comprehensive discussion of the quantum limit on the added noise of a continuous position detector, following the quantum linear response approach; a detailed discussion of the role of noise correlations, and how these can be achieved in an optomechanical cavity (by using squeezed input light, or by modifying the choice of measured output quadrature); and a discussion of back-action evading measurements of a mechanical quadrature, discussing how this can be achieved in a two-tone driven cavity system. The chapter ends with a quick introduction to the theory of conditional continuous quantum measurement, and a discussion on how a back-action evading measurement can be used to produce conditional mechanical squeezed states.