Review of methods of measurement There are two different variants of the task of measuring electromagnetic energy: i) measuring the energy in a mode of an electromagnetic resonator, and ii) measuring the energy of a traveling electromagnetic wave. This chapter deals with the first variant; the next chapter, with the second. The general principles underlying QND measurements of energy were formulated in chapter IV. Let us recall that the main principles are: i) the response of the measuring device must be directly proportional to the energy (and not, for example, to the strength of the electric field or the charge on the capacitor); and ii) the response must contain no information about the phase of the electromagnetic oscillations. Chapter IV analyzed the simplest example of such a device: a ponderomotive sensor that registers the electromagnetic pressure on the resonator's wall. The weakness of the electromagnetic pressure produced by a small number of quanta makes it unlikely that this ponderomotive method can be realized in practice. Because of this, several other schemes for QND energy measurements have been proposed. Most are based on nonlinear effects in dielectrics. Their practical realization is a rather complicated task because, when the energy density is low, the nonlinear effects hardly work at all, and the response of the measuring device is correspondingly small.
Review of methods of measurement There are two different variants of the task of measuring electromagnetic energy: i) measuring the energy in a mode of an electromagnetic resonator, and ii) measuring the energy of a traveling electromagnetic wave. This chapter deals with the first variant; the next chapter, with the second. The general principles underlying QND measurements of energy were formulated in chapter IV. Let us recall that the main principles are: i) the response of the measuring device must be directly proportional to the energy (and not, for example, to the strength of the electric field or the charge on the capacitor); and ii) the response must contain no information about the phase of the electromagnetic oscillations. Chapter IV analyzed the simplest example of such a device: a ponderomotive sensor that registers the electromagnetic pressure on the resonator's wall. The weakness of the electromagnetic pressure produced by a small number of quanta makes it unlikely that this ponderomotive method can be realized in practice. Because of this, several other schemes for QND energy measurements have been proposed. Most are based on nonlinear effects in dielectrics. Their practical realization is a rather complicated task because, when the energy density is low, the nonlinear effects hardly work at all, and the response of the measuring device is correspondingly small.
The history of the theory of quantum nondemolition (QND) measurements from the 1920s until today is reviewed. The definition and main principles of QND measurements are outlined. Achievements in the experimental realization of QND measurements and several new promising schemes of QND measurements are described. A list of the most important problems (from the authors' point of view) in the area of QND measurements is presented. The problem of measurement of a quantum oscillator phase is considered. A new method of phase measurement is proposed. Examples of possible solutions of fundamental physical problems using QND methods are given.
We analyze schemes for performing quantum nondemolition (QND) measurements in optical cavities. We consider three schemes: (1) measurement of a quadrature phase amplitude using a parametric process, (2) measurement of a quadrature phase amplitude using the optical Kerr effect in a nonlinear fiber, and (3) measurement of the photon number also using the Kerr effect in a fiber. We show that in the second scheme an enhancement of the QND effect may be obtained by making the cavity finesse for the signal larger than that for the probe.
In recent years advances in technology, especially in the field of quantum optics, have made possible repetitive quantum measurements on a single quantum object, an issue that remained an academic curiosity until 25 years ago. This book, first published in 1992, provides a tutorial introduction to this field. The book is coauthored by an experimentalist and a theorist: the former - Braginsky - was the first experimentalist after Weber to develop techniques for gravitational-wave detection; the latter - Khalili - contributed with Braginsky to develop the formal apparatus to describe the experimental devices for measurement schemes. After the first part, intended as a pedagogical appendix of a standard textbook, and devoted to the modern approach to quantum measurement, the second part is a review of some new methods of quantum measurements for macroscopic objects. The central problem is the detection of a small time-dependent classical force that acts on a quantum probe. This is the typical situation that occurs in gravitational-wave detection, where the probe is a macroscopic moving mass weakly coupled to a classical wave. The problem of gravitational-wave detection and the limitation to accuracy of measurement of position of a moving mass - the so-called standard quantum limit (SQL), which is a consequence of the random back-action of the measuring device on the mass - triggered the development of new quantum nondemolition (QND) measurements, i.e. measurements where the measured quantity is left unperturbed after the measurement. SQL and QND measurements (both Braginsky inventions) permeate the whole book as a transversal chapter. The aim is to give simple criteria and recipes to design a measuring device in order to optimize the accuracy of the measurement, and, at the same time, to minimize the perturbation that the device exerts on the measured object, so that it will not interfere with successive measurements on the same object. The continuous measurements are regarded as the limiting case of a sequence of repeated discrete measurements, when the interval between the measurements is much smaller than the characteristic time-scale on which the measured quantity changes. The analysis of continuous measurements give to the authors the opportunity to illustrate the watchdog effect and the quantum Zeno paradox (real effect), while deriving the master equation for the evolution of the object state. Then, an entire chapter is devoted to the analysis of devices for continuous measurement of small mechanical displacements. The book is easy to understand, and can serve as a complementary reading for graduate level courses in quantum mechanics. The material of the book can be useful for all scientists and engineers involved in researches related to high-sensitive quantum measurements, quantum information theory and quantum computation. The reader, however, should be aware that this book presents a very particular point of view, with no room left to very relevant concepts in quantum measurement theory, as, for example, squeezing, quantum amplification, joint-measurements of non-commuting observables and phase-measurements. More regrettably, the list of references is unsatisfactorily incomplete, with no mention of alternative developments, for example, regarding the SQL for measuring the position of a free mass. Here, schemes to breach the SQL have been proposed in the literature (Yuen H P 1983 Phys. Rev. Lett. 51 719, Ozawa M 1988 Phys. Rev. Lett. 60 385), whereas from the book one could conclude that such SQL would be unsurpassable. Also, a criticism is in order of the way of deriving some SQL's, based solely on the Heisenberg relation, even in consideration that the aim of such derivations is mostly pedagogical.
A quantum-nondemolition (QND) measurement is one which does not disturb the quantity it measures. Any measurement can be regarded as a QND measurement followed by an additional back-action which may disturb that quantity. By feeding back the measurement result to control the system dynamics, this back-action can always be eliminated in principle. A practical device based on homodyne measurement of the x quadrature of a cavity mode is investigated. First, a ${\mathrm{\ensuremath{\chi}}}^{(2)}$ crystal is inserted in the cavity so that it becomes a degenerate parametric oscillator at threshold. Then the photocurrent is used to control coherent driving of the cavity, so as to give positive feedback with unit gain. This feedback, combined with the nonlinearity, turns the homodyne measurement into a QND measurement of x. The device can also be used to measure the x quadrature of a traveling wave, and gives near-perfect QND correlations over a large bandwidth. However, the quality of the measurement is badly degraded by even slightly inefficient photodetectors.
The monitoring of a quantum-mechanical harmonic oscillator on which a classical force acts is important in a variety of high-precision experiments, such as the attempt to detect gravitational radiation. This paper reviews the standard techniques for monitoring the oscillator, and introduces a new technique which, in principle, can determine the details of the force with arbitrary accuracy, despite the quantum properties of the oscillator. The standard method for monitoring the oscillator is the "amplitude-and-phase" method (position or momentum transducer with output fed through a narrow-band amplifier). The accuracy obtainable by this method is limited by the uncertainty principle ("standard quantum limit"). To do better requires a measurement of the type which Braginsky has called "quantum nondemolition." A well known quantum nondemolition technique is "quantum counting," which can detect an arbitrarily weak classical force, but which cannot provide good accuracy in determining its precise time dependence. This paper considers extensively a new type of quantum nondemolition measurement---a "back-action-evading" measurement of the real part X1 (or the imaginary part X2) of the oscillator's complex amplitude. In principle X1 can be measured "arbitrarily quickly and arbitrarily accurately," and a sequence of such measurements can lead to an arbitrarily accurate monitoring of the classical force. The authors describe explicit Gedanken experiments which demonstrate that X1 can be measured arbitrarily quickly and arbitrarily accurately. In these experiments the measuring apparatus must be coupled to both the position (position transducer) and the momentum (momentum transducer) of the oscillator, and both couplings must be modulated sinusoidally. For a given measurement time the strength of the coupling determines the accuracy of the measurement; for arbitrarily strong coupling the measurement can be arbitrarily accurate. The "momentum transducer" is constructed by combining a "velocity transducer" with a "negative capacitor" or "negative spring." The modulated couplings are provided by an external, classical generator, which can be realized as a harmonic oscillator excited in an arbitrarily energetic, coherent state. One can avoid the use of two transducers by making "stroboscopic measurements" of X1, in which one measures position (or momentum) at half-cycle intervals. Alternatively, one can make "continuous single-transducer" measurements of X1 by modulating appropriately the output of a single transducer (position or momentum), and then filtering the output to pick out the information about X1 and reject information about X2. Continuous single-transducer measurements are useful in the case of weak coupling. In this case long measurement times are required to achieve good accuracy, and continuous single-transducer measurements are almost as good as perfectly coupled two-transducer measurements. Finally, the authors develop a theory of quantum nondemolition measurement for arbitrary systems. This paper (Paper I) concentrates on issues of principle; a sequel (Paper II) will consider issues of practice.
Aspects of quantum limits for the detection of a classical force The development of the theory of QND measurements was triggered by the problem of detecting a small classical force that acts on a quantum probe―e.g. on a free mass or an oscillator, see the review article and the monograph. This problem arose most especially in gravitational-wave experiments. Astrophysical estimates indicate that the gravitational waves from cosmic sources should be so strong that they behave almost perfectly classically. During the measuring time the number of gravitons that pass within one wavelength of the measuring device is enormously larger than unity; and correspondingly, as would be the case if the waves were electromagnetic, quantum effects in the radiation are utterly negligible. A central feature of gravitational waves is their extremely weak interaction with matter. Because of this weakness, the response of the measuring device's probe object to the waves is so small that it may be comparable to the object's quantum mechanical uncertainties. At the same time, the weakness of the interaction guarantees that there is almost no back action of the probe on the gravitational-wave field. As a result, the gravitational wave acts on the probe as through it were a precisely classical force (i.e. a force that is independent of the probe's quantum state).