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Reviews of Modern Physics

On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle

Carlton M. Caves, Kip S. Thorne, Ronald W. P. Drever, Vernon D. Sandberg, Mark Zimmermann

Published Apr 1, 1980• 1,235 citations
Publisher

Abstract

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 ${X}_{1}$ (or the imaginary part ${X}_{2}$) of the oscillator's complex amplitude. In principle ${X}_{1}$ 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 ${X}_{1}$ 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 ${X}_{1}$, in which one measures position (or momentum) at half-cycle intervals. Alternatively, one can make "continuous single-transducer" measurements of ${X}_{1}$ by modulating appropriately the output of a single transducer (position or momentum), and then filtering the output to pick out the information about ${X}_{1}$ and reject information about ${X}_{2}$. 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.

DOI
10.1103/revmodphys.52.341

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On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress

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Several high-precision physics experiments are approaching a level of sensitivity at which the intrinsic quantum nature of the experimental apparatus is the dominant source of fluctuations limiting the sensitivity of the measurements. This quantum limit is embodied by the Heisenberg uncertainty principle, which prohibits arbitrarily precise simultaneous measurements of two conjugate observables of a system but allows one-time measurements of a single observable with any precision. The dynamical evolution of a system immediately following a measurement limits the class of observables that may be measured repeatedly with arbitrary precision, with the influence of the measurement apparatus on the system being confined strictly to the conjugate observables. Observables having this feature, and the corresponding measurements performed on them have been named quantum nondemolition or back-action evasion observables. In a previous review (Caves et al., 1980, Rev. Mod. Phys. 52, 341) a quantum-mechanical analysis of quantum nondemolition measurements of a harmonic oscillator was presented. The present review summarizes the experimental progress on quantum nondemolition measurements and the classical models developed to describe and guide the development of practical implementations of quantum nondemolition measurements. The relationship between the classical and quantum theoretical models is also reviewed. The concept of quantum nondemolition and back-action evasion measurements originated in the context of measurements on a macroscopic mechanical harmonic oscillator, though these techniques may be useful in other experimental contexts as well, as is discussed in the last part of this review. [S0034-6861(96)00103-1]

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The complex amplitude ${X}_{1}+i{X}_{2}\ensuremath{\equiv}(x+\frac{\mathrm{ip}}{m\ensuremath{\omega}}){e}^{i\ensuremath{\omega}t}$ of a harmonic oscillator is constant in the absence of driving forces. Although the uncertainty principle forbids precise measurements of ${X}_{1}$ and ${X}_{2}$ simultaneously ($\ensuremath{\Delta}{X}_{1}\ensuremath{\Delta}{X}_{2}>~\frac{\ensuremath{\hbar}}{2m\ensuremath{\omega}}$), ${X}_{1}$ alone can be measured precisely and continuously ("quantum nondemolition measurement"). Examples are given of measuring systems that do this job. Such systems might play a crucial role in gravitational-wave detection and elsewhere.

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Quantum nondemolition measurements on a double harmonic oscillator

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In the measurements on an oscillator to detect short bursts of gravitational radiation, one important demand is the ability to use short integration times in order to decrease the contribution of thermal noise. This requires a strong coupling of the antenna to the measuring apparatus, i.e., a high value of the so-called electromechanical coupling parameter ${\mathrm{\ensuremath{\beta}}}_{0}$. A well-known solution is in the use of a resonant transducer, which entails the observation of a system of two harmonic oscillators. We extend to this system the back-action evasion technique by showing that one can perform a quantum nondemolition measurement on a double harmonic oscillator with essentially the same configuration as for the single one.

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Quantum nondemolition measurements: Comment on recent developments

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The limitations of the detectability of extremely weak signals (gravitational radiation for instance) imposed by Heisenberg's uncertainty principle on the sequential determination of those signals have been explored recently. A variety of schemes have been proposed to circumvent these limitations. Although all of the earlier attempts have been proven fruitless a recent proposal seems to be quite promising. The scheme, consisting of two harmonic oscillators interacting with each other in a peculiar way, allows for an exact analytical solution which is derived here. If it can be assumed that the expectation value of one of the canonical variables of the total system suffices to monitor the weak signal it can be shown that, in the absence of thermal noise, arbitrarily weak signals can in principle be measured without interference from the uncertainty principle.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1978•1 citation

Detection of classical forces

Vladimir B. Braginsky, Farid Ya Khalili, Kip S. Thorne

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).

Cambridge University Press eBooks•Sep 24, 1992•0 citations

Quantum Measurement

Giacomo Mauro D’Ariano

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.

Measurement Science and Technology•Jul 1, 1997•0 citations
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{X}_{1}X1​ (or the imaginary part X2{X}_{2}X2​) of the oscillator's complex amplitude. In principle X1{X}_{1}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{X}_{1}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{X}_{1}X1​, in which one measures position (or momentum) at half-cycle intervals. Alternatively, one can make "continuous single-transducer" measurements of X1{X}_{1}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{X}_{1}X1​ and reject information about X2{X}_{2}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.
Several high-precision physics experiments are approaching a level of sensitivity at which the intrinsic quantum nature of the experimental apparatus is the dominant source of fluctuations limiting the sensitivity of the measurements. This quantum limit is embodied by the Heisenberg uncertainty principle, which prohibits arbitrarily precise simultaneous measurements of two conjugate observables of a system but allows one-time measurements of a single observable with any precision. The dynamical evolution of a system immediately following a measurement limits the class of observables that may be measured repeatedly with arbitrary precision, with the influence of the measurement apparatus on the system being confined strictly to the conjugate observables. Observables having this feature, and the corresponding measurements performed on them have been named quantum nondemolition or back-action evasion observables. In a previous review (Caves et al., 1980, Rev. Mod. Phys. 52, 341) a quantum-mechanical analysis of quantum nondemolition measurements of a harmonic oscillator was presented. The present review summarizes the experimental progress on quantum nondemolition measurements and the classical models developed to describe and guide the development of practical implementations of quantum nondemolition measurements. The relationship between the classical and quantum theoretical models is also reviewed. The concept of quantum nondemolition and back-action evasion measurements originated in the context of measurements on a macroscopic mechanical harmonic oscillator, though these techniques may be useful in other experimental contexts as well, as is discussed in the last part of this review. [S0034-6861(96)00103-1]
The complex amplitude ${X}_{1}+i{X}_{2}\ensuremath{\equiv}(x+\frac{\mathrm{ip}}{m\ensuremath{\omega}}){e}^{i\ensuremath{\omega}t}$ of a harmonic oscillator is constant in the absence of driving forces. Although the uncertainty principle forbids precise measurements of X1{X}_{1}X1​ and X2{X}_{2}X2​ simultaneously ($\ensuremath{\Delta}{X}_{1}\ensuremath{\Delta}{X}_{2}>~\frac{\ensuremath{\hbar}}{2m\ensuremath{\omega}}$), X1{X}_{1}X1​ alone can be measured precisely and continuously ("quantum nondemolition measurement"). Examples are given of measuring systems that do this job. Such systems might play a crucial role in gravitational-wave detection and elsewhere.
In the measurements on an oscillator to detect short bursts of gravitational radiation, one important demand is the ability to use short integration times in order to decrease the contribution of thermal noise. This requires a strong coupling of the antenna to the measuring apparatus, i.e., a high value of the so-called electromechanical coupling parameter ${\mathrm{\ensuremath{\beta}}}_{0}$. A well-known solution is in the use of a resonant transducer, which entails the observation of a system of two harmonic oscillators. We extend to this system the back-action evasion technique by showing that one can perform a quantum nondemolition measurement on a double harmonic oscillator with essentially the same configuration as for the single one.
The limitations of the detectability of extremely weak signals (gravitational radiation for instance) imposed by Heisenberg's uncertainty principle on the sequential determination of those signals have been explored recently. A variety of schemes have been proposed to circumvent these limitations. Although all of the earlier attempts have been proven fruitless a recent proposal seems to be quite promising. The scheme, consisting of two harmonic oscillators interacting with each other in a peculiar way, allows for an exact analytical solution which is derived here. If it can be assumed that the expectation value of one of the canonical variables of the total system suffices to monitor the weak signal it can be shown that, in the absence of thermal noise, arbitrarily weak signals can in principle be measured without interference from the uncertainty principle.
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).
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.