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arXiv:quant-ph/9801039 · Physical Review A

Standard quantum limits for broadband position measurement

Hideo Mabuchi

Published Jul 1, 1998• 23 citations
arXivHTMLPDF

Abstract

I utilize the Caves-Milburn model for continuous position measurements to formulate a broadband version of the standard quantum limit (SQL) for monitoring the position of a free mass and illustrate the use of Kalman filtering to recover the SQL for estimating a weak classical force that acts on a quantum-mechanical test particle under continuous observation. These derivations are intended to clarify the interpretation of SQL's in the context of broadband quantum measurement, with particular attention paid to the question of how it might be possible to verify that a given laboratory measurement does indeed achieve backaction-limited sensitivity. The method used to analyze force detection may be extended to the case of weak classical forces with arbitrary time dependence, including nonstationary and impulsive signals.

DOI
10.1103/physreva.58.123

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Kalman filtering and Standard Quantum Limits for broadband measurement

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I utilize the Caves-Milburn model for continuous position measurements to formulate a broadband version of the Standard Quantum Limit (SQL) for monitoring the position of a free mass, and illustrate the use of Kalman filtering to recover the SQL for estimating a weak classical force that acts on a quantum-mechanical test particle under continuous observation.

•Jan 20, 1998•0 citations

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The standard quantum limit (SQL) of a classical mechanical force detection results from quantum back action perturbing evolution of a mechanical system. In this paper we show that usage of a multidimensional optical transducer may enable a broadband quantum back action evading measurement. We study theoretically a corresponding technique of measurement of a resonant signal force acting on a linear mechanical oscillator coupled to a generic optical system with three optical modes with separation nearly equal to the mechanical frequency. The measurement is performed by optical pumping of the central optical mode and analyzing the light escaping the two other modes. By detecting optimal quadrature components of the optical modes and postprocessing the measurement results we are able to exclude the back action in a broad frequency band and characterize the force with sensitivity better than the SQL. We show that the proposed scheme is similar to the multidimensional system containing quantum-mechanics-free subsystems which can evade the SQL using the idea of the so-called negative mass [M. Tsang and C. M. Caves, Phys. Rev. X 2, 031016 (2012).].

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Broadband dichromatic variational measurement

Sergey P. Vyatchanin, Albert I. Nazmiev, Andrey B. Matsko

The standard quantum limit (SQL) of a classical mechanical force detection results from quantum backaction impinged by the meter on a probe mechanical transducer perturbed by the force of interest. In this paper we introduce a technique of continuous broadband backaction, avoiding measurements for the case of a resonant signal force acting on a linear mechanical oscillator supporting one of the mirrors of an optical Michelson-Sagnac interferometer (MSI). The interferometer with the movable mirror is an optomechanical transducer able to support the polychromatic probe field. The method involves a dichromatic optical probe resonant with the MSI modes and having frequency separation equal to the mechanical frequency. We show that analyzing each of the harmonics of the probe reflected from the mechanical system separately and postprocessing the measurement results allows excluding the backaction in a broad frequency band and measuring the force with sensitivity better than the SQL.

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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

Quantum measurements of position, detection of weak classical forces, and interferometric noise

Wei-Tou Ni

We discuss quantum measurements of position and repeated-measurement schemes for monitoring free-mass position to an arbitrary accuracy. Using these schemes, weak classical forces can be monitored to an arbitrary accuracy. To illustrate various measurement uncertainties and intrinsic uncertainties, we consider the case of a Michelson interferometer. In the low-intensity limit, two measurement uncertainties, photon counting error and radiation pressure error, are independent of the intrinsic uncertainty. The uses of squeezed states and number eigenstates are discussed.

IEEE Transactions on Instrumentation and Measurement•Jun 1, 1987•0 citations

Quantum Limit for Successive Position Measurements

M. Hossein Partovi, R. Blankenbecler

We apply a formalism for the description of multitime measurements to the derivation of the quantum limit on the precision with which a pair of successive position measurements can be performed on a free mass under conditions that would be optimal for detecting very weak forces. The result depends on the position resolution of the measuring device as well as on the time interval between the two measurements, and it spans a range of values which include a presently controversial result known as the standard quantum limit; any controversy is thereby resolved.

Physical Review Letters•Dec 8, 1986•8 citations
I utilize the Caves-Milburn model for continuous position measurements to formulate a broadband version of the standard quantum limit (SQL) for monitoring the position of a free mass and illustrate the use of Kalman filtering to recover the SQL for estimating a weak classical force that acts on a quantum-mechanical test particle under continuous observation. These derivations are intended to clarify the interpretation of SQL's in the context of broadband quantum measurement, with particular attention paid to the question of how it might be possible to verify that a given laboratory measurement does indeed achieve backaction-limited sensitivity. The method used to analyze force detection may be extended to the case of weak classical forces with arbitrary time dependence, including nonstationary and impulsive signals.
The standard quantum limit (SQL) of a classical mechanical force detection results from quantum back action perturbing evolution of a mechanical system. In this paper we show that usage of a multidimensional optical transducer may enable a broadband quantum back action evading measurement. We study theoretically a corresponding technique of measurement of a resonant signal force acting on a linear mechanical oscillator coupled to a generic optical system with three optical modes with separation nearly equal to the mechanical frequency. The measurement is performed by optical pumping of the central optical mode and analyzing the light escaping the two other modes. By detecting optimal quadrature components of the optical modes and postprocessing the measurement results we are able to exclude the back action in a broad frequency band and characterize the force with sensitivity better than the SQL. We show that the proposed scheme is similar to the multidimensional system containing quantum-mechanics-free subsystems which can evade the SQL using the idea of the so-called negative mass [M. Tsang and C. M. Caves, Phys. Rev. X 2, 031016 (2012).].
The standard quantum limit (SQL) of a classical mechanical force detection results from quantum backaction impinged by the meter on a probe mechanical transducer perturbed by the force of interest. In this paper we introduce a technique of continuous broadband backaction, avoiding measurements for the case of a resonant signal force acting on a linear mechanical oscillator supporting one of the mirrors of an optical Michelson-Sagnac interferometer (MSI). The interferometer with the movable mirror is an optomechanical transducer able to support the polychromatic probe field. The method involves a dichromatic optical probe resonant with the MSI modes and having frequency separation equal to the mechanical frequency. We show that analyzing each of the harmonics of the probe reflected from the mechanical system separately and postprocessing the measurement results allows excluding the backaction in a broad frequency band and measuring the force with sensitivity better than the SQL.
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.
We discuss quantum measurements of position and repeated-measurement schemes for monitoring free-mass position to an arbitrary accuracy. Using these schemes, weak classical forces can be monitored to an arbitrary accuracy. To illustrate various measurement uncertainties and intrinsic uncertainties, we consider the case of a Michelson interferometer. In the low-intensity limit, two measurement uncertainties, photon counting error and radiation pressure error, are independent of the intrinsic uncertainty. The uses of squeezed states and number eigenstates are discussed.
We apply a formalism for the description of multitime measurements to the derivation of the quantum limit on the precision with which a pair of successive position measurements can be performed on a free mass under conditions that would be optimal for detecting very weak forces. The result depends on the position resolution of the measuring device as well as on the time interval between the two measurements, and it spans a range of values which include a presently controversial result known as the standard quantum limit; any controversy is thereby resolved.