Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Quantum Limit for Successive Position Measurements

M. Hossein Partovi, R. Blankenbecler

Published Dec 8, 1986• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.57.2891

Related papers

Defense of the Standard Quantum Limit for Free-Mass Position

Carlton M. Caves

Measurements of the position $x$ of a free mass $m$ are thought to be governed by the standard quantum limit (SQL): In two successive measurements of $x$ spaced a time $\ensuremath{\tau}$ apart, the result of the second measurement cannot be predicted with uncertainty smaller than ${(\frac{\ensuremath{\hbar}\ensuremath{\tau}}{m})}^{\frac{1}{2}}$. Yuen has suggested that there might be ways to beat the SQL. Here I give an improved formulation of the SQL, and I argue for, but do not prove, its validity.

Physical Review Letters•Jun 10, 1985•82 citations

Proof of the standard quantum limit for monitoring free-mass position

Seiji Kosugi

The measurement result of the moved distance for a free mass $m$ during the time $\ensuremath{\tau}$ between two position measurements cannot be predicted with uncertainty smaller than $\sqrt{\ensuremath{\hbar}\ensuremath{\tau}/m}$. This is formulated as a standard quantum limit and it has been proven to always hold for the following position measurement: a probe is set in a prescribed position before the measurement. Just after the interaction of the mass with the probe, the probe position is measured, and using this value, the measurement results of the premeasurement and postmeasurement positions are estimated.

arXiv:1005.5328•Physical Review A•Aug 30, 2010•0 citations

Measurement breaking the standard quantum limit for free-mass position

Masanao Ozawa

An explicit interaction-Hamiltonian realization of a measurement of the free-mass position with the following properties is given: (1) The probability distribution of the readouts is exactly the same as the free-mass position distribution just before the measurement. (2) The measurement leaves the free mass in a contractive state just after the measurement. It is shown that this measurement breaks the standard quantum limit for the free-mass position in the sense sharpened by the recent controversy.

Physical Review Letters•Feb 1, 1988•102 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

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 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
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.
Measurements of the position xxx of a free mass mmm are thought to be governed by the standard quantum limit (SQL): In two successive measurements of xxx spaced a time $\ensuremath{\tau}$ apart, the result of the second measurement cannot be predicted with uncertainty smaller than ${(\frac{\ensuremath{\hbar}\ensuremath{\tau}}{m})}^{\frac{1}{2}}$. Yuen has suggested that there might be ways to beat the SQL. Here I give an improved formulation of the SQL, and I argue for, but do not prove, its validity.
The measurement result of the moved distance for a free mass mmm during the time $\ensuremath{\tau}$ between two position measurements cannot be predicted with uncertainty smaller than $\sqrt{\ensuremath{\hbar}\ensuremath{\tau}/m}$. This is formulated as a standard quantum limit and it has been proven to always hold for the following position measurement: a probe is set in a prescribed position before the measurement. Just after the interaction of the mass with the probe, the probe position is measured, and using this value, the measurement results of the premeasurement and postmeasurement positions are estimated.
An explicit interaction-Hamiltonian realization of a measurement of the free-mass position with the following properties is given: (1) The probability distribution of the readouts is exactly the same as the free-mass position distribution just before the measurement. (2) The measurement leaves the free mass in a contractive state just after the measurement. It is shown that this measurement breaks the standard quantum limit for the free-mass position in the sense sharpened by the recent controversy.
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.
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).
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.