Physical Review Letters

Defense of the Standard Quantum Limit for Free-Mass Position

Carlton M. Caves

Published Jun 10, 1985• 82 citations

Abstract

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.

DOI: 10.1103/physrevlett.54.2465

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Physical Review Letters

Defense of the Standard Quantum Limit for Free-Mass Position

Carlton M. Caves

Published Jun 10, 1985• 82 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.54.2465

Related papers

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.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:1005.5328Physical Review AAug 30, 20100 citations

Rigorous quantum limits on monitoring free masses and harmonic oscillators

Shasanka Mohan Roy

There are heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.There are heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.

arXiv:1711.10261Physical Review AMar 13, 20181 citation

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.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.

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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.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.

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Wei-Tou Ni

Physical Review AApr 1, 198622 citations

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Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone

arXiv:quant-ph/0412078ScienceNov 18, 20043,046 citations

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.

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.

There are heuristic arguments proposing that the accuracy of monitoring position of a free mass

m

is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.