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 A

Quantum theory of interferometric length measurement with inefficient photodetectors

Jessica Gifford, Craig M. Savage

Published Feb 1, 1991• 1 citation
Publisher

Abstract

We find an upper bound for the sensitivity of Fabry-P\'erot interferometric length measurements using inefficient photodetectors. This bound applies to all possible quantum-mechanical states of the light entering the interferometer. It applies to standard photodetection, differenced photodetection, and homodyne detection. In the limit of large photon number we identify the optimal detection scheme and input states. They are homodyne detection with a coherent state in one input port, and a nonvacuum quadrature squeezed state in the other. No other quantum states of the input light give better sensitivity, provided that the interferometer output is measured by either photodetection or homodyning, and that the photodetectors are inefficient.

DOI
10.1103/physreva.43.1484

Related papers

Squeezed states in phase-sensing interferometers

Roy S. Bondurant, Jeffrey H. Shapiro

The performance of phase-sensing interferometers employing squeezed states and homodyne detection is analyzed and compared to the performance of systems employing direct detection. Standard differenced direct-detection Michelson and Mach-Zehnder interferometers are shown to be suboptimal in the sense that an observation/measurement-noise coupling occurs, which can degrade performance. Homodyne-detection interferometers in which the phase shift in one arm is the conjugate of that in the other arm do not suffer from the preceding drawback. Overall, however, the performance of differenced direct-detection and homodyne-detection interferometers is similar in single-frequency operation. In particular, both detection schemes reach the standard quantum limit on position-measurement sensitivity in single-frequency interferometric gravity-wave detectors at roughly the same average photon number. This limit arises from back action in the form of radiation pressure fluctuations entering through the energy-phase uncertainty principle. Multifrequency devices can circumvent this uncertainty principle, as illustrated by the conceptual design given for a two-frequency interferometer which can greatly surpass the standard quantum limit on position sensing. This configuration assumes that ideal photodetectors respond to photon flux rather than energy flux.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1984•299 citations

Introduction

Miguel Orszag

Quantum noise reduction in optical systems and the determination of quasi-distribution functions, in particular Wigner functions, to describe quantum optical states, have been central issues in quantum optics for a number of years. The generation of sources of squeezed light has opened very exciting possibilities of ultrahigh precision measurements beyond the standard quantum limit, by either interferometric or spectroscopic means. In the present issue, the paper by Mundarain and Orszag discusses the interferometric detection of ultrasmall signals, in particular gravitational waves, when one injects a squeezed state in the unused port of the interferometer. The authors find that the sensitivity of the system is very strongly affected by the quantum efficiency of the non-ideal photodetectors, when the squeezed signal is injected, as opposed to the ordinary vacuum case where the effects are rather small. This imposes strong conditions on the quality of those photodetectors. The standard quantum limit and the corresponding minimum detectable gravitational amplitude is also discussed in general, for short time measurements.

Optics Express•Aug 17, 1998•0 citations

Homodyning and heterodyning the quantum phase

Giacomo Mauro D’Ariano, Chiara Macchiavello, Matteo G. A. Paris

The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.

•May 1, 1994•0 citations

Gaussian-state interferometry with passive and active elements

Carlo Sparaciari, Stefano Olivares, Matteo G. A. Paris

We address the precision of optical interferometers fed by quantum and semiclassical Gaussian states involving passive and/or active elements, such as beam splitters, photodetectors, and optical parametric amplifiers. We first address the ultimate bounds to precision by discussing the behavior of the quantum Fisher information. We then consider photodetection at the output and calculate the sensitivity of the interferometers taking into account the nonunit quantum efficiency of the detectors. Our results show that in the ideal case of photon number detectors with unit quantum efficiency the best configuration is the symmetric one, namely, a passive (active) interferometer with a passive (active) detection stage: in this case one may achieve Heisenberg scaling of sensitivity by suitably optimizing over Gaussian states at the input. On the other hand, in the realistic case of detectors with nonunit quantum efficiency, the performances of the passive scheme are unavoidably degraded, whereas detectors involving optical parametric amplifiers allow us to fully compensate for the presence of loss in the detection stage, thus restoring the Heisenberg scaling.

arXiv:1510.07716•Physical Review A•Feb 5, 2016•97 citations

Quantum Theory of High-resolution Length Measurement with a Fabry-Perot Interferometer

M. Ley, R. Loudon

The quantum limits on measurements of small changes in the length of a Fabry-Perot cavity are calculated. The cavity is modelled by a pair of dissimilar mirrors oriented perpendicular to a one-dimensional axis of infinite extent. The continuous spectrum of spatial modes of the system is derived, and the electromagnetic field is quantized in terms of a continuous set of mode creation and destruction operators. Coherent state and squeezed vacuum-state excitations of the field are characterized by energy flow, or intensity, variables. The determination of small changes in the cavity length by observations of fringe intensity is considered for schemes in which the cavity is simultaneously excited by coherent and squeezed vacuum-state inputs. The contributions to the limiting resolution from photocount and radiation-pressure length uncertainties are evaluated. These properties of the Fabry-Perot cavity are compared with the corresponding results for the Michelson interferometer.

Journal of Modern Optics•Feb 1, 1987•99 citations

Adaptive measurements and optimal states for quantum interferometry

Dominic William Berry, Howard M. Wiseman

Summary form only given. Interferometry is the basis of many high-precision measurements. The ultimate limit to the precision is due to quantum effects. This limit is most easily explored for a Mach-Zehnder interferometer The outputs of this device can be measured to yield an estimate /spl phi/ of the phase difference /spl phi/ between the two arms of the interferometer. It is well known that this can achieve the standard quantum limit for phase variance when an N-photon number state enters one input port. Several authors have proposed ways of reducing the phase variance to the Heisenberg limit. In this paper we show that there is a physical measurement scheme using photodetectors and feedback that is almost as good as the optimal measurement. The fundamental idea is that, in addition to the unknown phase we wish to measure, /spl phi/, in one arm of the interferometer, we introduce a known phase shift, /spl Phi/, into the other arm of the interferometer. After each photodetection we adjust this introduced phase shift in order to minimize the expected uncertainty of our best phase estimate /spl phi/ after the next photodetection. This feedback process is uniquely defined using Bayesian statistics, and can be solved exactly for moderate photon numbers.

Quantum Electronics and Laser Science Conference•Nov 13, 2002•2 citations
We find an upper bound for the sensitivity of Fabry-P'erot interferometric length measurements using inefficient photodetectors. This bound applies to all possible quantum-mechanical states of the light entering the interferometer. It applies to standard photodetection, differenced photodetection, and homodyne detection. In the limit of large photon number we identify the optimal detection scheme and input states. They are homodyne detection with a coherent state in one input port, and a nonvacuum quadrature squeezed state in the other. No other quantum states of the input light give better sensitivity, provided that the interferometer output is measured by either photodetection or homodyning, and that the photodetectors are inefficient.
The performance of phase-sensing interferometers employing squeezed states and homodyne detection is analyzed and compared to the performance of systems employing direct detection. Standard differenced direct-detection Michelson and Mach-Zehnder interferometers are shown to be suboptimal in the sense that an observation/measurement-noise coupling occurs, which can degrade performance. Homodyne-detection interferometers in which the phase shift in one arm is the conjugate of that in the other arm do not suffer from the preceding drawback. Overall, however, the performance of differenced direct-detection and homodyne-detection interferometers is similar in single-frequency operation. In particular, both detection schemes reach the standard quantum limit on position-measurement sensitivity in single-frequency interferometric gravity-wave detectors at roughly the same average photon number. This limit arises from back action in the form of radiation pressure fluctuations entering through the energy-phase uncertainty principle. Multifrequency devices can circumvent this uncertainty principle, as illustrated by the conceptual design given for a two-frequency interferometer which can greatly surpass the standard quantum limit on position sensing. This configuration assumes that ideal photodetectors respond to photon flux rather than energy flux.
Quantum noise reduction in optical systems and the determination of quasi-distribution functions, in particular Wigner functions, to describe quantum optical states, have been central issues in quantum optics for a number of years. The generation of sources of squeezed light has opened very exciting possibilities of ultrahigh precision measurements beyond the standard quantum limit, by either interferometric or spectroscopic means. In the present issue, the paper by Mundarain and Orszag discusses the interferometric detection of ultrasmall signals, in particular gravitational waves, when one injects a squeezed state in the unused port of the interferometer. The authors find that the sensitivity of the system is very strongly affected by the quantum efficiency of the non-ideal photodetectors, when the squeezed signal is injected, as opposed to the ordinary vacuum case where the effects are rather small. This imposes strong conditions on the quality of those photodetectors. The standard quantum limit and the corresponding minimum detectable gravitational amplitude is also discussed in general, for short time measurements.
The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.
We address the precision of optical interferometers fed by quantum and semiclassical Gaussian states involving passive and/or active elements, such as beam splitters, photodetectors, and optical parametric amplifiers. We first address the ultimate bounds to precision by discussing the behavior of the quantum Fisher information. We then consider photodetection at the output and calculate the sensitivity of the interferometers taking into account the nonunit quantum efficiency of the detectors. Our results show that in the ideal case of photon number detectors with unit quantum efficiency the best configuration is the symmetric one, namely, a passive (active) interferometer with a passive (active) detection stage: in this case one may achieve Heisenberg scaling of sensitivity by suitably optimizing over Gaussian states at the input. On the other hand, in the realistic case of detectors with nonunit quantum efficiency, the performances of the passive scheme are unavoidably degraded, whereas detectors involving optical parametric amplifiers allow us to fully compensate for the presence of loss in the detection stage, thus restoring the Heisenberg scaling.
The quantum limits on measurements of small changes in the length of a Fabry-Perot cavity are calculated. The cavity is modelled by a pair of dissimilar mirrors oriented perpendicular to a one-dimensional axis of infinite extent. The continuous spectrum of spatial modes of the system is derived, and the electromagnetic field is quantized in terms of a continuous set of mode creation and destruction operators. Coherent state and squeezed vacuum-state excitations of the field are characterized by energy flow, or intensity, variables. The determination of small changes in the cavity length by observations of fringe intensity is considered for schemes in which the cavity is simultaneously excited by coherent and squeezed vacuum-state inputs. The contributions to the limiting resolution from photocount and radiation-pressure length uncertainties are evaluated. These properties of the Fabry-Perot cavity are compared with the corresponding results for the Michelson interferometer.
Summary form only given. Interferometry is the basis of many high-precision measurements. The ultimate limit to the precision is due to quantum effects. This limit is most easily explored for a Mach-Zehnder interferometer The outputs of this device can be measured to yield an estimate /spl phi/ of the phase difference /spl phi/ between the two arms of the interferometer. It is well known that this can achieve the standard quantum limit for phase variance when an N-photon number state enters one input port. Several authors have proposed ways of reducing the phase variance to the Heisenberg limit. In this paper we show that there is a physical measurement scheme using photodetectors and feedback that is almost as good as the optimal measurement. The fundamental idea is that, in addition to the unknown phase we wish to measure, /spl phi/, in one arm of the interferometer, we introduce a known phase shift, /spl Phi/, into the other arm of the interferometer. After each photodetection we adjust this introduced phase shift in order to minimize the expected uncertainty of our best phase estimate /spl phi/ after the next photodetection. This feedback process is uniquely defined using Bayesian statistics, and can be solved exactly for moderate photon numbers.