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Technical advantages for weak value amplification: When less is more

Andrew N. Jordan, Julián Martínez-Rincón, John C. Howell

arXiv:1309.5011v1physics.opticsquant-ph

TL;DR

The paper examines when weak value amplification offers technical advantages despite using only a fraction of photons. Across technical-noise models, Fisher-information analysis, practical-estimator considerations, imaginary weak values, and photon recycling identify settings where weak-value methods match or exceed standard measurements.

  • Problem

    The paper asks when weak value amplification can provide technical advantages over standard measurements under optical technical noise, despite using only a small fraction of photons.

  • Method

    The authors analyze several optical technical-noise models using Fisher information, estimator practicality, imaginary weak values for beam deflection, and photon recycling.

  • Results

    Weak-value measurements can concentrate all Fisher information in a small detected fraction, match standard-measurement Fisher information, outperform conventional methods for some combined noise types, and gain further information through recycling.

  • Takeaways & Limitations

    Weak-value techniques can be technically advantageous when noise conditions and measurement implementation favor postselection, while signal-to-noise estimation remains more practical than an optimal correlated-noise estimator in typical experiments.

  • Takeaways & Limitations

    The technical advantage depends on the noise model: weak-value techniques can be disadvantaged or unaffected in some cases, and dephasing reduces weak-value amplification.

Abstract

from arXiv · show

The technical merits of weak value amplification techniques are analyzed. We consider models of several different types of technical noise in an optical context and show that weak value amplification techniques (which only use a small fraction of the photons) compare favorably with standard techniques (which uses all of them). Using the Fisher information metric, we demonstrate that weak value techniques can put all of the Fisher information about the detected parameter into a small portion of the events and show how this fact alone gives technical advantages. We go on to consider a time correlated noise model, and find that a Fisher information analysis indicates that while the standard method can have much larger information about the detected parameter than the postselected technique. However, the estimator needed to gather the information is technically difficult to implement, showing that the inefficient (but practical) signal-to-noise estimation of the parameter is usually superior. We also describe other technical advantages unique to imaginary weak value amplification techniques, focusing on beam deflection measurements. In this case, we discuss combined noise types (such as detector transverse jitter, angular beam jitter before the interferometer and turbulence) for which the interferometric weak value technique gives higher Fisher information over conventional methods. We go on to calculate the Fisher information of the recently proposed photon recycling scheme for beam deflection measurements, and show it further boosts the Fisher information by the inverse postselection probability relative to the standard measurement case.

I. INTRODUCTION

The paper examines when weak value amplification can provide technical advantages despite using only a fraction of the photons. It frames this question through Fisher information, estimators, and several technical-noise models.

  • Weak value amplification does not overcome the standard quantum limit but can make precision measurements technically easier with common equipment.
  • The paper addresses controversy over whether weak value techniques provide technical advantages against noise compared with standard measurements.
  • The paper analyzes uncorrelated displacement noise, time-correlated noise, turbulence, and combined noise in optical weak-value measurements.
  • Fisher information quantifies available information about an unknown parameter and bounds the minimum variance of any unbiased estimator through the Cramér–Rao bound.
  • For N independent Gaussian measurements with variance σ^2, the Cramér–Rao variance bound is σ^2/N and the minimum resolvable signal scales as σ/√N.

III. REAL WEAK VALUES AND POSTSELECTION

Real weak values amplify the postselected meter shift while leaving its width unchanged. The resulting Fisher information can remain comparable to standard measurement information despite retaining only postselected events.

  • Postselection shifts the meter by the weak value Aw with probability γ = |⟨f|i⟩|^2, while the meter width σ remains unchanged.
  • The non-postselected meter shift is aid, which can be much smaller than the weak-value shift Awd when the initial state is an eigenstate of A.
  • The postselected distribution contains N′ = γN measurement events, reflecting the fraction of photons retained after postselection.
  • The postselection probability γ cancels from the Fisher-information expression, allowing the weak-value events to contain all the Fisher information in the ideal case.
  • The weak-value efficient estimator rescales the postselected sample average by 1/AwN′.

IV. TYPE 1 TECHNICAL NOISE: DISPLACEMENT NOISE

For type 1 displacement noise, technical noise broadens the measured distribution and reduces Fisher information. Real weak values do not increase Fisher information, whereas imaginary weak values are identified as a possible exception.

  • Type 1 technical noise models independent Gaussian displacement jitter added to each measured signal, with covariance ⟨ξiξj⟩ = J^2δij.
  • Convolving the intrinsic measurement distribution with technical noise broadens the signal width and decreases Fisher information.
  • For postselected measurements, the parameter becomes Awd and the event count becomes N′ = γN, while the Fisher information is reduced by the postselection probability.
  • Real weak-value amplification offers no Fisher-information increase for type 1 technical noise, matching earlier analyses.
  • Reducing the meter width improves Fisher information only until σ is approximately J, after which technical noise dominates the variance.

V. TYPE 2 TECHNICAL NOISE: CORRELATED NOISE MODEL OF FEIZPOUR, XINGXING, AND STEINBERG

The correlated-noise analysis compares Fisher information with estimator practicality. Postselection can lose information contained in correlations with rejected photons, while the simple SNR estimator is often more feasible experimentally.

  • Type 2 technical noise treats detected photon measurements as variables with generally correlated noise covariance Cij.
  • For white noise, averaging gives variance C/N, whereas fully correlated noise gives variance C, so averaging helps only in the white-noise case.
  • Strong correlations can make Fisher information scale at most as N^2, exceeding the N scaling associated with independent measurements.
  • Postselection boosts the mean by Aw and reduces the sample size to γN; for highly correlated noise, Fisher information can scale as N^2γ and decrease by γ.
  • Postselection loses correlations between retained and rejected photons unless the rejected photons undergo further processing.
  • The optimal correlated-noise estimator requires covariance-aware, data-dependent weighting of every point, making it difficult to implement experimentally.
  • The SNR estimator is suboptimal statistically but usually more practical because it uses equal weighting and avoids complete noise-correlation characterization.

VI. TYPE 3 TECHNICAL NOISE - AIR TURBULENCE

Turbulence broadens and shifts optical beams, making it important in long-path measurements of extremely small deflections. Weak-value schemes have a distinct advantage because their detectors can use shorter optical paths.

  • Type 3 technical noise — air turbulence: Turbulence causes beam broadening through beam breathing and beam wander during propagation through a random medium.Beam breathing acts on short time scales, while beam wander becomes important on longer time scales.
  • Type 3 technical noise — air turbulence: Long optical paths between the deflection point and detector make turbulence especially problematic for tiny beam deflections.Beam wander must be considered over experiment durations ranging from seconds to hours.
  • Type 3 technical noise — air turbulence: Weak-value schemes benefit from short optical paths, giving them an advantage over standard beam-deflection measurements in turbulent air.The paper indicates that this geometric advantage is developed in the following section.

VII. IMAGINARY WEAK VALUES AND TECHNICAL ADVANTAGES

For beam-deflection measurements, imaginary weak-value interferometry reorganizes the measurement geometry and concentrates detection in postselected events. It can retain standard Fisher information without noise and remain robust against detector jitter, while also mitigating turbulence through short paths.

  • VII. Imaginary weak values and technical advantages: Imaginary weak values are emphasized as a technically advantageous approach for beam-deflection experiments, which commonly report technical benefits using imaginary weak values.The paper distinguishes this from related effects involving position or momentum noise.
  • VII. Imaginary weak values and technical advantages: The standard beam-deflection method converts mirror tilt k into detector displacement fk/k0 after propagation and focusing.The focused beam width is σf = f/2k0σ, linking the measurement geometry to the input beam width and focal length.
  • VII. Imaginary weak values and technical advantages: Imaginary weak-value interferometry produces a postselected Gaussian beam with mean 4kσ2/φ and width σ2 for a fraction γN of photons.The postselection probability is γ = φ2/4.
  • VII. Imaginary weak values and technical advantages: The weak-value and standard methods yield the same classical Fisher information in the small-angle regime sin2(φ/2) ≈ φ2/4.In this regime, all Fisher information is in photons leaving the interferometer’s dark port.
  • VII. Imaginary weak values and technical advantages: Detector jitter leaves imaginary weak-value Fisher information unchanged because the beam waist σ can be chosen much larger than the jitter scale J.The focused-beam approach instead requires long focal lengths to suppress detector-jitter effects.
  • VII. Imaginary weak values and technical advantages: Measurement geometry lets weak-value detectors sit immediately after the last beam splitter, mitigating turbulence and suppressing transverse detector jitter.The standard method, when configured to eliminate type 1 noise, remains exposed to type 3 turbulence noise.

VIII. TYPE 4 TECHNICAL NOISE - ANGULAR BEAM JITTER

Angular beam jitter is modeled as a random momentum kick before the interferometer or before the signal mirror in the standard method. The kick can arise from air turbulence or mirror jitter.

  • VIII. Type 4 technical noise — angular beam jitter: Angular beam jitter is represented by a random momentum kick q applied before measurement.The model places q before the interferometer in the weak-value setup or before the signal mirror in the standard setup.
  • VIII. Type 4 technical noise — angular beam jitter: The modeled angular jitter may originate from air turbulence or mirror jitter.

A. Standard method results for angular jitter

The standard beam-deflection method models angular jitter as a random momentum kick that shifts and broadens the detected beam, reducing Fisher information about k.

  • Standard optical model: The standard setup applies an initial random momentum kick q, followed by propagation, a signal kick k, a lens, and detection at a position-sensitive detector.The initial transverse state is Gaussian with variance σ2, and the optical elements are represented by unitary operators.
  • Standard optical model: The lens and propagation geometry can focus the beam when l2 = f, yielding a scaled Fourier transform of the intermediate state.The detector coordinate corresponds to a particular momentum value in that Fourier transform.
  • Angular-jitter statistics: Angular jitter q adds directly to the signal k, producing a Gaussian detector distribution with mean −kf/k0 and increased variance.The jitter has zero mean and variance Q2, and averaging over it broadens the position distribution.
  • Angular-jitter statistics: The resulting Gaussian distribution provides the Fisher information about k for N independent measurements.This Fisher information is computed from the jitter-averaged detector distribution.

B. Weak value treatment of angular jitter

The weak-value interferometer postselects a dark-port beam whose displacement contains an amplified signal, while diffraction contributes a smaller correction under the weak-value ordering.

  • Weak-value interferometer: The weak-value interferometer uses clockwise and counterclockwise photon states, a relative phase shift, and postselection at the opposite interferometer port.The transverse detector state is obtained from the transition amplitude between interferometer ports.
  • Postselection: The dark-port probability is γ = (φ/2)2 + k2σ2 + O(k2q2) + O(φkq) + . . . .The analysis drops terms compared with (φ/2)2, so φ controls postselection.
  • Weak-value expansion: The first-order k term is the usual weak-value contribution, amplified by 1/φ.Higher-order terms in k, q, and φ are suppressed in the expansion.
  • Angular-jitter response: The q(l1 + l2)/k0 contribution represents free propagation of the imparted deflection angle q/k0 over the total propagation length.Its dependence on q differs from the corresponding standard-measurement expression.
  • Angular-jitter response: For angular jitter, the weak-value method gains the diffraction suppression factor (l1 + l2)/(2k0σ2) ≪1 and outperforms the standard method.Angular jitter directly adds to the standard detected deflection but is only a small diffraction correction in the weak-value scheme.
  • Information concentration: In the large-WVA limit, all measurement information can be placed in the detected photons and optimal SNR estimation can be performed.The associated geometric terms can arise even when weak-value amplification is small.

IX. FISHER INFORMATION FOR RECYCLED PHOTONS

Photon recycling re-injects rejected bright-port photons so they repeatedly sample the parameter while preserving large weak-value amplification.

  • Recycling scheme: Photon recycling treats photon number as a resource and re-injects the rejected photons into the interferometer for additional measurements of k.If N′ = γN photons are rejected, the remaining N1 = N − N′ photons are re-injected.
  • Information accumulation: Because successive measurements are independent, their Fisher information contributions add.The total information is obtained by summing the information from repeated passes.
  • Information gain: The Fisher information is boosted by a factor of 1/γ relative to the standard or single-pass weak-value method.This comparison uses photon number N as the resource.
  • Practical considerations: Profile reshaping removes some information about k but leaves a simple split-detection SNR estimator, while collecting all photons permits further use of interphoton correlations.The practical benefit depends on laboratory constraints such as quiet times between laser pulses or detector power thresholds.

X. CONCLUSIONS

The conclusions identify technical advantages of weak-value measurements, while emphasizing that their benefit depends on the noise model and estimator practicality.

  • Conclusions: Weak-value measurements using a small fraction of the available light can have Fisher information as large as standard measurements using all the light.The remaining light can be sent to another experiment or recycled for still higher Fisher information.
  • Conclusions: Weak-value advantages depend on the technical-noise case: the technique can help in some cases, hurt in others, or provide no advantage.The authors state that different noise types should be investigated case by case.
  • Estimator practicality: An estimator that fails to reach the Cramér–Rao bound should not be rejected unless the optimal estimator is also practically implementable.For time-correlated noise, the practical postselection-based SNR estimator can be preferable to the technically difficult optimal estimator.

Appendix: Weak value for a two-level system

The appendix formulates weak-value amplification for a two-level system using preselection and postselection states on the Bloch sphere. It relates the angular choices of postselection to real and imaginary weak values and their signal-to-noise expressions.

  • Interaction and measurement: The interaction is modeled as U = exp(−i d ˆp ˆA), with d the unknown small parameter and x or p measured after postselection.Position-meter measurements yield an SNR proportional to |⟨f|i⟩Re(Aw)|, while momentum measurements use the corresponding imaginary component.
  • Two-level system: The two-level operator has eigenstates |+⟩ and |−⟩ with eigenvalues +1 and −1, forming an orthonormal basis.
  • Bloch-sphere states: The initial and postselection states are represented by points on the Bloch sphere, with θ and φ specifying deviations from the state orthogonal to |i⟩.For Θ = π/2, the preselection state lies on the equatorial plane, while the postselection state is displaced in two angular directions.
  • Postselection probability: The postselection probability is γ ≈ |⟨f|i⟩|2 = cos2 φ sin2 θ + sin2(Θ + θ) sin2 φ.
  • Weak-value components: The angles θ and φ generate the real and imaginary parts of the weak value, respectively.Setting θ = 0 produces a pure imaginary weak value with γ = sin2 Θ sin2 φ.
  • Weak-value bounds: Both |⟨f|i⟩Re(Aw)| and |⟨f|i⟩Im(Aw)| are upper bounded by unity as functions of θ and φ for Θ = π/2.
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