Increasing the dynamic range of weak measurement with two pointers

Zhaoxue Li, Linguo Xie, Qian Ti, Pi Duan, Zhiyou Zhang, and Changliang Ren

Phys. Rev. A 102, 023701 – Published 3 August 2020

Abstract

We propose a scheme called the weak measurement with two pointers to measure time delays in a large dynamic range with high precision. One pointer with the light frequency and the other pointer with the postselected probability play a complementary role in the dynamic range of the time-delay estimation. By inserting the main systematic errors, we study the antierror capability and the amplification behavior of two types of pointers, where the pointer with the postselected probability shows stronger robustness against systematic errors and larger working range in general. The proof-of-principle experiments together with the Fisher information metric analysis show that high precision can be effectively retained in a large dynamic range of the time-delay estimation. Further implementation strategy of this scheme is also provided, which may promote the precise estimation of other parameters in a large dynamic range.

Received 27 March 2020
Revised 7 July 2020
Accepted 9 July 2020

DOI:https://doi.org/10.1103/PhysRevA.102.023701

Physics Subject Headings (PhySH)

Optical interferometry Quantum description of light-matter interaction Quantum measurements

Atomic, Molecular & Optical

Authors & Affiliations

Zhaoxue Li¹, Linguo Xie², Qian Ti¹, Pi Duan¹, Zhiyou Zhang ^1,*, and Changliang Ren^3,†

¹College of Physics, Sichuan University, Chengdu 610064, China
²Key Laboratory of Photonic and Optical Detection in Civil Aviation and Atmospheric Lidar Institute, Civil Aviation Flight University of China, Guanghan 618300, China
³Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China

^*zhangzhiyou@scu.edu.cn
^†renchangliang@cigit.ac.cn

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Issue

Vol. 102, Iss. 2 — August 2020

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Images

Figure 1
Schematic diagram of the weak measurement scheme with two pointers. The system (with the measured quantity $A$ ) is weakly coupled with the meter (with the input variable $F = ω$ ) by the unitary transformation $U$ in Eq. (2). Then a projective measurement of the system variable $B$ is performed, as in Eq. (3). Finally, the pointer measurements are performed, which simultaneously involve two pointers, $R_{1} = ω$ and $R_{2} = p_{f}$ .
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Figure 2
Simulated results of the WM regime with the pointer of $R_{1} = ω$ . Left: changes of the output spectrum $F (ω)$ with the frequency $ω$ and the systematic error $[ν$ (a) or $ε$ (c)]. Right: changes of the average spectrum shift $δ 〈 ω 〉$ with the measured time delay $τ$ and the error $[ν$ (b) or $ε$ (d)]. In calculations, $ε = 1.4 \times 10^{- 3}$ rad in (a) and (b) and $ν = 335.7$ as in (c) and (d).
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Figure 3
Simulated results of the WM regime with the pointer of $R_{2} = p_{f}$ . Left: changes of the output spectrum $F (ω)$ with the frequency $ω$ and the systematic error $[ν$ (a) or $ε$ (c)]. Right: changes of ICR $η$ with the measured time delay $τ$ and the error $[ν$ (b) or $ε$ (d)]. In calculations, $ε = 1.4 \times 10^{- 3}$ rad in (a) and (b) and $ν = 335.7$ as in (c) and (d).
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Figure 4
Experimental setup. WL: white laser; GF: Gaussian filter with the central wavelength of 633 nm and the FWHM (full width at half maximum) of 10 nm. P1 and P2: polarizers with optical axes set to $45^{\circ}$ and $135^{\circ}$ , respectively; $λ / 2 - λ / 2$ : half-half wave-plate combination with optical axes perpendicular to each other; $λ / 4$ : quarter wave plate with the optical axis set to $135^{\circ}$ ; SA: spectrum analyzer; DP: data processing.
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Figure 5
Measured average spectrum shift $δ 〈 ω 〉$ (a) and ICR $η$ (b) as a function of the time delay $τ$ . Blue circles and red diamonds are experimental results with $| ϕ | = 0.004$ , 0.02 rad, in which the effective working regions are represented by blue and red solid lines on fitting curves, respectively.
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Figure 6
Precision analysis for WM schemes with two pointers. The first two panels show the theoretical Fisher information evaluation $I (τ)$ as a function of the measured time delay $τ$ for the pointers of $R_{2} = p_{f}$ (top) and $R_{1} = ω$ (middle). The first five sets of the postselected angle are shown in the order of red, black, green, magenta, and orange in the middle panel. The bottom panel shows the experimentally obtained precision $Δ τ$ of combining the two types of pointers with the same postselected angles set as the theoretical analysis, where black stars and colorful circles are used for representing the pointers of $R_{2} = p_{f}$ and $R_{1} = ω$ , respectively.
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