Fig. 1. Post-selected weak measurement scheme.

Fig. 2. Lower bound of estimation variances δα^2 with different mode numbers under the projective measurement method. The measured photons number is set as N=4.04×107, and the weak value is set as Aw=cot 5°≈11. The x axis and y axis are the mode numbers of m and n, respectively, and the z axis is the estimation variance of parameter α. The red line in this figure is the CCR bound at m=n, where the precision limit is improved fastest.

Fig. 3. Diagram of the experimental setup. (a) The mn-order HG beam is converted from an expanded Gaussian beam of an 780 nm laser by an SLM and a spatial filter system. The pre-selection is implemented by a Glan–Taylor polarizer (GTP) and an HWP. A polarized Sagnac interferometer is employed to implement the weak interaction procedure, where the inverse rotation signals are introduced by a Dove prism. Then a Soleil–Babinet compensator (SBC), an HWP, and a GTP are used to implement the post-selection. Finally, another SLM with a Fourier transfer lens is used to implement the projective measurement, where the successful projected photons are collected by an APD with an SMF. (b) Dove prism with PZT chips and generation method of the rotation signal. There are four PZT chips pasted on the reflection side of the prism with a 2×2 array, where the vertical distance of the PZT array is 10 mm.

Fig. 4. Detected electrical spectrum of HG11 to HG66 modes at 500 Hz to 5 kHz. The driving voltage of the PZT is 5 V, which corresponds to 22 μrad rotation signal. The first line is the spectrum of the electrical-noise floor of the APD detector, which is detected without input light on the APD.

Fig. 5. Experimental results. (a) Detected peak signal level of the HG11, HG33, and HG55 modes at 1 kHz. (b) Detected signal-to-noise ratio of the HG11, HG33, and HG55 modes. The driving voltage of the PZT increases from 0 to 2 V, which corresponds to 0–8.8 μrad rotation signal.

Fig. 6. Schematic of monitoring quantum bits.

Fig. 7. QCR bounds of the measurement parameters. (a) QCR bounds of parameter α. (b) QCR bounds of parameter θ. (c) QCR bounds of parameter ϕ. The y axis is the variance of estimator g^, and the x axis is the mode numbers. Mode numbers m and n simultaneously increase from 0 to 25. The red dotted line is the QCR bound of the Gaussian pointer with displacement coupling. The blue line is the QCR bound of the HO pointer with displacement coupling. The orange line is the QCR bound of the HO pointer with rotation coupling. Here the pre-selection state and post-selection state are chosen as |i⟩=|f⟩=12(|0⟩+eiπ4|1⟩), and values of all the parameters are α=0.001, θ=π/4, and ϕ=0. In addition, the value of σ0 is normalized to 1/2.

Fig. 8. QCR bound of α about post-selection angle ε. This is the QCR bound of α with a 2D HO pointer and rotation coupling. The x axis is the mode numbers m and n, which simultaneously increase from 1 to 25. The pre-selection state is |i⟩=12(|0⟩+eiπ4|1⟩), and the post-selection state is |f⟩=12(|0⟩−ei(π4+ε)|1⟩), where the angle ε varies from 0.1 to 0.01. In addition, we plot three QCR bounds at different values of ε. The green line is ε=0.1, the red line is ε=0.05, and the yellow line is ε=0.01.