Experimentally quantifying the advantages of weak-value-based metrology

We experimentally investigate the relative advantages of implementing weak-value-based metrology versus standard methods. While the techniques outlined herein apply more generally, we measure small optical beam deflections both using a Sagnac interferometer with a monitored dark port (the weak-value-based technique), and by focusing the entire beam to a split detector (the standard technique). By introducing controlled external transverse detector modulations and transverse beam deflection momentum modulations, we quantify the mitigation of these sources in the weak-value-based experiment versus the standard focusing experiment. The experiments are compared using a combination of deterministic and stochastic methods. In all cases, the weak-value technique performs the same or better than the standard technique by up to two orders of magnitude in precision for our parameters. We further measure the statistical efficiency of the weak-value-based technique. By postselecting on $1\%$ of the photons, we obtain $99\%$ of the available Fisher information of the beam deflection parameter.

Figure 1

(Experimental schematic) We use a cw Gaussian beam exiting a fiber. We compare the different experiments WVT (upper box) and the ST (lower box) to determine a beam deflection using split detector $1\left(\mathrm{SD}1\right)$. The WVT uses a Sagnac interferometer, and the ST focuses the deflected beam with focal length $f$. The piezo 50:50 beam splitter imparts a momentum kick $k$, which we determine from the beam shift on split detector 1. The two external modulations are labeled as $q$ and $d$. The $d$ refers to the transverse detector modulation and $q$ refers to the transverse momentum modulation. Polarizing beam splitters (PBS) are in orange and 50:50 beam splitters are in blue. The PBS work as mirrors given that the light is vertically polarized. The split detector $2\left(\mathrm{SD}2\right)$ is used to collect the bright port beam shift from the WVT.

Figure 2

A dBV spectrum comparison of the WVT (blue) and ST (green) with both external modulations, where $\mathrm{dBV}=20{log}_{10}\left(V/V_{\text{total}}\right)$ is plotted as a function of frequency. The factor 20 in the dBV definition comes from the convention of using voltages instead of power. The peak-to-peak signals are normalized to the detected power, $V_{\text{total}}$, in their respective experiment; either the WVT or the ST. From the plot, we see a signal, $V_{\text{pp}}$, from the “Signal” deflection corresponds to an angle of 48 nrad peak to peak at 7 Hz, an external transverse “Momentum Modulation” corresponds to an angle of $2.5\mu \mathrm{rad}$ peak to peak at 56 Hz, and an external transverse “Detector Modulation” corresponds to displacement of 230 nm peak to peak at 28 Hz. The Fourier spectrum illustrates both the weak-value enhancement of the signal of a factor of $3.2$ over the ST, as well as the suppression of the transverse detector modulation and the transverse momentum modulation. In addition, the suppression of the two external modulations is larger than the amplification factor as predicted theoretically in Eqs. (8). We note, the suppression of the external modulations from the signal, $V_{\text{pp}}$, collected from $\mathrm{SD}1$ are not direct deflection measurements [see Eqs. (8)].

Figure 3

A log-log plot of the ratio of the signal voltage to external modulation of the WVT, ${\mathcal{R}}^{\text{wv}}$, as a function of the ratio of the ST, ${\mathcal{R}}^{\text{st}}$, with varying external modulation strengths. In plot (a), external transverse detector modulation $d$ is applied. In plot (b), external transverse momentum modulation $q$ is applied. We use 12 points to demonstrate the constant-slope behavior of Eqs. (4). The postselection angle is $0.38$ rad and $L\approx 34$ cm. The dotted red lines are the linear fits of the data.

Figure 4

A plot of the theoretical minimum deviation given by the Cramér-Rao $\mathrm{bound},\mathrm{\Delta }{k}_B$, divided by the deviation of the measurements of $k,\mathrm{\Delta }{k}_B/\mathrm{\Delta }k=1/\sqrt{1+{\xi }_{\text{rms}}^2/\mathrm{\Delta }{k}_B^2}$ as a function of external modulation strength ${\xi }_{\text{rms}}\in \{d_{\text{rms}},q_{\text{rms}}\}$. Data comes from a signal $k$ of 16 and 48 nrad deflection with variable external modulation at a frequency of 28 Hz. Plot (a) is for transverse detector modulation and plot (b) is for transverse momentum modulation. The blue lines are the WVT theory and the green lines are the ST theory. We stress that ${\xi }_{\text{rms}}$ is not a noise source, but models one frequency component of a general noise source.

Figure 5

A plot of the geometric factor $f^{\prime }/2{\sigma }^2{k}_{0}cot(\varphi /2)$ from Eq. (4) as a function of beam radius $\sigma$ and focal length $f^{\prime }=f$ or distance $f^{\prime }=L$. The plot never surpasses 1, thus the ST will not outperform the WVT. The plot uses a postselection angle of $\varphi =0.4$ rad.

Figure 6

Fisher information vs postselection angle $\varphi$. Data is taken from the Fourier transform and averaged over equal numbers of samples. Angle $\varphi$ ranges from $0.22$ to $0.9$ rad. The confidence interval is $95\%$, and we see the fit break down as $\varphi$ becomes large. Most of the information is found to be in the dark port even for large $\varphi$. Both dark and bright ports follow ${cos}^2(\varphi /2)$ and ${sin}^2(\varphi /2)$ behavior, respectively, as in Eqs. (6).

Figure 7

A spectrum voltage comparison of the WVT (blue) and ST (green) with naturally occurring laser-beam-jitter noise, where $V_{\text{pp}}$ is the signal from the detector in volts. Without the fiber, the beam is shaped to $\sigma =1.12$ mm and is sent to the experiments. The beam jitter is found in the low frequency regime (under 1 kHz). We see about 20 dBV improvement in the WVT for low frequencies (under 300 Hz). We note similarly to Fig. 2, the comparison is not of beam shift, but of voltages from the laser-beam-jitter noise. Both data sets were taken by averaging 128 samples. The plots are normalized to the detected power, $V_{\text{total}}$ (either the WVT or the ST in their respective experiments). Note that this is a voltage comparison and not a deflection comparison of WVT and the ST.