Anomalous amplification in almost-balanced weak measurement for measuring spin Hall effect of light

Lan Luo; Yu He; Xiong Liu; Zhaoxue Li; Pi Duan; Zhiyou Zhang

doi:10.1364/OE.386017

1. Introduction

The spin Hall effect of light (SHEL) describes a transverse spin-dependent splitting when a light beam reflected or refracted at an interface between two media with different refractive indices [1,2]. The SHEL is closely related to the physical nature of the media, such as, refractive index and thickness of the layer medium. Therefore, the SHEL has been widely applied in the field of precision measurement, such as, optical physics [3–5], semiconductor [6,7], plasmonics [8,9], nano-metal film [10], and magnetic film [11,12]. However, the SHEL is so weak that it is difficult to observe with conventional method.

Recently, the weak measurement which can realize signal enhancement has stepped into a public spotlight [13]. In 2008, Hosten and Kwiat used the weak measurement technique to observe the SHEL, and achieved an incredible resolution of 0.1 nm [3]. In addition, since the weak measurement can effectively improve the measurement precision limited by technical noises [14–17], it is has been widely used for precision measurement of small parameters [18–29]. However, the near orthogonality of the pre- and post-select states results in a large amount of statistical data being discarded, so standard weak measurement (SWM) is considered a ’harmful’ measurement [30–32].

In view of these problems, Strübi and Bruder reported a new method, the so-called almost-balanced weak measurement (ABWM), which collects all of the information by using two orthogonal post-selected states [33]. In this scheme, the intensities of two outputs are almost balanced and the differencing of them gives a SWM-like amplification in the meter. It has been shown that the ABWM scheme is robust against the alignment error and readout noise [33,34]. The ABWM scheme has also been experimentally validated in time delay estimation and provides a higher signal-to noise ratio (SNR) than SWM [35–37].

In this paper, we propose to measure the spin splitting of the SHEL based on the ABWM technique. The theory model of the SHEL measurement based on ABWM is established. In addition, the experiment based on the SWM technique is also presented for comparison. Our results show that the amplification factor and sensitivity to the original shift of the ABWM scheme are higher than that of the SWM technique.

2. Theory

First, we give general theories of parameter estimation based on SWM and ABWM. Consider a typical weak measurement procedure involving a two-level system $|\psi _{i}\rangle =\left (|0\rangle +|1\rangle \right )/\sqrt {2}$ and a meter state $|\phi _{m}\rangle =\int dF\varphi (F)|F\rangle$. Here, $|0\rangle$ and $|1\rangle$ are the eigenvectors of observable of the system $\hat {A}$, $\hat {A}=|0\rangle \langle 0|-|1\rangle \langle 1|$. $\varphi (F)$ represents the wave function of the continuous meter variable $F$. The interaction between the system and the meter at the time $t_{0}$ is described by the Hamiltonian $H=\gamma \delta (t-t_{0})\hat {A}\otimes \hat {F}$. Here, $\gamma$ stands for the coupling strength of the interaction, and $\hat {F}$ denotes the operator acting on the input meter variable. The system and meter are interrelated through interaction and evolve into an entangled state:

(1)$$|\psi^{\prime}\rangle=\exp\left({-}i\gamma\hat{A}\otimes\hat{F}\right)|\psi_{i}\rangle \otimes|\phi_{m}\rangle.$$

For the SWM technique, the system is post-selected in $|\psi _{f}\rangle =\left (e^{i\epsilon }|0\rangle -e^{-i\epsilon }|1\rangle \right )/\sqrt {2}$, where $\epsilon$ is the post-selected angle which describes the deviation from the orthogonal state of the pre-selected state. The final state takes the form:

(2)$$\begin{aligned}|\Psi\rangle &=\left\langle\psi_{f}\left|\exp\left({-}i\gamma\hat{A}\otimes\hat{F}\right)\right|\psi_{i}\right\rangle\otimes|\phi_{m}\rangle\\ & =\langle\psi_{f}|\psi_{i}\rangle\left[\cos(\gamma F)-i A_{\omega}\sin(\gamma F) \right]|\phi_{m}\rangle\\ & =\left\langle\psi_{f}|\psi_{i}\right\rangle\exp\left({-}i\gamma A_{\omega}F\right)|\phi_{m}\rangle, \end{aligned}$$

where $A_{\omega }=\langle \psi _{f}|\hat {A}|\psi _{i}\rangle /\left \langle \psi _{f}|\psi _{i}\right \rangle =-i\cot \epsilon$ denotes the weak value of $\hat {A}$. The approximation condition in Eq. (2) is $\gamma A_{\omega }\ll 1$. If one measure the output observable $\hat {R}=\Sigma R|R\rangle \langle R|$ after post-selection, the average shift of the pointer can be obtained as

(3)$$\delta R =\dfrac{\langle\Psi|\hat{R}|\Psi\rangle}{\left\langle\Psi|\Psi\right\rangle}-\langle\phi_{m}|\hat{R}|\phi_{m}\rangle.$$

Suppose $R=F$, and the meter wave function $\varphi (F)$ is considered to be a Gaussian distribution with a standard deviation $\Delta F$ and expected value $\bar {F}=0$. The pointer shift is given by

(4)$$\delta F_{0}^{S}=\dfrac{\gamma\Delta F^{2}\sin2\epsilon}{e^{\gamma^{2}\Delta F^{2}}-\cos2\epsilon} \approx\gamma\Delta F^{2}\cot\epsilon,$$

and the post-selected intensity is given by

(5)$$P_{0}^{S}=\dfrac{1}{2}I_{0}(1-e^{\gamma^{2}\Delta F^{2}}\cos2\epsilon ) \approx I_{0}\sin^{2}\epsilon,$$

where $I_{0}$ is the intensity of the input light. The approximate condition in Eqs. (4) and (5) is $\gamma \Delta F A_{\omega }\ll 1$.

For the ABWM scheme, the system is post-selected by two almost balanced complementary states $|\psi _{f1,2}\rangle =\left (e^{i\epsilon }|0\rangle \pm i e^{-i\epsilon }|1\rangle \right )/\sqrt {2}$. Consequently, the final states evolve into

(6)$$\begin{aligned}|\Psi_{1,2}\rangle &= \langle\psi_{f1,2}|\exp({-}i\gamma\hat{A}\otimes\hat{F})|\psi_{i}\rangle|\phi_{m}\rangle\\ & \approx \langle\psi_{f1,2}|\psi_{i}\rangle\exp\left({-}i\gamma A_{\omega1,2} F\right)|\phi_{m}\rangle, \end{aligned}$$

and two almost balanced weak values are obtained with $A_{\omega 1,2}=\langle \psi _{f1,2}|\hat {A}|\psi _{i}\rangle / \left \langle \psi _{f1,2}|\psi _{i}\right \rangle \approx \mp i$. After measuring the observable $\hat {F}$, the probability distribution is given by

(7)$$P_{1,2}=\langle\Psi_{1,2}|\Psi_{1,2}\rangle=\dfrac{1}{2}(1\pm\sin(2\epsilon+2\gamma F))P(F),$$

where $P(F)=\langle \phi _{m}|\phi _{m}\rangle |$. On the detector, the pointer shifts of the two light beams can be expressed as $\delta F_{1,2}\approx 2\gamma \Delta F^{2}$. Based on ABWM, the difference probability distribution should be obtained in the form of $P_{-}=P_{1}-P_{2}=\sin (2\epsilon +2\gamma F))P(F)$. The corresponding average meter shift can be obtained as

(8)$$\delta F_{0}^{AB}=\gamma \Delta F^{2} \cot2\epsilon,$$

and the post-selected intensity is

(9)$$P_{0}^{AB}=I_{0}e^{\gamma^{2}\Delta F^{2}}\sin2\epsilon \approx I_{0}\sin2\epsilon.$$

Without any approximation in Eqs. (4) and (8), Fig. 1(a) shows the pointer shifts of two weak measurement systems changing with the post-selected angle. The coupling strength and the uncertainty respectively set as $\gamma \times 10^{-7}$ and $\Delta F=2\times 10^4$. The meter shift of the ABWM scheme first increases slowly with the decrease of the pre-selected angle, and increases rapidly when the pre-selected angle approaches zero, see red line in Fig. 1(a). In the SWM scheme, the meter shift increases until it reaches a maximum value and then reduces to zero quickly, see green line in Fig. 1(a). When the post-selected angle is extremely close to 0, it is interesting to note that the ABWM can theoretically achieve infinite amplification, while SWM is not the case. In addition, the pointer shift of the ABWM is very sensitive to the change of post-selected angle, which may have application value in sensing and other fields. Comparing Eqs. (5) and (9), we can find that the detectable intensity of the ABWM technique, $\sim \sin 2\epsilon$, is higher than that of the SWM scheme, $\sim \sin ^{2}\epsilon$. The post-selected probabilities varying with the post-selected angle are shown in Fig. 1(b).

Fig. 1. (a) show the meter shifts of the ABWM and SWM techniques for different post-selected angle. (b) show the normalization post-selected intensities as functions of the post-selected angle. The red and green lines denote the ABWM and SWM techniques, respectively.

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3. Experiment and results

We take the measurement of the spin splitting induced by SHEL as an example of our theory. Here, the weak coupling $\gamma$ is replaced by spin splitting. The transverse were vector $k_{y}$ and polarization degree of freedom are adopted as the meter variable and system, respectively. The experiment setup of the ABWM technique is shown in Fig. 1. A Gaussian beam with a wavelength of 632.8nm is emitted from the He-Ne laser. Then, the light beam passes through a half-wave plate (HWP) for adjusting the intensity of the incident light. After being slightly focused by the lens (L), the light beam transmits through a polarizer (P) which is used to control the incident polarization state of the light $|\psi _{i}\rangle$. Next, the light is reflected at the air-prism interface, and the SHEL takes place as a result of the spin-orbit interaction. The Wollaston prism (WP) is used to prepare two orthogonal post-selections $|\psi _{f1,2}^{AB}\rangle$, so the photons are spilt into two different post-selected states. Finally, the distributions of two almost balanced outputs are detected simultaneously by a charge coupled-device (CCD). The difference signal and the corresponding beam displacement are calculated by the computer. For the SWM technique, WP is replaced by the second polarizer whose polarization axis is nearly orthogonal to the optical axis of the first polarizer.

The theoretical models to describe the spin splitting estimation based on SWM and ABWM are established. First, the incident polarization of the light is pre-selected in

(10)$$|\psi_{i}\rangle= \cos\alpha|H\rangle+\sin\alpha|V\rangle =\dfrac{1}{\sqrt{2}}\left(e^{i\alpha}|+\rangle+e^{{-}i\alpha}|-\rangle\right)$$

where $|H\rangle$ and $|V\rangle$ respectively represent the horizontal and vertical polarization components. $|\pm \rangle =(|H\rangle \pm i|V\rangle )/\sqrt {2}$ denotes the left- and right-circular polarized components, respectively. $\alpha \ll 1$ is the post-selected angle. Besides, the meter is $|\phi _{m}\rangle =\int d k_{y}\varphi (k_{y})|k_{y}\rangle$, where $\varphi (k_{y})=(w/\sqrt {2\pi })\exp (-w^{2}k_{y}^{2}/4)$ is the transverse distribution of the wave-function of the meter. $w$ represents the beam waist. Upon reflection, the polarization state of the system evolves into

(11)$$|\Psi_{i}\rangle=\hat{U}\hat{R}|\psi_{i}\rangle\otimes|\phi_{m}\rangle,$$

where

(12)$$\hat{U}=\exp({-}i k_{y}\delta\hat{\sigma_{3}}),$$

denotes the unitary evolution to describe the spin splitting of the SHEL and

(13)$$\hat{R}=\dfrac{1}{2} \left[ \begin{array}{cc} r_{p}+ r_{s} & r_{p}-r_{s}\\ r_{p}-r_{s} & r_{p}+ r_{s}\\ \end{array} \right]$$

is the reflection matrix that describes the variations of the horizontally and vertically polarized components due to reflection. $\hat {\sigma _{3}}=|+\rangle \langle +|-|-\rangle \langle -|$ is the spin operator of the photon. $\delta =\left (r_{p}^{2}\delta _{H}+\alpha ^{2}r_{s}^{2}\delta _{V}\right )/\left (r_{p}^{2}+\alpha ^{2}r_{s}^{2}\right )$ represents the original spin splitting, where $\delta _{H}=\left (r_{p}+r_{s}\right )\cot \theta /r_{p}k$ and $\delta _{V}=\left (r_{p}+r_{s}\right )\cot \theta /r_{s}k$ are the spin splitting of the $H$ and $V$ polarization, respectively [24]. $\theta$ denotes the incident angle. $r_{p}$ and $r_{s}$ respectively represent the Fresnel reflection coefficients of the horizontal and vertical polarization. In our system, the incident angle is set as $\theta =30^{\circ }$. The refractive indices of BK7 prism is 1.515 and the beam waist $w=27um$. The corresponding reflection coefficients $r_p=0.16$ and $r_s=-0.25$. Since the post-selected angle $\alpha \ll 1$, the original split $\delta \approx 0.088um$ which is basically independent of $\alpha$. In addition, $|\psi _{i}^{\prime }\rangle =\hat {R}|\psi _{i}\rangle$ is regarded as the pre-selection in the weak measurement model.

In the SWM scheme, the light beam is post-selected in $|\psi _{f}^{S}\rangle =|V\rangle =i(|+\rangle -|-\rangle )/\sqrt {2}$. Therefore, a large weak value is obtained with

(14)$$A_{\omega}^{S}=\dfrac{\left\langle\psi_{f}^{S}\left|\hat{\sigma_{3}}\right|\psi_{i}^{\prime}\right\rangle}{\left\langle\psi_{f}^{S}|\psi_{i}^{\prime}\right\rangle} =i \dfrac{r_{p}\cot\alpha}{r_{s}},$$

and the probability distribution can be calculated as

(15)$$\begin{aligned}P^{S}&= \left|\left\langle\psi_{f}^{S}\left|\exp({-}i k_{y}\delta\hat{\sigma_{3}})\right|\psi_{i}^{\prime}\right\rangle|\phi_{m}\rangle\right|^{2}\\ &=\left|\left\langle\psi_{f}^{S}|\psi_{i}^{\prime}\right\rangle\right|^{2}\exp\left({-}2Im(A_{\omega}^{S})k_{y}\delta)\right) P(k_{y}), \end{aligned}$$

where, the $|\langle \psi _{f}^{S}|\psi _{i}^{\prime }\rangle |^{2}=r_{s}^{2}\sin ^{2}\alpha$ is the post-selected probability, and $P(k_{y})=|\varphi (k_{y})|^{2}$. The meter shift is $\Delta k^{S} \approx 2 Im(A_{\omega }^{S})\delta /w^{2}=2r_{p}\delta \cot \alpha /r_{s}w^{2}$. Therefore, the amplified beam displacement takes the form:

(16)$$\Delta y_{0}^{S}=\dfrac{z}{k} \Delta k^{S}\approx\dfrac{z}{r}\dfrac{r_{p}\cot\alpha}{r_{s}}\delta.$$

Here, $z$ and $k$ respectively denote the free propagation distance and central wave vector of the light beam. $r=k w^{2}/2$ represents the Rayleigh distance.

For the ABWM procedure, the post-selected states are given by

(17)$$|\psi_{f1,2}^{AB}\rangle=(|H\rangle\mp|V\rangle)/\sqrt{2} =(|+\rangle\mp i|-\rangle)/\sqrt{2}.$$

The corresponding weak values are obtained as

(18)$$A_{\omega1,2}^{AB}=\dfrac{\left\langle\psi_{f1,2}^{AB}|\hat{\sigma_{3}}|\psi_{i}^{\prime}\right\rangle}{\left\langle\psi_{f1,2}^{AB}|\psi_{i}^{\prime}\right\rangle} \approx{\pm} i,$$

and the power distributions of the system can been expressed as

(19)$$\begin{aligned}P_{1,2}^{AB}&=\left|\left\langle\psi_{f1,2}^{AB}|\exp({-}i k_{y}\delta\hat{\sigma_{3}})|\psi_{i}^{\prime}\rangle|\phi_{m}\right\rangle\right|^{2}\\ &=\left|\left\langle\psi_{f1,2}^{AB}|\psi_{i}^{\prime}\right\rangle\right|^{2}\exp\left[{-}2Im(A_{\omega1,2}^{AB})k_{y}\delta)\right] P(k_{y}), \end{aligned}$$

where $|\langle \psi _{f1,2}^{AB}|\psi _{i}^{\prime }\rangle |^{2}=\left (r_{p}\cos \alpha \pm r_{s}\sin \alpha \right )^{2}$. The corresponding pointer shift is $\Delta k_{1,2}^{AB} \approx \pm 2\delta /w^{2}$. The difference signal is obtained as

(20)$$\begin{aligned}P_{-}^{AB}(k_{y})&=P_{1}^{AB}(k_{y})-P_{2}^{AB}(k_{y})\\ & \approx r_{p}r_{s}\sin2\alpha\left(1+\dfrac{2r_{p}k_{y}\cot2\alpha}{r_{s}}\delta\right)P(k_{y}). \end{aligned}$$

The corresponding amplified beam displacement of the difference signal takes the form:

(21)$$\Delta y_{0}^{AB}=\dfrac{z}{k} \Delta k^{AB}=\dfrac{z}{r}\dfrac{r_{p}\cot2\alpha}{r_{s}}\delta.$$

From the Eqs. (16) and (21), the original shift is amplified by the propagation amplification $\dfrac {z}{r}$ and weak measurement system which depends on the post-selected angle $\alpha$. The amplification factor is defined as

(22)$$\Lambda^{AB,S}=\dfrac{\Delta y_{0}^{AB,S}}{\delta}.$$

The experimental results along with the theoretical predictions are shown in Fig. 2. The distributions of two post-selected light beam are detected by CCD simultaneously at the plane of $z = 250mm$. Therefore, the difference signal and its beam displacement changes are obtained. As shown in Fig. 3, with decreasing of the post-selected angle, the amplified shift of the ABWM technique monotonically increases, while the amplified displacement of the SWM scheme first reaches the maximum value and then decreases to zero. According to the above analysis, the ABWM scheme can theoretically realize infinite amplification when the post-selected angle is extremely close to 0. In our experiment, the amplification factor of the ABWM scheme $\Lambda ^{AB}$ reaches $\sim 10^{5}$, while the maximum amplification factor of the SWM technique $\Lambda ^{S}$ is $\sim 10^{4}$. The experimental results agree well with the theoretical predictions.

Fig. 2. Experiment setup of the ABWM scheme. Laser source, He-Ne laser; L, lens of focal length 50 mm; HWP, half wave plate; P, Glan polarizer for pre-selection; WP, Wollaston prism; CCD, charge-coupled-device.

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Fig. 3. (a) and (b) show the amplified displacements of the ABWM and SWM techniques changing with the post-selected angle. The solid lines and dots denote the theoretical predictions and the experiment results, respectively.

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According to the experiment results, we calculated the signal-noise ratios (SNR) of the ABWM and SWM schemes, see Fig. 4. It can be seen that when the post-selected angle is close to 0, SNR of the ABWM technique is comparable to that of the SWM scheme. The ABWM scheme shows a relatively consistent SNR for all data points, see red dots in Fig. 4. For the SWM scheme, the SNR decreases with the post-selected angle reducing. For the ABWM scheme, since the light is divided into two beams after being post-selected by two different states, the technical noises of two measured signals are different. The ABWM technique is not robust to the non-common noises of two post-selelcted signal. Therefore, SNR of the ABWM scheme obtained in the experiment is lower than that of the SWM technique. However, if the experimental environment can be further optimized, ABWM can achieve higher SNR, which will find applications in precision measurement.

Fig. 4. Signal-to-noise ratio (SNR) of the ABWM and SWM techniques. Red and green dots represent the ABWM and SWM schemes, respectively.

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The measurement precision of the original shift is calculated as $\sigma _{ori}=\sigma ^{AB,S}/(|\partial \Delta y^{AB,S}_{0}/\partial \delta |)$ here, $\sigma ^{AB,S}$ is the standard deviation of the amplified shift calculated from the statistical value of repeated measurements, and $\partial \Delta y^{AB,S}_{0}/\partial \delta$ represents the sensitivity of the amplified displacement $\Delta y^{AB,S}_{0}$ to the original shift $\delta$. The sensitivities of two techniques changing with the post-selected angle are shown in Fig. 5. With the decrease of post-selected angle $\alpha$, $|\partial \Delta y^{AB}_{0}/\partial \delta |$ increases rapidly, see Fig. 5(a). It is interesting to note that $|\partial \Delta y^{S}_{0}/\partial \delta |$ reduces to zero, while the amplified displacement reaches the maximum value, see Fig. 5(b). it is notable that the sensitivity of the ABWM scheme is much higher than that of the SWM scheme. The experimental standard deviations of the amplified displacement were obtained from the statistics of 20 measurements. When choose a small post-selected angle ($\alpha <2\times 10^{-3} rad$), the measurement precision of the ABWM is about $3\times 10^{-3} um$, and that of SWM is $10^{-3} um$. As results, when the post-selected angle is small, the ABWM scheme can obtain higher amplification factor and sensitivity than the SWM strategy, while the SNR and measurement precision are comparable to the SWM scheme. Therefore, when it is necessary to detect more weak signals, the ABWM technique has more application prospects than the SWM scheme.

Fig. 5. (a) and (b) show the sensitivity to the original shift of the ABWM and SWM techniques as functions of the post-selected angle, respectively. The dotted and solid lines denote the sensitivity and amplified displacement.

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4. Conclusions

In conclusion, we present general theories for the parameter estimation based on ABWM and SWM schemes. Compared with the SWM technique, the ABWM scheme can obtain a larger meter shift when the two signals of different post-selections are almost equal. In addition, we study the SHEL measurement in both theory and experiment, with the ABWM and SWM schemes. As results, the ABWM technique can be widely used in sensing field, because it has a larger amplification factor and higher sensitivity than the SWM scheme. When a small post-elected angle is chose, the measurement precision and SNR of the ABWM technique are comparable to those of the SWM scheme. Therefore, ABWM could be used to measure more tiny shift of the light beam. We believe that the ABWM scheme has a promising application prospect in the precision measurement.

Funding

National Natural Science Foundation of China (11674234); Innovation project of Sichuan University (2018SCUH0021).

Disclosures

The authors declare no conflicts of interest.

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Anomalous amplification in almost-balanced weak measurement for measuring spin Hall effect of light

Abstract

1. Introduction

2. Theory

3. Experiment and results

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (5)

Equations (22)

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