1. Introduction

It is well recognized that in addition to the spin angular momentum (SAM) characterized by its polarization, photon may carry orbital angular momentum (OAM). OAM of photons is a Hilbert space which is discrete, orthonormal and high dimensional. Photons carrying OAM have been the subject of much recent scientific attention [1–4]. The OAM eigenstates are characterized by helical phase fronts described by $\exp (il\varphi )$, where $l$ is the quanta number of OAM, $\varphi$ is the azimuthal angle [5]. Because there is no theoretical upper limit on the quantity of the quanta number of OAM a single photon can carry, OAM possesses potential values in quantum information [6,7], quantum communication [8–10], and quantum detection [11–13], especially in the precise angular rotation measurement [14,15].

Due to the characteristics of helical phase $\exp (il\varphi )$, photon orbital angular momentum acts as a ‘photonic gear’, converting a mechanical rotation of an angle $\theta$ into an amplified rotation of the optical polarization by $l\theta$, with super-resolution of $l$ times [16]. Meanwhile the sensitivity $\delta \theta$ of the angular rotation measurement can also be improved by photon OAM $l\hslash$, with the minimum sensitivity limit of $1/\sqrt{N}l$ for untangled photons ($1/Nl$ for entangled photons) [17]. Soon after, experimental results demonstrate that the entanglement of very high OAM can improve the sensitivity of angular rotation measurement, through transferring polarization entanglement to OAM with an interferometric scheme [18]. Moreover, scheme using photon OAM is relatively robust as each of the remaining photons still carries OAM of $l\hslash$no matter how many photons are lost [19].

However, our approach is different from all other OAM-related proposals. We use a quantum detection strategy—binary-outcome homodyne detection (BHD) method in angular rotation measurement scheme. In this manner, BHD method shows a quantum-enhanced performance, and obtains a narrower super-resolving signal peak compared with the existing intensity detection (ID) method. This paper is organized as follows. First, the scheme of the angular rotation measurement using BHD method is analyzed theoretically. Then, the resolution and sensitivity of the angular rotation measurement using BHD method are compared with those using the existing ID method. Finally, the impact of photon loss on the resolution and sensitivity of BHD method is discussed.

2. The theoretical analysis

The angular rotation measurement scheme is shown in Fig. 1, and it is similar to an interferometer with two Dove prisms [20]. A coherent state ${|\alpha 〉}_{a,+l}$ with amplitude $\alpha \text{=}\sqrt{N}$ goes through an interferometer from the port ‘$a$’, and the other port ‘$b$’ is left in vacuum${|0〉}_b$. Here, the subscript $\text{+}l$ denotes the quanta number of OAM. OAM can be generated by q-plate [21] and Spatial Light Modulator (SLM) [22–24]. The input state can be expressed as $|{\psi }_{in}〉={|\alpha 〉}_{a,+l}\otimes {|0〉}_b$, and the density operator of the input state is given by ${\widehat{\rho }}_{in}=|{\psi }_{in}〉〈{\psi }_{in}|$. The clockwise light path is the detection light path, labeled as ‘$b$’ and DP₂ is a rotating Dove prism. While the counter-clockwise light path is the reference light path, labeled as ‘$a$’, and DP₁ is a stationary Dove prism with constant $\theta =0$. Rotating DP₂, when two Dove prisms are orientated with respect to each other by an angle of $\theta$, there would be a relative phase difference between two arms in the interferometer of $2l\theta$ [25], expressed by $\exp (i2l\theta {\widehat{N}}_b)$with the number operator ${\widehat{N}}_b={\widehat{b}}^{†}\widehat{b}$. The role of DP₁ ensures that the helical wave front of each beam has the same sense, and thus interference between the beams is uniform across the whole aperture of detector [14].

 

Fig. 1 The angular rotation measurement scheme. (BS: Beam splitter; DP: Dove prism; M: Mirror; D: Detector).

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The fictitious beam splitter $\widehat{B}\left(T\right)$ is used to describe photon loss with transmission coefficient $T$ [26]. Two 50:50 beam splitters are described by the beam-splitter operator ${\widehat{B}}_{1/2}$ [27]. Through a simple derivation, the output states of the interferometer take the following form

where, $\beta =\sqrt{TN}\left[1-\exp (i2l\theta )\right]/2$ and $\gamma =i\sqrt{TN}\left[1+\exp (i2l\theta )\right]/2$. Signal intensities of two output ports are $I_a=〈{\widehat{n}}_a〉={\left|\alpha \right|}^{2}T\left[1-\cos (2l\theta )\right]/2$ and $I_b=〈{\widehat{n}}_b〉={\left|\alpha \right|}^{2}T\left[1\text{+}\cos (2l\theta )\right]/2$. Considering $T=1$ and ${\left|\alpha \right|}^2=N$, the result of $b$ path can be simplified as $I_b\stackrel{T=1,{\left|\alpha \right|}^2=N}{\to }N{\cos }^2(l\theta )$, which agrees with the results in [17]. This shows that $l$-times resolution enhancement is over the classical Malus law $N{\cos }^2\theta$.

Next, we use binary-outcome homodyne detection (BHD) method, instead of the existing intensity detection (ID) method, to process the output state ${\widehat{\rho }}_{out}$. Recently, Distante et al. [28] demonstrated the binary-outcome homodyne detection (BHD) method by dividing the total data of phase quadrature into binary outcomes: $\left|h\right|\le h_0$ and $\left|h\right|>h_0$, denoted by “$+$” and “$-$”, respectively. Here, a homodyne detection of the phase quadrature $\widehat{H}=\left(\widehat{a}-{\widehat{a}}^{†}\right)/2i$ at the output port ‘$a$’ is represented by the projection operators $\left\{|h〉〈h|\right\}$ with $\widehat{H}|h〉=h|h〉$. The probability for an output $h$ can be obtained by $P\left(h\right)=Tr\left({\widehat{\rho }}_{out}|h〉〈h|\right)$, and its explicit form is

where, we have used the wave function of a coherent state, with $〈h|\alpha 〉={(2/\pi )}^{1/4}\exp \left\{-{\left[p-\mathrm{Im}(\alpha )\right]}^2-2i\mathrm{Re}(\alpha )h+i\mathrm{Re}(\alpha )\mathrm{Im}(\alpha )\right\}$. Then, according to Eq. (2), the probability of $\left|h\right|\le h_0$ and $\left|h\right|>h_0$ can be expressed as

Note that the probability of $\left|h\right|\le h_0$ is the same as the interference output signal $〈{\widehat{H}}_{+}〉=Tr\left({\widehat{\rho }}_{out}{\widehat{H}}_{+}\right)$ with observable variables ${\widehat{H}}_{+}={\displaystyle {\int }_{-h_0}^{+h_0}|h〉〈h|}\text{d}h$ and ${\widehat{H}}_{+}^2={\widehat{H}}_{+}$. Now considering the limit $h_0\to 0$, phase quadrature with $h\text{=}0$ is detected. In this case, the observable variable becomes ${\widehat{H}}_{+}=|h_0=0〉〈h_0=0|$, and the interference output signal $〈{\widehat{H}}_{+}〉=Tr\left({\widehat{\rho }}_{out}{\widehat{H}}_{+}\right)$ is given by

According to the quantum estimation theory$\delta {\theta }_{{\widehat{H}}_{+}}=\Delta {\widehat{H}}_{+}/\left|\text{d}〈{\widehat{H}}_{+}〉/\text{d}\phi \right|$ [16], where $\Delta {\widehat{H}}_{+}={\left(〈{\widehat{H}}_{+}{}^2〉-{〈{\widehat{H}}_{+}〉}^2\right)}^{1/2}$, the angular sensitivity can be derived as

3. The resolution and sensitivity of BHD method

Figures 2(a)–2(c) present the normalized super-resolving signals of binary-outcome homodyne detection (BHD) method, obtained from Eq. (4), with the average photon number $N=1$(blue dotted line), $N=3$(green dot-dashed line), and $N=10$(red dashed line). The black solid line represents the results of the existing Intensity Detection (ID) method, presented for comparison reason. Figures 2(a)–2(c) are the cases of $l=3$, 6 and 9, respectively. Note that BHD method has narrower peaks and halves the peak-to-peak spacing of the existing ID method, showing better resolution. The reason of this feature can be explained through the comparative analysis of ID output signal $I_b=NT{\cos }^2(l\theta )$ and BHD output signal $〈{\widehat{H}}_{+}〉=\sqrt{2/\pi }\exp \left[-NT{\sin }^2(2l\theta )/2\right]$. The BHD output signal decreases exponentially, so the BHD method has narrower peaks; besides, by comparing${\sin }^2(2l\theta )$in the BHD output signal with ${\cos }^2(l\theta )$ in the ID output signal, it is easy to understand that BHD method halves the peak-to-peak spacing of the existing ID method. This agrees with the previous result in [28].

 

Fig. 2 The output signal for the proposed binary-outcome homodyne detection (BHD) and the existing intensity detection (ID), with lossless $T=1$. (a) $l=3$, (b) $l=6$, (c) $l=9$, (d) The FWHM curves.

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Moreover, with the increase of the average photon number $N$, signal peaks of BHD method become narrower. This feature agrees with the previous experiment result [29]. This feature is what the existing ID method doesn’t have, with the same signal peak of ID for $N\text{=}1,3,10$. Full Width at Half Maximum (FWHM) of signal peak is usually used to evaluate the resolution, and FWHMs of BHD and ID method are shown in Fig. 2(d). FWHMs of BHD and ID decrease with the increase of OAM quanta number, and FWHM of BHD is always less than that of ID. It can be seen that BHD method reduces FWHM and improves the resolution through increasing OAM quanta number $l$ and the average photon number $N$.

In addition, In Fig. 2(a), it is worth noting that the visibility of BHD output signal is less than 1, when the average photon number $N\text{=}1$ and $N\text{=}3$, where the visibility is defined as $V=\left({〈{\widehat{H}}_{+}〉}_{\max }-{〈{\widehat{H}}_{+}〉}_{\min }\right)/\left({〈{\widehat{H}}_{+}〉}_{\max }\text{+}{〈{\widehat{H}}_{+}〉}_{\min }\right)$, with ${〈{\widehat{H}}_{+}〉}_{\max }$ and ${〈{\widehat{H}}_{+}〉}_{\min }$ being maximum and minimum values of BHD output signal. Figure 3(a) presents the visibility curve of BHD output signal with different average photon numbers. This visibility depends only on the average photon number $N$ and not on OAM quanta number $l$, and therefore a curve for arbitrary $l$ is given. From Fig. 3(a), when the average photon number $N$ is more than 10, the visibility is close to 1. The imperfect visibility mainly impacts sensitivity $\delta \theta$, shown as Figs. 3(b)–3(d). When $N\text{=}1$, the sensitivity of BHD is far worse than that of ID. When $N\text{=}10$ and $N\text{=}20$, the best sensitivity of BHD is comparable with that of ID. The best sensitivity $\delta {\theta }_{\min }$ is defined as the minimum value of sensitivity curves that can be obtained by choosing the appropriate angle $\theta$.

 

Fig. 3 (a) The visibility of BHD output signal. (b), (c) and (d) are the sensitivity of BHD method for $N\text{=}1$, $N\text{=}10$and $N\text{=}20$, respectively.

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Next, we discuss the sensitivity of angular rotation measurement using BHD method according to Eq. (5). In Fig. 4(a), all curves are for the average photon number $N=10$. Note that the best sensitivity of BHD method is improved with the increase of OAM quanta number $l$ efficiently. More details about the best sensitivity of BHD with various $l$ and $N$ are shown in Fig. 4(b). It is observed that the best sensitivity depends on not only the average photon number $N$ but also OAM quanta number $l$. This feature agrees with the previous research result in [16]. Therefore they redefine the Heisenberg precision scaling for OAM enhancement angular rotation measurement as ${[2l\sqrt{\nu }N]}^{-1}$, where $\nu$ represents the repeated measurements of $\nu$ times.

 

Fig. 4 (a) The sensitivity $\delta \theta$of BHD method against detected angular $\theta$ and (b) the best sensitivity with various OAM quanta number $l$, with lossless $T=1$.

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4. The impact of photon loss

The unavoidable photon loss mainly comes from the atmospheric absorption and diffraction, the imperfect optical system. Next, we use Eqs. (4) and (5) with transmission coefficients $T\ne 1$ to analyze the impact of photon loss on the resolution and sensitivity of angular rotation measurement.

In Fig. 5, it can be observed that the FWHM of BHD output signal peak increases with the decrease of transmissivity $T$, showing that photon loss makes the resolution of angular rotation measurement worse. In the top right corner of Fig. 5, the output signal peaks of BHD method with $T\text{=}1,0.8,0.6$ are given, for a better understanding of the degradation of resolution with photon loss (imperfect transmissivity).

 

Fig. 5 The impact of photon loss on the resolution of angular rotation measurement using BHD method.

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Figure 6 shows that the best sensitivity of angular rotation measurement degrades with the decrease of transmissivity $T$(or in other words, with the increase of photon loss). Note that when $T>0.2$, the best sensitivity changes smoothly; when $T<0.2$, the best sensitivity becomes worse dramatically. In the top right corner of Fig. 6, the sensitivity curves of BHD method against detected angular with $T\text{=}1,0.8,0.6$ are depicted, in order to show the degradation of sensitivity with photon loss (imperfect transmissivity) more intuitively. Considering the laboratory rotational measurement system, transmissivity T commonly can reach 0.8 or higher, much greater than 0.2. Therefore, the super-resolving angular rotation measurement using binary-outcome homodyne detection is easy to implement.

 

Fig. 6 The impact of photon loss on the best sensitivity of angular rotation measurement using BHD method.

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5. Conclusion

This paper applies a quantum detection strategy—binary-outcome homodyne detection (BHD) method into OAM enhancement angular rotation measurement scheme. The analysis results show that this BHD method can obtain a quantum-enhanced performance. Compared with the existing ID method, signal peaks of the BHD method are narrower, showing better resolution. Besides, the resolution and sensitivity of the BHD method can be further improved by increasing the average photon number and the OAM quanta number. Finally, the impact of photon loss on the resolution and sensitivity of angular rotation measurement is discussed. Results show that the resolution and sensitivity of the BHD method degrade with the increase of photon loss. When $T>0.2$, the best sensitivity changes smoothly; when $T<0.2$, the best sensitivity becomes worse dramatically.