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Abstract
We demonstrate a tried-and-true binary strategy for angular displacement estimation, of which the measuring system is a modified Mach-Zehnder interferometer fed by a coherent state carrying orbital angular momentum, and two Dove prisms are embedded in two arms. Unlike previous protocols, in this paper, we use fidelity instead of standard deviation to evaluate the detection strategies. Two binary strategy candidates, parity detection and Z detection, are considered and compared. In addition, we study the effects of several realistic scenarios on the estimation protocol, including transmission loss, detection efficiency, dark counts, and those which are a combination thereof. Finally, we exhibit a proof-of-principle experiment, the results suggest a resolution enhancement effect with a factor of 3.72.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM) of the light [1] is an important degree of freedom, and paves the way for a number of applications, including optical communication [2–4], computational imaging [5,6], particle manipulation [7], quantum simulation [8–10]. Almost all of the above applications are related to a physical characteristic of OAM: a limitless orthogonal Hilbert space associated with each integer ℓ, which is referred to as quantum number or topological charge. This provides OAM with an advantage over spin angular momentum (SAM), which only has two independent states (spin up and spin down). Photons that are eigenstates of OAM, originate as a result of spatial wavefront distribution exp(iℓφ) with azimuthal angle φ, corresponding macroscopic embodiments are Laguerre-Gaussian beams and Bessel-Gaussian beams. To such a phase structure there corresponds a central dark nucleus in an OAM beam; the photons each carry angular momentum ℓħ, which can be used for encoding more information into a single photon.
Within the past decade, OAM-based angular displacement estimation has gained a lot of attention [11–15], for it can provide a method to correct the measurement bases of two parties in quantum teleportation. Theoretically, the Heisenberg limit is reachable by inputing some exotic quantum states, e.g., two-mode squeezed vacuum, N00N states, and entangled coherent states. However, from the view of sensitivity, many researches have illustrated that the available photon numbers of these states may downplay their quantum advantages [16,17]. Hence, some protocols by deploying OAM coherent states and binary detection strategies have been proposed [15]. In these protocols, the detection strategies provide single-fold super-resolution peak, and OAM scales single-fold peak to multi-fold peak in accordance with double quantum number ℓ, i.e., 2ℓ-fold super-resolution peak. Nevertheless, the sensitivities of two binary detection strategies––Z detection [18] and parity detection [19, 20] ––saturate the shot-noise limit, so that we cannot determine the pros and cons of these two strategies in the framework of standard deviation. On the other hand, some studies pointed out that standard deviation δθ may not be an adequate criterion to characterize the parameter sensitivity in an interferometer when multiple peaks are presented in the probability distribution [21–24]. Related to this, the protocols using OAM coherent states and binary detection strategies exactly belong to this category. Therefore, one is entitled to compare the two strategies using other evaluation approaches. In this paper, we utilize fidelity [25–27] (the Shannon mutual information between the angular displacement θ and measuring outcomes) to appraise the two binary detection strategies whose sensibilities calculated by standard deviation are on a par with each other.
The remainder of this paper is organized as follows. In Sec. 2, we briefly introduce our angular displacement estimation protocol and two binary detection strategies, and the comparison between these two strategies is exhibited. Section 3 analyzes the effects of several realistic scenarios on estimation protocol. In Sec. 4, we demonstrate a proof-of-principle experiment, the impact factors in the experiment are also discussed. Finally, we summarize our work with a conclusion in Sec. 5.
2. Fundamental principle
In this section, we direct our attention to the fundamental principle of estimation protocol in an ideal scenario. The working principle of our protocol and that of detection strategies are presented. In addition, we introduce the concept and corresponding fidelity calculations of these detection strategies, a tried-and-true strategy is given by comparison.
2.1. Angular displacement estimation protocol
We start off with introducing the protocol for angular displacement estimation. Consider a modified Mach-Zehnder interferometer whose input is a coherent state carrying OAM, and two Dove prisms are embedded in its two paths, as illustrated in Fig. 1, where, here and throughout, the counterclockwise and clockwise paths are mode A and mode B, respectively. A coherent state |α〉 launched from a laser is coupled into a single-mode fiber, then the state is collimated and is incident on a polarizer. The fiber and polarizer are responsible for purifying spatial mode and polarization of this state, respectively. The OAM degree of freedom of the state is added via a spiral phase plate, accordingly, the input state can be written as |0〉A|αℓ〉B with respect to quantum number ℓ. Subsequently, the polarization of state is set to diagonal direction through a half wave plate, and then it enters a polarizing Mach-Zehnder interferometer, where the estimated parameter is angular displacement difference θ between the two prisms. To such an angular displacement difference θ there corresponds a phase difference 2ℓθ between the two modes [28]. Without loss of generality, we assume that the prism in mode A is rotated with displacement θ and that in mode B is immobile, the evolution operator can be expressed as follows: Û = exp(i2ℓâ†âθ). After the interferometer, the second half wave plate rotates the polarization from horizontal (vertical) direction to diagonal (anti-diagonal) one. Finally, the two polarized modes interfere with each other at the third polarizing beam splitter. In terms of the above analysis, the output state is found to be
Noting that the angular displacement information provided by the two output ports is complementary. Thus, we can take either of two outputs to perform detection strategy, and as manifested in Fig. 1, output port B is selected throughout this paper.
Fig. 1 Schematic of an angular displacement estimation protocol. The inset is intensity distribution of modulated OAM beam obtained by a CCD camera. The optical elements are abbreviated as: L, laser; SMF, single-mode fiber; P, polarizer; FC, fiber coupler; SPP, spiral phase plate; HWP, half wave plate; PBS, polarizing beam splitter; DP, Dove prism; RM, reflection mirror; D, detector.
2.2. Binary detection strategy
In this section, we consider two detection strategies, parity detection and Z detection, for our estimation protocol. Both of them are excellent binary strategies, and are beyond caring about the exact number of photons in the output. Z detection, also known as zero-nonzero detection, only focuses on whether there are photons arriving; zero and nonzero counts are recorded as 1 and 0, respectively. Parity detection merely pays attention to the parity of measured photon number; even and odd counts are labeled as 1 and −1, respectively. In general, Z detection requires the use of an avalanche photodiode in Geiger mode (Gm-APD), whereas parity detection needs to use a photon-number-resolving detector [29–31].
For output port B, the operators for Z detection and parity detection can be expressed as Ẑ = |0〉BB 〈0| and Π̂ = exp(−iπb̂†b̂), respectively. The probability of count outcome can be calculated from the formula P = 〈ψ|Ô|ψ〉 with a projective operator Ô.
In order to calculate this probability, with using two-mode Fock basis, we rewrite the output state in Eq. (1) as follows:
where N = |αℓ|2 is the mean photon number inside the interferometer.Based on the operator Ẑ = |0〉BB 〈0| and output state in Eq. (2), the probabilities of zero and nonzero counts are given by
Similarly, we can calculate the probability of even counts and that of odd ones in parity detection,2.3. Fidelity calculation and strategy comparison
On the basis of aforementioned detection probabilities, here we calculate and compare the two detection strategies in the framework of fidelity appraisal. Noting that this evaluation approach is applicable when multiple peaks are presented in the probability distribution [21–24]. Hence, we simply validate multi-peak characteristic of the two strategies ahead of fidelity calculation.
According to Bayes’ rule, the conditional probability density p(θ|m) with respect to angular displacement θ and outcome m is given by [25]
where p(θ) denotes priori information of probability density for angular displacement θ, over the interval −π ≤ θ ≤ π. In this paper, we assume that there is no priori information about angular displacement, namely, p(θ) = 1/2π. By inserting the probabilities of Eqs. (3)−(6) into Eq. (7), one can calculate the conditional probability densities. In Fig. 2 we plot these probability densities, it can be seen that the number of peaks is equal to double quantum number ℓ in each cycle. That is, fidelity appraisal holds true for out protocol and detection strategies.Fig. 2 Conditional probability density p(θ|m) against angular displacement θ in the case of N = 1, ℓ = 3, and p(θ) = 1/2π. PD, parity detection; ZD, Z detection.
In what follows, we turn our attention to fidelity calculation and strategy comparison. To begin with we give the definition of fidelity [25], also known as Shannon mutual information,
A simple understanding for the mutual information is the amount of sender’s original information which can be obtained by receiver from the acquired distortion information. It follows that high fidelity is closely relevant to excellent parameter sensitivity, and we rely on the size of fidelity to determine a tried-and-true strategy from the two strategies.In views of the definition, Eqs. (3)−(6), and the assumption about priori information, we can calculate fidelities of the two strategies. Figure 3 presents the fidelity variation with the increase of the mean photon number. One can find that the fidelity of Z detection is superior to that of parity detection, namely, Z detection provides more information than parity detection regarding angular displacement. Meanwhile, the fidelity is independent of angular displacement, since, for a given detection strategy, it is a mean value over all possible values of angular displacement and over all probabilities of measuring outcomes.
Fig. 3 The fidelity H against the mean photon number N, where N ranges from 1 to 20, ℓ is an arbitrary integer, and p(θ) = 1/2π.
Overall, Z detection is greater than parity detection via analyzing parameter sensitivity from the perspective of fidelity. This is a conclusion to be congratulated, at least for the level of practical realization, because a commercial Gm-APD is available, while the technology of photon-number-resolving detectors is undeveloped. In the following sections, the discussion and experiment are only directed against Z detection.
3. The effects of realistic factors on angular displacement estimation protocol
For an ideal scenario, the above section calculates the fidelities of detection strategies, and gives a tried-and-true strategy by comparison. However, it is well known that an estimation system in practical applications is inevitably affected by various realistic scenarios [32]. With respect to Z detection, in this section, we separately study the effects of several realistic scenarios on the fidelity, like transmission loss, detection efficiency, and dark counts. Further, we consider a situation composed of all realistic scenarios, and use the corresponding result to provide theoretical basis for proof-of-principle experiment.
3.1. Transmission loss
Of the practical interferometric measurements, there always exists transmission loss due to device-induced reflection or environmental absorption. In general, one can model this scenario by inserting two fictitious beam splitters with transmissivity TA and TB in two paths before the second polarizing beam splitter [33, 34]; accordingly, LA = 1 − TA and LB = 1 − TB are lossy ratios of the two paths. At this point, the state of output port B turns out to be
Further, the probability of zero count can be expressed asIn terms of this probability and the definition of fidelity in Eq. (8), we show dependence of the fidelity on transmission losses of the two paths, as manifested in Fig. 4. A remarkable phenomenon is that the contour lines are convex compared to the diagonal lines, LA + LB = Constants. More physically, regarding identical total loss, the fidelity is better maintained when the two path losses are the same. Meanwhile, one can find that the fidelity is reduced to 0 when one of the two paths is completely lossy. The reason behind these phenomena is that the inconsistent losses erase the system’s path indistinguishability––the origin of interference––and then lead to the degeneration in the fidelity.
Fig. 4 The fidelity H against transmission losses of the two paths, LA and LB, where both LA and LB range from 0 to 1 and N = 3. The color bar represents the corresponding values of the fidelity.
3.2. Detection efficiency
Next, we address the second scenario where the detection process is imperfect as well. This scenario arises from a fact that effective trigger of the detector caused by output photons is probabilistic. The photons without successful trigger can be regarded as detection loss. This process is generally modeled by placing a fictitious beam splitter with transmissivity η in front of an ideal detector [35], where the parameter η can be regarded as detection efficiency, and 1 − η stands for lossy ratio in the detection process. Based on the output in Eq. (9), the probability of zero count at port B is written as
Further, according to Eq. (8), the fidelity can be calculated. In Fig. 5, we plot the variation of fidelity in the presence of imperfect detection efficiency. It can be seen that the fidelity is increased as the increase of detection efficiency; moreover, the relationship between detection efficiency and fidelity is approximately linear positive correlation. Notice that, regarding the same losses in the two paths (TA = TB = T), Eq. (10) has an identical form with Eq. (11) in the case of T = η. This implies that the effects of detection loss and transmission loss on fidelity are similar since both of them are linear lossy.
3.3. Dark counts
In addition to the above scenarios, here we consider another realistic scenario about detectors: dark counts. It refers to abnormal avalanche counts induced by thermally excited carriers. Let us assume that the rate of dark counts can be denoted as r. Accordingly, the probability of n dark counts obeys the following Poissonian distribution [36]
Due to the existence of dark counts, the probability of nonzero counts is increased. Only when the rate of dark counts is zero does the probability of zero count remain the same. Hence, at this point the probability of zero count arrives at
Under the present technical conditions, where applicable, the range of r is between 10−8 and 10−2 for a single APD. In accordance with this range, we give the variation on the fidelity in the presence of dark counts, as shown in Fig. 6. One can find that the rate of dark counts has almost no impact on the fidelity except for r = 10−2 (slight decrease). Furthermore, with different mean photon numbers, the variation trend of fidelity remains the same whatever the rate of dark counts is. Overall, Z detection is a robust strategy for withstanding the disturbance originating from dark counts.
Fig. 6 The fidelity H against the mean photon number N and the rate of dark counts r, the range of r is between 10−8 and 10−2, N ranges from 1 to 10.
3.4. A combination of all realistic scenarios
On the basis of above consequents, we briefly analyze a scenario including all above realistic factors. Hereon, we let TA = TB = T, in turn, the probability of zero count takes the form:
where Ne = TηN is the effective mean photons triggering the Gm-APD. This indicates that the number of photons which contribute to sensitivity is Ne, instead of the input photon number N. In addition, this probability cannot reach its maximum due to the term e−r, while for a large N, its minimum approaches 0 suggesting visibility approximatively reach 100%.4. Proof-of-principle experiment for angular displacement estimation protocol
In the previous section, we discuss some realistic scenarios that may lead to degradation in estimation performance. The results lay the foundation for experimental studies and provide the basis for probability expression in a realistic scenario. In what follows, we demonstrate a proof-of-principle experiment and perform Bayesian estimation onto measuring data.
With regard to Bayesian estimation, the conditional probability––posteriori distribution––of measuring sample is the core. It is easy to obtain from the Bayesian theorem, which states that p(θ|W) p(W) = p(W|θ) p(θ), where p(θ) is the priori distribution defined in Eq. (7), and p(W) is the overall probability of measuring sample. Therefore, we can write this posteriori distribution as
where M is a positive integer meaning the number of trials [36], G is responsible for the normalization, it can be calculated byOne knows that Bayesian estimation is asymptotically unbiased. For M ≫ 1, the posteriori distribution in Eq. (15) can be recast as
where θ* represents the actual value of angular displacement. Furthermore, Mp (zero|θ*) and Mp (nonzero|θ*) are the numbers of zero and nonzero counts in the trials of M, respectively.In our experiment, by rotating the Dove prism, we record the number of zero count detected by the Gm-APD at several values of angular displacement, where ℓ = 2 is used. The mean and standard deviation for each sampling point are also provided. Then, using least-square method and imitating Eq. (14), we give a fit to the measuring outcome of zero count, as illustrated in Fig. 7. The detailed expression turns out to be
Compared with Eq. (14), the effective photon number is found to be Ne = 4.11. According to the definition of visibility [37], one finds that the visibility in our experiment is 96.7%. Furthermore, the full width at half maximum is 0.4224 radians, which is a super-resolved signal enhancement by a factor of 3.72 compared to the Rayleigh diffraction limit.
Fig. 7 Probability of zero count (normalized measuring counts) against angular displacement. The solid red dots are experimental measuring data, while the solid blue line is a fit to the data. For each error bar, its mean value is calculated from the trials of M = 40000, its standard deviation is calculated by dividing these trials into 20 sets (M = 2000 in each set).
However, the maximum of signal is less than 1, this means that destructive interference is not thorough. According to the model in Ref. [38], Eq. (18) can be rewritten as
where nb = 0.093, corresponding to exp (−nb) = 0.911, is background noise, including dark counts of the Gm-APD and the photons without participating in interference.The physical reasons behind this phenomenon are as follows. (i) The conversion efficiency of the spiral phase plate is less than 100%, and non-OAM beam is not modulated by Dove prism. (ii) The extinction ratios of polarizing beam splitters are non-unit. (iii) The linear polarization is changed slightly due to rotation of the Dove prism, it introduces a component which is perpendicular to the input polarization.
For an actual value θ*, we perform three sets of measurements, where the values of M are 200, 500, and 1000, respectively. Combining with Eqs. (14) and (16), in Fig. 8 we plot the posteriori distribution with different values of M. Figure 8 indicates that the posteriori distribution becomes centered against the actual value with increasing the values of M, as to other values of θ, the probabilities of zero count gradually approach 0. For a large M, the posteriori distribution can be approximately evolved into p(θ|θ*) = δ (θ*). That is, increasing the number of trials can enable the estimated value to approach the actual value. This phenomenon also proves that Bayesian estimation is asymptotically unbiased.
Fig. 8 The posteriori distribution p(θ|θ*) against angular displacement θ, where actual value θ* sits at 0.0698. The numbers of trials for blue dashed line, red dash-dotted line, and green dotted line are 200, 500, and 100; the corresponding numbers of zero count are 167, 419, and 841, respectively.
5. Conclusion
In summary, we reported on a tried-and-true binary strategy for angular displacement estimation based on fidelity appraisal in place of standard deviation metric. Two binary detection strategies-––Z detection and parity detection––were analyzed, the results indicated that the fidelity of Z detection has an overwhelming advantage compared to that of parity detection. Additionally, aiming at Z detection, we discussed the effects of several realistic scenarios on estimation protocol, containing transmission loss, detection efficiency, dark counts, and those which are a combination thereof. Regarding transmission loss, the scenario of identical losses in two paths can provide a greater fidelity than that of inconsistent losses when the total loss remains the same. An approximately positive correlated relationship between the detection efficiency and fidelity was found by analysis. We also showed that the effect of dark counts on the fidelity is almost negligible. Finally, a proof-of-principle experiment was carried out, and we performed Bayesian estimation onto measuring data. An enhanced super-resolving signal with a factor of 3.72 was shown, and the actual value of angular displacement can be precisely estimated with increasing the number of trials. Overall, the experimental results were in good agreement with the theoretical analysis.
Funding
National Natural Science Foundation of China (Grant No. 61701139).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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