In this paper, we propose a protocol for the estimation of angular displacement based upon orbital angular momentum and an SU(1,1)-SU(2) hybrid interferometer. This interferometer consists of an optical parametric amplifier, a beam splitter, and reflection mirrors; the balanced homodyne detection is used as the detection strategy. The results indicate that super-resolution and super-sensitivity can be achieved with an ideal scenario. Additionally, we study the effect of photon loss on resolution and sensitivity, and the robustness of our protocol is also discussed. Finally, the advantage of our protocol compared with an SU(1,1) protocol is demonstrated, and the merits of orbital angular momentum-enhanced protocol are summarized.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In recent years, apart from phase estimation, much attention has been paid to the estimation of angular displacement [1–7]. Angular displacement can be divided into two distinct physical scenarios: one is the rotation of polarization [3,5], and the other is the relative rotation of light axis [1,2,4,6,7]. The rotation of polarization is often caused by the birefringence effect of wave plates or the Faraday magneto-optical rotation effect of optical crystals. The rotation of the light axis corresponds to the rotation of prism or, equivalently, the rotation of the reference coordinate system .
For the first case, in the field of classical metrology, Malus’s Law and shot-noise limit (SNL) are two typical parametric representations. Therefore, much work has been carried out to improve such sensitivity [3,5,8,9]. It has been shown that the sensitivity can be increased by using N00N states  or two-mode squeezed vacuum  along with a quantum detection strategy (projective or parity measurement). Of the angular displacement-type measurements, including spin angular momentum (SAM)  and orbital angular momentum (OAM) , OAM performs better than SAM. This is because SAM forms a two-dimensional Hilbert space (spin up or spin down), whereas OAM space is inherently infinite dimensional.
Recently, OAM-based Heisenberg-limited estimation protocols for angular displacement have been achieved in both SU(1,1) and SU(2) interferometers [6, 7]. Owing to the difficulty in preparing exotic non-Gaussian states, such as N00N and entangled coherent states, with large photon numbers, the estimation sensitivities obtained using these quantum states are even inferior to those of high-intensity coherent and coherent pumped squeezed states [12,13]. Therefore, the estimation of angular displacement based upon the coherent states in an SU(2) interferometer has also been extensively studied with the sensitivity being limited by the SNL. On the other hand, some studies have shown that the sensitivity of a single-coherent-beam-stimulated two-mode squeezed state can surpass the SNL [14, 15]. In this paper, we propose an OAM-enhanced protocol that uses an SU(1,1)-SU(2) hybrid interferometer to achieve sub-shot-noise-limited sensitivity, that is better than both SU(1,1) and SU(2) interferometers.
The remainder of this paper is organized as follows. In Sec. 2, we briefly introduce the structure and working principle of hybrid interferometer. Then, we discuss the balanced homodyne detection with an ideal situation in Sec. 3. In Sec. 4, we study the effects of photon loss on resolution and sensitivity, additionally; we compare our protocol and the SU (1,1) protocol. Finally, we conclude our work with a summary in Sec. 5.
2. Fundamental principle and device
Consider a schematic of angular displacement estimation protocol whose main body is an SU(1,1)-SU(2) hybrid interferometer, as illustrated in Fig. 1. This interferometer can be achieved by using an optical parametric amplifier (OPA) to replace the first beam splitter (BS) in an SU(2) interferometer, or, equivalently, by using a BS to replace the second OPA in an SU(1,1) interferometer . The OPA and BS are used as the entry and exit gates of the device, respectively. A coherent state is injected into the hybrid interferometer, and the beam’s OAM degree of freedom is added by two spiral phase plates (SPPs). The difference in angular displacement φ between the two Dove prisms (DPs) is estimated in our study. This difference introduces a phase difference 2ℓφ between the two modes . In what follows, it is assumed that OAM quantum number ℓ is a positive number. After the above process is completed, the two modes are recombined at BS1, and any output port can be selected to implement a balanced homodyne detection.
Because OPA increases the number of photons, the calculation of the mean photon number N in our protocol is consistent with the SU (1,1) case ,
One can find that the two limits are increased by a factor of 2ℓ. This indicates that our protocol offers a huge advantage over the protocols that do not utilize OAM as it is easy to prepare an OAM beam with ℓ ≤ 10 in the laboratory. Under this situation, our protocol provides an order of magnitude increase over the non-OAM protocols with the same mean photon number.
From the perspective of metrology, the devices after two DPs can be regarded as measuring devices. Thus, our protocol can also be considered as a modified protocol in terms of an SU(1,1) interferometer. In turn, the ultimate precision provided by the quantum Cramér-Rao bound (QCRB), or quantum Fisher information (QFI) of our protocol is the same as that of an SU(1,1) interferometer. The QRCB for our protocol is found to be ,
3. Balanced homodyne detection
In this section, we analyze the balanced homodyne detection. This is a preeminent method for quantum noise detection by detecting phase quadrature or amplitude quadrature. It can also be used to achieve parity detection for Gaussian states . Take port A as an example, the detection operators have the following forms(10) and (11) can be derived from the transformation matrices in Appendix A.
We plot the output signal 〈X̂A〉 of balanced homodyne detection with φ and θ in Fig. 2. One can see that the signal changes ℓ times from −π/2 to π/2, i.e., our protocol has 2ℓ-fold super-resolution characteristic in every period (2π), similar to the N00N states. Furthermore, the signal waveform is also modulated by angle θ. In terms of definition of visibility ,Eqs. (9) and (13), we obtain the fluctuation of X quadrature,
To intuitively observe the behavior of sensitivity, in Fig. 3 the variation in sensitivity on balanced homodyne detection with φ is shown in Fig. 3. The results indicate that the sensitivity of balanced homodyne detection can outperform the SNL and achieve sub-shot-noise-limited sensitivity. In addition, compared with other quantum strategies, the variation in sensitivity curve is slow, indicating that quasi-optimal sensitivity can be obtained in the vicinity of optimal sensitivity. However, the optimal sensitivity in this strategy should satisfy the phase matching condition,
Figure 4 shows the effect of θ on the sensitivity. For simplicity, only the regions where the sensitivity is better than the SNL (sub-shot-noise-limited regions) are plotted. The results show 2ℓ optimal positions for sensitivity in our protocol. From Eq. (15), one can find that the optimal sensitivity appears at 2ℓφ = (2k + 1)π and 2ℓφ + θ = kπ + π/2. Thus, the optimum combination of solution is (2k + 1)π/2ℓ and θ = ±π/2.
To quantitatively analyze the degree of super-sensitivity in our protocol, we plot Fig. 5 with our sensitivity and QCRB. We can find that the sensitivity saturates the QCRB with respect to a large g. Additionally, with the increase in |α|2, the sensitivity approaches QCRB quickly. This trend can be observed from the pointed arrow in Fig. 5. Hence, |α|2 ≫ 1 and sinh2 g ≫ 1 are two main conditions for saturating the QCRB. This also indicates that balanced homodyne detection is a locally optimal strategy for our protocol. The globally optimal strategies work optimally for any φ, and the truncated SU(1,1) interferometers may further optimize our detection strategy [22,23]. For simplifying the discussion here, we leave it for a future work.
For the case of |α|2 ≫ 1 and sinh2 g ≫ 1, we also provide a theoretical proof that the optimal sensitivity of our protocol always saturates the QCRB, i.e., the equivalence between the minimum of Eq. (15) and Eq. (6), see Appendix B for details.
4. Analysis and discussion of realistic factor
In this section, the effect of a common realistic factor, photon loss, on the sensitivity of our protocol is evaluated. A simplified illustration of photon loss is shown in Fig. 6. Suppose that the loss is linear and occurs after the two DPs. Such a linear loss is usually simulated by placing two virtual beam splitters with arbitrary transmissivity in two arms of interferometer . The lost photons are reflected in the environment via the virtual beam splitters. In the following derivation, we assume that the transmissivity of the two virtual beam splitters is T. As the lost photons enter the environment, the number of modes considered increases from two to four, i.e., two environment ports are added. Hence, the transmission matrices change from four-by-four to eight-by-eight; the details are provided in Appendix A. According to the transformation relationships, we can obtain the resolution and sensitivity.
Under the situation when the system suffers from photon loss, the expectation value of X quadrature is calculated as follows:
Equations (12) and (19) show that the visibility is not affected by photon loss, i.e., 100% visibility can be maintained. The robustness of balanced homodyne detection is exhibited by no change in the signal. Figure 7 shows the sensitivity of homodyne detection with photon loss. One can find that our protocol is robust, for it can resist 38% photon loss in the case of g = 2, ℓ = 1, and |α|2 = 100. Moreover, a merit of this protocol is that the change in sensitivity curve is still slow.
More generally, Fig. 8 shows the relationship between maximum allowable loss and g with different values of |α|2. It can be seen that the maximum allowable loss has an optimal position with respect to fixed |α|2, and the optimal position moves to the right with the increase in |α|2 (arrow direction). In addition, with increasing |α|2, the maximum allowable loss also increases and gradually becomes saturated. A significant phenomenon is that irrespective of the value of |α|2, the maximum allowable loss ultimately approaches a fixed value when g reaches a threshold value. Overall, the universal conclusion is that our protocol has excellent robustness and can effectively resist photon loss (∼30–40%).
For an SU(2) interferometer fed by a coherent state, its optimal sensitivity is the SNL. This is inferior to our protocol, sub-shot-noise-limited optimal sensitivity. It should be noted that the same mean photon number is a prerequisite for this conclusion, namely, the mean photon number of coherent state in the SU(2) interferometer is equal to the photon number in Eq. (1). Here, a brief discussion for the advantage of our protocol over SU(1,1) interferometer is provided, also with the same mean photon number . For the case of lossless balanced homodyne detection, the optimal sensitivity of Eq. (15) can be rewritten asAppendix B. The sensitivity of an SU(1,1) interferometer with coherent and vacuum states as the inputs is given by : with NOPA = 2 sinh2 g. From Eq. (20), it can be inferred that even without OAM’s participation, i.e., by removing the factor 1/2ℓ arising from OAM, the sensitivity of our protocol is greater than the SU(1,1) protocol by a factor of under the condition that sinh2g ≃ cosh2g = G with a large g and |α|2 ≫ 1. For a coherent state input, our protocol achieves the QCRB that SU(1,1) wants to achieve without success. Then, the addition of OAM significantly increases the sensitivity. Overall, for a regular SU(1,1) without using OAM, its sensitivity equals of the sensitivity of a regular non-OAM hybrid interferometer, also, of the sensitivity of an OAM-enhanced hybrid interferometer. The enhancement factor originates from the quantum effect in the OPA, whereas the enhancement factor 2ℓ is a classical effect that originates from OAM.
Finally, the merits of OAM are summarized. It plays three significant roles in our protocol. (1) OAM acts as the linear amplifier for angular displacement to increase the estimation sensitivity in a single trial. (2) OAM extends the super-resolved output signal from a single peak to a 2ℓ-fold peak. (3) The multiple optimal positions in sensitivity provided by OAM shorten the scanning range to 1/2ℓ.
In summary, we present a protocol for estimating angular displacement using an OAM beam and a hybrid interferometer. The balanced homodyne detection is studied, and the results show that both super-resolution and super-sensitivity can be achieved in a lossless scenario. The output signal has 100% visibility, and we demonstrate that a sub-shot-noise-limited optimal sensitivity is reachable. The optimal sensitivity is saturated by QCRB when both |α|2 and sinh2 g are much larger than 1. Additionally, the effects of photon loss on resolution and sensitivity are evaluated. The visibility is maintained in the loss scenarios, and our protocol can resist photon loss of more than 30%. We discuss the advantage of our protocol compared to the SU(1,1) protocol, and the results show that the sensitivity of our protocol is improved by a factor of . Compared with the non-OAM protocols, the enhanced effect arising from OAM is reflected by a factor of in sensitivity. Finally, the merits of OAM are briefly summarized.
A. Transformation matrices of optical processes in the phase space
In this part of Appendix, we provide the transformation matrices of optical processes in the phase space. The matrices for OPA, angular displacement and BS are given by
For the situation of photon loss, the transformation matrices transforms from four-by-four to eight-by-eight, an increase in dimensionality because of the introduction of the environment modes. The specific forms are as follows:
B. The proof of equivalence between optimal sensitivity of Eq. (15) and QCRB (Eq. (6)), and the proof process of Eq. (20)
Finally, we consider Eq. (20) with the same approximation, the proof is given by
National Natural Science Foundation of China (61701139).
The authors declare that there are no conflicts of interest related to this article.
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