Abstract

Super-resolved angular displacement estimation is of crucial significance to the field of quantum information processing. Here we report an estimation protocol based on a Sagnac interferometer fed by a coherent state carrying orbital angular momentum. In a lossless scenario, through the use of parity measurement, our protocol can achieve a 4ℓ-fold super-resolved output with quantum number ℓ; meanwhile, a shot-noise-limited sensitivity saturating the quantum Cramér-Rao bound is reachable. We also consider the effects of several realistic factors, including nonideal state preparation, photon loss, and inefficient measurement. Finally, with mean photon number $\bar N=2.297$ and ℓ = 1 taken, we experimentally demonstrate a super-resolved effect of angular displacement with a factor of 7.88.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

It is generally known that an optical beam can carry two forms of angular momenta: spin angular momentum (SAM) and orbital angular momentum (OAM). SAM corresponds to the polarization of the beam, and angular momentum of each photon is $\sigma \hbar$, where $\sigma = + 1$ and $\sigma = - 1$ stand for the left- and right-handed polarization, respectively. OAM is associated with spiral wavefront of the beam, and each photon in the OAM beam carries angular momentum $\ell \hbar$ with quantum number $\ell$. The fact that a beam with spiral phase, $\exp \left ( {i\ell \theta } \right )$, is capable of carrying OAM was put forward by Allen and co-workers [1]. Since then, OAM has attracted a great deal of attention [24] and played a significant role in the fields of optical communications, optical imaging, and high-precision sensing [58].

Due to the characteristic of the spiral phase, an OAM state can take the part of ‘angular amplifier’, which converts an angular displacement $\theta$ into an amplified displacement $\ell \theta$ [9]. Over the past few years, angular displacement estimation has been widely analyzed in quantum process tomography [10] and weak measurements [1113]. Related to this, plenty of studies focus on super-resolved estimation protocols. Jha et al. showed an angular displacement estimation with two-photon N00N states carrying OAM [14]. Later, employing other inputs to estimate angular displacements is studied, such as entangled coherent states [15] and two-mode squeezed states [16,17]. In addition, some estimation protocols using nonlinear interferometers are proposed [1820].

Most of the aforementioned protocols are based on exotic quantum states. However, almost without exception these states are susceptible to photon loss. On the other hand, they have a flaw in high-photon preparation. For a lossy channel, these drawbacks may downplay the advantages arising from quantum resources. As a consequence, using coherent states may be the most effective way in a realistic scenario. In this work, we experimentally demonstrate a super-resolved estimation protocol for angular displacements by inputting an OAM coherent state. We utilize parity measurement and a Sagnac interferometer (SI) instead of a conventional Mach-Zehnder interferometer (MZI). The effects of several realistic factors are considered, and the advantages of our protocol over previous protocols are discussed.

The remainder of this paper is organized as follows. In Sec. 2, we introduce the fundamental principle and measurement strategy of our protocol. Section 3 focuses on studying the effects of several realistic factors on our protocol. In Sec. 4, we discuss the fundamental sensitivity limit of our protocol by calculating quantum Fisher information (QFI), and compare it with conventional protocols. The experimental results are showed and discussed in Sec. 5. Finally, we summarize our work in Sec. 6.

2. Fundamental principle and measurement strategy of estimation protocol

To begin with, let us consider an angular displacement estimation protocol, of which the setup is an SI consisting of three mirrors and a 50-50 beam splitter arranged in a square, as illustrated in Fig. 1. A coherent state is generated from a laser, and its OAM degree of freedom is added by a phase-only spatial light modulator. A polarizer is used to filter the polarization which is not suitable for the spatial light modulator, and an iris is responsible for retaining the first-order diffraction. Regarding quantum number $\ell$, the input state can be described as $\left | {{\psi _{\textrm{in}}}} \right \rangle = {\left | {{\alpha _\ell }} \right \rangle _A}{\left | 0 \right \rangle _B}$, where ${\alpha _\ell }=\sqrt N$ with $N$ being the mean photon number of the coherent state. Then, the input enters the SI and is divided into two beams. Here we assume that clockwise and counterclockwise directions in the SI are path $A$ and path $B$, respectively. Upon leaving the beam splitter, the state becomes $\left | {{\psi _1}} \right \rangle = {\left | {{{{\alpha _\ell }} \mathord {\left / {\vphantom {{{\alpha _\ell }} {\sqrt 2 }}} \right . } {\sqrt 2 }}} \right \rangle _A}{\left | {{{i{\alpha _\ell }} \mathord {\left / {\vphantom {{i{\alpha _\ell }} {\sqrt 2 }}} \right . } {\sqrt 2 }}} \right \rangle _B}$. Subsequently, two beams pass through a Dove prism [21] with an angular displacement $\varphi$. After such an evolution process, the state can be expressed as $\left | {{\psi _2}} \right \rangle = {\left | {{{{\alpha _\ell }{e^{i2\ell \varphi }}} \mathord {\left / {\vphantom {{{\alpha _\ell }{e^{i2\ell \varphi }}} {\sqrt 2 }}} \right . } {\sqrt 2 }}} \right \rangle _A}{\left | {{{i{\alpha _\ell }{e^{ - i2\ell \varphi }}} \mathord {\left / {\vphantom {{i{\alpha _\ell }{e^{ - i2\ell \varphi }}} {\sqrt 2 }}} \right . } {\sqrt 2 }}} \right \rangle _B}$. Finally, two beams go through the beam splitter again, and the output has the following ket representation $\left | {{\psi _{\textrm{out}}}} \right \rangle = {\left | {i{\alpha _\ell }\cos \left ( {2\ell \varphi } \right )} \right \rangle _A}{\left | { - i{\alpha _\ell }\sin \left ( {2\ell \varphi } \right )} \right \rangle _B}$.

 figure: Fig. 1.

Fig. 1. Schematic of the angular displacement estimation protocol. The full names of the abbreviations in the figure: L, laser; P, polarizer; SLM, spatial light modulator; I, iris; BS, beam splitter; DP, Dove prism; M, mirror; PNRD, photon-number-resolving detector.

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Compared with an MZI-based protocol, our protocol has two main advantages: (i) Two beams in an SI are more likely to experience identical optical paths since SIs are self-balanced; (ii) The effective angular displacement in an SI is twice as much as that in an MZI [22].

In what follows, we turn our attention to measurement strategy. Parity measurement was originally discussed by Bollinger $et$ $al.$ for enhanced frequency measurement with an entangled state [23]; subsequently, Gerry and Campos applied it to optical interferometry [24,25]. Generally, the implementation of parity measurement requires a photon-number-resolving detector [2629]. In this strategy, the results are assigned as $+ 1$ or $- 1$ for even counts or odd counts, respectively. Therefore, the parity operator for output port $B$ can be written as ${\hat \Pi } = \exp ( {i\pi {{\hat b}^\dagger }\hat b} )$.

In two-mode Fock basis, the output state is recast as

$$\left| {{\psi _{\textrm{out}}}} \right\rangle = {e^{ - \frac{N}{2}}}\sum_{n = 0}^\infty {\frac{{{{\left[ {i{\alpha _\ell}\cos \left( {2\ell\varphi } \right)} \right]}^n}}}{{\sqrt {n!} }}} \sum_{m = 0}^\infty\frac{{{{\left[ { - i{\alpha _\ell}\sin \left( {2\ell\varphi } \right)} \right]}^m}}}{{\sqrt {m!} }}\left| {n,m} \right\rangle.$$
Further, the probability of simultaneously detecting $n$ photons at port $A$ and $m$ ones at port $B$ is found to be
$$P\left( {n,m} \right) = \frac{{{e^{ - {N}}}}}{{n!m!}}{\left[ {{N}{{\cos }^2}\left( {2\ell\varphi } \right)} \right]^n}{\left[ {{N}{{\sin }^2}\left( {2\ell\varphi } \right)} \right]^m}.$$
As a consequence, the conditional probabilities ${P_\textrm{even}}$ and ${P_\textrm{odd}}$ for port $B$ can be calculated through a series sum of $P\left ( {n,m} \right )$ over the port $A$,
$${P_{\textrm{even}}} = \frac{1}{2}\left\{ {1 + \exp \left[ { - 2{N}{{\sin }^2}\left( {2\ell\varphi } \right)} \right]} \right\},$$
$${P_{\textrm{odd}}} = \frac{1}{2}\left\{ {1 - \exp \left[ { - 2{N}{{\sin }^2}\left( {2\ell\varphi } \right)} \right]} \right\}.$$
In terms of the definition of parity operator, we can obtain its expectation value,
$$\hat {\left\langle { \Pi } \right\rangle} = \exp \left[ { - 2N{{\sin }^2}\left( {2\ell\varphi } \right)} \right].$$
Note that parity measurement is a binary strategy, the classical Fisher information can be written as
$${{\cal F}} = \frac{1}{{{P_{{\textrm{even}} }}}}{\left( {\frac{{\partial {P_{\textrm{even}}}}}{{\partial \varphi }}} \right)^2} + \frac{1}{{{P_{\textrm{odd}}}}}{\left( {\frac{{\partial {P_{\textrm{odd}}}}}{{\partial \varphi }}} \right)^2}.$$
The relationship between classical Fisher information and sensitivity is $\Delta \varphi = 1/\sqrt {\cal F}$. Further, the sensitivity can be expressed as follows
$$\Delta \varphi = \frac{{\sqrt {\langle {{\hat\Pi ^2}} \rangle - {{\langle \hat\Pi \rangle ^2}}} }}{{\left| {{{\partial \langle \hat \Pi \rangle } \mathord{\left/ {\vphantom {{\partial \langle \hat \Pi \rangle } {\partial \varphi }}} \right. } {\partial \varphi }}} \right|}},$$
where the property of parity operator, $\langle {{{\hat \Pi }^2}}\rangle = 1$, is used. Equation (7) is referred to as error propagation formula.

According to Eq. (7), the sensitivity of our protocol is given by

$$\Delta \varphi = \frac{{\sqrt {\exp \left[ {4{N}{{\sin }^2}\left( {2\ell\varphi } \right)} \right] - 1} }}{{\left| {4\ell{N}\sin \left( {4\ell\varphi } \right)} \right|}}.$$
With the first-order approximation, the sensitivity will sit at its minimum when $\varphi$ approaches 0,
$$\Delta {\varphi _{\min }} = {\left. {\frac{{\sqrt {1 + 4N{{\sin }^2}\left( {2\ell\varphi } \right) - 1} }}{{\left| {4\ell N\sin \left( {4\ell\varphi } \right)} \right|}}} \right|_{\varphi \to 0}} = \frac{1}{{4\ell\sqrt N }}.$$
For intuitively observing the variation in resolution caused by mean photon number $N$ and quantum number $\ell$, we plot Fig. 2. One can find that the number of oscillating output fringes increases with increasing $\ell$; meanwhile, each fringe gets narrower as the increase of $N$. Moreover, the maximum sits at 1 while the minimum is close to 0 for a large $N$, indicating that the visibility of output approaches 100%. Here the definition of visibility refers to [30]
$$V = \frac{{{\hat{\left\langle {\Pi } \right\rangle }_{\max }} - {\hat{\left\langle { \Pi } \right\rangle }_{\min }}}}{{{\hat{\left\langle { \Pi } \right\rangle }_{\max }} + {\hat{\left\langle { \Pi } \right\rangle }_{\min }}}}.$$

 figure: Fig. 2.

Fig. 2. (a) The output of parity measurement against angular displacement with different quantum numbers, where $N = 10$. (b) The output of parity measurement against angular displacement with different mean photon numbers, where $\ell = 3$.

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In Fig. 3, we show the super-resolution criterion, full widths at half maximum (FWHMs), with different values of $N$ and $\ell$. Only when the FWHM of an output is narrower than that of a classical output (coherent state input and intensity measurement) can the super-resolution be obtained. Figure 3 indicates that the increase of $N$ or $\ell$ can provide an enhanced resolution; a more apparent enhancement is reachable when $N$ and $\ell$ increase simultaneously. Regarding the sensitivity, Eq. (9) suggests that the optimal sensitivity of our protocol is shot-noise limit. The factor $4\ell$ in Eq. (9) stands for a classical amplification effect, which amounts to the increase of the number of trials.

 figure: Fig. 3.

Fig. 3. The FWHM of parity measurement against the mean photon number.

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3. Effects of realistic factors

In general, a practical system contains nonideal devices, and is inevitably immersed in its surrounding environment. These factors may lead to the degradation of system performance. In this section, we analyze the effects of several realistic factors on the resolution and sensitivity.

3.1 Nonideal state preparation

We start off with nonideal state preparation. In this scenario, the input must be described by a density matrix, rather than a state vector [31,32]. Let $\eta$ be the conversion efficiency of the spatial light modulator; accordingly, we can write the input density matrix as the following form

$${\rho _{\textrm{in}}} = \left[ {\eta \left| {{\alpha _\ell}} \right\rangle \left\langle {{\alpha _\ell}} \right| + \left( {1 - \eta } \right)\left| {{\alpha _0}} \right\rangle \left\langle {{\alpha _0}} \right|} \right] \otimes \left| 0 \right\rangle \left\langle 0 \right|.$$
Based on the evolution process mentioned, the reduced output density matrix for path $B$ can be obtained. Further, the expectation value of parity operator is equal to
$${\hat{\left\langle { \Pi } \right\rangle} _{1}} = \eta\exp \left[ { - 2 {N}{{\sin }^2}\left( {2\ell\varphi } \right)} \right]+1-\eta.$$
Using the first-order approximation again, we have the optimal sensitivity,
$$\Delta {\varphi _1} = {\left. {\frac{{\sin \left( {2\ell\varphi } \right)\sqrt {1 - \eta N{{\sin }^2}\left( {2\ell\varphi } \right)} }}{{\left| {2\ell\sqrt {\eta N} \sin \left( {4\ell\varphi } \right)} \right|}}} \right|_{\varphi \to 0}} = \frac{1}{{\sqrt \eta }}\frac{1}{{4\ell\sqrt N }}.$$
Eq. (11) suggests that only those photons carrying OAM are imprinted with information on the estimated angular displacement. Related to this, there is no interference between non-OAM photons. As a consequence, the minimum of output increases and the visibility reduces. In addition, the sensitivity degenerates by a factor of ${\sqrt \eta }$.

3.2 Photon loss

We next take into account the effect of a type of inevitable realistic factor, photon loss. In general, the way to theoretically simulate photon loss is to insert a virtual beam splitter in the interference loop, the transmission ratios with respect to two paths are ${{T_A}}$ and ${{T_B}}$ [33,34]. Accordingly, the parameters $L_A = 1-T_A$ and $L_B=1-T_B$ can be used to represent two path losses. On the basis of this theory, the output state for path $B$ reduces to $\left | {{\psi _{\textrm{out}}}} \right \rangle _B = {| {{{{\alpha _\ell }( {\sqrt {{T_B}} {e^{ - i2\ell \varphi }} - \sqrt {{T_A}} {e^{i2\ell \varphi }}} )}}/{2}} \rangle }$. Further, the expectation value of parity operator turns out to be

$${\hat{\left\langle { \Pi } \right\rangle} _{2}} = {\kern 1pt} \exp \left[ {N\sqrt {{T_A}{T_B}} \cos \left( {4\ell\varphi } \right)}{ - \frac{N}{2}\left( {{T_A} + {T_B}} \right)} \right],$$
and the corresponding sensitivity can be calculated as
$${\Delta \varphi _{2}} = \frac{\sqrt {\exp \left\{ {N\left[ {{T_A} + {T_B} - 2\sqrt {{T_A}{T_B}} \cos \left( {4\ell\varphi } \right)} \right]} \right\} - 1}} { {{\left| {4\ell\left( {{T_A} + {T_B}} \right)N\sin \left( {4\ell\varphi } \right)} \right|}}}.$$
From Eq. (14) we can find that only if the condition ${T_A} ={T_B}$ is satisfied does the maximum of output sit at 1. This is consistent with the interference condition in classical optics. In our protocol, ${T_A} \approx {T_B}$ is satisfied since two beams experience the same spatial path in an SI. Figure 4(a) manifests that the FWHM broadens with the decrease of transmission ratio, while the visibility remains the same. As for the sensitivity, Fig. 4(b) indicates that, for the identical total loss, the optimal sensitivity is reachable when photon losses of two paths are the same.

 figure: Fig. 4.

Fig. 4. (a) The output of parity measurement against angular displacement, where $N = 10$ and $\ell = 3$. (b) The optimal sensitivity of parity measurement against two path losses, where $N = 10$ and $\ell = 3$.

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3.3 Inefficient measurement

Detection efficiency, response-time delay, and dark counts are three typical nonideal factors for photon-number-resolving detectors [35]. Here, we study the effects originating from these factors.

3.3.1 Detection efficiency

In general, there is no guarantee that a detector keeps 100% efficiency. This process is also simulated by inserting a virtual beam splitter in front of the detector, where the transmission ratio is $\kappa$ [36], also known as detection efficiency.

Through the calculation, the reduced output state for path $B$ can be rewritten as ${\left | {{\psi _{\textrm{out}}}} \right \rangle _B} = | { - i\sqrt \kappa {\alpha _\ell }\sin \left ( {2\ell \varphi } \right )} \rangle$, and the expectation value of parity operator is

$${\hat {\left\langle {\Pi } \right\rangle} _{3}} = \exp \left[ { - 2\kappa {N}{{\sin }^2}\left( {2\ell\varphi } \right)} \right].$$
One can find that this equation is the same as Eq. (14) with $T_A=T_B=\kappa$. This suggests that the effect of detection efficiency is identical with that of photon loss. Therefore, the previous conclusions are still applicable. This phenomenon stems from the fact that coherent states can maintain their own photon distributions with linear photon loss. That is, a lossy SI fed by a coherent state completely equals a lossless SI fed by a weaker coherent state [37].

3.3.2 Response-time delay and dark counts

Of the practical measurements, response-time delay and dark counts also affect the performance of detector. The existence of response-time delay results in the increase of rate of dark counts. A thoughtful discussion about the effect of dark counts on output with parity measurement was proposed in [38]. The expectation value of parity operator is found to be

$${\hat {\left\langle {\Pi } \right\rangle} _{4}} = {e^{ - 2r}}\hat{\left\langle { \Pi } \right\rangle}$$
with $r$ being the rate of dark counts. Based on Eq. (17) and error propagation, the sensitivity can be calculated. Under the current technologies, the value of $r$ is generally between ${10^{ - 8}}$ and ${10^{ - 3}}$.

The increase of rate of dark counts stemming from response-time delay is generally less than one order of magnitude. Hence, we consider the following scenarios: $r={10^{ - 3}}$, only dark counts; $r={10^{ - 2}}$, dark counts along with response-time delay. The results in Fig. 5 exhibit that the degeneration of sensitivity is slight in the presence of dark counts; however, the degeneration of sensitivity will become obvious when response-time delay and dark counts exist simultaneously.

 figure: Fig. 5.

Fig. 5. The sensitivity of parity measurement against angular displacement with different rates. Where the curve of $r = 0$ is the ideal curve, the curves of $r = 10^{-3}$ and $r = 10^{-2}$ respectively correspond to the scenarios: dark counts, and dark counts along with response-time delay.

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4. Analysis of fundamental sensitivity limit

In Sec. 2, we discuss the optimal sensitivity of our protocol, whereas the fundamental sensitivity limit over all possible positive operate valued measures (POVMs) is not given. Here, from the perspective of the QFI, we compare our protocol and a conventional MZI-based protocol.

The current angular displacement estimation protocols using linear interferometers can be divided into two configurations: SI configuration and MZI configuration, as shown in Fig. 6. For the above two protocols, the rotation of Dove prism can be described by the operators ${\hat U_{\varphi 1}} = \exp ( {i4\ell {{\hat J}_z}\varphi })$ and ${\hat U_{\varphi 2}} = \exp \left ( {i2\ell {{\hat n}_a}\varphi } \right )$, respectively. The operator for a 50-50 beam splitter is ${\hat U_{\textrm{BS}}} =\exp ( {i{\pi }{{\hat J}_x}/2})$, where

$${{\hat J}_x} = \frac{1}{2}( {{\hat a^\dagger }\hat b + \hat a{\hat b^\dagger }}),$$
$${{\hat J}_y} = - \frac{i}{2}( {{\hat a^\dagger }\hat b -\hat a{\hat b^\dagger }} ),$$
$${{\hat J}_z} = \frac{1}{2}( {{\hat a^\dagger }\hat a - {\hat b^\dagger }\hat b})$$
are the angular momentum operators in the Schwinger representation [39], satisfying the cyclic commutation relations for Lie algebra of SU(2) group: $[ {{{\hat J}_x} {\kern 1pt}, {{\hat J}_y}} ] = i{\hat J_z}$, $[ {{{\hat J}_y}{\kern 1pt} , {\kern 1pt} {{\hat J}_z}}] = i{\hat J_x}$, and $[ {{{\hat J}_z}{\kern 1pt} ,{\kern 1pt}{{\hat J}_x}}] = i{\hat J_y}$. In addition, the input density matrix can be written as ${\rho _{\textrm{in}}} = {\rho _a} {\otimes} {\rho _b}$ with ${\rho _a} = \left | {{\alpha _\ell }} \right \rangle \left \langle {{\alpha _\ell }} \right |$ and ${\rho _b} = \left | 0 \right \rangle \left \langle 0 \right |$. Here we define the counterclockwise and clockwise paths as mode $a$ and mode $b$, respectively.

 figure: Fig. 6.

Fig. 6. Diagram of two angular displacement estimation protocols: a Sagnac interferometer and a Mach-Zehnder interferometer. The full names of the abbreviations in the figure: DP, Dove prism; D, detector; BS, beam splitter; SI, Sagnac interferometer; MZI, Mach-Zehnder interferometer.

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In accordance with the above analysis, we can give the QFI calculations for two scenarios. As for an SI-based protocol, the output density matrix evolves into ${\rho _{\textrm{out}}} = {\hat U_{\varphi 1}}{\hat U_{\textrm{BS}}}{\rho _{\textrm{in}}}\hat U_{\textrm{BS}}^\dagger \hat U_{\varphi 1}^\dagger$. We can obtain the estimator $R$ in terms of the equation ${{\partial {\rho _{\textrm{out}}}} \mathord {\left / {\vphantom {{\partial {\rho _{\textrm{out}}}} {\partial \varphi }}} \right . } {\partial \varphi }} = { - i[ {{\rho _{\textrm{out}}},{\kern 1pt} R} ]}$. For pure state inputs, the QFI is simplified as $4{\langle {{\Delta ^2}R} \rangle _{\textrm{in}}}$ [4042]. Therefore, the QFI of an SI-based protocol is calculated as

$${{\cal F}_{\textrm{S}}} = 4[ {\left\langle \psi \right|{{( {4\ell{{\hat J}_z}} )^2}}\left| \psi \right\rangle - \left\langle \psi \right|4\ell{{\hat J}_z}{{\left| \psi \right\rangle }^2}} ] = 16{\ell^2}{N},$$
where the formula ${\hat U_{\textrm{BS}}}\left | \alpha \right \rangle \left | 0 \right \rangle =| \alpha /{\sqrt 2 } \rangle | i \alpha /{\sqrt 2 } \rangle \equiv | \psi \rangle$ is used. The relationship between the optimal sensitivity $\Delta \varphi _\textrm{min}$ and QFI is $\Delta \varphi _\textrm{min} = 1/{\sqrt {{\cal F}} }$. Further, the optimal sensitivity of an SI-based protocol is found to be $\Delta {\varphi _{\textrm{SI}}} = 1/{4\ell \left | {{\alpha _\ell }} \right |}$.

For an MZI-based protocol, via a similar approach, we have

$${{\cal F}_{\textrm{M}}} = 4[ {\left\langle \psi \right|{{( {2\ell{{\hat n}_a}} )}^2}\left| \psi \right\rangle - \left\langle \psi \right|2\ell{{{\hat n}_a}}{{\left| \psi \right\rangle }^2}}]= 8{\ell^2}{N}.$$
It is obvious that the QFI of an SI-based protocol is superior to that of an MZI-based one. Furthermore, an intriguing and perplexing phenomenon with respect to ${{\cal F}_\textrm{M}}$ is that the optimal sensitivity corresponding to Eq. (22) is $\Delta {\varphi _{\textrm{MZI}}} =1 / {2\sqrt 2 \ell \left | {{\alpha _\ell }} \right |}$. After removing the factor 2$\ell$ originating from OAM, this implies a sub-shot-noise-limited sensitivity. In order to solve this confusion, we can use the phase-averaging approach [4345] to recalculate the sensitivity limit. This approach can give a lowdown on sensitivity limit with only using input resources. A simplified understanding is to disrupt the input state into a mixed state losing all phase references. Using this approach, the QFI of an MZI-based protocol is given by ${{\cal F}_{\bar \rho }} = 4{\ell ^2}N$ (see Appendix A for details). This QFI indicates the sensitivity limit is the shot-noise limit. Ascertaining the special additional resources which can assist an MZI-based protocol to achieve the sensitivity in Eq. (22) is a meaningful and challenging work, for many practical measurements can be classified as the MZI configuration. Overall, our protocol is more sensitive to angular displacements, and its sensitivity is twice as much as that of an MZI-based protocol.

5. Proof-of-principle experiment

As the last part of the work in this paper, we perform a proof-of-principle experiment with $\ell =1$. The working principle and measuring results about the photon-number-resolving detector are provided in Appendix B. Here we provide some details about our experiment.

At the preparation stage, the coherent state is generated from a pulse laser working at 532nm. The repetition rate is set by the signal generator and $f=2\textrm{kHz}$ is used in our experiment. We use a set of attenuator to control and reduce the intensity of the coherent state. The polarization is prepared by means of a half wave plate and a Glan-Taylor prism. Then the coherent state is sent to a phase-only spatial light modulator (HOLOEYE, LETO-VIS-009). The spatial light modulator shows a 1-fold spiral phase mask along with a blazed grating, and an iris retains the first-order diffraction.

At the measurement stage, we directly measure the output beam without demodulation of OAM. Since the photon-number-resolving detector (HAMAMATSU, C14455-1550GA) is based on spatial coupling, we focus the output beam on the detector with a lens. The sampling time for each measured point is 0.5s, and the number of trials is 1000.

We show the experimental results in Fig. 9(a), the mean and standard deviation of each measured point are calculated from 1000 trials. Additionally, we fit the expectation value of output in terms of experimental data,

$$\hat{\left\langle { \Pi } \right\rangle} = 0.9507\exp \left\{ { - 4.594{{\sin }^2}\left[ {2\left( {\varphi - 0.7022} \right)} \right]} \right\}.$$
This equation implies that the mean photon number is $\bar N=2.297$, and the visibility of output is 98%. Note that here $\bar N=T \kappa N$ is the effective number of photons arriving at the detector, instead of input photon number $N$. That is, the effect of photon loss is not observed in Eq. (23).

By calculating the FWHM, the experimental results demonstrate that our protocol has an enhanced resolution with a factor of 7.88. Moreover, the Eq. (23) can be recast as

$$\hat {\left\langle { \Pi } \right\rangle} = \exp \left[ { - 4.594{{\sin }^2}\left( {2\varphi } \right) }-0.0506 \right]$$
by implementing translation to Fig. 7(a). According to Eq. (24), the rate $r$ in the experiment is 0.0253, which may include dark counts and background noise. The maximum is less than 1 as a result of these realistic factors, as shown in Fig. 7(a).

 figure: Fig. 7.

Fig. 7. Experimental data against angular displacement with $\ell = 1$. (a) The blue line is a fit to the output. Error bars are one standard deviation due to propagated Poissonian statistics. (b) The red line is the sensitivity deduced from the fit of output, blue dots are the sensitivities calculated from the experimental data, and the black dashed line is the shot-noise limit defined in accordance with $\bar N$.

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Figure 7(b) presents the sensitivities calculated from the experimental data. The results mean that the sensitivities are in agreement with the theoretical analysis and the fit to output. In addition, the optimal sensitivity is slightly inferior to the shot-noise limit due to noise photons. This is similar to the discussion about dark counts in Sec. 3.

Finally, we would like to point out that our protocol is not only suitable for a low-intensity output. There are two alternative measurement schemes for a bright output. The first scheme is to utilize time-multiplexing technique [26,46]. The basic idea of this technique is to divide the output pulse into $M$ separated pulses. These pulses are assigned to $M$ fibers of different transmission lengths, and the transmission length of the $m$-th fiber is ${L_m} = L + m\Delta L$. In general, there are two restrictions as follows: ${{\Delta L} \mathord {\left /{\vphantom {{\Delta L} c}} \right .} c} \gg \tau$ and ${{{L_M}} \mathord {\left / {\vphantom {{{L_N}} c}} \right . } c} + \tau \ll \Delta t$, where $\tau$ is dead time of the APD, $c$ is transmission speed, and $\Delta t$ is time interval between two input laser pulses. The total photon number is given by a processing system, which accumulates the number of photons measured by each fiber. The second scheme is based on balanced homodyne measurement. Since the input of our protocol is a Gaussian state, the parity of output can be obtained by balanced homodyne measurement rather than a photon-number-resolving detector [47]. This is due to the fact that the parity of output is related to its Wigner function, and balanced homodyne measurement can be used to reconstruct this function.

6. Conclusions

In conclusion, we have proposed an SI-based protocol, which can obtain super-resolved angular displacement estimation using parity measurement. In a lossless scenario, by inputting a coherent state carrying OAM, we can obtain 4$\ell$-fold output and shot-noise-limited sensitivity. The resolution and sensitivity can be improved by increasing the mean photon number or quantum number. We also discuss the effects of several realistic factors on our estimation protocol. Nonideal preparation efficiency brings the deterioration in the resolution, visibility, and sensitivity. Regarding photon loss, with identical total loss, the optimal sensitivity will be obtained when the two path losses are the same. The effects of dark counts and response-time delay on the sensitivity are unconspicuous, and the resolution is not affected by them. Additionally, the fundamental sensitivity limit of our protocol and that of an MZI-based protocol are given by calculating the QFI. The results suggest that our sensitivity is twice as much as sensitivity of an MZI-based protocol. Finally, a proof-of-principle experiment is performed, the experimental data are in agreement with the theoretical analysis. For the mean number of measured photons $\bar N=2.297$, we realize nearly shot-noise-limited sensitivity and a super-resolved output with a factor of 7.88.

Appendices

A. QFI of the MZI-based protocol using phase-averaging approach

In this part of Appendix, we give a detailed process for the QFI calculation with using phase-averaging approach. In this framework, phase initialization is required for the input density matrix, i.e.,

$${{\bar \rho }_1} = \frac{1}{{2\pi }}\int_0^{2\pi } {\exp \left( {i\delta {{\hat n}_a}} \right)} {\rho _a} \otimes {\rho _b}\exp \left( { - i\delta {{\hat n}_a}} \right)d\delta = \sum_{n = 0}^\infty {{p_n}\left| {{n}} \right\rangle \left\langle {{n}} \right| \otimes \left| 0 \right\rangle \left\langle 0 \right|}.$$
Where ${p_n} = {{{{N}^{n}}\exp \left ( { - {N}} \right )} / n}!$ is the probability of emerging $n$ photons in the coherent state $\left | {{\alpha _\ell }} \right \rangle$. It is easy to find that all off-diagonal elements of density matrix are zero, suggesting that the coherence information is erased.

Then the density matrix passes through the first beam splitter and becomes

$${{\bar \rho }_2}= {{\hat U}_{\textrm{BS}}}{{\bar \rho }_1}\hat U_{\textrm{BS}}^\dagger = \sum_{n = 0}^\infty {p_n}\sum_{m = 0}^n {C_n^m } \left| {{n} - {m}} \right\rangle \left\langle {{n} - {m}} \right| \otimes \left| {{m}} \right\rangle \left\langle {{m}} \right| ,$$
where $C_n^m$ is the binomial coefficient. In terms of the orthogonality of Fock state, $\left \langle {n} {\left | {\vphantom {n m}} \right . }{m} \right \rangle = {\delta _{nm}}$, the QFI of entire mixed state equals the sum of that of each Fock state according to weight factor ${p_n}$. For a two-mode Fock state input $\left | {{n}} \right \rangle \left | 0 \right \rangle$ and a unitary evolution process ${\hat U_{\varphi 2}}{\hat U_\textrm{BS}}$, the corresponding QFI can be calculated as
$${{\cal F}_{\textrm{Fock}}}= 4\left[ {\left\langle {{{\hat U}_{\textrm{BS}}}{{\left( {2\ell{{{\hat n}_a}}} \right)}^2}\hat U_{\textrm{BS}}^\dagger } \right\rangle - {{\left\langle {{{\hat U}_{\textrm{BS}}}\left( {2\ell{{{\hat n}_a}}} \right)\hat U_{\textrm{BS}}^\dagger } \right\rangle }^2}} \right]= 4{\ell^2}n.$$
The expectation values are taken over the state $\left | {{n}} \right \rangle \left | 0 \right \rangle$, the unitary property of the operator ${\hat U_{\textrm{BS}}}$ and Baker-Hausdorff lemma $\exp ( {-i{\pi }{{\hat J}_x}/2}){\hat {a}^\dagger }\hat a\exp ( {i{\pi }{{\hat J}_x}/2}) = ( {\hat {a}^\dagger }\hat a + {\hat {b}^\dagger }\hat b ) /2 - {\hat J_y}$ are used in the calculation. Consequently, the QFI of the input density matrix in Eq. (25) goes to
$${{\cal F}_{\bar \rho }} = \sum_{n = 0}^\infty {{p_n}4{\ell^2}} n = 4{\ell^2}N.$$
Note that the above result is the shot-noise limit, that is, the optimal sensitivity of an MZI-based protocol fed by a coherent state is the shot-noise limit without additional resources.

B. Working principle and measuring results of the photon-number-resolving detector

The photon-number-resolving detector used in the experiment is a Geiger mode avalanche photodiode (Gm-APD) array. Each APD only responses to the presence or absence of photons at one of the output ports, i.e., there is no knowledge of exact photon number. For low-intensity output, it is a small probability event that multiple photons trigger the same APD unit. Therefore, the total photon number in each measurement is the sum of all APD counts. As can be seen from Fig. 8(a), to each count there corresponds an analog voltage of $\sim$0.02V. We calculate the mean photon number from the experimental output, and provide the Poissonian distribution with the same mean photon number. The theoretical and experimental probability distributions are shown in Fig. 8(b). We take advantage of the credibility defined as $H = \sum \nolimits _i {\sqrt {{x_i}{y_i}} }$ to quantify the similarity between the experimental and theoretical probability distributions in Fig. 8(b). The calculation result is $H=0.9914$, implying that the detector has superb credibility.

 figure: Fig. 8.

Fig. 8. (a) The analog voltage signals displayed by oscillograph, and each signal is converted from single statistical trigger counts. (b) The probability distribution of output photon state and a fit based on Poissonian distribution.

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Funding

National Natural Science Foundation of China (61701139).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]  

2. N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef]  

3. J. Tabosa and D. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999). [CrossRef]  

4. M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002). [CrossRef]  

5. Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014). [CrossRef]  

6. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014). [CrossRef]  

7. G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012). [CrossRef]  

8. O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019). [CrossRef]  

9. V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013). [CrossRef]  

10. X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2(6), 510–516 (2015). [CrossRef]  

11. O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014). [CrossRef]  

12. G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016). [CrossRef]  

13. L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015). [CrossRef]  

14. A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011). [CrossRef]  

15. Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017). [CrossRef]  

16. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, C. You, S. Adhikari, J. P. Dowling, and Y. Zhao, “Orbital-angular-momentum-enhanced estimation of sub-Heisenberg-limited angular displacement with two-mode squeezed vacuum and parity detection,” Opt. Express 26(13), 16524–16534 (2018). [CrossRef]  

17. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Heisenberg-scaling angular displacement estimation with tunable squeezed bell states,” J. Opt. 21(3), 035201 (2019). [CrossRef]  

18. J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017). [CrossRef]  

19. J.-D. Zhang, C.-F. Jin, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Super-sensitive angular displacement estimation via an SU(1,1)-SU(2) hybrid interferometer,” Opt. Express 26(25), 33080–33090 (2018). [CrossRef]  

20. J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018). [CrossRef]  

21. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef]  

22. Z. Zhang, T. Qiao, K. Ma, L. Cen, J. Zhang, F. Wang, and Y. Zhao, “Ultra-sensitive and super-resolving angular rotation measurement based on photon orbital angular momentum using parity measurement,” Opt. Lett. 41(16), 3856–3859 (2016). [CrossRef]  

23. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996). [CrossRef]  

24. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000). [CrossRef]  

25. C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005). [CrossRef]  

26. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004). [CrossRef]  

27. L. Cohen, D. Istrati, L. Dovrat, and H. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014). [CrossRef]  

28. O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019). [CrossRef]  

29. C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

30. J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008). [CrossRef]  

31. T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011). [CrossRef]  

32. J. Zhang, Z. Zhang, L. Cen, M. Yu, S. Li, F. Wang, and Y. Zhao, “Effects of imperfect elements on resolution and sensitivity of quantum metrology using two-mode squeezed vacuum state,” Opt. Express 25(21), 24907–24916 (2017). [CrossRef]  

33. X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014). [CrossRef]  

34. M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010). [CrossRef]  

35. N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012). [CrossRef]  

36. A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011). [CrossRef]  

37. L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007). [CrossRef]  

38. Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017). [CrossRef]  

39. Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014). [CrossRef]  

40. P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007). [CrossRef]  

41. S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011). [CrossRef]  

42. J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013). [CrossRef]  

43. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012). [CrossRef]  

44. M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017). [CrossRef]  

45. C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019). [CrossRef]  

46. M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003). [CrossRef]  

47. W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010). [CrossRef]  

References

  • View by:

  1. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  2. N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
    [Crossref]
  3. J. Tabosa and D. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
    [Crossref]
  4. M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002).
    [Crossref]
  5. Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
    [Crossref]
  6. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
    [Crossref]
  7. G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012).
    [Crossref]
  8. O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
    [Crossref]
  9. V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
    [Crossref]
  10. X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2(6), 510–516 (2015).
    [Crossref]
  11. O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
    [Crossref]
  12. G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
    [Crossref]
  13. L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
    [Crossref]
  14. A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
    [Crossref]
  15. Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
    [Crossref]
  16. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, C. You, S. Adhikari, J. P. Dowling, and Y. Zhao, “Orbital-angular-momentum-enhanced estimation of sub-Heisenberg-limited angular displacement with two-mode squeezed vacuum and parity detection,” Opt. Express 26(13), 16524–16534 (2018).
    [Crossref]
  17. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Heisenberg-scaling angular displacement estimation with tunable squeezed bell states,” J. Opt. 21(3), 035201 (2019).
    [Crossref]
  18. J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
    [Crossref]
  19. J.-D. Zhang, C.-F. Jin, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Super-sensitive angular displacement estimation via an SU(1,1)-SU(2) hybrid interferometer,” Opt. Express 26(25), 33080–33090 (2018).
    [Crossref]
  20. J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
    [Crossref]
  21. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
    [Crossref]
  22. Z. Zhang, T. Qiao, K. Ma, L. Cen, J. Zhang, F. Wang, and Y. Zhao, “Ultra-sensitive and super-resolving angular rotation measurement based on photon orbital angular momentum using parity measurement,” Opt. Lett. 41(16), 3856–3859 (2016).
    [Crossref]
  23. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
    [Crossref]
  24. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
    [Crossref]
  25. C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
    [Crossref]
  26. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
    [Crossref]
  27. L. Cohen, D. Istrati, L. Dovrat, and H. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014).
    [Crossref]
  28. O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
    [Crossref]
  29. C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).
  30. J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
    [Crossref]
  31. T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
    [Crossref]
  32. J. Zhang, Z. Zhang, L. Cen, M. Yu, S. Li, F. Wang, and Y. Zhao, “Effects of imperfect elements on resolution and sensitivity of quantum metrology using two-mode squeezed vacuum state,” Opt. Express 25(21), 24907–24916 (2017).
    [Crossref]
  33. X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
    [Crossref]
  34. M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
    [Crossref]
  35. N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
    [Crossref]
  36. A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
    [Crossref]
  37. L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
    [Crossref]
  38. Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
    [Crossref]
  39. Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
    [Crossref]
  40. P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007).
    [Crossref]
  41. S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
    [Crossref]
  42. J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
    [Crossref]
  43. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
    [Crossref]
  44. M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
    [Crossref]
  45. C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
    [Crossref]
  46. M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
    [Crossref]
  47. W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
    [Crossref]

2019 (4)

O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
[Crossref]

J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Heisenberg-scaling angular displacement estimation with tunable squeezed bell states,” J. Opt. 21(3), 035201 (2019).
[Crossref]

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
[Crossref]

2018 (3)

2017 (5)

Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
[Crossref]

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

J. Zhang, Z. Zhang, L. Cen, M. Yu, S. Li, F. Wang, and Y. Zhao, “Effects of imperfect elements on resolution and sensitivity of quantum metrology using two-mode squeezed vacuum state,” Opt. Express 25(21), 24907–24916 (2017).
[Crossref]

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
[Crossref]

2016 (2)

Z. Zhang, T. Qiao, K. Ma, L. Cen, J. Zhang, F. Wang, and Y. Zhao, “Ultra-sensitive and super-resolving angular rotation measurement based on photon orbital angular momentum using parity measurement,” Opt. Lett. 41(16), 3856–3859 (2016).
[Crossref]

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

2015 (2)

L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
[Crossref]

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2(6), 510–516 (2015).
[Crossref]

2014 (6)

O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

L. Cohen, D. Istrati, L. Dovrat, and H. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014).
[Crossref]

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
[Crossref]

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

2013 (2)

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
[Crossref]

2012 (3)

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012).
[Crossref]

N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
[Crossref]

2011 (4)

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
[Crossref]

T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
[Crossref]

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

2010 (2)

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

2008 (1)

J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

2007 (2)

L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
[Crossref]

P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007).
[Crossref]

2005 (1)

C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
[Crossref]

2004 (1)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

2003 (1)

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

2002 (2)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002).
[Crossref]

2000 (1)

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

1999 (1)

J. Tabosa and D. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
[Crossref]

1997 (1)

1996 (1)

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Achilles, D.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

Adhikari, S.

Agarwal, G. S.

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
[Crossref]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

Ahmed, N.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Allen, L.

M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002).
[Crossref]

N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Anisimov, P. M.

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

Aolita, L.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Bahder, T. B.

T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
[Crossref]

Banaszek, K.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

Banker, J.

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

Bao, C.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Benmoussa, A.

C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
[Crossref]

Berry, D. W.

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

Bhusal, N.

C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

Bollinger, J. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref]

Bouwmeester, D.

L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
[Crossref]

Boyd, R. W.

O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
[Crossref]

O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
[Crossref]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
[Crossref]

Busch, K.

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

Cable, H.

Campos, R.

C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
[Crossref]

Cao, Y.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Cen, L.

Cen, L.-Z.

Chalich, Y.

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

Cohen, L.

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

d. J. León-Montiel, R.

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

D’ambrosio, V.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Datta, A.

L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

de J. León-Montiel, R.

C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

Del Re, L.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Demkowicz-Dobrzanski, R.

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

Dholakia, K.

Dong, C.

C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

Dorner, U.

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

Dovrat, L.

Dowling, J. P.

C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
[Crossref]

J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, C. You, S. Adhikari, J. P. Dowling, and Y. Zhao, “Orbital-angular-momentum-enhanced estimation of sub-Heisenberg-limited angular displacement with two-mode squeezed vacuum and parity detection,” Opt. Express 26(13), 16524–16534 (2018).
[Crossref]

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
[Crossref]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

Durkin, G. A.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

Eisenberg, H.

Feng, X.

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

Fitch, M. J.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

Franke-Arnold, S.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Franson, J. D.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

Gao, H.

J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
[Crossref]

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

Gerrits, T.

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

Gerry, C. C.

C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
[Crossref]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

Giacobino, E.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Gibilisco, P.

P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007).
[Crossref]

Giner, L.

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Giovannetti, V.

N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
[Crossref]

Heinzen, D.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
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Hermosa, N.

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O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

Veissier, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Vitelli, C.

N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
[Crossref]

Walborn, S. P.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Walmsley, I.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

Walmsley, I. A.

L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

Wang, F.

Wang, X.

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
[Crossref]

Wasilewski, W.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

Wei, D.

J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
[Crossref]

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

Whittaker, R.

Willner, A. E.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Wineland, D. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref]

Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Xie, G.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Yan, L.

Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
[Crossref]

Yan, Y.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Yang, W.

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

You, C.

C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
[Crossref]

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, C. You, S. Adhikari, J. P. Dowling, and Y. Zhao, “Orbital-angular-momentum-enhanced estimation of sub-Heisenberg-limited angular displacement with two-mode squeezed vacuum and parity detection,” Opt. Express 26(13), 16524–16534 (2018).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
[Crossref]

C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

Yu, M.

Zhang, J.

Zhang, J.-D.

Zhang, L.

L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

Zhang, Z.

Zhang, Z.-J.

Zhao, Y.

Zhao, Z.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Zhou, X.-Q.

Contemp. Phys. (1)

J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

J. Math. Phys. (1)

P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007).
[Crossref]

J. Mod. Opt. (1)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

J. Opt. (2)

J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
[Crossref]

J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Heisenberg-scaling angular displacement estimation with tunable squeezed bell states,” J. Opt. 21(3), 035201 (2019).
[Crossref]

J. Opt. B: Quantum Semiclassical Opt. (1)

M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002).
[Crossref]

Nat. Commun. (2)

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Nat. Photonics (2)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

New J. Phys. (1)

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

npj Quantum Inf. (1)

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

Opt. Commun. (1)

Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Optica (1)

Photonics Res. (1)

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

Phys. Rev. A (16)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
[Crossref]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
[Crossref]

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
[Crossref]

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
[Crossref]

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
[Crossref]

C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
[Crossref]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
[Crossref]

Phys. Rev. Lett. (8)

L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
[Crossref]

N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
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J. Tabosa and D. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
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O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
[Crossref]

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
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L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
[Crossref]

G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012).
[Crossref]

Rep. Prog. Phys. (1)

O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
[Crossref]

Other (1)

C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the angular displacement estimation protocol. The full names of the abbreviations in the figure: L, laser; P, polarizer; SLM, spatial light modulator; I, iris; BS, beam splitter; DP, Dove prism; M, mirror; PNRD, photon-number-resolving detector.
Fig. 2.
Fig. 2. (a) The output of parity measurement against angular displacement with different quantum numbers, where $N = 10$. (b) The output of parity measurement against angular displacement with different mean photon numbers, where $\ell = 3$.
Fig. 3.
Fig. 3. The FWHM of parity measurement against the mean photon number.
Fig. 4.
Fig. 4. (a) The output of parity measurement against angular displacement, where $N = 10$ and $\ell = 3$. (b) The optimal sensitivity of parity measurement against two path losses, where $N = 10$ and $\ell = 3$.
Fig. 5.
Fig. 5. The sensitivity of parity measurement against angular displacement with different rates. Where the curve of $r = 0$ is the ideal curve, the curves of $r = 10^{-3}$ and $r = 10^{-2}$ respectively correspond to the scenarios: dark counts, and dark counts along with response-time delay.
Fig. 6.
Fig. 6. Diagram of two angular displacement estimation protocols: a Sagnac interferometer and a Mach-Zehnder interferometer. The full names of the abbreviations in the figure: DP, Dove prism; D, detector; BS, beam splitter; SI, Sagnac interferometer; MZI, Mach-Zehnder interferometer.
Fig. 7.
Fig. 7. Experimental data against angular displacement with $\ell = 1$. (a) The blue line is a fit to the output. Error bars are one standard deviation due to propagated Poissonian statistics. (b) The red line is the sensitivity deduced from the fit of output, blue dots are the sensitivities calculated from the experimental data, and the black dashed line is the shot-noise limit defined in accordance with $\bar N$.
Fig. 8.
Fig. 8. (a) The analog voltage signals displayed by oscillograph, and each signal is converted from single statistical trigger counts. (b) The probability distribution of output photon state and a fit based on Poissonian distribution.

Equations (28)

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| ψ out = e N 2 n = 0 [ i α cos ( 2 φ ) ] n n ! m = 0 [ i α sin ( 2 φ ) ] m m ! | n , m .
P ( n , m ) = e N n ! m ! [ N cos 2 ( 2 φ ) ] n [ N sin 2 ( 2 φ ) ] m .
P even = 1 2 { 1 + exp [ 2 N sin 2 ( 2 φ ) ] } ,
P odd = 1 2 { 1 exp [ 2 N sin 2 ( 2 φ ) ] } .
Π ^ = exp [ 2 N sin 2 ( 2 φ ) ] .
F = 1 P even ( P even φ ) 2 + 1 P odd ( P odd φ ) 2 .
Δ φ = Π ^ 2 Π ^ 2 | Π ^ / Π ^ φ φ | ,
Δ φ = exp [ 4 N sin 2 ( 2 φ ) ] 1 | 4 N sin ( 4 φ ) | .
Δ φ min = 1 + 4 N sin 2 ( 2 φ ) 1 | 4 N sin ( 4 φ ) | | φ 0 = 1 4 N .
V = Π ^ max Π ^ min Π ^ max + Π ^ min .
ρ in = [ η | α α | + ( 1 η ) | α 0 α 0 | ] | 0 0 | .
Π ^ 1 = η exp [ 2 N sin 2 ( 2 φ ) ] + 1 η .
Δ φ 1 = sin ( 2 φ ) 1 η N sin 2 ( 2 φ ) | 2 η N sin ( 4 φ ) | | φ 0 = 1 η 1 4 N .
Π ^ 2 = exp [ N T A T B cos ( 4 φ ) N 2 ( T A + T B ) ] ,
Δ φ 2 = exp { N [ T A + T B 2 T A T B cos ( 4 φ ) ] } 1 | 4 ( T A + T B ) N sin ( 4 φ ) | .
Π ^ 3 = exp [ 2 κ N sin 2 ( 2 φ ) ] .
Π ^ 4 = e 2 r Π ^
J ^ x = 1 2 ( a ^ b ^ + a ^ b ^ ) ,
J ^ y = i 2 ( a ^ b ^ a ^ b ^ ) ,
J ^ z = 1 2 ( a ^ a ^ b ^ b ^ )
F S = 4 [ ψ | ( 4 J ^ z ) 2 | ψ ψ | 4 J ^ z | ψ 2 ] = 16 2 N ,
F M = 4 [ ψ | ( 2 n ^ a ) 2 | ψ ψ | 2 n ^ a | ψ 2 ] = 8 2 N .
Π ^ = 0.9507 exp { 4.594 sin 2 [ 2 ( φ 0.7022 ) ] } .
Π ^ = exp [ 4.594 sin 2 ( 2 φ ) 0.0506 ]
ρ ¯ 1 = 1 2 π 0 2 π exp ( i δ n ^ a ) ρ a ρ b exp ( i δ n ^ a ) d δ = n = 0 p n | n n | | 0 0 | .
ρ ¯ 2 = U ^ BS ρ ¯ 1 U ^ BS = n = 0 p n m = 0 n C n m | n m n m | | m m | ,
F Fock = 4 [ U ^ BS ( 2 n ^ a ) 2 U ^ BS U ^ BS ( 2 n ^ a ) U ^ BS 2 ] = 4 2 n .
F ρ ¯ = n = 0 p n 4 2 n = 4 2 N .

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