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Physical Review Applied

Optimal Detection Scheme for Shot-Noise-Limited Phase Estimation in Passive Classical-Light Interferometry

Vincent Michaud-Belleau, Jérôme Genest, Jean-Daniel Deschênes

Published Aug 17, 2018• 11 citations
Publisher

Abstract

Phase estimation is a central goal of optical interferometry, with applications ranging from gravitational-wave detection to fiber-optic sensing. Recent progress has focused on the use of nonclassical light to improve performance, but that approach is often unreasonably complicated. The authors reexamine the simple, passive Mach-Zehnder interferometer by linearizing the phase-estimation problem, for straightforward computation of an efficient estimator, which is then used to compare several detection schemes for two- and four-output interferometers. Independent monitoring of all available output ports leads to the best overall sensitivity, and allows cancellation of amplitude noise.

DOI
10.1103/physrevapplied.10.024025

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Phase estimation is a central goal of optical interferometry, with applications ranging from gravitational-wave detection to fiber-optic sensing. Recent progress has focused on the use of nonclassical light to improve performance, but that approach is often unreasonably complicated. The authors reexamine the simple, passive Mach-Zehnder interferometer by linearizing the phase-estimation problem, for straightforward computation of an efficient estimator, which is then used to compare several detection schemes for two- and four-output interferometers. Independent monitoring of all available output ports leads to the best overall sensitivity, and allows cancellation of amplitude noise.
When a photon interferes with itself while traversing a Mach-Zehnder interferometer, the output port where it emerges is influenced by the phase difference between the interferometer arms. This allows for highly precise estimation of the path length difference (delay) but is extremely sensitive to phase noise. By contrast, a delay between the arms of the two-photon Hong-Ou-Mandel interferometer directly affects the relative indistinguishability of the photon pair, affecting the rate of recorded coincidences. This likewise allows for delay estimation, notably less precise but with the advantage of being less sensitive to perturbations of the photons' phase. Focusing on two-photon input states, we here investigate to what degree of noise Mach-Zehnder interferometry retains its edge over Hong-Ou-Mandel interferometry. We also explore the competing benefits of different two-photon inputs for a Mach-Zehnder interferometer and under what parameter regimes each input performs best.
We study a Mach-Zehnder interferometer fed by a coherent state in one input port and vacuum in the other. We explore a Bayesian phase estimation strategy to demonstrate that it is possible to achieve the standard quantum limit independently from the true value of the phase shift and specific assumptions on the noise of the interferometer. We have been able to implement the protocol by using parallel operation of two photon-number-resolving detectors and multiphoton coincidence logic electronics at the output ports of a weakly illuminated Mach-Zehnder interferometer. This protocol is unbiased, saturates the Cramer-Rao phase uncertainty bound, and, therefore, is an optimal phase estimation strategy.
We consider a passive and active hybrid interferometer for phase estimation, which can reach the sub-shot-noise limit in phase sensitivity with only the cheapest coherent sources. This scheme is formed by adding an optical parametric amplifier before a Mach–Zehnder interferometer. It is shown that our hybrid protocol can obtain a better quantum Cramer–Rao bound than the pure active (e.g., SU(1,1)) interferometer, and this precision can be reached by implementing the parity measurements. Furthermore, we also draw a detailed comparison between our scheme and the scheme suggested by Caves [ Phys. Rev. D 23 1693 (1981)], and it is found that the optimal phase sensitivity gain obtained in our scheme is always larger than that in Caves’ scheme.
We present in this paper a new scheme for optical interferometry. We utilize coherent-beam-stimulated two-mode squeezed light, which interacts with a phase shifter and is then squeezed again before detection. Our theoretical device has the potential to reach far below the shot-noise limit in phase sensitivity. This new proposal avoids the pitfalls of other setups, such as difficulty in creating the required resource. Furthermore, our scheme requires no complicated detection protocol, relying instead only on simple intensity measurement. Also, bright, coherent sources 'boost' squeezed light, creating a very sensitive device. This hybrid scheme relies on no unknown components and can be constructed with current technology. In this paper, we present our analysis of this relatively straightforward device, using the operator propagation method. We derive the phase sensitivity and provide a simple numerical example of the power of our new proposal. Sensitivity to unknown phase shifts scales as a shot-noise-limited Mach–Zehnder interferometer, multiplied by a sub-Heisenberg contribution from the squeezed light.
Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.
The use of an acousto-optic modulator in an interferometer enables heterodyne detection and can alleviate technical issues such as laser intensity noise and photodetector flicker noise. However, it also introduces at least a 3 dB penalty to the shot-noise limited phase sensitivity when compared to that attainable through homodyne detection. In this paper, we show theoretically that this penalty can be lifted by implementing detection and demodulation schemes that exploit the cyclostationary nature of shot noise observed at the outputs of the self-heterodyne interferometer. We also discuss how expected departures from ideality, namely imperfect interferometric visibility, non-zero stationary noise, and finite detection bandwidth, affect the performance of these schemes and propose a simplified procedure relaxing the requirement for a rigorous instrument characterization. While two independent detectors are required to reach the true (classical-light) shot-noise limit for phase, single-detector instruments can also benefit from the introduced ideas to achieve up to 3 dB improvement in phase sensitivity.