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arXiv:1809.04821 · Physical Review A

Nonlinear phase estimation: Parity measurement approaches the quantum Cramér-Rao bound for coherent states

Jiandong Zhang, Zijing Zhang, Longzhu Cen, Jun-Yan Hu, Yuan Zhe Zhao

Published Feb 6, 2019• 26 citations
arXivHTMLPDF

Abstract

Quantum-enhanced phase estimation paves the way for precision measurements and is of great realistic significance. In this paper we theoretically investigate the second-order nonlinear phase estimation using a coherent state and parity measurement. A numerical expression of the parity signal is derived, and its visibility is analyzed. The resolution and sensitivity of the signal are compared to a linear phase estimation protocol with the same input and measurement. Additionally, by using the phase-averaging approach, we make an attempt at unveiling the lowdown on a conclusive sensitivity limit without external phase reference. Finally, the effects of several realistic scenarios on the resolution and sensitivity are studied, including photon loss, imperfect detector, those which are a combination thereof, and unbalanced beam splitter.

DOI
10.1103/physreva.99.022106

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Quantum-enhanced phase estimation paves the way for precision measurements and is of great realistic significance. In this paper we theoretically investigate the second-order nonlinear phase estimation using a coherent state and parity measurement. A numerical expression of the parity signal is derived, and its visibility is analyzed. The resolution and sensitivity of the signal are compared to a linear phase estimation protocol with the same input and measurement. Additionally, by using the phase-averaging approach, we make an attempt at unveiling the lowdown on a conclusive sensitivity limit without external phase reference. Finally, the effects of several realistic scenarios on the resolution and sensitivity are studied, including photon loss, imperfect detector, those which are a combination thereof, and unbalanced beam splitter.
Estimating nonlinear phase shifts is useful to analyze many dynamical processes. In this work, with respect to the second-order nonlinear phase shifts, we propose an estimation scheme with a coherent state as the input and balanced homodyne detection as the readout strategy. The measurement setup is a nonlinear-linear hybrid interferometer composed of an optical parametric amplifier and a beam splitter. We analyze the precision and determine the optimal parameters of our scheme. The precision is compared with the quantum Cramér–Rao bound as well as the precision of a conventional nonlinear interferometer. In view of the fact that photon loss is ubiquitous, the effects of transmission loss and detection loss on the precision are addressed. At low gain, our scheme is superior to the scheme using a conventional nonlinear interferometer, and the precision can saturate the quantum Cramér–Rao bound.
Abstract The precision of quantum measurements can be effectively improved by using both photon-added non-Gaussian operations and Kerr nonlinear phase shift. Here, we employ a coherent state mixed with a photon-added squeezed vacuum state as the input to a Mach–Zehnder interferometer with parity detection, thereby achieving a significant enhancement in phase measurement precision. Our research focuses on the phase sensitivity of linear phase shift under both ideal conditions and photon loss, as well as quantum Fisher information (QFI). The results demonstrate that employing the photon addition operations can markedly enhance phase sensitivity and QFI, and under optimal conditions, the measurement precision can approach the Heisenberg limit for the linear phase shift case. In addition, we delve deeper into the scenario of replacing the linear phase shift with a Kerr nonlinear one and systematically analyze the QFI under both ideal and photon loss conditions. By comparison, it is evident that employing both the photon addition operations and the Kerr nonlinear phase shift can further significantly enhance phase measurement precision while effectively improving the system’s robustness against photon loss. These findings are instrumental in facilitating the development and practical application of quantum metrology.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
We propose a protocol for the second-order nonlinear phase estimation with a coherent state as input and balanced homodyne detection as measurement strategy. The sensitivity is sub-Heisenberg limit, which scales as N−3/2N^{-3/2}N−3/2 for NNN photons on average. By ruling out hidden resources in quantum Fisher information, the fundamental sensitivity limit is recalculated and compared to the optimal sensitivity of our protocol. In addition, we investigate the effect of photon loss on sensitivity, and discuss the robustness of measurement strategy. The results indicate that our protocol is nearly optimal and robust.
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.
Nonlinear phase measurement plays an important role in many dynamic processes. In this paper, we propose a single-mode measurement scheme for second-order nonlinear phase shifts that utilizes coherent states and homodyne measurement. The sensitivity at operating point φ = 0 is 1 / 4 N 3 / 2 with N photons on average and approaches the quantum Cramér–Rao bound. In addition, sensitivity with a small value of φ is discussed. For practical purposes, we analyze the effects of photon losses, thermal photon noise, and phase diffusion on sensitivities with φ = 0 as well as a small value of φ . The sensitivity still scales as 1 / N 3 / 2 in the presence of photon losses and thermal photon noise but degrades to the scaling of 1 / N with phase diffusion.