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Association between probability distribution of optical phase and relative phase detection in photonic quantum metrology

Felipe F. Braz

Published Nov 8, 2022• 0 citations
OpenAlex

Abstract

The measurement of a phase difference between two optical modes in an interferometer has several important applications in metrology. This measurement can be useful in several contexts, such as in the detection of gravitational waves. The quantum nature of light imposes a limit on how accurately a phase difference can be measured, this limit being known as the Heisenberg limit (HL). On the other hand, photonic quantum metrology studies alternatives to reach the HL through entangled or squeezed quantum states. According to the theory of parameter estimation, the minimum uncertainty possible in the estimation of a phase $\theta$ considering classical light scales with $1/\sqrt{\Bar{N}}$, where $\Bar{N}$ is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with $1/{\Bar{N}}$, known as HL. In this thesis, starting from the relative phase probability distribution P($\phi$) introduced by Luis and S\'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of $0,1\%$. Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.

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The measurement of a phase difference between two optical modes in an interferometer has several important applications in metrology. This measurement can be useful in several contexts, such as in the detection of gravitational waves. The quantum nature of light imposes a limit on how accurately a phase difference can be measured, this limit being known as the Heisenberg limit (HL). On the other hand, photonic quantum metrology studies alternatives to reach the HL through entangled or squeezed quantum states. According to the theory of parameter estimation, the minimum uncertainty possible in the estimation of a phase θ\thetaθ considering classical light scales with 1/\BarN1/\sqrt{\Bar{N}}1/\BarN​, where \BarN\Bar{N}\BarN is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with 1/\BarN1/{\Bar{N}}1/\BarN, known as HL. In this thesis, starting from the relative phase probability distribution P(ϕ\phiϕ) introduced by Luis and S'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of 0,1%0,1\%0,1%. Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.
Optical phase measurement through its application in quantum metrology has pushed the precision limit with which some physical quantities can be measured accurately. At the very fundamental level, the laws of quantum mechanics dictate that the uncertainty in phase estimations scales as 1/N, where N is the number of quantum resources employed in the protocol [1]. This is the well known Heisenberg limit (HL) which is quadratically better than the traditional precision limit known as the standard quantum limit (SQL) with uncertainty asymptotically scaling as 1/sqrt{N} [1]. Several experiments have demonstrated that the SQL can be beaten by using an entangled state as the probe and a specific measurement scheme for ab initio estimation of unknown phases [2,3]. It has also been shown experimentally that even in the absence of the entanglement one can measure an unknown phase with imprecision scaling at the HL [4]. In this work we first present a new protocol able to estimate an optical phase at the Heisenberg limit, and then experimentally explore fundamental and practical issues in generating high-quality novel entangled states, for use in this protocol and beyond. Our aim in this study is to measure an unknown phase in the interval [0,2pi) with uncertainty attaining the exact HL. There is a condition that should be met to address this objective: preparation of an optimal state [5]. This would cover part of the presentation through which we explain how to experimentally realise such an optimal state with the current technological limitations and the feasibility of the scheme. In particular, we generate an entangled 3-photon (2-photon) state of specific superposition of GHZ (Bell) states. Our numerical simulation of the phase measurement gate together with the experimental outcomes show that the created state should have a high fidelity and purity to be able to have the phase uncertainty achieving the exact HL. Therefore, we briefly explain the modelling for experimental imperfections and finally present the results of experimental phase measurements.
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 fundamental precision limit of an interferometer is crucial since it bounds the best possible sensitivity one could achieve using such a device. This thesis will focus on several different interferometers and try to give the ultimate precision bounds by carefully counting all the resources used in the interferometers. The thesis begins with the basics of the quantum state of light. The fundamentals of quantum metrology are also reviewed and discussed. More specifically, the terminology of classical and quantum Cram\'er-Rao bound and classical and quantum Fisher information are introduced. Chapter 3 discusses the conclusive precision bounds in two-mode interferometer such as Mach-Zehnder interferometer (MZI) and SU(1,1) interferometer. I revisit the quantum Fisher information approach of these two interferometers and show the discrepancy of phase sensitivity on the physically same setup. Then I establish fundamental precision estimation bounds for such device, due to the reason that many works of literature fail to accurately count the resources and knowledge of phase-to-be-estimated used in the interferometers. The analysis suggests that for a MZI, one can never do better than SNL in phase sensitivity, when an input to one of the two ports is the vacuum. If one does not allow the detector to use any external phase reference or power resource, then the precision is limited by the SNL. For a SU(1,1) interferometer, firstly, when one of the input states is restricted to be a vacuum state, I showed that by using either the phase-averaging method or the quantum Fisher information matrix method, different phase configurations of the SU(1,1) interferometer result in the same QFI. Secondly, I compared the results of the phase-averaging method and the quantum Fisher information matrix method, and then I argued that for an SU(1,1) interferometer, phase averaging or quantum Fisher information matrix method is generally required, and they are equivalent. Finally, I used the quantum Fisher information matrix method to calculate the precision limit for other common input states, such as two coherent state inputs or coherent state with squeezed vacuum inputs. In chapter 4, I will consider a passive multi-mode interferometer for multiparameter phase estimation. It was suggested that optical networks with relatively inexpensive overhead---single photon Fock states, passive optical elements, and single photon detection---can show significant improvements over classical strategies for single-parameter estimation, when the number of modes in the network is small. In this chapter, I analytically compute the quantum Cramer-Rao bound to show these
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 consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.