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arXiv:1910.00857 · Physical Review Letters

Quantum Optical Metrology of Correlated Phase and Loss

Patrick M. Birchall, Euan J. Allen, Thomas M. Stace, Jeremy L. O’Brien, Jonathan C. F. Matthews, Hugo V. Cable

Published Apr 8, 2020• 28 citations
arXivHTMLPDF

Abstract

Optical absorption measurements characterize a wide variety of systems from atomic gases to in vivo diagnostics of living organisms. Here we study the potential of nonclassical techniques to reduce statistical noise below the shot-noise limit in absorption measurements with concomitant phase shifts imparted by a sample. We consider both cases where there is a known relationship between absorption and a phase shift, and where this relationship is unknown. For each case we derive the fundamental limit and provide a practical strategy to reduce statistical noise. Furthermore, we find an intuitive correspondence between measurements of absorption and of lossy phase shifts, which both show the same analytical form for precision enhancement for bright states. Our results demonstrate that nonclassical techniques can aid real-world tasks with present-day laboratory techniques.

DOI
10.1103/physrevlett.124.140501

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Optical absorption measurements characterize a wide variety of systems from atomic gases to in vivo diagnostics of living organisms. Here we study the potential of nonclassical techniques to reduce statistical noise below the shot-noise limit in absorption measurements with concomitant phase shifts imparted by a sample. We consider both cases where there is a known relationship between absorption and a phase shift, and where this relationship is unknown. For each case we derive the fundamental limit and provide a practical strategy to reduce statistical noise. Furthermore, we find an intuitive correspondence between measurements of absorption and of lossy phase shifts, which both show the same analytical form for precision enhancement for bright states. Our results demonstrate that nonclassical techniques can aid real-world tasks with present-day laboratory techniques.
Understanding the fundamental limits on the precision to which an optical phase can be estimated is of key interest for many investigative techniques utilized across science and technology. We study the estimation of a fixed optical phase shift due to a sample which has an associated optical loss, and compare phase estimation strategies using classical and nonclassical probe states. These comparisons are based on the attainable (quantum) Fisher information calculated per number of photons absorbed or scattered by the sample throughout the sensing process. We find that for a given number of incident photons upon the unknown phase, nonclassical techniques in principle provide less than a 2020%20 reduction in root-mean-square error (RMSE) in comparison with ideal classical techniques in multipass optical setups. Using classical techniques in a different optical setup that we analyze, which incorporates additional stages of interference during the sensing process, the achievable reduction in RMSE afforded by nonclassical techniques falls to only $\ensuremath{\simeq}4%$. We explain how these conclusions change when nonclassical techniques are compared to classical probe states in nonideal multipass optical setups, with additional photon losses due to the measurement apparatus.
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.
Optical quantum metrology uses specially-prepared states of light to probe sensitive samples. Choosing a suitable probe state allows one to extract more information per probe photon about an unknown parameter than is possible using any classical technique. In effect, these techniques improve the sensitivity of the measurement whenever the number of probe photons is constrained in some way. This thesis presents experimental and theoretical results on the use of pairs of time-frequency entangled photons in precision measurements. It begins with an overview of necessary background knowledge about quantum optics and quantum information. This is followed by a theoretical discussion on the estimation of a single delay using two-photon interference. We show that constraints on the probe photon's bandwidth complicate the measurement problem, requiring suitable optimization of the two-photon quantum state to avoid losing precision. We then consider the use of two-photon interference for linear spectroscopic measurements. Linear spectroscopy relates the absorption and phase spectrum of a sample to unknown electronic parameters. We describe a proof-of-principle experiment demonstrating that two-photon interferometric spectroscopy can be used for simultaneously measurement of both absorption and phase. This is followed by an information-theoretic discussion of the precision of this technique. For low losses, we show that the method provides phase sensitivity similar to classical interferometry while approaching the optimal absorption precision of any possible measurement. If the absorption and phase values are both related to a single unknown parameter, then our technique provides improved sensitivity to this parameter compared to both classical measurements and other quantum techniques in the presence of realistic levels of loss. We conclude with a discussion of quantum-optical coherence tomography, which is a two-photon interferometric method for determining the material structure of a sample. We present an ongoing experiment which aims to apply the method to tissue-like samples, where the quantum measurement is limited by a poor signal-to-noise ratio. Preliminary results demonstrate that this noise can be largely reduced by suitable frequency-domain filtering. This allows us to precisely determine the position of a scattering center using a few hundred detected photon pairs despite the presence of large amounts of noise. We follow this with a theoretical discussion of signal artifacts which are known to complicate the measured interferogram. We show these artifacts are due to a distinct interferometric effect, which can be distinguished from the signal of interest using the relative time delay between detected photons. We conclude with a brief discussion on possible techniques for artifact removal.
Abstract Recent theoretical and experimental advances have shown that squeezed states of light can be engineered to enhance the resolution of optical measurements. Extending such quantum-enhanced techniques to nonlinear optical processes has therefore attracted considerable interest. In particular, two-photon absorption (TPA), which has wide-ranging applications in physics, chemistry, and biology, has emerged as a promising platform for exploiting these quantum advantages. In this work, we investigate whether and how correlations between light modes can be used to improve optical resolution in TPA. We analyze non-degenerate TPA transmission measurements using two-mode squeezed light and compare different measurement strategies. We find that intensity correlation measurements offer better performance in terms of the achievable estimation error compared to normalized intensity correlations or measurements of the noise reduction factor (NRF). The achievable enhancement from intensity and NRF measurements are degraded by competing photon losses. In contrast, the normalized intensity correlation approach remains robust to loss, albeit at the cost of forfeiting the scaling enhancement at large photon numbers achievable with nonclassical light fields. Thus, we establish a trade-off between robustness to loss and the achievable quantum advantage. We further compute the corresponding Cramér–Rao bound, which sets the ultimate lower limit on the achievable estimation error for the photon-counting-based measurement strategies investigated in this work.
Quantum optical metrology deals with estimation of an unknown parameter by exploiting the non-classical properties of the light. The unknown parameter that we are trying to estimate is the optical phase. Precise optical phase measurement has been a well-known problem and has many applications, most notably the gravitational wave detection. In this thesis we investigate the interferometric measurement schemes. We consider the parity detection for a class of input states that have been shown to exhibit sub-shot noise limited phase estimate with their respective detection schemes. Our results indicate that the parity detection applies to all these strategies with various input states and thus acts as a unified detection scheme towards the goal of interferometric phase estimates beyond the shot-noise limit. We also consider the performance of the so-called optimal state with the canonical phase measurement scheme that was proposed by Sanders and Milburn [Phys. Rev. Lett. 75, 2944 (1995)] in presence of photon loss. The model for photon loss is a generic fictitious beam splitter and the analytical treatment requires density matrix approach rather than the state-vector formalism. We present full density-matrix calculations. Our results indicate that, for a given amount of loss, the phase estimate saturates but does not diverge as one would expect with increasing the loss. Finally, we study the continuous measurement and feedback scheme with optical homodyne detection for a single optical qubit. We found a protocol that speeds up the rate of increase of the average purity of the system and generates a deterministic evolution for the purity in the limit of strong feedback.
Optical interferometry is a cornerstone technique for precise phase measurements across various fields. In many applications, for example, biological imaging, it often necessitates stringent limits on light intensity to prevent adverse effects on light-sensitive samples, a condition known as dose-limited regimes. Maximizing the precision per dose is therefore crucial. In quantum metrology, quantum correlations enable high precision in phase estimation while adhering to dose constraints. Nevertheless, photon loss, including absorption by a sample, substantially diminishes the benefits of quantum enhancement in interferometry. In this work, we experimentally investigate a dose-efficient approach to quantum phase estimation using sequential strategies in the presence of loss. Performance of sequential strategies with and without control is evaluated through quantum Fisher information (QFI) per dose. Experimental results show that both sequential strategies exceed the classical limit and outperform the parallel strategy using unbalanced N00N states. Notably, the control-enhanced sequential strategy attains superior QFI per dose, approaching the quantum limit. These results highlight the promise of sequential strategy for imaging and sensing in resource-constrained scenarios, marking a significant step toward practical and efficient quantum metrology in lossy environments.