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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Fisher Information with Continuous Variable Quantum Resources

George S. Atkinson, Euan J. Allen, Giacomo Ferranti, Jonathan C. F. Matthews

Published Jun 1, 2019• 0 citations
Publisher

Abstract

Squeezed states of light are a crucial resource for improving the precision of parameter estimation [1], and this has made an impact in areas as wide ranging as biological imaging [2] and gravitational wave detection [3]. However, the optimal bounds of the Fisher information (FI) in continuous variable (CV) quantum metrology have, so far, only been abstractly defined and do not account for many important experimental parameters, such as the quantum efficiency and bandwidth of the detector [4]. We show that classical FI calculations can be used to compute these optimal bounds with more applicability to experimental measurement. This is demonstrated in the case of single parameter estimation of a modulated optical loss, using a bright amplitude squeezed state as a probe.

DOI
10.1109/cleoe-eqec.2019.8872844

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Squeezed states of light are a crucial resource for improving the precision of parameter estimation [1], and this has made an impact in areas as wide ranging as biological imaging [2] and gravitational wave detection [3]. However, the optimal bounds of the Fisher information (FI) in continuous variable (CV) quantum metrology have, so far, only been abstractly defined and do not account for many important experimental parameters, such as the quantum efficiency and bandwidth of the detector [4]. We show that classical FI calculations can be used to compute these optimal bounds with more applicability to experimental measurement. This is demonstrated in the case of single parameter estimation of a modulated optical loss, using a bright amplitude squeezed state as a probe.
The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.
Abstract Two-photon absorption (TPA) is a nonlinear optical process with wide-ranging applications from spectroscopy to super-resolution imaging. Despite this, the precise measurement and characterisation of TPA parameters are challenging due to their inherently weak nature. We study the potential of single-mode quantum light to enhance TPA parameter estimation through the quantum Fisher information (QFI). Discrete variable quantum states (defined to be a finite superposition of Fock states) are optimised to maximise the QFI for given absorption, revealing a quantum advantage compared to both the coherent state (classical) benchmark and the single-mode squeezed vacuum state. For fixed average energy n ¯ ∈ 2 N , the Fock state is shown to be optimal for large TPA parameters, while a superposition of vacuum and a particular Fock state is optimal for small absorption for all n ¯ . This differs from single-photon absorption where the Fock state is always optimal. Notably, photon counting is demonstrated to offer optimal or nearly optimal performance compared to the QFI bound for all levels of TPA parameters for the optimised quantum probes, and their quantum advantage is shown to be robust to single-photon loss. Our findings provide insight into known limiting behaviours of Gaussian probes and their different FI scalings under photon counting ( ∝ n ¯ 2 for squeezed vacuum states versus n ¯ 3 for coherent states). The squeezed state outperforms coherent states for small TPA parameters but underperforms in the intermediate regime, becoming comparable in the large absorption limit. This can be explained through fundamental differences between behaviours of even and odd number Fock states: the former’s QFI diverges in both large and small absorption limits, while the latter diverges only in the small absorption limit, dominating at intermediate scales.
Two-photon absorption (TPA) is a nonlinear optical process with wide-ranging applications from spectroscopy to super-resolution imaging. Despite this, the precise measurement and characterisation of TPA parameters are challenging due to their inherently weak nature. We study the potential of single-mode quantum light to enhance TPA parameter estimation through the quantum Fisher information (QFI). Discrete variable (DV) quantum states (defined to be a finite superposition of Fock states) are optimised to maximise the QFI for given absorption, revealing a quantum advantage compared to both the coherent state (classical) benchmark and the single-mode squeezed vacuum state. For fixed average energy nˉ∈2N\bar{n} \in 2\mathbb{N}nˉ∈2N, the Fock state is shown to be optimal for large TPA parameters, while a superposition of vacuum and a particular Fock state is optimal for small absorption for all nˉ\bar{n}nˉ. This differs from single-photon absorption where the Fock state is always optimal. Notably, photon counting is demonstrated to offer optimal or nearly optimal performance compared to the QFI bound for all levels of TPA parameters for the optimised quantum probes. Our findings provide insight into known limiting behaviours of Gaussian probes and their different Fisher information (FI) scalings under photon counting (∝nˉ2\propto \bar{n}^2∝nˉ2 for squeezed vacuum states versus nˉ3\bar{n}^3nˉ3 for coherent states). The squeezed state outperforms coherent states for small TPA parameters but underperforms in the intermediate regime, becoming comparable in the large absorption limit. This can be explained through fundamental differences between behaviours of even and odd number Fock states: the former's QFI diverges in both large and small absorption limits, while the latter diverges only in the small absorption limit, dominating at intermediate scales.
Abstract Quantum mechanics establishes the ultimate limit to the scaling of the precision on any parameter, by identifying optimal probe states and measurements. While this paradigm is, at least in principle, adequate for the metrology of quantum channels involving the estimation of phase and loss parameters, we show that estimating the loss parameters associated with a quantum channel and a realistic quantum detector are fundamentally different. While Fock states are provably optimal for the former, we identify a crossover in the nature of the optimal probe state for estimating detector imperfections as a function of the loss parameter using Fisher information as a benchmark. We provide theoretical results for on-off and homodyne detectors, the most widely used detectors in quantum photonics technologies, when using Fock states and coherent states as probes.
Enhancing the precision of a measurement requires maximizing the information that can be gained about the quantity of interest from probing a system. For optical-based measurements, such an enhancement can be achieved through two approaches, increasing the number of photons used to interrogate the system and using quantum states of light to increase the amount of quantum Fisher information gained per photon. Here we consider the use of quantum states of light with a large number of photons, namely the bright single-mode squeezed state and the bright two-mode squeezed state, which take advantage of both of these approaches for the problem of transmission estimation. We show that, in the limit of large squeezing, these states approach the maximum possible quantum Fisher information per photon for transmission estimation that is achieved with the Fock state and the vacuum two-mode squeezed state. Since the bright states we consider can be generated at powers much higher than those of the quantum states that achieve the maximum quantum Fisher information per photon, they can achieve a much higher absolute precision as quantified by the quantum Cram\'er-Rao bound. We discuss the effects of losses external to the system on the precision of transmission estimation and identify simple measurement techniques that can saturate the quantum Cram\'er-Rao bound for the bright squeezed states even in the presence of such external losses.
As one of the main pillars of quantum technologies, quantum metrology aims to improve measurement precision using techniques from quantum information. The two main strategies to achieve this are the preparation of nonclassical states and the design of optimized measurement observables. We discuss precision limits and optimal strategies in quantum metrology and sensing with a single mode of quantum continuous variables. We focus on the practically most relevant cases of estimating displacements and rotations and provide the sensitivities of the most important classes of states that includes Gaussian states and superpositions of Fock states or coherent states. Fundamental precision limits that are obtained from the quantum Fisher information are compared to the precision of a simple moment-based estimation strategy based on the data obtained from possibly sub-optimal measurement observables, including homodyne, photon number, parity and higher moments. Finally, we summarize some of the main experimental achievements and present emerging platforms for continuous-variable sensing. These results are of particular interest for experiments with quantum light, trapped ions, mechanical oscillators, and microwave resonators.