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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Metrology with entangled states

Morgan W. Mitchell

Published Aug 18, 2005• 17 citations
Publisher

Abstract

It is well known that classical states of light exhibit shot noise, characteristic of independent or uncorrelated particles. For phase estimation problems, this leads to a shot-noise limited uncertainty of 1/sqrt[N], where N is the number of particles detected. It is also well known that the shot-noise limit is not fundamental: squeezed states and entangled states can be used for sub-shot-noise phase measurements. The uncertainty principle sets a fundamental limit of 1/N, known as the "Heisenberg" limit. We have recently demonstrated a method, using parametric downconversion and post-selection, to generate entangled "NOON" states suitable for sub-shot-noise phase measurements [M.W. Mitchell et al, Nature 429, 161 (2004)]. We generated a three-photon NOON state and demonstrated three-fold improvement in phase resolution with this state. The relationship between phase resolution and phase uncertainty depends on prior information about the phase being estimated. As in the case of phase measurements with squeezed states, extra precision in one dimension is gained at the cost of reduced precision in other dimensions. Only when prior information is incorporated can entangled-state metrology be applied to beat the shot-noise limit. We illustrate this relationship and discuss adaptive strategies for phase estimation and the possibility of reaching the Heisenberg limit.

DOI
10.1117/12.621353

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It is well known that classical states of light exhibit shot noise, characteristic of independent or uncorrelated particles. For phase estimation problems, this leads to a shot-noise limited uncertainty of 1/sqrt[N], where N is the number of particles detected. It is also well known that the shot-noise limit is not fundamental: squeezed states and entangled states can be used for sub-shot-noise phase measurements. The uncertainty principle sets a fundamental limit of 1/N, known as the "Heisenberg" limit. We have recently demonstrated a method, using parametric downconversion and post-selection, to generate entangled "NOON" states suitable for sub-shot-noise phase measurements [M.W. Mitchell et al, Nature 429, 161 (2004)]. We generated a three-photon NOON state and demonstrated three-fold improvement in phase resolution with this state. The relationship between phase resolution and phase uncertainty depends on prior information about the phase being estimated. As in the case of phase measurements with squeezed states, extra precision in one dimension is gained at the cost of reduced precision in other dimensions. Only when prior information is incorporated can entangled-state metrology be applied to beat the shot-noise limit. We illustrate this relationship and discuss adaptive strategies for phase estimation and the possibility of reaching the Heisenberg limit.
Quantum metrology enables us to estimate optical phase shifts beyond the shot noise limit. Applications include quantum-enhanced biological measurement [1], microscopy [2] and gravitational wave detection[3]. For two-mode N-photon system, the NOON state can achieve the highest possible phase sensitivity where multi-photon coherence between two states significantly contributes to improving the phase sensitivity. Another alternative approach to improve the phase sensitivity by use of squeezed states. Squeezing is usually characterized by reducing the quantum noise of desired observable at the expense of increasing the quantum noise for its non-commute partner. Because shot-noise limits the phase sensitivity of any classical states, the reduced quantum noise can be used for improving the phase sensitivity. Here we experimentally study the phase sensitivity of an optical spin-squeezed state by using linear optics. We generated the optical spin-squeezed state using one photon subtraction from a down converted photon source. To implement the sub-shot noise measurement, we used multiplexed photon number resolving detectors so that we can reconstruct the photon number statistics of the state. In the talk, we will analyse the generated state in terms of both squeezing and phase sensitivity by using Fisher information.
Quantum metrology exploits quantum correlations to perform measurements with precision higher than can be achieved with classical approaches [1]. Photonic approaches promise transformative advances in the family of interferometric phase measurement techniques, a vital toolset used to precisely determine quantities including distance, velocity, acceleration and materials properties. Without quantum enhancement, the precision limit in optical phase sensing (i.e. the minimum uncertainty) is the shot noise limit (SNL): 1/sqrt(N) where N is the number of resources (e.g. photons) used. Entangled photons promise sensitivity surpassing the shot noise limit achievable with classical probes. The maximally phase-sensitive state is the NOON state [2], a path-entangled state of definite photon number N. Despite theoretical proposals stretching back decades [3], no measurement using such photonic (i.e. definite photon number) states has unconditionally surpassed the shot noise limit: by contrast, all such demonstrations employed postselection to discount photon loss in the source, interferometer or detectors. Here, we use the state of art single photon generation and detection technology to respectively make and measure a two-photon NOON state, and use it to perform unconditional phase sensing beyond the shot noise limit — that is, without artificially correcting for loss or any other source of imperfection [4]. We performed a two-photon NOON state polarisation interferometry measurement on a birefringent test phase. We use photons generated from a high-heralding-efficiency, high purity source of telecom-wavelength photon pairs [5], and we employ high efficiency superconducting photon detectors. Unlike previous experiments, our apparatus does not require postselection to achieve phase uncertainty below that achievable in an ideal, lossless classical interferometer. Our results show a clear violation (for a range of phases) of the stringent SNL Fisher-information bound, F_{SNL} = 2.09635, that takes into account the information in unrecorded trials arising from loss and higher order terms — making our demonstration unconditional. We also performed a direct phase sensing measurement and observed phase uncertainties more than 10 standard deviations below the SNL [4]. Our results enable quantum-enhanced phase measurements at low photon flux and open the door to the next generation of optical quantum metrology advances. References [1] V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011). [2] J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008). [3] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kołodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015). [4] S. Slussarenko, M. M.Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, S.W. Nam, and G. J. Pryde, “Unconditional violation of the shot noise limit in photonic quantum metrology,” Nat. Photon. 11, 700-703 (2017). [5] M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K. Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko, and G. J. Pryde, “Efficient and pure femtosecond-pulse length source of polarization-entangled photons,” Opt. Express 24, 10,869–10,879 (2016).
Quantum metrology promises phase sensitivity surpassing the shot-noise limit by exploiting entanglement and photon-number correlations. NOON states-maximally path-entangled NNN-photon superpositions (∣N,0⟩+∣0,N⟩)/2(|N,0\rangle + |0,N\rangle)/\sqrt{2}(∣N,0⟩+∣0,N⟩)/2​ -achieve the Heisenberg limit 1/N1/N1/N for single-parameter estimation, as demonstrated experimentally by Afek et al. (2010) using hybrid coherent-plus-squeezed light up to N=5N=5N=5. We present an end-to-end differentiable quantum-optical framework-implemented in Strawberry Fields (Killoran et al., 2019) with a TensorFlow backend -that learns optimal circuit parameters by maximising the classical Fisher information (CFI) across all coincidence channels for N=2,3,4,5N=2,3,4,5N=2,3,4,5. Starting from proper numerical reproductions of the Afek et al. coincidence fringes, verified by FFT analysis and parity measurements, we apply gradient descent (Adam) to the eight trainable circuit parameters. Raw CFI improvements grow dramatically with photon number: +153%+153\%+153% (N=2N=2N=2), +834%+834\%+834% to +956%+956\%+956% (N=3N=3N=3), +829%+829\%+829% to +1598%+1598\%+1598% (N=4N=4N=4), and +1775%+1775\%+1775% (N=5N=5N=5), alongside post-selection rate improvements of +153%+153\%+153% to +3269%+3269\%+3269%, and an 8×8\times8× to 133×133\times133× improvement in useful measurement events per pulse across N=2N=2N=2-555. A fundamental inter-channel trade-off is identified at N=2N=2N=2 but weakens at higher NNN where the Afek initialisation is further from optimal. These results provide numerically rigorous benchmarks for adaptive single-parameter quantum sensing and demonstrate that the Afek working point is significantly suboptimal at N≥3N\geq 3N≥3. QFI calculations confirm that the optimised probe reaches 82%82\%82% of the Heisenberg limit at N=2N=2N=2 and improves from 36%36\%36% to 58%58\%58% at N=5N=5N=5, while useful measurement events per pulse improve by 8×8\times8× to 133×133\times133× across all NNN, making quantum-enhanced sensing at N≥3N\geq 3N≥3 experimentally practical.
Quantum metrology enables estimation of optical phase shifts with precision beyond the shot-noise limit. One way to exceed this limit is to use squeezed states, where the quantum noise of one observable is reduced at the expense of increased quantum noise for its complementary partner. Because shot-noise limits the phase sensitivity of all classical states, reduced noise in the average value for the observable being measured allows for improved phase sensitivity. However, additional phase sensitivity can be achieved using phase estimation strategies that account for the full distribution of measurement outcomes. Here we experimentally investigate a model of optical spin-squeezing, which uses post-selection and photon subtraction from the state generated using a parametric downconversion photon source, and we investigate the phase sensitivity of this model. The Fisher information for all photon-number outcomes shows it is possible to obtain a quantum advantage of 1.58 compared to the shot-noise value for five-photon events, even though due to experimental imperfection, the average noise for the relevant spin-observable does not achieve sub-shot-noise precision. Our demonstration implies improved performance of spin squeezing for applications to quantum metrology.
We investigate the utility of parity detection to achieve Heisenberg-limited phase estimation for optical interferometry. We consider the parity detection with several input states that have been shown to exhibit sub-shot-noise interferometry with their respective detection schemes. We show that with parity detection, all these states achieve the sub-shot-noise limited phase uncertainty. Thus making the parity detection a unified detection strategy for quantum optical metrology. We also consider quantum states that are a combination of a NOON state and a dual-Fock state, which gives a great deal of freedom in the preparation of the input state, and is found to surpass the shot-noise limit.
We experimentally demonstrate a general criterion to identify entangled states useful for the estimation of an unknown phase shift with a sensitivity higher than the shot-noise limit. We show how to exploit this entanglement on the examples of a maximum likelihood as well as of a Bayesian phase estimation protocol. Using an entangled four-photon state we achieve a phase sensitivity clearly beyond the shot-noise limit. Our detailed comparison of methods and quantum states for entanglement enhanced metrology reveals the connection between multiparticle entanglement and sub-shot-noise uncertainty, both in a frequentist and in a Bayesian phase estimation setting.