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

Phase super-resolution using a photonic crystal fibre

Srikanth Kannan, Bryn A. Bell, Alex S. Clark, W. J. Wadsworth, John Rarity

Published May 1, 2011• 0 citations
Publisher

Abstract

Measurement, whilst being physically fundamental, is also fundamentally physical. Classical interferometric measurements suffer from physical limitations where the shot noise limit forms an upper bound of √n on a phase shift Δφ that can be resolved, with n being the average number of photons detected, making their applicability in modern nanoscale experiments quite restricted. Exotic quantum states such as maximally entangled NOON states (|N0〉A;B+|0N〉A;B)/√2, where N is the number of entangled photons are of particular importance as they have repeatedly been shown to exhibit phase super-resolution. Interesting post selection schemes and time-reversed detection schemes have been investigated for phase estimation using NOON states. These methods are inherently non-deterministic at the preparation or measurement stage and thus do not optimise resources, for instance the number of photons passing through the sample greatly exceeds those used in the measurement.

DOI
10.1109/cleoe.2011.5943464

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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).

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Phase measurements exhibiting super sensitivity and super resolution features

Clemens Schäfermeier, Miroslav Ježek, Tobias Gehring +1 more

By using an optical squeezed state and a post-processed homodyne detection scheme we show that phase measurements can overcome Rayleigh’s resolution criterion and beat the quantum shot noise limit simultaneously.

Conference on Lasers and Electro-Optics•Jan 1, 2016•0 citations
Measurement, whilst being physically fundamental, is also fundamentally physical. Classical interferometric measurements suffer from physical limitations where the shot noise limit forms an upper bound of √n on a phase shift Δφ that can be resolved, with n being the average number of photons detected, making their applicability in modern nanoscale experiments quite restricted. Exotic quantum states such as maximally entangled NOON states (|N0〉A;B+|0N〉A;B)/√2, where N is the number of entangled photons are of particular importance as they have repeatedly been shown to exhibit phase super-resolution. Interesting post selection schemes and time-reversed detection schemes have been investigated for phase estimation using NOON states. These methods are inherently non-deterministic at the preparation or measurement stage and thus do not optimise resources, for instance the number of photons passing through the sample greatly exceeds those used in the measurement.
The ultimate sensitivity of optical measurements is a key element of many recent works. Classically, it is mainly limited by the shot noise limit. However, a measurement setup that incorporates quantum mechanical principles can surpass the shot noise limit and reach the Heisenberg limit. Nevertheless, many of those experiments fail to break even the classical shot-noise limit. Following a recent proposal, we present here the results of optical phase measurements with a photon-number resolving detector using coherent states of up to 4200 photons on average. An additional scheme that can be implemented using standard single-photon detectors is also presented, and the results of the two schemes are compared. These measurements present deterministic single-shot sub-wavelength super-resolution up to 288 better than the optical wavelength. The results follow the classically limited sensitivity, up to 86 times better than the wavelength.
We demonstrate phase super-resolution in the absence of entangled states. The key insight is to use the inherent time-reversal symmetry of quantum mechanics: our theory shows that it is possible to measure, as opposed to prepare, entangled states. Our approach is robust, requiring only photons that exhibit classical interference: we experimentally demonstrate high-visibility phase super-resolution with three, four, and six photons using a standard laser and photon counters. Our six-photon experiment demonstrates the best phase super-resolution yet reported with high visibility and resolution.
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 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 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).