Abstract

Multiparameter estimation is a general problem that aims at measuring unknown physical quantities, obtaining high precision in the process. In this context, the adoption of quantum resources promises a substantial boost in achievable performances with respect to the classical case. However, several open problems remain to be addressed in the multiparameter scenario. A crucial requirement is the identification of suitable platforms to develop and experimentally test novel efficient methodologies that can be employed in this general framework. We report the experimental implementation of a reconfigurable integrated multimode interferometer designed for simultaneous estimation of two optical phases. We verify the high-fidelity operation of the implemented device and demonstrate quantum-enhanced performances in two-phase estimation with respect to the best classical case, post-selected to the number of detected coincidences. This device can be employed to test general adaptive multiphase protocols due to its high reconfigurability level, and represents a powerful platform to investigate the multiparameter estimation scenario.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (10)

R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, “Multiparameter Gaussian quantum metrology,” Phys. Rev. A 98, 012114 (2018).
[Crossref]

E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M. G. Genoni, and M. Barbieri, “Entangling measurements for multiparameter estimation with two qubits,” Quantum Sci. Technol. 3, 01LT01 (2018).
[Crossref]

E. Roccia, V. Cimini, M. Sbroscia, I. Gianani, L. Ruggiero, L. Mancino, M. G. Genoni, M. A. Ricci, and M. Barbieri, “Multiparameter approach to quantum phase estimation with limited visibility,” Optica 5, 001171 (2018).
[Crossref]

W. Ge, K. Jacobs, Z. Eldredge, A. V. Gorshkov, and M. Foss-Feig, “Distributed quantum metrology and the entangling power of linear networks,” Phys. Rev. Lett. 121, 043604 (2018).
[Crossref]

M. Gessner, L. Pezzè, and A. Smerzi, “Sensitivity bounds for multiparameter quantum metrology,” Phys. Rev. Lett. 121, 130503 (2018).
[Crossref]

A. Lumino, E. Polino, A. S. Rab, G. Milani, N. Spagnolo, N. Wiebe, and F. Sciarrino, “Experimental phase estimation enhanced by machine learning,” Phys. Rev. Appl. 10, 044033 (2018).
[Crossref]

S. Atzeni, A. S. Rab, G. Corrielli, E. Polino, M. Valeri, P. Mataloni, N. Spagnolo, A. Crespi, F. Sciarrino, and R. Osellame, “Integrated sources of entangled photons at the telecom wavelength in femtosecond-laser-written circuits,” Optica 5, 000311 (2018).
[Crossref]

L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, “Quantum metrology with nonclassical states of atomic ensembles,” Rev. Mod. Phys. 90, 035005 (2018).
[Crossref]

P. A. Ivanov and N. V. Vitanov, “Quantum sensing of the phase space displacement parameters using a single trapped ion,” Phys. Rev. A 97, 032308 (2018).
[Crossref]

T. J. Proctor, P. A. Knott, and J. A. Dunningham, “Multiparameter estimation in networked quantum sensors,” Phys. Rev. Lett. 120, 080501 (2018).
[Crossref]

2017 (8)

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. Photonics 11, 700–703 (2017).
[Crossref]

L. Zhang and K. W. C. Chan, “Quantum multiparameter estimation with generalized balanced multimode noon-like states,” Phys. Rev. A 95, 032321 (2017).
[Crossref]

R. Yousefjani, R. Nichols, S. Salimi, and G. Adesso, “Estimating phase with a random generator: strategies and resources in multiparameter quantum metrology,” Phys. Rev. A 95, 062307 (2017).
[Crossref]

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. O’Brien, and M. G. Thompson, “Experimental Bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

H. Yuan and C.-H. Fung, “Quantum metrology matrix,” Phys. Rev. A 96, 012310 (2017).
[Crossref]

N. Liu and H. Cable, “Quantum-enhanced multi-parameter estimation for unitary photonic systems,” Quantum Sci. Technol. 2, 025008 (2017).
[Crossref]

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

2016 (6)

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
[Crossref]

P. A. Knott, T. J. Proctor, A. J. Hayes, J. F. Ralph, P. Kok, and J. A. Dunningham, “Local versus global strategies in multiparameter estimation,” Phys. Rev. A 94, 062312 (2016).
[Crossref]

C. N. Gagatsos, D. Branford, and A. Datta, “Gaussian systems for quantum-enhanced multiple phase estimation,” Phys. Rev. A 94, 042342 (2016).
[Crossref]

M. A. Ciampini, N. Spagnolo, C. Vitelli, L. Pezzè, A. Smerzi, and F. Sciarrino, “Quantum-enhanced multiparameter estimation in multiarm interferometers,” Sci. Rep. 6, 28881 (2016).
[Crossref]

J. Liu, X.-M. Lu, Z. Sun, and X. Wang, “Quantum multiparameter metrology with generalized entangled coherent state,” J. Phys. A 49, 115302 (2016).
[Crossref]

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys.: X 1, 621–639 (2016).
[Crossref]

2015 (6)

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. F. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2, 510–516 (2015).
[Crossref]

F. Flamini, L. Magrini, A. S. Rab, N. Spagnolo, V. D’Ambrosio, P. Mataloni, F. Sciarrino, T. Zandrini, A. Crespi, R. Ramponi, and R. Osellame, “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light Sci. Appl. 4, e355 (2015).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martin-Lopez, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
[Crossref]

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref]

D. Safranek, A. R. Lee, and I. Fuentes, “Quantum parameter estimation using multi-mode Gaussian states,” New J. Phys. 17, 073016 (2015).
[Crossref]

M. Altorio, M. G. Genoni, M. D. Vidrighin, F. Somma, and M. Barbieri, “Weak measurements and the joint estimation of phase and phase diffusion,” Phys. Rev. A 92, 032114 (2015).
[Crossref]

2014 (2)

Y. Gao and H. Lee, “Bounds on quantum multiple-parameter estimation with Gaussian state,” Eur. Phys. J. D 68, 347 (2014).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

2013 (4)

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

N. Spagnolo, C. Vitelli, L. Aparo, P. Mataloni, F. Sciarrino, A. Crespi, R. Ramponi, and R. Osellame, “Three-photon bosonic coalescence in an integrated tritter,” Nat. Commun. 4, 1606 (2013).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

2012 (3)

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100, 233704 (2012).
[Crossref]

C. E. Granade, C. Ferrie, N. Wiebe, and D. G. Cory, “Robust online Hamiltonian learning,” New J. Phys. 14, 103013 (2012).
[Crossref]

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, and F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

2011 (3)

L. M. Pham, D. L. Sage, P. L. Stanwix, T. K. Yeung, D. Glenn, A. Trifonov, P. Cappellaro, P. R. Hemmer, M. D. Lukin, H. Park, A. Yacoby, and R. L. Walsworth, “Magnetic field imaging with NV ensembles,” New J. Phys. 13, 045021 (2011).
[Crossref]

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

2010 (1)

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref]

2009 (5)

A. Freise, S. Chelkowski, S. Hild, W. Del Pozzo, A. Perreca, and A. Vecchio, “Triple Michelson interferometer for a third-generation gravitational wave detector,” Class. Quantum Grav. 26, 085012 (2009).
[Crossref]

M. Tsang, “Quantum imaging beyond the diffraction limit by optical centroid measurements,” Phys. Rev. Lett. 102, 253601 (2009).
[Crossref]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[Crossref]

L. Pezzè and A. Smerzi, “Entanglement nonlinear dynamics, and the Heisenberg limit,” Phys. Rev. Lett. 102, 100401 (2009).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

2007 (1)

J. Kahn, “Fast rate estimation of a unitary operation in SU,” Phys. Rev. A 75, 022326 (2007).
[Crossref]

2006 (2)

M. Hayashi, “Parallel treatment of estimation of SU(2) and phase estimation,” Phys. Lett. A 354, 183–189 (2006).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref]

2004 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

2003 (1)

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2002 (1)

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2001 (2)

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2000 (1)

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1994 (1)

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1974 (1)

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1973 (1)

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Y. Yang, G. Chiribella, and M. Hayashi, “Attaining the ultimate precision limit in quantum state estimation,” arXiv:1802.07587 (2018).

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Daria, V.

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Datta, A.

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E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M. G. Genoni, and M. Barbieri, “Entangling measurements for multiparameter estimation with two qubits,” Quantum Sci. Technol. 3, 01LT01 (2018).
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Adv. Phys.: X (1)

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Supplementary Material (1)

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» Supplement 1       This supplemental material provides more details on the characterization of the integrated interferometer.

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Figures (7)

Fig. 1.
Fig. 1. Experimental apparatus. (a) Layout of the integrated reconfigurable device. Three straight waveguide segments are included between two multiport splitters UA and UB. The dynamical control of the phases is achieved by thermo-optic phase shifters. Central inset: conceptual scheme of the interferometer. Top left inset: layout of the multiport splitters UA,B, each composed of three directional couplers (T1,2,3A,B, green regions) and a dynamically reconfigurable phase (ϕTA,B, red). By appropriately tuning ϕTA,B, the two multiport splitters can be set to operate as balanced tritters. (b) Parametric down-conversion source for generation of single-photon and two-photon states. The dotted path is employed to inject classical light into the device for device alignment. The generated photons (p1 and p2) are coupled in single-mode fibers and sent to the integrated device. (c) Coupling and detection stage. Photons are coupled to the device by an input fiber array (single-mode operation), and collected with a second fiber array (multimode operation). For single-photon inputs, photon (p2) is directly measured to act as a trigger. For two-photon inputs, both photons are injected in the interferometer, and the output state is measured by adding a set of fiber beam splitters to detect bunching events. PDC, parametric down-conversion; SHG, second-harmonic generation; DM, dichroic mirror; HWP, half-wave plate; PBS, polarizing beam splitter; IF, interference filter; PC, polarization controller; FBS, fiber beam splitter; APD, avalanche photodiode.
Fig. 2.
Fig. 2. (a)–(f) Two-photon probabilities P(23ij) as a function of the phase differences Δϕ1 and Δϕ2. The latter are varied, changing the dissipated powers on resistors R1 and R2. In all plots, dots are experimental data, while surfaces are the theoretical expectations from the circuit characterization process. Error bars are standard deviations due to the Poissonian statistics of the measured single-photon counts and two-photon coincidences. The good agreement between model and experimental data is quantified by the average R2 value over all output combinations R2=0.835. In the model, photon indistinguishability of V=0.95 is taken into account.
Fig. 3.
Fig. 3. Two-photon measurements P(23kl) for an input state with a single photon on modes (2,3) as a function of the relative time delay δτ, normalized over the photon Hong–Ou–Mandel width σ. (a) Phase values set at Δϕ1=1.745 and Δϕ2=0.349. (b) Phase values set at Δϕ1=1.048 and Δϕ2=2.444. Points are experimental data, while dashed lines are predictions from the reconstructed parameters. [Red, output (1,2); green, output (1,3); blue, output (2,3); black, output (1,1); cyan, output (2,2); purple, output (3,3)]. Photon indistinguishability is introduced in the predictions by mixing the probability with indistinguishable and distinguishable photons with a parameter e(δτ/σ)2. Error bars are standard deviations due to the Poissonian statistics of the measured two-photon coincidences.
Fig. 4.
Fig. 4. Cramer–Rao bound Tr(I1) for multiphase estimation with two-photon input states. (a)–(c) CRB for the implemented device evaluated from the reconstructed parameters. (a) Input (1,2); (b) input (1,3); and (c) input (2,3). (d)–(f) CRB for the ideal three-mode interferometer. (d) Input (1,2); (e) input (1,3); and (f) input (2,3). In the ideal interferometer case, points where the Fisher information matrix is singular are not shown. Regions included within white closed curves highlight the presence of improved performances with respect to the QCRB with two distinguishable single-photon inputs.
Fig. 5.
Fig. 5. Results of a maximum likelihood estimator for local phase estimation at (Δϕ1,Δϕ2)=(1.159,2.810) with input (2,3). Points, experimental data, obtained by averaging over 100 random sequences of m coincidence events (2m photons) drawn from the measured Nev=1230 two-photon events. Top plot, red dashed line corresponds to Tr(I1), black dashed line to the optimal sensitivity Tr(H1) with 2m distinguishable single-photon inputs, and black dotted line to the optimal sensitivity when the phases are estimated separately with classical inputs. Bottom plot, green points (data) and line (I1)111/2 correspond to δ(Δϕ1); blue points (data) and line (I1)221/2 correspond to δ(Δϕ2).
Fig. 6.
Fig. 6. (a) Conceptual layout employed to tune the input and output transformations UA and UB. (b) Experimental single-photon probability measurements (blue bars) at (Δϕ1,Δϕ2)=(0,0), compared with the identity corresponding to the ideal case (red bars). (c) Experimental two-photon probability measurements for input (1,3) and output (1,3) as a function of (Δϕ1,Δϕ2) by tuning voltages applied to resistors R1 and R2. (b), (c) Transformations UA and UB are set to reach the condition UBUA=I (up to a set of output phases) as described in the main text.
Fig. 7.
Fig. 7. Cramer–Rao bound Tr(I1) for multiphase estimation with a three-photon input state (1,2,3). (a) CRB for the implemented device evaluated from the reconstructed parameters, and (b) CRB for the ideal three-mode interferometer. In the ideal interferometer case, points where the Fisher information matrix is singular are not shown. Regions included within white closed curves highlight the presence of improved performances with respect to the QCRB with three optimal distinguishable single-photon inputs.

Equations (2)

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Δϕj=i=16(αjiPRi+αjiNLPRi2),
i=1nVar(Δϕi)Tr[I1(ϕ)]mTr[H1(ϕ)]m,