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Entanglement-enhanced measurement of a completely unknown optical phase



Precise interferometric measurement is vital to many scientific and technological applications. Using quantum entanglement allows interferometric sensitivity that surpasses the shot-noise limit (SNL)1,2. To date, experiments demonstrating entanglement-enhanced sub-SNL interferometry3,4,5,6, and most theoretical treatments7,8,9,10,11,12,13, have addressed the goal of increasing signal-to-noise ratios. This is suitable for phase-sensing—detecting small variations about an already known phase. However, it is not sufficient for ab initio phase-estimation—making a self-contained determination of a phase that is initially completely unknown within the interval [0, 2π). Both tasks are important2, but not equivalent. To move from the sensing regime to the ab initio estimation regime requires a non-trivial phase-estimation algorithm14,15,16,17. Here, we implement a ‘bottom-up’ approach, optimally utilizing the available entangled photon states, obtained by post-selection5,6. This enables us to demonstrate sub-SNL ab initio estimation of an unknown phase by entanglement-enhanced optical interferometry.

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Figure 1: A Mach–Zehnder interferometer with the phase shift to be measured, φ, in one arm, and a controllable phase shift, θ, in the other arm.
Figure 2: Experimental layout.
Figure 3: Fits (lines) of measured (points) photon counts corresponding to fringes given by equations (1) to (4).
Figure 4: Standard deviations δ φ of phase estimates for varying total photon number N.

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  1. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004).

    Article  ADS  Google Scholar 

  2. Wiseman, H. M. & Milburn, G. J. Quantum Measurement and Control (Cambridge Univ. Press, 2010).

    MATH  Google Scholar 

  3. Meyer, V. et al. Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions. Phys. Rev. Lett. 86, 5870–5873 (2001).

    Article  ADS  Google Scholar 

  4. Leibfried, D. et al. Creation of a six-atom ‘Schrödinger cat’ state. Nature 438, 639–642 (2005).

    Article  ADS  Google Scholar 

  5. Nagata, T., Okamoto, R., O'Brien, J. L., Sasaki, K. & Takeuchi, S. Beating the standard quantum limit with four-entangled photons. Science 316, 726–729 (2007).

    Article  ADS  Google Scholar 

  6. Okamoto, R. et al. Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers. New J. Phys. 10, 073033 (2008).

    Article  ADS  Google Scholar 

  7. Caves, C. M. Quantum-mechanical noise in an interferometer. Phys. Rev. D 23, 1693–1708 (1981).

    Article  ADS  Google Scholar 

  8. Yurke, B., McCall, S. L. & Klauder, J. R. SU(2) and SU(1,1) interferometers. Phys. Rev. A 33, 4033–4054 (1986).

    Article  ADS  Google Scholar 

  9. Summy, G. S. & Pegg, D. T. Phase optimized quantum states of light. Opt. Commun. 77, 75–79 (1990).

    Article  ADS  Google Scholar 

  10. Holland, M. J. & Burnett, K. Interferometric detection of optical phase shifts at the Heisenberg limit. Phys. Rev. Lett. 71, 1355–1358 (1993).

    Article  ADS  Google Scholar 

  11. Sanders, B. C. & Milburn, G. J. Optimal quantum measurements for phase estimation. Phys. Rev. Lett. 75, 2944–2947 (1995).

    Article  ADS  Google Scholar 

  12. Lee, H., Kok, P. & Dowling, J. P. A quantum Rosetta stone for interferometry. J. Mod. Opt. 49, 2325–2338 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  13. Steuernagel, O. De Broglie wavelength reduction for a multiphoton wave packet. Phys. Rev. A 65, 033820 (2002).

    Article  ADS  Google Scholar 

  14. Braunstein, S. L. How large a sample is needed for the maximum likelihood estimator to be approximately Gaussian? J. Phys. A 25, 3813–3826 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  15. Durkin, G. A. & Dowling, J. P. Local and global distinguishability in quantum interferometry. Phys. Rev. Lett. 99, 070801 (2007).

    Article  ADS  Google Scholar 

  16. Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393–396 (2007).

    Article  ADS  Google Scholar 

  17. Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009).

    Article  ADS  Google Scholar 

  18. Sanders, B. C. Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement. Phys. Rev. A 40, 2417–2427 (1989).

    Article  ADS  Google Scholar 

  19. Cable, H. & Dowling, J. P. Efficient generation of large number–path entanglement using only linear optics and feed-forward. Phys. Rev. Lett. 99, 163604 (2007).

    Article  ADS  Google Scholar 

  20. Walther, P. et al. De Broglie wavelength of a non-local four-photon state. Nature 429, 158–161 (2004).

    Article  ADS  Google Scholar 

  21. Mitchell, M. W., Lundeen, J. S. & Steinberg, A. M. Super-resolving phase measurements with a multiphoton entangled state. Nature 429, 161–164 (2004).

    Article  ADS  Google Scholar 

  22. Afek, I., Ambar, O. & Silberberg, Y. High-NOON States by mixing quantum and classical light. Science 328, 879–881 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  23. Berry, D. W. & Wiseman, H. M. Optimal states and almost optimal adaptive measurements for quantum interferometry. Phys. Rev. Lett. 85, 5098–5101 (2000).

    Article  ADS  Google Scholar 

  24. Berry, D. W., Wiseman, H. M. & Breslin, J. K. Optimal input states and feedback for interferometric phase estimation. Phys. Rev. A 63, 053804 (2001).

    Article  ADS  Google Scholar 

  25. Thomas-Peter, N. L., Smith, B. J., Dorner, U. & Walmsley, I. A. Real-world quantum sensors: evaluating resources for precision measurement. Preprint at (2010).

  26. Kacprowicz, M. et al. Experimental quantum-enhanced estimation of a lossy phase shift. Nature Photon. 4, 357–360 (2010).

    Article  ADS  Google Scholar 

  27. Cable, H. & Durkin, G. A. Parameter estimation with entangled photons produced by parametric down-conversion. Phys. Rev. Lett. 105, 013603 (2010).

    Article  ADS  Google Scholar 

  28. Lee, T.-W. et al. Optimization of quantum interferometric metrological sensors in the presence of photon loss. Phys. Rev. A 80, 063803 (2009).

    Article  ADS  Google Scholar 

  29. Knysh, S., Smelyanskiy, V. N. & Durkin, G. A. Scaling laws for precision in quantum interferometry and bifurcation landscape of optimal state. Preprint at (2010).

  30. Politi, A., Matthews, J., Thompson, M. G. & O'Brien, J. L. Integrated quantum photonics. IEEE J. Sel. Top. Quant. Electron. 15, 1673–1684 (2009).

    Article  ADS  Google Scholar 

  31. Aspachs, M., Calsamiglia, J., Muñoz-Tapia, R. & Bagan, E. Phase estimation for thermal Gaussian states. Phys. Rev. A 79, 033834 (2009).

    Article  ADS  Google Scholar 

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The authors thank J. O'Brien for helpful discussions. This work was supported by the Australian Research Council.

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Authors and Affiliations



G.J.P. devised and designed the experiment. H.M.W. devised and analysed the theoretical approach. G.Y.X. constructed and operated the experiment, and collected the data. B.L.H. automated and experimentally implemented the feedback algorithm, and contributed to the operation of the apparatus. D.W.B. constructed and performed numerical simulations of the algorithm, and performed numerical processing of results. All authors contributed to the manuscript.

Corresponding authors

Correspondence to H. M. Wiseman or G. J. Pryde.

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The authors declare no competing financial interests.

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Xiang, G., Higgins, B., Berry, D. et al. Entanglement-enhanced measurement of a completely unknown optical phase. Nature Photon 5, 43–47 (2011).

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