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Direct Measurement of the Photon’s Spatial Wave Function

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Quantum Photonics: Pioneering Advances and Emerging Applications

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 217))

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Abstract

We overview recent progress in the tomography of structured light fields, with an emphasis on the method of direct measurement. Direct measurement provides a scalable and easy- to-implement approach for characterizing the transverse structure of single photons. This protocol is particularly attractive in light of the emerging role of high-dimensional optical states as a resource for encoding quantum information. We present a summary of various implementations of this technique that aim to characterize the spatial degree of freedom of the optical field.

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References

  1. W.K. Wootters, W.H. Zurek, A single quantum cannot be cloned. Nature 299(5886), 802 (1982)

    Article  ADS  Google Scholar 

  2. D. Dieks, Communication by EPR devices. Phys. Lett. A 92(6), 271 (1982)

    Article  ADS  Google Scholar 

  3. P.W. Milonni, M.L. Hardies, Photons cannot always be replicated. Phys. Lett. A 92(7), 321 (1982)

    Article  ADS  Google Scholar 

  4. B. Kanseri, T. Iskhakov, I. Agafonov, M. Chekhova, G. Leuchs, Three-dimensional quantum polarization tomography of macroscopic Bell states. Phys. Rev. A 85(2), 022126 (2012)

    Article  ADS  Google Scholar 

  5. M. Cramer, M.B. Plenio, S.T. Flammia, R. Somma, D. Gross, S.D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.K. Liu, Efficient quantum state tomography. Nat. Commun. 1(9), 149 (2010)

    Article  ADS  Google Scholar 

  6. M. Hofheinz, H. Wang, M. Ansmann, R.C. Bialczak, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, J. Wenner, J.M. Martinis, A.N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459(7246), 546 (2009)

    Article  ADS  Google Scholar 

  7. K. Resch, P. Walther, A. Zeilinger, Full characterization of a three-photon Greenberger-Horne-Zeilinger state using quantum state tomography. Phys. Rev. Lett. 94(7), 070402 (2005)

    Article  ADS  Google Scholar 

  8. M. Beck, C. Dorrer, I. Walmsley, Joint quantum measurement using unbalanced array detection. Phys. Rev. Lett. 87(25), 253601 (2001)

    Article  ADS  Google Scholar 

  9. D.F.V. James, P.G. Kwiat, W.J. Munro, A.G. White, Measurement of qubits. Phys. Rev. A 64(5), 052312 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  10. M.G. Raymer, M. Beck, D. McAlister, Complex wave-field reconstruction using phase-space tomography. Phys. Rev. Lett. 72(8), 1137 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  11. D. Smithey, M. Beck, M. Raymer, A. Faridani, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum. Phys. Rev. Lett. 70(9), 1244 (1993)

    Article  ADS  Google Scholar 

  12. K. Vogel, H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40(5), 2847 (1989)

    Article  ADS  Google Scholar 

  13. M. Agnew, J. Leach, M. McLaren, F.S. Roux, R.W. Boyd, Tomography of the quantum state of photons entangled in high dimensions. Phys. Rev. A 84(6), 062101 (2011)

    Article  ADS  Google Scholar 

  14. J.S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, Direct measurement of the quantum wavefunction. Nature 474(7350), 188 (2011)

    Article  Google Scholar 

  15. J.Z. Salvail, M. Agnew, A.S. Johnson, E. Bolduc, J. Leach, R.W. Boyd, Full characterization of polarization states of light via direct measurement. Nature Photon. 7(4), 316 (2013)

    Article  ADS  Google Scholar 

  16. M. Malik, M. Mirhosseini, M.P.J. Lavery, J. Leach, M.J. Padgett, R.W. Boyd, Direct measurement of a 27-dimensional orbital-angular-momentum state vector. Nat. Commun. 5, 3115 (2014)

    Article  ADS  Google Scholar 

  17. J. Fischbach, M. Freyberger, Quantum optical reconstruction scheme using weak values. Phys. Rev. A 86(5), 052110 (2012)

    Article  ADS  Google Scholar 

  18. S. Wu, State tomography via weak measurements. Sci. Rep. 3, (2013)

    Google Scholar 

  19. A. Di Lorenzo, Sequential measurement of conjugate variables as an alternative quantum state tomography. Phys. Rev. Lett. 110(1), 010404 (2013)

    Article  Google Scholar 

  20. Y. Aharonov, D. Albert, L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60(14), 1351 (1988)

    Article  ADS  Google Scholar 

  21. J.S. Lundeen, K. Resch, Practical measurement of joint weak values and their connection to the annihilation operator. Phys. Lett. A 334, 337 (2005)

    Article  ADS  Google Scholar 

  22. J. Dressel, S. Agarwal, A.N. Jordan, Contextual values of observables in quantum measurements. Phys. Rev. Lett. 104(24), 240401 (2010)

    Article  ADS  Google Scholar 

  23. P. Dixon, D. Starling, A. Jordan, J. Howell. Ultrasensitive beam deflection measurement via interferometric weak value amplification. Phys. Rev. Lett. 102(17) (2009)

    Google Scholar 

  24. N. Ritchie, J. Story, R. Hulet, Realization of a measurement of a “weak value”. Phys. Rev. Lett. 66(9), 1107 (1991)

    Article  ADS  Google Scholar 

  25. O. Hosten, P. Kwiat, Observation of the spin hall effect of light via weak measurements. Science 319(5864), 787 (2008)

    Article  ADS  Google Scholar 

  26. O.S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, R.W. Boyd, Amplification of angular rotations using weak measurements. Phys. Rev. Lett. 112(20), 200401 (2014)

    Article  ADS  Google Scholar 

  27. N. Brunner, C. Simon, Measuring small longitudinal phase shifts: weak measurements or standard interferometry? Phys. Rev. Lett. 105(1), 010405 (2010)

    Article  ADS  Google Scholar 

  28. N. Brunner, V. Scarani, M. Wegmüller, M. Legré, N. Gisin, Direct measurement of superluminal group velocity and signal velocity in an optical fiber. Phys. Rev. Lett. 93(20), 203902 (2004)

    Article  ADS  Google Scholar 

  29. A. Feizpour, X. Xing, A.M. Steinberg, Amplifying single-photon nonlinearity using weak measurements. Phys. Rev. Lett. 107(13), 133603 (2011)

    Article  ADS  Google Scholar 

  30. M. Mirhosseini, O.S. Magaña-Loaiza, S.M. Hashemi Rafsanjani, R.W. Boyd, Compressive direct measurement of the quantum wave function. Phys. Rev. Lett. 113(9), 090402 (2014)

    Article  ADS  Google Scholar 

  31. C. Bamber, B. Sutherland, A. Patel, C. Stewart, J.S. Lundeen, Measurement of the transverse electric field profile of light by a self-referencing method with direct phase determination. Opt. Express 20(3), 2034 (2012)

    Article  ADS  Google Scholar 

  32. J. Lundeen, C. Bamber, Procedure for direct measurement of general quantum states using weak measurement. Phys. Rev. Lett. 108(7) (2012)

    Google Scholar 

  33. C. Bamber, J.S. Lundeen, Observing dirac’s classical phase space analog to the quantum state. Phys. Rev. Lett. 112, 070405 (2014)

    Article  ADS  Google Scholar 

  34. G.S. Thekkadath, L. Giner, Y. Chalich, M.J. Horton, J. Banker, J.S. Lundeen. Direct measurement of the density matrix of a quantum system. Phys. Rev. Lett. 117(12), 120401 (2016)

    Google Scholar 

  35. G.A. Howland, D.J. Lum, J.C. Howell, Compressive wavefront sensing with weak values. Opt. Express 22(16), 18870 (2014)

    Article  ADS  Google Scholar 

  36. E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett, M. Padgett, Fourier relationship between angular position and optical orbital angular momentum. Opt. Express 14(20), 9071 (2006)

    Article  ADS  Google Scholar 

  37. B. Jack, M.J. Padgett, S. Franke-Arnold, Angular diffraction. New J. Phys. 10(10), 103013 (2008)

    Article  ADS  Google Scholar 

  38. M. Mirhosseini, O.S. Magaña-Loaiza, C. Chen, S.M.H. Rafsanjani, R.W. Boyd, Wigner distribution of twisted photons. Phys. Rev. Lett. 116(13), 130402 (2016)

    Article  ADS  Google Scholar 

  39. M. Mirhosseini, M. Malik, Z. Shi, R.W. Boyd, Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun. 4, 2781 (2013)

    Article  ADS  Google Scholar 

  40. M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, M.P.J. Lavery, M.J. Padgett, D.J. Gauthier, R.W. Boyd, High-dimensional quantum cryptography with twisted light. New J. Phys. 17(3), 033033 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  41. A. Shabani, R.L. Kosut, M. Mohseni, H. Rabitz, M.A. Broome, M.P. Almeida, A. Fedrizzi, A.G. White, Efficient measurement of quantum dynamics via compressive sensing. Phys. Rev. Lett. 106(10), 100401 (2011)

    Article  ADS  Google Scholar 

  42. G. Howland, J. Howell, Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera. Phys. Rev. X 3(1), 011013 (2013)

    Google Scholar 

  43. W.T. Liu, T. Zhang, J.Y. Liu, P.X. Chen, J.M. Yuan, Experimental quantum state tomography via compressed sampling. Phys. Rev. Lett. 108(17), 170403 (2012)

    Article  ADS  Google Scholar 

  44. D. Gross, Y.K. Liu, S.T. Flammia, S. Becker, J. Eisert, Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105(15), 150401 (2010)

    Article  ADS  Google Scholar 

  45. O. Katz, Y. Bromberg, Y. Silberberg, Compressive ghost imaging. Appl. Phys. Lett. 95(13), 131110 (2009)

    Article  ADS  Google Scholar 

  46. R.G. Baraniuk, Single-pixel imaging via compressive sampling. IEEE Sig. Process Mag. (2008)

    Google Scholar 

  47. J. Romberg, Imaging via compressive sampling. IEEE Sig. Process Mag. 25(2), 14 (2008)

    Article  ADS  Google Scholar 

  48. E. Candes, J. Romberg, Sparsity and incoherence in compressive sampling. Inverse Probl. 23(3), 969 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  49. P. Zerom, K.W.C. Chan, J.C. Howell, R.W. Boyd, Entangled-photon compressive ghost imaging. Phys. Rev. A 84(6), 061804 (2011)

    Article  ADS  Google Scholar 

  50. K. Lyons, J. Dressel, A.N. Jordan, J.C. Howell, P.G. Kwiat, Power-recycled weak-value-based metrology. Phys. Rev. Lett. 114(17), 170801 (2015)

    Article  ADS  Google Scholar 

  51. C. Bamber, J.S. Lundeen, Observing Dirac’s classical phase space analog to the quantum state. Phys. Rev. Lett. 112(7), 070405 (2014)

    Google Scholar 

  52. Z. Shi, M. Mirhosseini, J. Margiewicz, M. Malik, F. Rivera, Z. Zhu, R.W. Boyd, Scan-free direct measurement of an extremely high-dimensional photonic state. Optica 2(4), 388–392 (2015)

    Article  Google Scholar 

  53. D.F. McAlister, M. Beck, L. Clarke, A. Mayer, M.G. Raymer, Optical phase retrieval by phase-space tomography andfractional-order fourier transforms. Opt. Lett. 20(10), 1181 (1995)

    Article  ADS  Google Scholar 

  54. T. Durt, B.G. Englert, I. Bengtsson, On mutually unbiased bases. Int. J. Quantum Inf. 08(04), 535 (2010)

    Article  Google Scholar 

  55. H.F. Hofmann, Complex joint probabilities as expressions of reversible transformations in quantum mechanics. New J. Phys. 14(4), 043031 (2012)

    Article  ADS  Google Scholar 

  56. E. Arthurs, J. Kelly, On the simultaneous measurement of a pair of conjugate observables. Bell System Tech. J. 44, 725 (1965)

    Article  Google Scholar 

  57. J.H. Shapiro, S.S. Wagner, Phase and amplitude uncertainties in heterodyne detection. IEEE J. Quantum Electron. 20(7), 803 (1984)

    Article  ADS  Google Scholar 

  58. U. Leonhardt, H. Paul, Phase measurement and q function. Phys. Rev. A 47(4), R2460 (1993)

    Article  ADS  Google Scholar 

  59. U. Leonhardt, Quantum-state tomography and discrete Wigner function. Phys. Rev. Lett. 74(21), 4101 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  60. U. Leonhardt, Discrete Wigner function and quantum-state tomography. Phys. Rev. A 53(5), 2998 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  61. E. Haapasalo, P. Lahti, J. Schultz, Weak versus approximate values in quantum state determination. Phys. Rev. A 84(5), 052107 (2011)

    Article  ADS  Google Scholar 

  62. G. Vallone, D. Dequal, Strong measurements give a better direct measurement of the quantum wave function. Phys. Rev. Lett. 116(4), 040502 (2016)

    Article  ADS  Google Scholar 

  63. R. Okamoto, M. Iefuji, S. Oyama, K. Yamagata, H. Imai, A. Fujiwara, S. Takeuchi, Experimental demonstration of adaptive quantum state estimation. Phys. Rev. Lett. 109, 130404 (2012)

    Article  ADS  Google Scholar 

  64. D.H. Mahler, L.A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A.M. Steinberg, Adaptive quantum state tomography improves accuracy quadratically. Phys. Rev. Lett. 111, 183601 (2013)

    Article  ADS  Google Scholar 

  65. E. Bolduc, G. Gariepy, J. Leach, Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices. Nat. commun. 7, (2016)

    Google Scholar 

  66. U. Leonhardt, Measuring the Quantum State of Light. Cambridge Studies in Modern Optics (Cambridge University Press, 1997)

    Google Scholar 

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Correspondence to Mohammad Mirhosseini .

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Mirhosseini, M., Lundeen, J.S., Boyd, R.W. (2019). Direct Measurement of the Photon’s Spatial Wave Function. In: Boyd, R., Lukishova, S., Zadkov, V. (eds) Quantum Photonics: Pioneering Advances and Emerging Applications. Springer Series in Optical Sciences, vol 217. Springer, Cham. https://doi.org/10.1007/978-3-319-98402-5_2

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