Article | Published:

Direct measurement of a 27-dimensional orbital-angular-momentum state vector

Nature Communications volume 5, Article number: 3115 (2014) | Download Citation

Abstract

The measurement of a quantum state poses a unique challenge for experimentalists. Recently, the technique of ‘direct measurement’ was proposed for characterizing a quantum state in situ through sequential weak and strong measurements. While this method has been used for measuring polarization states, its real potential lies in the measurement of states with a large dimensionality. Here we show the practical direct measurement of a high-dimensional state vector in the discrete basis of orbital angular momentum. Through weak measurements of orbital angular momentum and strong measurements of angular position, we measure the complex probability amplitudes of a pure state with a dimensionality, d=27. Further, we use our method to directly observe the relationship between rotations of a state vector and the relative phase between its orbital-angular-momentum components. Our technique has important applications in high-dimensional classical and quantum information systems and can be extended to characterize other types of large quantum states.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1.

    Mathematical Foundations of Quantum Mechanics Princeton University (1955).

  2. 2.

    Wheeler J. A., Zurek W. H. (eds)Quantum Theory and Measurement Princeton University (1983).

  3. 3.

    , & Quantum tomography. Adv. Imag. Elect. Phys. 128, 205–308 (2003).

  4. 4.

    , , & Direct measurement of the quantum wavefunction. Nature 474, 188–191 (2011).

  5. 5.

    , & 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, 1351–1354 (1988).

  6. 6.

    , , , & Understanding Quantum Weak Values: Basics and Applications. Preprint at (2013).

  7. 7.

    et al. Full characterization of polarization states of light via direct measurement’. Nat. Photonics 7, 316–321 (2013).

  8. 8.

    , & Twisted photons. Nat. Phys. 3, 305–310 (2007).

  9. 9.

    , , & Interferometric measurement of the helical mode of a single photon. New J. Phys. 13, 053017 (2011).

  10. 10.

    & Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photon. 3, 161–204 (2011).

  11. 11.

    et al. Quantum entanglement of high angular momenta. Science 338, 640–643 (2012).

  12. 12.

    , , , & Experimental quantum cryptography with qutrits. New J. Phys. 8, 75 (2006).

  13. 13.

    et al. Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding. Opt. Express 20, 13195–13200 (2012).

  14. 14.

    , , , & Quantum key distribution using multilevel encoding: security analysis. J. Phys. A-Math. Gen. 35, 10065–10076 (2002).

  15. 15.

    , , & Entanglement of the orbital angular momentum states of photons. Nature 412, 313–316 (2001).

  16. 16.

    et al. Quantum correlations in optical angle-orbital angular momentum variables. Science 329, 662–665 (2010).

  17. 17.

    , , , & Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities. Nat. Phys. 7, 677–680 (2011).

  18. 18.

    Principles of Quantum Mechanics Plenum Press (1994).

  19. 19.

    , , , & Fourier relationship between angular position and optical orbital angular momentum. Opt. Express 14, 9071–9076 (2006).

  20. 20.

    , & Angular diffraction. New J. Phys. 10, 103013 (2008).

  21. 21.

    , , & Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun. 4, 2781 (2013).

  22. 22.

    , , , & Efficient sorting of orbital angular momentum states of light. Phys. Rev. Lett. 105, 153601 (2010).

  23. 23.

    et al. Refractive elements for the measurement of the orbital angular momentum of a single photon. Opt. Express 20, 2110–2115 (2012).

  24. 24.

    , , & Optimized kinoform structures for highly efficient fan-out elements. Appl. Optics 31, 5706–5711 (1992).

  25. 25.

    , , & Near-perfect sorting of orbital angular momentum and angular position states of light. Opt. Express 20, 24444–24449 (2012).

  26. 26.

    , , & Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator. Appl. Optics 39, 1549–1554 (2000).

  27. 27.

    , , & Pixelated phase computer holograms for the accurate encoding of scalar complex fields. J. Opt. Soc. Am. A 24, 3500–3507 (2007).

  28. 28.

    & Quantum theory of rotation angles. Phys. Rev. A 41, 3427–3435 (1990).

  29. 29.

    & Procedure for direct measurement of general quantum states using weak measurement. Phys. Rev. Lett. 108, 070402 (2012).

  30. 30.

    On the analogy between classical and quantum mechanics. Rev. Mod. Phys. 17, 195–199 (1945).

  31. 31.

    et al. Wigner–Weyl correspondence in quantum mechanics for continuous and discrete systems—a Dirac-inspired view. J. Phys. A-Math. Gen. 39, 1405–1423 (2006).

  32. 32.

    , , , & Measurement of the transverse electric field profile of light by a self-referencing method with direct phase determination. Opt. Express 20, 2034–2044 (2012).

  33. 33.

    , , & Qudit quantum-state tomography. Phys. Rev. A 66, 012303 (2002).

  34. 34.

    , , , & Tomography of the quantum state of photons entangled in high dimensions. Phys. Rev. A 84, 062101 (2011).

  35. 35.

    et al. Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon. Proc. SPIE 7948, 79480L–1–79480L–6 (2011).

  36. 36.

    et al. Rapid generation of light beams carrying orbital angular momentum. Opt. Express 21, 30196–30203 (2013).

  37. 37.

    & Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings. J. Opt. Soc. Am. A 24, 2280–2295 (2007).

  38. 38.

    & Practical measurement of joint weak values and their connection to the annihilation operator. Phys. Lett. A 334, 337–344 (2005).

Download references

Acknowledgements

This work was supported by the DARPA InPho program, the Canadian Excellence Research Chair (CERC) program, the Engineering and Physical Sciences Research Council (EPSRC), the Royal Society and the European Commission through a Marie Curie fellowship. M.Malik would like to thank Dr Justin Dressel for helpful discussions.

Author information

Affiliations

  1. The Institute of Optics, University of Rochester, Rochester, New York 14627, USA

    • Mehul Malik
    • , Mohammad Mirhosseini
    •  & Robert W. Boyd
  2. Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

    • Mehul Malik
  3. School of Physics and Astronomy, University of Glasgow, Glasgow, UK

    • Martin P. J. Lavery
    •  & Miles J. Padgett
  4. School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK

    • Jonathan Leach
  5. Department of Physics, University of Ottawa, Ottawa, Ontario, Canada, K1N 6N5

    • Jonathan Leach
    •  & Robert W. Boyd

Authors

  1. Search for Mehul Malik in:

  2. Search for Mohammad Mirhosseini in:

  3. Search for Martin P. J. Lavery in:

  4. Search for Jonathan Leach in:

  5. Search for Miles J. Padgett in:

  6. Search for Robert W. Boyd in:

Contributions

M.Ma. devised the concept of the experiment. M.Ma. and M.Mi. performed the experiment and analysed data. M.P.J.L. assisted with the experiment. J.L. advised on early aspects of experimental design. R.W.B. and M.J.P. supervised the project. M.Ma. wrote the manuscript with contributions from all authors.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Mehul Malik.

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/ncomms4115

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.