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Direct measurement of the quantum wavefunction


The wavefunction is the complex distribution used to completely describe a quantum system, and is central to quantum theory. But despite its fundamental role, it is typically introduced as an abstract element of the theory with no explicit definition1,2. Rather, physicists come to a working understanding of the wavefunction through its use to calculate measurement outcome probabilities by way of the Born rule3. At present, the wavefunction is determined through tomographic methods4,5,6,7,8, which estimate the wavefunction most consistent with a diverse collection of measurements. The indirectness of these methods compounds the problem of defining the wavefunction. Here we show that the wavefunction can be measured directly by the sequential measurement of two complementary variables of the system. The crux of our method is that the first measurement is performed in a gentle way through weak measurement9,10,11,12,13,14,15,16,17,18, so as not to invalidate the second. The result is that the real and imaginary components of the wavefunction appear directly on our measurement apparatus. We give an experimental example by directly measuring the transverse spatial wavefunction of a single photon, a task not previously realized by any method. We show that the concept is universal, being applicable to other degrees of freedom of the photon, such as polarization or frequency, and to other quantum systems—for example, electron spins, SQUIDs (superconducting quantum interference devices) and trapped ions. Consequently, this method gives the wavefunction a straightforward and general definition in terms of a specific set of experimental operations19. We expect it to expand the range of quantum systems that can be characterized and to initiate new avenues in fundamental quantum theory.

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Figure 1: Direct measurement of the photon transverse wavefunction.
Figure 2: The measured single-photon wavefunction, Ψ(x ), and its modulus squared and phase.
Figure 3: Measurements of modified wavefunctions.
Figure 4: Phase modification of the wavefunction.

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  1. Cohen-Tannoudji, C., Diu, B. & Laloe, F. Quantum Mechanics Vol. 1, 19 (Wiley-Interscience, 2006)

    MATH  Google Scholar 

  2. Mermin, N. D. What's bad about this habit. Phys. Today 62, 8–9 (2009)

    Article  ADS  Google Scholar 

  3. Landau, L. D. & Lifshitz, E. M. Course of Theoretical Physics Vol. 3, Quantum Mechanics: Non-Relativistic Theory 3rd edn, 6 (Pergamon, 1989)

    Google Scholar 

  4. Vogel, K. & Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847–2849 (1989)

    Article  ADS  CAS  Google Scholar 

  5. Smithey, D. T., Beck, M., Raymer, M. G. & Faridani, A. 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, 1244–1247 (1993)

    Article  ADS  CAS  Google Scholar 

  6. Breitenbach, G., Schiller, S. & Mlynek, J. Measurement of the quantum states of squeezed light. Nature 387, 471–475 (1997)

    Article  ADS  CAS  Google Scholar 

  7. White, A. G., James, D. F. V., Eberhard, P. H. & Kwiat, P. G. Nonmaximally entangled states: production, characterization, and utilization. Phys. Rev. Lett. 83, 3103–3107 (1999)

    Article  ADS  CAS  Google Scholar 

  8. Hofheinz, M. et al. Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459, 546–549 (2009)

    Article  ADS  CAS  Google Scholar 

  9. Aharonov, Y., Albert, D. Z. & Vaidman, L. 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)

    Article  ADS  CAS  Google Scholar 

  10. Ritchie, N. W. M., Story, J. G. & Hulet, R. G. Realization of a measurement of a “weak value”. Phys. Rev. Lett. 66, 1107–1110 (1991)

    Article  ADS  CAS  Google Scholar 

  11. Resch, K. J., Lundeen, J. S. & Steinberg, A. M. Experimental realization of the quantum box problem. Phys. Lett. A 324, 125–131 (2004)

    Article  ADS  CAS  Google Scholar 

  12. Smith, G. A., Chaudhury, S., Silberfarb, A., Deutsch, I. H. & Jessen, P. S. Continuous weak measurement and nonlinear dynamics in a cold spin ensemble. Phys. Rev. Lett. 93, 163602 (2004)

    Article  ADS  Google Scholar 

  13. Pryde, G. J., O'Brien, J. L., White, A. G., Ralph, T. C. & Wiseman, H. M. Measurement of quantum weak values of photon polarization. Phys. Rev. Lett. 94, 220405 (2005)

    Article  ADS  CAS  Google Scholar 

  14. Mir, R. et al. A double-slit ‘which-way’ experiment on the complementarity-uncertainty debate. N. J. Phys. 9, 287 (2007)

    Article  Google Scholar 

  15. Hosten, O. & Kwiat, P. Observation of the spin Hall effect of light via weak measurements. Science 319, 787–790 (2008)

    Article  ADS  CAS  Google Scholar 

  16. Dixon, P. B., Starling, D. J., Jordan, A. N. & Howell, J. C. Ultrasensitive beam deflection measurement via interferometric weak value amplification. Phys. Rev. Lett. 102, 173601 (2009)

    Article  ADS  Google Scholar 

  17. Lundeen, J. S. & Steinberg, A. M. Experimental joint weak measurement on a photon pair as a probe of Hardy's paradox. Phys. Rev. Lett. 102, 020404 (2009)

    Article  ADS  CAS  Google Scholar 

  18. Aharonov, Y., Popescu, S. & Tollaksen, J. A time-symmetric formulation of quantum mechanics. Phys. Today 63, 27–32 (2010)

    Article  Google Scholar 

  19. Bridgman, P. The Logic of Modern Physics (Macmillan, 1927)

    MATH  Google Scholar 

  20. Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nature 299, 802–803 (1982)

    Article  ADS  CAS  Google Scholar 

  21. Trebino, R. Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Springer, 2002)

    Google Scholar 

  22. Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001)

    Article  ADS  CAS  Google Scholar 

  23. Duan, L. M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001)

    Article  ADS  CAS  Google Scholar 

  24. Aharonov, Y. & Vaidman, L. Properties of a quantum system during the time interval between two measurements. Phys. Rev. A 41, 11–20 (1990)

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  26. Jozsa, R. Complex weak values in quantum measurement. Phys. Rev. A 76, 044103 (2007)

    Article  ADS  Google Scholar 

  27. Mukamel, E., Banaszek, K., Walmsley, I. A. & Dorrer, C. Direct measurement of the spatial Wigner function with area-integrated detection. Opt. Lett. 28, 1317–1319 (2003)

    Article  ADS  Google Scholar 

  28. Smith, B. J., Killett, B., Raymer, M. G., Walmsley, I. A. & Banaszek, K. Measurement of the transverse spatial quantum state of light at the single-photon level. Opt. Lett. 30, 3365–3367 (2005)

    Article  ADS  Google Scholar 

  29. Itatani, J. et al. Tomographic imaging of molecular orbitals. Nature 432, 867–871 (2004)

    Article  ADS  CAS  Google Scholar 

  30. Dudovich, N. et al. Measuring and controlling the birth of attosecond XUV pulses. Nature Phys. 2, 781–786 (2006)

    Article  ADS  CAS  Google Scholar 

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This work was supported by the Natural Sciences and Engineering Research Council and the Business Development Bank of Canada.

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Authors and Affiliations



The concept and the theory were developed by J.S.L. All authors contributed to the design and building of the experiment and the text of the manuscript. J.S.L, B.S. and C.B. performed the measurements and the data analysis.

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Correspondence to Jeff S. Lundeen.

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The authors declare no competing financial interests.

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Lundeen, J., Sutherland, B., Patel, A. et al. Direct measurement of the quantum wavefunction. Nature 474, 188–191 (2011).

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