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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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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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