Determining the Quantum Expectation Value by Measuring a Single Photon (and other recent applications of weak measurements)

Quantum mechanics exhibits several peculiar properties, differentiating it from classical mechanics. One of the most intriguing is that variables might not have definite values. A complete quantum description provides only probabilities for obtaining various eigenvalues of a quantum variable. The eigenvalues and corresponding probabilities specify the expectation value of a physical observable, but they are known to be statistical properties of large ensembles. In contrast to this paradigm, we demonstrate a unique method allowing to measure the expectation value of a physical variable on a single particle, namely, the polarization of a single protected photon. This is the first realization of quantum protective measurements [1,2], which are based on a combination of weak measurements and the quantum Zeno effect. Before discussing these issues, I will review the notion of weak measurements [3-5] and discuss their realization by presenting our previous experiment [6], where we measured two non-commuting observables, on one and the same photon, using sequential weak measurements. I will conclude by discussing a few applications of these methods, both in metrology and in the study of foundational questions.

References
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[2] Y. Aharonov, E. Cohen, Protective measurement, Post-selection and the Heisenberg representation, in Protective measurement and quantum reality: Towards a new understanding of quantum mechanics, Shan Gao (Ed.), Cambridge University Press (2014), arXiv: 1403.1084.
[3] Y. Aharonov, D.Z. 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, 1351 (1988).
[4] Y. Aharonov, E. Cohen, A.C. Elitzur, Foundations and applications of weak quantum measurements, Phys. Rev. A 89, 052105 (2014).
[5] Y. Aharonov, E. Cohen, A.C. Elitzur, Can a future choice affect a past measurement's outcome?, Ann. Phys. 355, 258-268 (2015).
[6] F. Piacentini M.P. Levi, A. Avella, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, M. Gramegna, G. Brida, I.P. Degiovanni, M. Genovese, Measuring incompatible observables of a single photon, Phys. Rev. Lett.. 117, 170402 (2016).