Quantum Measurement (1): Directly measuring the wavefunction

There has recently been a number of quite astonishing experiments measuring quantum systems, some of which seem of relevance, or at least of interest, to philosophers of physics. So I have resolved to present some of these results in a loose series entitled ‘Quantum Measurement’. I plan not so much to offer the definitive philosophical appraisal of these experiments than simply to report the results and link to the relevant publications.

I start the new series with an article which appeared in Nature in June (Vol 474, 188-191; cf. also the lead article in the same Vol, 170f) and reports the results of a Canadian group presenting a method for directly measuring the quantum wavefunction!

The usual ‘quantum-state tomography’ starts out from an ensemble of identically prepared quantum systems and then proceeds to making a series of measurements of each of a number of different properties of the system. From the obtained probability distributions of these properties, the wavefunction can then be reconstructed.

Not so Lundeen et al: they propose a method for directly measuring the real and the imaginary parts of the wavefunction! And they show that the method works by ‘weakly’ measuring the transverse spatial wavefunction of single photons. A weak measurement is a quantum measurement after which the different states of the measuring device–the pointer positions–differ by less than the quantum uncertainty of the pointer positions. To determine the weak value, i.e., the center of pointer’s position–its average–in the limit of zero coupling between system and device, a measurement needs to be repeated many times on identically prepared systems.

If I understand what Lundeen et al are saying correctly, then their way of getting around having to repeat measurements many times AND having to infer the wavefunction quite indirectly is to first weakly measure a system’s position and then to (strongly) measure its momentum. Conditioning on the outcome of the second measurement yielding a zero value for the momentum, the real and imaginary values of the system’s spatial wavefunction at that position can then be directly determined.

The optical experiments they perform use photons and their polarization. They identically prepare the photons to have the same transverse spatial wavefunction and then weakly measure these photons polarization. After this, photons are ‘post-selected’ by a strong measurement to have zero momentum. Finally, the polarization of these post-selected photons is analyzed–the average rotation of the polarization is proportional to the real part of the transverse spatial wavefunction and its average change in ellipticity is proportional to the imaginary value at the weakly measured position. (This is possible because the weak value, unlike the expectation value, is a complex number.) You repeat this for different spatial positions in the transverse direction to reconstruct the entire wavefunction at a longitudinal position. Done!

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