Experimental Demonstration of Commutation Relations Using Intensity Correlations

Abstract

The canonical commutation relation is a cornerstone of quantum theory and underlies the Heisenberg uncertainty principle. Although uncertainty relations have been extensively tested, direct verifications of the underlying commutation relation itself have remained elusive. We report an experimental demonstration of the bosonic commutation relation for optical field operators based on measurements of two distinct intensity correlation functions. From these measurements, we extract the expectation value of the field-operator commutator for both a single-photon state and coherent state. In both cases, the measured values are consistent with unity, in quantitative agreement with quantum theory.

Figures (3)

• Figure 1: a) Sketch of the parabolic mirror with inserted trap assembly and $^{174}\mathrm{Yb}^{+}$ energy levels. The light scattered towards the detection setup is either emitted by the ion at the focus of the parabolic mirror (ion-focus distance not to scale) or laser light scattered from one of the trap electrodes. b) and c) Sketches of the used detection methods.
• Figure 2: Measured correlation functions $g^{(2)}_{\mathrm{auto}}$ and $g^{(2)}_{\mathrm{cross}}$ of a train of single photons emitted by a single trapped ion. Both functions are nearly identical and only differ for $\tau = 0$, as expected from theory. The characteristic antibunching dip for a single-photon source is also apparent from the correlation functions.
• Figure 3: Measured correlation functions $g^{(2)}_{\mathrm{auto}}$ and $g^{(2)}_{\mathrm{cross}}$ of coherent laser light. The dip in the auto-correlation function for values close to $\tau =0$ is caused by the pulse width of 1.5 ns of the detectors and the dead time of 2 ns of the time-to-digital converter used to record the signals. Aside from the dip and the auto-correlation peak, both functions show a constant value of one, as expected for a coherent source.