Experimental Demonstration of the Bosonic Commutation Relation via Superpositions of Quantum Operations on Thermal Light Fields

We present the experimental realization of a scheme, based on single-photon interference, for implementing superpositions of distinct quantum operations. Its application to a thermal light field (a well-categorized classical entity) illustrates quantum superposition from a new standpoint and provides a direct and quantitative verification of the bosonic commutation relation between creation and annihilation operators. By shifting the focus towards operator superpositions, this result opens interesting alternative perspectives for manipulating quantum states.

Figure 1

Experimental setup. Pump pulses for parametric down-conversion are obtained by frequency doubling the laser output in a LBO—lithium triborate—crystal. The conditionally prepared signal state is mixed with a strong reference coherent field (LO, obtained from a portion of the main laser output) on a 50-50 beam splitter whose outputs are detected by two photodiodes (Hamamatsu PIN S3883). The balanced homodyne detection (BHD) signal is acquired and stored by a digital oscilloscope (Tektronix TDS7104) on a pulse-to-pulse basis triggered by a coincidence (C) between clicks from the Da and Ds single-photon detectors (Perkin Elmer model SPCM AQR-14).

Figure 2

(a) Sequence of histograms corresponding to about 250 000 raw quadrature data for the final state as a function of the normalized quadrature $x$ and of the superposition phase $\varphi$. Data have been obtained in a 16-hour measurement, and have been binned in 9 phase intervals between 0 and $\pi$. A thermal field with a mean photon number $\overline{n}=0.9(1)$ is used as the initial state. (b) Histogram of raw quadrature data (solid dots) for the anticommutator setup at $\varphi =\pi$. Also shown are theoretical curves calculated for the actual experimental parameters (total efficiency $\eta =0.61$, $\overline{n}=0.9$) and different values of the commutator ($K=0$: dashed orange; $K=2$: dotted green; $K=3$: dash-dotted blue). The solid red curve is the result of the best fit to the experimental data.

Figure 3

Experimentally reconstructed Wigner functions of the original thermal state and of those resulting from the application of the commutator and anticommutator superpositions. The state is not changed for $\varphi =0$. About ${10}^4$ (${10}^5$) quadrature data points have been acquired in the (anti)commutator case.