Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons

We report an experimental test of quantum complementarity with single-photon pulses sent into a Mach-Zehnder interferometer with an output beam splitter of adjustable reflection coefficient $R$. In addition, the experiment is realized in Wheeler’s delayed-choice regime. Each randomly set value of $R$ allows us to observe interference with visibility $V$ and to obtain incomplete which-path information characterized by the distinguishability parameter $D$. Measured values of $V$ and $D$ are found to fulfill the complementarity relation $V^2+D^2\le 1$.

Figure 1

Delayed-choice complementarity-test experiment. A single-photon pulse is sent into a Mach-Zehnder interferometer, composed of a $50/50$ input beam splitter (BS) and a variable output beam splitter (VBS). The reflection coefficient is randomly set either to the null value or to an adjustable value $R$, after the photon has entered the interferometer. The single-photon photodetectors $P_1$ and $P_2$ allow to record both the interference and the WPI.

Figure 2

Variable output beam splitter (VBS) implementation. The optical axis of the polarization beam splitter (PBS) and the polarization eigenstates of the Wollaston prism (WP) are aligned, and make an angle $\beta$ with the optical axis of the EOM. The voltage $V_{\mathrm{EOM}}$ applied to the EOM is randomly chosen accordingly to the output of a Quantum Random Number Generator (QRNG), located at the output of the interferometer and synchronized on the 4.2-MHz clock that triggers the single-photon emission.

Figure 3

Interference visibility $V$ measured in the delayed-choice regime for different values of $V_{\mathrm{EOM}}$. (a)–(c) correspond to $V_{\mathrm{EOM}}\approx 150\mathrm{V}$ ($R=0.43$ and $V=93\pm 2\%$), $V_{\mathrm{EOM}}\approx 40\mathrm{V}$ ($R=0.05$ and $V=42\pm 2\%$), and $V_{\mathrm{EOM}}=0$ ($R=0$ and $V=0$). Each point is recorded with 1.9 s acquisition time. Detectors dark counts, corresponding to a rate of $60{\mathrm{s}}^{-1}$ for each, have been substracted to the data.

Figure 4

Delayed choice test of complementarity with single-photon pulses. (a) Wavelike information $V^2$ and which-path information $D^2$ as a function of the EOM voltage corresponding to a given value $R$ of the VBS reflectivity. The solid lines are the theoretical expectations, with $\beta =24°$ and $V_{\pi }=217\mathrm{V}$, using Eqs. (2, 3, 7). (b) $V^2+D^2$ as a function of the EOM voltage.