Direct observation of phase-sensitive Hong-Ou-Mandel interference

The quality of individual photons and their ability to interfere are traditionally tested by measuring the Hong-Ou-Mandel photon bunching effect. However, this phase-insensitive measurement only tests the particle aspect of the quantum interference, leaving out the phase-sensitive aspects relevant for continuous-variable processing. To overcome these limitations we formulate a witness capable of recognizing both the indistinguishability of the single photons and their quality with regard to their continuous-variable utilization. We exploit the conditional nonclassical squeezing and show that it can reveal both the particle and the wave aspects of the quantum interference in a single set of direct measurements. We experimentally test the witness by applying it to a pair of independent single photons retrieved on demand.

Figure 1

(a) Conceptual scheme of the witness. The phase randomized (PR) states are mixed on a balanced beam splitter (BS) and two fixed quadratures are then measured by a pair of homodyne detectors (HD). (b) Experimental setup. Two single-photon states are independently generated and stored in the memory cavities. After synchronization, they are released at the same time, allowed to interfere on the balanced beam splitter, and recorded by the pair of homodyne detectors. Here EOM denotes electro-optic modulator, LO local oscillator, and APD avalanche photodiode.

Figure 2

Experimentally measured quadrature distribution of the state after interference with $\mathrm{\Delta }\theta =\mathrm{\Delta }{\theta }_{\text{sq}}$. (a) Joint probability distribution $P(X_1,X_2)$. (b) Normalized quadrature distribution $P\left(X_1\right)$. An occurrence rate of $X_1$ with the window $1.9\le |X_2|\le 2.5$ is shown in histogram. Lines are Gaussian distributions with the variance of conditioned $P\left(X_1\right)$ (red solid line) and that of vacuum fluctuation for reference (blue dashed line), respectively.

Figure 3

Conditioned second moment $E_{\min ,\mathrm{\Delta }}\left[X_1^2\right]$ (red line) and its upper confidence band (green line) relative to the width of the postselection interval $\mathrm{\Delta }$. The position of the conditioning interval given by the measured value of $X_2$ was optimized over in order to obtain minimal certifiable variance given by the upper confidence interval.

Figure 4

Conditional second moment $E\left[X_1^2\right|X_2\in B\left(x_2\right)]$, where $B\left(x_2\right)$ is the set of values satisfying $x_2-\mathrm{\Delta }/2\le \left|X_2\right|\le x_2+\mathrm{\Delta }/2$. The width of the postselection interval was chosen to be $\mathrm{\Delta }=0.6$. The blue dashed line marks the theoretical prediction for the interference of two ideal single-photon states. The red solid line shows the prediction for interference of two imperfect photons produced by the cavities. Red dots represent experimental data with the light red zone marking the $3\sigma$ confidence band. The inset shows detailed behavior in the area with the highest certifiable squeezing.

Figure 5

Conditional second moment for different input states.

Figure 6

Range of parameters for states ${\rho }_1={\rho }_2=p_0|0\rangle \langle 0|+p_1|1\rangle \langle 1|+p_2|2\rangle \langle 2|$, in which squeezing can be witnessed. For the conditioning value $x_2=2.2$ squeezing can be observed in both the light blue and dark red areas. For the conditioning value $x_2=1.5$ squeezing can be observed only in the dark red area.