Interferometric Measurement of the Biphoton Wave Function

Interference between an unknown two-photon state (a “biphoton”) and the two-photon component of a reference state gives a phase-sensitive arrival-time distribution containing full information about the biphoton temporal wave function. Using a coherent state as a reference, we observe this interference and reconstruct the wave function of single-mode biphotons from a low-intensity narrow band squeezed vacuum state.

Figure 1

Experimental setup. AOM (EOM): acousto- (electro-) optic modulator. PBS: polarizing beam splitter. QWP (HWP): quarter- (half-)wave plate. PZT: piezoelectric actuator. SMF: single mode fiber. PMF: polarization maintaining fiber. FBS: fiber beam splitter.

Figure 2

Arrival-time distributions showing interference of two-photon amplitudes. Main graph shows coincidence rates $g_{AB(\kappa )}^{(2)}(0)$ (circles) for delay $\tau =0$ versus analysis phase $\varphi$ for a coincidence windows of 8 ns. The dashed line is $A[1+V\cos (2\varphi )]$, where $V=0.54$ is the expected visibility and $A$ comes from a fit to the data. The sinusoidal behavior in good agreement with the data reveals two-photon interference as predicted by Eq. (7). Error bars show $\pm 1\sigma$ (standard deviation) statistical uncertainty, assuming Poisson statistics. Insets show $g_{AB(\kappa )}^{(2)}(\tau )$ for the values of $\varphi$ indicated with arrows. These clearly show the passage from constructive interference at $\varphi =0$, where a peak is visible, to destructive interference at $\varphi =\pi /2$, where a dip appears.

Figure 3

Squared amplitude (above) and phase (below) of the reconstructed two-photon wave function for the squeezed vacuum state. The solid line shows the predicted, double exponential amplitude describing an ideal squeezed vacuum state from our OPO with an independently measured 8.1 MHz bandwidth, with the amplitude and the offset fitted to the data. Error bars show $\pm 1\sigma$ statistical uncertainty assuming Poisson statistics and using propagation of error through Eqs. (8) and (9).