Analysis of the time-energy entanglement of down-converted photon pairs by correlated single-photon interference

The time-energy entanglement of down-converted photon pairs is particularly difficult to characterize because direct measurements of photon arrival times are limited by the temporal resolution of photon detection. Here, we explore an alternative possibility of characterizing the temporal coherence of two-photon wave functions using single-photon interference with weak coherent light. Specifically, this method makes use of the fact that down conversion generates a coherent superposition of vacuum and two-photon states, so that the coincidence count rates for photon pairs after interference with weak coherent light are given by a superposition of the two-photon wave function from the down conversion with the product wave function defined by the weak coherent references. By observing the frequency-dependent interference pattern, it is possible to reconstruct the amplitudes and the phases of the two-photon wave function within the bandwidth of the reference pulses used in the single-photon interferences. The correlated single-photon interferences therefore provide a direct map of the entangled two-photon wave function generated in the down-conversion process.

Figure 1

Illustration of the measurement of single-photon interference between the signal field and the reference field. Interference occurs because the origin of the single photon observed in the detector cannot be identified.

Figure 2

Illustration of correlated measurements of single-photon interference between down-converted light and two separate weak coherent references. The two-photon coincidence rate is sensitive to interference because the detection does not distinguish between photon pairs originating from the down-conversion source and photon pairs originating from the two references.

Figure 3

Illustration of coincidence counts obtained for Fourier-limited down-converted light with uncertainties of $\delta ({\omega }_1+{\omega }_2)=0.2{\sigma }_r$ and $\delta ({\omega }_1-{\omega }_2)=2{\sigma }_r$. The reference pulses have a delay time difference of $t_{r_1}-t_{r_2}=10/{\sigma }_r$, centered around the peak time of the pump pulse used in the down conversion ($t_{r_1}+t_{r_2}=0$). (a) The contour plot, (b) the coincidence counts for ${\omega }_1+{\omega }_2={\omega }_{\mathrm{pump}}$ and the envelop functions indicating the maximal and minimal values of the interference fringes, and (c) a plot of the phase dependence of ${\psi }_{-}$ that can be determined from the periodicity of the interference pattern in (b).

Figure 4

Illustration of dispersion effects observed in the correlated single-photon interferences. Parameters are the same as in Fig. 3, but a phase dispersion had been added, so that $\delta (t_1-t_2)$ is now equal to $1/\delta ({\omega }_1+{\omega }_2)$ and the correlations of arrival times and photon energies do not exceed the uncertainty limit of separable states. (a) The contour plot, (b) the coincidence counts for ${\omega }_1+{\omega }_2={\omega }_{\mathrm{pump}}$ and the envelop functions indicating the maximal and minimal values of the interference fringes, and (c) a plot of the phase dependence of ${\psi }_{-}$ that can be determined from the periodicity of the interference pattern in (b).