Einstein-Podolsky-Rosen Energy-Time Entanglement of Narrow-Band Biphotons

We report the direct characterization of energy-time entanglement of narrow-band biphotons produced from spontaneous four-wave mixing in cold atoms. The Stokes and anti-Stokes two-photon temporal correlation is measured by single-photon counters with nanosecond temporal resolution, and their joint spectrum is determined by using a narrow linewidth optical cavity. The energy-time entanglement is verified by the joint frequency-time uncertainty product of $0.063\pm 0.0044$, which does not only violate the separability criterion but also satisfies the continuous variable Einstein-Podolsky-Rosen steering inequality.

Figure 1

Narrow-band energy-time entangled biphoton generation and characterization. (a) Experiment setup. In the presence of the pump (${\omega }_p$) and coupling (${\omega }_c$) laser beams, counterpropagating paired Stokes (${\omega }_s$) and anti-Stokes (${\omega }_{\text{as}}$) photons are produced via spontaneous four-wave-mixing (SFWM) process from a cold ${}^{85}\mathrm{Rb}$ atomic ensemble in a 2D magneto-optical trap (MOT). The photon pairs are then coupled into a pair of opposing single-mode fibers (SMF). To determine the joint spectral distribution, the Stokes and anti-Stokes photon frequencies are shifted by two acousto-optic modulators (${\mathrm{AOM}}_1$ and ${\mathrm{AOM}}_2$) before they are injected into the high-finesse optical cavity. Two dichroic mirrors (${\mathrm{DM}}_1$ and ${\mathrm{DM}}_2$) are employed to combine and separate these photons. The transmitted photon pairs are detected by two single-photon counting modules (SPCMs). Two etalon filters are inserted in front of the SPCMs to further filter out the scattering from the pump and coupling laser beams. (b) ${}^{85}\mathrm{Rb}$ atomic energy level diagram for the SFWM process. $|1⟩=|5S_{1/2},F=2⟩$, $|2⟩=|5S_{1/2},F=3⟩$, $|3⟩=|5P_{1/2},F=3⟩$, and $|4⟩=|5P_{3/2},F=3⟩$.

Figure 2

Temporal correlation of energy-time entangled biphotons. (a) Two-photon temporal correlation measured as coincidence counts with a time bin of 1 ns and total data collection time 60 s. The circle data points are experimentally measured coincidence counts. The solid line is calculated from theory. (b) The conditional autocorrelation $g_c^{(2)}$ of the heralded anti-Stokes photons as a function of the integration time width $\delta t$. The circle points with error bars are experimental data, and the dashed line indicates the two-photon event threshold value of 0.5.

Figure 3

Joint spectral distribution of Stokes and anti-Stokes photons. (a) The 2D joint spectral distribution measured as spectral coincidence counts between Stokes and anti-Stokes photons transmitted through the optical cavity. Each data point is taken with a two-photon coincidence time window $3\mu \mathrm{s}$ and acquisition time 3000 s. (b) The single-photon power spectrums of anti-Stokes and Stokes photons. The histogram data are directly measured using the high-finesse optical cavity. The dot data are calculated from the autocorrelation measurements. (c) The normalized spectral distribution as a function of the two-photon frequency sum $\delta ({\omega }_{\text{as}}+{\omega }_s)$. (d) The normalized two-photon joint spectrum with the accidental coincidence counts subtracted. The inset is the enlargement for the frequency sum range of $\pm 1\mathrm{MHz}$.