Optical Precursor of a Single Photon

We report the direct observation of optical precursors of heralded single photons with step- and square-modulated wave packets passing through cold atoms. Using electromagnetically induced transparency and the slow-light effect, we separate the single-photon precursor, which always travels at the speed of light in vacuum, from its delayed main wave packet. In the two-level superluminal medium, our result suggests that the causality holds for a single photon.

Figure 1

Schematics of the experimental setup. In the presence of a counterpropagating pump (${\omega }_p$) and coupling (${\omega }_c$) beams, narrow-band paired Stokes (${\omega }_s$) and anti-Stokes (${\omega }_{as}$) photons are produced from a cold ${}^{85}\mathrm{Rb}$ atomic ensemble in MOT1 and coupled into opposing single-mode fibers (SMF). The anti-Stokes photons are passed through an EOM driven by a function generator triggered by the detection of Stokes photons at the single-photon detector $D_1$. We then send the heralded anti-Stokes photons with amplitude modulation to the second cold ${}^{85}\mathrm{Rb}$ atomic ensemble at MOT2. A 50%-50% beam splitter (BS) and two detectors ($D_2$ and $D_3$) are used to confirm the single-photon quantum nature. The ${}^{85}\mathrm{Rb}$ atomic energy levels are chosen as $|1⟩=|5S_{1/2},F=2⟩$, $|2⟩=|5S_{1/2},F=3⟩$, $|3⟩=|5P_{1/2},F=3⟩$ and $|4⟩=|5P_{3/2},F=3⟩$.

Figure 2

Separation of optical precursors without modulation. Two-photon coincidences are measured after the anti-Stokes photons passing through (a) vacuum without atoms present, (b) the EIT system (${\Omega }_{c2}=3.5{\gamma }_{13}$, $\mathrm{OD}=14$) and (c) the two-level system (${\Omega }_{c2}=0$, $\mathrm{OD}=14$) in MOT2. The Stokes and anti-Stokes two-photon coincidence counts are collected for $T=900\mathrm{s}$ with $\Delta t=1\mathrm{ns}$ bin width. The theoretical curve in a is the best fitting using $\eta |{\psi }_0(\tau ){|}^{2}T\Delta t$. Then the ${\psi }_0(\tau )$ is used to calculate the theoretical curves in b and c without any adjustable parameters following Eqs. (1, 2). (d) The relative time delay (respected to the vacuum) of the precursor and main wave packet as functions of OD. The red solid line is the calculated EIT group delay.

Figure 3

Single-photon optical precursors from a step amplitude modulation. (a) The heralded anti-Stokes photon waveform with a step modulation. (b) and (c) are two-photon coincidences after the anti-Stokes photons passing through the EIT system (${\Omega }_{c2}=3.8{\gamma }_{13}$, $\mathrm{OD}=18$) and two-level system (${\Omega }_{c2}=0$, $\mathrm{OD}=2.5$) in MOT2, respectively. Inset (d) shows a Gaussian pulse propagation in the two-level system with a peak advancement of about 40 ns (the lower curve) compared to the reference pulse (the up curve).

Figure 4

Single-photon optical precursors from a square amplitude modulation. (a) The heralded anti-Stokes photon waveform with a square modulation. (b) and (c) are two-photon coincidences after the anti-Stokes photons passing through the EIT system (${\Omega }_{c2}=3.5{\gamma }_{13}$, $\mathrm{OD}=10$) and two-level system (${\Omega }_{c2}=0$, $\mathrm{OD}=10$) in MOT2, respectively. (d) The relative time delay (respected to the vacuum) of the precursor and main wave packet as functions of OD. The red solid line is the calculated EIT group delay.