Engineering biphoton wave packets with an electromagnetically induced grating

We propose to shape biphoton wave packets with an electromagnetically induced grating in a four-level double-$\Lambda$ cold atomic system. We show that the induced hybrid grating plays an essential role in directing the new fields into different angular positions, especially for the zeroth-order diffraction. A number of interesting features appears in the shaped two-photon wave forms. For example, broadening or narrowing the spectrum would be possible in the proposed scheme even without the use of a cavity.

Figure 1

Shaping biphoton wave packets with an EIG. (a) The level structure, where in the presence of a cw probe (${\omega }_p$) and control (${\omega }_c$) fields, paired Stokes (${\omega }_s$) and anti-Stokes (${\omega }_{\mathit{as}}$) photons are spontaneously created from the four-wave mixing processes in the low-gain regime. (b) The backward generation geometry, where two strong control beams symmetrically displace with respect to $z$ and form a standing wave along $x$. (c) The standing wave formed by control fields.

Figure 2

A typical transmission profile of the anti-Stokes field as a function of $x$. Parameters are chosen as $d=2\hspace{0.5em}\mathrm{\mu }\mathrm{m}$, the optical depth about 5, ${\gamma }_{31}=2\pi \times 3\hspace{0.5em}\mathrm{MHz}$, and ${\gamma }_{21}=0.6{\gamma }_{31}$.

Figure 3

Two-photon temporal coincidences exhibit damped and overdamped oscillations with the space period $d=2\hspace{0.5em}\mathrm{\mu }\mathrm{m}$. Other parameters are the same as in Ref. [16].

Figure 4

Two-photon temporal coincidences: (a) modulated by an EIG with the space period $d=2\hspace{0.5em}\mathrm{\mu }\mathrm{m}$. Other parameters are chosen as $L/v_0=800\hspace{0.5em}\mathrm{ns}$, ${\Omega }_c=5{\gamma }_{31}$, ${\gamma }_{31}=2\pi \times 3\hspace{0.5em}\mathrm{MHz}$, and ${\gamma }_{21}=0.6{\gamma }_{31}$. (b) Without the induced grating. Same parameters are chosen as in (a).