Phase modulation induced by cooperative effects in electromagnetically induced transparency

We analyze the influence of dipole-dipole interactions in an electromagnetically induced transparency set up for a density at the onset of cooperative effects. To this end, we include mean-field models for the influence of local-field corrections and radiation trapping into our calculation. We show both analytically and numerically that the polarization contribution to the local field strongly modulates the phase of a weak pulse. We give an intuitive explanation for this local-field-induced phase modulation and demonstrate that it distinctively differs from the nonlinear self-phase-modulation that a strong pulse experiences in a Kerr medium.

Figure 1

EIT level scheme with probe field ${\Omega }_{31}$ (red dashed arrow) and control field ${\Omega }_{32}$ (blue solid arrow). The green wiggly arrows indicate the radiative decay ${\gamma }_{31}$, ${\gamma }_{32}$ and the incoherent pump rates $R_{31}$, $R_{32}$, while ${\gamma }_{\mathrm{deph}}$, ${\gamma }_s$ denote the ground-state dephasing and the ground-state population transfer, respectively.

Figure 2

Real part (blue dashed line) and imaginary part (red solid line) of the EIT susceptibility given by Eq. (8) for (a) negligible LFC ($L={10}^{-5}$) and (b) strong LFC ($L=4$).

Figure 3

Probe-field absorption in dependence of the radiation trapping rate $r_a/r_e$ and the ground-state transfer rate ${\gamma }_s$. The shading indicates the peak value of the propagated probe-field amplitude ${\Omega }_{31}(z)$ relative to the initial amplitude ${\Omega }_{31}(0)$. The three contour lines indicate parameters at which $50\%$, $25\%$, or $1\%$ of the initial amplitude leave the sample. The results are obtained using the numerical solution of the full set of EOM. All other parameters remain fixed and are chosen as discussed in Sec. 3c.

Figure 4

(a) Real part (blue dashed line) and imaginary part (red solid line) of a Gaussian pulse with LFC-induced phase modulation. Both parts are scaled to arbitrary but equal units. The parabola (black dash-dotted line) shows the time dependent phase shift in radian measure (right axis) and the dashed straight line is the corresponding instantaneous frequency. It exhibits a linear frequency chirp with slope ${\alpha }_{\mathrm{LFC}}$ over the total extent of the pulse. (b) Envelope and carrier wave of the pulse (black upper line), the polarization induced directly by the pulse (blue middle line), and the additional polarization induced by dipole-dipole interactions (red lower line). The amplitudes are not drawn to scale and the carrier wavelength has been strongly exaggerated to make phase relations clearly visible. (c) The nonlinear self-phase-modulation (black dash-dotted line) of a Gaussian pulse. The frequency chirp is approximately linear only in the pulse center (black dashed straight line) with slope ${\alpha }_{\mathrm{NSM}}$.