# Observation of an Electromagnetically Induced Grating in Cold \({}^{85}\)Rb Atoms
## 3 Results and Discussion  
We consider a cold \({}^{85}\)Rb atomic medium with a length of \(L\) in the V-type system shown in Figure 1a, where \(\Omega_{p}\) and \(\Omega_{c}\) are the Rabi frequencies of the weak probe field and strong coupling field, respectively. By using the density matrix and perturbation theory [26], the susceptibility of probe field is given by [1]  
\[\chi=-\frac{N|\mu_{12}|^{2}}{\epsilon_{0}\hbar}\cdot\frac{(\Delta_{c}-\Delta_{p }-\ i\gamma_{23})+|\bar{\Omega}_{c}|^{2}/(\Delta_{c}-\ i\gamma_{13})}{(\Delta_{ p}+\ i\gamma_{21})(\Delta_{c}-\Delta_{p}-\ i\gamma_{23})+|\bar{\Omega}_{c}|^{2}} \tag{1}\]  
where \(N\) is the atomic density, \(\mu_{12}\) denotes the transition dipole momentum between the stats \(|1\rangle\) and \(|2\rangle\), \(\epsilon_{0}\) is the permittivity of free space, \(\gamma_{ij}=\frac{1}{2}(\Gamma_{i}+\Gamma_{j})\) (\(i,j=1,2,3\)) and \(\Gamma_{i}\) is the decay rate from the corresponding level. Here, \(\Delta_{p}=\omega_{p}-\omega_{12}\) and \(\Delta_{c}=\omega_{c}-\omega_{13}\) are the probe and coupling frequency detunings, respectively. In addition, \(\omega_{ij}\) are the frequency differences between the states \(|i\rangle\) and \(|j\rangle\). We assume the standing-wave field has a sinusoidal profile as \(\bar{\Omega}_{c}=\Omega_{c}sin(\pi x/d)\). Considering \(\chi=\chi^{\prime}+i\chi^{\prime\prime}\), we simulate the dispersion (\(\chi^{\prime}\)) and absorption (\(\chi^{\prime\prime}\)) of the probe field with the probe laser frequency detuning and coupling laser Rabi frequency, which are shown in Figure 2a,b. From the simulation, we find that not only the probe laser detuning, but also the coupling laser Rabi frequency can influence the dispersion and absorption of the probe field. It is observed that the width of EIT windows is proportional to the Rabi frequency of the coupling field. By using the Maxwell equation, the propagation dynamics of the probe field in the ensemble can be calculated analytically and the transmission at the output probe surface at \(z=L\) can be written as  
\[E_{p}(x,L)=E_{p}(x,0)\exp[-\frac{k_{p}\chi^{\prime\prime}}{2}L+\ i\frac{k_{p} \chi^{\prime}}{2}L] \tag{2}\]  
where \(E_{p}(x,0)\) is the input probe profile and \(k_{p}=2\pi/\lambda_{p}\). When \(\Delta_{p}=\Delta_{c}=0\), there only has the amplitude modulation (\(\chi^{\prime}=0\), \(\chi^{\prime\prime}\
eq 0\)), which results in a periodic amplitude modulation on the probe profile. A phase modulation (\(\chi^{\prime}\), \(\chi^{\prime\prime}\
eq 0\)) will appear in the output profile if we detune the probe field frequency away from the resonance transition. Figure 2c shows a typical hybrid grating when \(\Delta_{p}=1.5\Gamma_{2}\) and \(\Delta_{c}=0\), the blue and the red curves are the transmission function \(E_{p}(x,L)\) and  
Figure 1: **(a)** the energy-level diagram of the V-type system of \({}^{85}\)Rb cold atoms; **(b)** the geometry of the beam configuration in a cold atom medium, where the cooling and repumping lasers are omitted. The beam profile of probe laser is detected by a CCD; **(c)** schematic diagram of the standing-wave field formed in a cold atoms ensemble. the corresponding phase \(\Phi\), respectively. It can be found that there exists a rapid phase change of the probe field at nodes, which will not appear in an amplitude grating [27]. The experimental EIT configuration was observed and confirmed by a photodiode detector to detect the absorption of the probe laser. Figure 3a shows the saturation absorption spectroscopy of probe laser (above) and the absorption spectrum of probe laser in the presence of a coupling laser of the V-type system (below) in \({}^{85}\)Rb cold atoms. The SAS without Doppler background is obtained by scanning the probe laser frequency near the hyperfine transition. We can get all three hyperfine transitions \(5S_{1/2}(F=3)-5P_{3/2}(F^{\prime}=2,3,4)\).