Theoretical analysis of dipole-induced electromagnetic transparency

We present a detailed, realistic analysis of the implementation of a proposal for dipole-induced electromagnetic transparency (DIET) [R. Puthumpally-Joseph, M. Sukharev, O. Atabek, and E. Charron, Phys. Rev. Lett. 113, 163603 (2014)] using an ensemble of cold atoms at high density. Using both direct numerical simulations and simple analytical models, we show how, in a realistic $N$-level quantum system, narrow transparency windows can appear at large densities. The existence of such windows is attributed to quantum interference effects in overlapping resonances. Our analysis is applied to the $D_1$ transition of Rb atoms, and we show that, at high densities, Rb can behave like a simple three-level emitter exhibiting all the properties of DIET. Some interesting effects such as slow light are also presented, and their limits in the context of DIET are discussed

Figure 1

The schematic setup of simulations: a slab of thickness $\ell$ composed of interacting quantum emitters is exposed to linearly polarized EM fields acting normal to the interface.

Figure 2

Reflection spectra as a function of the reduced detuning $\delta =(\omega -{\omega }_{01})/\gamma$ for a slab of interacting two-level emitters in different interaction regimes. The thickness is $\ell =600$ nm and the transition dipole ${\mu }_1=2\mathrm{D}$. The radiationless decay rate and the pure dephasing rate are $\Gamma =0.5$ THz and ${\gamma }^{*}=10$ THz, respectively. Reflections for smaller ${\Delta }_1/\gamma$ values are weak and hence they are multiplied by some constant factors $({\Delta }_1/\gamma =0.001$ is by ${10}^4,{\Delta }_1/\gamma =0.01$ is by ${10}^3,{\Delta }_1/\gamma =0.1$ is by 50, and ${\Delta }_1/\gamma =1$ is by 2) for better comparison.

Figure 3

Extinction (a) and transmission (b) probabilities as functions of the reduced detuning $\delta$. The solid blue curves (left vertical scale) represent the results for ${\Delta }_1/\gamma ={10}^{-3}$ and the dotted green lines (right vertical scale) show the data for ${\Delta }_1/\gamma =12$. All other parameters are the same as in Fig. 2.

Figure 4

Relative error in the absolute values of the electric susceptibility using the semiclassical approximation with respect to the numerical integration of the Maxwell-Bloch equations as a function of the reduced detuning for parameters used in Fig. 3. The blue solid line shows the relative error for weakly interacting dipoles, and the red dashed line presents the same for strongly interacting dipoles.

Figure 5

Reflection as a function of the detuning for $\ell =600$ nm. The dashed blue line is the reflection probability calculated by integrating the Maxwell-Bloch equations, and the solid red line is the same as that obtained from the semiclassical approximation [Eq. (23)]. Other parameters are ${\Delta }_1/\gamma =12,{\mu }_1=2\mathrm{D}$, and $\gamma =10.5$ THz.

Figure 6

The blue solid line is the reflection as a function of the reduced detuning of three-level emitters with ${\Delta }_1/\gamma ={\Delta }_2/\gamma =12$. The red dotted line is the reflection from the same system in the absence of coupling between the dipoles. The excited states are separated by ${\omega }_2-{\omega }_1=4.6\gamma$. The reduced detuning, $\delta$, is defined with respect to the transition frequency of the first excited state of this three-level system.

Figure 7

Susceptibility [panel (a), log scale] and reflection probability [panel (b), linear scale] as a function of the reduced detuning with ${\Delta }_0/\gamma =3.8\times {10}^{-3}$. The reduced detuning $\delta$ is defined with respect to the transition frequency of $5{}^2{S}_{1/2}$ to $5{}^2{P}_{1/2}$ in the absence of hyperfine splitting $(377.107$ THz) and $\gamma =36.13$ MHz. The positions of transitions $F=2,3\to F^{\prime }=2,3$ are indicated by the red dashed lines marked with $(F,F^{\prime })$.

Figure 8

Susceptibility [panel (a), log scale], reflection [panel (b), linear scale], and transmission [panel (c), linear scale] as functions of the reduced detuning of ${}^{85}\mathrm{Rb}$ with ${\Delta }_0/\gamma =21$. Vertical red dashed lines show the positions of destructive interference between the different transition dipoles.

Figure 9

Susceptibility [panel (a), log scale] of a slab of ${}^{85}\mathrm{Rb}$ atoms with ${\Delta }_0/\gamma =200$. Reflection spectra [panels (b) and (c)] for thickness $\ell =600$ nm and $\ell =2.5$ $\mu m$ and corresponding transmission spectra in panels (d) and (e) as functions of the reduced detuning. A zoom of the susceptibility is given in the inset of panel (a) for a better view of the faint Fano profiles associated with the holes in the reflection spectra.

Figure 10

Reflected [panels (a) and (b)] and transmitted [panels (c) and (d)] pulse shapes from the layer of ${}^{85}\mathrm{Rb}$ atoms at ${\Delta }_0/\gamma =200$ as functions of the reduced detuning $\delta$. The blue dashed lines in the panels are the normalized incident pulse, and the red lines are the reflected and transmitted pulses envelopes. Panels (a) and (c) are for a slab of $\ell =600$ nm, and panels (b) and (d) are for $\ell =2.5$ $\mu \mathrm{m}$.

Figure 11

Slow light effect associated with DIET in $D_1$ transitions of ${}^{85}\mathrm{Rb}$ for ${\Delta }_0/\gamma =200$. Panel (a) shows the refractive index: the solid blue line is the imaginary part and the dashed red line is the real part. Panel (b) shows the group index of the layer, and panel (c) shows the group velocity as functions of the reduced detuning $\delta$.