Photon wave-packet manipulation via dynamic electromagnetically induced transparency in multilayer structures

We present a theoretical study of the dynamics of a light pulse propagating through a multilayer system consisting of alternating blocks of electromagnetically induced transparency (EIT) media and vacuum. We study the effect of a dynamical modulation of the EIT control field on the shape of the wave packet. Interesting effects due to the group velocity mismatch at the interfaces are found. Modulation schemes that can be realized in ultracold atomic samples with standard experimental techniques are proposed and discussed. Calculations are performed using a modified slowly varying envelope approximation of the Maxwell-Bloch equations and are compared to an effective description based on a continuity equation for the polariton flow.

Figure 1

(a) Scheme of the three-level $\Lambda$ configuration of atomic levels and laser beams. (b) and (c) Frequency dispersion and decay rate of the three polariton branches (green, upper; blue, dark; red, lower). Parameters of the system: $\sqrt{D}=0.1$, ${\Omega }_c/{\omega }_{\mathit{eg}}=0.04$, ${\gamma }_e/{\omega }_{\mathit{eg}}=0.01$, resonant dressing ${\omega }_c={\omega }_e-{\omega }_m$. The solid lines are the MSVEA predictions using ${\omega }_p={\omega }_{\mathit{eg}}$ and and assuming an Erf shape for the $f(k)$ function with a half-bandwidth of $0.5\hspace{0.5em}{\omega }_{\mathit{eg}}$. The dashed line is the exact EIT dispersion.

Figure 2

Pictorial scheme of a double-layer EIT chain. The shown configuration with orthogonal probe and control beams [17] allows for independent modulation of ${\Omega }_c$ in the different layers, but the predictions of the present article are straightforwardly extended to other geometries.

Figure 3

Examples of modulation of a pulse via a space- and time-dependent EIT. The propagation of the pulse is calculated by solving the MB equations (6). In all of the panels, the pulse moves from left to right and the shaded regions correspond to a EIT medium. The dashed curves show the initial pulses. (a) Propagation across a static interface. Solid line, pulse shape while being spatially compressed in entering into a medium with $v_{\mathrm{gr}}^i=0.11\hspace{0.5em}c$; dotted line, pulse shape once it has completely entered the EIT medium. (b) Wave-packet propagation in a homogeneous EIT layer. Dotted line, propagation without modulation; solid line, result of a slow-down ramp with $v_{\mathrm{gr}}^f=v_{\mathrm{gr}}^i/2$; dot-dashed line, result of a speed-up ramp $v_{\mathrm{gr}}^f=1.8\hspace{0.5em}{v}_{\mathrm{gr}}^i$. (c) Exit from a EIT medium into vacuum. Dotted line, propagation without modulation; solid line, result of a slow-down ramp with $v_{\mathrm{gr}}^f=v_{\mathrm{gr}}^i/2$. All of the panels have been calculated for a pulse carrier exactly on Raman resonance ${\delta }_e=0$, a ramp time ${\omega }_p\tau =100$, and a pulse width ${\omega }_p{\sigma }_t=400$. Material parameters: $D=0.01$, ${\gamma }_e={10}^{-3}{\omega }_p$.

Figure 4

Modulation of a wave packet using a vacuum defect. In the EIT medium, the group velocity is decreased from $v_{\mathrm{gr}}^{+}=0.11\hspace{0.5em}c$ to $v_{\mathrm{gr}}^{-}=0.02\hspace{0.5em}c$ and then increased back to $v_{\mathrm{gr}}^{+}$ as shown in the inset. (Main panel) Pulse at the beginning (dotted blue line) and at the end of the process (solid blue line); comparison with the results of the continuity equation for polariton flow (24) (black dot-dashed line) and the numerical solution of the effective equation (29) (red dashed line). (Inset) Temporal dependence of ${\Omega }_c$. (Bottom panels) Three snapshots during the propagation time. (a) Slow-down ramp; (b) storage time; (c) speed-up ramp. Parameters: ${\gamma }_e={10}^{-3}{\omega }_p$, pulse length $k_p{\mathrm{\sigma ̄}}_x=1600$, defect thickness $k_p{L}_d=6400$, ramp time ${\omega }_p\tau =100$, storage time ${\omega }_p{\tau }_s=6$0 000.

Figure 5

Snapshots of the pulse profile during a light-storage process in a single EIT layer. ${\Omega }_c$ is modulated in time from ${\Omega }_c^{+}=0.07\hspace{0.5em}{\omega }_p$ to ${\Omega }_c^{-}=0$ with the same shape as shown in the inset of Fig. 4. The storage time is ${\omega }_p{\tau }_s=1350$. The initial group velocity in the EIT layer is $v_{\mathrm{gr}}=0.11\hspace{0.5em}c$. The pulse has a Gaussian shape with $k_p{\sigma }_x=540$ in vacuum and the defect has a length $k_p{L}_d=10$. (a) Initial pulse (dashed line), pulse hitting the defect and splitting immediately after the stopping ramp (solid line), and counterpropagating wave packets during the storage time (dotted line). (b) Emerging wave packets after the retrieval ramp. The arrows indicate the propagation directions of each pulse. Blue lines in panels (a) and (b) are in the absence of spontaneous emission ${\gamma }_e=0$. The green line in panel (b) is for ${\gamma }_e=0.07\hspace{0.5em}{\omega }_p$.

Figure 6

Examples of wave-packet manipulation in several realizations of the EIT chain. All curves have been obtained using the model of Eq. (29). Solid red lines show the modulated pulse, while dashed black lines result from static cases. (a) Single EIT layer [$L=200$ $\mu$m, $v_{\mathrm{gr}}={10}^{-7}\hspace{0.5em}c$, $\mathcal{D}=6\times {10}^{-7}\hspace{0.5em}({\omega }_p/k_p^2)$] with a Gaussian pulse (${\sigma }_t=10$ $\mu s$, ${\ell }_{\mathrm{abs}}\approx 500{\mathrm{\sigma ̄}}_x$). Effect of a double ramp (as, e.g., in Fig. 4): $v_{\mathrm{gr}}^{+}/v_{\mathrm{gr}}^{-}=10$, $\tau =3.5$ $\mu$s, and storage time ${\tau }_s=8$ $\mu$s. (b) EIT double-layer structure [$L=30$ $\mu$m (each layer), interlayer distance $\Delta L=3\times {10}^7$ $\mu$m, $v_{\mathrm{gr}}=5\times {10}^{-7}\hspace{0.5em}c$, $\mathcal{D}=3\times {10}^{-6}\hspace{0.5em}{\omega }_p/k_p^2$] with a Gaussian pulse (${\sigma }_t=1$ $\mu$s, ${\ell }_{\mathrm{abs}}\approx 400\hspace{0.5em}{\mathrm{\sigma ̄}}_x$). Slow-down ramp in both layers: $\Delta v=-0.5\hspace{0.5em}{v}_{\mathrm{gr}}$, $\tau =50$ ns. (c) EIT chain with four layers [same as (b) but $\Delta L=6\times {10}^7$ $\mu$m] with Gaussian pulse [same as (b)]. Effect of a single slow-down ramp: $\Delta v=-0.7\hspace{0.5em}{v}_{\mathrm{gr}}$, $\tau =50$ ns.