Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps

We study both the steady and the dynamic optical response of cold atoms trapped in an optical lattice and driven to the three-level $\mathrm{\Lambda }$ configuration. These atoms are found to exhibit gain without population inversion when an incoherent pump is applied to activate spontaneously generated coherence. Gain-assisted double photonic band gaps characterized by reflectivities over 100% then grow up near the probe resonance due to the periodic distribution of the atomic density. These band gaps along with the neighboring allowed bands of transmissivities over 100% can be tuned by modulating the control field in amplitude, frequency, and, especially, phase. Consequently it is viable to realize a reflector, an amplifier, or a splitter when a weak incident light pulse is totally reflected in the photonic band gaps, totally transmitted in the allowed bands, or equally reflected and transmitted in the intersecting regions. Our results have potential applications in all-optical networks with respect to fabricating dynamically switchable devices for manipulating photon flows at low-light levels.

Figure 1

(a) Schematic of a three-level atomic system driven by a weak probe field of Rabi frequency ${\mathrm{\Omega }}_p$, a strong control field of Rabi frequency ${\mathrm{\Omega }}_c$, and a weak incoherent field of pumping rate $\mathrm{\Lambda }$. (b) Nonorthogonal arrangement of field polarizations and dipole moments in the probe $\left|3\right.〉\leftrightarrow \left|1\right.〉$ and control $\left|3\right.〉\leftrightarrow \left|2\right.〉$ transitions. (c) Density distribution of cold atoms in periodic traps of a one-dimensional optical lattice with the probe field applied close to the lattice axis.

Figure 2

(a) Reflectivity and (c) transmissivity vs ${\mathrm{\Delta }}_p$ and $G_{c0}$ with ${\mathrm{\Delta }}_c=0$; (b) reflectivity and (d) transmissivity vs ${\mathrm{\Delta }}_p$ and ${\mathrm{\Delta }}_c$ with $G_{c0}=13\mathrm{\Gamma }$ in the absence of an incoherent pump ($\mathrm{\Lambda }=0$). Other parameters are $G_{p0}=0.015\mathrm{\Gamma }, \theta =\pi /4, {\mathrm{\Gamma }}_{31}={\mathrm{\Gamma }}_{32}=6$ MHz, $d_{13}=d_{23}=1.0\times {10}^{-29}\mathrm{C}·\mathrm{m}, D_0=5.0\times {10}^{11}{\mathrm{cm}}^{-3}, L=4$ mm, $\alpha =0.2^{\circ }$, and $dz=a/5$.

Figure 3

Imaginary part of the (a) probe susceptibility, (b) reflectivity, and (c) transmissivity vs ${\mathrm{\Delta }}_p$ with $\mathrm{\Phi }=0$ (dashed blue line), $\mathrm{\Phi }=\pi /3$ (dotted red line), and $\mathrm{\Phi }=\pi /2$ (solid black line). Imaginary part of the (d) probe susceptibility, (e) reflectivity, and (f) transmissivity vs ${\mathrm{\Delta }}_p$ with $\mathrm{\Phi }=\pi /2$ (solid black line), $\mathrm{\Phi }=4\pi /7$ (dotted red line), and $\mathrm{\Phi }=\pi$ (dashed blue line). Other parameters are the same as for Fig. 2 except $G_{c0}=13\mathrm{\Gamma }$ and ${\mathrm{\Delta }}_c=0$ in the presence of an incoherent pump ($\mathrm{\Lambda }=0.002\mathrm{\Gamma })$.

Figure 4

Reflectivity (a) vs ${\mathrm{\Delta }}_p$ and $\mathrm{\Lambda }$ with $\mathrm{\Phi }=0, G_{c0}=13\mathrm{\Gamma }$, and ${\mathrm{\Delta }}_c=0$; (b) vs ${\mathrm{\Delta }}_p$ and $G_{c0}$ with $\mathrm{\Lambda }=0.005\mathrm{\Gamma }, \mathrm{\Phi }=0$, and ${\mathrm{\Delta }}_c=0$; (c) vs ${\mathrm{\Delta }}_p$ and ${\mathrm{\Delta }}_c$ with $\mathrm{\Lambda }=0.005\mathrm{\Gamma }, \mathrm{\Phi }=0$, and $G_{c0}=13\mathrm{\Gamma }$; and (d) vs ${\mathrm{\Delta }}_p$ and $\mathrm{\Phi }$ with $\mathrm{\Lambda }=0.005\mathrm{\Gamma }, G_{c0}=13\mathrm{\Gamma }$, and ${\mathrm{\Delta }}_c=0$. Other parameters are the same as for Fig. 2.

Figure 5

Scaled intensities of incident, reflected, and transmitted pulses ${\left|E_{It}/E_{0t}\right|}^2$ (solid black line), ${\left|E_{Rt}/E_{0t}\right|}^2$ (dashed blue line), and ${\left|E_{Tt}/E_{0t}\right|}^2$ (dotted red line) vs time $t$ with (a) ${\mathrm{\Delta }}_0=16.5\mathrm{\Gamma }$, (b) ${\mathrm{\Delta }}_0=25\mathrm{\Gamma }$, and (c) ${\mathrm{\Delta }}_0=40\mathrm{\Gamma }$. Other parameters are the same as for Fig. 2 except $\mathrm{\Lambda }=0.009\mathrm{\Gamma }, G_{c0}=13\mathrm{\Gamma }, {\mathrm{\Delta }}_c=0, \mathrm{\Phi }=0, t_0=235/\mathrm{\Gamma }$, and $\tau =230/\mathrm{\Gamma }$.

Figure 6

Scaled intensities of incident, reflected, and transmitted pulses ${\left|E_{It}/E_{0t}\right|}^2$ (solid black line), ${\left|E_{Rt}/E_{0t}\right|}^2$ (dashed blue line), and ${\left|E_{Tt}/E_{0t}\right|}^2$ (dotted red line) vs time $t$ with (a) $G_{c0}=7.3\mathrm{\Gamma }$, (b) $G_{c0}=10.0\mathrm{\Gamma }$, and (c) $G_{c0}=15.3\mathrm{\Gamma }$. Other parameters are the same as for Fig. 2 except $\mathrm{\Lambda }=0.002\mathrm{\Gamma }, {\mathrm{\Delta }}_0=-1.95\mathrm{\Gamma }, {\mathrm{\Delta }}_c=0, \mathrm{\Phi }=\pi , t_0=235/\mathrm{\Gamma }$, and $\tau =230/\mathrm{\Gamma }$.