Low-loss nonlinear polaritonics

We propose a large low-loss cross-phase modulation between two coupled surface polaritons propagating through a double electromagnetically induced transparency medium situated close to a negative-index metamaterial. In particular, a mutual $\pi$ phase shift is attainable between the two pulses at the single-photon level.

Figure 1

Two SPs, shown as dark (red) and white pulses created between two media $i=1,2$ with permittivities ${\varepsilon }_i$ and permeabilities ${\mu }_i$. Medium 1 is in the region $z>0$ and medium 2 in the region $z<0$. The pulses propagate forward in the $+x$ direction and are exponentially confined to the interface $z=0$. The white spots represent multilevel atoms in the double $\Lambda$ configuration and are confined to a region between $z=0$ and $z=z_0$.

Figure 2

Characterizing LLSPs as a function of $\omega /{\omega }_{\mathrm{e}}$, with ${\omega }_{\mathrm{e}}=1.37\times {10}^{16}\hspace{0.5em}{\text{s}}^{-1}$, for confinement factor ${\zeta }_1$ (blue-dashed), and mode length (purple-dotted) normalized to the Rb $D_2$ wavelength $\lambda =780\mathrm{nm}$ and absorption (red-solid) ($\times 5.{10}^3$ ${\mathrm{m}}^{-1}$).

Figure 3

Fractional energies in the NIMM part (lower, brown curve), in the dielectric part (middle, blue line) and total energy (upper, purple line) as functions of $\omega /{\omega }_{\mathrm{e}}$.

Figure 4

Frequencies of two slow LLSP fields $E_{a,b}$, of the control LLSP field ${\Omega }_c$ and energy diagram for the $D_2$ line of ${}^{87}\mathrm{Rb}$.

Figure 5

Spatial and temporal diagram of excitation and interaction of two slow LLSP pulses propagating with group velocities $v_b>v_a$, with the $a$ pulse in white and the $b$ pulse in dark red, in the presence of a control field ${\Omega }_c$ shown in light blue. The second (dark red) LLSP pulse outraces the first (white) LLSP pulse in the medium output. The dashed lines on the left and right indicate conceptually how the fields are directed into and out of the interaction region, and the solid lines on other side represent the waveguide that brings the field in and out.

Figure 6

Third-order susceptibility ${\chi }_{\mathrm{a}}^{(3)}$ (dashed line) due to the DEIT scheme ($\times {10}^5$) as a function of probe field frequency $\omega /{\omega }_{\mathrm{e}}$, and the corresponding phase shift ${\phi }_b$ (solid line) due to SP cross-phase modulation in the DEIT scheme (in units $\pi$) as a function of probe field frequency $\omega /{\omega }_{\mathrm{e}}$.