Dark-state polaritons in single- and double-$\Lambda$ media

We derive the properties of polaritons in single-$\Lambda$ and double-$\Lambda$ media using a microscopic equation-of-motion technique. In each case, the polaritonic dispersion relation and composition arise from a matrix eigenvalue problem for arbitrary control field strengths. We show that the double-$\Lambda$ medium can be used to up- or down-convert single photons while preserving quantum coherence. The existence of a dark-state polariton protects this single-photon four-wave mixing effect against incoherent decay of the excited atomic states. The efficiency of this conversion is limited mainly by the sample size and the lifetime of the metastable state.

Figure 1

(a) Three-level $\Lambda$-type medium. (b) Double-$\Lambda$ medium.

Figure 2

Polaritonic dispersion curve for ${\omega }_L={\omega }_{ac}$. The solid lines show the exact polariton solutions given by Eq. (14); the dashed line shows the Fleischhauer-Lukin solution, Eq. (15).

Figure 3

Polaritonic dispersion curve for the double-$\Lambda$ medium. The solid lines show the exact polariton solutions given by Eq. (22); the dashed line shows the linearized solution given by Eq. (23). The horizontal asymptotes occur at ${\omega }_{cb}$ and ${\omega }_{cb}\pm \sqrt{{\mid \Omega \mid }^2+{\mid {\Omega }^{\prime }\mid }^2}$.

Figure 4

Numerical solutions of $\mid ⟨0\mid b_{k-k_L+k_L^{\prime }}\mid \psi \left(t\right)⟩\mid$ against $t$, where $\mid \psi \left(t\right)⟩$ is the quantum state at time $t$ after inserting a photon $a_k^{†}$ with $k={\omega }_{ab}∕c$. Here, ${\omega }_{cb}={10}^4{\mathrm{cm}}^{-1}$, $\mid g\mid =\mid g^{\prime }\mid =0.1{\mathrm{cm}}^{-1}$, and $\mid \Omega \mid =\mid {\Omega }^{\prime }\mid =1{\mathrm{cm}}^{-1}$. (a) No excited state decay, ${\Gamma }_a={\Gamma }_d=0$. (b) ${\Gamma }_a={\Gamma }_d=0.02{\mathrm{cm}}^{-1}$.

Figure 5

Photon frequency conversion. The $a$-photon amplitude $\mid {\varphi }_4(z,t)\mid$ (dashed line) and $b$-photon amplitude $\mid {\varphi }_5(z,t)\mid$ (solid line) are plotted at three instants. The abscissa is $z∕\cos \theta$, where $k_L\widehat{z}=\cos \theta$. The second control beam is pointed that $k_L^{\prime }\widehat{z}=0.9\cos \theta$. Both control beams are cw. The effective thickness of the double-$\Lambda$ medium, $z_0=60\mathrm{cm}$, is indicated. Within the medium, $\mid g\mid =\mid g^{\prime }\mid =0.1{\mathrm{cm}}^{-1}$, ${\Gamma }_a={\Gamma }_d=0.2g$, $\mid \Omega \mid =1{\mathrm{cm}}^{-1}$, and $\mid {\Omega }^{\prime }\mid =3\{1+\tanh [4(z∕\cos \theta -0.5)∕z_0]\}{\mathrm{cm}}^{-1}$. Outside the sample lies vacuum. The amplitudes are computed by integrating Eq. (36) numerically for the initial conditions in Eq. (38), where $\beta =0.8{\mathrm{m}}^{-2}$ and $z_2=-50\mathrm{cm}$.

Figure 6

Frequency conversion efficiency, parametrized by the converted photon amplitude ${\Phi }^5$ normalized to the input photon amplitude, as a function of the sample length $z_0$. Within the sample, the control beam ${\Omega }^{\prime }$ varies as $\mid {\Omega }^{\prime }\mid =3\{1+\tanh [4(z∕\cos \theta -0.5)∕z_0]\}{\mathrm{cm}}^{-1}$. Curves for incoherent decay rates ${\Gamma }_a={\Gamma }_d=0.2g$, $0.4g$, and $0.6g$ are shown. All other parameters are the same as in Fig. 5.