Master equation approach for interacting slow- and stationary-light polaritons

A master equation approach for the description of dark-state polaritons in coherently driven atomic media is presented. This technique provides a description of light-matter interactions under conditions of electromagnetically induced transparency (EIT) that is well suited for the treatment of polariton losses. The master equation approach allows us to describe general polariton-polariton interactions that may be conservative, dissipative, or a mixture of both. In particular, it enables us to study dissipation-induced correlations as a means for the creation of strongly correlated polariton systems. Our technique reveals a loss mechanism for stationary-light polaritons that has not been discussed so far. We find that polariton losses in level configurations with nondegenerate ground states can be a multiple of those in level schemes with degenerate ground states.

Figure 1

Considered setup of $N$ atoms confined to an interaction volume of length $L$ and transverse area $A$. ${\Omega }_{\pm }$ are the Rabi frequencies of the classical control fields, and ${\mathrm{Ê}}_{\pm }$ are the quantum probe fields. A typical intensity profile of the quantum fields is shown for the case of stationary light where the control fields have the same intensity (${\Omega }_{+}={\Omega }_{-}$).

Figure 2

Atomic level scheme. ${\gamma }_{\mathit{ij}}$ is the full decay rate on the $|i\rangle \leftrightarrow |j\rangle$ transition, $\delta$ and $\Delta$ label the detuning of the probe fields with states $|3\rangle$ and $|4\rangle$, respectively, and $\varepsilon$ is the two-photon detuning.

Figure 3

Decay of the $k=0$ polariton mode for nondegenerate ground states $|1\rangle$ and $|2\rangle$. The solid line shows the decay according to Eq. (62); the corresponding result from a numerical integration of Maxwell-Bloch equations is indicated by red dots. The parameters are $\Delta \omega =150\Gamma$, $\cos {}^2\theta =2.5\times {10}^{-4}$, ${\Omega }_0=90\Gamma$, $\delta =0$, and $\varepsilon$ is fixed by Eq. (37). The initial spin coherence $R_{21}$ (see Appendix D) has a Gaussian shape $R_{21}\propto \exp [-(z-z_0){}^2/(2{\sigma }^2)]$ with $\sigma =24.3\Gamma c/{\Omega }_0^2$, and the control fields are ramped up according to ${\Omega }_c(t)=0.5[1+\tanh (t-10/\Gamma )]\times \Gamma$. The initial value $\langle {\psi }_0^{†}{\psi }_0\rangle {}_{\mathrm{init}}$ refers to $\langle {\psi }_0^{†}{\psi }_0\rangle$ at $t=20/\Gamma$ where the control fields have attained their maximal value of ${\Omega }_c=\Gamma$.

Figure 4

The number of dark-state polaritons is shown as a function of time. The results of Eqs. (64) and (65) are represented by the solid and the dotted line, respectively. The good agreement indicates that the polariton losses are described correctly by the master equation (56). The number of polaritons follows the dashed line if the additional decay term ${\mathcal{L}}_2{\varrho }_{\mathrm{D}}$ in Eq. (56) is omitted. The parameters are the same as in Fig. 3.

Figure 5

Alternative level scheme that gives rise to the master equation for polaritons described in Sec. 4. All symbols are defined in the same way as in Fig. 2.