Bose-Einstein Condensation of Stationary-Light Polaritons

We propose and analyze a mechanism for Bose-Einstein condensation of stationary dark-state polaritons. Dark-state polaritons (DSPs) are formed in the interaction of light with laser-driven 3-level $\Lambda$-type atoms and are the basis of phenomena such as electromagnetically induced transparency, ultraslow, and stored light. They have long intrinsic lifetimes and in a stationary setup, a 3D quadratic dispersion profile with variable effective mass. Since DSPs are bosons, they can undergo a Bose-Einstein condensation at a critical temperature which can be many orders of magnitude larger than that of atoms. We show that thermalization of polaritons can occur via elastic collisions mediated by a resonantly enhanced optical Kerr nonlinearity on a time scale short compared to the decay time. Finally, condensation can be observed by turning stationary into propagating polaritons and monitoring the emitted light.

Figure 1

(a) Raman interaction of two counterpropagating control lasers ${\Omega }_{\pm }$ with opposite circular polarization coupling $|s⟩-|e_{\pm }⟩$ transitions of a double $\Lambda$ system generates stationary light of Stokes fields $E_{\pm }$. The stationary DSPs formed by this do not decay radiatively and have a 3D quadratic dispersion profile. (b) Off-resonant coupling of Stokes fields $E_{\pm }$ with other excited states leads to ac-Stark shift of $|s⟩$ resulting in a resonantly enhanced Kerr-type self-interaction of $E_{\pm }$.

Figure 2

Dispersion of stationary DSP (centermost line) and other quasiparticle solutions in propagation direction for ${\Delta }_{+}={\Delta }_{-}=\Delta$, and ${\Omega }_{+}={\Omega }_{-}=\Omega$, $\tan \theta =\tan \varphi =1$, $\Delta /\Omega =2$. Since only the real parts of the eigenvalues are of interest here, we set $\gamma =0$. One clearly recognizes a quadratic profile for the dark polariton around $k=0$. It should be noted that the polariton is defined by the slowly varying envelopes of field and matter variables and $k$ refers to spatial modulations of the slowly varying amplitudes.

Figure 3

(a) Detection of condensation: switching off one of the two control beams transforms the stationary into a propagating polariton which can be observed as emitted light pulse. (b) Transverse emission profile above (left) and much below (right) condensation. An ideal quasihomogeneous Bose-gas was assumed with transversal Gaussian distribution of width 178 ${\lambda }_{T,x}$ with ${\lambda }_{T,x}$ being the transversal de Broglie wavelength at temperature $T$.