Spin Squeezing and Light Entanglement in Coherent Population Trapping

We show that strong squeezing and entanglement can be generated at the output of a cavity containing atoms interacting with two fields in a coherent population trapping situation, on account of a nonlinear Faraday effect experienced by the fields close to a dark-state resonance in a cavity. Moreover, the cavity provides a feedback mechanism allowing to reduce the quantum fluctuations of the ground state spin, resulting in strong steady state spin squeezing.

Figure 1

Real and imaginary parts of the reflectivity coefficients of the cavity when the two-photon detuning $\delta$ is varied. Parameters: $C=100$, $I=1$, $\phi =0$. The dashed parts correspond to unstable solutions for the linearly polarized light.

Figure 2

Self-rotation induced on the field close to the CPT resonance: a nonzero detuning $\delta$ causes the spin to rotate in the equatorial plane of the Bloch sphere by an angle proportional to $\delta$. Because of the Faraday effect, the Stokes vector also tends to rotate by an angle ${\varphi }_{\mathrm{nl}}$ proportional to $\delta$ in the equatorial plane of the Poincaré sphere, but the detuned cavity brings it back along $x$. The ellipsoids represent the quantum fluctuations of the Bloch and Stokes vectors.

Figure 3

Squeezing spectra of the circularly and linearly polarized modes, $A_{1,2}$ and $A_{x,y}$ [as defined by Eq. (7)]. Parameters: $C=100$, $\kappa =2\gamma$, $\overline{\delta }=1$, $\phi =1$, $I=144$.

Figure 4

Maximal entanglement ${\mathcal{E}}^{*}$ that can be obtained by adequately mixing the outgoing fields vs sideband frequency $\omega /\gamma$. (a) Parameters as in Fig. 3. (b) $C=1000$, $\kappa =2\gamma$, $\overline{\delta }=0.1$, $\phi =2$, $I=49$.

Figure 5

$\Delta {\overline{J}}_z^2$ vs the cavity detuning $\phi$ for $2\delta =0.01\gamma$. (a) Analytical (13), (b) $C=1000$, (c) $C=100$. $\alpha$ is chosen to optimize the squeezing for each value of the cavity detuning.