Extreme spin squeezing in the steady state of a generalized Dicke model

We present a scheme to generate steady-state atomic spin squeezing in a cavity QED system using cavity-mediated Raman transitions to engineer effective atom-photon interactions, which include both linear and nonlinear (dispersive) atom-cavity couplings, on a potentially equal footing. We focus on a regime where the dispersive coupling is very large and find that the steady state of the system can in fact be a strongly spin-squeezed Dicke state $|N/2,0\rangle$ of the atomic ensemble. These states offer Heisenberg-limited metrological properties and feature genuine multipartite entanglement among the entire atomic ensemble.

Figure 1

Steady-state expectation values for $\{\omega ,{\omega }_0\}/\kappa =\{1.0,0.2\}$ and $N=10$, with $\lambda /\kappa =0.05$ (red solid line), $\lambda /\kappa =0.2$ (blue dashed line), and $\lambda /\kappa =0.4$ (green dash-dotted line). The black lines in the plot of ${\xi }_D$ are the standard quantum limit (solid) and the ideal limit of $1/(N+2)=1/12$ (dashed).

Figure 2

Steady-state expectation values for $\{\omega ,{\omega }_0,\lambda \}/\kappa =\{1.0,0.2,0.1\}$ with $N=4$ (red solid line), $N=8$ (blue dashed line), and $N=12$ (green dash-dotted line). Here the Dicke squeezing parameter ${\xi }_D$ is given in proportion to the ideal limit $1/(N+2)$.

Figure 3

Approximate level diagram for large positive $U\gg \omega ,{\omega }_0,\lambda$. The dominant evolution pathway is illustrated.

Figure 4

Steady-state Dicke squeezing parameter ${\xi }_D$ as a function of $U/\omega$ for $\{\omega ,{\omega }_0,\lambda \}/\kappa =\{100.0,0.2,0.2\}$, with $N=10$. The dashed lines are those predicted by Eq. (2).

Figure 5

Time evolution of $\langle {\widehat{S}}_z\rangle$ with photon detections (vertical lines) for a successful trajectory (blue) and a failure (red). The parameters are $\{{\omega }_0,\lambda ,U\}/\kappa =\{0.2,0.2,1000.0\}$, $N=20$, and $\omega$ is linearly varied as $\omega \left(t\right)/\kappa =470-0.2\kappa t$.

Figure 6

Properties of 985 successful trajectories out of 1000 total trajectories, with the parameters of $\{{\omega }_0,\lambda ,U\}/\kappa =\{0.2,0.2,500.0\}$, $N=10$, and time dependent $\omega$ discretely stepped such that it makes each transition $|0,-5\rangle \to |0,-1\rangle$ resonant in turn. The system is held at each step for $\kappa t_h=52$. (a) Time evolution of Dicke state populations with blue, green, red, cyan, and pink (or peaks from left to right) denoting $m=-5$, $-4$, $-3$, $-2$, and $-1$, respectively. (b) Histogram of final squeezing. The vertical lines are the ideal limit (dotted) and steady-state squeezing (dashed).

Figure 7

Level scheme for the implementation of the generalized Dicke model with ${}^{87}\mathrm{Rb}$ atoms. Transitions are driven with Raman transitions composed of a cavity mode (red solid line) and ${\sigma }_{+}$-polarized (green dash-dotted line) and ${\sigma }_{-}$-polarized (blue dashed line) laser fields. Raman transitions are detuned on either side of the excited manifold to maximize the nonlinear term. Note that the level diagram is not drawn to scale.

Figure 8

For a single trajectory with $N=1000$ and $\{\omega ,{\omega }_0,U,\lambda \}/\kappa =\{0.01,0.01,3000.0,0.316\}$, (a) squeezing with the times of photon detection events represented by red vertical dashed lines and (b) populations initially (top) and after the first jump (middle) and 12th jump (bottom).