Conditional Spin Squeezing of a Large Ensemble via the Vacuum Rabi Splitting

We use the vacuum Rabi splitting to perform quantum nondemolition measurements that prepare a conditionally spin squeezed state of a collective atomic psuedospin. We infer a 3.4(6) dB improvement in quantum phase estimation relative to the standard quantum limit for a coherent spin state composed of uncorrelated atoms. The measured collective spin is composed of the two-level clock states of nearly ${10}^6$ ${}^{87}\mathrm{Rb}$ atoms confined inside a low finesse $F=710$ optical cavity. This technique may improve atomic sensor precision and/or bandwidth, and may lead to more precise tests of fundamental physics.

Figure 1

(a) The quantum noise of a collection of pseudospin $\frac{1}{2}$ atoms can be represented as a classical probability distribution for the collective spin or Bloch vector with length $J_{\max }=N/2$. Probing the number of atoms in spin up and spin down measures the projection $J_z=(N_{\uparrow }-N_{\downarrow })/2$, as well as $J_{\max }$. For states near the equator, the polar angle is ${\theta }_B\approx J_z/J_{\max }$. The population measurement sequence consists of composite microwave rotations (dashed line) and QND population measurements (solid line). The evolution of the ensemble is represented by Bloch spheres 1, 2, and 3. The QND measurement projects the ensemble into a conditionally squeezed state (3), which is verified with the second measurement. (b) The observed variance of ${\theta }_B$ versus atom number confirms the predicted standard quantum limit. Two different rotations (circles and diamonds) are used to constrain added noise from the rotations [21].

Figure 2

(a) The clock states $|\uparrow ⟩$ and $|\downarrow ⟩$ form a pseudospin-$\frac{1}{2}$ system. The population $N_{\uparrow }$ is measured by probing near the $|\uparrow ⟩$ to $|e⟩$ transition, which couples to a degenerate cavity mode, creating a collective vacuum Rabi splitting ${\Omega }_{\uparrow }$. (b) An ensemble of ${10}^6$ atoms are tightly confined within the ${\mathrm{TEM}}_{00}$ mode using a 1D intracavity optical lattice at wavelength 823 nm. The cavity power and atomic population decay rates are $\kappa$ and $\Gamma$, respectively. A heterodyne interferometer (not shown) is used to probe the atom-cavity resonances. The collective nature of the Rabi splitting prevents loss of coherence as atoms remain in a superposition after the measurement. (c) The vacuum Rabi splitting ${\Omega }_{\uparrow }={\omega }_{+}-{\omega }_{-}$ is measured by simultaneously scanning the probe frequencies across the upper and lower atom-cavity resonances ${\omega }_{\pm }$. The size of the fitted splitting determines the population in $|\uparrow ⟩$ from $N_{\uparrow }=({\Omega }_{\uparrow }/2g{)}^2$. A single scan requires $\approx 70\mu \mathrm{s}$.

Figure 3

(a) The first and second QND measurements of $J_z$ exhibit correlated fluctuations that arise largely from projection noise. (b) The noise in the backaction quadrature is observed by inserting a rotation through angle $\psi$ between the QND measurements $J_{z1}$ and $J_{z2}$. The dashed curve is the calculated response for a minimum uncertainty squeezed state, while the solid curve with the gray 68% confidence interval is the predicted backaction from the intracavity probe vacuum noise.

Figure 4

(a) The degree of coherence remaining after the measurement $J_{z1}$ is determined using the sequence: ($\frac{\pi }{2}$)–(measure $N_{\uparrow }$)–($\pi$)–(measure $N_{\downarrow })$–($\frac{\pi }{2}$)–(measure $N_{\uparrow }$). The sequence is repeated and the final measured value of $N_{\uparrow }$ is plotted versus the phase of the final $\frac{\pi }{2}$ pulse. With no measurements (empty circles), the background contrast is ${\mathcal{C}}_i=0.97(1)$. Probing with $1.8\times {10}^5$ photons (filled circles), causes a small reduction in contrast despite a measurement sensitivity below the projection noise level. (b) Measured contrast versus probe photon number (solid circles), second order polynomial fit (solid line), and the predicted contrast loss due to free space scattering alone (dashed line).