Implementation of Cavity Squeezing of a Collective Atomic Spin

We squeeze unconditionally the collective spin of a dilute ensemble of laser-cooled ${}^{87}\mathrm{Rb}$ atoms using their interaction with a driven optical resonator. The shape and size of the resulting spin uncertainty region are well described by a simple analytical model [M. H. Schleier-Smith, I. D. Leroux, and V. Vuletić, arXiv:0911.3936 [Phys. Rev. A (to be published)]] through 2 orders of magnitude in the effective interaction strength, without free parameters. We deterministically generate states with up to 5.6(6) dB of metrologically relevant spin squeezing on the canonical ${}^{87}\mathrm{Rb}$ hyperfine clock transition.

Figure 1

Cavity squeezing [15]. (a) The atoms are trapped in a standing-wave dipole trap inside an optical resonator. (b) The probe laser is detuned from cavity resonance by half a linewidth, so that atom-induced shifts of the cavity frequency change the transmitted power. (c) The cavity is tuned halfway between the optical transition frequencies for the two clock states. (d) The $S_z$-dependent light shift shears the circular uncertainty region of the initial coherent spin state (red circle) into an ellipse (dotted). Photon shot noise causes phase broadening that increases the ellipse area (solid). The illustration is for a modest shearing $Q=3$ (see text).

Figure 2

Normalized variance ${\sigma }^2$ as a function of rotation angle $\alpha$ about the mean spin direction for states prepared with shearing $Q=0$ (black circles), $Q=1.2$ (red squares), $Q=7.7$ (green triangles), and $Q=30.7$ (blue diamonds). The curves are cosine fits. Statistical error bars are comparable to the symbol size. The shapes of the corresponding uncertainty regions are illustrated below the plot. Inset: Observed variance $\Delta S_z^2$ of the initial state as a function of $S_0$. The line is the projection noise limit as determined from cavity parameters.

Figure 3

${\sigma }_{\max }^2$ (top panel) is the normalized maximum variance, ${\sigma }_{\min }^2$ (middle panel) the normalized minimum variance, and ${\alpha }_0$ (bottom panel) the rotation angle for minimum variance for each measured ellipse as a function of shearing strength $Q$. Statistical error bars are given for ${\sigma }_{\min }^2$, and are smaller than the symbols for ${\sigma }_{\max }^2$ and ${\alpha }_0$. The blue dashed curves are theoretical predictions for an ideal system, while the black lines are predictions including separately measured technical imperfections. Shearing was varied by adjusting the photon number $p_0\approx 2200\times Q$.

Figure 4

Metrologically relevant squeezing $\zeta$ (red solid circles) and signal contrast $C$ (green diamonds, inset) as a function of shearing strength $Q$.