Robust Spin Squeezing via Photon-Mediated Interactions on an Optical Clock Transition

Cavity QED is a promising avenue for the deterministic generation of entangled and spin-squeezed states for quantum metrology. One archetypal scheme generates squeezing via collective one-axis twisting interactions. However, we show that in implementations using optical transitions in long-lived atoms the achievable squeezing is fundamentally limited by collectively enhanced emission into the cavity mode which is generated in parallel with the cavity-mediated spin-spin interactions. We propose an alternative scheme which generates a squeezed state that is protected from collective emission, and investigate its sensitivity to realistic sources of experimental noise and imperfections.

Figure 1

(a) Proposed experimental system: An ensemble of $N$ atoms trapped in a standing-wave lattice potential and optically coupled to the field within an optical cavity of linewidth $\kappa$. Selective population of the $m_F$ levels of the ${}^1{S}_0$ to ${}^3{P}_0$ transition of ${}^{87}\mathrm{Sr}$ [20] realizes independently controllable collective spins. (b) For both OAT and TSS, the Hamiltonian can be decomposed into a shearing term, ${\widehat{S}}_z^2$ (OAT) or ${\widehat{S}}_y^2$ (TSS), and negligible terms (gray) which do not contribute to dynamics. The shearing terms can be interpreted semiclassically as a precession driven by quantum fluctuations (green). (c) Schematic of squeezing protocols. (i)–(iii) For OAT, fluctuations along $S_z$ (example shown by green vector) drive a precession of the Bloch vector $\overrightarrow{S}$ (red, precession indicated in blue) about the $S_z$ axis, generating squeezing of the noise distribution [red in (i) and (iii)]. (iv)–(vi) For TSS, two back-to-back collective spins are prepared [(iv) light red], and common-mode fluctuations along $S_y$ [(v) light red] generate a weak coherence along $S_y$ (green), driving the precession of the Bloch vectors ${\overrightarrow{S}}_{1,2}$ (solid red, precession indicated in blue). Subsequent rotation about $S_y$ of one of the spins maps this precession into squeezing of the collective distribution [(vi) red].

Figure 2

(a) Comparison of squeezing with OAT and TSS protocols for $N=1000$. Faded lines are ideal $\mathrm{\Gamma }=0$ results for OAT (blue) and TSS (red dashed). Solid lines are $\mathrm{\Gamma }=0.1\chi$ results for OAT (blue) and TSS (red dashed). Shown inset is the best squeezing as a function of $\mathrm{\Gamma }/\chi$ (OAT: solid blue, TSS: dashed red). Dot-dashed blue line indicates squeezing for OAT and $N=200$, illustrating the invariance with $N$. (b) Dark region indicates ideal squeezed state, light region indicates distribution with added dissipative noise due to photon leakage from the cavity (arrows), which approximately aligns with the squeezed quadrature or antisqueezed quadrature.