Rapid Production of Many-Body Entanglement in Spin-1 Atoms via Cavity Output Photon Counting

We propose a simple and efficient method for generating metrologically useful quantum entanglement in an ensemble of spin-1 atoms that interacts with a high-finesse optical cavity mode. It requires straightforward preparation of $N$ atoms in the $m_F=0$ sublevel, tailoring of the atom-field interaction to give an effective Tavis-Cummings model for the collective spin-1 ensemble, and a photon counting measurement on the cavity output field. The photon number provides a projective measurement of the collective spin length $S$, which, for the chosen initial state, is heavily weighted around values $S\simeq \sqrt{N}$, for which the corresponding spin states are strongly entangled and exhibit Heisenberg scaling of the metrological sensitivity with $N$, as quantified by the quantum Fisher information.

Figure 1

Implementation of an effective Dicke model using the $F=1$ ground state of ${}^{87}\mathrm{Rb}$. Interactions are mediated by detuned Raman transitions on the $D_1$ line mediated by a cavity mode (red) and ${\sigma }_{-}$ (blue) and ${\sigma }_{+}$ (green) polarized lasers.

Figure 2

Populations $|c_S{|}^2$ for even $S$ in the collective Dicke states $|S,0⟩$ for the product state $|m_F=0{⟩}^{\otimes N}$ of $N=1000$ spin-1 atoms. Populations for states beyond $S=200$ are not shown as the total sum of these populations is $\sum _{S>200}|c_S{|}^2=1.34\times {10}^{-9}$.

Figure 3

Average QFI on the generator ${\widehat{Q}}_{xx}-{\widehat{Q}}_{yy}$ of states $|S,-S⟩$ weighted by initial populations $|c_S{|}^2$ of the states $|S,0⟩$ in Eq. (7). Even and odd numbers are represented differently due to their slightly different dependence on $N$. The inset shows the populations and QFI of the individual states $|S,-S⟩$ for $N=40$.

Figure 4

Average QFI on the generator ${\widehat{Q}}_{xx}-{\widehat{Q}}_{yy}$ of states ${\rho }_n$ weighted by the probability $p(n)$ of measuring $n$ photons, as given by Eqs. (13) and (14), for various photodetector efficiencies $\eta$.

Figure 5

Population as a function of spin length $S$ for a varying total number of output (multiphoton) pulses generated by a sequence of TC and anti-TC interactions. Each pulse is simulated with a binomially distributed registered photon number using a detection efficiency of $\eta =0.7$ and an assumed actual photon number based on a spin length $S=30$.

Figure 6

Sample mean QFI on generator ${\widehat{Q}}_{xx}-{\widehat{Q}}_{yy}$, for $N=16$, of 1000 states produced by randomly selecting a spin length $S$ and a sequence of binomially distributed measured photons given the initial state and a photon detection efficiency $\eta$, respectively.