Generating entangled spin states for quantum metrology by single-photon detection

We propose and analyze a probabilistic but heralded scheme to generate pure, entangled, non-Gaussian states of collective spin in large atomic ensembles by means of single-photon detection. One photon announces the preparation of a Dicke state, while two or more photons announce Schrödinger cat states. The method produces pure states even for finite photon detection efficiency and weak atom-photon coupling. The entanglement generation can be made quasideterministic by means of repeated trial and feedback, enabling metrology beyond the standard quantum limit.

Figure 1

Scheme for the heralded generation of nonclassical states. (a) Atoms with two spin states $\left|\uparrow \right.〉$ and $\left|\downarrow \right.〉$ are coupled to an electronic excited state $\left|e\right.〉$ via two degenerate circularly polarized modes. (b) Incident vertically polarized photons experience weak Faraday rotation as they traverse the ensemble. The detection of a horizontally polarized transmitted photon heralds the generation of a non-Gaussian entangled state of collective atomic spin. A cavity enhances the Faraday rotation and the state preparation probability.

Figure 2

Normalized Wigner quasiprobability distribution $W(\theta ,\varphi )/\sqrt{2\pi S}$ (left) and probability distributions of angular-momentum eigenvalues (right, solid line for $S_z$, dashed line for $S_y$) calculated for $N=50$ atoms for [(a), (b)] the input CSS, [(c), (d)] $n=1$ detected $\left|\text{h}\right.〉$ photon, [(e), (f)] $n=2$ detected $\left|\text{h}\right.〉$ photons, and [(g), (h)] $n=5$ detected $\left|\text{h}\right.〉$ photons. Wigner functions [(e), (g)] indicate the production of a Schrödinger cat state for $n\ge 2$ detected $\left|\text{h}\right.〉$ photons. The distributions of $S_y$ eigenvalues in panels (d), (f), and (h) consist of several peaks, narrower than the CSS width, enabling measurements surpassing the SQL.

Figure 3

Black squares show the measurement variance of the states $\left|{\psi }_n\right.〉$, normalized to the CSS variance, ${\left(\Delta S_z^{\left(w\right)}\right)}^2/(S/2)$, in dB, as a function of number of detected $\left|\text{h}\right.〉$ photons $n$, indicating substantial metrological gain for $n$ of a few. The dashed curve shows that for larger $n$ the normalized variance asymptotically approaches the value ${\left(\Delta S_z^{\left(w\right)}\right)}^2/(S/2)=0.64/n$ (see text). Due to the finite atom number, $N=100$, used for the calculations, points in the figure show slight deviations from the asymptotic behavior expected for large $N$.