Generating resonating-valence-bond states through Dicke subradiance

Dicke's original thought experiment with two spins (two-level atoms) coupled to a photon mode has recently been experimentally realized. We propose extending this experiment to many spins as a way to synthesize highly entangled states. We suggest a protocol in which we start with a direct product state of $M$ atoms in the excited state and $N$ atoms in the ground state, placed within a lossy cavity. A null observation for photon emission collapses the system onto a dark state which, remarkably, has resonating-valence-bond (RVB) character. We demonstrate this by taking advantage of the symmetry of the initial state under permutations of the $M$ excited atoms and of the $N$ unexcited atoms. Using angular momentum analysis, we reexpress the wave function of the dark state to illustrate its RVB character. We discuss two limiting cases in detail, with $M=1$ (one excited atom) and $M=N$ (equal number of excited and unexcited atoms). In the latter case, we show that the probability for null emission scales as $N^{-1}$, making it possible to generate highly entangled RVB states of 20 spins or more.

Figure 1

Explicit construction of the RVB state. The sum is over $P(N,M)$, the number of ways of assigning one partner from the bottom row for every spin in the top row, leading to ${}^N{P}_M$ terms. The quantity ${\beta }_{N,M}$ is a normalization constant. Up to an overall constant, this sum can be rewritten as a sum over all permutations of spins in the bottom row, with $N!$ terms. The extra constant has been absorbed into the normalization constant, ${\gamma }_{N,M}$.

Figure 2

Dark component of $(N+1)$-spin state with one spin up, for $N=3,4$. The dark state is obtained by action of the projection operator ${\widehat{\mathcal{P}}}_{S=(N-1)/2}$, followed by normalization. The result is a linear combination of single-dimer states. The normalization constant for general $N$ is $\sqrt{2/N(N+1)}$.

Figure 3

Dark component of $\left(2N\right)$-spin state with half the spins excited initially, for $N=3,4$. An RVB state results from the action of the singlet projection operator.