Multipartite Entangled States in Dipolar Quantum Simulators

The scalable production of multipartite entangled states in ensembles of qubits is a crucial function of quantum devices, as such states are an essential resource both for fundamental studies on entanglement, as well as for applied tasks. Here we focus on the U(1) symmetric Hamiltonians for qubits with dipolar interactions—a model realized in several state-of-the-art quantum simulation platforms for lattice spin models, including Rydberg-atom arrays with resonant interactions. Making use of exact and variational simulations, we theoretically show that the nonequilibrium dynamics generated by this Hamiltonian shares fundamental features with that of the one-axis-twisting model, namely, the simplest interacting collective-spin model with U(1) symmetry. The evolution governed by the dipolar Hamiltonian generates a cascade of multipartite entangled states—spin-squeezed states, Schrödinger’s cat states, and multicomponent superpositions of coherent spin states. Investigating systems with up to $N=144$ qubits, we observe full scalability of the entanglement features of these states directly related to metrology, namely, scalable spin squeezing at an evolution time $\mathcal{O}(N^{1/3})$ and Heisenberg scaling of sensitivity of the spin parity to global rotations for cat states reached at times $\mathcal{O}(N)$. Our results suggest that the native Hamiltonian dynamics of state-of-the-art quantum simulation platforms, such as Rydberg-atom arrays, can act as a robust source of multipartite entanglement.

Figure 1

(a) Cascade of entangled states observed in the OAT model and in this Letter—here we only indicate even-headed cat states, but $q$-headed states with odd $q$ exist as well, at times $t=2\pi I/q$. (b) Dynamics of the average spin $⟨J^x⟩(t)$ for the dipolar $XX$ model on the square lattice. (c) Fourier transform $⟨J^x⟩(\omega )$, for coherence with the $y$-axis label in the figure. (d) Effective moment of inertia for the square (Squ) and triangular (Tri) dipolar lattices as extracted from the TVMC dynamics (squares and triangles) and estimated from the low-energy spectrum (dashed lines). The specific combination $\pi I_N^{(\mathrm{eff})}\mathcal{J}$ represents the time of the $q=2$ cat state.

Figure 2

(a) Evolution of the spin-squeezing parameter for the dipolar $XX$ model on a square lattice, where the circles mark the optimum. (b) Scaling of the optimal squeezing value and optimal squeezing time (with Kac normalization $K_N^{(\alpha )}$ [47]), showing exponents $\nu =0.72$ and $\mu =0.36$ (to be compared with $\nu =2/3$ and $\mu =1/3$ for the OAT model).

Figure 3

(a) Sketch of the $q$-cat states studied here—the balls indicate the coherent states involved in the $q$-cat state. (b) Distributions $P(J^x)$ obtained at the times of formation of the cat states with $q=6$, 4, and 2, obtained via exact calculations on a $5\times 4$ square lattice—only the $P$ values for even $J^x$ are indicated, as the probability for the odd $J^x$ values vanish identically, since the Hamiltonian is parity conserving. (c) Log-lin plot of the tail of the peaks for the $q=2$ cat state; the dashed line is the fit to an exponential.

Figure 4

(a) Time evolution of $4\mathrm{Var}(J^x)/N^2$ compared to that of the inverse uncertainty on phase estimation from the evolution of the parity—the results are for a square lattice with $N=100$. (b) Size dependence of the maximal variance of $J^x$ during time evolution, for both square and triangular lattices.