Many-Body Spin Interactions and the Ground State of a Dense Rydberg Lattice Gas

We study a one-dimensional atomic lattice gas in which Rydberg atoms are excited by a laser and whose external dynamics is frozen. We identify a parameter regime in which the Hamiltonian is well approximated by a spin Hamiltonian with quasilocal many-body interactions which possesses an exact analytic ground state solution. This state is a superposition of all states of the system that are compatible with an interaction induced constraint weighted by a fugacity. We perform a detailed analysis of this state which exhibits a crossover between a paramagnetic phase with short-ranged correlations and a crystal. This study also leads us to a class of spin models with many-body interactions that permit an analytic ground state solution.

Figure 1

Comparison between the numerical results obtained for a lattice with $L=20$ sites and $\gamma =6$ (solid blue curve) and the analytical expressions (dashed red curve). (a) Energy per particle in the ground state, (b) mean density of Rydberg atoms on the lattice, (c) Mandel $Q$ parameter of the Rydberg number distribution. (d) Numerically calculated density-density correlation function. (e) Density-density correlation function obtained for the state (5). Note that at the same time as $\Delta$ the potential is also varied according to $V=2^{\gamma }\Omega \xi$. The values of $\xi$ are given underneath (c).

Figure 2

(a) Entanglement entropy $S$ of a single spin with the remaining ones. For large negative detuning the ground state becomes a GHZ state and $S$ reaches its maximum $\log 2$. (b) Density of excited atoms as a function of the detuning and the Rabi frequency. The black line represents the set of parameters where the Rydberg gas ground state is approximately given by Eq. (5). Dashed lines are used as a guide to the eye delimiting the regions where the Rydberg density is approximately $1/3$ and $1/2$. (c) The spin Hamiltonian (3) can be generalized to higher dimensions (here 2D) where the excitation on the $k$th site blocks the excitation of all sites contained in the set $G_k$.