Mobile bound states of Rydberg excitations in a lattice

Spin-lattice models play a central role in the studies of quantum magnetism and nonequilibrium dynamics of spin excitations—–magnons. We show that a spin lattice with strong nearest-neighbor interactions and tunable long-range hopping of excitations can be realized by a regular array of laser-driven atoms, with an excited Rydberg state representing the spin-up state and a Rydberg-dressed ground state corresponding to the spin-down state. We find exotic interaction-bound states of magnons that propagate in the lattice via the combination of resonant two-site hopping and nonresonant second-order hopping processes. Arrays of trapped Rydberg-dressed atoms can thus serve as a flexible platform to simulate and study fundamental few-body dynamics in spin lattices.

Figure 1

(a) Spectrum of two spin (Rydberg) excitations in a lattice versus the center-of-mass quasimomentum $K$. The scattering states form a continuum spectrum (black) [Eq. (7)]. The bound states for strong (red lines) and weak (blue lines) repulsive interactions are obtained from the spin-lattice Hamiltonian (dashed lines) [Eq. (9)] and from exact diagonalization of the Hamiltonian for the system sketched in (b) (solid lines). In the simulations, we used a lattice of size $L=100$ and periodic boundary conditions, with the spin model parameters $J_2/J_1=1/8$, $U_1/J_1=3.4,1.9$ for the red and blue dashed lines, respectively. The inset illustrates the motion of the bound pair via resonant two-site hopping $J_2$ and second-order hopping $J_1^2/U_1$. (b) Level scheme of atoms to realize a spin-lattice model. Atoms in Rydberg states $|e\rangle$ and $|s\rangle$ undergo dipole-dipole exchange interaction $|es\rangle \to |se\rangle$ with rate $D$. The atomic ground state $|g\rangle$ is dressed with the Rydberg state $|s\rangle$ by a nonresonant laser with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }\gg \mathrm{\Omega }$. The spin-up and spin-down states correspond to $|\uparrow \rangle =|e\rangle$ and $|\downarrow \rangle \simeq |g\rangle +\frac{\mathrm{\Omega }}{\mathrm{\Delta }}|s\rangle$. Interactions $V^{ee}$ between the atoms in state $|e\rangle$ lead to formation of mobile bound states of Rydberg excitations. The parameters in numerical simulations shown in (a) correspond to $\mathrm{\Delta }/\mathrm{\Omega }=10$, $D_1/\mathrm{\Omega }=1$, $V_1^{es}/\mathrm{\Omega }=-0.125$, and $V_1^{ee}/\mathrm{\Omega }=0.03,0.015$ for the red and blue solid lines, respectively.

Figure 2

(a) Diagram of values of ${\alpha }_K$ and ${\beta }_K$ for the existence of bound states (light blue shaded region). (b) Wave function ${\psi }_K\left(r\right)$ versus the relative distance $r$ for several values of the center-of-mass quasimomentum $K$. The parameters are the same as in Fig. 1, with $U_1/J_1=3.4$ [left graph and red line in (a)], and $U_1/J_1=1.9$ [right graph and blue line in (a)], where the bound state does not exist in the vicinity of $K=0$.

Figure 3

Diagram of values of $J_2/J_1$, for fixed $U_1=3J_1$ (upper panel), and $U_1/J_1$, for fixed $J_2=J_1/8$ (lower panel), versus $K$, for the existence (white regions) and absence (black regions) of the bound states.

Figure 4

Scattering (black) and bound-state (red solid line) spectra obtained by exact numerical diagonalization of Hamiltonian with long-range interactions $J_d=J_1/d^3$ and $U_r=U_1/r^6$ ($U_1/J_1=4$). The bound-state energy $E_K$ of Eq. (A12) (dashed blue line), obtained with truncated interactions ($d_J=2$ and $d_U=1$), is nearly indistinguishable from the exact result.

Figure 5

(a) Diagram of transitions for retrieving the perturbative energy shifts and excitation hopping rates for two excited $|e\rangle$ and one ground $|g\rangle$ state atoms. The atomic positions are $x,y,z$. The red shaded region denotes the high-energy subspace, $\mathrm{\Delta }\gg \mathrm{\Omega },D$, which is eliminated adiabatically. (b) Illustration of three virtual processes contributing to the energy shift of $|eeg\rangle$, as per Eq. (B16). (c) Two possible paths for the hopping process $|eeg\rangle \leftrightarrow |gee\rangle$ given by Eq. (B17).

Figure 6

Comparison of the low-energy spectra of the exact Hamiltonian (B14) including the interactions $V_r^{ee}$ (solid lines), and the effective Hamiltonian (B18) (dashed lines). The positions of the first and second atoms are fixed, $x=0$ and $y=a$, while the position of the third atom varies: $z\ge 2a$. Black lines at $E\ge 0$ show the exact spectrum for $\mathrm{\Omega }=0$, corresponding to the bare states $|ege\rangle$, $|gee\rangle$, and $|eeg\rangle$. Blue lines show the spectra for the dressed states with the parameters $\mathrm{\Delta }/\mathrm{\Omega }=10$, $D_1/\mathrm{\Omega }=1$, $V_1^{se}/\mathrm{\Omega }=-1/8$, and $V_1^{ee}/\mathrm{\Omega }=0.03$ ($D_r\propto 1/r^3$, $V_r\propto 1/r^6$).

Figure 7

(a) One- and two-site hopping rates $J_1\left(r\right)$ and $J_2\left(r\right)$ versus distance $r$ between the two excitations. (b) Interaction potential $U_r$ of Eq. (B23), for $V_1^{ee}=0$. The parameters are $\mathrm{\Delta }/\mathrm{\Omega }=10$, $D_1/\mathrm{\Omega }=1$, and $V_1^{se}/\mathrm{\Omega }=1$.

Figure 8

Scattering and bound spectrum for two $|e\rangle$ excitations in a lattice, with the parameters as in Fig. 7.