Designing Frustrated Quantum Magnets with Laser-Dressed Rydberg Atoms

We show how a broad class of lattice spin-$1/2$ models with angular- and distance-dependent couplings can be realized with cold alkali atoms stored in optical or magnetic trap arrays. The effective spin-$1/2$ is represented by a pair of atomic ground states, and spin-spin interactions are obtained by admixing van der Waals interactions between fine-structure split Rydberg states with laser light. The strengths of the diagonal spin interactions as well as the “flip-flop,” and “flip-flip” and “flop-flop” interactions can be tuned by exploiting quantum interference, thus realizing different spin symmetries. The resulting energy scales of interactions compare well with typical temperatures and decoherence time scales, making the exploration of exotic forms of quantum magnetism, including emergent gauge theories and compass models, accessible within state-of-the-art experiments.

Figure 1

(a) Atoms loaded in a kagome lattice driven by laser light ($L$) propagating along the $z$ axis defined by the magnetic field ($B$). (b) Atomic level scheme: ${}^{87}\mathrm{Rb}$ atoms with hyperfine ground states $|g_{\sigma }⟩$ (representing spin $1/2$) coupled to $n^2{P}_{1/2}$ Rydberg states with ${\sigma }^{\pm }$ polarized light and interacting via VDW interactions. (c) Spin interactions $J_{\alpha }$ of Eq. (1) as a function of distance $\rho$ realizing quantum spin ice on a kagome lattice [10]. Red (gray) dotted lines indicate NN and NNN interactions [red (gray) arrows in panel (a)]. Here $|r_{\pm }⟩=|60^2{P}_{1/2},\pm \frac{1}{2}⟩$ with Rabi frequencies ${\mathrm{\Omega }}_{-}={\mathrm{\Omega }}_{+}/4=2\pi \times 2.5\mathrm{MHz}$ and detunings ${\mathrm{\Delta }}_{-}=-{\mathrm{\Delta }}_{+}=2\pi \times 50\mathrm{MHz}$ (so that $J_{+-}=0$—see text).

Figure 2

(a) $C_6$ coefficients of Eq. (2) in atomic units for Rb atoms vs principal quantum number $n$. (b) Ratios of diagonal and the $m$-changing $C_6$. (c) Eigenenergies $E_{\sigma {\sigma }^{\prime }}(\rho )$ (thick solid lines), energies of states with a single Rydberg excitation in $r_{+}$ (gray dashed line) and $r_{-}$ (gray dotted line) vs $\rho /r_c$ with $r_c=(c_{++}/2|{\mathrm{\Delta }}_{+}|{)}^{1/6}$. Asymptotic energies and eigenstates are indicated on the right. Here, ${\mathrm{\Delta }}_{+-}\equiv {\mathrm{\Delta }}_{+}+{\mathrm{\Delta }}_{-}$.

Figure 3

Path of perturbative couplings between the states (a) $|g_{+}{g}_{+}⟩$ and $|g_{-}{g}_{-}⟩$ and (b) $|g_{+}{g}_{-}⟩$ and $|g_{-}{g}_{+}⟩$ visualizing the perturbative expressions behind $J_{++}$ and $J_{+-}$ of Eqs. (5), respectively. The energies $E_{\sigma {\sigma }^{\prime }}$ are plotted for a specific interatomic distance $\rho$ (with abbreviation ${\overline{\mathrm{\Omega }}}_{\sigma }\equiv {\mathrm{\Omega }}_{\sigma }/\sqrt{2}$). Yellow and blue dotted paths can interfere destructively (see text).

Figure 4

(a)–(c) Effective spin-spin interactions $J_{\alpha }$ [Eq. (1)] as a function of $r$ and $\vartheta$ for a laser propagating along the $z$ axes and atoms in the $zx$ plane. Here $|r_{\pm }⟩=|60^2{P}_{1/2},\pm \frac{1}{2}⟩$ with ${\mathrm{\Omega }}_{-}={\mathrm{\Omega }}_{+}/2=2\pi \times 5\mathrm{MHz}$ and $({\mathrm{\Delta }}_{-},{\mathrm{\Delta }}_{+})=2\pi \times (-60,40)\mathrm{MHz}$. (d) Cut through the energy surfaces $E_{\sigma {\sigma }^{\prime }}$ along the $z$ axis. In contrast to Fig. 2 resonances appear, indicated with a star where $J_{\alpha }$ becomes singular as shown in panels (a)–(c). (e) Cut through panels (a)–(c) for $\vartheta =0$ (solid lines) and $\vartheta =\pi /8$ (dotted lines).