Emerging Two-Dimensional Gauge Theories in Rydberg Configurable Arrays

Solving strongly coupled gauge theories in two or three spatial dimensions is of fundamental importance in several areas of physics ranging from high-energy physics to condensed matter. On a lattice, gauge invariance and gauge-invariant (plaquette) interactions involve (at least) four-body interactions that are challenging to realize. Here, we show that Rydberg atoms in configurable arrays realized in current tweezer experiments are the natural platform to realize scalable simulators of the Rokhsar-Kivelson Hamiltonian—a 2D U(1) lattice gauge theory that describes quantum dimer and spin-ice dynamics. Using an electromagnetic duality, we implement the plaquette interactions as Rabi oscillations subject to Rydberg blockade. Remarkably, we show that by controlling the atom arrangement in the array we can engineer anisotropic interactions and generalized blockade conditions for spins built of atom pairs. We describe how to prepare the resonating valence bond and the crystal phases of the Rokhsar-Kivelson Hamiltonian adiabatically and probe them and their quench dynamics by on-site measurements of their quantum correlations. We discuss the potential applications of our Rydberg simulator to lattice gauge theory and exotic spin models.

Popular Summary

Predicting the outcome of strong interactions among fundamental particles is very hard to compute. While the masses of protons and other particles can be calculated with high precision on traditional supercomputers, to know what happens in a collision between two protons, for example, requires a quantum simulator, a system that exploits quantum behavior of real particles to simulate a specific physics problem. However, building a quantum simulator of proton collisions remains a challenge. We take an intermediate step and propose a scalable quantum simulator of a particular model of 2D electromagnetism that describes quantum magnets.

Our proposed device relies on Rydberg atoms—atoms with one or more highly excited electrons—trapped in optical tweezers. We engineer charge conservation and magnetic interactions by reformulating the underlying theory in terms of atomic transitions between the ground and very excited state of the Rydberg atoms. Such atoms strongly interact with one another and their atomic transitions depend on the states of the surrounding atoms. Our discovery is that we have only to properly arrange the atoms with optical tweezers to reproduce the model and study it in current atomic experiments.

Our proposed quantum simulator will allow researchers, for the first time, to “watch” in real time how photons evolve when they strongly interact in two dimensions.

Figure 1

Spin gauge theories and the Rokhsar-Kivelson model. (a) Spin-$1/2$ lattice gauge theories are the simplest gauge theories with a continuous gauge symmetry group. The gauge degrees of freedom are spins $1/2$ that live on the oriented links of the lattice, labeled as $s$, $\mu$, where $s$ is the starting site and $\mu$ is the direction, e.g., $x$ or $y$ in a square lattice. They display the main features of gauge theories in $D>1$: (i) the Gauss law that determines the allowed spin configurations on the links in terms of the charges and (ii) the magnetic interactions acting on the links around a plaquette. On a square lattice, these operators correspond to both four-body spin operators indicated by the green and violet rhombi, respectively. (b) In a system without charges, we depict the effect of the Gauss law by coloring in red the links in $|\uparrow ⟩$ and not coloring the ones in $|\downarrow ⟩$ that represent the electric states. Physical states are a superposition of endless red strings going up and right [see (c) and Fig. 2]. Such electric configurations correspond to position states, and ${\widehat{\mathcal{S}}}_s^{†}+{\widehat{\mathcal{S}}}_s$ acts as a kinetic term on them. (c) The square of the plaquette operator $({\widehat{\mathcal{S}}}_s^{†}+{\widehat{\mathcal{S}}}_s{)}^2$ is diagonal in the electric basis and plays the role of a potential term that counts flippable plaquettes. The competition of the kinetic and potential terms in the Rokhsar-Kivelson Hamiltonian (4) gives rise to a rich phase diagram, with the resonating valence bond solid (RVBS) phase separating two crystal phases, the Néel and the columnar phases (see the description in the main text). The resonant plaquettes are depicted by diagonal double arrows. The alternated green and pink arrows reflect that the resonant plaquettes are correlated within the same sublattice and anticorrelated with the ones in the opposite sublattice.

Figure 2

Graphical illustration of the duality. (a) Definition of the link spins (blue dots) in terms of the dual plaquette spins (black dots), as in Eq. (5). The dual spins live in the dual lattice formed by the centers of the plaquettes. The value of the link spin is determined by the plaquette spins of the two plaquettes sharing the link, as indicated by the purple ellipses for the links $p$, $x$ and $p$, $y$. The orientation of the plaquette spins is chosen such that the fully flippable state $|\mathrm{\Omega }⟩$ is mapped in the dual basis into the ferromagnetic state (b). (b)–(d) We represent the configurations that are relevant to understand the generalized blockade condition simultaneously in the original (above) and the dual (below) basis. In the former, we represent the link spins with the color convention in Fig. 1. In the latter, we represent the dual plaquette spins as arrows placed in the center of the corresponding plaquettes. (b) The reference state $|\mathrm{\Omega }⟩$: We identify it with the ferromagnetic state with all plaquette spins up. (c) After flipping one plaquette of $|\mathrm{\Omega }⟩$, all the neighboring plaquettes are blocked. (d) After flipping all its neighboring plaquettes, the plaquette $p$ is flippable.

Figure 3

Dual RK model and its phases. (a) Phase diagram of the 2D RK model on the square lattice without background charges. In the boxes, we plot the spin-spin correlations $⟨S_p^{\mu }{S}_{p^{\prime }}^{\mu }⟩$ obtained via DMRG [87] on square lattices $N_x\times N_y=8\times 8$ and for three values of $\lambda =-1$, 0, 2 that belong to the ferromagnetic (Néel), RVBS, and columnar phases, respectively. The correlations are represented with respect to the center, $p=(4,4)$, for varying $p^{\prime }$. Note that in the $\lambda \to \infty$ limit there are multiple degenerate unflippable states. The particular configuration shown here in the right panel corresponds to one of them. (b),(c) Structure factors $S_k[\mu ]=4/(N_x{N}_y{)}^2{\sum }_{p,p^{\prime }}\exp [i(p-p^{\prime })k]⟨{\widehat{S}}_p^{\mu }{\widehat{S}}_{p^{\prime }}^{\mu }⟩$, $\mu =x$, $z$ for $N_x\times N_y=8\times 8$ [(b)] and for different system sizes $N=N_x{N}_y=N_x^2$ [(c)]. The raising of the structure factors $S_{(\pi ,\pi )}[x,z]$ identifies the RVBS phase. In particular, $S_{(\pi ,\pi )}[z]$ equals $S_{(0,0)}[z]$ in the RVBS region. (d),(e) Structure factors on the periodic ladder of $N_x\times 2$ sizes, without background charges, with $N_x=32$ [(d)] and $N_x=32$, 64, 128 [(e)].

Figure 4

RVBS in decorated Rydberg arrays. (a) Implementation of the RK Hamiltonian in decorated Rydberg arrays: Oriented pairs of close-by atoms with blue-detuned Rabi oscillations between the ground $|g⟩$ and Rydberg $|r⟩$ states are equivalent to oscillating spins $1/2$ (see inset) with Ising interactions between them. By controlling the pair orientation relative to the lattice and the lattice geometry, we can engineer the spin-spin interactions such to achieve the generalized blockade conditions and realize the RK Hamiltonian in the dual formulation with a modified RK potential. We name such Hamiltonians as Rydberg-RK Hamiltonians. On the left we show the square lattice construction and on the bottom right the implementation for a ladder. The same principle applies to a generic 2D lattice. (b),(c) The phase diagrams of the Rydberg RK Hamiltonian on the square lattice (18) [$N=8\times 8$, (b)] and on the (periodic) ladder (15) [$N=32\times 2$, (c)] as a function of $\lambda$, calculated via DMRG. The two phase diagrams differ qualitatively from the ones of the original RK Hamiltonians shown in Figs. 3 and 3, respectively, for $\lambda \gtrsim 1$. In particular, the ferromagnetic and the RVBS phases remain, while the columnar phase is substituted by a “glassy” phase. (d) In order to compare the phase diagrams of the effective RK Hamiltonian with the full spin description of the decorated Rydberg array, we compare the spin-spin correlations on a $6\times 2$ decorated ladder, for $|\eta |=0.38a_x$ and periodic boundary conditions, via exact diagonalization. The structure factors for the ground state of the effective RK Hamiltonian and the corresponding eigenstates of the Rydberg array Hamiltonian (up to next-nearest-neighbor interactions) are shown, respectively, as dashed and solid lines. We included a small detuning $\delta =0.1C_6/a_x^6\approx 0.11|\mathrm{\Lambda }|$ within the atoms of each pair to select a specific ferromagnetic state as a reference. (e) Adiabatic preparation of the RVBS phase for the same parameters as (d). By smoothly ramping up the Rabi frequency $J$ (inset), we can enter the RVBS phase within the validity of generalized blockade condition, $J\ll G$.

Figure 5

Relation between gauge magnets and the quantum dimer model. (a) By reversing the spins of the links at the bottom-left (or at the top-right) corner of the plaquette, the plaquette operator becomes a dimer move; see Eq. (a1). (b) In the dimer basis, obtained by reversing the spins of the links at the bottom-left corner of the odd plaquettes, the fully flippable background $|\mathrm{\Omega }⟩$ appears as a columnar state. (c) The physical states of the quantum dimer model are obtained for a staggered distribution of static charges, $Q_s=(-1{)}^s$. Such states can be constructed by applying the plaquette operator on the maximally flippable background depicted in the figure. It is obtained by superimposing a string covering associated to the charges on $|\mathrm{\Omega }⟩$. Because of the charges, only half of the plaquettes are flippable.

Figure 6

The ground state of the 1D $PXP$ model. (a) Structure factors for $z$ spin-spin correlations for $k=0,\pi ,2\pi /3$. (b) Structure factors for $x$ spin-spin correlations for $k=0,\pi ,2\pi /3$. Parameters: $N_x=60$ and maximum bond dimension $D=128$.