Prediction of Toric Code Topological Order from Rydberg Blockade

The physical realization of ${\mathbb{Z}}_2$ topological order as encountered in the paradigmatic toric code has proven to be an elusive goal. We predict that this phase of matter can be realized in a two-dimensional array of Rydberg atoms placed on the ruby lattice, at specific values of the Rydberg blockade radius. First, we show that the blockade model—also known as a “$PXP$” model—realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term. This model can be interpreted as a ${\mathbb{Z}}_2$ gauge theory whose dynamics is generated by monomer fluctuations. We obtain its phase diagram using the numerical density matrix renormalization group method and find a topological quantum liquid (TQL) as evidenced by multiple measures including (i) a continuous transition between two featureless phases, (ii) a topological entanglement entropy of $\ln 2$ as measured in various geometries, (iii) degenerate topological ground states, and (iv) the expected modular matrix from ground state overlap. Next, we show that the TQL persists upon including realistic, algebraically decaying van der Waals interactions $V(r)\sim 1/r^6$ for a choice of lattice parameters. Moreover, we can directly access topological loop operators, including the Fredenhagen-Marcu order parameter. We show how these can be measured experimentally using a dynamic protocol, providing a “smoking gun” experimental signature of the TQL phase. Finally, we show how to trap an emergent anyon and realize different topological boundary conditions, and we discuss the implications for exploring fault-tolerant quantum memories.

Popular Summary

Phases of matter encountered in everyday life, such as solids and magnets, can be diagnosed through local measurements such as of density or magnetization. However, under special circumstances and at low temperatures, quantum effects can stabilize new phases of matter whose identification requires “nonlocal measurements,” which involve probing a large number of atoms at the same time. Unfortunately, the experimental realization of such phases, which provide a route to robust quantum computers, has been exceedingly difficult. We propose how the tunability of certain existing cold-atom arrays could be used to stabilize a paradigmatic “quantum spin liquid” phase and even directly measure several of its nonlocal properties.

In a quantum spin liquid, the interacting spins of the particles fail to organize into clear magnetic patterns even at zero temperature. Instead, the quantum entanglement in the system gives rise to a sort of hidden order that can be probed only by the aforementioned nonlocal measurements. One of the original conceptualizations of this phase of matter envisions atoms on a lattice paired into dimers, reminiscent of a chemical bond, with quantum fluctuations exploring all possible dimer configurations. Alas, this has not been turned into an experimental fact.

We identify a precise set of conditions under which the strongly repulsive forces of atoms in highly excited (Rydberg) states can naturally encode dimer configurations on a kagome lattice, a lattice of corner-sharing triangles. We predict that a macroscopic quantum superposition between such coverings will arise upon exciting these atoms with lasers, and we provide a blueprint for creating protected quantum bits in this novel state of matter.

Our theoretical proposal shows how engineered quantum matter—where interacting atoms are individually controlled using light—can realize and detect phenomena that have thus far eluded more traditional approaches.

Figure 1

Rydberg blockade model and relation to dimer model. (a) Hard-core bosons on the links of the kagome lattice (forming the ruby lattice) are strongly repelling, punishing double occupation within the disk $r\le r_3=2a$. (b) An example of a state consistent with the Rydberg blockade at maximal filling. (c) Since the blockade forbids occupation of any two touching bonds, we can equivalently draw the configuration as a dimer covering on the kagome lattice.

Figure 2

Phase diagram of Rydberg blockade model on the links of the kagome lattice. The trivial phase at small $\delta /\mathrm{\Omega }$ is separated from the valence bond solid (VBS) at large $\delta /\mathrm{\Omega }$ by an intermediate phase which has a large entanglement plateau. We show an exemplary density plot for each of the three phases, which shows that the intermediate phase is featureless. The VBS phase has a 36-site unit cell (72 atoms on the links) highlighted by the gray shaded region—this pattern is studied in Refs. [82, 83, 113] in the context of the spin-$1/2$ Heisenberg model on the kagome lattice. Numerical results are for a cylinder with XC-8 geometry, as depicted.

Figure 3

Detecting phase transitions via filling fraction. These data are obtained for an infinitely long cylinder with XC-8 geometry. (a) The filling fraction has a singular behavior upon transitioning from the trivial phase into the spin liquid, after which the system enters a regime where $⟨n⟩\approx 0.25$, consistent with it being an approximate dimer state. (b) The spin liquid and VBS phase are separated by a first-order transition.

Figure 4

Topological entanglement entropy. We determine the offset $\gamma$ in the area law $S=\alpha L-\gamma$ [27, 28, 114, 115]. (a) For the trivial phase, this value is zero. (b) As we increase $\delta /\mathrm{\Omega }$, we enter the spin liquid where $\gamma \approx \ln 2$. Here, we plot $S_{L=8}-2S_{L=4}$, where $S_{L=n}$ is the bipartition entanglement entropy for the XC-$n$ geometry. (c) For an exemplary point in the spin liquid, we extract $\gamma$ for two distinct geometries (up to XC-12 and YC-8). Note that XC-$n$ (YC-$n$) has circumference $L_{\text{circ}}/a=\sqrt{3}n$ ($2n$).

Figure 5

Topological string operators. (a) The two different string operators are defined by their action on a single triangle. We call the diagonal and off-diagonal string operators $P$ and $Q$, respectively. (b) An example of the action of the string operators on a classical dimer state. (c) The definition of the Fredenhagen-Marcu order parameter [89, 90, 91, 92, 93] is shown for the diagonal string, $⟨P{⟩}_{\mathrm{FM}}$, which measures the condensation of the $m$ anyon. The analogous definition for $⟨Q{⟩}_{\mathrm{FM}}$ (not shown) measures an $e$ condensate.

Figure 6

Diagnosing phases in terms of topological string operators. Top: The Fredenhagen-Marcu (FM) string order parameters show that the trivial phase is an $e$ condensate ($=\text{Higgs phase}$) and the VBS phase is an $m$ condensate ($=\mathrm{c}\text{onfined phase}$). These string order parameters decay to zero in the spin liquid, confirming that it is the deconfined phase of the ${\mathbb{Z}}_2$ lattice gauge theory. We sketch the strings used for calculating the FM order parameter [the transparent strings show the closed loop used to normalize the string; see Fig. 5]. Bottom: Long-range order in the FM string for $P$ ($Q$) suppresses the value of a closed $Q$ ($P$) loop around the circumference. In the spin liquid, both loops are nonzero (we plot the absolute value: Their signs label degenerate ground states; see Fig. 8). As in Fig. 2, results are for XC-8 (depicted) with the vertical (horizontal) direction being periodic (infinite).

Figure 7

Ground states and modular transformations. From the ground states on the infinitely long cylinder, we can obtain minimally entangled ground states on the torus geometry. For the smaller geometry, we show that, while the $\pi /3$ rotation acts trivially on the trivial ($\delta /\mathrm{\Omega }=1$) or symmetry-breaking ($\delta /\mathrm{\Omega }=2.5$) phases, it leads to a nontrivial overlap in the spin liquid ($\delta /\mathrm{\Omega }=1.7$). We confirm for the larger torus (96 sites) that the overlaps for distinct ground states agree with the prediction (3) based on the modular transformation of a ${\mathbb{Z}}_2$ spin liquid. The overlaps are shown as a function of the Monte Carlo sweeps, converging toward the value of approximately 0.5.

Figure 8

Topological ground state degeneracy. In the topological phase, we obtain the 1 and $e$ ground states from DMRG with random initial states. The $m$ and $f$ states are obtained by starting from fixed-point resonating dimer states and subsequently applying imaginary time evolution. The energies shown are for $\delta /\mathrm{\Omega }=1.7$. In light gray, we also show the eigenvalues of the $P$ and $Q$ loop operators around the circumference (for YC-4); the four ground states are characterized by the signs of these numbers.

Figure 9

The ruby lattice. (a) Atoms on the links of the kagome lattice form the vertices of a ruby lattice where the rectangle has an aspect ratio $\rho =\sqrt{3}$. (b) The ruby lattice with $\rho =1$. (c) The ruby lattice with $\rho =3$. The colored disks show seven distinct interaction distances; the phase diagram in Fig. 10 is obtained by including $V(r)=\mathrm{\Omega }(R_b/r{)}^6$ for 16 distinct distances, coupling each site to 44 other sites.

Figure 10

Spin liquid on ruby lattice ($\rho =3$) with $V(r)\sim 1/r^6$. We consider the lattice in Fig. 9 for blockade radius $R_b=3.8a$, keeping all interactions within a radius $r\le 9a$. There is a phase transition between two featureless phases, the latter having a large entanglement plateau. The spin liquid is characterized by the simultaneous vanishing of the off-diagonal string operator (i.e., the trivial phase is an $e$ condensate) and emergence of a large signal for the parity loop around the circumference. The latter also labels two of the degenerate ground states, as annotated on the density plot. The fact that this transition approximates a dimer model is evidenced by, e.g., $⟨n⟩\approx 0.249$ for $\delta /\mathrm{\Omega }=5.3$. To emphasize the connection to a dimer model, we plot the density on the links of the kagome lattice (i.e., $\rho =\sqrt{3}$). The correlation length is also expressed for this kagome geometry (i.e., the length of a triangle of the kagome lattice is $2a$).

Figure 11

Modular transformations on ruby lattice ($\rho =3$) with $V(r)\sim 1/r^6$. For blockade radius $R_b=3.8a$ and detuning $\delta /\mathrm{\Omega }=5.5$, we consider two topologically distinct ground states on the YC-8 torus as shown (see the main text and Appendix pp3 for details). The overlaps after a $\pi /3$ rotation agree with the universal value predicted for a ${\mathbb{Z}}_2$ spin liquid. [As in Fig. 10, the simulation faithfully represents $V(r)$ within a distance $r\le 9a$.]

Figure 12

Measuring the off-diagonal string operator through a quench protocol. (a) The vertical direction is periodic on the cylinder. The off-diagonal string $Q$ (blue wiggly line) can be obtained by measuring the diagonal string $P_1{P}_2$ (orange dashed lines) after a time evolution with a nearest-neighbor Rydberg blockade Hamiltonian to time $\mathrm{\Omega }t=4\pi /(3\sqrt{3})$ [see Eq. (6)]. (b) Starting from the spin liquid ground state of the Rydberg Hamiltonian with $R_b=3.8a$ and $\delta /\mathrm{\Omega }=5.3$ on a ruby lattice with aspect ratio $\rho =3$ (see Fig. 10), we measure $⟨P_1{P}_2⟩$ after time evolving with either the nearest-neighbor Rydberg blockade model (solid black line) or the ground state Hamiltonian quenched to $R_b=2$ (red dashed line). The horizontal gray dashed line denotes the ground state value for $⟨Q⟩$.

Figure 13

A trapping potential for $e$ anyons. The ground state (here, on a cylinder) for a lattice where four sites around a vertex are removed captures an $e$ anyon. This result can be read off from the expectation value of the parity loops (dashed orange lines) around the circumference: If two neighboring loops have opposite sign, then a charge is enclosed.

Figure 14

Boundary phase diagram of the blockade model. We consider an infinitely long strip of the XC-8 geometry: The bulk is the spin liquid at $\delta /\mathrm{\Omega }=1.7$, but we tune $\delta$ on the outermost boundary links. (a) The correlation length diverges at two boundary phase transitions; in the intermediate shaded regime, the entanglement is increased. (b) The small and large ${\delta }_{\mathrm{bdy}}$ phases have a classical-like dimer filling at the boundary, whereas the intermediate regime has a compressible boundary. (c) By calculating the string operators from boundary to boundary, we diagnose the small and large (intermediate) ${\delta }_{\mathrm{bdy}}$ phases as having $m$-condensed ($e$-condensed) boundaries. (d) Density plots $⟨n⟩$ in the three boundary regimes. The strip is infinitely long (finite) in the horizontal (vertical) direction.

Figure 15

Topological degeneracy in planar geometry. (a) Alternating $e$- and $m$-condensed boundaries imply a twofold degeneracy. One way of understanding this implication is in terms of the Majorana zero modes (red dots) that live at the points where the boundary condition changes [142]; due to the global emergent fermion parity having to be unity, these four Majorana modes give rise to only a twofold degeneracy. If we label states using the $P$ string connecting the left and right boundaries (see the main text), then pulling an $e$ anyon out of one $e$-condensed boundary to another effectively toggles the states in this two-level system. (b) An annulus geometry with $m$-condensed boundaries also has a twofold degeneracy. Moving the $e$ anyon around the hole toggles the states. Since $e$ anyons can be created only in pairs, there is another $e$ anyon which we do not move (not shown).

Figure 16

Readout of a topological ground state. We consider the blockade model for $\delta /\mathrm{\Omega }=1.7$ on a finite sample with open boundaries, as shown. Moreover, we change the laser detuning on the top and bottom boundary to ${\delta }_{\mathrm{bdy}}=0.48\delta$. From the boundary phase diagram in Fig. 14, we know that this change realizes the $e$-condensed boundary, whereas the left and right boundaries are $m$ condensates. For the ground state of this system, we show the values for the two types of topological string operators which connect their corresponding condensates. Upon using the FM normalization [see Eq. (7)], the readout for the logical variables gives a state that lies along the $x$ axis of the Bloch sphere. Note that both string operators can be experimentally measured using the prescription in Sec. 3b.

Figure 17

Connectivity graph for van der Waals interactions on the ruby lattice. Black dots denote the ruby lattice with $\rho =3$ [also see Fig. 9]. Each line represents a coupling in $V(r)=\mathrm{\Omega }(R_b/r{)}^6$ that is included in the numerics for the phase diagram in Fig. 10. The gray dashed lines denote how this coupling is wrapped into an XC-8 cylinder; any site of the ruby lattice outside this region can be identified with a site inside this region. The cylinder is infinitely long in the horizontal direction.

Figure 18

The FM string order parameter in the blockade model. (a) The lattice shown is the XC-12 cylinder (periodic along the vertical direction, infinite along the horizontal direction). The blue lines denote the $Q$ strings defining the FM order parameter $⟨Q{⟩}_{\mathrm{FM}}^{(n\times n)}$ ($n=1$, 2, 3), which is normalized by the square root of the closed string [see Fig. 5 for the general definition]. (b) Values for the FM order parameters in the blockade model on XC-12 deep in the spin liquid, $\delta /\mathrm{\Omega }=1.7$. Both strings decay to zero exponentially with the length of the string, a property that is unique to the deconfined phase.

Figure 19

The FM string order parameter in the $V(r)\sim 1/r^6$ model. Similar to Fig. 18, except (a) we work on the YC-8 cylinder and (b) the DMRG results for the FM order parameters are now for the ruby lattice model discussed in Sec. 3a, in particular, the blockade radius is $R_b=3.8a$, the detuning $\delta /\mathrm{\Omega }=5.5$, and we faithfully represent $V(r)=\mathrm{\Omega }(R_b/r{)}^6$ within a distance $r\le 9a$.