Quantum-Coherent Regime of Programmable Dipolar Spin Ice

Abstract

Frustrated spin-ice systems support emergent gauge fields and fractionalized quasiparticles that act as magnetic monopoles. Although artificial platforms have enabled their direct visualization, access to their quantum-coherent dynamics has remained limited. Here we realize a programmable dipolar square spin-ice model using a superconducting-qubit quantum annealer, providing access to a previously unexplored quantum-coherent regime of artificial spin ice. By implementing a direct one-to-one mapping between lattice spins and physical qubits, together with engineered extended couplings, we realize effective dipolar interactions on frustrated lattices comprising more than 400 vertices. Tuning transverse-field fluctuations enables us to probe the real-time dynamics of Dirac-string defects and interacting monopole plasmas. We observe super-diffusive monopole transport, with scaling exponents intermediate between classical diffusion and ballistic motion, indicating dynamics beyond classical stochastic relaxation and consistent with coherent propagation within an emergent gauge manifold. These results establish programmable quantum spin ice as a scalable platform for investigating fractionalized excitations and emergent gauge dynamics in engineered quantum matter.

Figures (9)

• Figure 1: Schematic of the artificial spin ice model and its ground states. a: Each yellow dot represents a position of single point dipole (qubit). Straight black arrows indicate the dipole orientations. In subsequent panels dipoles that are flipped compared to this configuration are highlighted in yellow. Four types of interaction terms are included, representatives are pictured as curved arrows. Interactions values are set to ones corresponding to truncation of two‑dimensional dipolar interactions to the first four terms. (b-d): Ice‑rule ground states obtained from annealing for different model parameters. Holes in the lattice correspond to device defects. In all ground states, each vertex satisfies the ice rule, indicated by two yellow and two black colored arrows. Vertex statistics and correlations extracted from annealing are compared with the six‑vertex ice ground states (c), in order to identify the point $J_\perp=J_\parallel$. In the following, we focus on the configuration shown in (b) and investigate the dynamics of its excitations.
• Figure 2: Schematic illustration of measurement sequence. The black colors on plots here are plotted, when the dipole state is opposite to the ground state configuration (Fig. \ref{['fig1']} b). Circles correspond to the vertices with $\pm2Q$ charge (3-1 in-out or 1-3 in-out). Vertices with charges $\pm4Q$ were very suppressed. Initially the system dynamics are frozen by high values of interaction terms. We change the parameters as fast as possible decreasing the $ZZ$ interaction terms and increasing the $\Gamma X$ - fluctuation terms. The system is kept fixed $\Delta t=t_2-t_1=100~\text{ns}$ in the evolving condition. We vary the fluctuations strength $\Gamma$ to obtain different effective interactions time $t_{\text{eff}}=\Gamma\Delta t$. The $J_\perp,J_\parallel,J_3,J_4$ parameters are rescaled accordingly. The ice rules are always enforced at $45\%$ of total available value of $ZZ$ interactions. For example for $t_{\text{eff}}=100$ the energy associated with generation of pair of charges is still large $E_{2Q}=20\Gamma$, but for longer effective times ice rules are enforced less strictly.
• Figure 3: Results of the dynamics measurement of non-interacting Dirac String defect. We present the log-log plots of the mean square displacement of the positive monopole at the end of the Dirac String (a) as well as the mean square distance between ends of the string (b). The distance is given in the lattice units. Each point corresponds to an average of 4000 measurements over four runs of 1000 consecutive measurements, performed on different dates. The error bars correspond to the standard deviation for each point. The horizontal lines between 10 and 100 show the range of data used to fit the exponential function. The range is selected to discard the faster, transient initial dynamics and the dynamics, where fraction of samples with conserved charges is begging to fall. We recover two surprisingly different exponents about 1.5 for displacement and 1.25 for distance. (c): The number of observed configurations with different numbers of charges at the point $t_{eff}=100$. The 3725 corresponds to the correct number of charges - 1 positive and 1 negative monopole - mostly corresponding to the conserved number of charges, but also including samples coming from e.g. pair annihilation and subsequent pair creation. The plot is mainly diagonal, showing the charges are predominantly annihilated and created in opposite pairs. (d): The fraction of samples with the correct number of charges after different durations of dynamics. The dynamics shows initially a slow non-exponential decay.
• Figure 4: Results of non-interacting $J=0$ monopole plasma dynamics running for $t_{eff}=100$. (a) & (b) The average positive and negative charge densities. The 40 initial charges of each type were placed uniformly in the areas highlighted by the green dashed rectangles. Each square pixel of the image corresponds to the charge density calculated at a single vertex. The gray $X$ symbols highlight the lattice defects - missing vertices. (c) The distribution of experiments with given final number of positive and negative charges. (d) The average correlation between positions of positive charges. The center of the image corresponds to the distance 0,0 and each pixel corresponds to single lattice unit distance. (e) Similar plot for the correlations between positive and negative charges. (f) Central vertical and horizontal cross-sections of the correlation functions shown in (d) & (e).
• Figure 5: Array of positive charge density plots. Top row depicts the average of the 50 initial states. Subsequent rows show average of states after $t_{eff}=100$ of evolution, at different 2D dipolar interaction strengths. Different columns correspond to different number of initial charges.