Loops and Strings in a Superconducting Lattice Gauge Simulator

We propose an architecture for an analog quantum simulator of electromagnetism in $2+1$ dimensions, based on an array of superconducting fluxonium devices. The encoding is in the integer (spin-1) representation of the quantum link model formulation of compact $U(1)$ lattice gauge theory. We show how to engineer Gauss’ law via an ancilla mediated gadget construction, and how to tune between the strongly coupled and intermediately coupled regimes. The witnesses to the existence of the predicted confining phase of the model are provided by nonlocal order parameters from Wilson loops and disorder parameters from ’t Hooft strings. We show how to construct such operators in this model and how to measure them nondestructively via dispersive coupling of the fluxonium islands to a microwave cavity mode. Numerical evidence is found for the existence of the confined phase in the ground state of the simulation Hamiltonian on a ladder geometry.

Figure 1

$U(1)$ quantum link model engineered in a fluxonium array. (a) “Electric” ${\widehat{E}}_{\alpha ,\beta }$, and “magnetic” ${\widehat{U}}_{\alpha ,\beta }$, degrees of freedom are associated with links $⟨\alpha ,\beta ⟩$ of a square lattice. The link degrees of freedom (red circles) are encoded in eigenstates of the fluxonia. The ancillae (blue diamonds) on vertices are inductively coupled to neighboring link islands to mediate the Gauss constraint and plaquette interactions are obtained via link nearest neighbor capacitive coupling. (b) Superconducting circuit elements used to build and couple components of the simulation. The link devices have local phase ${\widehat{\varphi }}_{\mathrm{link}}$ and capacitive, inductive, and flux-biased Josephson energies $E_C$, $E_L$, and $E_J$, respectively, and similarly for the ancilla devices. The capacitive and inductive coupling energies are $E_C^c$ and $E_L^c$. (c) A minimal quasi-1D “ladder” implementation embedded in a microwave cavity (black box), in which a ’t Hooft string of link fluxonia (green circles) can be measured via an ancilla coupled to the cavity (green triangle).

Figure 2

Expectation value of the ’t Hooft string $\widehat{\mathrm{\Upsilon }}(\phi )$ which inserts a flux $\phi$ at the plaquette in the middle of a ladder [see Fig. 1]. Left panel: value in the ground state of the pure gauge model ${\widehat{H}}_{\mathrm{QLM}}$ as a function of the electric coupling $g_{\mathrm{elec}}^2$ with a perturbative value of the magnetic coupling $(1/g_{\mathrm{mag}}^2)\sim (2J^2/U)\to (2/75)$. Right panel: value in the ground state of the two-body Hamiltonian ${\widehat{H}}_{\mathrm{imp}}$ as a function of the on-site energy $V$, with $J=1$ and $U=75$. Numerics were performed for a size $L=29$ rung ladder with 85 spins using DMRG calculation with 300 states and a truncation error estimated at $<{10}^{-12}$. From the plots, $⟨\widehat{\mathrm{\Upsilon }}⟩\ge 0.5$ is indicative of a confining phase. The equivalence between the implemented model (3) and the gauge invariant model (4) in the strongly coupled and intermediately coupled regimes is evident.

Figure 3

(a) Link fluxonium devices operate as qutrits with ${\varphi }_{\mathrm{off}}=0$. The potential $P({\varphi }_{\mathrm{link}})$ (red) and the eigenfunctions are plotted for qutrit states $|0⟩,|\pm 1⟩$, which represent electric flux, and the next energy state $|s⟩$. A cavity field couples states $|0⟩\leftrightarrow |s⟩$ with Rabi frequency $\mathrm{\Omega }$ and detuning $\delta$ to tune the electric coupling in the QLM. (b) Ancilla fluxonium potential $P({\varphi }_a)$ (blue) with ${\varphi }_{\mathrm{off}}=\pi$, and eigenfunctions $|g⟩$ and $|e⟩$. Ancillae are initialized in state $|e⟩$ and generate the Gauss constraint on link devices through an effective interaction.