Superconducting Circuits for Quantum Simulation of Dynamical Gauge Fields

We describe a superconducting-circuit lattice design for the implementation and simulation of dynamical lattice gauge theories. We illustrate our proposal by analyzing a one-dimensional U(1) quantum-link model, where superconducting qubits play the role of matter fields on the lattice sites and the gauge fields are represented by two coupled microwave resonators on each link between neighboring sites. A detailed analysis of a minimal experimental protocol for probing the physics related to string breaking effects shows that, despite the presence of decoherence in these systems, distinctive phenomena from condensed-matter and high-energy physics can be visualized with state-of-the-art technology in small superconducting-circuit arrays.

Figure 1

(a) Pictorial view of a 1D quantum-link model, where the operators ${\widehat{\psi }}_{\ell }$ on even (odd) sites represent matter (antimatter) fields and the spin operators ${\widehat{S}}_{\ell ,\ell +1}$ residing on each link represent the gauge fields. (b) Equivalent physical implementation, where two-level systems replace the fermionic matter fields and two oscillators with a fixed total number of excitations $N$ encode a spin $S=N/2$ on each link. (c) Superconducting-circuit implementation. Neighboring superconducting qubits on the sites of a 1D lattice are connected via two nonlinear $LC$ resonators.

Figure 2

(a) Schematic representation of the states $|\mathrm{\text{meson}}⟩$ (left) and $|\mathrm{\text{string}}⟩$ (right) for a lattice of $L=4$ sites. The spins (matter-antimatter excitations) at the end of the chain are considered fixed and the gauge-invariant dynamics in this minimal setting only involves a single unit cell with two sites and one link as indicated by the dashed box. (b) Spectrum of the microscopic Hamiltonian [Eq. (5)] and effective model [Eq. (1)] for a single unit cell. The thick solid lines show the energies of the states $|\mathrm{\text{meson}}⟩$ and $|\mathrm{\text{string}}⟩$, which transform into each other via an avoided crossing (symmetric and antisymmetric superpositions of these states) at $m\approx g/2$. Other lines correspond to spin combinations that for the boundary conditions defined in (a) are not consistent with the Gauss law ${\widehat{G}}_{\ell }|\psi ⟩=0$. The parameters for this plot are ${\Omega }_a={\Omega }_b=2\pi \times 200\mathrm{MHz}$. ${\Omega }_{ab}=2\pi \times 420\mathrm{MHz}$, $\lambda =2\pi \times 30\mathrm{MHz}$, $W=({\Omega }_a+{\Omega }_b+{\Omega }_{ab})/4=2\pi \times 205\mathrm{MHz}$. The energy splitting is given by $2\sqrt{2}J$, and the effective parameters are $|J|\approx 2{\lambda }^2/W=2\pi \times 8.78\mathrm{MHz}$ and $g={\Omega }_{ab}-{\Omega }_a-{\Omega }_b=2\pi \times 20\mathrm{MHz}$.

Figure 3

Parameters as in Fig. 2c and values in the legends in $2\pi \times \mathrm{MHz}$. (a) Fidelity of the state $|{\Psi }_f⟩\approx |\mathrm{\text{string}}⟩$ [eigenstate at $m/(2\pi )=50\mathrm{MHz}$] after a Landau-Zener sweep from the state $|{\Psi }_i⟩\approx |\mathrm{\text{meson}}⟩$ [eigenstate at $m/(2\pi )=-30\mathrm{MHz}$]. $m$ is changed proportionally to a constant speed $v$. (b) Meson-string transition, shown by the average value of the spin on the link, choosing $v/(2\pi )=2\pi \times 100\mathrm{MHz}/\mu \mathrm{s}$ and starting from the state $|{\Psi }_i⟩\approx |\mathrm{\text{meson}}⟩$. The result from a Landau-Zener sweep compares well with the static case of the microscopic and effective models (solid lines). Oscillations are present in the string phase due to the nonadiabaticity of the sweep. (c) Gauss-law violation through the sweep, choosing $v/(2\pi )=2\pi \times 100\mathrm{MHz}/\mu \mathrm{s}$, which compares well with the static case (solid line). The effective model has, by construction, $⟨{\widehat{G}}^2⟩=0$.