Engineering a U(1) lattice gauge theory in classical electric circuits

Lattice gauge theories are fundamental to such distinct fields as particle physics, condensed matter, and quantum information science. Their local symmetries enforce the charge conservation observed in the laws of physics. Impressive experimental progress has demonstrated that they can be engineered in table-top experiments using synthetic quantum systems. However, the challenges posed by the scalability of such lattice gauge simulators are pressing, thereby making the exploration of different experimental setups desirable. Here, we realize a U(1) lattice gauge theory with five matter sites and four gauge links in classical electric circuits employing nonlinear elements connecting LC oscillators. This allows for probing previously inaccessible spectral and transport properties in a multisite system. We directly observe Gauss's law, known from electrodynamics, and the emergence of long-range interactions between massive particles in full agreement with theoretical predictions. Our paper paves the way for investigations of increasingly complex gauge theories on table-top classical setups, and demonstrates the precise control of nonlinear effects within metamaterial devices.

Figure 1

Engineering a classical gauge theory. (a) Structure of a lattice gauge theory. Matter fields reside on sites, gauge fields on the links in-between. (b) Circuit implementation with LC resonators for each site and link. The interaction is realized by three-wave mixers as described in the main text. (c) Measurement of Gauss's law $G_x$ through the oscillator energy on each matter site (blue) and its neighboring links (orange). Links that appear with negative sign in Gauss's law are shown inverted along $y$ axis (dashed). The smaller oscillation of the resulting observable (black) confirm that the local conservation laws are fulfilled. Curves are offset for clarity.

Figure 2

Site-resolved spectrum. (a) Level scheme of the Hamiltonian in the regime of large mass $m=2.5\mathrm{kHz}$ compared to the coupling strength $J=0.92\mathrm{kHz}$. The bare frequencies of the matter sites (blue) have a resonance at $f_{\text{rot}}=-{(-1)}^{x}m$ and the links (orange) at $f_{\text{rot}}=0$, shown as straight lines. They are dressed by the gauge-invariant coupling, which allows excitations to move between matter sites (green arrows). The coupling results in weak spectral lines at $f_{\text{rot}}={(-1)}^{x}m$ for the matter field, which are mirrored on the links through weak spectral lines at $f_{\text{rot}}=-2m$ due to the Gauss law. (b) Observed frequency spectrum of oscillator voltages. The positions of spectral lines compare well to the predicted level structure from perturbation theory.

Figure 3

Mass dependence of the lattice gauge theory spectrum. A spectral analysis of our model in Eq. (1) with five matter sites, as a function of the matter field mass $m$, is presented in terms of experimental measurements and numerical simulations. All spectral features observed in the experiment are well reproduced by the numerics. Agreement with first-order perturbation theory in the thermodynamic limit (black lines) confirms that our implementation is sufficiently large to render finite-size effects insignificant.

Figure 4

Long-range interactions between matter sites. (a) The gauge invariant coupling of the matter sites leads to long-range interactions between them (green lines). To observe the resulting frequency shift, we populate the first site in a controlled fashion by driving with an alternating current of frequency $f_{\text{drv}}$. (b) The first matter site has the frequency response of a harmonic oscillator with resonance frequency $m=2.5\left(2\right)\mathrm{k}\mathrm{Hz}$ and its signal is off-resonantly coupled into the second matter site with a coupling strength $J=0.92\left(5\right)\mathrm{kHz}$. The response of the second matter site has a marked frequency shift, which is well explained through perturbation theory by the interaction with the first matter site (red line). (c) We measured the frequency shifts $f_U$ as a function of lattice distance to the first site. The observed independence of distance within the experimental uncertainties is in good agreement with perturbation theory. The uncertainties are systematically limited by the frequency resolution.

Figure 5

Setup. Photograph of the circuit with five sites. Milled circuit boards can be chained together for arbitrarily large lattices. Inside the handmade coils, secondary feedback and tuning coils are visible. The illustration below indicates the general content and connections on the circuit boards.

Figure 6

Spectrum of the five-site lattice depending on detuning $\mathrm{\Delta }={\mathrm{\Delta }}_x$. Dashed lines indicate frequency components of first-order perturbation theory. Simulation of the U(1) Hamiltonian shows higher order components which are also discernible in the measurements. Dotted lines in sites indicate $\pm 3m$ frequency components and in links $+2m$ components from higher orders of perturbation theory.

Figure 7

Observed exponential decline of oscillator voltage amplitudes along lattice when driving first site and comparison to the next-neighbor coupling mediated by links calculated for the nondriven system. This exponential decline is used as hierarchy for the perturbation theory in the driven case.

Figure 8

Comparison of measurements and simulations of the driven lattice, which is also the basis for Fig. 4. As a guide for the eye, the staggered levels from perturbation theory in the large mass limit are indicated in black. Without driving, amplitudes and the frequency shift diverge when driving on resonance. With the empirical dissipation term, features in the simulation match the measurements.