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

Hannes Riechert,1 Jad C. Halimeh,2 Valentin Kasper,3, 4 Landry Bretheau,5 Erez Zohar,6 Philipp Hauke,2 and Fred Jendrzejewski1 1Universität Heidelberg, Kirchhoff-Institut für Physik, Im Neuenheimer Feld 227, 69120 Heidelberg, 2INO-CNR BEC Center and Department of Physics, University of Trento, Via Sommarive 14, I-38123 Trento, Italy 3ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and , Av. 3, 08860 Castelldefels (Barcelona), Spain 4Department of Physics, Harvard University, Cambridge, MA, 02138, USA 5Laboratoire de Physique de la Matière condensée, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France 6Racah Institute of Physics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (Dated: August 4, 2021) 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 engi- neered 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 multi-site 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 work paves the way for investigations of increasingly complex gauge theories on table-top clas- sical setups, and demonstrates the precise control of nonlinear effects within devices.

Local symmetries provide a mathematical framework noise-assisted energy transport [23, 24]. to describe emergent behavior from a small set of mi- In the material, electric fields propagate through a croscopic rules. Paradigmatic examples are topological chain of LC oscillators, and the U(1) symmetric cou- phases of matter, which can emerge as ground states pling is engineered from three-wave mixers. The - of an extensive set of commuting local operators [1, 2]. circuit is then described by nonlinear differ- In the Standard Model of particle physics all interac- ential equations, similar to the ones known from non- tions between elementary particles are mediated by gauge linear . Based on this approach, we experimentally bosons [3]. Recently, there has been a flurry of proposals demonstrate the engineering of a lattice with nine LC os- and experimental implementations of lattice gauge theo- cillators that represent five matter fields and four gauge ries in quantum many-body platforms [4–14]. Yet, gauge links, see Fig. 1. We demonstrate the high tunability invariance is as fundamental in classical physics and ap- of the setup, and confirm its faithful representation of plications thereof. A famous example is classical elec- the desired model using analytical models and numerical trodynamics, where gauge invariance appears as Gauss’s benchmark calculations. The ease of use and low cost of law. Its presence in Maxwell’s equations has been a guid- such classical compared to quantum ex- ing principle for [15, 16] based on periments open new ways to employ gauge symmetries a variational approach [17], which has led to the experi- for a wide range of materials from acoustics [25] over mental realization of intriguing devices such as metama- [26] to mechanical pendulums [27]. terials with negative indices of refraction [18] and invis- ibility cloaks [19]. In this it is natural for gauge Our material can be described within the language of invariance to be investigated using classical setups that a classical lattice gauge theory as sketched in Fig. 1A. are usually less expensive and simpler to implement com- In this framework, matter fields reside on discrete sites arXiv:2108.01086v1 [cond-mat.mes-hall] 2 Aug 2021 pared to their counterparts in quantum synthetic matter. x of a 1-dimensional lattice and are coupled through links, which host the gauge fields. Within our model- Here, we engineer a complex metamaterial with a lo- ing, we implement both the matter and the gauge fields cal U(1) gauge symmetry—the simplest continuous gauge by harmonic oscillators described by the complex num- symmetry and the basis of quantum electrodynamics— bers ax and bx. In the appropriate rotating frame, the x which embodies the invariance of the matter sites ax have staggered ( 1) m (see − under a local phase transformation. We base our setup on Appendix A). Two consecutive matter sites are coupled classical electric circuits, which have proven to be a pow- by a gauge field bx on the link connecting them. The erful platform for studying topological lattice structures term is a three-wave mixing term with the in- and multidimensional metamaterials [20–22], as well as teraction frequency J. This system is described by the 1 2

real (imaginary) part of the complex fields ax. Dividing out a typical energy and time scale allows us to write the Hamiltonian in units of frequency and in terms of dimensionless fields ax and bx. The Hamiltonian is invariant under the local U(1) iθx transformation ax ax e where the gauge field ab- x → i( 1) (θx θx+1) sorbs the difference bx bx e − − . With the → gauge symmetry comes a local conserved quantity Gx, which is the generator of Gauss’s law. Writing ρx = ax∗ ax x and Ex = ( 1) b∗ bx yields the expression Gx = − − x ρx + Ex 1 Ex. Interpreting ρx and Ex as the lo- − − cal oscillators’ energies, Gauss’s law describes the con- servation of the total energy on a matter site and its neighboring sites and gauge fields. Gauss’s law is im- plemented through the with a ring of three appropriately connected multipliers (see Site x = 1 Site x = 2 Site x = 3 Appendix B). The frequencies of the LC oscillators are designed such that the sum of the odd matter sites and 0.2 the links is approximately equal to the frequency of the even matter sites. The small frequency detuning m can 0.0 then be isolated from the fast timescales in the rotating frame. By averaging over fast timescales on the order of 0.2 the free frequency ωx, gauge-violating terms in − Oscillator energy (dim.less) the coupling are removed [14, 28, 29], which allows the 0 5 0 5 0 5 Time / ms U(1) gauge invariant Hamiltonian of Eq. (1) to emerge. The LC oscillators realizing the matter sites have a Figure 1. Engineering a classical gauge theory. free resonance frequency of 31.0(5) kHz (86(1) kHz) for (A) Structure of a lattice gauge theory. Matter fields reside odd (even) sites (see Appendix B). The links have a free on sites, gauge fields on the links in-between. (B) Circuit im- resonance frequency of 55 kHz with the possibility to be plementation with LC for each site and link. The tuned. This setup results effectively in a controllable de- interaction is realized by three-wave mixers as described in tuning of 0 kHz m 4 kHz with a precision of 200 Hz. ≤ ≤ the main text. (C) Measurement of Gauss’s law. The Gauss The coupling strength, depending on free resonance fre- law Gx can be measured through the oscillator energy on each quencies, is J = 0.92(5) kHz at m = 2.5 kHz. With a typ- matter site (blue) and its neighboring links (orange). Links that appear with negative sign in Gauss’s law are shown in- ical quality factor of 50 the in the resonators verted along y-axis (dashed). The smaller of the takes effect before the U(1) interaction. As a remedy, a resulting observable (black) confirm that the local conserva- positive current controlled by a pickup coil in tion laws are fulfilled. Curves are offset for clarity (see Ap- the is added, resembling regenerative pendix C). receivers of early radio technology. We observe a non- trivial energy exchange between matter sites as shown in Fig. 1C. The local symmetry enforces concerted dynam- classical Hamiltonian ics of matter sites and their neighboring links, such that the measured Gauss law has only small variations. This observation quantifies the weak violation of local gauge = + H J ax∗ bx∗ ax+1 c. c. invariance in our metamaterial. ( x odd X  To analyze the non-trivial dynamics, we investigate the spectral properties of the chain. We set the initial condi- + ax∗ bx ax+1 + c. c. tions to Qx(t = 0) = 0 and the initial flux is chosen such x even ) X  that the oscillators start with an of 0.7 V. l ∼ x After initialization, 8.3 ms long time traces of the voltage m ( 1) a∗ ax, (1) x signals of all resonators are available for spectral analy- − x=1 − X sis as shown in Fig. 2. The observed spectra can be well with equations of motion understood perturbatively as they are obtained in the regime of large detuning. The spectra of the matter sites ∂H ˙ ∂H contain two frequency components of different strengths. ia˙ x = , ibx = . (2) − ∂ax∗ − ∂bx∗ The stronger frequency originates from the free resonance of the LC oscillators. The second and weaker one orig- We realize the Hamiltonian (1) through an array of inates from the pertubative interactions with the gauge LC oscillators, whose charge Qx (flux Φx) represents the links. Gauss’s law implies that the gauge field has to 1 3

Simulation Odd sites Links Even sites 10

5

/ kHz 0 rot f 5 − 6 10 4 − Measurements 2 10

/ kHz 0

rot 5 f 2 −

4 / kHz 0 − rot 6 f − 5 V1 1 V2 2 V3 3 V4 4 V5 − V V Lattice positionV V 10 − 0 2 4 0 2 4 0 2 4 3 1 3 1 10− 10− 10− 10− m / kHz Voltage amplitude / V Voltage amplitude / V

3 1 3 1 Figure 2. (A) Level scheme of 10− 10− 10− 10− Site-resolved spectrum. Voltage amplitude / V Voltage amplitude / V the Hamiltonian in the regime of large mass m = 2.5 kHz compared to the coupling strength J = 0.92 kHz. The bare frequencies of the matter sites (blue) have a resonance at Figure 3. Mass dependence of the lattice gauge the- x A spectral analysis of our model in Eq. (1) frot = −(−1) m and the links (orange) at frot = 0, shown as ory spectrum. straight lines. They are dressed by the gauge invariant cou- with five matter sites, as a function of the matter field mass pling, which allows excitations to move between matter sites m, is presented in terms of experimental measurements and (green arrows). The coupling results in weak spectral lines numerical simulations. All spectral features observed in the x experiment are well reproduced by the numerics. Agreement at frot = (−1) m for the matter field, which are mirrored on with first-order perturbation theory in the thermodynamic the links through weak spectral lines at frot = −2m due to the Gauss law. (B) Observed frequency spectrum of oscillator limit (black lines) confirms that our implementation is suffi- . The positions of spectral lines compare well to the ciently large to render finite-size effects insignificant. predicted level structure from perturbation theory.

these shifts, we populate the first matter site by driving match the appearance of a second frequency component it with an . The first site has the typ- in the matter field. This appearance is clearly visible in ical frequency response of a driven our spectra and is observed at the predicted frequency. with resonance at fdrv = +m, where fdrv is the driving We employed the circuit to systematically investigate frequency in the rotating frame of this first site as ex- the dependence of the spectrum on the mass, as shown in plained in Appendix F. of the other matter Fig. 3. For large mass, we observe experimentally a linear sites are initialized to zero, while the amplitudes of dy- dependence of the spectral lines, which is well explained namical links are initialized as previously. We observe by first-order perturbation theory in the thermodynamic a significant shift of the resonance frequency on the sec- limit (see Appendix E). This agreement indicates that ond matter site as shown in Fig. 4B. We extended this even at few matter sites, our lattice gauge theory faith- measurement of the frequency shift fU to all matter sites fully reproduces the thermodynamic limit. Similar be- as shown in Fig. 4C. We observe no significant decay of havior has also been seen in a quantum-link-model lattice fU as a function of distance from the first site. The role gauge theory, where few lattice sites can capture the dy- of interactions is confirmed by good quantitative agree- namics of local observables in the thermodynamic limit ment between our observations and perturbation theory [30]. For smaller mass, we observe deviations from the without free parameters. These findings show, that our perturbative predictions derived around the large-mass experimental platform provides high control over long- limit, while non-perturbative numerical simulations of range interactions through the engineering of local sym- the spectra yield a quantitative agreement for the salient metries. experimental observations over the full regime. This work opens the door towards the investigation of In our metamaterial, the Gauss law implies long-range gauge theories in electrical circuits. The realization of interactions between matter sites [7] as visualized in the Hamiltonian (1) is directly transferable to the quan- Fig. 4A (see Appendix D for the derivation). They man- tum realm using superconducting circuits architectures ifest as shifts of the resonance frequency. To investigate cooled-down to 10 mK, which can be manipulated and 41

Frequency shift on matter sites 2.5

2.0 / kHz U f 1.5 Driving V1 Spectrum V2 4 0 Perturb. theory 10 1.0

/V 2 / kHz

1 2 V drv f 2 Frequency shift 1 fU 10− 0.5

− Perturbation theory

RMS 0 rot Measured peaks f 0 0.0 Voltage amplitude / V 0 2 4 0 2 4 1 2 3 4 fdrv / kHz fdrv / kHz Lattice distance

Figure 4. Long range interactions between matter sites. (A) The existence of the links introduces 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 fdrv.(B) The first matter site has the frequency response of a harmonic oscillator with resonance frequency m = 2.5(2) kHz and its signal is off-resonantly coupled into the second matter site with a coupling strength J = 0.92(5) 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 fU 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. readout with signals while maintaining long AEI (“Severo Ochoa” Center of Excellence CEX2019- enough quantum coherence. In such a platform, the re- 000910-S, Plan National FIDEUA PID2019-106901GB- quired gauge invariant three-wave mixing interaction can I00/10.13039 / 501100011033, FPI, QUANTERA MAQS be implemented using Josephson ring modulators based PCI2019-111828-2 / 10.13039/501100011033), Fundació on superconducting tunnel junctions [31]. Our work also Privada Cellex, Fundació Mir-Puig, Generalitat de directly offers the possibility to implement systems in Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA higher following models that were proposed program, QuantumCAT U16-011424, co-funded by for 2D [32–39] or non-Abelian systems [40–48] with the ERDF Operational Program of Catalonia 2014-2020), exciting prospect of directly observing confinement pre- EU Horizon 2020 FET-OPEN OPTOLogic (Grant No dicted in theses theories. 899794), and the National Science Centre, Poland (Symfonia Grant No. 2016/20/W/ST4/00314), Marie Skłodowska-Curie grant STREDCH No 101029393, ACKNOWLEDGMENTS “La Caixa” Junior Leaders fellowships (ID100010434), and EU Horizon 2020 under Marie Skłodowska-Curie The authors are grateful for fruitful discussions with grant agreement No 847648 (LCF/BQ/PI19/11690013, T. Gasenzer, J. Berges, and the members of the SynQS LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012). seminar. This work is part of and supported by the F. J. acknowledges the DFG support through the Emmy- DFG Collaborative Research Centre “SFB 1225 (ISO- Noether grant (project-id 377616843). P. H. acknowl- QUANT)”, the ERC Advanced Grant “EntangleGen” edges the Google Research Scholar Award ProGauge. (Project-ID 694561), the ERC Starting Grant “StrEn- E. Z. was supported by the Israel Science Founda- QTh” (Project-ID 804305), Quantum Science and Tech- tion (grant No. 523/20). V. K. received support nology in Trento (Q@TN), the Provincia Autonoma di from the ”la Caixa” Foundation (ID 100010434) and Trento, and the Excellence Initiative of the German from the European Union’s Horizon 2020 research and federal government and the state governments – fund- innovation programme under the Marie Skłodowska- ing line Institutional Strategy (Zukunftskonzept): DFG Curie grant agreement No 847648 with fellowship code project number ZUK 49/Ü. ICFO group acknowledges LCF/BQ/PI20/11760031. support from ERC AdG NOQIA, State Research Agency

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SUPPLEMENTAL MATERIAL

Appendix A: Hamiltonian derivation are ax, a∗ = iδxy, requiring the additional i in Hamil- { y} ton’s equations (Eq. (2) in main text). The Hamiltonian The equations of motion for the circuit sketched in in the new variables is Fig. 1B written as Kirchhoff current laws are l l 1 − = + 1 H ωx ax∗ ax ωx0 bx∗ bx I = Φ + 2CΦ¨ x=1 x=1 x x x X X (A5) Lx l 1 C d d − ˙ ˙ ˙ ˙ + Jx (ax + ax∗ )(bx + bx∗ )(ax+1 + ax∗+1) Φx0 1Φx 1 + Φx0 Φx+1 , (A1a) − V dt − − dt x=1 ref   X 1 ¨ C d ˙ ˙ Ix0 = Φx0 + 2CΦx0 ΦxΦx+1 . (A1b) with coupling strength Lx0 − Vref dt 1 V0 ( ) Jx = ωxωx0 ωx+1/2 . (A6) On the left hand side are external currents Ix0 , where 4 Vref primed quantities denote the links. The right hand side p ( ) is formulated in terms of magnetic fluxes Φx0 which are Only two of the coupling terms in Eq. (A5) are invariant integrals of the voltage signal Φ˙ = V . The index x under the local U(1) transformation introduced in the runs from 1 to l for sites and from 1 to l 1 for links. main text and discussed in Appendix C). All other terms − Vref = 10.2(3) V is the internal reference of the voltage can be separated in a rotating frame with the non-unique ( ) staggered tuning multipliers. Upon setting Ix0 to zero, which holds true when no external drive is connected, these equations arise x ωx ωx+1 = ( 1) (ω0 δx) (A7) from the following Lagrange function with generalized co- − − x − ( ) ordinates Φx0 : requiring δx to be much smaller than the resonance fre- quencies. The rotating frame is then L 1 1 1 − C ˙ 2 2 ˙ ˙ ˙ x = 2CΦ Φ ΦxΦ0 Φx+1. i(ωx+( 1) mx)τ x x x ax ax e − , (A8a) L 2 0 − Lx − Vref x,x   x=1 → 0 X X iωxτ bx bx e , (A8b) (A2) → Because the interaction is in the terms, the conjugate momenta Q = ∂ /∂Φ˙ , which are needed using site based detunings mx+1 = δx mx, m1 = δ1. L − to transform to the Hamiltonian, are non-trivial to invert The simplest allowed configuration is all m = mx = δx for Φ(˙ Q). In the small coupling approximation, i.e., as- having the same value. In the rotating frame, and with J = J m = m x suming typical voltages Φ˙ to be much smaller than the x and x independent of , the Hamiltonian h i takes on the form of Eq. (1) in the main text. By dividing reference voltage Vref, the conjugate momentum is the ˙ the equations of motion by the time scale 1/ω0, all mea- local oscillator’s Qx 2CΦx. ≈ surements of the Hamiltonian are in units of frequency: After defining the scales V0 = 1 V for voltage and f0 = ω0/2π = 60 kHz for time, we change to dimen- d d ¯ ¯ ∂H ∂H sionless flux Φ = ω0 Φ/V0 and charge Q = Q/2CV0. The ax = i ax = iω0 . (A9) dτ ∂a ⇔ dt ∂a corresponding Hamiltonian can be expressed in dimen- x∗ x∗ 2 sionless units after dividing by the energy scale 2CV0 : Similarly the time scale can be absorbed into the mass m and coupling J to give them and all figure axes intuitive ¯ 1 2 ¯ 2 ¯2 V0 ¯ ¯ ¯ H = ωx Φx + Qx + QxQx0 Qx+1 . (A3) units. 2 2Vref x,x0 x Driving. For the Hamiltonian description, we have set X  X ( ) the external currents Ix0 (t) to zero, but for the driven Here, the free resonance frequencies 1/√2CLx of the LC model, these have to be taken into account. This is oscillators are normalized to ωx = 1/(ω0 √2CLx). Using achieved by adding a term to the equations of motion dimensionless time τ = tω0 allows Hamilton’s equations (2): to appear unchanged. The complex variables are defined from dimensionless dax ∂H ˆ = i + Ix(τ) , (A10a) quantities as dτ ∂ax∗ dbx ∂H 1 1 ˆ = ( ¯ + Φ¯ ) = ( ¯ + Φ¯ ) = i + Ix0 (τ) . (A10b) ax √2ω Qx iωx x , bx 0 Qx0 iωx0 x0 , x √2ωx dτ ∂bx∗ (A4) ( ) with complex conjugates as canonical momenta. The Here, Iˆx0 (τ) is the dimensionless external current in the Poisson brackets with respect to the old variables (Φ,Q) rotating frame. For the first site, which is the only one 71 in our experiment that is connected to an external drive, the calculation from Ix(t) is as follows:

I1(τ/ω0) ˆ i(ω1 m1)τ I1(τ) = e− − . (A11) 2√2ω1 ω0CV0

A harmonic driving signal I1(t) = I sin(2πfextt) takes in the rotating wave approximation the form

I fext i(ω1 m1)τ Iˆ1 = sin τ e− − f0 √2ω1 ω02CV0 1   I (fext/f0 ω1+m1)τ e − . (A12) ≈ 2i √2ω1 ω02CV0

In the main text, we use fdrv = fext/f0 ω1 + m1 to de- − note the external driving frequency, in the rotating frame. Dissipation. We want to inspect features of the system with external driving close to resonance. Without tak- ing into account dissipation, numerical results diverge in these regimes. As a remedy, we add an empirical dissi- pation term to the equations of motion,

dax ∂H ˆ = i + Ix(τ) kax , (A13a) Figure 5. Setup. Photograph of the circuit with 5 sites. dτ ∂ax∗ − Milled circuit boards can be chained together for arbitrarily dbx ∂H large lattices. Inside the handmade coils, secondary feedback = i + Iˆ (τ) kb . x0 x (A13b) and tuning coils are visible. The illustration below indicates dτ ∂bx∗ − the general content and connections on the circuit boards. The dissipation term is used only for numerical results of the driven system shown in Fig. 8. Simulation. The equations of motion above, includ- 31.0(5) kHz 85(1) kHz ing driving and dissipation terms, are integrated with to . The main inductors are multi- 4.0(3) cm standard numerical solvers starting from initial condi- layer coils with a diameter of and lengths be- 5 cm 9 cm tions like in the experiment. The parameters of mass tween to . The winding numbers range between 80 a 270 and coupling are based on tuning parameters. Only the nd Inside the coils secondary coils for the feed- dissipation strength is treated as free parameter and ad- back circuit are mounted. The can be rotated to tune the justed to k = 0.0045 such that observations match the amount of feedback. The inductors for link resonators simulations in Fig. 8. Numerical results have negligible hold next to the feedback coil another coil, that is con- violation of local conservation laws. nected in series with the outer coil, and allows the links to be tuned from 53 kHz to 64 kHz. Mixer core. The interaction term of the U(1) Hamil- Appendix B: Properties of the LC oscillators and tonian is a QxQx0 Qx+1 coupling (see Eq. A2), which is multipliers symmetric under exchange and therefore the implemen- tation must have the same symmetry under exchange This section describes implementation details of the of its connectors. Furthermore the voltage of all addi- circuit and lists some device properties. The overall de- tional nodes introduced by the interaction need to be sign frequency of the circuit between 10 kHz and 100 kHz fully determined by the site variables, in order to not has a number of advantages: it allows cheap integrated introduce additional canonical variables. The Kirchhoff circuits to be used, does not require high-frequency aware current laws in Eq. (A1) state the interaction term as design of the circuit, and simplifies the recording of fully voltage multiplication (since Φ˙ = V ) that is coupled ca- sampled time traces of the dynamics. pacitively into the site with the additional time deriva- . All capacitors in the circuit schematic tive. In our circuit the voltage multiplication is achieved (Fig. 1B) are polypropylene film capacitors and have the by explicitly inserting analog voltage multipliers (IC part same value of C = 10 nF 2%. Similar to the design of number AD633), also called mixers. The voltage multi- ± some topological metamaterials [20], the coupling capac- pliers in our circuit support an output voltage between itance doubles as the on-site and the small 10 V. Multipliers have an internal scale appearing as ± coupling limit is achieved by a scale factor in the multi- Vref = 10 V in the equations of motion. Each interaction pliers. term in the KCL requires its own multiplier which sums Inductors. The inductors Lx are handmade to match up to three multipliers per link. The resulting coupling resonance frequencies ω = 1√2CLx in the range of strength J is given by Eq. (A6) and depends on Vref as 8 well as the free resonance frequencies of the resonators. Fig. 1C shows measurements in our circuit. The tun- For the measurement of the interactions in Fig. 4 with ing for this measurement is m1...5 = 0.9(2), 0.9(2), 1.3(2), m = 2.5(2) kHz, it evaluates to J = 0.92(5) kHz. 1.0(2), 1.7(2)kHz. Measurements of the circuit produce Initializers. Initial conditions of the circuit are set by voltage signals of the oscillators. Peaks in the voltage sig- cutting of external currents at t = 0. External current nal are used to reconstruct an envelope, which is also the are supplied by one MOSFET per oscillator that isolates amplitude in the rotating frame. The voltage amplitudes the circuit from the external source at t > 0. Initial are used to infer the oscillator energies. An offset and currents are tunable with on-board potentiometers. an exponentially decaying background are removed from Feedback. By adding feedback, dissipation effects can the energies, in order to center energy variations around be reduced. Typical quality factors of our oscillators are zero. The energy variations x are then expressed in E 2 between 50 to 80. Longer time traces than this dissi- dimensionless terms using a∗ ax = f0 x/2CV fx (links x E 0 pation time scale are required to reach the frequency analogously). The results in Fig. 1C show that the con- resolution of the spectra shown above. Each oscillator servation is much better on odd sites than on even sites, is connected to a biased bipolar junction transistor that because even sites are more sensitive to the feedback cir- couples the signal picked up by the feedback coil back into cuit. The violation on odd sites and on short time scales the oscillator. Careful tuning of the feedback strength is faster than 0.5 kHz is consistent with the violation ex- required to achieve both long time traces and small gauge pected from the rotating wave approximation. violation.

Appendix D: Long-range interactions Appendix C: Conservation laws Starting from Hamiltonian (1), we make the canonical As discussed in the main text, our Hamiltonian has variable change the following local continuous symmetry, parameterized by the phase θx: iφx bx = nx0 e (D1)

iθx ax ax e , (C1a) p → x on the links only. Sites ax stay unchanged. We then have i( 1) (θx θx+1) bx bx e − − . (C1b) the Hamiltonian → This symmetry transformation is a canonical transfor- iφx mation of the complex variables and as such can be ex- H = J ax∗ nx0 e− ax+1 + c. c. pressed using Poisson brackets w. r. t. a generator func- ( x odd X p  tion Gx, that depends on the complex variables local to +iφx x [49], + ax∗ nx0 e ax+1 + c. c. x even ) X i ∂z = z, Gx dθx , (C2) l p  − { } x m ( 1) ax∗ ax. (D2) where z denotes a vector of all canonical variables ax and − x=1 − bx (not including their complex counterpart), and ∂z is X their change under the infinitesimal gauge transforma- The phases φx can be eliminated by performing a canon- tion. For this purpose, the Poisson brackets are defined ical transformation on the matter fields [7, 50, 51]. The as transformation is controlled by the phase φx of the gauge l fields: ∂f ∂g ∂f ∂g f, g = y ∂a ∂a ∂a ∂a ( 1) iφy { } x=1 x x∗ − x∗ x ax ax e− − . (D3) X   (C3) → l 1 y

Appendix E: Perturbation theory for non-driven system

We now carry out perturbation theory to provide an analytic footing for our experimental results. We start with the Hamiltonian (1), and assume we are in the thermodynamic limit. This reduces our model to a unit cell of two matter sites and two gauge links with periodic boundary conditions described by the Hamiltonian

H = m a∗a1 a∗a2 + J a∗ b∗ + b∗ a2 + a1 b1 + b2 a∗ . (E1) 1 − 2 1 1 2 2 We see that the dynamics of the bfield is the same  whether on an odd or even link. As such, we can rewrite our Hamilton’s equations simply as

idτ a1 = ma1 2Jb∗a2, (E2a) − − idτ a2 = ma2 2Jba1, (E2b) − idτ b = Ja∗a2. (E2c) − 1 The full solution of the b field in terms of the a fields is τ b(τ) = + iJ ds a∗(s)a2(s). (E3) B 1 Z0 Employing perturbation theory with J/m as small parameter, we now solve up to third order for the a and b fields. The zeroth-order contribution in J is b(0)(τ) = (E4a) B (0) imτ a (τ) = 1e , (E4b) 1 A (0) imτ a (τ) = 2e− . (E4c) 2 A We can now solve for the first-order contribution to the b field as follows: τ τ (1) (0) (0) 2ims J 2imτ b (τ) = iJ ds a ∗(s)a (s) = iJ ∗ 2 ds e− = ∗ 2 1 e− . (E5) 1 2 A1A 2mA1A − Z0 Z0  The first-order contributions to the a fields can be found by solving

(1) (1) (0) (0) (1) imτ idτ a = ma 2Jb ∗a = ma 2J 2 ∗e− , (E6a) 1 − 1 − 2 − 1 − A B (1) (1) (0) (0) (1) imτ idτ a = ma 2Jb a = ma 2J 1 e , (E6b) 2 2 − 1 2 − A B the solutions to which are

(1) J imτ imτ a (τ) = 2 ∗ e e− , (E7a) 1 mA B − (1) J imτ imτ  a (τ) = 1 e e− . (E7b) 2 mA B −  10

Consequently, the second-order contribution to the b field is τ (2) (0) (1) (1) (0) b (τ) = iJ ds a1 ∗a2 + a1 ∗a2 0 2Z τ h i J 2 2 2ims = i ds 1 2 1 e− m B |A | − |A | − Z0 2   J 2 2 1 2imτ = 1 2 iτ 1 e− , (E8) m B |A | − |A | − 2m −     which again carries the same frequency as the first-order contribution. We now continue to the second-order contri- bution in the a fields by solving 2 (2) (2) (0) (1) (1) (0) (2) J 2 2 imτ imτ idτ a = ma 2J b ∗a + b ∗a = ma + 1 2 2 e e− , (E9a) 1 − 1 − 2 2 − 1 m A |A | − |B| − h i 2 (2) (2) (0) (1) (1) (0) (2) J 2 2 imτ imτ  idτ a = ma 2J b a + b a = ma 2 1 + 2 e e− , (E9b) 2 1 − 1 1 2 − m A |A | |B| − h i the solutions for which are   2 (2) J 2 2 imτ imτ a (τ) = 1 2 2 (2imτ 1)e + e− , (E10a) 1 −2m2 A |A | − |B| − 2 (2) J 2 2  imτ imτ  a (τ) = 2 1 + 2 e (2imτ + 1)e− . (E10b) 2 2m2 A |A | |B| − The third-order contribution to the b field is    τ (3) (0) (2) (1) (1) (2) (0) b (τ) = iJ ds a1 ∗a2 + a1 ∗a2 + a1 ∗a2 . (E11) Z0   The third-order contribution to the a fields is obtained by solving

(3) (3) (0) (2) (1) (1) (2) (0) idτ a = ma 2J b ∗a + b ∗a + b ∗a 1 − 1 − 2 2 2 (3) 3imτ imτ imτ = ma + c1e + c2e + (c3t + c4)e−  , (E12a) − 1 (3) (3) (0) (2) (1) (1) (2) (0) idτ a = ma 2J b a + b a + b a 2 1 − 1 1 1 (3) imτ imτ 3imτ = ma1 + (d1t + d2)e + d3e− + d4e− , (E12b) 3 J 2 c1 = ∗ , (E12c) m2 A1A2B 3 J 2 2 2 2 c2 = 2 ∗ 2 1 2 + 2 + 2 ∗ , (E12d) −m2 A B |A | − |A | |B| A1A2B 3 J  2 2 2   c3 = 2i 2 ∗ 2 1 2 + 2 , (E12e) m A B |A | − |A | |B| 3 J 2 2 2  2 c4 = 2 ∗ 2 1 2 + 2 + ∗ , (E12f) m2 A B |A | − |A | |B| A1A2B 3 J 2 2 2  d1 = 2i 1 1 2 2 + 2 , (E12g) − m A B |A | − |A | |B| 3 J 2 2 2  2 d2 = 1 1 2 2 + 2 ∗ ∗ , (E12h) m2 A B |A | − |A | |B| − A1A2B 3 J 2 2 2 2 d3 = 1 1 2 2 + 2 2 ∗ ∗ , (E12i) −m2 A B |A | − |A | |B| − A1A2B 3 J  2   d4 = ∗ ∗, (E12j) −m2 A1A2B the solutions for which are 3 (3) c1 3imτ J 2 2 2 imτ 2mc3τ + 2mc4 ic3 imτ a (τ) = e ic2t + 2 ∗ 2 1 2 + 2 e + − e− , (E13a) 1 −2m − m3 A B |A | − |A | |B| 4m2   3  (3) 2md1τ + 2md2 + id1 imτ J 2 2 2 imτ d4 3imτ a (τ) = e id3τ 1 1 2 2 + 2 e− + e− . (E13b) 2 − 4m2 − − m3 A B |A | − |A | |B| 2m    111

Measurements V V V V V 1 V1 2 V2 3 V3 4 V4 5 8 6 4 1 10− 2

/ kHz 0

rot 2 f − 4 10 3 − − 6 − Voltage amplitude / V 8 − 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 a1 b1 a2 b2 MassSimulationma3/ kHz b3 a4 b4 a5 8 6 4 1 10− 2

/ kHz 0

rot 2 f − 4 10 3 − − 6 − Voltage amplitude / V 8 − 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Mass m / kHz

Figure 6. Spectrum of the 5-site lattice depending on detuning ∆ = ∆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 ±3m frequency components and in links +2m components from higher orders of perturbation theory.

The results of the perturbation theory are compared to the experimental data and the numerical predictions of the spectrum in Fig. 6.

Appendix F: Perturbation theory for driven system d A J/2m 1 x=1 Bx Measured amplitudes 100 Q For the perturbation theory in the driven case, we as- sume a setup as described in the main text: Only the 1 links are initialized at t = 0 and sites are left empty. The 10− first site is driven with a harmonic signal, see Eq. (A10a).

We know from measurements that the driving signal Voltage amplitude / V 2 propagates onto the other sites with exponentially de- 10− creasing amplitude over distance (Fig. 7). The reason for this are the staggered site frequencies, such that the 0 1 2 3 4 driving is off-resonant at least at every second site. This Distance d to first site motivates the approximation that the back-action of the second onto the first site and the back-action of the third Figure 7. Observed exponential decline of oscillator voltage onto the second site can be neglected. Then the equa- amplitudes along lattice when driving first site and compari- tions of motion of the second site become linear. With son to the next neighbor coupling mediated by links calculated the exponential structure of the amplitudes, this proce- in Appendix E. This exponential decline is used as hierarchy dure can be repeated along the lattice. for the perturbation theory in the driven case.

Without back-action of the second site, the first site iντ relaxes to match the driving signal a1 = Ae , ν = With the variable change fdrv/f0. The equations of motion for the link and second 1 2 i(m+ν)τ a bx bx0 b = e− b0 (F2a) site x+1 (without terms including the third site) form a → two-level system: 1 i(m ν)τ ax+1 a0 ax+1 = e− 2 − a0 (F2b) → x+1 x+1 the time dependence and energy offset are removed: ˙ ˙ 1 bx 0 Jax∗ (τ) bx bx0 + 2 (ν + m) JA∗ bx0 = i . (F1) = i 1 . (F3) a˙ x+1 Jax(τ) m ax+1 a˙ JA (ν + m) a0    −     x0 +1  − 2   x+1 12

Eigenvalues of the matrix are Eigenvalues of the matrix are 1 1 2 2 2 2 2 2 µo(ν, A) = (ν + m) + 4J A (F4) µe(ν, A) = (m ν) 4J A . (F9) ±2 | | ±2 − − | | p p and solutions have the shape and the solutions have the frequency components i(µ + 1 (m ν))τ) i( µ + 1 (m ν))τ) 1 1 e 2 e 2 +i(µo (m+ν))τ i(µo+ (m+ν))τ bx = e − + e − − , (F10a) bx = c1e − 2 + c2e− 2 , (F5a) i(µ + 1 (m+ν))τ) i(µ 1 (m+ν))τ) 1 1 e 2 e 2 i(µo+ (m ν))τ +i(µo (m ν))τ ax+1 = e + e − . (F10b) ax+1 = c3e− 2 − + c4e − 2 − . (F5b) The prediction for the frequency shift in Fig. 4 is based ci ν + m JA Prefactors depend on parameters , , and the on measured site amplitudes Ax shown in Fig. 7 (made bx ax+1 initial state of and . We do not calculate them dimensionless with scale V0 = 1 V). The red line in the here explicitly. spectrum of V2 is The calculation above holds for any given odd x along the chain to calculate the behavior of its following even fU2(ν)/f0 = µo(ν, A1) (m ν)/2 ν. (F11) site. Next we look at the other case: given a solution − − − iντ ax = Ae with even x, we calculate the dynamics of Predictions for the long range interactions in Fig. 4 use the next or previous odd site ax 1: the iterated approach: The frequency of the third site ν3 ± is based on the solution of the second site: ˙ b∗ 0 Jax∗ (τ) bx∗ x = i − . (F6) = ( ) ( ) 2 a˙ Jax(τ) m ax+1 ν2 µo ν, A1 m ν / , (F12a)  x+1     − − ν3 = µe(ν2,A2) (m ν2)/2, (F12b) − − The appropriate variable change is ν4 = µo(ν3,A3) (m ν3)/2, (F12c) − − 1 ν5 = µe(ν2,A2) (m ν2)/2. (F12d) i 2 (m ν)τ bx∗ bx∗0 b∗ = e− − b∗0 (F7a) − − → 1 +i 2 (m+ν)τ The frequency shifts are calculated relative to the driving: ax+1 ax0 +1 ax+1 = e ax0 +1 (F7b) → fU = νf0 fdrv. Fig. 8 shows the full spectrum from − to reach the expression which Fig 4 is calculated. Also shown are comparisons to numerical simulations of Hamilton’s equations in the ˙ 1 rotating frame with and without the empirical dissipation bx∗0 2 (m ν) JA∗ bx∗0 = i − − 1− . (F8) term. a˙ JA + (m ν) a0  x0 +1  2 −   x+1 131

Measurements V V V V V 1 V1 2 V2 3 V3 4 V4 5 6 4 2

/ kHz 0 rot f 2 − 4 − 6 − 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 − − − − − − − − − fdrv / kHz

4 3 2 1 0 1 4 3 2 1 0 1 10− 10− 10− 10− 10 10 10− 10− 10− 10− 10 10 Voltage amplitude / V Voltage amplitude / V Simulation with term a1 b1 a2 b2 a3 b3 a4 b4 a5 6 4 2

/ kHz 0 rot f 2 − 4 − 6 − 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 − − − − − − − − − fdrv / kHz Simulation without damping term a1 b1 a2 b2 a3 b3 a4 b4 a5 6 4 2

/ kHz 0 rot f 2 − 4 − 6 − 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 − − − − − − − − − fdrv / kHz

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.