Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator

Abstract

The modern description of elementary particles, as formulated in the standard model of particle physics, is built on gauge theories¹. Gauge theories implement fundamental laws of physics by local symmetry constraints. For example, in quantum electrodynamics Gauss’s law introduces an intrinsic local relation between charged matter and electromagnetic fields, which protects many salient physical properties, including massless photons and a long-ranged Coulomb law. Solving gauge theories using classical computers is an extremely arduous task², which has stimulated an effort to simulate gauge-theory dynamics in microscopically engineered quantum devices^3,4,5,6. Previous achievements implemented density-dependent Peierls phases without defining a local symmetry^7,8, realized mappings onto effective models to integrate out either matter or electric fields^9,10,11,12, or were limited to very small systems^13,14,15,16. However, the essential gauge symmetry has not been observed experimentally. Here we report the quantum simulation of an extended U(1) lattice gauge theory, and experimentally quantify the gauge invariance in a many-body system comprising matter and gauge fields. These fields are realized in defect-free arrays of bosonic atoms in an optical superlattice of 71 sites. We demonstrate full tunability of the model parameters and benchmark the matter–gauge interactions by sweeping across a quantum phase transition. Using high-fidelity manipulation techniques, we measure the degree to which Gauss’s law is violated by extracting probabilities of locally gauge-invariant states from correlated atom occupations. Our work provides a way to explore gauge symmetry in the interplay of fundamental particles using controllable large-scale quantum simulators.

Main

Quantum electrodynamics (QED), the paradigmatic example of a gauge-invariant quantum field theory, has fundamentally shaped our understanding of modern physics. Gauge invariance in QED—described as a local U(1) symmetry of the Hamiltonian—ties electric fields E and charges ρ to each other through Gauss’s law, $\nabla \cdot {\bf{E}}=\rho$. The standard model of particle physics, including, for example, quantum chromodynamics, has been designed on the basis of this principle of gauge invariance. However, despite impressive feats^17,18, it remains extremely difficult for classical computers to solve the dynamics of gauge theories^3,4,5,6. Quantum simulation offers the tantalizing prospect of sidestepping this difficulty by microscopically engineering gauge-theory dynamics in table-top experiments, based on, for example, trapped ions, superconducting qubits and cold atoms^{7,8,9,10,11,12,13,14,15,16}. In the quest for experimentally realizing gauge-theory phenomena, a large quantum system is essential to mitigate finite-size effects irrelevant to the theory in the thermodynamic limit. Moreover, while Gauss’s law in QED holds fundamentally, it is merely approximate when engineered in present-day cold-atom experiments keeping both fermionic matter and dynamical gauge fields explicitly^15,16. Thus, it is a crucial challenge to determine the reliability of gauge invariance in large-scale quantum simulators¹⁹.

Here we verify Gauss’s law in a many-body quantum simulator. To this end, we devise a mapping from a Bose–Hubbard model (BHM) describing ultracold atoms in an optical superlattice to a U(1) lattice gauge theory with fermionic matter. We exploit the formalism of quantum link models (QLMs)^3,20, which incorporate salient features of QED, in particular Coleman’s phase transition in one spatial dimension (1D) at topological angle θ = π (ref. ²¹). Here, gauge-invariant ‘matter–gauge field’ interactions emerge through a suitable choice of Hubbard parameters, effectively penalizing unwanted processes. Experimentally, we prepare large arrays of atoms in high-fidelity staggered chains, realize the quantum phase transition by slowly ramping the lattice potentials, and observe the characteristic dynamics via probing of site occupancies and density–density correlations. In our model, Gauss’s law constrains boson occupations over sets of three adjacent sites in the optical lattice. By tracking the coherent evolution of the state in these elementary units, we detect the degree of local violation of Gauss’s law.

Our target model is a U(1) gauge theory on a 1D spatial lattice with ℓ = 0, 1, …, N − 1 sites, described by the Hamiltonian (see Methods)

$${\hat{H}}_{{\rm{Q}}{\rm{L}}{\rm{M}}}=\sum _{{\ell }}\left[-\frac{{\rm{i}}\mathop{t}\limits^{ \sim }}{2}({\hat{\psi }}_{{\ell }}\hat{S}{}_{{\ell },{\ell }+1}^{+}{\hat{\psi }}_{{\ell }+1}-{\rm{h}}.{\rm{c}}.)+m\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{\psi }}_{{\ell }}\right].$$

(1)

Using the QLM formalism, the gauge field is represented by spin-1/2 operators $\hat{S}{}_{{\ell },{\ell }+1}^{z}$ on links connecting neighbouring lattice sites, ${\hat{E}}_{{\ell },{\ell }+1}\equiv {(-1)}^{{\ell }+1}\hat{S}{}_{{\ell },{\ell }+1}^{z}$, corresponding to an electric field coarse-grained to two values (red and blue arrows in Fig. 1). Further, h.c. denotes hermitian conjugate. Using staggered fermions²², matter fields ${\hat{\psi }}_{{\ell }}$ represent particles and antiparticles on alternating sites, with alternating electric charge ${\hat{Q}}_{{\ell }}={(-1)}^{{\ell }}\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{\psi }}_{{\ell }}$. By tuning the fermion rest mass m, we can drive the system across a quantum phase transition from a charge-dominated disordered phase to an ordered phase, characterized by the spontaneous breaking of charge and parity (C/P) symmetries^21,23; see Fig. 1a. During the transition, owing to the term proportional to $\mathop{t}\limits^{ \sim }$ (gauge–matter coupling strength), particle–antiparticle pairs annihilate accompanied by the correct adjustment of the electric field according to Gauss’s law.

Fig. 1: Quantum simulation of a U(1) lattice gauge theory.

a, A quantum phase transition separates a charge-proliferated phase from a C/P symmetry-breaking phase where the electric field (triangles) passes unhindered through the system (sketched at particle rest mass m → −∞ and +∞, respectively). The Feynman diagram, depicted as wavy lines, describes the gauge-invariant annihilation of particles and antiparticles (circles with charges) with a coupling strength of $\tilde{t}$. The transition leads to two opposite configurations in terms of the directions of the electric field. b, Gauss’s law strongly restricts the permitted gauge-invariant configurations of charges and neighbouring electric fields. The matter field consists of antiparticle and particle sites. The mapping from QLM to BHM is sketched in the shaded diagrams, where the eigenvalues in Gauss’s law are labelled below each site. c, Simulation of the model on a 71-site Bose–Hubbard system consisting of ultracold atoms in an optical superlattice. See main text for nomenclature. We sweep through the quantum phase transition by controlling the Hubbard parameters over time t. Particle–antiparticle annihilation is realized by atoms initially residing on even (shallow) sites binding into doublons on odd (deep) sites. The upper and lower panel depict the initial and final state, respectively. Insets are their corresponding atomic densities.

Source data.

Gauss’s law requires the generators of the U(1) gauge transformations,

$${\hat{G}}_{{\ell }}={(-1)}^{{\ell }+1}(\hat{S}{}_{{\ell },{\ell }+1}^{z}+\hat{S}{}_{{\ell }-1,{\ell }}^{z}+\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{\psi }}_{{\ell }}),$$

(2)

to be conserved quantities for each matter site ${\ell }$. We choose, as is usual, to work in the charge-neutral sector, where the state |ψ⟩ fulfils ${\sum }_{{\ell }}{\hat{Q}}_{{\ell }}|\psi \rangle =0$, and in the Gauss’s law sector specified by ${\hat{G}}_{{\ell }}|\psi \rangle =0$, $\forall {\ell }$. Ensuring adherence to this local conservation law is the main experimental challenge, as it intrinsically constrains matter and electric fields across three neighbouring sites (see Fig. 1b).

We simulate this QLM with ultracold bosons in a 1D optical superlattice as sketched in Fig. 1c (see Methods for details). The experiment is governed by the BHM

$${\hat{H}}_{{\rm{B}}{\rm{H}}{\rm{M}}}=\sum _{j}\left[-J(\hat{b}{}_{j}^{\dagger }{\hat{b}}_{j+1}+{\rm{h.}}{\rm{c.}})+\frac{U}{2}{\hat{n}}_{j}({\hat{n}}_{j}-1)+{\varepsilon }_{j}{\hat{n}}_{j}\right],$$

(3)

where $\hat{b}{}_{j}^{\dagger },{\hat{b}}_{j}$ are creation and annihilation operators, ${\hat{n}}_{j}=\hat{b}{}_{j}^{\dagger }{\hat{b}}_{j}$, J is the tunnelling strength, and U is the on-site interaction. The energy offset ${\varepsilon }_{j}={(-1)}^{j}\,\delta /2+j{\Delta }$ consists of a linear tilt Δ to suppress long-range tunnelling along the 1D chain, and a staggered superlattice potential δ. Here, the even sites j of the superlattice correspond to the matter sites ${\ell }$ in the lattice gauge theory, while we identify odd sites j with link indices ${\ell },{\ell }+1$. Choosing δ ≫ J and on-site interaction U ≈ 2δ effectively constrains the system to the relevant subspace limited to the number states |0⟩, |2⟩ on odd (gauge) sites and |0⟩, |1⟩ on even (matter) sites. On this subspace, we can hence identify the operators as $\hat{S}{}_{{\ell },{\ell }+1}^{+}\simeq (\hat{b}{}_{j=2{\ell }+1}^{\dagger }){}^{2}/\sqrt{2}$ and similarly ${\hat{\psi }}_{{\ell }}\hat{S}{}_{{\ell },{\ell }+1}^{+}{\hat{\psi }}_{{\ell }+1}\simeq {\hat{b}}_{2{\ell }}(\hat{b}{}_{2{\ell }+1}^{\dagger }){}^{2}{\hat{b}}_{2{\ell }+2}/\sqrt{2}$ (using a Jordan–Wigner transformation for the matter sites), see Methods. This term can be physically realized by atoms on neighbouring matter sites combining into a doublon (that is, two indistinguishable atoms residing in one site). The rest mass corresponds to m = δ − U/2, which enables us to cross the phase transition by tuning m < 0 → m > 0. The strength of the gauge-invariant coupling ($\mathop{t}\limits^{ \sim }\approx 8\sqrt{2}{J}^{2}/U\approx 70\,{\rm{H}}{\rm{z}}$ at resonance m ≈ 0) is much larger than the dissipation rate, enabling a faithful implementation in a large many-body system.

The experiment starts with a quasi two-dimensional Bose–Einstein condensate of about 100,000 ⁸⁷Rb atoms in the x–y plane. We implement a recently demonstrated cooling method in optical lattices to create a Mott insulator with a filling factor of 0.992(1) (ref. ²⁴). Figure 2a shows a uniform area containing 10,000 lattice sites, from which a region of interest (ROI) with 71 × 36 sites is selected for simulating the gauge theory. A lattice along the y axis with depth 61.5(4)E_r isolates the system into copies of 1D chains. Here, ${E}_{{\rm{r}}}={h}^{2}/(2{m}_{{\rm{Rb}}}{\lambda }_{{\rm{s}}}^{2})$ is the recoil energy, with λ_s = 767 nm the wavelength of a ‘short lattice’ laser, h the Planck constant, and m_Rb the atomic mass. The near-unity filling enables the average length of defect-free chains to be longer than the 71 sites. Even without a quantum gas microscope, the size of our many-body system is confirmed by counting the lattice sites with single-site resonance imaging (see Methods). Along the x direction, another lattice, with wavelength λ_l = 2λ_s (the ‘long lattice’), is employed to construct a superlattice that divides the trapped atoms into odd and even sites. Two different configurations of the superlattice are used here. First, to manipulate quantum states in isolated double wells, which we use for state initialization and readout, the superlattice phase is controlled to match the positions of the intensity maxima of the short and long lattices. Second, in contrast, when performing the phase transition, overlapping the intensity minima of the lattices enables the production of identical tunnelling strength between neighbouring sites.

Fig. 2: Probing the many-body dynamics.

a, Experimental sequence. Starting from a near perfect Mott insulator in the ‘short’ lattice, the initial staggered state is prepared by removing the atoms on odd sites. We drive the phase transition by ramping the mass m = δ − U/2 and the tunnelling J. Afterwards, the occupation probabilities ${p}^{({\rm{m}}/{\rm{g}})}(n)$ are identified for even and odd sites by engineering the atomic states with measurement schemes (i)–(iii), see Methods section ‘State preparation and detection’. b, c, Time-resolved observation of the C/P-breaking phase transition. As revealed by the probabilities (b), atoms initially residing on even sites (upper panel) bind into doublons on odd sites (lower panel), corresponding to an annihilation of particles on matter sites and a deviation of the electric field, quantified by ${\sum }_{{\ell }}{(-1)}^{{\ell }}\langle {\hat{E}}_{{\ell },{\ell }+1}(t)-{\hat{E}}_{{\ell },{\ell }+1}(0)\rangle /(2N)$. c, Experimental observables and their correspondence in the QLM. Measured results agree well with theoretical predictions (solid curves) from the time-adaptive density matrix renormalization group (t-DMRG) method, where our numerics takes into account spatial inhomogeneity and sampling over noisy experimental parameters (see Methods). Error bars and shaded regions, s.d. The dashed lines represent the exact evolution of the ideal QLM (see Methods).

Source data.

To prepare the initial state, we selectively address and flip the hyperfine state of the atoms residing on odd sites²⁴, followed by their removal using resonant light. The remaining atoms on the even sites of the 1D chains correspond to an overall charge neutral configuration. They form the ground state of our target gauge theory, equation (1), at m → −∞ in the ${\hat{G}}_{{\ell }}|\psi \rangle =0$ sub-sector.

The phase transition is accessed by slowly tuning the superlattice structure in terms of the Hubbard parameters. The linear potential Δ = 57 Hz per site (formed by the projection of gravity) as well as the main contribution to the staggered potential δ = 0.73(1) kHz (arising from the depth of the long lattice) are kept constant during the 120-ms transition process. This ramp speed has been chosen to minimize both non-adiabatic excitations when crossing the phase transition and undesired heating effects. As shown in Fig. 2a, the tunnelling strength J/U is ramped from 0.014 up to 0.065 and back to 0.019. Simultaneously, we linearly lower the z-lattice potential to ramp the on-site interaction U from 1.82(1) kHz to 1.35(1) kHz. This ramp corresponds to driving the system from a large and negative m, through its critical point at m ≈ 0, to a large and positive value deep within the C/P-broken phase.

To probe the system dynamics, we ramp up the lattice barriers after evolution time t and extract the probability distributions ${p}_{j}^{({\rm{m}}/{\rm{g}})}(n)$ of the occupation number n. With our optical resolution of about 1 μm, in situ observables average the signal over a small region around site j. Our measurements distinguish between even matter sites (m) and odd gauge-field sites (g). We illustrate the procedure for ${p}_{j}^{({\rm{g}})}(n)$. To extract it for n ≤ 3, we combine the three schemes sketched in Fig. 2a ((i)–(iii); see Methods for a detailed application of (i)–(iii) to obtain the ${p}_{j}^{({\rm{m}}/{\rm{g}})}(n)$). (i) The mean occupation of gauge-field sites is recorded by in situ absorption imaging after applying a site-selective spin flip in the superlattice, which gives ${\bar{n}}^{({\rm{g}})}={\sum }_{n}n{p}_{j}^{({\rm{g}})}(n)$ with natural numbers n. (ii) We use a photoassociation (PA) laser to project the occupancy into odd or even parity. Unlike selecting out doublons via Feshbach resonances^25,26, the PA-excited molecule decays spontaneously and gains kinetic energy to escape from the trap. After this parity projection, the residual atomic density is ${\bar{n}}_{{\rm{c}}}^{({\rm{g}})}={\sum }_{n}{{\rm{mod}}}_{2}(n){p}_{j}^{({\rm{g}})}(n)$. (iii) A further engineering of atoms in double wells allows us to measure the probabilities of occupancies larger than two. We first clean the matter sites and then split the atoms into double wells. After a subsequent parity projection via illumination with PA light, the remaining atomic density is ${\bar{n}}_{{\rm{c}}}^{({\rm{g}})}+2{p}_{j}^{({\rm{g}})}(2)$. From the population, we find that high-energy excitations, such as n = 3, are negligible throughout our experiment.

As the data for ${p}_{j}^{({\rm{m}}/{\rm{g}})}(n)$ in Fig. 2b, c show, after the ramp through the phase transition, on average 80(±3)% of the atoms have left the even sites and 39(±2)% of double occupancy is observed on the odd sites (we checked the coherence and reversibility of the process by ramping back from the final state, see Methods). This corresponds to the annihilation of 78(±5)% of particle–antiparticle pairs. From the remaining 22(±5)% of particles that have not annihilated, we estimate the average size of ordered domains after the ramp to be 9 ± 2 sites. The formation of ordered domains can be further confirmed by measuring density–density correlations $C(i,j)=\langle {\hat{n}}_{i}{\hat{n}}_{j}\rangle$ (refs. ^27,28,29). We extract the correlation functions in momentum space after an 8-ms time of flight. For a bosonic Mott state with unity filling, the correlation function shows a bunching effect at momentum positions of ±2ħk, where k = 2π/λ_s is the wave vector. Two more peaks at ±ħk appear in the correlation function of our initial state owing to the staggered distribution, as shown in Fig. 3a. The width of these peaks is mainly determined by the spatial resolution of the absorption imaging. The correlation function of the final state in Fig. 3 shows two broader peaks at ±0.5ħk, which indicates the emergence of a new ordering with a doubled spatial period. The finite correlation length ξ of the final state broadens the interference pattern. Assuming exponential decay of density–density correlations, $C(i,j)\propto \exp (\,-|i-j|/\xi )$, we obtain the correlation length of the final state as $\xi ={4.4}_{-1.0}^{+2.0}$ sites (see Methods). Thus, we can achieve many-body regions with spontaneously broken C/P symmetry.

Fig. 3: Density–density correlation.

a, Left, idealized sketches of the initial (top) and final (bottom) state. The domain length of the final state equals the distance between two unconverted atoms, which are removed from the system before measurement. Right, measured interference patterns in the initial and final states (averaged over 523 and 1,729 images, respectively). The x lattice defining the 1D chains is tilted by 4° relative to the imaging plane. b, Single-pixel sections along the x direction through the centre of the patterns in a. In the final state, additional peaks at ±0.5ħk appear, indicating the emergence of a new ordering.

Source data.

Finally, we quantify the violation of Gauss’s law, for which we monitor the probabilities ${p}_{|\ldots {n}_{j-1}{n}_{j}{n}_{j+1}\ldots \rangle }$ of the three allowed gauge-invariant Fock states sketched in Fig. 1b, $|\ldots {n}_{j-1}{n}_{j}{n}_{j+1}\ldots \rangle =|\ldots 010\ldots \rangle$, $|\ldots 200\ldots \rangle$, and $|\ldots 002\ldots \rangle$, j even. To achieve this, we have developed a method to measure the density correlations between neighbouring lattice sites within double wells. Unlike the approach in Fig. 2a, which does not give access to correlations between sites, here we distinguish different states by their dynamical features (Fig. 4a). In particular, we use the characteristic tunnelling frequency to distinguish the target states from the others. For example, to detect the state $|\ldots 010\ldots \rangle$, we perform tunnelling sequences between double wells in two mirrored superlattice configurations (setting the parameters to J/h = 68.9(5) Hz and U/h = 1.71(1) kHz to avoid frequency overlap between different processes). The tunnelling frequency 2J/h for the state |10⟩ in a double well is one order of magnitude higher than the superexchange frequency 4J²/(hU) for the states |20⟩ or |11⟩. Thus, the oscillation amplitudes at frequency 2J/h yield the probabilities ${p}_{|\ldots 01{n}_{j+1}\ldots \rangle }$ and ${p}_{|\ldots {n}_{j-1}10\ldots \rangle }$. In addition, the probability ${p}_{|\ldots {n}_{j-1}1{n}_{j+1}\ldots \rangle }$ equals ${p}_{j}^{({\rm{m}})}(1)$ (see Fig. 2b, c). With these, we can deduce a lower bound ${p}_{|\ldots 010\ldots \rangle }\ge {p}_{|\ldots 01{n}_{j+1}\ldots \rangle }+{p}_{|\ldots {n}_{j-1}10\ldots \rangle }-{p}_{|\ldots {n}_{j-1}1{n}_{j+1}\ldots \rangle }$. We obtain the population of the states $|\ldots 002\ldots \rangle$ and $|\ldots 200\ldots \rangle$ in a similar fashion (see Methods).

Fig. 4: Fulfilment of Gauss’s law.

a, Correlated measurements detect gauge-invariant states $|\ldots {n}_{j-1}{n}_{j}{n}_{j+1}\ldots \rangle$, j even, within gauge–matter–gauge three-site units. For probing $|\ldots 010\ldots \rangle$ (left), we first flip the hyperfine levels of the atoms on odd sites. Then, we change the superlattice into two kinds of double well structures and monitor the tunnelling of the middle atoms. For $|\ldots 002\ldots \rangle$ and $|\ldots 200\ldots \rangle$ (right), we split the doublons into two sites and mark them by the hyperfine levels. Their state populations correlate to the oscillation amplitudes of tunnelling dynamics. b, The state populations of the gauge-invariant states are plotted in the upper graph, where the initial and final phases of the QLM are sketched in the shaded diagrams. From these probabilities, we extract the gauge violation ${\epsilon }(t)=1-({p}_{|\ldots 010\ldots \rangle }+{p}_{|\ldots 002\ldots \rangle }+{p}_{|\ldots 200\ldots \rangle })$ as shown in the bottom graph. While the inversion between the Fock states after the phase transition is stronger in the ideal QLM (exact numerics, orange and blue curves), fulfilment of Gauss’s law and a high level of gauge invariance is retained throughout. The experimental results are in quantitative agreement with t-DMRG calculations for our isolated Bose–Hubbard system (red curve). Error bars and shading, s.d.

Source data.

From these measurements, we can obtain the degree of gauge violation ϵ(t), defined as the spatial average of $1-\langle \psi (t)|{P}_{{\ell }}|\psi (t)\rangle$, where ${P}_{{\ell }}$ projects the system state |ψ(t)⟩ onto the local gauge-invariant subspace. As shown in Fig. 4b, throughout our entire experiment the summed probabilities of gauge-invariant states remains close to 1. Thus, our many-body quantum simulator retains gauge invariance to an excellent degree, even during and after a sweep through a quantum phase transition.

In conclusion, we have developed a fully tunable many-body quantum simulator for a U(1) gauge theory and demonstrated that it faithfully implements gauge invariance, the essential property of lattice gauge theories. Future extensions may give access to other symmetry groups and gauge theories in higher dimensions. The main challenge for the latter is to combine the model with a plaquette term that has been demonstrated previously in the present apparatus¹³. Importantly, our results enable the controlled analysis of gauge theories far from equilibrium, which is notoriously difficult for classical computers^3,4,5,6. A plethora of target phenomena offers itself for investigation, including false vacuum decay^30,31, dynamical transitions related to the topological θ-angle^32,33,34, and thermal signatures of gauge theories under extreme conditions³⁵.

Methods

Target model

Our experiment is motivated by the lattice Schwinger model of QED in one spatial dimension in a Kogut–Susskind Hamiltonian formulation³⁶,

$$\begin{array}{cc}{\hat{H}}_{{\rm{Q}}{\rm{E}}{\rm{D}}} & =\frac{a}{2}\sum _{{\ell }}(\hat{E}{}_{{\ell },{\ell }+1}^{2}+m{(-1)}^{{\ell }}\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{\psi }}_{{\ell }})\\ & -\frac{{\rm{i}}}{2a}\sum _{{\ell }}(\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{U}}_{{\ell },{\ell }+1}{\hat{\psi }}_{{\ell }+1}-{\rm{h}}.{\rm{c}}.),\end{array}$$

(4)

with lattice spacing a, gauge coupling e, and where we have set ħ and c to unity for notational brevity. Gauge links and electric fields fulfil the commutation relations $[{\hat{E}}_{{\ell },{\ell }+1},{\hat{U}}_{m,m+1}]=e{\delta }_{{\ell },m}{\hat{U}}_{{\ell },{\ell }+1}$, while fermion field operators obey canonical anti-commutation relations $\{\hat{\psi }{}_{{\ell }}^{\dagger },{\hat{\psi }}_{m}\}={\delta }_{{\ell }m}$. Here, we use ‘staggered fermions’²², which are an elegant way to represent oppositely charged particles and antiparticles, using a single set of spin-less fermionic operators, but at the expense of alternating signs on even and odd sites.

Gauge transformations are expressed in terms of the local Gauss’s law operators

$${\hat{G}}_{{\ell }}={\hat{E}}_{{\ell },{\ell }+1}-{\hat{E}}_{{\ell }-1,{\ell }}-e\frac{\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{\psi }}_{{\ell }}+{(-1)}^{{\ell }}}{2}.$$

(5)

These generate local U(1) transformations parametrized by real numbers ${\alpha }_{{\ell }}$, under which an operator $\hat{{\mathcal{O}}}$ transforms as $\hat{{\mathcal{O}}}{}^{{\prime} }=\hat{V}{}^{\dagger }\hat{{\mathcal{O}}}\hat{V}$, with $\hat{V}=\exp [{\rm{i}}{\sum }_{{\ell }}{\alpha }_{{\ell }}{\hat{G}}_{{\ell }}]$. Explicitly, the matter and gauge fields transform according to $\hat{\psi }{}_{{\ell }}^{{\prime} }=\exp (-{\rm{i}}e{\alpha }_{{\ell }}){\hat{\psi }}_{{\ell }}$, $\hat{U}{}_{{\ell },{\ell }+1}^{{\prime} }=\exp (-{\rm{i}}e{\alpha }_{{\ell }}){\hat{U}}_{{\ell },{\ell }+1}\exp ({\rm{i}}e{\alpha }_{{\ell }+1})$ and ${\hat{E}}_{{\ell },{\ell }+1}^{{\prime} }={\hat{E}}_{{\ell },{\ell }+1}$. In the absence of external charges, a physical state |ψ(t)⟩ is required to be invariant under a gauge transformation, that is, $\hat{V}|\psi (t)\rangle =|\psi (t)\rangle$. Thus, gauge invariance under Hamiltonian time evolution is equivalent to ${\hat{G}}_{{\ell }}|\psi (t)\rangle =0$, $\forall {\ell },t$, that is, $[{\hat{G}}_{{\ell }},{\hat{H}}_{{\rm{QED}}}]=0$ and the ${\hat{G}}_{{\ell }}$ are conserved charges. In our experiments, we achieve explicit probing of this local conservation law (see Fig. 4b and further below).

Using the QLM formalism²⁰, we represent U(1) gauge fields in our analogue quantum simulator by spin-1/2 operators ${\hat{U}}_{{\ell },{\ell }+1}\to \frac{2}{\sqrt{3}}\hat{S}{}_{{\ell },{\ell }+1}^{+}$ ($\frac{2}{\sqrt{3}}\hat{S}{}_{{\ell },{\ell }+1}^{-}$) for  odd (even) ${\ell }$, as well as ${\hat{E}}_{{\ell },{\ell }+1}\to e{(-1)}^{{\ell }+1}\hat{S}{}_{{\ell },{\ell }+1}^{z}$. In this spin-1/2 QLM representation, the electric field energy term proportional to ${\hat{E}}^{2}$ represents a constant energy offset, and hence drops out. Therefore, without loss of generality, we may set e → 1 in the following. With the rather untypical sign conventions in the above QLM definition, and an additional particle–hole transformation on every second matter site (${\hat{\psi }}_{{\ell }}\leftrightarrow \hat{\psi }{}_{{\ell }}^{\dagger }$, ${\ell }$ odd), the alternating signs of the staggered fermions are cancelled, yielding a simpler homogeneous model^30,31. The Hamiltonian takes the form of equation (1) and Gauss’s law is represented by equation (2).

For large negative values of the mass, m → −∞, the ground state is given by fully occupied fermion sites and an alternating electric field $\hat{E}$. However, for large positive masses, the absence of fermions is energetically favourable. In this configuration there are no charges, and hence the electric fields are aligned, in a superposition of all pointing to the left and all pointing to the right (see Fig. 1). In between these two extreme cases, the system hosts a second-order quantum phase transition, commonly termed Coleman’s phase transition^21,37. While the quantum link Hamiltonian, equation (1), is invariant under a transformation³⁷ of parity (P) and charge conjugation (C) the ground state does not always respect these symmetries: The vacuum state for m → −∞ is C- and P-invariant, but the respective vacua in the m → ∞ phase are C- and P-broken. An order parameter for the transition is given by the staggered change of the electric fields with respect to the initial configuration, ${\sum }_{{\ell }}\langle {\hat{E}}_{{\ell },{\ell }+1}(t)-{\hat{E}}_{{\ell },{\ell }+1}(0)\rangle /(2N)$.

Mapping to Bose–Hubbard simulator

Starting from the Hamiltonian of equation (1), we employ a Jordan–Wigner transformation with alternating minus signs,

$$\hat{\psi }{}_{{\ell }}^{\dagger }={(-1)}^{{\ell }}\exp [{\rm{i}}{\rm{\pi }}\mathop{\sum }\limits_{{{\ell }}^{{\prime} }=0}^{{\ell }-1}({(-1)}^{{{\ell }}^{{\prime} }}{\hat{\sigma }}_{{{\ell }}^{{\prime} }}^{z}+1)/2]{\hat{\sigma }}_{{\ell }}^{+},$$

(6a)

$${\hat{\psi }}_{{\ell }}={(-1)}^{{\ell }}\exp [-{\rm{i}}{\rm{\pi }}\mathop{\sum }\limits_{{{\ell }}^{{\prime} }=0}^{{\ell }\,-1}({(-1)}^{{{\ell }}^{{\prime} }}{\hat{\sigma }}_{{{\ell }}^{{\prime} }}^{z}+1)/2]{\hat{\sigma }}_{{\ell }}^{-}$$

(6b)

$$\hat{\psi }{}_{{\ell }}^{\dagger }{\hat{\psi }}_{{\ell }}=\frac{\hat{\sigma }{}_{{\ell }}^{z}+1}{2},$$

(6c)

replacing the fermionic operators ${\hat{\psi }}_{{\ell }}/\hat{\psi }{}_{{\ell }}^{\dagger }$ by local spin-1/2 operators ${\hat{\sigma }}_{{\ell }}^{\pm }$ and non-local strings involving ${\hat{\sigma }}_{{{\ell }}^{{\prime} } < {\ell }}^{z}$. We further identify the eigenstates of ${\hat{\sigma }}_{{\ell }}^{z}$ with two bosonic harmonic oscillator eigenstates ${|0\rangle }_{{\ell }}$ and ${|1\rangle }_{{\ell }}$. Projecting to the subspace ${ {\mathcal H} }_{{\ell }}={\rm{span}}\{{|0\rangle }_{{\ell }},{|1\rangle }_{{\ell }}\}$, we then realize the spin operators in terms of bosonic creation/annihilation operators ${\hat{a}}^{\dagger }/\hat{a}$ as follows:

$${\hat{\sigma }}_{{\ell }}^{-}={{\mathcal{P}}}_{{\ell }}{\hat{a}}_{{\ell }}{{\mathcal{P}}}_{{\ell }},$$

(7a)

$${\hat{\sigma }}_{{\ell }}^{+}={{\mathcal{P}}}_{{\ell }}{\hat{a}}_{{\ell }}^{\dagger }{{\mathcal{P}}}_{{\ell }},$$

(7b)

$${\hat{\sigma }}_{{\ell }}^{z}={{\mathcal{P}}}_{{\ell }}(2{\hat{a}}_{{\ell }}^{\dagger }{\hat{a}}_{{\ell }}-1){{\mathcal{P}}}_{{\ell }},$$

(7c)

where ${{\mathcal{P}}}_{{\ell }}$ is the projector onto ${ {\mathcal H} }_{{\ell }}$. The bosonic commutation relations $[{\hat{a}}_{{\ell }},{\hat{a}}_{{\ell }}^{\dagger }]=1$ when restricted to ${ {\mathcal H} }_{{\ell }}$ imply the required algebra of the Pauli matrices as given by $[{\hat{\sigma }}_{{\ell }}^{z},{\hat{\sigma }}_{{\ell }}^{\pm }]=\pm 2{\hat{\sigma }}_{{\ell }}^{\pm }$ and $[{\hat{\sigma }}_{{\ell }}^{+},{\hat{\sigma }}_{{\ell }}^{-}]={\hat{\sigma }}_{{\ell }}^{z}$. Similarly, we identify the two eigenstates of the ‘gauge’ spins $\hat{S}{}_{{\ell },{\ell }+1}^{z}$ with two eigenstates ${|0\rangle }_{{\ell },{\ell }+1}$ and ${|2\rangle }_{{\ell },{\ell }+1}$ associated with further bosonic operators, ${\hat{d}}_{{\ell },{\ell }+1}$ and ${\hat{d}}_{{\ell },{\ell }+1}^{\dagger }$ located at the links. Projecting to the subspace ${ {\mathcal H} }_{{\ell },{\ell }+1}={\rm{span}}\{{|0\rangle }_{{\ell },{\ell }+1},{|2\rangle }_{{\ell },{\ell }+1}\}$, we have

$$\hat{S}{}_{{\ell },{\ell }+1}^{-}=\frac{1}{\sqrt{2}}{{\mathcal{P}}}_{{\ell },{\ell }+1}{({\hat{d}}_{{\ell },{\ell }+1})}^{2}{{\mathcal{P}}}_{{\ell },{\ell }+1},$$

(8a)

$$\hat{S}{}_{{\ell },{\ell }+1}^{+}=\frac{1}{\sqrt{2}}{{\mathcal{P}}}_{{\ell },{\ell }+1}{(\hat{d}{}_{{\ell },{\ell }+1}^{\dagger })}^{2}{{\mathcal{P}}}_{{\ell },{\ell }+1},$$

(8b)

$$\hat{S}{}_{{\ell },{\ell }+1}^{z}=\frac{1}{2}{{\mathcal{P}}}_{{\ell },{\ell }+1}(\hat{d}{}_{{\ell },{\ell }+1}^{\dagger }{\hat{d}}_{{\ell },{\ell }+1}-1){{\mathcal{P}}}_{{\ell },{\ell }+1},$$

(8c)

fulfilling the desired angular momentum algebra. With these replacements, the Hamiltonian, equation (1), becomes

$${\hat{H}}_{{\rm{Q}}{\rm{L}}{\rm{M}}}={\mathcal{P}}\sum _{{\ell }}\left\{m{\hat{a}}_{{\ell }}^{\dagger }{\hat{a}}_{{\ell }}+\frac{\mathop{t}\limits^{ \sim }}{2\sqrt{2}}[{\hat{a}}_{{\ell }}{(\hat{d}{}_{{\ell },{\ell }+1}^{\dagger })}^{2}{\hat{a}}_{{\ell }+1}+{\rm{h.}}{\rm{c.}}]\right\}{\mathcal{P}},$$

(9)

where ${\mathcal{P}}=\prod _{{\ell }}{{\mathcal{P}}}_{{\ell }}{{\mathcal{P}}}_{{\ell },{\ell }+1}$. In the main text, the projection ${\mathcal{P}}$ is implied in the notation $\hat{A}\simeq \hat{B}$, which abbreviates the equality $\hat{A}={\mathcal{P}}\hat{B}{\mathcal{P}}$ for two operators $\hat{A}$ and $\hat{B}$. We emphasize that even though equation (9) is written in terms of bosonic operators, the projectors together with the Jordan–Wigner transform ensure—at the level of the Hamiltonian and diagonal observables—the equivalence with the original lattice gauge theory including fermionic matter.

The Hamiltonian in equation (9) is generated effectively in our Bose–Hubbard system through a suitable tuning of the parameters described in equation (3). As a preceding step, matter sites are identified with even sites of the optical superlattice (${\hat{a}}_{{\ell }}\to {\hat{b}}_{j=2{\ell }},{\ell }=0,\ldots ,N-1$) and gauge links with odd sites of the superlattice (${\hat{d}}_{{\ell },{\ell }+1}\to {\hat{b}}_{j=2{\ell }+1},{\ell }=0,\ldots ,N-2$). For our quantum simulator, we have N = 36 matter sites and 35 gauge links, yielding a total of 71 bosonic sites. The principle for generating the gauge-invariant dynamics is then conveniently illustrated in a three-site building block consisting of optical-lattice sites j = 0, 1, 2, as shown in Extended Data Fig. 1. The system is initialized in a state where all matter sites are singly occupied, while gauge links are empty, that is, the system starts in the boson occupation state |101⟩. By choosing U, δ ≫ J and U ≈ 2δ, the two states |101⟩ and |020⟩ form an almost degenerate energy manifold α (not the absolute ground-state manifold of the BHM). This manifold is well-separated from the states |110⟩ and |011⟩, shown in the middle of Extended Data Fig. 1. Hence, direct tunnelling of the bosons into (and out of) the deep gauge-link well is energetically off-resonant and suppressed. The effective dynamics between the states within the manifold α is then described by degenerate perturbation theory³⁸, leading to the Hamiltonian of equation (9) acting on the subspace indicated by the projectors ${\mathcal{P}}$. An explicit calculation yields the effective coupling

$$\mathop{t}\limits^{ \sim }=\sqrt{2}{J}^{2}\left(\frac{1}{\delta +\varDelta }+\frac{1}{U-\delta +\varDelta }+\frac{1}{\delta -\varDelta }+\frac{1}{U-\delta -\varDelta }\right),$$

(10)

which reduces to a simple relation close to resonance, $\tilde{t}\mathop{\to }\limits^{U\approx 2\delta }8\sqrt{2}{J}^{2}/U$. A key ingredient for this manner of generating the term proportional to $\mathop{t}\limits^{ \sim }$ was the particle–hole transformation^30,31. It enabled us to rewrite this term, which is usually interpreted as a kinetic hopping term, as the simultaneous motion of two bosons on neighbouring matter sites into the gauge link in between (and back). Note that our approach of constraining to an energy manifold α from the total Hilbert space is different from previous work, where the authors proposed to implement gauge symmetry in the ground state manifold by adding a term proportional to ${\sum }_{x}{G}_{x}^{2}$ to the Hamiltonian^19,30,39,40.

The above result, where we include couplings of our initial-state manifold to other manifolds at order J, is valid for both the building block as well as for the extended system close to resonance. In our many-body system, at this order in perturbation theory, the mass is represented by the energy imbalance of on-site interaction and staggering, m = δ − U/2, such that the gauge-invariant particle creation/annihilation becomes resonant once fermions are massless. In the chosen parameter configuration, occupations other than the desired |0⟩, |1⟩ (even) and |0⟩, |2⟩ (odd sites) are highly suppressed, as we confirmed through numerics and direct measurement (see Fig. 2b). We also include a linear tilt potential to suppress the tunnelling of atoms to their next-nearest neighbouring sites, for example, $|02001\rangle \leftrightarrow |02100\rangle$ (here j = 0, …, 4), as such processes are also generated at second order and would break gauge invariance.

State preparation and detection

The experiment begins with a quasi two-dimensional quantum gas of ~8.6 × 10⁴ atoms prepared by adiabatically loading a nearly pure Bose–Einstein condensate into a single well of a pancake-shaped standing wave. The pancake trap is generated by interfering two blue-detuned laser beams at wavelength λ_s = 767 nm, which provides the confinement along the z axis. We implement a staggered-immersion cooling for the quantum gas to create a Mott insulator with near-unity filling²⁴. The cooling is performed within an optical superlattice where the atoms are separated into superfluid and Mott-insulator phases with a staggered structure. The superlattice potential can be written as

$$V(x)={V}_{{\rm{s}}}{\cos }^{2}(kx)-{V}_{{\rm{l}}}{\cos }^{2}(kx/2+\phi ).$$

(11)

Here, V_s and V_l are the depths of the short and long lattices, respectively. The relative phase φ determines the superlattice configuration, which is controlled by changing the relative frequency of these lasers. At φ = 0, the atoms on odd and even sites of the double wells experience the same trap potential. In the cooling stage, we keep the phase at φ = 7.5(7) mrad to generate a staggered energy difference for the odd and even sites. After cooling, the final temperature (T_f) of the Mott-insulator sample with $\bar{n}=2$ is k_BT_f = 0.046(10)U. Then, we freeze the atomic motions and remove the high-entropy atoms. Based on the low-entropy sample, a Mott insulator with 99.2(±0.1)% of single occupancy is prepared by separating the atom pairs within the double-well superlattice. Figure 2a shows such a two-dimensional sample with a homogeneous regime containing 10⁴ lattice sites.

The technique of site-selective addressing is widely used in our experiment^24,41. For the Mott insulator, all the atoms are prepared in the hyperfine level of |↓⟩ = |F = 1, m_F = −1⟩. We define another pseudo-spin state as |↑⟩ = |F = 2, m_F = −2⟩. When the direction of the bias field is along x and the phase of the electro-optical modulator is set to π/3, the energy splitting between |↓⟩ and |↑⟩ has a 28-kHz difference for the odd and even sites. We edit the microwave pulse and perform a rapid adiabatic passage to selectively flip the hyperfine states of atoms on odd or even sites, achieving an efficiency of 99.5(±0.3)%. For state initialization, we flip the atomic levels of odd sites and then remove these atoms with a resonant laser pulse. This site-selective addressing is also employed for state readout, as shown in Fig. 1b. Combining such techniques with absorption imaging, we record the atomic densities of odd and even sites successively in a single experimental sequence.

We use a parity projection of the atom number to probe the distribution of site occupancies. The basic idea is to remove the atom pairs by exciting them to a non-stable molecular state via the photoassociation (PA) process. The laser frequency is 13.6 cm⁻¹ red-detuned to the D2 line of the ⁸⁷Rb atom, which drives a transition to the v = 17 vibrational state in the ${0}_{g}^{-}$ channel. The decay rate of the atom pairs is 5.6(2) kHz in the laser intensity of 0.67 W cm⁻². After applying this PA light for 20 ms, the recorded atom loss equals the ratio of atom pairs. For detecting the filling number of more than double occupancy, we first engineer the atoms in the double wells and then detect the number parity with PA collision. As shown in Fig. 2a, we remove the atoms occupying the even sites and then separate the atoms of odd sites into double wells. If the occupancy is more than two, we can observe some atom loss after applying the PA light. The remaining atom number after this operation is ${\bar{n}}_{t}^{({\rm{g}})}={\bar{n}}_{c}+2{p}^{({\rm{g}})}(2)$. From these measurements, we obtain the upper bound of the probabilities for these highly excited states, such as three or four atoms. Here, we consider the excitations up to three atoms with the probability ${p}^{({\rm{m}}/{\rm{g}})}(3)$. Hence, the probabilities of matter or gauge sites derived from these detections are:

$$\begin{array}{c}{p}^{({\rm{m}}/{\rm{g}})}(0)=1-\frac{1}{2}\left[\phantom{\frac{1}{2}}\,{\bar{n}}_{c}^{({\rm{m}}/{\rm{g}})}+{\bar{n}}_{t}^{({\rm{m}}/{\rm{g}})}\right],\\ {p}^{({\rm{m}}/{\rm{g}})}(1)={\bar{n}}_{c}^{({\rm{m}}/{\rm{g}})}-\frac{1}{2}\left[\phantom{\frac{1}{2}}\,{\bar{n}}^{({\rm{m}}/{\rm{g}})}-{\bar{n}}_{t}^{({\rm{m}}/{\rm{g}})}\right],\\ {p}^{({\rm{m}}/{\rm{g}})}(2)=\frac{1}{2}\left[\phantom{\frac{1}{2}}\,{\bar{n}}_{t}^{({\rm{m}}/{\rm{g}})}-{\bar{n}}_{c}^{({\rm{m}}/{\rm{g}})}\right],\\ {p}^{({\rm{m}}/{\rm{g}})}(3)=\frac{1}{2}\left[\phantom{\frac{1}{2}}\,{\bar{n}}^{({\rm{m}}/{\rm{g}})}-{\bar{n}}_{t}^{({\rm{m}}/{\rm{g}})}\right].\end{array}$$

(12)

These probabilities refer to the observables given by the detection methods (i)–(iii) in the main text.

Imaging individual sites

We develop a technique to detect individual atoms at any specific site residing in the 1D optical lattice, without requiring a quantum gas microscope. By lifting the energy degeneracy of the transition frequency in each lattice site, we can flip the atomic state with a locally resonant microwave pulse. The potential used for shifting the energy levels is provided by a homogeneous magnetic gradient. We set the magnetic axis along the x direction with a 7.3 G bias field, and meanwhile apply a ~70 G cm⁻¹ gradient field along this axis. In such a magnetic field, the energy level for the |↓⟩ → |↑⟩ transition is split by 5.6 kHz per lattice site. Addressing individual sites is realized by flipping the atomic internal level from |↓⟩ to |↑⟩ with a square π pulse. Afterwards, the atom number on the corresponding site is recorded on a CCD camera with in situ absorption imaging.

This method enables the imaging of atoms with a spatial resolution better than the optical resolution of our imaging system. Instead, the resolution is determined by the energy splitting between lattice sites and the Fourier broadening of the microwave transition. To achieve such a high precision, we improve the stability of the magnetic field and of the position of the optical lattice. At an arbitrary microwave frequency, the position of the flipped stripe changes from shot-to-shot with a standard deviation of 0.11 μm. We set the Rabi frequency for the transition to 1.9 kHz, to make sure the Fourier broadening is smaller than the splitting between the neighbouring sites. The number occupations on lattice sites are measured by scanning the microwave frequency. The frequency starts at 6.819104 GHz and ends at 6.819532 GHz, covering 75 sites of the optical lattice.

To benchmark our method, we perform this site-resolved imaging in a staggered state, as shown in Extended Data Fig. 2. Two essential features are captured by our measurement. First, the detected atomic density oscillates with the same period as the site occupancy of the staggered state. Second, the central position of the flipped atoms follows the staircase behaviour of the discrete lattice sites. We can clearly locate each individual lattice site in the 1D chain with this site-resolved imaging technique.

Building blocks and state reversibility

Before performing the phase transition, the elementary parameters of the Hubbard model are calibrated precisely. The lattice depths are measured by applying a parametric excitation to the ground-band atoms. Then, we derive the Wannier functions of the atoms at certain lattice depths, from which the on-site interaction U and tunnelling J are obtained by integrating the overlap of Wannier functions. The linear potential Δ = 57 Hz per site is formed by the projection of gravity along the x axis. The staggered offset δ is generated by the long lattice at a superlattice phase of φ = π/2.

We investigate the building block of our model and observe the coherent dynamics. To prepare a sample with isolated units, we quench the short lattice to 11.8(1) E_r for 3 ms starting from the staggered initial state. During such a short time, some of the atoms start to bind into doublons and enter the odd sites. We remove the majority atoms residing on the even sites, thereby creating a dilute sample with isolated building blocks. Afterwards, the superlattice is reshaped into the configuration of φ = π/2 and the dynamics of the atoms are monitored at the resonant condition with U = 1.17(1) kHz, J = 105(1) Hz. The atoms oscillate between the state |020⟩ and |101⟩ via second-order hopping, forming an effective two-level system. In this building block, the self-energy correction shifts the resonant point to U = 2δ − 4J²/U. Extended Data Fig. 3a shows a Rabi oscillation with negligible decay in this dilute sample, indicating excellent coherence of the system. The amplitude of the oscillation is determined by the preparation fidelity of the building block.

Another characteristic feature of a coherent adiabatic transition is its reversibility. Figure 2 shows a quantum phase transition from the charge-dominated phase to the C/P-broken phase. In our sample with 71 sites, we compensate part of the residual potential of the blue-detuned lattice with a red-detuned dipole trap, which reduces the spatial inhomogeneity. The coherence of the system allows us to recover the particle–antiparticle phase in another 120 ms. We ramp the mass m and tunnelling J in a reversed way as compared to the curves given in Fig. 2a, thereby decreasing $m/\tilde{t}$ from 11.6 to −39.8 in order to return to the charge-dominated phase. The occupancy of even sites is recovered to 0.66(3) in Extended Data Fig. 4, which is attributed mainly to the non-adiabaticity of the ramping process.

Numerical calculations

The dynamics of our 71-site quantum simulator can hardly be computed up to the times we are interested in by classical numerical methods. However, we can calculate results at smaller system sizes and then check for convergence. To understand the quantum phase transition, we use exact diagonalization (ED) to calculate the QLM, and the time-adaptive density matrix renormalization group method (t-DMRG) to simulate the dynamics governed by the BHM.

To compute the dynamics in the ideal QLM, we use the mass m and coupling strength $\tilde{t}$ as deduced from the Hubbard parameters. The time-dependent dynamics in the QLM fully obey Gauss’s law. Using this conservation law to restrict our calculations to the implemented Hilbert space, we perform numerically exact diagonalizations for system sizes ranging from L = 8 up to L = 52 sites (see Extended Data Fig. 5a). Owing to finite-size effects, the dynamics for smaller systems (such as L = 8) show strong oscillations after crossing the critical point. We find that the non-adiabaticity caused by the ramping reduces the fidelity of our final state with increasing system size, owing to the closure of the minimal gap at the critical point. The discrepancy between the curves for L = 40 and L = 52 is of the order of 10⁻³, indicating the volume convergence of our calculations. In Fig. 4b, the orange and blue curves for state populations are the ED results for system size of L = 52.

We apply t-DMRG^42,43 to calculate the full dynamics of the 1D Bose-Hubbard chain. For our simulations, convergence is achieved at a time step of 10⁻⁴ s, a truncation threshold of 10⁻⁶ per time step, and a maximum occupation of 2 bosons per site. Finite-size effects are also investigated for several chain lengths. Similar to the behaviour in the QLM, the dynamics becomes smooth with increasing chain length. Extended Data Fig. 5b shows volume convergence between the results for system sizes L = 32 and L = 40 sites. The theoretical predictions in Fig. 2c and Fig. 4b are obtained with system size L = 32. Moreover, some imperfections of our system are taken into account in our t-DMRG calculations. Owing to the inhomogeneity of the Gaussian-shaped y-lattice, the on-site interaction at the edge of the 71-site chain is about 10 Hz smaller than at the central site. Also, fluctuations of the depth of the long lattice lead to about ±4.5 Hz uncertainty in the staggered energy δ. Including these influences into our model, we estimate experimental observables with ±1σ confidence intervals (equal to the standard deviations).

These two numerical methods show consistent behaviour at the converged system sizes, which means the discrepancies between our experiments and numerical calculations are not caused by finite-size effects. We attribute the remaining deviations to heating due to off-resonant excitations of the atoms by the optical lattice beams. Although the correlation length is ${4.4}_{-1.0}^{+2.0}$ sites and the domain size is 9(2) sites, both the ED and t-DMRG calculations converge only once the system size is above about 40 sites, showing the essential role of many-body effects in the observed phenomena.

Density–density correlations

Constrained by the finite resolution of our microscope, we are not able to extract density–density correlations from the in situ images. However, we can measure the correlation function by mapping the atomic distributions into momentum space. After a free expansion, the relation between the initial momentum k_x and real-space position x is k_x = m_Rbx/t. One characteristic momentum corresponding to the unity-filling Mott insulator is k_x = 2ħk, which is related to the real-space position of x₀ = ht/(m_Rbλ/2). Then, the correlation function for a long chain with N_ddc sites is

$${C}_{k}(x)=1+\frac{1}{{N}_{{\rm{d}}{\rm{d}}{\rm{c}}}^{2}}\sum _{i,j}\exp [\,-{\rm{i}}2{\rm{\pi }}x(i-j)/{x}_{0}]{n}_{i}{n}_{j}.$$

(13)

Here, the position x can be discretized into the pixels of the imaging plane. From this relation, we can easily find that the interference patterns emerge at a multiple of x₀/d, where d is the periodicity of an ordered site occupation. Hence, the initial state with d = 2 has first-order peaks at k_x = ħk, and the states with d = 4 would have peaks at k_x = 0.5ħk.

To detect the density–density correlations, we release the cloud and let it expand in the x–z plane for 8 ms. The lattice depth along the y axis is 25.6(2)E_r, which blocks the crosstalk of different 1D chains. Loosening the confinement along the z axis strongly reduces the interaction between atoms, but also degrades the optical resolution. The characteristic length is x₀ = 105 μm, which allows us to observe the new ordering with the microscope. We find that the initial size of the sample is much smaller than that of the cloud after expansion. The exposure time for the absorption imaging is 10 μs, thereby making photon shot noise the major source of fluctuations on the signal.

The pattern in Fig. 3a is obtained by calculating the correlation function as defined in equation (13). For each image, the density correlation is the autocorrelation function. When we have a set of images, we perform this procedure using two different routes²⁸. One is calculating the autocorrelation function for each image and then averaging them. Another is first averaging the images and then calculating the autocorrelation function once, which is used for normalizing the signal. Then we obtain the normalized density–density correlation. Such a method enables the extraction of correlations from noisy signals and is also robust to the cloud shape. The patterns in Fig. 3a are averaged over 523 and 1,729 images, respectively. In the horizontal direction of the imaging plane, some stripes appear around y = 0, which is caused by the fluctuations of the atomic centre and total atom number²⁸. Unlike the in situ images, the atoms outside the region of interest still contribute to signals in momentum space.

The correlation length is obtained from the width of the interference peak. For an entirely ordered state, such as the initial state, the amplitude of the density correlation is inversely proportional to the atom number, and the width is determined by the imaging resolution. However, spontaneous symmetry breaking in the phase transition induces the formation of domains. At finite correlation length ξ, the peak width becomes broader. Assuming the correlation function decays exponentially in this 1D system, we can deduce ξ from the peak width. To extract the peak width, we first subtract the background profile from the correlation function. The background is a single-pixel section through the pattern centre, whose direction is along −4° with respect to the horizontal plane. As shown in Extended Data Fig. 6, we apply a Lorentzian fitting to the curve and find the width is 4.5 ± 1.1 μm. The peaks at ±ħk and ±2ħk have widths of 2.0(2) μm and 1.9(4) μm respectively, which corresponds to the imaging resolution. Considering the broadening due to optical resolution, we obtain the correlation length as $\xi ={4.4}_{-1.0}^{+2.0}$ sites.

Potential violations of Gauss’s law

In our setup, potential gauge-violation terms arising from coherent processes are suppressed owing to suitably engineered energy penalties. We can estimate the effect of these error terms in a three-site building block consisting of two—initially occupied—matter sites and the gauge link in between, described by the initial state $|\ldots {n}_{j}{n}_{j+1}{n}_{j+2}\ldots \rangle =|\ldots 101\ldots \rangle$, j even. The main cause of gauge violation stems from the desired matter–gauge-field coupling, which requires a second-order process of strength ∝J²/δ involving the gauge-violating bare tunnelling J. Similarly to a detuned Rabi oscillation, the population of the gauge-violating states $|\ldots 110\ldots \rangle$ and $|\ldots 011\ldots \rangle$ is of the order of (J/δ)². At the highest coupling strength, which we reach at t = 60 ms, we have J/δ = 0.13, that is, gauge-violating states have at most a few per cent of population. Rather than an incoherent dynamics that leads to accumulation of gauge violation over time, this bare tunnelling is a coherent process that strongly mitigates the increasing of the induced gauge violation. In Fig. 4b, the oscillations in the t-DMRG calculations are caused by such a detuned tunnelling process. We further theoretically calculate the gauge violation of our system at long evolution time, as shown in Extended Data Fig. 7b. The gauge violation does not increase substantially even when the ramping time is about one order of magnitude longer than our experimental timescale.

At the same order of perturbation theory, direct tunnelling between matter sites can occur, with a coupling strength of the order of J²/δ. This second-order tunnelling is energetically suppressed by U ± 2Δ when we consider the initial filling and the linear potential. The coherent oscillation in Fig. 3a indicates that the atoms perform only the desired conversion between matter and gauge-field sites, and otherwise reside in their respective building blocks. Likewise, the staggered and linear potential suppress any long-range transport, which is also confirmed by the nearly constant size of the atomic cloud measured in absorption imaging.

If all gauge-violating many-body states experience such energy penalties, a deformed symmetry emerges that is perturbatively close to the ideal original one⁴⁴, and which indefinitely suppresses gauge invariance-violating processes¹⁹. In the present case, the leading gauge-violating processes are energetically penalized, but violations at distant sites may in principle energetically compensate each other. This may lead to a slow leakage out of the gauge-invariant subspace through higher-order processes. On the experimentally relevant timescales, these processes are, however, irrelevant.

Our theoretical calculations, which are based on unitary time evolution, capture only coherent sources of gauge violations such as those mentioned above. As the agreement with our measured data suggests (see Fig. 4b), coherent processes contribute substantially to the weak gauge violation ϵ(t), especially the first-order tunnelling J. In addition, there may appear dissipative processes that violate Gauss’s law. Pure dephasing processes that couple to the atom density commute with Gauss’s law and thus do not lead to gauge violations. In contrast, atom loss might affect gauge invariance. The lifetime characterizing the atom loss in optical lattices is about 10 s, two orders magnitude longer than the duration of our sweep through the phase transition. Finally, the finite lifetime of Wannier–Stark states⁴⁵ caused by the lattice tilt is also much longer than experimentally relevant times. Our direct measurements of the violation of local gauge invariance corroborate the weakness of the various potential error sources over our experimental timescales.

Measurement of Gauss’s law violations

We can verify the local fulfilment of Gauss’s law—without the need for full state tomography—by measuring the probabilities of the gauge-invariant states. Considering the relevant three-site units, we can couple the central site with its left or right site by isolating the atoms into double wells. Then the sensing is achieved by the atom which can discriminate the filling number of its neighbour via subsequent dynamics. For the state |10⟩ in a double well, the atom can tunnel and evolve to another state |01⟩. The frequency is dramatically different from the state |20⟩, which would tunnel in a pair with a strength of 4J²/U. Since we mark the atoms with hyperfine levels, atoms in the |↓↑⟩ would exchange their hyperfine state in a superexchange process. As shown in Extended Data Fig. 8, we observe the dynamics of these states by initially preparing them in double wells. The superexchange frequency is 4J²/hU = 11 Hz. However, the atom pairs cannot tunnel freely because this dynamics requires a further stabilization of the superlattice phase φ. Even though the superexchange interaction can drive the evolution, such a process does not contribute to the oscillation amplitude at the frequency 2J/h. In addition, the state |0n_j⟩ does not contribute to the desired signal because we flip the hyperfine levels of the atoms on odd sites before implementing the tunnelling sequence. We fit the oscillation with a function y = y₀ + Ae^−t/τsin(2πf + ϕ₀). The frequency f, initial phase ϕ₀, offset value y₀, and damping rate τ are fixed in the fitting. The signal is identified not only by the atom population but also by the characteristic frequency. Therefore, we can establish relations between the oscillation amplitude and the state probability.

Extended Data Figure 9 shows the measurements for determining the population of gauge-invariant states. As illustrated in Fig. 4a, we monitor the oscillation of tunnelling at four different sequences. After an evolution time t, the state detections begin by ramping the short lattice to 51.3(4)E_r. Then we tune the superlattice phase φ from π/2 to 0 or π and consequently divide the atoms into isolated double wells. In the procedure for detecting the state |010⟩, we address and flip the hyperfine level of atoms residing the odd sites to |↑⟩, and thereby mark the sites by their hyperfine levels. Afterwards, we quench the depth of the short- and long-lattice to 18.7(1)E_r and 10.0(1)E_r simultaneously. The atoms tunnel from even to odd sites within each double well, whose expectation value is recorded by performing absorption imaging. As shown in Extended Data Fig. 9a, b, the oscillation amplitudes are almost equal to the ratios of even-site atoms.

For detecting the state $|\ldots 002\ldots \rangle$ and $|\ldots 200\ldots \rangle$, the procedure consists of more operations because the doublons cannot tunnel easily. Before the splitting of doublons, we remove the atoms residing on the even sites to ensure a 99.3(±0.1)% efficiency of atom splitting. For instance, the state |12⟩ in the double well would disturb the separation of doublons and also influence the following signal. Next, we perform the state flip operation and tune the superlattice phase from 0 (π) to π (0) to reach another configuration. In these double wells, the oscillations corresponding to atom tunnelling are shown in Extended Data Fig. 9c, d. However, we should exclude the probability of other kinds of states, such as $|\ldots 012\ldots \rangle$, because we remove the central particle and project it into $|\ldots 002\ldots \rangle$. To clarify the process by which we derive the final probability, the states that may contribute to the signals are listed in a square array in Extended Data Fig. 10. Using seven experimental observables, we can extrapolate the population of the state |002⟩ and |200⟩. Actually, the other high-energy excitations, such as four particles per site, are also eliminated from this calculation. After performing the error propagation, the errors of the total probabilities mainly arise from the shot noise of the absorption imaging. In Fig. 4b, the probabilities of the state |010⟩ at t = 0, 30 ms represent the gauge-invariant terms with smaller errors.

Through these measurements, we are thus able to measure the probabilities of the states $|\ldots {n}_{j-1}{n}_{j}{n}_{j+1}\ldots \rangle =|\ldots 010\ldots \rangle$, $|\ldots 002\ldots \rangle$, and $|\ldots 200\ldots \rangle$, j even, from which we can compute the local projectors onto the gauge-invariant states, ${P}_{{\ell }}=|010\rangle \langle 010|+|002\rangle \langle 002|+|200\rangle \langle 200|$, where ${\ell }=j/2$ denotes the central matter site. These measurements enable us to certify the adherence to gauge invariance in our U(1) lattice-gauge quantum simulator.

Extended data figures and tables

Extended Data Fig. 1 Level structure of a three-site building block (matter–gauge–matter).

The energy manifold of interest is given by the state on the left, which represents a particle pair, and the one on the right, where particles have annihilated while changing the in-between gauge field configuration. In the middle, we show the detuned intermediate processes and states by which these ‘physical’ (that is, gauge-invariant) states are coupled. See main text for nomenclature used in this figure.

Extended Data Fig. 2 Single-site resolved imaging.

a, A staggered-filled 1D chain as our initial state is sketched, which begins with the first site (left part) and ends up with the 75th site (right part). The energy levels are split by a linear magnetic gradient field. Therefore, the internal states |↓⟩ (wavefunction in blue colour) and |↑⟩ (wavefunction in orange colour) of each site can be coupled by a local resonant microwave field. b, Spectroscopic measurement of the site occupation. At each frequency, we average the data over five repetitions, and integrate the signal along the y axis. For simplicity, we show only the beginning site and a few sites at the end of the chain. The blue circles are central positions of the atomic densities along x. According to the spatial position of the image, we plot the staircase structure of the lattice sites in cyan. The lower panel shows the atomic density averages over the measurements on the upper panel, where the amplitude is normalized to the maximum atomic density. The sinusoidal fitting (orange-dashed line) shows the positions of the sites and the staggered structure.

Extended Data Fig. 3 Dynamics in building blocks.

a, Observation of coherent evolution in the building blocks, as sketched above the data. We monitored the dynamics at two different coupling strengths. The solid curves are sinusoidal fitting results, which give oscillation frequencies of 89.3(3) Hz (upper plot) and 57.1(3) Hz (lower plot). Error bars, s.d. The small atom number in the dilute sample leads to larger statistical errors as compared to our many-body experiment reported in the main text. As these data show, the decay of oscillations is insignificant over a range of coupling strengths $\tilde{t}$, even for values larger than the greatest coupling strength used in the phase transition ($\mathop{t}\limits^{ \sim }=70\,{\rm{H}}{\rm{z}}$). b, The oscillation frequency and J²/U have an almost linear relation, which is in excellent quantitative agreement with the theoretical prediction based on the BHM (solid curve).

Extended Data Fig. 4 Quantum phase transition and revival.

Over 240 ms, we ramp the mass as follows: first, from negative to positive; second, from positive to negative, back to the symmetry-unbroken charge-proliferated phase. Error bars, s.d. The recovery of the atoms on even sites indicates the reversibility of this phase transition. The solid curve is a guide for the eye.

Extended Data Fig. 5 Numerical simulations of the phase transition dynamics.

These are calculated by ED (a) and DMRG (b) methods. We monitor the evolution of the deviation of the electric field, which corresponds to the double occupancy (‘doublons’) of the odd sites. a, Simulations of the ideal, fully gauge-invariant QLM, using ED calculations under periodic boundary conditions. b, Simulations of a 1D Bose–Hubbard system modelling our experiment, using the t-DMRG method. The insets show the differences between results for different system sizes and the curve at the largest size (L = 52 for ED and L = 40 for t-DMRG), demonstrating finite-size convergence well below the range of experimental errors.

Extended Data Fig. 6 Correlation length.

Density–density correlation of the final state is plotted against the distance in momentum space. We select the central region with two interfering peaks, where the background has been subtracted from the signal. The solid curve is a Lorentzian fitting of the data. The inset shows the relation between the peak width and the correlation length. The solid curve is calculated for a 1D system with N_ddc = 100 sites. The red point represents the correlation length of the final state as shown in Fig. 3a, where the error bars are s.d.

Extended Data Fig. 7 Ramping speed and gauge violation.

a, The phase transition is driven by ramping the mass m and the effective coupling $\tilde{t}$. We start from a large negative value of mass $m/\tilde{t}$, retain stronger coupling around the critical point, and end up with a large positive mass. b, Gauge violation against total ramping time calculated with the t-DMRG method in a system with 16 (red), 24 (blue) and 32 (orange) optical-lattice sites. Using the same shape of the ramping curve in a, we change the ramping speed by constraining the total ramping time. The squares points are the maxima of ϵ(t) throughout the dynamics, while the circles represent the gauge violations of the final states. Owing to the coherence of our many-body system, ϵ(t) reaches its maximum around the critical point, and decreases after crossing the critical point (see Fig. 4b).

Extended Data Fig. 8 Dynamics in double wells.

Under the same superlattice configuration, we measure the evolution of three different states in double wells. The initial states are |10⟩ (blue squares), |11⟩ (orange triangles) and |20⟩ (red circles). The state |10⟩ oscillates with almost the full amplitude. The superexchange interaction drives the spin exchange process as expected. In contrast, the atom population remains constant for the state |20⟩. Error bars, s.d. The solid curves are exponentially damped sinusoidal fittings, where the frequency, phase and decay rate are fixed. We find that the oscillation amplitude of tunnelling is almost three orders of magnitude larger than the other two fitting values.

Extended Data Fig. 9 Detecting the gauge-invariant states.

We divide the atoms into double wells and then measure atom tunnelling within each two-site unit. a–d, The dynamics of tunnelling for four different experimental sequences, as sketched in the insets. Five different moments during the phase transition (t = 0, 30, 60, 90, 120 ms) are selected for detecting the gauge-invariant states. Error bars, s.d. We fit the data with a sinusoidal damping function, which has a period of 7.2 ms and an exponential decay constant of 96 ms. The amplitudes of the oscillations in a–d refer to ${A}_{|10\rangle }^{(1)}$, ${A}_{|01\rangle }^{(1)}$, ${A}_{|10\rangle }^{(2)}$ and ${A}_{|01\rangle }^{(2)}$, respectively. These amplitudes are then used for calculating the state probabilities, where the error bars are s.d.

Extended Data Fig. 10 Resolving the population of the states.

For the detection of states |002⟩ and |200⟩, we extract their probabilities from several measurements. There are 64 states that may contribute to the oscillations, which are listed from |000⟩ to |333⟩ as an 8 × 8 square array (left). The amplitudes of these states according to our detection procedures are given by distinct colours (key at bottom right). For example, the state |002⟩ in the third column of the first row only contributes to the first observable ${A}_{|01\rangle }^{(2)}$+${A}_{|10\rangle }^{(2)}$ with a factor of 1, while the state |013⟩ at the end of the first row will be recorded by all these observables with the colour-denoted factors. We use seven terms to deduce the lower bound for the probabilities as ${p}_{|\ldots 002\ldots \rangle }+{p}_{|\ldots 200\ldots \rangle }\ge {A}_{|01\rangle }^{(2)}+{A}_{|10\rangle }^{(2)}+{A}_{|01\rangle }^{(1)}+{A}_{|10\rangle }^{(1)}-{\bar{n}}_{c}^{{\rm{o}}}-0.5{\bar{n}}^{{\rm{e}}}-1.5{\bar{n}}_{{\rm{c}}}^{{\rm{e}}}$. Such a relation can be captured from the chequerboard diagram.

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