## Gauge invariance with cold atoms

There is considerable interest in developing quantum computational technologies that can simulate a series of physical phenomena inaccessible by classical computers. Mil *et al.* propose a modular scheme for quantum simulation of a U(1) lattice gauge theory based on heteronuclear spin-changing collisions in a mixture of two bosonic quantum gases isolated in single wells of a one-dimensional optical lattice. They engineered the elementary building block for a single well and demonstrate its reliable operation that preserves the gauge invariance. The potential for scalability of the proposed scheme opens up opportunities to address challenges in quantum simulating the continuum limit of the gauge theories.

*Science*, this issue p. 1128

## Abstract

In the fundamental laws of physics, gauge fields mediate the interaction between charged particles. An example is the quantum theory of electrons interacting with the electromagnetic field, based on U(1) gauge symmetry. Solving such gauge theories is in general a hard problem for classical computational techniques. Although quantum computers suggest a way forward, large-scale digital quantum devices for complex simulations are difficult to build. We propose a scalable analog quantum simulator of a U(1) gauge theory in one spatial dimension. Using interspecies spin-changing collisions in an atomic mixture, we achieve gauge-invariant interactions between matter and gauge fields with spin- and species-independent trapping potentials. We experimentally realize the elementary building block as a key step toward a platform for quantum simulations of continuous gauge theories.

Gauge symmetries are a cornerstone of our fundamental description of quantum physics as encoded in the standard model of particle physics. The presence of a gauge symmetry implies a concerted dynamics of matter and gauge fields that is subject to local symmetry constraints at each point in space and time (*1*). To uncover the complex dynamical properties of such highly constrained quantum many-body systems, enormous computational resources are required. This difficulty is stimulating great efforts to quantum simulate these systems, i.e., to solve their dynamics using highly controlled experimental setups with synthetic quantum systems (*2*–*4*). First experimental breakthroughs have used quantum-computer algorithms that implement gauge invariance exactly, but which are either limited to one spatial dimension (*5*, *6*), restrict the dynamics of the gauge fields (*7*, *8*), or require classical preprocessing resources that scale exponentially with system size (*9*). Recently, the dynamics of a discrete *10*–*12*). Despite these advances, the faithful realization of large-scale quantum simulators describing the continuum behavior of gauge theories remains highly challenging.

Our aim is the development of a scalable and highly tunable platform for a continuous U(1) gauge theory, such as realized in quantum electrodynamics. In the past years, ultracold atoms have become a well-established system for mimicking condensed-matter models with static electric and magnetic fields (*13*) and even dynamical background fields for moving particles (*14*–*16*). These systems possess global U(1) symmetries related to the conservation of total magnetization and atom number (*17*). However, a gauge theory is based on a local symmetry, which we enforce here through spin-changing collisions in atomic mixtures. This promising mechanism to protect gauge invariance has been put forward in various proposals (*18*–*21*) but not yet demonstrated experimentally. We demonstrate the engineering of an elementary building block in a mixture of bosonic atoms, demonstrate its high tunability, and verify its faithful representation of the desired model.

We further propose an extended implementation scheme in an optical lattice, where each lattice well constitutes an elementary building block that contains both matter and gauge fields. Repetitions of this elementary unit can be connected using Raman-assisted tunneling (*22*). Gauge and matter fields are spatially arranged in such a way that the spin-changing collisions occur within single-lattice wells, in contrast to previous proposals (*18*–*21*) where the gauge and matter fields were spatially separated and spin-changing collisions had to be accompanied by hopping across different sites of the optical lattice.

We specify our proposal for a one-dimensional gauge theory on a spatial lattice, as visualized in Fig. 1A. Charged matter fields reside on the lattice sites *n*, with gauge fields on the links in between the sites (*23*). We consider two-component matter fields labeled “p” and “v”, which are described by the operators *24*–*26*), where the gauge fields are replaced by quantum mechanical spins *z*-component *20*).

Physically, this system of charged matter and gauge fields can be realized in a mixture of two atomic Bose–Einstein condensates (BECs) with two internal components each (in our experiment, we use ^{7}Li and ^{23}Na). An extended system can be obtained by use of an optical lattice. In our scheme, we abandon the one-to-one correspondence between the sites of the simulated lattice gauge theory and the sites of the optical-lattice simulator. This correspondence characterized previous proposals and necessitated physically placing the gauge fields in-between matter sites (*18*–*21*). Instead, as illustrated in Fig. 1B, here one site of the physical lattice hosts two matter components, each taken from one adjacent site (

The enhanced physical overlap in this configuration decisively improved time scales of the spin-changing collisions, which until now were a major limiting factor for experimental implementations. Moreover, a single well of the optical lattice already contains the essential processes between matter and gauge fields and thus represents an elementary building block of the lattice gauge theory. These building blocks can be coupled by Raman-assisted tunneling of the matter fields [see supplementary materials (SM)].

The Hamiltonian

We implemented the elementary building block Hamiltonian ^{23}Na states are labeled as *27*, *28*). We label the ^{7}Li states as “particle” *29*).

The resulting setup is highly tunable, as we demonstrated experimentally on the building block. We achieved tunability of the gauge field through a two-pulse Rabi coupling of the Na atoms between ^{7}Li atoms in

When the gauge-invariant coupling was turned off by removing the Na atoms from the trap, we observed no dynamics in the matter sector beyond the detection noise. By contrast, once the gauge field was present, the matter sector clearly underwent a transfer from *30*). This observation demonstrated the controlled operation of heteronuclear spin-changing collisions implementing the gauge-invariant dynamics in the experiment.

To quantify our observations, we extracted the ratio ^{23}Na condensate, the expected corresponding change in

We display

We compare the experimental results to the mean-field predictions of Hamiltonian (Eq. 1) for chosen χ, λ, and *31*) of the atomic clouds within the trapping potential, which renormalized the model parameters. Moreover, the mean-field approximation was not able to capture the decoherence observed in Fig. 3 at later times. However, the features of the resonance data in Fig. 4 were more robust against the decoherence as it probed the initial rise of particle production.

We included the decoherence into the model phenomenologically by implementing a damping term characterized by

Our results demonstrated the controlled operation of an elementary building block of a *5*, *9*) are challenging to scale up. This difficulty makes analog quantum simulators, as treated here, highly attractive, because they can be scaled up and still maintain excellent quantum coherence (*6*–*8*, *32*–*34*). Proceeding to the extended system requires optical lattices and Raman-assisted tunneling [see SM and (*22*)]. The resulting extended gauge theory will enable the observation of relevant phenomena, such as plasma oscillations or resonant particle production in strong-field quantum electrodynamics (*35*). Along the path to the relativistic gauge theories realized in nature, we will replace bosonic ^{7}Li with fermionic ^{6}Li, which will allow for the recovery of Lorentz invariance in the continuum limit (see SM).

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**

**Funding:**We acknowledge funding from the DFG Collaborative Research Centre “SFB 1225 (ISOQUANT),” the ERC Advanced Grant “EntangleGen” (Project-ID 694561), the ERC Starting Grant “StrEnQTh” (Project-ID 804305), and the Excellence Initiative of the German federal government and the state governments - funding line Institutional Strategy (Zukunftskonzept): DFG project number ZUK 49/Ü. F.J. acknowledges the DFG support through the project FOR 2724, the Emmy-Noether grant (project-id 377616843) and from the Juniorprofessorenprogramm Baden-Württemberg (MWK). P.H. acknowledges support by Provincia Autonoma di Trento, Quantum Science and Technology in Trento.

**Author contributions:**A.M., A.H., A.X. and R.P.B. set up the experiment and performed the measurements; A.M. and T.V.Z. performed the data analysis; T.V.Z., P.H., and J.B. developed the theory; P.H., J.B., M.K.O., and F.J. supervised the project; all authors took part in writing the manuscript.

**Competing Interests:**The authors declare no competing interests.

**Data and materials availability:**The data are available on the Dryad database (

*36*).