D. J. Webb

Quantum simulations of lattice gauge theories (LGTs) with both dynamical matter and gauge fields provide a promising approach to studying strongly coupled problems beyond classical computational reach. Yet, implementing gauge-invariant encodings and real-time evolution remains experimentally challenging. Here, we demonstrate a resource-efficient encoding of a $\mathbb{Z}_2$ LGT using a hybrid qubit-oscillator trapped-ion quantum device, where qubits represent gauge fields and vibrational modes naturally encode bosonic matter fields. This architecture utilises synthetic dimensions to construct higher-dimensional lattice geometries and combines digital and analogue techniques to prepare initial states, realise gauge-invariant real-time evolution, and measure the relevant observables. We experimentally probe dynamics obeying Gauss's law in a $\mathbb{Z}_2$ link and extend this to a loop geometry, marking the first steps towards higher-dimensional LGTs. In this quasi-2D setup, we observe Aharonov-Bohm interference for the first time with dynamical gauge fields encoding magnetic flux, demonstrating the interplay between charge and flux. Our results chart a promising path for scalable quantum simulations of bosonic gauge theories and outline a roadmap for realising exotic LGTs in higher dimensions.

Full coherent control and generation of superpositions of the quantum harmonic oscillator are not only of fundamental interest but are crucial for applications in quantum simulations, quantum-enhanced metrology and continuous-variable quantum computation. The extension of such superpositions to nonclassical states increases their power as a resource for such applications. Here, we create arbitrary superpositions of nonclassical and non-Gaussian states of a quantum harmonic oscillator using the motion of a trapped ion coupled to its internal spin states. We interleave spin-dependent nonlinear bosonic interactions and mid-circuit measurements of the spin that preserve the coherence of the oscillator. These techniques enable the creation of superpositions between squeezed, trisqueezed, and quadsqueezed states, which have never been demonstrated before, with independent control over the complex-valued squeezing parameter and the probability amplitude of each constituent, as well as their spatial separation. We directly observe the nonclassical nature of these states in the form of Wigner negativity following a full state reconstruction. Our methods apply to any system where a quantum harmonic oscillator is coupled to a spin.

Quantum harmonic oscillators model a wide variety of phenomena ranging from electromagnetic fields to vibrations of atoms in molecules. Their excitations can be represented by bosons such as photons, single particles of light, or phonons, the quanta of vibrational energy. Linear interactions that only create and annihilate single bosons can generate coherent states of light or motion. Introducing nth-order nonlinear interactions, that instead involve n bosons, leads to increasingly complex quantum behaviour. For example, second-order interactions enable squeezing, used to enhance the precision of measurements beyond classical limits, while higher-order interactions create non-Gaussian states essential for continuous-variable quantum computation. However, generating nonlinear interactions is challenging, typically requiring higher-order derivatives of the driving field or specialized hardware. Hybrid systems, where linear interactions couple an oscillator to an additional spin, offer a solution and are readily available across many platforms. Here, using the spin of a single trapped ion coupled to its motion, we employ two linear interactions to demonstrate up to fourth-order bosonic interactions; we focus on generalised squeezing interactions and demonstrate squeezing, trisqueezing, and quadsqueezing. We characterise these interactions, including their spin dependence, and reconstruct the Wigner function of the resulting states. We also discuss the scaling of the interaction strength, where we drive the quadsqueezing interaction more than 100 times faster than using conventional techniques. Our method presents no fundamental limit in the interaction order n and applies to any platform supporting spin-dependent linear interactions. Strong higher-order nonlinear interactions unlock the study of fundamental quantum optics, quantum simulation, and computation in a hitherto unexplored regime.

All laser-driven entangling operations for trapped-ion qubits have hitherto been performed without control of the optical phase of the light field, which precludes independent tuning of the carrier and motional coupling. By placing $^{88}$Sr$^+$ ions in a $λ=674$ nm standing wave, whose relative position is controlled to $\approxλ/100$, we suppress the carrier coupling by a factor of $18$, while coherently enhancing the spin-motion coupling. We experimentally demonstrate that the off-resonant carrier coupling imposes a speed limit for conventional traveling-wave Mølmer-Sørensen gates; we use the standing wave to surpass this limit and achieve a gate duration of $15\ μ$s, restricted by the available laser power.