Observing dynamical phases of BCS superconductors in a cavity QED simulator.

In conventional Bardeen-Cooper-Schrieffer superconductors1, electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. Although superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system parameters are abruptly changed2-8. The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms, but not all have yet been directly observed. Here we realize an alternative way to generate the proposed dynamical phases using cavity quantum electrodynamics (QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in 88Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity9,10. To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of the subsequent dynamics of the superconducting order parameter using nondestructive measurements. We observe regimes in which the order parameter decays to zero (phase I)3,4, assumes a non-equilibrium steady-state value (phase II)2,3 or exhibits persistent oscillations (phase III)2,3. This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor11,12, and for increasing the coherence time for quantum sensing.

Quantum-enhanced sensing on optical transitions through finite-range interactions.

The control over quantum states in atomic systems has led to the most precise optical atomic clocks so far1-3. Their sensitivity is bounded at present by the standard quantum limit, a fundamental floor set by quantum mechanics for uncorrelated particles, which can-nevertheless-be overcome when operated with entangled particles. Yet demonstrating a quantum advantage in real-world sensors is extremely challenging. Here we illustrate a pathway for harnessing large-scale entanglement in an optical transition using 1D chains of up to 51 ions with interactions that decay as a power-law function of the ion separation. We show that our sensor can emulate many features of the one-axis-twisting (OAT) model, an iconic, fully connected model known to generate scalable squeezing4 and Greenberger-Horne-Zeilinger-like states5-8. The collective nature of the state manifests itself in the preservation of the total transverse magnetization, the reduced growth of the structure factor, that is, spin-wave excitations (SWE), at finite momenta, the generation of spin squeezing comparable with OAT (a Wineland parameter9,10 of -3.9 ± 0.3 dB for only N = 12 ions) and the development of non-Gaussian states in the form of multi-headed cat states in the Q-distribution. We demonstrate the metrological utility of the states in a Ramsey-type interferometer, in which we reduce the measurement uncertainty by -3.2 ± 0.5 dB below the standard quantum limit for N = 51 ions.

Unitary p-wave interactions between fermions in an optical lattice.

Exchange-antisymmetric pair wavefunctions in fermionic systems can give rise to unconventional superconductors and superfluids1-3. The realization of these states in controllable quantum systems, such as ultracold gases, could enable new types of quantum simulations4-8, topological quantum gates9-11 and exotic few-body states12-15. However, p-wave and other antisymmetric interactions are weak in naturally occurring systems16,17, and their enhancement via Feshbach resonances in ultracold systems has been limited by three-body loss18-24. Here we create isolated pairs of spin-polarized fermionic atoms in a multiorbital three-dimensional optical lattice. We spectroscopically measure elastic p-wave interaction energies of strongly interacting pairs of atoms near a magnetic Feshbach resonance. The interaction strengths are widely tunable by the magnetic field and confinement strength, and yet collapse onto a universal curve when rescaled by the harmonic energy and length scales of a single lattice site. The absence of three-body processes enables the observation of elastic unitary p-wave interactions, as well as coherent oscillations between free-atom and interacting-pair states. All observations are compared both to an exact solution using a p-wave pseudopotential and to numerical solutions using an ab initio interaction potential. The understanding and control of on-site p-wave interactions provides a necessary component for the assembly of multiorbital lattice models25,26 and a starting point for investigations of how to protect such systems from three-body recombination in the presence of tunnelling, for instance using Pauli blocking and lattice engineering27,28.

Exploring dynamical phase transitions with cold atoms in an optical cavity.

Interactions between atoms and light in optical cavities provide a means of investigating collective (many-body) quantum physics in controlled environments. Such ensembles of atoms in cavities have been proposed for studying collective quantum spin models, where the atomic internal levels mimic a spin degree of freedom and interact through long-range interactions tunable by changing the cavity parameters1-4. Non-classical steady-state phases arising from the interplay between atom-light interactions and dissipation of light from the cavity have previously been investigated5-11. These systems also offer the opportunity to study dynamical phases of matter that are precluded from existence at equilibrium but can be stabilized by driving a system out of equilibrium12-16, as demonstrated by recent experiments17-22. These phases can also display universal behaviours akin to standard equilibrium phase transitions8,23,24. Here, we use an ensemble of about a million strontium-88 atoms in an optical cavity to simulate a collective Lipkin-Meshkov-Glick model25,26, an iconic model in quantum magnetism, and report the observation of distinct dynamical phases of matter in this system. Our system allows us to probe the dependence of dynamical phase transitions on system size, initial state and other parameters. These observations can be linked to similar dynamical phases in related systems, including the Josephson effect in superfluid helium27, or coupled atomic28 and solid-state polariton29 condensates. The system itself offers potential for generation of metrologically useful entangled states in optical transitions, which could permit quantum enhancement in state-of-the-art atomic clocks30,31.

Squeezing Multilevel Atoms in Dark States via Cavity Superradiance.

We describe a method to create and store scalable and long-lived entangled spin-squeezed states within a manifold of many-body cavity dark states using collective emission of light from multilevel atoms inside an optical cavity. We show that the system can be tuned to generate squeezing in a dark state where it will be immune to superradiance. We also show more generically that squeezing can be generated using a combination of superradiance and coherent driving in a bright state, and subsequently be transferred via single-particle rotations to a dark state where squeezing can be stored. Our findings, readily testable in current optical cavity experiments with alkaline-earth-like atoms, can open a path for dissipative generation and storage of metrologically useful states in optical transitions.

Manipulating Growth and Propagation of Correlations in Dipolar Multilayers: From Pair Production to Bosonic Kitaev Models.

We study the nonequilibrium dynamics of dipoles confined in multiple stacked two-dimensional layers realizing a long-range interacting quantum spin 1/2 XXX model. We demonstrate that strong in-plane interactions can protect a manifold of collective layer dynamics. This then allows us to map the many-body spin dynamics to bosonic models. In a bilayer configuration we show how to engineer the paradigmatic two-mode squeezing Hamiltonian known from quantum optics, resulting in exponential production of entangled pairs and generation of metrologically useful entanglement from initially prepared product states. In multilayer configurations we engineer a bosonic variant of the Kitaev model displaying chiral propagation along the layer direction. Our study illustrates how the control over interactions, lattice geometry, and state preparation in interacting dipolar systems uniquely afforded by AMO platforms such as Rydberg and magnetic atoms, polar molecules, or trapped ions allows for the control over the temporal and spatial propagation of correlations for applications in quantum sensing and quantum simulation.

Robust Nuclear Spin Entanglement via Dipolar Interactions in Polar Molecules.

We propose a general protocol for on-demand generation of robust entangled states of nuclear and/or electron spins of ultracold ^{1}Σ and ^{2}Σ polar molecules using electric dipolar interactions. By encoding a spin-1/2 degree of freedom in a combined set of spin and rotational molecular levels, we theoretically demonstrate the emergence of effective spin-spin interactions of the Ising and XXZ forms, enabled by efficient magnetic control over electric dipolar interactions. We show how to use these interactions to create long-lived cluster and squeezed spin states.

Validating Phase-Space Methods with Tensor Networks in Two-Dimensional Spin Models with Power-Law Interactions.

Using a recently developed extension of the time-dependent variational principle for matrix product states, we evaluate the dynamics of 2D power-law interacting XXZ models, implementable in a variety of state-of-the-art experimental platforms. We compute the spin squeezing as a measure of correlations in the system, and compare to semiclassical phase-space calculations utilizing the discrete truncated Wigner approximation (DTWA). We find the latter efficiently and accurately captures the scaling of entanglement with system size in these systems, despite the comparatively resource-intensive tensor network representation of the dynamics. We also compare the steady-state behavior of DTWA to thermal ensemble calculations with tensor networks. Our results open a way to benchmark dynamical calculations for two-dimensional quantum systems, and allow us to rigorously validate recent predictions for the generation of scalable entangled resources for metrology in these systems.

Bosonic Pair Production and Squeezing for Optical Phase Measurements in Long-Lived Dipoles Coupled to a Cavity.

We propose to simulate bosonic pair creation using large arrays of long-lived dipoles with multilevel internal structure coupled to an undriven optical cavity. Entanglement between the atoms, generated by the exchange of virtual photons through a common cavity mode, grows exponentially fast and is described by two-mode squeezing of effective bosonic quadratures. The mapping between an effective bosonic model and the natural spin description of the dipoles allows us to realize the analog of optical homodyne measurements via straightforward global rotations and population measurements of the electronic states, and we propose to exploit this for quantum-enhanced sensing of an optical phase (common and differential between two ensembles). We discuss a specific implementation based on Sr atoms and show that our sensing protocol is robust to sources of decoherence intrinsic to cavity platforms. Our proposal can open unique opportunities for next-generation optical atomic clocks.

Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments.

We overview the concept of dynamical phase transitions (DPTs) in isolated quantum systems quenched out of equilibrium. We focus on non-equilibrium transitions characterized by an order parameter, which features qualitatively distinct temporal behavior on the two sides of a certain dynamical critical point. DPTs are currently mostly understood as long-lived prethermal phenomena in a regime where inelastic collisions are incapable to thermalize the system. The latter enables the dynamics to substain phases that explicitly break detailed balance and therefore cannot be encompassed by traditional thermodynamics. Our presentation covers both cold atoms as well as condensed matter systems. We revisit a broad plethora of platforms exhibiting pre-thermal DPTs, which become theoretically tractable in a certain limit, such as for a large number of particles, large number of order parameter components, or large spatial dimension. The systems we explore include, among others, quantum magnets with collective interactions,ϕ4quantum field theories, and Fermi-Hubbard models. A section dedicated to experimental explorations of DPTs in condensed matter and AMO systems connects this large variety of theoretical models.

Measuring Correlations from the Collective Spin Fluctuations of a Large Ensemble of Lattice-Trapped Dipolar Spin-3 Atoms.

We perform collective spin measurements to study the buildup of two-body correlations between ≈10^{4} spin s=3 chromium atoms pinned in a 3D optical lattice. The spins interact via long range and anisotropic dipolar interactions. From the fluctuations of total magnetization, measured at the standard quantum limit, we estimate the dynamical growth of the connected pairwise correlations associated with magnetization. The quantum nature of the correlations is assessed by comparisons with analytical short- and long-time expansions and numerical simulations. Our Letter shows that measuring fluctuations of spin populations for s>1/2 spins provides new ways to characterize correlations in quantum many-body systems.

Disentangling Pauli Blocking of Atomic Decay from Cooperative Radiation and Atomic Motion in a 2D Fermi Gas.

The observation of Pauli blocking of atomic spontaneous decay via direct measurements of the atomic population requires the use of long-lived atomic gases where quantum statistics, atom recoil, and cooperative radiative processes are all relevant. We develop a theoretical framework capable of simultaneously accounting for all these effects in the many-body quantum degenerate regime. We apply it to atoms in a single 2D pancake or arrays of pancakes featuring an effective Λ level structure (one excited and two degenerate ground states). We identify a parameter window in which a factor of 2 extension in the atomic lifetime clearly attributable to Pauli blocking should be experimentally observable in deeply degenerate gases with ∼10^{3} atoms. We experimentally observe a suppressed excited-state decay rate, fully consistent with the theory prediction of an enhanced excited-state lifetime, on the ^{1}S_{0}-^{3}P_{1} transition in ^{87}Sr atoms.

Effect of Active Photons on Dynamical Frustration in Cavity QED.

We study the far-from-equilibrium dynamical regimes of a many-body spin-boson model with disordered couplings relevant for cavity QED and trapped ion experiments, using the discrete truncated Wigner approximation. We focus on the dynamics of spin observables upon varying the disorder strength and the frequency of the photons, finding that the latter can considerably alter the structure of the system's dynamical responses. When the photons evolve at a similar rate as the spins, they can induce qualitatively distinct frustrated dynamics characterized by either logarithmic or algebraically slow relaxation. The latter illustrates resilience of glassylike dynamics in the presence of active photonic degrees of freedom, suggesting that disordered quantum many-body systems with resonant photons or phonons can display a rich diagram of nonequilibrium responses, with near future applications for quantum information science.

Quantum Enhanced Cavity QED Interferometer with Partially Delocalized Atoms in Lattices.

We propose a quantum enhanced interferometric protocol for gravimetry and force sensing using cold atoms in an optical lattice supported by a standing-wave cavity. By loading the atoms in partially delocalized Wannier-Stark states, it is possible to cancel the undesirable inhomogeneities arising from the mismatch between the lattice and cavity fields and to generate spin squeezed states via a uniform one-axis twisting model. The quantum enhanced sensitivity of the states is combined with the subsequent application of a compound pulse sequence that allows us to separate atoms by several lattice sites. This, together with the capability to load small atomic clouds in the lattice at micrometric distances from a surface, make our setup ideal for sensing short-range forces. We show that for arrays of 10^{4} atoms, our protocol can reduce the required averaging time by a factor of 10 compared to unentangled lattice-based interferometers after accounting for primary sources of decoherence.

Collective P-Wave Orbital Dynamics of Ultracold Fermions.

We consider the nonequilibrium orbital dynamics of spin-polarized ultracold fermions in the first excited band of an optical lattice. A specific lattice depth and filling configuration is designed to allow the p_{x} and p_{y} excited orbital degrees of freedom to act as a pseudospin. Starting from the full Hamiltonian for p-wave interactions in a periodic potential, we derive an extended Hubbard-type model that describes the anisotropic lattice dynamics of the excited orbitals at low energy. We then show how dispersion engineering can provide a viable route to realizing collective behavior driven by p-wave interactions. In particular, Bragg dressing and lattice depth can reduce single-particle dispersion rates, such that a collective many-body gap is opened with only moderate Feshbach enhancement of p-wave interactions. Physical insight into the emergent gap-protected collective dynamics is gained by projecting the Hamiltonian into the Dicke manifold, yielding a one-axis twisting model for the orbital pseudospin that can be probed using conventional Ramsey-style interferometry. Experimentally realistic protocols to prepare and measure the many-body dynamics are discussed, including the effects of band relaxation, particle loss, spin-orbit coupling, and doping.

Dynamical Generation of Spin Squeezing in Ultracold Dipolar Molecules.

We study a bulk fermionic dipolar molecular gas in the quantum degenerate regime confined in a two-dimensional geometry. Using two rotational states of the molecules, we encode a spin 1/2 degree of freedom. To describe the many-body spin dynamics of the molecules, we derive a long-range interacting XXZ model valid in the regime where motional degrees of freedom are frozen. Because of the spatially extended nature of the harmonic oscillator modes, the interactions in the spin model are very long ranged, and the system behaves close to the collective limit, resulting in robust dynamics and generation of entanglement in the form of spin squeezing even at finite temperature and in the presence of dephasing and chemical reactions. We discuss how the internal state structure can be exploited to realize time reversal and enhanced metrological sensing protocols.

Cavity-QED Quantum Simulator of Dynamical Phases of a Bardeen-Cooper-Schrieffer Superconductor.

We propose to simulate dynamical phases of a BCS superconductor using an ensemble of cold atoms trapped in an optical cavity. Effective Cooper pairs are encoded via the internal states of the atoms, and attractive interactions are realized via the exchange of virtual photons between atoms coupled to a common cavity mode. Control of the interaction strength combined with a tunable dispersion relation of the effective Cooper pairs allows exploration of the full dynamical phase diagram of the BCS model as a function of system parameters and the prepared initial state. Our proposal paves the way for the study of the nonequilibrium features of quantum magnetism and superconductivity by harnessing atom-light interactions in cold atomic gases.

Generating Multipartite Spin States with Fermionic Atoms in a Driven Optical Lattice.

We propose a protocol for generating generalized Greenberger-Horne-Zeilinger (GHZ) states using ultracold fermions in 3D optical lattices or optical tweezer arrays. The protocol uses the interplay between laser driving, on site interactions and external trapping confinement to enforce energetic spin- and position-dependent constraints on the atomic motion. These constraints allow us to transform a local superposition into a GHZ state through a stepwise protocol that flips one site at a time. The protocol requires no site-resolved drives or spin-dependent potentials, exhibits robustness to slow global laser phase drift, and naturally makes use of the harmonic trap that would normally cause difficulties for entanglement-generating protocols in optical lattices. We also discuss an improved protocol that can compensate for holes in the loadout at the cost of increased generation time. The state can immediately be used for quantum-enhanced metrology in 3D optical lattice clocks, opening a window to push the sensitivity of state-of-the-art sensors beyond the standard quantum limit.

Spin Squeezing with Short-Range Spin-Exchange Interactions.

We investigate many-body spin squeezing dynamics in an XXZ model with interactions that fall off with distance r as 1/r^{α} in D=2 and 3 spatial dimensions. In stark contrast to the Ising model, we find a broad parameter regime where spin squeezing comparable to the infinite-range α=0 limit is achievable even when interactions are short ranged, α>D. A region of "collective" behavior in which optimal squeezing grows with system size extends all the way to the α→∞ limit of nearest-neighbor interactions. Our predictions, made using the discrete truncated Wigner approximation, are testable in a variety of experimental cold atomic, molecular, and optical platforms.

Simulation of XXZ Spin Models Using Sideband Transitions in Trapped Bosonic Gases.

We theoretically propose and experimentally demonstrate the use of motional sidebands in a trapped ensemble of ^{87}Rb atoms to engineer tunable long-range XXZ spin models. We benchmark our simulator by probing a ferromagnetic to paramagnetic dynamical phase transition in the Lipkin-Meshkov-Glick model, a collective XXZ model plus additional transverse and longitudinal fields, via Rabi spectroscopy. We experimentally reconstruct the boundary between the dynamical phases, which is in good agreement with mean-field theoretical predictions. Our work introduces new possibilities in quantum simulation of anisotropic spin-spin interactions and quantum metrology enhanced by many-body entanglement.