Spinor Self-Ordering of a Quantum Gas in a Cavity

A dissipation-induced superradiant transition in a strontium cavity-QED system

Driven-dissipative many-body systems are ubiquitous in nature and a fundamental resource for quantum technologies. However, they are also complex and hard to model because they cannot be described by the standard tools in equilibrium statistical mechanics. Probing nonequilibrium critical phenomena in pristine setups can illuminate fresh perspectives on these systems. Here, we use an ensemble of cold 88Sr atoms coupled to a driven high-finesse cavity to study the cooperative resonance fluorescence (CRF) model, a classic driven-dissipative model describing coherently driven dipoles superradiantly emitting light. We observe its nonequilibrium phase diagram characterized by a second-order phase transition. Below a critical drive strength, the atoms quickly reach the so-called superradiant steady state featuring a macroscopic dipole moment; above the critical point, the atoms undergo persistent Rabi-like oscillations. At longer times, spontaneous emission transforms the second-order transition into a discontinuous first-order transition. Our observations pave the way for harnessing robust entangled states and exploring boundary time crystals in driven-dissipative systems.

Fluctuation-Induced Bistability of Fermionic Atoms Coupled to a Dissipative Cavity

We investigate the steady state phase diagram of fermionic atoms subjected to an optical lattice and coupled to a high finesse optical cavity with photon losses. The coupling between the atoms and the cavity field is induced by a transverse pump beam. Taking fluctuations around the mean-field solutions into account, we find that a transition to a self-organized phase takes place at a critical value of the pump strength. In the self-organized phase the cavity field takes a finite expectation value and the atoms show a modulation in the density. Surprisingly, at even larger pump strengths two self-organized stable solutions of the cavity field and the atoms occur, signaling the presence of a bistability. We show that the bistable behavior is induced by the atoms-cavity fluctuations and is not captured by the mean-field approach.

Frustration Elimination and Excited State Search in Coherent Ising Machines

Frustration, that is, the impossibility of satisfying the energetic preferences between all spin pairs simultaneously, underlies the complexity of many fundamental properties in spin systems, including the computational difficulty in determining their ground states. Coherent Ising machines (CIMs) have been proposed as a promising analog computational approach to efficiently find different degenerate ground states of large and complex Ising models. However, CIMs also face challenges in solving frustrated Ising models: frustration not only reduces the probability of finding good solutions, but it also prohibits the leveraging of quantum effects in doing so. To circumvent these detrimental effects of frustration, we show how frustrated Ising models can be mapped to frustration-free CIM configurations by including ancillary modes and modifying the coupling protocol used in current CIM designs. Such frustration elimination may empower current CIMs to benefit from quantum effects in dealing with frustrated Ising models. In addition, these ancillary modes can also enable error detection and searching for excited states.

Spin Self-Organization in an Optical Cavity Facilitated by Inhomogeneous Broadening

We study the onset of collective spin self-organization in a thermal ensemble of driven two-level atoms confined in an optical cavity. The atoms spontaneously form a spin pattern above a critical driving strength that sets a threshold and is determined by the cavity parameters, the initial temperature, and the transition frequency of the atomic spin. Remarkably, we find that inhomogeneous Doppler broadening facilitates the onset of spin self-organization. In particular, the threshold is nonmonotonic when increasing the spin transition frequency and reaches a minimum when the Doppler broadening is of similar magnitude. This feature emerges due to Doppler-induced resonances. Above the threshold, we find cooperative dynamics of spin, spatial, and momentum degrees of freedom leading to density modulations, fast reduction of kinetic energy, and the emergence of nonthermal states. More broadly, our work demonstrates how broadening can facilitate strong light-matter interactions in many-body systems.

Improving metrology with quantum scrambling in a spin-1 Bose-Einstein condensate coupled to a cavity

Spinor Bose-Einstein condensate is an ideal candidate for implementing the many-body entanglement, quantum measurement and quantum information processing owing to its inherent spin-mixing dynamics. Here we present a system of an 87Rb atomic spin-1 Bose-Einstein condensate coupled to an optical ring cavity, in which cavity-mediated nonlinear interactions give rise to saddle points in the semiclassical phase space, providing a general mechanism for exponential fast scrambling and metrological gain augment. We theoretically study metrological gain and fidelity out-of-time-ordered correlator based on time-reversal protocols and demonstrate that exponential rapid scrambling dynamics can enhance quantum metrology. In addition, we use the out-of-time-ordered correlator to probe dynamical phase transitions. This work is useful to understand the intrinsic relation between the concepts from different subfields of quantum science.

Momentum-exchange interactions in a Bragg atom interferometer suppress Doppler dephasing

Large ensembles of laser-cooled atoms interacting through infinite-range photon-mediated interactions are powerful platforms for quantum simulation and sensing. Here we realize momentum-exchange interactions in which pairs of atoms exchange their momentum states by collective emission and absorption of photons from a common cavity mode, a process equivalent to a spin-exchange or XX collective Heisenberg interaction. The momentum-exchange interaction leads to an observed all-to-all Ising-like interaction in a matter-wave interferometer. A many-body energy gap also emerges, effectively binding interferometer matter-wave packets together to suppress Doppler dephasing in analogy to Mössbauer spectroscopy. The tunable momentum-exchange interaction expands the capabilities of quantum interaction-enhanced matter-wave interferometry and may enable the realization of exotic behaviors, including simulations of superconductors and dynamical gauge fields.

Conventional and Unconventional Dicke Models: Multistabilities and Nonequilibrium Dynamics

The Dicke model describes the collective behavior of a subwavelength-size ensemble of two-level atoms (i.e., spin-1/2) interacting identically with a single quantized radiation field of a cavity. Across a critical coupling strength it exhibits a zero-temperature phase transition from the normal state to the superradiant phase where the field is populated and the collective spin acquires a nonzero x component, which can be imagined as ferromagnetic ordering of the atomic spins along x. Here we introduce a variant of this model where two subwavelength-size ensembles of spins interact with a single quantized radiation field with different strengths. Subsequently, we restrict ourselves to a special case where the coupling strengths are opposite (which is unitarily equivalent to equal-coupling strengths). Because of the conservation of the total spin in each ensemble individually, the system supports two distinct superradiant states with x-ferromagnetic and x-ferrimagnetic spin ordering, coexisting with each other in a large parameter regime. The stability and dynamics of the system in the thermodynamic limit are examined using a semiclassical approach, which predicts nonstationary behaviors due to the multistabilities. At the end, we also perform small-scale full quantum-mechanical calculations, with results consistent with the semiclassical ones.

Superradiant and Subradiant Cavity Scattering by Atom Arrays

We realize collective enhancement and suppression of light scattered by an array of tweezer-trapped ^{87}Rb atoms positioned within a strongly coupled Fabry-Pérot optical cavity. We illuminate the array with light directed transverse to the cavity axis, in the low saturation regime, and detect photons scattered into the cavity. For an array with integer-optical-wavelength spacing each atom scatters light into the cavity with nearly identical scattering amplitude, leading to an observed N^{2} scaling of cavity photon number as the atom number increases stepwise from N=1 to N=8. By contrast, for an array with half-integer-wavelength spacing, destructive interference of scattering amplitudes yields a nonmonotonic, subradiant cavity intensity versus N. By analyzing the polarization of light emitted from the cavity, we find that Rayleigh scattering can be collectively enhanced or suppressed with respect to Raman scattering. We observe also that atom-induced shifts and broadenings of the cavity resonance are precisely tuned by varying the atom number and positions. Altogether, tweezer arrays provide exquisite control of atomic cavity QED spanning from the single- to the many-body regime.

Optimal Generators for Quantum Sensing

We propose a computationally efficient method to derive the unitary evolution that a quantum state is most sensitive to. This allows one to determine the optimal use of an entangled state for quantum sensing, even in complex systems where intuition from canonical squeezing examples breaks down. In this paper we show that the maximal obtainable sensitivity using a given quantum state is determined by the largest eigenvalue of the quantum Fisher information matrix (QFIM) and the corresponding evolution is uniquely determined by the coinciding eigenvector. Since we optimize the process of parameter encoding rather than focusing on state preparation protocols, our scheme is relevant for any quantum sensor. This procedure naturally optimizes multiparameter estimation by determining, through the eigenvectors of the QFIM, the maximal set of commuting observables with optimal sensitivity.

Nonreciprocal Dicke Model

We investigate the physics of an open two-component Dicke model, where the light field mediates nonreciprocal interactions between two spin species. We show that the model, which we dub nonreciprocal Dicke model, exhibits a discrete parity-time (PT) symmetry and we characterize the emergence of a nonstationary phase, so far explained in terms of dissipation-induced instability, as spontaneous breaking of PT symmetry. We further show that such PT symmetry breaking embodies an instance of a nonreciprocal phase transition, a concept recently introduced by Fruchart et al. [Nature (London) 592, 363 (2021)NATUAS0028-083610.1038/s41586-021-03375-9]. Remarkably, the phase transition in our model does not necessitate the presence of any underlying broken symmetry or exceptional points in the spectrum, both believed to be essential requirements for nonreciprocal phase transitions. Our results establish driven-dissipative light-matter systems as a new avenue for exploring nonreciprocal phase transitions and contribute to the theory of nonreciprocal collective phenomena.

Density-wave ordering in a unitary Fermi gas with photon-mediated interactions

A density wave (DW) is a fundamental type of long-range order in quantum matter tied to self-organization into a crystalline structure. The interplay of DW order with superfluidity can lead to complex scenarios that pose a great challenge to theoretical analysis. In the past decades, tunable quantum Fermi gases have served as model systems for exploring the physics of strongly interacting fermions, including most notably magnetic ordering¹, pairing and superfluidity², and the crossover from a Bardeen-Cooper-Schrieffer superfluid to a Bose-Einstein condensate³. Here, we realize a Fermi gas featuring both strong, tunable contact interactions and photon-mediated, spatially structured long-range interactions in a transversely driven high-finesse optical cavity. Above a critical long-range interaction strength, DW order is stabilized in the system, which we identify via its superradiant light-scattering properties. We quantitatively measure the variation of the onset of DW order as the contact interaction is varied across the Bardeen-Cooper-Schrieffer superfluid and Bose-Einstein condensate crossover, in qualitative agreement with a mean-field theory. The atomic DW susceptibility varies over an order of magnitude upon tuning the strength and the sign of the long-range interactions below the self-ordering threshold, demonstrating independent and simultaneous control over the contact and long-range interactions. Therefore, our experimental setup provides a fully tunable and microscopically controllable platform for the experimental study of the interplay of superfluidity and DW order.

Optomechanics and quantum phase of the Bose-Einstein condensate with the cavity mediated spin-orbit coupling

We investigated the optomechanical dynamics and explored the quantum phase of a Bose-Einstein condensate in a ring cavity. The interaction between the atoms and the cavity field in the running wave mode induces a semiquantized spin-orbit coupling (SOC) for the atoms. We found that the evolution of the magnetic excitations of the matter field resembles that of an optomechanical oscillator moving in a viscous optical medium, with very good integrability and traceability, regardless of the atomic interaction. Moreover, the light-atom coupling induces a sign-changeable long-range interatomic interaction, which reshapes the typical energy spectrum of the system in a drastic manner. As a result, a new quantum phase featuring a high quantum degeneracy was found in the transitional area for SOC. Our scheme is immediately realizable and the results are measurable in experiments.

Raman laser induced self-organization with topology in a dipolar condensate

We investigate the ground states of a dipolar Bose-Einstein condensate (BEC) subject to Raman laser induced spin-orbit coupling with mean-field theory. Owing to the interplay between spin-orbit coupling and atom-atom interactions, the BEC presents remarkable self-organization behavior and thus hosts various exotic phases including vortex with discrete rotational symmetry, stripe with spin helix, and chiral lattices with C₄ symmetry. The peculiar chiral self-organized array of square lattice, which spontaneously breaks both U(1) and rotational symmetries, is observed when the contact interaction is considerable in comparison with the spin-orbit coupling. Moreover, we show that the Raman-induced spin-orbit coupling plays a crucial role in forming rich topological spin textures of the chiral self-organized phases by introducing a channel for atoms to turn on spin flipping between two components. The self-organization phenomena predicted here feature topology owing to spin-orbit coupling. In addition, we find long-lived metastable self-organized arrays with C₆ symmetry in the case of strong spin-orbit coupling. We also present a proposal to observe these predicted phases in ultracold atomic dipolar gases with laser-induced spin-orbit coupling, which may stimulate broad theoretical as well as experimental interest.

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.

Self-oscillating pump in a topological dissipative atom-cavity system

Pumps are transport mechanisms in which direct currents result from a cyclic evolution of the potential^1,2. As Thouless showed, the pumping process can have topological origins, when considering the motion of quantum particles in spatially and temporally periodic potentials³. However, the periodic evolution that drives these pumps has always been assumed to be imparted from outside, as has been the case in the experimental systems studied so far^4-12. Here we report on an emergent mechanism for pumping in a quantum gas coupled to an optical resonator, where we observe a particle current without applying a periodic drive. The pumping potential experienced by the atoms is formed by the self-consistent cavity field interfering with the static laser field driving the atoms. Owing to dissipation, the cavity field evolves between its two quadratures¹³, each corresponding to a different centrosymmetric crystal configuration¹⁴. This self-oscillation results in a time-periodic potential analogous to that describing the transport of electrons in topological tight-binding models, such as the paradigmatic Rice-Mele pump¹⁵. In the experiment, we directly follow the evolution by measuring the phase winding of the cavity field with respect to the driving field and observing the atomic motion in situ. The observed mechanism combines the dynamics of topological and open systems, and features characteristics of continuous dissipative time crystals.

Frustrated Superradiant Phase Transition

Frustration occurs when a system cannot find a lowest-energy configuration due to conflicting constraints. We show that a frustrated superradiant phase transition occurs when the ground-state superradiance of cavity fields due to local light-matter interactions cannot simultaneously minimize the positive photon hopping energies. We solve the Dicke trimer model on a triangle motif with both negative and positive hopping energies and show that the latter results in a sixfold degenerate ground-state manifold in which the translational symmetry is spontaneously broken. In the frustrated superradiant phase, we find that two sets of diverging time and fluctuation scales coexist, one governed by the mean-field critical exponent and another by a novel critical exponent. The latter is associated with the fluctuation in the difference of local order parameters and gives rise to site-dependent photon number critical exponents, which may serve as an experimental probe for the frustrated superradiant phase. We provide a qualitative explanation for the emergence of unconventional critical scalings and demonstrate that they are generic properties of the frustrated superradiant phase at the hand of a one-dimensional Dicke lattice with an odd number of sites. The mechanism for the frustrated superradiant phase transition discovered here applies to any lattice geometries where the antiferromagnetic ordering of neighboring sites are incompatible and therefore our work paves the way toward the exploration of frustrated phases of coupled light and matter.

Observing Dynamical Currents in a Non-Hermitian Momentum Lattice

We report on the experimental realization and detection of dynamical currents in a spin-textured lattice in momentum space. Collective tunneling is implemented via cavity-assisted Raman scattering of photons by a spinor Bose-Einstein condensate into an optical cavity. The photon field inducing the tunneling processes is subject to cavity dissipation, resulting in effective directional dynamics in a non-Hermitian setting. We observe that the individual tunneling events are superradiant in nature and locally resolve them in the lattice by performing real-time, frequency-resolved measurements of the leaking cavity field. The results can be extended to a regime exhibiting a cascade of currents and simultaneous coherences between multiple lattice sites, where numerical simulations provide further understanding of the dynamics. Our observations showcase dynamical tunneling in momentum-space lattices and provide prospects to realize dynamical gauge fields in driven-dissipative settings.

Spin Entanglement and Magnetic Competition via Long-Range Interactions in Spinor Quantum Optical Lattices

Quantum matter at ultralow temperatures offers a test bed for analyzing and controlling desired properties in strongly correlated systems. Under typical conditions the nature of the atoms fixes the magnetic character of the system. Beyond classical light potentials leading to optical lattices and short-range interactions, high-Q cavities introduce novel dynamics into the system via the quantumness of light. Here we propose a theoretical model and we analyze it using exact diagonalization and density matrix renormalization group simulations. We explore the effects of cavity mediated long-range magnetic interactions and optical lattices in ultracold matter. We find that global interactions modify the underlying magnetic character of the system while introducing competition scenarios. Antiferromagnetic correlated bosonic matter emerges in conditions beyond what nature typically provides. These allow new alternatives toward the design of robust mechanisms for quantum information purposes, exploiting the properties of magnetic phases of strongly correlated quantum matter.

An optical lattice with sound

Quantized sound waves-phonons-govern the elastic response of crystalline materials, and also play an integral part in determining their thermodynamic properties and electrical response (for example, by binding electrons into superconducting Cooper pairs)^1-3. The physics of lattice phonons and elasticity is absent in simulators of quantum solids constructed of neutral atoms in periodic light potentials: unlike real solids, traditional optical lattices are silent because they are infinitely stiff⁴. Optical-lattice realizations of crystals therefore lack some of the central dynamical degrees of freedom that determine the low-temperature properties of real materials. Here, we create an optical lattice with phonon modes using a Bose-Einstein condensate (BEC) coupled to a confocal optical resonator. Playing the role of an active quantum gas microscope, the multimode cavity QED system both images the phonons and induces the crystallization that supports phonons via short-range, photon-mediated atom-atom interactions. Dynamical susceptibility measurements reveal the phonon dispersion relation, showing that these collective excitations exhibit a sound speed dependent on the BEC-photon coupling strength. Our results pave the way for exploring the rich physics of elasticity in quantum solids, ranging from quantum melting transitions⁵ to exotic 'fractonic' topological defects⁶ in the quantum regime.

Cavity-induced quantum spin liquids

Quantum spin liquids provide paradigmatic examples of highly entangled quantum states of matter. Frustration is the key mechanism to favor spin liquids over more conventional magnetically ordered states. Here we propose to engineer frustration by exploiting the coupling of quantum magnets to the quantized light of an optical cavity. The interplay between the quantum fluctuations of the electro-magnetic field and the strongly correlated electrons results in a tunable long-range interaction between localized spins. This cavity-induced frustration robustly stabilizes spin liquid states, which occupy an extensive region in the phase diagram spanned by the range and strength of the tailored interaction. This occurs even in originally unfrustrated systems, as we showcase for the Heisenberg model on the square lattice.

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.

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.

Feedback induced magnetic phases in binary Bose-Einstein condensates

Weak measurement in tandem with real-time feedback control is a new route toward engineering novel nonequilibrium quantum matter. Here we develop a theoretical toolbox for quantum feedback control of multicomponent Bose-Einstein condensates (BECs) using backaction-limited weak measurements in conjunction with spatially resolved feedback. Feedback in the form of a single-particle potential can introduce effective interactions that enter into the stochastic equation governing system dynamics. The effective interactions are tunable and can be made analogous to Feshbach resonances-spin independent and spin dependent-but without changing atomic scattering parameters. Feedback cooling prevents runaway heating due to measurement backaction and we present an analytical model to explain its effectiveness. We showcase our toolbox by studying a two-component BEC using a stochastic mean-field theory, where feedback induces a phase transition between easy-axis ferromagnet and spin-disordered paramagnet phases. We present the steady-state phase diagram as a function of intrinsic and effective spin-dependent interaction strengths. Our result demonstrates that closed-loop quantum control of Bose-Einstein condensates is a powerful tool for quantum engineering in cold-atom systems.

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.

Numerically Exact Treatment of Many-Body Self-Organization in a Cavity

We investigate the full quantum evolution of ultracold interacting bosonic atoms on a chain and coupled to an optical cavity. Extending the time-dependent matrix product state techniques and the many-body adiabatic elimination technique to capture the global coupling to the cavity mode and the open nature of the cavity, we examine the long time behavior of the system beyond the mean-field elimination of the cavity field. We investigate the many-body steady states and the self-organization transition for a wide range of parameters. We show that in the self-organized phase the steady state consists in a mixture of the mean-field predicted density wave states and excited states with additional defects. In particular, for large dissipation strengths a steady state with a fully mixed atomic sector is obtained crucially different from the predicted mean-field state.

Finite-Component Multicriticality at the Superradiant Quantum Phase Transition

We demonstrate the existence of finite-component multicriticality in a qubit-boson model where biased qubits collectively coupled to a single-mode bosonic field. The interplay between biases and boson-qubit coupling produces a rich phase diagram which shows multiple superradiant phases and phase boundaries of different orders. In particular, multiple phases become indistinguishable in appropriate bias configurations, which is the signature of multicriticality. A series of universality classes characterizing these multicritical points are identified. Moreover, we present a trapped-ion realization with the potential to explore multicritical phenomena experimentally using a small number of ions. The results open a novel way to probe multicritical universality classes in experiments.

Photoinduced Electron Pairing in a Driven Cavity

We demonstrate how virtual scattering of laser photons inside a cavity via two-photon processes can induce controllable long-range electron interactions in two-dimensional materials. We show that laser light that is red (blue) detuned from the cavity yields attractive (repulsive) interactions whose strength is proportional to the laser intensity. Furthermore, we find that the interactions are not screened effectively except at very low frequencies. For realistic cavity parameters, laser-induced heating of the electrons by inelastic photon scattering is suppressed and coherent electron interactions dominate. When the interactions are attractive, they cause an instability in the Cooper channel at a temperature proportional to the square root of the driving intensity. Our results provide a novel route for engineering electron interactions in a wide range of two-dimensional materials including AB-stacked bilayer graphene and the conducting interface between LaAlO_{3} and SrTiO_{3}.

Photon-Mediated Peierls Transition of a 1D Gas in a Multimode Optical Cavity

The Peierls instability toward a charge density wave is a canonical example of phonon-driven strongly correlated physics and is intimately related to topological quantum matter and exotic superconductivity. We propose a method for realizing an analogous photon-mediated Peierls transition, using a system of one-dimensional tubes of interacting Bose or Fermi atoms trapped inside a multimode confocal cavity. Pumping the cavity transversely engineers a cavity-mediated metal-to-insulator transition in the atomic system. For strongly interacting bosons in the Tonks-Girardeau limit, this transition can be understood (through fermionization) as being the Peierls instability. We extend the calculation to finite values of the interaction strength and derive analytic expressions for both the cavity field and mass gap. They display nontrivial power law dependence on the dimensionless matter-light coupling.

Extended Bose-Hubbard Model with Cavity-Mediated Infinite-Range Interactions at Finite Temperatures

We consider the finite-temperature properties of the extended Bose-Hubbard model realized recently in an ETH experiment [Nature 532, 476 (2016)]. Competing short- and global-range interactions accommodate fascinating collective phenomena. We formulate a self-consistent mean-field theory to describe the behaviors of the system at finite temperatures. At a fixed chemical potential, we map out the distributions of the superfluid order parameters and number densities with respect to the temperatures. For a charge density wave, we find that the global-range interaction enhances the charge order by increasing the transition temperature at which the charge order melts out, while for a supersolid phase, we find that the disappearance of the charge order and the superfluid order occurs at different temperature. At a fixed number-density filling factor, we extract the temperature dependence of the thermodynamic functions such as internal energy, specific heat and entropy. Across the superfluid phase transition, the specific heat has a discontinuous jump.

Overlapping two standing waves in a microcavity for a multi-atom photon interface

We develop a light-matter interface enabling strong and uniform coupling between a chain of cold atoms and photons of an optical cavity. This interface is a fiber Fabry-Perot cavity, doubly resonant for both the wavelength of the atomic transition and for a geometrically commensurate red-detuned intracavity trapping lattice. Fulfilling the condition of a strong and uniform atom-photon coupling requires optimization of the spatial overlap between the two standing waves in the cavity. In a strong-coupling cavity, where the mode waists and Rayleigh range are small, we derive the expression of the optimal trapping wavelength, taking into account the Gouy phase. The main parameter controlling the overlap of the standing waves is the relative phase shift at the reflection on the cavity mirrors between the two wavelengths, for which we derive the optimal value. We have built a microcavity optimized according to these results, employing custom-made mirrors with engineered reflection phase for both wavelengths. We present a method to measure with high precision the relative phase shift at reflection, which allows us to determine the spatial overlap of the two modes in this cavity.

Unraveling the Quantum Nature of Atomic Self-Ordering in a Ring Cavity

Atomic self-ordering to a crystalline phase in optical resonators is a consequence of the intriguing nonlinear dynamics of strongly coupled atom motion and photons. Generally the resulting phase diagrams and atomic states can be largely understood on a mean-field level. However, close to the phase transition point, quantum fluctuations and atom-field entanglement play a key role and initiate the symmetry breaking. Here we propose a modified ring cavity geometry, in which the asymmetry imposed by a tilted pump beam reveals clear signatures of quantum dynamics even in a larger regime around the phase transition point. Quantum fluctuations become visible both in the dynamic and steady-state properties. Most strikingly we can identify a regime where a mean-field approximation predicts a runaway instability, while in the full quantum model the quantum fluctuations of the light field modes stabilize uniform atomic motion. The proposed geometry thus allows to unveil the "quantumness" of atomic self-ordering via experimentally directly accessible quantities.

Feedback-Induced Quantum Phase Transitions Using Weak Measurements

We show that applying feedback and weak measurements to a quantum system induces phase transitions beyond the dissipative ones. Feedback enables controlling essentially quantum properties of the transition, i.e., its critical exponent, as it is driven by the fundamental quantum fluctuations due to measurement. Feedback provides the non-Markovianity and nonlinearity to the hybrid quantum-classical system, and enables simulating effects similar to spin-bath problems and Floquet time crystals with tunable long-range (long-memory) interactions.

Spin-Wave Multiplexed Atom-Cavity Electrodynamics

We introduce multiplexed atom-cavity quantum electrodynamics with an atomic ensemble coupled to a single optical cavity mode. Multiple Raman dressing beams establish cavity-coupled spin-wave excitations with distinctive spatial profiles. Experimentally, we demonstrate the concept by observing spin-wave vacuum Rabi splittings, selective superradiance, and interference in the cavity-mediated interactions of two spin waves. We highlight that the current experimental configuration allows rapid, interchangeable cavity coupling to 4 profiles with an overlap parameter of less than 10%, enough to demonstrate, for example, a quantum repeater network simulation in the cavity. With further improvements to the optical multiplexing setup, we infer the ability to access more than 10^{3} independent spin-wave profiles.

P-Band Induced Self-Organization and Dynamics with Repulsively Driven Ultracold Atoms in an Optical Cavity

We investigate a Bose-Einstein condensate strongly coupled to an optical cavity via a repulsive optical lattice. We detect a stable self-ordered phase in this regime, and show that the atoms order through an antisymmetric coupling to the P band of the lattice, limiting the extent of the phase and changing the geometry of the emergent density modulation. Furthermore, we find a nonequilibrium phase with repeated intense bursts of the intracavity photon number, indicating nontrivial driven-dissipative dynamics.

Emergent Quasicrystalline Symmetry in Light-Induced Quantum Phase Transitions

The discovery of quasicrystals with crystallographically forbidden rotational symmetries has changed the notion of the ordering in materials, yet little is known about the dynamical emergence of such exotic forms of order. Here we theoretically study a nonequilibrium cavity-QED setup realizing a zero-temperature quantum phase transition from a homogeneous Bose-Einstein condensate to a quasicrystalline phase via collective superradiant light scattering. Across the superradiant phase transition, collective light scattering creates a dynamical, quasicrystalline optical potential for the atoms. Remarkably, the quasicrystalline potential is "emergent" as its eightfold rotational symmetry is not present in the Hamiltonian of the system, rather appears solely in the low-energy states. For sufficiently strong two-body contact interactions between atoms, a quasicrystalline order is stabilized in the system, while for weakly interacting atoms the condensate is localized in one or few of the deepest minima of the quasicrystalline potential.

Dynamical Spin-Orbit Coupling of a Quantum Gas

We realize the dynamical 1D spin-orbit coupling (SOC) of a Bose-Einstein condensate confined within an optical cavity. The SOC emerges through spin-correlated momentum impulses delivered to the atoms via Raman transitions. These are effected by classical pump fields acting in concert with the quantum dynamical cavity field. Above a critical pump power, the Raman coupling emerges as the atoms superradiantly populate the cavity mode with photons. Concomitantly, these photons cause a backaction onto the atoms, forcing them to order their spin-spatial state. This SOC-inducing superradiant Dicke phase transition results in a spinor-helix polariton condensate. We observe emergent SOC through spin-resolved atomic momentum imaging and temporal heterodyne measurement of the cavity-field emission. Dynamical SOC in quantum gas cavity QED, and the extension to dynamical gauge fields, may enable the creation of Meissner-like effects, topological superfluids, and exotic quantum Hall states in coupled light-matter systems.

Treelike Interactions and Fast Scrambling with Cold Atoms

We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on nonlocal interactions that couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, nonrandom couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system.

Cavity-Mediated Unconventional Pairing in Ultracold Fermionic Atoms

We investigate long-range pairing interactions between ultracold fermionic atoms confined in an optical lattice which are mediated by the coupling to a cavity. In the absence of other perturbations, we find three degenerate pairing symmetries for a two-dimensional square lattice. By tuning a weak local atomic interaction via a Feshbach resonance or by tuning a weak magnetic field, the superfluid system can be driven from a topologically trivial s wave to topologically ordered, chiral superfluids containing Majorana edge states. Our work points out a novel path towards the creation of exotic superfluid states by exploiting the competition between long-range and short-range interactions.

Dynamical Phase Transitions to Optomechanical Superradiance

We theoretically analyze superradiant emission of light from an ultracold gas of bosonic atoms confined in a bad cavity. A metastable dipolar transition of the atoms couples to the cavity field and is incoherently pumped, and the mechanical effects of cavity-atom interactions tend to order the atoms in the periodic cavity potential. By means of a mean-field model we determine the conditions on the cavity parameters and pump rate that lead to the buildup of a stable macroscopic dipole emitting coherent light. We show that this occurs when the superradiant decay rate and the pump rate exceed threshold values of the order of the photon recoil energy. Above these thresholds superradiant emission is accompanied by the formation of stable matter-wave gratings that diffract the emitted photons. Outside of this regime, instead, the optomechanical coupling can give rise to dephasing or chaos, for which the emitted light is respectively incoherent or chaotic. These behaviors exhibit the features of a dynamical phase transitions and emerge from the interplay between global optomechanical interactions, quantum fluctuations, and noise.

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

The Hamiltonian mean-field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in one dimension and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE), which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wave function and the depth of the mean-field potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the mean-field potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.

Spatiotemporal Spin Noise Spectroscopy

We report on the potential of a new spin noise spectroscopy approach by demonstrating all-optical probing of spatiotemporal spin fluctuations. This is achieved by homodyne mixing of a spatially phase-modulated local oscillator with spin-flip scattered light, from which the frequency and wave vector dependence of the spin noise power is unveiled. As a first application of the method we measure the spatiotemporal spin noise in weakly n-doped CdTe layers, from which the electron spin diffusion constant and spin relaxation rates are determined. The absence of spatial spin correlations is also shown for this particular system.

Sign-Changing Photon-Mediated Atom Interactions in Multimode Cavity Quantum Electrodynamics

Sign-changing interactions constitute a crucial ingredient in the creation of frustrated many-body systems such as spin glasses. We present here the demonstration of a photon-mediated sign-changing interaction between Bose-Einstein-condensed atoms in a confocal cavity. The interaction between two atoms is of an unusual, nonlocal form proportional to the cosine of the inner product of the atoms' position vectors. This interaction arises from the differing Gouy phase shifts of the cavity's degenerate modes. The interaction drives a nonequilibrium Dicke-type phase transition in the system leading to atomic checkerboard density-wave order. Because of the Gouy phase anomalies, the checkerboard pattern can assume either a sinelike or cosinelike character. This state is detected via the holographic imaging of the cavity's superradiant emission. Together with a companion paper [Y. Guo, V. D. Vaidya, R. M. Kroeze, R. A. Lunney, B. L. Lev, and J. Keeling, Emergent and broken symmetries of atomic self-organization arising from Gouy phases in multimode cavity QED, Phys. Rev. A 99, 053818 (2019)PLRAAN2469-992610.1103/PhysRevA.99.053818], we explore this interaction's influence on superradiant phase transitions in multimode cavities. Employing this interaction in cavity QED spin systems may enable the creation of artificial spin glasses and quantum neural networks.

Dissipation-Induced Instabilities of a Spinor Bose-Einstein Condensate Inside an Optical Cavity

We investigate the dynamics of a spinor Bose-Einstein condensate inside an optical cavity, driven transversely by a laser with a controllable polarization angle. We focus on a two-component Dicke model with complex light-matter couplings, in the presence of photon losses. We calculate the steady-state phase diagram and find dynamical instabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes that generate nonreciprocal couplings between the two collective spins. Our predictions can be readily tested in state-of-the-art experiments and open up the study of nonreciprocal many-body dynamics out of equilibrium.

Cavity-Quantum-Electrodynamical Toolbox for Quantum Magnetism

The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas-cavity systems. Based on this platform, we introduce a generic scheme for the implementation of long-range quantum spin Hamiltonians composed of various types of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. Our model is composed of an effective two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double Λ scheme. Cavity photons mediate the long-range spin-spin interactions with spatially modulated coupling coefficients, where the latter ones can be tuned by modifying spatial profiles of the pump lasers. As experimentally relevant examples, we demonstrate that by properly choosing the spatial profiles of the pump lasers achiral domain-wall antiferromagnetic and chiral spin-spiral orders emerge beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss extensions of our scheme for the implementation of other classes of spin Hamiltonians.

Photon-Mediated Spin-Exchange Dynamics of Spin-1 Atoms

We report direct observations of photon-mediated spin-exchange interactions in an atomic ensemble. Interactions extending over a distance of 500  μm are generated within a cloud of cold rubidium atoms coupled to a single mode of light in an optical resonator. We characterize the system via quench dynamics and imaging of the local magnetization, verifying the coherence of the interactions and demonstrating optical control of their strength and sign. Furthermore, by initializing the spin-1 system in the m_{f}=0 Zeeman state, we observe correlated pair creation in the m_{f}=±1 states, a process analogous to spontaneous parametric down-conversion and to spin mixing in Bose-Einstein condensates. Our work opens new opportunities in quantum simulation with long-range interactions and in entanglement-enhanced metrology.