Quantum zigzag transition in ion chains

Two-dimensional supersolidity in a dipolar quantum gas

Supersolid states simultaneously feature properties typically associated with a solid and with a superfluid. Like a solid, they possess crystalline order, manifesting as a periodic modulation of the particle density; but unlike a typical solid, they also have superfluid properties, resulting from coherent particle delocalization across the system. Such states were initially envisioned in the context of bulk solid helium, as a possible answer to the question of whether a solid could have superfluid properties^1-5. Although supersolidity has not been observed in solid helium (despite much effort)⁶, ultracold atomic gases provide an alternative approach, recently enabling the observation and study of supersolids with dipolar atoms^7-16. However, unlike the proposed phenomena in helium, these gaseous systems have so far only shown supersolidity along a single direction. Here we demonstrate the extension of supersolid properties into two dimensions by preparing a supersolid quantum gas of dysprosium atoms on both sides of a structural phase transition similar to those occurring in ionic chains^17-20, quantum wires^21,22 and theoretically in chains of individual dipolar particles^23,24. This opens the possibility of studying rich excitation properties^25-28, including vortex formation^29-31, and ground-state phases with varied geometrical structure^7,32 in a highly flexible and controllable system.

Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum

We consider a zero-temperature one-dimensional system of bosons interacting via the soft-shoulder potential in the continuum, typical of dressed Rydberg gases. We employ quantum Monte Carlo simulations, which allow for the exact calculation of imaginary-time correlations, and a stochastic analytic continuation method, to extract the dynamical structure factor. At finite densities, in the weakly interacting homogeneous regime, a rotonic spectrum marks the tendency to clustering. With strong interactions, we indeed observe cluster liquid phases emerging, characterized by the spectrum of a composite harmonic chain. Luttinger theory has to be adapted by changing the reference lattice density field. In both the liquid and cluster liquid phases, we find convincing evidence of a secondary mode, which becomes gapless only at the transition. In that region, we also measure the central charge and observe its increase towards c=3/2, as recently evaluated in a related extended Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter. For two-particle clusters, we then interpret such observations in terms of the compresence of a Luttinger liquid and a critical transverse Ising model, related to the instability of the reference lattice density field towards coalescence of sites, typical of potentials which are flat at short distances. Even in the absence of a true lattice, we are able to evaluate the spatial correlation function of a suitable pseudospin operator, which manifests ferromagnetic order in the cluster liquid phase, exponential decay in the liquid phase, and algebraic order at criticality.

Revealing Quantum Statistics with a Pair of Distant Atoms

Quantum statistics have a profound impact on the properties of systems composed of identical particles. At the most elementary level, Bose and Fermi quantum statistics differ in the exchange phase, either 0 or π, which the wave function acquires when two identical particles are exchanged. In this Letter, we demonstrate that the exchange phase can be directly probed with a pair of massive particles by physically exchanging their positions. We present two protocols where the particles always remain spatially well separated, thus ensuring that the exchange contribution to their interaction energy is negligible and that the detected signal can only be attributed to the exchange symmetry of the wave function. We discuss possible implementations with a pair of trapped atoms or ions.

Observation of optomechanical buckling transitions

Correlated phases of matter provide long-term stability for systems as diverse as solids, magnets and potential exotic quantum materials. Mechanical systems, such as buckling transition spring switches, can have engineered, stable configurations whose dependence on a control variable is reminiscent of non-equilibrium phase transitions. In hybrid optomechanical systems, light and matter are strongly coupled, allowing engineering of rapid changes in the force landscape, storing and processing information, and ultimately probing and controlling behaviour at the quantum level. Here we report the observation of first- and second-order buckling transitions between stable mechanical states in an optomechanical system, in which full control of the nature of the transition is obtained by means of the laser power and detuning. The underlying multiwell confining potential we create is highly tunable, with a sub-nanometre distance between potential wells. Our results enable new applications in photonics and information technology, and may enable explorations of quantum phase transitions and macroscopic quantum tunnelling in mechanical systems.

Optomechanical systems could form logic gates, but key requirements are two stable static states and the ability to switch between them. Here, the authors observe radiation-pressure induced buckling transitions in an optomechanical system, and control this transition by varying laser power and detuning.

Realization of Translational Symmetry in Trapped Cold Ion Rings

We crystallize up to 15 ^{40}Ca^{+} ions in a ring with a microfabricated silicon surface Paul trap. Delocalization of the Doppler laser-cooled ions shows that the translational symmetry of the ion ring is preserved at millikelvin temperatures. By characterizing the collective motion of the ion crystals, we identify homogeneous electric fields as the dominant symmetry-breaking mechanism at this energy scale. With increasing ion numbers, such detrimental effects are reduced. We predict that, with only a ten-ion ring, uncompensated homogeneous fields will not break the translational symmetry of the rotational ground state. This experiment opens a door towards studying quantum many-body physics with translational symmetry at the single-particle level.

Crossover from Classical to Quantum Kibble-Zurek Scaling

The Kibble-Zurek (KZ) hypothesis identifies the relevant time scales in out-of-equilibrium dynamics of critical systems employing concepts valid at equilibrium: It predicts the scaling of the defect formation immediately after quenches across classical and quantum phase transitions as a function of the quench speed. Here, we study the crossover between the scaling dictated by a slow quench, which is ruled by the critical properties of the quantum phase transition, and the excitations due to a faster quench, where the dynamics is often well described by the classical model. We estimate the value of the quench rate that separates the two regimes and support our argument using numerical simulations of the out-of-equilibrium many-body dynamics. For the specific case of a ϕ^{4} model we demonstrate that the two regimes exhibit two different power-law scalings, which are in agreement with the KZ theory when applied to the quantum and classical cases. This result contributes to extending the prediction power of the Kibble-Zurek mechanism and to providing insight into recent experimental observations in systems of cold atoms and ions.

Exploring structural phase transitions of ion crystals

Phase transitions have been a research focus in many-body physics over past decades. Cold ions, under strong Coulomb repulsion, provide a repealing paradigm of exploring phase transitions in stable confinement by electromagnetic field. We demonstrate various conformations of up to sixteen laser-cooled (40)Ca(+) ion crystals in a home-built surface-electrode trap, where besides the usually mentioned structural phase transition from the linear to the zigzag, two additional phase transitions to more complicated two-dimensional configurations are identified. The experimental observation agrees well with the numerical simulation. Heating due to micromotion of the ions is analysed by comparison of the numerical simulation with the experimental observation. Our investigation implies very rich and complicated many-body behaviour in the trapped-ion systems and provides effective mechanism for further exploring quantum phase transitions and quantum information processing with ultracold trapped ions.

Subcriticality of the zigzag transition: A nonlinear bifurcation analysis

When repelling particles are confined by a transverse potential in quasi-one-dimensional geometry, the straight line equilibrium configuration becomes unstable at small confinement, in favor of a staggered row that may be inhomogeneous or homogeneous. This conformational phase transition is a pitchfork bifurcation called the zigzag transition. We study the zigzag transition in infinite and periodic finite systems with short-range interactions. We provide numerical evidence that in this case the bifurcation is subcritical since it exhibits phase coexistence and hysteretic behavior. The physical mechanism responsible for the change in the bifurcation character is the nonlinear coupling between the transverse soft mode at the transition and the longitudinal Goldstone mode linked to the translational or rotational invariance of the zigzag pattern. An asymptotic analysis, near the bifurcation threshold and assuming an infinite system, gives an explicit expression for the normal form of the bifurcation. We establish the subcriticality, and we describe with excellent precision the inhomogeneous zigzag patterns observed in the simulations. A direct test of the physical mechanism responsible for the bifurcation character evidences a quantitative agreement.

Two-dimensional spectroscopy for the study of ion coulomb crystals

Ion Coulomb crystals are currently establishing themselves as a highly controllable test bed for mesoscopic systems of statistical mechanics. The detailed experimental interrogation of the dynamics of these crystals, however, remains an experimental challenge. In this work, we show how to extend the concepts of multidimensional nonlinear spectroscopy to the study of the dynamics of ion Coulomb crystals. The scheme we present can be realized with state-of-the-art technology and gives direct access to the dynamics, revealing nonlinear couplings even in the presence of thermal excitations. We illustrate the advantages of our proposal showing how two-dimensional spectroscopy can be used to detect signatures of a structural phase transition of the ion crystal, as well as resonant energy exchange between modes. Furthermore, we demonstrate in these examples how different decoherence mechanisms can be identified.

Linear instability of a zigzag pattern

Interacting particles confined in a quasi-one-dimensional channel are physical systems which display various equilibrium patterns according to the interparticle interaction and the transverse confinement potential. Depending on the confinement, the particles may be distributed along a straight line, in a staggered row (zigzag), or in a configuration in which the linear and zigzag phases coexist (distorted zigzag). In order to clarify the conditions of existence of each configuration, we have studied the linear stability of the zigzag pattern. We find an acoustic transverse mode that destabilizes the zigzag configuration for short-range interaction potentials, and we calculate the interaction range above which this instability disappears. In particular, we recover the unconditional stability of zigzag patterns for Coulomb interactions. We show that the domain of existence for the distorted zigzag patterns is accurately described by our linear stability analysis. We also emphasize the complexity of finite size effects. Last, we provide a criterion for the onset of instability in the thermodynamic limit and propose a biphasic model that explains some characteristics of the distorted zigzag patterns.

Longitudinal and Transverse Single File Diffusion in Quasi-1D Systems

We review our recent results on Single File Diffusion (SFD) of a chain of particles that cannot cross each other, in a thermal bath, with long ranged interactions, and arbitrary damping. We exhibit new behaviors specifically associated to small systems and to small damping. The fluctuation dynamics is explained by the decomposition of the particles' motion in the normal modes of the chain. For longitudinal fluctuations, we emphasize the relevance of the soft mode linked to the translational invariance of the system to the long time SFD behavior. We show that close to the zigzag threshold, the transverse fluctuations also exhibit the SFD behavior, characterized by a mean square displacement that increases as the square root of time. This cannot be explained by the single file ordering, and the SFD behavior results from the strong correlation of the transverse displacements of neighbouring particles near the bifurcation. Extending our analytical modelization, we demonstrate the existence of this subdiffusive regime near the zigzag transition, in the thermodynamic limit. The zigzag transition is a supercritical pitchfork bifurcation, and we show that the transverse SFD behavior is closely linked to the vanishing of the frequency of the zigzag transverse mode at the bifurcation threshold.

[Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in files with longitudinal movements. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics,28 expanding files,63 and files with force and advanced formulations.29

Aharonov-Bohm effect in the tunnelling of a quantum rotor in a linear Paul trap

Quantum tunnelling is a common fundamental quantum mechanical phenomenon that originates from the wave-like characteristics of quantum particles. Although the quantum tunnelling effect was first observed 85 years ago, some questions regarding the dynamics of quantum tunnelling remain unresolved. Here we realize a quantum tunnelling system using two-dimensional ionic structures in a linear Paul trap. We demonstrate that the charged particles in this quantum tunnelling system are coupled to the vector potential of a magnetic field throughout the entire process, even during quantum tunnelling, as indicated by the manifestation of the Aharonov-Bohm effect in this system. The tunnelling rate of the structures periodically depends on the strength of the magnetic field, whose period is the same as the magnetic flux quantum φ0 through the rotor [(0.99 ± 0.07) × φ0].

Spatiotemporal oscillation patterns in the collective relaxation dynamics of interacting particles in periodic potentials

We demonstrate the emergence of self-organized structures in the course of the relaxation of an initially excited, dissipative, and finite chain of interacting particles in a periodic potential towards its many particle equilibrium configuration. Specifically, we observe a transition from an in phase correlated motion via phase randomized oscillations towards oscillations with a phase difference π between adjacent particles thereby yielding the growth of long time transient spatiotemporal oscillation patterns. Parameter modifications allow for designing these patterns, including steady states and even states that combine in phase and correlated out of phase oscillations along the chain. The complex relaxation dynamics is based on finite size effects together with an evolution running from the nonlinear to the linear regime, thereby providing a highly unbalanced population of the center-of-mass and relative motion.

Control of the conformations of ion Coulomb crystals in a Penning trap

Laser-cooled atomic ions form ordered structures in radiofrequency ion traps and in Penning traps. Here we demonstrate in a Penning trap the creation and manipulation of a wide variety of ion Coulomb crystals formed from small numbers of ions. The configuration can be changed from a linear string, through intermediate geometries, to a planar structure. The transition from a linear string to a zigzag geometry is observed for the first time in a Penning trap. The conformations of the crystals are set by the applied trap potential and the laser parameters, and agree with simulations. These simulations indicate that the rotation frequency of a small crystal is mainly determined by the laser parameters, independent of the number of ions and the axial confinement strength. This system has potential applications for quantum simulation, quantum information processing and tests of fundamental physics models from quantum field theory to cosmology.

Emulating solid-state physics with a hybrid system of ultracold ions and atoms

We propose and theoretically investigate a hybrid system composed of a crystal of trapped ions coupled to a cloud of ultracold fermions. The ions form a periodic lattice and induce a band structure in the atoms. This system combines the advantages of high fidelity operations and detection offered by trapped ion systems with ultracold atomic systems. It also features close analogies to natural solid-state systems, as the atomic degrees of freedom couple to phonons of the ion lattice, thereby emulating a solid-state system. Starting from the microscopic many-body Hamiltonian, we derive the low energy Hamiltonian, including the atomic band structure, and give an expression for the atom-phonon coupling. We discuss possible experimental implementations such as a Peierls-like transition into a period-doubled dimerized state.

Zigzag phase transition in quantum wires

We study the quantum phase transition of interacting electrons in quantum wires from a one-dimensional (1D) linear configuration to a quasi-1D zigzag arrangement using quantum Monte Carlo methods. As the density increases from its lowest values, first, the electrons form a linear Wigner crystal, then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase, and, finally, the electrons form a disordered liquidlike phase. We show that the linear to zigzag phase transition is not destroyed by the strong quantum fluctuations present in narrow wires; it has characteristics which are qualitatively different from the classical transition.

Trapping of topological-structural defects in Coulomb crystals

We study experimentally and theoretically structural defects which are formed during the transition from a laser cooled cloud to a Coulomb crystal, consisting of tens of ions in a linear radio frequency trap. We demonstrate the creation of predicted topological defects ("kinks") in purely two-dimensional crystals and also find kinks which show novel dynamical features in a regime of parameters not considered before. The kinks are always observed at the center of the trap, showing a large nonlinear localized excitation, and the probability of their occurrence saturates at ∼0.5. Simulations reveal a strong anharmonicity of the kink's internal mode of vibration, due to the kink's extension into three dimensions. As a consequence, the periodic Peierls-Nabarro potential experienced by a discrete kink becomes a globally confining potential, capable of trapping one cooled defect at the center of the crystal.

Structural transitions of ion strings in quantum potentials

We analyze the stability and dynamics of an ion chain confined inside a high-finesse optical resonator. When the dipolar transition of the ions strongly couples to one cavity mode, the mechanical effects of light modify the chain properties close to a structural transition. We focus on the linear chain close to the zigzag instability and show that linear and zigzag arrays are bistable for certain strengths of the laser pumping the cavity. For these regimes the chain is cooled into one of the configurations by cavity-enhanced photon scattering. The excitations of these structures mix photonic and vibrational fluctuations, which can be entangled at steady state. These features are signaled by Fano-like resonances in the spectrum of light at the cavity output.

Spin Peierls quantum phase transitions in Coulomb crystals

The spin Peierls instability describes a structural transition of a crystal due to strong magnetic interactions. Here, we demonstrate that cold Coulomb crystals of trapped ions provide an experimental test bed in which to study this complex many-body problem and to access extreme regimes where the instability is triggered by quantum fluctuations alone. We present a consistent analysis based on different analytical and numerical methods, and we provide a detailed discussion of its experimental feasibility.

Experimental quantum simulations of many-body physics with trapped ions

Direct experimental access to some of the most intriguing quantum phenomena is not granted due to the lack of precise control of the relevant parameters in their naturally intricate environment. Their simulation on conventional computers is impossible, since quantum behaviour arising with superposition states or entanglement is not efficiently translatable into the classical language. However, one could gain deeper insight into complex quantum dynamics by experimentally simulating the quantum behaviour of interest in another quantum system, where the relevant parameters and interactions can be controlled and robust effects detected sufficiently well. Systems of trapped ions provide unique control of both the internal (electronic) and external (motional) degrees of freedom. The mutual Coulomb interaction between the ions allows for large interaction strengths at comparatively large mutual ion distances enabling individual control and readout. Systems of trapped ions therefore exhibit a prominent system in several physical disciplines, for example, quantum information processing or metrology. Here, we will give an overview of different trapping techniques of ions as well as implementations for coherent manipulation of their quantum states and discuss the related theoretical basics. We then report on the experimental and theoretical progress in simulating quantum many-body physics with trapped ions and present current approaches for scaling up to more ions and more-dimensional systems.