Table of contents

A witch ball is a reflecting sphere of glass. Looking into the disk that it subtends, the whole sky can be seen at one glance. This feature can be exploited to see and photograph the squint Moon illusion, in which the direction normal to the illuminated face of the Moon—its 'attitude vector'—does not appear to point towards the Sun. The images of the Sun and Moon in the disk, the geodesic connecting them, the Moon's attitude, and the squint angle (distinct from the tilt), can be calculated and simulated, for all celestial configurations and viewing inclinations. The Moon direction antipodal to the Sun, corresponding to full Moon, is a singularity of the attitude vector field, with index +1. The main features of the witch ball images also occur in other ways of imaging the squint Moon.

Understanding the role of correlations in quantum systems is both a fundamental challenge as well as of high practical relevance for the control of multi-particle quantum systems. Whereas a lot of research has been devoted to study the various types of correlations that can be present in the states of quantum systems, in this work we introduce a general and rigorous method to quantify the amount of correlations in the dynamics of quantum systems. Using a resource-theoretical approach, we introduce a suitable quantifier and characterize the properties of correlated dynamics. Furthermore, we benchmark our method by applying it to the paradigmatic case of two atoms weakly coupled to the electromagnetic radiation field, and illustrate its potential use to detect and assess spatial noise correlations in quantum computing architectures.

The conical shape of a shuttlecock allows it to flip on impact. As a light and extended particle, it flies with a pure drag trajectory. We first study the flip phenomenon and the dynamics of the flight and then discuss the implications on the game. Lastly, a possible classification of different shots is proposed.

We present the hybrid anti-symmetrized coupled channels method for the calculation of fully differential photo-electron spectra of multi-electron atoms and small molecules interacting with strong laser fields. The method unites quantum chemical few-body electronic structure with strong-field dynamics by solving the time dependent Schrödinger equation in a fully anti-symmetrized basis composed of multi-electron states from quantum chemistry and a one-electron numerical basis. Photoelectron spectra are obtained via the time dependent surface flux (tSURFF) method. Performance and accuracy of the approach are demonstrated for spectra from the helium and beryllium atoms and the hydrogen molecule in linearly polarized laser fields at wavelengths from 21 to 400 nm. At long wavelengths, helium and the hydrogen molecule at equilibrium inter-nuclear distance can be approximated as single channel systems whereas beryllium needs a multi-channel description.

We analyze electron paths in a solid-state double-slit interferometer based on two-dimensional electron gas and mapping by scanning gate microscopy (SGM). A device with a quantum point source contact of a split exit and a drain contact for electron detection is considered. We study the SGM maps of source-drain conductance (G) as functions of the probe position, and we find that for a narrow drain, the classical electron paths are clearly resolved without any trace of double-slit interference. The latter is only present in the SGM maps of backscattering (R) probability. Double-slit interference is found in the G maps for a wider drain contact, but at the expense of a loss of information on the electron trajectories. We discuss the interplay of Young's interference and interference effects between various electron paths introduced by the tip and the electron detector. The stability of the G and R maps versus the geometry parameters of the scattering device is also discussed.

We report the experimental demonstration of femtosecond electron diffraction using high-brightness MeV electron beams. High-quality, single-shot electron diffraction patterns for both polycrystalline aluminum and single-crystal 1T-TaS₂ are obtained utilizing a 5 fC (∼3 × 10⁴ electrons) pulse of electrons at 2.8 MeV. The high quality of the electron diffraction patterns confirms that electron beam has a normalized emittance of ∼50 nm rad. The transverse and longitudinal coherence length is ∼11 and ∼2.5 nm, respectively. The timing jitter between the pump laser and probe electron beam was found to be ∼100 fs (rms). The temporal resolution is demonstrated by observing the evolution of Bragg and superlattice peaks of 1T-TaS₂ following an 800 nm optical pump and was found to be 130 fs. Our results demonstrate the advantages of MeV electrons, including large elastic differential scattering cross-section and access to high-order reflections, and the feasibility of ultimately realizing below 10 fs time-resolved electron diffraction.

Lattice gauge theory has provided a crucial non-perturbative method in studying canonical models in high-energy physics such as quantum chromodynamics. Among other models of lattice gauge theory, the lattice gauge–Higgs model is a quite important one because it describes a wide variety of phenomena/models related to the Anderson–Higgs mechanism, such as superconductivity, the standard model of particle physics, and the inflation process of the early Universe. In this paper, we first show that atomic description of the lattice gauge model allows us to explore real-time dynamics of the gauge variables by using the Gross–Pitaevskii equations. Numerical simulations of the time development of an electric flux reveal some interesting characteristics of the dynamic aspect of the model and determine its phase diagram. Next, to realize a quantum simulator of the U(1) lattice gauge–Higgs model on an optical lattice filled by cold atoms, we propose two feasible methods: (i) Wannier states in the excited bands and (ii) dipolar atoms in a multilayer optical lattice. We pay attention to the constraint of Gauss's law and avoid nonlocal gauge interactions.

In the present paper, recent experimental results on large-scale coherent steady states observed in experimental von Kármán flows are revisited from a statistical mechanics perspective. The latter is rooted on two levels of description. We first argue that the coherent steady states may be described as the equilibrium states of well-chosen lattice models, which can be used to define global properties of von Kármán flows, such as their temperatures. The equilibrium description is then enlarged, in order to reinterpret a series of results about the stability of those steady states and their susceptibility to symmetry breaking, in the light of a deep analogy with the statistical theory of ferro-magnetism.

The slime mold Dictyostelium discoideum is a well known model system for the study of biological pattern formation. In the natural environment, aggregating populations of starving Dictyostelium discoideum cells may experience fluid flows that can profoundly change the underlying wave generation process. Here we study the effect of advection on the pattern formation in a colony of homogeneously distributed Dictyostelium discoideum cells described by the standard Martiel–Goldbeter model. The external flow advects the signaling molecule cyclic adenosine monophosphate (cAMP) downstream, while the chemotactic cells attached to the solid substrate are not transported with the flow. The evolution of small perturbations in cAMP concentrations is studied analytically in the linear regime and by corresponding numerical simulations. We show that flow can significantly influence the dynamics of the system and lead to a flow-driven instability that initiate downstream traveling cAMP waves. We also show that boundary conditions have a significant effect on the observed patterns and can lead to a new kind of instability.

Quantum Key Distribution is a quantum communication technique in which random numbers are encoded on quantum systems, usually photons, and sent from one party, Alice, to another, Bob. Using the data sent via the quantum signals, supplemented by classical communication, it is possible for Alice and Bob to share an unconditionally secure secret key. This is not possible if only classical signals are sent. While this last statement is a long standing result from quantum information theory it turns out only to be true in a non-relativistic setting. If relativistic quantum field theory is considered we show it is possible to distribute an unconditionally secure secret key without sending a quantum signal, instead harnessing the intrinsic entanglement between different regions of space–time. The protocol is practical in free space given horizon technology and might be testable in principle in the near term using microwave technology.

A three-orbital itinerant-electron model involving d_xz, d_yz and d_xy Fe 3d orbitals is proposed for iron pnictides towards understanding the ($\pi ,0$) ordered magnetism and magnetic excitations in these materials. It is shown that this model at half filling yields a gapped ($\pi ,0$) magnetic state, and simultaneously reproduces several experimentally observed features such as the electronic structure, spin excitations, as well as the ferro orbital order between the d_xz and d_yz orbitals.

We theoretically investigate the dependence of the enhancement of optical near-fields at nanometric tips on the shape, size, and material of the tip. We confirm the strong dependence of the field enhancement factor on the radius of curvature. In addition, we find a surprisingly strong increase of field enhancement with increasing opening angle of the nanotips. For gold and tungsten nanotips in the experimentally relevant parameter range (radius of curvature $\geqslant 5\;{\rm nm}$ at $800\;{\rm nm}$ laser wavelength), we obtain field enhancement factors of up to $\sim 35$ for Au and $\sim 12$ for W for large opening angles. We confirm this strong dependence on the opening angle for many other materials featuring a wide variety in their dielectric response. For dielectrics, the opening angle dependence is traced back to the electrostatic force of the induced surface charge at the tip shank. For metals, the plasmonic response strongly increases the field enhancement and shifts the maximum field enhancement to smaller opening angles.

The passage of cold cesium 49S$_{1/2}$ Rydberg atoms through an electric-field-induced multi-level avoided crossing with nearby hydrogen-like Rydberg levels is employed to prepare a cold, dipolar Rydberg atom gas. When the electric field is ramped through the avoided crossing on time scales on the order of 100 ns or slower, the 49S$_{1/2}$ population adiabatically transitions into high-l Rydberg Stark states. The adiabatic state transformation results in a cold gas of Rydberg atoms with large electric dipole moments. After a waiting time of about 1 μs and at sufficient atom density, the adiabatically transformed highly dipolar atoms become undetectable, enabling us to discern adiabatic from diabatic passage behavior through the avoided crossing. We attribute the state-selectivity to m-mixing collisions between the dipolar atoms. The data interpretation is supported by numerical simulations of the passage dynamics and of binary m-mixing collisions.

In ultracold atoms settings, inelastic light scattering is a preeminent technique to reveal static and dynamic properties at nonzero momentum. In this work, we investigate an array of one-dimensional trapped Bose gases, by measuring both the energy and the momentum imparted to the system via light scattering experiments. The measurements are performed in the weak perturbation regime, where these two quantities—the energy and momentum transferred—are expected to be related to the dynamic structure factor of the system. We discuss this relation, with special attention to the role of in-trap dynamics on the transferred momentum.

A number of novel experimental and theoretical results have recently been obtained on active soft matter, demonstrating the various interesting universal and anomalous features of this kind of driven systems. Here we consider the adhesion difference-driven segregation of actively moving units, a fundamental but still poorly explored aspect of collective motility. In particular, we propose a model in which particles have a tendency to adhere through a mechanism which makes them both stay in touch and synchronize their direction of motion—but the interaction is limited to particles of the same kind. The calculations corresponding to the related differential equations can be made in parallel, thus a powerful GPU card allows large scale simulations. We find that in a very large system of particles, interacting without explicit alignment rule, three basic segregation regimes seem to exist as a function of time: (i) at the beginning the time dependence of the correlation length is analogous to that predicted by the Cahn–Hilliard theory, (ii) next rapid segregation occurs characterized with a separation of the different kinds of units being faster than any previously suggested speed, finally, (iii) the growth of the characteristic sizes in the system slows down due to a new regime in which self-confined, rotating, splitting and re-joining clusters appear. Our results can explain recent observations of segregating tissue cells in vitro.

We revisit the problem of the surface plasmon polariton (SPP) propagation at a metal–magneto-optical dielectric interface and predict a conceptually different regime of one-way surface wave propagation. We show that in the presence of magnetization, the nonreciprocal crossing of the light-line by dispersion curve of SPPs is possible. This leads to the formation of leaky-wave surface states in one of the propagation directions, whereas in other direction SPPs are confined to the interface. We show that such a regime of surface plasmon propagation is fundamentally different from one-way regimes of surface waves predicted earlier.

We report the observation of a universal behavior of ultracold quantum critical (QC) Bose gases trapped in a one-dimensional optical lattice. We extract the density probability distributions from the measured atomic density of the rubidium Bose gases after a free expansion time. The density probability is found to follow a simple exponential law once the lattice system has entered the QC region above the Berezinskii–Kosterlitz–Thouless transition. We show that, in addition to relative phase fluctuations between the subcondensates in different lattice wells, there also exist spatial phase fluctuations within single lattice wells in this QC region. The universal density-probability distribution can thus be well understood by a simple theoretical model taking into account these two kinds of phase fluctuations.

In recent years inelastic spin-flip spectroscopy using a low-temperature scanning tunneling microscope has been a very successful tool for studying not only individual spins but also complex coupled systems. When these systems interact with the electrons of the supporting substrate correlated many-particle states can emerge, making them ideal prototypical quantum systems. The spin systems, which can be constructed by arranging individual atoms on appropriate surfaces or embedded in synthesized molecular structures, can reveal very rich spectral features. Up to now the spectral complexity has only been partly described. This manuscript shows that perturbation theory enables one to describe the tunneling transport, reproducing the differential conductance with surprisingly high accuracy. Well established scattering models, which include Kondo-like spin–spin and potential interactions, are expanded to enable calculation of arbitrary complex spin systems in reasonable time scale and the extraction of important physical properties. The emergence of correlations between spins and, in particular, between the localized spins and the supporting bath electrons are discussed and related to experimentally tunable parameters. These results might stimulate new experiments by providing experimentalists with an easily applicable modeling tool.

The emergence of an effective field theory out of equilibrium is studied in the case in which a light field—the system—interacts with very heavy fields in a finite temperature bath. We obtain the reduced density matrix for the light field, its time evolution is determined by an effective action that includes the influence action from correlations of the heavy degrees of freedom. The non-equilibrium effective field theory yields a Langevin equation of motion for the light field in terms of dissipative and noise kernels that obey a generalized fluctuation dissipation relation. These are completely determined by the spectral density of the bath which is analyzed in detail for several cases. At T = 0 we elucidate the effect of thresholds in the renormalization aspects and the asymptotic emergence of a local effective field theory with unitary time evolution. At $T\ne 0$ new 'anomalous' thresholds arise, in particular the decay of the environmental heavy fields into the light field leads to dissipative dynamics of the light field. Even when the heavy bath particles are thermally suppressed this dissipative contribution leads to the thermalization of the light field which is confirmed by a quantum kinetics analysis. We obtain the quantum master equation and show explicitly that its solution in the field basis is precisely the influence action that determines the effective non-equilibrium field theory. The Lindblad form of the quantum master equation features time dependent dissipative coefficients. Their time dependence is crucial to extract renormalization effects at asymptotically long time. The dynamics from the quantum master equation is in complete agreement with that of the effective action, Langevin dynamics and quantum kinetics, thus providing a unified framework to effective field theory out of equilibrium.

We describe the realization of multimode phononic correlations that arise from nonlinear interactions in a mechanical nondegenerate parametric amplifier. The nature of these correlations differs qualitatively depending on the strength of the driving field in relation to the threshold for parametric instability. Below this threshold, the correlations are manifest in a combined quadrature of the coupled mechanical modes. In this regime, the system is amenable to back-action evading measurement schemes for the detection of weak forces. Above threshold, the correlations are manifest in the amplitude difference between the two mechanical modes, akin to intensity difference squeezing observed in optical parametric oscillators. We discuss the crossover of correlations between these two regimes and applications of this quantum-compatible mechanical system to nonlinear metrology and out-of-equilibrium dynamics.

We develop a general theoretical approach to describing the interaction of metamaterials with optical beams. The metamaterials are allowed to be anisotropic, chiral, noncentrosymmetric, and spatially dispersive. Unlike plane waves, beams can change their field distributions upon interaction with metamaterials, which can reveal new optical effects. Our method is based on a vector form of the angular spectrum representation and a technique to calculate the wave parameters for all required directions of wave propagation. Applying the method to various metamaterial designs, we discover a new optical phenomenon: the conversion of light polarization by spatial dispersion. Because of this phenomenon, the refractive index and impedance cannot be introduced for many metamaterial designs. In such cases, we propose an alternative approach to treating the beam–metamaterial interaction. This work takes a step forward in describing optical metamaterials by moving from unphysical plane waves to realistic optical beams.

Quantum parameter estimation is central to many fields such as quantum computation, communications and metrology. Optimal estimation theory has been instrumental in achieving the best accuracy in quantum parameter estimation, which is possible when we have very precise knowledge of and control over the model. However, uncertainties in key parameters underlying the system are unavoidable and may impact the quality of the estimate. We show here how quantum optical phase estimation of a squeezed state of light exhibits improvement when using a robust fixed-interval smoother designed with uncertainties explicitly introduced in parameters underlying the phase noise.

We study the spreading of correlations and other physical quantities in quantum lattice models with interactions or hopping decaying like ${r}^{-\alpha }$ with the distance r. Our focus is on exponents α between 0 and 6, where the interplay of long- and short-range features gives rise to a complex phenomenology and interesting physical effects, and which is also the relevant range for experimental realizations with cold atoms, ions, or molecules. We present analytical and numerical results, providing a comprehensive picture of spatio-temporal propagation. Lieb–Robinson-type bounds are extended to strongly long-range interactions where α is smaller than the lattice dimension, and we report particularly sharp bounds that are capable of reproducing regimes with soundcone as well as supersonic dynamics. Complementary lower bounds prove that faster-than-soundcone propagation occurs for $\alpha \lt 2$ in any spatial dimension, although cone-like features are shown to also occur in that regime. Our results provide guidance for optimizing experimental efforts to harness long-range interactions in a variety of quantum information and signaling tasks.

We demonstrate that the deposition of a self-assembled monolayer of alkanethiolates on a 1 nm thick cobalt ultrathin film grown on Au(111) induces a spin reorientation transition from in-plane to out-of-plane magnetization. Using ab initio calculations, we show that a methanethiolate layer changes slightly both the magnetocrystalline and shape anisotropy, both effects almost cancelling each other out for a 1 nm Co film. Finally, the change in hysteresis cycles upon alkanethiolate adsorption could be assigned to a molecular-induced roughening of the Co layer, as shown by STM. In addition, we calculate how a methanethiolate layer modifies the spin density of states of the Co layer and we show that the spin polarization at the Fermi level through the organic layer is reversed as compared to the uncovered Co. These results give new theoretical and experimental insights for the use of thiol-based self-assembled monolayers in spintronic devices.

Using ab initio molecular dynamics simulation, we investigate the supercritical phenomenon associated with the liquid–liquid phase transition of hydrogen by studying the isothermal response functions, such as electric conductivity, molecular dissociation coefficient and isothermal compressibility, with respect to pressure. We find that, along each isotherm in the supercritical region, each of these response functions shows a maximum, the location of which is different for different response functions. As temperature decreases, the loci of these maxima asymptotically converge to a line of zero ordering field, known as the Widom line along which the magnitude of the response function maxima becomes larger and larger until it diverges as the critical point is approached. Thus, our study provides a possible way to locate the liquid–liquid critical point of hydrogen from the supercritical region at lower pressures. It also indicates that the supercritical phonomenon near the critical point of hydrogen is a rather general feature of second-order phase transition, it is not only true for classical systems with weak interactions but also true for highly condensed system with strong inter-atomic interactions.

Temperature dependent Raman spectra and photoluminescence, as well the Raman mapping of different parts in an individual Sn-doped CdS comb-like nanostructure reveal that the stronger electron–phonon coupling exist at the trunk-branch junctions than other parts. The Huang–Rhys factor was calculated and further confirms that the strong electron–phonon correlation at the junction. The localized deformation from the Sn dopant in the lattice leads to strong electron–phonon coupling at the local junction, which is proved by the scanning transmission electron microscope and energy dispersion spectrum. Moreover, the lifetime of near-band-edge emission and deep-level emission drastically increase with decreasing temperature, which both relate to the localized electron–phonon coupling and relaxation process. This work provides a clear description of the localized carrier correlation in the cross junction part of these branched nanostructures, which can be used to modulate the optoelectronic performance of micro/nanodevices.

Interdependent networks in areas ranging from infrastructure to economics are ubiquitous in our society, and the study of their cascading behaviors using percolation theory has attracted much attention in recent years. To analyze the percolation phenomena of these systems, different mathematical frameworks have been proposed, including generating functions and eigenvalues, and others. These different frameworks approach phase transition behaviors from different angles and have been very successful in shaping the different quantities of interest, including critical threshold, size of the giant component, order of phase transition, and the dynamics of cascading. These methods also vary in their mathematical complexity in dealing with interdependent networks that have additional complexity in terms of the correlation among different layers of networks or links. In this work, we review a particular approach of simple, self-consistent probability equations, and we illustrate that this approach can greatly simplify the mathematical analysis for systems ranging from single-layer network to various different interdependent networks. We give an overview of the detailed framework to study the nature of the critical phase transition, the value of the critical threshold, and the size of the giant component for these different systems.

Homogeneous and heterogeneous nucleations in a reduced-dimensional system undergoing a first-order structural phase transition were examined by using low electron energy diffraction and scanning tunneling microscopy. The high-temperature 4 × 1 phase of a Si(111)-In surface was supercooled at temperatures below the transition temperature (${T}_{{\rm{c}}}$) and evolved slowly into a low-temperature 8 × 2 phase with time. The transition rate decreased significantly as the temperature approached ${T}_{{\rm{c}}}$. The kinetics of the observed homogeneous nucleation was analyzed by classical nucleation theory. The introduction of oxygen atoms reduced the hysteresis and accelerated nucleation significantly, showing that the ${T}_{{\rm{c}}}$-raising oxygen impurity plays the role of a nucleation seed for heterogeneous nucleation.

We present a classical linear response theory for a magneto–dielectric material and determine the polariton dispersion relations. The electromagnetic field fluctuation spectra are obtained and polariton sum rules for their optical parameters are presented. The electromagnetic field for systems with multiple polariton branches is quantized in three dimensions and field operators are converted to 1–dimensional forms appropriate for parallel light beams. We show that the field–operator commutation relations agree with previous calculations that ignored polariton effects. The Abraham (kinetic) and Minkowski (canonical) momentum operators are introduced and their corresponding single–photon momenta are identified. The commutation relations of these and of their angular analogues support the identification, in particular, of the Minkowski momentum with the canonical momentum of the light. We exploit the Heaviside–Larmor symmetry of Maxwell's equations to obtain, very directly, the Einsetin–Laub force density for action on a magneto–dielectric. The surface and bulk contributions to the radiation pressure are calculated for the passage of an optical pulse into a semi–infinite sample.

We report results of Reynolds-number measurements, based on multi-point temperature measurements and the elliptic approximation (EA) of He and Zhang (2006 Phys. Rev. E 73 055303), Zhao and He (2009 Phys. Rev. E 79 046316) for turbulent Rayleigh–Bénard convection (RBC) over the Rayleigh-number range ${10}^{11}\lesssim {\text{}}{Ra}\lesssim 2\times {10}^{14}$ and for a Prandtl number Pr ≃ 0.8. The sample was a right-circular cylinder with the diameter D and the height L both equal to 112 cm. The Reynolds numbers Re_U and Re_V were obtained from the mean-flow velocity U and the root-mean-square fluctuation velocity V, respectively. Both were measured approximately at the mid-height of the sample and near (but not too near) the side wall close to a maximum of Re_U. A detailed examination, based on several experimental tests, of the applicability of the EA to turbulent RBC in our parameter range is provided. The main contribution to Re_U came from a large-scale circulation in the form of a single convection roll with the preferred azimuthal orientation of its down flow nearly coinciding with the location of the measurement probes. First we measured time sequences of Re_U(t) and Re_V(t) from short (10 s) segments which moved along much longer sequences of many hours. The corresponding probability distributions of Re_U(t) and Re_V(t) had single peaks and thus did not reveal significant flow reversals. The two averaged Reynolds numbers determined from the entire data sequences were of comparable size. For ${\text{}}{Ra}\lt {\text{}}{{Ra}}_{1}^{*}\simeq 2\times {10}^{13}$ both Re_U and Re_V could be described by a power-law dependence on Ra with an exponent ζ close to 0.44. This exponent is consistent with several other measurements for the classical RBC state at smaller Ra and larger Pr and with the Grossmann–Lohse (GL) prediction for Re_U (Grossmann and Lohse 2000 J. Fluid. Mech.407 27; Grossmann and Lohse 2001 86 3316; Grossmann and Lohse 2002 66 016305) but disagrees with the prediction $\zeta \simeq 0.33$ by GL (Grossmann and Lohse 2004 Phys. Fluids16 4462) for Re_V. At ${\text{}}{Ra}={\text{}}{{Ra}}_{2}^{*}\simeq 7\times {10}^{13}$ the dependence of Re_V on Ra changed, and for larger Ra${\text{}}{{Re}}_{V}\sim {\text{}}{{Ra}}^{0.50\pm 0.02}$, consistent with the prediction for Re_U (Grossmann and Lohse 2000 J. Fluid. Mech.407 27; Grossmann and Lohse Phys. Rev. Lett. 2001 86 3316; Grossmann and Lohse Phys. Rev. E 2002 66 016305; Grossmann and Lohse 2012 Phys. Fluids24 125103) in the ultimate state of RBC.

Supply-demand processes take place on a large variety of real-world networked systems ranging from power grids and the internet to social networking and urban systems. In a modern infrastructure, supply-demand systems are constantly expanding, leading to constant increase in load requirement for resources and consequently, to problems such as low efficiency, resource scarcity, and partial system failures. Under certain conditions global catastrophe on the scale of the whole system can occur through the dynamical process of cascading failures. We investigate optimization and resilience of time-varying supply-demand systems by constructing network models of such systems, where resources are transported from the supplier sites to users through various links. Here by optimization we mean minimization of the maximum load on links, and system resilience can be characterized using the cascading failure size of users who fail to connect with suppliers. We consider two representative classes of supply schemes: load driven supply and fix fraction supply. Our findings are: (1) optimized systems are more robust since relatively smaller cascading failures occur when triggered by external perturbation to the links; (2) a large fraction of links can be free of load if resources are directed to transport through the shortest paths; (3) redundant links in the performance of the system can help to reroute the traffic but may undesirably transmit and enlarge the failure size of the system; (4) the patterns of cascading failures depend strongly upon the capacity of links; (5) the specific location of the trigger determines the specific route of cascading failure, but has little effect on the final cascading size; (6) system expansion typically reduces the efficiency; and (7) when the locations of the suppliers are optimized over a long expanding period, fewer suppliers are required. These results hold for heterogeneous networks in general, providing insights into designing optimal and resilient complex supply-demand systems that expand constantly in time.

Gravitational waves (GWs) imprint apparent Doppler shifts on the frequency of photons propagating between an emitter and detector of light. This forms the basis of a method to detect GWs using Doppler velocimetry between pairs of satellites. Operating in the micro-hertz to milli-hertz gravitational frequency band, this method could lead to the direct detection of GWs. The crucial component in such detectors is the frequency standard on board the emitting and receiving satellites. Recent developments in atomic frequency standards have led to devices that are approaching the sensitivity required to detect GWs from astrophysically interesting sources. The sensitivity of satellites equipped with optical frequency standards for Doppler velocimetry is examined, and a design for a robust, space-capable optical frequency standard is presented.

We study optimal control strategies to optimize the relaxation rate towards the fixed point of a quantum system in the presence of a non-Markovian (NM) dissipative bath. Contrary to naive expectations that suggest that memory effects might be exploited to improve optimal control effectiveness, NM effects influence the optimal strategy in a non trivial way: we present a necessary condition to be satisfied so that the effectiveness of optimal control is enhanced by NM subject to suitable unitary controls. For illustration, we specialize our findings for the case of the dynamics of single qubit amplitude damping channels. The optimal control strategy presented here can be used to implement optimal cooling processes in quantum technologies and may have implications in quantum thermodynamics when assessing the efficiency of thermal micro-machines.

Using the time-dependent Ginzburg–Landau approach, we analyze vortex states and vortex dynamics in type-I superconductor films with a thickness gradient in one direction. In the thinnest part of the structures under consideration, the equilibrium states manifest the typical type-II vortex patterns with only singly-quantized vortices. At the same time, in the regions with larger thickness the singly-quantized and giant vortices coexist, in a qualitative agreement with our scanning Hall probe microscopy measurements on relatively thick Pb films. In the presence of an external current applied perpendicularly to the thickness gradient direction, the singly-quantized vortices, which enter the wedge through its thinnest edge, merge into giant vortices when propagating to the thicker parts of the structure. Remarkably, the results of our simulations imply that at moderate external current densities a regime is possible where the winding number of giant vortices, formed as a result of vortex coalescence, takes preferentially (or even exclusively) the values given by positive integer powers of two.

We present a thorough analysis of soliton solutions to the quasi-one-dimensional (quasi-1D) nonlinear Dirac equation (NLDE) for a Bose–Einstein condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds to a quasi-1D reduction of the honeycomb lattice along the zigzag direction, in direct analogy to graphene nanoribbons. Excitations in the remaining large direction of the lattice correspond to the linear subbands in the armchair nanoribbon spectrum. Analytical as well as numerical soliton Dirac spinor solutions are obtained. We analyze the solution space of the quasi-1D NLDE by finding fixed points, delineating the various regions in solution space, and through an invariance relation which we obtain as a first integral of the NLDE. We obtain spatially oscillating multi-soliton solutions as well as asymptotically flat single soliton solutions using five different methods: by direct integration; an invariance relation; parametric transformation; a series expansion; and by numerical shooting. By tuning the ratio of the chemical potential to the nonlinearity for a fixed value of the energy–momentum tensor, we can obtain both bright and dark solitons over a nonzero density background.

The nonlinear Dirac equation for Bose–Einstein condensates (BECs) in honeycomb optical lattices gives rise to relativistic multi-component bright and dark soliton solutions. Using the relativistic linear stability equations, the relativistic generalization of the Boguliubov-de Gennes equations, we compute soliton lifetimes against quantum fluctuations and classify the different excitation types. For a BEC of ${}^{87}\mathrm{Rb}$ atoms, we find that our soliton solutions are stable on time scales relevant to experiments. Excitations in the bulk region far from the core of a soliton and bound states in the core are classified as either spin waves or as a Nambu–Goldstone mode. Thus, solitons are topologically distinct pseudospin-$1/2$ domain walls between polarized regions of ${S}_{z}=\pm 1/2$. Numerical analysis in the presence of a harmonic trap potential reveals a discrete spectrum reflecting the number of bright soliton peaks or dark soliton notches in the condensate background. For each quantized mode the chemical potential versus nonlinearity exhibits two distinct power law regimes corresponding to the free-particle (weakly nonlinear) and soliton (strongly nonlinear) limits.

When a quantum state is subjected to frequent measurements, the time evolution of the quantum state is frozen. This is called the quantum Zeno effect. Here, we observe such an effect by performing frequent discrete measurements in a macroscopic quantum system, a superconducting quantum bit. The quantum Zeno effect induced by discrete measurements is similar to the original idea of the quantum Zeno effect. By using a Josephson bifurcation amplifier pulse readout, we have experimentally suppressed the time evolution of Rabi oscillation using projective measurements, and also observed the enhancement of the quantum state holding time by shortening the measurement period time. This is a crucial step to realize quantum information processing using the quantum Zeno effect.

High conversion efficiency between electrical and optical power is highly desirable both for high peak and high average power radiation sources. In this paper we discuss a new mechanism based on stimulated superradiant emission in a strongly tapered undulator whereby a prebunched electron beam and focused laser are injected into an undulator with an optimal tapering calculated by dynamically matching the resonant energy variation to the ponderomotive decelerating gradient. The method has the potential to allow the extraction of a large fraction (∼50%) of power from a relativistic electron beam by converting it into coherent narrow-band tunable radiation, and shows a clear path to very high power radiation sources of EUV and hard x-rays for applications such as lithography and single molecule x-ray diffraction. Finally, we discuss a technique using chicane delays to suppress the sideband instability, improving radiation generation efficiencies for interaction lengths many synchrotron wavelengths long.

We introduce an operational discord-type measure for quantifying nonclassical correlations in bipartite Gaussian states based on using Gaussian measurements. We refer to this measure as operational Gaussian discord (OGD). It is defined as the difference between the entropies of two conditional probability distributions associated to one subsystem, which are obtained by performing optimal local and joint Gaussian measurements. We demonstrate the operational significance of this measure in terms of a Gaussian quantum protocol for extracting maximal information about an encoded classical signal. As examples, we calculate OGD for several Gaussian states in the standard form.

We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form $D(t)\simeq 1/t$. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.

A natural way to generalize tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled-pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class' definition must be Lorentz-invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.

We show that all Dirichlet series, linear combinations of them and their analytical continuations represent probability amplitudes for measurements on time-dependent quantum systems. In particular, we connect an arbitrary Dirichlet series to the time evolution of an appropriately prepared quantum state in a non-linear oscillator with logarithmic energy spectrum. However, the realization of a superposition of two Dirichlet sums and its analytical continuation requires two quantum systems which are entangled, and a joint measurement. We illustrate our approach of implementing arbitrary Dirichlet series in quantum systems using the example of the Riemann zeta function and relate its non-trivial zeros to the interference of two quantum states reminiscent of a Schrödinger cat.

We envision that the quantum spin Hall effect should be observed in (111)-oriented thin films of SnSe and SnTe topological crystalline insulators. Using a tight-binding approach supported by first-principles calculations of the band structures, we demonstrate that in these films the energy gaps in the two-dimensional band spectrum depend in an oscillatory fashion on the layer thickness. These results as well as the calculated topological invariant indexes and edge state spin polarizations show that for films ∼20–40 monolayers thick a two-dimensional topological insulator phase appears. In this range of thicknesses in both SnSe and SnTe, (111)-oriented films edge states with Dirac cones with opposite spin polarization in their two branches are obtained. While in the SnTe layers a single Dirac cone appears at the projection of the ${\boldsymbol{}}\bar{\Gamma }$ point of the two-dimensional Brillouin zone, in the SnSe (111)-oriented layers three Dirac cones at ${\boldsymbol{}}\bar{M}$ points projections are predicted.

Quantum sensors based on coherent matter-waves are precise measurement devices whose ultimate accuracy is achieved with Bose–Einstein condensates (BECs) in extended free fall. This is ideally realized in microgravity environments such as drop towers, ballistic rockets and space platforms. However, the transition from lab-based BEC machines to robust and mobile sources with comparable performance is a challenging endeavor. Here we report on the realization of a miniaturized setup, generating a flux of $4\times {{10}^{5}}$ quantum degenerate ⁸⁷Rb atoms every 1.6 s. Ensembles of $1\times {{10}^{5}}$ atoms can be produced at a 1 Hz rate. This is achieved by loading a cold atomic beam directly into a multi-layer atom chip that is designed for efficient transfer from laser-cooled to magnetically trapped clouds. The attained flux of degenerate atoms is on par with current lab-based BEC experiments while offering significantly higher repetition rates. Additionally, the flux is approaching those of current interferometers employing Raman-type velocity selection of laser-cooled atoms. The compact and robust design allows for mobile operation in a variety of demanding environments and paves the way for transportable high-precision quantum sensors.

A previously developed approach for the numerical treatment of two particles that are confined in a finite optical-lattice potential and interact via an arbitrary isotropic interaction potential has been extended to incorporate an additional anisotropic dipole–dipole interaction (DDI). The interplay of a model but realistic short-range Born–Oppenheimer potential and the DDI for two confined particles is investigated. A variation of the strength of the DDI leads to diverse resonance phenomena. In a harmonic confinement potential some resonances show similarities to s-wave scattering resonances while in an anharmonic trapping potential like the one of an optical lattice additional inelastic confinement-induced dipolar resonances occur. The latter are due to a coupling of the relative and center-of-mass motion caused by the anharmonicity of the external confinement.

We report results for reaction and vibrational quenching of the collision D with para-H₂($v,j=0$) and ortho-H₂($v,j=1$) at cold and ultracold temperatures. We investigate the effect of nuclear spin symmetry for barrier dominated processes ($0\leqslant v\leqslant 4$) and for one barrierless case (v = 5). We find resonant structures for energies in the range corresponding to 0.01–10 K, which depend on the nuclear spin of H₂, arising from contributions of specific partial waves. We discuss the implications on the results in this benchmark system for ultracold chemistry.

Stochastic efficiency is evaluated in five case studies: driven Brownian motion, effusion with a thermo-chemical and thermo-velocity gradient, a quantum dot and a model for information to work conversion. The salient features of stochastic efficiency, including the maximum of the large deviation function at the reversible efficiency, are reproduced. The approach to and extrapolation into the asymptotic time regime are documented.

Fluctuation theorems impose constraints on the probability of observing negative entropy production in small systems driven out of equilibrium. The range of validity of fluctuation theorems has been extensively tested for transitions between equilibrium and non-equilibrium stationary states, but not between general non-equilibrium states. Here we report an experimental verification of the detailed fluctuation theorem for the total amount of entropy produced in the isothermal transition between two non-equilibrium states. The experimental setup is a parallel RC circuit driven by an alternating current. We investigate the statistics of the heat released, of the variation of the entropy of the system, and of the entropy produced for processes of different durations. We show that the fluctuation theorem is satisfied with high accuracy for current drivings at different frequencies and different amplitudes.

A genuine feature of projective quantum measurements is that they inevitably alter the mean energy of the observed system if the measured quantity does not commute with the Hamiltonian. Compared to the classical case, Jacobs proved that this additional energetic cost leads to a stronger bound on the work extractable after a single measurement from a system initially in thermal equilibrium (2009 Phys. Rev. A 80 012322). Here, we extend this bound to a large class of feedback-driven quantum engines operating periodically and in finite time. The bound thus implies a natural definition for the efficiency of information to work conversion in such devices. For a simple model consisting of a laser-driven two-level system, we maximize the efficiency with respect to the observable whose measurement is used to control the feedback operations. We find that the optimal observable typically does not commute with the Hamiltonian and hence would not be available in a classical two level system. This result reveals that periodic feedback engines operating in the quantum realm can exploit quantum coherences to enhance efficiency.

The change of the von Neumann entropy of a set of harmonic oscillators initially in thermal equilibrium and interacting linearly with an externally driven quantum system is computed by adapting the Feynman–Vernon influence functional formalism. This quantum entropy production has the form of the expectation value of three functionals of the forward and backward paths describing the system history in the Feynman–Vernon theory. In the classical limit of Kramers–Langevin dynamics (Caldeira–Leggett model) these functionals combine to three terms, where the first is the entropy production functional of stochastic thermodynamics, the classical work done by the system on the environment in units of k_BT, and the second and the third other functionals which have no analogue in stochastic thermodynamics.

We investigate the fundamental limitations imposed by thermodynamics for creating correlations. Considering a collection of initially uncorrelated thermal quantum systems, we ask how much classical and quantum correlations can be obtained via a cyclic Hamiltonian process. We derive bounds on both the mutual information and entanglement of formation, as a function of the temperature of the systems and the available energy. While for a finite number of systems there is a maximal temperature allowing for the creation of entanglement, we show that genuine multipartite entanglement—the strongest form of entanglement in multipartite systems—can be created at any finite temperature when sufficiently many systems are considered. This approach may find applications, e.g. in quantum information processing, for physical platforms in which thermodynamic considerations cannot be ignored.

Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems. Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings. However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited. In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one- and two-point correlations in one-dimensional (1D) systems. Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable. We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system. Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent. Our predictions are relevant for a broad range of systems including solid state materials, atom–photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms. We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.

We theoretically consider transport properties of a normal metal (N)-superconducting semiconductor nanowire (S)-normal metal (N) structure (NSN) in the context of the possible existence of Majorana bound states in semiconductor–superconductor hybrid systems with spin–orbit coupling and external magnetic field. We study in detail the transport signatures of the topological quantum phase transition (TQPT) as well as the existence of the Majorana bound states in the electrical transport properties of the NSN structure. Our treatment includes the realistic non-perturbative effects of disorder, which is detrimental to the topological phase (eventually suppressing the superconducting gap completely), and the effects of the tunneling barriers (or the transparency at the tunneling NS contacts), which affect (and suppress) the zero bias conductance peak associated with the zero-energy Majorana bound states. We show that in the presence of generic disorder and barrier transparency the interpretation of the zero bias peak as being associated with the Majorana bound state is problematic since the non-local correlations between the two NS contacts at two ends may not manifest themselves in the tunneling conductance through the whole NSN structure. We establish that a simple modification of the standard transport measurements using conductance differences (rather than the conductance itself as in a single NS junction) as the measured quantity can allow direct observation of the non-local correlations inherent in the Majorana bound states. We also show that our proposed analysis of transport properties of the NSN junction enables the mapping out of the topological phase diagram (even in the presence of considerable disorder) by precisely detecting the TQPT point. We propose direct experimental studies of NSN junctions (rather than just a single NS junction) in order to establish the existence of Majorana bound states and the topological superconducting phase in semiconductor nanowires of current interest. Throughout the work we emphasize that the NSN transport properties are sensitive to both the bulk topological phase and the end Majorana bound states, and thus the NSN junction is well-suited for studying the non-local correlations between the end Majorana modes as well as the bulk TQPT itself.

We present a model for a feedback-controlled ratchet consisting of a Brownian particle and a moving, finite barrier that is shifted by an external agent depending on the position of the particle. By modifying the value of a single parameter of the feedback protocol, the model can act either as a pure rectifier, a power-stroke (PS) motor, or a combination of both. Interestingly, in certain situations the motor reaches a maximum efficiency for an intermediate value of that parameter, i.e., for a combination of the information ratchet and the PS mechanisms. We relate our results to the biological motors kinesin, myosin II, and myosin V, finding that these motors operate in a regime of length scales and forces where the efficiency is maximized for a combination of rectification and PS mechanisms.

The fluxonium qubit has arisen as one of the most promising candidate devices for implementing quantum information in superconducting devices, since it is both insensitive to charge noise (like flux qubits) and insensitive to flux noise (like charge qubits). Here, we investigate the stability of the quantum information to quasiparticle tunneling through a Josephson junction. Microscopically, this dephasing is due to the dependence of the quasiparticle transmission probability on the qubit state. We argue that on a phenomenological level the dephasing mechanism can be understood as originating from heat currents, which are flowing in the device due to possible effective temperature gradients, and their sensitivity to the qubit state. The emerging dephasing time is found to be insensitive to the number of junctions with which the superinductance of the fluxonium qubit is realized. Furthermore, we find that the dephasing time increases quadratically with the shunt-inductance of the circuit which highlights the stability of the device to this dephasing mechanism.

We revisit the Ornstein–Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic 'equation of state' of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.

We consider the dynamics of rotational excitations placed on a single molecule in spatially disordered one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) ensembles of ultracold molecules trapped in optical lattices. The disorder arises from incomplete populations of optical lattices with molecules. This leads to a model corresponding to a quantum particle with long-range tunnelling amplitudes moving on a lattice with the same on-site energy but with forbidden access to random sites (vacancies). We examine the time and length scales of Anderson localization for this type of disorder with realistic experimental parameters in the Hamiltonian. We show that for an experimentally realized system of KRb molecules on an optical lattice this type of disorder leads to disorder-induced localization in 1D and 2D systems on a time scale $t\sim 1$ s. For 3D lattices with 55 sites in each dimension and vacancy concentration 90%, the rotational excitations diffuse to the edges of the lattice and show no signature of Anderson localization. We examine the role of the long-range tunnelling amplitudes allowing for transfer of rotational excitations between distant lattice sites. Our results show that the long-range tunnelling has little impact on the dynamics in the diffusive regime but affects significantly the localization dynamics in lattices with large concentrations of vacancies, enhancing the width of the localized distributions in 2D lattices by more than a factor of 2.

A unique property of Zeeman effect based manipulation of paramagnetic particle's motion is the ability to control velocities of both atoms and molecules. In particular the moving magnetic trap decelerator is capable of slowing and eventually trapping mixtures of both cold atoms and cold molecules generated in a supersonic expansion. Here we report the deceleration of molecular oxygen together with metastable argon atoms. The cold mixture with temperature below 1 K is slowed from an initial velocity of 430 m s⁻¹ down to 100 m s⁻¹. Our decelerator spans 2.4 m and consists of 480 quadrupole traps. Our results pave the way for the study of sympathetic cooling of molecules by laser cooled atoms.

The order parameter of a quantum-coherent many-body system can include a phase degree of freedom, which, in the presence of an electromagnetic field, depends on the choice of gauge. Because of the relationship between the phase gradient and the velocity, time-of-flight measurements reveal this gradient. Here, we describe such measurements of initially trapped Bose–Einstein condensates (BECs) subject to an artificial magnetic field. Vortices nucleated in the BEC for artificial field strengths above a critical value, which represented a structural phase transition. By comparing to superfluid-hydrodynamic and Gross–Pitaevskii calculations, we confirmed that the transition from the vortex-free state gives rise to a shear in the released BEC's spatial distribution, representing a macroscopic method to measure this transition, distinct from direct imaging of vortex entry. Shear is also affected by an artificial electric field accompanying the artificial magnetic field turn-off, which depends on the details of the physical mechanism creating the artificial fields, and implies a most natural choice of gauge. Measurements of this kind offer opportunities for studying phase in less-well-understood quantum gas systems.

It is argued that gravity should cause a breakdown of quantum mechanics, at low energies, accessible to table-top experiments. It is then shown that one can formulate a theory of quantum gravity in which gravitational correlations exist between worldline or worldsheet paths, for the particle or field of interest. Using a generalized equivalence principle, one can give a unique form for the correlators, yielding a theory with no adjustable parameters. A key feature of the theory is the 'bunching' of quantum trajectories caused by the gravitational correlations—this is not a decoherence or a 'collapse' mechanism. This bunching causes a breakdown of the superposition principle for large masses, with a very rapid crossover to classical behaviour at an energy scale which depends on the physical structure of the object. Formal details, and applications of the theory, are kept to a minimum in this paper; but we show how physical quantities can be calculated, and give a detailed discussion of the dynamics of a single particle.

We propose a trapped ion scheme en route to realize spin Hamiltonians on a Kagome lattice which, at low energies, are described by emergent ${{\mathbb{Z}}}_{2}$ gauge fields, and support a topological quantum spin liquid ground state. The enabling element in our scheme is the hexagonal plaquette spin–spin interactions in a two-dimensional ion crystal. For this, the phonon-mode spectrum of the crystal is engineered by standing-wave optical potentials or by using Rydberg excited ions, thus generating localized phonon-modes around a hexagon of ions selected out of the entire two-dimensional crystal. These tailored modes can mediate spin–spin interactions between ion-qubits on a hexagonal plaquette when subject to state-dependent optical dipole forces. We discuss how these interactions can be employed to emulate a generalized Balents–Fisher–Girvin model in minimal instances of one and two plaquettes. This model is an archetypical Hamiltonian in which gauge fields are the emergent degrees of freedom on top of the classical ground state manifold. Under realistic situations, we show the emergence of a discrete Gauss's law as well as the dynamics of a deconfined charge excitation on a gauge-invariant background using the two-plaquettes trapped ions spin-system. The proposed scheme in principle allows further scaling in a future trapped ion quantum simulator, and we conclude that our work will pave the way towards the simulation of emergent gauge theories and quantum spin liquids in trapped ion systems.

We study, both theoretically and experimentally, the dynamical polarizability $\alpha (\omega )$ of ${\mathrm{Rb}}_{2}$ molecules in the rovibrational ground state of ${a}^{3}{\Sigma }_{u}^{+}$. Taking all relevant excited molecular bound states into account, we compute the complex-valued polarizability $\alpha (\omega )$ for wave numbers up to 20 000 ${{\rm{cm}}}^{-1}$. Our calculations are compared to experimental results at $1064.5\;\mathrm{nm}$ ($\sim 9400\;{{\rm{cm}}}^{-1}$) as well as at $830.4\;\mathrm{nm}$ ($\sim 12\;000\;{{\rm{cm}}}^{-1}$). Here, we discuss the measurements at $1064.5\;\mathrm{nm}$. The ultracold Rb₂ molecules are trapped in the lowest Bloch band of a 3D optical lattice. Their polarizability is determined by lattice modulation spectroscopy which measures the potential depth for a given light intensity. Moreover, we investigate the decay of molecules in the optical lattice, where lifetimes of more than $2\;{\rm{s}}$ are observed. In addition, the dynamical polarizability for the ${X}^{1}{\Sigma }_{g}^{+}$ state is calculated. We provide simple analytical expressions that reproduce the numerical results for $\alpha (\omega )$ for all vibrational levels of ${a}^{3}{\Sigma }_{u}^{+}$ as well as ${X}^{1}{\Sigma }_{g}^{+}$. Precise knowledge of the molecular polarizability is essential for designing experiments with ultracold molecules as lifetimes and lattice depths are key parameters. Specifically the wavelength at $\sim 1064\;\mathrm{nm}$ is of interest, since here, ultrastable high power lasers are available.