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Persistent currents flowing in spatially closed tracks define one of the most iconic concepts in mesoscopic physics. They have been studied in solid-state platforms such as superfluids, superconductors and metals. Cold atoms trapped in magneto-optical toroidal circuits and driven by suitable artificial gauge fields allow us to study persistent currents with unprecedented control and flexibility of the system's physical conditions. Here, we review persistent currents of ultracold matter. Capitalizing on the remarkable progress in driving different atomic species to quantum degeneracy, persistent currents of single or multicomponent bosons/fermions, and their mixtures can be addressed within the present experimental know-how. This way, fundamental concepts of quantum science and many-body physics, like macroscopic quantum coherence, solitons, vortex dynamics, fermionic pairing and BEC-BCS crossover can be studied from a novel perspective. Finally, we discuss how persistent currents can form the basis of new technological applications like matter-wave gyroscopes and interferometers.

Atomtronics deals with matter-wave circuits of ultra-cold atoms manipulated through magnetic or laser-generated guides with different shapes and intensities. In this way, new types of quantum networks can be constructed, in which coherent fluids are controlled with the know-how developed in the atomic and molecular physics community. In particular, quantum devices with enhanced precision, control and flexibility of their operating conditions can be accessed. Concomitantly, new quantum simulators and emulators harnessing on the coherent current flows can also be developed. Here, we survey the landscape of atomtronics-enabled quantum technology and draw a roadmap for the field in the near future. We review some of the latest progresses achieved in matter-wave circuits design and atom-chips. Atomtronic networks are deployed as promising platforms for probing many-body physics with a new angle and a new twist. The latter can be done both at the level of equilibrium and non-equilibrium situations. Numerous relevant problems in mesoscopic physics, like persistent currents and quantum transport in circuits of fermionic or bosonic atoms, are studied through a new lens. We summarize some of the atomtronics quantum devices and sensors. Finally, we discuss alkali-earth and Rydberg atoms as potential platforms for the realization of atomtronic circuits with special features., Comment: 113 latex pages, 83 figs. To be published in AVS Quantum Science

Topological pumping and duality transformations are paradigmatic concepts in condensed matter and statistical mechanics. In this paper, we extend the concept of topological pumping of particles to topological pumping of quantum correlations. We propose a scheme to find pumping protocols for highly-correlated states by mapping them to uncorrelated ones. We show that one way to achieve this is to use dualities, because they are non-local transformations that preserve the topological properties of the system. By using them, we demonstrate that topological pumping of kinks and cluster-like excitations can be realized. We find that the entanglement of these highly-correlated excitations is strongly modified during the pumping process and the interactions enhance the robustness against disorder. Our work paves the way to explore topological pumping beyond the notion of particles and opens a new avenue to investigate the relation between correlations and topology., Comment: 8 Pages, 1 Table, 7 Figures. Accepted for publication in Physical Review Research

Complex circuits of cold atoms can be exploited to devise new protocols for the diagnostics of cold-atoms systems. Specifically, we study the quench dynamics of a condensate confined in a ring-shaped potential coupled with a rectilinear guide of finite size. We find that the dynamics of the atoms inside the guide is distinctive of the states with different winding numbers in the ring condensate. We also observe that the depletion of the density, localized around the tunneling region of the ring condensate, can decay in a pair of excitations experiencing a Sagnac effect. In our approach, the current states of the condensate in the ring can be read out by inspection of the rectilinear guide only, leaving the ring condensate minimally affected by the measurement. We believe that our results set the basis for definition of new quantum rotation sensors. At the same time, our scheme can be employed to explore fundamental questions involving dynamics of bosonic condensates., Comment: Figures are enlarged. Section IV is added. Journal reference added

We study a physical system consisting of a Bose-Einstein condensate confined to a ring shaped lattice potential interrupted by three weak links. The system is assumed to be driven by an effective flux piercing the ring lattice. By employing path integral techniques, we explore the effective quantum dynamics of the system in a pure quantum phase dynamics regime. Moreover, the effects of the density's quantum fluctuations are studied through exact diagonalization analysis of the spectroscopy of the Bose-Hubbard model. We demonstrate that a clear two-level system emerges by tuning the magnetic flux at degeneracy. The lattice confinement, platform for the condensate, is realized experimentally employing a spatial light modulator., Comment: Accepted in New Journal of Physics

We investigate the entanglement content of the ground state of a system characterized by effective elementary degrees of freedom with fractional statistics. To this end, we explicitly construct the ground state for a chain of $N$ spins with inverse square interaction (the Haldane-Shastry model) in the presence of an external uniform magnetic field. For such a system at zero temperature, we evaluate the entanglement in the ground state both at finite size and in the thermodynamic limit. We relate the behavior of the quantum correlations with the spinon condensation phenomenon occurring at the saturation field., Comment: 11 pages, 7 eps figures. Submitted to New. J. Phys, special issue "Quantum Information and Many-Body Theory"(co-edited by J Eisert and M Plenio)

We study the adiabatic dynamics of Majorana fermions across a quantum phase transition. We show that the Kibble-Zurek scaling, which describes the density of bulk defects produced during the critical point crossing, is not valid for edge Majorana fermions. Therefore, the dynamics governing an edge state quench is nonuniversal and depends on the topological features of the system. Besides, we show that the localization of Majorana fermions is a necessary ingredient to guaranty robustness against defect production., Comment: Submitted to the Special Issue on "Dynamics and Thermalization in Isolated Quantum Many-Body Systems" in New Journal of Physics. Editors:M. Cazalilla, M. Rigol. New references and some typos corrected

We study the influence of topology on the quench dynamics of a system driven across a quantum critical point. We show how the appearance of certain edge states, which fully characterise the topology of the system, dramatically modifies the process of defect production during the crossing of the critical point. Interestingly enough, the density of defects is no longer described by the Kibble-Zurek scaling, but determined instead by the non-universal topological features of the system. Edge states are shown to be robust against defect production, which highlights their topological nature., Comment: Phys. Rev. Lett. (to be published)

We analyze the effective Hamiltonian arising from a suitable power series expansion of the overlap integrals of Wannier functions for confined bosonic atoms in a 1d optical lattice. For certain constraints between the coupling constants, we construct an explicit relation between such an effective bosonic Hamiltonian and the integrable spin-$S$ anisotropic Heisenberg model. Therefore the former results to be integrable by construction. The field theory is governed by an anisotropic non linear $\sigma$-model with singlet and triplet massive excitations; such a result holds also in the generic non-integrable cases. The criticality of the bosonic system is investigated. The schematic phase diagram is drawn. Our study is shedding light on the hidden symmetry of the Haldane type for one dimensional bosons., Comment: 5 pages; 1 eps figure. Revised version, to be published in New. J. Phys

We study the multi-spin entanglement for the 1D anisotropic XY model concentrating on the simplest case of three-spin entanglement. As compared to the pairwise entanglement, three-party quantum correlations have a longer range and they are more robust on increasing the temperature. We find regions of the phase diagram of the system where bound entanglement occurs, both at zero and finite temperature. Bound entanglement in the ground state can be obtained by tuning the magnetic field. Thermal bound entanglement emerges naturally due to the effect of temperature on the free ground state entanglement., Comment: 7 pages, 3 figures; some typos corrected, references added

We construct commuting transfer matrices for models describing the interaction between a single quantum spin and a single bosonic mode using the quantum inverse scattering framework. The transfer matrices are obtained from certain inhomogeneous rational vertex models combining bosonic and spin representations of SU(2), subject to non-diagonal toroidal and open boundary conditions. Only open boundary conditions are found to lead to integrable Hamiltonians combining both rotating and counter-rotating terms in the interaction. If the boundary matrices can be brought to triangular form simultaneously, the spectrum of the model can be obtained by means of the algebraic Bethe ansatz after a suitable gauge transformation; the corresponding Hamiltonians are found to be non-hermitian. Alternatively, a certain quasi-classical limit of the transfer matrix is considered where hermitian Hamiltonians are obtained as members of a family of commuting operators; their diagonalization, however, remains an unsolved problem., Comment: 16 pages, 2 eps figures

For the anisotropic XY model in transverse magnetic field, we analyze the ground state and its concurrence-free point for generic anisotropy, and the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. A transmutation from singlet-like to triplet-like states is observed during the propagation. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The ``entanglement wave'' evolving from a Bell state on the ground state turns out to be very localized in space-time. Our findings agree with a recently formulated conjecture on entanglement sharing; some results are interpreted in terms of this conjecture., Comment: 14 pages, 32 figures, revtex4

We study the pairwise entanglement close to separable ground states of a class of one dimensional quantum spin models. At T=0 we find that such ground states separate regions, in the space of the Hamiltonian parameters, which are characterized by qualitatively different types of entanglement, namely parallel and antiparallel entanglement; we further demonstrate that the range of the Concurrence diverges while approaching separable ground states, therefore evidencing that such states, with uncorrelated fluctuations, are reached by a long range reshuffling of the entanglement. We generalize our results to the analysis of quantum phase transitions occurring in bosonic and fermionic systems. Finally, the effects of finite temperature are considered: At T>0 we evidence the existence of a region where no pairwise entanglement survives, so that entanglement, if present, is genuinely multipartite., Comment: 6 pages, 4 figures

We study the dynamics of quantum correlations in a class of exactly solvable Ising-type models. We analyze in particular the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. Singlet-like states are favored during the propagation, in the sense that triplet-like states change their character during the propagation under certain circumstances. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The ``entanglement wave'' evolving from a Bell state on the ground state turns out to be very localized in space-time. Further support to a recently formulated conjecture on entanglement sharing is given., Comment: 25 pages, 21 figures; revtex

For the exactly solved reduced BCS model an electrostatic analogy exists; in particular it served to obtain the exact thermodynamic limit of the model from the Richardson Bethe ansatz equations. We present an electrostatic analogy for a wider class of integrable Hamiltonians with pairing force interactions. We apply it to obtain the exact thermodynamic limit of this class of models. To verify the analytical results, we compare them with numerical solutions of the Bethe ansatz equations for finite systems at half-filling for the ground state., Comment: 14 pages, 6 figures, revtex4. Minor changes

In this Letter we discuss the entanglement near a quantum phase transition by analyzing the properties of the concurrence for a class of exactly solvable models in one dimension. We find that entanglement can be classified in the framework of scaling theory. Further, we reveal a profound difference between classical correlations and the non-local quantum correlation, entanglement: the correlation length diverges at the phase transition, whereas entanglement in general remains short ranged., Comment: 4 pages, 4 figures, revtex. Stylistic changes and format modified

A class of integrable one-dimensional models presented by Shastry and Schulz is consequently extended to the whole class of one-dimensional Hubbard- or XXZ-type models with correlated gauge-like hopping. A complete characterization concerning solvability by coordinate Bethe ansatz of this class of models is found., Comment: 13 pages, 3 figures, revtex. Minor changes, references added

The transition line of superconducting arrays of holes exhibits a rich field structure due to the interference of superconducting states nucleated at the holes edges. We studied by means of resistance measurements their effect on the T*c(H) line as a function of transverse magnetic field using regular arrays of nanofabricated micron size holes. The arrays transition fields are higher than for the bulk. Moreover we found a nontrivial field modulation of the T*c(H) line with an inversion, with increasing field, of the modulation concavity which we assigned to a crossover from a collective to an isolated edge state regime. The high field regime is well described by the nucleation at a single hole in an infinite film. The modulation at low fields was found to be dominated by the interference of neighbor edge states when the inter-hole distance w becomes comparable to the temperature dependent coherence length. A comparison between arrays of different hole shape shows the influence of geometry on the type of interaction established, which can described either as a superconducting wire network or as a weak link array., Comment: revised version, fig.4 changed 23 pages, 7 figures, submitted to Journal of Low Temperature Physics

We define one-dimensional particles with generalized exchange statistics. The exact solution of a Hubbard-type Hamiltonian constructed with such particles is achieved using the Coordinate Bethe Ansatz. The chosen deformation of the statistics is equivalent to the presence of a magnetic field produced by the particles themselves, which is present also in a ``free gas'' of these particles., Comment: 4 pages, revtex. Essentially modified version

Quantum dynamics of the Bose-Hubbard Model is investigated through a semiclassical hamiltonian picture provided by the Time-Dependent Variational Principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ll 1$. The phase diagram is in good agreement with Quantum Monte Carlo results and third order strong coupling perturbative expansion., Comment: 4 pages Revtex, 2 figures .eps, to be published in PRL