"Bottom-up" approaches to the many-body physics of fermions have demonstrated recently precise number and site-resolved preparations with tunability of interparticle interactions in single-well, SW, and double-well, DW, nano-scale confinements created by manipulating ultracold fermionic atoms with optical tweezers. These experiments emulate an analoguesimulator mapping onto the requisite microscopic hamiltonian, approaching realization of Feynmans' vision of quantum simulators that "will do exactly the same as nature". Here we report on exact benchmark configuration-interaction computational microscopy solutions of the hamiltonian, uncovering the spectral evolution, wave function anatomy, and entanglement properties of the interacting fermions in the entire parameter range, including crossover from a SW to a DW confinement and a controllable energy imbalance between the wells. We demonstrate attractive pairing and formation of repulsive, highly-correlated, ultracold Wigner molecules, well-described in the natural orbital representation. The agreement with the measurements affirms the henceforth gained deep insights into ultracold molecules and opens access to the size-dependent evolution of nano-clustered and condensed-matter phases and ultracold-atoms quantum information.Key words: ultracold atoms, double-well nano-confinement, Wigner molecule, configuration interaction, entanglement, strong correlated matter *Corresponding Author: uzi.landman@physics.gatech.edu arXiv:1508.02308v3 2 Ingress to the origins of complex physical phenomena often requires experiments whereby theories are tested or suggested through artificial manipulations of physical circumstances. During the past decade, a cornucopia of new tools have emerged resulting from the discovery and advancement of methods for the preparation and trapping of ultracold atomic gases, controlled tuning of the interparticle interactions (via magnetic manipulation of the Feshbach resonance), and the creation of synthetic gauge fields through atom-light interactions in optical lattices of varied geometries and topologies 1,2 . The remarkable pristine nature of these systems, and the exquisite level of control that can be exercised over them, brought forth a realization of Richard Feynman's vision 3 for the construction of physical quantum simulators, capable of an exact simulation, of systems or situations that are computationally or analytically intractable. Indeed, in the past several years we witnessed a surge of realizations of such exact simulations addressing diverse fields (see reviews in refs.1 and 2), including in particular the behavior of strongly interacting fermions where computations are precluded because of the "fermion sign problem." 4 . These systems range from high-Tc superconductivity 1,2 , collosal magnetoresistance 5 and quantum Hall effects 2 to atomic frequency resonators 6 , interferometry 7,8 , matter wave gyroscopes 9 and the development of scalable quantum computers with neutral atoms 10,11 .Progress aiming at a "bottom-up" app...

Identification and understanding of the evolution of interference patterns in two-particle momentum correlations as a function of the strength of interatomic interactions are important in explorations of the nature of quantum states of trapped particles. Together with the analysis of two-particle spatial correlations, they offer the prospect of uncovering fundamental symmetries and structure of correlated many-body states, as well as opening vistas into potential control and utilization of correlated quantum states as quantum information resources. With the use of the secondorder density matrix constructed via exact diagonalization of the microscopic Hamiltonian, and an analytic Hubbard-type model, we explore here the systematic evolution of characteristic interference patterns in the two-body momentum and spatial correlation maps of two entangled ultracold fermionic atoms in a double well, for the entire attractive-and repulsive-interaction range. We uncover statistics-governed bunching and antibunching, as well as interaction-dependent interference patterns, in the ground and excited states, and interpret our results in light of the Hong-Ou-Mandel interference physics, widely exploited in photon indistinguishability testing and quantum information science.

Spatial and momentum correlations are important in the analysis of the quantum states and different phases of trapped ultracold atom systems as a function of the strength of interatomic interactions. Identification and understanding of spin-resolved patterns exhibited in two-point correlations, accessible directly by experiments, are key for uncovering the symmetry and structure of the many-body wavefunctions of the trapped system. Using the full configuration interaction method for exact diagonalization of the many-body Hamiltonian of N = 2 − 4 fermionic atoms trapped in single, double, triple, and quadruple wells, we analyze both two-point momentum and space correlations, as well as associated noise distributions, for a broad range of interparticle contact repulsion strengths and interwell separations, unveiling characteristics allowing insights into the transition, via an intermediate phase, from the non-interacting Bose-Einstein condensate to the weakly interacting quasi-Bose-Einstein regime, and from the latter to the strong-repulsion Tonks-Girardeau (TG) one. The ab-initio numerical predictions are shown to agree well with the results of a constructed analytical model employing localized displaced Gaussian functions to represent the N fermions. The two-point momentum correlations are found to exhibit damped oscillatory diffraction behavior. This diffraction behavior develops fully for atoms trapped in a single well with strong interatomic repulsion in the TG regime, or for atoms in well-separated multi-well traps. Additionally, the two-body momentum correlation and noise distributions are found to exhibit "shortsightedness", with the main contribution coming from nearest-negihboring particles.arXiv:1710.07853v2 [cond-mat.quant-gas]

Advances with trapped ultracold atoms intensified interest in simulating complex physical phenomena, including quantum magnetism and transitions from itinerant to non-itinerant behavior.Here we show formation of antiferromagnetic ground states of few ultracold fermionic atoms in single and double well (DW) traps, through microscopic Hamiltonian exact diagonalization for two DW arrangements: (i) two linearly oriented one-dimensional, 1D, wells, and (ii) two coupled parallel wells, forming a trap of two-dimensional, 2D, nature. The spectra and spin-resolved conditional probabilities reveal for both cases, under strong repulsion, atomic spatial localization at extemporaneously created sites, forming quantum molecular magnetic structures with non-itinerant character. These findings usher future theoretical and experimental explorations into the highly correlated behavior of ultracold strongly repelling fermionic atoms in higher dimensions, beyond the fermionization physics that is strictly applicable only in the 1D case. The results for four atoms are well described with finite Heisenberg spin-chain and cluster models. The numerical simulations of three fermionic atoms in symmetric DWs reveal the emergent appearance of coupled resonating 2D Heisenberg clusters, whose emulation requires the use of a t-J-like model, akin to that used in investigations of high T c superconductivity. The highly entangled states discovered in the microscopic and model calculations of controllably detuned, asymmetric, DWs suggest three-cold-atom DW quantum computing qubits. New J. Phys. 18 (2016) 073018 C Yannouleas et al New J. Phys. 18 (2016) 073018 C Yannouleas et al ⎟ ⎟ ⎟ 18 J J J t t t J J J J t t t J J J J t t t t t t J J J t t t J J J J t t t J J J J Figure B1. Schematics indicating the four-site numbering convention in the Heisenberg Hamiltonian. (a) The case of formation of a rectangular parallelogram (ring topology). The Heisenberg exchange parameters = = J J s 14 23 and = = J J r 12 34 . (b) The linear arrangement of the four sites which results from (a) by opening the ring through setting = J 0 34 , = J r 12 , = = J J s 14 23 . ( )  Due to the reflection symmetry in x and y,  H RP,gen has only two different exchange constants = = J J s 14 23 and = = J J r 12 34

Aiming at elucidating similarities and differences between quantum-optics biphoton interference phenomena and the quantum physics of quasi-one-dimensional double-well optically-trapped ultracold neutral bosonic or fermionic atoms, we show that the analog of the optical biphoton jointcoincidence spectral correlations, studied with massless non-interacting biphotons emanating from EPR-Bell-Bohm single-occupancy sources, corresponds to a distinct contribution in the total secondorder momentum correlations of the massive, interacting, and time-evolving ultracold atoms. This single-occupancy contribution can be extracted from the total second-order momentum correlation function measured in time-of-flight experiments, which for the trapped atomic system contains, in general, a double-occupancy, NOON, component. The dynamics of the two-particle system are modeled by a Hubbard Hamiltonian. The general form of this partial coincidence spectrum is a cosine-square quantum beating dependent on the difference of the momenta of the two particles, while the corresponding coincidence probability proper, familiar from its role in describing the Hong-Ou-Mandel coincidence dip of overlapping photons, results from an integration over the particle momenta. Because the second-order momentum correlations are mirrored in the time-of-flight spectra in space, our theoretical findings provide impetus for time-of-flight experimental protocols for emulating with (massive) ultracold atoms venerable optical interferometries that use two space-time separated and entangled (massless) photons or double-slit optical sources. The implementation of such developments will facilitate testing of fundamental aspects and enable applications of quantum physics with trapped massive ultracold atoms, that is, investigations of nonlocality and violation of Bell inequalities, entanglement, and quantum information science. arXiv:1812.05977v2 [cond-mat.quant-gas]