Knowing when a physical system has reached sufficient size for its macroscopic properties to be well described by many-body theory is difficult. We investigated the crossover from few- to many-body physics by studying quasi-one-dimensional systems of ultracold atoms consisting of a single impurity interacting with an increasing number of identical fermions. We measured the interaction energy of such a system as a function of the number of majority atoms for different strengths of the interparticle interaction. As we increased the number of majority atoms one by one, we observed fast convergence of the normalized interaction energy toward a many-body limit calculated for a single impurity immersed in a Fermi sea of majority particles.

We develop an analytical many-body wave function to accurately describe the crossover of a onedimensional bosonic system from weak to strong interactions in a harmonic trap. The explicit wave function, which is based on the exact two-body states, consists of symmetric multiple products of the corresponding parabolic cylinder functions, and respects the analytically known limits of zero and infinite repulsion for arbitrary number of particles. For intermediate interaction strengths we demonstrate, that the energies, as well as the reduced densities of first and second order, are in excellent agreement with large scale numerical calculations. Introduction Ultracold dilute quantum gases represent an amazingly rich platform for the realization of strongly interacting many-body systems [1]. Extensive control of the trapping geometry via external fields as well as tuning of the interaction properties of ultracold atomic ensembles is nowadays routinely possible [2,3]. Theoretical models of quantum many-body systems that were considered as rough idealizations in the past, can now be prepared in a very pure way in order to study, for example, quantum phase transitions [4]. One very fundamental model of this type is explored in the present work: an ensemble of bosons at zero temperature, confined to a one-dimensional (1D) harmonic trap and interacting via a repulsive contact potential of arbitrary strength. Although conceptually simple, this system is in general not analytically solvable. However, the experimental preparation of the trap is possible and the interaction strength can be tuned to any desired value by adjusting the strength of magnetic fields in the vicinity of a Feshbach resonance [3], or by employing confinement induced resonances [5], a tool specific to quasi-1D waveguide-like systems.

Properties of a single impurity in a one-dimensional Fermi gas are investigated in homogeneous and trapped geometries. In a homogeneous system we use McGuire's expression [J. B. McGuire, J. Math. Phys. 6, 432 (1965)] to obtain interaction and kinetic energies, as well as the local pair correlation function. The energy of a trapped system is obtained (i) by generalizing McGuire expression (ii) within local density approximation (iii) using perturbative approach in the case of a weakly interacting impurity and (iv) diffusion Monte Carlo method. We demonstrate that a closed formula based on the exact solution of the homogeneous case provides a precise estimation for the energy of a trapped system for arbitrary coupling constant of the impurity even for a small number of fermions. We analyze energy contributions from kinetic, interaction and potential components, as well as spatial properties such as the system size. Finally, we calculate the frequency of the breathing mode. Our analysis is directly connected and applicable to the recent experiments in microtraps.Comment: 5 pages, 4 figure

We extend the concept of quantum speed limit -the minimal time needed to perform a driven evolution -to complex interacting many-body systems. We investigate a prototypical many-body system, a bosonic Josephson junction, at increasing levels of complexity: (a) within the two-mode approximation corresponding to a nonlinear two-level system, (b) at the mean-field level by solving the nonlinear Gross-Pitaevskii equation in a double well potential, and (c) at an exact many-body level by solving the time-dependent many-body Schrödinger equation. We propose a control protocol to transfer atoms from the ground state of a well to the ground state of the neighbouring well. Furthermore, we show that the detrimental effects of the inter-particle repulsion can be eliminated by means of a compensating control pulse, yielding, quite surprisingly, an enhancement of the transfer speed because of the particle interaction -in contrast to the self-trapping scenario. Finally, we perform numerical optimisations of both the nonlinear and the (exact) many-body quantum dynamics in order to further enhance the transfer efficiency close to the quantum speed limit.