We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.The correspondence between Bethe Ansatz / Ordinary Differential Equation (ODE) has been found some time ago 1 on the basis of similarity between T-Q system of the Bethe Ansatz and functional relations including spectral determinants on the ODE side. For extensive review we refer to 2 . The correspondence is related to the Langlands correspondence where the ODE part has to do with socalled Miura opers 3 . It has many interesting links with conformal field theory, quasi-exact solvability, supersymmetry and PT-symmetric quantum mechanics. It was also used it in the context of physics of cold atoms and quantum impurity problem 4 . Here we develop further this remarkable correspondence to facilitate the solving of otherwise difficult-to-solve Gaudin systems and implement it to several physically interesting situations.Indeed, numerous integrable models derived from a generalized Gaudin algebra have been used to describe the properties of physical systems. The Richardson fermionic pairing Hamiltonian 5 has explained most properties of superconducting nanograins 6,7 , numerous studies of the decoherence of a single electron spin trapped in a quantum dot rely on the central spin model 8 while the inhomogeneous Dicke models 9 have been used to study light-matter interaction in many cavity-based systems.The quantum integrability of such systems brings major simplifications to the structure of their eigenstates. In a system containing N degrees of freedom, they can be fully described by a number M of complex parameters we call rapidities. M is here a number of the same order as N despite the exponentially large Hilbert space. The possible values of this set of parameters can be obtained by finding solutions to a set of M coupled non-linear algebraic equation: the Bethe equations. However, analytically finding solutions to the Bethe equations remains impossible except in very specific cases. The problem is still numerically approachable but solving non-linear equations remains a challenge that only iterative methods can tackle.While previous efforts in designing algorithms have shown promising results [10][11][12][13][14], methods which are sufficiently fast and stable for systematically finding a large number of eigenstates remained elusive. As was shown by one of the authors, computing efficiently only a small fraction of the complete Hilbert space can be sufficient to access static 15 , dynamical 16 and even non-equilibrium dyna...

We propose a simple scheme for mimicking the physics of one-dimensional anyons in an optical-lattice experiment. It relies on a bosonic representation of the anyonic Hubbard model to be realized via lattice-shaking-induced resonant tunneling against potential offsets, which are created by a combination of a lattice tilt and strong on-site interactions. No lasers additional to those used for the creation of the optical lattice are required. We also discuss experimental signatures of the continuous interpolation between bosons and fermions when the statistical angle θ is varied from 0 to π. Whereas the real-space density of the bosonic atoms corresponds directly to that of the simulated anyonic model, this is not the case for the momentum distribution. Therefore, we propose to use Friedel oscillations in the density as a probe for continuous fermionization of the bosonic atoms.

Periodic driving of optical lattices has enabled the creation of novel band structures not realizable in static lattice systems, such as topological bands for neutral particles. However, especially driven systems of interacting bosonic particles often suffer from strong heating. We have systematically studied heating in an interacting Bose-Einstein condensate in a driven one-dimensional optical lattice. We find interaction dependent heating rates that depend on both the scattering length and the driving strength and identify the underlying resonant intra- and interband scattering processes. By comparing the experimental data and theory, we find that, for driving frequencies well above the trap depth, the heating rate is dramatically reduced by the fact that resonantly scattered atoms leave the trap before dissipating their energy into the system. This mechanism of Floquet evaporative cooling offers a powerful strategy to minimize heating in Floquet engineered quantum gases.

We report on the observation of multiphoton interband absorption processes for quantum gases in shaken light crystals. Periodic inertial forcing, induced by a spatial motion of the lattice potential, drives multiphoton interband excitations of up to the ninth order. The occurrence of such excitation features is systematically investigated with respect to the potential depth and the driving amplitude. Ab initio calculations of resonance positions as well as numerical evaluation of their strengths exhibit good agreement with experimental data. In addition our findings could make it possible to reach novel phases of quantum matter by tailoring appropriate driving schemes.

We propose a cold-atom realization of a zigzag ladder. The two legs of the ladder correspond to a “synthetic” dimension given by two internal (spin) states of the atoms, so that tunneling between them can be realized as a laser-assisted process. The zigzag geometry is achieved by employing a spin-dependent optical lattice with the site position depending on the internal atomic state, i.e., on the ladder’s leg. The lattice offers a possibility to tune the single-particle dispersion from a double-well to a single-minimum configuration. In contrast to previously considered semisynthetic lattices with a square geometry, the tunneling in the synthetic dimension is accompanied by spatial displacements of atoms. Therefore, the atom-atom interactions are nonlocal and act along the diagonal (semisynthetic) direction. We investigate the ground-state properties of the system for the case of strongly interacting bosons. In particular, we find that the interplay between the frustration induced by the magnetic field and the interactions gives rise to an interesting gapped phase at fractional filling factors corresponding to one particle per magnetic unit cell.