Box 1: Example Platforms for Analogue Quantum Simulation Analogue Quantum Simulations are today performed on a variety of platforms [15][16][17][18][19][20], each of which offer distinct features that make them more suitable for specific simulation tasks. Ultracold atoms in optical lattices -Currently up to 3000 atoms in optical potentials with single atom detection and control via so-called quantum gas microscopes. These uniquely implement models of interacting fermionic particles (such as the Hubbard model, see Box 2) or its bosonic variant using bosonic atoms. Spin models can also be engineered, with tailored local parameters and potentials shaped by spatial light modulators [21,22]. Trapped Ions -Currently ca. 50 ions in linear chains (Paul traps) or 2D arrays (Penning traps), with local addressing. Using laser excitations and the collective motion of the array to generate effective interactions, a range of magnetic spin models can be engineered, with tailored long-range interactions. They benefit from readout techniques and manipulation developed for digital quantum computing in ion traps [17]. Atom arrays with Rydberg interactions -Currently ca. 256 atoms held in optical tweezers, with spin models generated by excitation to high-lying electronic states. They benefit from arrangements in arbitrary geometries, and tailored magnetic spin models from different choices in laser excitation to excited states [20]. Superconducting circuits -Currently ca. 50-130 superconducting resonators, which can be used as qubits (spins) or anharmonic oscillators (interacting bosons), with couplings adjusted on the level of individual resonators. As with trapped ions, they benefit directly from architectures and readout techniques in superconducing quantum computers [19,23]. Photonic waveguides or beamsplitter arrays -Currently up to ca. 50-100 channels and photons, either chip-based or with large-scale beamsplitter arrays. They generally implement models for non-interacting bosons, including boson sampling, which is exponentially difficult to reproduce using classical calculations [18,24].Box 2: The Hubbard model A particularly good example of the complex problems addressed in quantum simulation is the determination of low-energy properties and dynamics in a Hubbard model, which is a prototypical model for describing strongly interacting electrons in solids [25]. Although in one dimension the Hubbard model is exactly solvable [26], in two dimensions, it has been a long-term challenge even to find the lowest energy states of this model, although a lot of recent progress has been made [27,28].

We experimentally study the excitation modes of bright matter-wave solitons in a quasi-one-dimensional geometry. The solitons are created by quenching the interactions of a Bose-Einstein condensate of cesium atoms from repulsive to attractive in combination with a rapid reduction of the longitudinal confinement. A deliberate mismatch of quench parameters allows for the excitation of breathing modes of the emerging soliton and for the determination of its breathing frequency as a function of atom number and confinement. In addition, we observe signatures of higher-order solitons and the splitting of the wave packet after the quench. Our experimental results are compared to analytical predictions and to numerical simulations of the one-dimensional Gross-Pitaevskii equation.The dispersionless propagation of solitary waves is one of the most striking features of nonlinear dynamics, with multiple applications in hydrodynamics, nonlinear optics and broadband long-distance communications [1]. In fiber optics, one-dimensional (1D) "bright" solitons, i.e. solitons presenting a local electric field maximum with one-dimensional propagation, have been observed [2]. They exhibit a dispersionless flow and excitation modes such as breathing or higher-order modes [2][3][4]. Matter waves can also display solitary dispersion properties. Typically, bright matter-wave solitons are created in quasi-1D systems by quenching the particle interaction in a Bose-Einstein condensate (BEC) from repulsive to attractive [5]. Recent experiments demonstrated the collapse [6], collisions [7], reflection from a barrier [8], and the formation of trains [9-11] of bright solitons.In this letter, we experimentally study the excitation modes of a single bright matter-wave soliton. In previous studies, other dynamical properties have been observed, such as center-of-mass oscillations of solitons in an external trap [7], excitations following the collapse of attractive BECs [6,12], and quadrupole oscillations of attractive BECs in three dimensions (3D) [13]. Here, we probe the fundamental breathing mode of a single soliton by measuring its oscillation frequency and the time evolution of its density profile. In addition, we observe signatures of higher-order matter-wave solitons, which can be interpreted as stable excitations with periodic oscillations of the density profile and phase, or as a bound state of overlapping modes [3,14].The shape-preserving evolution of a matter-wave soliton is due to a balancing of dispersive and attractive terms in the underlying 3D Gross-Pitaevskii equation (GPE) [15]. For quasi-1D systems with tight radial confinement, we can approximate the matter wave in the 3D-GPE by the product of a Gaussian wave function for the radial direction and a function f (z) for the longitudinal direction (see [16]). Depending on the ansatz for the Gaussian with either constant or varying radial sizes, f (z) satisfies either the 1D-GPE or the non-polynomial Schrödinger equation (NPSE) [17]. We reference to the analytical solutions of the 1D-G...

Capturing non-Markovian dynamics of open quantum systems is generally a challenging problem, especially for strongly interacting many-body systems. In this Letter, we combine recently developed non-Markovian quantum state diffusion techniques with tensor network methods to address this challenge. As a first example, we explore a Hubbard-Holstein model with dissipative phonon modes, where this new approach allows us to quantitatively assess how correlations spread in the presence of non-Markovian dissipation in a 1D many-body system. We find regimes where correlation growth can be enhanced by these effects, offering new routes for dissipatively enhancing transport and correlation spreading, relevant for both solid state and cold atom experiments.

We propose and analyze a mechanism for rectification of spin transport through a small junction between two spin baths or leads. For interacting baths we show that transport is conditioned on the spacial asymmetry of the quantum junction mediating the transport, and attribute this behavior to a gapped spectral structure of the lead-system-lead configuration. For non-interacting leads a minimal quantum model that allows for spin rectification requires an interface of only two interacting two-level systems. We obtain approximate results with a weak-coupling Born-masterequation in excellent agreement with matrix-product-state calculations that are extrapolated in time by mimicking absorbing boundary conditions. These results should be observable in controlled spin systems realized with cold atoms, trapped ions, or in electrons in quantum dot arrays.

There is a growing interest in using cold-atom systems to explore the effects of strong interactions in topological band structures. Here we investigate interacting bosons in a Cruetz ladder, which is characterised by topological flat energy bands where it has been proposed that interactions can lead to the formation of bound atomic pairs giving rise to pair superfluidity. By investigating realistic experimental implementations, we understand how the lattice topology enhances the properties of bound pairs giving rise to relatively large effective pair-tunnelling in these systems which can lead to robust pair superfluidity, and we find lattice supersolid phases involving only pairs. We identify schemes for preparation of these phases via time-dependent parameter variation and look at ways to detect and characterise these systems in a lattice. This work provides a starting point for investigating the interplay between the effects of topology, interactions and pairing in more general systems, with potential future connections to quantum simulation of topological materials.