Considering a two-dimensional (2D) Bose-Hubbard spinor lattice with weak nearest-neighbor interactions and no particle transfer between sites, we theoretically study the transport of energy from one initially excited dimer to the rest of the lattice. Beyond a critical interaction strength, low-energy on-site excitations are quickly dispersed throughout the array, while stronger excitations are self-trapped, resulting in localized energy breathers and solitons. These structures are quasiparticle analogs to the discrete 2D solitons in photonic lattices. Full many-body simulations additionally demonstrate the localization of one-particle entropy.
Considering a two-dimensional (2D) Bose-Hubbard spinor lattice with weak nearest-neighbor interactions and no particle transfer between sites, we theoretically study the transport of energy from one initially excited dimer to the rest of the lattice. Beyond a critical interaction strength, low-energy on-site excitations are quickly dispersed throughout the array, while stronger excitations are self-trapped, resulting in localized energy breathers and solitons. These structures are quasiparticle analogs to the discrete 2D solitons in photonic lattices. Full many-body simulations additionally demonstrate the localization of one-particle entropy.
We investigate theoretically the Bose-Hubbard version of the celebrated Su-Schrieffer-Heeger topological model, which essentially describes a one-dimensional dimerized array of coupled oscillators with on-site interactions. We study the physics arising from the whole gamut of possible dimerizations of the chain, including both the weakly and the strongly dimerized limiting cases. Focusing on two-excitation subspace, we systematically uncover and characterize the different types of states which may emerge due to the competition between the inter-oscillator couplings, the intrinsic topology of the lattice, and the strength of the on-site interactions. In particular, we discuss the formation of scattering bands full of extended states, bound bands full of two-particle pairs (including so-called 'doublons', when the pair occupies the same lattice site), and different flavors of topological edge states. The features we describe may be realized in a plethora of systems, including nanoscale architectures such as photonic cavities, optical lattices and qubits, and provide perspectives for topological two-particle and many-body physics.
We study the temporal expansion of an ultracold Bose gas in two-dimensional square optical lattices. The gas is described by the Bose-Hubbard model deep in the superfluid regime, with initially all bosons condensed in the central site of the lattice. We use the previously developed nonequilibrium propagator method for capturing the time evolution of an interacting bosonic system, where the many-body Hamiltonian is represented in an appropriate local basis and the corresponding field operators are separated into the classical [Bose-Einstein condensate (BEC)] part and quantum mechanical fluctuations. After a quench, i.e., after a sudden switch of the lattice nearest-neighbor hopping, the expanding bosonic cloud separates spatially into a fast ballistic forerunner and a slowly expanding central part controlled by self-trapping. We show that the forerunner expansion is driven by the coherent dynamics of the BEC and that its velocity is consistent with the Lieb-Robinson bound. For smaller lattices we analyze how quasiparticle collisions lead to enhanced condensate depletion and oscillation damping.
The Bose-Hubbard model is used to study bosons in optical lattices. In this thesis we will use an extended Bose-Hubbard model to study a type of completely localized solutions, called compactons. The compactons are a special case of the much studied solitons. The soliton is a familiar concept in non-linear physics. It is a stable, localized wave-solution, found in a range of different systems; from DNA-molecules to optical fibers. The compacton is a soliton that is completely localized, i.e. strictly zero outside a given area. The dynamics of the (extended) Bose-Hubbard model is based on the tunneling of particles between the lattice sites. The ordinary Bose-Hubbard model only accounts for one-particle tunneling processes. We will consider a model that also takes some two-particle tunneling processes into account, basically by considering long-range effects of the particle interaction. The aim of this thesis is to find and study the quantum analog of the compactons found in an extended Discrete Non-Linear Schrödinger equation. We will study analytical solutions and try to find if and under which conditions specific compactons exist. Numerical calculations are made to study the properties of the compactons and to study how compacton solutions arise in the classical limit.
Superconducting circuits constitute a promising platform for future implementation of quantum processors and simulators. Arrays of capacitively coupled transmon qubits naturally implement the Bose-Hubbard model with attractive on-site interaction. The spectrum of such many-body systems is characterized by low-energy localized states defining the lattice analog of bright solitons. Here, we demonstrate that these bright solitons can be pinned in the system, and we find that a soliton moves while maintaining its shape. Its velocity obeys a scaling law in terms of the combined interaction and number of constituent bosons. In contrast, the source-to-drain transport of photons through the array occurs through extended states that have higher energy compared to the bright soliton. For weak coupling between the source or drain and the array, the populations of the source and drain oscillate in time; for chains of even length, their population remains low at all times, while it can reach half the number of total bosons in odd chains. Implications of our results for actual experimental realizations are discussed.
Ultracold atoms in optical lattices are a powerful platform for the study of quantum many-body physics. The combination of a high degree of isolation from the environment and external control over all relevant parameters makes these systems ideal candidates for the quantum simulation of fundamental lattice models. However, since the atoms are trapped in standing waves of interfering laser beams, the available trap geometries are constrained to regular lattices and single-site control is limited. In this thesis, an alternative experimental platform is investigated. Here, the combination of a microlens array and a spatial light modulator is used to provide a two-dimensional optical microtrap array for ultracold atoms. This setup allows for versatile trap geometries and comprehensive single-site control. The experimental feasibility of the described platform is investigated in the following way. First, the light field generating the microtrap array is simulated using a detailed model of the optical setup. The computed intensity distribution is proportional to the optical dipole potential for the atoms. Second, the simulation results are used to obtain the Hubbard parameters for multiple alkalies from numerical calculations as well as approximative analytical methods. It is shown that the strongly correlated regimes of the Bose-Hubbard model can be reached at sufficiently large tunneling rates. In addition, the impact of fluctuations in the trap parameters is investigated. Third, two approaches are considered for the preparation of low-entropy many-body states. On the one hand, a loading scheme is investigated which starts from a Bose-Einstein condensate and is used in optical lattice experiments. Here, the depth of the microtrap array is increased adiabatically. On the other hand, an array of isolated traps, which is initialized with one atom per site in the respective motional ground state, is considered as starting point. The itinerant regime of the Hubbard model is accessed by an adiabatic decrease of the trap depth. An analysis of ramp-induced excitations and external heating processes shows the feasibility of both approaches. Demonstrating the potential of the investigated platform, two applications are described. On the one hand, the tunneling dynamics of ultracold atoms between weakly coupled ring lattices is analyzed. Controlled by the interaction strength, multiple phenomena can be observed: Josephson oscillations exhibiting collapse and revival, inter-action-induced self-trapping, and tunneling resonances. On the other hand, the implementation of a scheme for universal quantum computing based on time-continuous quantum walks of interacting particles is proposed. Here, the information is encoded into the position of atomic wave packets moving through a planar graph which is built from optical microtraps and implements a quantum circuit. Details of an experimental implementation are discussed for both applications using the results derived in the preceding parts of this thesis.
We analyze the disorder-driven localization of the two-dimensional Bose-Hubbard model by evaluating the full low-energy quasiparticle spectrum via a recently developed fluctuation operator expansion method.For any strength of the local interaction we find a mobility edge that displays an approximately exponential decay with increasing disorder strength.We determine the finite-size scaling collapse and exponents at this critical line finding that the localization of excitations is characterized by weak multifractality and a thermal-like critical gap ratio.A direct comparison to a recent experiment yields an excellent match of the predicted finite-size transition point and scaling of single particle correlations.