One-dimensional Bose-Hubbard models provide a variety of intriguing predictions including the formation of repulsively bound two-boson pairs, their collapse and revival and the realization of the two-boson topological states. While probing such systems in quantum regime is quite a challenging task, there is another appealing option to explore the properties of repulsively bound bosons based on mapping of the one-dimensional interacting system onto the two-dimensional classical model described by the standard coupled-mode equations. Adopting the latter approach, we develop the procedure to probe the energy of bound boson pairs as well as to perform the tomography of the bound pair quantum state in realistic dissipative systems.
One-dimensional Bose-Hubbard models provide a variety of intriguing predictions including the formation of repulsively bound two-boson pairs, their collapse and revival and the realization of the two-boson topological states. While probing such systems in quantum regime is quite a challenging task, there is another appealing option to explore the properties of repulsively bound bosons based on mapping of the one-dimensional interacting system onto the two-dimensional classical model described by the standard coupled-mode equations. Adopting the latter approach, we develop the procedure to probe the energy of bound boson pairs as well as to perform the tomography of the bound pair quantum state in realistic dissipative systems.
We study the system of multibody interacting bosons on a two-dimensional optical lattice and analyze the formation of bound bosonic pairs in the context of the Bose-Hubbard model. Assuming a repulsive two-body interaction we obtain the signatures of pair formation in the regions between the Mott insulator lobes of the phase diagram for different choices of higher-order local interactions. Considering the most general Bose-Hubbard model involving local multibody interactions we investigate the ground state properties utilizing the cluster mean-field theory approach and further confirm the results by means of sophisticated infinite projected entangled pair states calculations. By using various order parameters, we show that the choice of higher-order interaction can lead to pair superfluid phase in the system between two different Mott lobes. We also analyze the effect of temperature and density-dependent tunneling to establish the stability of the PSF phase.
This thesis investigates the physics of strongly interacting driven–dissipative bosonic systems in one dimension, combining field-theoretical approaches with various numerical simulation techniques. While weakly interacting condensates are accurately described by a generalized Gross–Pitaevskii equation, the strongly correlated regime of the same microscopic models remains far less understood, particularly out of equilibrium. This thesis aims to bridge this gap by developing controlled quantum-many-body simulations for open systems governed by Lindblad dynamics, as well as investigating the corresponding action under the functional renormalization group.In the first part, we formulate a field-theoretical description of driven–dissipative one-dimen-sional bosons using the Schwinger–Keldysh formalism and the functional renormalization group (fRG).Two models are analyzed: (i) a driven-dissipative condensate whose phase dynamics is expected to exhibit emergent Kardar–Parisi–Zhang (KPZ) behavior, for which we search the emergent KPZ fixed point starting from the bosonic field theory; and (ii) a two-photon–driven system with two-body losses realizing dissipative cat qubits. For the latter, we determine the phase diagram and identify a continuous phase transition between symmetry-broken and symmetric phases, which motivates the construction of an fRG Ansatz capturing this critical behavior.The second part of the thesis focuses on the open system dynamics using exact diagonalization and tensor-network methods. For the Bose–Hubbard model with incoherent pump and loss, we characterize the steady state, and study spectral functions across the full parameter space. We show that key ground-state features are absent under dissipation, while the high-temperature unitary spectrum captures aspects of the open-system dynamics. In an extended model with additional two-body and non-local losses, we establish the semi-classical emergence of KPZ scaling and extract the KPZ dynamical exponent from the spectral function. A finite-size analysis exposes the current limitations of the tensor-network methods for reaching the high occupancy regime in the full quantum evolution.Finally, a last project develops a two-loop perturbative RG analysis of a spin-1 chain at its continuous symmetry breaking quantum critical point, revealing critical exponents distinct from the Ising universality class.Together, these results advance the understanding of strongly interacting open quantum systems, elucidating their symmetries, effective field theories, and dynamical properties, and outlining key directions for future theoretical and numerical work.
We propose an optical simulation of dissipation-induced correlations in one-dimensional (1D) interacting bosonic systems, using a two-dimensional (2D) array of linear photonic waveguides and only classical light. We show that for the case of two bosons in a 1D lattice, one can simulate on-site two-body dissipative dynamics using a linear 2D waveguide array with lossy diagonal waveguides. The intensity distribution of the propagating light directly maps out the wave function, allowing one to observe the dissipation-induced correlations with simple measurements. Beyond the on-site model, we also show that a generalised model containing nearest-neighbour dissipative interaction can be engineered and probed in the proposed set-up.
Quantum statistics in low-dimensional systems predicts anyonic particles with fractional exchange statistics which are neither that of bosons nor fermions. While anyons are typically found in two dimensions as excitations of topologically-ordered states of matter, anyon-like exchange statistics has also been discussed in one dimension, for instance, in the context of the anyon-Hubbard model (AHM), the physics of which has recently been observed in experiment [Kwan et al., arXiv:2306.01737; Dhar et al., arXiv:2412.21131; and Bakkali-Hassani et al., arXiv:2602.20421]. The AHM can be formulated in terms of bosons featuring density-dependent Peierls phases, described by a statistical phase angle θ, which controls asymmetric transport and the formation of dynamically bound pairs at finite momentum. Here, we show theoretically that the AHM also hosts exact two-body bound states in the continuum (BICs) for arbitrary θ=0, and genuine three- and four-body bound states. Unlike conventional bound states stabilized by attractive (or repulsive) interactions, which are energetically localized with a large effective mass, these clusters here are bound by a purely kinematic mechanism endowing them with fast chiral transport properties. We provide a simple variational approximation to the three-body bound states and explain their binding mechanism. Moreover, we show that the signatures of three-body bound states in the AHM can be directly probed experimentally from the expansion dynamics starting from three localized particles.
We relate the quantum dynamics of the Bose–Hubbard model (BHM) to the semiclassical nonlinear equations that describe an array of interacting Bose condensates by implementing a standard variational procedure based on the coherent state method. We investigate the dynamics of the two-site BHM from the purely quantum viewpoint by recasting first the model within a spin picture and using then the related dynamical algebra. The latter allows us to study thoroughly the energy spectrum structure and to interpret quantally the classical symmetries of the two-site dynamics. The energy spectrum is also evaluated through various approximations relying on the coherent state approach.
The Bose-Hubbard model provides a foundational paradigm for exploring quantum phase transitions in many-body systems. This thesis presents a systematic investigation into this model and its extensions, designed to provide a clear path for understanding both the rich underlying physics and the sophisticated numerical methods required for its study. Our approach is sequential: we begin with the standard one-dimensional Bose-Hubbard model, introduce complexity via the extended model with long-range interactions, and conclude by analyzing the impact of dissipation in a cavity-coupled open quantum system.To navigate these regimes, we implement and apply a suite of powerful computational techniques, including Density Matrix Renormalization Group (DMRG), finite-temperature Quantum Monte Carlo (QMC), and Mean-Field Theory (MFT). A significant component of this work is the detailed exposition of the implementation and logic of these numerical methods. Through their comparative application, we precisely characterize the equilibrium phase diagrams and benchmark the capabilities of each method. Crucially, our analysis of the open-system dynamics reveals that dissipation replaces the equilibrium phase diagram with one governed by non-equilibrium steady states (NESS), where the system evolves to either a stable solution or a persistent limit cycle.As such, this thesis is intended to serve a dual purpose: it is both a rigorous physical study of quantum phase transitions in a condensed matter system and a selfcontained pedagogical guide for researchers beginning to explore the phenomenology of quantum phase transitions through numerical schemes.