We study the quantum dynamics of the central-depleted-well (CDW) regime in a three-mode Bose Hubbard model subject to a confining parabolic potential. By introducing a suitable set of momentum-like modes we identify the microscopic variables involved in the quantization process and the dynamical algebra of the model. We describe the diagonalization procedure showing that the model reduces to a double oscillator. Interestingly, we find that the parameter-space domain where this scheme entails a discrete spectrum well reproduces the two regions where the classical trimer excludes unstable oscillations. Spectral properties are examined in different limiting cases together with various delocalization effects. These are shown to characterize quantum states of the CDW regime in the proximity of the borderline with classically unstable domains.
We study the quantum dynamics of the central-depleted-well (CDW) regime in a three-mode Bose Hubbard model subject to a confining parabolic potential. By introducing a suitable set of momentum-like modes we identify the microscopic variables involved in the quantization process and the dynamical algebra of the model. We describe the diagonalization procedure showing that the model reduces to a double oscillator. Interestingly, we find that the parameter-space domain where this scheme entails a discrete spectrum well reproduces the two regions where the classical trimer excludes unstable oscillations. Spectral properties are examined in different limiting cases together with various delocalization effects. These are shown to characterize quantum states of the CDW regime in the proximity of the borderline with classically unstable domains.
We study the Bose-Hubbard model for three sites in a symmetric, triangular configuration and search for quantum signatures of the classical regime of oscillatory instabilities, known to appear through Hamiltonian Hopf bifurcations for the "single-depleted-well" family of stationary states in the discrete nonlinear Schrödinger equation. In the regimes of classical stability, single quantum eigenstates with properties analogous to those of the classical stationary states can be identified already for rather small particle numbers. On the other hand, in the instability regime the interaction with other eigenstates through avoided crossings leads to strong mixing, and no single eigenstate with classical-like properties can be seen. We compare the quantum dynamics resulting from initial conditions taken as perturbed quantum eigenstates and SU(3) coherent states, respectively, in a quantum-semiclassical transitional regime of 10-100 particles. While the perturbed quantum eigenstates do not show a classical-like behavior in the instability regime, a coherent state behaves analogously to a perturbed classical stationary state, and exhibits initially resonant oscillations with oscillation frequencies well described by classical internal-mode oscillations.
We consider a Bose–Hubbard (BH) trimer, i.e. an ultracold Bose gas populating three quantum states. The latter can be either different sites of a triple-well potential or three internal states of the atoms. The bosons can tunnel between different states with variable tunnelling strength between two of them. This will allow us to study; (i) different geometrical configurations, i.e. from a closed triangle to three aligned wells and (ii) a triangular configuration with a π -phase, i.e. by setting one of the tunnellings negative. By solving the corresponding three-site BH Hamiltonian we obtain the ground state of the system as a function of the trap topology. We characterize the different ground states by means of the coherence and entanglement properties. For small repulsive interactions, fragmented condensates are found for the π -phase case. These are found to be robust against small variations of the tunnelling in the small interaction regime. A low-energy effective many-body Hamiltonian restricted to the degenerate manifold provides a compelling description of the π -phase degeneration and explains the low-energy spectrum as excitations of discrete semifluxon states.
In order to explore a quantum version of a discrete nonlinear Schrodinger equation (DNLS), we quantize one nonlinear Schrodinger model, which is used to study different physical systems, e.g. coupled Bose-Einstein condensates. We will focus on small systems, like Dimer and Trimer.In our efforts to solve this quantum problem, we develop a Mathematica routine that implements the Number State Method and solves the corresponding Schrodinger equation. We calculate analytically and numerically the energy spectrum of the Dimer and Trimer systems. Those eigenenergies depend on the parameter set Q=Q1, Q2, Q3, Q4, Q5 and by adjusting this set Q, we can obtain the desired results and examine their effects. After the quantization of the extended DNLS we obtain a quantum DNLS, also known as an extended Bose-Hubbard (BH) model. The aim of this Master's thesis is to study the differences and similarities between the classical DNLS and the extended BH model, and what happens when we approach from the quantum regime to the classical one. Taking into account that the Hamiltonian has an important conserved quantity, the number operator, enables the total Hamiltonian to be block-diagonalized. This can be accomplished by taking advantage of additional symmetries, such as translational symmetry, which will simplify the analysis of the Hamiltonian matrix. In our results we discuss several effects that break the lattice symmetry, as the intersection between symmetric and antisymmetric states. We also compare our results with those obtained in previous works for the classical model, and we find some similarities, e.g. the transition of the highest-energy state from a one-site solution to a two-site solution depending on which Q parameters we vary, but also differences, as the appearance of a three-site solution, in a Trimer system.
We study the quantum dynamics of the Bose-Hubbard model on a ladder formed by two rings coupled by the tunneling effect. By implementing the Bogoliubov approximation scheme, we prove that, despite the presence of the inter-ring coupling term, the Hamiltonian decouples in many independent sub-Hamiltonians ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ associated with momentum-mode pairs $\ifmmode\pm\else\textpm\fi{}k$. Each sub-Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ is then shown to be part of a specific dynamical algebra. The properties of the latter allow us to perform the diagonalization process, to find the energy spectrum and the conserved quantities of the model, and to derive the time evolution of important physical observables. We then apply this solution scheme to the simplest possible closed ladder, the double trimer. After observing that the excitations of the system are weakly populated vortices, we explore the corresponding dynamics by varying the initial conditions and the model parameters. Finally, we show that the inter-ring tunneling determines a spectral collapse when approaching the border of the dynamical-stability region.
The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites.In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies.Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer.These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis.In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories.Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.
The triangular Bose-Hubbard trimer is topologically the minimal model for a BEC superfluid circuit. As a dynamical system of two coupled freedoms it has mixed phase space with chaotic dynamics. We employ a semiclassical perspective to study triangular trimer physics beyond the conventional picture of the superfluid-to-insulator transition. From the analysis of the Peierls-Nabarro energy landscape, we deduce the various regimes in the $(\ensuremath{\Omega},u)$ parameter space, where u is the interaction and $\ensuremath{\Omega}$ is the superfluid rotation-velocity. We thus characterize the superfluid stability and chaoticity of the many-body eigenstates throughout the Hilbert space.