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.
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.
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.
We propose a scheme for investigating the quantum dynamics of interacting electron models by means of a time-dependent variational principle and spin coherent states of space lattice operators. We apply such a scheme to the one-dimensional Hubbard model, and solve the resulting equations in different regimes. In particular, we find that at low densities the dynamics is mapped into two coupled nonlinear Schr\"odinger equations, whereas near half-filling the model is described by two coupled Josephson-junction arrays. Focusing then to the case in which only the phases of the spin variables are dynamically active, we examine a number of different solutions corresponding to the excitations of few macroscopic modes. Based on fixed-point equations of the simpler among them, we show that the standard one-band ground-state phase space is found.
Quantum dynamics of the Bose-Hubbard model is investigated through a semiclassical Hamiltonian picture provided by the time-dependent variational principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ensuremath{\ll}1$. The phase diagram is in good agreement with quantum Monte Carlo results and third order strong coupling perturbative expansion.
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.
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 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.