The implementation of a vibronic coupling in the context of trapped ions, and its diagona- lization are reported. The specific spin-boson-like Hamiltonian model is studied exploiting its symmetries and supersymmetries. Applications are discussed. The study of symmetries plays a central role in the analysis of trapped ion dynamical prob- lems. Indeed, in most of the cases, symmetries provide an elegant and effective way for finding Hamiltonian eigensolutions. In this paper we describe a trapped ion physical situation described by a Hamiltonian model possessing interesting and useful symmetries. In detail, conservation of total excitation num- ber and rotational invariance are found. Moreover a Second Order Hidden Supersymmetry is discovered. Symmetry based diagonalization provides the possibility of writing down the time evolution concerning simple initial conditions. Interesting nonclassical behaviors like the gene- ration of GHZ (7) states are predicted.
We investigate the spectral properties of the Hamiltonian function that describes an ion confined within a nonlinear trap. The Hamiltonian is then particularized for the case of dynamic traps and we introduce algebraic models with the aim to characterize the associated dynamics in such traps. The coherent states formalism for dynamic groups and the time-dependent variational principle are applied in order to study the semiclassical behaviour of the confined ion. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined (Paul and Penning) trap. We infer the equations of motion for the ion in a semiclassical approach. We suggest an algorithm by means of which we can associate a classical Hamiltonian with the quantum Hamiltonian that describes the ion. The classical Hamiltonian implicitly contains spectral information on the quantum system, which means we can dequantify the system.
We investigate nonergodic behavior of the one-dimensional Bose-Hubbard model, which emerges in the unitary quantum dynamics starting with initial-state $|\ensuremath{\psi}(0)\ensuremath{\rangle}=|\ensuremath{\cdots}2020\ensuremath{\cdots}\ensuremath{\rangle}$ in the presence of a trapping potential. We compute the level spacing statistic, the time evolution of the number imbalance between the odd and the even sites, and the entanglement entropy in order to show that the system exhibits nonergodicity in a strongly interacting regime. The trapping potential enhances nonergodicity even when the trapping potential is weak compared to the hopping energy. We derive the effective spin-1/2
XXZ Hamiltonian for the strongly interacting regimes by using a perturbation method. On the basis of the effective Hamiltonian, we show that the trapping potential is effectively strengthened by the on-site interaction, leading to the enhancement of the nonergodic behavior. We also calculate the real-time dynamics under the effective Hamiltonian and find that the entanglement entropy grows logarithmically in time.