Fast Control Of Exact Quantum States Of Trapped Ions In The Lamb-Dicke Region And Related Applications

In recent years,with the continuous development of laser technology,we can coherently and rapidly manipulate the internal electron states and the external motional ’states of trapped ions with extremely high precision by coupling the trapped ions to laser fields,which provides a reliable foundation for the quantum computation,quantum information processing and quantum dynamics based on trapped ions.Because the trapped ion system has a very long coherent manipu-lation time and scalability,it becomes one of the promising system for large-scale quantum computing.The thesis is divided into five chapters.In the first chapter,we give a brief introduction about the trapping and laser-cooling of ions,some commonly approx-imations in laser-ion interaction,the delta kicked optical lattice system and the preparation of quantum states and quantum computation based on trapped ions.In the second chapter,we study quantum motion of a laser-kicked two-level ion in the Lamb-Dicke regime and obtain a set of exact solutions of the generalized coherent states.A new stability region of parameter space where classical stability criterion fully agrees with the fidelity treatment of quantum ground-state stabil-ity is found,which unusually contains the well-known resonance frequency for a weak kick and larger kick strength for a far-off-resonance frequency.When the field parameters in the stability region are applied,the ion’s wave-packet train-s continuously oscillate in the Lamb-Dicke regime,while for the parameters in the instability region,they collapse and spread to far away from the Lamb-Dicke regime,resulting in the crossover from linearity to nonlinearity.Meanwhile the laser kicks bring hopping phases of the exact solutions,and lead to the stable or unstable hops of the expectation values of ion’s momentum and energy.We also obtain the probability for the ion being in different pseudospin states when we apply a π/2 pulse of Ramsey type experiment to the ion to rotate the exact gen-eralized coherent states,which reveals that the ultrafast entanglement generation and population transfer of the system can be analytically controlled by managing the laser pulses.In chapter three,we investigate quantum dynamics of a single trapped ion with two stable electronic ground states and a lager energy gap from the rest of the spectrum,held in the Lamb-Dicke regime of a driven optical lattice.A set of de-generate Schrodinger’s cat states with the same expected energy is found in which the quasiparticle excitations above the ground eigenstate of a undriven n-phonon system are referred to "live" and "dead" states of the cat and obey non-Abelian statistics.The degeneracy is not based on simple symmetry considerations,so the braiding operations based on the interchanges of the macroscopically separated non-Abelian quasiparticles may be insensitive to microscopic perturbations and noise from the environment.The controlled transitions between different instanta-neous degenerate ground states are illustrated for an array of δ-shaped laser pulses.The results could be justified experimentally in the existing setups and may be use-ful in engineering quantum dynamics of many single ions for topological quantum computation.In chapter four,we investigate the quantum dynamics of two ions confined in a one-dimensional linear trap under the effect of state-dependent forces.The Hamiltonian of the two trapped ions can be expressed as two independent har-monic oscillators describing the center of mass motion and the relative motion respectively.The trial solution method that we use is applied to obtain the ions’center of mass motion and the relative motion exactly for arbitrarily integrable,time-dependent forces.The exact analytically solvable dynamics provide a reliable recipe for tuning system parameters to achieve the phase control,because of which we realize a fast two-qubit phase gate based on the exact solutions.Considering that state-dependent forces are simple sinusoidal functions,a short predetermined operation time reversely determine intensities and frequencies of the laser induced,state-dependent forces.Such a smooth continuous force can reduce the experimen-tal compared with abrupt kicks.The scheme of the two-qubit phase gate can be simply extended to the cases of ions of different species and arbitrary proportion-alities between the state-dependent forces.In chapter five,a summary of the work and an outlook of this topic are given.