Engineering Macroscopic Quantum State Of A Single Trapped Ion: High-order Lamb-Dicke Approximations

Trapped cold ions are important candidates for building quantumcomputers in the future and provide attractive platforms to test certainfundamental laws such as quantum non-locality in quantum mechanics. Inparticular, a trapped ion driven by classical laser beams can providevarious physically realizable interactions.Even and odd coherent states are typically macroscopic quantum states;they are formed by superposing two coherent states |α＞and|-α＞. Forcertain aparameters, e.g., |α|≤0.5, even and odd coherent states showvery different quantum statistical properties: even coherent states showsqueezed but not anti-bunching effects, while the odd coherent statesshow anti-bunching but not squeezed effects. Thus, generating andcontrolling these states are always paid much attention in variousexperimental quantum-state engineerings. Indeed, most of the approachesto prepare motional quantum states of a single trapped cold ion operatedunder the Lamb-Dicke approximation (LDA), only the effects of the firstorder of LD parameterηare important. LDA is not always satisfiedstrictly, e. g., the corresponding LD parameter is about 0.25 in a recentexperiment, the higher-order term ofηshould be taken into account forthe precise quantum engineering. Specially, for a single trapped ion thisparameter could, in principle, be further enhanced by adjusting theapplied laser beams. Sequentially, strong laser-ion interactions outsidethe LD regime could be achieved.In this paper we deducted the time evolvement functor of the systemby the Hamiltonian. And obtain the state evolvement equation of the systemin the LDA and NLDA. And calculated squeezed, anti-bunching effects andfidelity in the LDA and NLDA according to the state evolvement equationof the system.By the numerical computation we find that if the LD parameterηisrelatively small, then the usual even/odd coherent states could be approached well. While, for a relatively large LD parameters, e.g.,η＞0.2, the high-order LD approximations are significant and the effectsofη~k with k≥2 could be revealed more obviously in the quantumstatistical properties of the generated states superposed by the infiniteeven/odd number states with different weights.