Engineering superpositions of displaced number states of a trapped ion

We present a protocol that permits the generation of a subtle superposition with 2ℓ+1 displaced number states on a circle in phase space as target state for the center-of-mass motion of a trapped ion. Through a sequence of ℓ cycles involving the application of laser pulses and no-fluorescence measurements, explicit expressions for the total duration of laser pulses employed in the sequence and probability of getting the ion in the upper electronic state during the ℓ cycles are obtained and analyzed in detail. Furthermore, assuming that the effective relaxation process of a trapped ion can be described in the framework of the standard master equation for the damped harmonic oscillator, we investigate the degradation of the quantum interference effects inherent to superpositions via Wigner function