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We examine the time evolution of a two level ion interacting with a lightfield in harmonic oscillator trap and in a trap with anharmonicities. Theanharmonicities of the trap are quantified in terms of the deformationparameter $\tau $ characterizing the q-analog of the harmonic oscillator trap.Initially the ion is prepared in a Schr\"{o}dinger cat state. The entanglementof the center of mass motional states and the internal degrees of freedom ofthe ion results in characteristic collapse and revival pattern. We calculatenumerically the population inversion I(t), quasi-probabilities $Q(t),$ andpartial mutual quantum entropy S(P), for the system as a function of time.Interestingly, small deformations of the trap enhance the contrast betweenpopulation inversion collapse and revival peaks as compared to the zerodeformation case. For \beta =3 and $4,(% \beta $ determines the average numberof trap quanta linked to center of mass motion) the best collapse and revivalsequence is obtained for \tau =0.0047 and \tau =0.004 respectively. For largevalues of \tau decoherence sets in accompanied by loss of amplitude ofpopulation inversion and for \tau \sim 0.1 the collapse and revival phenomenondisappear. Each collapse or revival of population inversion is characterized bya peak in S(P) versus t plot. During the transition from collapse to revivaland vice-versa we have minimum mutual entropy value that is S(P)=0. Successiverevival peaks show a lowering of the local maximum point indicating adissipative irreversible change in the ionic state. Improved definition ofcollapse and revival pattern as the anharminicity of the trapping potentialincreases is also reflected in the Quasi- probability versus t plots.Comment: Revised version, 16 pages,6 figures. Revte

The Schrödinger cat state of trapped ions in harmonic and anharmonic oscillator traps