  A scheme to implement quantum logic gates by manipulating trapped ions
through interaction with monochromatic external laser field and quantized
cavity field, beyond the Lamb-Dicke regime, is presented. Characteristic times,
for implementing ionic state transitions using non-resont laser pulse or
quantized cavity field, shows a sharp decline for large Lamb-Dicke parameter
value of $\eta_{L}=\eta_{c}=0.2$, and is seen to decrease further with increase
in number of initial state vibrational quanta $m$.
