  Although the initial proposal for ion trap quantum computation made use of an
auxiliary internal level to perform logic between ions, this resource is not
necessary in principle. Instead, one may perform such operations directly using
sideband laser pulses, operating with an arbitrary (sufficiently small)
Lamb-Dicke parameter. We explore the potential of this technique, showing how
to perform logical operations between the internal state of an ion and the
collective motional state and giving explicit constructions for a
controlled-not gate between ions.
