Page 177 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 165

H = Ș ȍ [e g ae − iij + g e a + e iij ], (6)
n, q q n n q
N 2
+
where a and a are the creation an annihilation operators of CM phonons,
respectively, Ω is the Rabi frequency, φ is the phase of the laser field at the

mean position of the ion, q = , 0 1 levels involved in the energy transition

excited by the laser, and =Ș k = 2 2MȞ << 1 is the LDL parameter, with
ș x
k = kcosș , k the laser wavevector, θ the angle between the direction of
ș
propagation of the laser and the x-axis of motion of the qubit, and M the ion
G
⋅
mass. The Rabi frequency, Ω = − E ↑ d ε ˆ ↓ = 2 , characterizes the
0
L
transition frequency between the ground and metastable states produced by a
laser with electric field amplitude E and polarization vector İ ˆ in an ion
0
L
K
of electric dipole operator d .
The evolution of the system upon being impinged by a laser beam pulse
of time duration t = kʌ (ȍȘ N ) on the n-th ion is described by the
unitary operator,
ª ʌ º
ij
U k, q () = exp −ik ( e g ae −iij + g e a + e iij ) . (7)
n « q n q »
¬ 2 ¼
Application of this unitary operator on the various states of the n-th qubit
yields the results of Table 4-1. 0 and 1 represent the population of the
CM mode with zero and one phonon, respectively.

Table 4-1. Effect of Ion-Trap Unitary Operator on State Evolution
Operator Initial State Final State

U n k, q g 0 g 0
n n
U k, q g 1 cos (kʌ 2 ) g 1 − ie iij sin (kʌ 2 ) e 0
n n n q
n
U k, q e 0 cos (kʌ 2 ) e 0 − ie − iij sin (kʌ 2 ) g 1
n n q n n

The above interaction is amenable to the implementation of a two-bit gate.
In particular, Cirac and Zoller [191] have shown that this is accomplished by
0
following three steps: 1) Apply a π laser pulse with polarization q = and
ˆ
φ
ˆ
0
phase = to the m-th ion to create the evolution U m 0 , 1 ≡ U m 0 , 1 () 0 ; 2) Turn
on the laser directed to the n-th ion for a time duration π2 and polarization