Page 178 - Principles and Applications of NanoMEMS Physics

166 Chapter 4

φ
and phase q = 1 and phase = , respectively. This creates the evolution
0
ˆ
operator U 1 , 2 , which exclusively changes the sign of the sate g 1 via a
n n
rotation through the state e 0 ; 3) Apply again a π laser pulse with
1 n
φ
0
0
polarization q = and phase = to the m-th ion to create the evolution
ˆ
ˆ
U 0 , 1 ≡ U 0 , 1 () 0 . Since these operators act on non-interacting ions, the
m m
ˆ
ˆ
ˆ
ˆ
overall effect is given by the product U ≡ U 0 , 1 U 1 , 2 U 0 , 1 in Eq. (8) below.
m n , m n m
Comparison of the first and last columns reveals that the effect of the
composite operation is to change the sign of the state only when both ions
are initially excited, thus, Eq. (8) embodies a C-NOT gate.
ˆ
ˆ
ˆ
U 0 , 1 U 1 , 2 U 0 , 1
m n m
g g 0 → g g 0 → g g 0 → g g 0
m n m n m n m n
g e 0 → g e 0 → g e 0 0 → g e 0 0
m 0 m 0 m m . (8)
e g 0 → − gi g 1 → i g g 1 → e g 0
0 m n m n m n 0 m n
e e 0 → − gi e 1 →− gi e 1 → − e e 0
0 m 0 n m 0 n m 0 n 0 m 0 n
Many successful implementations of ion-trap qubits have been
experimentally demonstrated [192]. Key to these experimental
demonstrations are techniques to address a variety of issues, most notably: 1)
Mitigating the decoherence of the ion trap, which is due to the spontaneous
decay of the internal atomic states and the motion damping; 2) Suppressing
spontaneous emission; 3) Obtaining highly efficient read-out schemes. A
thorough discussion of problems and solutions regarding ion-trap qubits is
given by Wineland et al [192].

4.3.1.2 The Nuclear Magnetic Resonance (NMR) Qubit

As is well known, some atoms exhibit an intrinsic nuclear magnetic
K
G
moment µ and an angular momentum I = , and these are related through the
gyromagnetic ratio γ by [28],
G
G
µ = I = γ . (9)
Since the angular momentum is quantized [60], with values
m = I, I − 1,...,− I , a nucleus with an intrinsic angular momentum of half a
I
unit, i.e., =I 1 2 , will have the allowed values of m = ± 1 2 . Thus, in the
G I
presence of a magnetic field B = B z ˆ , the energy of interaction between the
0
magnetic moment and the field,