Page 176 - Principles and Applications of NanoMEMS Physics

164 Chapter 4

the interaction may be particularized, to the state of one of the ions in the
motional qubit, by causing the magnetic field gradient to exist on it. This is
accomplished by focusing a laser beam on the ion in question, see Figure 4-
2
1.
Analytically, the inner workings of the ion-trap qubit were described in
detail by Cirac and Zoller [191] as follows. The two states of a particular
ion, namely, its ground and excited states, are denoted by g ≡ 0 and
n n
e ≡ 1 , respectively. The 3-dimensional motion confinement of the
n n
ions is described by an anisotropic harmonic potential characterized by
frequencies Ȟ << Ȟ Ȟ , . The typical energy level scheme contemplated for
x y z
the ion trap is shown in Fig. 4-12(b). When the extent of the ion’s motion is
much less than the inverse wavevector of the laser field, the so-called Lamb-
Dicke limit (LDL), the oscillations of the ground state become normal
modes. Under these circumstances, a laser beam with frequency
ω = ω − ν , or detuning equal to minus the CM mode frequency,
L 0 x
į = − Ȟ , will excite the common mode exclusively. This is the situation in
x
n
which transitions ↓ → ↑ lead to motional mode (phonon number)
transitions n → n − 1 . On the other hand, if ω = ω + ν , then the
L 0 x
transition ↓ → ↑ leads to n → n + 1 transitions. Finally, when
ω = ω the induced transitions ↓ → ↑ leave n unchanged. Thus,
L 0
the relationship between laser detuning, į and motional frequency, and the
fact that the frequencies of the different normal modes are well separated in
the excitation spectrum, allows the control of interaction between ions via
the CM motion and, in fact constitutes the coupling of two qubits which is
necessary to produce quantum gates.
After the quantum qubits are manipulated to effect a quantum
computation, the result must be read. In the case of the ion trap this is
accomplished by measuring the spin-dependent scattered light when a laser
beam impinges upon an ion. Exploiting the fact that scattering is
substantially greater for the ↓ spin than for the ↑ spin, the state of the
spin is inferred.
The manipulation of the state of an N-ion-trap qubit by a laser beam is
driven by the interaction between an ion and the electric field of the laser.
Starting with the Hamiltonian for the n-th ion, H , in the ground state and
0
in the absence of any laser field, and choosing the laser frequency as above,
i.e., į = − Ȟ , and the ion position to coincide with a node of the laser
n x
standing wave, the system is described by,