Page 183 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 171

ω − ω >> 2π J , which may be obtained when dealing with
i j ij
heteronuclear spins or with small homonuclear molecules, it reduces to
[194],
N
j
H = = ¦ 2 πJ I i I . (19)
J ij z z
i < j
Eq. (19) captures the circumstance that, in addition to a constant externally
G
applied magnetic field, B , the actual field at a given spin location includes a
z ˆ
static field along ± , which is elicited by spins in its neighborhood. The
consequence of this additional field is to shift the spin’s energy levels and
manifests as a change in the Larmor frequency. For instance, a neighboring
spin j in state 0 will shift the frequency of spin i by J− 2 , whereas if
ij
spin j is in state 1 , it will shift the frequency of spin i by J + J ij 2 . In
ij
general, it turns out that, when in the presence of neighboring spins, the
spectrum of a given spin would show, instead of a single line at its Larmor
frequency, two lines for every neighboring spin, the lines being separated by
the coupling strength J and located equidistantly above and below the
ij
Larmor frequency.
In the majority of NMR-based QC experiments, the system Hamiltonian
realized is described by the simplified Hamiltonians [194], i.e.,
j
H = − ¦ = Ȧ i I i + = ¦ 2ʌJ I i I , (20)
sys 0 z ij z z
i i < j
where the first term arises from the energy of isolated spins, and the second
from the energy of interacting (coupled) spins.
To effect the manipulation of qubits in NMR-based QC [194], it is
necessary to apply a magnetic field that will rotate the state of the spin-1/2
nuclei, see Fig. 4-1 . This is accomplished by adding to the static z ˆ -directed
6
magnetic field, B , a time-varying (RF) electromagnetic field oriented in
0
the x ˆ − y ˆ plane, of a frequency ω close to the spin precession frequency
RF
ω . This RF field gives rise to the control Hamiltonian which, for a single
0
spin, is given by [194],
+
I
H = −= ȖB [cos (Ȧ t + ij )I + sin (Ȧ t φ ) ], (21)
RF 1 RF x RF y
where B is the applied RF field amplitude and φ its phase. For liquid-state
1
NMR, B ≈ 50 KHz ≡ ω . In the presence of N spins, the total control
γ
1 1
Hamiltonian is the sum of the terms such as Eq. (21) of each spin. The
implementation of quantum gates in NMR-based QC exploits the ability to