Spin qubits for quantum information processing

Chapter 14
Spin Qubits for Quantum
Information Processing
In this chapter we will review the basic principles of manipulating spin qubits for quantum
information processing. The history of spin manipulation (magnetic resonance) techniques
for nuclear spins and electron spins in solids and liquids was dated back to early 1940s [1].
The first nuclear magnetic resonance (NMR) experiments were performed independently
by E.M. Purcell’s group at Harvard [2] and by F. Bloch’s group at Stanford [3] in 1946.
The first electron spin resonance (ESR) experiment was performed by E. Zavoisky in
1945 [4]. After those breakthrough experiments, people quickly recognized that the newly
developed experimental techniques provide an indispensable ultra-small probe inside solids
and liquids. Today, magnetic resonance techniques have many application areas in physics,
chemistry, biology, medicine and engineering [5]. More recently, a single electron spin or
a single nuclear spin is expected to play as a robust qubit for quantum computing. This
is because the quantum state of a spin is relatively stable against external perturbations,
compared to other degrees of freedom such as an electric charge.
In particular, an electron spin is suitable for a qubit in quantum processor, because an
electron spin can be manipulated and also can be coupled to other electron spins with a
much shorter time scale than a nuclear spin. On the other hand, a nuclear spin is suitable
for a qubit in quantum memory, because a nuclear spin has a much longer coherence
time than an electron spin. We will describe in this chapter haw to manipulate a single
electron spin and nuclear spin with an alternating magnetic field and also how to interface
an electron spin and a nuclear spin, i.e. transferring a qubit of information between the
two, using a hyperfine coupling.
14.1 Simple magnetic resonance theory
A nucleus (or electron) possesses a magnetic moment(vector operator),
µ̂ = h̄γ ˆ
I , (14.1)
where γ is a gyromagnetic ratio and ˆ
I is a dimension-less angular momentum operator
( ˆ
J = h̄ˆ
I is a standard angular momentum operator).
1

A magnetic interaction Hamiltonian for a spin in a dc magnetic field H0 (along z-axis)
is
Ĥ = −µ̂ ·
−
→
k H0 = −h̄γH0
ˆ
Iz , (14.2)
where
−
→
k is a unit vector along z-axis.
The squared total angular momentum ˆ
I2 and the z-component of the total angular
momentum ˆ
Iz commute [6]: h
ˆ
I2
, ˆ
Iz
i
= 0 . (14.3)
Therefore, a simultaneous eigenstate of ˆ
I2 and ˆ
Iz exists and can be defined by
ˆ
I2
|I, mi = I(I + 1)|I, mi , (14.4)
ˆ
Iz|I, mi = m|I, mi . (14.5)
The eigenvalues take I = 1
2 , 1, 3
2, · · · and m = I, I − 1, · · · − I, respectively. The eigen-
energies corresponding to (14.2) for such eigenstates are given by
E = −h̄γH0m . (14.6)
Figure 14.1 shows the Zeeman spectrum of the eigen-energies for an electron spin with
I = 1/2 and Cu nuclear spin with I = 3
2 . The gyromagnetic ratio γ for an electron spin is
negative so that m = −1/2 state is the ground state as shown in (a), while that of a Cu
nuclear spin is positive so that m = 3/2 state is the ground state as shown in (b).
(a) (b)
(a) (b)
Figure 14.1: The eigen-energies of an electron spin (a) and a Cu nuclear spin (b).
If we apply an alternating magnetic field H1 cos ωt along a perpendicular direction, for
instance x-axis, in addition to the static field H0, the new Hamiltonian is
ĤI = −h̄γH1 cos ωtˆ
Ix . (14.7)
The x-component ˆ
Ix of the angular momentum operator is expressed as
ˆ
Ix =
1
2
³
ˆ
I+ + ˆ
I−
´
, (14.8)
where ˆ
I+ and ˆ
I− are called a rasing and lowering operator because of the following recursion
relation:
ˆ
I+|I, mi =
q
I(I + 1) − m(m + 1)|I, m + 1i , (14.9)
2

ˆ
I−|I, mi =
q
I(I + 1) − m(m − 1)|I, m − 1i . (14.10)
From the above argument, we understand the transverse ac magnetic field H1 cos ωt has
the capability to change the eigenvalue m, if the frequency ω is close to the resonance with
the Zeeman splitting,
ω ' |γ|H0 . (14.11)
The change of the eigenvalue m by ±1, of course, must accompany the absorption or
emission of a single photon at frequency ω to satisfy energy conservation law. This is the
principle of magnetic resonance.
14.2 Gyromagnetic ratio and g-factor
If a charged particle with a mass m makes a circular motion as shown in Fig. 14.2, the
corresponding angular momentum is
−
→
J = −
→
p × −
→
r = m
2πr2
T
−
→
z , (14.12)
where T and r are a period and radius of circular motion. The magnetic moment is
−
→
µ = iA−
→
z =
e
T
³
πr2
´
−
→
z , (14.13)
where i is a current and A is an effective area. The gyromagnetic ratio γ is thus obtained
from (14.12) and (14.13) as
γ =
−
→
µ
−
→
J
=
e
2m
. (14.14)
Since the nuclear mass is heavier than the electron mass by three to four orders of mag-
nitude, the electron gyromagnetic ratio is lager than the nuclear gyromagnetic ratio by a
factor of 103 ∼ 104. Therefore, at a dc magnetic field of H0 = 1T, the nuclear Zeeman
frequency is in the range of 5 ∼ 50 MHz while the electron Zeeman frequency is ∼ 50GHz.
Figure 14.2: A circular motion of a changed particle.
The above classical picture addresses a so-called orbital angular momentum as shown
in Fig. 14.2, but cannot address properly a spin angular momentum, which is intrinsically
a quantum effect. An electron in free space has a new gyromagnetic ratio:
γe = −ge
µ
e
2m
¶
, (14.15)
3

where ge = 2.002319 · · · is an electron g-factor [6]. An electron is solids acquires a different
value for a g-factor due to the presence of spin-orbit coupling [5, 6]. The gyromagnetic
ratio of a nuclear spin is defined by
µ̂n = gn
e
2Mp
h̄ˆ
I = γnh̄ˆ
I , (14.16)
where Mp is a proton mass. The gyromagnetic ratio γn = gn
e
2Mp
(> 0) is independent of
environments and a universal value for a nucleus, because the nuclear wavefunction is so
tiny and protected by surrounding valence electrons that the environment hardly affects
it.
14.3 Thermal population and saturation
In a magnetic resonance experiment, a spin system continuously absorbs a photon from
an alternating magnetic field and dissipates its energy to thermal reservoirs, as shown in
Fig. 14.3. The steady state (thermal equiliblium) populations of the two spin states, N−
and N+, and those of the two reservoir states, Na and Nb, satisfy the detailed balance:
spin system external reservoir
Figure 14.3: Absorption of photon and energy dissipation of a spin system.
N−NbW−b→+a = N+NaW+a→−b , (14.17)
where the two transition matrix elements are identical, since
W−b→+a =
2π
h̄
¯
¯
¯h+, a|ĤI|−, bi
¯
¯
¯
2
(14.18)
=
2π
h̄
¯
¯
¯h−, b|ĤI|+, ai
¯
¯
¯
2
= W+a→−b.
Therefore, the spin population ratio is identical to the reservoir population ratio (Boltzman
factor) as it should be at thermal equilibrium:
N−
N+
=
Na
Nb
= e−E/kBT
. (14.19)
If we begin with a non-thermal population, the transient behavior is descried by the
following rate equation:
d
dt
N+ = N−W↓ − N+W↑ , (14.20)
4

d
dt
N− = N+W↑ − N−W↓ , (14.21)
where W↓ = NbW−b→+a > W↑ = NaW+a→−b. The population difference n = N+ − N− is
governed by the rate equation:
d
dt
n = −
n − n0
T1
. (14.22)
Here n0 = N
W↓−W↑
W↓+W↑
is a thermal equilibrium population difference and T1 = (W↓ + W↑)−1
is a spin relaxation time. Equation (14.22) indicates that any departure of n from n0 can
be removed with a time constant T1 due to the coupling of the spin system to external
reservoirs.
The rate equation for photon absorption and emission processes is given by
d
dt
N+ = W−→+N− − W+→−N+ , (14.23)
d
dt
N− = W+→−N+ − W−→+N− . (14.24)
The two transition matrix elements are also identical with each other, W−→+ = W+→− =
W, so that the population difference n is governed by
d
dt
n = −2Wn . (14.25)
Equation (14.25) means that the population difference n disappears (population satura-
tion) if only absorption and emission of photons from the alternating magnetic field is
taken into account and the dissipation to external reservoirs is neglected.
If we combine the two effects, we have a complete rate equation:
d
dt
n = −2Wn −
n − n0
T1
. (14.26)
The steady state solution of the above equation is
n =
n0
1 + 2WT1
. (14.27)
The energy absorption rate per second is now calculated as
d
dt
E = h̄ωnW =
h̄ωn0W
1 + 2WT1
=



h̄ωn0W
³
W ¿ 1
2T1
´
h̄ωn0
2T1
³
W À 1
2T1
´ . (14.28)
The energy absorption rate dE
dt initially increases with the incident field energy W but
saturates at the value determined by T1 time constant. A short T1 is essential to detect a
weak absorption signal.
5

14.4 Manipulation of a single spin
14.4.1 Heisenberg equation of motion
The motion of an isolated spin in a static magnetic field is governed by the Zeeman
Hamiltonian,
Ĥ = −h̄γH0
ˆ
Iz . (14.29)
Using the commutation relations,
h
ˆ
Ix, ˆ
Iy
i
= iˆ
Iz , (14.30)
h
ˆ
Iy, ˆ
Iz
i
= iˆ
Ix , (14.31)
h
ˆ
Iz, ˆ
Ix
i
= iˆ
Iy , (14.32)
we obtain the Heisenberg equations of motion:
d
dt
ˆ
Ix =
1
ih̄
h
ˆ
Ix, Ĥ
i
= γH0
ˆ
Iy , (14.33)
d
dt
ˆ
Iy = γH0
ˆ
Ix , (14.34)
d
dt
ˆ
Iz = 0 . (14.35)
Combining (14.33), (14.34) and (14.35), we have the famous operator torque equation:
d
dt
ˆ
I =
−
→
i
dˆ
Ix
dt
+
−
→
j
dˆ
Iy
dt
+
−
→
k
dˆ
Iz
dt
= γH0
ˆ
I ×
−
→
k , (14.36)
or equivalently
d
dt
µ̂ = γH0µ̂ ×
−
→
k , (14.37)
where
−
→
i ,
−
→
j ,
−
→
k are the unit vectors along x, y, z directions and µ̂ = γh̄ˆ
I is the magnetic
moment operator. Equation (14.36) or (14.37) suggest that the spin precesses along a
static magnetic field with right-handed circulation for γ < 0 or left-handed circulation for
γ > 0 as shown in Fig. 14.4.
6

Figure 14.4: Spin precession about a static magnetic field with a Larmor frequency ωL =
|γ|H0.
14.4.2 Rotating reference frame
Suppose a fictitious frame (
−
→
i ,
−
→
j ,
−
→
k ) rotates along z-axis (or
−
→
k ) with an angular frequency
−
→
Ω, their dynamics are described by
d
dt
−
→
i =
−
→
Ω ×
−
→
i , (14.38)
d
dt
−
→
j =
−
→
Ω ×
−
→
j , (14.39)
d
dt
−
→
k =
−
→
Ω ×
−
→
k , (14.40)
Then a dynamical variable
−
→
F =
−
→
i Fx +
−
→
j Fy +
−
→
k Fz follows the equation,
d
dt
−
→
F =
−
→
i
dFx
dt
+
−
→
j
dFy
dt
+
−
→
k
dFz
dt
+ Fx
d
−
→
i
dt
+ Fy
d
−
→
j
dt
+ Fz
d
−
→
k
dt
(14.41)
=
∂
−
→
F
∂t
+
−
→
Ω ×
−
→
F .
Here d
−
→
F
dt and ∂
−
→
F
∂t are considered as the time rates of change of
−
→
F with respect to a (fixed)
laboratory frame and (rotating) fictitious frame (
−
→
i ,
−
→
j ,
−
→
k ), respectively.
If we substitute
−
→
F = µ̂ in (14.41) and use (14.37), we have
d
dt
µ̂ =
∂µ̂
∂t
+
−
→
Ω × µ̂ = γH0µ̂ ×
−
→
k . (14.42)
The above result leads to the Heisenberg equation of motion for µ̂ in a rotating frame:
∂µ̂
∂t
= µ̂ ×
³
γH0
−
→
k +
−
→
Ω
´
(14.43)
= γHeffµ̂ ×
−
→
k ,
7

where Heff
−
→
k = H0
−
→
k +
−
→
Ω/γ is an effective field. If a fictiones frame rotates with an
angular frequency
−
→
Ω = −γH0
−
→
k , the effective field Heff disappears and the spin remains
fixed, ∂µ̂
∂t = 0, in this rotating frame.
Alternatively, the spin µ̂ rotates along a static field
³−
→
k
´
with an angular frequency
−
→
Ω = −γH0
−
→
k in the (fixed) laboratory frame. This oscillation frequency is called a Larmor
frequency and given by
ωL = |γ|H0 . (14.44)
14.4.3 Effect of alternating magnetic fields
Suppose an alternating magnetic field is applied along x-axis
³
−
→
i
´
,
Hx(t) =
−
→
i Hx0 cos ωt . (14.45)
This ac field can be decomposed into two circularly ratating fields:
HR(t) = H1
³
−
→
i cos ωt +
−
→
j sin ωt
´
, (14.46)
HL(t) = H1
³
−
→
i cos ωt −
−
→
j sin ωt
´
, (14.47)
where H1 = Hx0/2. If HR(t) rotates synchronously as the precession of the spin, then
HL(t) rotates in an opposite direction so that we can neglect the effect of HL(t) on the
spin.
The torque equation in the laboratory frame is now
d
dt
µ̂ = µ̂ × γ
h
H0
−
→
k + H1
³
−
→
i cos ωzt +
−
→
j sin ωzt
´i
. (14.48)
The above equation can be translated into the one in the rotating frame with an angular
frequency ωz:
∂
∂t
µ̂ = µ̂ ×
h
(γH0 + ωz)
−
→
k + γH1
−
→
i
i
(14.49)
= µ̂ × γ
·µ
H0 +
ωz
γ
¶
−
→
k + γH1
−
→
i
¸
In this rotating frame, HR(t) is a static field along x-axis while the field along z-axis is
offset by ωz
γ , so that the effective static is now
Heff =
µ
H0 +
ωz
γ
¶
−
→
k + H1
−
→
i . (14.50)
At resonance condition, ωz = −γH0, the effective static field Heff along z-axis disappears
and only H1
−
→
i survives. In such a case, the spin simply rotates about x-axis in the rotating
frame (Fig. 14.5(a)). If there is a slight detuning, ωz 6= −γH0, the effective static field
Heff =
³
H0 + ωz
γ
´ −
→
k + H1
−
→
i is in the x − z plane and the spin precesses in a cone of fixed
angle about the direction of Heff with an angular frequency γHeff (Fig. 14.5(b)).
8

spin
spin
(a) (b)
Figure 14.5: Spin precession at resonance condition, ωz = −γH0 (a), and at off-resonance
condition ωz 6= −γH0 (b), where γ > 0 is assumed.
14.4.4 Bloch equations
A macroscopic magnetization consisting of many spins can be described by the phenomeno-
logical model called “Bloch equations”. The magnetization along z-axis is expressed as an
ensemble averaged quantity:
Mz = h̄γhˆ
IziN = h̄γ
n
2
, (14.51)
where N is the total number of spins in a sample and n = N+ − N− is the population
difference. The equation of motion for Mz is driven by the spin precession term due to an
external field and spin relaxation term due to dissipative coupling to heat reservoirs:
d
dt
Mz = γ
³−
→
M ×
−
→
H
´
z
−
Mz − M0
T1
, (14.52)
where
−
→
M =
−
→
i Mx +
−
→
j My +
−
→
k Mz, M0 is the thermal equilibrium magnetization and T1 is
the spin relaxation time. Similarly, the equation of motions for Mx and My are given by
d
dt
Mx = γ
³−
→
M ×
−
→
H
´
x
−
Mx
T2
, (14.53)
d
dt
My = γ
³−
→
M ×
−
→
H
´
y
−
My
T2
, (14.54)
where T2 is the spin decoherence time.
When there is a static field H0 along z-axis and an alternating field H1 with a slight
detuning h0 = H0 + ωz
γ 6= 0, (14.52)-(14.54) are reduced to
d
dt
Mz = −γH1My −
Mz − M0
T1
, (14.55)
9

d
dt
Mx = γh0My −
Mx
T2
, (14.56)
d
dt
My = γH1Mz − γh0Mx −
My
T2
, (14.57)
in the rotating frame.
If H1 is small so that saturation along x-axis is negligible (weak probing case), we have
the approximate solution for Mz:
Mz ' M0 . (14.58)
We now introduce
M+ = Mx + iMy = h̄γhˆ
I+iN , (14.59)
which satisfies the equation of motion
d
dt
M+ = −
µ
iγh0 +
1
T2
¶
M+ + iγH1M0 . (14.60)
The steady state solution of (14.60) is given by
M+ =
γM0
γh0 − i 1
T2
H1 . (14.61)
The in-phase and quadrature-phase responses to the small external probe field H1 are now
expressed as
Mx = χ0ω0T2
(ω0 − ω) T2
1 + (ω0 − ω)2
T2
2
H1 , (14.62)
My = χ0ω0T2
1
1 + (ω0 − ω)2
T2
2
H1 , (14.63)
where χ0 = M0/H0 is the magnetic susceptibility at dc and ω0 = γH0 is the spin Larmor
frequency. Equations (14.62) and (14.63) represent the linear dispersion and absorption
spectrum for a small external probe field.
14.5 Optical manipulation of semiconductor spin qubits
S.M. Clark et al., PRL 99, 040501 (2007)
D. Press et al., Nature 456, 218 (2008)
14.6 Double resonance
14.6.1 Hyperfine interaction
Let us consider a magnetic coupling between two spins, say a nuclear spin and an electron
spin, as shown in Fig. 14.6
Suppose a nucleus with a magnetic moment µn at −
→
r = 0 has a finite radius ρ0, there
exist two kinds of magnetic fields: uniform internal field Hi inside the nucleus and external
dipole field He outside the nucleus, as shown in Fig. 14.7. The continuity of magnetic flux
10

(nuclear spin)
(electron spin)
Figure 14.6: A magnetic coupling between a nuclear spin ˆ
I and electron spin Ŝ.
requires the integrated magnetic fields inside and outside the nucleus exactly cancel with
each other out:
φi (ρ0) + φe (ρ0) = 0. . (14.64)
The internal flux is given by
φi (ρ) = πρ2
0Hi , (14.65)
while the external flux is
φe (ρ) = 2π
Z ∞
ρ0
µ
−
µn
r3
¶
rdr (14.66)
= −µn
2π
ρ0
.
From (14.64), (14.65) and (14.66), we have the internal field
Hi =
2µn
ρ2
0
. (14.67)
If an electron wavefunction overlaps with a nuclear wavefunction, the effective nuclear
magnetic field which an electron spin feels is given by
Hz = Hi
µ
4
3
πρ3
0
¶
|ue(0)|2
, (14.68)
where |ue(0)|2 is the electron density at −
→
r = 0 and thus
³
4
3 πρ3
0
´
|ue(0)|2 is the probability
of finding an electron inside the nucleus. The magnetic interaction between the nucleus and
the electron outside the nucleus is identically equal to zero, when the electron wavefunction
spreads isotropically to a much larger space than the nuclear wavefunction, which is the
case for a s-wave symmetry electron wavefunction. Then, the space integration cancels
out exactly due to the continuity of magnetic flux. Form (14.67) and (14.68), the effective
interaction Hamiltonian.
Ĥ = −µeHz = −
8π
3
µ̂e · µ̂n|ue(0)|2
(14.69)
=
8π
3
γeγnh̄2 ˆ
I · Ŝ|ue(0)|2
,
11

Figure 14.7: The internal and external magnetic fields created by a single nuclear spin
with a magnetic moment µn.
where µ̂e = −h̄γeŜ and µ̂n = h̄γn
ˆ
I are used. The Hamiltonian (14.69) is called a con-
tact Hyperfine interaction, which exists only for the case that the electron and nucleus
wavefunctions overlap.
If an electron is distant from a nucleus or an electron and a nucleus are co-located but
the electron wavefunction is a p-wave, d-wave or other symmetries with non-zero angular
momentum, the space integration for an external dipole field He is non-zero. We have a
new interaction Hamiltonian [6]:
Ĥ =
µ̂e · µ̂n
r3
−
3 (µ̂e · −
→
r ) (µ̂n · −
→
r )
rs
(14.70)
=
γeγn
r3
h̄2


3
³
Ŝ · −
→
r
´ ³
ˆ
I · −
→
r
´
r2
− Ŝ · ˆ
I

 ,
The Hamiltonian (14.70) has the terms such as −γeγn
r3 h̄2
Ŝx
ˆ
Ix and γeγn
r5 h̄2
Ŝx
ˆ
Iyxy. If
ˆ
Ix = 1
2
³
ˆ
I+ + ˆ
I−
´
and ˆ
Iy = 1
2i
³
ˆ
I+ − ˆ
I−
´
and similar expressions for Ŝx and Ŝy are used
in (14.70) and we introduce a polar coordinate (r, θ, ϕ) as shown in Fig. 14.8, we have the
new expression:
Ĥ = −
γeγn
r3
h̄2
(A + B + C + D + E + F) , (14.71)
where
A = Ŝz
ˆ
Iz
³
1 − 3 cos2
θ
´
, (14.72)
B = −
1
4
³
Ŝ+
ˆ
I− + Ŝ−
ˆ
I+
´ ³
1 − 3 cos2
θ
´
, (14.73)
C = −
3
2
³
Ŝ+
ˆ
Iz + Ŝz
ˆ
I+
´
sin θ cos θe−iφ
, (14.74)
D = −
3
2
³
Ŝ−
ˆ
Iz + Ŝz
ˆ
I−
´
sin θ cos θe−iφ
, (14.75)
E = −
3
4
Ŝ+
ˆ
I+ sin2
θe−2iφ
, (14.76)
12

F = −
3
4
Ŝ−
ˆ
I− sin2
θe2iφ
. (14.77)
Figure 14.8: A polar coordinate for an electron spin and a nuclear spin.
If there is a strong static field H0 along z-axis, the Zeeman Hamiltonian
Ĥz = −γnh̄H0
ˆ
Iz + γeh̄H0Ŝz , (14.78)
dominates over the hyperfine interaction Hamiltonian. In this case we can solve the Zee-
man problem first and then treat the hyperfine coupling as a perturbation. The eigenen-
ergy of the unperturbed state is
Ez = −γnh̄H0mI + γeh̄H0ms , (14.79)
where mI and ms are the eigenvalues of ˆ
Iz and Ŝz. The energy level diagram is shown
in Fig. 14.9 for mI = ±1/2 and ms = ±1/2. The matrix elements A to F connect these
Zeeman eigenstates, as shown in Fig. 14.10
13

(+ +)
(+ –)
(– +)
(– –)
Figure 14.9: The Zeeman energy level diagram for a pair of nuclear and election spins
(mI = ±1/2, ms = ±1/2)
14.6.2 Spin Hamiltonian and hyperfine spectrum
When a nucleus is located at −
→
r = 0 and an electron wavefunction χ(r) is located at an
arbitrary position, both contact and dipolar hyperfine coupling terms are non-negligible,
the over-all electron-nuclear interaction Hamiltonian is given by
ĤIS =
Z
|ψ(r)|2
½
8π
3
γeγnh̄2 ˆ
I · Ŝδ(r) +
γeγn
r3
h̄2
(14.80)
×

3
³
ˆ
I · −
→
r
´ ³
Ŝ · −
→
r
´
r2
− ˆ
I · Ŝ





= AxŜx
ˆ
Ix + AyŜy
ˆ
Iy + AzŜz
ˆ
Iz .
In most cases of interst, the electron Zeeman energy γeh̄H0 is much larger than the
hyperfine interaction energy Ax, Ay, Az. Therefore, the z-component of the electron spin
operator commutes with the spin Hamiltonian:
h
Ĥ, Ŝz
i
' 0. . (14.81)
This fact leads to a reduced spin Hamiltonian
Ĥ = γnh̄H0
ˆ
Iz + γeh̄H0Ŝz + AzŜz
ˆ
Iz . (14.82)
The eigenstates and corresponding eigen-energies of the above Hamiltonian are
|msi|mIi (ms = ±1/2, mI = ±1/2) , (14.83)
E = −γnh̄H0mI + γeh̄H0ms + AzmsmI . (14.84)
14

(+ +)
(– +)
(– –)
(+ –)
E, F
(+ +)
(– +)
(– –)
(+ –)
(+ +)
(– +)
(– –)
(+ –)
(+ +)
(– +)
(– –)
(+ –)
C, D
A B
Figure 14.10: The Zeeman energy eigenstates connected by the matrix elements A-F.
The hyperfine energy diagrams for a high field case (γnh̄H0 > Az) and for a low field case
(γnh̄H0 < Az) are shown in Fig. 14.11, where Az > 0 is assumed.
The ESR resonance spectrum splits into double peaks,
ωESR = γeH0 +
Az
h̄
mI , (14.85)
while the NMR resonance spectrum splits into the two kinds of doublets depending on the
field strength:
ωNMR =
(
γnH0 + Az
h̄ ms (high field)
Az
2h̄ ± γnH0 (low field)
. (14.86)
Figure 14.12 shows the ESR spectrum and the two kinds of NMR spectra. If the
alternating transverse field at ωESR = γeH0 ± Az
2h̄ is sent to the coupled electron-nuclear
spin system, the conditional rotation of the electron spin depending on the nuclear spin
state is realized. Similarly, if the alternating transverse field at ωNMR = γnH0 ± Az
2h̄ is
sent, the conditional rotation of the nuclear spin depending on the electron spin state
is realized. This is a basic principle of double resonance, which can be extended to an
electron-electron coupling and nuclear-nuclear coupling, too.
15

(+ +)
(– +)
(– –)
(+ –)
(a)
(+ +)
(– +)
(– –)
(+ –)
(b)
Figure 14.11: The hyperfine energy diagrams for (a) high field case (γnh̄H0 > Az) and (b)
low field case (γnh̄H0 < Az) .
14.6.3 Optical control of two semiconductor spins
T.D. Ladd and Y. Yamamoto, arXiv:0910.4988v1 (2009)
16

(a)
(b)
(c)
Figure 14.12: The ESR and two kinds of NMR spectra for a hyperfine coupling electron-
nuclear spin system.
17

Bibliography
[1] C.J. Gorter et al., Physica 9, 591 (1942).
[2] E.M. Purcell et al., Phys. Rev. 69, 37 (1946).
[3] F. Bloch et al., Phys. Rev. 69, 127 (1946).
[4] E. Zavoicky, Fiz. Zh 9, 211, 245 (1945).
[5] P. Schlichter, Principles of Magnetic Resonance, Springer, Berlin (1990).
[6] C. Cohen-Tannoudji, B. Diu, and F. Laoë, Quantum Mechanics John Wiley & Sons,
New York (1977).
18

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