The Bloch-Siegert shift is a phenomenon in quantum physics that becomes important for driven two-level systems when the driving gets strong (e.g. atoms driven by a strong laser drive or nuclear spins in NMR, driven by a strong oscillating magnetic field).
When the rotating wave approximation (RWA) is invoked, the resonance between the driving field and a pseudospin occurs when the field frequency
ω
is identical to the spin's transition frequency
ω
0
. The RWA is, however, an approximation. In 1940 Bloch and Siegert showed that the dropped parts oscillating rapidly can give rise to a shift in the true resonance frequency of the dipoles.
In RWA, when the perturbation to the two level system is
H
a
b
=
V
a
b
2
cos
(
ω
t
)
, a linearly polarized field is considered as a superposition of two circularly polarized fields of the same amplitude rotating in opposite directions with frequencies
ω
,
−
ω
. Then, in the rotating frame(
ω
), we can neglect the counter-rotating field and the Rabi frequency is
Ω
=
1
2
(
|
V
a
b
/
ℏ
|
)
2
+
(
ω
−
ω
0
)
2
.
Consider the effect due to the counter-rotating field. In the counter-rotating frame (
ω
c
r
=
−
ω
), the effective detuning is
Δ
ω
c
r
=
ω
+
ω
0
and the counter-rotating field adds a driving component perpendicular to the detuning, with amplitude
Ω
c
r
=
|
V
a
b
/
ℏ
|
. The counter-rotating field effectively dresses the system, where we can define a new quantization axis slightly tilted from the original one, with an effective detuning
Δ
ω
e
f
f
=
±
(
|
V
a
b
/
ℏ
|
)
2
+
(
ω
+
ω
0
)
2
Therefore, the resonance frequency (
ω
r
e
s
) of the system dressed by the counter-rotating field is
Δ
ω
e
f
f
away from our frame of reference, which is rotating at
−
ω
ω
r
e
s
+
ω
=
±
(
|
V
a
b
/
ℏ
|
)
2
+
(
ω
+
ω
0
)
2
and there are two solutions for
ω
r
e
s
ω
r
e
s
=
ω
0
[
1
+
1
4
(
V
a
b
ℏ
ω
0
)
2
]
and
ω
r
e
s
=
−
1
3
ω
0
[
1
+
3
4
(
V
a
b
ℏ
ω
0
)
2
]
.
The shift from the RWA of the first solution is dominant, and the correction to
ω
0
is known as the Bloch-Siegert shift:
δ
ω
B
−
S
=
1
4
(
V
a
b
)
2
ℏ
2
ω
0
As the Bloch-Siegert shift is caused by the off-resonant component of the oscillating field, the shift can be generalized to any non-resonant field of the form
H
o
r
=
V
o
r
2
cos
(
ω
o
r
t
)
perturbing the spin. With a similar treatment as above, and invoking the RWA on the off-resonant field, the resulting Bloch-Siegert shift is
δ
ω
B
−
S
=
1
2
(
V
a
b
)
2
ℏ
2
(
ω
0
−
ω
o
r
)
