522 Measurement Techniques of Femtosecond Spectroscopy
vibration is easily analyzed through diffraction of a probe pulse by the standing
wave pattern.
10.10.3. Theoretical Framework
The same density matrix formalism as in Chapters 3 and 4 can be used to
describe impulsive stimulated Raman scattering. As in Fig. 10.20, we will con-
sider Raman transitions between a ground state |1and a first excited state |2
of a vibrational mode with frequency ω
21
. The states |1 and |2 are infrared
inactive, i.e., there is no dipole allowed transition |1→|2. Coupling between
these two states can occur via any electronic state |. All states |connected
to |1 and |2 via a dipole transition will contribute to the Raman transition. We
assume the optical field E to be off-resonant with all single-photon transitions.
For this assumption to hold, the detuning of the intermediate levels | has to
exceed several pulse bandwidths.
The evolution of the system is described by the density matrix equations (4.1).
For the particular level system being considered:
∂ρ
12
∂t
− iω
21
ρ
12
=−
i
˜
E
+
(ρ
1
p
2
− p
1
ρ
2
)
∂ρ
22
∂t
=−
i
˜
E
+
(ρ
2
p
2
− p
2
ρ
2
)
∂ρ
1
∂t
− iω
1
ρ
1
=−
i
˜
E
+
j
(ρ
1j
p
j
− p
1j
ρ
j
), (10.33)
where the sum over j applies to any level connected to | by a dipole transition,
including levels 1 and 2. A similar equation applies for ρ
2
.
As we have seen in Chapter 4, it is more convenient to decompose the off-
diagonal matrix elements ρ
1
and ρ
2
into an envelope and fast varying phase
term. For instance:
ρ
1
=
1
e
iω
t
(10.34)
and a similar equation for ρ
2
. Substituting Eq. (10.34) into the third equa-
tion (10.33), and keeping only the levels 1 and 2 as levels that are dipole
connected to levels :
∂
1
∂t
+ i(ω
− ω
1
)
1
=−
i
˜
E
2
(ρ
11
p
1
+ ρ
12
p
2
). (10.35)
Impulsive Stimulated Raman Scattering 523
The assumption of the intermediate levels | being off-resonance enables us to
use the adiabatic approximation. This is a standard approximation used routinely
in the context of deriving interaction equations in condition of two- (and more)
photon resonance [29]. A detailed analysis of the use of the adiabatic approxi-
mation in the context of two-photon transitions can be found in [29]. Essentially,
the second term in the left-hand side of Eq. (10.35) dominates, and we can
approximate
1
by its steady-state value:
1
=
−
˜
E(ρ
11
p
1
+ ρ
12
p
2
)
2(ω
− ω
1
)
, (10.36)
and a similar equation for
2
. Substituting into the first equation (10.33), we
find the evolution equation for the coherent Raman excitation:
∂ρ
12
∂t
− iω
21
ρ
12
=−
i
4
2
˜
E
˜
E
∗
ρ
11
p
1
p
2
ω
− ω
1
−
p
1
p
2
ω
− ω
2
ρ
22
. (10.37)
Of particular interest is the amplitude of the off-diagonal element ρ
12
. Let
us define a (complex) amplitude
12
similarly as in Eq. (10.34): ρ
12
=
12
exp(iω
12
t). In addition, to simplify the discussion, let us assume, that there
is only one level | that dominates the interaction. We note that ω
− ω
2
=
(ω
− ω
1
) [1 +ω
21
/(ω
− ω
1
)]. Substituting in Eq. (10.37) yields:
∂
12
∂t
e
iω
21
t
=
ip
1
p
2
4
2
(ω
− ω
1
)
˜
E
˜
E
∗
(
22
− ρ
11
)
(10.38)
where
22
= ρ
22
/[1 + ω
12
(ω
− ω
1
)]. We recognize in Eq. (10.38) a Rabi
frequency similar to the two-photon Rabi frequency discussed in Chapter 4:
p
1
p
2
4
2
(ω
− ω
1
)
˜
E(t)
˜
E
∗
(t) =
r
12
2
˜
E(t)
˜
E
∗
(t). (10.39)
The evolution equations for the density matrix components can be rewritten:
∂
12
∂t
= i
r
12
2
˜
E
˜
E
∗
e
−iω
21
t
[
22
− ρ
11
]
∂ρ
22
∂t
=−2Im
r
12
2
˜
E
˜
E
∗
12
e
iω
21
t
. (10.40)

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