Multiphoton Coherent Interaction 245
(a) (b)
Figure 4.7 The problem of reaching a high level with a harmonic ladder (a), and the multiple-pulse
multiphoton coherent solution (b).
section, Section 4.4.3. we will proceed from this more general situation to the
particular approximation that leads to the multiphoton analog of Bloch’s optical
equations.
4.3.2. Multiphoton Multilevel Transitions
General Formalism
The complex atomic–molecular system is represented by the unperturbed
Hamiltonian H
0
to which corresponds a set of eigenstates ψ
k
of energy ω
k
.
In the presence of an electric field E, the state of the atomic–molecular system is
described by the wavefunction ψ, a solution of the time–dependent Schrödinger
equation:
Hψ = i
∂ψ
∂t
, (4.53)
246 Coherent Phenomena
with the total Hamiltonian given by:
H = H
0
+ H
= H
0
− p · E(t) (4.54)
where p is the dipole moment. In the standard technique for solving time-
dependent problems, the wave function ψ is written as a linear combination
of the basis functions ψ
k
:
ψ(t) =
k
a
k
(t)ψ
k
. (4.55)
This expression for ψ is inserted in the time-dependent Schrödinger equation (4.53).
Taking into account the normalization conditions for the basis functions ψ
k
, one
finds the coefficients a
k
have to satisfy the following set of differential equations:
da
k
dt
=−
i
ω
k
a
k
+
j
i
p
k,j
E(t) cos[ω
t + ϕ(t)]a
j
=−iω
k
a
k
+
j
i
2
p
k,j
[
˜
Ee
iω
t
+
˜
E
∗
e
−iω
t
]a
j
(4.56)
where p
kj
are the components of the dipole coupling matrix
2
for the transition
k → j, and a
k
are the amplitudes of the eigenstates.
Apart from the basic assumption that the time scale is short enough for all
phase and amplitude relaxation to be negligible, Eq. (4.56) is of a quite gen-
eral nature, and it is ideally suited to numerical integration. It can be used to
solve the most complex problem of light–matter interaction, because no assump-
tions have been made as to the transitions, resonances, detuning, or degeneracies.
However, the complete numerical treatment leaves little room for physical insight.
We will therefore consider some simplified cases for which general trends emerge
from the solution. Even though numerical analysis is still required, it seems
possible to gain some intuition as to the response of the system.
As a first simplifying assumption, we will consider only the levels that can be
connected directly by a single photon at ω
. The separations of the successive
energy levels k from the ground state 0 are ω
0k
. We designate with an index k
a state at an energy close to kω
(the ground state being thus labeled with the
index 0). The important levels to consider will be those sketched in Figure 4.8(a)
for which the detuning:
k
= ω
0k
− kω
(4.57)
2
We will use both the notation p
ik
and p
i,k
depending on the complexity of the subscripts.
Multiphoton Coherent Interaction 247
(a)
(b)
(c)
ᐉ
ᐉ
ᐉ
ᐉ
ᐉ
01
01
23
t
1
2
3
4
t
2
t
3
t
4
03
4
3
2
1
0
∆
4
∆
3
∆
2
∆
1
Time
Frequency
Figure 4.8 (a) Energy level diagram, (b) spectral representation, and (c) sequence of exciting
pulses.
is at most comparable to the sum of the radiation bandwidth and the transition
rate. Such a near-resonance condition is illustrated by the sketch of Fig. 4.8(b)
where the dashed line indicates the pulse spectrum overlapping the successive
single photon transitions. Consistent with this approximation, we replace the set
of coefficients a
k
, which have temporal variations at optical frequencies, by the
“slowly varying” set of coefficients c
k
, using the transformation:
a
k
= e
−ikω
t
c
k
. (4.58)
This is the same type of transformation to a “rotating set of coordinates” as we
applied to the set of density matrix equations (4.2) and (4.3) in the section on
two-level systems.
Selecting from the sum in Eq. (4.56) the particular pairs of levels that matches
best the resonance condition for an incident field at frequency ω
, leads to the
simpler form:
dc
k
dt
=−i
k
c
k
+
i
2
p
k−1,k
˜
E
∗
(t)c
k−1
+
i
2
p
k,k+1
˜
E(t)c
k+1
. (4.59)
The use of amplitudes c
k
rather than elements of the density matrix is more
convenient in dealing with multilevel systems. It is easier, however, to get a
physical picture from a density matrix representation. The density matrix ele-
ments can be calculated from ρ
ii
= a
i
a
∗
i
and ρ
ij
=
ij
e
iω
t
= a
i
a
∗
j
e
iω
t
for i = j.
To verify that the two descriptions are equivalent, let us look at the equations for
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