Coherent Interactions with Two-Level Systems 225
If the source is an atom, spherical waves are emitted with identical fluctuations
at all points at equal distance from the source. Macroscopically, we are used to
looking at a volume average of randomly distributed dipoles. Some new sources
have the emitted dipoles arranged in a regular fashion. For instance, multiple
quantum well lasers are made of “sheets” of emitting atoms which are thinner
than the wavelength of the light. This particular arrangement of emitters has
important implications for the macroscopic properties of the source [5, 6].
4.2. COHERENT INTERACTIONS WITH
TWO-LEVEL SYSTEMS
4.2.1. Maxwell–Bloch Equations
Whether we are dealing with molecular or atomic transitions, the situation
can arise where the ultrashort duration of the optical pulse becomes compara-
ble with—or even less than—the phase-relaxation time of the excitation. In the
frequency domain, the pulse spectrum is broader than the homogeneous linewidth
defined in the first section of Chapter 3. If the pulse is so short that its spectrum
becomes much larger than the inhomogeneous linewidth, the medium response
becomes similar to that of a single atom. It may seem like a simplified situa-
tion when the excitation occurs in a time shorter than all interatomic interaction.
It is in fact quite to the contrary; in dealing with longer pulses, the faster phase-
relaxation time of the induced excitation simplifies the light–matter response. One
is accustomed to dealing with a steady state rather than the “transient” response
of light–matter interaction.
We will start from the semiclassical equations for the interaction of near
resonant radiation with an ensemble of two-level systems inhomogeneously
broadened around a frequency ω
ih
. The extension to multilevel systems will
be discussed in the next section. We refer to the book by Allen and Eberly [7]
for more detailed developments.
We summarize briefly the results of Chapter 3 for a two-level system, of
ground state |0 and upper state |1, excited by the field E(t). The density matrix
equation for this two-level system is:
˙ρ =
1
i
[
H
0
− pE, ρ
]
, (4.1)
where H
0
is the unperturbed Hamiltonian, and p the dipole moment that is parallel
to the polarization direction of the field. Introducing the complex field through
E =
˜
E
+
+
˜
E
−
in Eq. (4.1) leads to the following differential equations for the
226 Coherent Phenomena
diagonal and off-diagonal matrix elements:
˙ρ
11
−˙ρ
00
=
2p
6
iρ
01
˜
E
−
− iρ
10
˜
E
+
7
(4.2)
˙ρ
01
= iω
0
ρ
01
+
ip
˜
E
+
[
ρ
11
− ρ
00
]
, (4.3)
where ω
0
is the resonance frequency of the two-level system. It is generally
convenient to define a complex “pseudo-polarization” amplitude
˜
Q by
iρ
01
p
¯
N =
1
2
˜
Q exp(iω
t) (4.4)
where
¯
N =
¯
N
0
g
inh
(ω
0
−ω
ih
) and
¯
N
0
is the total number density of the two-level
systems. The real part of
˜
Q will describe the attenuation (or amplification for
an initially inverted system) of the electric field. Note that
˜
Q = i
˜
P where
˜
P
is the slowly varying polarization envelope defined in Eq. (3.28). Further we
introduce a normalized population inversion:
w = p
¯
N(ρ
11
− ρ
00
). (4.5)
The complete system of interaction and propagation equations can now be
written as:
˙
˜
Q = i(ω
0
− ω
)
˜
Q − κ
˜
Ew −
˜
Q
T
2
(4.6)
˙w =
κ
2
[
˜
Q
∗
˜
E +
˜
Q
˜
E
∗
]−
w − w
0
T
1
(4.7)
∂
˜
E
∂z
=−
µ
0
ω
c
2n
∞
0
˜
Q(ω
0
)g
inh
(ω
0
− ω
ih
)dω
0
. (4.8)
The quantity κE with κ = p/ is the Rabi frequency. T
1
and T
2
are respectively
the energy and phase-relaxation times. Most of the energy conserving relaxations
are generally lumped in the phase-relaxation time T
2
. Equation (4.8) has been
obtained from Eq. (3.30) by integrating over the polarization of subensembles
with resonance frequency ω
0
. The set of Eqs. (4.6)–(4.8) is generally designated
as Maxwell–Bloch equations.
Another common set of notations to describe the light–matter interaction uses
only real quantities, such as the in-phase (ν ) and out-of-phase (u) components
Coherent Interactions with Two-Level Systems 227
of the pseudo-polarization
˜
Q, and, for the electric field
˜
E, its (real) amplitude E
and its phase ϕ. Defining
˜
Q = (iu + ν)e
iϕ
(4.9)
and substituting in the above system of equations leads to the usual form of
Bloch equations
1
for the subensemble of two-level systems having a resonance
frequency ω
0
.
˙u = (ω
0
− ω
−˙ϕ)ν −
u
T
2
(4.10)
˙v =−(ω
0
− ω
−˙ϕ)u −κE w −
ν
T
2
(4.11)
˙w = κEν −
w − w
0
T
1
(4.12)
where the initial value for w at t =−∞is
w
0
= p
¯
N(ρ
(e)
11
− ρ
(e)
00
). (4.13)
The propagation equation, [Eq. (4.8)] in terms of
˜
E and ϕ, becomes
∂E
∂z
=−
µ
0
ω
c
2n
∞
0
ν (ω
0
)g
inh
(ω
0
− ω
ih
)dω
0
(4.14)
∂ϕ
∂z
=−
µ
0
ω
c
2n
∞
0
u(ω
0
)
E
g
inh
(ω
0
− ω
ih
)dω
0
. (4.15)
The vector representation of Feynman et al. [9] for the interaction equations
is particularly useful in the description of coherent phenomena. The repre-
sentation is a cinematic representation of the set of equations (4.10), (4.11),
and (4.12). For simplicity, we consider first an undamped isolated two-level
system (T
1
= T
2
= T
3
=∞), and construct a fictitious vector
P of components
(u, v, w), and a pseudo-electric field vector
E of components (κE ,0,−ω). The
detuning is defined as ω = ω
0
− ω
−˙ϕ. The system of Eqs. (4.10)–(4.12)
are then the cinematic equations describing the rotation of a pseudo-polarization
vector
P rotating around the pseudo-electric vector
E with an angular velocity
1
These equations are the electric dipole analog of equations derived by F. Bloch [8] to describe
spin precession in magnetic resonance.
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