3.3 The Two-State Model: Rabi Oscillations
87
produce a full population inversion. The product Δt HWHM ω is π , in agreement with
what we found in Sect. 3.2 concerning the properties of a finite radiation pulse.
The RWA is a poor approximation when the detuning Δω is of the same order
of magnitude as |ω 21 |, because then one cannot assume ||ω 21 | − ω| | ||ω 21 | + ω|.
However, even in this case Eq. (3.31) yields P max 1 when Δω HWHM ω . So,
the RWA absorption profile as a function of Δω is close to the exact one, as far as
HWHM ω |ω 21 |, i.e., |W | | |Δε|. Figure 3.1 illustrates the accuracy of the RWA
in various cases, by comparison with virtually exact numerical calculations. The
narrow peaks that appear at approximately ω 21 /3 and ω 21 /5 are due to multiphoton
resonances that cannot be accounted for by the RWA in the form we have applied it
(see Ref. [1] for this topic that exceeds the scope of this book).
3.4 Time-Dependent Perturbation Theory
Let us recall here the set of coupled equations (3.6) that were derived without approximations:
dc i
dt
= −
i
j
c j (t) e
i(ε i −ε j )t/
ψ
(0)
i
ˆ
V
ψ
(0)
j
∀ i .
In order to find a general solution for an arbitrary number of states, we now introduce
the first-order perturbative approximation that consists in evaluating the RHS of
these equations by replacing the c j (t) coefficients with their initial values. This
approximation is only valid if the perturbation is applied for a short time. As we shall
see, how short this time interval must be depends on the strength of the perturbation,
i.e., on the magnitude of the relevant V i j matrix elements. The main advantage of
first-order time-dependent perturbation theory (TDPT) is to decouple Eq. (3.6), i.e.,
to enable us to evaluate each time- dependent coefficient independently of the others.
In fact, time integration yields
c i (t) = c i (t 0 ) −
i
j
c j (t 0 )
t
t 0
e
iω i j t
V i j (t
) dt
(3.33)
where ω i j ≡ hν i j = ε i − ε j . The initial time t 0 can be any time at which the coefficients c j (t 0 ) are known.
Let’s assume for simplicity that one state, ψ
(0)
0 , is initially populated, i.e.,
c 0 = 1
c i = 0 ∀ i = 0 .
(3.34)
Then, from Eq. (3.33) we get
87
produce a full population inversion. The product Δt HWHM ω is π , in agreement with
what we found in Sect. 3.2 concerning the properties of a finite radiation pulse.
The RWA is a poor approximation when the detuning Δω is of the same order
of magnitude as |ω 21 |, because then one cannot assume ||ω 21 | − ω| | ||ω 21 | + ω|.
However, even in this case Eq. (3.31) yields P max 1 when Δω HWHM ω . So,
the RWA absorption profile as a function of Δω is close to the exact one, as far as
HWHM ω |ω 21 |, i.e., |W | | |Δε|. Figure 3.1 illustrates the accuracy of the RWA
in various cases, by comparison with virtually exact numerical calculations. The
narrow peaks that appear at approximately ω 21 /3 and ω 21 /5 are due to multiphoton
resonances that cannot be accounted for by the RWA in the form we have applied it
(see Ref. [1] for this topic that exceeds the scope of this book).
3.4 Time-Dependent Perturbation Theory
Let us recall here the set of coupled equations (3.6) that were derived without approximations:
dc i
dt
= −
i
j
c j (t) e
i(ε i −ε j )t/
ψ
(0)
i
ˆ
V
ψ
(0)
j
∀ i .
In order to find a general solution for an arbitrary number of states, we now introduce
the first-order perturbative approximation that consists in evaluating the RHS of
these equations by replacing the c j (t) coefficients with their initial values. This
approximation is only valid if the perturbation is applied for a short time. As we shall
see, how short this time interval must be depends on the strength of the perturbation,
i.e., on the magnitude of the relevant V i j matrix elements. The main advantage of
first-order time-dependent perturbation theory (TDPT) is to decouple Eq. (3.6), i.e.,
to enable us to evaluate each time- dependent coefficient independently of the others.
In fact, time integration yields
c i (t) = c i (t 0 ) −
i
j
c j (t 0 )
t
t 0
e
iω i j t
V i j (t
) dt
(3.33)
where ω i j ≡ hν i j = ε i − ε j . The initial time t 0 can be any time at which the coefficients c j (t 0 ) are known.
Let’s assume for simplicity that one state, ψ
(0)
0 , is initially populated, i.e.,
c 0 = 1
c i = 0 ∀ i = 0 .
(3.34)
Then, from Eq. (3.33) we get