Page 275 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
field vector . Insert this into (7.82) and drive the electric field in perfect
E
resonance, the solution is given by
c t() = cos ( Ωt)
1
(7.83)
c t() = – iexp – ( iϕ)sin ( Ωt)
2
⁄
Rabi with Ω = E µ ( 2—) . Ω is called the Rabi frequency. It is best inter-
0
Frequency preted by looking at the square modulus of the coefficients:
c t() 2 = ( cos ( Ωt)) 2 and c t() 2 = ( sin ( Ωt)) 2 . These square moduli
2
1
are the probabilities for the electron to occupy either quantum state 1 or
2. Thus the absorption process is going on in a time window from t = 0
⁄
⁄
to t = π ( 2Ω) . For t > π ( 2Ω) the probability for the electron to
occupy quantum state 2 decreases. This can be interpreted as an emission
process. In order to have a well defined final state the external field exci-
tation pulse has to be of finite duration. It is interesting to observe that the
transition dipole matrix element oscillates with the frequency ω 0 and is
modulated by harmonic function with twice the Rabi frequency.

In the case where there are multiple electrons in the system distributed
onto the two energy levels the square moduli of the coefficients denote
the number of electrons to be found in the respective level
c t() 2 = N 1 , c t() 2 = N 2 , with the total number of electrons given
2
1
by N = N + N 2 . Let us calculate the dipole moment of the electron
1
charge distribution given by the wavefunction
∫
(
(
(
,
,
p = ψ∗ r t) – e)rψ r t) V
d
(7.84)
– iω 0 t iω 0 t
= ( – c ∗ c e µµ µ µ + c c ∗ e µµ µ µ )
1 2
12
21
2
1
– iω 0 t
With the definition α t() = c ∗ c e , we have for the dipole moment
1
2
– p = α t()µµ µµ + α∗ t()µµ µµ 21 and an equation of motion for α t() to read
12
i
γ
α ˙ = – ( iω + )α + ---V D (7.85)
0 21
—
272 Semiconductors for Micro and Nanosystem Technology