In the current section, we present numerical results from the solution of the nonlinear Bloch equations, Equations 1 to 3, for a specific semiconductor quantum well system. We consider a GaAs/AlGaAs double quantum well. The structure consists of two GaAs symmetric square wells with a width of 5.5 nm and a height of 219 meV. The wells are separated by a AlGaAs barrier with a width of 1.1 nm. The form of the quantum well structure and the corresponding envelope wavefunctions are presented in Figure
.

This system has been studied in several previous works
[

20,

24,

25,

28,

35-

38]. The electron sheet density takes values between 10

^{9}and 7 × 10

^{11}cm

^{−2}. These values ensure that the system is initially in the lowest subband, so the initial conditions can be taken as

*S*_{1}(0) =

* S*_{2}(0) = 0 and

*S*_{3}(0) = −1. The relevant parameters are calculated to be

*E*_{1 }−

* E*_{0 }= 44.955 meV and

*z*_{01 }= −3.29 nm. Also, for electron sheet density

*N *= 5 × 10

^{11}cm

^{−2}, we obtain

*Π**e*^{2}*N*(

*L*_{1111}−

*L*_{0000})/2

*ε *= 1.03 meV,

*γ *= 0.2375 meV, and

meV. In all calculations, we include the population decay and dephasing rates with values

*T*_{1}= 10 ps and

*T*_{2}= 1 ps. Also, in all calculations, the angular frequency of the field is at exact resonance with the modified frequency

*ω*_{10}, i.e.,

*ω *=

* ω*_{10}.

In Figure
, we present the time evolution of the inversion

*S*_{3}(

*t*) for different values of the electron sheet density for a Gaussian-shaped pulse with

. Here,

*t*_{p}= 2

*Π**n*_{p}/

*ω* is the duration (full width at half maximum) of the pulse, where

*n*_{p }is the number of cycles of the pulse and can be a noninteger number. The computation is in the time period [0,4

*t*_{p}] for pulse area

*θ *=

* Π*. For electron sheet density

*N *= 10

^{9}cm

^{−2}, which is a small electron sheet density, Equations 1 to 3 are very well approximated by the atomic optical Bloch equations; therefore, a

*Π *pulse leads to some inversion in the system in the case that the pulse contains several cycles. However, the inversion is not complete as the relaxation processes are included in the calculation and

*T*_{2} is smaller than the pulse duration. In Figure
a, that is for

*n*_{p}= 10, we see that the electron sheet densities have a very strong influence in the inversion dynamics. For example, for

*N *= 3 × 10

^{11}cm

^{−2}, the population inversion evolves to a smaller value, and for larger values of electron sheet density, the final inversion decreases further and even becomes nonexistent.

A quite different behavior is found in Figure
b,c,d for pulses with smaller number of cycles. In Figure
b, we see that essentially the inversion dynamics differs slightly for

*N *= 10

^{9}cm

^{−2},

*N *= 3 × 10

^{11}cm

^{−2}, and

*N *= 5 × 10

^{11}cm

^{−2} and all of these values lead to essentially the same final inversion. There is only a small difference in the inversion dynamics for the case of

*N *= 7 × 10

^{11}cm

^{−2} that leads to slightly smaller inversion. For even smaller number of cycles, Figure
c,d, the inversion dynamics differs slightly for all the values of electron sheet density, and the final value of inversion is practically the same, independent of the value of electron sheet density. We note that the largest values of inversion are obtained for

*n*_{p }= 2 and

*n*_{p }= 3 and not for

*n*_{p }= 1, as one may expect, as in the latter case the influence of the decay mechanisms will be weaker. However, the second term on the right-hand side of the electric field of Equation 8 influences the dynamics for

*n*_{p }= 1, and in this case, the pulse area

*θ *=

* Π *does not lead to the largest inversion
[

42].

Similar results to that of Figure
are also obtained for the case of sin-squared pulse shape with

that are presented in Figure
. In this case, the computation is in the time period [0,2

*t*_{p}] and the pulse area is again

*θ *=

* Π*. We have also found similar results for other pulse shapes, e.g., for hyperbolic secant pulses. These results show that the present findings do not depend on the actual pulse shape, as long as a typical smooth pulse shape is used.

In order to explore further the dependence of the inversion in pulse area, we present in Figure
the final inversion, i.e., the value of the inversion at the end of the pulse, as a function of the pulse area *θ *for a sin-squared pulse. We find that for pulses of several cycles, e.g., *n*_{p }= 10, the pulse areas for maximum inversion can be quite different than *Π* depending on the value of electron sheet density. For example, for *N *= 5 × 10^{11}cm^{−2}, the pulse area is about 1.5*Π*, and for *N *= 7 × 10^{11}cm^{−2}, the pulse area is about 2.1*Π*. Similar results have also been obtained for other pulse shapes, e.g., Gaussian and hyperbolic secant pulses. The displayed dependence explains the results of Figure
a (and of Figure
a), as one may see that a *Π* pulse area leads to some final inversion for *N *= 10^{9}cm^{−2} and *N *= 3 × 10^{11}cm^{−2} but gives very small final inversion for *N *= 5 × 10^{11}cm^{−2} and *N *= 7 × 10^{11}cm^{−2}. However, for pulses with 3 cycles or with a smaller number of cycles, the maximum inversion occurs for pulse area *Π* or very close to *Π* (and odd multiples of *Π *if the figures are extended in higher pulse areas) independent of the value of electron sheet density.

An interesting effect in the interaction of an ultrashort electromagnetic pulse with a multi-level system is the influence of the carrier envelope phase

on the populations of the quantum states
[

40,

43,

45,

46,

48]. In Figure
, we present the dependence of the final inversion on the carrier envelope phase

for a sin-squared pulse with

*θ *=

* Π *for different number of cycles and electron sheet densities. We find that there is a dependence of the final inversion on the carrier envelope phase and this dependence is strongest for larger sheet electron densities and for pulses with smaller number of cycles.