Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

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3.2 Introduction to quantum dots and various derivations 69
(a)
2.5
(b)
1
0.8
δmax/˜λ2
2
1.5
1
Pmax
0.6
0.4
0.2
0
0 2 4 6 8 10
0 0.2
0.4
0.6
0.8
1
˜λ 1 /˜λ 2
˜λ 1 /˜λ 2
Figure 3.6: (a) Optimal detuning δ max /˜λ 2 to find maximum in amplitude of probability
oscillations as a function of ˜λ 2 /˜λ 1 . (b) Maximal amplitude of oscillation
P max as a function of ˜λ 2 /˜λ 1 . For a change in λ by a factor 5 the maximal amplitude
of oscillation is ∼45%.
From Eq. (3.36) and (3.39) we see that the amplitude of oscillation does not
depend on the sequence, i.e. whether we change from α 1 → α 2 or vice versa.
However, the sequence is important for the frequency, see Eq. (3.34). In addition,
the detuning plays an important role concerning the amplitude and frequency
of oscillations. For δ = 0 (resonant JCM) eigenfunctions are independent of α
[Eq. (3.29)],
|ψ ± 〉 = 1 √
2
(|n,↑〉 ± |n + 1,↓〉). (3.42)
Thus, for δ = 0, a non-adiabatic change of α does not lead to any oscillations
because the system stays in a stationary state. Conversely, for δ ≫ ω, ˜λ 1 , ˜λ 2 the
amplitude is also strongly suppressed because
δ ≫ ω, ˜λ 1 , ˜λ 2
⇒ γ + → δ, γ − → 1 δ
(3.43)
⇒ ∆θ + → π 2 , ∆θ − → 0 (3.44)
⇒ P max = 0 by Eq. (3.36). (3.45)
Thus, for a given sequence α 1 → α 2 the maximum amplitude of probability oscillations
can be found at a non-zero detuning δ max . The evolution of δ max /˜λ 2 as
function of ˜λ 2 /˜λ 1 is shown in Fig. 3.6a.
The maximal amplitude of probability oscillation at δ = δ max as a function of
˜λ 2 /˜λ 1 is shown in Fig. 3.6b. Since the amplitude does not depend on the sequence