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

68 Rashba spin-orbit coupling in quantum dots
For zero detuning we find Ω = Ω 0 := ˜λ 2 . In quantum optics Ω 0 is called Rabi
frequency. It describes the periodic energy exchange between an atomic pseudospin
and the resonant radiation field. In our model we will call the frequency of
CO Rabi frequency – even for finite detuning.
From Eq. (3.34) we see that the frequency of oscillation is only determined
by the coupling strength which is present during the coherent evolution (˜λ 2 ). The
amplitude of oscillation, however, depends on the ratio ˜λ 1 /˜λ 2 and the detuning δ.
In Eq. (3.33) completeness yields
c 2 + + c 2 − = cos 2 ∆θ + + cos 2 ∆θ − = 1. (3.35)
The maximum amplitude in P(t p ) is found for t p such that exp(−2iΩt p ) = −1,
leading to
P max = 1 − |cos 2 ∆θ + − cos 2 ∆θ − | 2 . (3.36)
If cos 2 ∆θ ± ≈ 1/2 we expect probability oscillations of order 100%. To translate
this into a condition for δ, ˜λ 1 and ˜λ 2 we express
leading to
tanθ ± α 1 ,α 2
=: χ ± 1,2 , (3.37)
∆θ ± = arctanχ ± 2 − arctanχ− 1
(3.38)
= arctan χ± 2 − χ− 1
1 + χ ± 2 χ− 1
=: arctanγ ± , (3.39)
by using Eq. (4.4.34) in Ref. [130]. To find probability oscillation of ≈ 100% we
need
c 2 ± ≈ 1 2 ⇒ ∆θ ≈ ± { 1
4 π, 5 4 π }
⇒ γ ≈ ±1. (3.40)
Trivially, we have γ ≈ ±1 in the limit
˜λ 1 ≪ δ ≪ ˜λ 2 ⇒ P(t p ) ≈ cos 2 Ωt p , (3.41)
corresponding to CO in the resonant JCM limit in quantum optics. However,
for the QD condition (3.41) corresponds to the limit of non-adiabatic switching
between almost zero and strong SO coupling. Unfortunately, in real physical systems
this limit is unlikely to be fulfilled. Thus, in contrast to optical systems,
only probability oscillations with amplitudes less than 100% are feasible in SOinteracting
QDs.