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

66 Rashba spin-orbit coupling in quantum dots
finement. This decouples the ω + modes from the rest of the system, giving
H = ω + n + + H JC , (3.26)
with energies rescaled by ω 0 and the Jaynes–Cummings model (JCM) H JC given
by Eq. (3.6),
H JC = ω − a † −a − + 1 )
2 E zσ z + λ
(a − σ + + a † −σ − , (3.27)
with λ = l0 2γ −/2˜l l SO . This model is completely integrable (see appendix. A) has
ground state |0,↓〉 with energy E G = −E z /2 and excited energies
(
E α (n,±) = n + 1 )
ω − ± ∆ n
2 2 , (3.28)
with detuning δ ≡ ω − − E z and ∆ n ≡ √ δ 2 + 4λ 2 (n + 1), corresponding to the
eigenstates
|ψ (n,±)
α
〉 = cosθ (n,±)
α
|n,↑〉 + sinθ (n,±)
α |n + 1,↓〉, (3.29)
with tanθ α
(n,±) = (δ±∆ n )/2λ √ n + 1. For our parameters, this model describes the
energy levels of the SO-interacting QD to within 10% of the typical anticrossing
width (given by ∆ n in the JCM) and 1% of ω 0 . This small discrepancy is shown
in Fig. 3.1b. As a characteristic measure of the quality of the RWA we can use
the prediction of the JCM that the anticrossing width increases with α √ n + 1.
In Fig. 3.1c this width ∆ is plotted against its central energy. The JCM (solid
line) is compared with the exact numerical result (circles). For a QD of size
l 0 = 150nm, almost perfect coincidence is found for α between (0.3...1.5) ×
10 −12 eVm, corresponding to experimentally found values in InGaAs [25]. For
larger values the anticrossing width is underestimated in the JCM, as indicated
in uppermost curve in Fig. 3.1c for the value α = 2.0 × 10 −12 eVm. Therefore,
we can conclude that the JCM is a reasonable approximation for SO-interacting
QDs where the confinement is stronger than the SO coupling. In this regime the
integrability of the JCM provides us with the analytical eigenstates and energies
which we use in the following to discuss coherent oscillations in a QD.
3.2.4 Coherent oscillations
In this section we investigate coherent oscillations (CO) in a SO-coupled QD system.
This illustrates the fruitfulness of the analogy with quantum optics. Although
the origin of coherent oscillations is simply the fact that the time evolution of a