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

56 Rashba spin-orbit coupling in quantum dots
requires a suitably large change in α. In experiments with 2DEGs, changes in α
of a factor of 2 are reported, and in a recent Letter by Koga et al., α was shown
to vary in the range ≈ (0.3 − 1.5) × 10 −12 eVm (a factor of 5) in one InGaAs
sample [25]. Grundler [27] has shown that the large back-gate voltages usually
used to change α can be drastically reduced by placing the gates closer to the
2DEG. Thus, it is conceivable that changes in α of a factor between 2 and 5 could
be produced with voltages small enough to be pulsed with rise times substantially
shorter than a typical coherent oscillation period.
In Fig. 3.3a we plot time-traces of the transition probability P(t p ) calculated
for the first anticrossing as a function of magnetic field. We have used the values
α 1 = 1.5×10 −12 and α 2 = 0.3×10 −12 eVm from the Koga experiment [25]. The
amplitude of the oscillations P max for different ratios of α 2 /α 1 is presented in
Fig. 3.3b, which shows a node at B = B 0 (δ = 0). This is because, for δ = 0, the
eigenstates of JCM are 2 −1/2 (|n,↑〉 ± |n + 1,↓〉) for all α ≠ 0. Therefore, a finite
detuning is required to obtain the maximum amplitude, which concurs with δ >
k B T,Γ R to overcome broadening effects. Both the amplitude P max and frequency
Ω show non-trivial dependencies on α 1 and α 2 as well as on the magnetic field.
This latter behaviour stems from the parametric dependence on B of all three
parameters in H JC .
For our model parameters with α 2 /α 1 = 1/5 and with the detuning set such
that the amplitude is maximised, we have P max ≈ 0.45 with a Rabi frequency of
Ω = 2GHz, which corresponds to a period of about 3 ns. This is within accessible
range of state-of-the-art experimental technique. Note that the period can be
extended by using weaker confinement and SO coupling.
For both the observation of coherent oscillations, and the operation as a qubit,
it is essential that the lifetime of state ψ + α is long. This is the case for a pure
electronic spin in a QD [37, 110], and we now show that the hybridisation of
the spin with the orbitals, and the ensuing interaction phonons, does not affect
this. We assume a piezo-electric coupling to acoustic phonons via the potential
V ep = λ q e iq·r (b q + b † −q), with phonon operators b q and |λ q | 2 = P/2ρcqV , with
coupling P, mass density ρ, speed of sound c, and volume V [111]. For n = 0, a
Golden Rule calculation yields the rate
Γ ep /ω 0 =
√
mP 2l0
8π(ω s ) 2 sin
ρl 0
˜l
2 θ + sin 2 θ − ξ 5 I(ξ), (3.9)
with ω s = c/l 0 , ξ = 2 −1/2 (˜l/l 0 )(∆/ω s ), and I(ξ) ≤ 8/15. Close to B 0 , ξ ≪ 1,
and thus the rate is extremely small Γ ep ≈ 10 4 s −1 (Fig. 3.3c). Therefore, the
robustness of spin qubits is not significantly weakened by the SO hybridisation.
In general, residual relaxation affects our measurement scheme in two ways.
During the oscillation (Fig. 3.2c), the system may relax to the eigenstate ψ − α 2
. This