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

3.3 Effects of relaxation 73
The relaxation during read-out phase corresponds to the transition Ψ + α 1
→
Ψ − α 1
and this occurs with probability [1 − exp(−Γ 2 t o )]P(t p ). Thus, the current
[Eq. (3.47)] is reduced to
(
I =eΓ R
[1 − 1 − e − Γ 2
Γ R
)]P(t p ) (3.48)
=eΓe − Γ 2
Γ R P(t p ) ≈ eΓ R
(1 − Γ )
2
P(t p ), for Γ 2 ≪ Γ R . (3.49)
Γ R
This type of relaxation simply reduces the amplitude of oscillation. To obtain a
significant oscillation signal in the current we require Γ 2 ≪ Γ R .
The effect of the relaxation during the coherent evolution (Fig. 3.2c) has more
serious consequences because it leads to dephasing. Although the relaxed ground
state Ψ − α 2
is an eigenstate of H JC (α 2 ), when flipping back to α 1 after the pulse time
t p the system starts Rabi oscillations – this time during the read-out phase – and
thus effectively scrambling the signal. The probability to relax during the pulse is
[1 − exp(−Γ t t p )]. When returning to the original coupling strength α 2 → α 1 the
relaxation induced coherent oscillations have the same amplitude as the wanted
ones but a different frequency Ω 2 ≈ Ω 1 α 1 /α 2 (for δ ≈ 0). During the long readout
phase (t o ∼ Γ −1
R ) this oscillation cycles through many periods because Γ R ≪ Ω.
Thus, the average probability to find the system in state Ψ + α 1
after relaxation (Γ 1 )
took place is P max /2, leading to the total probability
P = e −Γ 1t p
P(t p )
} {{ }
no relax.
+ ( 1 − e −Γ 1t p
) P max
2
} {{ }
relax.
. (3.50)
Therefore, the dephasing during the rotation phase leads to an exponential damping
of the oscillation features in the current towards the average value eΓ R P max /2.
Since P(t p ) is a periodic function of frequency Ω, oscillations should be observable
for Γ 1 ≪ Ω.
3.3.2 Phonon induced relaxation rates
Due to the hybridisation of spin and orbital wavefunction in the JCM, the question
arises whether dissipation to phonons which couple to the orbital wavefunction
might lead to relaxation from the excited state Ψ + to the ground state Ψ − , and thus
cancelling the robustness of pure spin qubits. We now show that the hybridisation
indeed leads to an addition relaxation channel, but that the corresponding rate is
small enough that coherent oscillation still may be observed.