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This will most likely take forbiddingly long, making it necessary to find a different
approach to understanding the dynamics.
3.3 Interaction with the Qubit
The qubit is controlled by varying its potential via changes ∆I dc (t) in the
current bias I dc .
As described above, the effect of this can be understood by
splitting the resulting Ĥ(t) into:
Ĥ(t) = Ĥ0 + ˆV (t) = Ĥ0 − Φ 0
2π ∆I dc(t) δ (3.27)
This changes the level occupations according to:
i ∂ t a n (t) = ∑ m
a m (t) e i(E n−E m )t/ 〈 n | V (δ, t) | m 〉 (3.28)
where:
∫
〈 n | V (δ, t) | m 〉 = ψn(δ) r ∗ V (δ, t) ψm(δ) r dδ
= − Φ ∫
0
2π ∆I dc(t) δ ψn(δ) r ∗ ψm(δ) r dδ
= ∆I dc (t) T nm (3.29)
T nm gives the strength of the coupling between levels n and m. If T nm = 0 for
certain n and m, no direct transitions can be driven between the levels.
Figure 3.2b shows |T nm | for the harmonic oscillator potential shown with its
eigenstates in Figure 3.2a. This figure shows that a perturbation of the form c(t) x
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