1.
Mario Palma

2. Motivation Quasiparticles (QPs) poisoning :
Counting error in Superconducting Qubit
Decrease the coherence of Superconducting Qubit
Decoherence of Majorana Qubit
Gap engineering
Vortex trap
Normal metal trap
J. Aumentado et al., PRL 92, (2004).
M. Taupin et al., Nat. Commun. 7,10977 (2016)

3. Model NIS system electrons in N with energy ξn Tunneling Hamiltonian
QPs in S with energy 𝜖 𝑛 = 𝜉 𝑛 2 + Δ 2
QPs tunneling from S into N
Electrons escape from N into S
c electron in the normal metal and d the electron in the superconductor, which is related by the Bogoliubov’ s transformation to the quasi-particles operators.
Γ 𝑡𝑟 =2𝜋| 𝑡 | 2 𝜈 𝑠0 / Ω 𝑠
Γ 𝑒𝑠𝑐 𝜖 =2𝜋| 𝑡 | 2 𝜈 𝑠0 𝜈 𝑠 (𝜖)/ Ω 𝑁
BCS density of the states
Energy independent

4. Model 𝜖≫∆ Γ 𝑒𝑠𝑐 (𝜖)→ Γ 𝑒𝑠𝑐 𝑑 𝑠 ≈ 𝑑 𝑁 𝜈 𝑠0 ≈ 𝜈 𝑁0 Γ 𝑒𝑠𝑐 ≈ Γ 𝑡𝑟
Γ tr ~8𝑥 𝑠 −1
𝜖≥∆
Γ 𝑟 relaxation rate of the electron in the normal metal
due to electron-electron interaction or
electron-phonon interaction
𝑇≫Δ
Γ r →0
𝑇≪Δ
There are many unoccupied state below ∆ in N
Γ 𝑟 ≠0

5. Model
Rate equations for the probability density in normal metal and in the superconductor
Assuming steady-state distribution of the electrons in the normal metal
Normalize QPs density
𝑝 𝑁 =0
We have to compare the relaxation rate with the escape rate
They normalize the density to V_s0
Fast relaxation rate
Slow relaxation
Excitation in the normal metal fast relax at energy below the gap and cannot return in the super conductor

6. Real case Assume that the diffusion time is bigger than 1/Γr
The QPs density distribution can be describe through a diffusion equation:
The trap component
Pair breaking mechanisms
The scale over which the density decay due to the trapping
Source of QPs
tinj =time that the source is on
t = time after the source is switched off

7. Device and experimental set up
3D transmon qubit
Josephson junction Al/AlOx /Al
Superconductive cavity
Cu trap
20 µm<d<400 µm
Reference
Antenna
C. Wang et al., Nat.Commun. 5,5836 (2014)

8. Decay rate w/o trap T1 = 19 µs
10 µs<T1 <22µs µm < d < 80 µm T1 is qualitative the same
T1 =5 µs →d = 200 µm & T1 =7 µs →d = 400 µm T1 is reduce
The time dependent part of the qubit si directly proportional to the QPs density at the junction
𝑡> 𝑡 𝐿

9. short & long trap Long trap
Short trap the model predict: linear dependence of the characteristic time scale 1/𝜏 𝑤 on the trap length
Short trap
Long trap saturation behavior
1/ 𝑡 𝐿 ~1/ 𝜏 𝑤
Tfr =13 mK 𝑙 0 =41.2±17.1𝜇𝑚 𝑡 𝐿 =184±29 𝜇𝑠 Γeff =2.42x105 s-1
Tfr =50 mK 𝑙 0 =45.8±16.7𝜇𝑚 𝑡 𝐿 =125±29 𝜇𝑠 Γeff =3.74x105 s-1

10. Conclusion
Γeff is energy dependent for time scale shorter than electron relaxation rate
Evacuation time depend linearly on the length of trap and saturated for long trap
The decay rate increase with length of the trap
For short trap Γeff increases with temperature indicating back flow of QPs

11. Gap engineering