1) Long relaxation times were observed in a c-shunt flux qubit coupled to a 3D microwave cavity. Relaxation times were longer than previous 2D qubit designs due to the cleaner electromagnetic environment and lower dielectric losses of the 3D cavity. 2) Qubit energy relaxation is due to quasiparticle tunneling or interactions with two-level system defects. Dephasing results from charge noise or critical current fluctuations near the optimal point and 1/f flux noise away from the optimal point. 3) Dynamical decoupling techniques such as CPMG pulse sequences can be used to reach T1-limited coherence by filtering out low-frequency noise sources.

1. Long relaxation times of a c-shunt flux qubit
coupled to a 3D cavity
Leonid Abdurakhimov, I. Mahboob, H. Toida, K. Kakuyanagi, S. Saito
NTT Basic Research Laboratories, NTT Corporation, Japan
APS 2020 March Meeting
W08.00014

2. 1
3D c-shunt flux qubit
2DFQ[2]
2DFQ[1]
2DCSFQ[4]
2D/3DFQ[5]
2DCSFQ[6]
ourresults
FQ = flux qubit
CSFQ = c-shunt flux qubit
[4] APL 99, 181906 (2011)
[5] PRL 113, 123601 (2014)
[6] PRL 120, 260504 (2018)
[1] Science, 299, 5614 (2003)
[2] PRL 95, 257002 (2005)
[3] PRL 97, 167001 (2006)
Evolution of relaxation times
in 3JJ flux qubits
coplanar
shunt capacitor
Capacitively-shunted (c-shunt) flux qubit
embedded in a 3D microwave cavity
Details: Abdurakhimov et al., APL 115, 262601 (2019) [arXiv:1911.04635]
Photo of the 3D cavityDevice schematic
flux qubit loop
(3 Josephson junctions, 3JJs) • cleaner electromagnetic environment
• lower surface participation ratio
(lower dielectric losses)
2DFQ[3]

3. 2
C-shunt flux qubit: Hamiltonian
𝑈 𝜑 𝑝, 𝜑 𝑚,Φ = 2𝐸𝐽 1 − cos 𝜑 𝑝 cos 𝜑 𝑚 + 𝛼𝐸𝐽 1 − cos 2𝜋
Φe + 𝛿Φ
Φ0
+ 2𝜑 𝑚
𝐸 𝑝 = 2𝐸𝑐 = 𝑒2/𝐶𝐽, 𝐸 𝑚 =
𝐸 𝑝
1 + 2 𝛼 +
𝐶𝑆
𝐶𝐽
• the value of 𝛼 parameter should be small to reduce the effect of magnetic flux noise:
• shunt capacitance 𝐶𝑆 should be large to reduce the effect of charge fluctuations:
𝐻(𝜑 𝑝, 𝜑 𝑚) = 𝐸 𝑝 𝑛 𝑝 − 𝛿𝑁 𝑎
2
+ 𝐸 𝑚 𝑛 𝑚 − 𝛿𝑁𝑏 − 𝛿𝑁𝑐
2 + 𝑈 𝜑 𝑝, 𝜑 𝑚, Φ 𝑒 + 𝛿Φ
J.Q. You et al., PRB 75, 140515(R) (2007)
𝑛 𝑝 = −𝑖𝜕/𝜕𝜑 𝑝
In the rotated frame (𝜑 𝑚 = (𝜑1 − 𝜑2)/2, 𝜑 𝑝 = (𝜑1 + 𝜑2)/2) :
charge noise magnetic flux noise
𝑛 𝑚 = −𝑖𝜕/𝜕𝜑 𝑚

4. 3
C-shunt flux qubit: numerical simulations
Fourier transform method (J.R. Johansson, http://jrjohansson.github.io/)
Potential energy 𝑈 𝜑 𝑝, 𝜑 𝑚,Φ = 0.5Φ0
𝛼 𝐶S 𝐶J 𝐼c 𝐸J 𝐸C
0.437 60 fF 6 fF 275 nA 136.75 GHz 3.2 GHz
Qubit parameters (used to reproduce the experimental data)
Qubit spectrum
𝑓01
𝑓12
𝑓23
optimal point
≈ 0.8 GHz

5. 4
C-shunt flux qubit: perturbation theory
Δ = 4𝐸 𝐶 𝑆
𝐸𝐽(1 − 2𝛼) +
8𝛼 − 1
4(1 − 2𝛼)
𝐸 𝐶 𝑆
𝜀 = 2 2𝜋𝛼𝐸𝐽
𝐸 𝐶 𝑆
𝐸𝐽(1 − 2𝛼)
1/4
Φ
Φ0
− 0.5
Hamiltonian can be reduced to a one-dimensional form by
neglecting terms with 𝜑 𝑝
𝐻(𝜑 = 𝜑 𝑚) =
𝑒2
2𝐶𝑆
𝑛2 + 2𝐸𝐽 1 − cos 𝜑 + 𝛼𝐸𝐽 1 + cos2𝜑
Using perturbation theory, the qubit frequency is expressed as:
ℏ𝜔01 = Δ +
2𝜀2
Δ
Qubit frequency at the optimal point (𝐸 𝐶 𝑆
= 𝑒2/2𝐶𝑆):
Flux-dependent term:
𝛼 ≈ 0.41
𝐶𝑆 ≈ 78 fF
𝐸𝐽 ≈ 85 GHz
(Steffen et al, PRL 105, 100502(2010), DiVincenzo et al, PRB 74, 014514 (2006)) Qubit spectrum
anharmonicity

6. 5
Experimental details: dispersive readout
𝐻 =
1
2
ℏ𝜔01
′
𝜎𝑧 + (ℏ𝜔 𝑟
′
+ ℏ𝜒 𝜎𝑧) 𝑎+
𝑎
cavity-resonance frequency
depends on the qubit state
Coupling strengths: 𝑔01/2𝜋 = 73 MHz, 𝑔12/2𝜋 = 115 MHz
ground
state
excited
state
𝜒 = 𝜒01 −
𝜒12
2
𝜒𝑖𝑗 =
𝑔𝑖𝑗
2
𝜔𝑖𝑗 − 𝜔 𝑐0
Anharmonicity: 𝐴/2𝜋 = 780 MHz
Qubit spectrumCavity spectrum
3D c-shunt flux qubit
2𝜒/2𝜋 ≈1.8 MHz2𝜒

7. 6
Energy-relaxation time T1
𝑛 𝑞𝑝 ≤ 0.7 µm-3
Quasiparticle generation
by stray infrared photons,
environmental radioactive
materials and cosmic rays
(Vepsäläinen et al,
arXiv:2001.09190 (2020))
Lisenfeld et al, Sci. Rep. 6, 23786 (2016)
Interacting TLS defects
Al film: Δ ≈ 50 GHz
Energy relaxation mechanisms
Quasiparticle tunneling
Γ = Γ𝑛𝑒𝑞,1→0 + Γ𝑒𝑞,1→0 1 + 𝑒−ℏ𝜔01/𝑘 𝐵 𝑇
Catelani et al, PRL 106, 077002 (2011)
etc.
Weak coupling to a bath of two-level-system
defects
Muller et al, PRB 92, 035442 (2015)
etc.
-tunneling atoms
-trapped electrons
-dangling bonds
-hydroxide defects
-etc

8. Decoherence due to flux noise:
Ithier et al, PRB 72 134519 (2005)
etc.
7
Coherence time T2
𝐹 = 𝑒−𝑡/2𝑇1 𝑒− Γ 𝜑 𝑡
2
Γ 𝜑𝐸 = 𝐴Φln2|𝜕𝜔01/𝜕𝑓|
𝑨 𝚽 = 𝟏. 𝟖𝝁𝚽 𝟎
𝟐
Γ 𝜑𝑅 = 𝐴Φln(1/𝜔𝑖𝑟 𝑡)
𝜕𝜔01
𝜕𝑓
Echo Echo
Ramsey Ramsey
dephasing
due to
1/f flux noise
Echo
𝐺 𝐸 = 𝑒−𝑡/𝑇2𝐸
dephasing
due to
1/f flux noise
Ramsey
𝐺 𝑅 = 𝑒−𝑡/𝑇2𝑅

9. 8
CPMG coherence time
Filter function of the CPMG sequence:
𝑔 𝑁 𝜔, 𝑡 =
1
𝜔𝜏 2 1 + −1 𝑁+1 𝑒 𝑖𝜔𝜏 + 2
𝑗=1
𝑁
−1 𝑗 𝑒 𝑖𝜔𝛿 𝑗 𝜏
cos
𝜔𝜏 𝜋
2
2
Biercuk et al, PRA 79, 062324 (2009)
Bylander et al, Nature Physics 7, 565 (2011)

10. 9
Conclusions
Details and references:
Abdurakhimov et al., APL 115, 262601 (2019) (Editor’s Pick) [arXiv:1911.04635]
• 3D c-shunt flux qubits demonstrate long relaxation times
• qubit energy relaxation is due to quasiparticle tunneling or TLS defects
• qubit dephasing mechanisms:
a) charge noise or critical-current fluctuations at the optimal point
b) 1/f flux noise far from the optimal point
• a 𝑻 𝟏-limited coherence time can be reached using dynamical decoupling
This work was partially supported by JST CREST (JPMJCR1774).
Postdoc positions available – contact shiro.saito.bx@hco.ntt.co.jp