Superconducting transmission line resonatorsfor quantum information processing
Gleb Fedorov
Methods of experimental physics March 31, 2021
1 IntroductionSuperconducting qubits and circuitQEDPhysical quantum oscillatorsLC-oscillatorTheory of transmission linesTransmission line resonators (TLR)Equivalent parameters of TLRS-parametersSome numerical examples
More complex features of realdevices
2 Designing TLRHow the samples are madeEDA softwareSimulating your design
3 Measuring TLRSample connectionLab equipmentMeasurement processData processing in Python
Superconducting qubits and circuit QEDIntroduction
What do people call qubits?
Formally, any two-state system can be usedas a qubit:|ψ〉 = α|a〉+ β|b〉, α, β ∈ C,|α|2 + |β|2 = 1, global phase does notmatter
But how to measure one?
Depends on the system. We will talk aboutSC quits.
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Superconducting qubits and circuit QEDIntroduction
Current technique is based on cavity QED1:
The Hamiltonian of the system:
H = ~ωr(a†a+
1
2
)
+~Ω
2σz + ~g(a†σ− + aσ+)
(1)
Dispersive limit:
H ≈ ~[ωr +
g2
∆σz]a†a+
~2
[Ω +
g2
∆
]σz.
(2)
1A. Blais et al., Physical Review A 69, 062320 (2004)3 / 34
Superconducting qubits and circuit QEDIntroduction
In such configuration, the transmission of the Fabri-Perot cavity will depend on thequbit state:
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Superconducting qubits and circuit QEDIntroduction
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Physical quantum oscillatorsIntroduction
Since 2004, many types of devices were used for the role of a cavity:
Notch (shunting) type Lumped-element LC
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Physical quantum oscillatorsIntroduction
Since 2004, many types of devices were used for the role of a cavity:
Transmission Coax stub
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LC-oscillatorIntroduction
A common feature of all electrical resonators is that they may be represented asLC-circuits (in terms of their impedance)
Resonance angular frequency:
ω0 = 1/√LC = 2πf0
Series impedance:
Zs(ω) = jLω2 − ω2
0
ω
Parallel impedance:
Zp(ω) = −j 1
C
ω
ω2 − ω20
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LC-oscillatorIntroduction
Quality factors: internal and external
R∗ =1 + ω2C2
κ(Z0/2)2
ω2C2κ(Z0/2)
,
C∗ =Cκ
1 + ω2C2κ(Z0/2)2
≈ Cκ (in our case).
Q−1l = Q−1i +Q−1e ,
Qi = ω(C + C∗)Rin,
Qe = ω(C + C∗)R∗.9 / 34
Theory of transmission linesIntroduction
2
∂
∂xv(z, t) = −L ∂
∂ti(z, t)−R i(z, t) (3)
∂
∂xi(z, t) = −C ∂
∂tv(z, t)−G v(z, t) (4)
2D. M. Pozar, Microwave engineering, (John Wiley & Sons, 2009)10 / 34
Theory of transmission linesIntroduction
For harmonic signals i(z, t) = I(z)eiωt, V (z, t) = V (z)eiωt follow the wave equations:
∂2V
∂t2− u2∂
2V
∂z2= 0 (5)
∂2I
∂t2− u2∂
2I
∂z2= 0, (6)
where u – the speed of light in the line, u = α+ jβ =√
(R+ jωL)(G+ jωC).Voltage across the line is v(z, t) = V +
0 cos(ωt− βz)e−αz + V −0 cos(ωt+ βz)eαz, were
V+(−)0 are the waves travelling backward and forward along z axis.
Z0 =
√R+ jωL
G+ jωC
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Theory of transmission linesIntroduction
The impedance of a line terminated by some impedance ZL (load):
Zin = Z0ZL + jZ0 tanul
Z0 + jZL tanul
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Transmission line resonators (TLR)Introduction
Using the formula above, we can find the equivalent parameters of the resonatorsbased on transmission lines:
Halfwave open
Zin =Z0
αl + j(∆ωπ/ω0)
Halfwave shorted
Zin = R+ 2jL∆ω
Quarterwave
Zin =1
1/R+ 2k∆ωC13 / 34
Equivalent parameters of TLRIntroduction
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Equivalent parameters of TLRIntroduction
Fundamental mode equivalent parameters:
Quarter wave Halfwave short Halfwave open
l λ/4 λ/2 λ/2
ω0 = 2π β/λ πβ/2l πβ/l πβ/l
Rr Z0/αl Z0αl Z0/αl
Lr 1/ω20Cr Z0π/2ω0 1/ω2
0Cr
Cr π/4ω0Z0 1/ω20Lr π/2ω0Z0
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S-parametersIntroduction
Calculating S-parameters from voltages andcurrents determined by Kirchgoff laws:
V1,2 = V +1,2 + V −1,2,
I1,2 = I+1,2 + I−1,2 =V +1,2 − V
−1,2
Z0,
V ±1,2 =1
2(V1,2 ± Z0I1,2).(
V −1V −2
)=
(S11 S12S21 S22
)(V +1
V +2
) If V = 1,V +1 = 1
2(V1 +Z0I1) = 12(V −Z0I1 +
Z0I1) = 1/2.V −2 = 1
2(V2 − Z0I2) = V2.S21 = 2V2
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Some numerical examplesIntroduction
Using LT-Spice:https://www.analog.com/en/design-center/
design-tools-and-calculators/ltspice-simulator.html
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More complex features of real devicesIntroduction
Bifurcation effects from nonlinearity (kinetic inductance)
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More complex features of real devicesIntroduction
Interaction with dipole two-level defects
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More complex features of real devicesIntroduction
Power-dependence of quality factors (saturation of two-level defects)
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1 IntroductionSuperconducting qubits and circuitQEDPhysical quantum oscillatorsLC-oscillatorTheory of transmission linesTransmission line resonators (TLR)Equivalent parameters of TLRS-parametersSome numerical examples
More complex features of realdevices
2 Designing TLRHow the samples are madeEDA softwareSimulating your design
3 Measuring TLRSample connectionLab equipmentMeasurement processData processing in Python
How the samples are madeDesigning TLR
Mask lithography: Direct lase write:
3M Goppl et al., Journal of Applied Physics 104, 113904 (2008)21 / 34
How the samples are madeDesigning TLR
Mask lithography: Example result3:
3M Goppl et al., Journal of Applied Physics 104, 113904 (2008)22 / 34
EDA softwareDesigning TLR
Everything from IC design, but simplified.
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EDA softwareDesigning TLR
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EDA softwareDesigning TLR
Scripting for design automation:
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Simulating your designDesigning TLR
Several numeric packages existhttps://en.wikipedia.org/wiki/Comparison_of_EM_simulation_software
We use HFSS and Sonnet Suite.
Example results from Sonnet HFSS interface
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1 IntroductionSuperconducting qubits and circuitQEDPhysical quantum oscillatorsLC-oscillatorTheory of transmission linesTransmission line resonators (TLR)Equivalent parameters of TLRS-parametersSome numerical examples
More complex features of realdevices
2 Designing TLRHow the samples are madeEDA softwareSimulating your design
3 Measuring TLRSample connectionLab equipmentMeasurement processData processing in Python
Sample connectionMeasuring TLR
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Lab equipmentMeasuring TLR
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Measurement processMeasuring TLR
Remote control of the VNA:1. https://en.wikipedia.org/wiki/Standard_Commands_for_Programmable_
Instruments
2. https://www.ni.com/ru-ru/support/documentation/supplemental/06/
ni-visa-overview.html
3. https://pyvisa.readthedocs.io/en/1.7/index.html
4. One of our Python SCPI-based “drivers”: https:
//github.com/vdrhtc/Measurement-automation/blob/master/drivers/znb.py
5. ZNB manual: https://www.rohde-schwarz.com/webhelp/ZNB_ZNBT_HTML_
UserManual_en/ZNB_ZNBT_HTML_UserManual_en.htm
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Measurement processMeasuring TLR
1. Making a wide frequency scan
znb = Znb("VISA ADDRESS FROM NI-MAX SOFTWARE")
znb.select_S_param("S21")
znb.set_freq_limits(1e9, 10e9)
znb.set_power(-60)
znb.prepare_for_stb()
znb.sweep_single()
znb.wait_for_stb()
freqs, data = znb.get_frequencies(), znb.get_sdata()
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Measurement processMeasuring TLR
For example, here one can see a scan (amplitude shown) of 6 notch-type resonators:
Resonator areas are shown with coloured crosses
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Measurement processMeasuring TLR
2. Scanning each peak for a range of powers
powers = linspace(-60, -10, 51)
full_data =
for resonator_area in resonator_areas:
znb.set_freq_limits(*resonator_area)
resonator_data = []
for power in powers:
znb.set_power(power)
znb.prepare_for_stb()
znb.sweep_single()
znb.wait_for_stb()
resonator_data.append(znb.get_sdata())
full_data[resonator_area] = resonator_data32 / 34
Data processing in PythonMeasuring TLR
Fitting the model for the notch-type resonator3:https://github.com/vdrhtc/resonator_tools
Snotch21 (fp) = aeiαe2πifpτ[1− Ql/Q
′e
1 + 2iQl(fp/fr − 1)
],
3S Probst et al., Review of Scientific Instruments 86, 024706 (2015)33 / 34
Data processing in PythonMeasuring TLR
Extracting the quality factors vs power:
104
105
106
Q-fa
ctor
s
I
Qe = 0.50, Qsfi /Qhp
i = 4.47/10.33 (x105)
Ql
Qe
Qi
fr
104
105
106
II
Qe = 0.54, Qsfi /Qhp
i = 3.27/9.79 (x105)104
105
106
III
Qe = 0.89, Qsfi /Qhp
i = 2.66/9.14 (x105)
60 50 40 30 20 10 0Power [dBm]
104
105
106
Q-fa
ctor
s
IV
Qe = 0.82, Qsfi /Qhp
i = 2.41/8.44 (x105)
60 50 40 30 20 10 0Power [dBm]
104
105
106
V
Qe = 1.06, Qsfi /Qhp
i = 2.40/8.34 (x105)
60 50 40 30 20 10 0Power [dBm]
104
105
106
VI
Qe = 0.91, Qsfi /Qhp
i = 2.73/8.35 (x105)
7.0
7.5
8.0
8.5
9.0
9.5+6.95682×106
2
3
4
5+7.05436×106
1.5
1.0
0.5
0.0
0.5
Freq
uenc
y [k
Hz]
+7.1554×106
1.5
1.0
0.5
0.0
0.5+7.2469×106
1.0
1.5
2.0
2.5+7.35846×106
2.0
2.5
3.0
3.5
4.0
Freq
uenc
y [k
Hz]
+7.45644×106
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