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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