Superconducting Quantum Circuits Superconducting Qu… · Nanotechnology meets Quantum Information...

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Superconducting Quantum Circuits

Rudolf Gross

Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften

andTechnische Universität München

Summer SchoolNanotechnology meets Quantum Information - NanoQI 2017

24 – 28th July 2017, San Sebastian, Spain

http://www.we-heraeus-stiftung.de/index.html

24-28.07.2017/RG - 2www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI

Research Campus Garching

Walther-Meißner-Institute

FRM II

Physics-Department

Mechanical Engineering

Informatics

Mathematics

LRZMPQ

ESOAstrophysics

Plasma Physics

Extraterrestr. Physics

ZAE

GRS


Frank DeppeKirill FedorovHans Huebl Achim Marx

Michael FischerMatthias PernpeintnerHannes Maier-FlaigStefan Klingler

Stephan PogorzalekDaniel SchwienbacherPhilipp SchmidtEdwar Xie

Jan Goetz (Aalto University, Finland)Elisabeth Hoffmann (attocube)Matteo Mariantoni (Waterloo, Canada)Edwin P. Menzel (Rohde & Schwarz)Tomasz Niemczyk (BMW Group)

WMI team & partners

• postdocs:

• PHD students: Peter EderDaniel SchwienbacherPhilipp SchmidtEdwar Xie

• former PHD students:

Manuel Schwarz (IAV GmbH)Thomas Weißl (Inst. Néel, Grenoble)Karl-Friedrich Wulschner (U. of Vienna)Ling Zhong (Yale University)Christoph Zollitsch (UC London)

• financialsupport:


WMI team & partners

http://www.wmi.badw.de/teaching/Lecturenotes/

see also notes and slides to lectures on Applied SuperconductivitySuperconductivity & Low Temperature Physics

supplementary material

memos (remind you to some basic relations)

these slides provideadditional information

&derivations

http://www.wmi.badw.de/teaching/Lecturenotes/


WMI Mission in QST

• develop physical foundations of

quantum electronics, fluxonics and spintronics

• develop required experimental techniques

• low temperature technology

• nanotechnology

• microwave technology

• develop required materials technology

• thin film technology for superconducting and

magnetic materials

• single crystal growth of quantum materials

(2003 – 2015)

(2006 – 2018)

(2019 – 2033)


single/few

electron, spin, fluxon, photon

devices

near future far future

quantifiable,but not quantum

single electron transistor

PTB

multi


devices

today

classicaldescription

65 nm process 2005

Intel

• quantumconfinement

• tunneling• …

... solid state circuits go quantum


single/few


devices

quantum


devices

near future far future


quantumdescription

superconducting qubitsingle electron transistor

PTB

multi


devices

today


65 nm process 2005

Intel WMI 20072 µm



multi


devices

single/few


devices

quantum


devices

today near future far future



quantumdescription

65 nm process 2005 superconducting qubitsingle electron transistor

PTBIntel

• superposition of states• entanglement• quantized em-fields

WMI 20072 µm


quantum1.0 quantum2.0


conventional electronic circuits

• classical physics• no quantization of fields• no superposition of states• no entanglement

... from conventional to quantum electronics

𝑯 =𝚽𝟐

𝟐𝑳+𝑸𝟐

𝟐𝑪

2

1

quantum electronic circuits

• quantum mechanics• quantization of fields• coherent superposition of states• entanglement

2

1

Y. Nakamura et al., Nature 398, 786 (1999)

𝑯 =𝚽𝟐

𝟐𝑳+𝑸𝟐

𝟐𝑪= ℏ𝝎 ෝ𝒂† ෝ𝒂 +

𝟏

𝟐

𝚽, 𝑸 = 𝒊ℏ

LC oscillator


Superconducting Quantum Circuits

© WMI


Vesuvius 3:512 superconducting qubits,operating temperature: 30 mK

Quantum computing @ mK temperature


http://web.physics.ucsb.edu/~martinisgroup/photos/BBCReZQu1103.jpg

quantumcomputing

http://research.physics.illinois.edu/QI/Photonics/research/

quantumcommunication

quantumsensing

Application fields

……. and more to come

quantummatter

https://www.mpq.mpg.de/4572004/profil

quantumsimulation

Credit: Francis Pratt / ISIS / STFC

quantummetrology

http://www.npl.co.uk/news/


contents

I. Superconductivity in a nutshell

II. Josephson Junctions

III. Superconducting Quantum Circuits

IV. Superconducting Resonators & Qubits

V. Circuit Quantum Electrodynamics (QED)

VI. Experimental Techniques

VII. Qubit: control, decoherence, etc.

VIII.Continuous-variable propagating quantum

microwaves

IX. Summary

Par

t I

Par

t II

Par

t II

I

Part I

I. Superconductivityin a nutshell



©WMI


attractive interaction among conductionelectrons

Cooper pairs (𝒌 ↑, −𝒌 ↓)

Cooper pairs condense into coherent quantumstate

description bymacroscopic wave function

𝚿 𝐫, 𝒕 = 𝚿𝟎𝐞𝒊𝜽(𝐫,𝒕)

𝚿 𝐫, 𝒕 𝟐 = 𝒏𝒔 𝐫, 𝒕

typical interaction range (phonon mediated) 100 nm

typical size of Cooper pairs 𝑽𝐂𝐏 ≃ 𝟏𝟎𝟎 𝐧𝐦 𝟑

electron density 𝒏 ≃ 𝟏𝟎𝟔/𝑽𝐂𝐏


(Fritz London, 1948)



• Bosonic coherent state (Schrödinger 1926, Glauber 1963)

• BCS ground state |𝜳𝐁𝐂𝐒 = ς𝐤(𝒖𝒌 + 𝒗𝒌𝑷𝒌†) |𝟎 as a fermionic coherent state

|𝛼 = e−|𝛼2|/2

𝑛=0

∞𝛼𝑛

𝑛!|𝜙𝑛 with Fock states |𝜙𝑛 =

1

𝑛!𝑎†

𝑛|0

|𝜶 = e−|𝛼2|/2

𝑛=0

∞𝛼𝑎†

𝑛

𝑛!|0 = 𝒆−|𝜶

𝟐|/𝟐 𝒆 𝜶𝒂† |𝟎

• Fermionic coherent state

replace 𝛼𝑎† by sum over pair creation operators: σ𝑘 𝛼𝑘𝑃𝑘† with 𝑃𝑘

† = 𝑐𝑘↑† 𝑐−𝑘↓

†

take care about Pauli principle: : 𝑃𝑘†𝑃𝑘

† |0 = 0

|ΨBCS = 𝑐 ⋅ eσ𝑘 𝛼𝑘𝑃𝑘

†

|0 = 𝑐 ⋅ෑ

𝑘

e𝛼𝑘𝑃𝑘

†

|0 = 𝑐 ⋅ෑ

𝑘

(1 + 𝛼𝑘𝑃𝑘†) |0

|ΨBCS = ෑ

𝑘

(𝑢𝑘 + 𝑣𝑘𝑃𝑘†) |0 with 𝑢𝑘 =

1

1+ 𝛼𝑘2

and 𝑣𝑘 =𝛼𝑘

1+ 𝛼𝑘2

𝑃𝑘† = 𝑐𝑘↑

† 𝑐−𝑘↓†


I. Superconductivity in a nutshell• Madelung transformation:

insert 𝚿 𝒓, 𝒕 = 𝚿𝟎𝐞𝒊𝜽(𝒓,𝒕) into Schrödinger equation for charged particle

𝟏

𝟐𝒎𝒔

ℏ

𝒊𝛁 − 𝒒𝒔𝐀 𝐫, 𝒕

𝟐

𝚿 𝐫, 𝐭 + 𝒒𝐬𝝓 𝐫, 𝒕 + 𝝁 𝐫, 𝒕 𝚿 𝐫, 𝒕 = 𝒊ℏ𝝏𝚿 𝐫, 𝒕

𝝏𝒕

electro-chemical potentialvector potential 𝑚𝑠 = 2𝑚𝑒

𝑞𝑠 = −2𝑒

…. 5 pages of calculation (see supplementary material)

𝐉𝐬 𝐫, 𝐭 = 𝒒𝐬𝒏𝒔 𝐫, 𝒕ℏ

𝒎𝒔𝛁𝜽 𝐫, 𝒕 −

𝒒𝒔𝒎𝒔

𝐀 𝐫, 𝒕

ℏ𝝏𝜽 𝐫, 𝐭

𝝏𝒕= −

𝟏

𝟐𝒏𝒔𝚲𝐉𝐬

𝟐 𝐫, 𝒕 + 𝒒𝒔𝝓 𝐫, 𝒕 + 𝝁(𝐫, 𝒕)

• current-phase relation:

• energy-phase relation:

Λ =𝑚𝑠

𝑛𝑠𝑞𝑠2

𝜆𝐿 =Λ

𝜇0

London penetrationdepth

London parameter

sup

ple

men

tary

mat

eria

l


Supplement: Madelung Transformation

• we start from Schrödinger equation:

electro-chemical potential

• we use the definition and obtain with

sup

ple

men

tary

mat

eria

l



sup

ple

men

tary

mat

eria

l


• equation for real part:

energy-phase relation (term of order ²ns is usually neglected)

∆Supplement: Madelung Transformation

sup

ple

men

tary

mat

eria

l


• interpretation of energy-phase relation

corresponds to action

in the quasi-classical limes the energy-phase-relation becomesthe Hamilton-Jacobi equation


sup

ple

men

tary

mat

eria

l


• equation for imaginary part:

continuity equation for probabilitydensityandprobability current density

conservation law for probability density



I. Superconductivity in a nutshell• derive London equations, fluxoid quantization, ….

take curl

𝛁 × 𝚲𝐉𝐬 + 𝛁 × 𝑨 = 𝛁 × 𝚲𝐉𝐬 + 𝐁 = 𝟎

take time derivative

𝝏

𝝏𝒕𝚲𝐉𝐬 = −

𝝏𝐀

𝝏𝒕−ℏ

𝒒𝒔𝛁

𝝏𝜽

𝝏𝒕

𝝏

𝝏𝒕𝚲𝐉𝐬 = 𝐄 −

𝟏

𝒏𝒔𝒒𝒔𝛁

𝟏

𝟐𝚲 𝐉𝐬

𝟐

use energy-phase relation and𝐄 = −𝜕𝐀/𝜕𝑡 − 𝛁(𝜙 + 𝜇/𝑞𝑠)


𝝏𝒕= −

𝟏


𝟐 + 𝒒𝒔𝝓+ 𝝁

1. London equation

2. London equation

take ring integral

ර

𝑪

.

𝚲𝐉𝐬 ⋅ 𝒅ℓ + න

𝑭

.

𝛁 × 𝐀 ⋅ ෝ𝒏 𝑑𝐹 =ℏ

𝒒𝒔ර

𝑪

.

𝛁𝜽 ⋅ 𝒅ℓ

ර

𝑪

.


𝑭

.

𝐁 ⋅ ෝ𝒏 𝑑𝐹 = 𝑛𝒉

𝒒𝒔= 𝒏𝚽𝟎 fluxoid quantization

fluxoid

use 𝛁 × 𝐀 = 𝐁 and

𝛁𝜽ׯ ⋅ 𝒅ℓ = 𝒏 𝟐𝝅

𝚲𝐉𝐬 𝐫, 𝐭 = − 𝐀 𝐫, 𝒕 −ℏ

𝒒𝒔𝛁𝜽 𝐫, 𝒕

𝚽𝟎 =𝒉

𝟐𝒆= 𝟐. 𝟎𝟔𝟖 ⋅ 𝟏𝟎−𝟏𝟓 𝐕𝐬


I. Superconductivity in a nutshell derive Josephson equations


𝝏𝒕= −

𝟏



replace gauge invariant phase gradient by phase difference

𝐉𝐬 𝐫, 𝐭 = 𝒒𝐬𝒏𝒔ℏ

𝒎𝒔𝛁𝜽 −

𝒒𝒔𝒎𝒔

𝐀

𝐉𝐬 𝐫, 𝐭 =𝒒𝐬𝒏𝒔ℏ


𝒒𝒔ℏ𝐀

𝝋 𝐫, 𝒕 = න

𝟏

𝟐

𝛁𝜽 −𝒒𝒔ℏ𝐀 ⋅ 𝒅ℓ = 𝜽𝟏 𝐫, 𝒕 − 𝜽𝟐 𝐫, 𝒕 −

𝟐𝝅

𝚽𝟎න

𝟏

𝟐

𝐀(𝐫, 𝒕) ⋅ 𝒅ℓ

two weakly coupledsuperconductors


= 12𝛻 ෨𝜙 ⋅ 𝑑ℓ

I. Superconductivity in a nutshell1. Josephson equation (current-phase relation)

𝝋 = 𝜽𝟏 − 𝜽𝟐 −𝟐𝝅

𝚽𝟎න

𝟏

𝟐

𝐀 ⋅ 𝒅ℓ

𝐽𝑠 𝜑 = 𝐽𝑠 𝜑 + 𝑛 ⋅ 2𝜋 (2𝜋 periodicity)

𝐽𝑠 𝜑 = 0 = 𝐽𝑠 𝜑 = 𝑛 ⋅ 2𝜋 = 0

𝐽𝑠 𝜑 = 𝐽𝑐 sin𝜑 +

𝑚=2

∞

𝐽𝑐,𝑚 sin 𝑚𝜑

𝑱𝒔 𝒓, 𝒕 = 𝑱𝒄(𝒓) 𝐬𝐢𝐧𝝋 (𝒓, 𝒕) 1. Josephson equation

can usally be neglected

2. Josephson equation (energy-phase relation)

𝝏𝝋

𝝏𝒕=𝝏𝜽𝟏𝝏𝒕

−𝝏𝜽𝟐𝝏𝒕

−𝟐𝝅

𝚽𝟎

𝝏

𝝏𝒕න

𝟏

𝟐

𝐀 ⋅ 𝒅ℓ use ℏ𝝏𝜽 𝐫,𝐭

𝝏𝒕= −

𝟏



𝝏𝝋

𝝏𝒕= −

𝟏

ℏ

𝚲

𝟐𝐧𝐬𝑱𝒔𝟐 𝟐 − 𝑱𝒔

𝟐 𝟏 + 𝒒𝒔 𝝓 𝟐 − 𝝓 𝟏 + 𝝁 𝟐 − 𝝁 𝟏 −𝟐𝝅

𝚽𝟎

𝝏

𝝏𝒕න

𝟏

𝟐

𝐀 ⋅ 𝒅ℓ

= 0

𝝏𝝋

𝝏𝒕=𝟐𝝅

𝚽𝟎න

𝟏

𝟐

𝐄 ⋅ 𝒅ℓ =𝟐𝝅

𝚽𝟎𝑽 =

𝟐𝒆𝑽

ℏ2. Josephson equation

𝐄 = −𝛁 ෨𝜙 −𝜕𝐀

𝜕𝑡

𝝎/𝟐𝝅

𝑽=

𝟏

𝚽𝟎= 𝟒𝟖𝟑

𝐆𝐇𝐳

𝐦𝐕



Summary

• superconducting ground state can be described by macroscopic wave function

𝚿 𝐫, 𝒕 = 𝚿𝟎𝐞𝒊𝜽(𝐫,𝒕) with 𝚿 𝐫, 𝒕 𝟐 = 𝒏𝒔 𝐫, 𝒕

• Madelung transformation yields

• current-phase relation 𝐉𝐬 𝐫, 𝐭 =𝒒𝐬𝒏𝒔ℏ


𝒒𝒔

ℏ𝐀

• energy-phase relation ℏ𝝏𝜽 𝐫,𝐭

𝝏𝒕= −

𝟏



• London equations:𝝏

𝝏𝒕𝚲𝐉𝐬 = 𝐄 (1)

𝛁 × 𝚲𝐉𝐬 + 𝐁 = 𝟎 (2)

• fluxoid quantization 𝑪ׯ.𝚲𝐉𝐬 ⋅ 𝒅ℓ + 𝑭

.𝐁 ⋅ ෝ𝒏 𝑑𝐹 = 𝑛

𝒉

𝒒𝒔= 𝒏𝚽𝟎

• Josephson equations 𝑱𝒔 𝐫, 𝒕 = 𝑱𝒄(𝐫) 𝐬𝐢𝐧𝝋 (𝐫, 𝒕)

𝝏𝝋

𝝏𝒕=

𝟐𝝅

𝚽𝟎𝑽 =

𝟐𝒆𝑽

ℏ

II. Josepson Junctions



• we consider only small (zero-dimensional) Josephson Junctions (JJs)

spatial dimensions in 𝑦𝑧 − plane smallcompared to Josephson penetration depth

𝜆𝐽 ≡𝛷0

2𝜋𝜇0𝑡𝐵𝐽c

example: 𝐽𝑐 = 106A/m2, 𝑡𝐵 = 100 nm 𝜆𝐽 ≃ 50 𝜇𝑚

small junctions: 𝝋 𝒚, 𝒛 = 𝒄𝒐𝒏𝒔𝒕. for 𝑩 = 𝟎

large junctions𝝏𝟐𝝋

𝝏𝒕𝟐=

𝟏

𝝀𝑱𝟐 𝐬𝐢𝐧𝝋(𝒚, 𝒛) Sine-Gordon equation



Josephson Coupling Energy (binding energy of two weakly coupled superconductors)

𝑬𝑱 = න

𝟎

𝒕

𝑰𝒔𝑽 𝒅𝒕′ = න

𝟎

𝒕

𝑰𝒄 𝐬𝐢𝐧𝝋ℏ

𝟐𝒆

𝒅𝝋

𝒅𝒕𝒅𝒕′ =

𝟐𝝅

𝚽𝟎න

𝟎

𝝋

𝑰𝒄 𝐬𝐢𝐧𝝋′ 𝒅𝝋′ =

𝚽𝟎𝑰𝒄𝟐𝝅

𝟏 − 𝐜𝐨𝐬𝝋

𝑬𝑱 =𝚽𝟎𝑰𝒄𝟐𝝅

𝟏 − 𝐜𝐨𝐬𝝋 = 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬𝝋 Josephson Coupling Energy

example: 𝐼𝑐 = 1 mA ⇒ 𝐸𝐽0 = 3 ⋅ 10−19 J = 𝑘B ⋅ 20 000 K

Josephson Inductance

𝒅𝑰𝒔𝒅𝒕

= 𝑰𝒄 𝐜𝐨𝐬𝝋𝒅𝝋

𝒅𝒕= 𝑰𝒄 𝐜𝐨𝐬𝝋

𝟐𝝅

𝚽𝟎𝑽 in general 𝑉 = 𝐿

𝑑𝐼

𝑑𝑡

𝑳𝑱 =𝚽𝟎

𝟐𝝅𝑰𝒄 𝐜𝐨𝐬𝝋=

𝑳𝒄𝐜𝐨𝐬𝝋

with 𝑳𝒄 =𝚽𝟎

𝟐𝝅𝑰𝒄Josephson Inductance

negative values for𝜋

2+ 2𝜋𝑛 < 𝜑 <

3𝜋

2+ 2𝜋𝑛

JJ can be considered as a lossless nonlinear inductor

example: 𝐼𝑐 = 1 mA ⇒ 𝐿𝑐 = 0.3 pH, 𝐼𝑐= 1 𝜇A ⇒ 𝐿𝑐 = 300 pH



𝑳𝑱

𝑹𝒏

𝑪

Josephson junction

equivalent circuit of Josephson tunnel junction

a. characteristic energies

𝑬𝑱𝟎 =𝚽𝟎𝑰𝒄𝟐𝝅

∝ 𝑨 𝐴 = junction area

𝑬𝑪 =𝟐𝒆 𝟐

𝟐𝑪∝𝟏

𝑨𝐶 = 𝜖𝜖0𝐴/𝑑 = junction capacitance

ℏ𝝎𝒑 =ℏ

𝑳𝒄𝑪=

𝟐𝒆ℏ𝑰𝒄𝑪

=𝟐𝒆ℏ𝑱𝒄෩𝑪

= 𝟐𝑬𝑪𝑬𝑱𝟎 ≃ 𝒄𝒐𝒏𝒔𝒕

plasma frequency, 𝜔𝑝

2𝜋≃ 30 GHz @

𝐽𝑐 = 100A

cm2, ሚ𝐶 ≃ 100fF

μm2

b. characteristic times

𝝉𝒑 = 𝑳𝒄𝑪 = ℏ෩𝑪/𝟐𝒆𝑱𝒄 𝝉𝒄 =𝑳𝒄𝑹𝒏

=𝟐𝒆𝑰𝒄𝑹𝒏

ℏ𝝉𝑹𝑪 =

𝟏

𝑹𝒏𝑪=𝝉𝒄

𝝉𝒑𝟐

c. quality factor

𝑸 =𝝎𝒑

𝝎𝑹𝑪= 𝟐𝒆𝑰𝒄𝑹𝒏

𝟐𝑪/ℏ = 𝜷𝑪 𝛽𝐶 = Stewart-McCumber parameter



Josephson tunnel junction with flux-tunable critical current

𝑳𝑱

𝑹𝒏

𝑪

Josephson junction

𝟐𝑳𝑱

𝑹𝒏/𝟐

𝟐𝑪

dc-SQUID

𝟐𝑰𝒄

𝚽𝐜𝐨𝐧𝐭𝐫

controlflux

𝑰𝒔 𝚽𝐜𝐨𝐧𝐭𝐫 = 𝟐𝑰𝒄 𝐜𝐨𝐬 𝝅𝚽𝐜𝐨𝐧𝐭𝐫

𝚽𝟎

• supercurrent can be tuned by control flux Φcontr through SQUID loop of inductance 𝐿

for 𝛽𝐿 =2𝐿𝐼𝑐

Φ0≪ 1

𝑳𝑱 =𝑳𝒄 𝚽𝐜𝐨𝐧𝐭𝐫

𝐜𝐨𝐬𝝋with 𝑳𝒄 𝚽𝐜𝐨𝐧𝐭𝐫 =

𝚽𝟎

𝟐𝝅𝑰𝒔 𝚽𝐜𝐨𝐧𝐭𝐫

• tuneable nonlinear inductance



classical variables:

phase 𝝋 and charge 𝑸 = 𝑪𝑽 ∝𝒅𝝋

𝒅𝒕

as classical variables, (𝑸,𝝋) are assumed to be measurable simultaneously

classical energies:

potential energy 𝑼(𝝋)(Josephson coupling energy 𝑬𝑱 / Josephson inductance 𝑳𝑱)

kinetic energy 𝑲 ሶ𝝋

(charging energy 𝑸𝟐

𝟐𝑪=

𝟏

𝟐𝑪𝑽𝟐 ∝

𝒅𝝋

𝒅𝒕

𝟐/ junction capacitance 𝑪)

first quantization:current- & voltage-phase relation are derived from macroscopic quantum model

quantum origin

primary macroscopic quantum effects

second quantization:

treat (𝑸,𝝋) as quantum variables (commutation relations, uncertainty)

secondary macroscopic quantum effects

classical vs. quantum treatment of Josephson junctions



𝑬𝑱

j

𝟐𝑬𝐉𝟎

classical vs. quantum treatment of Josephson junctions

classical treatment valid for 𝟐𝑬𝑱𝟎

ℏ𝝎𝒑≃

𝑬𝑱𝟎

𝑬𝑪

𝟏/𝟐≫ 𝟏 (level spacing ≪ 𝑘B𝑇, potential depth)

enter quantum regime by decreasing junction area 𝑨 and reducing 𝑻

harmonic oscillator potentialclose to minimum- level spacing: ℏ𝜔p- lowest energy: ℏ𝜔p/2

≃ ℏ𝝎𝐩/𝟐

≃ ℏ𝝎𝐩

𝑬𝑱 = 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬𝝋

ℏ𝝎𝒑 = 𝟐𝑬𝑪𝑬𝑱𝟎

𝑬𝑱𝟎 =𝚽𝟎𝑰𝒄𝟐𝝅

∝ 𝑨

𝑬𝑪 =𝟐𝒆 𝟐

𝟐𝑪∝𝟏

𝑨

nanotechnology & low temperatures required


1E-4 1E-3 0.01 0.1 1 1010

-2

10-1

100

101

102

10-2

10-1

100

101

102

Ai (m

2)

Ec / k

B

EJ0 / k

B


𝑬𝑱𝟎 ∝ 𝑨

𝑬𝑪 ∝ 𝟏/𝑨

𝑬𝑪 > 𝑬𝑱𝟎

𝑬𝑪 < 𝑬𝑱𝟎


II. Josephson JunctionsExample 1junction area 𝐴 = 10 μm2

barrier thickness 𝑑 = 1 nm

휀 = 10, 𝐽c = 100A

cm2

𝐶 = 0𝐴

𝑑≃ 1 pF

𝐸J0 = 3 ⋅ 10−21 J

𝐸J0/ℎ = 4500 GHz

𝐸𝐶 = 2 ⋅ 10−26 J 𝐸𝐶/ℏ = 30 MHz (∼ 1 mK)

classical junction

Example 2junction area 𝐴 = 0.02 μm2

barrier thickness 𝑑 = 1 nm

휀 = 10, 𝐽c = 100A

cm2

𝐶 ≃ 1 fF

𝐸𝐶 ≃ 𝐸J0 ≃ 6 ⋅ 10−24 J (≃ 0.5 K)

quantum junction

quantum effects observable only at 𝑻 ≪ 𝟎. 𝟓 𝑲for 𝒌𝑩𝑻 ≪ 𝑬𝑱𝟎, 𝑬𝑪 !

sup

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II. Josephson Junctions• classical treatment: Josephson junction with applied current

Kirchhoff‘s law: 𝑰 = 𝑰𝐬 + 𝑰𝐍 + 𝑰𝐃 + 𝑰𝑭

voltage-phase relation: 𝒅𝝋

𝒅𝒕=

𝟐𝒆𝑽

ℏ

nonlinear differential equation withnonlinear coefficients

complex behavior, numerical solution

super-current

normalcurrent

displace-ment

current

noisecurrent

𝑹𝒏 𝑪

𝑰

𝑽

𝑰 = 𝑰𝒄 𝐬𝐢𝐧𝝋 +𝑽

𝑹𝒏+ 𝑪

𝒅𝑽

𝒅𝒕+ 𝑰𝑭

𝑰 = 𝑰𝒄 𝐬𝐢𝐧𝝋 +ℏ

𝟐𝒆

𝟏

𝑹𝒏

𝒅𝝋

𝒅𝒕+ 𝑪

ℏ

𝟐𝒆

𝒅𝟐𝝋

𝒅𝒕𝟐+ 𝑰𝑭

motion of „phase particle“ of mass 𝑴 in thetilted washboard potential

𝑳𝑱 =𝑳𝒄

𝐜𝐨𝐬 𝝋

𝑳𝒄 =𝚽𝟎

𝟐𝝅𝑰𝒄



kinetic energy: 𝑬𝐤𝐢𝐧 =𝑸𝟐

𝟐𝑪=

𝟏

𝟐𝑪𝑽𝟐 =

𝟏

𝟐𝑪

ℏ

𝟐𝒆

𝟐 𝒅𝝋

𝒅𝒕

𝟐=

𝟏

𝟐

𝑬𝑱𝟎

𝝎𝒑𝟐

𝒅𝝋

𝒅𝒕

𝟐

total energy: 𝑬 = 𝑬𝐤𝐢𝐧 + 𝑬𝐩𝐨𝐭 = 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬 𝝋 +𝟏

𝟐

ሶ𝝋

𝝎𝐩

𝟐

energy due to extra charge 𝑄 = 2𝑒 on junction capacitor

consider 𝑬 𝝋, ሶ𝝋 as junction Hamiltonian, rewrite kinetic energy

𝐸kin = 𝑝2/2𝑀

𝑝 =ℏ

2𝑒𝑄

position coordinate is associated with phase: 𝒙 ↔ 𝝋

momentum is associated to charge 𝒑 ↔ ℏ𝑸

𝟐𝒆

• Hamiltonian of a strongly underdamped junction (with 𝑑𝜑

𝑑𝑡≠ 0)

potential energy: 𝑬𝐩𝐨𝐭 = 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬 𝝋 =𝚽𝟎𝑰𝒄

𝟐𝝅𝟏 − 𝐜𝐨𝐬 𝝋 ≃

𝚽𝟎𝟐

𝟐𝑳𝒄

𝝋

𝟐𝝅

𝟐

energy due to extra flux Φ = Φ0 in Josephson inductor

𝑬𝐤𝐢𝐧 =𝑸𝟐

𝟐𝑪=𝟏

𝟐

𝟏

ℏ/𝟐𝒆 𝟐𝑪

ℏ

𝟐𝒆𝑸

𝟐

mass momentum



• canonical quantization (operator replacement)

with # of Cooper pairs 𝑁 =𝑄

2𝑒:

Hamiltonian:

𝑁 ≡𝑄

2𝑒: deviation of # of CP

in electrodes from equilibrium

commutation rules for the operators:

Heisenberg uncertainty relation:

ℏ𝑸

𝟐𝒆→ −𝒊ℏ

𝝏

𝝏𝝋

𝑵 → −𝒊𝝏

𝝏𝝋, 𝑸 = 𝟐𝒆𝑵 → −𝒊 𝟐𝒆

𝝏

𝝏𝝋

ℋ =𝑄2

2𝐶+ 𝐸𝐽0 1 − cos𝜑 = −

2𝑒 2

2𝐶

𝜕

𝜕𝜑

2

+ 𝐸𝐽0 1 − cos𝜑 𝐸𝐶 =(2𝑒)2

2𝐶

𝓗 = −𝑬𝑪𝝏

𝝏𝝋

𝟐

+ 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬𝝋

𝝋,𝑸 = 𝒊 𝟐𝒆 , 𝝋,𝑵 = 𝒊 , 𝝋, ℏ𝑸

𝟐𝒆= 𝒊ℏ

𝚫𝑸 ⋅ 𝚫𝝋 ≥ 𝟐𝒆, 𝚫𝑵 ⋅ 𝚫𝝋 ≥ 𝟏,ℏ

𝟐𝒆𝚫𝑸 ⋅ 𝚫𝝋 ≥ ℏ

Hamiltonian in phase basis 𝝋,𝝏

𝝏𝝋



• Hamiltonian in the flux basis:

circuit variables are now quantized

design superconducting quantum circuits

Hamiltonian:

ℋ =𝑄2

2𝐶+ 𝐸𝐽0 1 − cos𝜑 = −

2𝑒 2

2𝐶

ℏ

2𝑒

2𝜕

𝜕𝜙

2

+ 𝐸𝐽0 1 − cos 2𝜋𝜙

Φ0

𝓗 = −ℏ

𝟐𝑪

𝝏

𝝏𝝓

𝟐

+ 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬𝟐𝝅𝝓

𝜱𝟎

Hamiltonian in flux basis 𝝓,𝝏

𝝏𝝓

𝝓,𝑸 = 𝒊ℏ

commutation rules for the operators:

𝜙 and Q are canonically conjugate (analogous to 𝑥 and 𝑝)

Heisenberg uncertainty relation:

𝚫𝑸 ⋅ 𝚫𝝓 ≥ ℏ

𝝓 =ℏ

𝟐𝒆𝝋 =

𝚽𝟎

𝟐𝝅𝝋



small phase fluctuations Δ𝜑negligible Δ𝜑 ⇒ classical treatment of phase dynamics is often a

good approximation

large charge fluctuations of 𝑄 on junction electrodes since Δ𝑄 ⋅ Δ𝜑 ≥ 2𝑒very small EC ⇒ pairs can easily fluctuate, large Δ𝑄

• The phase regime: ℏ𝝎𝐩 ≪ 𝑬𝐉𝟎 , 𝑬𝑪 ≪ 𝑬𝐉𝟎

• phase 𝝋 (position) is a good quantum number!

• lowest energy levels localized nearbottom of potential wells at 𝝋𝒏 = 𝟐𝝅 𝒏

• Taylor expansion of 𝑬𝐩𝐨𝐭 𝝋

harmonic oscillator frequency 𝝎𝐩

eigenenergies 𝑬𝒏 = ℏ𝝎𝒑 𝒏 +𝟏

𝟐

• ground state: narrowly peaked wavefunction at 𝝋 = 𝝋𝒏, very small 𝚫𝝋

𝑬

𝟐𝑬𝐉𝟎

ℏ𝝎𝐩

𝝋𝒏 𝝋

ℏ𝜔𝑝 = 2𝐸𝐶𝐸𝐽0

sup

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mat

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l



• 1D problem numerical solution straightforward variational approach for approximate ground state

𝑬𝑪

𝑬𝐉𝟎= 𝟎. 𝟎𝟐𝟓

𝑬𝐦𝐢𝐧 = 𝟎. 𝟏 𝑬𝐉𝟎


vary 𝜎 to find minimum energy𝚿 𝝋 ∝ 𝐞𝐱𝐩 −𝝋𝟐

𝟒𝝈𝟐

𝑬𝐦𝐢𝐧 = 𝑬𝑱𝟎 𝟏 − 𝟏 −𝑬𝑪𝟐𝑬𝑱𝟎

𝟐

= 𝑬𝑱𝟎 𝟏 − 𝟏 −ℏ𝝎𝒑

𝟐𝑬𝑱𝟎

𝟐

• trial function for 𝐸𝐶 ≪ 𝐸J0:


sup

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mat

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l



𝑬𝑪

𝑬𝐉𝟎= 𝟎. 𝟎𝟐𝟓

𝑬𝐦𝐢𝐧 = 𝟎. 𝟏 𝑬𝐉𝟎


𝑬𝐦𝐢𝐧 = 𝑬𝑱𝟎 𝟏 − 𝟏 −𝑬𝑪𝟐𝑬𝑱𝟎

𝟐

= 𝑬𝑱𝟎 𝟏 − 𝟏 −ℏ𝝎𝒑

𝟐𝑬𝑱𝟎

𝟐


tunneling coupling ∝ exp −2𝐸J0−𝐸

ℏ𝜔p very small since ℏ𝜔p ≪ 𝐸J0

bandwidth of lowest bands is exponentially small negligible dispersion of 𝐸 𝑄



small charge fluctuations of 𝑄 on junction electrodes since Δ𝑄 ⋅ Δ𝜑 ≥ 2𝑒large 𝐸𝐶 ⇒ pair number on junction electrode can not fluctuate, small Δ𝑄

• The charge regime: ℏ𝝎𝐩 > 𝑬𝐉𝟎 , 𝑬𝑪 ≫ 𝑬𝐉𝟎

• charge 𝑄 (momentum) is good quantum number!

• kinetic energy ∝ 𝑬𝒄𝒅𝝋

𝒅𝒕

𝟐dominates

complete delocalization of phase wave function should approach

constant value, 𝛹 𝜑 ≃ 𝑐𝑜𝑛𝑠𝑡.

𝑬

𝟐𝑬𝐉𝟎

𝑬𝒄

𝝋𝒏 𝝋

large phase fluctuations Δ𝜑small 𝐸𝐽0 ⇒ phase difference across junction can take arbitrary values,

large Δ𝜑

sup

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• The charge regime: ℏ𝝎𝐩 > 𝑬𝐉𝟎 , 𝑬𝑪 ≫ 𝑬𝐉𝟎

vary 𝛼 (𝛼 ≪ 1) to find minimum energy𝚿 𝝋 ∝ 𝟏 − 𝜶 𝐜𝐨𝐬 𝝋• trial function:

• Hamiltonian: 𝓗 = −𝑬𝑪𝝏

𝝏𝝋

𝟐

+ 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬𝝋

𝑬𝐦𝐢𝐧 = 𝑬𝑱𝟎 𝟏 −𝑬𝑱𝟎

𝟐𝑬𝑪= 𝑬𝑱𝟎 𝟏 −

𝑬𝑱𝟎𝟐

ℏ𝝎𝒑𝟐 ℏ𝝎𝒑 = 𝟐𝑬𝑪𝑬𝑱𝟎

𝑬𝑪𝑬𝑱𝟎

= 𝟏𝟎

𝑬𝐦𝐢𝐧 = 𝟎. 𝟗𝟓 𝑬𝑱𝟎

sup

ple

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mat

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l



• The charge regime: ℏ𝝎𝐩 ≫ 𝑬𝐉𝟎 , 𝑬𝑪 ≫ 𝑬𝐉𝟎

𝑬𝐦𝐢𝐧 = 𝑬𝑱𝟎 𝟏 −𝑬𝑱𝟎

𝟐𝑬𝑪= 𝑬𝑱𝟎 𝟏 −

𝑬𝑱𝟎𝟐

ℏ𝝎𝒑𝟐 ℏ𝝎𝒑 = 𝟐𝑬𝑪𝑬𝑱𝟎

𝑬𝑪𝑬𝑱𝟎

= 𝟏𝟎

𝑬𝐦𝐢𝐧 = 𝟎. 𝟗𝟓 𝑬𝑱𝟎

periodic potential is weak: 𝐸J0 ≪ 𝐸𝐶 (𝐸kin ≫ 𝐸pot)

delocalization in 𝜑 space (analog to particle in weak periodic potential) formation of broad bands, strong dispersion of 𝐸 𝑄


Summary

• Josephson junction can be described by nonlinear lossless inductor 𝑳𝑱 =𝚽𝟎

𝟐𝝅𝑰𝒄 𝐜𝐨𝐬 𝝋

• analog circuit: parallel connection of 𝑳𝑱 and 𝑪 (and 𝑹𝒏)

• characteristic energies: 𝑬𝑱𝟎 =𝚽𝟎𝑰𝒄

𝟐𝝅, 𝑬𝑪 =

𝟐𝒆 𝟐

𝟐𝑪, ℏ𝝎𝒑 =

ℏ

𝑳𝒄𝑪= 𝟐𝑬𝑪𝑬𝑱𝟎

• classical teatment if 𝑬𝑱 ≫ 𝑬𝑪, 𝒌𝑩𝑻 ≫ 𝑬𝑪 (always the case @ large junction area)

classical eqn. of motion of phase particle in tilted washboard potential

• quantum treatment if 𝑬𝑱 ∼ 𝑬𝑪, 𝒌𝑩𝑻 < 𝑬𝑪, 𝑬𝑱

• Hamiltonian (phase basis): 𝓗 = −𝑬𝑪𝜕

𝜕𝜑

2+ 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬𝝋

𝝋,𝑸 = 𝒊 𝟐𝒆, 𝝋, 𝑵 = 𝒊, 𝚫𝑸 ⋅ 𝚫𝝋 ≥ 𝟐𝒆, 𝚫𝑵 ⋅ 𝚫𝝋 ≥ 𝟏

• Hamiltonian (flux basis): 𝓗 = −ℏ

𝟐𝑪

𝜕

𝜕𝜙

2+ 𝑬𝑱𝟎 𝟏 − 𝐜𝐨𝐬 𝟐𝝅

𝝓

𝜱𝟎

𝝓,𝑸 = 𝒊ℏ, 𝚫𝑸 ⋅ 𝚫𝝓 ≥ ℏ


III. SuperconductingQuantum Circuits



key physical ingredients

𝑱𝒔 = 𝑱𝒄 𝐬𝐢𝐧𝝋 ,𝒅𝝋

𝒅𝒕=𝟐𝒆𝑽

ℏර

𝑪

.


𝑭

.

𝑩 ⋅ ෝ𝒏 𝑑𝐹 = 𝒏𝚽𝟎

𝚿 𝐫, 𝐭 = 𝚿𝟎𝒆𝒊𝜽 𝒓,𝒕

𝚿 𝐫, 𝐭 𝟐 = 𝒏𝒔 𝒓, 𝒕

𝝓,𝑸 = 𝒊ℏ


harmonic LC oscillator

E

|0>|1>|2>|3>|4>|5>

|g>|e>

„artificial solid-state atom“„artificial solid-state photon“

„quantum optics“ on a chip

quantum2-levelsystem

=qubit

𝑯 = ℏ𝝎 ෝ𝒂†ෝ𝒂 +𝟏

𝟐𝑳𝑱 𝚽 =

𝚽𝟎

𝟐𝝅𝑰𝒄 𝐜𝐨𝐬 𝟐𝝅𝚽𝚽𝟎

tunable, anharmonic LC oscillator

E

tunablenonlinearlossless

inductance

𝚽

𝑰

Josephsonjunctions



75 µm

photon box:

microwave resonator

artificial atom:

solid state quantum circuit

A. Wallraff et al., Nature 431, 162 (2004).S. Girvin, R. Schoelkopf, Nature 451, 664-669 (2008) .

anharmonic level structure(quantum two-level system: qubit)

quantum coherence(coherence time: < 100 µs)

e.g. persistent current flux qubit

e.g. coplanar waveguide (CPW) resonator

small mode volume(Vmod/l3 10-5 – 10-6)

high quality factor(Q 104 – 106)

Circuit QED



2D / h ≈ 100 GHz – 1 THz

ℏ𝝎𝐠𝐞 ≈ 1 – 10 GHz

normal metal superconductorEF

E E E

D >> kBT

|g>|e>co

ntinuum

of

exc

itations

ℏ𝝎𝐠𝐞


• advantages of superconducting systems

1. Macroscopic quantum nature of superconducting ground state 2. Energy gap in excitation spectrum


• exploit macroscopic quantum nature of sc ground state andgap in excitation spectrum long coherence time

M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)

Moore‘s Law for QubitLifetime



2000 2004 2008 2012 201610-9

10-8

10-7

10-6

10-5

10-4

10-3

coh

ere

nce

tim

e (s

)

year

best T2 times

reproducible T2 times

CPB

quantronium

cQED

transmon

3D transmon

fluxonium



fabricate tailor-made quantum circuits

2 µm

flux qubit(Al)

transmon qubit(Al)

3D coplanarwaveguideresonator(Al)

coplanar waveguideresonator (Nb)


3. Established fabrication technology: thin film & nanotechnology

4. Superb design flexibility, tunability and scalability


quantum circuit with 8 resonators and 3 qubits coupled to each rersonator



interferometer

nano-electromechanical circuit

Si3N4 nanobeam coupled to CPW resonator



M. Mariantoni et al. Phys. Rev. B 78, 104508 (2008)A. Baust et al., Phys. Rev. B 91, 014515 (2015); Phys. Rev. B 93, 214501 (2016)

Superconducting Quantum Switch



State preservation by repetitive error detection in a superconducting quantum circuit,J. Kelly et al., Nature 519, 66-69 (2015)

UCSB&

chip with9 X-mon qubits



interaction energy = dipole moment ⋅ respective field

ℏ𝒈 = 𝐩 ⋅ 𝐄𝐫𝐦𝐬

ℏ𝒈 = 𝛍 ⋅ 𝐁𝐫𝐦𝐬

make electric (𝒑) or magnetic dipolemoment (𝝁) as big as possible

„big atoms“µm-sized circuits

„small cavities“quasi 1D cavities

make mode volume of cavity as small aspossible

𝑬𝐫𝐦𝐬𝐯𝐚𝐜 =

ℏ𝝎

𝝐𝟎𝑽𝐦𝐨𝐝

𝑩𝐫𝐦𝐬𝐯𝐚𝐜 =

𝝁𝟎ℏ𝝎

𝑽𝐦𝐨𝐝

5. Strong and ultrastrong coupling due to large dipole moments6. Fast manipulation by control pulses



superconducting resonator

T. Niemczyk et al., Nature Phys. 6, 772 (2010)

flux qubit

X. Zhou, et al., Nature Physics 9 , 179 (2013)

Si3N4 nanomechanical beam


7. Realize hybrid quantum systems by combination with other degrees of freedom (e.g. spin, photonic, phononic, plasmonic, ….)

examples from WMI

Ch. Zollitsch et al., Appl. Phys. Lett. 107, 142105 (2015)

paramagnetic spins

phosphorousdonors in Si

H. Huebl et al., PRL 111, 127003 (2013)

ferrimagneticspin ensemble

YIG



• drawbacks of superconducting systems

resonator atom

𝝎𝒓 𝝎𝐠𝐞

𝝎𝒓

𝟐𝝅≃

𝝎𝐠𝐞

𝟐𝝅≃ few GHz

1 GHz ↔ 50 mK

ℏ𝝎𝒓 ≃ 10-24 J

ultra-low temperatures

ultra-sensitive µ-wave experiments

challenges

nano-fabrication

1. Low energy scales



2. Strong coupling to environment

protection against thermal microwave fieldse.g. cold attenuators, circulators, „Purcell filtering“ by cavity, ….

reduction of two-level fluctuatorse.g. substrate cleaning, avoid oxide layers, ….

strategies

optimum choice of operation pointe.g. operation @ qubit „sweet spot“, …


Summary

superconducting quantum circuits

• make use of

macroscopic quantum nature of superconductivity

fluxoid quantization

Josephson effect

• offer superb advantages (strong coupling, established fabrication technology,

design flexibility, tunability, scalability, …)

• challenge experimentalists

ultra-low temperature

nanotechnology

ultra-sensitive microwave measurements


Part II


contents









microwaves

IX. Summary

IV. Superconducting

Resonators & Qubits


IV. SC Resonators & Qubits

superconductingquantum circuits

resonators qubits

couplers interferometers

switches JPAs

hybrid systems

resonators


𝑯𝐋𝐂 = 𝑬𝐤𝐢𝐧 + 𝑬𝐩𝐨𝐭 =𝑸𝟐

𝟐𝑪+𝜱𝟐

𝟐𝑳=𝑸𝟐

𝟐𝑪+𝟏

𝟐𝑪

𝟏

𝑳𝑪𝜱𝟐

L C

general Hamiltonian

𝑯𝐇𝐎 = 𝑬𝐤𝐢𝐧 + 𝑬𝐩𝐨𝐭 =ෝ𝒑𝟐

𝟐𝒎+𝟏

𝟐𝒎𝝎𝒓

𝟐ෝ𝒙𝟐

𝒎

𝑘

𝑳𝑪 resonant circuit

𝑥

e.g., 𝜔𝑟 = 𝑘/𝑚 for mass-spring system

momentum ෝ𝒑 ↔ charge 𝑸position ෝ𝒙 ↔ flux 𝚽mass 𝒎 ↔ capacitance C

resonance frequency 𝝎𝐫 ↔ 𝝎𝐫 = Τ𝟏 𝑳𝑪

continuous-variable operators

IV. SC Resonators & Qubits𝑳𝑪 resonant circuit as a harmonic oscillator


L C

continuous-variable operatorsmomentum Ƹ𝑝 ↔ charge 𝑄position ො𝑥 ↔ flux 𝛷mass 𝑚 ↔ capacitance C

resonance frequency 𝜔r ↔ 𝜔r = Τ1 𝐿𝐶

parabolic potential

ෝ𝒒 and 𝜱 form a conjugate pair such as ෝ𝒙 and ෝ𝒑

Heisenberg uncertainty: 𝚫𝑸 𝚫𝜱 ≥ℏ

𝟐

commutation relation: 𝜱, 𝑸 = −𝒊ℏ

Τ𝑬𝐩𝐨𝐭ℏ

𝒒, 𝜱


𝑯𝐋𝐂 = 𝑬𝐤𝐢𝐧 + 𝑬𝐩𝐨𝐭 =𝑸𝟐

𝟐𝑪+𝜱𝟐

𝟐𝑳=𝑸𝟐

𝟐𝑪+𝟏

𝟐𝑪𝝎𝒓

𝟐 𝜱𝟐


L C

photon number operator: ො𝑛 ≡ ො𝑎† ො𝑎

eigenstates are Fock states: 𝐻𝐿𝐶 𝑛 = 𝐸𝑛 𝑛

eigenvalues: 𝐸𝑛 = ℏ𝜔r 𝑛 +1

2

linear system equidistant level spacing

𝑛 = 0 finite vacuum energy: 𝐸0 =ℏ𝜔r

2

𝑛 is the Fock or number basis

0

𝝎𝐫

ෝ𝒂 ≡𝝎𝐫𝑪𝜱+𝒊𝑸

𝟐𝝎𝐫𝑪ℏannihilation operator


𝒒

𝑯𝑳𝑪 = ℏ𝝎𝐫 ෝ𝒂†ෝ𝒂 +𝟏

𝟐

ෝ𝒂† ≡𝝎𝐫𝑪𝜱−𝒊𝑸

𝟐𝝎𝐫𝑪ℏcreation operator

,Τ

𝑬𝒏ℏ

1

2

3

⋮

𝝎𝐫/2

IV. SC Resonators & Qubitsmomentum Ƹ𝑝 ↔ charge 𝑄position ො𝑥 ↔ flux 𝛷mass 𝑚 ↔ capacitance C


discrete basis (Fock states)


L C

when applied to a Fock state, ෝ𝒂 annihilates a photon inside the oscillator

𝝎𝐫


𝒒

,Τ

𝑬𝒏ℏ

⋮

ෝ𝒂†

ෝ𝒂ෝ𝒂 𝒏 = 𝒏|𝒏 − 𝟏⟩

ෝ𝒂† 𝒏 = 𝒏 + 𝟏|𝒏 + 𝟏⟩

when applied to a Fock state, ෝ𝒂† creates a photon inside the oscillator




𝟐





𝝎𝐫/𝟐

annihilation & creation operator

sup

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mat

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L C


𝒒

,Τ

𝑬𝒏ℏ

⋮

ෝ𝒂†

ෝ𝒂eigenstates of ෝ𝒂: ෝ𝒂 𝜶 = 𝜶|𝜶⟩, 𝜶 ∈ ℂ

ෝ𝒂, ෝ𝒂† = 𝟏 bosonic communation relation

coherent states: 𝜶 𝜶 = 𝒆−𝜶 𝟐

𝟐 σ𝒏𝜶𝒏

𝒏!|𝒏⟩

{|𝜶⟩} normal but not orthogonal




𝟐





𝝎𝐫

𝝎𝐫/𝟐

coherent states

sup

ple

men

tary

mat

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l


L C


𝒒

,Τ

𝑬𝒏ℏ

⋮

ෝ𝒂†

ෝ𝒂

ෝ𝒂 𝜶 = 𝜶|𝜶⟩, 𝜶 ∈ ℂ

𝑨(𝒕) ≡𝟏

𝟐(ෝ𝒂 + ෝ𝒂†) bosonic field amplitude operator

for an intuitive understanding

move to interaction picture

𝑈 † = 𝑒(−)𝑖𝜔r𝑡 ො𝑎† ො𝑎

መ𝐴𝐼 𝑡 ≡ 𝑈 መ𝐴𝑈† =1

2ො𝑎𝑒−𝑖𝜔r𝑡 + ො𝑎†e+𝑖𝜔r𝑡



𝝎𝐫

𝝎𝐫/𝟐

practical importance of coherent states


𝟐

sup

ple

men

tary

mat

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l


L C


𝒒

መ𝐴(𝑡) ≡1

2ො𝑎𝑒−𝑖𝜔r𝑡 + ො𝑎†e+𝑖𝜔r𝑡

,Τ

𝑬𝒏ℏ

⋮

ෝ𝒂†

ෝ𝒂

classical limit

𝛼 መ𝐴(𝑡) 𝛼 =𝛼

2𝛼 𝛼 e−𝑖𝜔r𝑡 +

𝛼⋆

2𝛼 𝛼 e+𝑖𝜔r𝑡

= 1 = 1

=𝛼

2e−𝑖 𝜔r𝑡+𝜙 + e+𝑖 𝜔r𝑡+𝜙

= 𝛼 cos 𝜔r𝑡 + 𝜙

oscillating field with amplitude 𝛼 and phase 𝜙

coherent statemost classical quantum states(expectation values obey classical eqns of motion)

IV. SC Resonator & Qubitsmomentum Ƹ𝑝 ↔ charge 𝑄position ො𝑥 ↔ flux 𝛷mass 𝑚 ↔ capacitance C


𝝎𝐫

𝝎𝐫/𝟐

ො𝑎 𝛼 = 𝛼|𝛼⟩, 𝛼 ∈ ℂ

practical importance of coherent states


𝐻𝐿𝐶 = ℏ𝜔r ො𝑎† ො𝑎 +1

2

75 µm75 µm

𝒌𝐁𝑻 ≪ ℏ𝝎𝒏

LC quantum harmonic oscillator

𝝀/𝟐 coplanar waveguide resonator (quasi-1D)

eachmode 𝒏

standing waves (“modes”)

𝑯𝐓𝐋 = ℏ

𝒏

𝝎𝒓,𝒏ෝ𝒂𝒏†ෝ𝒂𝒏


multimode Hamiltonian


L C

• how to measure spectrum of 𝑯𝑳𝑪 ? 𝐻𝐿𝐶 = ℏ𝜔r ො𝑎† ො𝑎 +1

2

we must consider coupling to external channel loss rates 𝛾1 and 𝛾2 simplification 𝛾 ≡ 𝛾1 = 𝛾2

𝜸𝟏 𝜸𝟐

measurement requires two components input probe field ෝ𝒂𝐢𝐧 detected output field ෝ𝒂𝐨𝐮𝐭

ෝ𝒂𝐢𝐧ෝ𝒂𝐨𝐮𝐭

properties of ො𝑎in and ො𝑎out free (propagating) multi-mode fields field ො𝑎 inside the resonator is a single mode-field transition mediated by coupling capacitors borrow input-output formalism from quantum optics!

D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 2008)

challenge:describe interaction between a single modefield and a continuum of modes!



L C

𝜸𝟏𝜸𝟐

ෝ𝒂𝐢𝐧

ෝ𝒂𝐨𝐮𝐭

ෝ𝒂 measure transmission 𝑻 and/orreflection coefficient 𝚪 by vectornetwork analyzer

example: 𝑏in = 0, 𝛾1 = 𝛾2 ≡ 𝛾

ෝ𝒂𝐢𝐧 𝝎 + ෝ𝒂𝐨𝐮𝐭 𝝎 = 𝜸𝟏ෝ𝒂 𝝎

ෝ𝒂 𝒕 =𝟏

𝟐𝝅න−∞

∞

𝒅𝝎𝒆−𝒊𝝎 𝒕−𝒕𝟎 ෝ𝒂 𝝎

𝑻 ≡𝒃𝐨𝐮𝐭ෝ𝒂𝐢𝐧

=𝜸

𝜸 − 𝒊 𝝎 − 𝝎𝐫

𝒃𝐨𝐮𝐭

𝒃𝐢𝐧

transmitted power ∝ 𝑻 𝟐 is Lorentzian!(equals the classical result)

reflection and transmission (two ports) 𝜞 + 𝑻 = 𝟏

𝝎𝝎𝐫

𝜟𝝎 = 𝟐𝜸

𝑻 𝟐

D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 2008)


𝐻𝐿𝐶 = ℏ𝜔r ො𝑎† ො𝑎 +1

2 characterization of 𝝀/𝟐-resonator (two ports)


𝝎𝝎𝐫

𝜟𝝎 ≡ 𝜿 = 𝟐𝜸

𝑻

𝑇 =𝛾

𝛾 − 𝑖 𝜔 − 𝜔r

• photon storage time is given by resonator quality factor:

• energy-time uncertainty 𝛥𝐸𝛥𝑡 ≃ ℏ Δ𝜔Δ𝑡 ≃ 1:

• identify 2𝛥𝑡 with dephasing time 𝑻𝟐 (𝑇2 = 2𝑇1):

IV. SC Resonators & Qubits characterization of 𝝀/𝟐-resonator (two ports)

L C

𝜸𝜸

ෝ𝒂𝐢𝐧

ෝ𝒂𝐨𝐮𝐭

ෝ𝒂

𝒃𝐨𝐮𝐭

𝒃𝐢𝐧

𝑸 ≡𝝎𝐫

𝜟𝝎=𝝎𝐫

𝜿=𝝎𝐫

𝟐𝜸

𝚫𝒕 =𝟏

𝜟𝝎=𝟏

𝜿=

𝟏

𝟐𝜸

𝑻𝟐 =𝟐

𝜟𝝎=𝟐

𝜿=𝟏

𝜸


𝑸 ∝ 𝟏/𝟐𝜸 quality factor of ideal resonator determined by loss through coupling capacitors

• for real resonator many types of loss channels coupling capacitors: 𝑄c internal dissipative/dielectric losses: 𝑄i radiation losses: 𝑄rad …

loaded quality factor

𝑇 =𝛾

𝛾 − 𝑖 𝜔 − 𝜔r

IV. SC Resonators & Qubits characterization of 𝝀/𝟐-resonator (two ports)

𝝎𝝎𝐫

𝜟𝝎 ≡ 𝜿 = 𝟐𝜸

𝑻

L C

𝜸𝜸

ෝ𝒂𝐢𝐧

ෝ𝒂𝐨𝐮𝐭

ෝ𝒂

𝒃𝐨𝐮𝐭

𝒃𝐢𝐧

𝟏

𝑸=

𝟏

𝑸𝐜+𝟏

𝑸𝐢+

𝟏

𝑸𝐫𝐚𝐝+⋯

J. Goetz et al., J. Appl. Phys. 119, 015304 (2016)



𝑻𝟐 times of 2D superconductingresonators

• MBE grown (epitaxially) Al on sapphiresubstrate @ mK temperatures

𝑓0 = 6.121 GHz 𝑄i = 1.7 × 106, 𝑄c = 4 × 105

𝑻𝟐 ≲ 𝟏𝟎𝟎𝛍𝐬Megrant et al., APL 100, 113510 (2012)

• Niobium on SiO2-coated high-resistivity Si substrate @mK temperatures

𝑓0 = few GHz 𝑄i ≈ 105

𝑻𝟐 ≲ 𝟐𝟎 𝛍𝐬

𝑇2 =2

𝛥𝜔=2

𝜅=1

𝛾

J. Burnett et al, SUST 29, 044008 (2016)J. Goetz et al., JAP 119, 015304 (2016)


• alternative: 3D (cavity type) superconducting resonators – losses

no more dielectrics negligible amount of TLS in cavity 𝑸𝐢 ≈ 𝟏𝟎𝟕 − 𝟏𝟎𝟖

𝑻𝟐 ≲ 𝟏𝟎𝐦𝐬 M. Reagor et al., Appl. Phys. Lett. 102, 192604 (2013)

reduced relevance of interface lossesfor embedded circuits

3D transmon qubit 𝑻𝟏 ≲ 𝟏𝟓𝟎𝛍𝐬

Paik et al., PRL 107, 240501 (2011)

• losses in 2D (planar) superconducting resonators

resistive or QP losses superconductivity & low temperatures

radiation losses clever design

problem: dielectric losses from material defects (spurious TLS)

TLS in bulk substrate use clean single crystal (sapphire, intrinsic Si)

TLS at substrate-metal interface

required: clean materials & growth processes

𝑻𝟐 times of 3D superconducting resonators 𝑇2 =2

𝛥𝜔=2

𝜅=1

𝛾


sup

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tary

mat

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• applications of quantum harmonic oscillators

quantum HO: linear circuit, not a qubit, not directly useful for quantum computation!

quantumsimulation of

manybodyHamiltonians

L C

ancilla qubit/nonlinearity explore quantumphysics (Fock states,

squeezing etc.)

typically longcoherence times

quantum memory

mediate couplingbetween qubits quantum bus

qubit readout(„dispersivereadout“)

identifydecoherence sources in


indirect use





resonators qubits

couplers interferometers

switches JPAs

hybrid systems

qubits


quantum bit (qubit) superposition of two computational basis states

𝑎 𝑡 , 𝑏 𝑡 ∈ ℂ with 𝑎(𝑡) 2 + 𝑏 𝑡 2 = 1 all states can be visualized on the

surface of a sphere

Bloch sphere representation

𝜳 𝒕 = 𝒂 𝒕 𝐠 + 𝒃 𝒕 𝐞

classical bit deterministic, either in ground state “g” or in excited state “e”

𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)

𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧

𝜽(𝒕)

𝟐𝐠

𝝋(𝒕) phase coherence

𝜽 𝒕 amplitude energy, population

• definition of a quantum bit

𝒙

𝒛

|𝐠⟩

|𝐞⟩

𝝋(𝒕)

𝜽(𝒕)|𝜳(𝒕)⟩

𝒚

Bloch angles:

IV. SC Resonators & QubitsIntro

𝜑 𝑡 =𝐸𝑒 − 𝐸𝑔

ℏ𝑡

sup

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linear algebra notation of operators and state vectors

qubit states can be written as vectors

𝒂 𝐞 + 𝒃|𝐠⟩ 𝐚𝟏𝟎

+ 𝒃𝟎𝟏

=𝒂𝒃

qubit operators (gates) can be written as matrices

𝒂 𝐞 𝐞 + 𝒃 𝐠 𝐠 + 𝒄 𝐞 𝐠 + 𝒅 𝐠 𝐞

𝒂𝟏𝟎

𝟏 𝟎 + 𝒃𝟎𝟏

𝟎 𝟏 + 𝒄𝟏𝟎

𝟎 𝟏 + 𝒂𝟎𝟏

𝟏 𝟎 =𝒂 𝒄𝒅 𝒃


sup

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unitary operations

𝑼|𝜳⟩ expressed via the Hermitian Pauli spin matrices 𝟏, ෝ𝝈𝒙, ෝ𝝈𝒚, ෝ𝝈𝒛

ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎

ෝ𝝈𝒚 ≡𝟎 −𝒊𝒊 𝟎

ෝ𝝈𝒛 ≡𝟏 𝟎𝟎 −𝟏

𝟏 ≡𝟏 𝟎𝟎 𝟏

|𝐠⟩ and |𝐞⟩ are the eigenvectors of ෝ𝝈𝒛

pseudo spin

|𝜳⟩ is equivalent to spin wave function in external magnetic field

pseudo spin and Pauli matrices



𝜽(𝒕)

𝟐𝐠


sup

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ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎


ෝ𝝈𝒛 ≡𝟏 𝟎𝟎 −𝟏

𝟏 ≡𝟏 𝟎𝟎 𝟏

conventions: Pauli matrices and Bloch sphere

these definitons contain several conventions, such as

the global scaling factor the positon of the minus sign in 𝜎𝑧 here, we show two examples with fixed 𝜎𝑧

physics convention 𝐠 ≡𝟎𝟏

, 𝐞 ≡𝟏𝟎

ground state energy negative (more „physical“)



𝜽(𝒕)

𝟐𝐠


𝟐𝟎 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧

𝜽(𝒕)

𝟐𝟏

information theory (IT) convention

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press

𝟎 ≡𝟏𝟎

, 𝟏 ≡𝟎𝟏

ground state energy positive („unphysical“) easily generalized (more „logical”)

unless otherwise mentioned physics convention! formal resolution equate g to 1 and e to 0 used in this lecture!



important states on the Bloch sphere

𝒙

𝒚

𝒛

|𝐠⟩

|𝐞⟩

𝐞 + 𝐠

𝟐

𝐞 − 𝒊 𝐠

𝟐

𝐞 − 𝐠

𝟐

𝐞 + 𝒊 𝐠

𝟐

𝒙

𝒚

𝒛

|𝟏⟩

|𝟎⟩

𝟎 + 𝟏

𝟐

𝟎 − 𝒊 𝟏

𝟐

𝟎 − 𝟏

𝟐

𝟎 + 𝒊|𝟏⟩

𝟐

𝚿 𝒕 = 𝐜𝐨𝐬𝜽

𝟐𝐞 + 𝒆𝒊𝝋 𝐬𝐢𝐧

𝜽

𝟐𝐠 𝚿 𝒕 = 𝐜𝐨𝐬

𝜽

𝟐𝟎 + 𝒆𝒊𝝋 𝐬𝐢𝐧

𝜽

𝟐𝟏

ITphysics


sup

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ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎


ෝ𝝈𝒛 ≡𝟏 𝟎𝟎 −𝟏

𝟏 ≡𝟏 𝟎𝟎 𝟏

interpretation of the Pauli matrices

1 = |g⟩⟨g| + e e

ො𝜎𝑥 = ො𝜎− + ො𝜎+

ො𝜎𝑧 = |e⟩⟨e| − g g

ො𝜎𝑦 = 𝑖 ො𝜎− − ො𝜎+

• Pauli matrices can expressed in terms of projection operators

ො𝜎− = g e

ො𝜎+ = e g

induce transitions between |g⟩ and |e⟩

puts an excitation into the qubit

removes an excitation from the qubit

⟨ ො𝜎𝑧⟩ gives the qubit population

reflects normalization

• combination of basis definition and operator description in terms of projection operators matrix form of operators

• in this lecture, we fix the matrix definitions of the Pauli matrices “physical” intuition in g , e -notation notation consistent with Nielsen & Chuang and most physics papers!

g

e

g

e

g

e

g

e? ?


sup

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• single qubit gate

unitary operation 𝑈 on state |𝛹⟩ described by rotations on Bloch sphere + global phase

• rotation matrices

about x-axis 𝑅𝑥 𝛼 ≡ e−𝑖𝛼ෝ𝜎𝑥2 =

cos𝛼

2−𝑖 sin

𝛼

2

−𝑖 sin𝛼

2cos

𝛼

2

about y-axis 𝑅𝑦 𝛼 ≡ e−𝑖𝛼ෝ𝜎𝑦

2 =cos

𝛼

2−sin

𝛼

2

sin𝛼

2cos

𝛼

2

about z-axis 𝑅𝑧 𝛼 ≡ e−𝑖𝛼ෝ𝜎𝑧2 = 𝑒−𝑖 Τ𝛼 2 0

0 𝑒𝑖 Τ𝛼 2

Why? In general unitary expressed by rotations

𝑈 = 𝑒𝑖𝛼 𝑅𝑧 𝛽 𝑅𝑦 𝛾 𝑅𝑧 𝛿 with 𝛼, 𝛽, 𝛾, 𝛿 ∈ ℝ

Z-Y decomposition (others possible) 𝛼 is a global phase (unobservable)

definition of a single qubit gate


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examples for 1-qubit gates

NOT

graphical representation example

matrix representation (taken from QI theroy books) typically follow IT convention!


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Hadamard gate 𝑯 is of particular importance

𝑯 𝐠 =𝟏

𝟐( 𝐞 − |𝐠⟩)

𝑯 𝐞 =𝟏

𝟐(|𝐞⟩ + |𝐠⟩)

𝑯 ≡𝟏

𝟐

𝟏 𝟏𝟏 −𝟏

=𝟏

𝟐ෝ𝝈𝒙 + ෝ𝝈𝒛

𝒙

𝒚

𝒛

|𝐠⟩

|𝐞⟩

𝐞 + 𝐠

𝟐

𝐞 − 𝐠

𝟐

• physics convention

• applied to one of the basis states |g⟩ or |e⟩, it results in a superposition state ofthe basis states


𝑯 =𝟏

𝟐𝒆 𝒆 − 𝒈 𝒈 + 𝒆 𝒈 + 𝒈 𝒆

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M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013); DOI:10.1126/science.1231930

we do not go beyond this point in this lecture

status in 2013



• interaction with environment for control

uncontrolled interactions (noise) also exist quantum effects (population oscillations, quantum interference,

superpositions, entanglement) unobservable after characteristic time after decoherence time 𝑻𝐝𝐞𝐜, quantum effects have decayed to Τ1 𝑒 of their

original level term “decoherence” originally only referred to phase nowadays sloppily comprises both phase and amplitude effects

quantum coherence



𝜽(𝒕)

𝟐𝐠



• ideal quantum system

completely isolated in reality, however, …



• population energy relaxation time 𝑇1 or 𝑇r decay from |e⟩ to |g⟩ nonadiabatic (irreversible) processes induced by high-frequency fluctuations (𝜔 ≈ 𝜔ge)

• phase pure dephasing time 𝑇𝜑 adiabatic (reversible) processes induced by low-frequency fluctuations (𝜔 → 0) often encountered: 1/f-noise real measurements always contain 𝑇1-effects

𝑻𝟐−𝟏 = 𝟐𝑻𝟏

−𝟏 + 𝑻𝝋−𝟏

nomenclature not very consistent in literature!

energy relaxation and dephasing




𝜽(𝒕)

𝟐𝐠



𝝋 𝒕 =𝑬𝒆 − 𝑬𝒈

ℏ𝒕

𝛿𝜑 =𝛿𝐸

ℏ𝑇𝜑 ≃ 2𝜋



|g>

|e>

quantum2-levelsystem

=qubit𝑳𝑱 𝚽 =

𝚽𝟎

𝟐𝝅𝑰𝒄 𝐜𝐨𝐬 𝟐𝝅𝚽𝚽𝟎

E

tunable Josephson inductance

𝚽

𝑰

qubit = anharmonic superconducting quantum circuit

design flexibility leads to a

plethora of superconducting qubits

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• coherence time: 𝑻𝐝𝐞𝐜 ≈𝟏

𝜹𝝎𝟎𝟏=

𝝎𝟎𝟏

𝜹𝝎𝟎𝟏

𝟏

𝝎𝟎𝟏= 𝑸

𝟏

𝝎𝟎𝟏

• 1 bit operation time: 𝒕𝐨𝐩 >𝟏

𝚫𝝎(otherwise 1 → 2 -transitions are induced!)

• # of 1 bit operations:𝑻𝐝𝐞𝐜

𝒕𝐨𝐩≈

𝑸

𝝎𝟎𝟏 𝒕𝐨𝐩< 𝑸

𝚫𝝎

𝝎𝟎𝟏anharmonicity

how much anharmonicity is required ?


𝚫𝝎

ℏ𝝎𝟎𝟏

ℏ𝝎𝟏𝟐

𝝎𝟎𝟏 𝝎𝟎𝟐𝝎

𝜹𝝎𝟎𝟏 𝜹𝝎𝟎𝟐𝒕𝒐𝒑

sometimes trade-off between

anharmonicity ⇔ qubit decoherence


phase qubit

(EJ >> EC)current biased JJ

flux qubit

(EJ > EC)fluxon boxes

charge qubit

(EJ < EC)Cooper pair boxes

I I

I

V

J. Martinis (NIST) H. Mooij (Delft) V. Bouchiat (Quantronics)

nowadays superconducting qubit zoo is larger transmon, camel-back, capacitively shunted 3JJ-FQB, quantronium, fluxonium… “traditional” classification via 𝐸𝐽/𝐸𝐶 is increasingly difficult



phase qubit

(EJ >> EC)current biased JJ

flux qubit

(EJ > EC)fluxon boxes

charge qubit

(EJ < EC)Cooper pair boxes

I I

V


engineered qubit potential


𝑳𝑱

𝑪

add inductance

• qubit design by potential engineering

𝑯𝐉 = −𝑬𝑪𝝏

𝝏𝝋

𝟐

+ 𝑬𝐉 (𝟏 − 𝐜𝐨𝐬 ෝ𝝋)

unsuitable for TLS!

add junctions

• flux/phase engineering

add bias current

add gate capacitor

• charge engineering

𝐸J naturally induces

anticrossings

add shunt capacitor change curvature ofcharge parabola

(3JJ flux qubit)

(rf SQUID &phase qubit)

(phase qubit)

(charge qubit)

(transmon qubit)


𝑳𝑱

𝑪

𝑳𝑱 𝜶𝑳𝑱

𝑳𝑱

𝑪

𝑳

𝑳𝑱

𝑪

𝑰

𝑵

𝑬𝑱𝟎𝑪𝐠

𝑽𝐠

𝑪𝑱


additional „force term“ due to current source

𝝋

|𝟎⟩|𝟏⟩

arcsin(𝐼𝑥/𝐼𝑐)

tilted washboard potential significant anharmonicity

Um

|𝟐⟩

G0

G1

G2

levels 𝟎 , 𝟏 form the qubitoscillator states differ in phase

phase qubit𝛤2 ≫ 𝛤1, 𝛤0 pump 𝜔12 for readout readout detects running phase (voltage)

𝑯 = 𝑬𝑪𝑵𝟐 + 𝑬𝑱 𝟏 − 𝐜𝐨𝐬𝝋 +

ℏ

𝟐𝒆𝑰𝒙𝝋


𝑳𝑱

𝑪

phase qubit

current bias 𝑰𝐱

very small

𝑨 > 𝟏 × 𝟏 𝛍𝐦𝟐

𝑬𝑱𝟎

𝑬𝑪∝ 𝑨𝟐 ≫ 𝟏𝟎𝟒

𝝋 ≈ classical

𝑵 = −𝒊𝝏

𝝏𝝋

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IV. SC Resonators & Qubits phase qubit – first Rabi oscillations

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parameters similarto RF SQUID qubit

better decoupling from readout electronics significantly longer decoherence preferred over current-biased version


𝑳𝑱

𝑪

phase qubit (with flux bias)

current bias 𝑰𝐱

𝑳𝑱

𝑪

𝑳replace current source 𝑰𝒙 by superconducting loop with

applied 𝚽𝒙

𝑬𝑱𝟎 ≫ 𝑬𝑪ℏ𝝎𝒑 > 𝒌𝑩𝑻

𝑰𝒄𝑳 ≃ 𝚽𝟎oppositecirculating current dc SQUID readout

𝚽𝒙

flux bias 𝚽𝐱


circulating current due to Φ𝑥: 𝑰𝑳 =𝚽𝐱

𝑳= 𝒇

𝚽𝟎

𝑳

𝑬𝐉 ≫ 𝑬𝑪 (phase/flux regime)

ℏ𝝎𝐩 ≫ 𝒌𝐁𝑻 (ℏ𝝎𝒑 = 𝟐𝑬𝑱𝑬𝑪 )

𝑰𝑪𝑳 ≈ 𝜱𝟎

MQT causes level splitting 𝜟 𝟎 , |𝟏⟩ are symmetric and antisymmetric

superpositions of +𝐼𝐿 , −𝐼𝐿

theoretical prediction: Leggett (1984)experimental realization: Friedman et al. (2000)

2𝜋Φ

Φ0

|𝟎⟩|𝟏⟩

𝜑 + 2𝜋𝑓= 2𝜋

𝒇 =𝟏

𝟐

𝜟

+𝐼𝐿 −𝐼𝐿

𝜑 + 2𝜋𝑓= 0

IV. SC Resonators & Qubits RF SQUID qubit

𝑳𝑱

𝑪

𝑳

flux bias𝚽𝐱

𝚽𝟎= 𝒇 ≃ 𝟏/𝟐

𝑯 = 𝑬𝑪𝑵𝟐 + 𝑬𝑱 𝟏 − 𝐜𝐨𝐬𝝋 + 𝑬𝑳

𝝋− 𝟐𝝅𝒇 𝟐

𝟐

very small

𝑵 = −𝒊𝝏

𝝏𝝋

𝐸𝐿 ≡ Φ02/2𝐿 𝑓 ≡ Φx/Φ0

requires large 𝐿 large „antenna“ for flux noise


𝑳𝑱

𝑪

𝑳𝑱 𝑳𝑱

3-JJ persistent current flux qubit

𝑳𝐥𝐨𝐨𝐩 ≪ 𝑳𝐉 ⇒ 𝜷𝑳 = 𝟐𝑳𝐥𝐨𝐨𝐩𝑰𝒄/𝚽𝟎 ≪ 𝟏

two junctions have identical sizethird junction smaller by factor 𝛼

𝑬𝐉 ≡ 𝑬𝐉𝟏 = 𝑬𝐉𝟐 and 𝑬𝑪 ≡ 𝑬𝑪𝟏 = 𝑬𝑪𝟐

𝑬𝐉𝟑 = 𝜶𝑬𝐉 and 𝑬𝑪𝟑 =𝑬𝑪

𝜶

𝑬𝐉 > 𝑬𝑪 (phase regime) & ℏ𝝎𝐩 ≪ 𝒌𝐁𝑻

control knob

external flux Φx applied to loop

still 2 quantum degrees of freedom left after fluxoid quantization

more than one JJ in the loop to overcome drawback ofRF SQUID flux qubit

J.E. Mooij et al., Science 285, 1036 (1999)T. P. Orlando et al., PRB 60, 15399-15413 (1999)


flux bias𝚽𝐱

𝚽𝟎= 𝒇 ≃ 𝟏/𝟐

𝑳𝐥𝐨𝐨𝐩

𝜑1 𝜑2

𝛼

+𝐼p −𝐼p

𝜱𝐱

𝑳𝐥𝐨𝐨𝐩

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𝑬𝐩𝐨𝐭 𝝋𝟏, 𝝋𝟐, 𝝋𝟑 = 𝑬𝐉 𝟐 − 𝐜𝐨𝐬𝝋𝟏 − 𝐜𝐨𝐬𝝋𝟐 + 𝜶 𝟏 − 𝐜𝐨𝐬𝝋𝟑

𝑬𝐩𝐨𝐭 = 𝑬𝐉𝟎 𝟏 − 𝐜𝐨𝐬𝝋

fluxoid quantization:

𝝋𝟏 − 𝝋𝟐 + 𝝋𝜶 = −𝟐𝝅𝒇 with frustration 𝒇 ≡𝚽𝐱

𝚽𝟎

signs are mere convention!

𝑬𝐩𝐨𝐭 𝝋𝟏, 𝝋𝟐 =

𝑬𝐉 𝟐 + 𝜶 − 𝐜𝐨𝐬𝝋𝟏 − 𝐜𝐨𝐬𝝋𝟐 − 𝜶𝐜𝐨𝐬 𝟐𝝅𝒇 + 𝝋𝟏 − 𝝋𝟐

color code: 𝐸pot 𝜑1,𝜑2

𝐸JΦx

Φ0= 𝑓 =

1

2

𝛼 = 0.8

𝜑1/2𝜋

𝜑2/2𝜋

−1

0

1

−2

2

−1 0 1−2 2

𝜑1/2𝜋

𝜑2/2𝜋

0

−1

1

0−1 1

double-well potential in each unit cell !



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𝜑1/2𝜋

𝜑2/2𝜋

0

−1

1

0−1 1

• double well rotated by 45° in the 𝜑1𝜑2-plane variable transformation

two stable minima at 𝝋∗, −𝝋∗ and (−𝝋∗, 𝝋∗), where 𝐜𝐨𝐬𝝋∗ ≡𝟏

𝟐𝜶

𝜑+ ≡1

2𝜑1 + 𝜑2

𝜑− ≡1

2𝜑1 − 𝜑2

𝑬𝐩𝐨𝐭 𝝋+, 𝝋− =

𝑬𝐉 [𝟐 + 𝜶 − 𝟐𝐜𝐨𝐬𝝋+ 𝐜𝐨𝐬𝝋− − 𝜶𝐜𝐨𝐬 𝟐𝝅𝒇 + 𝟐𝝋−

𝜑+/2𝜋

𝜑−/2𝜋

0

−0.5

0.5

0−0.5 0.5

variable relevant for qubit dynamics 𝜑−



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𝜑+/2𝜋

𝜑−/2𝜋

0

−0.5

0.5

0−0.5 0.5

𝚽𝐱

𝚽𝟎= 𝒇 = 𝒏 +

𝟏

𝟐

symmetric double-well potential

no tunneling degenerate ground state left/right well correspond to clockwise/anticlockwise circulating persistent current ±𝑰𝐩

𝜑−

𝐸pot

𝐸J0

ቤ𝑰𝒑 = −𝝏𝑬𝐩𝐨𝐭(𝝋− = −𝝋∗)

𝝏𝚽𝒙 𝚽𝒙=𝚽𝟎/𝟐

thermodynamics: 𝑰 = −𝝏𝑬𝐩𝐨𝐭

𝝏𝜱

sin(𝑥 + 𝑦) = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦 sin 2𝑥 = 2 sin 𝑥 cos𝑦

= −𝟐𝑰𝒄𝜶𝐬𝐢𝐧𝟐𝝋∗ 𝐜𝐨𝐬𝝅

cos𝜑∗ ≡1

2𝛼

= ቤ−𝑬𝐉𝟎𝜶 −𝐬𝐢𝐧 𝟐𝝅𝒇 − 𝟐𝝋∗𝟐𝝅

𝜱𝟎 𝜱𝒙=𝜱𝟎/𝟐

= 𝑰𝒄𝜶𝐬𝐢𝐧 𝝅 − 𝟐𝝋∗ = 𝟐𝑰𝒄𝜶𝐬𝐢𝐧𝝋∗ 𝐜𝐨𝐬𝝋∗ = 𝑰𝒄 𝟏 −

𝟏

𝟐𝜶

𝟐

𝜑∗−𝜑∗

+𝐼p −𝐼p



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𝚽𝒙

𝚽𝟎= 𝒇 ≠ 𝒏 +

𝟏

𝟐 tilted double-well potential

flux bias induces energy bias 𝜺 𝚽𝐱

near Φx/Φ0 = 𝑛 +1

2: 𝜺 𝚽𝒙 = 𝟐𝑰𝐩𝚽𝟎 𝒇 − 𝒏 −

𝟏

𝟐

휀 Φx

𝝋−

𝚽𝐱

𝚽𝟎= 𝒇 < 𝒏 +

𝟏

𝟐

𝐸pot

𝐸J0

+𝑰𝒑

−𝑰𝒑


𝜑+/2𝜋

𝜑−/2𝜋

0

−0.5

0.5

0−0.5 0.5

𝚽𝐱

𝚽𝟎= 𝒇 = 𝒏 +

𝟏

𝟐

symmetric double-well potential

no tunneling degenerate ground state circulating persistent current ±𝑰𝐩

𝝋−

𝐸pot

𝐸J0

thermodynamics: 𝑰 = −𝝏𝑬𝐩𝐨𝐭

𝝏𝜱

𝜑∗−𝜑∗

+𝑰𝒑 −𝑰𝒑


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𝑯 = 𝑬𝑪 𝑵𝟏𝟐 + 𝑵𝟐

𝟐 + 𝑬𝑱 [𝟐 + 𝜶 − 𝟐𝐜𝐨𝐬 ෝ𝝋+ 𝐜𝐨𝐬 ෝ𝝋− − 𝜶𝐜𝐨𝐬 𝟐𝝅𝒇 + 𝟐ෝ𝝋− ]

𝝋+ ≡𝟏

𝟐𝝋𝟏 +𝝋𝟐 𝝋− ≡

𝟏

𝟐𝝋𝟏 −𝝋𝟐𝑬𝑪 ≡

(𝟐𝒆)𝟐

𝟐𝑪

𝑵𝟏𝟐 + 𝑵𝟐

𝟐

−𝒊=

𝝏

𝝏𝝋𝟏

𝟐

+𝝏

𝝏𝝋𝟐

𝟐

𝑵𝟏,𝟐 ≡ −𝒊𝝏

𝝏𝝋𝟏,𝟐

• task convert 𝑁1 and 𝑁2 into 𝑁+ and 𝑁−

=𝝏

𝝏𝝋+

𝝏𝝋+

𝝏𝝋𝟏+

𝝏

𝝏𝝋−

𝝏𝝋−

𝝏𝝋𝟏

𝟐

+𝝏

𝝏𝝋+

𝝏𝝋+

𝝏𝝋𝟐+

𝝏

𝝏𝝋−

𝝏𝝋−

𝝏𝝋𝟐

𝟐

=𝟏

𝟒

𝝏

𝝏𝝋++

𝝏

𝝏𝝋−

𝟐

+𝝏

𝝏𝝋+−

𝝏

𝝏𝝋−

𝟐

=𝟏

𝟐

𝝏

𝝏𝝋+

𝟐

+𝝏

𝝏𝝋−

𝟐


3-JJ persistent current flux qubit: quantum treatment

with

=

𝟏𝟐

𝑵+𝟐 + 𝑵−

𝟐

−𝒊

• flux qubit Hamiltonian

𝑯 =𝟏

𝟐𝑬𝑪 𝑵+

𝟐 + 𝑵−𝟐 + 𝑬𝑱 𝟐 + 𝜶 − 𝟐𝐜𝐨𝐬 ෝ𝝋+ 𝐜𝐨𝐬 ෝ𝝋− − 𝜶𝐜𝐨𝐬 𝟐𝝅𝒇 + 𝟐ෝ𝝋−

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2𝑓2𝑓

𝐸𝑛/ℏ

(GH

z)

𝐸𝑛/ℏ

(GH

z)

numerical diagonalization eigenenergies 𝐸𝑛

near Φx

Φ0= 𝑓 = 𝑛 +

1

2: approximation as two-level system with linear energy bias!

𝒇 = 𝜱𝐱/𝜱𝟎


3-JJ persistent current flux qubit: quantum treatment

𝑯 =𝟏

𝟐𝑬𝑪 𝑵+

𝟐 + 𝑵−𝟐 + 𝑬𝑱 𝟐 + 𝜶 − 𝟐𝐜𝐨𝐬 ෝ𝝋+ 𝐜𝐨𝐬 ෝ𝝋− − 𝜶𝐜𝐨𝐬 𝟐𝝅𝒇 + 𝟐ෝ𝝋−

𝑬𝑱𝟎𝑬𝑪

≃ 𝟑𝟓

𝜺 𝚽𝒙 = 𝟐𝑰𝐩𝚽𝟎 𝒇 − 𝒏 −𝟏

𝟐

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⟨𝑰𝒑ෝ𝝈𝒛⟩

0 𝛿Φx/Φ0

0

𝐼p

−𝐼p

𝑬𝟎,𝟏

0

𝛿Φx/Φ00

Δ

𝟏

𝟐+𝑰𝒑 + −𝑰𝒑

𝟏

𝟐+𝑰𝒑 − −𝑰𝒑

• energy bias 𝜺 𝚽𝒙 = 𝟐𝑰𝐩𝜹𝚽𝒙

+𝑰𝒑 and −𝑰𝒑 are eigenstates of 𝜺 𝚽𝒙 ෝ𝝈𝒛

𝛿Φx ≡ Φ0 𝑓 − 𝑛 −1

2

• tunneling rate 𝜟/𝒉

tunnel splitting 𝜟 ∝ 𝐞𝐱𝐩 − Τ𝑬𝑱 𝑬𝑪

𝑯 = 𝜺 𝚽𝒙 ෝ𝝈𝒛 + 𝚫ෝ𝝈𝒙

𝑬𝟏 − 𝑬𝟎 ≡ ℏ𝝎𝐪 𝜱𝒙 = 𝜺𝟐 𝚽𝒙 + 𝜟𝟐

• Bloch angle 𝜽 denotes operation point Φx

sin 𝜃 ≡Δ

ℏ𝜔q Φxand cos 𝜃 ≡

Φx

ℏ𝜔q Φx

• persistent circulating current 𝑰𝒑ෝ𝝈𝒛 depends on Φx

𝐼𝑝 ො𝜎𝑧 = 𝐼p cos 𝜃



sup

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⟨𝑰𝒑ෝ𝝈𝒛⟩

0 𝛿Φx/Φ0

0

𝐼p

−𝐼p

𝑬𝟎,𝟏

0

𝛿Φx/Φ00

Δ

𝟏

𝟐+𝑰𝒑 + −𝑰𝒑

𝟏

𝟐+𝑰𝒑 − −𝑰𝒑



readout of 𝑰𝒑ෝ𝝈𝒛 by inductively coupled dc-SQUID

Example: 𝐸𝐽0/𝐸𝐶 ≃ 50

JJs: 𝐼c ≃ 750 nA, 𝐶 ≃ 3 fF, 𝛼 ≃ 0.7 loop: 𝑑 ≃ 10 μm 𝐿 ≈ 𝜇0𝑑 ≃ 10 pH

flux signal 𝑳𝑰𝐩 ≈ 𝑳 ⋅ 𝜶𝑰𝐜 ≃ 𝟑𝐦𝜱𝟎

±𝐼p can be distinguished

by an on-chip dc SQUID

sup

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𝑬

0

𝜹𝚽𝒙/𝚽𝟎0

ℏ𝝎

microwave pulse sequence

adiabatic fluxshift pulse

readoutpulse sequence

timeprepare

& readout

Prepare

Readout

K. Kakuyanagi et al., Phys. Rev. Lett. 98, 047004 (2007)F. Deppe et al., Phys. Rev. B 76, 214503 (2007)


3-JJ persistent current flux qubit: pulsed readout at the degeneracy point


𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉 𝟏 − 𝐜𝐨𝐬 ෝ𝝋

• gate charge 𝑵𝐠 ≡𝑪𝐠𝑽𝐠

𝟐𝒆

induced by gate voltage 𝑽𝐠

adds/removes excess CP to/fromisland

classical quantity

may assume fractional values!

charge qubit – the Cooper pair box (CPB)

charge regime 𝑬𝑪 ≳ 𝑬𝐉𝟎 charge is good quantum number

superconducting island


𝑵


𝑽𝐠

𝑪𝑱

additional term due gate voltage small

𝑵 = −𝒊𝝏

𝝏𝝋

• charge energy: 𝑬𝑪 ≡(𝟐𝒆)𝟐

𝟐 𝑪𝒈+𝑪𝑱

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𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉 𝟏 − 𝐜𝐨𝐬 ෝ𝝋

• gate charge 𝑵𝐠 ≡𝑪𝐠𝑽𝐠

𝟐𝒆

induced by gate voltage 𝑽𝐠

adds/removes excess CP to/fromisland

classical quantity

may assume fractional values!

Schrödinger equation 𝑯𝐂𝐏𝐁 𝜳𝒌 = 𝑬𝒌 𝜳𝒌

Mathieu equation for ෩𝜳𝒌 ≡ 𝜳𝒌 𝐞−𝒊𝑵𝐠𝝋

𝝏𝟐 ෩𝜳𝒌

𝝏𝜶𝟐− 𝟐

𝟐𝑬𝐉

𝑬𝑪𝐜𝐨𝐬 𝜶 ෩𝜳𝒌 =

𝟒𝑬𝒌𝑬𝑪

෩𝜳𝒌

𝛼 ≡ 𝜑/2

numerical solution eigenenergies 𝐸𝑘

charge qubit – the Cooper pair box (CPB)

charge regime 𝑬𝑪 ≳ 𝑬𝐉𝟎 charge is good quantum number

superconducting island


𝑵


𝑽𝐠

𝑪𝑱

additional term due gate voltage small

𝑵 = −𝒊𝝏

𝝏𝝋

• charge energy: 𝑬𝑪 ≡(𝟐𝒆)𝟐

𝟐 𝑪𝒈+𝑪𝑱


tunable JJ (dc SQUID)

𝑪𝐉/𝟐

readout

• dc SQUID with 𝜷𝑳 ≪ 𝟏 tunable JJ with effective 𝑬𝑱𝟎 and 𝑬𝑪

• gate voltage 𝑽𝐠 is control knob

• readout with additional JJ

detect number of excess Cooper pairs on island

• Josephson energy 𝑬𝑱𝟎 𝐜𝐨𝐬 ෝ𝝋

couples charge states/parabolas

avoided level crossings

island

𝑪𝐠

𝑽𝐠

typical prameters:𝐸𝐶/ℎ ≃ 5 GHz, 𝐸J0/ℎ ≃ 5 GHz

𝑪𝐉/𝟐


charge qubit – the split Cooper pair box (CPB)

𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉𝟎 𝟏 − 𝐜𝐨𝐬 ෝ𝝋

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

E / E

C

CVe / 2e

E

E+

E

𝑬𝐉𝟎

𝑬𝑪= 𝟎. 𝟎𝟔

𝑬𝐉𝟎

𝑵𝐠

𝐸/𝐸

𝐶

E+

𝑵=𝟎

𝑵=𝟏

𝑵=𝟐

sup

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𝑪𝐉/𝟐

readout

island

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

E / E

C

CVe / 2e

E

E+

E

𝑬𝐉𝟎

𝑬𝑪= 𝟎. 𝟎𝟔

𝑬𝐉𝟎

𝑵𝐠

𝐸/𝐸

𝐶

𝑪𝐠

𝑽𝐠


𝑪𝐉/𝟐




E+

theory questions

• why is coupling exactly 𝑬𝐉𝟎?

• near 𝒏𝐠 =𝟏

𝟐, energy levels look

hyperbolic. Is this correct?

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two-level-representation of the CPB

goal: express 𝑯𝐂𝐏𝐁 = 𝑬𝑪 ෝ𝒏 − 𝒏𝐠𝟐+ 𝑬𝐉 𝐜𝐨𝐬 ෝ𝝋 as TLS near 𝒏𝐠 =

𝟏

𝟐

charge states 𝑛 ො𝑛 𝑛 = 𝑛 𝑛

= 𝑛 − 𝑛g2

𝑛

𝑛 𝑛ො𝑛 − 𝑛g2= ො𝑛 − 𝑛g

𝑛

𝑛 𝑛 ො𝑛 − 𝑛g

commutation relations ො𝑛, ො𝜑 = 1

𝑛 =1

2𝜋න0

2𝜋

𝑑𝜑 exp −𝑖𝑛 ො𝜑 𝜑

exp 𝑖𝑝 ො𝜑 𝑛 =1

2𝜋න0

2𝜋

𝑑𝜑 exp −𝑖 𝑛 + 𝑝 ො𝜑 𝜑 = 𝑛 + 𝑝

exp ±𝑖 ො𝜑 𝑛 = 𝑛 ± 1 cos ො𝜑 =1

2exp 𝑖 ො𝜑 + exp −𝑖 ො𝜑

=1

2

𝑛

𝑛 𝑛 exp 𝑖 ො𝜑 + exp −𝑖 ො𝜑

𝑛

𝑛 𝑛

=1

2

𝑛

𝑛 𝑛 + 1 + 𝑛 + 1 𝑛


sup

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𝐻CPB =𝐸el2

ො𝜎𝑧 +𝐸J2ො𝜎𝑥

𝐸el ≡ 4𝐸𝐶 𝑛 − 𝑛g2

𝑛 ො𝑛 𝑛 = n 𝑛

exp ±𝑖 ො𝜑 𝑛 = 𝑛 ± 1

1

2

𝑛

𝑛 𝑛 exp 𝑖 ො𝜑 + exp −𝑖 ො𝜑

𝑛

𝑛 𝑛

TLS 𝒏 ∈ {𝟎, 𝟏}

=1

20 0 + 1 1 exp 𝑖 ො𝜑 + exp −𝑖 ො𝜑 0 0 + 1 1

=1

2( 0 0 exp 𝑖 ො𝜑 0 0 + 0 0 exp 𝑖 ො𝜑 1 1

+ 1 1 exp 𝑖 ො𝜑 0 0 + 1 1 exp 𝑖 ො𝜑 1 1+ 0 0 exp −𝑖 ො𝜑 0 0 + 0 0 exp −𝑖 ො𝜑 1 1

=1

2( 0 0 1 0 + 0 0 2 1 + 1 1 1 0 + 1 1 2 1

+ 0 + 0 0 0 1 + 0 + 1 1 0 1 )

𝑛 − 𝑛g2

𝑛

𝑛 𝑛 =1

2𝑛 − 𝑛g

2ො𝜎𝑧

𝐻CPB = 4𝐸𝐶 ො𝑛 − 𝑛g2+ 𝐸J cos ො𝜑

=1

2ො𝜎𝑥


two-level-representation of the CPB

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𝑪𝐉/𝟐

readout

island

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

E / E

C

CVe / 2e

E

E+

E

𝑬𝐉𝟎

𝑬𝑪= 𝟎. 𝟎𝟔

𝑬𝐉𝟎

𝑵𝐠

𝐸/𝐸

𝐶

𝑪𝐠

𝑽𝐠


𝑪𝐉/𝟐




E+

theory questions




hyperbolic. Is this correct?

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𝑪𝐉/𝟐

readout

island

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

E / E

C

CVe / 2e

E

E+

E

𝑬𝐉𝟎

𝑬𝑪= 𝟎. 𝟎𝟔

𝑬𝐉𝟎

𝑵𝐠

𝐸/𝐸

𝐶

𝑪𝐠

𝑽𝐠


𝑪𝐉/𝟐



𝑯𝐂𝐏𝐁 =𝑬𝐞𝐥𝟐ෝ𝝈𝒛 +

𝑬𝑱𝟎𝟐

ෝ𝝈𝒙, 𝑬𝐞𝐥≡ 𝑬𝑪 𝑵−𝑵𝒈𝟐

E+

theory questions


because of the two-level representation



hyperbolic. Is this correct ?

yes, because 𝐸el 𝑉g can be linearized

for 𝑁 −𝑁g ≪ 1


from the Cooper pair box to the transmon qubit

advantages of the CPB:

• simple design (2JJ, 𝛽𝐿 ≪ 1)

• level splitting Δ = 𝐸J0 ∝ 𝐼c(flux qubit: 𝛥 ∝ exp − Τ𝐸𝐽0 𝐸𝐶 )

• voltages convenient for coupling to other qubits coupling to readout circuitry coupling to control signals

• large anharmonicity (few GHz)

• in first order insensitive to charge fluctuations at „sweet spot“ 𝑁g = 𝑛 +1

2

disadvantages:

• coherence times short due to susceptibility to 1/𝑓 charge noise

• in practice: coherence times of a few tens of nanoseconds even at the sweet spot!(typical charge noise magnitude ≫ typical flux noise magnitude)

• idea flatten energy dispersion


0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

E / E

C

CVe / 2e

E

E+

E

𝑬𝐉𝟎

𝑬𝑪= 𝟎. 𝟎𝟔

𝑬𝐉𝟎

𝑵𝐠

𝐸/𝐸

𝐶


J. Koch et al., PRA 76, 042319 (2007).

The transmon qubit

take a CPB geometry andincrease Τ𝑬𝐉𝟎 𝑬𝑪 by shunt capacitor

charge dispersion decreases exponentially with Τ𝐸J0 𝐸𝐶 anharmonicity decreases only polynomially with Τ𝐸J0 𝐸𝐶 optimum trade-off for Τ𝐸J 𝐸𝐶 ≈ 50

few hundreds of MHz anharmonicity left charge no longer good quantum number not tunable via gate voltage anymore tune via flux (dc SQUID)

transmission line shuntedplasma oscillation qubit


𝑵𝒈 𝑵𝒈

𝑵

𝑬𝑱𝑪𝐠

𝑽𝐠

𝑪𝑱

𝑪𝒔superconductingisland


J. Koch et al., Phys. Rev. A 76, 042319 (2007).

The transmon qubit

take a CPB geometry and increase Τ𝑬𝐉𝟎 𝑬𝑪

charge dispersion decreases exponentially with Τ𝐸J0 𝐸𝐶 less sensitive to charge noise

anharmonicity decreases only polynomially with Τ𝐸J0 𝐸𝐶 optimum trade-off for Τ𝐸J 𝐸𝐶 ≈ 50

few hundreds of MHz anharmonicity left charge no longer good quantum number not tunable via gate voltage anymore tune via flux (dc SQUID)

transmission line shunted plasmaoscillation qubit


𝑬𝒎 𝑵𝒈 ≈ 𝑬𝒎 𝑵𝒈 =𝟏

𝟒−𝝐𝒎𝟐𝐜𝐨𝐬𝟐𝝅𝒏𝒈

𝝐𝒎 ≈ −𝟏 𝒎𝑬𝑪𝟐𝟒𝒎+𝟓

𝒎

𝟐

𝝅

𝑬𝑱𝟎𝟐𝑬𝑪

𝒎𝟐+𝟑𝟒

𝒆−

𝟖𝑬𝑱𝟎𝑬𝑪

with 𝑬𝑪 = 𝒆𝟐/𝟐𝑪 𝑁𝑔 𝑁𝑔


• embed into a resonator for readout filtering control

2D geometries: 𝟏𝟎 − 𝟒𝟎 𝛍𝐬3D geometries: up to 𝟏𝟎𝟎 𝛍𝐬

• the transmon is currentlymost successful qubit withrespect to coherence times

• coherence of transmonsmostly limited by spuriousTLS (defects) in substrateand metal-substrate interface

IV. SC Resonators & Qubits The transmon qubit

J. Koch et al., Phys. Rev. A 76, 042319 (2007).


J. Q. You et al., Phys. Rev. B 75, 140515(R) (2007).M. Steffen et al., Phys. Rev. Lett 105, 100502 (2010).

The C-shunted flux qubit

• decreasing 𝑬𝐉𝟎/𝑬𝑪 and/or 𝜶 reduces

influence of flux noise by level flattening

• however, sensitivity to charge noise on islands a,b,c is increased

• suppress charge noise by shuntcapacitance 𝑪𝐬𝐡 = 𝜷 − 𝜶 𝑪𝐉

• typically, 𝐶sh ≃ 100 fF ≫ 𝐶J ≃ 5 fF

• first promising results 𝑇2∗ ≈ 𝑇1 ≃ 1.5 μs

𝑪𝐬𝐡

𝑵𝐛

𝑵𝐚

𝑵𝐜


𝑬𝐉𝟎, 𝑪𝑱 𝑬𝐉𝟎, 𝑪𝑱

𝜶𝑬𝐉𝟎, 𝜶𝑪𝑱

+𝐼p −𝐼p

𝚽𝐱

E

sup

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• optimize thin film fabrication as in transmon anharmonicity 800 MHz slightly larger

than for typical transmon qubits neverthless, transmon-like design

• noise sources limiting 𝑻𝟏 ≲ 𝟓𝟓 𝛍𝐬 @ 𝚽𝟎/𝟐 resonator loss Ohmic charge noise 1/𝑓 flux noise temporal variations attributed to

quasiparticles

• noise sources limiting 𝑻𝟐 ≃ 𝟖𝟓 𝛍𝐬 @ 𝚽𝟎/𝟐 photon shot noise from residual thermal

photons in the readout resonator

F. Yan et al., arXiv:1508.06299 (2015).


The C-shunted flux qubit𝑪𝐬𝐡

𝑵𝐛

𝑵𝐚

𝑵𝐜

𝑬𝐉𝟎, 𝑪𝑱 𝑬𝐉𝟎, 𝑪𝑱

𝜶𝑬𝐉𝟎, 𝜶𝑪𝑱

+𝐼p −𝐼p

𝚽𝐱


Summary

• superconducting resonators can be fabricated in various geometries with high

quality factors

thin film based resonators: 𝑻𝟐 ≤ 𝟐𝟎 𝛍𝐬 (Nb on Si)

𝑻𝟐 ≤ 𝟏𝟎𝟎 𝛍𝐬 (Al on sapphire)

3D (bulk based) resonators: 𝑻𝟐 ≤ 𝟏𝟎𝐦𝐬 (Al)

• large variety of different qubits due to flexible potential engineering

different qubits offer different advantages and disadvantages:

coherence time, tunability, anharmonicity, controllability, …..

transmon qubits presently show best coherence times: 𝑻𝐝𝐞𝐜 ≤ 𝟏𝟓𝟎 𝛍𝐬


V. Circuit QED


e.g. Kimble and Mabuchi groups at CaltechRempe group at MPQ Garching, ….

cavity QED natural atom in optical cavity

Rempe group

circuit QED solid state circuit in µ-wave cavity

e.g. Wallraff (ETH), Martinis (UCSB), Schoelkopf (Yale), Nakamura (Tokyo), ….

WMI

resonator QED

IV. Circuit QED


Conference on Resonator QED 2017


superconductingqubit

striplineresonator

quantumdot

photonic crystalresonator

Rydberg atom µ-wave Fabry-Pérot resonator

alkali atom optical Fabry-Pérot resonator

+ manymore

besides

IV. Circuit QEDCavity and Circuit QED Systems


ℏ𝝎

𝜸 = 𝝎𝐪/𝑸𝐪: spontaneous emission rate

ext.driving 𝜿 = 𝝎𝒓/𝑸𝒓

optical cavity QED

A. Wallraff et al., Nature (2004)

superconducting circuit QED

• make g, k as small as possible• „low loss“ atoms and resonators

strong coupling/large cooperativity: 𝒈 > 𝜸, 𝜿 𝒈𝟐/𝜸𝜿 > 𝟏

• make g as large as possible• atoms with large dipole moments,

cavities with small mode volumes

ultra-strong coupling: 𝒈𝟐/𝝎𝐪𝝎𝒓 ∼ 𝟏

𝒈𝟐/𝜸𝜿 ≫ 𝟏 < 𝟏𝟎𝟔 𝒈𝟐/𝝎𝐪𝝎𝒓 > 𝟎. 𝟎𝟏

IV. Cavity vs. Circuit QED

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A. Wallraff et al., Nature 431, 162 (2004)

vacuum Rabi-mode splitting

I. Chiorescu et al., Nature 431, 159 (2004)

coherent flux qubit / SQUID coupling

D. Schuster et al., Nature 445, 515 (2007)

photon number splitting

M. Hofheinz et al., Nature 454, 310 (2008)

M. Hofheinz et al., Nature 459, 546 (2009)

Fock & arbitrary photon states

J. Fink et al., Nature 454, 315 (2008)

J. Fink et al., PRL 103, 083601 (2009)

n – nonlinearity & N – nonlinearity

F. Deppe et al., Nature Physics 4, 686 (2008)

controlled symmetry breaking

L. DiCarlo et al., Nature 460, 240 (2009)

L. DiCarlo et al., Nature 467, 574 (2010)

quantum algorithms & GHZ, W states

M. Sillanpää et al., Nature 449, 438 (2007)

H. Majer et al., Nature 449, 443 (2007)

quantum bus

O. Astafiev et al., Nature 449, 588 (2007)

single artificial atom masing

A. Houck et al., Nature 449, 328 (2007)

single photon source

T. Niemczyk et al., Nature Physics 6, 772 (2010)G. Günter et al., Nature 458, 178 (2009)

circuit QED in the ultrastrong-coupling regime

……

…

IV. Circuit QED – an ongoing success story


qubit resonator coupling

qubitrelaxation

resonatordecay

+ 𝐻𝛾 + 𝐻𝜅 + 𝐻driveexternal

drive

ෝ𝒂†, ෝ𝒂: photon creation/annihilation operatorෝ𝝈+, ෝ𝝈−: qubit raising/lowering operatorෝ𝝈𝒛: Pauli matrix

Jaynes-Cummings model

E.T. Jaynes, F.W. Cummings, Proc. IEEE 51, 89 (1963).D. Walls, G. Milburn, Quantum Optics, Spinger-Verlag (1994)

(i) strong coupling regime: 𝝎𝒒, 𝝎𝒓 ≫ 𝒈 > 𝜸, 𝜿

coupling

total number of excitations is conserved

coupled resonator – two-level system

IV. Circuit QED


qubit resonator coupling

ෝ𝒂†, ෝ𝒂: photon creation/annihilation operatorෝ𝝈+, ෝ𝝈−: qubit raising/lowering operatorෝ𝝈𝒛: Pauli matrix

coupled resonator – two-level system

IV. Circuit QED

(ii) ultra-strong coupling regime: 𝝎𝒒, 𝝎𝒓 ∼ 𝒈 ≫ 𝜸, 𝜿

G. Günther et al., Nature 458, 178 (2009).T. Niemczyk et al., Nature Physics 6, 772 (2010).

C. Ciuti et al., Phys. Rev. A 74, 033811 (2006).

total number of excitations is not conserved

coupling ≃ ℏ𝒈 ෝ𝒂 + ෝ𝒂† ෝ𝝈− + ෝ𝝈+

J. Casanova et al., Phys. Rev. Lett. 105, 263603 (2010).

D. Zueco et al., Phys. Rev. A 80, 033846 (2009)J. Bourassa et al., Phys. Rev. A 80, 032109 (2009).B. Peropadre et al., Phys. Rev. Lett. 105, 023601 (2010).


solid-state atom


𝒈/𝟐𝝅 ≃ 630 MHz

with superconducting quantum circuits:

GHz resonator

transmission spectrum

IV. Circuit QED – ultrastrong coupling


𝟑

𝟐𝝀 - mode 𝝀 - mode

IV. Circuit QED – ultrastrong coupling

T. Niemczyk,PhD Thesis, TU Munich (2011)

sup

ple

men

tary

mat

eria

l


qualitative explanationof datawithinJC-model

onlyquantitativedeviations

qualitative deviations from JC-model

evidence for ultra-strong coupling regime


3l/2-mode

USC in superconducting cQED


resonant case

𝝎𝒓 ≃ 𝝎𝒒

strong coupling regime:

𝒈 ≫ 𝜸, 𝜿

E.T. Jaynes, F.W. Cummings, Proc. IEEE 51, 89 (1963). S. Haroche and J.M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, Oxford Univ. Press (2006) J. Ye, H. J. Kimble, H. Katori, Science 320, 1734 -1738 (2008)

dressed states (polaritons):

𝝎𝒓 𝝎𝒒

quantumnonlinearity

photon blockade effective photon-photon interaction nonlinear resonator photon transistor

U

quantum nonlinearities

IV. Circuit QED

|±, 𝒏 =|𝒈, 𝒏 ± |𝒆, 𝒏 − 𝟏

𝟐


example: bottom-up construction of many-body Hamiltonians

… …

Bose-Hubbard or JC chain driven dissipative dynamics, scaling behavior

M. Leib et al., NJP (2010)

tunablenonlinearities

U

tunablecoupling J

M. Leib, et al., NJP 14, 075024 (2012)

realize analog superconducting quantum simulator

Prospects

M. Mariantoni et al. Phys. Rev. B 78, 104508 (2008)A. Baust et al., PRB 91, 014515 (2015); PRB 93, 214501 (2016)


𝑯𝐒𝐁 =ℏ𝝎𝐪 𝚽𝐱

𝟐𝝈𝒛 +

𝒌

ℏ𝝎𝒌 𝒂𝒌†𝒂𝒌 + ℏ𝐬𝐢𝐧𝜽𝝈𝒙

𝒌

𝒈𝒌 𝒂𝒌† +𝒂𝒌

bosonicbath

tunablespin

interaction(Ohmic)

I.-C.Hoi et al., arXiv:1410.8840J.-T. Shen and S. Fan, Phys. Rev. Lett. 95, 213001 (2005).O. Astafiev et al., Science 327, 840 (2010).

𝝎flux qubit

𝝎𝐪

superconducting qubits in open transmission lines

spin-boson Hamiltonian in circuit QED

Prospects

VI. ExperimentalTechniques


• drawbacks of superconducting systems:

resonator atom

𝝎𝒓 𝝎𝐠𝐞

𝝎𝒓

𝟐𝝅≃

𝝎𝐠𝐞

𝟐𝝅≃ few GHz

1 GHz ↔ 50 mK

ℏ𝝎𝒓 ≃ 10-24 J

ultra-low temperatures

ultra-sensitive µ-wave experiments

challenges

nano-fabrication

Low energy scales




ultra-low Ttechniques

microwavetechnology

nano-technology

key physical and technological ingredients



mK

tech

no

logy

fo

r sc

qu

antu

m c

ircu

its


1 GHz ≃ 50 mK

ħωr ≃ 10-24 J

Optical “table” @ mK temperature

mK

tech

no

logy

fo

r sc

qu

antu

m c

ircu

its


Optical “table” @ mK temperature

1 GHz ≃ 50 mK

ħωr ≃ 10-24 J

mK

tech

no

logy

fo

r sc

qu

antu

m c

ircu

its

sup

ple

men

tary

mat

eria

l



WMI-made microwave-ready dilution refrigerators

55

cm

30

cm

40 cm

mK technology for sc quantum circuits @ WMI



µ-wave technology @ mK temperatures



µ-w

ave

te

chn

olo

gy @

mK

tem

pe

ratu

res


• materials for superconducting circuits

• Typical superconductors Nb

type-II superconductor, 𝑇c ≈ 9K fast measurements at 4K possible shadow evaporation for nanoscale junction not possible (without hard mask)

Al type-I superconductor, 𝑇c ≈ 1.2 K measurements require millikelvin temperatures shadow evaporation possible (stable oxide)

• Normal metals mainly Au (no natural oxide layer) for on-chip resistors and passivation layers

• Dielectric substrates silicon, sapphire contribute to dielectric losses (𝑇1)



• micro- and nanopatterning of superconducting circuits

• Lithography define pattern optical lithography (UV) electron beam lithography (EBL)

• Thin-film deposition deposit materials DC sputtering (metals, e.g. Nb) RF sputtering (insulators) electron beam evaporation (metals, e.g. Al) epitaxial growth (molecular beam epitaxy, higher substrate temperatures)

• Processing positive pattern Lift-off

deposit material only where you want it negative pattern Etching

deposit material everywhere remove what you don‘t want



WMI EBL system

• Nanobeam nb5• up to 100 kV acceleration voltage

strongly reduced „natural“ undercut frombackscattered electrons

undercut now deliberately designedduring the process

• large beam current fast• few nm resolution (in practice mostly resist

limited)• heavily automated (operated „from the office“)

advantage: fewer user-dependentparameers in the process

better reproducibility


electron beam lithography (EBL)


resist mask first layer second layer tunnel junction

ghost

structures

small

junctions

large

junction

J. Schuler, PhD ThesisTU Munich (2005)

key fabrication technique for Al/AlOx/Al Josephson junctions with submicron lateral dimensions


qubit fabrication by shadow evaporation technique



qubit fabrication by shadow evaporation technique


substrate

resist layer 1

resist layer 2

Co

urt

esy

of

J. S

chu

ler



evaporation of thefirst Al layer

Co

urt

esy

of

J. S

chu

ler



oxidation of thefirst Al layer

Co

urt

esy

of

J. S

chu

ler



evaporation ofthe second Al layer

Co

urt

esy

of

J. S

chu

ler



After resistremoval (liftoff)

Co

urt

esy

of

J. S

chu

ler



500 μm

ground

center

20 μm

1 μm

Nb Si3N4 Si

Fredrik Hocke et al., New J. Phys. 14 , 123037 (2012)

Xiaoqing Zhou, et al., Nature Physics 9 , 179 (2013)

Matthias Perpeintner, et al., APL 105, 123106 (2014)

Fredrik Hocke, et al., APL 105, 133102 (2014)

M. Abdi et al., PRL 114, 173602 (2015)


CPW resonator coupled to nanomechanical beam


CPW resonator with inductively coupled beam


Part III

some recent results of WMI research in QST


contents









microwaves

IX. Summary

VII:Qubits:

Control & Decoherence


• superconducting qubits strongly couple to electromagnetic fields decoherence due to environmental fluctuations

• place qubit in cavity: „Purcell filtering“

𝝎𝒓 𝝎𝝎𝒒

large detuning𝛿 = 𝜔𝑟 − 𝜔𝑞 ≫ 𝑔

strongly reduced „photon DOS“ @ 𝝎𝒒

𝜔𝑞

(GH

z)

𝜆 = 𝛿Φ/Φ0

𝜹𝝎𝒒

𝜹𝝎𝒒

• operate qubit @ sweet spot: 1st order coupling to noise vanishes

𝜹𝝎𝒒 =𝝏𝝎𝒒

𝝏𝝀𝜹𝝀 +

𝟏

𝟐

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐𝜹𝝀𝟐 +⋯

1st ordercoupling

2nd ordercoupling

VII.Qubits – Control & Decoherence



𝜽(𝒕)

𝟐𝐠

𝛿𝜑 𝑡 ∝ 𝛿𝜔𝑞𝑡



example: effect of thermal noise fields on qubit decoherence

experiment: transmon qubit in 𝝀/𝟒 resonator

How to generate well-controlled thermal microwave fields ?


J. Goetz PhD Thesis, TU Munich (2017)

Tant

Tx

Tant

Tx

𝒏(𝝎, 𝑻) =𝟏

𝐞𝐱𝐩൫ Τℏ𝝎 )𝒌𝐁𝑻 − 𝟏

𝑺𝟏/𝟒(𝟏)

𝝎,𝑻 =𝟏

𝟒⋅ 𝟒ℏ𝝎 𝒏 +

𝟏

𝟐=𝟏

𝟒⋅ 𝟐ℏ𝝎𝐜𝐨𝐭𝐡

ℏ𝝎

𝟐𝒌𝐁𝑻

thermal noise source:

50 mK ≤ 𝑻 ≤ 1.5 K



175

𝒏(𝝎, 𝑻) =𝟏


𝑺𝟏/𝟒(𝟏)

𝝎, 𝑻 =𝟏

𝟒⋅ 𝟒ℏ𝝎 𝒏 +

𝟏

𝟐=𝟐ℏ𝝎

𝟒𝐜𝐨𝐭𝐡

ℏ𝝎

𝟐𝒌𝐁𝑻

study qubit decay and dephasing induced by noise photons

𝜹𝝎𝒒


thermal noise source: 50 mK ≤ 𝑻 ≤ 1.5 K


𝝏𝝀𝜹𝝀 +

𝟏

𝟐

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐𝜹𝝀𝟐 +⋯

1st ordercoupling

dominates


study qubit decay and dephasing induced by noise photons


@ flux sweet spot:

𝑺𝟏/𝟒𝟐

𝝎 = 𝝎ℏ𝟐𝝎𝟐+𝟒𝝅𝟐𝒌𝑩

𝟐𝑻𝟐

𝟏𝟐𝝅𝐜𝐨𝐭𝐡

ℏ𝝎

𝟐𝒌𝐁𝑻

𝛿𝜔𝑞

flux sweet spot



𝝏𝝀𝜹𝝀 +

𝟏

𝟐

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐𝜹𝝀𝟐 +⋯

2nd ordercoupling

dominates


experimental setup

Tant

Tx


J. Goetz, PhD Thesis, TU Munich (2017)


𝑇2 ≃ 2𝑇1transmon qubit is

T1-limited

Ramsey decay 2 MHz

spin-echo decay 2 MHz

relaxation 4 MHz

J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)



Ramsey decay 2 MHz

spin-echo decay 2 MHz

relaxation 4 MHz

Tant

Tx





𝝏𝝀× 𝜹𝝀 =

𝟏

ℏ

𝝏𝑯𝒒 𝝀

𝝏𝝀×𝑴𝒂

𝚽𝟎

𝜹𝑽

𝒊𝝎𝑳𝒂

additional dephasing:

𝛾𝜑,𝑎(1)

=𝝏𝝎𝒒

𝝏𝝀

𝑀𝑎

Φ0

2𝑆 𝜔

2𝑍0≈

𝝏𝝎𝒒

𝝏𝝀

𝑀𝑎

Φ0

2𝑘𝐵𝑻

𝑍0

𝜸𝟏,𝒂

Tant

Tx

𝜹𝝎𝒒

𝑺𝟏/𝟒(𝟏)

𝝎 → 𝟎 =𝟐ℏ𝝎


ℏ𝝎

𝟐𝒌𝐁𝑻∝ 𝑻

VII.Qubits – Control & Decoherence first order coupling

𝛿𝜆 = 𝛿Φ/Φ0



additional dephasing:

𝛾𝜑,𝑎(1)

=𝝏𝝎𝒒

𝝏𝝀

𝑀𝑎

Φ0

2𝑆 𝜔

2𝑍0≈

𝝏𝝎𝒒

𝝏𝝀

𝑀𝑎

Φ0

2𝑘𝐵𝑻

𝑍0

𝜸𝟏,𝒂

Tant

Tx

𝑺𝟏/𝟒(𝟏)

𝝎 → 𝟎 =𝟐ℏ𝝎


ℏ𝝎

𝟐𝒌𝐁𝑻∝ 𝑻



𝝏𝝀× 𝜹𝝀 =

𝟏

ℏ

𝝏𝑯𝒒 𝝀

𝝏𝝀×𝑴𝒂

𝚽𝟎

𝜹𝑽

𝒊𝝎𝑳𝒂


first order coupling


Tant

Tx

(𝛿𝜆2 = Τ𝛿Φ2 Φ02)𝜹𝝎𝒒 =

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐× Τ𝜹𝝀𝟐 𝟐 =

𝟏

ℏ

𝝏𝟐𝑯𝒒(𝝀)

𝝏𝝀𝟐×𝜹𝚽𝟐

𝟐𝚽𝟎𝟐

additional dephasing: 𝛾𝜑,𝑎(2)

=1

2

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐𝑀𝑎

2

𝐿𝑙

2𝑆(2) 𝜔

ℏ𝜔𝑞,02𝑍02~𝑻𝟑

𝜸𝝋,𝒂(𝟐)

𝜹𝝎𝒒

𝑺𝟏/𝟒𝟐

𝝎 → 𝟎 = 𝝎ℏ𝟐𝝎𝟐 + 𝟒𝝅𝟐𝒌𝑩

𝟐𝑻𝟐

𝟏𝟐𝝅𝐜𝐨𝐭𝐡

ℏ𝝎

𝟐𝒌𝐁𝑻∝ 𝑻𝟑



𝛿𝜆 = 𝛿Φ/Φ0

2nd order coupling (@ qubit sweet spot)


Tant

Tx

𝜸𝝋,𝒂(𝟐)


(𝛿𝜆2 = Τ𝛿Φ2 Φ02)𝜹𝝎𝒒 =

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐× Τ𝜹𝝀𝟐 𝟐 =

𝟏

ℏ

𝝏𝟐𝑯𝒒(𝝀)

𝝏𝝀𝟐×𝜹𝚽𝟐

𝟐𝚽𝟎𝟐

additional dephasing: 𝛾𝜑,𝑎(2)

=1

2

𝝏𝟐𝝎𝒒

𝝏𝝀𝟐𝑀𝑎

2

𝐿𝑙

2𝑆(2) 𝜔

ℏ𝜔𝑞,02𝑍02~𝑻𝟑


2nd order coupling (@ qubit sweet spot)

sup

ple

men

tary

mat

eria

l


cavity filter

Tant

Tx

𝜸𝟏𝒅 = 𝜸𝟏 𝟏 −

𝝌

𝜹𝟐𝒏𝒓 + 𝟏 + 𝜸𝑷 𝟐𝒏𝒒 + 𝟏 +

𝟒 𝝌

𝜹

𝑺 𝜹

ℏ(𝟐𝒏𝒓 + 𝟏)

M. Boissonneault et al.,Phys. Rev. A 79, 013819 (2009)

sideband decaycavity filterdispersive coupling

𝜸𝟏𝒅

Purcell decay rate

dispersiveshift

𝛾1: intrinsic decay rate


sup

ple

men

tary

mat

eria

l


Tant

Tx

𝜹𝜸𝟏,𝒓 𝒏𝒓 = 𝟐𝒏𝒓𝝌

𝜹

𝟒𝑺 𝜹

ℏ− 𝜸𝟏

𝜸𝟏𝒅

M. Boissonneault et al.,Phys. Rev. A 79, 013819 (2009)

selective drive at 𝜔𝑟 (coherent field/narrow band shot noise):

reduction the relaxation rate for: Τ4𝑆 𝛿 ℏ < 𝛾1 𝛾1: intrinsic decay rate

VII.Qubits – Control & Decoherence cavity filter


Tant

Tx

𝜸𝟏,𝒂

low frequency variations of qubit relaxationmechanism: TLSthermal field TLS fluctuation rate low-frequency fluctuations of

noise power spectral density 𝑆(𝜔𝑞, 𝑇)




Tant

Tx

𝜸𝟏,𝒂




Evaluate Photon Statistics with Qubit

Bose-Einstein statisticsfor thermal field at 𝝎:

𝒏 𝑻 =𝟏

𝒆ℏ𝝎𝒌𝑩𝑻 − 𝟏

thermal field classical limit Poissonian

Var(𝑛) 𝑛2 + 𝑛 𝑛2 𝑛

control parameter

relevant regime: 0.05 ≤ n ≤ 1 100 mK ≤ T ≤ 1 K (@ 6 GHz)

J. Goetz et al., PRL 118, 103602 (2017)


sup

ple

men

tary

mat

eria

l


Field Correlation Measurements

Dual-path state reconstruction signal moments up to 4th order

w/o JPA confirms 𝒏𝟐 + 𝒏 dependence large scatter of datause Josephson parametric amplifier (JPA)

E.P. Menzel et al., PRL 105, 100401 (2010)K. Fedorov et al., PRL 117, 020502 (2016)

𝑔 2 0 = ො𝑎 ො𝑎 2𝑔(2) 0 = Var 𝑛 − 𝑛 + 𝑛2

𝑔 2 0 = 2𝑛2 for thermal fields


J. Goetz et al., PRL 118, 103602 (2017)

sup

ple

men

tary

mat

eria

l


w/o JPA

with JPAs

Dual-path state reconstruction signal moments up to 4th order

JPA adds noise(thermal + non-thermal contribution)

E.P. Menzel et al., PRL 105, 100401 (2010)K. Fedorov et al., PRL 117, 020502 (2016)


J. Goetz et al., PRL 118, 103602 (2017)

• Field Correlation Measurements


193

dispersive regime: 𝒈

𝜹≪ 𝟏,

𝒈

𝟐𝝅≃ 𝟔𝟕𝐌𝐇𝐳

resonator:𝝎𝒓

𝟐𝝅≃ 𝟔. 𝟎𝟕 𝐆𝐇𝐳

qubit: 𝝎𝒒

𝟐𝝅≃ 𝟔. 𝟗𝟐 𝐆𝐇𝐳

dispersive shift:𝝌

𝟐𝝅≃ 𝟑. 𝟏𝟓 𝐌𝐇𝐳

use qubit to analyze photon statistics



𝒏(𝝎, 𝑻) =𝟏


𝑺𝟏/𝟒 𝝎,𝑻 =𝟏

𝟒⋅ 𝟒ℏ𝝎 𝒏 +

𝟏

𝟐=𝟏

𝟒⋅ 𝟐ℏ𝝎𝐜𝐨𝐭𝐡

ℏ𝝎

𝟐𝒌𝐁𝑻



derive photon statistics from qubit decay

sup

ple

men

tary

mat

eria

l


calibration of photon number

use photon number dependent ac-Stark shift

Tant

Tx

𝛿𝜔𝑞

approx.3 photons @ 1.5 K


J. Goetz et al., PRL 118, 103602 (2017)


• Photon Statistics from Dephasing

Tant

Tx

𝑪 𝝉 ∝ 𝐕𝐚𝐫(𝒏)

𝜸𝝋𝒏 ∝ 𝐕𝐚𝐫(𝒏𝒓)

𝝌 𝒏𝒓⟨ෝ𝝈𝒛⟩

thermal field

classicallimit

Poissonian

𝐕𝐚𝐫(𝐧) 𝒏𝟐 + 𝐧 𝒏𝟐 𝐧

𝜸𝝋𝒏𝐭𝐡 𝒏𝒓 = 𝜿𝒙𝜽𝟎

𝟐 𝒏𝒓𝟐 + 𝒏𝒓

Ramsey

𝜃0 = tan−12𝜒

𝜅𝑥


J. Goetz et al., PRL 118, 103602 (2017)


Tant

Tx

𝑪 𝝉 ∝ 𝐕𝐚𝐫(𝒏)

𝜸𝝋𝒏 ∝ 𝐕𝐚𝐫(𝒏𝒓)

𝝌𝒏𝒓⟨ෝ𝝈𝒛⟩

𝜸𝝋𝒏𝐭𝐡 𝒏𝒓 = 𝜿𝒙𝜽𝟎

𝟐 𝒏𝒓𝟐 + 𝒏𝒓

𝜸𝝋𝒏𝐜𝐨𝐡 𝒏𝒓 = 𝟐𝜿𝒙𝜽𝟎

𝟐 𝒏𝒓𝜸𝝋𝒏𝐬𝐡𝐨𝐭 𝒏𝒓 = 𝜿𝒙𝜽𝟎

𝟐 𝒏𝒓

thermal field

classicallimit

Poissonian

𝐕𝐚𝐫(𝐧) 𝒏𝟐 + 𝐧 𝒏𝟐 𝐧


J. Goetz et al., PRL 118, 103602 (2017)

VIII.CV Propagating

Quantum Microwaves

theory support by U. Las Heras, M. Sanz, E. Solano


199199

AluminumΤ𝛥 ℎ ≃ 50 GHz

superconductingquantumcircuits

emit

propagating quantummicrowaves

coherence?

VIII.CV Propagating Microwaves


discrete variables (DV)

| 𝟎

| 𝟏

| 𝟎 + | 𝟏

𝟐

0

1

classical bit quantum bit (Qubit)

𝑨𝒄𝒐𝒔 𝝎𝒕 + 𝝋 = 𝑷𝒄𝒐𝒔 𝝎𝒕 + 𝑸𝒔𝒊𝒏 𝝎𝒕

continuous variables (CV)

𝑷

𝑸

𝒕

analogy to mechanics𝑸, 𝑷 ⇔ ෝ𝒙, ෝ𝒑



𝑃𝑄

𝑷𝑸

Wignerfunction

𝑷, 𝑸 = 𝒊 ⇔ 𝚫𝑷 𝜟𝑸 ≥𝟏

𝟒

𝚫𝑷 𝟐 ≥𝟏

𝟒𝐚𝐧𝐝 𝜟𝑸 𝟐 ≥

𝟏

𝟒

𝚫𝑷 𝟐 ≤𝟏

𝟒𝐗𝐎𝐑 𝜟𝑸 𝟐 ≤

𝟏

𝟒

non-classical



R. Di Candia et al., EPJ Quantum Technology 2, 25 (2015)

quantum communication quantum illumination

Phys. Rev. Lett. 101, 250501 (2008)

digital quantum computinganalog quantum computingS. L. Braunstein and P. van Loock,Rev. Mod. Phys. 77, 513 (2005)



flux-tunable inductance𝐿SQUID 𝛷dc +Φrf

for parametric drive

flux-driven Jospehsonparametric amplifier(JPA)

𝑓0 2𝑓0



coil current (arb. units)

freq

ue

ncy

(GH

z)

reflectionphase

-180°

180°

0°

5.2

5.4

5.6

5.0

-300 300-150 1500

pump power (dBm)

gain

(dB

)

squ

eez

ing

(dB

)

0

2

4

6

8

5

10

15

0

20

-40 -20 0

S. Pogorzalek et al., arXiv:1609.09041.K. G. Fedorov et al.,PRL 117, 020502 (2016).

9.41 photons

6.40 dB

p

-10 0 10

q

-10

0

10

0 0.1 0.2

9.41 photons

6.40 dB

-1 0 1

-1

0

1

(a)

Tnoise ≈ 300 mK


squeezing level:


• The dual-path tomography scheme

microwave photons low photon energies linear amplification & signal recovery

E. P. Menzel et al., Phys. Rev. Lett. 105, 100401 (2010).L. Zhong et al., New. J. Phys. 15, 125013 (2013).R. Di Candia et al., New J. Phys. 16, 015001 (2014) .

𝑃1, 𝑄1

𝑃2, 𝑄2

Signal+ Noise2

Signal+ Noise1

Signal

beamsplitter

correlations

⟨𝑷𝟏𝒌𝑷𝟐

𝒍𝑸𝟏𝒎𝑸𝟐

𝒏⟩

all signal anddetector noisemoments



L. Zhong et al., New. J. Phys. 15, 125013 (2013).E. P. Menzel et al., Phys. Rev. Lett. 105, 250502 (2010).

vacuum & coherent states

squeezed vacuum

squeezed thermal states

squeezedcoherentstates



• path entanglement

E. P. Menzel et al., Phys. Rev. Lett. 109, 250502 (2012).

𝑃1, 𝑄1

𝑃2, 𝑄2

Signal+ Noise2

Signal+ Noise1

beamsplitter

pathentangle-

ment

maximum negativity 0.55 ↔ 3.2 dB TMS



• displacement of propagating quantum microwaves

directional coupler acts as displacer

K. G. Fedorov et al., PRL 117, 020502 (2016).

𝑃1, 𝑄1

𝑃2, 𝑄2

Signal + Noise2

Signal + Noise1

beamsplitter

pathentangle-

ment

displacement is CV quantum gate required in feedforward schemes



165.74 photons

6.62 dB

p

-10 0 10

-10

0

10

0 0.1 0.2

165.74 photons

6.62 dB

-9 -8

8

99.41 photons

6.40 dB

p

-10 0 10

-10

0

10

0 0.1 0.2

9.41 photons

6.40 dB

-1 0 1

-1

0

1

(c)

166.23 photons

6.82 dB

p

-10 0 10

q

-10

0

10

0 0.1 0.2

166.23 photons

6.82 dB

8 9

8

9

high degree of control overangle and magnitude

hundreds of displacementphotons referred to400 kHz bandwidth

squeezing and negativitynearly unchanged




-155 -150 -145 -140 -135 -130 -1250.0

0.5

1.0

-155 -150 -145 -140 -135 -130 -1250.0

0.5

1.0

-155 -150 -145 -140 -135 -130 -125

8

6

4

2

0

displacement power Pdisp

(dBm)

sq

ue

ezin

g le

ve

l S (

dB

)

0

40

80

120

160

ph

oto

n n

um

be

r

°

(b)

°

ne

ga

tivity N

reference state method

dual-path method

(a)

reference state method

dual-path methodne

ga

tivity N

displacement power Pdisp

(dBm)

direct experimental evidenceconfirms:

strong displacement doesnot destroy squeezing

both squeezing anddisplacement contribute tothe photon number

these contributions arequalitatively different (resultin path entanglement ornot)

contributions areindependent


VIII. CV Propagating Microwaves


Path Entanglement of TMS States

JPA 2

Self-correlations (local) Cross-correlations (non-local)

JPA 1

Entanglingbeam splitter

Input Output

No local correlations (each path looks thermal)

Only non-local correlations

Resource state for quantum communication & sensingK. G. Fedorov et al., Scientific Reports 8, 6416 (2018)

VIII. CV Propagating Microwaves


217

Outlook

projective measurement and feedforward for Remote-State Preparation (RSP) and teleportation

quantum microwave communication, illumination, …

TheoryU. Las Heras, M. Sanz, E. Solano

• toolbox of cv propagating quantummicrowaves for quantum communication, quantum computing, and quantumillumination

tomography established

single-mode squeezing ≃ 8 dB

finite-time correlations 𝑔 2 𝜏

displacement gate

balanced two-mode squeezing

“finite-time” entanglement


Summary

IX.Summary


The future looks bright !

IX. Summary


The WMI team

Thank you !

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