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http://www.wmi.badw.de
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
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
24-28.07.2017/RG - 3www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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:
24-28.07.2017/RG - 4www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 5www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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)
24-28.07.2017/RG - 6www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
single/few
electron, spin, fluxon, photon
devices
near future far future
quantifiable,but not quantum
single electron transistor
PTB
multi
electron, spin, fluxon, photon
devices
today
classicaldescription
65 nm process 2005
Intel
• quantumconfinement
• tunneling• …
... solid state circuits go quantum
24-28.07.2017/RG - 7www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
single/few
electron, spin, fluxon, photon
devices
quantum
electron, spin, fluxon, photon
devices
near future far future
quantifiable,but not quantum
quantumdescription
superconducting qubitsingle electron transistor
PTB
multi
electron, spin, fluxon, photon
devices
today
classicaldescription
65 nm process 2005
Intel WMI 20072 µm
... solid state circuits go quantum
24-28.07.2017/RG - 8www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
multi
electron, spin, fluxon, photon
devices
single/few
electron, spin, fluxon, photon
devices
quantum
electron, spin, fluxon, photon
devices
today near future far future
quantifiable,but not quantum
classicaldescription
quantumdescription
65 nm process 2005 superconducting qubitsingle electron transistor
PTBIntel
• superposition of states• entanglement• quantized em-fields
WMI 20072 µm
... solid state circuits go quantum
quantum1.0 quantum2.0
24-28.07.2017/RG - 9www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 10www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
Superconducting Quantum Circuits
© WMI
24-28.07.2017/RG - 11www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
Vesuvius 3:512 superconducting qubits,operating temperature: 30 mK
Quantum computing @ mK temperature
24-28.07.2017/RG - 12www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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/
24-28.07.2017/RG - 13www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 16www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
I. Superconductivity in a nutshell
©WMI
24-28.07.2017/RG - 17www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © 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
I. Superconductivity in a nutshell
(Fritz London, 1948)
24-28.07.2017/RG - 18www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
I. Superconductivity in a nutshell
• 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↓†
24-28.07.2017/RG - 19www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 20www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 21www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
Supplement: Madelung Transformation
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 22www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
• equation for real part:
energy-phase relation (term of order ²ns is usually neglected)
∆Supplement: Madelung Transformation
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 23www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
• interpretation of energy-phase relation
corresponds to action
in the quasi-classical limes the energy-phase-relation becomesthe Hamilton-Jacobi equation
Supplement: Madelung Transformation
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 24www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
• equation for imaginary part:
continuity equation for probabilitydensityandprobability current density
conservation law for probability density
Supplement: Madelung Transformation
24-28.07.2017/RG - 25www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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)
ℏ𝝏𝜽 𝐫, 𝐭
𝝏𝒕= −
𝟏
𝟐𝒏𝒔𝚲𝐉𝐬
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
.
𝚲𝐉𝐬 ⋅ 𝒅ℓ + න
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
24-28.07.2017/RG - 26www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
I. Superconductivity in a nutshell derive Josephson equations
ℏ𝝏𝜽 𝐫, 𝐭
𝝏𝒕= −
𝟏
𝟐𝒏𝒔𝚲𝐉𝐬
𝟐 + 𝒒𝒔𝝓+ 𝝁
replace gauge invariant phase gradient by phase difference
𝐉𝐬 𝐫, 𝐭 = 𝒒𝐬𝒏𝒔ℏ
𝒎𝒔𝛁𝜽 −
𝒒𝒔𝒎𝒔
𝐀
𝐉𝐬 𝐫, 𝐭 =𝒒𝐬𝒏𝒔ℏ
𝒎𝒔𝛁𝜽 −
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
24-28.07.2017/RG - 27www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
= 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
𝟐𝒏𝒔𝚲𝐉𝐬
𝟐 + 𝒒𝒔𝝓+ 𝝁
𝝏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
24-28.07.2017/RG - 28www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
I. Superconductivity in a nutshell
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
𝒎𝒔𝛁𝜽 −
𝒒𝒔
ℏ𝐀
• energy-phase relation ℏ𝝏𝜽 𝐫,𝐭
𝝏𝒕= −
𝟏
𝟐𝒏𝒔𝚲𝐉𝐬
𝟐 + 𝒒𝒔𝝓+ 𝝁
• 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
24-28.07.2017/RG - 30www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson 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
24-28.07.2017/RG - 31www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
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
24-28.07.2017/RG - 32www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
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
24-28.07.2017/RG - 33www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
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
24-28.07.2017/RG - 34www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
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
24-28.07.2017/RG - 35www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. 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
24-28.07.2017/RG - 36www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
II. Josephson Junctions
𝑬𝑱𝟎 ∝ 𝑨
𝑬𝑪 ∝ 𝟏/𝑨
𝑬𝑪 > 𝑬𝑱𝟎
𝑬𝑪 < 𝑬𝑱𝟎
24-28.07.2017/RG - 37www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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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24-28.07.2017/RG - 38www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 39www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
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
24-28.07.2017/RG - 40www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
• 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
24-28.07.2017/RG - 41www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
• 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
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II. Josephson Junctions
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
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24-28.07.2017/RG - 43www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
• 1D problem numerical solution straightforward variational approach for approximate ground state
𝑬𝑪
𝑬𝐉𝟎= 𝟎. 𝟎𝟐𝟓
𝑬𝐦𝐢𝐧 = 𝟎. 𝟏 𝑬𝐉𝟎
• The phase regime: ℏ𝝎𝐩 ≪ 𝑬𝐉𝟎 , 𝑬𝑪 ≪ 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:
ℏ𝝎𝒑 = 𝟐𝑬𝑪𝑬𝑱𝟎
sup
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24-28.07.2017/RG - 44www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
𝑬𝑪
𝑬𝐉𝟎= 𝟎. 𝟎𝟐𝟓
𝑬𝐦𝐢𝐧 = 𝟎. 𝟏 𝑬𝐉𝟎
• The phase regime: ℏ𝝎𝐩 ≪ 𝑬𝐉𝟎 , 𝑬𝑪 ≪ 𝑬𝐉𝟎
𝑬𝐦𝐢𝐧 = 𝑬𝑱𝟎 𝟏 − 𝟏 −𝑬𝑪𝟐𝑬𝑱𝟎
𝟐
= 𝑬𝑱𝟎 𝟏 − 𝟏 −ℏ𝝎𝒑
𝟐𝑬𝑱𝟎
𝟐
ℏ𝝎𝒑 = 𝟐𝑬𝑪𝑬𝑱𝟎
tunneling coupling ∝ exp −2𝐸J0−𝐸
ℏ𝜔p very small since ℏ𝜔p ≪ 𝐸J0
bandwidth of lowest bands is exponentially small negligible dispersion of 𝐸 𝑄
24-28.07.2017/RG - 45www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
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
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24-28.07.2017/RG - 46www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
• 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
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24-28.07.2017/RG - 47www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
II. Josephson Junctions
• 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
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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
II. Josephson Junctions
III. SuperconductingQuantum Circuits
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III. Superconducting Quantum Circuits
key physical ingredients
𝑱𝒔 = 𝑱𝒄 𝐬𝐢𝐧𝝋 ,𝒅𝝋
𝒅𝒕=𝟐𝒆𝑽
ℏර
𝑪
.
𝚲𝐉𝐬 ⋅ 𝒅ℓ + න
𝑭
.
𝑩 ⋅ ෝ𝒏 𝑑𝐹 = 𝒏𝚽𝟎
𝚿 𝐫, 𝐭 = 𝚿𝟎𝒆𝒊𝜽 𝒓,𝒕
𝚿 𝐫, 𝐭 𝟐 = 𝒏𝒔 𝒓, 𝒕
𝝓,𝑸 = 𝒊ℏ
24-28.07.2017/RG - 51www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
III. Superconducting Quantum Circuits
24-28.07.2017/RG - 52www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
III. Superconducting Quantum Circuits
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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
III. Superconducting Quantum Circuits
• advantages of superconducting systems
1. Macroscopic quantum nature of superconducting ground state 2. Energy gap in excitation spectrum
24-28.07.2017/RG - 54www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
• 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
III. Superconducting Quantum Circuits
24-28.07.2017/RG - 55www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
III. Superconducting Quantum Circuits
24-28.07.2017/RG - 56www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
fabricate tailor-made quantum circuits
2 µm
flux qubit(Al)
transmon qubit(Al)
3D coplanarwaveguideresonator(Al)
coplanar waveguideresonator (Nb)
III. Superconducting Quantum Circuits
3. Established fabrication technology: thin film & nanotechnology
4. Superb design flexibility, tunability and scalability
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III. Superconducting Quantum Circuits
24-28.07.2017/RG - 58www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
quantum circuit with 8 resonators and 3 qubits coupled to each rersonator
III. Superconducting Quantum Circuits
24-28.07.2017/RG - 59www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
interferometer
nano-electromechanical circuit
Si3N4 nanobeam coupled to CPW resonator
III. Superconducting Quantum Circuits
24-28.07.2017/RG - 60www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
III. Superconducting Quantum Circuits
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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
III. Superconducting Quantum Circuits
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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
III. Superconducting Quantum Circuits
24-28.07.2017/RG - 63www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
III. Superconducting Quantum Circuits
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
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III. Superconducting Quantum Circuits
• 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
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III. Superconducting Quantum Circuits
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“, …
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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
III. Superconducting Quantum Circuits
Part II
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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
IV. Superconducting
Resonators & Qubits
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IV. SC Resonators & Qubits
superconductingquantum circuits
resonators qubits
couplers interferometers
switches JPAs
hybrid systems
resonators
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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
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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
IV. SC Resonators & Qubits
𝑯𝐋𝐂 = 𝑬𝐤𝐢𝐧 + 𝑬𝐩𝐨𝐭 =𝑸𝟐
𝟐𝑪+𝜱𝟐
𝟐𝑳=𝑸𝟐
𝟐𝑪+𝟏
𝟐𝑪𝝎𝒓
𝟐 𝜱𝟐
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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
Τ𝑬𝐩𝐨𝐭ℏ
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
resonance frequency 𝜔r ↔ 𝜔r = Τ1 𝐿𝐶
discrete basis (Fock states)
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L C
when applied to a Fock state, ෝ𝒂 annihilates a photon inside the oscillator
𝝎𝐫
Τ𝑬𝐩𝐨𝐭ℏ
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
IV. SC Resonators & Qubitsmomentum Ƹ𝑝 ↔ charge 𝑄position ො𝑥 ↔ flux 𝛷mass 𝑚 ↔ capacitance C
resonance frequency 𝜔r ↔ 𝜔r = Τ1 𝐿𝐶
𝑯𝑳𝑪 = ℏ𝝎𝐫 ෝ𝒂†ෝ𝒂 +𝟏
𝟐
ෝ𝒂 ≡𝝎𝐫𝑪𝜱+𝒊𝑸
𝟐𝝎𝐫𝑪ℏannihilation operator
ෝ𝒂† ≡𝝎𝐫𝑪𝜱−𝒊𝑸
𝟐𝝎𝐫𝑪ℏcreation operator
𝝎𝐫/𝟐
annihilation & creation operator
sup
ple
men
tary
mat
eria
l
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L C
Τ𝑬𝐩𝐨𝐭ℏ
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
IV. SC Resonators & Qubitsmomentum Ƹ𝑝 ↔ charge 𝑄position ො𝑥 ↔ flux 𝛷mass 𝑚 ↔ capacitance C
resonance frequency 𝜔r ↔ 𝜔r = Τ1 𝐿𝐶
𝑯𝑳𝑪 = ℏ𝝎𝐫 ෝ𝒂†ෝ𝒂 +𝟏
𝟐
ෝ𝒂 ≡𝝎𝐫𝑪𝜱+𝒊𝑸
𝟐𝝎𝐫𝑪ℏannihilation operator
ෝ𝒂† ≡𝝎𝐫𝑪𝜱−𝒊𝑸
𝟐𝝎𝐫𝑪ℏcreation operator
𝝎𝐫
𝝎𝐫/𝟐
coherent states
sup
ple
men
tary
mat
eria
l
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L C
Τ𝑬𝐩𝐨𝐭ℏ
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
IV. SC Resonators & Qubitsmomentum Ƹ𝑝 ↔ charge 𝑄position ො𝑥 ↔ flux 𝛷mass 𝑚 ↔ capacitance C
resonance frequency 𝜔r ↔ 𝜔r = Τ1 𝐿𝐶
𝝎𝐫
𝝎𝐫/𝟐
practical importance of coherent states
𝑯𝑳𝑪 = ℏ𝝎𝐫 ෝ𝒂†ෝ𝒂 +𝟏
𝟐
sup
ple
men
tary
mat
eria
l
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L C
Τ𝑬𝐩𝐨𝐭ℏ
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
resonance frequency 𝜔r ↔ 𝜔r = Τ1 𝐿𝐶
𝝎𝐫
𝝎𝐫/𝟐
ො𝑎 𝛼 = 𝛼|𝛼⟩, 𝛼 ∈ ℂ
practical importance of coherent states
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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
IV. SC Resonators & Qubits
multimode Hamiltonian
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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!
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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)
IV. SC Resonators & Qubits
𝐻𝐿𝐶 = ℏ𝜔r ො𝑎† ො𝑎 +1
2 characterization of 𝝀/𝟐-resonator (two ports)
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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
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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)
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IV. SC Resonators & Qubits
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)
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• 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
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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
superconductingquantum circuits
indirect use
IV. SC Resonators & Qubits
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IV. SC Resonators & Qubits
superconductingquantum circuits
resonators qubits
couplers interferometers
switches JPAs
hybrid systems
qubits
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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
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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
IV. SC Resonators & QubitsIntro
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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
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
IV. SC Resonators & QubitsIntro
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ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎
ෝ𝝈𝒚 ≡𝟎 −𝒊𝒊 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“)
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
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!
IV. SC Resonators & QubitsIntro
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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
IV. SC Resonators & QubitsIntro
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ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎
ෝ𝝈𝒚 ≡𝟎 −𝒊𝒊 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? ?
IV. SC Resonators & QubitsIntro
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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
IV. SC Resonators & QubitsIntro
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
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• 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
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝝋(𝒕) phase coherence
𝜽 𝒕 amplitude energy, population
• ideal quantum system
completely isolated in reality, however, …
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• 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
IV. SC Resonators & QubitsIntro
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝝋(𝒕) phase coherence
𝜽 𝒕 amplitude energy, population
𝝋 𝒕 =𝑬𝒆 − 𝑬𝒈
ℏ𝒕
𝛿𝜑 =𝛿𝐸
ℏ𝑇𝜑 ≃ 2𝜋
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IV. SC Resonators & Qubits
|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 ?
IV. SC Resonators & Qubits
𝚫𝝎
ℏ𝝎𝟎𝟏
ℏ𝝎𝟏𝟐
𝝎𝟎𝟏 𝝎𝟎𝟐𝝎
𝜹𝝎𝟎𝟏 𝜹𝝎𝟎𝟐𝒕𝒐𝒑
sometimes trade-off between
anharmonicity ⇔ qubit decoherence
24-28.07.2017/RG - 101www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
IV. SC Resonators & Qubits
24-28.07.2017/RG - 102www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
phase qubit
(EJ >> EC)current biased JJ
flux qubit
(EJ > EC)fluxon boxes
charge qubit
(EJ < EC)Cooper pair boxes
I I
V
IV. SC Resonators & Qubits
engineered qubit potential
24-28.07.2017/RG - 103www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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)
IV. SC Resonators & Qubits
𝑳𝑱
𝑪
𝑳𝑱 𝜶𝑳𝑱
𝑳𝑱
𝑪
𝑳
𝑳𝑱
𝑪
𝑰
𝑵
𝑬𝑱𝟎𝑪𝐠
𝑽𝐠
𝑪𝑱
24-28.07.2017/RG - 104www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
IV. SC Resonators & Qubits
𝑳𝑱
𝑪
phase qubit
current bias 𝑰𝐱
very small
𝑨 > 𝟏 × 𝟏 𝛍𝐦𝟐
𝑬𝑱𝟎
𝑬𝑪∝ 𝑨𝟐 ≫ 𝟏𝟎𝟒
𝝋 ≈ classical
𝑵 = −𝒊𝝏
𝝏𝝋
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 105www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
IV. SC Resonators & Qubits phase qubit – first Rabi oscillations
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 106www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
parameters similarto RF SQUID qubit
better decoupling from readout electronics significantly longer decoherence preferred over current-biased version
IV. SC Resonators & Qubits
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
24-28.07.2017/RG - 107www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
24-28.07.2017/RG - 108www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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)
IV. SC Resonators & Qubits
flux bias𝚽𝐱
𝚽𝟎= 𝒇 ≃ 𝟏/𝟐
𝑳𝐥𝐨𝐨𝐩
𝜑1 𝜑2
𝛼
+𝐼p −𝐼p
𝜱𝐱
𝑳𝐥𝐨𝐨𝐩
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 109www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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 !
IV. SC Resonators & Qubits
3-JJ persistent current flux qubit
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 110www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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−
IV. SC Resonators & Qubits
3-JJ persistent current flux qubit
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 111www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
IV. SC Resonators & Qubits
3-JJ persistent current flux qubit
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 112www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
IV. SC Resonators & Qubits
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
3-JJ persistent current flux qubit
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 113www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
IV. SC Resonators & Qubits
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−
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 114www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
IV. SC Resonators & Qubits
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
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 115www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
⟨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
IV. SC Resonators & Qubits
3-JJ persistent current flux qubit
sup
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 116www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
⟨𝑰𝒑ෝ𝝈𝒛⟩
0 𝛿Φx/Φ0
0
𝐼p
−𝐼p
𝑬𝟎,𝟏
0
𝛿Φx/Φ00
Δ
𝟏
𝟐+𝑰𝒑 + −𝑰𝒑
𝟏
𝟐+𝑰𝒑 − −𝑰𝒑
IV. SC Resonators & Qubits
3-JJ persistent current flux qubit
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
ple
men
tary
mat
eria
l
24-28.07.2017/RG - 117www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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)
IV. SC Resonators & Qubits
3-JJ persistent current flux qubit: pulsed readout at the degeneracy point
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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!
charge qubit – the Cooper pair box (CPB)
charge regime 𝑬𝑪 ≳ 𝑬𝐉𝟎 charge is good quantum number
superconducting island
IV. SC Resonators & Qubits
𝑵
𝑬𝑱𝟎𝑪𝐠
𝑽𝐠
𝑪𝑱
additional term due gate voltage small
𝑵 = −𝒊𝝏
𝝏𝝋
• charge energy: 𝑬𝑪 ≡(𝟐𝒆)𝟐
𝟐 𝑪𝒈+𝑪𝑱
sup
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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
IV. SC Resonators & Qubits
𝑵
𝑬𝑱𝟎𝑪𝐠
𝑽𝐠
𝑪𝑱
additional term due gate voltage small
𝑵 = −𝒊𝝏
𝝏𝝋
• charge energy: 𝑬𝑪 ≡(𝟐𝒆)𝟐
𝟐 𝑪𝒈+𝑪𝑱
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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
IV. SC Resonators & Qubits
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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tunable JJ (dc SQUID)
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
typical prameters:𝐸𝐶/ℎ ≃ 5 GHz, 𝐸J0/ℎ ≃ 5 GHz
𝑪𝐉/𝟐
IV. SC Resonators & Qubits
charge qubit – the split Cooper pair box (CPB)
𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉𝟎 𝟏 − 𝐜𝐨𝐬 ෝ𝝋
E+
theory questions
• why is coupling exactly 𝑬𝐉𝟎?
• near 𝒏𝐠 =𝟏
𝟐, energy levels look
hyperbolic. Is this correct?
sup
ple
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tary
mat
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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
IV. SC Resonators & Qubits
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
IV. SC Resonators & Qubits
two-level-representation of the CPB
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tunable JJ (dc SQUID)
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
typical prameters:𝐸𝐶/ℎ ≃ 5 GHz, 𝐸J0/ℎ ≃ 5 GHz
𝑪𝐉/𝟐
IV. SC Resonators & Qubits
charge qubit – the split Cooper pair box (CPB)
𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉𝟎 𝟏 − 𝐜𝐨𝐬 ෝ𝝋
E+
theory questions
• why is coupling exactly 𝑬𝐉𝟎?
• near 𝒏𝐠 =𝟏
𝟐, energy levels look
hyperbolic. Is this correct?
sup
ple
men
tary
mat
eria
l
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tunable JJ (dc SQUID)
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
typical prameters:𝐸𝐶/ℎ ≃ 5 GHz, 𝐸J0/ℎ ≃ 5 GHz
𝑪𝐉/𝟐
IV. SC Resonators & Qubits
charge qubit – the split Cooper pair box (CPB)
𝑯𝐂𝐏𝐁 =𝑬𝐞𝐥𝟐ෝ𝝈𝒛 +
𝑬𝑱𝟎𝟐
ෝ𝝈𝒙, 𝑬𝐞𝐥≡ 𝑬𝑪 𝑵−𝑵𝒈𝟐
E+
theory questions
• why is coupling exactly 𝑬𝐉𝟎?
because of the two-level representation
• near 𝒏𝐠 =𝟏
𝟐, energy levels look
hyperbolic. Is this correct ?
yes, because 𝐸el 𝑉g can be linearized
for 𝑁 −𝑁g ≪ 1
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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
IV. SC Resonators & Qubits
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
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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
IV. SC Resonators & Qubits
𝑵𝒈 𝑵𝒈
𝑵
𝑬𝑱𝑪𝐠
𝑽𝐠
𝑪𝑱
𝑪𝒔superconductingisland
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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
IV. SC Resonators & Qubits
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
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• 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).
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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
IV. SC Resonators & Qubits
𝑬𝐉𝟎, 𝑪𝑱 𝑬𝐉𝟎, 𝑪𝑱
𝜶𝑬𝐉𝟎, 𝜶𝑪𝑱
+𝐼p −𝐼p
𝚽𝐱
E
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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).
IV. SC Resonators & Qubits
The C-shunted flux qubit𝑪𝐬𝐡
𝑵𝐛
𝑵𝐚
𝑵𝐜
𝑬𝐉𝟎, 𝑪𝑱 𝑬𝐉𝟎, 𝑪𝑱
𝜶𝑬𝐉𝟎, 𝜶𝑪𝑱
+𝐼p −𝐼p
𝚽𝐱
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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
IV. SC Resonators & Qubits
V. Circuit QED
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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
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Conference on Resonator QED 2017
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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
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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
sup
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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
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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
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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).
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solid-state atom
T. Niemczyk et al., Nature Phys. 6, 772 (2010)
𝒈/𝟐𝝅 ≃ 630 MHz
with superconducting quantum circuits:
GHz resonator
transmission spectrum
IV. Circuit QED – ultrastrong coupling
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𝟑
𝟐𝝀 - mode 𝝀 - mode
IV. Circuit QED – ultrastrong coupling
T. Niemczyk,PhD Thesis, TU Munich (2011)
sup
ple
men
tary
mat
eria
l
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qualitative explanationof datawithinJC-model
onlyquantitativedeviations
qualitative deviations from JC-model
evidence for ultra-strong coupling regime
T. Niemczyk et al., Nature Phys. 6, 772 (2010)
3l/2-mode
USC in superconducting cQED
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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
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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)
24-28.07.2017/RG - 146www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
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• 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
VI. Experimental Techniques
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VI. Experimental Techniques
ultra-low Ttechniques
microwavetechnology
nano-technology
key physical and technological ingredients
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VI. Experimental Techniques
mK
tech
no
logy
fo
r sc
qu
antu
m c
ircu
its
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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
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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
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VI. Experimental Techniques
WMI-made microwave-ready dilution refrigerators
55
cm
30
cm
40 cm
mK technology for sc quantum circuits @ WMI
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VI. Experimental Techniques
µ-wave technology @ mK temperatures
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VI. Experimental Techniques
µ-w
ave
te
chn
olo
gy @
mK
tem
pe
ratu
res
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• 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)
VI. Experimental Techniques
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• 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
VI. Experimental Techniques
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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
VI. Experimental Techniques
electron beam lithography (EBL)
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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
VI. Experimental Techniques
qubit fabrication by shadow evaporation technique
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VI. Experimental Techniques
qubit fabrication by shadow evaporation technique
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substrate
resist layer 1
resist layer 2
Co
urt
esy
of
J. S
chu
ler
VI. Experimental Techniques
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evaporation of thefirst Al layer
Co
urt
esy
of
J. S
chu
ler
VI. Experimental Techniques
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oxidation of thefirst Al layer
Co
urt
esy
of
J. S
chu
ler
VI. Experimental Techniques
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evaporation ofthe second Al layer
Co
urt
esy
of
J. S
chu
ler
VI. Experimental Techniques
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After resistremoval (liftoff)
Co
urt
esy
of
J. S
chu
ler
VI. Experimental Techniques
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VI. Experimental Techniques
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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)
VI. Experimental Techniques
CPW resonator coupled to nanomechanical beam
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CPW resonator with inductively coupled beam
VI. Experimental Techniques
Part III
some recent results of WMI research in QST
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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
VII:Qubits:
Control & Decoherence
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• 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
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝛿𝜑 𝑡 ∝ 𝛿𝜔𝑞𝑡
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VII.Qubits – Control & Decoherence
example: effect of thermal noise fields on qubit decoherence
experiment: transmon qubit in 𝝀/𝟒 resonator
How to generate well-controlled thermal microwave fields ?
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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
VII.Qubits – Control & Decoherence
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175
𝒏(𝝎, 𝑻) =𝟏
𝐞𝐱𝐩൫ Τℏ𝝎 )𝒌𝐁𝑻 − 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
VII.Qubits – Control & Decoherence
thermal noise source: 50 mK ≤ 𝑻 ≤ 1.5 K
𝜹𝝎𝒒 =𝝏𝝎𝒒
𝝏𝝀𝜹𝝀 +
𝟏
𝟐
𝝏𝟐𝝎𝒒
𝝏𝝀𝟐𝜹𝝀𝟐 +⋯
1st ordercoupling
dominates
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study qubit decay and dephasing induced by noise photons
VII.Qubits – Control & Decoherence
@ flux sweet spot:
𝑺𝟏/𝟒𝟐
𝝎 = 𝝎ℏ𝟐𝝎𝟐+𝟒𝝅𝟐𝒌𝑩
𝟐𝑻𝟐
𝟏𝟐𝝅𝐜𝐨𝐭𝐡
ℏ𝝎
𝟐𝒌𝐁𝑻
𝛿𝜔𝑞
flux sweet spot
thermal noise source: 50 mK ≤ 𝑻 ≤ 1.5 K
𝜹𝝎𝒒 =𝝏𝝎𝒒
𝝏𝝀𝜹𝝀 +
𝟏
𝟐
𝝏𝟐𝝎𝒒
𝝏𝝀𝟐𝜹𝝀𝟐 +⋯
2nd ordercoupling
dominates
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experimental setup
Tant
Tx
VII.Qubits – Control & Decoherence
J. Goetz, PhD Thesis, TU Munich (2017)
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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)
VII.Qubits – Control & Decoherence
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Ramsey decay 2 MHz
spin-echo decay 2 MHz
relaxation 4 MHz
Tant
Tx
VII.Qubits – Control & Decoherence
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
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𝜹𝝎𝒒 =𝝏𝝎𝒒
𝝏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
𝟒𝐜𝐨𝐭𝐡
ℏ𝝎
𝟐𝒌𝐁𝑻∝ 𝑻
VII.Qubits – Control & Decoherence first order coupling
𝛿𝜆 = 𝛿Φ/Φ0
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
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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
𝟒𝐜𝐨𝐭𝐡
ℏ𝝎
𝟐𝒌𝐁𝑻∝ 𝑻
VII.Qubits – Control & Decoherence
𝜹𝝎𝒒 =𝝏𝝎𝒒
𝝏𝝀× 𝜹𝝀 =
𝟏
ℏ
𝝏𝑯𝒒 𝝀
𝝏𝝀×𝑴𝒂
𝚽𝟎
𝜹𝑽
𝒊𝝎𝑳𝒂
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
first order coupling
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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
VII.Qubits – Control & Decoherence
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
𝛿𝜆 = 𝛿Φ/Φ0
2nd order coupling (@ qubit sweet spot)
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Tant
Tx
𝜸𝝋,𝒂(𝟐)
VII.Qubits – Control & Decoherence
(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
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
2nd order coupling (@ qubit sweet spot)
sup
ple
men
tary
mat
eria
l
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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
VII.Qubits – Control & Decoherence
sup
ple
men
tary
mat
eria
l
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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
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VII.Qubits – Control & Decoherence
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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)
VII.Qubits – Control & Decoherence
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
24-28.07.2017/RG - 189www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
Tant
Tx
𝜸𝟏,𝒂
VII.Qubits – Control & Decoherence
J. Goetz et al., Quantum Sci. Technol. 2, 025002 (2017)
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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)
VII.Qubits – Control & Decoherence
sup
ple
men
tary
mat
eria
l
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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
VII.Qubits – Control & Decoherence
J. Goetz et al., PRL 118, 103602 (2017)
sup
ple
men
tary
mat
eria
l
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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)
VII.Qubits – Control & Decoherence
J. Goetz et al., PRL 118, 103602 (2017)
• Field Correlation Measurements
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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
VII.Qubits – Control & Decoherence
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𝒏(𝝎, 𝑻) =𝟏
𝐞𝐱𝐩൫ Τℏ𝝎 )𝒌𝐁𝑻 − 𝟏
𝑺𝟏/𝟒 𝝎,𝑻 =𝟏
𝟒⋅ 𝟒ℏ𝝎 𝒏 +
𝟏
𝟐=𝟏
𝟒⋅ 𝟐ℏ𝝎𝐜𝐨𝐭𝐡
ℏ𝝎
𝟐𝒌𝐁𝑻
thermal noise source: 50 mK ≤ 𝑻 ≤ 1.5 K
VII.Qubits – Control & Decoherence
derive photon statistics from qubit decay
sup
ple
men
tary
mat
eria
l
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calibration of photon number
use photon number dependent ac-Stark shift
Tant
Tx
𝛿𝜔𝑞
approx.3 photons @ 1.5 K
VII.Qubits – Control & Decoherence
J. Goetz et al., PRL 118, 103602 (2017)
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• 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
VII.Qubits – Control & Decoherence
J. Goetz et al., PRL 118, 103602 (2017)
24-28.07.2017/RG - 197www.wmi.badw.de Nanotechnology meets Quantum Information – NanoQI 2017 – San Sebastian – © WMI
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
VII.Qubits – Control & Decoherence
J. Goetz et al., PRL 118, 103602 (2017)
VIII.CV Propagating
Quantum Microwaves
theory support by U. Las Heras, M. Sanz, E. Solano
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199199
AluminumΤ𝛥 ℎ ≃ 50 GHz
superconductingquantumcircuits
emit
propagating quantummicrowaves
coherence?
VIII.CV Propagating Microwaves
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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
VIII.CV Propagating Microwaves
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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
VIII.CV Propagating Microwaves
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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)
VIII.CV Propagating Microwaves
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flux-tunable inductance𝐿SQUID 𝛷dc +Φrf
for parametric drive
flux-driven Jospehsonparametric amplifier(JPA)
𝑓0 2𝑓0
VIII.CV Propagating Microwaves
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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
VIII.CV Propagating Microwaves
squeezing level:
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• 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
VIII.CV Propagating Microwaves
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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
VIII.CV Propagating Microwaves
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• 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
VIII.CV Propagating Microwaves
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• 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
VIII.CV Propagating Microwaves
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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
K. G. Fedorov et al., PRL 117, 020502 (2016).
VIII.CV Propagating Microwaves
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-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
K. G. Fedorov et al., PRL 117, 020502 (2016).
VIII. CV Propagating Microwaves
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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
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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
VIII.CV Propagating Microwaves
Summary
IX.Summary
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The future looks bright !
IX. Summary
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The WMI team
Thank you !