/19 2017/6/301

T. J. Suzuki, PRB 95, 241302(R) (2017)

~ 2017 3

4 ~

Ph.D.

Postdoc

/19 2017/6/302

/19 2017/6/303

C. Marcus Group HarvardDong-In Chang et al..

Nat. Phys. 4, 205 (2008) Groupe Physique

Mesoscopique, LPS, Orsay

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𝑡 → 𝑡𝒥0 Τ𝜖𝐴𝐶 Ω 𝑈 → 𝑈𝒥0 Τ𝜖𝐴𝐶 Ω

𝑉exc

J. Gabelli et al., Science 313, 499 (2006)

G. Fève et al., Science 316, 1169 (2007) A. G. Grushin et al., PRL 112, 156801 (2014)

N. Tsuji et al., PRL 106,

236401 (2011).

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ℋ = ℋ𝐿 +ℋ𝑅 +ℋ𝑇

ℋ𝛼=𝐿,𝑅 =

𝒌

𝜖𝛼𝒌𝑐𝛼𝒌† 𝑐𝛼𝒌

ℋ𝑇 =

𝒌

𝑡 𝑒−𝑖𝜑 𝑡 𝑐𝑅𝒌† 𝑐𝐿𝒌 + H. c. 𝜑 𝑡 =

𝑒

ℏන−∞

𝑡

𝑑𝑡′𝑉 𝑡′

𝑡𝑒±𝑖𝜑 𝑡

𝑉 𝑡

𝑒−𝑖𝜑 𝑡 =

𝑒−𝑖 Τ𝑒𝑉𝑡 ℏ

𝑛=−∞

∞

𝒥𝑛 𝑞 𝑒𝑖𝑛Ω𝑡

𝑛=−∞

∞𝑞 𝑒𝑖𝜋𝑞 − 1

𝜋𝑖 𝑛2 − 𝑞2𝑒𝑖𝑛Ω𝑡

𝑞 ≡𝑒𝑉𝐴𝐶ℏΩ

𝑉 𝑡 = 𝑉

𝑉 𝑡 = 𝑉𝐴𝐶 cos Ω𝑡

𝑉 𝑡 = ቐ𝑉𝐴𝐶 0 ≤ 𝑡 < Τ𝑇𝑝 2

−𝑉𝐴𝐶 Τ𝑇𝑝 2 ≤ 𝑡 < 𝑇𝑝

L. S. Levitov et al.,

J. Math. Phys. 37, 4845 (1996)

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𝑇𝑝 = Τ2𝜋 Ω

𝜑 𝑡 = 𝜑 𝑡 + 𝑇𝑝𝑒−𝑖𝜑 𝑡 =

𝑛

𝑢𝑛𝑒𝑖𝑛Ω𝑡

𝑢𝑛 = න− Τ𝑇𝑝 2

Τ𝑇𝑝 2 𝑑𝑡

𝑇𝑝𝑒𝑖𝑛Ω𝑡𝑒−𝑖𝜑 𝑡

= ර𝑧 =1

𝑑𝑧

2𝜋𝑖𝑧𝑛−1𝐹 𝑧

= ර𝑧 =1

𝑑𝑧

2𝜋𝑖𝑧𝑛−1 𝐹+ 𝑧 + 𝐹− 𝑧

𝑧 ≡ 𝑒𝑖Ω𝑡 ,

𝐹 𝑧 ≡ 𝑒−𝑖𝜑 𝑡

𝑉 𝑡

𝐹± 𝑧 =

𝑛=0

∞

𝑎𝑛±𝑧±𝑛

𝑧 = 1 /

L. S. Levitov et al.,

J. Math. Phys. 37, 4845 (1996)

𝑛 > 0 / 𝑛 < 0

/19 2017/6/307

𝐹 𝑧 =ෑ

𝑖=1

𝑁1 − 𝑎𝑖

∗𝑧

𝑧 − 𝑎𝑖

𝑧 > 1

𝑎𝑖=1,2,⋯,𝑁 < 1

𝑢𝑛<0 = 0

𝑎𝑖

𝑧

𝑎𝑁𝑎1

𝑎2

Re

Im

𝑎𝑖 > 1 → 𝑢𝑛>0 = 0

𝑃𝑆𝐿 2,ℝ

𝑎𝑖 = 𝑒Ω −𝜏𝑖+𝑖𝑡𝑖

𝑒𝑉 𝑡 = 𝑖ℏ𝑑

𝑑𝑡ln𝐹 𝑧 = 𝑒𝑖Ω𝑡 =

𝑖=1

𝑁

𝑒𝑉𝑖 𝑡

𝑒𝑉𝑖 𝑡 =

𝑚=−∞

∞𝜏𝑖𝑇𝑝/𝜋

𝑡 − 𝑡𝑖 −𝑚𝑇𝑝2+ 𝜏𝑖

2

𝑇𝑝

𝜏𝑖

𝑡𝑖

/19 2017/6/308

L. S. Levitov and coauthrs

J. Math. Phys. 37, 4845 (1996), PRB 56, 6839 (1997), PRL 97, 116403 (2006).

or

/19 2017/6/309

/19 2017/6/3010

/19 2017/6/3011

𝑇 = 15~900mK

𝑉gl = −413mV

W. G. van der Wiel et al. Science 289, 2105 (2000)

/19 2017/6/3012

/19 2017/6/3013

ℋ =

𝜎=↑,↓

𝜖𝑑𝑑𝜎†𝑑𝜎 +

𝒌

𝛼=𝐿,𝑅

𝜖𝛼𝒌 − 𝑒𝑉𝛼 𝑡 𝑐𝛼𝒌𝜎† 𝑐𝛼𝒌𝜎

+

𝛼𝜎𝒌

𝑡 𝑑𝜎†𝑐𝛼𝒌𝜎 + 𝑐𝛼𝒌𝜎

† 𝑑𝜎 + 𝑈𝑑↑†𝑑↑𝑑↓

†𝑑↓

𝑉𝐿 𝑡 =

𝑚=−∞

∞𝑉ACΩ

2𝜏𝑤

𝑡 − 𝑚𝑇𝑝2+ 𝜏𝑤

2

𝑡𝑡

Τ𝑉AC 𝜋𝜏

𝜏 ≡ Τ𝜏𝑤 𝑇𝑝 ≪ 1

𝑇𝑝

𝜏𝑤

𝑈

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𝑒−𝑖𝜑 𝑡 = 𝑒−𝑖𝑒𝑉A𝐶𝑡

𝑛

𝑢𝑛𝑒𝑖𝑛Ω𝑡𝜑 𝑡 =

𝑒

ℏන−∞

𝑡

𝑑𝑡′𝑉𝐿 𝑡′

𝑉𝐿 𝑡

𝑞 =𝑒𝑉ACℏΩ

✓ 𝑞 ∈ ℤ

𝑢𝑙 =

𝑘=max 0,−𝑙

∞Γ 𝑘 + 𝑙 + 𝑞 Γ 𝑘 − 𝑞 𝑒−2𝜋𝜏 2𝑘+𝑙

Γ 𝑞 Γ 𝑘 + 𝑙 + 1 Γ −𝑞 Γ 𝑘 + 1

for 𝑙 < −1for 𝑙 = −1for 𝑙 > −1

𝑢𝑙 = ቐ0

𝑒−2𝜋𝜏

𝑒−2𝜋𝑙𝜏 1 − 𝑒−4𝜋𝜏

𝑞 = 1 No Holes

cf. 正弦波𝑢𝑙 = 𝒥𝑙 𝑞

/19 2017/6/3015

𝐺𝑟 𝑡, 𝑡′ ≡ −𝑖𝜃 𝑡 − 𝑡′ 𝑑 𝑡 𝑑† 𝑡′

𝑮𝑚𝑛𝑟 𝜔 ≡ න

−∞

∞

𝑑𝑡𝑟𝑒𝑖𝜔𝑡𝑟න

−𝑇𝑝2

𝑇𝑝2𝑑𝑇 𝑒𝑖 𝑚−𝑛 Ω ҧ𝑡𝐺𝑟 𝑇 +

𝑡𝑟2, 𝑇 −

𝑡𝑟2

Ω = Τ2𝜋 𝑇𝑝 , 𝑡𝑟 = 𝑡 − 𝑡′, 𝑇 =𝑡 + 𝑡′

2

N. Tsuji et al., PRB 78, 235124 (2008)

𝑚Ω 𝑛Ω𝜔

Ω Ω Ω=

⋱ ⋮ ⋱⋯ 𝑮𝑚𝑛

𝑟 ⋯⋱ ⋮ ⋱

近似1. Truncation

∼ 201

∼ 201

近似2. 𝑈2摂動論

Σ𝑈 = 𝑈 𝑈

S. Hershfield et al., PRB 46, 7046 (1992)

/19 2017/6/3016

𝛤 = 1,𝜖𝑑 = −4,𝑈 = 8,𝛽 = 100,𝜏 = 0.02ℏΩ = 3,

ҧ𝜌 𝜔 ≡ −1

𝜋Imන

− Τ𝑇𝑝 2

Τ𝑇𝑝 2 𝑑𝑇

𝑇𝑝න−∞

∞

𝑑𝑡𝑟 𝑒𝑖𝜔𝑡𝑟𝐺𝑟 𝑇 +

𝑡𝑟2, 𝑇 −

𝑡𝑟2

𝑞 = 𝑛 ∈ ℤ

Τ𝑘𝐵𝑇𝐾 Γ≃ 0.086

( ∼ 𝑒−4𝜋𝜏𝑙)

/19 2017/6/3017

DC

𝜏 ≪ 1( 𝑃−𝑙 ∼ 𝑒−4𝜋𝜏𝑙 at 𝑞 = 𝑙 ∈ ℤ )

𝛤 = 1,𝜖𝑑 = −3,𝑈 = 6,𝛽 = 100,ℏΩ = 2,

Τ𝑘𝐵𝑇𝐾 ℏΩ ≃ 0.082

𝜕𝐼

𝜕𝑉ACΤ𝑒2 ℎ

/19 2017/6/3018

Τ𝜕𝐼𝜕𝑉 A

CΤ

𝑒2ℎ

/19 2017/6/3019

1. or

2.

1.

2.

3.

/

TJS, PRB 95, 241302(R) (2017)

/19 2017/6/3020

Towards ExperimentsRecent ExperimentLeviton Kondo effect

J. Dubois et al., Nature 502, 659 (2013)

𝑇 ≤ 𝑇𝐾 ≃ 700mK 𝑇𝑒 ≃ 35mK

𝜏 ≤ Τ1 4𝜋 ≃ 0.08 𝜏 ≃ 0.09

Ω ≥ 𝑇𝐾 ≃ 15GHz Ω = 6GHz

Typical vales of 𝑇𝐾 for lateral QD from

Y. Yamauchi et al., PRL 106, 176601 (2011)

OK!

should be larger

should be sharper

/19 2017/6/30

𝑞 ∈ ℤ

21

𝑡𝑒−𝑖𝜑 𝑡

Δ𝜑 =𝑒

ℏන−∞

∞

𝑑𝑡′𝑉ACΩ

2𝜏𝑤𝑡2 + 𝜏𝑤

2= 2𝜋𝑞

𝑉𝐿 𝑡

𝑞 =𝑒𝑉𝐴𝐶ℏΩ

Δ𝜑

𝜑 𝑡 =𝑒

ℏන−∞

𝑡

𝑑𝑡′𝑉𝐿 𝑡′

𝑞 ∈ ℤ Δ𝜑

𝑞 ∉ ℤL. S. Levitov, H. Lee, and G. B. Lesovik, J. Math. Phys. 37, 4845 (1996)

/19 2017/6/3022

𝑊 𝜔, ҧ𝑡 ≡Γ𝐿Γ𝑅Γ𝐿 + Γ𝑅

Imන−∞

∞

𝑑𝑡𝑟𝑒𝑖𝜔𝑡𝑟𝐺< ҧ𝑡 +

𝑡𝑟2, ҧ𝑡 −

𝑡𝑟2

𝐺< 𝑡, 𝑡′ ≡ 𝑖 𝑑† 𝑡′ 𝑑 𝑡

D. Ferraro, et al., PRB 88, 205303 (2013)

𝜖𝐹 < 𝜔 𝑞 = 1

/19 2017/6/3023

Irradiating Kondo QD

Quasiparticle

=Local Fermi Liquid

𝑉SD

AC field with

a few GHz ∼ 𝑇𝐾

New spectroscopy?

Additional excitation?

Dynamical control?

Tunability of QD parameters

Probe the Kondo resonance in a controlled manner

/19 2017/6/3024

AC Kondo Effect (Theory)

R. López et al., PRL 81, 4688 (1998)

𝑈 = 0.2𝜖0 = −0.15Γ = 0.025

𝜖0

𝑉AC cos Ω𝑡

Kondo peak

satellites

Suppression

ℏΩ = 0.35

/19 2017/6/3025

AC Kondo Effect (Exp.)A. Kogan et al., Science 304, 28 (2004)

✓ Single-photon satellites

observed for 𝑒𝑉SD ∼ ℏΩ > 𝑘𝐵𝑇𝐾

but quite delicate…

satellites

/19 2017/6/3026

Twofold aspects of AC fields

Photon-assisted satellites of the Kondo peak

Suppression of the Kondo resonance

A better way to probe the Kondo resonance?

Electron Quantum Optics !

P. Nordlander et al., PRB 61, 2146 (2000)Kaminski et al., PRL 83, 384 (1999)

Ionizing QDBreaking resonant spin scattering

or

Kondo regime

/19 2017/6/3027

Electron Quantum OpticsElectron

QHE Chiral Edge

Quantum Point Contact

Electron Source

E. Bocquillon et al., Science 339, 1054 (2013)

Photon

Waveguide

Beam splitter

Photon source

“Hong-Ou-Mandel Interferometers”

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