Non-equilibrium Non-equilibrium Fluctuations in Strongly - - PowerPoint PPT Presentation

Non-equilibrium Fluctuations in Strongly Correlated Quantum Liquids Non-equilibrium Fluctuations in Strongly Correlated Quantum Liquids

International Symposium in Honor of Professor Nambu for the 10th Anniversary of his Nobel Prize in Physics Osaka City University, December 12-13, 2018

Kensuke Kobayashi

Graduate School of Science, Osaka University Center for Spintronics Research Network, Osaka University Quantum Information and Quantum Biology Division, Institute for Open and Transdisciplinary Research Initiatives, Osaka University

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Outline

• 1. Mesoscopic systems
• 2. Fluctuation = Noise
• 3. Kondo effect & Fluctuation
• Quantum Liquid
• Symmetry crossover

On-chip collision experiment to probe non-equilibrium quantum liquids On-chip collision experiment to probe non-equilibrium quantum liquids

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Mesoscopic Systems

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Mesoscopic System Solid-state device where quantum mechanics manifests itself. 1980ʼs- Stage for fundamental physics Exotic materials

nanotube, graphene, topological…

MEMS

micro-electro mechanical systems

Charge Spin Phase Coherence Interaction …

Spintronics etc.

Example Electron interference Single electron control etc.

Quantum computing

Strongly-correlated physics / Quantum liquid

this talk

Webb et al. PRL 54, 2696 (1985)

~ 0.8μm

 Degree of freedom in the design  Controllability

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Landauer formula

Conductance gives you information on the electronic properties of single site quantum systems (interference, single-level transport, Kondo physics…).

lead lead lead lead

Mesoscopic system Mesoscopic system

electron electron

transmit

Rolf Landauer (1927–1999)

“Conductance is Transmission.”

• ∼ .
• 5

Quantum Point Contact (QPC)

van Wees et al., PRL 60, 848 (1988).

Constriction width ~ Fermi wave length of electrons (~50 nm) → Conductance quantized due to formation of perfect quantum channel.

i-GaAs i-AlGaAs n-AlGaAs

electron gas

Landauer Formula 2

• 1st channel

0 → 100 % 2nd channel 0 → 100 %

Width increases.

• plateau

plateau

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Quantum Dot (Artificial Atom)

Electrons in a small box ↓ Charging effect & Confinement ↓ Discrete energy levels in QD. # of electrons in QD is fixed.

Electron can pass QD only when the level coincides with those of the leads.

Gate Voltage

QD

Lead Lead

Electron

Lead Lead

QD

Gate

Gate voltage (V)

Current

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Mesoscopic Systems (from our work) Mesoscopic Systems (from our work)

Wave nature Particle nature Atom in Interferometer

• Phys. Rev. Lett. 88, 256806 (2002); 92, 176802

(2004); 95, 066801 (2005); Phys. Rev. B 68, 235304 (2003); 70, 035319 (2004); 73, 195329 (2006).

• Phys. Rev. Lett. 106, 176601 (2011);
• J. Phys. Soc. Jpn 73, L3235 (2004)
• Phys. Rev. Lett 104, 080602 (2010); Phys. Rev. B 79,

161306 (R) (2009); 83, 155431 (2011); J. Phys. Soc. Jpn 71, L2094 (2002); Physica E 42, 1091 (2010).

Quantum Dot (artificial atom) Electron Interferometer Wave-particle duality



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Fluctuation = Noise

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Some electrons cannot transmit

lead lead lead lead

Mesoscopic system Mesoscopic system

electron electron

transmit

Reflected

You can’t avoid this. Noise = Current fluctuation

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Noise = Variance of # of electrons

Resistor

V R = 1/G I

I t

FFT

Current noise power spectral density [/]

Review: Blanter-Büttiker, Phys. Rep. 336, 1 (2000).

Current: Noise:

• Current:

Noise:

• ≡
• Different Unit = New Info.

Different Unit = New Info.

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Shot Noise = Noneq. Current Fluctuation

Schottky (1918)

Poisson process: indep. event

• Ave. = Var. → 〈 〉

2〈 〉

• 2〈 〉
• 2〈〉
• 2〈〉

〈〉 〈〉

• A

inject transmit reflected barrier : time Noise Current 〈〉 ∝ 〈〉

Not so simple! Not so simple!

Electrons are correlated.

Fano factor

• 〈〉

Effective charge 2∗〈〉 12

Shot noise in QPC

Blanter and Büttiker, Phys. Rep. 336, 1 (2000).

transmitted reflected → shot noise

Exp.

Nakamura, et al. PRB 79, 201308(R) (2009).

Fano factor

Noise

• Current
• ∑
• 1
• ∑
• 13

Shot noise current （pA）

Saminadayar et al., PRL 79, 2526 (1997); de-Picciotto et al., Nature 389, 162 (1997).

∗ ∗

Cooper pair in SN junction Fractional quantum Hall effect

Jehl et al., Nature 405, 50 (2000).

Shot noise current （mA）

Super.

Nobel Prize in Physics (1998)

Normal

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“The noise is the signal.”

Spin polarization Spin polarization

Kohda et al. Nature Comm. 3, 1082 (2012). Nishihara et al. APL 100, 203111 (2012).

Stern–Gerlach on chip

Fluctuation Theorem Fluctuation Theorem

Nakamura et al., PRL 104, 080602 (2010); PRB 83, 155431 (2011)

Mesoscopic nonequilibrium statistical physics.

Reservoir

System

Spin shot noise Spin shot noise

Arakawa et al., PRL 114, 016601 (2015); Iwakiri, Niimi, Kobayashi, APEX 10, 053001 (2017).

Edge dynamics Edge dynamics

Matsuo et al. Sci. Rep. 5, 11723 (2015). Matsuo et al. et al. Nature Comm. 6, 8066 (2015).

Edge mixing in graphene pn junction in QH regime

Rolf Landauer (1927–1999)

Nature 392, 658 (1998)

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Ferrier et al., Nature Physics 12, 230 (2016). Teratani et al., J. Phys. Soc. Jpn. 85, 094718 (2016). Ferrier et al., Phys. Rev. Lett. 118, 196803 (2017). Hata et al., Phys. Rev. Lett. (in press): Noise of Andreev-Kondo effect

Kondo effect & Fluctuation

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Collaborators

Experiment:

• T. Arakawa, T. Hata, S. –H. Lee, R. Fujiwara

(Osaka University)

• M. Ferrier, R. Delagrange, R. Weil, R. Deblock,
• H. Bouchiat (CNRS, Université Paris sud)

Theory:

• R. Sakano (University of Tokyo)
• Y. Teratani, A. Oguri (Osaka City University)

Financial support: Grant-in-Aid for Scientific Research (S) (26220711) JSPS KAKENHI (26400319, 16K17723, 15K17680, 25103003, 15H05854) Yazaki Memorial Foundation for Science and Technology Financial support: Grant-in-Aid for Scientific Research (S) (26220711) JSPS KAKENHI (26400319, 16K17723, 15K17680, 25103003, 15H05854) Yazaki Memorial Foundation for Science and Technology

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Quantum liquid

Quantum liquid: Essentially different from a single particle ex. liquid He, BEC, supercondutivity…. Non-equilibrium quantum liquid: strongly-correlated systems, cold-atom physics, spintronics …

~de Broglie length interaction

Single particle Many particles

electron, neutron, atom, molecule…

Quantum liquid

Particle flow Particle flow

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Kondo effect 1964 Kondo effect 1964

Magnetic impurity in metals Magnetic impurity in metals

High T Low T High T Low T

T R T G

Quantum dot Quantum dot

• J. Kondo 1930-
• J. Kondo 1930-

Kondo effect in QD: Gordhaber-Gordon et al. Nature 391, 156 (1998); Cronenwett et al., Science 281, 540 (1998); Schmid et al. Physica B 256-258, 182 (1998). van der Wiel et al., Science 289, 2105 (2000).

Kondo state = Quantum liquid A localized spin screened by many electrons Kondo state = Quantum liquid A localized spin screened by many electrons

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Why Kondo effect in QD?

Realization of Kondo effect in a QD (1998-)

→Ideal test bed to explore nonequibilirium

quantum many-body systems Realization of Kondo effect in a QD (1998-)

→Ideal test bed to explore nonequibilirium

quantum many-body systems

Gordhaber-Gordon et al. Nature 391, 156 (1998); Cronenwett et al., Science 281, 540 (1998); Schmid et al. Physica B 256-258, 182 (1998). van der Wiel et al., Science 289, 2105 (2000).

Impact of Kondo effect (1964)  A monument in solid state physics  Typical quantum many body effect  Extensively studied & understood (but limited within equilibrium to linear response regime) Impact of Kondo effect (1964)  A monument in solid state physics  Typical quantum many body effect  Extensively studied & understood (but limited within equilibrium to linear response regime)

(2) External control (3) Eq. ↔ Non-eq. tunable (1) Single Kondo state

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Kondo shot noise

Theory：Sela, Oreg, von Oppen, and Koch, PRL 97, 086601 (2006); Golub, PRB 73, 233310 (2006); Gogolin and Komnik, PRL 97, 016602 (2006); Mora, Leyronas, and Regnault PRL 100, 036604 (2008); Vitushinsky, Clerk, and Le Hur, PRL 100, 036603 (2008). Fujii, JPSJ 79, 044714 (2010); Sela and Malecki, PRB 80, 233103 (2010); Sakano, Fujii, and Oguri, PRB 83, 075440 (2011)… Only a few experiments : Zarchin et al., PRB 77, 241303 (2008); Delattre et al.,

• Nat. Phys. 5, 208 (2009); Yamauchi, KK et al., PRL 106, 176601 (2011)

Theory predicts enhanced shot noise due to Kondo correlation.

Kondo state

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Device & Measurements

Drain Drain Source Source

Gate electrode Gate electrode

Dilution fridge: 15 ~ 800 mK Dilution fridge: 15 ~ 800 mK

Magnetic field Magnetic field

Measure G Measure G

Mixer Mixer

V V

• Measure

Measure

1 MΩ 1 MΩ

Pd/Al Pd/Al

Carbon nanotube quantum dot Carbon nanotube quantum dot

• 22

Ideal Kondo state realized

4N 4N+1 4N+2 4N+3 4N+4

 Parity effect with 4-fold degeneracy  Low-T conductance reaches 2/ (unitary limit)  Zero-bias anomaly = Kondo resonance  Parity effect with 4-fold degeneracy  Low-T conductance reaches 2/ (unitary limit)  Zero-bias anomaly = Kondo resonance

Kondo resonance ∼

• 23

1.0 0.5 0.0 G (2e

2/h)

23 22 21 20 Vg (V) 16 mK 1.0 0.5 0.0 G (2e

2/h)

23 22 21 20 Vg (V) 16 mK

Shot noise

1.0 0.8 0.6 0.4 0.2 0.0 dI/dV (2e

2/h)

2

• 2

I (nA) 1.0 0.5 0.0 SI (10

• 27A

2/Hz)

Coulomb blockade (non-Kondo) Coulomb blockade (non-Kondo)

1.0 0.8 0.6 0.4 0.2 0.0 dI/dV (2e

2/h)

10 5

• 5
• 10

I (nA) 2 1 SI (10

• 27A

2/Hz)

TK=2K

Kondo regime Kondo regime

As Kondo state is broken, noise is enhanced.

Conventional shot noise 2| |

No noise = perfect Kondo.

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Effective charge

1.0 0.8 0.6 0.4 0.2 0.0 dI/dV (2e

2/h)

10 5

• 5
• 10

I (nA) 2 1 SI (10

• 27A

2/Hz)

TK=2K

2

• reflected current

enhanced shot noise

Kondo correlation appears in nonlinear non-eq. regime. ∗/ is close to 5/3 as theoretically predicted.

Theory：Sela, Oreg, von Oppen, and Koch, PRL 97, 086601 (2006); Golub, PRB 73, 233310 (2006); Gogolin and Komnik, PRL 97, 016602 (2006); Mora, Leyronas, and Regnault PRL 100, 036604 (2008); Vitushinsky, Clerk, and Le Hur, PRL 100, 036603 (2008). Fujii, JPSJ 79, 044714 (2010); Sela and Malecki, PRB 80, 233103 (2010); Sakano, Fujii, and Oguri, PRB 83, 075440 (2011). Theory：Sela, Oreg, von Oppen, and Koch, PRL 97, 086601 (2006); Golub, PRB 73, 233310 (2006); Gogolin and Komnik, PRL 97, 016602 (2006); Mora, Leyronas, and Regnault PRL 100, 036604 (2008); Vitushinsky, Clerk, and Le Hur, PRL 100, 036603 (2008). Fujii, JPSJ 79, 044714 (2010); Sela and Malecki, PRB 80, 233103 (2010); Sakano, Fujii, and Oguri, PRB 83, 075440 (2011).

2 1 SK (10

• 27A

2/Hz)

• 4
• 2

2 4

IK(nA)

∗ shot noise reflected current

• 2

. .

reflection (Back scattered)

Current (nA) Noise

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Fermi Liquid

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Fermi liquid theory

Describes the low-energy physics of interacting Fermion systems.

Applicable to liquid He, usual metals, heavy fermion systems, neutron stars...

Single particle excitations are quasi-particles with energies ̃, in

• ne-to-one correspondence with those of non-interacting system ,

̃,

,

• , 1

2

• ,

, ,,,

• ,

, ̃,

,

• , 1

2

• ,

, ,,,

• ,

,

• ,: momentum distribution
• ,: deviation from the ground

state

Residual interaction: characterized by Wilson ratio R

You can treat the system as if it is a non-interacting one. Kondo state = locally formed Fermi liquid.

Yamada, Prog. Theor. Phys. 53, 970 (1975); Yosida and K. Yamada, ibid. 53, 1286 (1975). Nozières, and, Blandin, J. Physique 41, 193 (1980)

Landau, 1957

• L. Landau
• K. G. Wilson

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Wilson ratio

• N

S N S

Magnetic susceptibility

Free electron gas = simplest quantum liquid

Specific heat C

• 3
• → Charge and spin “united”
• Wilson ratio

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Wilson ratio

 Free electron gas at

 Magnetic (Pauli para.):  Thermal (Sp. heat):

•  Wilson ratio
• Wilson ratio = measure of quantum fluctuation

Nozières (1974)

charge susceptibility spin susceptibility → 0: charge frozen with spin fluctuated

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Meaning of

Two scattering processes at Kondo state Single charge

a free particle is reflected.

Double charge

two particles interact via “residual” interaction.

Kondo state

∗

• → In the Kondo limit (

),

holds. Probability

• Kondo state

Probability

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Wilson ratio determined

≅ 1 ̅

• Γ
• ̅
• Γ
• ̅
• Γ
• ̅ 1 2 1

3 ̅ 1 5 1 4 ̅ 1 4

Costi et al, J. Phys.: Condens. Matter 6 (1994); Oguri, PRB 64, 153305 (2001)

Conductance scales to Temperature, Magnetic Field, and Bias Voltage.

 Wilson ratio is determined 1.9 ∼ 2.  Strict quantitative test of quantum many body theory in the nonequilibrium regime.

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Symmetry Crossover

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SU(4) Kondo effect

Kondo effect of even # of electrons

Coulomb oscillation

Low T High T

N=1

N=2

N=3

Ferrier et al., Nature Physics 12, 230 (2016)

Spin-orbital Kondo effect

∼

• cf. Conventional: SU(2) Kondo effect

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Symmetry and Noise

• M. Ferrier, et al., Phys. Rev. Lett. 118, 196803 (2017);

Y . Teratani et al., J. Phys. Soc. Jpn. 85, 094718 (2016).

∗ 5 3 ∗ 3 2

SU(4) Kondo SU(2) Kondo

Reflected current (nA) Noise Reflected current (nA) Noise

Spin & Orbital

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SU(N) Kondo effect: exp. vs theory

Sakano-Fujii-Oguri, PRB 83, 075440 (2011).

∗ 1 9 1 1 1 5 1 1

(Kondo limit: → ∞)

→

• 1

Symmetry effective charge ∗/ Wilson ratio

• SU2

5/3 2 SU4 3/2 4/3

… … …

SU∞ 1 1 (free particle) Wilson ratio : quantum fluctuations effective charge ∗/ : interaction

Link between interaction and quantum fluctuations across symmetry crossover Link between interaction and quantum fluctuations across symmetry crossover

Theory predicts…

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Conclusion

 Realize ideal Kondo state  Strict test of non-eq. quantum

many body theory

 Fluctuation increases as

symmetry lowers: SU(4) → SU(2)

 Connect different symmetries

cf: BEC-BCS crossover, symmetry in SC

On-chip collision experiment to probe non-equilibrium quantum liquids On-chip collision experiment to probe non-equilibrium quantum liquids

Ferrier et al., Nature Physics 12, 230 (2016). Teratani et al., J. Phys. Soc. Jpn. 85, 094718 (2016). Ferrier et al., Phys. Rev. Lett. 118, 196803 (2017). Hata et al., Phys. Rev. Lett. (in press).

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