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$: The Capri Spring School on Transport in Nanostructures 2020 - … · CAPRI, 2014 CPT, Marseille C.Wahletal.,PRL112,046802(’14) 1 / 31 Interactions and charge fractionalization in$
Non-interacting picture Charge fractionalization Interferometry at ν = 2

Interactions and charge fractionalization in anelectronic Hong-Ou-Mandel interferometer

C. Wahl, J. Rech, T. Jonckheere and T. Martin

CAPRI, 2014

CPT, Marseille

C. Wahl et al., PRL 112, 046802 (’14)

1 / 31Interactions and charge fractionalization in an electronic Hong-Ou-Mandel interferometer

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Electronic Quantum Optics

Photons → Electrons

Light beam → Chiral edge QHE

Beam splitter → QPC

Coherent Light source →Single Electron Source


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Single Electron Sources

D

1D

bb

V(t)

VG

VG

Macmillan Publishers Limited. All rights reserved©2013

Energy| (t)|2

(t) ~1

t + iW

t

V(t)

V(t) = h

eπ t2 + W2

W

a

c


Lorentzian voltage pulses

[Dubois et al., Nature (’13)] 17

Fig1Flying electrons on surface acoustic waves

[Hermelin et al., Nature 477, 435 (’11)][McNeil et al., Nature 477, 439 (’11)]

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�Charge pumps, quantum turnstiles

[Giblin et al., Nature Comm. 3, 930 (’12)]

Mesoscopic capacitor

[Fève et al., Science 316, 1169 (’07)]

opens the way to all sorts of interference experiments!


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D

1D

bb

V(t)

VG

VG


Energy| (t)|2

(t) ~1

t + iW

t

V(t)

V(t) = h

eπ t2 + W2

W

a

c



[Dubois et al., Nature (’13)]

17



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D

1D

bb

V(t)

VG

VG


Energy| (t)|2

(t) ~1

t + iW

t

V(t)

V(t) = h

eπ t2 + W2

W

a

c



[Dubois et al., Nature (’13)] 17



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The LPA mesoscopic capacitor

∆

D∆ Quantum dot

Level spacing: ∆

Level width: D∆tuned by Vg

Time-dependent excitation voltage Vexc(t)

Ù 1 e + 1 h

Injected exponantial wave-packet

Vexc

Vexc(t)

t

∆

,〈I (t)〉

[Fève et al., Science 316, 1169 (’07); Mahé et al., PRB 82,201309 (’10)]


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∆

D∆

Quantum dot

Level spacing: ∆



Ù 1 e + 1 h


Vexc

Vexc(t)

t

∆

,〈I (t)〉



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∆

D∆ Quantum dot

Level spacing: ∆



Ù 1 e + 1 h


Vexc

Vexc(t)

t

∆

,〈I (t)〉



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∆

D∆ Quantum dot

Level spacing: ∆



Ù 1 e + 1 hInjected exponantial wave-packet

Vexc

Vexc(t)

t

∆

,〈I (t)〉



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∆

D∆ Quantum dot

Level spacing: ∆


Time-dependent excitation voltage Vexc(t) Ù 1 e + 1 h


Vexc

Vexc(t)

t

∆

,〈I (t)〉



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∆

D∆ Quantum dot

Level spacing: ∆


Time-dependent excitation voltage Vexc(t) Ù 1 e + 1 hInjected exponantial wave-packet

Vexc

Vexc(t)

t

∆

,〈I (t)〉



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HOM interferometry

Electrons

anti-bunchingeffect of interactions ?

Photons

bunching


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HOM interferometry

Electrons

anti-bunching

effect of interactions ?

Photons

bunching


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HOM interferometry

Electrons

ν = 2

anti-bunching

effect of interactions ?

Photons

bunching


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HOM interferometry

Electrons

ν = 2

anti-bunchingeffect of interactions ?

Photons

bunching


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Table of Contents

1 Non-interacting picture

2 Charge fractionalization

3 Interferometry at ν = 2


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Table of Contents





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HOM experiment with photons

Interference experimentnon-linear crystal generatesphoton pairs

[Hong, Ou and Mandel, PRL 59, 2044 (’87)]

Coincidence rateHOM dip is observed at 0time delay

signatures of incoming wavepackets


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HOM with electrons

Setup2 single electron sources

counter-propagatingchannels coupled at QPC

I outR

I outL

Zero-frequency cross-correlations

SoutRL =

∫dtdt ′

[〈I out

R (x, t)I outL (x ′, t ′)〉 − 〈I out

R (x, t)〉〈I outL (x ′, t ′)〉

]


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HOM with electrons

Setup2 single electron sources

counter-propagatingchannels coupled at QPC

Differences with photonsfermionic statistics: Fermisea, holes, ...

thermal effects

Coulomb interaction

Zero-frequency cross-correlations

SoutRL =

∫dtdt ′

[〈I out

R (x, t)I outL (x ′, t ′)〉 − 〈I out

R (x, t)〉〈I outL (x ′, t ′)〉

]


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Filling factor ν = 1[Jonckheere et al., Phys. Rev. B 86, 125425 (’12)]

−120 −100 −80 −60 −40 −20 0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

δT

SH

OM/2S

HB

T

D = 0.2D = 0.5D = 0.8

Collision of identical wave-packetsLarge δT2SHBT < 0 → Hanbury-Brown andTwiss

δT = 0SHOM(0) = 0 → Pauli dip

∆t

H L

-100 -50 0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

∆t

SHOM H∆t L

2 SHBT

D1 = 0.2D2 = 0.5

D1 = 0.1D2 = 0.8

Different packets → asymmetry0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

∆t

SHOM H∆t L

2 SHBT

Ε

V=V+

EFΕ

V=V-

EF

ε = 0

ε = ∆/2

Electron-hole collision → peak


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−120 −100 −80 −60 −40 −20 0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

δT

SH

OM/2S

HB

T

D = 0.2D = 0.5D = 0.8



∆t

H L

-100 -50 0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

∆t

SHOM H∆t L

2 SHBT

D1 = 0.2D2 = 0.5

D1 = 0.1D2 = 0.8

Different packets → asymmetry

0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

∆t

SHOM H∆t L

2 SHBT

Ε

V=V+

EFΕ

V=V-

EF

ε = 0

ε = ∆/2



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−120 −100 −80 −60 −40 −20 0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

δT

SH

OM/2S

HB

T

D = 0.2D = 0.5D = 0.8



∆t

H L

-100 -50 0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

∆t

SHOM H∆t L

2 SHBT

D1 = 0.2D2 = 0.5

D1 = 0.1D2 = 0.8

Different packets → asymmetry0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

∆t

SHOM H∆t L

2 SHBT

Ε

V=V+

EFΕ

V=V-

EF

ε = 0

ε = ∆/2



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HOM with electrons: experimental results

Experimental setup[Bocquillon et al., Science 339, 1054 (’13)]

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Results12

Time delay ⌧ [ps]

Correlations�q

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Random par**oning

Pauli dip

FIG. 3: Excess noise ∆q between both sources switched on and both sources switched off as a function

of the delay τ . The noise values are normalized by the value on the plateau observed for long delays. The

blurry blue line represents the sum of the partition noise of both sources. The blue trace is an exponential

fit by ∆q = 1− γe−|t−τ0|/τe . The red trace is obtained using Floquet scattering theory.

Pauli dipSHOM(0) 6= 0

Decoherence at ν = 2!!


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HOM with electrons: experimental results

Experimental setup[Bocquillon et al., Science 339, 1054 (’13)]

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Results12

Time delay ⌧ [ps]

Correlation

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Random par**oning

Pauli dip

FIG. 3: Excess noise ∆q between both sources switched on and both sources switched off as a function

of the delay τ . The noise values are normalized by the value on the plateau observed for long delays. The

blurry blue line represents the sum of the partition noise of both sources. The blue trace is an exponential

fit by ∆q = 1− γe−|t−τ0|/τe . The red trace is obtained using Floquet scattering theory.

Pauli dipSHOM(0) 6= 0

Decoherence at ν = 2!!


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Beyond ν = 1: interactions!

between counter-propagatingchannels, near the QPC

strong interactions within achannel: Fractional QHE

between co-propagating channelsat filling ν = 2

interactin

g region

(work in progress)

(this talk)

injection

propagation

tunneling

ν = 2

prepared state |φ〉

bosonized H(+ diagonalization)

scattering matrix

Bosonization Keldysh


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(to be published)

(this talk)

injection

propagation

tunneling

ν = 2



scattering matrix



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Beyond ν = 1: interactions!between counter-propagatingchannels, near the QPC



(this talk)

→ known to be present in the experimental setup!

injection

propagation

tunneling

ν = 2



scattering matrix



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(this talk)

injection

propagation

tunneling

ν = 2



scattering matrix



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Table of Contents





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Injecting a single electron

|φR〉 =∫

dxϕR(x)ψ†1,R(x − L, 0)|0〉

∫dxϕ(x)ψ†(x)

prepared state |φR〉

Exponential packetϕR(x) =

√2Γeiε0xeΓxθ(−x)

wide in energy spaceγ = ε0/Γ = 1ε0 = 175mK

energy-resolvedγ = 8ε0 = 0.7K


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|φR〉 =∫

dxϕR(x)ψ†1,R(x − L, 0)|0〉






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|φR〉 =∫

dxϕR(x)ψ†1,R(x − L, 0)|0〉

−1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

ε0 = 0.175K

ε(K)

|ϕ̂L

(ε)|2

−1 0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

x(µm)

|ϕL

(x)|2






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|φR〉 =∫

dxϕR(x)ψ†1,R(x − L, 0)|0〉

−1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

ε0 = 0.7K

ε(K)

|ϕ̂L

(ε)|2

−1 0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

x(µm)

|ϕL

(x)|2






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Bosonization frameworkFinite temperature: Θ = 0.1K

one outer and one inner (j = 1, 2) channels per edge (r = R,L):

r = R

r = L

j = 1

j = 2

j = 2

j = 1

Bosonization identity:

ψj,r(x, t) = Ur√2πa

eiφj,r (x,t)

Ur : Klein factora: short-distance cutoffφj,r : chiral bosonic field

J. von Delft and H. Schoeller, Annalen Phys., 1998


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Bosonization frameworkFinite temperature: Θ = 0.1Kone outer and one inner (j = 1, 2) channels per edge (r = R,L):

r = R

r = L

j = 1

j = 2

j = 2

j = 1



eiφj,r (x,t)

Ur : Klein factora: short-distance cutoffφj,r : chiral bosonic field



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r = R

r = L

j = 1

j = 2

j = 2

j = 1



eiφj,r (x,t)Ur : Klein factor

a: short-distance cutoffφj,r : chiral bosonic field



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r = R

r = L

j = 1

j = 2

j = 2

j = 1



eiφj,r (x,t)Ur : Klein factora: short-distance cutoff

φj,r : chiral bosonic field



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r = R

r = L

j = 1

j = 2

j = 2

j = 1



eiφj,r (x,t)Ur : Klein factora: short-distance cutoffφj,r : chiral bosonic field



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Hamiltonian

Propagation along the edges:

Hkin = ~π

∑r=R,L

∫dx v1 (∂xφ1,r)2 + v2 (∂xφ2,r)2

Intra-channel interaction:

Hintra = ~π

U∑

r=R,L

∑j=1,2

∫dx(∂xφj,r)2

Inter-channel interaction:

Hinter = 2~π

u∑

r=R,L

∫dx(∂xφ1,r)(∂xφ2,r)

parameters0 ≤ u ≤ UU & vF


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Mixing angle θ

H = ~π

∑r=R,L

∑j

(vj + U )(∂xφj,r)2 + 2u(∂xφ1,r)(∂xφ2,r)

Diagonalization Ù Mixing angle θ: tan(2θ) = 2u/(v1 − v2)Eigenmodes φ+,r and φ−,r

Eigenvelocities v±

No coupling: θ = 0φ+,r = φ1,rφ−,r = φ2,rv+ = v1 + Uv− = v2 + U

Strong coupling: θ = π/4φ+,r =

√2/2(φ1,r + φ2,r)

φ−,r =√2/2(φ1,r − φ2,r)

v1 = v2 = vFv± = vF + U ± u


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Mixing angle θ

H = ~π

∑r=R,L

∑j

(vj + U )(∂xφj,r)2 + 2u(∂xφ1,r)(∂xφ2,r)

Diagonalization Ù Mixing angle θ: tan(2θ) = 2u/(v1 − v2)

Eigenmodes φ+,r and φ−,r

Eigenvelocities v±



√2/2(φ1,r + φ2,r)

φ−,r =√2/2(φ1,r − φ2,r)

v1 = v2 = vFv± = vF + U ± u


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Mixing angle θ

H = ~π

∑r=R,L

∑j

(vj + U )(∂xφj,r)2 + 2u(∂xφ1,r)(∂xφ2,r)


Eigenvelocities v±

φ+,r = sin θφ1,r + cos θφ2,r

φ−,r = cos θφ1,r − sin θφ2,r



√2/2(φ1,r + φ2,r)

φ−,r =√2/2(φ1,r − φ2,r)

v1 = v2 = vFv± = vF + U ± u


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Mixing angle θ

H = ~π

∑r=R,L

∑j

(vj + U )(∂xφj,r)2 + 2u(∂xφ1,r)(∂xφ2,r)


Eigenvelocities v±

v± = v1 + v2 + 2U2 ±

√(v1 − v2)2

4 + u2



√2/2(φ1,r + φ2,r)

φ−,r =√2/2(φ1,r − φ2,r)

v1 = v2 = vFv± = vF + U ± u


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Mixing angle θ

H = ~π

∑r=R,L

[v+(∂xφ+,r)2 + v−(∂xφ−,r)2

]


Eigenvelocities v±



√2/2(φ1,r + φ2,r)

φ−,r =√2/2(φ1,r − φ2,r)

v1 = v2 = vFv± = vF + U ± u


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Charge fractionalization

qs,L(x, t) = e/π〈∂xφs,L(x, t)〉φL

Outer injection

q2,L

10 15 20−0.200.2q1,L

x(µm)

fast charged mode: ⊕ ⊕slow neutral mode: ⊕

E. Berg et al., PRL 102 (2009) P. Degiovanni et al., PRB 81 (2010)


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Outer injection

+ −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)




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Outer injection

+ −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)

fast charged mode: ⊕ ⊕

slow neutral mode: ⊕



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Outer injection

+ −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)




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Table of Contents





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HBT setup

setup 1

coupling at QPC: channels 1

setup 2

channels 2

S =∫

dtdt ′〈Is(t)Is(t ′)〉 − 〈Is(t)〉〈Is(t ′)〉averages performed on |φL〉

⊗ |φR〉

H. Lee and L. Levitov, arXiv:cond-mat/9312013 (1993)

G. Burkard and D. Loss, PRL 91 (2003)Ol’khovskaya et al., PRL 101 (2008)


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HBT setup

setup 1

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)


setup 2

channels 2

S =∫


⊗ |φR〉




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HBT setup

setup 1

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)


setup 2

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)

channels 2

S =∫


⊗ |φR〉




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HBT setup

setup 1

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)


setup 2

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)

channels 2

SHBT =∫


⊗ |φR〉




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HOM setup

setup 1

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)


setup 2

I1

I2 + −q2,L

10 15 20−0.200.2

+ + q1,L

x(µm)

channels 2

SHOM =∫

dtdt ′〈Is(t)Is(t ′)〉 − 〈Is(t)〉〈Is(t ′)〉averages performed on |φL〉 ⊗ |φR〉G. Burkard and D. Loss, PRL 91 (2003)Ol’khovskaya et al., PRL 101 (2008)


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Performing the calculation

Quantity of interestSHBT and SHOM

Operators involvedIs,r(x, t)

I outs,r (x, t)

I outs,r (0, t)

ψs,rout

(0, t)

ψs,rin

(0, t)

φins,r(0, t)

φin±,r(0, t)

Linear dispersion

Electric current

Scattering matrix

Bosonization

Diagonalization

ψs,rin

(0, t) = Ur√2πa expiφin

s,r (0,t)


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I outs,r (x, t)

I outs,r (0, t)

ψs,rout

(0, t)

ψs,rin

(0, t)

φins,r(0, t)

φin±,r(0, t)

Linear dispersion

Electric current

Scattering matrix

Bosonization

Diagonalization

∫dt I out

s,r (x, t) =∫

dt I outs,r (0, t)

ψs,rin


s,r (0,t)


N





I outs,r (x, t)

I outs,r (0, t)

ψs,rout

(0, t)

ψs,rin

(0, t)

φins,r(0, t)

φin±,r(0, t)

Linear dispersion

Electric current

Scattering matrix

Bosonization

Diagonalization

I outs,r (0, t) = −ev : ψ†s,r

out(0, t)ψs,r

out(0, t) :

ψs,rin


s,r (0,t)


N





I outs,r (x, t)

I outs,r (0, t)

ψs,rout

(0, t)

ψs,rin

(0, t)

φins,r(0, t)

φin±,r(0, t)

Linear dispersion

Electric current

Scattering matrix

Bosonization

Diagonalization

(ψs,R(0, t)ψs,L(0, t)

)out

=(√T i

√R

i√R√T

)(ψs,R(0, t)ψs,L(0, t)

)in

ψs,rin


s,r (0,t)


N





I outs,r (x, t)

I outs,r (0, t)

ψs,rout

(0, t)

ψs,rin

(0, t)

φins,r(0, t)

φin±,r(0, t)

Linear dispersion

Electric current

Scattering matrix

Bosonization

Diagonalization

ψs,rin


s,r (0,t)


N




Operators involvedIs,r(x, t) → φin

±,r(0, t)

I outs,r (x, t)

I outs,r (0, t)

ψs,rout

(0, t)

ψs,rin

(0, t)

φins,r(0, t)

φin±,r(0, t)

Linear dispersion

Electric current

Scattering matrix

Bosonization

Diagonalization

ψs,rin


s,r (0,t)


N


HBT Result

SHBT = − 2e2RT(2πa)3N Re

{∫dyLdzL ϕL(yL)ϕ∗L(zL)g(0, zL − yL)∫

dt dτ Re[g(τ, 0)2

] [ hs(t; yL + L, zL + L)hs(t + τ ; yL + L, zL + L) − 1

]}(1)

g(t, x) =

sinh(i πaβv+

)sinh

(ia+v+t−xβv+/π

) sinh(i πaβv−

)sinh

(ia+v−t−xβv−/π

)1/2

hs(t; x, y) =

sinh(

ia−v+t+xβv+/π

)sinh

(ia+v+t−yβv+/π

)

12sinh

(ia−v−t+xβv−/π

)sinh

(ia+v−t−yβv−/π

)

32−s


N


HBT Result



dt dτ Re[g(τ, 0)2


]}(1)

N = 〈φL|φL〉

g(t, x) =

sinh(i πaβv+

)sinh

(ia+v+t−xβv+/π

) sinh(i πaβv−

)sinh


)1/2

hs(t; x, y) =

sinh(

ia−v+t+xβv+/π

)sinh

(ia+v+t−yβv+/π

)

12sinh


)sinh


)

32−s


N


HBT Result



dt dτ Re[g(τ, 0)2


]}(1)

g(t, x) =

sinh(i πaβv+

)sinh

(ia+v+t−xβv+/π

) sinh(i πaβv−

)sinh


)1/2

hs(t; x, y) =

sinh(

ia−v+t+xβv+/π

)sinh

(ia+v+t−yβv+/π

)

12sinh


)sinh


)

32−s


N


HOM Result

SHOM(δT ) = − 2e2RT(2πa)4N Re

{∫dyLdzL ϕL(yL)ϕ∗L(zL)g(0, zL − yL)

×∫

dyRdzR ϕR(yR)ϕ∗R(zR)g(0, yR − zR)∫

dτ Re[g(τ, 0)2

]×∫

dt[ hs(t; yL + L, zL + L)

hs(t + τ ; yL + L, zL + L)hs(t + τ − δT ; L − yR,L − zR)

hs(t − δT ; L − yR,L − zR) − 1]}

(2)


N


HOM Result



×∫


dτ Re[g(τ, 0)2

]×∫



hs(t − δT ; L − yR,L − zR) − 1]}

(2)

N = 〈φL|φL〉〈φR|φR〉


N


HOM Result



×∫


dτ Re[g(τ, 0)2

]×∫



hs(t − δT ; L − yR,L − zR) − 1]}

(2)

Numerics: quasi Monte Carlo algorithm

T. Hahn, Comput. Phys. Commun., 2005


N


Interferometry

Setup 1 Setup 2

δT = 0Pauli dip Pauli dip

δT = +2Lu/(1− u2)SHBT + asymmetric dip SHBT + asymmetric peak

δT = −2Lu/(1− u2)SHBT + asymmetric dip SHBT + asymmetric peak


N


Interferometry

Setup 1 Setup 2

δT = 0

Pauli dip Pauli dip




N


Interferometry

Setup 1 Setup 2

δT = 0Pauli dip

Pauli dip




N


Interferometry

Setup 1 Setup 2





N


Interferometry

Setup 1 Setup 2


δT = +2Lu/(1− u2)

SHBT + asymmetric dip SHBT + asymmetric peak



N


Interferometry

Setup 1 Setup 2


δT = +2Lu/(1− u2)SHBT + asymmetric dip

SHBT + asymmetric peak



N


Interferometry

Setup 1 Setup 2





N


Wide Packets: Setup 1

Wide packets in energyε0 = 175mK γ = 1

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

12|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

L = 5µmL = 2.5µm

setup 1

SHOM(0) 6= 0

velocity mismatch:asymmetry +loss of contrast


N




∆t

H L

-100 -50 0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

∆t

SHOM H∆t L

2 SHBT

SHOM(0) 6= 0



N




−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

12|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

L = 5µmL = 2.5µm

setup 1

SHOM(0) 6= 0



N




−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

12|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

L = 5µmL = 2.5µm

setup 2

SHOM(0) 6= 0unchanged

side peaks:constructiveinterference


N


Thin packets: Setups 1 and 2

Energy resolved packetsε0 = 0.7K γ = 8

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.5

1

1.52|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

L = 5µmL = 2.5µm

setup 1

dramatic loss ofcontrast!!agreement withexperiment

no peaks


N


Thin packets: Setups 1 and 2

Energy resolved packetsε0 = 0.7K γ = 8

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.5

1

1.52|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

L = 5µmL = 2.5µm

setup 2

dramatic loss ofcontrast!!agreement withexperiment

no peaks


N


Electron-hole collision

Wide packets in energyε0 = ±175mK γ = 1

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

2|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

ε0 = 175mKγ = 1

L = 5µme-h setup1

⊕/→ constructive

⊕/⊕ or /→ destructive

⊕/ interferenceweaker


N


Electron-hole collision

Wide packets in energyε0 = ±175mK γ = 1

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

2|SHBT|

δT (ns)

|SH

OM

(δT

)|(e2RT

)

ε0 = 175mKγ = 1

L = 5µme-h setup1

e-h setup 2

⊕/→ constructive

⊕/⊕ or /→ destructive

⊕/ interferenceweaker


N


Energy width dependency

I2

I1

−L L

escape timeτe = α/Γα = 3.85psK

parametersε0 = 0.7Kv+ = 2v−L = 2.45µm



Very good agreement!!


N



0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

τe(ps)

SH

OM

(δT

)/(2S

HB

T)(e2

RT

)

theorie







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0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

τe(ps)

SH

OM

(δT

)/(2S

HB

T)(e2

RT

)

theorie

experiment







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Conclusion

Interactions account for the observed loss of contrast!

The contrast depends on the energy resolution and theinjection energy of the packet.

Fast and slow excitations interference → lateral dips/peaks


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Thank you for yourattention!!!


N

The Capri Spring School on Transport in Nanostructures 2020 - … · CAPRI, 2014 CPT, Marseille...

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Transcript of The Capri Spring School on Transport in Nanostructures 2020 - … · CAPRI, 2014 CPT, Marseille...