world of 2d electrons is exciting modulation doping

– high purity enables ‘ballistic transport’

– easy electrostatic control gates close to surface

– unique excitations (e.g., fractional statistics)

‘exchange statistics’ in the

2d world

richer than the 3d world

exchange statistics in 3d

fermions bosons

ψ ψ→ + ψ ψ→ −

both

ψ ψ→

exchange statistics in 2d → abelian (Laughlin qp’s)

anyons

ψ ie θψ→ψ 2ie θψ→

ψ 2 4i ie eθ θψ→

2 4 4 2i i i ie e e eθ θ θ θ=

2ie θ

4ie θ

exchange statistics in 2d → non-abelian

degenerate ground state

i ii

a aψ ψ ψ= = ⋅∑

aψ ψ= ⋅ ( )a ψ→ ⋅

U

exchange → unitary

ψ1 2ψ→U U

1 2 2 1U U U U≠

non-abelian anyons 1U

2U

statistics revealed

via interference of quasiparticles

however, interference of fractional charges

was never observed by us

looking for interference of fractional charges

we stumbled upon…

Itamar Sivan, Hyungkook Choi, Amir Rosenblatt Vladimir Umansky Diana Mahalu M. Heiblum

Unexpected ‘Pairing’ in the IQHE Regime in interference

our 2d world

high mobility 2DEG

in GaAs-AlGaAs

e

e

B

applying quantizing magnetic field…QHE

skipping orbits

2d electron layer

high magnetic field

ν = number of filled LL = number of electrons per flux quantum

φ0=h/e

EF cω

energy

*c meB

=ω

Ne

e

EF

Ne

ν =2

)( 21+= nE cn ω

2 =

Bl eB

in the bulk

simplistic view of LL’s near the edge

an approximation there are inter-channel interactions

current carrying edge states

edge channels

edge channels immune to back scattering

1d edge channel carries VheI

2

=

Ef +eV

Ef Ef

interference with edge channels

• incompressible bulk; current carried by edge channels

• edge channels encloses a definite area minimizes phase averaging, high visibility fringes

• electrons directed along definite paths flexible design of interferometers

• no back-scattering

insensitive to impurities

interfering edge channels

two-path Mach-Zehnder & many-path Fabry-Perot

B X

every interferometer needs a beam splitter Quantum Point Contact (QPC)

Vgate

Vsource

r λF

t

QPC 0 < t < 1

preferential backscattering of edge channels

reflected higher LLs

transmitted lower LLs

partitioned LL

Vgate

ne =1-2.5 x 1011 cm-2 B = 2-9 T T =20-30 mK

interference experiments

Fabry – Perot interferometer

S BS

optical FPI

electronic FPI

in the limit of only two-path interference

etched

quantum point contact (w/bridge)

gates

2DEG in GaAs-AlGaAs

actual realization

area modulation gate

v = 2

(2,0)

increasing B → lowers Area

small capacitance keeps # electrons constant

bare FPI – is Coulomb dominated (CD)

interfering the lowest Landau level

screening Coulomb interaction in FPI

effective screening, AB >~4 µm2 effective screening, AB all sizes

tested FPI areas……..2 – 16 µm2

FPI from CD to AB

adding screening: -- grounded ohmic contact -- top gate

our experiments:

interference of outer edge channel

v =3

𝑒𝑒2𝜋𝜋𝑖𝑖∙𝐴𝐴𝐴𝐴/(ℎ𝑒𝑒) 𝑒𝑒2𝜋𝜋𝑖𝑖∙𝐴𝐴𝐴𝐴/( ℎ2𝑒𝑒)

2.25 µm2

surprising AB interference

2.25 µm2

1 ABh / e

δ − = 1 2 ABh / e

δ − =

area ~ 2.3 µm2

periodicity in B

periodicity in B …. large 12.5µm FPI

12.5 µm2

h /e h /2e h /e

12.5 µm2

periodicity in VMG …. large 12.5µm FPI

doubled slope….e*=2e

can it be related to preferred even windings ?

unlikely two windings we measured h /2e ~ 50%

AB transmission…

𝐴𝐴 2 = 𝐴𝐴0 + 𝐴𝐴1 cos 2𝜋𝜋 ∙ 𝐴𝐴𝐴𝐴/𝜙𝜙0 + 𝐴𝐴2 cos 2𝜋𝜋 ∙ 2𝐴𝐴𝐴𝐴/𝜙𝜙0 + ⋯

h /e

h /2e

measured h/e…

h /e

h /2e

ideally…

4 x 4µm2

w/ dephasing

independence on transmission coefficient

ℎ/2𝑒𝑒 regime

large FPI ℎ/𝑒𝑒 regime

coherence @ dephasing

preferential dephasing of channels

VC

VC

adding a ‘center QPC’

preferential dephasing of channels

VC

VC

ground

role of second channel @ vB =2

dephasing of second channel is irrelevant

h/e

VC VC VC

inner edge grounded

outer edge grounded

dephasing inner channel fully dephases the outer channel

role of second channel @ vB =3

h/ 2e

VC VC VC

‘two-channel entanglement’

second edge grounded third edge grounded

summarizing interference :

h /e & h /2e independent of FPI pinching

h /2e not due to preferred even windings

appearance of h /2e depends on ff (not on B or ne)

h /2e only when outer channel interferes

no inter-channel tunneling

are 2e charges interfere ?

shot noise

hot filament

cathode anode + -

emitted electrons

noisy current in vacuum tubes

classical shot noise

Schottky, 1918

it started with - noise in vacuum tubes

Vapplied

~h/eVapplied

zero temperature ordered electrons are noiseless !

shot noise =0 …. full Fermi sea (non-partitioned electrons)

Khlus, 1987 Lesovik, 1989

shot noise - single channel

t <<1 poissonian S =2eI Schottky formula

incoming transmitted

binomial S =2eI (1-t )

t

Khlus, 1987 Lesovik, 1989

spectral density of current fluctuations *)i(

i eI)(S 2

∝≡ ><

ν∆ν ν∆∆

(A2/Hz)

experimental setup

* frequency above 1/f noise corner of preamplifier; * capacitance compensated by resonant circuit;

QPC

cooled preamp

L R C

calibration signal

spectrum analyzer

f0 ,∆f0 C<<

50 Ω ; 300 K

1 GΩ

warm preamp

voltage gain = 1000

coax

averaging time,τ

noise <i2> DC current

VDC

cryostat

‘home made’

kHzRCπ

f,MHzLCπ

f 30≈2

1=Δ4→2≈

21

= 00

shot noise in partitioning QPC

0

2

4

6

0 1 2 3

curr

ent n

oise

, S

i (1

0 -28 A

2 /Hz)

current, I (nA)

T = 57mK t = 0.37

−

−+=

eVTk

TkeVcoth)t(eITgk)(S B

BBi

22

1240

e

Reznikov et al. PRL 1995

-20 -10 0 10 20

0

1

2

3

4

5

6

ISD

(nA)

SI(0

) (×1

0-27 A

2 Hz-1

)

3

0 128 254-276 -135V

SD (µV)

Cooper pairs

e*=e

e*=2e

shot noise in a superconductor

Das et al., Nature Comm. 2012

0 20 400

1

2

3

4T=9mK

e/3

Shot

Noi

se, S

(10-3

0 A2 /Hz)

Back Scattered Current, IB (pA)

Chung et. al. PRL 2003

fractional charge

shot noise, charge

quasiparticle charge e* = 2e @ h/2e regime

charge evolution @ vB ~ 3

VR

VR

forming the FPI by pinching QPCR

h/2e

QPCR

QPCR

highly non-linear transmission

charge: dephasing h/2e …… 2e → e

no interference

no interference

dephasing by grounding the second edge channel

charge drops to e

– a screened FPI …. AB interference

– AB periodicity vB ~1 - 2.5 …… 𝜙𝜙𝑜𝑜 = ℎ𝑒𝑒

• quasi-particles charge: 𝑒𝑒∗~𝑒𝑒

– AB periodicity: vB ~3 - 4.5 … . .𝜙𝜙𝑜𝑜∗ = ℎ2𝑒𝑒

• quasi-particles charge: 𝑒𝑒∗~2𝑒𝑒

– two edge-channels entanglement

summarizing :

screening Coulomb domination

revealed inter-channel interaction

leading to unexpected pairing of electrons

presently, we do not understand the effect