HANBURY BROWN & TWISS* INTERFEROMETRY WITH ANYONS

HANBURY BROWN & TWISS*
INTERFEROMETRY WITH
ANYONS
G. Campagnano , O. Zilberberg, YG, I.Gornyi D.E. Feldman and A. Potter
(PRL 2012, Editor’s suggestion)
G. Campagnano , O. Zilberberg, YG, I.Gornyi (to be published)
*Hanbury Brown & Twiss , Nature 1956

one particle: interferometry (e.g. with AB flux)
two particles: entanglement; quantum statistics
(of identical particles)

Motivation: Two–particle interference
S1
D4
D2
D3
D1
# Photons (bosons)
Electrons (fermions)
See I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu,
and V. Umansky, Nature 448, 333 (2007).
Statistics + Entanglement
S4

Hanbury Brown & Twiss with
Fermions, Bosons, Classical
Colliding beams of Anyons (heuristic)
Aharonov-Bohm with Anyons

Fermions, Bosons, Classical particles….

S1
Buttiker, Blanter
r
t
D2
D1
Classical
t’
r’
S2
P(2,0) P(1,1) P(0,2)

Fermions:
b
⎛ D1
⎞ ˆ S1
⎜ ⎟=
S ⎜ ⎟
bD
2 aS
2
Buttiker, Blanter
⎛a⎞ ⎝ ⎠ ⎝ ⎠
nˆ = b b
1
†
D1 D1
nˆ = b b
2
†
D2 D2
S1
r
t
D2
P(1,1) = Ψ nn ˆˆ Ψ = 0 a a b b b b a a 0
P T R
† † † †
1 2 S2 S1 D1 D1 D2 D2 S1 S2
2
(1,1) = ( + ) =
1
D1
t’
r’
S2
P(2,0) P(1,1) P(0,2)
Fermions 0 1 0

S1
“bunching”
=enhanced
noise
“anti-bunching”
=reduced noise
Buttiker, Blanter
r
t
D2
t’
r’
S2
D1 P(2,0) P(1,1) P(0,2)
Classical
Bosons
Fermions 0 1 0

… adding another handle: A-B flux
S1
ɸ
D1
D2
S2
Samuelson,Sukhorukov, Buttiker 2004
I I |
A + A | ⋅ | A + A |
2 2
D1 D2
1 4 2 3
A ⋅A ⋅A ⋅A
* *
1 4 3 2

HBT
HBT with AB
SUMMARY Fermions/Bosons
⎧fermions
⎪
⎨
⎪
⎩bosons
⎧
⎪
⎨
⎪
⎪
⎩
anti − bunching
bunching
fermions
bosons
φ
#cos(2 π )
φ
0

ealization of HBT + AB
Mach-Zehnder interferometer
2 Mach-Zehnder interferometers

Mach-Zehnder photonic interferometer
S
M2
BS1
D2
M1
no back scattering
BS2
D1
intensity (a.u.)
1.0
0.5
0.0
0
1
2
D1
phase difference ( π)
3
4
D2
5

D
# 2
QPC
M#
2
# G
G
# 1
# air bridge
G
# 3
D
# 1
G
# 2
QPC
# 1
D2 S2
S2 D2
S1 D1
# S
Heiblum et al

introducing ‘two-particle’ interference
Neder, Ofek,
Heiblum, …

actual sample
Neder, Ofek,
Heiblum …
Nature 2007

ANYONS– why are they non trivial?
no “second quantization”
(cannot assign “Klein factors”)

e
iπν
i −
e πν
KLEIN FACTORS (=statistical factors) cannot be
can be assigned to ANYONIC FIELD OPERATORS

TWO WAYS TO OVERCOME THE PROBLEM:
1. assign statistical flux
2. Assign a Klein factor to a tunneling operator
…. example anyonic Mach-Zehnder
Law, Feldman, YG
Feldman, YG, Kitaev, Stern

S1
D2 S2
S2 D2

S1
D2 S2
S2 D2

S1
D2 S2
S2 D2

S1
D2 S2
S2 D2

zero temp. kinetic eq.
0
p
0
p 1
2
p
2
1
#qp trapped
transition rates
( a more detailed analysis: Keldysh)
ν = 1/3
Law, Feldman, YG

Hanbury Brown & Twiss with
Ferrmions, Bosons, Classical
Colliding beams of Anyons
Aharonov-Bohm with Anyons

A CARICATURE WITH (Abelian) ANYONS

A CARICATURE WITH (Abelian) ANYONS
P(2,0)
P(1,1)
r
t
1
(1+ cos( πν ))
4
1
(1− cos( πν ))
2
t’
r’
P(2, 0) + P(1,1) + P(0,
2) = 1
⎧ 1 fermions
⎪
P(1,1)
= ⎨ 0 bosons
⎪⎩
1/ 2 classical

Hanbury Brown & Twiss with
Ferrmions, Bosons, Classical
Colliding beams of Anyons
Aharonov-Bohm with Anyons

2 anyons
S1
ɸ
D1
D2
S2
4
Γ
AHARONOV-BOHM PERIOD 3Φ0
???
Compare
Thouless & YG

3 anyons + 3 anyons interferometry
Φ periodicity
0
Byers-Yang
S1
ɸ
D1
D2
12 th order pert. theory !
S2

THE MAIN RESULT
| ΓΓΓΓ | Ω
I I cos(2 π / )
3 3
D1 D1 = 2
6(| Γ A |
A B C D
⋅ 2 2 2 2 2
+ | Γ B | + | Γ C | + | ΓD
| ) γ
ΦAB Φ0
microscopic periodicity parameters
calculated from Keldysh
not 12
Γ
+ sign !! BOSON-LIKE !!

ingredients of the analysis
3 state kinetics ( cf. Mach-Zehnder)
# statistical fluxes mod(3)

ingredients of the analysis
3 state kinetics ( cf. Mach-Zehnder)
1- particle and 2-particle processes

One &
Two
Qp processes

S1
ɸ
D1
D
2
S2

RATE EQUATION TREATMENT:
three-state kinetics
number of
trapped
fluxes
mod(3)

ingredients of the analysis
3 state kinetics ( cf. Mach-Zehnder)
1- particle and 2-particle processes
how to tackle 12 th order ?
12 th order 8 th order

qp-electron tunneling
hybrids
qp
electron

SUMMARY: signatures of fractional statistics
boson- like
boson-like
12 th order 8 th order