Laughlin quasiparticle interferometer:Laughlin quasiparticle interferometer:Observation of Aharonov-Bohm superperiod

and fractional statistics

F.E. Camino, W. Zhou and V.J. Goldman

Stony Brook UniversityStony Brook University

OutlineOutline

Exchange statistics in 2D, Berry’s phase and AB effect

Review of quantum antidot results

Laughlin quasiparticle interferometer

Samples and quantum Hall effect (QHE) transport

Results in the integer QHE: device calibration with electrons

FQHE results: direct observation of AB “superperiod”

Fractional statistics in quasiparticle interferometer

Questions and Answers

Next stage – application to topological quantum computation

Exchange statistics

• Ψ(r1, r2) acquires phase γ = πΘ upon exchange:

do again:

single-valuedness of Ψ ⇒⇒

, thus Θ = j (integer)

e.g., fermions:fermions: ΘF = 1 , bosons:bosons: ΘB = 0

),(),( 212

21 rrerr i Ψ=Ψ Θπ

),(),( 1221 rrerr i Ψ=Ψ Θπ

12 +=Θπie

Exchange statistics in 2D• exchange = half loop + translation

(exchange)2 = complete loop

• in 3D:in 3D: loop with particle inside is NOT distinctfrom loop with no particle inside ⇒⇒ Θ = j

• in 2D:in 2D: loop with particle inside is topologicallydistinct from loop with no particle inside

⇒⇒ NO requirement for Θ to be an integere.g., Θ can be any real number

““anyonsanyons””Leinaas, Myrheim 1977; F. Wilczek 1982

Q:Q: are there such particles in Nature?are there such particles in Nature?A:A: collective excitations of a manycollective excitations of a many--electron 2D systemelectron 2D system

Adiabatic transport: Berry’s phase

• electronselectrons

electron at ℜ adiabatically executes closed path C⇒⇒ wave function acquires Berry’s phase

jjjj yixzz +=ℜΨ where),;(

∫ ℜΨ∂ℜ∂

ℜΨℜ=C

jj zzdiT );();()(γ

M.V. Berry 1984

)exp( γi

Adiabatic transport in magnetic field

• electronselectrons

wave function of encircling electron acquires phase

two contributions: Aharonov-Bohm + statistics

ee Neq ,1, =Θ=

⇒⇒ statistical contribution is NOT observable: exp[i2πΘeNe]=1

ABAB:: Y. Aharonov & D. Bohm 1959

⇒⇒ Byers-Yang theorem: AB period ∆Φ = h/je where j = 1, 2, …

Gauge invarianceGauge invariance:: N. Byers and C.N. Yang 1961; C.N. Yang 1962

Θ+

Φ=Θ+Φ= eeee N

ehNeT

/22)( ππγ

h

Φ

• ff ==1/3 1/3 Laughlin quasiholeLaughlin quasihole

Ψ of encircling quasihole acquires phase

change between m and m+1

• q = e/3, when flux changes by ∆Φ=Φ0=h/e, Ψ acquires AB phase ∆γ=2π/3⇒⇒ need Θ1/3 =2/3 for single-valued Ψ (exp. period is Φ0, NOT 3Φ0 !)

in in FQHEFQHE:: Halperin 1984; Arovas, Schrieffer, Wilczek 1984; W.P. Su 1986e/3 and e/5 chargee/3 and e/5 charge:: Laughlin 1983, Haldane 1984, exp.exp.:: Goldman, Su 1995

Fractional statistics in 2D

mNqqhm ππγ 22 3/1 =Θ+Φ=

h

qhNeq ,3/2,3/ 3/1 =Θ=Φ

ππγ 22 3/1 =∆Θ+∆Φ=∆ qhNqh

• large 2D electron systemlarge 2D electron system(include donors = neutral)

quantum number variable

e.g., change B, electron density n is fixed ⇒⇒ ν changes; remains neutral

How can one make FQH quasiparticles?

hef XY

/2σ

=

f

BB f

=

=

ν"fillingexact"

ffB

B

BB

f

f

>=

<

ν ffB

B

BB

f

f

<=

>

ν

∆QHole

∆QElectron

FQH condensate at f

µFQHgap

eBhn

=ν

Laughlin’s QH and QE in f=1/3 condensate

charge density profile calculation: CF particle-hole pair at f=1/3 (Ne=48)Park & Jain 2001

charge hole in condensate extra charge on top of condensate

bizarrebizarre (fractional charge & long-range statistical interaction) particlesparticles

Resonant Tunneling via a Quantum Antidot

EF

E

VX

XIX

XIX

2/ XYXXT RRG ≈

XXXX IVR /=

Experiments on Experiments on single single quantum antidotquantum antidotGoldman et al. 1995, 1997, 2001, 2005

BGBG VVd

Sq ∆∝∆= 0εε

5/3/

5/2

3/1

eqeq

==⇒⇒

⇓

Experiments on Experiments on single single quantum antidotquantum antidotGoldman et al. 1995, 1997, 2001, 2005

πππγ 232

31223/

3/13/1 =

+=∆Θ+∆Φ=∆ Ne

h

3/1

/

eqN

ehBS

==∆

=∆Φ=∆

consistent with Θ1/3 = 2/3 …, but ensured by Byers-Yang theorem theory:theory: Kivelson, Pokrovsky et al. 1989 - 1993, Chamon et al. 1998

Fabry-Perot electron interferometer

• chiral edge channels are coupled by tunneling (dotted)

× vary perpendicular B : change enclosed Φ ⇒⇒Aharonov-Bohm effect interference

× vary perpendicular E (back gate VBG): change number of enclosed particles ⇒⇒interference due to statistical phase

• four front gates (VFG1-4) to fine-tune symmetry, tunneling amplitude

Φ

Laughlin quasiparticle interferometer: Idea

• an e/3 QP encircles the f = 2/5 island; δΦ=h/2e creates an e/5 QP; detect by e/3 tunneling

• if e/3 QP path is quantum-coherent, expect to see an interference pattern, e. g., Aharonov-Bohm vs. Φthrough inner island

• expect to observe effects of fractional statistical phase (neither bosons nor fermions produce an observable statistical effect)

Laughlin quasiparticle interferometer: Samples

2D electrons ≈300 nm below

surface in these low n, high µGaAs/AlGaAs heterojunctions

suitable for FQHE

XXXX IVR /=

lithographic island radius ≈1,050 nm

2/ XYXXT RRG ≈

Cn

The electron density profile of the island

Density profile of island defined by etched annulus of inner R=1,050 nm, nB=1.2×1011 cm−2 ; following a B=0 model of Gelfand and Halperin, 1994depletion length parameter W=245 nm (VFG=0)

C = “constriction”B = “bulk”, 2D

20.13/1

5/2 =nn

donors2D electrons

3/1nm685

==f

r

5/2nm570

== f

rexperimental,

from AB period

mesaAlGaAsGaAs

Quantization of RXX ⇒⇒ determine nB and nC

VFG ≈ 0T = 10.2 mKC = “constriction”, B = “bulk”, 2D

Quantum Hall transportQuantum Hall transport

)/1/1)(/()B()C( 2BCXYXYXX ffehRRR −=−=

fC and fB plateaus may overlap in a range of B-field ⇒⇒

C = “constriction”, B = “bulk”, 2D edge:edge: X-G. Wen 1990 - 1994

BC ff = BC ff <

AharonovAharonov--Bohm interference of electrons Bohm interference of electrons ⇒⇒ outer ring outer ring rr

mT81.21 =∆B

mT85.22 2 =∆B

nm685/

;/

1

12

≈

∆=

=∆

Behr

ehBr

π

π⇓⇓

Interference of electrons in the outer ring Interference of electrons in the outer ring vsvs. backgate. backgate

V1upon0017.0/ ≈nnδ

unlike QAD,need to calibrate

∆Q = e, ∆VBG

⇓⇓

BGVQ

δδ

small perturbation:

Aharonov-Bohm interference of e/3 Laughlin quasiparticles in the inner ring circling the island of the f = 2/5 FQH fluid

Observation of Aharonov-Bohm “superperiod”

∆Φ = 5 h/e !

Observation of Aharonov-Bohm “superperiod”

Aharonov-Bohm superperiod of ∆Φ > h/ehas never been reported before

Derivation of the Byers-Yang theorem uses a singular gauge transformation at the center of the AB ring, where electrons are

excluded

Present interferometer geometry has no electron vacuum within the AB path ⇒⇒ BY theorem is not applicable (no “violation” of BY theorem)

N. Byers and C.N. Yang, PRL 1961; C.N. Yang, RMP 1962

∆Φ = 5h/e

⇓⇓creation of ten

e/5 QPsin the island

backgate voltageperiod of

∆Q = 10(e/5) = 2e

AB “superperiod”

∆Φ = 5 h/e !

Flux and charge periods

151

351

101

21035223/

3/15/2

3/15/25/2

3/15/2

−=

−=Θ

=

Θ+=∆Θ+∆Φ=∆ πππγ Ne

h

• f =1/3 quasiparticles: • f =2/5 quasiparticles:

⇒⇒ an e/3 encircling ∆N=10 of f = 2/5 QPs:

relative statistics isrelative statistics is

∗ Inputs: q‘s , but NOT Θ‘s

† Integer statistics would allow addition of one quasiparticle per period,fractional statistics forces period of ten quasiparticles!

‡ Exchange of charge ∆Q=3(e/3)=5(e/5)=1e (neglecting statistics)would result in ∆Φ=(5/2)h/e – NOT what is observed

Fractional statistics of Laughlin quasiparticlesFractional statistics of Laughlin quasiparticles3/eq =

5/eq =

Q:Q: How do we know the island filling?A:A: The ratio of the oscillations periods

is independent of the edge ring area S

Questions & Answers

⇒⇒ no depletion model is used to establish island filling

feNN

VSBS

BG

1≡∝

∆∆ Φ

25

37

Ratios fall on straight lineforced through (0,0) and the f = 1 data point

⇒⇒ island filling is f = 2/5

Questions & Answers

C = constrictionB = bulk, 2D

Q:Q: How do we know the fC = 1/3 FQH fluid surrounds the island?A:A: The quantized plateau at 12.35 T

(fB = 2/5) confirms conductionthrough uninterrupted fC = 1/3

22 2//5/2

13/1

1)( ehehBRXX =

−=

• no fit to a detailed model is necessary:an f = 1/3 QP encircling one f = 2/5 QP andsingle-valuedness of wave function requiresrelative statistics to be fractional:

• direct: experiment closely models definition of fractional statistics in 2D

• the only input: Laughlin QP charge e/3, has been measured directly in quantum antidots

• adiabatic manipulation of wave functions’ phase factorsis exactly what is needed for quantum computation

1513/1

5/2 −=Θ

Direct observation of fractional statisticsDirect observation of fractional statistics

Thanks for attention!

“no quasiparticles created” model Jain et al., 1993

• addition of flux Φ0 = h/e to island decreases island area by 2π l 2⇒⇒ island charges by +2e/5 , surrounding 1/3 charges by −2e/5

• the 1/3 and the 2/5 condensates must contain integer number of electrons

• addition of (5/2)h/e to island charges it by 5(e/5) = e• e = 3(e/3) can be transferred to the surrounding 1/3 fluid to restore island

⇒⇒ wrong periods: ∆Φ = (5/2)h/e , ∆Q = 1e

1/3 2/5 1/3

“no quasiparticles created” counterarguments:

eBhn

=νelectrons

donorsincrease B, no qp allowed

⇒⇒ ν = f = fixedmust increase n = feB/hbut donor charge same

• In above scenario, increasing B systematically shifts the 1/3-2/5 boundary ⇒⇒ net charging of island and the boundary ⇒⇒ huge Coulomb energy

e.g., for r = 600 nm, at 10 periods from exact filling, need 1,000 K much more than 100 K to excite 100 quasiholes

• Deriving Eqs. 5 and 7, Jain et al. implicitly assume flux is quantized in units of Φ0 , therefore ∆Φ/Φ0 must be an integer – no reason whatsoever

⇒⇒ integer ∆Φ/Φ0 put in by hand, not derived

neutral

−Q+Q

“no statistics, just charge transfer” unattributable

argument:• addition of Φ0 pushes 2e/5 out of the island• the outer boundary of the 1/3 fluid pushes e/3 to the contacts• the 1/3-2/5 boundary is thereby charged by 2e/5 – e/3 = e/15• the 1/3 fluid restores its state every e/3 ⇒⇒ 5Φ0 restores original state

(a la Laughlin gedanken experiment?) Φ0

2/5

1/3

“nothing but charge transfer” counterarguments:• Here’s NOT Laughlin gedanken experiment geometry: flux is real, 2D electrons in uniform field B, not gauge-transformed in electron vacuum

• Addition of flux does not “push charge”: each Φ0 excites +fe in quasihole charge out of condensate, but condensate charge changes by –fe

⇒⇒ total FQH fluid is neutral (net charging = huge Coulomb energy)

• Even assuming charge is “pushed”, quasiparticle charge is e/5 (not 2e/5); each Φ0/2 excites charge +e/5 quasihole

E.g., 1st qh “charges boundary” by e/5; 2nd to 2e/5, but e/3 goes to contacts, e/15 remain; 3rd to 4e/15; 4th to 7e/15, e/3 to contacts, 2e/15 remain; 5th to 5e/15 = e/3, goes to contacts, zero remain = period.

⇒⇒ periods are still wrong: ∆Φ = (5/2)h/e , ∆Q = 1e

FabryFabry--Perot electron interferometerPerot electron interferometer

f = 1

up to 250 AB interference oscillations; also observe oscillations at f = 2

FabryFabry--Perot quasiparticle interferometerPerot quasiparticle interferometer