Laughlin quasiparticle interferometer: Observation of Aharonov-Bohm superperiod and fractional statistics

Transcription

1 Laughlin quasiparticle interferometer: Observation of Aharonov-Bohm superperiod and fractional statistics F.E. Camino, W. Zhou and V.J. Goldman Stony Brook University

2 Outline 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

3 Exchange statistics Ψ(r 1, r 2 ) acquires phase γ = πθ upon exchange: iπ Θ Ψ ( r1, r2 ) = e Ψ ( r2, r1 ) do again: i2π Θ Ψ ( r1, r2 ) = e Ψ ( r1, r2 ) single-valuedness of Ψ i e 2π Θ = + 1, thus Θ = j (integer) e.g., fermions: Θ F = 1, bosons: Θ B = 0

4 Exchange statistics in 2D exchange = half loop + translation (exchange) 2 = complete loop in 3D: loop with particle inside is NOT distinct from loop with no particle inside Θ = j in 2D: loop with particle inside is topologically distinct from loop with no particle inside NO requirement for Θ to be an integer e.g., Θ can be any real number anyons Leinaas, Myrheim 1977; F. Wilczek 1982 Q: are there such particles in Nature? A: collective excitations of a many collective excitations of a many-electron 2D system

5 Adiabatic transport: Berry s phase electrons Ψ( R; z ), where z = x + j j j iy j electron at R adiabatically executes closed path C wave function acquires Berry s phase exp( iγ ) γ ( T ) = i C d R Ψ( R; z j ) Ψ( R; z j R ) M.V. Berry 1984

6 Adiabatic transport in magnetic field electrons q = e, Θ =1, e N e Φ wave function of encircling electron acquires phase e Φ γ ( T ) = Φ + 2π ΘeNe = 2π + ΘeNe h h / e two contributions: Aharonov-Bohm + statistics statistical contribution is NOT observable: exp[i2πθ e N e ]=1 Byers-Yang theorem: AB period Φ = h/je where j = 1, 2, AB: Y. Aharonov & D. Bohm 1959 Gauge invariance: N. Byers and C.N. Yang 1961; C.N. Yang 1962

7 Fractional statistics in 2D f =1/3 Laughlin quasihole q = e 3, Θ 1 = 2/3, / / 3 N qh Ψ of encircling quasihole acquires phase q γ m = Φ + 2π Θ 1 / 3Nqh = 2π m h Φ change between m and m+1 q γ = Φ + 2π Θ1 / 3 Nqh = 2π h q = e/3, when flux changes by Φ=Φ 0 =h/e, Ψ acquires AB phase γ=2π/3 need Θ 1/3 =2/3 for single-valued Ψ (exp. period is Φ 0, NOT 3Φ 0!) in FQHE: Halperin 1984; Arovas, Schrieffer, Wilczek 1984; W.P. Su 1986 e/3 and e/5 charge: Laughlin 1983, Haldane 1984, exp.: Goldman, Su 1995

8 How can one make FQH quasiparticles? large 2D electron system (include donors = neutral) FQH gap QElectron QHole µ quantum number variable f = σ e XY 2 / h ν = hn eb FQH condensate at f e.g., change B, electron density n is fixed ν changes; remains neutral ν = B < B f B B f f > f B = B "exact filling " ν = f f ν = B > B f B B f f < f

9 Laughlin s QH and QE in f=1/3 condensate charge hole in condensate extra charge on top of condensate bizarre (fractional charge & long-range statistical interaction) particles charge density profile calculation: CF particle-hole pair at f=1/3 (N e =48) Park & Jain 2001

10 Resonant Tunneling via a Quantum Antidot X IX X IX R = V / I XX X X G R T XX 2 / RXY V X E E F

11 Experiments on single quantum antidot Goldman et al. 1995, 1997, 2001, 2005 q εε S d = 0 V BG V BG q q 1/3 2/5 = e/3 = e/5

12 Experiments on single quantum antidot Goldman et al. 1995, 1997, 2001, 2005 S B = Φ = h/ e N = 1 q = e/3 e / γ = Φ + 2π Θ1 / 3 N1/ 3 = 2π + = 2π h 3 3 consistent with Θ 1/3 = 2/3, but ensured by Byers-Yang theorem theory: Kivelson, Pokrovsky et al , Chamon et al. 1998

13 Fabry-Perot electron interferometer chiral edge channels are coupled by tunneling (dotted) vary perpendicular B: change enclosed Φ Aharonov-Bohm effect interference Φ vary perpendicular E (back gate V BG ): change number of enclosed particles interference due to statistical phase four front gates (V FG1-4 ) to fine-tune symmetry, tunneling amplitude

14 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 quantumcoherent, expect to see an interference pattern, e. g., Aharonov-Bohm vs. Φ through inner island expect to observe effects of fractional statistical phase (neither bosons nor fermions produce an observable statistical effect)

15 Laughlin quasiparticle interferometer: Samples 2D electrons 300 nm below surface in these low n, high µ GaAs/AlGaAs heterojunctions suitable for FQHE lithographic island radius 1,050 nm R = V / I XX X X G R T XX 2 / RXY

16 The electron density profile of the island mesa AlGaAs GaAs donors 2D electrons n C r = 685nm f = 1/3 r = 570 nm f = 2 /5 C = constriction B = bulk, 2D n n 2 /5 = 1/ experimental, from AB period Density profile of island defined by etched annulus of inner R=1,050 nm, n B = cm 2 ; following a B=0 model of Gelfand and Halperin, 1994 depletion length parameter W=245 nm (V FG =0)

17 Quantization of R XX determine n B and n C C = constriction, B = bulk, 2D V FG 0 T = 10.2 mk

18 Quantum Hall transport f C = f B f C < fb f C and f B plateaus may overlap in a range of B-field R XX = R XY (C) R XY (B) = ( h / e 2 )(1/ f C 1/ f B ) C = constriction, B = bulk, 2D edge: X-G. Wen

19 Aharonov-Bohm interference of electrons outer ring r 2 B2 = 2.85 mt B 1 = 2.81 mt πr r = 2 B 1 = h/ e; h/ π e B 685 nm 1

20 Interference of electrons in the outer ring vs.. backgate small perturbation: δ n / n upon 1V unlike QAD, need to calibrate Q = e, V BG δ Q δ V BG

21 Observation of Aharonov-Bohm superperiod Φ = 5 h/e! Aharonov-Bohm interference of e/3 Laughlin quasiparticles in the inner ring circling the island of the f = 2/5 FQH fluid

22 Observation of Aharonov-Bohm superperiod Aharonov-Bohm superperiod of Φ > h/e has 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

23 Flux and charge periods AB superperiod Φ = 5 h/e! Φ = 5h/e creation of ten e/5 QPs in the island backgate voltage period of Q = 10(e/5) = 2e

24 Fractional statistics of Laughlin quasiparticles f =1/3 quasiparticles: f =2/5 quasiparticles: q = e / 3 q = e / 5 an e/3 encircling N =10 of f = 2/5 QPs: γ = e / 3 h Φ + 2π Θ 1/ 3 2 / 5 N 2 / 5 relative statistics is = Θ 2π 1/ 3 2 / 5 = Θ / 3 2 / = = 2π 1 15 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

25 Questions & Answers Q: How do we know the island filling? A: The ratio of the oscillations periods is independent of the edge ring area S S B S V BG NΦ Ne 1 f Ratios fall on straight line forced through (0,0) and the f = 1 data point island filling is f = 2/5 no depletion model is used to establish island filling

26 Questions & Answers Q: How do we know the f C = 1/3 FQH fluid surrounds the island? A: The quantized plateau at T (f B = 2/5) confirms conduction B) = h / e 1/3 2/5 through uninterrupted f C = 1/3 R XX ( = h /2e 2 C = constriction B = bulk, 2D

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28 Direct observation of fractional statistics no fit to a detailed model is necessary: an f = 1/3 QP encircling one f = 2/5 QP and single-valuedness of wave function requires relative statistics to be fractional: direct: experiment closely models Θ definition of fractional statistics in 2D 1/ 3 2 / 5 = 1 15 the only input: Laughlin QP charge e/3, has been measured directly in quantum antidots adiabatic manipulation of wave functions phase factors is exactly what is needed for quantum computation

29 Thanks for attention!

30 no quasiparticles created model Jain et al., 1993 addition of flux Φ 0 = h/e to island decreases island area by 2πl 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

31 no quasiparticles created counterarguments: ν = hn eb electrons donors increase B, no qp allowed ν = f = fixed must increase n = feb/h but donor charge same +Q neutral 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 Φ 0, therefore Φ/Φ 0 must be an integer no reason whatsoever Q integer Φ/Φ 0 put in by hand, not derived

32 argument: no statistics, just charge transfer unattributable addition of Φ 0 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 5Φ 0 restores original state (a la Laughlin Φ gedanken experiment?) 0 1/3 2/5

33 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 Φ 0 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 Φ 0 /2 excites charge +e/5 quasihole E.g., 1 st qh charges boundary by e/5; 2 nd to 2e/5, but e/3 goes to contacts, e/15 remain; 3 rd to 4e/15; 4 th to 7e/15, e/3 to contacts, 2e/15 remain; 5 th to 5e/15 = e/3, goes to contacts, zero remain = period. periods are still wrong: Φ = (5/2)h/e, Q= 1e

34 Fabry-Perot electron interferometer f = 1 up to 250 AB interference oscillations; also observe oscillations at f = 2

35 Fabry-Perot quasiparticle interferometer