FRACTIONAL CHARGE & FRACTIONAL STATISTICS & HANBURY BROWN & TWISS INTERFEROMETRY WITH ANYONS

Transcription

1 FRACTIONAL CHARGE & FRACTIONAL STATISTICS & HANBURY BROWN & TWISS INTERFEROMETRY WITH ANYONS with G. Campagnano (WIS), O. Zilberberg (WIS) I.Gornyi (KIT) also D.E. Feldman (Brown) and A. Potter (MIT)

2 QUANTUM STATISTICS (Fermions, Bosons)

3 Ψ e e e i 2π n Ψ

4 QUANTUM STATISTICS OF PARTICLES identical e e.g. electrons e Indistinguishable particles fermions Ψ (particle 1,particle 2)= Ψ (particle 2,particle 1) bosons Ψ (particle 1,particle 2)=Ψ (particle 2,particle 1)

5 anyons 3-D 2-D { QUANTUM STATISTICS (Fermions, Bosons, Other) Leinaas, Myrheim 1977; F. Wilczek 1982 (+ cyclic 1-D)

6 the fractional quantum Hall effect Laughlin s ground state wave function: Ψ1/ m = quasi-hole localized at j <k ( z j zk ) m 1 exp( 4 z0 Ψ1/+ zm0 = (...) ( zl z0 ) Ψ1/ m l zl 2 ) l

7 BERRY (GEOMETRIC) PHASE a short digression: closed contour: Ψ iα e Ψ α = E (t )dt + γ Berry phase γ = i d Ψ (t ) Ψ (t ) dt

8 Laughlin wave function + qh Berry Phase (constant density) : Aharonov-Bohm Phase φ 2π ν φ0 ur r φ * (e / hc) Agd l = 2π (e / e) φ0 * e =ν e *

9 Berry Phase an extra term φ 2π ν + 2π ν φ0 Θ stat = ν

10 TWO OPTIONS to VISUALIZE a qh RESULT IS THE SAME gate voltage generated qh VG Φ0 flux generated qh

11 ANDERSON: WHAT IS THE Aharonov-Bohm PERIODICITY * WITH e = e / 3? 2h π φ 2 En = (n ) = 2 ml φ0 2 φ e e e = 3 * 2 2h π eφ 2 = (n ) 2 ml hc 2 2 periodicity hc φ0 = e 3φ0? But that would contradict Byers-Yang??

12 1 ν= 3 V Y.G. & D. J. Thouless Φ

13 electron tunneling qp tunneling Φ1 periodicity Φ0 quasi-particle tunneling periodicity 3Φ 0 electron tunneling (cf. Gefen and Thouless) CRUCIAL QUESTION: MATRIX ELEMENTS

14 Goldman et. al Φ0 oscillations with adding flux= or charge= 2e to central island

15 NOTE : e =ν e * Θ stat = 2π ν

16 Chamon, Freed, Kivelson, Sondhi, Wen (97); (Fabry-Perot interferometer) T. Martin ; Kane; A. Stern ; Stone ; Kim Fradkin, Nayak, Tsvelik, Wilczek; Das Sarma, Freedman, Nayak; Stern Halperin; Kitaev Shtengel; Hou, Chamon; Fradkin et al Law, Feldman, Y.G. : MACH-ZEHNDER INTERFEROMETER possibly the leading candidate

17 mesa D1 S1 D2 QPC1 QPC2 D2 S2 S1 MG2 MG1 S2 D2 D1 air bridge D 2 S D 1 QPC M2 G G 1 G QPC 1 G 2 Neder, Heiblum et al

18 Mach-Zehnder Photonic Interferometer D1 D2 M1 BS1 S D1 BS2 M2.)intensity (a.u D2 TS D2 = t BS1t BS 2 + rbs1rbs 2 e Visibility= T1 / T (phase difference ) π i Φ 2 = T0 + T1 cos Φ 5

19 why is a Mach-Zehnder interferometer a detector for (anyonic) quantum statistics? why is it a good detector? what are the signatures of anyonic statistics?

20 D2 S2 S2 S1 D2 D1

21 D2 S2 S2 S1 D2 D1

22 D2 S2 S2 S1 D2 D1

23 D2 S2 S2 S1 D2 D1

24 why is a Mach-Zehnder interferometer a detector for (anyonic) quantum statistics? why is it a good detector? what are the signatures of anyonic statistics?

25 ν = 1/ 3 zero temp. kinetic eq. p0 0 1 p1 p2 #qp trapped transition rates 2 ( a more detailed analysis: Keldysh) Law, Feldman, YG

26 D2 S2 S2 = 1/νD2 tunneling prob.ms1 periodic in S1 S1 S2 D2 D2 Γ1 pn = cﾰ1 ( Γ1 2 + Γ 2 2 ) + (cﾰ2 Γ1Γ 2* exp( iφ% ) + c.c.) ﾰ / Φ = 2πνΦ / Φ + 2πν n φ% = 2πν Φ 0 0 Aharonov-Bohm of fractional charge n statistical fluxoids Γ2 D1D1

27 WHY IS THIS ROBUST against fluctuations (of q.p., area )? dynamical averaging over q.p. # you cannot kill a dead horse more complicated rate flows for non-abelian Feldman, YG, Kitaev, Stern

28 compare with Fabry-Perot interferometer Chamon, Freed, Kivelson, Sondhi, and Wen (1997) Φ sensitivity to # of qp s

29 why is a Mach-Zehnder interferometer a detector for (anyonic) quantum statistics? why is it a good detector? what are the signatures of anyonic statistics?

30 D2 S2 S2 Γ1 S1 I = I0 + IΦ D2 Γ2 D1 generalization to finite T,V (kinetic, Keldysh) π /Θ [ I Φ (Γ 2 ) I Φ (Γ 2 = 0)] = [ I 0 (Γ 2 ) I 0 (Γ 2 = 0)]

31 NOISE (kinetic eq. ; zero temp) also Keldysh S = δ Q2 / t N tunneling events t = N ( + + ) ν p0 p1 p2 average square of each tunneling event: δ t 2n = 2 pn2 1 fluctuations of total time : δ t = N pn2 δq noise S=e I charge Q0 = Ne transmitted after t=t1+ δ 2t ( ) charge transmitted after t Q0 I δptn 2 1 pn2 FANO factor > ν maximum value =1 ( one of the p n is small) non- Abelian case: max. Fano factor >1

32 Abelian: invariance under V - V Non- Abelian : asymmetry under V -V (V>T)

33 OTHER APPROACHES topological algebraic few more details

34 WHAT DOES IT MEAN TO MEASURE THE STATISTICAL PHASE? accepted definition OBSERVABLES THAT explicitly DEPEND ON KLEIN FACTORS

35 Ψ Ψ same branch (edge) -Bosonic fields guarantee antisymmetrization

36 OPPOSITE BRANCHES (EDGES) WE NEED KLEIN FACTORS Ψη =gggkη iφη ﾵ f (Nη ) e {Kη, Kη ' } = 2δηη '

37 iπν iπν e

38 NO UNIQUE / CONSISTENT WAY TO ASSIGN A KLEIN FACTOR TO A FRACTIONALLY CHARGED QUASI-PARTICLE ASSIGN ASSIGN A A KLEIN KLEIN FACTOR FACTOR TO TO A A TUNNELING TUNNELING OPERATOR OPERATOR cf. Kane 2003 see also Ponomarenko & Averin

39 D2 S2 S1 S2 D2 D1

40 D2 S2 S1 S2 D2 T1 T2 D1 T1 = K1Γ1 exp(i[φ1 (0, t )ν φ2 (0, t )ν ]) + h.c Γ1 : exp( iν evt ) T2 = K 2 Γ 2 exp(ggg) Γ 2 : exp( iν evt ) exp(i 2πνΦ / Φ 0 )

41 Tr[(K1 ) n1 ( K1 ) m1 ( K 2 ) n 2 ( K 2 ) m2 ]= D2 S2 Tr[( K 2 ) n 2 ( K 2 ) m2 (K1 ) n1 ( K1 ) m1 ] S1 S2 D2 Γ1 (K1 ) n1 ( K1 ) m1 ( K 2 ) n 2 ( K 2 ) m2 = exp[i 2πν (n1 m1) 2 ] ( K 2 ) n 2 ( K 2 ) m2 (K1 ) n1 ( K1 ) m1 INTEGER Γ2 D1

42 Summary: π /Θ [ I Φ (Γ 2 ) I Φ (Γ 2 = 0)] = [ I 0 (Γ 2 ) I 0 (Γ 2 = 0)] FANO factor : Abelian : max 1 non-abelian: >1 Abelian: invariance under V - V Non- Abelian : asymmetry under V -V (V>T) MZI : robust

43 experimental doldrums very difficult to obtain interferometry signal (see however Willett et al for FP) even the measured fractional charge is a puzzle

44 ν = 5/ 2 ν = 7/3 similar results for ν = 1/ 3 1 T=12mK effective charge e* 0.9 νb=7/ Average transmission t7/3-2 1 Dolev, Heiblum et al

45 I = I0 + IΦ T > ev (linear response) T < ev 1/ν (3ν 2) /(2ν 2 ν ) GΦ (T ) : [G0 (T )] I Φ (V ) : [ I 0 (V )] D2 S2 S1 Γ1 S2 D2 Γ2 D1 2ν [ I 0 (Γ 2 ) I 0 (Γ 2 = 0)] = [ I Φ (Γ 2 ) I Φ (Γ 2 = 0)]

46 SUMMARY 2ν [ I 0 (Γ 2 ) I 0 (Γ 2 = 0)] = [ I Φ (Γ 2 ) I Φ (Γ 2 = 0)] DEPENDS ON KLEIN FACTORS temperature dephasing visibility vs. arm asymmetry

47 END OF INTRODUCTION

48 HANBURY BROWN & TWISS INTERFEROMETRY WITH ANYONS

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

50 D2 t r S1 r t D1 Buttiker, Blanter S2

51 Fermions: bd1 ˆ as 1 = S bd 2 as 2 nˆ1 = bd 1bD1 D2 t r S1 S2 r t ˆn2 = bd 2bD 2 P (1,1) = Ψ nˆ1nˆ2 Ψ = 0 as 2 as 1bD 1bD1 b b a a D1 D 2 D 2 S 1 S 2 0 P(1,1) = (T + R ) 2 = 1 P(2,0) Buttiker, Blanter Fermions 0 P(1,1) 1 P(0,2) 0

52 D2 t r S1 r t bunching =enhanced noise anti-bunching =reduced noise Buttiker, Blanter D1 S2

53 adding another handle: A-B flux Samuelson,Sukhorukov, Buttiker 2004 S1 ɸ D1 D 2 S2 I D1 I D 2 : A1 + A4 2 A2 + A3 2 * 4 * 2 A1 A A3 A

54 SUMMARY Fermions/Bosons HBT HBT with AB fermions anti bunching bosons bunching fermions φ # cos(2π ) φ0 bosons

55 realization of HBT + AB Mach-Zehnder interferometer 2 Mach-Zehnder interferometers

56 D2 S2 S1 S2 D2 D1 air bridge D 2 S D 1 QPC M2 G G 1 G QPC 1 G 2 Heiblum et al

57 introducing two-particle interference Neder, Ofek, Heiblum,

58 actual sample Neder, Ofek, Heiblum Nature 2007

59 A CARICATURE WITH (Abelian) ANYONS

60 A CARICATURE WITH (Abelian) ANYONS P(2,0) r t t r 1 (1 + cos(πν ))(2,0) + P 4 P(1,1) 1 (1 cos(πν )) 2 P(1,1) + P(0, 2) = 1 1 fermions P (1,1) = 0 bosons 1 / 2 classical

61 S1 ɸ D1 D 2 S2

62 One & Two Qp processes

63 S1 ɸD D1 2 S2

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

65 1 4 focus on AB component of current-current correlation: two questions: What is the order in Γ of leading harmonics? What is its sign?

66 What is the order in Γ of leading harmonics? What is its sign?

67

68 calculated by Keldysh

69 What is the order in Γ of leading harmonics? What is its sign? positive boson-like in agreement with the caricature

70 SUMMARY: 1.current-current non-analytic in single-particle rate

71