Quantum optics with edge states

Transcription

1 Quantum optics with edge states Bernard Plaçais Gwendal Fève Jean-Marc Berroir Theory : P. Degiovanni et al. (ENS-Lyon), T. Martin et al. (CPT-Marseille) M. Butttiker et al. (Uni Genève)

2 Summary (quantum optics with edge states) Part 3 : Introduction to electrons optics o Electron optics principles o Single electron sources o Mesoscopic capacitor : average current and noise Part 4 : Quantum optics experiments o Partitioning single electrons o Colliding indistinguishable electrons o Decoherence and charge fractionalization Electron quantum optics in ballistic chiral conductors, Bocquillon et al., Annalen der Physik 526, 1 (2014)

3 Basic tools for optics What do we need to realize an interferometer? mirrors source detector ballistic 1D propagation beam splitters

4 Mach-Zehnder inteferometry with a biased contact (two-path interferometer) B V I Y. Ji et al., Nature 422, 415 (2003) Coherent source = biased contact V G «Natural source» not noisy Coherence length = contrast LL φφ = 20 μμμμ aaaa 20 mmmm ττ φφ 200 pppp for vv DD 10 5 mmss 1

5 Optics basic experiments Coherence properties of light Wavelike description encoded in the first order coherence of the electromagnetic field Source 1 : mirror τ mirror Detector : I(t) GG 1 (tt, tt + ττ) EE tt EE(tt + ττ) beam splitter Measured by e.g. Mach-Zehnder interferometry Source 2 : Mach-Zehnder interferometer: Statistical properties of light Corpuscular description encoded in intensity corrrelations GG 2 (tt, tt + ττ) II ppp tt II ppp (tt + ττ) Measured by Hanbury-Brown and Twiss (HBT) interferometry Source 1 : Detector 1 : Source 2 : Detector 2 : Aalto University, Fri. April 25th. 2014, Electron Quantum Optics with Edge States, Parts 3&4, BP I 1 ( t) I ( t 2 ') I1( t) I 2( t') Hanbury-Brown and Twiss interferometer

6 Electron optics in edge channels : formalism Electron field operator ψψ xx = 1 ddεε aa εε ee iiiiii ħvv FF hvv FF + ψ (x) F Chiral, 1D, propagation x ψψ xx, tt = ψψ xx vv FF tt ψψ xx EE + xx ψψ ϯ xx EE xx Charge density and electrical currrent II eeee x, t = evv FF ψψ ϯ xx, tt ψψ xx, tt II ppp x, t EE + xx, tt EE xx, tt GG 1 εε, εε = aa ϯ εε aa εεε Vacuum is the Fermi sea! transport subtracts the Fermi sea GG 1 xx, xx = GG 1 FF xx, xx + GG 1 xx, xx GG 1 FF εε, εε = θθ εε εε FF δδ(εε εε ) II(xx) = evv FF GG 1 xx, xx

7 Electron optics in edge channels : formalism Electron field operator ψψ xx = 1 ddεε aa εε ee iiiiii ħvv FF hvv FF + ψ (x) F Chiral, 1D, propagation x ψψ xx, tt = ψψ xx vv FF tt ψψ xx EE + xx ψψ ϯ xx EE xx Charge density and electrical currrent ε ' ε = ε II eeee x, t = evv FF ψψ ϯ xx, tt ψψ xx, tt II ppp x, t EE + xx, tt EE xx, tt GG 1 εε, εε = aa ϯ εε aa εεε Vacuum is the Fermi sea! transport subtracts the Fermi sea ε F ε F ε ' GG 1 xx, xx = GG 1 FF xx, xx + GG 1 xx, xx GG 1 FF εε, εε = θθ εε εε FF δδ(εε εε ) II(xx) = evv FF GG 1 xx, xx

8 Stationary versus single electron emitters Stationary emitters : ε ' ε = ε GG 1 xx, xxx = GG 1 xx xxx V GG 1 εε, εε δ εε εε cccccccccccccc oooo ttttt pppppppppppppppppppppp (nnnn oooooo dddddddddddddddd eeeeeeeeeeeeee iiii GG!) Single electron state, wave packet φφ xx, «a Landau QP» B ev ' ε ψψ ϯ [φφ] FF = 1 hvv FF ddxx φφ xx ψψ ϯ xx FF GG 1 GG 1 xx, xxx = φφ xx φφ xx εε, εε = φφ εε φφ εε γ ε ' ε = ε γ eeeeeeeeeeeeeeeeee εε > 0 hoooooooo (εε < 0) aaaaaaaaaa tttt ttttt ccccccccccccccccc (ffffffffffffff oooooo dddddddddddddddd tttttttttt iiii GG) ' ε

9 Electron pumps Triggerred electron emitters (2DEGs) J. Pekola et al., Rev. Mod. Phys. 85, 1421 (2013) M.D. Blumenthal et al., Nature Physics 3, 343 (2007) F. Hohls et al., Phys. Rev. Lett. 109, (2012) J. Fletcher et al., Phys. Rev. Lett. 111, (2013) Electrons flying on SAW (flying Q-dot) R. McNeil et al., Nature 477 (7365), 439 (2011) S. Hermelin et al., Nature 477 (7365), 435 (2011) Lorentzian voltage pulse (Levitons) D. A. Ivanov, et al., Phys. Rev B 56, 6839 (1997) J. Dubois et al., Nature 502, 659 (2013) Mesoscopic capacitor (electron/hole, energy resolved) G. Fève et al., Science 316, 1169 (2007)

10 Mesoscopic capacitor : A single eletron/hole emitter V G ev exc( t ) V G

11 Cartoon of single charge injection γ=d Dot QPC e 2D Gas B V(t) transmission D V(t) C 2eVexc t

12 Cartoon of single charge injection γ=d Dot QPC e 2D Gas B V(t) transmission D V(t) C 2eVexc t

13 Cartoon of single charge injection γ=d emission Dot QPC e 2D Gas B V(t) transmission D V(t) C 2eVexc τ ~ h D typically 100 ps t

14 Cartoon of single charge injection Dot QPC e 2D Gas B V(t) transmission D V(t) C 2eVexc τ ~ h D t

15 Cartoon of single charge injection Dot QPC e 2D Gas B V(t) transmission D V(t) C 2eVexc τ ~ h D t

16 Statistical averaging Energy scale (1 Kelvin 86 µev 20.8 GHz) : 2 KKKKKKKKKKKK Energy scale : ττ = ħ DD ps Dot QPC e 2D Gas B V(t) transmission D V(t) C 2eVexc It () f0 =1. 5GHz 2ef nA τ ~ h D t

17 Single electron current pulses Measuring single electron current pulses? Use a slow 32 MHz clock and a 1GHz fast averaging card (Acquiris) e D D eV exc = τ = 09. ns e τ =10 ns Time (ns)

18 Cartoon or reality? Quantization of the AC current? Use homodyne detection at large clock frequency : II ωω = (2iiiiii) (1 iiωωωω) 3 f = 180 MHz D=0.2 2eVexc I ω ( ef ) 2 1 Q t = e t eV exc /

19 Cartoon or reality? Subnanosecond control of the emission time Use homodyne detection at large clock frequency : II ωω = (2iiiiii) (1 iiωωωω)

20 Short time current correlations Single electron emission is characterized by current correlator II tt II(tt ) = QQ2 TT δδ tt tt AC noise correlations δδii tt δδii(tt ) = II tt II(tt ) II tt II(tt ) τ T T t' where II tt II(tt ) = QQ2 TT ee (tt tt)/ττ Whence the AC noise spectrum SS IIII (ττ, ωω) = 2 ee2 TT (ωωωω) (ωωωω) 2 ττ S II ( e 2 f 0 ) f p=0.02 τ=1.5x T/2 p=0.06 τ=0.5x T/2 p= 0.1 τ=0.3x T/2 p =0.5 τ=0.05x T/2 Experiment SS IIII (ττ, ωω = 2ππff 0 ) f/f 0 Aalto University, Fri. April 25th. 2014, Electron Quantum Optics with Edge States, Parts 3&4, BP

21 Cartoon or reality? Single electron source, beyond average current, average noise

22 Beyond current and noise, 2 particle interference aaaaaaaaaaaaaa cccccccccccccc II(tt) = ee φφ(tt) 2 ϕ(t) 2 cccccccccccccc nnnnnnnnnn II(tt)II(tt + ττ) = ee 2 φφ(tt) 2 δδ ττ t sensitive to waaaaaa pppppppppppp eeeeeeeeeeeeee φφ(tt) 2 How to access wave coherence? ρρ tt, tt = φφ tt φφ ttt ε ' ε = ε γ Asses decoherence: ρρ tt, tt = φφ tt φφ tt DD(tt tt ) ' ε

23 Part 4 (quantum optics with edge states) Part 3 : Introduction to electrons optics o Electron optics principles o Single electron sources o Mesoscopic capacitor : average current and noise Part 4 : Quantum optics experiments o Partitioning single electrons o Colliding indistinguishable electrons o Decoherence and charge fractionalization

24 Hanbury-Brown and Twiss partioning (classical AC source) Aalto University, Fri. April 25th. 2014, Electron Quantum Optics with Edge States, Parts 3&4, BP Input channel 1 noise QQ 1 = NN ee NN h δδδδ 1 2 = 0 SS 11 (ωω = 0) = AC source is noise free at DC DC noise correlations at outputs 3 2 δδqq 3 δδqq 4 = ee 2 TT(1 TT) NN ee + NN h Measures the average number of electron-hole pairs Low frequency noise spectrum SS 33 ωω = 0 = SS 34 ωω = 0 = 4ee 2 ff TT(1 TT) NN ee/h Enough to measure noise at output 3 1 Source 3 4 Beamsplitter 2

25 Exemples of QPC noise Thermal crossover SS II = 2eeee tanh 1 eeee 2kkθθ ee partition noise : SS II TT 1 TT T B PIB=f(Vbias), T=1/2, nu=3, B=2,75T Titre 7e 09 7e 09 2e 09 6e 09 6e 09 5e 09 5e 09 1,5e 09 Bruit 4e 09 4e 09 Transmission PIB (Vrms^2) 3e 09 3e 09 PIB (Vrms^2) 1e 09 Bruit 2e 09 2e 09 5e 10 1e 09 1e e 05 4e 05 6e 05 V bias (µv) V(V) Thermal noise partition noise

26 Partitioning single electrons : Experiment? HBT noise is smaller than classical prediction, why???? classical?

27 The explanation : Antibunching with thermal excitations emitted by auxiliary entry 4!!!! Tel= 150 mk fermions δδnn HHHHHH = NN ee,h NN ee + NN h 2 = ddεε δδδδ ee,h εε 0 ddεε δδnn ee εε + δδnn εε 0 ff εε

28 Two indistinguishable particles interference a b d c + Splitter scattering matrix : aa bb = cc dd aa + bb + 0,0 aaaa = 1 2 cc+ + dd + cc + dd + 0,0 cccc bosons (destructive 2 part. interference) : aa + bb + 0,0 aaaa = 1 2 2,0 cccc 0,2 cccc fermions (constructive 2 part. interference): aa + bb + 0,0 aaaa = 1,1 cccc

29 Two particle Hong-Ou-Mandel interferences 3 S34( t, t') S 1 I ( t 3 ) 4 Source 1 Source 2 S 2 I ( t 4 ') Hanbury Brown & Twiss experiment antibunching bunching

30 How indistinguisahble are photons? Undistinguishable photons Photons pairs : C. Hong et al., PRL 59(18), 2044 (1987) Independent emitters : J. Beugnon et al., Nature 440, 779 (2006) P. Maunz et al., Nature Physics 3, 538 (2007) E. B. Flagg et al., PRL 104, (2010) C. Lang et al., Nature Physics 9, 345 (2013)

31 2 particle interferences with 2 single electron sources e Single electron emitter e Time (ns) FFFFFFFF jjjjjjjjjj pppppppppppppppppppppp : P 1,1 = φφ 1 ee φφ 2 ee TTTT nnnnnnnnnnnnnnnnnnnn nnnnnnnnnn : SS HHHHHH SS HHHHHH = 1 dddd φφ 1 (tt + ττ)φφ 2 (tt) 2 = 1 ee ττ /ττ ee First order coherence

32 First electronic HOM Experiment (2013) D 1 = D2 0.4 τ e 58 ps τ /τ e τ e 62 ps e

33 Decoherence of the single electron wave packet!! HHHHHH eeeeeeeeeeeeeeeeeeee cccccccccccccccc pppppppppppppppp aaaaaa wwwwwwww aaaaaaaaaaaaaa!! SS HHHHHH SS HHHHHH = 1 dddd φφ 1 tt + ττ DD ττ φφ 2 tt 2 1 ee ττ /ττ ee ττ ee ττ φφ Aalto University, Fri. April 25th. 2014, Electron Quantum Optics with Edge States, Parts 3&4, BP

34 Autopsy of the quasi-particle (QP) HOM in the environment QQQQQQ aaaaaa nnnnnn eeeeeeeeeeeeeeeeeeeeee iiii 1DD; ttttttt dddddddddddddd iiiiiiii cccccccccccccccccccc ee h eeeeeeeeeeeeeeeeeeeeee IIIIIIIIII oooo ttttt iiiiiiiiiiiiiiiiiiiiii rrrrrrrrrrrr ssssssssssss eeeeeeeeeeeeeeee ssssssssss iiii oooooooooo ccccccccccccc ψψ iiii = dddd φφ xx λ(xx) oooooooooo ) 0 iiiiiiiiii OOOOOOpppppp ttttt eeeeeeeeeeeeeeee gggggggg eeeeeeeeeeeeeeeeee wwwwwww eeeeeeeeeeeeeeeeeeeeee ee h eeeeeeeeeeeeeeeeeeeeee ψψ oooooo = dddd φφ xx tt λ(xx) oooooooooo ) rr λ(xx) iiiiiiiiii

35 HOM experiment in the environment evidence for electron-fractionalization Aalto University, Fri. April 25th. 2014, Electron Quantum Optics with Edge States, Parts 3&4, BP OOOOOOpppppp ttttt eeeeeeeeeeeeeeee gggggggg eeeeeeeeeeeeeeeeee wwwwwww eeeeeeeeeeeeeeeeeeeeee eeeeeeeeeeeeeeeeeeeeee ψψ oooooo = dddd φφ xx tt λ(xx) oooooooooo ) rr λ(xx) iiiiiiiiii HOM-system eeeeeeeeeeeeeeee hoooooo aaaaaaaaaaaaaaaa. HOM-environment eeeeeeeeeeeeeeee eeeeeeeeeeeeeeee ssssssssssssssssssss

36 Conclusion o Is single particle description relevant? (YES, to some extent!) o Learn about particle statistics and coherence o Role of the Fermi sea o Many-body effects perspectives o Quantum tomography of the single particle states o Implementation in other systems (graphene? TI s?) o Use for quantum information (Flying qubits?)