Interactions and Charge Fractionalization in an Electronic Hong-Ou-Mandel Interferometer

Transcription

1 Interactions and Charge Fractionalization in an Electronic Hong-Ou-Mandel Interferometer Thierry Martin Centre de Physique Théorique, Marseille in collaboration with C. Wahl, J. Rech and T. Jonckheere Phys. Rev. Lett. 112, (2014) I out R I out L 1 / 22

2 Electronic quantum optics in quantum Hall systems Quantum optics analogs with electrons, i.e. the controlled preparation, manipulation and measurement of single excitations in ballistic conductors INGREDIENT LIST Photons Electrons Light beam Chiral edge QHE Beam-splitter Point contact Coherent light source Single electron source 2 / 22

3 Single electron sources Lorentzian voltage pulses b (t) 2 a Energy (t) ~ VG 1 t + iw Charge pumps, quantum turnstiles D 1D VG V(t) V(t) V(t) = h W t eπ t2 + W2 c [Dubois et al., Nature ( 13)] 17 [Giblin et al., Nature Comm. 3, 930 ( 12)] Fig1 Flying electrons on surface acoustic waves [Hermelin et al., Nature 477, 435 ( 11)] [McNeil et al., Nature 477, 439 ( 11)] Ù Mesoscopic capacitor [Fève et al., Science 316, 1169 ( 07)] opens the way to all sorts of interference experiments! 3 / 22

4 Single electron source: the LPA mesoscopic capacitor 10 Setup [Fève et al., Science 316, 1169 ( 07); Mahé et al., PRB 82, ( 10)] quantum dot quantum dot coupled to edge V G D discrete levels spaced by t r tunable dot transmission via V g V G sets level broadening V exc Operating the source time-dependent excitation voltage V exc (t) well described through Floquet scattering theory emission of one electron + one hole per period V exc(t), I(t) t V exc Injected wave-packet? exponential shape 4 / 22

5 Hong-Ou-Mandel interference experiment Two-photon interferences Interference experiment [Hong, Ou and Mandel, PRL 59, 2044 ( 87)] non-linear crystal generates photon pairs measure the coincidence rate Coincidence rate two identical photons sent on a beam-splitter necessarily exit by the same output channel signature of bosonic statistics counts occurrences of photons present in the two output channels dip is observed when photons arrive at the same time signatures of incoming wave packets 5 / 22

6 HOM with electrons: general principle Setup 2 single electron sources counter-propagating channels coupled at QPC measure output currents I out L I out R Zero-frequency cross-correlations of output currents SRL out = dtdt [ IR out (x, t)il out (x, t ) IR out (x, t) IL out (x, t ) ] Differences with photons they obey fermionic statistics existence of a Fermi sea, hole-like excitations,... thermal effects do matter they interact via Coulomb interaction 6 / 22

7 HOM with electrons: theoretical results for ν = 1 [Jonckheere et al. Phys. Rev. B 86, ( 12)] SHOM/2SHBT Collision of identical wave-packets When electrons arrive independently S 12 < 0 : sum of the partition noise flat background contribution Hanbury-Brown and Twiss 0.2 D = 0.2 D = 0.5 D = δt More exotic situations When electrons arrive simultaneously S 12 = 0 HOM/Fermi/Pauli dip signatures of injected object D 1 = 0.1 D 2 = SHOM t 2SHBT ɛ = /2 ɛ = S HOM t D 1 = 0.2 D 2S 2 = 0.5 HBT t Different packets asymmetry 0.6 Ε E F Ε E F V V 0.0 V V Electron-hole collision peak t 7 / 22

8 HOM with electrons: experimental results Experimental setup two independent sources synchronized electron emission, and collision at the beamsplitter observe two-particle interferences? [Bocquillon et al., Science 339, 1054 ( 13)] Correlations q Main results 12 Pauli dip Time delay Random par**oning [ps] FIG. 3: Excess noise q between both sources switched on and both sources switched off as a function Something of the delay τ. The noise values are special normalized by the value happens on the plateau observed for long when delays. The we go beyond the simple ν = 1 picture blurry blue line represents the sum of the partition noise of both sources. The blue trace is an exponential fit by q = 1 γe t τ0 /τe. The red trace is obtained using Floquet scattering theory. As expected Flat background contribution (random partiotioning) Pauli dip for simultaneous injection But... How come it does not reach 0? decoherence 8 / 22

9 Beyond ν = 1: interactions! Different possible types of interactions between counter-propagating channels, near the QPC strong interactions within a channel: Fractional QHE between co-propagating channels at filling ν = 2 (this talk) Formalism and methods ν = 2 injection prepared state ϕ Bosonization Keldysh propagation bosonized H (+ diagonalization) tunneling scattering matrix 9 / 22

10 Injection Simplified model of injection: prepared state injection in the past at t = T 0 : ϕ = O ( T 0 ) preparation operator O = O R O L with preparation operator O R,L = dkϕ R,L (k)ψ R,L (k; t = T 0) 0 ground-state dkϕ(k)ψ k True one shot injection of electron or hole ϕ L (x) Versatile: any wave-packet 2Γ Exponential wave-packets ϕ R,L (x) = e ±(iɛ0+γ)x/v F θ( x) v F x(µm) ˆϕ L (ɛ) ɛ 0 = 0.7K ɛ(k) Tunable resolution γ = ɛ 0 /Γ } ɛ 0 = 0.175K γ = 1 Γ = 0.175K } ɛ 0 = 0.7K γ = 8 Γ = K 10 / 22

11 Propagation ν = 2 QHE Two channels: outer and inner (j = 1, 2) Bosonization identity: ψ j,r (x) = U j,r 2πa exp (i φ j,r (x)) Hamiltonian H = H 0 + H intra + H inter j = 1 Propagation along the edge r = R j = 2 H 0 = v (0) j dx ( x φ j,r ) 2 π j=1,2 r=r,l j = 2 j = 1 r = L 0 u U Typically U v F Intra-channel interaction H intra = π U dx ( x φ j,r ) 2 j=1,2 r=r,l Inter-channel interaction H inter = 2 π u dx ( x φ 1,r ) ( x φ 2,r ) r=r,l 11 / 22

12 Charge fractionalization Hamiltonian: H = π r=r,l Diagonalization mixing angle θ tan θ = dx [v + ( x φ +,r ) 2 + v ( x φ,r ) 2] 2u v 1 v 2 Rotated fields and eigen-velocities { φ1 = cos θ φ + + sin θ φ φ 2 = sin θ φ + cos θ φ and v ± = v (v1 ) v 2 v 2 ± + u qinj,l Propagating excitations γ = 8, L = 5µm qcopr,l x x injection channel co-propagating channel γ = 8, L = 5µm Average charge density q s,r (x, t) = e π xφ s,r (x, t) ϕ Strong interaction: θ = π/4 Excitations characterized by the charge they carry / Free propagation of two modes: fast charged φ + and slow neutral φ 12 / 22

13 Tunneling QPC couples counter-propagating channels two possibilities Two setups s = 1, I1 I I1 I SETUP 1 Tunneling Hamiltonian SETUP 2 [ ] H tun = Γ ψ s,r (0)ψ s,l(0) + ψ s,l (0)ψ s,r(0) Scattering matrix: ( ) outgoing ( ) ( ψs,r T i R ψs,r = ) incoming ψ s,l i R T T is the transmission and R the reflexion probability ψ s,l 13 / 22

14 Performing the calculation I out s,r (x, t) I out s,r (0, t) ψ j,r (0, t) out ψ j,r (0, t) in φ in j,r (0, t) φ in ±,r(0, t) Zero-frequency crossed correlations of outgoing currents SRL out = dtdt [ Is,R(x, out t)is,l out (x, t ) ϕ Is,R(x, out t) ϕ Is,L out (x, t ] ) ϕ Linear dispersion dt Ij,r out (x, t) = dt I out j,r (0, t) Electric current: Ij,r out(0, t) = ev : ψ j,r (0, t)ψ j,r (0, t) : out out ( ) ( ) ψj,r (0, t) ψj,r (0, t) Scattering matrix: = S ψ j,l (0, t) ψ out j,l (0, t) in Bosonization: ψ j,r (0, t) = U j,r exp ( iφ in j,r (0, t) ) in 2πa { φ in Diagonalization: 1,r (0, t) = cos θ φ in +,r (0, t) + sin θ φ in,r (0, t) φ in 2,r (0, t) = sin θ φ in +,r (0, t) cos θ φ in,r (0, t) Quantity of interest: [ ] SRL out Is,r out (x, t) [ ] SRL out φ in ±,r(0, t) 14 / 22

15 Performing the calculation: final expression Focus on the following situation different setups s = 1, I1 I2 strong interaction θ = π 4 symmetric injection ±L I1 I2 SETUP identical packets ϕ(x) time delay δt SETUP Final expression of noise for the Hong-Ou-Mandel experiment { S HOM = 2S 0 Re dτre [ g(τ, 0) 2] ϕ R (y R )ϕ R dy R dz (z R) ϕ L (y L )ϕ L R (2πa) 2 g(0, y R z R ) dy L dz (z L) L N R (2πa) 2 g(0, z L y L ) N L [ ]} hs (t; y L + L, z L + L)h s (t + τ δt ; L y R, L z R ) dt h s (t + τ; y L + L, z L + L)h s (t δt ; L y R, L z R ) 1 [ ( ) ( ) sinh i πa sinh i πa 2 [ βv + sinh( ia v+ ) t+x ] 1 2 [ βv βv ) )]1 h s(t; x, y) = + sinh( ia v t+x )] s 3 2 /π βv /π ) ) g(t, x) = sinh( ia+v+ t x βv + /π sinh( ia+v t x βv /π sinh( ia+v+ t y βv + /π sinh( ia+v t y βv /π 15 / 22

16 + Interference pattern: expected structures time delay δt = 0 time delay δt = ±L v+ v v +v interference of excitations with same charge and velocity interference of excitations with different velocity and possibly different charge SETUP 1 I1 I2 v v + v + v v v + v + v SETUP 2 I1 I2 v + v + v + v v v v v + flat background contribution from non-interfering excitations 16 / 22

17 Results: setup 1 SHOM(δT ) (e 2 RT ) S HBT L = 2.5µm L = 5µm ɛ0 = 175mK, γ = 1 setup δt (ns) Central dip noise reduction destructive interference of / excitations loss of contrast due to interactions, strong dependence on resolution Side dips -excitations with different velocities destructive interference velocity mismatch : asymmetry + smaller than half central dip SHOM(δT ) (e 2 RT ) S HBT L = 2.5µm L = 5µm ɛ0 = 0.7K, γ = 8 setup δt (ns) 3-dip structure + flat background contribution (no interference) 17 / 22

18 Results: setup 2 1 SHOM(δT ) (e 2 RT ) S HBT L = 2.5µm L = 5µm ɛ0 = 175mK, γ = 1 setup δt (ns) Central dip identical for the 2 setups interference independent of the charge carried by the excitations, both in sign and amplitude Side peaks excitations with opposite charge constructive interference vanish as the resolution increases velocity mismatch: asymmetry SHOM(δT ) (e 2 RT ) S HBT L = 2.5µm L = 5µm ɛ0 = 0.7K, γ = 8 setup δt (ns) peak-dip-peak structure + flat background contribution (no interference) 18 / 22

19 Results η ν = 1 Central dip vs. packet broadening bottom of dip sinks deeper for wider packets γ Contrast η = 1 S HOM(δT =0) 2S HBT dramatic reduction as resolution increases Electron-hole collision confirms the pattern / or / destructive / constructive side peaks smaller than dips / interference is weaker SHOM(δT ) (e 2 RT ) S HBT L = 5µm ɛ0 = 175mK γ = 1 e-h setup 1 e-h setup δt (ns) 19 / 22

20 Refined model for the source More experimental results are now becoming available for setup 1 Injection problematic for spatially extended packets From the SES In our model Source Propagation length L QPC Refined setup and contrast Propagation length L Propagation length L + v F τ e QPC I1 I2 η(ɛ 0, τ e, β, L, v +, v ) L +L η(ɛ 0, τ e, β, τ s ) All these are given by the experiment No adjustable parameters! Allows for a more careful comparison with experimental results 20 / 22

21 Comparing with experimental results Contrast Experimental data Theoretical predictions Contrast η Parameters: ɛ 0 = 0.7 K 1/(k B β) = 100 mk τ s = 70 ps f = 1 GHz τ e (ps) Possible sources of decoherence? Differences between emitters Environmental noise Coulomb interaction Only the last scenario can account for all observed data! 21 / 22

22 Conclusions Our interacting model recovers the main experimental features Detailed quantitative comparison is under way! Strong coupling between channels accounts for a sensible loss of contrast of the HOM central dip The contrast strongly depends on the energy resolution of the injected wave-packet Fast and slow modes interfere and produce, depending on the charge carried by the colliding excitations, smaller asymmetric dips or peaks Interactions and charge fractionalization in an electronic HOM interferometer Claire Wahl, Jérôme Rech, Thibaut Jonckheere, Thierry Martin Phys. Rev. Lett. 112, (2014) 22 / 22

23 Floquet scattering theory Stationary scattering matrix ˆb(ɛ) = S(ɛ)â(ɛ) relates outgoing fermionic operators ˆb(ɛ) to incoming ones â(ɛ): Dynamic scattering matrix S(ɛ) S(ɛ 1, ɛ 2 ) two energy arguments! The energy of incoming and scattered electron can be different Floquet scattering matrix extends this to time-periodic problems, by expressing the absorbtion/emission of a quantized number of energy quanta Ω (Ω is the frequency of the periodic potential): ˆb(ɛ) = m U m (ɛ) = n U m (ɛ)â(ɛ m ) c n c n+ms(ɛ n ) with ɛ ±m = ɛ ± m Ω, and c n the Fourier coeff. of the periodic potential Back to TALK 1 / 2

24 Tunneling and refermionization Tunnel Hamiltonian Refermionization Ψ p± (x) = U p± 2πa e iφ p±x [ ] H tun = Γ ψ 1,R (0)ψ 1,L(0) + ψ 1,L (0)ψ 1,R(0) where φ A± = ± (φ 1,R φ 1,L ) ± (φ 2,R φ 2,L ) 2 φ S± = ± (φ 1,R + φ 1,L ) ± (φ 2,R + φ 2,L ) 2 Full Hamiltonian is now quadratic! H = i v σ dxψ pσ(x) x Ψ pσ (x) ΓΨ A+ (0)Ψ A (0) p,σ Scattering matrix r 0 = cos ϕ and t 0 = sin ϕ with ϕ = Γ/( v + v ) ( ) outgoing ( ) ( ) incoming ΨA+ t0 ir = 0 ΨA+ Ψ A ir 0 t 0 Ψ A Outgoing current? exact same expression up to defining r 0 = R and t 0 = T Back to main 2 / 2