Time-bin entanglement of quasi-particles in semiconductor devices. Luca Chirolli, Vittorio Giovannetti, Rosario Fazio, Valerio Scarani

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1 Time-bin entanglement of quasi-particles in semiconductor devices Luca Chirolli, Vittorio Giovannetti, Rosario Fazio, Valerio Scarani

2 The concept of energy-time entanglement τ 1 τ 2 total energy well defined ω 1 + ω 2 = E state entangled in energy ψ dωâ 1 (ω)â 2 ( E/ ω) 0 since τ 1 is very short the two photons are emitted approximately at equal time ψ dtê( ) 1 (t)ê( ) 2 (t) 0 coherent superposition of times at which the two photons have been emitted

3 Franson interferometer second order coherence length Φ 1 Φ 2 introduce a time delay first order coherence length quantum interference between L 1 L 2 and S 1 S 2 amplitudes give rise to coincidence events

4 Biphoton state field operator a(x, t) = dω 2πc a(ω)e ik(ω)x ωt k(ω) =±ω/c a(x, t) =a(x ct) a(x t ) Mach Zenhder: ã α (x α )= 1 2 [a α(x α )+a α (x α + δ α )] output field input field path difference length Coincidence events: two-photon process I(x L ; x R ) ã L (x L)ã R (x R)ã R (x R )ã L (x L ) State produced by the source Ψ = dxdyψ(x, y)a R (x)a L (y) 0 biphoton I(x L ; x R )= 0 ã R (x R )ã L (x L ) Ψ Ψ(x R,x L )+Ψ(x R + δ R,x L + δ L ) 2 coherent superposition of two photon states short path long path

5 Electronic implementation of optics setup photonic wave guide edge states in the Integer Quantum Hall Effect beam splitter Quantum Point Contact (QPC) Optical Mach-Zenhder interferometer Electronic Mach-Zenhder interferometer in the IQHE single photon source single electron source n-photon coincidence measurements measurement of current, noise and higher order momenta two-photon source?

6 Edge states in the IQHE 2D system in a perpendicular magnetic field B H = 2 2m 2 y + 1 2m ( i x eby) 2 + V c (y) Landau gauge confinement potential V c (y) eey ψ nk (x, y) =e ikx χ nk (y) E n (y) ω c n V c (y n ) y 0 = k/ 2 B linear dispersion ψ(x) = 1 L k e ikx c k (i t v F k)c k (t) =0 fermionic field operator i( t + v F x )ψ(x, t) =0 linear dispersion ψ(x, t) = 1 L k e ik(x v F t) c k constant drift velocity

7 Electronic Mach-Zenhder interferometer Y. Ji et al. Nature 422, 415 (2003)

8 Single electron emitter energy spacing between dot levels /h D(V g ) attempt frequency transmission τ 1 = D /h escape rate I(t) = q τ e t/τ 0 t T/2 I ω = 2qf 1 iωτ Féve et al. Science 316, 1169 (2007)

9 Electron-hole pair dot driving sequence: two periods weak tunneling regime electron emission is probabilistic but quantized H = D d d + k k c k c k + k V k d c k + h.c. dot: driven single fermionic level D = D (t) φ 0 = 1 Θ F one electron in the dot + free Fermi sea φ(t) = U(t) φ 0 expand at second order in the tunneling Hamiltonian

10 Electron-hole pair from charge conservation: 1) dot above the Fermi energy electron emission in the first half-period of the first or second period 2) dot below the Fermi energy hole emission in the second halfperiod of the first or second period 3) an electron can be absorbed only when the dot is empty a hole can be emitted only after an electron is emitted at second order we expect a coherent superposition + +

11 we define a proper qubit basis Time-bin entanglement time-bin = period 1, 0 e,h 0, 1 e,h state with an electrn (hole) in the first time-bin state with an electrn (hole) in the second time-bin we can rewrite the state of the system at second order as Ψ 1, 0 e 1, 0 h + 1, 0 e 0, 1 h + 0, 1 e 0, 1 h entangled two-qubit state violates a Clauser-Horne-Shimony-Holt (CHSH) inequality

12 Electronic Franson interferometer driven QPC we separate electron and hole with a time-dependent QPC electron right hole left electron and hole time/space shifted v F τ length difference larger than single particle coherence length 2v F τ ϕ L ϕ R Aharonov-Bohm phase of the left L and right R Mach-Zenhder

13 Electronic Franson interferometer particles via long path in the MZs acquire an AB phase difference length in MZs set to compensate one time-bin electron: hole: positive AB phase negative AB phase + e i ϕ R + e -i ϕ L + e i ϕ R -i ϕ L + e i ϕ R + e -i ϕ L + e i ϕ R -i ϕ L + e i ϕ R + e -i ϕ L + e i ϕ R -i ϕ L time

14 Current cross-correlations we detect coincidence events by measuring current cross-correlations δc ij (t, t )=δi Li (t)δi Rj (t ) δi αi (t) =I αi (t) I αi (t) we are interested in coincidence in the second time-bin 1, 0 e 1, 0 h e iϕ R iϕ L 0, 1 e 0, 1 h , 0 e 0, 1 h e iϕ R 0, 1 e 0, 1 h , 1 e 0, 1 h 0, 1 e 0, 1 h +... δc R1,L1 = Γ eiϕ R + e iϕ R iϕ L 2 Γ V 2 tun

15 Effective description tunneling restricted to a small set of k-states around the dot level H T = V d c k + h.c, k D = D /v F k [k D, k] ψ kd (x) 1 L effective field operator k [k D, k] e ikx c k if x y 1/ k {ψ kd (x), ψ k D (y)} δ(x y) {ψ kd (x), ψ k D (y)} 0 coarse graining of the position resolution

16 Effective field if k D k/2 >k F + ψ (x) dot k D Δk well defined electon above the Fermi sea k F ψ e (x) ψ kd (x) if k D + k/2 <k F well defined hole in the Fermi sea ψ h (x) ψ k D (x) k F ψ(x) -k D dot Δk

17 Effective description the dot driving induces a time dependence in the field operator ψ kd (t)(x) the average momentum follows the dot level in the interaction picture H tun (t) =V Le iϕ D(t) d ψ kd (t)( v F t)+h.c. ϕ D (t,t ) dot dynamical phase Ψ t dt t dt e iϕ D(t,t ) ψ kd (t )( v F t )ψ k D (t ) ( v F t ) Θ F electron-hole wavefunction adds an electron if k D (t ) > 0 adds a hole if k D (t ) < 0 Ψ dxdy Ψ(x, y)ψ h (x)ψ e(y) 0

18 Current cross-correlations treat time dependent QPC in the Landauer Buettiker formalism electrons will always be transmitted holes will always be reflected drop from description current operator effective field I(x) =v F ψ (x)ψ(x) ψ(x) ψ e (x)+ψ h (x) current cross-correlator δc(x e ; y h )= v 2 F ψ (x e )ψ(x h )