Quantum shot noise: From Schottky to Bell
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1 Quantum shot noise: From Schottky to Bell Markus Büttiker University of Geneva Peter Samuelsson Eugene V. Sukhorukov Vanessa Chung Valentin Rytchkov Micheal Polianski Micheal Moskalets Niels Bohr Summer Institute 2005 EKO 2
2 From Schottky to Bell Classical shot noise: W. Schottky, Ann. Phys. (Leipzig) 57, 541 (1918) 2 Quantum shot noise: Khlus (1987), Lesovik (1989), Yurke and Kochanski (1989), Buttiker (1990) r t Entangled States J. Bell, Physics 1, 195 (1964)
3 Content of lecture Scattering theory of electron noise Two-particle Aharonov-Bohm effect Orbital quasi-particle entanglement [Entanglement test: Bell Inequality] Dynamic generation of entangled states
4 Scattering Theory of Electron Noise 4
5 Scattering channels 5 asymptotic perfect translation invariant potential seprable wave function energy of transverse motion channel threshold energy for transverse and longitudnial motion scattering channel
6 Occupation number and current amplitudes 6 Buttiker, PRB 46, (1992) Incident current at kt = 0 Incident current at kt > 0 Occupation number Creation and annihilation operators < > = statistical average «Incident current» «Current amplitude»
7 The scattering matrix and the Fermi-field 7 L ^ a L ^ a R R T μ L L sample T μ R R ^ b L ^ b R Fermi-field in contact Elimination of Equilibrium. Stat. Mech.
8 The current matrix Buttiker, PRB 46, (1992) 8 Remark: particle current versus total current dc-transport, ac-transport, noise
9 Conductance matrix Butttiker, PRL 57, 1761 (1986), IBM J. Res. Developm. 2, 17 (1988)
10 Shot-noise: two-terminal 10 Consider kt = 0, V>0, and a two-terminal conductor: Quantum partition noise If all Schottky (Poisson) Fano factor Khlus (1987) Lesovik (1989) Buttiker (1990)
11 Shot-noise: Qunatum point contact 11 A. Kumar, L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, PRL 76, 2778 (1996) M. I. Reznikov, M. Heiblum, H. Shtrikman, D. Mahalu, PRL 75, 40 (1996) Ideally only one channel contributes
12 Shot-noise: Metallic diffusive wire Beenakker and Buttiker, PRB 46, 1889 (1992) 12 Henny et al. PRB 59, 2871 (1999)
13 Shot-noise: Chaotic cavity Jalabert, Pichard and Beenakker, Europhys. Lett. 27, 255 (1994) 1 for symmetric cavity with Oberholzer et al., PRL 86, 2114 (2001)
14 Shot-noise: correlations 14 Consider multi-terminal conductor at kt = 0, M source contacts with distribution at voltage All other contacts grounded with distribution Correlation measured bewteen two grounded contacts: M =1, partition noise M =2, exchange effects, two particle Aharonov-Bohm effect, orbital entanglement, violation of Bell inequality
15 Current correlations, Oberholzer et. al. experiment Oberholzer et al. Physica E6, 14 (2) 15 Bias configuration: κ=1 λ T=1 R
16 Optical and Electrical Mach-Zehnder- 2 Interferometer One particle Aharonov-Bohm effect 4 ϕ B Φ ϕ A Ji et al, Nature 422, 415 (200)
17 Two-particle Aharonov-Bohm effect 17
18 The quantum phase of current correlations Mesoscopic conductor with N contacts S= s s s s s s s s s 1 2 s 1N At kt = 0, N s N1 s NN 4 At kt = 0, M contacts with N-M contacts at M=1, partition noise M > 1, relative phase of scattering matrix elements becomes important Exchange interference effects: Buttiker, PRL 68, 84 (1992)
19 19
20 HBT-Intensity Interferometer Hanbury Brown and Twiss, Nature 177, 27 (1956) 20 Interference not of amplitudes but of intensities Optics: classical interpretation possible Quantum mechanical explanation: Purcell, Nature 178, 1449 (1956) Indistinguishable particles: star Statistics, exchange amplitudes θ d I (t) I (t+ τ) A B
21 Electrical HBT Interferometer 21 Samuelsson, Sukhorukov, Buttiker, PRL 92, (2004) A C D B A Φ C D B All elements of the conductance matrix are independent of AB-flux
22 Two-particle Aharonov-Bohm Effect A B C D Φ D C 4 2 A B Fourth-order interference: Current-current correlation Samuelsson, Sukhorukov, Buttiker, PRL 92, (2004) For 22
23 Orbital Entanglement 2
24 Two-particle sources Superconducting-normal hybrid structures 24 Bogoliubov-de Gennes picture Pair-tunneling picture μ S μ N = μ S ev μ S μ N Normal-conductors Electron picture of tunneling Samuelsson, Sukhorukov and Buttiker, PRL 91, (200) Electron-hole picture ev R T μ μ L μr Beenakker et al, PRL 91, (200) P. Samuelsson, E.V. Sukhorukov, M. Buttiker cond-mat/050016
25 Two-particle entanglement Electron side Hole side A B C D Tunnel limit: orbitally entangled e-h-state incident state tunneling limit Samuelsson, Sukhorukov, Buttiker, PRL 92, (2004) 25
26 Thermal photon sources C B 8 5 A 4 D 2 7 Sources: black body Energy window: narrow band filters 4 No violation: In contrast to electron injection through a single quantum channel where in each time-slot only one particle is injected, in the bosonic case, many particles can be injected.
27 Summary of recent advances 27 Orbital entanglement P. Samuelsson, E. V. Sukhorukov, M. Buttiker, PRL 91, (200) Zero-frequency measurement (Bell Inequality) X. Maitre, W. D. Oliver, Y. Yamamoto, Physica E6, 01 (2) N.M. Chtchelkatchev et al., Phys. Rev. B 66, (2002) P. Samuelsson, E.V. Sukhorukov and M. Buttiker, PRL 91, (200) Normal components: e-h-pairs C.W.J. Beenakker et al, PRL 91, (200); P. Samuelsson, E.V. Sukhorukov and M. Buttiker, PRL 92, (2004). Controllable geometries P. Samuelsson, E.V. Sukhorukov and M. Buttiker, PRL 92, (2004)
28 Entanglement on demand 28
29 Electron-hole generation: oscillating barrier potentials P. Samuelsson and M. Buttiker, Phys. Rev. B 71, (2005) C 29 (a) (b) 2 V C(t) 4 A B (c) (d) 1 V (t) D D C. W. J. Beenakker, M. Titov and B. Trauzettel, PRL 94, (2005)
30 0 Summary Scattering theory conductance and noise Two-particle Aharonov-Bohm effect Orbital quasi-particle entanglement [Entanglement test: Bell Inequality ] Dynamic generation of entangled states
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