Lecture 3 Shot noise correlations: The two-particle haronv-bohm effect 1 6 1 C 3 B 8 5 4 D 3 4 7 Markus Buttiker University of Geneva IV-th Windsor Summer School on Condensed Matter Theory, organized by B. ltshuler, P. Littlewood and J. von Delft. Cumberland Lodge, Windsor Royal Park, Windsor, UK, 06-19 ugust, 007.
Shot Noise: Multi-terminal 4 Mesoscopic conductor with N contacts 3 1 S= s s s s s s 11 1 13 1 3 s s s 31 3 33 s 1N t kt = 0, N s N1 s NN 4 t 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, 843 (199)
Intensity-Interferometers 5 Hanbury Brown and Twiss, Nature 177, 7 (1956) Mandel, Phys. Rev. 8, 99 (1983) star θ d 0 1 0 1 I (t) I (t+ τ) B Interference of independent photons: R. Kaltenbach et al. (Zeilinger), PRL 96, 4050 (006) (synchronization) M. Halder, et al. (Gisin), unpublished. (down conversion)
HBT-Intensity Interferometer Hanbury Brown and Twiss, Nature 177, 7 (1956) 6 Interference not of amplitudes but of intensities Optics: classical interpretation possible Quantum mechanical explanation: Purcell, Nature 178, 1449 (1956) Indistinguishable particles: Statistics, exchange amplitudes θ star d 0 1 0 1 I (t) I (t+ τ) B "The Quantum Phase of Flux Correlations in Wave Gudies", Buttiker, Physica B175, 199 (1991); PRL 68, 843, (199).
Optical and Electrical Mach-Zehnder- Interferometer 0 1 0 1 1 7 1 3 One particle haronov-bohm effect 4 ϕ B 4 0 1 0 1 0 1 0 1 00 11 0 1 0 1 Φ ϕ Ji et al (Heiblum), Nature 4, 415 (003) 3
Electrical Mach-Zehnder-Interferometer Ji et al (Heiblum), Nature 4, 415 (003) 8 0 1 0 1 1 ϕ B 4 Φ 0 1 0 1 0 1 0 1 00 11 ϕ 0 1 0 1
Two-particle interferometer Samuelsson, Sukhorukov, Buttiker, PRL 9, 06805 (004) 9 6 5 1 4 1 C D 3 B 8 7 5 1 00 11 0 1 0 1 00 11 1 6 4 0 1 0 1 00 11 Φ 00 11 C 0 1 0 1 00 11 0 1 D 00 11 B 7 4 3 00 11 00 11 8 3 4 Yurke and Stoler, PR 46, 9 (199) 3 00 11 ll elements of the conductance matrix are independent of B-flux
Two-particle haronov-bohm Effect 0 00 0 0 00 1 11 1 1 11 00 0 0 11 1 1 00 11 00 11 00 11 0 0 00 1 1 11 00 11 1 6 5 7 8 3 4 1 3 4 B C D Φ 1 3 4 5 6 7 8 1 D C 3 4 B Samuelsson, Sukhorukov, Buttiker, PRL 9, 06805 (004) For 10 Spin polarized [N > ; Sim and Sukhorukov (006)]
Two-particle haronov-bohm effect: Experiment I I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu and V. Umansky, Nature 448, 333 (007). 11
gate voltage Two-particle haronov-bohm effect: Experiment II I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu and V. Umansky, Nature 448, 333 (007). 1 flux
Spin entanglement 13 + + S a b B..a kind of correlation quite different from previously known correlations.. Quantum communication Quantum computation? Here: Orbital entanglement
Orbital entanglement 14 U + + B a D b NS-structures Pair-tunneling picture Normal conductors Electron-hole picture Samuelsson, Sukhorukov, Büttiker, PRL 91, 15700 (003) Beenakker, Emary, Kindermann, van Velsen, PRL 91, 147901 (003)
Two-particle entanglement Electron side Hole side 0 0 1 1 00 11 00 11 00 11 0 0 00 1 1 11 00 11 00 0 11 1 1 6 5 7 8 3 4 1 3 4 B C D Tunnel limit: orbitally entangled e-h-state incident state tunneling limit Samuelsson, Sukhorukov, Buttiker, PRL 9, 06805 (004) 15
Entanglement test: Bell Inequality 16 Comparison of classical local theory with quantum mechanical prediction. Here: entanglement test Bell Inequality: Clauser et al, PRL 3, 880 (1969) 16 measurements Orbital + 1 B S B + Samuelsson et al. PRL 91, 15700 (003) Chtchelkatchev et al, PRB 66, 16130 (00); Faoro, Taddei, Fazio, PRB 69, 1536 (004) violation of BI implies entanglement but not all entgangled states violate BI not invarinant under local rotations
Electron-electron entanglement through postselection Symmetric interferometer Electron-hole picture not appropriate Incident electron state is a product state: no intrinsic entanglement Two-particle effects nevertheless persists Bell Inequality can be violated Explanation: Entanglement through ``postselection'' (measurement) Joint detection probability 17 Bell parameter (Bell Inequality): α 3 β α 3 β Short time statistics: Pauli principle leads to injection of at most one electron in a short time interval: only two-particle transmission probability enters
Black body radiation sources 18 1 6 1 C 3 B 8 5 4 D 7 Sources: black body Energy window: narrow band filters 3 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.
Quantum Tomography with current and shot noise measurements complete reconstruction of one and two particle density matrices with current and shot-noise mesurements Samuelsson, Büttiker, Phys. Rev. B 73, 041305 (006) 19 Matrix elements Entanglement determined by
Quantum state tomography with quantum shot noise P. Samuelsson and M. Buttiker, Phys. Rev. B 73, 041305 (006) 0 + 1 S B1 B B B + Reconstruction of one-particle d.m. Setting I II III IV Parameters T =0, arb. T =1, arb. T =1/, =0 T =1/, =π/ 1 1 3 S 0 1 0 1 0 1 0 1 0 1 0 1 0 1 4 0 1 0 1 0 1 0 1 0 1 0 1 0 1 B1 B
Quantum State Tomography with shot noise P. Samuelsson and M. Buttiker, Phys. Rev. B 73, 041305 (006) 1 Setting I II III IV Parameters T =0, arb. T =1, arb. T =1/, =0 T =1/, =π/ reduced density matrix single particle 1 1 3 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 S 4 B1 B two particle 8 current measurements 16 current correlation measurements ( same as for BI but in contrast to BI, completely determines entanglement )
Summary Interference of independently emitted electrons Two-particle haronov-bohm effect Orbital entanglement Bell test of orbital entanglement Orbital quantum state tomography Pumping of entanglement
Quantum electronics: from Schottky to Bell Markus Büttiker University of Geneva Coherence and Quantum Optics 9 Rochester, NY, June 13, 007 EKO
Quantum optics/electronics Experiments and theory of quantum coherent electron transport resemble more and more quantum optics Important difference: Electrons are Fermions and carry charge Counting statistics Quantum measurements Generation of entangled states
Quantum Point Contact + Edge 5 States = Beam Splitter Quantum Point Contact Solid State Beam Splitter Edge state Transmission probability is a function of gate voltage
Opening the tool box.. Making quantum mechanics visible in man made structures Looking at individual quantum systems Webb et al. (1985) Persistent currents haronov-bohm effect Universal conductance fluctuations diffusive Conductance quantization. ballistic
Spin entanglement + + S a b B 4 S N N Dot & superconductor entanglers Recher et al. PRB 63 165314 (001) Lesovik et al. EPJB 4, 87 (001) Taddei and Fazio, PRB 65, 075317 (00) Oliver et al PRL 88 037901 (00) Bena et al PRL 89 037901 (00) Saraga and Loss, PRL 90 166803 (003)... + Long spin coherence length - Difficult to manipulate/detect
Quantum versus classical shot noise 6 Classical shot noise: W. Schottky, nn. Phys. (Leipzig) 57, 541 (1918) Quantum shot noise: Khlus (1987), Lesovik (1989), Yurke and Kochanski (1989), Buttiker (1990), Beenakker and van Houten (1991) r t
Shot Noise: Two-terminal 7 L ^ a L ^ a R R T μ L L sample T μ R R b ^ L ^ b R Quantum partition noise: kt = 0, V>0, If all Buttiker (1990) Fano factor Schottky (Poisson) Experiments: Reznikov (Heiblum) et al. PRL 75, 3340 (1995) Kumar, (Glattli) et al. PRL 76, 778 (1996)
Partition noise of fermions Oberholzer, Henny, Strunk et al, Physica E6, 314 () 9 κ=1 λ 1 3 00 11 T=1 R See also: Henny, et al., Science 84, 96 (1999); Oliver et al. Science 84, 99 (1999)
Why interference from independent sources? 11 Probably the most interesting effect of an independent particle description of transport Exchange can be made visible Vital * for implementation of quantum networks and quantum computing schemes: quantum repeaters Knill-Laflamme-Milburn (linear) quantum computing schemes In opitcs interference of independent sources is only possible with active synchronization of the sources. *Kaltenbaek (Zeilinger) et al. PRL 96, 4050 (006)
Visibility and violation of Bell Inequality (Dephasing) 0 0 0 0 1 1 1 1 0 0 1 1 00 11 0 0 1 1 1 6 5 7 8 3 4 1 3 4 B C D Φ Spatially seprated sources: qubit protectet against relaxation:
Two-particle Intensity Interferometers 0 Glattli et al. Schoenenberger Heiblum et al. 004 but no success!
Electrical Mach-Zehnder Interferometer II Neder, Heiblum, Levinson, Mahalu, Umansky, PRL 96, 016804 (006) 1 old!
Electrical Mach-Zehnder Interferometer III Litvin, Tranitz, Wegscheider and Strunk, PRB 75, 033315 (007) Chung, Samuelsson, and Buttiker, PRB 7, 1530 (005)
Electrical Mach-Zehnder Interferometer IV Roulleau, Portier, Glattli, Roche, Faini, Gennser, and D. Mailly, cond-mat/0704.0746 phase fluctuations! 3
Failure to see two-particle B-effect: Is there something wrong with theory? 4 Theory missing Luttinger like physics Resonant dephasing of the electronic Mach-Zehnder interferometer E.V. Sukhorukov, V. V. Cheianov, cond-mat/060988 Electrons in different reservoirs already entangled Suppression of Visibility in a Two-Electron Mach-Zehnder Interferometer Ya. M. Blanter, Yuval Gefen, cond-mat/0703186
Tomography: Medical 7 Cormack and Hounsfield J. Radon, 1917
Pauli s question M. G. Raymer, Contemporary Physics, 38, 343 (1997) Pauli s question (1933) : Can the wave function be determined uniquely from the distribution of position and momentum? No: such data is not tomographically complete. Pauli question generalized: Can one infer the whole complex function from some series of measurements on a large collection of identically prepared particles? Yes. J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1997)
Continuous variable tomography 3 Wigner function measure with Radon transformation obtain Wigner function from Wigner obtain via inverse Fourier transform taking gives
Quantum State Tomography: Experiments 4 ngular momentum state of an electron in hydrogene atom J.R. shburn et al, Phys. Rev. 41, 407 (1990). Quantum state of squeezed light D.T. Smithey et al, Phys. Rev. Lett. 70, 144 (1993). Vibrational state o a molecule T.J. Dunn, et al, Phys. Rev. Lett. 74, 884 (1995). Trapped ions D. Liebfried et al, Phys. Rev. Lett. 77, 481 (1996). tomic wave packets Ch. Kurtsiefer, T. Pfau, and Mlynek, Nature 386, 150 (1997). Polarization entangled photons P.G. Kwiat, et al, Nature 409, 1014 (001); T. Yamamoto et al ibid 41, 343 (003). Entanglement of two superconducting qubits Matthias Steffen et al, Science 313, 143 (006)
Dynamic electron-hole generation Samuelsson and Buttiker, Phys. Rev. B 7, 15536 (005) C 0 (a) (b) V C(t) 4 (c) (d) 1 V (t) D 3 B D 1 e h e 3 V 1 (t) Δ V (t) e h Rychkov, Polianski, Buttiker, Phys. Rev. B 7, 15536 (005) h 4 C. W. J. Beenakker, M. Titov, and B. Trauzettel, Phys. Rev. Lett. 94, 186804 (005) Multi-particle correlations of an oscillating scatterer, M. Moskalets, M. Buttiker, PRB 73, 15315 (006).
Summary 8 Shot noise correlations probe two-particle physics Two-particle haronov-bohm interferometer Orbital entanglement Bell test of orbital entanglement Orbital quantum state tomography Shot noise measurements determine the reduced two-particle density matrix up to local rotations Tomography delivers not only a criteria for entanglement but allows to experimentally quantify entanglement