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First things first 40 100 24,980

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Outline Physics = surprise Physics = surprise Electron counting = field theory Electron counting = field theory Full Counting Statistic of a scatterer Full Counting Statistic of a scatterer Two conductors together Two conductors together Saddle point Saddle point Quantum effects = Coulomb blockade Quantum effects = Coulomb blockade Ohmic Renormalization Ohmic Renormalization Quantum renormalization Quantum renormalization

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Full counting statistics of a coherent conductor Coherent conductor = scatterer = {T p } (Landauer-Buttiker) Coherent conductor = scatterer = {T p } (Landauer-Buttiker) Probability to transfer N electrons during time interval t = characteristic function (of ). Probability to transfer N electrons during time interval t = characteristic function (of ). Levitov ’93 Levitov ’93 Electrons gamble Electrons gamble Charge quantization LR

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Two conductors together Connect ‘m in series Connect ‘m in series Ohm’s law Ohm’s law Noises add as Noises add as FCS? FCS?

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make a field theory Fixed field(s) in the terminals (Keldysh = 2) Fixed field(s) in the terminals (Keldysh = 2) Fluctuating field(s) in the node Fluctuating field(s) in the node Composition law for the action Composition law for the action FCS!!!

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Predictions and experiments Slow realizations, saddle-point solution Slow realizations, saddle-point solution A-noisy, B- not(environment), shot noise limit A-noisy, B- not(environment), shot noise limit Beyond noise Beyond noise Pioneering measurement Pioneering measurement Of the 3 rd cumulant Of the 3 rd cumulant M. Kindermann, Yu. V. Nazarov and C. W. J. Beenakker, Phys. Rev. Lett. 90, 246805 (2003). Reulet BReulet B, Senzier J, Prober DE PHYS. REV. LETT. 91, 196601 (2003) Senzier JProber DE

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Properties of the field theory Low frequencies: always saddle-point solution Low frequencies: always saddle-point solution Physically: I-V curves, Full Counting Statistics Physically: I-V curves, Full Counting Statistics High frequencies: quantum effects High frequencies: quantum effects Coulomb blockade Coulomb blockade R A+B ≠ R A + R B R A+B ≠ R A + R B Renormalization loop Renormalization loop High frequency fluctuations renormalize the properties of low- frequency fluctuations High frequency fluctuations renormalize the properties of low- frequency fluctuations How big are renormalizations? How big are renormalizations? Universally, G Q /G Universally, G Q /G

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Coulomb blockade (G ≪ G Q ) Shifts potential = induces charge CgVg VgVg CCgCg N

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Where’s the capacitance? Never gone Never gone Defines high-energy cut-off Defines high-energy cut-off Traditional Coulomb blockade = at the cut-off Traditional Coulomb blockade = at the cut-off Island

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Renormalizations by Ohmic connector Complicated action – but in terms of transmission eigenvalues Complicated action – but in terms of transmission eigenvalues ZG Q – small, but log-divergent at low energies ZG Q – small, but log-divergent at low energies Program: renormalization of transmission eigenvalues Program: renormalization of transmission eigenvalues M. Kindermann and Yu. V. Nazarov, Phys. Rev. Lett. 91, 136802 (2003)

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Transmission flow Key to environment Key to environment Consequencies: Consequencies: two classes of coherent conductors two classes of coherent conductors universality of resonant tunneling universality of resonant tunneling

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Renormalizations in a quantum dot G ≫ G Q G ≫ G Q Is there Coulomb blockade? Is there Coulomb blockade? Extra parameter Extra parameter S –mean level spacing S –mean level spacing E C ≫ S E C ≫ S D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. Lett 94, 056801 (2005)

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Transmission flow Very similar, with its own conductance Very similar, with its own conductance Conductance itself subject to renormalization Conductance itself subject to renormalization That stops at That stops at

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Two scenarios Blockade with reduced Coulomb energy Blockade with reduced Coulomb energy Alternative : finite renormalization Alternative : finite renormalization

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© 2010 Eric Pop, UIUCECE 598EP: Hot Chips Conductance Quantization One-dimensional ballistic/coherent transport Landauer theory The role of contacts Quantum.

© 2010 Eric Pop, UIUCECE 598EP: Hot Chips Conductance Quantization One-dimensional ballistic/coherent transport Landauer theory The role of contacts Quantum.

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