1. Markus Büttiker University of Geneva Haifa, Jan. 12, 2007 Mesoscopic Capacitors

2. The elementary system Mesoscopic physics focuses on a few elementary geometries which illustrate best the effect we are interested in: Closed ringsPersistent currents Rings with leads Aharonov-Bohm effect Quantum point contacts Conductance quantization............ Cavity connected to one leadRC-time Mesoscopic physics = Wave nature of electrons is important

3. The mesoscopic capacitor single potential U geometrical capacitance C Buttiker, Thomas, Prêtre, Phys. Lett. A 180, 364 (1993) Gabelli, Fève, Berroir, Plaçais, Cavanna, Etienne, Jin, Glattli, Science 313, 499 (2006). What is the RC-time?

4. Classical versus quantum charge relaxation Classical circuit Mesoscopic capacitor For a single, spin-polarized channel is universal !! Buttiker, Thomas, Pretre, Phys. Lett. A 180, 364 (1993)

5. Dynamic external and internal response Internal response Invariance under arbitrary potential shift single potential U geometrical capacitance C Buttiker, Thomas, Pretre, Phys. Lett. A 180, 364 (1993) External response 7

6. Capacitance and Charge Relaxation electrochemical capacitance charge relaxation resistance Eigen channels of s; Universal for n =1; Buttiker, Thomas, Pretre, Phys. Lett. A180, 364 (1993)

7. Universal for n =1; For k degenerate channels Spin less electrons Spin degenerate channel Ideally coupled Carbon Nanotube Chaotic cavity coupled to two QPC (N channel) Chaotic cavity coupled to two QPC (one channel) Quantized charge relaxation resistances Brouwer and M. B., Europhys. Lett. 37, 441 (1997). Pedersen, van Langen, M. B., Phys. Rev. B 57, 1838 (1998). -------------------------------------------------------------------------------

8. Experimentalists model density of states assumption 1: uniform level spacing Gabelli (thesis), Gabelli et al, Science 313, 499 (2006) assumption 2: voltage dependence of transmission through QPC

9. Gabelli, Feve, Berroir, Placais, Cavanna, Etienne, Jin, Glattli Science 313, 499 (2006). Mesoscopic Capacitor: Experiment

10. Role of coherence: S. Nigg and M. Buttiker, (unpublished)

11. Role of coherence: S. Nigg and M. Buttiker, (unpublished)

12. Role of charge quantization M. Buttiker and S. E. Nigg, Nanotechnolgy 18, 044029 (2007) [S. E. Nigg, R. Lopez and M. Buttiker, PRL 97, 206804 (2006)]

13. Role of Interactions S. E. Nigg, R. Lopez and M. Buttiker, PRL 97, 206804 (2006) For poarized spin channel for “arbitrary” interactions!!

14. Coulomb blockade and spin degeneracy S. E. Nigg, R. Lopez, MB, Phys. Rev. Lett. 97, 206804 (2006) two levels low magnetic fields coupling strongly blockaded weakly blockaded

15. Quantized dynamic charge injection G. Feve, Thesis, ENS, Paris, Dec. 23, 2006

16. Summary For a single spin-polarized channel, self-consistent scattering theory predicts a universal charge relaxation resistance of half a resistance quantum A seminal experiment by Gabelli et al. supports this prediction Quantized dynamic charge emission and absorption Role of dephasing Quantized charge relaxation resistance Role of charge quantization Role of inetractions

17. Works on pumping

18. Pumping (w. M. Moskalets) Time-resolved noise of adiabatic quantum pumps M. Moskalets, M. Buttiker, Phys. Rev. B 75, 035315 (2007) ● Multiparticle correlations of an oscillating scatterer M. Moskalets and M. Büttiker, Phys. Rev. B 73, 125315 (2006) Magnetic-field symmetry of pump currents of adiabatically driven mesoscopic structures M. Moskalets and M. Büttiker, Phys. Rev. B 72, 035324 (2005) Scattering Theory of Dynamic Electrical Transport M. Buttiker, M. Moskalets, Lect. Notes Phys. 690, 33 (2006) Floquet scattering theory for current and heat noise in large amplitude adiabatic pumps M. Moskalets and M. Büttiker, Phys. Rev. B 70, 245305 (2004) Adiabatic quantum pump in the presence of external ac voltages M. Moskalets and M. Büttiker, Phys. Rev. B 69, 205316 (2004) Quantum pumping: Coherent rings versus open conductors M. Moskalets and M. Büttiker, Phys. Rev. B 68, 161311 (2003)

19. Pumping (w. M. Moskalets) Hidden quantum pump effects in quantum coherent rings M. Moskalets and M. Büttiker, Phys. Rev. B 68, 075303 (2003) Floquet states and persistent-current transitions in a mesoscopic ring M. Moskalets and M. Büttiker, Phys. Rev. B 66, 245321 (2002) Floquet scattering theory of quantum pumps M. Moskalets and M. Büttiker, Phys. Rev. B 66, 205320 (2002) Dissipation and noise in adiabatic quantum pumps M. Moskalets and M. Büttiker, Phys. Rev. B 66, 035306 (2002) Effect of inelastic scattering on parametric pumping M. Moskalets and M. Büttiker, Phys. Rev. B 64, 201305 (2001)

20. Pumping ● Leggett-Garg Inequality with a Kicked Quantum Pump A. N. Jordan, A. N. Korotkov, and M. Büttiker, Phys. Rev. Lett. 97, 026805 (2006) Shot noise of photon-excited electron-hole pairs in open quantum dots M. L. Polianski, P. Samuelsson, and M. Büttiker, Phys. Rev. B 72, 161302 (2005) ● Dynamic generation of orbital quasiparticle entanglement in mesoscopic conductors P. Samuelsson and M. Büttiker, Phys. Rev. B 71, 245317 (2005) ● Photon-assisted electron-hole shot noise in multiterminal conductors V. S. Rychkov, M. L. Polianski, and M. Büttiker, Phys. Rev. B 72, 155326 (2005) Noise-assisted classical adiabatic pumping in a symmetric periodic potential O. Usmani, E. Lutz, and M. Büttiker, Phys. Rev. E 66, 021111 (2002) Scattering theory of photon-assisted electron transport M. H. Pedersen and M. Büttiker, Phys. Rev. B 58, 12993 (1998)