1. Quantum charge fluctuation in a superconducting grain Manuel Houzet SPSMS, CEA Grenoble In collaboration with L. Glazman (University of Minnesota) D. Pesin (University of Washington) A. Andreev(University of Washington) Ref: Phys. Rev. B 72, 104507 (2005)

2. Isolated superconducting grains In "large" grains, conventional Bardeen-Cooper-Schrieffer theory applies: The gap in the grain obeys the self consistency equation: Thermal fluctuations (Ginzburg-Levanyuk criterion): Same criterion: Mean level spacing: Bulk gap at Anderson, 1959 gapped spectrum normal spectrum gap in the grain at

3. Parity effect in isolated superconducting grains The number of electrons in the grain is fixed → parity effect Parity effect subsists till ionisation temperature: Averin and Nazarov, 1992 Tuominen et al., 1992 Free energy difference at low temperature:

4. N Coulomb blockade in almost isolated grains SN Charge transfered in the grain: Energy: Coulomb blockade requires low temperature large barrier

5. Lafarge et al., 1993Junction Al/Al 2 O 3 /Cu Experiment Finite temperature: vanishes at The thermal width remains small

6. Quantum charge fluctuations at finite coupling Even side SN 22 Odd side S N e Competing states near degeneracy point SNSN e "vaccum corrections" to ground state energy are different: We calculate them in perturbation theory with Hamiltonian: This gives a correction to the step position (odd plateaus are narrower) e h e h

7. SN e h 22 Effective Hamiltonian for low energy processes near(even side) Tunnel coupling Quasiparticule scattering Electron-hole pair creation in the lead Schrieffer-Wolf transformation: Even state(0 electron = 0 q.p.) odd state (1 electron = 1 q.p.) Shape of the step (1)

8. Simplification : For a large junction, only the states with 0 ou 1 electron/hole pair are important in all orders. The difficulty : creates n electron/hole pairs diverges at Perturbation theory diverges in any finite order Shape of the step (2) Fermi sea in lead Analogy with Fano problem: Continuum of states with excitation energies: Discrete state with energy U < 0 without coupling

9. 1/2 0 3/2 2e 1 e Scenario for even/odd transition quantum mechanics for a single particule in 3d space + potential well The bound state forms only if the well is deep enough: Its energy dependence is close to Quantum width of the step: Corrections are small for large junctions

10. ionisation temperature of the bound state Finite temperature Step position width Step position hardly changes at T<T q Width behaves nonmonotonically with T Excited Fermi sea in lead Continuum of statesDiscrete state

11. not sufficient to test Matveev’s prediction: Conclusion quantum phase transition in presence of electron-electron interactions N-I-Nmultichannel Kondo problem (idem for S-I-N at Δ>E c ) Matveev, 1991 S-I-SJosephson coupling → avoided level crossing Bouchiat, 1997 N-I-Sabrupt transition Matveev and Glazman, 1998 S-I-Nat Δ<E c = new class: charge is continuous, differential capacitance is not Physical picture of even/odd transition: bound state formed by an electron/hole pair across the tunnel barrier. Experimental accuracy? Lehnert et al, 2003 N