Full counting statistics of incoherent multiple Andreev reflection

1. Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland

2. Outline • Voltage biased Josephson junctions, multiple Andreev reflections. • Coherent and incoherent transport. • Noise and full counting statistics, stochastic path integral approach. • Examples: double barrier and diffusive wire junctions. • Low voltage - energy space diffusion. • Conclusions.

3. Josephson effect Voltage biased superconducting tunnel junction Josephson, Phys. Lett. 1, 251 (1962) I S S 1 2 3 V • Josephson current • Dc-component Cohen, Falicov, Philips, PRL 8, 316 (1962)

4. Subharmonic gap structure and excess current Additional features in IV-curve Cooper pair tunneling Schrieffer, Wilkins, PRL 10, 17 (1963) Taylor, Burstein, PRL 10, 14 (1963) • Subharmonic gap structures • Excess current Van der Post et al, PRL 73, 2611 (1994)

5. Multiple Andreev reflections Boltzmann approach (incoherent), weak link Klapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1083). Klapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983). S S e h V Current Gives • subharmonics • excess current

6. Quantum point contacts Coherent transport, single mode contact, transparency D Theory Arnold, J. Low. Temp. Phys. 68 1 (1987). Bratus et al, PRL 74, 2110 (1995). Averin, Bardas, PRL 75, 1831 (1995). Cuevas, Martin-Rodero, Levy-Yeyati, PRB 74, xxxx (1996). Atomic point contacts Scheer et al, PRL 78, 3535 (1998). Scheer et al, Nature 394, 154 (1998). Ludoph et al, PRB 61, 8561 (2000).

7. Noise: multiple charges Theory Cuevas, Martin-Rodero, Levy-Yeyati, PRL 82, 4086 (1999), Naveh, Averin, PRL 82, 4090 (1999). Zero frequency noise Fano factor Experiment Cron et al PRL 86, 4104 (1999). Quanta of multiple charge Dieleman et al, PRL 79, 3486 (1997).

8. Full counting statistics Full distribution of transported charge • Long measurement time • Charge Cumulant generating function Cumulants [ non Gaussian fluctuations]

9. Coherent transport Theory Johansson, Samuelsson, Ingerman, PRL 91, 187002 (2003), Cuevas, Belzig, PRL 91, xxx (2003); PRB xx, xxx (2004). Cumulant generating function n-particle scattering probability

10. S S S S S S Incoherent transport Strong phase breakingsuppressed proximity effect diffusive ballistic chaotic Experimentally important regime (noise) Jehl et al, PRL 83, 1660 (1999), Hoss et al, PRB 62 4079 (2000), Roche et al Physica C 352, 73 (2001), Hoffmann, Lefloch, Sanquer, EPJB 29 629 (2002). Current and noise theory (incoherent) Bezuglyi et al, PRL 83, 2050 (1999), Nagaev, PRL 86, 3112 (2001), Bezuglyi et al, PRB 63, 100501 (2001), Samuelsson et al, PRB 65, 180514 (2002). No theory for full counting statistics!

11. Incoherent full counting statistics Stochastic path integral approach, semiclassics Pilgram et al, PRL 90, 206801 (2003), Jordan, Sukhorukov, Pilgram, J. Math. Phys. 45, 4386 (2004). • Separation of time scales: Nagaev, xxxx. • fast quasiparticle scattering, • slow dynamics of distribution functions, f f f f L R t generalized Boltzmann-Languevin approach Related approaches: Kindermann, Beenakker, Nazarov, PRB, xxxx, Bodineau, Derrida, PRL 92, 180601 (2004), Gutman, Mirlin, Gefen, xxxx.

12. Our work Pilgram, Samuelsson, PRL 94, 086806 (2005) Example: ballistic SNS-junction, interface barriers Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983). h N S S S e Generating function, NS-interface Muzukantskii, Khmelnitskii, PRB 50, 3982 (1994). Andreev / normal reflection probability Composed from elementary scattering probabilities

13. Stochastic path integral approach • Formulate as path integral over possible internal • charge configugurations • Integrate out fast charge fluctuations effective generating function in slow variables . For

14. Saddle point equations Semiclassical limit path integral in saddle point approximation Solution inserted back into Cumulants gives OTBK (No simple expression...) .....

15. Cumulants Numerical evaluation, differential cumulants • Subharmonic gap structure • diverges at low

16. Probability distribution Stationary phase approximation With from Conditional distribution functions

17. Low voltage limit Low voltage limit , finite normal back scattering E t Quasiparticle diffusion in energy space Generating function Diffusive wire with renormalized charge Jordan, Sukhorukov, Pilgram, J. Math. Phys. 45, 4386 (2004).

18. Generating function, saddle point solution Low voltage cumulants , diverges for ...... Holds for large class of junctions, only different general incoherent low voltage behavior Coherent junctions, diverges for Naveh, Averin, PRL 82, 4090 (1999),Johansson, Samuelsson, Ingerman, PRL 91, 187002 (2003), Cuevas, Belzig, PRB xxx • Theory breaks down at , inelastic scattering cuts off divergence. • Effect of environment not considered. Reulet et al, PRL xx, xx (xxxx), Kindermann, Beenakker, Nazarov PRL ...

19. S S Diffusive wire • Diffusive normal region • Normal conductance • Negligiable interface resistance Recent experiments on third cumulantReulet, Les Houches.

20. Arbitrary voltage approach 2e 3e For a voltage Injection energies • electron charges transfered • Effective conductance Injection energies • electron charges transfered • Effective conductance

21. Generating function – adding up the two processes First cumulants Nagaev, PRL 86, 3112 (2001), Bezuglyi et al, PRB 63, 100501 (2001). shows subharmonic gap structure Excess generating function

22. Conclusions