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

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.

Josephson effect Voltage biased superconducting tunnel junction Josephson, Phys. Lett. 1, 251 (1962)  Josephson current  Dc-component Cohen, Falicov, Philips, PRL 8, 316 (1962) SS I V 123

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

Multiple Andreev reflections Boltzmann approach (incoherent), weak link Klapwijk, Blonder, Tinkham, Physica B+C, (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, (1083). Gives  subharmonics  excess current Current SS V Klapwijk, Blonder, Tinkham, Physica B+C, (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, (1983). e h

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

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

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

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

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

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

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

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

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

Cumulants Numerical evaluation, differential cumulants  Subharmonic gap structure  diverges at low

Probability distribution Stationary phase approximation With from Conditional distribution functions

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).

general incoherent low voltage behavior Low voltage cumulants, diverges for Holds for large class of junctions, only different Generating function, saddle point solution  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... Coherent junctions, diverges for Naveh, Averin, PRL 82, 4090 (1999), Johansson, Samuelsson, Ingerman, PRL 91, (2003), Cuevas, Belzig, PRB xxx

Diffusive wire SS  Diffusive normal region  Normal conductance  Negligiable interface resistance Recent experiments on third cumulant Reulet, Les Houches.

 electron charges transfered Arbitrary voltage approach For a voltage Injection energies  electron charges transfered  Effective conductance Injection energies  Effective conductance 3e 2e

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

Conclusions