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 effectVoltage biased superconducting tunnel junctionJosephson, Phys. Lett. 1, 251 (1962)

Josephson current

Dc-component Cohen, Falicov, Philips, PRL 8, 316 (1962)

S S

I

V1 2 3

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 reflectionsBoltzmann approach (incoherent), weak linkKlapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1083).

Gives subharmonics excess current

Current

S S

V

Klapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983).

e

h

Quantum point contactsCoherent transport, single mode contact, transparency D

Atomic point contacts

TheoryArnold, 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).

Scheer et al, PRL 78, 3535 (1998).Scheer et al, Nature 394, 154 (1998).Ludoph et al, PRB 61, 8561 (2000).

Noise: multiple chargesTheory 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 statisticsFull distribution of transported charge

Long measurement time Charge

Cumulant generating function

Cumulants

[ non Gaussian fluctuations]

Coherent transportTheoryCuevas, Belzig, PRL 91, xxx (2003); PRB xx, xxx (2004).

Johansson, Samuelsson, Ingerman, PRL 91, 187002 (2003),

Cumulant generating function

n-particle scattering probability

Incoherent transport Strong phase breaking suppressed proximity effect

S S S S S S

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).

ballistic diffusive chaotic

No theory for full counting statistics!

Incoherent full counting statisticsStochastic path integral approach, semiclassicsPilgram 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,

generalized Boltzmann-Languevin approach

fL fRf

t

f

Related approaches: Kindermann, Beenakker, Nazarov, PRB, xxxx, Bodineau, Derrida, PRL 92, 180601 (2004), Gutman, Mirlin, Gefen, xxxx.

Our work

Generating function, NS-interface

S S

Example: ballistic SNS-junction, interface barriersOctavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983).

e

h

Muzukantskii, Khmelnitskii, PRB 50, 3982 (1994).

S

Composed from elementary scattering probabilities

Andreev / normal reflection probability

N

Pilgram, Samuelsson, PRL 94, 086806 (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 equationsSemiclassical limit path integral in saddle point approximation

Solution inserted back into

Cumulants givesOTBK

.....

(No simple expression...)

CumulantsNumerical evaluation, differential cumulants

Subharmonic gap structure diverges at low

Probability distributionStationary phase approximation

With from

Conditional distributionfunctions

Low voltage limitLow voltage limit , finite normal back scattering

E

t

Quasiparticle diffusion in energy space

Generating function

Diffusive wire with renormalized chargeJordan, 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, 187002 (2003), Cuevas, Belzig, PRB xxx

Diffusive wire

S S 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, 100501 (2001).

shows subharmonic gap structure

Excess generating function

Conclusions