Slide 1

Transport through ballistic chaotic cavities in the classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Support: NSF, Packard Foundation Humboldt Foundation With: Saar Rahav Wesleyan October 26 th, 2008

Slide 2

Ballistic chaotic cavities: Energy levels level density mean level density: depends on size Conjecture: Fluctuations of level density are universal and described by random matrix theory Bohigas, Giannoni, Schmit (1984) valid if L

Slide 3

Spectral correlations Correlation function Random matrix theory in units of Altshuler and Shklovskii (1986) This expression for 1 only; Exact result for all is known. b: magnetic field

Slide 4

Ballistic chaotic cavities: transport level densityconductance G

Slide 5

Ballistic chaotic cavities: transport level densityconductance G G is random function of (Fermi) energy and magnetic field b Marcus group

Slide 6

Ballistic chaotic cavities: transport level densityconductance G Conjecture: Fluctuations of the conductance of an open ballistic chaotic cavity are universal and described by random matrix theory Blmel and Smilansky (1988)

Slide 7

Ballistic chaotic cavities: transport level densityconductance G Conjecture: Fluctuations of the conductance of an open ballistic chaotic cavity are universal and described by random matrix theory Blmel and Smilansky (1988) Requirement for universality: Additional time scale in open cavity: dwell time D ( )

Slide 8

2 1 Conductance autocorrelation function Correlation function Random matrix theory Jalabert, Baranger, Stone (1993) Efetov (1995) Frahm (1995) in units of P j : probability to escape through opening j L

Slide 9

This talk Semiclassical calculation of autocorrelation function for ballistic cavity Role of the Ehrenfest time E Recover random matrix theory if E > D (classical limit).

Slide 10

Semiclassics conductance = transmission = 1 reflection R j : total reflection from opening j : classical trajectories A : stability amplitude S : classical action Miller (1971) Blmel and Smilansky (1988)

Slide 11

Semiclassics conductance = transmission = 1 reflection R j : total reflection from opening j Miller (1971) Blmel and Smilansky (1988) : classical trajectories A : stability amplitude S : classical action

Slide 12

Conductance fluctuations Need to calculate fourfold sum over classical trajectories. But: Trajectories 1, 1, 2, 2 contribute only if total action difference S is of order h systematically

Slide 13

Conductance fluctuations 1 21 2 1 21 2 1 11 1 2 22 2 Need to calculate fourfold sum over classical trajectories. But: Trajectories 1, 1, 2, 2 contribute only if total action difference S is of order h systematically Sieber and Richter (2001)

Slide 14

Conductance fluctuations 1 21 2 1 21 2 1 11 1 2 22 2 1 11 1 2 22 2 1 21 2

Slide 15

1 21 2 1 21 2 1 11 1 2 22 2 1 11 1 2 22 2 1 21 2 This contribution vanishes for chaotic cavity

Slide 16

Conductance fluctuations EE EE Duration of small angle encounter with action difference S ~ h is Ehrenfest time E : : Lyapunov exponent Aleiner and Larkin (1996)

Slide 17

Conductance fluctuations EE EE random matrix theory if E