Transport through ballistic chaotic cavities in the classical limitPiet BrouwerLaboratory of Atomic and Solid State PhysicsCornell UniversitySupport: NSF, Packard Foundation Humboldt Foundation With: Saar RahavWesleyanOctober 26th, 2008

Ballistic chaotic cavities:Energy levelslevel densitymean level density:depends on sizeConjecture:Fluctuations of level density are universal and described by random matrix theoryBohigas, Giannoni, Schmit (1984)valid ifL

Spectral correlationsCorrelation functionRandom matrix theoryAltshuler and Shklovskii (1986)This expressionfor e 1 only;Exact result for all e is known.b: magnetic field

Ballistic chaotic cavities: transportlevel densityconductance G

Ballistic chaotic cavities: transportlevel densityconductance GG is random function of (Fermi) energy e and magnetic field bMarcus group

Ballistic chaotic cavities: transportlevel densityconductance GConjecture:Fluctuations of the conductance of an open ballistic chaotic cavity are universal and described by random matrix theoryBlmel and Smilansky (1988)

Ballistic chaotic cavities: transportlevel densityconductance GConjecture:Fluctuations of the conductance of an open ballistic chaotic cavity are universal and described by random matrix theoryBlmel and Smilansky (1988)Requirement for universality:Additional time scale in open cavity: dwell time tD( )

Conductance autocorrelation function21Correlation functionRandom matrix theoryJalabert, Baranger, Stone (1993)Efetov (1995)Frahm (1995)Pj: probability to escape through opening jL

This talk Semiclassical calculation of autocorrelation function for ballistic cavity Role of the Ehrenfest time tE

Recover random matrix theory if tE > tD (classical limit).

Semiclassicsconductance = transmission = 1 reflectionRj: total reflection from opening jaa, b: classical trajectoriesAa: stability amplitudeSa: classical actionMiller (1971)Blmel and Smilansky (1988)

Semiclassicsconductance = transmission = 1 reflectionRj: total reflection from opening jMiller (1971)Blmel and Smilansky (1988)a, b: classical trajectoriesAa: stability amplitudeSa: classical actiona

Conductance fluctuationsNeed to calculate fourfold sum over classical trajectories.

But: Trajectories a1, b1, a2, b2 contribute only if total action difference DS is of order h systematicallya

Conductance fluctuationsa1 b2b1 a2a1 b1b2 a2Need to calculate fourfold sum over classical trajectories.

But: Trajectories a1, b1, a2, b2 contribute only if total action difference DS is of order h systematicallySieber and Richter (2001)

Conductance fluctuationsa1 b2b1 a2a1 b1b2 a2a1 b1b2 a2a1 b2

Conductance fluctuationsa1 b2b1 a2a1 b1b2 a2a1 b1b2 a2a1 b2This contribution vanishes for chaotic cavity

Conductance fluctuationstEtEDuration of small angle encounter with action difference DS ~ h is Ehrenfest time tE:l: Lyapunov exponentAleiner and Larkin (1996)

Conductance fluctuationstEtErandom matrix theory if tE

Conductance fluctuationst1t2a1 b2b1 a2t = t1, t2action differences accumulated between encounters:

Conductance fluctuationst1t2a1 b2b1 a2probabilities to enter/escape through contacts 1,2action differencesurvival probability

Conductance fluctuationst1t2a1 b2b1 a2action differencesurvival probabilityJalabert, Baranger, Stone (1993)Brouwer, Rahav (2006)Heusler et al. (2007)

Classical LimittEtEl: Lyapunov exponentrandom matrix theory if tE

Classical LimittEtETwo encounter give factor

Classical LimittEtEtEtEtE

Classical LimittEtEtEtEtOverlapping encounters give factortEBrouwer and Rahav (2006)

Classical LimittEtEtEaction differencefactor from encounterstpsurvival probability

Classical LimittEtEtEBrouwer and Rahav (2007)random matrix theoryclassical limittp

Conductance fluctuations

Tworzydlo et al. (2004)Jacquod and Sukhurukov (2004)2 var g102 103 104M~kFLBrouwer and Rahav (2006)Obtain var G by setting e = e, b = b:var G in classical limit still given by random matrix theory(but not correlation function!)

Summary: Classical LimitOther quantum effects Weak localization Shot noise Statistics of energy levels Proximity effect Quantum pumps Interaction effects

Anderson localization from classical trajectories (tE=0)Altshuler-Aronov correctionSNgBConductance fluctuations of an open ballistic chaotic cavity remain universal in the classical limit kFL , but they are not described by random matrix theory

Weak localization: semiclassical theoryLandauer formula Sa,b: classical action Aa,b: stability amplitudesJalabert, Baranger, Stone (1990)ab|Aa,b|2: probabilitytransmission matrix t Green function path integral stationary phase approximation a, b: classical trajectories; a and b have equal angles upon entrance/exitappendix 1

Weak localization: semiclassical theoryd g: Trajectories with small-angle self intersectionSieber, Richter (2001)appendix 1

Weak localization: semiclassical theoryd g: Trajectories with small-angle self intersectionSieber, Richter (2001)a,ba,bappendix 1

Weak localization: semiclassical theoryd g: Trajectories with small-angle self intersectionSieber, Richter (2001)a,ba,bStretch where trajectories are correlated: encounterappendix 1

Weak localization: semiclassical theoryd g: Trajectories with small-angle self intersectionSieber, Richter (2001)a,ba,bPoincar surface of sectionstable, unstable phase space coordinatesStretch where trajectories are correlated: encounterencounter: |u| < umax, |s| < smaxasmaxubumaxsappendix 1

Weak localization: semiclassical theoryd g: Trajectories with small-angle self intersectionSieber, Richter (2001)a,ba,bPoincar surface of sectionstable, unstable phase space coordinatesStretch where trajectories are correlated: encounterencounter: |u| < umax, |s| < smaxasmaxuumaxsbbappendix 1

Weak localization: semiclassical theorya,ba,basmaxuumaxssu: invariantsymplectic areabappendix 1

Weak localization: semiclassical theorya,ba,basmaxuumaxssu: invariantsymplectic areaSpehner (2003)Turek and Richter (2003)Heusler et al. (2006)bappendix 1

Weak localization: semiclassical theoryAB A, B: Phase space points (x,y,f) at beginning, end of self encounter Parameterize encounter using action difference DS (= symplectic area)t12 : typical classical actionBrouwer (2007)Exact in limit kFL at fixed tE/tD.appendix 1