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

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

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

Ballistic chaotic cavities: transport level densityconductance G

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

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 Blümel and Smilansky (1988)

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 Blümel and Smilansky (1988) Requirement for universality: Additional time scale in open cavity: dwell time  D ( )

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

This talk Semiclassical calculation of autocorrelation function for ballistic cavity Role of the “Ehrenfest time”  E Recover random matrix theory if  E <<  D Different, but still universal autocorrelation function if  E >>  D (“classical limit”).

Semiclassics “conductance” = “transmission” = “1 – reflection” R j : total reflection from opening j   : classical trajectories A  : stability amplitude S  : classical action Miller (1971) Blümel and Smilansky (1988)

Semiclassics “conductance” = “transmission” = “1 – reflection” R j : total reflection from opening j Miller (1971) Blümel and Smilansky (1988)  : classical trajectories A  : stability amplitude S  : classical action 

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 

Conductance fluctuations 1 21 2 1 21 2 1 11 1 2 22 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)

Conductance fluctuations 1 21 2 1 21 2 1 11 1 2 22 2 1 11 1 2 22 2 1 21 2

1 21 2 1 21 2 1 11 1 2 22 2 1 11 1 2 22 2 1 21 2 This contribution vanishes for chaotic cavity

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

Conductance fluctuations EE EE random matrix theory if  E <<  D : Lyapunov exponent Aleiner and Larkin (1996) Duration of small angle encounter with action difference  S ~ h is “Ehrenfest time”  E :

Conductance fluctuations 11 22 1 21 2 1 21 2  =  1,  2 action differences accumulated between encounters:

Conductance fluctuations 11 22 1 21 2 1 21 2 probabilities to enter/escape through contacts 1,2 action difference survival probability

Conductance fluctuations 11 22 1 21 2 1 21 2 action difference survival probability Jalabert, Baranger, Stone (1993) Brouwer, Rahav (2006) Heusler et al. (2007)

Classical Limit EE EE : Lyapunov exponent random matrix theory if  E <<  D In classical limit k F L : Dwell time  D unaffected (because classical), But  E Condition  E <<  D violated!

Classical Limit EE EE Two encounter give factor

Classical Limit EE EE EE EE EE

EE EE EE EE ’’ Overlapping encounters give factor EE Brouwer and Rahav (2006)

Classical Limit EE EE EE action difference factor from encounters pp survival probability

Classical Limit EE EE EE Brouwer and Rahav (2007) random matrix theory classical limit pp

Conductance fluctuations Tworzydlo et al. (2004) Jacquod and Sukhurukov (2004) 2 var g M~kFLM~kFL Brouwer and Rahav (2006) Obtain var G by setting  =  ’, b = b’: var G in classical limit still given by random matrix theory (but not correlation function!)

Summary: Classical Limit Other quantum effects Weak localization Shot noise Statistics of energy levels Proximity effect Quantum pumps Interaction effects Anderson localization from classical trajectories (  E =0) “Altshuler-Aronov correction” SN 11 22 I g B Conductance fluctuations of an open ballistic chaotic cavity remain universal in the classical limit k F L, but they are not described by random matrix theory

Weak localization: semiclassical theory Landauer formula S  : classical action A  : stability amplitudes Jalabert, Baranger, Stone (1990)   |A  | 2 : probability transmission matrix t … Green function … path integral … stationary phase approximation … ,  : classical trajectories;  and  have equal angles upon entrance/exit appendix 1

Weak localization: semiclassical theory     g: Trajectories with small-angle self intersection Sieber, Richter (2001) appendix 1

Weak localization: semiclassical theory   g: Trajectories with small-angle self intersection Sieber, Richter (2001)   appendix 1

Weak localization: semiclassical theory   g: Trajectories with small-angle self intersection Sieber, Richter (2001)   Stretch where trajectories are correlated: “encounter” appendix 1

Weak localization: semiclassical theory   g: Trajectories with small-angle self intersection Sieber, Richter (2001)   Poincaré surface of section stable, unstable phase space coordinates Stretch where trajectories are correlated: “encounter” encounter: |u| < u max, |s| < s max  s max u  u max s   appendix 1

Weak localization: semiclassical theory   g: Trajectories with small-angle self intersection Sieber, Richter (2001)   Poincaré surface of section stable, unstable phase space coordinates Stretch where trajectories are correlated: “encounter” encounter: |u| < u max, |s| < s max  s max u u max s      appendix 1

Weak localization: semiclassical theory    s max u u max s su: invariant Poincaré surface of section stable, unstable phase space coordinates encounter: |u| < u max, |s| < s max   g: Trajectories with small-angle self intersection Stretch where trajectories are correlated: “encounter” Sieber, Richter (2001) “symplectic area”   appendix 1

Weak localization: semiclassical theory    s max u u max s su: invariant Poincaré surface of section stable, unstable phase space coordinates encounter: |u| < u max, |s| < s max “symplectic area” Spehner (2003) Turek and Richter (2003) Heusler et al. (2006)   appendix 1

Weak localization: semiclassical theory A B A, B: Phase space points (x,y,  ) at beginning, end of “self encounter” Parameterize encounter using action difference  S (= symplectic area) t 1 2 : typical classical action Brouwer (2007) Exact in limit k F L at fixed  E /  D. appendix 1