Transport through ballistic chaotic cavities in the classical limit

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 Wesleyan October 26th, 2008 With: Saar Rahav

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

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

4. Ballistic chaotic cavities: transport level density conductance G

5. Ballistic chaotic cavities: transport level density conductance G G is random function of (Fermi) energy e and magnetic field b Marcus group

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

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

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

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

10. Semiclassics Rj: total reflection from opening j “conductance” = “transmission” = “1 – reflection” a a, b: classical trajectories Aa: stability amplitude Sa: classical action Miller (1971) Blümel and Smilansky (1988)

11. Semiclassics Rj: total reflection from opening j “conductance” = “transmission” = “1 – reflection” a a, b: classical trajectories Aa: stability amplitude Sa: classical action Miller (1971) Blümel and Smilansky (1988)

12. Conductance fluctuations Need to calculate fourfold sum over classical trajectories. But: Trajectories a1, b1, a2, b2 contribute only if total action difference DS is of order hsystematically a

13. Conductance fluctuations b2a2 a1b1 a1b2 b1a2 Need to calculate fourfold sum over classical trajectories. But: Trajectories a1, b1, a2, b2 contribute only if total action difference DS is of order hsystematically Sieber and Richter (2001)

14. Conductance fluctuations b2a2 a1b1 a1b1 b2a2 a1b2 a1b2 b1a2

15. Conductance fluctuations b2a2 a1b1 a1b1 b2a2 a1b2 a1b2 b1a2 This contribution vanishes for chaotic cavity

16. Conductance fluctuations Duration of small angle encounter with action difference DS ~ h is “Ehrenfest time” tE: tE tE l: Lyapunov exponent Aleiner and Larkin (1996)

17. Conductance fluctuations Duration of small angle encounter with action difference DS ~ h is “Ehrenfest time” tE: tE tE l: Lyapunov exponent random matrix theory if tE << tD Aleiner and Larkin (1996)

18. Conductance fluctuations b1a2 t1 t2 a1b2 action differences accumulated between encounters: t = t1, t2

19. Conductance fluctuations b1a2 t1 t2 a1b2 action difference survival probability probabilities to enter/escape through contacts 1,2

20. Conductance fluctuations b1a2 t1 t2 a1b2 action difference survival probability Jalabert, Baranger, Stone (1993) Brouwer, Rahav (2006) Heusler et al. (2007)

21. Classical Limit random matrix theory if tE << tD tE l: Lyapunov exponent tE In classical limitkFL : Dwell time tD unaffected (because classical), But tE Condition tE << tD violated!

22. Classical Limit tE Two encounter give factor tE

23. Classical Limit tE tE tE tE tE

24. Classical Limit tE tE tE tE t’ tE Overlapping encounters give factor Brouwer and Rahav (2006)

25. Classical Limit tE tp tE tE action difference survival probability factor from encounters

26. Classical Limit tE tp tE tE classical limit random matrix theory Brouwer and Rahav (2007)

27. Conductance fluctuations Obtain varG by setting e = e’, b = b’: 2 var g var G in classical limit still given by random matrix theory (but not correlation function!) M~kFL 102 103 104 Tworzydlo et al. (2004) Jacquod and Sukhurukov (2004) Brouwer and Rahav (2006)

28. F1 F2 I Summary: Classical Limit Conductance fluctuations of an open ballistic chaotic cavity remain universal in the classical limit kFL , but they are not described by random matrix theory g B • Other 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 correction” S N

29. appendix 1 Weak localization: semiclassical theory b Landauer formula transmission matrix t … Green function … path integral … stationary phase approximation … a Jalabert, Baranger, Stone (1990) • a, b: classical trajectories; • a and b have equal angles upon entrance/exit • Sa,b:classical action • Aa,b: stability amplitudes |Aa,b|2: probability

30. a b appendix 1 Weak localization: semiclassical theory dg: Trajectories with small-angle self intersection Sieber, Richter (2001)

31. appendix 1 Weak localization: semiclassical theory a,b a,b dg: Trajectories with small-angle self intersection Sieber, Richter (2001)

32. appendix 1 Weak localization: semiclassical theory a,b a,b dg: Trajectories with small-angle self intersection Sieber, Richter (2001) Stretch where trajectories are correlated: “encounter”

33. a b appendix 1 Weak localization: semiclassical theory a,b a,b dg: Trajectories with small-angle self intersection Sieber, Richter (2001) Stretch where trajectories are correlated: “encounter” Poincaré surface of section umax stable, unstable phase space coordinates u encounter: |u| < umax, |s| < smax b a smax s

34. a a b b appendix 1 Weak localization: semiclassical theory a,b a,b dg: Trajectories with small-angle self intersection Sieber, Richter (2001) Stretch where trajectories are correlated: “encounter” Poincaré surface of section b umax stable, unstable phase space coordinates u encounter: |u| < umax, |s| < smax b a smax s

35. dg: Trajectories with small-angle self intersection Sieber, Richter (2001) Stretch where trajectories are correlated: “encounter” Poincaré surface of section stable, unstable phase space coordinates encounter: |u| < umax, |s| < smax a b appendix 1 Weak localization: semiclassical theory a,b a,b b umax su: invariant “symplectic area” u a smax s

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

37. appendix 1 Weak localization: semiclassical theory 2 1 A • A, B: Phase space points (x,y,f) at • beginning, end of “self encounter” B t • Parameterize encounter using action • differenceDS (= symplectic area) • :typical classical action Exact in limit kFL at fixed tE/tD. Brouwer (2007)