Transport through chaotic cavities: RMT reproduced from - PowerPoint PPT Presentation

Transport through chaotic cavities: RMT reproduced from semiclassics P.B., joint work with Stefan Heusler, Sebastian Müller and Fritz Haake Universität Duisburg-Essen

channels N 2 Transport problem chaotic cavity N  N 1  N 2 channels N 1

S, t, and r-matrices Inside leads: E   2  k   2   k ‖  2   C sin  k  y  exp  ik ‖ x  , fixed. 2 m In-(1) and out- (2) channels ( w = width):   m i  m i  1 … N i  i  1,2  w i , k i In- and out- states in the transport problem:  1   ∑ l 1  1   2   ∑ l 2  1   1    m 1  2  N 2 t m 1 l 2  l 2  1  ∗ N 1 r m 1 l 1  l 1 , S -matrix: r t S  t ′ r ′

Transport properties ~ 〈 Tr tt   • Conductance • Conductance variance ~  Tr tt   2 − 〈 Tr tt   2 ~ 〈 Tr tt  − Tr tt  tt   • Shot noise • … (Averaging done over energy interval)

Conductance, random-matrix prediction unitary case N 1 N 2 (with magnetic field, N no time - reversal invariance) G  orthogonal case N 1 N 2 N 1 N 2 − N 1 N 2 N 1 N 2 − …   (without magnetic field, N  1 N 2 N 3 N time - reversal invariance) why true for individual systems?

Semiclassical approach ∑  A  e iS  /   t m 1 m 2 Van Vleck: In- channel Out- channel Entrance and exit angles fixed by channel numbers 

Semiclassical approach Conductance 〈  T H ∑ ∗ e i  S  − S   /  † t   1 G  Tr  t ′ ′ A  A ′  ,   Heisenberg time T H   − 1 Need pairs of trajectories with small action difference

Diagonal approximation identical trajectories    ´ T H 〈 ∑  | A  | 2  1 G diag   dT e − TH T  N T H  N 1 N 2 N 1 N 2  N 0 probability of staying inside up to time T N = escape rate T H

Higher orders • Created by pairs of trajectories-partners composed of same pieces traversed in different order / with different sense • Switches of motion: encounters • l -encounter: avoided crossing in phase space of l stretches of same trajectory, or trajectory and its time reversed, or of different trajectories • 2-encounter, viewed in configuration space: small- angle crossing / narrow avoided crossing

Richter / Sieber pairs t 1 t 2 t enc t 3 • dwell time T  t 1  t 2  t 3  2 t enc • if no escape on first stretch, no escape on second stretch either survival probability − TH  t 1  t 2  t 3  t enc   e N − N TH T e Encounters hinder escape into leads.

Richter / Sieber pairs Pairs characterized by • link durations 0  t 1 , t 2 , t 3   • phase-space separations inside encounter s,u ∗ e i Δ S  ′ /  T H ∑   ,  ′  RS A  A  ′ 1 G RS  Integration gives − TH  t 1  t 2  t 3  t enc  N  1 N 1 N 2 i Δ S /   dt 1 dt 2 dt 3 ds du e e  t enc T H survival “ergodic density probability of encounters” 3  − N T H  T H N  2  N 1 N 2  − N 1 N 2  N 2 T H three links one encounter

Higher orders in 1 / N Reconnections may not lead to periodic orbits splitting off Survival probability TH   loops t loop  enc t enc   e − N − N TH T e

Diagrammatic rules for trajectory pairs • 1/ N for every link • (- N ) for every encounter • Multiply by the number of in- and out- channels ( N 1 N 2 ) • Sum over all families (topological versions) of trajectory pairs.

Higher orders in 1 / N Each family of trajectory pairs contributes  − 1  #encounters N 1 N 2 N #loops − #encounters Summation gives with magnetic field N 1 N 2 (not TR invariant) N G  without magnetic field − N 1 N 2 − … N 1 N 2 N 1 N 2 N 1 N 2  (time-reversal invariant) N  1 N 2 N 3 N in agreement with RMT

Shot noise Fluctuations of current through a cavity: P ~ Tr  t † t t † t t † t  − Semiclassically Tr tt  tt   ∑ m 1 , n 1 ∗ exp i S p  S r − S q − S t ∑ m 2 , n 2 ∑ pqrt A p A q N 1 N 2 ∗ A r A t  Sum over quadruplets of classical trajectories p, q, r, t , connecting channels m 1 m 2 , n 1 m 2 , n 1 n 2 , m 1 n 2 Contributing quadruplets must have S p  S r ≈ S q  S t i.e. the pairs ( q, t) and (p, r) must be partners

Leading term Schanz, Schanz, Puhlmann, Puhlmann, Geisel 03 Geisel 03 Using diagrammatic rules (1/ N per link, – N per encounter), 1 2 N 2 2 P  − N 4   − N   N 1

Shot noise Example for higher orders:

Shot noise semiclassical prediction 2 N 2 2 N 1 unitary case N  N 2 − 1  P  N 1  N 1  1  N 2  N 2  1  orthogonal case N  N  1  N  3  O(N) and O(1) agree with RMT higher orders first obtained semiclassically (later confirmed in RMT)

GOE / GUE crossover Weak magnetic field B: trajectories unchanged. Additional action: 2 mc  L z  t  dt Θ  eB ; Angular momentum e i  S  − S  ′  /  → e i  S  − S  ′ Θ  − Θ  ′  /  Magnetic phase on elements of γ and γ ’ traversed in same direction cancels, in opposite directions is doubled .

Diagrammatic rules under crossover • For a loop changing direction : N -1 (1 + ξ) -1 preserving direction : N -1 • • For an encounter with μ stretches changing direction : - N (1 + μ 2 ξ )    B /   2 is the crossover parameter. (Per one in- and one out-channel; for each topologic family)

GOE / GUE crossover  1 …  N 1 N 2 1 4 3 2 N  − N  1    4   3   13   2  1 1 − G   2 2 3 5 N  1    N  1    →  0 orthogonal case    B /   2  unitary case Coincides with RMT (Weidenmüller e.a., 1995)

Wigner delay time (Cuipers, Sieber 2007) • Approach: similar, leads to RMT results. • Equivalence proven of delay time representation as sums over trajectory pairs and periodic orbits of open resonator

Conclusions • Diagrammatic rules found leading to RMT results for all examined transport properties. Based on: a) partnership of trajectories differing in encounters; b) increase of dwell time in orbits with encounters • Applicability limited by Ehrenfest-time corrections (case N>>1) and diffraction effects (N~1)