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

Transport through chaotic cavities: RMT reproduced from semiclassics

P.B., joint work with

Universität Duisburg-Essen

Stefan Heusler, Sebastian Müller and Fritz Haake

Transport problem

chaotic cavity

N  N1  N2

channels

N1

channels

N2

S, t, and r-matrices

In- and out- states in the transport problem:

1  m 1

1  ∑ l11 N 1 r m 1l1l1 1∗

2  ∑l21

N 2 tm 1l2 l2 2

,

S  r t t′ r′

S-matrix:

ki

  m i w i ,

mi  1… N i i  1,2

In-(1) and out- (2) channels (w = width):

E  2 k 2k ‖2

2m

fixed. Inside leads:

  Csink yexpik ‖x,

Transport properties

• Conductance
• Conductance variance
• Shot noise
• …

~〈Tr tt

~ Tr tt2 − 〈Tr tt2

~〈Tr tt − Trtt tt

(Averaging done over energy interval)

Conductance, random-matrix prediction

N1N2 N

unitary case

(with magnetic field, no time-reversal invariance)

N1N2

why true for individual systems?

N 1

• rthogonal case

(without magnetic field, time-reversal invariance)

G 



N1N2 N

− N1N2

N2



N1N2 N3

−…

Semiclassical approach



Van Vleck:

In- channel Out- channel

tm 1m 2

∑ AeiS/



Entrance and exit angles fixed by channel numbers

Semiclassical approach

Conductance

Need pairs of trajectories with small action difference

Heisenberg time TH  −1



〈





, 

G  Trt

† t 

1 TH ∑

′ AA ′

∗ eiS−S/

′

Diagonal approximation

Gdiag 

1 TH 〈∑|A|2

TH 

N



N1N2 dT e − TH T 

N1N2 N

probability of staying inside up to time T = escape rate

N TH

identical trajectories  ´

Higher orders

• Created by pairs of trajectories-partners composed
• f 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

• if no escape on first stretch,
• dwell time T  t1  t2  t3  2tenc

no escape on second stretch either Encounters hinder escape into leads. survival probability

N

e

− TH t1t2t3tenc  e − N

TH T

t1 t2 t3 tenc

Richter / Sieber pairs

Pairs characterized by

• phase-space separations inside encounter s,u

e

H

du 

T



 tenc

N

N1N2

dt 1dt2 dt3 ds

1

e

− TH t1t2t3tenc iΔS/

“ergodic density

• f encounters”

survival probability three links

• ne encounter

 − N1N2

N2



N1N2 TH  TH N  3− N T H

2 

• link durations 0  t1, t2, t3  

Integration gives

GRS 

1 TH ∑,′RS AA′ ∗ ei ΔS′ / 

Higher orders in 1/N

Survival probability

e

− N

TH loops tloop enc tenc   e

− N

TH T

Reconnections may not lead to periodic

• rbits splitting off

Diagrammatic rules for trajectory pairs

• 1/N for every link
• (-N) for every encounter
• Multiply by the number of in- and out-

channels (N1N2)

• Sum over all families (topological versions)
• f trajectory pairs.

Higher orders in 1/N

Each family of trajectory pairs contributes

−1#encountersN1N2 N#loops−#encounters

Summation gives

N1N2 N

with magnetic field (not TR invariant) without magnetic field (time-reversal invariant)

N1N2 N

− N1N2

N2



N1N2 N3

−…

N1N2 N1

in agreement with RMT

G 

Shot noise

Fluctuations of current through a cavity:

− P ~ Tr t† t t†t t †t

Sp  Sr ≈ S q  St

Contributing quadruplets must have i.e. the pairs (q, t) and (p, r) must be partners Sum over quadruplets of classical trajectories p, q, r, t , connecting channels m1m2 , n1m2 , n1n2 , m1n2 Semiclassically

Tr tt tt  ∑ m 1,n1

N1

∑ m 2,n2

N 2

∑ pqrt ApAq

∗ArAt ∗ exp i SpSr−Sq−St 

Leading term

1 N 4 −N

P  −

N1

2N2 2

Using diagrammatic rules (1/N per link, –N per encounter), Schanz, Puhlmann, Geisel 03 Schanz, Puhlmann, Geisel 03

Shot noise

Example for higher orders:

Shot noise

semiclassical prediction

O(N) and O(1) agree with RMT higher orders first obtained semiclassically (later confirmed in RMT)

N1N11N2N21 NN1N3

• rthogonal case

N1

2N2 2

NN2−1

unitary case

P 

GOE/GUE crossover

eiS−S′/ → eiS−S′Θ −Θ ′/

Magnetic phase on elements of γ and γ’ traversed in same direction cancels, in opposite directions is doubled.

Angular momentum

Θ 

eB 2mc Lztdt

Weak magnetic field B: trajectories unchanged. Additional action:

;

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+μ2ξ )

is the crossover parameter. (Per one in- and one out-channel; for each topologic family)

  B/2

GOE/GUE crossover

Coincides with RMT (Weidenmüller e.a., 1995)

N  − N1 N 1

−

N 1

G 

N1N2 1 1



1

2 2

4 3 13 2

3 5 4 3 2

1 …

  B/2 → 

• rthogonal case



unitary case

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)