Transcript The classical limit of quantum transport

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
Orsay
June 26th, 2008
With: Saar Rahav
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
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
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.
in units of
in units of
in units of
Ballistic chaotic cavities: transport
level density
conductance G
Ballistic chaotic cavities: transport
level density
conductance G
G is random function
of (Fermi) energy e
and magnetic field b
Marcus group
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)
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:
(
)
Conductance autocorrelation function
Correlation function
Random matrix theory
Jalabert, Baranger, Stone (1993)
Efetov (1995)
Frahm (1995)
2
in units of
in units of
in units of
1
L
Pj: probability to escape
through opening j
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”).
Semiclassics
Rj: total reflection from opening j
“conductance” = “transmission”
= “1 – reflection”
a, b: classical trajectories
Aa: stability amplitude
Sa: classical action
Miller (1971)
Blümel and Smilansky (1988)
a
Semiclassics
Rj: total reflection from opening j
“conductance” = “transmission”
= “1 – reflection”
a, b: classical trajectories
Aa: stability amplitude
Sa: classical action
Miller (1971)
Blümel and Smilansky (1988)
a
Conductance fluctuations
Need to calculate fourfold sum
over classical trajectories.
a
But: Trajectories a1, b1, a2, b2
contribute only if total action
difference DS is of order h
systematically
Conductance fluctuations
a1 b1
a1 b2
b2 a2
b1 a2
Need to calculate fourfold sum
over classical trajectories.
Sieber and Richter (2001)
But: Trajectories a1, b1, a2, b2
contribute only if total action
difference DS is of order h
systematically
Conductance fluctuations
a1 b1
a1 b2
b2 a2
b1 a2
a1 b1
a1 b2
b2 a2
Conductance fluctuations
a1 b1
a1 b2
b2 a2
a1 b1
b1 a2
This contribution
vanishes for
chaotic cavity
a1 b2
b2 a2
Conductance fluctuations
tE
Duration of small angle
encounter with action difference
DS ~ h is “Ehrenfest time” tE:
tE
l: Lyapunov exponent
Aleiner and Larkin (1996)
Conductance fluctuations
tE
Duration of small angle
encounter with action difference
DS ~ h is “Ehrenfest time” tE:
tE
l: Lyapunov exponent
random matrix theory
if tE << tD
Aleiner and Larkin (1996)
Conductance fluctuations
t1
a1 b2
b1 a2
t2
action differences accumulated
between encounters:
t = t1, t2
Conductance fluctuations
t1
a1 b2
b1 a2
t2
survival probability
action difference
probabilities to enter/escape through contacts 1,2
Conductance fluctuations
t1
a1 b2
b1 a2
t2
survival probability
action difference
Jalabert, Baranger, Stone (1993)
Brouwer, Rahav (2006)
Heusler et al. (2007)
Classical Limit
tE
random matrix theory if tE << tD
l: Lyapunov exponent
tE
In classical limit kFL   :
Dwell time tD unaffected (because classical),
But tE  
Condition tE << tD violated!
Classical Limit
tE
tE
Two encounter give
factor
Classical Limit
tE
tE
tE
tE
tE
Classical Limit
tE
tE
tE
tE
t’
tE
Overlapping encounters give factor
Brouwer and Rahav (2006)
Classical Limit
tE
tE
factor from encounters
tp
tE
action difference
survival probability
Classical Limit
tE
tp
tE
tE
classical limit
random matrix theory
Brouwer and Rahav (2007)
Conductance fluctuations
Obtain var G by setting e = e’, b = b’:
2 var g
var G in classical limit still given by
random matrix theory
(but not correlation function!)
Brouwer and Rahav (2006)
M~kFL
102
103
104
Tworzydlo et al. (2004)
Jacquod and Sukhurukov (2004)
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
Other quantum effects
• Weak localization
• Shot noise
• Statistics of energy levels
• Proximity effect
• Quantum pumps
• Interaction effects
F1 F
B
2
I
“Altshuler-Aronov correction”
• Anderson localization from
classical trajectories (tE=0)
g
S
N
appendix 1
Weak localization: semiclassical theory
Landauer formula
transmission matrix t …
Green function …
path integral …
stationary phase approximation …
b
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
appendix 1
Weak localization: semiclassical theory
d g: Trajectories with small-angle self
intersection
Sieber, Richter (2001)
a
b
appendix 1
Weak localization: semiclassical theory
a,b
d g: Trajectories with small-angle self
intersection
Sieber, Richter (2001)
a,b
appendix 1
Weak localization: semiclassical theory
a,b
d g: Trajectories with small-angle self
intersection
Sieber, Richter (2001)
Stretch where trajectories are correlated:
“encounter”
a,b
appendix 1
Weak localization: semiclassical theory
a,b
a,b
d g: 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
umax
u
b
a
a
s
b smax
appendix 1
Weak localization: semiclassical theory
a,b
a,b
d g: 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
b
umax
a
u
b
a b
a
s
b smax
appendix 1
Weak localization: semiclassical theory
a,b
a,b
d g: Trajectories with small-angle self
intersection
Sieber, Richter (2001)
Stretch where trajectories are correlated:
“encounter”
Poincaré surface of section
area”
su:
invariant
stable,
unstable“symplectic
phase space
coordinates
encounter: |u| < umax, |s| < smax
b
umax
a
u
a b
s
smax
appendix 1
Weak localization: semiclassical theory
a,b
a,b
Poincaré surface of section
stable, unstable phase space coordinates
encounter: |u| < umax, |s| < smax
su: invariant “symplectic area”
b
umax
a
u
Spehner (2003)
Turek and Richter (2003)
Heusler et al. (2006)
a b
s
smax
appendix 1
Weak localization: semiclassical theory
1
• A, B: Phase space points (x,y,f) at
beginning, end of “self encounter”
• Parameterize encounter using action
difference DS (= symplectic area)
•
: typical classical action
Exact in limit kFL
Brouwer (2007)

at fixed tE/tD.
A
B
2
t