Institut Laue Langevin Grenoble, France.

Noise of quantum chaotic systems in the classical limit Robert S Whitney

R.W. Phys. Rev. B 75

235404 (2007)

R.W. and Ph. Jacquod, Phys. Rev. Lett. 96 206804 (2006) Ph. Jacquod and R.W. Phys. Rev. B 73 195115 (2006) R.W. and Ph. Jacquod, Phys. Rev. Lett. 94 116801 (2005)

OUTLINE

Quantum particles: Noise in current

y0

!

Classical particles: No noise in current y

...and classical limit of quantum? Random matrix theory (RMT)

⇒ “noise in classical limit” NEW REGIME in classical limit — Not RMT Should revisit old questions for new regime:

Tunnel-barriers on leads

NOISE DUE TO TUNNEL-BARRIER Multi-mode quantum wire:

Independent processes ⇒ central limit theorum Fano factor, F = (1− transmission prob.)

"δI 2 # = F × e × I

Classical limit: wavelength λF & other scales barriers ⇒ impenetrable/transparent ⇒ noiseless

CLASSICAL DETERMINISM

NOISE DUE TO CHAOTIC SYSTEM

λF & W & L

Chaotic system: L "F W

Bohigas-Giannoni-Schmit (1984): obeys random matrix theory??

"δI 2 # = F × e × I

Fano factor, F = 1/4

Classical limit (λF & W, L) : noise in I remains

NO CLASSICAL DETERMINISM ...similar for integrable system (i.e. rectangle): not RMT, but expect noise

EXPERIMENT SAYS “NOT RMT”

"δI 2 # =

Oberholtzer et al, Nature (2002)

F × eI

...fits theory for smooth disorder

F

Aleiner-Larkin (1996,1997) Agam-Aleiner-Larkin (2000)

W/λF

R

λF & R

...introduced Ehrenfest time

RAY-OPTICS for the 21st century Landauer-B¨ uttiker:

scattering matrix

⇒ Fano-factor

S=

!

r t t! r!

semiclassics (vanVleck/Gutzwiller):

tnm =

y0

!

y

&

%

"

F =

# $ Bnm × Aγ exp iSγ /¯ h

γ classical action/stability

⇒ Sγ , Aγ

join lead-modes

y0

y

!2 y0 y’0

y y’

!4 !2 !3

⇒ Bnm

!2 =!1 average

y0

average

y0 y’0

!1

tr[t† tt† t] =

$

over classical paths from mode m to n

!1

tr[t† t] =

#

tr t† t−t† tt† t tr[t† t]

y

!1 !4 !2 !3 encounter

y y’

DYNAMICS OF CLASSICAL PATHS Relative dynamics of paths y0

hyperbolic chaos: Lyapunov exponent =

(r,p)

hyperbolic dynamics uncorr. correlated escape escape

(r,p)

rmin

W

y0

λ

random dynamics ~L

W

T (rmin ) Log. timescale: T (rmin ) = 2λ−1 ln[W/rmin ] Semiclassics: Integrate over exp[iδS(rmin )/¯ h]

⇒ Encounter size ∼

√

λF

⇒ T (rmin ) ∼ Ehrenfest time =λ−1 ln[L/λF ]

CALCULATING THE NOISE Closed system: RMT level-statistics Sieber(2001)

⇒ non-perturb.

Haake group (2006)

L

L

encounter

encounter L

W

Open system: RMT weak-localization Richter-Sieber(2001)

~ #E

#E W

W

encounter

~#E W

W

encounter

encounter

< #E

⇒ RMT result classical limit

encounter Braun-Heusler-M¨ uller-Haake (2006), Whitney-Jacquod (2006)

Hand-waving argument for noise

1/2

"F

W wavepacket escapes in pieces

⇒ RMT NOISE

1/2

"F

W

wavepacket escape as whole

⇒ NO NOISE

Suppression in classical limit W

W

W

encounter

encounter

< #E

W

encounter

Classical limit: escapes without diverging to ) W

encounter

# $ F ∝ tr t† t − t† tt† t → zero

Recover CLASSICAL DETERMINISM Classical limit noiseless ⇒ not random matrix theory (RMT)

proposed in Beenakker, van Houten (1991)

NEW CLASSICAL REGIME Closed system level−statistics, etc

1 Open system Noise, cond. fluct., etc

RMT regime

system specific 10 system specific

system size/wavelength RMT regime

classical regime

# E #D

Dwell time, τD , is time to escape system

Weight of non-classical contributions = probability to escape before Ehrenfest time, τE = exp[−τE //τD ]

⇒ RMT-to-CLASSICAL cross-over powerlaw in L/λF , exponent= (λτD )−1 & 1

PHASE-SPACE (PS) BASIS

• complete & orthonormal basis:

states localized in r and p

p/p F

States on large bands (area > ¯ h) ~WR /L exp[−"t 1 ]

1-to-1: ingoing ↔ outgoing

~WR /L exp[−" t2 ]

WL /L

phase−space

phase−space

L lead

R lead

q/L

Classical limit: DIAGONALIZE scattering matrix ALL eigenvalues = 0,1 ALL cummulants of noise = 0

TUNNEL-BARRIERS ON LEADS Whitney (2007)

system ⇒ classical limit barriers ⇒ quantum tunnel probability

= ρj

V 1

for lead j

2

⇒ usual semiclassics

PS-basis not useful

I

3

[A] a quantum contribution

⇒ Random matrix result chaotic system

[B] a classical contribution

path 1

Lead m0

Lead m

path 3

chaotic system

??

Lead m0

??

path 1

Lead m

path 3

Exhausive list of contributions m

(b)

m0 path 1

path 1 path 3

path 3

m

path 1

0

m0

(e) m 0

path 3

m

(f) m 0

path 1

path 3 path 3 CLASSICAL limit

m0 (g) m 0 path (a) 1 path 3

path 1 $ path 3

m

(d) m 0 path 1

m

path 1

m

path 3

(b) m 0

m

path 1

m

(e) m 0 path 1

m

path 3

(h) m 0

m

path 3

(f) m 0 path 1

m

path 3

m

path 1

m

path 3

path 3

path 1

(c) m 0 path 1

path 3

path 3

(g) m 0

m path 1

er

(d) m

(c)

m

ov s−

m0

os

(a)

cr

RMT limit

(i) m 0

m path 1 path 3

EXAMPLE: Tunnel-barrier on third lead

F

V 1 2 3

1/2 I

4/9

RMT 1/4

classical 0 0

61/2 5−12 = 0.088 61/2 6−12

(61/2−2) =0.42

1

%

CONCLUSIONS Noiseless transport in classical limit • “wavepacket” escapes as a whole before spreading to lead width Closed system

RMT regime

system specific

level−statistics, etc

1 Open system Noise, cond. fluct., etc

10 system specific

system size/wavelength RMT regime

classical regime

# E #D

F 1/2 4/9

RMT 1/4

classical 0 0

1/2

(6 −2) =0.42

61/2 5−12 = 0.088 61/2 6−12

1

%