14. april 2003 quantum mechanics on the large scale banff, alberta 1 relaxation and decoherence in...

14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta

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Relaxation and Decoherence in Quantum Impurity Models: From Weak to Strong Tunneling

Relaxation and Decoherence in Quantum Impurity Models: From Weak to Strong Tunneling

Ulrich WeissInstitute for Theoretical Physics

University of Stuttgart

H. Saleur (USCLA)A. Fubini (Florence)H. Baur (Stuttgart)

Quantum impurity models (spin-boson, Kondo, Schmid, BSG, ....) Dynamics From weak to strong tunneling

Quantum relaxation Decoherence

14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta

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solvent

donor acceptor

Electron transfer (ET):

bath dynamics

dissipationdecoherence

dynamicse-

tunneling

biological electron transportmolecular electronicsquantum dotsmolecular wirescharge transport in nanotubes

classical rate theoryMarcus theory of ET activationless ET inverted regimenonadiabatic ETadiabatic ET

14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta

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Spin-boson model with ultracold atoms:Recati et al. 2002

a b

bV

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System

Heat bath TIBS HHHH

Physical baths: Phonons Conduction electrons (Fermi liquid) 1d electrons (Luttinger liquid) BCS quasiparticles Electromagn env. (circuits, leads) Nuclear spins Solvent Electromagnetic modes

Spectral density of the coupling:

sJ )0(

Global system:

s

> 1 super-Ohmic

= 1 Ohmic

< 1 sub-Ohmic

phonons (d > 1)

e-h excitations

RC transmission line

14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta

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Truncated double well:TSS:

stochastic force:

driven TSS:

)()( 2221

221

21

T21

2

xmtxcH m

p

zzx 0)( Tt

0

1 )]sin()cos()2/)[coth(J()0()( titTdt T

Spin-boson Hamiltonian:

)(t stochastic force

T

T

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Anisotropic Kondo model

)(21

||41

,,

,FK

ccccJccJckcvH zkk

k

conduction band

spin polarizationconserved

spin flipscattering

Correspondence with spin-boson model:

)(cos K2

c

T

J

2K )/21( K

)4/arctan()( ||||K JJ

universal in the regime

1|1|||;1/ ||c KJJ

ferromagnetic Kondo regime

antiferromagn. Kondo regime

)1(0|| KJ

)1(0|| KJ

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Schmid model: particle in a tilted cosine potential

TB limit nn

nnn

n aantaaH

])([- )h.c.( 21

1S21

S

Current-biased Josephson junction (charge-phase duality) Impurity scattering in 1d quantum wire Point contact tunneling between quantum Hall edges

Boundary sine-Gordon model Exact selfduality in the Ohmic scaling limit Scaling function for transport and noise at T=0 is known in analytic form

A. Schmid, Phys. Rev. Lett. 51, 1506 (1983)P.Fendley, A.W.W. Ludwig, and H. Saleur, Phys. Rev. B 52, 8934 (1995)

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Density matrix:

Global system: |)()(|)( ttptW kkk k Reduced description: )(tr)( B tWt partial trace

time-local interactions time-nonlocal interactions

reduced dynamics: )(tfull dynamics: W(t)

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m

j

j

iiji

l

j

m

ijij

l

j

j

iijijijlm ttQuvttQvvttQuuqq

2

1

1 1 12

1

1

)'()''()(exp]',[F

Tight-binding model:1nP

nP

1nP

'q

q

1v

1u1u

1v

charges 1ju 1jv

nvnutPj ji in ;:)(0

Influence functional:

Absorption and emission of energy according to detailed balance

)()/( tQTitQ

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Keldysh contour

nvnui ij j ;

Laplace representation in the limit : 0

)(,oncontributi rate cluster eirreducibl n

2u 1mu mu

2v1v 3v lv2lv

1u

1lv

q

q

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Ohmic scaling limit:

c2)( KJ

Pair interaction between tunneling transitions:

sgn(t)

2Tt)sinh(

Tln2)()()( c

iKtQitQtQ

])sgn()ln(2 c

0

tKitKT

Kondo scale: T2

)1(K

1T11

KK

c

K

K

K

S)(K1S1

1

2

KK

c

K

K

Spectral density:

at fixed Kondo scale ST K)sin(2

TSS model

Schmid model

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LP

RPRL

LR

q

´q

N=5:N=2:

1;;11

j

j

kkjj p charges:

scaling limit: ])(exp[]exp[2

2

1

1

12

12 kjk

m

j

j

kjj

m

jjm ttQpKi

F

friction noise (Gaussian filter)

phase factor noise integral

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Incoherent tunneling:

golden rule limit: )(22 p

is probability for transfer of energy tofrom

the bath)( p

12

c

2

21 )(4

)(:0

K

c

T

K

K

;0;K)2(2

:012

cc

2

K

T

{ }

0

)(1)(

0

1 e)sin(d)sin(t)cos(d)cos()( tQtQ ttKetKp

phase factor noise integral noise integralphase factor

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-

- -

-

+ c.c. =

T/e + c.c.

-

-

-+ c.c. =

T/e + c.c.

-

-

- -+ c.c. =

T/e + c.c.

= T/2e -

--

-

Order :4

:)(1

:)(2

-

)(1

/T)(1 e

)(2

/T2)(2 e

(1)

(2)

(3)

(4)

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Noise integrals: 0T

irred0

2

1c12 )ln(2exp)()(

l

ijjiijll KI

D

12

1cos

l

j jjp

12

1sin

l

j jjp

Formidable relations between the variousnoise integrals of same order l

Up-hill partial rates are zero

Scaling property sign same have all if)()tan()( jll pIKlI

0, n general!

particular!

__

2 3 12 l22 l l21

22 l 12 l21

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Results:

Only minimal number of transitions contribute to the rate )( 2nn O

contributescancelled

Schmid model:

All rates can be reconstructed from the known mobility n nn

Knowledge of all statistical fluctuations (full probability distribution)

TSS model:

Exact relations between rates of the Schmid and TSS model

nn

n Kn )(sin4)1(~ 21

H. Saleur and U.Weiss, Phys. Rev. B 63, 201302(R) (2001)

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mK

m

vmK

KmKm

m)22(

23

1 ])1([

])2cos(1)[(

!

1

2~

22222 1e1d

2Re~ KKKKiK vzzvzz

z

z

i

C

Weak-tunneling expansion

Integral representation

Re(z)

Im(z)C

K/v

H. Baur, A. Fubini, and U.Weiss, cond-mat/0211046

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K021

0 )(~

Kb

12

0

)(2

~;~~

nn

nnn vKb

Strong-tunneling expansion

The case K<1:

:131 K

])[()(

])[(

!

1)()(

11

21

21

121

K

KK

nn nn

n

nKdKb

:31K )()]([sin2)( 2

11

2 KdnKb nKK

n

Leading asymptotic term:

:0

K/v

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mK

mm vKc )2/2(

1

)(2

~

The case K>1:

])1([

)sin(]sin[)(2

!

)1()(

123

1

mK

mm

mKc

K

Kp

Kp

Kmm

m

Strong-tunneling expansion

K/v

pKintegerp

10

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weak tunnelinglarge bias

strong tunnelingsmall bias

10 K

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weak tunnelingsmall bias

strong tunnelinglarge bias

1K

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Decoherence 21K

0

decdec

nn

21

31 K

31K

n

dec

~n 2

1

)]([sin 21

12 nK

K

1221

12 )]([sin

nKK vn

])[()(

])[(

!

1

2 11

21

21

121

0

dec

K

KK

n nn

n

n

Strong-tunneling expansion:

Conjecture: holds in all known special cases