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

1

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


2

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


3

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

a b

bV


4

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


5

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


6

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


7

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)


8

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)


9

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


10

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


11

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


12

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


13

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


14

-

- -

-

+ 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)


15

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


16

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)


17

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


18

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


19

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


20

weak tunnelinglarge bias

strong tunnelingsmall bias

10 K


21

weak tunnelingsmall bias

strong tunnelinglarge bias

1K


22

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

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

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