14. April 2003 Quantum Mechanics on the Large Scale Banff, Alberta 1 Relaxation and Decoherence in...
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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
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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
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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
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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