Tunneling in Complex Systems: From Semiclassical Methods to Monte
Carlo simulations
Joachim AnkerholdTheoretical condensed matter physics
University of FreiburgGermany
„Challenges in Material Sciences“Hanse-Kolleg, February 16/17, 2006
( )R E( )T E
2 2 2 ( ) /out
in
( )( ) ( ) e
( )bm V E Lj E
T E t Ej E
L
bV
:bV E
Barrier transmission: Scattering
1R T e e eikx ikx ikxr t
L
inv2 ( ) / ( ) /( ) e e
dx m V E p x dxT E
Semiclassics (WKB):Action of a periodic path in the inverted barrier with Energy -E
( ) ( )V x V x
Equivalent: inv( ) 2 [ ( )] ( ) ( )p x m E V x i p x ip x
A A-4 * 4Z Z-2 2X Y He E
Alpha-Decay (Gamow)
42 He
R
L
outpop
1 ( )dE j E
Z
Tunneling rate:
Density of statesProbability distribution
Incoherent tunneling from a reservoir
out out( ) ( ) v ( ) ( ) ( )j E T E E E P E
Total rate:
Scanning tunneling microscope
SiC (0001) 33 surface
Tip
Sample
0V
x0 d
Tunneling current (Temperature = 0)
( ) v( ) ( , 0)F
F
E
E eV
I e dE T E E E x
( )T
VI
R d
02 2 /1( , 0) v e
( )d mV
F FT
e E xR d
Tunneling resistance:
Tunneling resistance
Exponential sensitivity
Tunneling in NH3
x
Friedrich Hund 1926:Friedrich Hund 1926:
Coherent tunneling
H
N
H
H
/ 2 / 2 0 ,
/ 2 0 / 2
10 0 0
21
0 0 02
a L R
s L R
E[1/cm] Energy doublets
0
0 ,
L
R
Incoherent tunneling
in presence of a dissipative environment
Example: Josephson-junction
RL
phase difference
V()
Applied current:
Potential energy:
(Josephson 1961)
Particle in a periodic potential
Macroscopic quantum tunneling
phase difference
Tunneling of a collective degree of freedom
• Squids• Vortices• Nanomagnets• Superfluids• Bose-Einstein Condensates
potential energy
1m
Environment: Electromagnetic modes
Groupe Quantronique, CEA Saclay
Decay rate of metastable systems
FIm
Tunneling rate in presence of thermal environment:
(Leggett et al)
1lnF Z
Decay channels:
thermal activation
quantum tunneling
2
/ 2
e n
n n n
tn
E i
Open quantum systems
, ( )T J
SH + IH RH+
System + reservoir: reduced density
R
1Tr e H
Z
Path integrals
Feynman: ( )
//
(0)
e ef
i
q t qiS qiHt
f i
q q
q q D q
iq
fq
mt /
“Sum over all paths“
Path integrals
Feynman: ( )
//
(0)
e ef
i
q t qiS qiHt
f i
q q
q q D q
iq
fq
mt /
“Sum over all paths“
itH :e Density matrix:
Influence functional
[ ]/ [ ]/
periodicorbits ininterval
[ ] e eES q qZ D q
Influence functional:describes interactionwith environment
RTr
, 1/ c
, ( )T J ( )q ( )q
Path integral in imaginary time:
Semiclassics:
Periodic orbits in the inverted barrier with period
q|
well barrier
02 /
0 e2
bVcl
Thermal activation
Semiclassics:
Periodic orbits in the inverted barrier with period
q
)(qV
|
well barrier
q|
well barrier
02 / 02 /
0 e2
bVcl
/0 e
2BS
q
Quantum tunnelingThermal activation
|Ln( ) |
const
Devoret et al,1988
Experiment
|Ln( ) |
const
Thermal activation
Quantum tunneling
Experiment
Rate processes
Rate theory in JJ equivalent to rate theory for
chemical reactions
diffusion of interstitials in metals
collaps of BECs with attractive interactions
proton transfer
JJ as detectors for: read-out in quantum bit devices measurement of non-Gaussian electrical noise
Tunneling of a qubit: Crossing of surfaces
?
Flip: Smaller barrier larger rate ?
2
2
( )2
( )2
pV
mHp
Vm
Landau-Zener transitions „under“ the barrier: MQT of a Spin
JA et al, PRL 91, 016803 (2003)Vion et al & JA, PRL 94, 057004 (2005)
Tunneling in the system and
Tunneling in the phonon environment
Large Molecules: Photosynthesis
2 nm
Photosynthesis: Reaction center
2 nm
Photosynthesis: Reaction center
Electron transfer
fast: ~ 3ps
efficient: 95%
2 nm
„Bottom up“ instead of „top down“: Molecular electronics
Reed et al, 2002
Classical Marcus theory
++
+
+
+Polar environment:Fluctuating polarization
2 e
electronic tunnelingactivation energy
Marcus et al, 1985
Classical Marcus theory
++
+
+
+Polar environment:Fluctuating polarization
2 e
electronic coupling activation energy
Low T: Nuclear tunneling
Open quantum systems: Nonequilibrium dynamics
, ( )T J
SH + IH RH+
System + reservoir: reduced dynamics
/ /R
1( ) Tr e (0) eiHt iHtt W
Z
ts
Reduced dynamics
( , )D A t paths
Path integrals: Paths in real and imaginary time
' ' ' ' '( , , ) ( , , , , ) ( , ,0)f f i i f f i i i iq q t dq dq J q q t q q q q
ts
Reduced dynamics
( , )D A t paths
Influence functional: self-interactions non-local in time
In general no simpleequation of motion !
Mak, Egger, JCP 1995; Mühlbacher & JA, JCP 2004, 2005
,1/ c
Redfield-Equation
)()(,)( 2 ttH
i
dt
tdS
R
2. order perturbation theory in coupling
powerful method for many chemical systems
numerically efficient
weak friction, higher temperatures
sufficiently fast bath modes
How to evaluate high-dimensional integrals?
MC
P
K
kk
N fxfK
xPxfxd 1
)(1
)()(
Monte Carlo: Stochastic evaluation (numerically exact)
MC weight
Distributed according to MC weight
(K >> 1)
Electron transfer along molecular wires: Tight binding system
Davis, Ratner et al, Nature 1998
D A
In general: d localized states
Real-time Quantum Monte Carlo
Dicretization of time (Trotter)
t N
/ / /e e ... e , /iHt iH iH t N
Real-time Quantum Monte Carlo
t N
System: d orthonormal states
At each time step: d different configurations possible
d-possible orientations at each time step= configurationsNd 153, 30 10d N
Real-time Quantum Monte Carlo
t N
System: d orthonormal states
At each time step: d different configurations possible
Important sampling over spin chains
Tr ( )( )
Tr ( )
A tA t
t
i i is P s G s
distr.
distr.
i iP
iP
A s G s
G s
53, 30 10d N Convergence:
Real-time Quantum Monte Carlo
Integrand oscillates: Dynamical sign problem
Treat subspace exactly: Reduction of Hilbert space to be sampled
Mak et al, PRB 50, 15210 (1994); Mühlbacher & JA, JCP 121, 12696 (2004); ibid 122, 184715 (2005)
Quantum mechanicslives from interferences !
Wave mechanics lives from interferences
Coherent / Incoherent dynamics
0.1
0.75
1
1
300 cm
0...4500 cm
T 300 ...60 K
20simt
Assembling of molecular wires
Davis, Ratner et al, Nature 1998
D ANot an ab initio method: Structure Dynamics
Population dynamics:
I
Molecular wire: Diffusion versus Superexchange
I
0.5
/I
qmclass
Molecular wire: Phonon tunneling vs. Superexchange
Mayor et al, Angew. Chemie 2002Mühlbacher & JA, JCP 122, 184715 (2005)
0.5
/I
qmclass
Park et al, Science 2002
Tunneling in presence of Charging effects:
Coulomb-blockade
3+Co 2+Co
Quantum dots: artificial molecules
Dissipative Hubbard system
Two charges with opposite spin:
0
†0 1
, ,
ˆ ˆ ˆ ˆ ˆ ( . .)
ˆ
I R
S S
k k k k k jk S k j S
I
H H H H
H n a a h c U n n
H P c X
Polarization operator
Non-Boltzmann equilibrium
Charges on same site U > 0
Charges on different sites
???
Non-Boltzmann equilibrium
Mühlbacher, JA, Komnik, PRL 95, 220404 (2005)
0U
0U
Non-Boltzmann equilibrium
Mühlbacher, JA, Komnik, PRL 95, 220404 (2005)
0U Invariant subspace
bosons
„Coherent“ channels for faster transfer
0 0, 0H
Summary and Conclusions
Nanosystems show a variety of tunneling phenomena
Strongly influenced by the surrounding
Semiclassics: very successful for mesoscopics
Exact reduced dynamics: Real-time Monte Carlo
L. MühlbacherM. DuckheimH. LehleM. Saltzer
Thanks