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