Joël Peguiron Department of Physics and Astronomy, University of Basel, Switzerland

Work done with Milena Grifoni atKavli Institute of Nanoscience, Delft University of Technology, The Netherlands

Institut I - Theoretische Physik, Universität Regensburg, Germany

Dynamics of Quantum Dissipative Systems:The Example of Quantum Brownian Motors

CCP6 Workshop on Open Quantum Systems Bangor, Wales, UK, 25 August 2006

August 25, 2006 2

Aim of the Talk

Illustrate some techniquesused by

theoretical physiciststo describe

quantum dissipative systemswith the example of

quantum Brownian motors(also known as quantum ratchets)

August 25, 2006 3

What is a Brownian Motor / Ratchet ?

A ratchet system is:

• periodic• asymmetric• designed and driven to extract work from fluctuating forces

August 25, 2006 4

Ratchets: a Physicist´s Model

A ratchet system is:

• periodic• asymmetric• designed and driven to extract work from fluctuating forces

Natural case: interaction with adissipative thermal environment

Driving force: tilting of the potential

• maintains a non-equilibrium situation → work extraction allowed by the 2nd Principle of Thermodynamics

• unbiased → e.g. rocking force

V(q) – F q

→ q

work released ↔ particle current

A single particle in a1D periodic asymmetric potential

→ q

V(q)

August 25, 2006 5

The Classical Ratchet EffectΓforward

Γbackward

force +F force -F

• Current:

• Thermal rate:TkU

thBe

∆−∝Γ

backwardforward Γ−Γ∝∞DCv

The barrier height ΔU may be different in opposite tilted situations→ ratchet effect: net average current )(v)(v DCDC FF −−≠+ ∞∞

→ This current may be tuned through the parameters of the environment

August 25, 2006 6

The Quantum Ratchet Effect

TkU

thBe

∆−∝Γ

backwardforward Γ−Γ∝∞DCv

tunneling

?=Γtun

Γforward

Γbackward

force +F force -F

The interaction with adissipative environment is crucial

• Current:

• Thermal rate:

The barrier parameters may be different in opposite tilted situations→ quantum ratchet effect: net average current )(v)(v DCDC FF −−≠+ ∞∞

→ This current may be tuned through the parameters of the environment→ Current reversals may occur

P. Reimann, M. Grifoni and P. Hänggi, PRL 79, 10 (1997)

August 25, 2006 7

Some Introductory Literature on(Quantum) Ratchets

Brownian motors,R. D. Astumian and P. Hänggi, Phys. Today 55(11), 33 (2002)

Ratchets and Brownian motors: Basics, experiments and applications,special issue edited by H. Linke, Appl. Phys. A 75, 167 (2002)

Brownian motors: Noisy transport far from equilibrium,P. Reimann, Phys. Rep. A 361, 57 (2002)

Fundamental aspects of quantum Brownian motionP. Hänggi and G. L. Ingold, Chaos 15, 026105 (2005)

Quantum Ratchets,J.P., PhD thesis (2005)available from http://theorie5.physik.unibas.ch/peguiron/publications.html

August 25, 2006 8

Vortices in circular Josephson junction arrays

F. Falo et al.,Appl. Phys. A 75, 263 (2002)

Quantum Ratchet Experiments

Electrons in asymmetrically confined 2DEG

H. Linke et al.,Science 286, 2314 (1999)

Vortices in annular long Josephson junctions A. Ustinov, University of Nürnberg-Erlangen

Vortices in Josephson junction arrays• quasi 1D dynamics• quantum regime ( EJ / EC ≈ 11 )

• potential designed at will

J . B. Majer et al., PRL 90, 056802 (2003)

force ↔ ↔ velocity

August 25, 2006 9

Model

Bath1D Ratchet system

Driving force

Quantity of interest: the particle current at long times

bathdrivingratchettot HtHHtH ++= )()(

Ratchet

1D system, asymmetric potential

)(2

2

qVmpH Rratchet +=

Driving

unbiased

)( tFqH driving =

0)(12 =+ tF n

How to introduce dissipation in a quantum framework ?

August 25, 2006 10

Other solution:contact to reservoirs ofnon-interacting electrons

J. Lehmann et al., PRL 88, 228305 (2002)

characterized by the spectral density

Bath of harmonic oscillators

A.O. Caldeira and A.J. Legget,Ann. Phys. 149, 374 (1983)

Quantum Dissipative Bath

∑=

−+=

N

i ii

iiii

i

ibath m

qcXmmPH

1

2

22

2

21

2 ωω

( ) cemcJ i

N

i ii

i ωωηωωωδω

πω /

1

2

2)( −

=

=−= ∑Ohmic

linear coupling to the system position q

viscosity η

August 25, 2006 11

initial preparation:temperature T = 1/βkB

Feynman-Vernon influence functionaltime-nonlocal Gaussian correlations between q and q’

propagator

Wanted: position

→ needed: diagonal elements

Integration of the Bath Degrees of Freedom

[ ](t)Trq(t) Ratchet ρq=

qtqtqP )(),( ρ=

],[][][)0(ˆdd),( FV**

iiiifi

i

f

i

qqFqAqAqDDqqqqqtqPq

q

q

q′′′′′= ∫∫∫∫

′

′ρ

[ ]B

B

eTre)0()0(

BathH

H

W β

β

ρ −

−

=

i

t

f qtHtiqqA

′′−= ∫0 R )(ˆdexp][

],[FV qqF ′

Feynman-Vernonpath integralstechniques

′′

′′= ∫∫

tttHtWtHtW

00)(diexp(0))(di-exp(t)

August 25, 2006 12

Reduced density matrixof the system

Density matrixof the system-plus-bath

Reduced Density Matrix

[ ](t)Tr)( Bath Wt =ρ

many degrees of freedomtime-local correlations

)((t) tot tHW ↔

few relevant degrees of freedomtime-nonlocal correlations

Slide design inspired by U. Weiss

August 25, 2006 13

continuous path integrals

Bloch theorem, truncation to the bands of lowest energy

tight-binding description

non-linearity of the potential

exact expansion

duality relation to atight-binding system

Treatment of the system dynamics

Problems,approaches:

in tunneling amplitude in potential amplitude

Feynman-Vernon influence functionaltime-nonlocal Gaussian correlations between q and q’

propagator

],[][][)0(ˆdd),( FV**

iiiifi

i

f

i

qqFqAqAqDDqqqqqtqPq

q

q

q′′′′′= ∫∫∫∫

′

′ρ

i

t

f qtHtiqqA

′′−= ∫0 R )(ˆdexp][

],[FV qqF ′

Seriesexpression:

or

August 25, 2006 14

tight-binding description

transition rates

02inter

0intra

10

5.01.0

ω

ω

µµ

µµ

−′

′

≈∆

−≈∆

Bloch theorem

band structure

1st Approach: Few Energy Bandstruncation tolower energy bands

Eigenbasis of the position operator(Discrete Variable Representation)

mm ′Γ ,,νµ

(M=3)

00B~,; ωω <Ω<< FLTk

1q 2q 3q

1ε

2ε3ε

cell m = -1 cell m = 0 cell m = +1

position

ener

gy

1=µ

2=µ3=µ

intraµµ ′∆

interµµ ′∆

0ω

localized states

[ ( ) ( )]∑ ∑∑∑′

′ +′++′∆+=m p

pR mpmpmmmmH

µµµµµ

µµµµµµµε

,

)( ,,,,2,,

August 25, 2006 15

Path-integral expression→ Generalized Master Equation

M. Grifoni, M. Sasseti and U. Weiss, PRE (1996)

Transition rates

For high dissipation or high temperature:→ analytical expressions up to 2nd order in tunneling amplitude

U. Weiss, Quantum Dissipative Systems

Few Energy Bands II

Stationary solution of the Generalized Master Equation→ averaged velocity at long times as a function of the transition rates∞v mm ′Γ ,

,νµ

∑∫′′

′′′′ ′′−′=m

t

mmmm tPttKttP,

0 ,,;,, )()(d)(µ

µµµµ

∫∞

=Γ0 ,;,

,, )(d ττ νµνµ nmnm K

AC driving:

PP ,,, all Γ→ΓΓ>>Ω

( )2,,

,,

mmmm ′′ ∆∝Γ νµνµ

)cos()( tFtF Ω=

August 25, 2006 16

Current inversion depending on the parameters:

Few Energy Bands: Results

M. Grifoni, M. S. Ferreira, J. Peguiron and J. B. Majer, PRL 89, 146801 (2002)

Ratchet current

Driving amplitude

Drawbacks of the method:- breakdown at large F → no comparison with experiment, cannot reach classical limit- impossible to go to low temperature or dissipation due to our Golden Rule approximation

strong dampingmoderate damping

August 25, 2006 17

Dissipative tight-binding modelDissipative ratchet system

harmonics ↔ couplings

expansion of the kind

Fisher and Zwerger, PRB (1985)

2nd Approach: Duality Relation

TBTB0

0 )()( tqFtpqtq −++ →ηη

initial preparation

∑∞

=−=

1)2cos()(

lll q

LlVqV ϕπ ( )∑ ∑

∞

−∞=

∞

=

+∆++∆=n l

ll lnnnlnH1

*TB

∑∞

−∞=

=n

nnLnq ~TB

lil leV ∆=ϕ

2∆1

∆2

2TB )(1)()(

γωηωωηωω

+=↔= JJL

LLηπ2~ =↔ periodicity length

spectral density

L~

- single band - non nearest-neighbors couplings

∑±=

−=σ

σππ LqLq i2exp)2cos(long times

rare transitions

August 25, 2006 18

Ratchet current to third order

vanishes for symmetric potentials

Bistable driving:

Relation to the time-independent case

power series in the potential harmonics

Tight-binding dynamics → use the techniques developed for the 1st approach!

→ solve the Generalized Master Equation

Application: Ratchet Current

)(v)(v)(v DCDC FFFR −+= ∞∞∞

∑∞

=−

∞ Γ−Γ=1

TB )(~vm

mmmL

∑∞

=−

∞ Γ−Γ−=1

DC )()(vm

mmmLFFαη

Γm

n n+m

)2sin()(v 1222

1R ϕϕ −∝∞ VVF

0)2sin( 12 =− ϕϕ

−=+=

→ −

+

FFFF

tF )(

Transition rates Γm: power series in the couplings ∆l

Duality relation

August 25, 2006 19

Function of temperature, Function of driving,

Stationary velocity and ratchet currentVFL ∆= 57.0 VTk ∆= 076.0B

Weak dissipation

V∆==

76.02.0

γα

Localization in the TB system No Maxwell daemon

Free system

0)( ≡qVηF=0v

J. Peguiron and M. Grifoni, PRE 71, 010101R (2005)

August 25, 2006 20

As a function of dissipation strength:

Stationary velocity and ratchet current II

Localization at low temperature

Delocalization at low temperature

πηα2

2L=

J. Peguiron and M. Grifoni, Chem. Phys. 322, 169 (2006)

August 25, 2006 21

• Generalization to any Ohmic spectral density→ diffusion coefficient, current noise?

• The case of zero temperature → further analytical results?

Conclusions• Two complementary methods to evaluate the ratchet current in

different parameter regimes

• Ratchet effect and current inversions depending on the parameters

• Proper classical limit (with the duality relation)

• Explicit dependence on the potential (with the duality relation) → experiments ?

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