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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