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

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Switzerland Work done with Milena Grifoni

University of Technology, The Netherlands

1 Dynamics of Quantum Dissipative Systems: The Example of Quantum Brownian Motors Joël Peguiron Department of Physics and Astronomy, University of Basel, Switzerland Work done with Milena Grifoni at Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands Institut I - Theoretische Physik, Universität Regensburg, Germany CCP6 Workshop on Open Quantum Systems Bangor, Wales, UK, 25 August 2006

2 Aim of the Talk Illustrate some techniques used by theoretical physicists to describe quantum dissipative systems with the example of quantum Brownian motors (also known as quantum ratchets) 2

3 What is a Brownian Motor / Ratchet? A ratchet system is: periodic asymmetric designed and driven to extract work from fluctuating forces 3

4 Ratchets: a Physicist s Model A single particle in a 1D periodic asymmetric potential Driving force: tilting of the potential V(q) F q V(q) q q maintains a non-equilibrium situation A ratchet system is: work extraction allowed by the 2nd Principle of Thermodynamics unbiased e.g. rocking force work released particle current Natural case: interaction with a periodic dissipative thermal environment asymmetric designed and driven to extract work from fluctuating forces 4

5 The Classical Ratchet Effect Γbackward Γforward force +F Current: force -F v DC Γ forward Γ backward Thermal rate: Γth e U k BT The barrier height ΔU may be different in opposite tilted situations ratchet effect: net average current v DC ( + F ) v DC ( F ) This current may be tuned through the parameters of the environment 5

6 The Quantum Ratchet Effect Γbackward force +F Γforward tunneling Current: force -F v DC Γ forward Γ backward Thermal rate: Γth e U k BT Γtun =? The interaction with a dissipative environment is crucial The barrier parameters may be different in opposite tilted situations quantum ratchet effect: net average current v DC ( + F ) v DC ( F ) 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) 6

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 motion P. Hänggi and G. L. Ingold, Chaos 15, (2005) Quantum Ratchets, J.P., PhD thesis (2005) available from 7

Quantum Ratchet Experiments Vortices in Josephson junction arrays quasi 1D

junction arrays potential designed at will F. Falo et al., Appl. Phys.

, Science 286, 2314 (1999) force velocity J. B. Majer et al.

8 Quantum Ratchet Experiments Vortices in Josephson junction arrays quasi 1D dynamics quantum regime ( EJ / EC 11 ) Vortices in circular Josephson junction arrays potential designed at will F. Falo et al., Appl. Phys. A 75, 263 (2002) Electrons in asymmetrically confined 2DEG H. Linke et al., Science 286, 2314 (1999) force velocity J. B. Majer et al., PRL 90, (2003) Vortices in annular long Josephson junctions A. Ustinov, University of Nürnberg-Erlangen 8

9 Model Quantity of interest: the particle current at long times H tot (t ) = H ratchet + H driving (t ) + H bath Ratchet H ratchet p2 = + VR ( q ) 2m 1D system, asymmetric potential Driving unbiased H driving = q F (t ) F 2 n +1 (t ) = 0 How to introduce dissipation in a quantum framework? Bath 1D Ratchet system Driving force 9

10 Quantum Dissipative Bath Bath of harmonic oscillators A.O. Caldeira and A.J. Legget, Ann. Phys. 149, 374 (1983) H bath 2 P2 1 ci q 2 i = + miωi X i 2 2 miωi i =1 2 mi N characterized by the spectral density π N ci2 J (ω ) = δ ( ω ω i ) = ηω e ω / ω c 2 i =1 miω i viscosity η linear coupling to the system position q Ohmic Other solution: contact to reservoirs of non-interacting electrons J. Lehmann et al., PRL 88, (2002) 10

11 Integration of the Bath Degrees of Freedom i t i t W (t) = exp - dt H (t ) W (0) exp dt H (t ) 0 0 Wanted: position Feynman-Vernon path integrals techniques q(t) = TrRatchet [ qρ (t) ] needed: diagonal elements P ( q, t ) = q ρ (t ) q initial preparation: e βh B W (0) = ρ (0) temperature T = 1/βkB TrBath e βh B [ P(qf, t ) = dqi dqi qi ρˆ (0) qi propagator i t A[q ] = q f exp dt Hˆ R (t ) qi 0 qf qi ] qi Dq D *q A[q ] A*[q ]FFV [q, q ] qi Feynman-Vernon influence functional FFV [ q, q ] time-nonlocal Gaussian correlations between q and q 11

density matrix of the system ρ (t ) = TrBath [W (t)] few relevant

12 Reduced Density Matrix Density matrix of the system-plus-bath W (t) H tot (t ) many degrees of freedom time-local correlations Reduced density matrix of the system ρ (t ) = TrBath [W (t)] few relevant degrees of freedom time-nonlocal correlations Slide design inspired by U. Weiss 12

13 Treatment of the system dynamics P (qf, t ) = dqi dqi qi ρˆ (0) qi propagator i t A[q ] = q f exp dt Hˆ R (t ) qi 0 Problems, approaches: qf qi Dq D *q A[q ] A*[q ]FFV [q, q ] qi Feynman-Vernon influence functional FFV [q, q ] time-nonlocal Gaussian correlations between q and q non-linearity of the potential continuous path integrals or Bloch theorem, truncation to the bands of lowest energy Series expression: qi exact expansion tight-binding description duality relation to a tight-binding system in tunneling amplitude in potential amplitude 13

1st Approach: Few Energy Bands truncation to lower energy bands Bloch theorem band structure ω 0 k BT << ω0 ; FL, Ω ~ < ω 0 Eigenbasis of the position operator (Discrete Variable Representation)

14 1st Approach: Few Energy Bands truncation to lower energy bands Bloch theorem band structure ω 0 k BT << ω0 ; FL, Ω ~ < ω 0 Eigenbasis of the position operator (Discrete Variable Representation) localized states cell m = -1 transition rates Γµm,ν,m HR = [ m µ (M=3) µ =2 cell m = +1 ε1 intra µ = 1 µµ intra µµ ω 0 2 inter ω 0 µµ 10 cell m = 0 energy tight-binding description inter µµ ε3 ε2 µ =3 q1 q2 q3 position ε µ µ, m µ, m + ( (µpµ) 2) ( µ, m µ, m + p + µ, m + p µ, m µ,µ p )] 14

15 Few Energy Bands II Path-integral expression Generalized Master Equation P µ,m (t ) = µ, m t 0 dt K µ,m ;µ,m (t t ) Pµ,m (t ) AC driving: F (t ) = F cos(ωt ) Ω >> all Γ, Γ, P Γ, P M. Grifoni, M. Sasseti and U. Weiss, PRE (1996) Stationary solution of the Generalized Master Equation m, m averaged velocity at long times v as a function of the transition rates Γµ,ν Transition rates Γ m,n µ,ν = dτ K µ,m;ν,n (τ ) 0 For high dissipation or high temperature: analytical expressions up to 2nd order in tunneling amplitude m, m µ,ν Γ ( ) m, m 2 µ,ν U. Weiss, Quantum Dissipative Systems 15

16 Few Energy Bands: Results Current inversion depending on the parameters: Ratchet current Driving amplitude moderate damping strong damping M. Grifoni, M. S. Ferreira, J. Peguiron and J. B. Majer, PRL 89, (2002) 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 16

2nd Approach: Duality Relation expansion of the kind long times q (t ) q0 + rare transitions cos(2πq L) = exp{ 2πiσq L} σ =± p0 Ft + qtb (t ) η η TB Fisher and Zwerger, PRB (1985) initial preparation

17 2nd Approach: Duality Relation expansion of the kind long times q (t ) q0 + rare transitions cos(2πq L) = exp{ 2πiσq L} σ =± p0 Ft + qtb (t ) η η TB Fisher and Zwerger, PRB (1985) initial preparation Dissipative tight-binding model Dissipative ratchet system 2π V (q) = Vl cos(l q ϕl ) L l =1 H TB = harmonics couplings Vl iϕ l e = l 2 ( n = l =1 * l n + l n + l n n + l qtb = ~ nl n - single band n = - non nearest-neighbors couplings 1 2 ~ L ~ 2π L L= ηl J (ω ) = ηω J TB (ω ) = periodicity length ηω 1 + (ω γ ) 2 spectral density 17 n )

18 Application: Ratchet Current Tight-binding dynamics use the techniques developed for the 1st approach! Γm solve the Generalized Master Equation v TB ~ = L m(γm Γ m ) m =1 Transition rates Γm: power series in the couplings l n n+m Duality relation F L v ( F ) = m(γm Γ m ) power series in the potential harmonics η α m =1 DC Bistable driving: F + = + F F (t ) F = F Relation to the time-independent case v R ( F ) = v DC ( F ) + v DC ( F ) Ratchet current to third order v R ( F ) V12V2 sin(ϕ 2 2ϕ1 ) vanishes for symmetric potentials sin(ϕ 2 2ϕ1 ) = 0 18

19 Stationary velocity and ratchet current Function of driving, k BT = V Function of temperature, FL = 0.57 V Free system V (q) 0 v0 = F η Localization in the TB system No Maxwell daemon Weak dissipation α = 0.2 γ = 0.76 V J. Peguiron and M. Grifoni, PRE 71, R (2005) 19

20 Stationary velocity and ratchet current II As a function of dissipation strength: ηl2 α= 2π Localization at low temperature Delocalization at low temperature J. Peguiron and M. Grifoni, Chem. Phys. 322, 169 (2006) 20

21 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? Future Projects Generalization to any Ohmic spectral density diffusion coefficient, current noise? The case of zero temperature further analytical results? 21

Preface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1

Preface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1 xi Contents Preface Preface to the Third Edition Preface to the Second Edition Preface to the First Edition v vii viii ix 1 Introduction 1 I GENERAL THEORY OF OPEN QUANTUM SYSTEMS 5 Diverse limited approaches: