Quantum thermodynamics:Thermodynamics at the nanoscale

Armen E. Allahverdyan (Amsterdam/Yerevan)Roger Balian (CEA-Saclay; Academie des Sciences)Theo M. Nieuwenhuizen (University of Amsterdam)

Outline

First law: what is work, what is heat, system energy.Second law: confirmation versus violations.

Maximal extractable work from a quantum system.Are adiabatic changes always optimal?

Efficiency of quantum enginesTowards a quantum fluctuation theorem?

First law: is there a thermodynamic description?

dWdQdU

HU where H is that part of the total Hamiltonian,that governs the unitary part of (Langevin) dynamics

dWWork: Energy-without-entropy added to the system macroscopic source induces classical functions of time.

dQ The rest: energy-without-work from the bathEnergy related to uncontrollable degrees of freedom

1) Just energy increase of work source2) Gibbs-Planck: energy of macroscopic degree of freedom

Caldeira-Leggett model

ii bath, linear ninteractio system

ixi im

i imip

ixi

icxxb

m

ptotH

)2222

2(2

22

2

Langevin equation if initially no correlation between S and B

)'()'()()()( ttKt t ,t xx bxm

ba xa

m

pH :nHamiltonia System ,2

22

2

0dm if 0 2m

dm TdQ :ineq Clausius

2)0(

22

2)(

2

2)(:

i i

i imic

J Ohmicquasi

Second law for finite quantum systemsNo thermodynamic limit Thermodynamics endangered Different formulations are inequivalent

Generalized Thomson formulation is valid:Cyclic changes on system in Gibbs equilibrium cannot yield work (Pusz+Woronowicz ’78, Lenard’78, A+N ’02.)

-Clausius inequality may be violated due to formation of cloud of bath modes

TdSdQ

TT

CdTS :eConsequenc 0 '

'

- Rate of energy dispersion may be negative

Classically: = T *( rate of entropy production ): non-negative

A+N, PRL 85, 1799 (’00) ; PRE 66, 036102 (’02), PRB 02, J. Phys A,02 experiments proposed for mesoscopic circuits and quantum optics.

0dm if 0 2m

dm TdQ :ineq Clausius

2)0(

Work extraction from finite Q-systems

Thermodynamics: minimize final energy at fixed entropyAssume final state is gibbsian: fix final T from S = const.Extracted work=(free) energy difference U(0)-TS(0)+T log Z(T)

But: Quantum mechanics is unitary, )()0()()( tUtUt So all n eigenvalues conserved: n-1 constraints: (Gibbs state typically unattainable for n>2) Optimal: eigenvectors of become those of H, if ordering

n

iiiUW

1)0( Maximally extractable work:

ergotropy

tiontransforma-work- ergotropy (Clausius) tiontransforma-in-entropy

tiontransformaturnwork

,

Couple to work source and do all possible work extractions

Ze in as ,d ,d /...21...21

Aspects of ergotropy

- Optimal unitary transformations unitary matrices U(t) do yield, in examples, explicit Hamiltonians for achieving optimal work extraction

-non-gibbsian states can be passive

-Comparison of activities: )()();0();0( SS but UU

Thermodynamic upper bounds: more work possible from But actual work may be largest from

-Coupling to an auxiliary system : if is less active thanThen can be more active than

-Thermodynamic regime reduced to states that majorize one another

1...nk for k

1jj

k

1jj if , majorizes sr

,

Are adiabatic processes always optimal?

One of the formulations of the second law:Adiabatic thermally isolated processes done on an equilibrium system are optimal (cost least work or yield most work)

In finite Q-systems: Work larger or equal to free energy difference But adiabatic work is not free energy

difference.A+N, 2003: -No level crossing : adiabatic theorem holds

-Level crossing: solve using adiabatic perturbation theory. Diabatic processes are less costly than adiabatic. Work = new tool to test level crossing.

Level crossing possible if two or more parameters are changed. Review expts on level crossing: Yarkony, Rev Mod Phys 1996

Q-enginesABN’04: System S coupled to two baths, with T1 < T2

Zetr

H

112

11

21

Baths Gibbsian

but correlated

Correlation entropy

2112

01221 SSSScorr

Theorem:

- No correlations: Carnot efficiency is optimum- Correlations can give more work: Scully group: phaseonium

SSQT

TT

fcorricorr ,,2

1

2

11

Photo-Carnot engineScully group, Texas A&M (Science, 2002)

The cavity has one movable mirror, temperature T_c

Volume is set by photon pressure on piston

Coherent atom beam ‘phaseonium’ interacts with photons: bath T_h

Efficiency exceeds Carnot value, due to correlations of atoms

Fluctuation relationClassical work in trajectory W = H(x(t),p(t),t)-H(x(0),p(0),0)

1)0(

)(

)0(

1)()(

)0(

1)0()0(

)0()0()0(

)(

)(

)]0()([)0(

Z

tZ

Ztdptdx

Zdpdx

Zdpdx

e

e

ee

e

tH

tH

HtHH

W

relation Jarzynski-KuzovlevBochkov We 1

-Trajectories with W<0 yield work; observed in small systems

What about Quantum regime? Is QM compatible with such a relation?How should work be defined for small Q-systems?

At t=0 preparation by measuring H: selection of subensembles.Can work

Frontiers of Quantum and Mesoscopic Thermodynamics26-29 July 2004, Prague, Czech Republic

Satellite Conference of the Condensed Matter Division, EPS, 19 - 23 July Prague

Tentative list of topics

Quantum, mesoscopic and (partly) classical thermodynamics •Quantum limits to the second law. •Quantum measurement •Quantum decoherence and dephasing •Mesoscopic and nanomechanical systems •Classical molecular motors, ratchet systems and rectified motion •Quantum Brownian motion •Quantum motors •Relevant experiments from the nano- to the macro-scale

Summary

New results for thermodynamics of small Q-systems:

-violation of Clausius inequality-optimal extractable work: ergotropy-adiabatic changes non-optimal if level crossing

-correlations enhance efficiency in Q- engines- Q-fluctuation relation

Q-thermodynamics: small system, large work source+bathDifferent formulations of the second law have different ranges of validityExperimental tests feasible e.g. in quantum optics