Quantum thermodynamics: Thermodynamics at the nanoscale

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

Frontiers of Quantum and Mesoscopic Thermodynamics

Prague, 26 July 2004

Session in memory of Vlada Capek

Outline

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

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

Position of works of Vlada Capek within quantum thermodynamics.

Introduction to quantum thermodynamics (Amsterdam-Paris-Yerevan view).

Introduction to quantum thermodynamics

Standard thermodynamics: large system + large bath + large work source

Classical thermodynamics: of bath only temperature T needed (and timescale for heat exchange)But consider for example:Mesoscopic ring: metal ring with size between micron and nanometer

1/10 000 cm 1/10 000 000 cm 0.1 hair 0.000 1 hair

Mesoscopic ring still has many atoms: many degrees of freedomStudy the electric current of such a ring at low temperature:one interesting degree of freedom coupled to many uninteresting ones

Quantum thermodynamics: small system, large bath, large worksource whole spectral density of coupling to bath needed

Caldeira-Leggett model 1983 (van Kampen’s thesis 1951; Ullersma’s thesis 1965)

soscillator harmonicmany ofbath n interactio particle

)22

( 22

22

22

2

iii

i i

ii

iitot x

m

m

pxcxx

b

m

pH

System-bath models: small quantum systems + large bath see book Uli Weiss 1993; 1998

Spin ½ : spin up or spin down = two level system

Capek models: coupled 2,3,4,5 two-level systems + their baths rich class of models

rich amount of physical phenomena

Spin-boson model: spin ½ + harmonic oscillator bath Leggett model + 10 coauthors: review 1983

Excursion to hill of Celts, April 2001

Is there a thermodynamic description?

dQdWdU HU where H is that part of the total Hamiltonian,

that governs the unitary part of (Langevin) dynamics

dW Work: Energy-without-entropy added to the system

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

1) Caratheodory: increase average energy of work source 2) Gibbs-Planck: energy of macroscopic degree of freedom

First law: Change in energy = work added + heat added

Internal energy in Caldeira-Leggett model

photons)or (phonons

bath oscillator harmonic n interactio particle

)22

( 22

22

22

2

iii

i i

ii

iitot x

m

m

pxcxx

b

m

pH

Taking together effects of bath yields: Langevin equation for particle

)'()'( )( ,)( )( ttKtttxxbxm

baxa

m

pH ,

222

2

22

2

2

) ( 2

) (

iiii

i

m

cJ

Ohm’s law for resistor: V = I R.

quasi-Ohmicspectral density

_______

Newton force defines system Hamiltonian:

Internal energy: U=<H> phonons: b renormalized to aphotons: a is the physical parameter

___________

“All” about work

Work = change of averge energy of system + bath

= minus (change of energy of work source)

= time-integral of rate of change of energy of system alone

Why does the average energy enter this definition? Thermodynamics does not apply to single systems

Quantum mechanics does not apply to single systems

What is special about macroscopic work source? It produces time-dependent parabeters e.g. m(t), b(t), V(t)

so it does not enlarge dimension of Hilbert space.

The second law of thermodynamicsHeat goes from high temperatures to low temperaturesNo cycles of work from bath (no perpetuum mobile): Thomson formul- Optimal changes are adiabatically slow ationEntropy of closed system cannot decreaseRate of entropy production is non-negative

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

Finite quantum systems: Thermodynamics endangeredNo thermodynamic limit : Different formulations become inequivalent Some may apply, others not

The Linus effect:The cloud goes where Linus goes

The appearence of clouds (“the Linus effect”)

Clausius inequality may be violatedTdSdQ

A+N, PRB 02, experiments proposed for mesoscopic circuits J. Phys A 02 expts for quantum optics.

0dm if 0 m

dm

2 2

dQ

In small quantum systems at not very high temperaturesa cloud of bath modes surrounds the central particleKondo cloud, polaron cloud

Such clouds must be attributed to bathNot part of standard thermodynamics: new effects in quantum thermo

A+N: 2000, 2002

Caldeira-Leggett at T = 0:

Negative rate of energy dispersion, though starting from equilibriumOut of equilibrium: work extraction cycles constructed (Finite yield)

Capek: electric currents, heat currents going in “wrong” direction

Work extraction from finite quantum systems

Thermodynamics: minimize final energy at fixed entropyAssume final state is gibbsian: fix final T from S = const.

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

ABN, EPL 2004: Properties of ergotropy

• Majorization: defines set of states within which thermodynamic relations are satisfied qualitatively.

• Other states: all kinds of thermodynamic surprises

Are adiabatic processes always optimal?

Minimal work principle (one of the formulations of the second law):Slow thermally isolated processes (“adiabatic 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 : minimal work principle 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

Summary

New results for thermodynamics of small Quantum-systems:

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

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

Vada Capek was strong forefighter of Quantum Thermodynamics

Vlada Capek was strong forefighter of Quantum Thermodynamics

Summary

Closing session

• Thanks to all those who contributed and why

1) Why of those2) All of those

Vlada Capek was a strong forefighter of Quantum Thermodynamics Capek models

In loving memory

The Linus effect:The cloud goes where Linus goes

Capek’s and our common issue in science

dampingrelaxationentanglementpurity

quantum thermodynamics = classical thermodynamics + Linus book with Daniel Sheehan

Thanks to all participants

• You all came here in good mood Contributed to the extremely high level of the meeting

• Even though we could provide no funding

• Even though we will ask you to contribute to the proceedings = equally fine as the meeting

• Thanks, thanks and (thanks)^2

• Special thanks to Toni Leggett

Thanks to our sponsors

• Czech Senate, Wallenstein palaceCzech Academy of SciencesCharles University

Masarykova kolej

• Local hotels, printing office, restaurant

• Czech press

Thanks to the scientific organizers etc

• Roger BalianMarlan ScullyDaniel SheehanMilena GrifoniVladimir Zakharov + Alexei NikulovVaclav SpickaTheo Nieuwenhuizen + Armen Allahverydan

• Our international organizer: Peter Keefe

• All chairwomen and chairmen (chairhumans)

Thanks to our many local organizers

• Jiri BokPetr ChovstaMichal FantaSona FialovaPavel HubikZdenek Kozisek

•Karla KuldovaJan KrajnikJiri MaresEvzen SubrtDavid VyskocilKarolina Vyskocilova

Thanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks

There is one special person to thank• Our friend and main organizer

• Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav

Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav