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Quantum thermodynamics: Thermodynamics at the nanoscale. Armen E. Allahverdyan (Amsterdam/Yerevan) Roger Balian (CEA-Saclay; Academie des Sciences) Theo M. Nieuwenhuizen (University of Amsterdam). Session in memory of Vlada Capek. - PowerPoint PPT Presentation

Quantum thermodynamics: Thermodynamics at the nanoscaleArmen 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 2004Session in memory of Vlada Capek

OutlineFirst 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 thermodynamicsStandard 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 hairMesoscopic 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 onesQuantum thermodynamics: small system, large bath, large worksource whole spectral density of coupling to bath needed

Caldeira-Leggett model 1983 (van Kampens thesis 1951; Ullersmas thesis 1965)System-bath models: small quantum systems + large bath see book Uli Weiss 1993; 1998Spin : spin up or spin down = two level systemCapek models: coupled 2,3,4,5 two-level systems + their baths rich class of models rich amount of physical phenomenaSpin-boson model: spin + harmonic oscillator bath Leggett model + 10 coauthors: review 1983

Excursion to hill of Celts, April 2001

Is there a thermodynamic description?where H is that part of the total Hamiltonian,that governs the unitary part of (Langevin) dynamicsWork: Energy-without-entropy added to the system 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 freedomFirst law: Change in energy = work added + heat added

Internal energy in Caldeira-Leggett modelTaking together effects of bath yields: Langevin equation for particleOhms law for resistor: V = I R. quasi-Ohmicspectral density_______Newton force defines system Hamiltonian:Internal energy: U= phonons: b renormalized to aphotons: a is the physical parameter___________

All about workWork = 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 ation Entropy of closed system cannot decreaseRate of entropy production is non-negativeBut: Generalized Thomson formulation is valid:Cyclic changes on system in Gibbs equilibrium cannot yield work (Pusz+Woronowicz 78, Lenard78, 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 violatedA+N, PRB 02, experiments proposed for mesoscopic circuits J. Phys A 02 expts for quantum optics.In small quantum systems at not very high temperatures a cloud of bath modes surrounds the central particle Kondo cloud, polaron cloudSuch clouds must be attributed to bathNot part of standard thermodynamics: new effects in quantum thermoA+N: 2000, 2002Caldeira-Leggett at T = 0:Negative rate of energy dispersion, though starting from equilibrium Out of equilibrium: work extraction cycles constructed (Finite yield) Capek: electric currents, heat currents going in wrong direction

Work extraction from finite quantum systemsThermodynamics: minimize final energy at fixed entropy Assume final state is gibbsian: fix final T from S = const.But: Quantum mechanics is unitary, So all n eigenvalues conserved: n-1 constraints: (Gibbs state typically unattainable for n>2) Optimal: eigenvectors of become those of H, if ordering Maximally extractable work:ergotropyCouple to work source and do all possible work extractions

ABN, EPL 2004: Properties of ergotropyMajorization: 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

SummaryNew results for thermodynamics of small Quantum-systems: -violation of Clausius inequality-optimal extractable work: ergotropy-adiabatic changes non-optimal if level crossingQ-thermodynamics: small system, macroscopicwork source+bathDifferent formulations of the second law have different ranges of validityExperimental tests feasible e.g. in quantum opticsVada Capek was strong forefighter of Quantum Thermodynamics

Vlada Capek was strong forefighter of Quantum ThermodynamicsSummary

Closing sessionThanks to all those who contributed and whyWhy of thoseAll of those

Vlada Capek was a strong forefighter of Quantum Thermodynamics Capek modelsIn loving memory

The Linus effect:The cloud goes where Linus goesCapeks and our common issue in sciencedamping relaxation entanglement purityquantum thermodynamics = classical thermodynamics + Linus book with Daniel Sheehan

Thanks to all participantsYou all came here in good mood Contributed to the extremely high level of the meetingEven though we could provide no fundingEven though we will ask you to contribute to the proceedings = equally fine as the meetingThanks, thanks and (thanks)^2Special thanks to Toni Leggett

Thanks to our sponsorsCzech Senate, Wallenstein palace Czech Academy of Sciences Charles University Masarykova kolej

Local hotels, printing office, restaurant Czech press

Thanks to the scientific organizers etcRoger Balian Marlan Scully Daniel Sheehan Milena Grifoni Vladimir Zakharov + Alexei Nikulov Vaclav Spicka Theo Nieuwenhuizen + Armen Allahverydan

Our international organizer: Peter Keefe All chairwomen and chairmen (chairhumans)

Thanks to our many local organizersJiri Bok Petr Chovsta Michal Fanta Sona Fialova Pavel Hubik Zdenek KozisekKarla Kuldova Jan KrajnikJiri Mares Evzen SubrtDavid Vyskocil Karolina VyskocilovaThanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks

There is one special person to thankOur 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