1. 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

2. 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).

3. 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 freedom Study 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

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

5. Excursion to hill of Celts, April 2001

6. Is there a thermodynamic description ? where H is that part of the total Hamiltonian, that governs the unitary part of (Langevin) dynamics Work: Energy-without-entropy added to the system The rest: energy-without-work from the bath Energy 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

7. Internal energy in Caldeira-Leggett model Taking together effects of bath yields: Langevin equation for particle Ohm’s law for resistor: V = I R. quasi-Ohmic spectral density _______ Newton force defines system Hamiltonian: Internal energy: U= phonons: b renormalized to a photons: a is the physical parameter ___________

8. “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.

9. The second law of thermodynamics Heat goes from high temperatures to low temperatures No cycles of work from bath (no perpetuum mobile): Thomson formul- Optimal changes are adiabatically slow ation Entropy of closed system cannot decrease Rate 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 endangered No thermodynamic limit : Different formulations become inequivalent Some may apply, others not

10. The Linus effect: The cloud goes where Linus goes

11. The appearence of clouds (“the Linus effect”) Clausius inequality may be violated A+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 cloud Such clouds must be attributed to bath Not 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 equilibrium Out of equilibrium: work extraction cycles constructed (Finite yield) Capek: electric currents, heat currents going in “wrong” direction

12. Work extraction from finite quantum systems Thermodynamics: 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: ergotropy Couple to work source and do all possible work extractions

13. 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

14. 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

15. 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+bath Different formulations of the second law have different ranges of validity Experimental tests feasible e.g. in quantum optics Vada Capek was strong forefighter of Quantum Thermodynamics

16. Vlada Capek was strong forefighter of Quantum Thermodynamics Summary

19. Closing session Thanks to all those who contributed and why 1)Why of those 2)All of those

20. Vlada Capek was a strong forefighter of Quantum Thermodynamics Capek models In loving memory

21. The Linus effect: The cloud goes where Linus goes Capek’s and our common issue in science damping relaxation entanglement purity quantum thermodynamics = classical thermodynamics + Linus book with Daniel Sheehan

22. 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

23. Thanks to our sponsors Czech Senate, Wallenstein palace Czech Academy of Sciences Charles University Masarykova kolej Local hotels, printing office, restaurant Czech press

24. Thanks to the scientific organizers etc Roger 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)

25. Thanks to our many local organizers Jiri Bok Petr Chovsta Michal Fanta Sona Fialova Pavel Hubik Zdenek Kozisek Karla Kuldova Jan Krajnik Jiri Mares Evzen Subrt David Vyskocil Karolina Vyskocilova Thanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks

26. There is one special person to thank Our friend and main organizer Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav