Quantum thermodynamics: Thermodynamics at the nanoscale

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)

2. 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 engines Towards a quantum fluctuation theorem?

3. First law: 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 macroscopic source induces classical functions of time. 1) Just energy increase of work source2) Gibbs-Planck: energy of macroscopic degree of freedom The rest: energy-without-work from the bath Energy related to uncontrollable degrees of freedom

4. Caldeira-Leggett model Langevin equation if initially no correlation between S and B

5. Second law for finite quantum systems No 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 A+N, PRL 85, 1799 (’00) ; PRE 66, 036102 (’02), PRB 02, J. Phys A,02 experiments proposed for mesoscopic circuits and quantum optics. - Rate of energy dispersion may be negative Classically: = T*( rate of entropy production ): non-negative

6. Work extraction from finite Q-systems Couple to work source and do all possible work extractions 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, 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

7. -non-gibbsian states can be passive -Comparison of activities: Aspects of ergotropy Thermodynamic upper bounds: more work possible from But actual work may be largest from -Coupling to an auxiliary system : if is less active than Then can be more active than -Thermodynamic regime reduced to states that majorize one another - Optimal unitary transformations unitary matrices U(t) do yield, in examples, explicit Hamiltonians for achieving optimal work extraction

8. 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

9. Q-engines ABN’04: System S coupled to two baths, with T1 < T2 Baths Gibbsian but correlatedCorrelation entropy Theorem: - No correlations: Carnot efficiency is optimum- Correlations can give more work: Scully group: phaseonium

10. Photo-Carnot engine Scully 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

11. Fluctuation relation Classical work in trajectory W = H(x(t),p(t),t)-H(x(0),p(0),0) -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

12. Frontiers of Quantum and Mesoscopic Thermodynamics 26-29 July 2004, Prague, Czech RepublicSatellite 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

13. Summary Q-thermodynamics: small system, large work source+bath Different formulations of the second law have different ranges of validity Experimental tests feasible e.g. in quantum optics • 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