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 source 2) 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, (’02), PRB 02, J. Phys A, 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 entropy Assume 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. Aspects of ergotropy -non-gibbsian states can be passive
-Comparison of activities:
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 correlated Correlation 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 Republic Satellite Conference of the Condensed Matter Division, EPS, 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