Quantum thermodynamics: Thermodynamics at the nanoscale

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Quantum thermodynamicsThermodynamics at the
nanoscale
Armen E. Allahverdyan (Amsterdam/Yerevan)Roger
Balian (CEA-Saclay Academie des Sciences)Theo
M. Nieuwenhuizen (University of Amsterdam)
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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?
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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
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Caldeira-Leggett model
Langevin equation if initially no correlation
between S and B
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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 (PuszWoronowicz 78, Lenard78,
AN 02.)

• Clausius inequality may
  be violated
• due to formation of cloud of bath modes

AN, 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
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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 ngt2) Optimal
eigenvectors of become those of H, if
ordering
Maximally extractable work ergotropy
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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
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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.
AN, 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
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Q-engines
ABN04 System S coupled to two baths, with T1 lt
T2
Baths Gibbsian
but correlatedCorrelation entropy
Theorem - No correlations Carnot efficiency
is optimum- Correlations can give more work
Scully group phaseonium
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Photo-Carnot engine
Scully group, Texas AM (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
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Fluctuation relation
Classical work in trajectory W
H(x(t),p(t),t)-H(x(0),p(0),0)
-Trajectories with Wlt0 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 t0 preparation by measuring H selection of
subensembles. Can work
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Frontiers of Quantum and Mesoscopic
Thermodynamics 26-29 July 2004, Prague, Czech
Republic Satellite 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

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Summary
Q-thermodynamics small system, large work
sourcebath 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