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)
Session in memory of Vlada Capek
Frontiers of Quantum and Mesoscopic
Thermodynamics Prague, 26 July 2004
2
Outline
Introduction to quantum thermodynamics
(Amsterdam-Paris-Yerevan view).
Position of works of Vlada Capek within quantum
thermodynamics.
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?
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 exampleMesoscopic 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 Kampens thesis
1951
Ullersmas thesis 1965)
System-bath models small quantum systems large
bath
see book Uli Weiss 1993 1998
Spin-boson model spin ½ harmonic oscillator
bath Leggett model 10
coauthors review 1983
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
5
Excursion to hill of Celts, April 2001
6
Is there a thermodynamic description?
First law Change in energy work added heat
added
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
1) Caratheodory increase average energy 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
7
Internal energy in Caldeira-Leggett model
Ohms law for resistor V I R.
quasi-Ohmicspectral density
Taking together effects of bath yields Langevin
equation for particle
_______
___________
Newton force defines system Hamiltonian
Internal energy UltHgt 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
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.
Why does the average energy enter this
definition? Thermodynamics does not apply to
single systems Quantum mechanics
does not apply to single systems
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
ationEntropy of closed
system cannot decrease Rate of entropy production
is non-negative
Finite quantum systems Thermodynamics
endangered No thermodynamic limit Different
formulations become inequivalent
Some may apply, others
not
But Generalized Thomson formulation is
valid Cyclic changes on system in Gibbs
equilibrium cannot yield work
(PuszWoronowicz 78, Lenard78, AN 02.)
10
The Linus effectThe cloud goes where Linus goes
11

• The appearence of clouds (the Linus effect)

In small quantum systems at not very high
temperaturesa cloud of bath modes surrounds the
central particleKondo cloud, polaron cloud
Such clouds must be attributed to bath Not part
of standard thermodynamics new effects in
quantum thermo
AN 2000, 2002
Clausius inequality may
be violated
Caldeira-Leggett at T 0
Negative rate of energy dispersion, though
starting from equilibriumOut of equilibrium
work extraction cycles constructed (Finite yield)
AN, PRB 02, experiments proposed for
mesoscopic circuits J. Phys A 02
expts for quantum optics.
Capek electric currents, heat currents going in
wrong direction
12
Work extraction from finite quantum 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.
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
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.
AN, 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
Q-thermodynamics small system, macroscopicwork
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
Quantum-systems -violation of Clausius
inequality -optimal extractable work
ergotropy -adiabatic changes non-optimal if level
crossing
Vada Capek was strong forefighter of Quantum
Thermodynamics
16
Summary
Vlada Capek was strong forefighter of Quantum
Thermodynamics
17
(No Transcript)
18
(No Transcript)
19
Closing session

• Thanks to all those who contributed and why

1. Why of those
2. All of those

20
In loving memory
Vlada Capek was a
strong forefighter of Quantum Thermodynamics
Capek models
21
The Linus effectThe cloud goes where Linus goes
Capeks and our common issue in science
dampingrelaxationentanglementpurity
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 palaceCzech Academy of
  SciencesCharles UniversityMasarykova kolej
• Local hotels, printing office, restaurant
• Czech press

24
Thanks to the scientific organizers etc

• Roger BalianMarlan ScullyDaniel SheehanMilena
  GrifoniVladimir Zakharov Alexei NikulovVaclav
  SpickaTheo Nieuwenhuizen Armen Allahverydan
• Our international organizer Peter Keefe
• All chairwomen and chairmen (chairhumans)

25
Thanks to our many local organizers

• Karla KuldovaJan Krajnik
• Jiri MaresEvzen Subrt
• David VyskocilKarolina Vyskocilova

• Jiri BokPetr ChovstaMichal FantaSona
  FialovaPavel HubikZdenek Kozisek

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 Vaclav Vaclav
  Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav Vaclav
  Vaclav Vaclav Vaclav