Quantum thermodynamics: Thermodynamics at the nanoscale

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27 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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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

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).

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

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

Excursion to hill of Celts, April 2001

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


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=<H> phonons: b renormalized to a





photons: a is the physical parameter

___________

“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.

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

The Linus effect:

The cloud goes where Linus goes

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

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

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


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

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

Vlada Capek was strong forefighter of Quantum Thermodynamics

Summary



Closing session


Thanks to all those who contributed

and why

1)
Why of those

2)
All of those


Vlada Capek


was a strong forefighter of Quantum Thermodynamics


Capek models

In loving memory

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

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


Thanks to our sponsors


Czech Senate, Wallenstein palace

Czech Academy of Sciences

Charles University


Masarykova kolej



Local hotels, printing office, restaurant



Czech press

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)

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

There is
one

special person to thank


Our friend and main organizer



Vaclav Vaclav Vaclav Vaclav
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