Nonequilibrium entropy production in open and closed quantum systems [Elektronische Ressource] / vorgelegt von Sebastian Deffner

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Nonequilibrium entropy production inopen and closed quantum systemsDissertationzur Erlangung des Doktorgrades der Naturwissenschaftenan der Mathematisch-Naturwissenschaftlichen Fakulta¨tder Universita¨t AugsburgvorgelegtvonDipl.-Phys. SebastianDeffnerausAugsburgAugsburg,Dezember2010Dissertation zur Promotion im Institut fu¨r Physik der Mathematisch-NaturwissenschaftlichenFakulta¨tanderUniversita¨tAugsburg(Doctorrerumnaturalium)1. Gutachter: Dr. EricLutz2. Gutachter: Prof. Dr. PeterHa¨nggi3. Gutachter: Prof. Dr. UdoSeifertTagdermu¨ndlichenPru¨fung: 07. Februar2011NonequilibriumentropyproductioninopenandclosedquantumsystemsLitterarumradices amaras esse,fructusiucundiores.(MarcusPorciusCato)IIIIVContents1 Prologue 11.1 Thermodynamics-Thetheoryofheatandwork . . . . . . . . . . . . . . . . . . . . 11.2 Organization ofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Classical systems far from equilibrium 52.1 Entropyproductioninthelinearregime . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Microscopicdynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Langevinequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Fokker-Planckequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Generalizations ofthesecondlawarbitrarily farfromequilibrium . . . . . . . . . 122.3.1 Jarzynski’sworkrelation . . . . . . . . . . . . . . . . . . . . . . .
Publié le : vendredi 1 janvier 2010
Lecture(s) : 38
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Source : D-NB.INFO/101192949X/34
Nombre de pages : 145
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Nonequilibrium entropy production in
open and closed quantum systems
Dissertation
zur Erlangung des Doktorgrades der Naturwissenschaften
an der Mathematisch-Naturwissenschaftlichen Fakulta¨t
der Universita¨t Augsburg
vorgelegtvon
Dipl.-Phys. SebastianDeffner
ausAugsburg
Augsburg,Dezember2010Dissertation zur Promotion im Institut fu¨r Physik der Mathematisch-Naturwissenschaftlichen
Fakulta¨tanderUniversita¨tAugsburg(Doctorrerumnaturalium)
1. Gutachter: Dr. EricLutz
2. Gutachter: Prof. Dr. PeterHa¨nggi
3. Gutachter: Prof. Dr. UdoSeifert
Tagdermu¨ndlichenPru¨fung: 07. Februar2011Nonequilibriumentropyproductioninopenandclosedquantumsystems
Litterarumradices amaras esse,
fructusiucundiores.
(MarcusPorciusCato)
IIIIVContents
1 Prologue 1
1.1 Thermodynamics-Thetheoryofheatandwork . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization ofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Classical systems far from equilibrium 5
2.1 Entropyproductioninthelinearregime . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Microscopicdynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Langevinequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Fokker-Planckequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Generalizations ofthesecondlawarbitrarily farfromequilibrium . . . . . . . . . 12
2.3.1 Jarzynski’sworkrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Crooks’fluctuationtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Generalizationtoarbitrary initialstates . . . . . . . . . . . . . . . . . . . . . 17
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Dynamical properties of nonequilibrium quantum systems 21
3.1 Geometricapproachtoisolatedquantumsystems . . . . . . . . . . . . . . . . . . . 21
3.1.1 Wootters’statisticaldistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Generalizationtomixedstates: TheBureslength . . . . . . . . . . . . . . . . 26
3.2 Measuringthedistancetoequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Green-Kuboformalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 FidelityforGaussianstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Theparameterizedharmonicoscillatorinthelinearregime . . . . . . . . . 32
3.3 Minimal quantumevolutiontime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Mandelstam-Tammtypebound . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Margolus-Levitintypebound . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.3 Quantumspeedlimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Unitary quantum processes in thermally isolated systems 41
4.1 Thermodynamics: Workandheatinquantummechanics . . . . . . . . . . . . . . . 41
4.1.1 Workisnotanobservable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Fluctuationtheoremforheatexchange . . . . . . . . . . . . . . . . . . . . . 43
4.2 GeneralizedClausius inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Irreversibleentropyproduction . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Lowerboundfortheirreversibleentropy . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Upperestimationoftherelative entropy . . . . . . . . . . . . . . . . . . . . 51
VContents
4.3 Maximal rateofentropyproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Illustrativeexample-theparameterizedoscillator . . . . . . . . . . . . . . . . . . . 53
4.4.1 Lowerboundonentropyproduction . . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Maximal rateofentropyproduction . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Experimentalrealization incoldiontraps . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5.1 Experimentalset-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5.2 VerifyingthequantumJarzynskiequality . . . . . . . . . . . . . . . . . . . . 57
4.5.3 Anharmoniccorrectionsandfluctuatingelectricfields . . . . . . . . . . . . 59
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Thermodynamics of open quantum systems 65
5.1 QuantumLangevinequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.1 Caldeira-Leggettmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.2 Freeparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.3 Harmonicpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Thermodynamicsintheweakcouplinglimit . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Quantumentropyproduction . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Particularprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.3 Jarzynskitypefluctuationtheorem . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Statisticalphysicsofopenquantumsystems . . . . . . . . . . . . . . . . . . . . . . 78
5.3.1 Markovianapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.2 QuantumBrownianmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.3 Hu-Paz-Zhangmasterequation. . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Strong coupling limit - a semiclassical approach 85
6.1 QuantumSmoluchowskidynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.1 Reduceddynamicsinpathintegralformulation . . . . . . . . . . . . . . . . 85
6.1.2 Quantumstrongfrictionregime . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.3 QuantumSmoluchowskiequation . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.4 Quantumenhancedescaperates . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Quantumfluctuationtheoremsinthestrongdampinglimit . . . . . . . . . . . . . 91
6.3 ExperimentalverificationinJosephsonjunctions . . . . . . . . . . . . . . . . . . . . 94
6.3.1 RCSJ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3.2 I-Vcharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3.3 Possiblemeasurementprocedure . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Epilogue 103
A Quantum information theory 105
A.1 Relativeentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.1.1 Inequalitiesininformationtheory . . . . . . . . . . . . . . . . . . . . . . . . 105
A.1.2 Quantumrelative entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
VIContents
A.2 Fisherinformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2.1 RelationtoKullback-Leiblerdivergence . . . . . . . . . . . . . . . . . . . . . 107
A.2.2 Crame´r-Rao bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.3 Buresmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.3.1 Explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.3.2 QuantumFisherinformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B Solution of the parametric harmonic oscillator 111
B.1 Theparametricharmonicoscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.2 Methodofgeneratingfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.3 Measureofadiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
B.4 Exacttransitionprobabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C Stochastic path integrals 117
C.1 Definitionandbasicproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C.2 Onsager-Machlup functionalforspacedependentdiffusion . . . . . . . . . . . . . 119
Bibliography 123
List of figures 133
Acknowledgments 135
Curriculum vitae 137
VIIContents
VIII1 Prologue
1.1 Thermodynamics - The theory of heat and work
Thermodynamicsisthephenomenologicaltheorydescribingtheenergyconversionofheatand
work. The Scottishphysicist Lord Kelvin was the first to formulate a concise definitionof ther-
modynamicswhenhestatedin1854 [Tho82]:
Thermodynamics is the subject of the relation of heat to forces acting between con-
tiguouspartsofbodies,andtherelationofheattoelectricalagency.
Atitsoriginsthetheoryofthermodynamicswasdevelopedtounderstandandimproveheaten-
gines. Hence, special interest lies on the dynamical properties of energy conversion processes.
However, the original theorywas only able to predict the behavior of physical systemsby con-
sideringtheirmacroscopic statefunctions(suchasentropy,temperature,pressureorvolume).
Equilibrium and nonequilibrium processes
Asystemisconsideredtobeinastationarystate,ifallrelaxationprocesseshavecometoanend.
Moreover,thermalequilibriumischaracterizedasastationarystateinwhichallthermodynamic
properties of the system of interest are time-independent. If the physical system changes very
slowly, and, hence, the system is in an equilibrium state at all times, the process is considered
to be quasistatic. All real physical processes, however, contain nonequilibrium contributions.
A thermodynamic system is out of equilibrium, if the system is time-dependent or fluxes are
present. Due to the importance of mass or energy fluxes at the system’s boundaries, it is not
possible to apply the thermodynamiclimit. Especially for small systemsizes it becomes neces-
sarytodescribethedynamicalpropertiesincludingthermalfluctuations.
Quantum thermodynamics
The modern trend of miniaturization leads to the development of smaller and smaller devices,
such as nanoengines and molecular motors [CZ03, Cer09, HM09]. On these very short length
scales, thermal as well as quantum fluctuations become important, and usual thermodynamic
quantities,suchasworkandheat,acquireastochasticnature. Moreover,inthequantumregime
a completely new theory had to be invented, since classical notions of work and heat are no
longer valid [LH07]. The present thesis contributes to this prevailing field by the research for
analytical expressions of the nonequilibrium entropy production in open and closed quantum
systems. Complementarytootherpublications[TH09a,TH09b,CH09]ourapproachdealswith
the reduced dynamics of the system only. We are motivated by an experimental point of view
11 Prologue
inthesensethatthesystemunderconsiderationcanalwaysbeseparatedintoanaccessiblesub-
systemand theenvironment. Since, generally,the environmentcan be arbitrarily large, e.g. the
universe, it is usually notexperimentally controllable. Hence,the presentthesisis interestedin
thethermodynamicpropertiesofthereducedsystemonly. Tothisend,wewillhavetodealwith
methodsandquantitiesofstatisticalphysics,conventionalthermodynamics,quantuminforma-
tiontheoryandthetheoryofopenquantumsystems.
1.2 Organization of the thesis
The scope of the present thesis is to draw a bow over a wide range of coupling strengths of a
quantum system to its thermal surroundings. Therefore, we start with an introductory chap-
ter 2, in which we summarize the main developments in recent statistical physics for classical
systemsarbitrarily far from equilibrium. In particular, we will briefly summarize the notion of
fluctuationtheorems[CM93,Jar97,Cro98,HS01,Sei05]andacoupleofexemplaryderivations.
Then, we will turn to quantum systems and discuss in chapter 3 the dynamical properties
of isolated quantum systems. Chapter 3 presents a detailed analysis of quantum peculiarities,
whichwillhaveanotableimpactonthethermodynamicspropertiesdiscussedinthesucceeding
chapter. In particular, chapter3 focuseson the descriptionand implications of the dynamics of
quantumsystems. Tothisend,wewillseethatageometricapproach[Rup95]andthedefinition
of statistical distances [Woo81] capture the dynamical properties. Moreover, we will propose
an appropriate measure to quantify how far from equilibrium an arbitrary process operates in
terms of the time averaged Bures length[Bur68, Bur69]. The definition of the Bures length will
also serve as our starting point for the derivation of the generalized Heisenberg uncertainty
relation[MT45,ML98,LT09]. Wewillderivetheminimal timethatanisolatedquantumsystem
needstoevolvefromonestatetoanother.
Inchapter4 weturnto a thermodynamicdiscussionofisolated quantumsystems. Quantum
mechanical work and heat, however, are not given by the eigenvalues of Hermitian operators
[LH07]. Hence, we will have to deal with the quantum probability distributions of work and
heat. Itwillturnoutthattheirreversibleentropyproductioncanbewrittenasarelativeentropy
[Kul78, Ume62] between the current, nonequilibrium state and the corresponding equilibrium
one. ThisidentificationwillleadtoageneralizedClausiusinequality,wherewewillfindasharp
lowerboundfortheirreversibleentropyproductionintermsoftheBureslength. Further,com-
biningthequantumspeedlimitfromchapter3withtheanalyticexpressionfortheentropypro-
ductionwe will derive the maximal rate of entropyproductionin an isolated quantum system.
The latter is a mere quantum result and a generalized version of the Bremermann-Bekenstein
bound[Bre67,Bek74,Bek81,BS90]. Thisboundisanupperlimit ontheentropy,orinformation,
that can be contained within a given finite regionof space which has a finite amount of energy.
Ininformationtheorythisimpliesthatthereisamaximumrateofcommunicationalongagiven
channel. The chapter will be completed by illustrating the rigorous results with the help of the
parameterizedharmonicoscillator[Def08]. Thelattermodelistheparadigmforanexperimental
verificationofourgeneralizedexpressionsofthesecondlawofthermodynamicsinmodulated,
coldiontraps[SSK08].
The next chapter introduces the quantum heat bath to the earlier considerations. We will
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