QuantumThermodynamics OldřichSedlák BACHELORTHESIS

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BACHELOR THESIS

Oldřich Sedlák

Quantum Thermodynamics

Department of Macromolecular Physics

Supervisor of the bachelor thesis: RNDr. Viktor Holubec, Ph.D.

Study programme: Physics

Study branch: General Physics

Prague 2018

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I declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources.

I understand that my work relates to the rights and obligations under the Act No.121/2000 Sb., the Copyright Act, as amended, in particular, the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act.

In ... date ...^Prague ^19.7.2018 signature of the author

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Title: Quantum Thermodynamics Author: Oldřich Sedlák

Department: Department of Macromolecular Physics

Supervisor: RNDr. Viktor Holubec, Ph.D., Department of Macromolecular Physics Abstract:

Quantum coherence is being viewed as a possible resource that could improve the per- formance of quantum technologies. This thesis analyzes a quantum heat engine model inspired by Dorfman et al. (PNAS vol. 110 no. 8) while using a standard Markovian quantum optical master equation in the Lindblad form. Steady-state coherence arises from the degeneracy of the two upper energy levels and its effects become significant for near-perfect alignment of the associated transition dipole moments. For the maximum alignment, the steady-state current becomes highly dependent on the relative phase and exhibits quantum interference. The performed numerical calculations show some promise of possible enhancement of the current above the classical limit.

Keywords: quantum coherence, quantum thermodynamics, open quantum system

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I would like to express my appreciation to my supervisor RNDr. Viktor Holubec, Ph.D. for going above and beyond in keeping me focused but also for allowing me to delve into parts I found most interesting. I am grateful for his permanent support and dedication in helping me complete this thesis.

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Introduction

Quantum coherence is being viewed as a possible resource that could improve the per- formance of quantum technologies and attempts are being made to precisely quantify its usefulness [1]. A discovery that sparked research interest in quantum effects in biology was the observation of long-lived coherence in natural light-harvesting systems [2]. Without a doubt, the study of the various quantum effects [3, 4, 5, 6] present in photosynthetic systems could be highly beneficial to the development of new enhanced artificial solar cells and nanotechnologies. For example, in the first phase of photosynthesis antennae complexes harvest solar energy which is used for charge separation carried out by the photosynthetic reaction centre. The quantum efficiency of this process can under certain conditions exceed 95 % [7]. In other words, each absorbed photon almost certainly reaches the reaction centre and drives the separation of charge.

Inspired by the observed coherence phenomena [2], Dorfman et al. have proposed a simple quantum heat engine (QHE) model [8] based on a pair of tightly coupled chloro- phylls at the heart of the reaction centre. In their model, coherence between two populations representing this special pair can boost the classical photocurrent by 27 %. While theoretical results look promising, they (Dorfman et al.) also comment, however, that the quantum coherence effects observed in photosynthetic experiments are studied with coherent laser radiation that might affect the natural conditions of excitation by solar incoherent light.

This thesis further explores the effects of coherence on a very similar model. The first two chapters serve as an introduction of the used Markovian quantum optical master equation, including its application to a simple system. The third chapter introduces the model of the QHE, the equations of motion and the physical quantities used for the analysis of the engine. Finally, the last chapter analyzes the effects of coherence, focusing mainly on the steady-state current in the QHE.

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1. T h e o r e ti c al b a c k g r o u n d

T his c h a pter c o nsists of t w o secti o ns. First, we brie ﬂ y re vie w t he c o nce pt of a n o p e n q u a nt u m s yste m a n d i n t he sec o n d secti o n, we t a ke a l o o k at t he necess ar y ass u m pti o ns re q uire d f or t he vali dit y of o ur M ar k o vi a n descri pti o n of t he d y n a mics.

1. 1 I nt r o d u c ti o n t o o p e n q u a nt u m s y s t e m s

T he m ai n m at he m atic al disti ncti o n b et wee n a n i de alize d cl ose d s yste m a n d a n o p e n q u a n- t u m s yste m is t h at t he H a milt o ni a n d y n a mics of cl ose d s yste ms is re versi ble a n d c a n b e re prese nte d b y u nit ar y tr a nsf or m ati o ns. T his is ge ner all y n ot t he c ase f or o p e n q u a nt u m s yste ms w here we c o nsi der t he i nter acti o ns wit h t he e n vir o n me nt. T he res ulti n g irre- versi ble d y n a mics is t he n ass o ci ate d wit h t he pr o d ucti o n of e ntr o p y. A n o p e n q u a nt u m s yste m is, t heref ore, a s yste m S i nter acti n g wit h a n ot her e xter n al s yste m B w hic h we ge ner all y c all t he e n vir o n m e nt . If a n e n vir o n me nt h as a n i n ﬁ nite n u m b er of de grees of free d o m it is c alle d a res er v oir a n d if a reser v oir is i n a t her m al e q uili bri u m st ate we c all it a h e at b at h or si m pl y a b at h [ 9].

T he i ntr o d ucti o n of t he e n vir o n me nt B t o t he s yste m S e x p a n ds t he Hil b ert s p ace, H = H S ⊗ H B .

C o nse q ue ntl y, t he o p er at or Ô pre vi o usl y acti n g i n t he Hil b ert s p ace H S nee ds t o b e e x p a n de d o ver t he c o m bi ne d Hil b ert s p ace H wit h t he i de ntit y o p er at or ÎB of t he e n vi- r o n me nt t o Ô ⊗ ÎB . T he e x p ect ati o n val ue is t he n o bt ai ne d as:

O (t) = tr{ ( Ô ⊗ ÎB ) ˆρ(t)} = trS{ Ô ˆρS(t)} ,

w here ˆρ is t he de nsit y m atri x of t he c o m bi ne d s yste m S + B a n d ˆρS is t he re d u ce d d e nsit y m atri x descri bi n g t he o p e n s yste m,

ˆρS (t) = trB { ˆρ(t)} .

T he d y n a mics of t he o p e n s yste m f oll o ws fr o m t he d y n a mics of t he c o m bi ne d s yste m w hic h is g o ver ne d b y its H a milt o ni a n ˆH a n d t he Sc hr ö di n ger e q u ati o n. T his H a milt o ni a n c o nsists of t he self- H a milt o ni a ns of t he o p e n s yste m a n d t he e n vir o n me nt, ˆHS a n d ˆHB , a n d t he H a milt o ni a n ˆHI descri bi n g t he i nter acti o n b et wee n t he t w o s yste ms [ 9],

ˆH = ˆHS ⊗ ˆIB + ˆIS ⊗ ˆHB + ˆHI.

T he e v ol uti o n fr o m ti me t0 t o ti me t c a n b e e x presse d wit h t he e v ol uti o n o p er at or ˆU (t, t0) [ 1 0],

ˆρ(t) = ˆU (t, t0) ˆρ(t0) ˆU ^†(t, t0).

I n t he c ase of a ti me-i n de p e n de nt H a milt o ni a n ˆH , t he o p er at or ˆU (t, t0) h as t he well- k n o w n f or m:

ˆU (t, t0) = e x p − i ˆH (t − t0) , w here is t he re d uce d Pl a nc k c o nst a nt.

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1. 2 M a r k o vi a n e v ol u ti o n a n d Li n d bl a d f o r m

I n ge ner al, t he e v ol uti o n of t he de nsit y m atri x ˆρS c a n de p e n d o n its hist or y (e arlier st ates of t he o p e n s yste m) b ec a use c h a n ges i n t he o p e n s yste m c a n b e re me m b ere d b y t he e n vir o n me nt w hic h c a n, i n t ur n, a ﬀect t he o p e n s yste m at a l ater ti me.

I n s o me sit u ati o ns, h o we ver, t hese me m or y e ﬀects c a n b e ne glecte d. I n t h at c ase, it is p ossi ble t o f or m a n a p pr o xi m ate descri pti o n w here t he c o n diti o n al e v ol uti o n t o f ut ure st ates, gi ve n t he prese nt st ate of t he o p e n s yste m ˆρS(t), d o es n ot de p e n d o n a n y p ast st ates b ut o nl y o n t he prese nt st ate. We c all t his i n t he c o nte xt of st o c h astic pr o cesses t he M ar k o vi a n pr o p ert y.

Let us ass u me t h at t he me m or y i n t he e n vir o n me nt (i nf or m ati o n i n t he f or m of c or- rel ati o ns) c a use d b y t he c h a n ges i n t he o p e n s yste m t h at we w a nt t o f oll o w is l ost after ti me ∆ t. T his c a n h a p p e n t hr o u g h str o n g e ﬀects of dec o here nce. T he ti me ∆ t t he n sets t he sc ale of t his s o-c alle d ti me c o arse- gr ai ni n g, i.e. t he s h ortest p eri o d of ti me o ver w hic h t he b e h a vi or of t he o p e n s yste m c a n b e deter mi ne d [ 1 1, 1 2]. It f oll o ws t h at we m ust als o ass u me t h at t he o p e n s yste m c h a n ges a p preci a bl y o ver a t y pic al ti mesc ale τS m uc h l ar ger t h a n t his ti me ∆ t:

τS ∆ t.

F urt her m ore, w he n deri vi n g t his M ar k o vi a n m aster e q u ati o n f or ˆρS, we w o ul d li ke t o preser ve t he pr o p erties of t he ge ner al Kr a us re prese nt ati o n [ 1 3] of t he e v ol uti o n. T hese pr o p erties i ncl u de li ne arit y, her micit y preser vati o n, tr ace preser vati o n a n d c o m plete p ositi vit y. For e x a m ple, t his c a n b e d o ne t hr o u g h t he pr o cess of ti me c o arse- gr ai ni n g descri b e d a b o ve al o n g wit h t he ass u m pti o n of we a k i nter acti o n (t o sec o n d or der) b et wee n t he o p e n s yste m a n d t he e n vir o n me nt [ 1 1, 1 2]. T he res ulti n g e q u ati o n is t he n i n t he Li n d bl a d f or m w hic h is t he m ost ge ner al M ar k o vi a n m aster e q u ati o n wit h t hese pr o p erties [ 9].

If we restrict o ursel ves t o a ti me-i n de p e n de nt H a milt o ni a n ˆHS a n d t he i nter acti o n H a milt o ni a n i n t he f or m ˆHI = ˆA ⊗ ˆB , t he Li n d bl a d f or m c a n b e e x press e d wit h t he use of ei ge n o p er at ors of t he o p e n s yste m,

ˆAω =

ω = −

| | ˆA | | ,

w here t he s u m r u ns o ver all e ner g y ei ge n val ues a n d of ˆHS s uc h t h at t heir di ﬀere nce c orres p o n ds t o t he tr a nsiti o n fre q ue nc y ω = ( − )/ . T he m aster e q u ati o n i n t he Li n d bl a d f or m e x presse d f or t he re d uce d de nsit y m atri x i n t he i nter acti o n pict ure ˆρSI re a ds:

d

dt ˆρSI(t) = − i[ ˆHU ( ∆t), ˆρSI(t)] +

ω, ω γω ω ( ∆t) ˆAω ˆρSI(t) ˆA^†_ω − 1

2 ˆA^†_ω ˆAω, ˆρSI(t) , ( 1. 1) w here it is re q uire d t h at ˆHU is her miti a n a n d t he r ates γω ω ( ∆t) f or m a p ositi ve m atri x [ 1 2].

T he ﬁrst ter m i n ( 1. 1) acc o u nts f or t he u nit ar y p art of t he e v ol uti o n of t he o p e n s yste m w hile t he rest ser ves t o descri b e t he n o n- u nit ar y d y n a mics. N ote t h at t he n o n- u nit ar y p art is s u m me d o ver t w o sets of p ossi ble tr a nsiti o n fre q ue ncies ( ω a n d ω ) a n d t h at t here are ter ms de p e n di n g o n t he c o arse- gr ai ni n g p ar a meter ∆ t.

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2. Q u a nt u m o p ti c al m a s t e r e q u a ti o n

Micr osc o pic a n d p he n o me n ol o gic al deri vati o ns of M ar k o vi a n m aster e q u ati o ns ofte n rel y o n f urt her ass u m pti o ns a n d t he res ulti n g m aster e q u ati o n m a y n ot c o me o ut i n t he Li n d- bl a d f or m. T his c a n le a d t o p at h ol o gic al b e h a vi o ur s uc h as vi ol ati o n of p ositi vit y [ 1 4]. For o ur m o del, we use a st a n d ar d q u a nt u m o ptic al m aster e q u ati o n ( Q O M E) w h ose deri vati o n c a n b e f o u n d i n m a n y te xt b o o ks, s uc h as T h e T h e or y of O pe n Q u a nt u m S yst e ms b y Bre uer a n d Petr ucci o ne [ 9]. I n o ur deri vati o n, t he Li n d bl a d f or m is e ns ure d wit h t he use of t he r ot ati n g w a ve a p pr o xi m ati o n ( R W A).

2. 1 Mi c r o s c o pi c a s s u m p ti o n s a n d d e ri v a ti o n

O ur q u a nt u m o ptic al m aster e q u ati o n ass u mes t he f oll o wi n g H a milt o ni a n:

ˆH = ^N

i ω i|i i| +

k λ = 1 ,2

ω kâ^†_λ(k ) âλ(k ) − e ˆD · Ê. ( 2. 1) T he o p e n s yste m h as a discrete e ner g y s p ectr u m wit h N well-s p ace d e ner g y le vels,

ˆHS = ^N

i ω i|i i| , ( 2. 2)

w here e ner g y ei ge nst ates |i h a ve ei ge n val ues E i pr o p orti o n al t o t he a n g ul ar fre- q ue ncies ω i (E i= ωi).

T he he at b at h is re prese nte d b y a q u a ntize d electr o m a g netic ﬁel d t h at is i n a t her m o d y n a mic e q uili bri u m at te m p er at ure T , i.e. it is a n i de al bl ac k b o d y r a di a- ti o n,

ˆHB =

k λ = 1 ,2

ω kˆa^†_λ(k ) ˆaλ(k ). ( 2. 3)

T he o p er at ors ˆaλ(k ) a n d ˆa^†_λ(k ) are t he cre ati o n a n d a n ni hil ati o n o p er at ors f or a p h ot o n i n t he p ol ariz ati o n st ate λ , wit h t he w a ve vect or k a n d t he ass o ci ate d a n g ul ar fre q ue nc y ω k = c|k |. T he ass u me d t her m o d y n a mic e q uili bri u m i m p oses t he B ose- Ei nstei n distri b uti o n o n t he a ver a ge p h ot o n o cc u p a nc y n u m b er:

ˆa^†_λ(k ) ˆaλ (k ) = δ_{k k} δλ λ N (ωk), N(ω ) = 1

e x p(_k_B^ω_T ) − 1, ( 2. 4) w here N (ω ) is t he a ver a ge p h ot o n o cc u p a nc y n u m b er at a n g ul ar fre q ue nc y ω a n d te m p er at ure T (kB is t he B oltz m a n n c o nst a nt).

T he i nter acti o n b et wee n t he b at h a n d t he o p e n s yste m is si m pli ﬁe d wit h t he di p ole a p pr o xi m ati o n [ 1 5],

ˆHI = − e ˆD · ˆE, ( 2. 5)

w here ˆD is t he tr a nsiti o n di p ole o p er at or w h ose c o m ple x vect or ele me nts i n t he e ner g y ei ge n b asis we de n ote as d ,

d = | ˆD | = e | ˆr | . ( 2. 6)

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T he o p er at or Ê is t he q u a ntize d electric fiel d o p er at or i n t he Sc hr ö di n ger pict ure, Ê = i

k λ = 1 ,2

2 π ω k

V âλ(k ) − â^†_λ(k ) eλ, ( 2. 7) w here V is t he fi nite v ol u me of t he res o n at or (c a vit y) c o nt ai ni n g t he Bl ac k b o d y r a di ati o n.

T he deri vati o n of t he m aster e q u ati o n fr o m t he v o n Ne u m a n n e q u ati o n le a ds u n der t he ass u m pti o ns of we a k c o u pli n g a n d t he ass u m pti o ns of t he c o arse- gr ai ni n g pr o ce d ure brie ﬂ y descri b e d i n t he pre vi o us c h a pter (sec. 1. 2) t o t he f oll o wi n g ( Re d ﬁel d) e q u ati o n (see e q.

3. 1 8 8 i n [ 9]):

d

dt ˆρSI(t) =

ω, ω j e x p [ i(ω − ω )t] γ (ω ) Âj(ω ) ˆρSI(t) Â^†_j(ω ) − Â^†_j(ω ) Âj(ω ) ˆρSI(t) + h.c.

( 2. 8) We c a n see t h at t his e q u ati o n is n ot yet i n t he Li n d bl a d f or m ( 1. 1) d ue t o t he f act ors e x p [ i(ω − ω )t]. T his pr o ble m c a n b e s ol ve d wit h t he r ot ati n g w a ve a p pr o xi m ati o n w here we ass u me t h at t he e ﬀects of t he f ast r ot ati n g ter ms ( ω = ω ) e ﬀecti vel y a ver a ge o ut. To b e c o nsiste nt wit h t he pr o cess of c o arse- gr ai ni n g, we nee d t o m a ke s ure t h at t he ti mesc ales τR W A = ( ω − ω )^{− 1} c orres p o n di n g t o t he all o we d tr a nsiti o n fre q ue ncies s atisf y:

τS ∆ t m a x( τR W A ). ( 2. 9)

Ne glecti o n of t hese ter ms t h us re q uires t h at all of t he e ner g y le vels are well s p ace d. We c a n n o w t hi n k of t he p ar a meter ∆ t als o as t he stre n gt h of a ﬁlter w hic h deter mi nes t he ti mesc ales o ver w hic h t he b e h a vi o ur of t he o p e n s yste m c a n b e a ﬀecte d.

Wit h t he r ot ati n g w a ve a p pr o xi m ati o n, t he Q O M E ( 2. 8) c a n b e c ast i nt o t he Li n d bl a d f or m:

T h e q u a nt u m o p ti c al m a s t e r e q u a ti o n d

dt ˆρSI(t) =

ω j γ (ω ) Âj(ω ) ˆρSI(t) Â^†_j(ω ) − Â^†_j(ω ) Âj(ω ) ˆρSI(t) + h.c. ( 2. 1 0) Â (ω ) =

ω = − | | ˆD | | ( 2. 1 1)

γ (ω ) = ω ³

6 π 0 c³( 1 + N (ω ) + i S(ω )) ( 2. 1 2)

S (ω ) = 1 ω ³π P ^∞

0 d ωkω _k³ 1 + N (ωk)

ω − ωk + N (ω k)

ω − ω k ( 2. 1 3)

T he Q O M E is e x presse d f or t he de nsit y m atri x ˆρSI i n t he i nter acti o n pict ure a n d t he first s u m r u ns o ver all ( p ositi ve a n d ne g ati ve) a n g ul ar fre q ue ncies ω ij c orres p o n di n g t o t he e ner g y di ffere nces of t he all o we d tr a nsiti o ns E j− E i= ω ij. T he o p er at ors Âj(ω ) dec o m- p ose t he tr a nsiti o n di p ole m o me nt o p er at or ˆDj w here j desi g n ates t he s p ati al c o m p o ne nts.

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Fi n all y, t he i nte gr al S (ω ) i n t he r ate c o e ﬃcie nts γ (ω ) acc o u nts f or t he L a m b a n d St ar k s hifts a n d c o ntri b utes t o t he u nit ar y e v ol uti o n [ 9]. T he di ver ge nce of t hes e ter ms S (ω ) nee ds t o b e a me n de d b y re n or m aliz ati o n. H o we ver, t he sizes of t hese ter ms are ge ner all y at or b el o w t he or der of t he ne glecte d ter ms i n t he we a k c o u pli n g a p pr o xi m ati o n a n d t heref ore we h a ve t o ne glect t he m.

2. 2 A p pli c a ti o n t o V- t y p e s y s t e m i n si n gl e b a t h

Let us c alc ul ate t he e q u ati o ns of m oti o n f or a t hree-le vel de ge ner ate o p e n s yste m c o u ple d t o a si n gle he at b at h de picte d i n t he ﬁ g ure 2. 1. D ue t o t he li ne arit y of t he Q O M E, t his c alc ul ati o n will act u all y ser ve as a b uil di n g bl o c k f or f or mi n g t he e q u ati o ns of m oti o n f or t he Q H E (see ﬁ g. 3. 1).

Fi g ure 2. 1: T he o p e n s yste m c o nsists of t hree st ates l a b ele d { 0 , 1 , 2 } w hic h are re prese nte d b y t he bl ac k s oli d h oriz o nt al li nes wit h t he u p p er st ates h a vi n g hi g her e ner g y. T he gree n d as he d li nes re prese nt t he all o we d tr a nsiti o ns b et wee n t hese st ates. T hese tr a nsiti o ns are c o u ple d t o a si n gle p h ot o n b at h at te m p er at ure T w hic h is ill ustr ate d wit h gree n arr o ws p oi nti n g t o t hese tr a nsiti o ns.

At ﬁst, we rec all t he Q O M E prese nte d i n t he precee di n g secti o n, d

dt ˆρSI(t) =

ω j γ (ω ) Âj(ω ) ˆρSI(t) Â^†_j(ω ) − Â^†_j(ω ) Âj(ω ) ˆρSI(t) + h.c., ( 2. 1 4) a n d t a ke a l o o k at t he o p er at or Â (ω ).

ˆAj(ω ) =

ω = − | | ˆD j| | =

ω = − | d _j | . ( 2. 1 5)

T he ri g ht- h a n d si de of e q. ( 2. 1 4) is a s u m of ter ms w here e ac h ter m c o ntri b utes t o o ne s p ati al c o m p o ne nt. For bre vit y, we will o mit t he i n de x j , w or k i n o ne di me nsi o n a n d a d d t he t w o re m ai ni n g s p ati al c o m p o ne nts at t he e n d. I n t he c ase of t he ele me nt d _j, we will

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als o o mit t he b ol d f o nt i n dic ati n g a vect or a n d write d : ˆA (ω ) =

ω = − | | ˆD | | =

ω = − | d | . ( 2. 1 6)

F urt her m ore, we n otice t h at Â (− ω ) = Â^†(ω ). T his f or m ul a all o ws us t o re alize t h at w he n we p erf or m t he s u m o ver t he tr a nsiti o n fre q ue ncies { ω, − ω } , w here ω = E 1 2− E 0, we e nc o u nter o nl y t w o di ffere nt o p er at ors Â a n d its her miti a n c o nj u g ate Â^†:

Â = Â (ω ) = Â^†(− ω ) = |0 0 | ˆD |1 1 | + |0 0 | ˆD |2 2 | = d0 1 |0 1 | + d0 2 |0 2 | , Â^† = Â (− ω ) = Â^†(ω ) = |1 1 | ˆD |0 0 | + |2 2 | ˆD |0 0 | = d^∗_{0 1} |1 0 | + d^∗_{0 2} |2 0 | . I n t his n ot ati o n, t he Q O M E i n o ne s p ati al di me nsi o n re a ds:

d

dt ˆρSI = γ (ω )( 2 Â ˆρSI Â^†− Â^† Â ˆρSI− ˆρSI Â^† Â )

+ γ (− ω )( 2 Â^†ˆρSI Â − Â Â^†ˆρSI− ˆρSI Â Â^†). ( 2. 1 7) As t he ne xt ste p, we c alc ul ate t he f oll o wi n g ter ms c o nt ai ne d i n ( 2. 1 7) w hile de n oti n g t he ele me nts i| ˆρSI |j as rij:

ˆA ˆA^† =( |d0 1|² + |d0 2|²) |0 0 | ,

Â^† Â = |d0 1|² |1 1 | + |d0 2|² |2 2 | + d^∗_{0 1}d0 2 |1 2 | + d0 1d^∗_{0 2} |2 1 | , Â ˆρSI Â^† = |d0 1|² r1 1 + |d0 2|² r2 2 + d0 1d^∗_{0 2}r1 2 + d^∗_{0 1}d0 2r2 1 |0 0 | ,

ˆA^†ˆρSI ˆA = r0 0 |d0 1|² |1 1 | + |d0 2|² |2 2 | + d^∗_{0 1}d0 2 |1 2 | + d0 1d^∗_{0 2} |2 1 | .

( 2. 1 8)

B y c aref ul o bser vati o n of t hese ter ms, we n otice t h at t he ele me nts { r0 1, r0 2} a n d c o nse- q ue ntl y t heir c o m ple x c o nj u g ates { r1 0, r2 0} e v ol ve i n de p e n de ntl y of t he ot her ele me nts i ncl u di n g t he p o p ul ati o ns ( di a g o n al ele me nts),

d

dtr0 1 = f1(r0 1, r0 2), d

dtr0 2 = f2(r0 1, r0 2). ( 2. 1 9) T his me a ns t h at we c a n i g n ore t heir e v ol uti o n a n d set t he m t o zer o. F urt her m ore, it m a kes o ur w or k of tr a nsf or mi n g b ac k i nt o t he Sc hr ö di n ger pict ure e as y b ec a use t he ele me nts { r0 0,r1 1,r2 2,r1 2,r2 1} w hic h we are i ntereste d i n are i de ntic al i n b ot h pict ures. To see t his, let us de n ote t he ele me nts i n t he Sc hr ö di n ger pict ure as ρij, i.e. ρij= i| ˆρS |j , a n d de n ote t he a n g ul ar fre q ue nc y pr o p orti o n al t o t he e ner g y E i i n t he f or m ul a E = ω as ωi. T he f or m ul as f or tr a nsf or mi n g b et wee n t he Sc hr ö di n ger a n d t he i nter acti o n pic ut ure are t he n:

ρij = i| e^{− i ˆH}^S^{t /} ˆρIS e^{i ˆH}^S^{t /} |j = e^i(ω^j^{− ω}ⁱ^)trij, d

dtρij = i(ωj − ω i)ρij+ e^i(ω^j^{− ω}ⁱ^)t d dtrij.

Si nce t he u p p er e ner g y le vels are de ge ner ate, E 1 = E 2 = E 1 2, t he ele me nts we are i ntereste d i n, i.e. { r0 0,r1 1,r2 2,r1 2,r2 1} , are i n dee d i n vari a nt u n der t his tr a nsf or m ati o n.

We n o w pr o cee d t o i ncl u de t he ot her t w o s p ati al di me nsi o ns c o nt ai ne d i n t he Q O M E, e q. ( 2. 1 4), i.e. s u m o ver t he o m mite d i n de x j . I n or der t o gr o u p t his l ar ge n u m b er of ter ms, we will use t he st a n d ar d a bs ol ute val ue o p er ati o n:

|a |² =

j aja^∗_j,

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a n d t he or di n ar y d ot pr o d uct s uc h t h at f or t w o c o m ple x vect ors a a n d b t he d ot pr o d uct is s y m metric b ut ge ner all y c o m ple x:

a · b = b · a =

j ajb j.

S u m m ati o n o ver t he s p ati al c o m p o ne nts (x, y, z ) usi n g t he e q u ati o ns ( 2. 1 4), ( 2. 1 7) a n d ( 2. 1 8) yiel ds:

d

dt ˆρSI = 2 γ (ω ) |0 0 | |d 0 1|² ρ1 1+ |d0 2|² ρ2 2+ d 0 1·d ^∗_{0 2}ρ1 2+ d ^∗_{0 1}·d 0 2ρ2 1

+ 2 γ (− ω )ρ0 0 |d 0 1|² |1 1 |+ |d 0 2|² |2 2 |+ d ^∗_{0 1}·d 0 2 |1 2 |+ d 0 1·d ^∗_{0 2} |2 1 |

− γ (ω ) |d0 1|² |1 1 |+ |d 0 2|² |2 2 |+ d ^∗_{0 1}·d 0 2 |1 2 |+ d 0 1·d ^∗_{0 2} |2 1 | ˆρSI

− γ (ω ) ˆρSI |d 0 1|² |1 1 |+ |d 0 2|² |2 2 |+ d ^∗_{0 1}·d 0 2 |1 2 |+ d 0 1·d ^∗_{0 2} |2 1 |

− γ (− ω ) |d 0 1|²+ |d 0 2|² |0 0 | ˆρSI− γ (− ω ) ˆρSI |0 0 | |d 0 1|²+ |d 0 2|² .

Si nce t he u ni nteresti n g d y n a mics of t he ele me nts { ρ0 1, ρ0 2, ρ1 0, ρ2 0} c a n b e o mitte d, we re arr a n ge t he Q O M E i nt o a si m ple m aster e q u ati o n,

d

dtρ = ˆM ρ,

w here ˆM is a 5 × 5 tr a nsiti o n m atri x acti n g o n a vect or ρ w hic h c o nt ai ns t he ﬁ ve rel- e va nt ele me nts { ρ0 0, ρ1 1, ρ2 2, ρ1 2, ρ2 1} . F urt her m ore, we si m plif y t he tr a nsiti o n m atri x ele me nts b y m a ki n g use of t he pr o p ert y of t he o cc u p ati o n n u m b er ( de ﬁ ne d i n e q. ( 2. 4)):

N (− ω ) = − 1 − N (ω ), a n d b y i ntr o d uci n g t he f oll o wi n g c o nst a nts:

α = ( 3 π 0 c³)^{− 1}, γ1 2 = α ω ³d 0 1 · d ^∗_{0 2},

γ1 = α ω ³ |d 0 1|² , n = N (ω ),

γ2 = α ω ³ |d 0 2|² . ( 2. 2 0)

Wit h t hese c o nst a nts, t he m aster e q u ati o n re a ds:







˙ρ0 0

˙ρ1 1

˙ρ2 2

˙ρ1 2

˙ρ2 1







=







− (γ1+ γ2)n γ1( 1 + n ) γ2( 1 + n ) γ1 2( 1 + n ) γ^∗_{1 2}( 1 + n ) γ1n − γ1( 1 + n ) 0 − ^γ^{1 2}₂ ( 1 + n ) − ^γ₂^{1 2}^∗ ( 1 + n ) γ2n 0 − γ2( 1 + n ) − ^γ^{1 2}₂ ( 1 + n ) − ^γ₂^{1 2}^∗ ( 1 + n ) γ_{1 2}^∗ n − ^γ₂^{1 2}^∗ ( 1 + n ) − ^γ₂^∗^{1 2}( 1 + n ) − ^γ¹^{+ γ}₂ ²( 1 + n ) 0 γ1 2n − ^γ₂^{1 2}( 1 + n ) − ^γ₂^{1 2}( 1 + n ) 0 − ^γ¹^{+ γ}₂ ²( 1 + n )













ρ0 0

ρ1 1

ρ2 2

ρ1 2

ρ2 1







.

( 2. 2 1) We c a n i ntr o d uce a n e ve n m ore c o ncise n ot ati o n:

( 1 + n )γ1 = T1, n γ 1 = t1,

( 1 + n )γ2 = T2, n γ 2 = t2,

( 1 + n )γ1 2 = 2 T1 2, n γ1 2 = 2 t1 2. ( 2. 2 2)

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In this notation, the QOME takes the form:







ρ˙00

˙ ρ₁₁

˙ ρ₂₂ ρ˙12

˙ ρ₂₁







=







−t₁−t₂ T₁ T₂ 2T₁₂ 2T₁₂^∗

t₁ −T₁ 0 −T₁₂ −T₁₂^∗

t₂ 0 −T₂ −T₁₂ −T₁₂^∗

2t^∗₁₂ −T₁₂^∗ −T₁₂^∗ −(T₁+T₂)/2 0 2t₁₂ −T₁₂ −T₁₂ 0 −(T₁+T₂)/2













ρ₀₀ ρ₁₁ ρ₂₂ ρ₁₂ ρ₂₁







.

Lastly, let us note that the probability in the populations of the open system is conserved.

We can verify this by summation of the first three rows of the transition matrix, d

dt (ρ₀₀+ρ₁₁+ρ₂₂) = 0.

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3. Quantum heat engine

We proceed by presenting the QHE model. First, we briefly describe its state cycle and outline the formation of the equations of motion from the simpler V-type system. Next, we examine the master equation and the underlying fundamental processes. This is then followed by the study of the probability currents, the energy currents and the power output of the QHE. In the last section of this chapter, we take a look at the entropy production and the efficiency.

3.1 Introduction

A quantum heat engine is a microscopic power-generating device coupled to two or more reservoirs at different temperatures. In our case, the working medium of the QHE is modeled by a five-level open system (see figure 3.1) which operates between two baths: a hot (red) and a cold (blue) bath.

Figure 3.1: A diagram of the QHE. The symbols γ_1h,γ_2h, γ_1c, γ_2c, Γ_c represent the rates of transitions coupled to the heat baths while the rate Γ represents the one-way power output of the QHE (as explained in sec. 3.9).

Our model corresponds to an exactly degenerate version of a quantum heat engine proposed by Dorfman et al. [8]. Their quantum heat engine is inspired by photosynthesis in plants and bacteria and models the light-dependent cycle of two tightly coupled chloro- phylls at the heart of the photosynthetic reaction center. Dorfman et al. show that this model can represent artificial quantum heat engines such as a laser or a photocell [8, 14].

The open system is represented by two donor molecules and one acceptor molecule and the cycle of states of this open system (see fig. 3.1) goes as follows:

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T he c ycle b e gi ns wit h t w o d o n or m olec ules s urr o u n di n g a n acce pt or m olec ule i n t he gr o u n d st ate |b .

A n a bs or pti o n of a s ol ar p h ot o n res ults i n t he e xcite d st ates of t he d o n ors: |1 a n d

|2 , w here t he cre ate d h ole a n d electr o n p air are still i n t he d o n ors.

T he e xcite d electr o n is t he n tr a nsferre d fr o m t he d o n ors t o t he acce pt or m olec ule wit h t he e missi o n of a p h o n o n, res ulti n g i n t he c h ar ge-se p ar ate d st ate |α . T he electr o n is i n t he acce pt or m olec ule n o w a n d t he h ole re m ai ns i n t he d o n ors.

Ne xt, t he tr a nsiti o n t o |β c orres p o n ds t o t he electr o n c o mi n g o ut of t he acce pt or m olec ule t o s o me l o wer e ner g y st ate a n d p erf or mi n g usef ul w or k. We m o del t his pr o cess b y t he p he n o me n ol o gic al r ate Γ a n d t he a m o u nt of a vail a ble w or k i n t his tr a nsiti o n de p e n ds o n t he r ati o of t he ste a d y-st ate p o p ul ati o ns ρα α a n d ρβ β. Fi n all y, t he p ositi ve c h ar ge is ne utr alize d b y a n electr o n s o urce wit h t he e missi o n of a p h o n o n. T he o p e n s yste m ret ur ns t o t he gr o u n d st ate |b .

It t ur ns o ut t h at t heir pr o p ose d m o del s u ﬀers fr o m a vi ol ati o n of p ositi vit y a n d di ver- ge nt b e h a vi or [ 1 4]. We a d o pt t he s a me sc he me of t he Q H E wit h t he di ﬀere nce t h at t he u p p er e ner g y le vels are p erfectl y de ge ner ate a n d we use a Q O M E i n t he Li n d bl a d f or m.

T his m o del t he n e ns ures t he pr o p erties of t he de nsit y m atri x.

3. 2 E q u a ti o n s of m o ti o n

Let us pr o cee d b y c alc ul ati n g t he e q u ati o ns of m oti o n f or t he Q H E. First, we t a ke a l o o k at t he all o we d tr a nsiti o ns i n ﬁ g. 3. 1 w hic h deter mi ne t he f or m of t he tr a nsiti o n di p ole m o me nt o p er at or ˆD :

ˆD =







0 d b 1 d b 2 0 d b β

d ^∗_{b 1} 0 0 d α 1 0 d ^∗_{b 2} 0 0 d α 2 0 0 d ^∗_{α 1} d ^∗_{α 2} 0 0

d ^∗_{b β} 0 0 0 0







. ( 3. 1)

N ote t h at we or der t he st ate b asis i n t he m atri x re prese nt ati o n cl o c k wise i n t he Q H E c ycle st arti n g wit h t he gr o u n d st ate b, i.e. { b, 1 , 2 , α, β} . T he tr a nsiti o n b et wee n t he st ates { α, β } is ne glecte d f or n o w a n d will b e a d de d at t he ver y e n d.

T he all o we d tr a nsiti o ns deter mi ne t he set of t he tr a nsiti o n fre q ue ncies w hic h we s u m o ver i n t he Q O M E. T hese i ncl u de t he a n g ul ar fre q ue ncies { ω b 1 2, ωα 1 2, ωb β} w here ωA B

de n otes t he fre q ue nc y c orres p o n di n g t o t he e ner g y di ffere nce E B − E A = ωA B . We c a n t hi n k of t he s u m o ver di ffere nt tr a nsiti o n fre q ue ncies as a s u m o ver di ffere nt s u bs yste ms.

I n o ur c ase, we h a ve t w o V-t y p e s yste ms: ˆK (ω b 1 2, Th) a n d ˆK (ωα 1 2, Tc) o p er ati n g b et wee n t he st ates { b, 1 , 2 } a n d { α, 1 , 2 } res p ecti vel y, a n d a t w o-le vel s yste m ˆK (ω b β, Tc) o p er ati n g b et wee n { b, β } . T hese s yste ms are c o u ple d t o t heir res p ecti ve b at hs, i.e. t he h ot (Th) or t he c ol d (Tc) b at h, a n d a d d u p t o f or m t he Q H E m o del:

d

dt ˆρS = ˆK (ω b 1 2, Th) + ˆK (ωα 1 2, Tc) + ˆK (ω b β, Tc), ˆK (ω, T ) =γ (ω, T )( 2 Â ˆρSI Â^† − Â^† Â ˆρSI − ˆρSI Â^† Â )

+ γ (− ω, T )( 2 Â^†ˆρSI Â − Â Â^†ˆρSI − ˆρSI Â Â^†).

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N ote t h at t he e x pressi o n f or ˆK (ω, T ) is writte n i n o ne di me nsi o n, as i n e q. ( 2. 1 7). I n or der t o arri ve at t he e q u ati o ns f or t he t w o-le vel s yste m, it is c o n ve nie nt t o c o nsi der t he e q u ati o ns f or a t hree-le vel V-t y p e s yste m a n d set o ne of t he tr a nsiti o n di p ole m o me nts t o zer o.

Besi des t he p o p ul ati o ns, t he rele va nt re d uce d de nsit y m atri x ele me nts of t he s u bs yste ms o nl y i ncl u de t he c o here nces ρ1 2 a n d ρ2 1 ( w hic h arise i n t he c alc ul ati o n f or t he V-t y p e s yste m):

ˆρS =







ρb b ρb 1 ρb 2 ρb α ρb β

ρ1 b ρ1 1 ρ1 2 ρ1 α ρ1 β

ρ2 b ρ2 1 ρ2 2 ρ2 α ρ2 β

ρα b ρα 1 ρα 2 ρα α ρα β

ρβ b ρβ 1 ρβ 2 ρβ α ρβ β







. ( 3. 2)

We will arr a n ge t he rele va nt ele me nts ( bl ac k) i nt o a m aster e q u ati o n a n d or der t he ele- me nts i n t he st ate vect or ρ s uc h t h at t he c o here nces are at t he e n d:

d

dtρ = ˆM ρ, ρ = ( ρb b, ρ1 1, ρ2 2, ρα α, ρβ β , ρ1 2, ρ2 1)^T . ( 3. 3) For a c o ncise descri pti o n, let us i ntr o d uce c o nst a nts a n al o g o us t o t he o nes f or t he V-t y p e s yste m ( 2. 2 0):

γ1 h = α ω _{b 1 2}³ |d b 1|² , γ1 2 h = α ω _{b 1 2}³ d b 1 · d ^∗_{b 2}, nh = N (ωb 1 2, Th), γ2 h = α ω _{b 1 2}³ |d b 2|² , γ1 2 c = α ω _{α 1 2}³ d α 1 · d ^∗_{α 2}, nc = N (ωα 1 2, Tc), γ1 c = α ω _{α 1 2}³ |d α 1|² , Γ c = α ω _{b β}³ |d b β|² , Nc = N (ωb β, Tc),

γ2 c = α ω _{α 1 2}³ |d α 2|² , α = ( 3 π 0 c³)^{− 1}. ( 3. 4)

We f urt her gr o u p t hese c o nst a nts t o get her:

( 1 + nh)γ1 h = H 1, ( 1 + nc)γ1 c = C 1, ncγ1 c = c1, nhγ1 h = h1, ( 1 + nh)γ2 h = H 2, ( 1 + nc)γ2 c = C 2, ncγ2 c = c2, nhγ2 h = h2, ( 1 + nh)γ1 2 h = 2 H 1 2, ( 1 + nc)γ1 2 c = 2 C 1 2, ncγ1 2 c = 2 c1 2, nhγ1 2 h = 2 h1 2,

( 1 + N c) Γc = C 0, NcΓ c = c0. ( 3. 5) I n a d diti o n t o t he tr a nsiti o ns g o ver ne d b y t he Q O M E, we descri b e t he o ne- w a y tr a nsiti o n b et wee n t he st ates { α, β } b y a n o n- ne g ati ve tr a nsiti o n r ate Γ . D uri n g t his tr a nsiti o n, p art of t he e ner g y tr a nsfer c orres p o n ds t o usef ul w or k a n d t heref ore t he r ate Γ c o ntr ols t he p o wer o ut p ut of t he Q H E, see ﬁ g. 3. 1 a n d sec. 3. 9. To test t he st a bilit y of t he res ult, we als o i ncl u de a p he n o me n ol o gic al dec o here nce r ate m o deli n g t he dec a y of t he c o here nce ele me nts ρ1 2 a n d ρ2 1. T hese t w o p he n o me n ol o gic al e ﬀects c orres p o n d t o t he f oll o wi n g d y n a mics:

˙ρα α

˙ρβ β = − Γ 00 + Γ ρα α

ρβ β , ˙ρ1 2

˙ρ2 1 = − Λ / 2 0

0 − Λ / 2 ρ1 2

ρ2 1 . ( 3. 6)

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T he tr a nsiti o n m atri x of t he t he res ulti n g m aster e q u ati o n re a ds:

ˆM =







− c0− h1− h2 H 1 H 2 0 C 0 2 H 1 2 2 H _{1 2}^∗

h1 − C 1− H 1 0 c1 0 ^{− C}^{1 2}^{− H} ^{1 2} ^{− C}^∗^{1 2}^{− H} ^∗^{1 2}

h2 0 − C 2− H 2 c2 0 ^{− C}^{1 2}^{− H} ^{1 2} ^{− C}^{1 2}^∗ ^{− H} ^{1 2}^∗

0 C 1 C 2 − c1− c2− Γ 0 ^{2 C} ^{1 2} ^{2 C}^{1 2}^∗

c0 0 0 Γ − C 0 0 0

2 h^∗_{1 2} − C_{1 2}^∗ − H _{1 2}^∗ − C_{1 2}^∗ − H _{1 2}^∗ 2 c^∗_{1 2} 0 _{+ H}^{− (C}₁_{+ H}¹^{+ C}₂_{+ Λ) / 2}²⁺ 0

2 h1 2 − C1 2− H 1 2 − C1 2− H 1 2 2 c1 2 0 0 _{+ H}^{− (C}₁_{+ H}¹^{+ C}₂_{+ Λ) / 2}²⁺







.

( 3. 7) L astl y , we see t h at t he pr o b alit y i n t he p o p ul ati o ns _iρii is c o nser ve d:

i

d

dtρii(t) = 0. ( 3. 8)

3. 3 S ol u ti o n s t o t h e m a s t e r e q u a ti o n

T he o bt ai ne d m aster e q u ati o n is a s yste m of ﬁrst- or der li ne ar h o m o ge ne o us di ﬀere nti al e q u ati o ns:

d

dtρ = ˆM ρ. ( 3. 9)

T he e v ol uti o n of a s p eci ﬁc i niti al st ate ρ (t0) fr o m ti me t0 t o t is g o ver ne d b y t he pr o p a g at or ˆUM (t, t0):

ρ (t) = ˆUM (t, t0)ρ (t0) = e x p M (t − t0) ρ (t0). ( 3. 1 0) We c a n o bt ai n t he ste a d y-st ate s ol uti o ns ρst fr o m t he n ull s p ace of t he tr a nsiti o n m atri x ker( ˆM ):

d

dtρst = ˆM ρst = 0 . ( 3. 1 1)

T he ste a d y-st ate vect ors ρst d o n ot e v ol ve i n ti me a n d t heref ore de p e n d o nl y o n t he p ar a meters deter mi ni n g t he m atri x ˆM . T hese c a n b e e x presse d usi n g t he r ates Γ a n d Λ , t he te m p er at ures Th a n d Tc, t he e ner gies E b, Eα, E1 2 a n d E β, a n d t he tr a nsiti o n di p ole m o me nts d b 1, d b 2, dα 1, d α 2 a n d d b β. We s h all c all t he sets of p ar a meters deter mi ni n g t he ste a d y-st ate vect or

ρst = ρst( Γ, Th, Tc, Eb, E1 2, Eα, Eβ, d b 1, d b 2, d α 1, d α 2, d b β, Λ) ,

t he m o d el p ar a m et ers . Us u all y, o ne e x p ects t he n ulls p ace of ˆM t o b e o ne- di me nsi o n al.

H o we ver, as we s h all see l ater, c o nstr ai nts o n t he c o e ﬃciets of t he tr a nsiti o n m atri x ˆM c a n res ult i n a m ore t h a n o ne- di me nsi o n al n ull s p ace. T he si g ni ﬁc a nce of t hese a d diti o n al ste a d y-st ate s ol uti o ns will b e disc usse d se p ar atel y (see secs. 4. 1, 4. 3, 4. 6).

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3. 4 F u n d a m e nt al p r o c e s s e s

To i nter pret t he e q u ati o ns of m oti o n, we rec all t he Ei nstei n c o e ﬃcie nts [ 1 6] A a n d B c orres p o n di n g t o t he pr o cesses of s p o nt a ne o us e missi o n a n d sti m ul ate d e missi o n a n d a b- s or bti o n. T he f oll o wi n g e q u ati o ns descri b e t hese pr o cesses i n a s yste m wit h N 0 + N 1 at o ms i nter acti n g wit h electr o m a g netic r a di ati o n i n t her m o d y n a mic e q uili bri u m w here N 0 is t he n u m b er of at o ms i n t he gr o u n d st ate a n d N 1 is t he n u m b er of at o ms i n t he e xcite d st ate:

d N0

dt _{a b s} = − u (ω, T )B N 1 = − d N1

dt _{a b s} , ( 3. 1 2)

d N1

dt _{sti m} = − u (ω, T )B N 1 = − d N0

dt _{sti m}, ( 3. 1 3)

d N1

dt _{s p o nt} = − A N 1 = − d N0

dt _{s p o nt} . ( 3. 1 4)

T he f u ncti o n u (ω, T ) is t he s p ectr al e ner g y de nsit y at a n g ul ar fre q ue nc y ω a n d te m p er a- t ure T gi ve n b y Pl a nc k’s l a w of bl ac k b o d y r a di ati o n,

u (ω, T ) = 2 ω ³ π c³

1

e x p _k_b^ω_T − 1 = A

B N (ω ). ( 3. 1 5)

It is a n i nteresti n g f act t h at t he s p o nt a ne o us e missi o n is te m p er at ure i n de p e n de nt a n d c a uses a n e x p o ne nti al dec a y of t he e xcite d st ate, w here as t he sti m ul ate d pr o cesses de p e n d o n t he s p ectr al e ner g y de nsit y. F urt her m ore, t he s p o nt a ne o us e missi o n c a n b e re g ar de d as t he sti m ul ate d e missi o n pr o cess c a use d b y a virt u al p h ot o n c o mi n g fr o m vac u u m ﬂ uct u ati o ns. T he p h ase, t he p ol ariz ati o n a n d t he directi o n of pr o p a g ati o n of t he e mitte d p h ot o n are t he n r a n d o m, u nli ke i n t he pr o cess of sti m ul ate d e missi o n w here t he e mitte d p h ot o n is cre ate d i n t he s a me st ate as t he re al sti m ul ati n g p h ot o n [ 1 6].

S u m m ati o n of t he pr o cesses ( 3. 1 2 − 3. 1 4) a n d a re arr a n g me nt yiel ds:

d N1

dt = − d N0

dt = d N1

dt _{a b s} + d N1

dt _{sti m} + d N1

dt _{s p o nt}

= A [B

A u (ν )N 0 − ( 1 + B

A u (ν ))N 1]

= A [N (ω )N 0 − ( 1 + N (ω ))N 1], w hic h c a n b e p ut i nt o a m atri x f or m:

d dt N 0

N 1 = A − N (ω ) ( 1 + N (ω ))N (ω ) − ( 1 + N (ω )) N 0

N 1 . ( 3. 1 6)

T his 2 × 2 m atri x all o ws us t o see t he f o ot pri nts of t he pr o cesses ( 3. 1 2 − 3. 1 4) i n t he tr a nsiti o n m atri x:

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Mˆ =







ρbb ρ11 ρ22 ραα ρββ ρ₁₂ ρ₂₁

˙

ρ_bb −c0−h1−h2 H₁ H₂ 0 C₀ ^2H12 2H₁₂^∗

˙

ρ11 h1 −C1−H1 0 c1 0 ^−C12−H12 −C₁₂^∗−H^∗₁₂

˙

ρ22 h2 0 −C2−H2 c2 0 ^−C12−H12 −C₁₂^∗−H^∗₁₂

˙

ραα 0 C₁ C₂ −c₁−c₂−Γ 0 ^2C12 2C₁₂^∗

˙

ρ_ββ c₀ 0 0 Γ −C₀ 0 0

ρ˙12 2h^∗₁₂ −C₁₂^∗−H₁₂^∗ −C₁₂^∗ −H₁₂^∗ 2c^∗₁₂ 0 _+H^−(C¹^+C²⁺

1+H₂+Λ)/2 0

˙

ρ21 2h12 −C12−H12 −C12−H12 2c₁₂ 0 0 _+H^−(C¹^+C²⁺

1+H₂+Λ)/2







.

(3.17) Specifically, we can compare the form of the equation (3.16) with the coefficients in the upper left 5×5 part of the transition matrix in eq. (3.17).¹ From the definition of the coefficients, see eqs. (3.4) and (3.5), we check that the processes are indeed contained in the transition matrix. For example, the coefficients in red correspond to the transition between the states {b,1} where the rate H₁ is the rate of the emission process (both stimulated and spontaneous emission) and the rate h₁ is the rate of absorption.

The presence of these processes can be further supported by checking the standard result that in the dipole approximation the rate of spontaneous emission is proportional to the square of the transition dipole moment, see eqs. (3.4) and (3.5).

Notice that the rates of emission{C₁,C₂,H₁,H₂} from the upper states contribute to the decay of the coherences.

3.5 Alignment of transition dipole moments

Besides the dynamics in the upper left 5×5 part of the transition matrix in eq. (3.17), we have terms describing the coherent part of the fundamental processes arising from the degeneracy of the upper energy levels. The rate constants {c₁₂, C₁₂} and {h₁₂, H₁₂} are proportional to the dot products of the transition dipole moments d_α1·d^∗_α2 and d_b1·d^∗_b2 respectively, see eqs. (3.4) and (3.5). The values of these dot products clearly affect the behaviour of the coherences and if we set them to zero, the open system is always in a non-coherent state (ρ₁₂=ρ₂₁= 0). We can also push the open system towards a non- coherent state by increasing the decoherence rate Λ. Therefore, we call either of these cases where c₁₂=C₁₂=h₁₂=H₁₂= 0 or Λ→ ∞the no coherence case.

3.6 Probability currents

The conservation of probability tells us that the changes in the populations are caused by the flow of probability, i.e. probability currents. Let us denote a probability current

1The added top row and left column in eq. (3.17) serve for better orientation in the transition matrix.

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fr o m t he p o p ul ati o n ρii t o ρk k as ji k. T he c o nser vati o n of pr o b a bilit y t he n re a ds:

d

dtρii(t) = −

k = i

ji k(t) =

k = i

jki(t), ( 3. 1 8)

w here we h a ve use d t he pr o p ert y ji k(t) = − jki(t) of c urre nts. We are de fi ni n g pr o b a bilit y c urre nts b et wee n p o p ul ati o ns o nl y a n d t h us we c a n restrict o ursel ves t o t he first fi ve r o ws of t he tr a nsiti o n m atri x:²







ρb b ρ1 1 ρ2 2 ρα α ρβ β ρ1 2 ρ2 1

˙ρb b − c0− h1− h2 H 1 H 2 0 C 0 2 H 1 2 2 H _{1 2}^∗

˙ρ1 1 h1 − C 1− H 1 0 c1 0 ^{− C} ^{1 2}^{− H} ^{1 2} ^{− C} ^∗1 2− H _{1 2}^∗

˙ρ2 2 h2 0 − C 2− H 2 c2 0 ^{− C} ^{1 2}^{− H} ^{1 2} ^{− C} ^∗1 2− H _{1 2}^∗

˙ρα α 0 C 1 C 2 − c1− c2− Γ 0 ^{2 C} ^{1 2} ^{2 C} 1 2^∗

˙ρβ β c0 0 0 Γ − C 0 0 0







.

I n t he n o c o here nce c ase, we c a n i de ntif y t he pr o b a bilit y c urre nts, i.e. tr ace t he c h a n ges i n t he p o p ul ati o ns, b y f oll o wi n g t he f u n d a me nt al pr o cesses. T he pr o b a bilit y c urre nt ji k

t he n h as t he f or m:

ji k(|d i k|²) = γ1(|d i k|²)ρii − γ2(|d i k|²)ρk k,

w here t he r ates γ1 a n d γ2 are pr o p orti o n al t o t he s q u are of t he tr a nsiti o n di p ole m o me nt

|d i k|². As a n e x a m ple, t he pr o b a bilit y c urre nt b et wee n t he st ates { b, β } re a ds:

jb β = c0ρb b − C 0ρβ β

= α ω_{b β}³ |d b β|² (N cρb b − ( 1 + N c)ρβ β).

T he s p eci al c ase is t he tr a nsiti o n b et wee n t he st ates { α, β } w hic h is n ot m o dele d wit h si m ple a bs or pti o n a n d e missi o n pr o cesses. It is m o dele d s o t h at t he r ate of e xcit ati o n fr o m β t o α is zer o a n d t he r ate of dee xcit ati o n is g o ver ne d b y t he r ate Γ ,

jα β = Γ ρα α.

I n t he ge ner al c ase, t he p o p ul ati o ns are als o a ﬀecte d b y ter ms pr o p orti o n al t o t he d ot pr o d ucts of t he tr a nsiti o n di p ole m o me nts, d α 1 · d ^∗_{α 2} a n d d b 1 · d ^∗_{b 2}, a n d t he c o here nces ρ1 2 a n d ρ2 1. T hese ter ms arise b ec a use of c o here nt e xcit ati o n t o t he u p p er de ge ner ate le vels { 1 , 2 } , i.e. w he n t he o p e n s yste m is e xcite d i nt o a s u p er p ositi o n of t hese st ates.

T he c o nser vati o n of pr o b a bilit y tells us t h at t hese ter ms m ust m o dif y t he s u ms of t he pr o b a bilit y c urre nts jb 1 + jα 1 a n d jb 2 + jα 2 c o n necte d t o t hese le vels:

˙ρ1 1 = jb 1 + jα 1 = h1ρb b + c1ρα α − (C 1 + H 1)ρ1 1 − (C 1 2 + H 1 2)ρ1 2 − (C _{1 2}^∗ + H _{1 2}^∗ )ρ2 1,

˙ρ2 2 = jb 2 + jα 2 = h2ρb b + c2ρα α − (C 2 + H 2)ρ1 1 − (C 1 2 + H 1 2)ρ1 2 − (C _{1 2}^∗ + H _{1 2}^∗ )ρ2 1. We c a n s plit t hese ne w ter ms b y disti n g uis hi n g b et wee n t he h ot a n d c ol d V-t y p e s yste ms, s uc h t h at t he c urre nts jb 1 a n d jb 2 n o w als o h a ve ter ms pr o p orti o n al t o db 1 · d ^∗_{b 2}, a n d jα 1

a n d jα 2 ter ms pr o p orti o n al t o dα 1 · d^∗_{α 2}. Fi n all y, we write d o w n t he pr o b a bilit y c urre nts

2T h e a d d e d t o p r o w a n d l eft c ol u m n s er ve f or b ett er ori e nt ati o n i n t h e tr a nsiti o n m atri x.

QuantumThermodynamics OldřichSedlák BACHELORTHESIS

BACHELOR THESIS

Oldřich Sedlák