Quantum Many-Body Effects in the Optoelectronic Response of Plasmonic Nanostructures and their Coupling to Quantum Emitters

EUSKAL HERRIKO UNIBERTSITATEA
THE UNIVERSITY OF THE BASQUE COUNTRY
Depa men o Elec ici y and Elec onics
CAMPUS OF
INTERNATIONAL
EXCELLENCE
Quan um Many-Body E ec s in he
Op oelec onic Response o Plasmonic
Nanos uc u es and hei Coupling o
Quan um Emi e s
Thesis by
An on Babaze Aizpu ua
Supe ised by
P o . Ja ie Aizpu ua I iazabal
and
D . Rubén Es eban Llo en e
Donos ia-San Sebas ián, Sep embe 2022
(cc)2022 ANTTON BABAZE AIZPURUA (cc by-sa 4.0)
Ama e a Ai a i,
esi honen oina i sendoak eza zeaga ik.
ACKNOWLEDGMENTS
Han pasado casi cinco años desde el 2 de oc ub e de 2017, p ime día que acudía a
la o icina de Rubén, po aquel en onces en el DIPC. Ja i es aba en algún iaje de
los suyos. Aquel día empezaba mi p ime abajo “se io”, donde me pagaban po
“es udia ”. Me pagaban poco pa a mi gus o, pe o eso es o o ema. Ese p ime día
Rubén me imp imió unos 10 a ículos, no sé cuán as hojas en o al (muchas), “pa a
que ayas leyendo y haciéndo e una idea”, me decía. An es de imp imi los, cla o,
me explicó lo mejo que pudo de qué iban esos a ículos y cómo se elacionaban
con lo que iba a se mi esis. Tesis que culmino aho a, cinco años más a de, con
es as palab as de ag adecimien o. También me mos ó uno po uno los pos e s
que había colgados en la pa ed del pasillo en el CFM. C eo que ingí bas an e
bien que más o menos es aba siguiendo lo que me decía, pe o he de con esa que,
en ealidad, no es aba en endiendo nada. No sé si Ruben e a conscien e de ello,
puede que sí. Ta dé unos cuan os meses en sabe , po ejemplo, qué e a un plasmón.
C eo que aho a puedo a i ma que en iendo los a ículos que me imp imió hace
cinco años como e e encia. En g an pa e, g acias al cons an e ap endizaje que
he ecibido po pa e de Rubén. No sabes cuán o iempo a dé en a e igua cómo
se elacionaban las pelo i as de me al iluminadas con láse con el pa de muelles
acoplados que oscilan siguiendo las ecuaciones que ap endimos en segundo de
ca e a. Además, apa e de habe sido capaz de en ende aquellos a ículos, me
sien o bas an e o gulloso de habe con ibuido, aunque sea mínimamen e, en el
campo de la plasmónica. Con ibuciones que no se ían posibles sin u conocimien o,
abajo incansable, y es ue zo po enseña me. G acias, Rubén.
A las pocas semanas de empeza mi doc o ado en ó Ja i en escena. Fue
en un ca é en el DIPC. De p ime as supe que iba a se muy sencillo a a con
él como supe iso . Así ue. Po supues o, no sabía ni que e a el di ec o del
CFM ni que e a un ío an conocido en el campo de la Nano o ónica, con un
índice h de no sé cuán o. Me hacía mucha g acia cuando, en las con e encias,
odo el mundo me decía: “O cou se I know Ja ie Aizpu ua”. Al p incipio no
sabía muy bien po qué. Tampoco sabía la impo ancia que Ja i iba a ene
en mi esis, que desde luego ha sido mucha. U e haue an zeha sa i an izan
du bu uan Pe nando Amezke a a en ma azki bizidune an behinola en zundako
esaldia: “Galde a zailik ez dago, e an zunak di a zailak”. Denon a ean, ni e
esian hain e azak ez zi en e an zun gu xi ba zuk ema ea lo u dugu. Ez dago
gaizki. Ho e an, pisu handia izan duzu zuk, Ja i. Eske ik asko u e haue an

i aka si ako guz ia enga ik: zien zian e a, esango nuke, ba ez e e zien zia ik kanpo
i aka si akoaga ik. Pun a-pun ako alde ba ean ike zeko auke a ema eaga ik.
Ni e ike ke a-lana maisuki gida zeaga ik. A azo xikien au ean be i i enbide ik
hobe ena bila zeaga ik. Ni e lana au e a a e a zeko egindako es o zua enga ik.
Ni e alde egindako apus ua enga ik. Eske ik asko, Ja i.
In May 2018, jus se en mon hs a e I s a ed wi h my PhD hesis, I had
he oppo uni y o mee And ei Bo iso in ISMO, O say. And ei has been my
hi d supe iso , along wi h Ruben and Ja i. He has been he key pe son in his
hesis. The one who has augh me e e y hing I know abou TDDFT and quan um
plasmonics. All he esea ch in his hesis has been ca ied ou in ex emely close
collabo a ion wi h And ei. Thank you o you pa ience, guidance, and willingness
o each me. Willingness o discuss wi h me. So many discussions. So many calls. I
ha e checked ha we ha e exchanged 321 e-mails du ing he las ou yea s. Thank
you o sha ing wi h me you WPP codes. This hesis is also you s. Only one
ema k: we shouldn´ e e o sodium as a simple me al. Thank you, And ei.
Si he enido sue e con los supe iso es de mi esis, he enido aún mucha más
sue e con los compañe os de abajo que he enido en el CFM. Compañe os de
abajo que son aho a amigos. No ecue do en qué esis leí que los ag adecimien os
son la pa e más di ícil de esc ibi . Pa a nada. Me esul a e dade amen e ácil
esc ibi es as líneas de o ma hones a. Habéis sido la hos ia. No c eo que nunca
enga ningún abajo donde aya odos los días con an as ganas a abaja . Ha
sido súpe di e ido habe compa ido an os momen os con oso os. Den o del
abajo, y po supues o, ue a. Con el iesgo de ol ida me de alguien impo an e,
p ocedo a nomb a a a ios de oso os.
Empiezo po Ál a o, Al a iño, mi masajis a pa icula . Compañe o de
doc o ado desde el p incipio has a el inal. Compañe o en las clases magis ales
de Rubén. Cómo me ali iaba sabe que ú ampoco e en e abas de nada. O de
muy poco. No sé en qué pun o empezamos a in es iga cosas an dis in as, cuando
al p incipio pa ecía que es udiábamos exac amen e lo mismo: el dipolo jun o a la
pelo i a. Ha sido un place habe hecho odo el doc o ado con igo, habe cha lado
sob e ciencia, y, sob e odo, habe cha lado sob e an as o as cosas ue a de la
ciencia. Muchos buenos momen os ue a de la o icina. Viajes. También den o de
la o icina. Muchas isas. Muchas es upideces. Mucho iempo “pe dido”, pe o con
mucho gus o. Sue e con u esis, ío. En nada lo ienes.
Con inúo con los demás muchachos de la o icina. Ca los, Cha ly, Ca li os.
Teo emán. Qué capacidad, macho. Siemp e que he enido alguna duda sob e ísica
o ma emá ica, ahí es abas ú con us eo emas. Sé que muchos e an in en ados,
aunque nunca lo econozcas. A e cuándo me pones de au o en uno de us
pape s. Una máquina con inkscape y blende . Cuando nadie e e, cla o. Nunca
ol ida é odos esos días de cha le a en la o icina cuando odo el mundo ya se había
ma chado a casa. O los skypes du an e la pandemia. O aquellos días p e ios a
Na idad, cuando nos quedábamos los dos solos abajando y ú ponías illancicos
a odo olumen ingiendo se eliz. Yo me lo pasaba bien. Me aleg o de que
mexicanos como ú c ucen el cha co pa a qui a nos el abajo, wey. Jon, Jone o,
qué ío. Siemp e dispues o a ayuda en lo que sea, pe o ambién a saca le pun a
i
a odo lo que yo digo. Be din du nik ze esan, be i egongo za ela kon an. Un
i án. Dibe iga ia izan da jokoa, bene an. Ja ai u ho ela, mo el. B uno, el
más lis o del CFM. Segu amen e ambién el más ápido con el coche. Haciendo
TDDFT como yo, pe o bien. Zo e on dok o e za ekin. Ad ián, el más g acioso,
o o que se quedaba haciendo gua dia has a bien a de en la o icina del CFM.
Jona han, con ese humo an pa icula , siemp e “hue eando”. Mikel, Mikela s,
ni e Honda ibiko bizilaguna. Tipo bikaina. Dene an ona. Plaze a izan da zu ekin
denbo a guz i hau pa eka zea. Zo e on zu i e e esia ekin, au ki buka uko duzula.
Jo ge, o o g ande. A e cuándo e sacas el í ulo de euske a. Un place los aciles
mu uos que hemos enido, y espe o segui eniendo. Robe o, Robe , ídolo. Me he
eído mucho con igo, e a asko eske zen du euska a ekin egi en duzun ahalegina.
Ez nuen espe o As u ia ba ekin euska az hi z egi en buka uko nuenik. Ike , mi
Bilbaino a o i o. G acias po habe me dejado acila e an o. Lás ima la úl ima
copa. G acias a o os muchos ambién: Ma ín, Mi iam, Ma in, Fe nando, Josu,
Joseba, Ma io, Raulillo, Txemikel... Todos oso os habéis conseguido que sea un
e dade o place eni odos los días al CFM.
Lanean egindako lagunez gain, lane ik kanpo di udan lagunak e e aipa u
nahi ni uzke. Le o gu xie an bada e e. I ungo be iko lagunak. A alekuko
Hin xak. Mue e a los Geólogos. Simon-eneako Co ije oak. Hulk. Kemenkideak.
Dul zaine oak. “Noiz buka zen duzu p oiek ua?”; “Ze moduz da amazu mas e -
a?”; “Baina, i a, zu ea ez da “lana-lana”, ez?”. Eske ik asko denoi.
Buka zeko, le o ba zuk eskaini nahi dizkie esi honen bene ako euska i izan
za e enei: E xekoei. Tesi hau zuena da. Zuek gabe ezingo nuke esi hau inoiz
au e a a e a. Ama, ho enbes e gauza i akas eaga ik. Nik ha u ako e abakiak be i
babes eaga ik. Ni e helbu uak be e zeko be i zu e esku dagoen guz ia egi eaga ik.
Ez du sekula ahaz uko momen u zaile an egindako es o zua. Ondo dakizu ze az
a i naizen, noski. Ai a, ni egan be i ja e a k i ikoa bul za zeaga ik. Gauzak
es o zua ekin a e a zen di ela i akas eaga ik. Bizi zan bide one ik noala behin
e a be i o azpima a zeaga ik. Ni egan e aku si ako kon ian zaga ik. Iñaki,
zu egandik jaso ako animo e a laudo ioenga ik. Ni e esiaz, e a bizi zaz, ho enbes e
a du a zeaga ik. Gu e bizi ze an azal zeaga ik. Peio, be i animoak ema eko
p es ego eaga ik. Ni e lo pen xikiekin ni bezain bes e poz eaga ik. Azken
u ee an lane ik kanpoko bizi za e azagoa egi eaga ik. Jokin, inoiz ai o u ez a en,
xiki a ik gauza asko an ni e e edu iza eaga ik. Ho enbes e gauza i akas eaga ik.
Mundu akademikoan sa ze a anima zeaga ik. Aizpea, ze esan. Ezagu zen dudan
pe sona ik onena iza eaga ik. Bizi zan ausa dia izan beha dela i akas eaga ik.
Be i ni e ondoan ego eaga ik. Biho z-biho zez, eske ik asko denoi.
I un-Donos ia, 2022ko i aila.
An on Babaze
ii
LABURPENA
A gia en e a ma e ia en a eko elka ekin za aspaldi ik izan da az e gai
zien ziala ien za . Esa e ako, 1850eko hama kadan, Michael Fa aday-k au ki u
zuen amaina nanome ikoko u ezko pa ikulek e a bolumen handiko u ezko
egi u ek p opie a e op iko oso desbe dinak di uz ela [
1
]. P opie a e op iko be ezi
ho iek di a, adibidez, u e-e ubi bei a en kolo e go i dis i a sua so zen du enak
[
2
,
3
]. Au kikun za ha en azalpen isikoa Gus a Mie-k eman zuen zenbai
u e ge oago [
4
], 1908an, James Cle k Maxwell-en eo ia elek omagne ikoa [
5
]
e abiliz u ezko nanopa ikula xikien sakabana ze-p opie a eak az e u zi uenean
[
6
–
8
]. Nanopa ikula me aliko xiki hauek a gia maiz asun jakin ba zue an
(no malean espek o ikusgaian) modu e aginko ean ba eia zen du ela au ki u
zuen Mie-k; maiz asun hauek ma e iala en, pa ikula en amaina en e a ingu une
dielek ikoa en p opie a een a abe akoak izanik [
9
]. Maiz asun jakin ho iek
gainazaleko plasmoi lokaliza uei dagozkie [
10
–
12
], a gia en bidez ki zika dai ezkeen
nanopa ikula me alikoen gainazaleko ka ga-oszilazioen e esonan ziei, alegia [
13
–
16].
Azken u ee an, nanopa ikula me alikoen plasmoi e esonan ziek in e es handia
piz u du e Nano o onika alo ean [
17
–
19
], uhin elek omagne iko e aso zailea en
anpli udea a eago zeko e a a gia uhin-luze a baino eskualde xikiagoe an
lokaliza zeko du en ahalmena dela-e a [
20
–
22
]. Adibidez, bi nanopa ikula
me aliko en a eko eskualde nanome ikoan (nanoba unbe plasmoniko dei u ikoa,
nanogap edo a nanoca i y ingelesez), e emu elek omagne ikoa en anpli udea 100-
1000 aldiz handi u dai eke [
23
]. Ondo ioz, gau egun plasmoi e esonan ziak
asko e abil zen di a hainba espek oskopia e a mik oskopia eknike an, hala
nola gainazalak a eago u ako Raman espek oskopian (su ace-enhanced Raman
spec oscopy) [
24
], gainazalak a eago u ako luo eszen zian (su ace-enhanced
luo escence) [
25
–
27
], edo a molekula baka en de ekzioan (single-molecule imaging)
[
28
,
29
]. Gaine a, plasmoi e esonan ziek aplikazio i xa open suak di uz e, bes eak
bes e, biomedikun zan [
30
–
32
], ene gia en bil egi a zean [
33
–
35
], edo a op ika
ez-linealean [36,37].
E ek u plasmonikoak elek omagne ismo klasikoa en eo ia en es uingu uan
az e u izan di a ba ik ba [
38
–
45
], non a gia en e a ma e ia en a eko elka ekin za
Maxwell-en ekuazioek desk iba zen du en e a sis ema en e an zun op ikoa, o o
ha , lineal za jo zen den [
6
]. Hala e e, egungo kon igu azio espe imen ale an
(nanopa ikula bene an xikiak, pa ikulen a eko dis an ziak azpi-nanome ikoak
ix
Con en s
Acknowledgmen s
Labu pena ix
Lis o abb e ia ions x
In oduc ion 1
1 Classical desc ip ion o ligh –ma e in e ac ion 5
1.1 Maxwell’s equa ions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 The local- esponse app oxima ion . . . . . . . . . . . . . . . 7
1.1.2 Bounda y condi ions . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 The non e a ded app oxima ion . . . . . . . . . . . . . . . . 11
1.2 Plasmonics ............................... 16
1.2.1 Bulkplasmons ......................... 16
1.2.2 Su ace plasmons and su ace plasmon pola i ons . . . . . . 17
1.2.3 Localized su ace plasmon pola i ons (LSPPs) . . . . . . . . 18
1.3 Nonlinea op ical esponse o small nanos uc u es . . . . . . . . . 24
1.4 Plexci onics: Quan um emi e exci ons coupled o plasmons . . . 27
1.4.1 The poin -dipole app oxima ion . . . . . . . . . . . . . . . . 28
1.4.2 The sel -in e ac ion dyadic G een’s unc ion . . . . . . . . . 28
1.4.3 Coupled ha monic-oscilla o model . . . . . . . . . . . . . . 31
1.5 Summa y ................................ 34
2 Quan um many-body desc ip ion o ligh –ma e in e ac ion 35
2.1 Fundamen als o densi y unc ional heo y (DFT) . . . . . . . . . . 36
2.1.1 The local-densi y app oxima ion (LDA) . . . . . . . . . . . 38
2.1.2 The jellium model o ee-elec on me als . . . . . . . . . . 39
2.2
Fundamen als o ime-dependen densi y unc ional heo y (TDDFT)
41
2.2.1 The wa e-packe p opaga ion (WPP) me hod . . . . . . . . 42
2.3
Linea op ical esponse o canonical plasmonic nanos uc u es
add essed wi hin TDDFT . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Indi idual sphe ical me allic nanopa icles . . . . . . . . . . 49
2.3.2 Dime s o sphe ical me allic nanopa icles . . . . . . . . . . 55
2.3.3 Cylind ical me allic nanowi es . . . . . . . . . . . . . . . . 61
x i

2.4
Nonlinea e ec s in he op ical esponse o sphe ical plasmonic
nanopa icles add essed wi hin TDDFT . . . . . . . . . . . . . . . 65
2.5 Semiclassical su ace- esponse o malism (SRF) . . . . . . . . . . . 69
2.5.1 Op ical esponse o sphe ical nanos uc u es using he SRF 72
2.6 Summa y ................................ 75
3
Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles 77
3.1 Sys emandme hods.......................... 78
3.1.1 Time-dependen densi y unc ional heo y (TDDFT) . . . . 80
3.1.2 Classical local- esponse app oxima ion (LRA) . . . . . . . . 82
3.1.3 Semiclassical su ace- esponse o malism (SRF) . . . . . . . 82
3.2 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.1 Quan um TDDFT s. classical LRA . . . . . . . . . . . . . 84
3.2.2 Quan um TDDFT s. semiclassical nondispe si e SRF . . . 87
3.2.3
In e p e a ion o he quan um e ec s wi hin he nondispe si e
SRF and i s limi a ions . . . . . . . . . . . . . . . . . . . . 88
3.2.4
Quan um TDDFT s. nondispe si e SRF in a nanopa icle
dime .............................. 91
3.3 Summa y ................................ 94
4
Dispe si e su ace- esponse o malism o add ess op ical
nonlocali y in si ua ions o ex eme plasmonic ield con inemen 97
4.1 Sys emandme hods.......................... 98
4.2 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.1
TDDFT s udy o he ene gy dispe sion o mul ipola plasmon
esonances in a me allic nanowi e . . . . . . . . . . . . . . . 101
4.2.2 Calcula ion o he dispe si e Feibelman pa ame e d⊥(ω, k∥)103
4.2.3 Valida ion o he dispe si e SRF . . . . . . . . . . . . . . . 107
4.3 Summa y ................................ 112
5
Elec onic exci on–plasmon coupling in a nanoca i y beyond
he elec omagne ic in e ac ion pic u e 115
5.1 Sys emandme hods.......................... 116
5.1.1 Cha ac e iza ion o he model quan um emi e (QE) . . . 118
5.2 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.1 In luence o he QE exci on quenching a op ical equencies 120
5.2.2 Quan um ini e-size e ec s s. elec onic QE–MNPs coupling 127
5.2.3 Cha ge- ans e esonances a low equencies . . . . . . . . 128
5.3 Summa y ................................ 130
6
Second-ha monic gene a ion om a quan um emi e coupled
o a me allic nanopa icle 131
6.1 Sys emandme hods.......................... 132
6.2 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 134
x ii
6.2.1 TDDFT esul s......................... 134
6.2.2 TDDFT s. semi-analy ical model . . . . . . . . . . . . . . 138
6.2.3
In luence o he in insic losses and he posi ion o he QE
on he e iciency o SHG . . . . . . . . . . . . . . . . . . . . 141
6.2.4 Pola iza ion con e sion o he second-ha monic ield . . . . 146
6.3 Summa y ................................ 147
Conclusions and Ou look 149
Appendices
A A omic uni s 155
B
Elec ic nea ield induced by a sphe ical nanopa icle wi hin
he classical local- esponse app oxima ion (LRA) 157
C Dynamics o quan um emi e s in he ime domain 159
D
Dispe si e Feibelman pa ame e ob ained o cylind ical
nanowi es o di e en size 163
E
Plasmon esonances sus ained by a cylind ical nanowi e wi hin
he su ace- esponse o malism 165
F
Rela ionship be ween he esul s ob ained unde ini e Gaussian
pulses and plane-wa e exci a ion 169
Lis o publica ions 173
Bibliog aphy 175
x iii
INTRODUCTION
The in e ac ion be ween ligh and ma e has been a he ocus o he scien i ic
communi y o e he las cen u ies. Fo example, in he 1850s, Michael Fa aday
disco e ed ha nanoscopic gold colloids exhibi op ical p ope ies di e en o hose
in bulk me als [
1
], gi ing ise, e.g., o he b igh ed colo o gold- uby glass [
2
,
3
].
Some yea s la e , in 1908, Gus a Mie p o ided he physical explana ion o his
e ec [
4
] when he applied he elec omagne ic heo y de eloped by James Cle k
Maxwell [
5
] o s udy he sca e ing p ope ies o small gold nanopa icles [
6
–
8
].
Mie ound ha such small me allic nanopa icles e icien ly sca e ligh a speci ic
equencies ( ypically in he isible ange) which depend upon he ma e ial and size
o he nanopa icle as well as upon he p ope ies o he dielec ic en i onmen [
9
].
I would be la e known ha hese speci ic equencies co espond o he so-called
localized su ace plasmons [
10
–
12
], he esonances o he su ace cha ge oscilla ion
sus ained by he collec i e oscilla ion o he ee elec ons in me allic nanopa icles
ha can be exci ed by ligh [13–16].
In he las yea s, he exci a ion o plasmon esonances in me allic nanopa icles
has a ac ed g ea in e es in Nanopho onics [
17
–
19
] due o hei capabili y o
enhance and squeeze inciden elec omagne ic ields in o subwa eleng h egions
[
20
–
22
]. Fo example, i is possible o ob ain elec omagne ic ields up o 100-
1000 imes la ge han he inciden ield in he nanome ic egion be ween wo
me allic nanopa icles ( e e ed o as a plasmonic gap o plasmonic nanoca i y)
[
23
]. As a esul , plasmon esonances a e widely used in a a ie y o spec oscopy
and mic oscopy echniques such as su ace-enhanced Raman spec oscopy [
24
],
su ace-enhanced luo escence [
25
–
27
], o single-molecule imaging [
28
,
29
], and
enable p omising applica ions in biomedicine [30–32], ene gy s o age [33–35], and
nonlinea op ics [36,37], among o he s.
Theo e ically, plasmonic e ec s ha e been usually s udied wi hin he con ex o
classical elec omagne ism [
38
–
45
], whe e he in e ac ion be ween ligh and ma e is
desc ibed by Maxwell’s equa ions and he op ical esponse o he sys em is gene ally
conside ed o be linea [
6
]. Howe e , cu en expe imen al con igu a ions in ol ing
nanome e -sized me allic nanopa icles and in e pa icle dis ances push ligh –ma e
in e ac ion o he limi whe e quan um many-body phenomena in luence op ical
p ope ies [
46
–
52
]. In hese ex eme si ua ions, classical desc ip ions a e no longe
alid [
53
–
55
], and al e na i e heo e ical app oaches allowing o inco po a ing
nonlocali y [
56
–
60
], elec onic spill in/ou [
61
–
63
], su ace-enabled Landau damping
1
In oduc ion
[
64
–
66
], and elec on unneling [
67
,
68
] a e equi ed o desc ibe he op ical p ope ies
o minia u ized plasmonic s uc u es co ec ly.
In his con ex , ime-dependen densi y unc ional heo y (TDDFT) [
69
–
72
] is
he me hod o choice o add ess he op ical esponse o nanoscale sys ems in his
hesis, since his heo y accoun s o he quan um na u e o he elec on dynamics
om i s p inciples by add essing he ime e olu ion o he elec on densi y in
me allic nanopa icles subjec ed o ex e nal illumina ion [
73
]. In addi ion, he use o
TDDFT calcula ions is no es ic ed o ob aining only he linea op ical esponse,
bu i also enables o di ec ly ob ain he nonlinea esponse o plasmonic sys ems
[
74
–
78
]. TDDFT is hus a e y powe ul ool o cap u e quan um many-body
phenomena in ol ed in he op ical and elec onic (i.e., op oelec onic) esponse,
and i se s up he co ne s one o his hesis. On he o he hand, since TDDFT is
limi ed o add essing small sys ems ha con ain a ew housands o a oms due o i s
compu a ional complexi y [79,80], less-demanding semiclassical models ha e also
been de eloped o cap u e a ious quan um e ec s [
81
,
82
]. He e we can men ion,
o ins ance, he quan um-co ec ed model accoun ing o elec on unneling
in subnanome ic me allic junc ions [
83
–
87
], se e al hyd odynamic desc ip ions
inco po a ing nonlocali y o he op ical esponse o me allic nanos uc u es [
88
–
94
], and he su ace- esponse o malism (SRF) [
95
–
97
] based on he inclusion o
quan um su ace- esponse co ec ions ( he so-called Feibelman pa ame e s) a he
me al–dielec ic bounda ies [
98
]. Toge he wi h TDDFT, in his hesis we also
pe o m simula ions based on he SRF, which cap u es impo an quan um e ec s
such as he spill in/ou o he induced cha ge densi y and su ace-enabled Landau
damping, bu canno accoun e.g. o elec on unneling be ween wo nanopa icles
in close p oximi y.
TDDFT and he a o emen ioned semiclassical models a e o en used o s udy
he op ical esponse o isola ed plasmonic nanos uc u es, bu hey can also
be used o analyze quan um e ec s in he op oelec onic in e ac ion be ween
plasmonic nanos uc u es and quan um emi e s such as a oms, quan um do s o
o ganic molecules. Indeed, he op ical esponse o a coupled emi e –plasmonic
nanos uc u e sys em has been widely s udied in Nanopho onics. The plasmonic
ield enhancemen and con inemen allow quan um emi e s o in e ac much mo e
e icien ly wi h ligh , leading o a ple ho a o in e es ing phenomena [
99
,
100
].
Some o hese e ec s can be desc ibed by classical calcula ions, such as he
enhancemen o he spon aneous emission a e (Pu cell e ec [
101
,
102
]) and
he modi ica ion o he esonan equency (Lamb shi [
103
,
104
]) o a quan um
emi e loca ed nea a plasmonic nanos uc u e. On he o he hand, he coupling
be ween a plasmonic ca i y and a quan um emi e can also in ol e quan um-
mechanical in e ac ions ha can a ec he chemical p ope ies o he emi e [
105
],
induce elec on ans e be ween he nanopa icles and he emi e [
106
–
109
], and
e en ually modi y d as ically he nonlinea op ical esponse o he coupled sys em
[110,111].
This hesis aims a heo e ically s udying no el quan um many-body phenomena
in he op oelec onic esponse o plasmonic nanos uc u es and hei in e ac ion
wi h quan um emi e s. In pa icula , we seek a deepe undamen al knowledge
2
In oduc ion
in o quan um-mechanical e ec s ha occu in plasmonic ca i ies o nanome ic size
o smalle , whe e nonlocali y, nonlinea i y, o elec on- ans e e ec s can all play
an impo an ole. To his end, quan um TDDFT simula ions a e used p ima ily,
bu classical calcula ions based on he local- esponse app oxima ion wi hin he
dielec ic amewo k as well as semiclassical models based on he SRF a e also
employed o comple e he analysis and p o ide addi ional insigh s.
In Chap e 1, we e iew he undamen als o classical ligh –ma e in e ac ion
and explain he key concep s o his hesis om a classical pe spec i e, ocusing
on he classical desc ip ion o localized su ace plasmon pola i ons sus ained by
sphe ical me allic nanopa icles and hei dime s as well as on he elec omagne ic
in e ac ion be ween plasmonic nanos uc u es and quan um emi e s. In Chap e 2,
we desc ibe he quan um TDDFT me hodology and he semiclassical SRF adop ed
in his hesis o accoun o quan um many-body e ec s p esen in he ligh –ma e
in e ac ion, and in oduce quan um-mechanical concep s such as elec on unneling,
su ace-enabled Landau damping, and elec on spill ou wi h he examples o he
op ical esponse o canonical plasmonic nanos uc u es. In Chap e 3, we use
TDDFT o s udy he in luence o quan um su ace e ec s in he elec omagne ic
in e ac ion be ween quan um emi e s and me allic nanopa icles. We iden i y
he dominan e ec s when elec on unneling and he elec onic coupling be ween
he quan um emi e and he me allic nanopa icle a e negligible. We u he
analyze he o igin o he obse ed e ec s wi h semiclassical calcula ions based
on a SRF ha neglec s he op ical nonlocali y in he di ec ion pa allel o he
me allic su ace, as usually implemen ed in he li e a u e, and es ablish he alidi y
ange o his s anda d implemen a ion o he semiclassical SRF. In Chap e 4, we
show ha he s anda d implemen a ion o he SRF can be ex ended by explici ly
accoun ing o he nonlocali y o he op ical esponse in he pa allel di ec ion along
he me allic su ace in he calcula ion o he Feibelman pa ame e s. The inclusion
o his nonlocali y has no been so a conside ed in he li e a u e o he bes o
ou knowledge, and p oduces a b oade ange o alidi y o he SRF including
ex eme subnanome ic con igu a ions. In Chap e 5, we analyze he e ec o
elec onic coupling be ween he elec onic s a es o a quan um emi e and hose o
a me allic nanopa icle dime when he dis ance be ween he nanopa icles and he
emi e is subnanome ic. This e ec has been neglec ed in Chap e s 3and 4, which
ocused on he elec omagne ic in e ac ion. We demons a e ha , in such si ua ions,
he elec onic in e ac ion be ween he emi e and he nanopa icles d as ically
modi ies he op ical esonances o he coupled sys em. Finally, in Chap e 6, we
s udy he nonlinea op ical esponse o a sys em consis ing o a quan um emi e
and a sphe ical me allic nanopa icle, and show ha he elec omagne ic emi e –
nanopa icle coupling can enable s ong nonlinea second-ha monic gene a ion,
o he wise o bidden due o symme y cons ains i an isola ed sphe ical nanopa icle
we e conside ed.
The con en o his hesis hus p esen s a quan um mechanical many-body
app oach o he op ical esponse o plasmonic ca i ies, which p o ides new
insigh s in o coupling wi h emi e s, elec on ans e p ocesses and nonlinea
e ec s. All hese e ec s a e o pa amoun impo ance in nowadays s a e-o - he-a
3

In oduc ion
Nanopho onics.
4
Chap e
1
CLASSICAL DESCRIPTION OF
LIGHT–MATTER INTERACTION
1.1 Maxwell’s equa ions
The in e ac ion be ween ligh and ma e is well add essed om a classical and
mac oscopic poin o iew by he ime-dependen Maxwell’s equa ions [6,22],
∇×E( , ) = −∂
∂ B( , ),(1.1a)
∇×H( , ) = ∂
∂ D( , ) + Jex ( , ),(1.1b)
∇·D( , )=4πρex ( , ),(1.1c)
∇·B( , )=0,(1.1d)
which ep esen a se o coupled pa ial di e en ial equa ions whe e ligh is
desc ibed as an elec omagne ic wa e wi h elec ic
E
(
,
)and magne ic
H
(
,
) ield
componen s.
1
The ime-dependen Maxwell’s equa ions in Eq.
(1.1)
de e mine he
dynamics o he elec omagne ic wa e (ligh ) a a posi ion
and ins an o ime
h ough a gi en dielec ic ma e ial in esponse o an ex e nal cha ge densi y
ρex
(
,
)
and cu en densi y
Jex
(
,
) ha ac as sou ces o elec omagne ic adia ion. Fo
he nonmagne ic ma e ials ha we conside in his hesis, he magne ic induc ion
B
(
,
)is s aigh o wa dly linked o
H
(
,
)by
B
(
,
) =
1
c2H
(
,
), wi h
c
he speed
o ligh in a acuum. Finally, he displacemen ec o ,
D
(
,
), is ela ed o he
elec ic ield
E
(
,
)wi hin he linea - esponse egime by he ollowing cons i u i e
1
A omic uni s (au) a e used h oughou his manusc ip unless o he wise s a ed (see Appendix
A).
5
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
ela ionship
D( , ) = Z∞
−∞ Z∞
−∞
d ′d ′ε( − ′, − ′)E( ′, ′),(1.2)
whe e
ε
(
− ′, − ′
)is he dielec ic unc ion desc ibing op ical exci a ions in he
ma e ial. We assume in Eq.
(1.2)
ha he medium is iso opic and homogeneous
by conside ing ha he dielec ic unc ion
ε
is a scala quan i y ha spa ially
depends on
− ′
. Mo eo e , he causali y o he dielec ic esponse is in oduced
by imposing ε( − ′, − ′)=0 o any ′> .
Acco ding o Eq.
(1.2)
, he displacemen ec o
D
a a pa icula posi ion
and ins an o ime
depends on he alue o he elec ic ield
E
a all posi ions
′
and imes
′<
. Thus, he dielec ic esponse o a ma e ial is, in gene al, nonlocal
bo h in space and ime. To deal wi h he empo al nonlocali y, i is con enien o
use he ime- o- equency Fou ie ans o m
2
and exp ess Eq.
(1.2)
in he domain
o he angula equency ωo he elec omagne ic ield,
D( , ω) = Z+∞
−∞
d ′ε( − ′, ω)E( ′, ω).(1.4)
Consequen ly, by applying he same Fou ie ans o m o Eq.
(1.1)
, Maxwell’s
equa ions can be exp essed in he equency domain as:
∇×E( , ω) = iωB( , ω),(1.5a)
∇×H( , ω) = −iωD( , ω) + Jex ( , ω),(1.5b)
∇·D( , ω)=4πρex ( , ω),(1.5c)
∇·B( , ω)=0.(1.5d)
Equa ions
(1.1)
and
(1.5)
a e, o cou se, equi alen , and can be sol ed by adop ing
di e en echniques. Among nume ical me hods o sol e Maxwell’s equa ions, we
can ci e he Bounda y Elemen Me hod [
112
], he Fini e-Di e ence Time-Domain
me hod [
113
], he Disc e e-Dipole App oxima ion [
114
] o he Fini e-Elemen
Me hod [115].
On he o he hand, he in luence o he spa ial nonlocali y o he dielec ic
unc ion p esen in Eq.
(1.4)
on he op ical esponse o di e en ma e ials is
cu en ly an ac i e esea ch opic by i sel [
81
], and many e o s ha e been de o ed
2
In his hesis, we use he ollowing de ini ions o he ime- o- equency (
F
) and o he
equency- o- ime (F−1) Fou ie ans o ms o a unc ion :
(ω) = F[ ( )] = Z+∞
−∞
( )eiω d ,
( ) = F−1[ (ω)] = 1
2πZ+∞
−∞
(ω)e−iω dω,
(1.3)
6
1.1. Maxwell’s equa ions
o he de elopmen o nume ical me hods ha co ec ly accoun o such nonlocali y
[
59
,
60
,
88
,
89
,
91
,
92
,
94
,
116
–
123
]. Using hese me hods, i has been shown ha
spa ial nonlocali y can play a signi ican ole in de ining he p ope ies o he op ical
esponse o me allic nanos uc u es wi h cha ac e is ic dimensions below
∼
10
nm
[
56
,
86
,
124
], as i is he case o he sys ems s udied in his hesis. Indeed, we discuss
in Chap e s 2,3,4and 5 he ole ha spa ial nonlocali y (and o he quan um
phenomena) plays in di e en scena ios in ol ing small me allic nanopa icles when
hey in e ac wi h ligh and wi h quan um emi e s. We accoun o such e ec s
by using quan um ime-dependen densi y unc ional heo y (TDDFT) simula ions
(see Chap e 2), and, a a di e en le el o app oxima ion, by a semiclassical su ace-
esponse o malism (SRF) inco po a ing he Feibelman pa ame e s ob ained ab
ini io (Sec ion 2.5). In con as , he classical calcula ions pe o med wi hin his
hesis a e based on a local desc ip ion o he op ical esponse o me als, so ha
nonlocal e ec s can only be pa ially in oduced wi h he use o phenomenological
pa ame e s [58,125,126], as we de ail in he ollowing Subsec ion 1.1.1.
1.1.1 The local- esponse app oxima ion
The local- esponse app oxima ion (LRA) neglec s he spa ial nonlocali y o he
dielec ic unc ion
ε
(
− ′, ω
)by assuming ha he dielec ic esponse o a ma e ial
a a posi ion
is independen o he esponse a any o he posi ion
′
=
. This
assump ion can be exp essed as
ε( − ′, ω) = ε( , ω)δ( − ′),(1.6)
wi h
δ
(
− ′
) he Di ac del a unc ion. The displacemen ec o
D
(
, ω
) hus
ans o ms om Eq. (1.4) o a much simple cons i u i e ela ionship,
D( , ω) = Z+∞
−∞
d ′ε( , ω)δ( − ′)E( ′, ω)
=ε( , ω)E( , ω).
(1.7)
Despi e i s simplici y, he LRA has success ully desc ibed many physical phenomena
o in e es in he con ex o ligh –ma e in e ac ion a he nanoscale, pa icula ly
in si ua ions whe e he cha ac e is ic leng h scale o he s udied nanos uc u es is
subs an ially la ge han he Fe mi wa eleng h o elec ons in me als.
The D ude esponse model o ee-elec on me als
The simples way o es ima e he ( equency-dependen ) local dielec ic unc ion
ε
(
, ω
)o a me al is o adop he D ude model [
127
,
128
]. This model conside s
ha conduc ion elec ons eely mo e wi hin a homogeneous gas in esponse o
an ex e nal elec ic ield
E
(
)while he hea y me al ions emain immobile. The
equa ion o mo ion o a conduc ion elec on is hen
¨
x( ) + γp˙
x( ) = −E( ),(1.8)
7
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
Figu e 1.2:(a) Sphe ical coo dina e sys em (
, θ, φ
)wi h he o igin a he cen e o a sphe ical
nanopa icle o adius
a
cha ac e ized by a dielec ic unc ion
ε
(
ω
).(b) Coo dina e sys em
employed o add ess he op ical esponse o a sphe ical dime o med by wo nanopa icles o
adius
a1
and
a2
sepa a ed by a gap o size
D
. The nanopa icles a e cha ac e ized wi h a
dielec ic unc ion
ε1
(
ω
)and
ε1
(
ω
). The coo dina es a e exp essed in sphe ical coo dina es
(
1, θ1, φ1
)and (
2, θ2, φ2
)wi h he o igin a he cen e o each nanopa icle. In bo h indi idual
nanopa icle and dime s uc u es, he sys em is su ounded by a dielec ic ma e ial cha ac e ized
wi h ϵd.
Since a poin dipole o ampli ude
p
(
ω
)loca ed a he cen e o he coo dina e
sys em c ea es an elec os a ic po en ial a a posi ion
gi en by
ϕind
(
, ω
) =
·p(ω)
| |3
,
p(ω)can be ob ained as
p(ω) =
m=1
X
m=−1
b1m(ω)a2Ym
1(π
2,0) ˆ
x+Ym
1(π
2,π
2)ˆ
y+Ym
1(0,0) ˆ
z,(1.29)
whe e {ˆ
x,ˆ
y,ˆ
z}a e he uni ec o s along he {x, y, z}-axes.
Op ical esponse o a sphe ical dime
We now conside he case o a dime consis ing o wo sphe ical nanopa icles
o adius
a1
and
a2
sepa a ed by a gap o size
D
(see Figu e 1.2b). Each
nanopa icle is cha ac e ized by a dielec ic unc ion
ε1
(
ω
)and
ε2
(
ω
), and he
en i e sys em is su ounded by a dielec ic ma e ial cha ac e ized by
εd
. The
nume ical implemen a ion desc ibed in his sec ion o ob ain he op ical esponse
o his sys em is based on a coupled-mul ipole me hod [
131
,
132
], which uses he
solu ion o he indi idual sphe ical nanopa icle explained abo e.
The po en ial
ϕind
dime
(
, ω
)induced by he dime in esponse o an ex e nal
po en ial
ϕex
(
, ω
)is gi en by he sum o he po en ial induced by each nanopa icle
14

1.1. Maxwell’s equa ions
(1 and 2),
ϕind
dime ( , ω) = ϕind
1( 1, ω) + ϕind
2( 2, ω),(1.30)
whe e he ec o s
1≡
(
1sinθ1cosφ1, 1sinθ1sin φ1, 1cosθ1
)and
2≡
(
2sinθ2cosφ2, 2sinθ2sin φ2, 2cosθ2
)a e w i en in sphe ical coo dina es wi h
he o igins a he cen e o he co esponding nanopa icle, as shown in Figu e 1.2b.
We de ine he coo dina es such ha he cen e s o he nanopa icles a e loca ed a
he
z
-axis sepa a ed by a dis ance
δ
=
a1
+
a2
+
D
. Thus,
1
and
2
a e ela ed by
2=q 2
1+δ2−2δ 1cosθ1,
cosθ2= ( 1cosθ1−δ)/ 2,
φ2=φ1.
(1.31)
The elec os a ic po en ial
ϕind
i
(
i, ω
)induced by he nanopa icle
i
(wi h
i
= 1
,
2) ollows iden ical exp ession as ha o he indi idual nanopa icle explained
abo e [Eq. (1.26)],
ϕind
1( 1, ω) =
ℓmax
X
ℓ=0
ℓ
X
m=−ℓ
bℓm
1(ω)Ym
ℓ(θ1, φ1) ℓ
1/aℓ
1 1< a1
aℓ+1
1/ ℓ+1
1 1> a1,,
ϕind
2( 2, ω) =
ℓmax
X
ℓ=0
ℓ
X
m=−ℓ
bℓm
2(ω)Ym
ℓ(θ2, φ2) ℓ
2/aℓ
2 2< a2
aℓ+1
2/ ℓ+1
2 2> a2,,
(1.32)
wi h, in his case,
bℓm
1(ω) = ξ1(ω, ℓ)Zd˜
Ω1[Ym
ℓ(θ1, φ1)]∗∂
∂ 1ϕex ( 1, ω) + ϕind
2( 2, ω) 1=a1
,
bℓm
2(ω) = ξ2(ω, ℓ)Zd˜
Ω2[Ym
ℓ(θ2, φ2)]∗∂
∂ 2ϕex ( 2, ω) + ϕind
1( 1, ω) 2=a2
,
(1.33)
whe e
ξ1(ω, ℓ) = −a1(ε1−εd)
εd(ℓ+ 1) + ε1ℓ,
ξ2(ω, ℓ) = −a2(ε2−εd)
εd(ℓ+ 1) + ε2ℓ.
(1.34)
The in eg als in Eq.
(1.33)
ake in o accoun ha he po en ial induced by one
nanopa icle ac s as ex e nal po en ial o he o he one. F om Eq.
(1.32)
, he
coe icien s
bℓm
1
(
ω
)and
bℓm
2
(
ω
)in Eq.
(1.33)
can be ound om he ollowing
exp ession gi en in ma ix o m:
bℓm
1(ω) = I−T2→1T1→2−1bℓm,ex
1(ω) + T2→1bℓm,ex
2(ω),
bℓm
2(ω) = I−T1→2T2→1−1bℓm,ex
2(ω) + T1→2bℓm,ex
1(ω).
(1.35)
15
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
He e,
bℓm
1
(
ω
)and
bℓm
2
(
ω
) ep esen column ec o s con aining he coe icien s
bℓm
1
(
ω
)
and
bℓm
2
(
ω
),
I
is he iden i y ma ix, and
bℓm,ex
1
and
bℓm,ex
2
a e column ec o s
wi h he ollowing elemen s:
bℓm,ex
1(ω) = ξ1(ω, ℓ)Zd˜
Ω1[Ym
ℓ(θ1, φ1)]∗∂
∂ 1
ϕex ( 1, ω) 1=a1
,
bℓm,ex
2(ω) = ξ2(ω, ℓ)Zd˜
Ω2[Ym
ℓ(θ2, φ2)]∗∂
∂ 2
ϕex ( 2, ω) 2=a2
.
(1.36)
Finally, he elemen s (ℓm, ℓ′m′)o ma ices T2→1and T1→2a e gi en by
T2→1
ℓm,ℓ′m′=ξ1(ω, ℓ)Zd˜
Ω1[Ym
ℓ(θ1, φ1)]∗∂
∂ 1 Ym′
ℓ′(θ2, φ2)aℓ′+1
2
ℓ′+1
2! 1=a1
,
T1→2
ℓm,ℓ′m′=ξ2(ω, ℓ)Zd˜
Ω2[Ym
ℓ(θ2, φ2)]∗∂
∂ 2 Ym′
ℓ′(θ1, φ1)aℓ′+1
1
ℓ′+1
1! 2=a2
.
(1.37)
Once he coe icien s
bℓm
1
(
ω
)and
bℓm
2
(
ω
)a e ob ained by sol ing Eq.
(1.35)
, he
physical quan i ies o in e es such as he induced elec ic ield
Eind
(
, ω
)o he
dipole momen
p
(
ω
)o he dime s uc u e can be de e mined as he sum o he
co esponding con ibu ions o each nanopa icle [Eqs. (1.28) and (1.29)].
1.2 Plasmonics
Plasmons a e collec i e oscilla ions o he ee-elec on gas inside a me al [
22
,
133
].
F om a classical elec omagne ic poin o iew, plasmons mani es hemsel es as
abso p ion and sca e ing esonances ob ained om he solu ions o Maxwell’s
equa ions and he associa ed bounda y condi ions discussed in Sec ion 1.1. On
he o he hand, wi hin quan um-elec odynamics, he wo d plasmon e e s o he
quan um o he me allic elec on plasma oscilla ion [134,135].
Depending on he geome y o he sys em, plasmons can be exci ed in he
bulk o a he me al–dielec ic bounda y, and, in he la e si ua ion, hey can
be ei he p opaga ing o localized elec on-densi y oscilla ions. In he ollowing,
we desc ibe some canonical examples o plasmonic exci a ions wi hin he classical
elec omagne ic heo y. In addi ion o p o iding he undamen al concep s o
plasmonics, his classical desc ip ion es ablishes a e e ence poin o compa e wi h
he esul s ob ained wi hin a quan um many-body app oach, hus allowing us o
iden i y he o igin o he e ec s o in e es in his hesis.
1.2.1 Bulk plasmons
Bulk o olume plasmons a e longi udinal exci a ions consis ing in he cohe en
oscilla ion o he elec on gas p opaga ing in an in ini ely ex ended me al, whe e
he exci ed elec ons mo e collec i ely oscilla ing a he same equency. These
16
1.2. Plasmonics
longi udinal elec omagne ic wa es (
∇·E
(
, ω
)

= 0) can be desc ibed wi hin he
local- esponse app oxima ion (see Subsec ion 1.1.1), and can only exis i
ε
(
ω
) = 0
[
54
,
57
]. Thus, assuming a lossless D ude dielec ic unc ion [Eq.
(1.10)
, wi h
γp
= 0], he bulk plasmon equency is equal o he plasma equency,
ωbulk
=
ωp
[Eq. (1.12)].
1.2.2 Su ace plasmons and su ace plasmon pola i ons
When we conside a semi-in ini e plana me al slab (
ε
(
, ω
) =
ε
(
ω
)
, z <
0) in
con ac wi h acuum (
ε
(
, ω
) =
εd
= 1
, z >
0), he ansla ional in a iance o he
bulk ma e ial along he
z
-di ec ion is b oken. As a consequence, a new ype o
plasmon eme ges associa ed wi h an oscilla ing cha ge densi y p opaga ing along
he me al–dielec ic in e ace. These densi y oscilla ions a e e e ed o as su ace
plasmons. The esonance condi ion o a su ace plasmon can be de i ed wi hin
he non e a ded app oxima ion om he sel -sus ained induced po en ial
ϕind
(
, ω
)
[Eq. (1.22), wi h ϕex ( , ω)=0]. ϕind( , ω)is gi en in his case by [10]
ϕind( , ω) = 


ϕ<(ω)eik∥· ∥e|k∥|zz < 0
ϕ>(ω)eik∥· ∥e−|k∥|zz > 0
,(1.38)
whe e
∥
and
k∥
a e he wo-dimensional posi ion and momen um ec o s in he
plane o he me al su ace, espec i ely, and
ϕ<
(
ω
)and
ϕ>
(
ω
)a e equency-
dependen coe icien s o be de e mined om he bounda y condi ions gi en by
Eq.
(1.23)
. F om he con inui y o he po en ial a
z
= 0 [Eq.
(1.23a)
] i is ob ained
ha
ϕ<
(
ω
) =
ϕ>
(
ω
). Fu he , om he con inui y o he no mal componen o
D
(
, ω
) =
−ε
(
, ω
)
∇ϕ
(
, ω
)a
z
= 0 [Eq.
(1.23b)
], he esonance condi ion o he
non e a ded su ace plasmon is gi en by
ε
(
ω
) =
−
1. Thus, assuming a lossless
D ude dielec ic unc ion [Eq.
(1.10)
, wi h
γp
= 0], he long-wa eleng h limi o
he su ace plasmon equency is equal o
ωsp =ωp/√2.(1.39)
On he o he hand, a su ace plasmon pola i on (SPP) is a hyb id mode ha
p opaga es a he me al–dielec ic bounda y and ha esul s om he coupling o a
su ace plasmon and an elec omagne ic wa e [
136
]. These SPPs a e cha ac e ized
by elec omagne ic ields ha a e e anescen in he di ec ion no mal o he su ace,
hus exponen ially decaying wi h inc easing dis ance om he in e ace (see ske ch
in Figu e 1.3a). Conside ing an elec omagne ic wa e ha sa is ies Maxwell’s
equa ions in his sys em [Eq.
(1.5)
], and applying he bounda y condi ions gi en
by Eqs.
(1.16)
and
(1.18)
as well as he conse a ion o momen um ec o
k∥
along
he p opaga ion di ec ion [
22
], he ollowing dispe sion ela ionship o SPPs is
ob ained ( o a lossless D ude dielec ic unc ion),
ω2
spp(k∥) = ω2
p/2 + c2k2
∥−qω4
p/4 + c4k4
∥,(1.40)
17
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
+ + +
− − −
Figu e 1.3:(a) Schema ic ep esen a ion o a SPP wi h a wa enumbe
kspp
p opaga ing along he
me al– acuum in e ace loca ed a
z
= 0. The elec omagne ic ield
E
associa ed wi h he SPP
is ep esen ed by b own lines. (b) F equency o a SPP ob ained in his sys em om Eq.
(1.40)
(
ωspp
, blue line), o ligh in ee space (
ωc
, ed line), and o a long-wa eleng h non e a ded su ace
plasmon (
ωsp
, g een line) as a unc ion o he wa enumbe
k∥
pa allel o he me al– acuum
in e ace. The equency
ω
in he e ical axis is measu ed in uni s o
ωp
, and
k∥
in he ho izon al
axis in uni s o ωp/c, wi h c he speed o ligh in acuum.
wi h
c
he speed o ligh in a acuum. The ampli ude o he momen um ec o
k∥
in Eq.
(1.40)
co esponds o he wa enumbe
kspp
o a SPP,
kspp
=
k∥
, as
schema ically depic ed in Figu e 1.3a.
As shown in Figu e 1.3b, o small
k∥
, he SPP equency
ωspp
(blue line)
app oaches he dispe sion line o ligh in ee space,
ωc
=
ck∥
( ed line). In
con as , o la ge
k∥
,
ωspp
yields he nondispe si e (
k∥
-independen ) equency o
he non e a ded su ace-plasmon equency in he long-wa eleng h limi [Eq.
(1.39)
],
ωsp
=
ωp/√2
(g een line) [
10
]. Impo an ly, o a gi en equency
ω
, he momen um
k∥
o ligh in ee space is always smalle han he momen um
kspp
o SPPs, and
he e o e, a lase beam inciden on an ideal su ace canno exci e SPPs because
momen um and ene gy canno be simul aneously p ese ed. Fo his eason, se e al
mechanisms ha e been adop ed o p o ide he ex a momen um needed o exci e
SPPs wi h ligh , such as he use o su ace oughness o g a ings [
137
], e anescen
ields [
138
,
139
], o sha p me allic ips placed on op o he me al–dielec ic su aces
[140].
1.2.3 Localized su ace plasmon pola i ons (LSPPs)
When me allic nanopa icles (MNPs) o ini e size a e conside ed, a new ype o
plasmon esonances eme ges, so-called localized su ace plasmon pola i ons (LSPPs)
[
11
,
141
]. The wo d localized is used because, unlike he SPP p e iously discussed,
in his case he su ace-cha ge oscilla ions a e no p opaga ing in space. Indeed, he
ansna ional in a iance o he sys em is b oken, and plasmons a e con ined in he
h ee dimensions o space. Thus, LSPPs can o en be unde s ood as con ined SPPs
wi h a quan ized wa enumbe
kspp
. Impo an ly, con a y o SPPs in semi-in ini e
18
1.2. Plasmonics
me al slabs, LSPPs in small MNPs can be exci ed by an inciden lase beam,
because he ini e geome y o he MNP p o ides he ex a momen um needed
o o e come he misma ch be ween he momen um o ligh and plasmons. Fo
simplici y, in his hesis we e e o LSPPs as “localized su ace plasmons (LSPs)”,
o simply “plasmon esonances” o “plasmons”.
In his hesis, we ocus on quan um e ec s associa ed wi h he exci a ion o
LSPPs in small MNPs, whe e special a en ion is paid o sphe ical nanopa icles
and nanopa icle dime s. In he ollowing, we desc ibe he basic p ope ies o
LSPPs sus ained in such geome ies using a classical elec omagne ic heo y.
LSPPs in sphe ical nanopa icles
We i s de i e he esonance condi ions ul illed by LSPPs in a sphe ical MNP o
adius
a
su ounded by acuum (dielec ic unc ion
εd
= 1). These condi ions a e
again de e mined om he sel -sus ained induced po en ial
ϕind
(
, ω
) ha sa is ies
Laplace’s equa ion [Eq.
(1.22)
], which o he sphe ical MNP is gi en by Eq.
(1.26)
.
Unde no ex e nal exci a ion,
ϕex
(
, ω
)=0, he coe icien s
bℓm
(
ω
)a e non-null
only a he poles o Eq.
(1.27)
, so ha he esonance condi ions o he non e a ded
LSPPs sus ained in a sphe ical MNP a e gi en by
ℓε(ω)+(ℓ+ 1) = 0,(1.41)
whe e
ℓ
is he mul ipole o de o he LSPP esonance. The e o e, assuming a
lossless D ude dielec ic unc ion [Eq.
(1.10)
, wi h
γp
= 0], he equencies
ωℓ
o
LSPPs in a sphe ical MNP a e gi en by:
ωℓ=ωp ℓ
2ℓ+ 1,(1.42)
which a e independen o he size o he nanopa icle. Howe e , his exp ession
is igo ous only wi hin he alidi y ange o he non e a ded and local- esponse
app oxima ions. Indeed, i is known ha he scale in a iance o Eq.
(1.42)
is
li ed when e a da ion e ec s a e conside ed in la ge MNPs (
a≳
15
nm
) [
7
,
142
],
as well as when quan um-size e ec s a e conside ed in e y small nanos uc u es
(
a≲
5
nm
) [
61
,
95
,
96
]. We e u n in Chap e s 2,3, and 4 o he impac o
quan um-size e ec s on he esonan equencies o LSPPs, ωℓ.
Figu e 1.4a shows he alues o he LSPP equencies
ωℓ
as a unc ion o he
mul ipole o de
ℓ
[Eq.
(1.42)
]. Fo small alues o
ℓ
, he esonance equencies
ωℓ
o di e en o de
ℓ
can be well di e en ia ed. Fo example, he dipola plasmon
(DP) equency (
ℓ
= 1) is gi en by
ωDP
=
ωℓ=1
=
ωp/√3
, he quad upola plasmon
(QP) equency (
ℓ
= 2) by
ωQP
=
ωℓ=2
=
ωpq2
5
, and he oc upola plasmon (OP)
equency (
ℓ
= 3) by
ωOP
=
ωℓ=3
=
ωpq3
7
. In con as , di e en LSPPs wi h
la ge
ℓ
ha e e y simila esonan equencies, and hus, due o he b oadening
o each esonance gi en by he losses (
γp
in a D ude me al), hey pile up in a
single b oad esonance, he so-called pseudomode [
143
,
144
] (see Chap e 3). The
19

Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
+
+
+
−
−
−
+−
Figu e 1.4:(a) F equencies
ωℓ
o he non e a ded LSPP esonances sus ained in a sphe ical
MNP as a unc ion o he mul ipole o de
ℓ
, conside ing a lossless D ude dielec ic unc ion
[Eq.
(1.10)
]. The equencies
ωℓ
in he e ical axis a e gi en in uni s o
ωp
. The g een dashed line
ep esen s he equency o he su ace plasmon,
ωsp/ωp
= 1
/√2
.(b) Ske ch o he su ace cha ge
densi y induced in a sphe ical MNP by an ex e nal elec omagne ic ield
Eex
co esponding o
plane-wa e illumina ion. The g een a ow ep esen s he oscilla ing dipole momen
p
induced a
he MNP. (c) Absolu e alue o he elec ic ield
Eind
induced by a sphe ical MNP in esponse
o an ex e nal plane wa e
Eex
, calcula ed wi hin he non e a ded app oxima ion. The MNP
has a adius
a
= 5
nm
, and i is cha ac e ized by a D ude dielec ic unc ion wi h
ωp
= 5
.
89
eV
and
γp
= 0
.
21
eV
. The colo map is shown a he DP esonance,
ωDP
= 3
.
4
eV
.(d) Ex inc ion
(blue) and sca e ing ( ed, mul iplied by 10) c oss-sec ion spec a o he same MNP as in panel
(c). Due o he small size o he MNP, he ex inc ion and abso p ion c oss sec ions a e almos
equi alen , σex (ω)≈σabs(ω).
equency o his pseudomode app oaches he equency o he su ace plasmon,
ωℓ→∞ ∼ωp/√2
=
ωsp
[Eq.
(1.39)
], since o la ge
ℓ
he local cu a u e o he
sphe ical MNP is e y la ge (app oxima ely la in e ace) as compa ed o he
wa eleng h o he exci ed LSPPs, λLSPP ∼2πa/ℓ.
We nex show ha a plane wa e illumina ing a sphe ical MNP can exci e
LSPPs. We conside a plane wa e linea ly pola ized along he
z
-di ec ion, which is
exp essed wi hin he non e a ded app oxima ion as
Eex
(
, ω
) =
E0ˆ
z
. In ui i ely,
Eex
(
, ω
)pola izes he MNP along he
z
-di ec ion, p oducing a displacemen o
he ee-elec on gas wi h espec o he posi i ely cha ged backg ound (see ske ch
in Figu e 1.4b). Then, seeking o es o e he equilib ium, he Coulomb in e ac ion
be ween he posi i e and nega i e cha ge densi ies p oduces a collec i e oscilla ion
o he ee-elec on gas. The LSPP co esponds o he esonan exci a ion o his
20
1.2. Plasmonics
collec i e oscilla ion.
A quan i a i e s udy o he exci a ion o LSPPs unde plane-wa e illumina ion
can be done by calcula ing he op ical esponse o he MNP ollowing he p ocedu e
explained in Subsec ion 1.1.3. The ex e nal po en ial
ϕex
(
, ω
)in his case is gi en
in sphe ical coo dina es by:
ϕex ( , ω) = −E0 cosθ, (1.43)
and he in eg al in Eq.
(1.27)
p o iding he coe icien s
bℓm
(
ω
)has an analy ical
solu ion [
145
], which is non-null only o
ℓ
= 1. This means ha , in con as wi h
SPPs, he DP esonance can be exci ed by plane-wa e illumina ion. The elec ic
ield Eind( , ω)induced in he p oximi y o he MNP is gi en by:
Eind( , ω) = −∇ϕind( , ω) = 








−E0ε(ω)−1
ε(ω)+2 (cosθˆ
−sinθˆ
θ)
| {z }
ˆ
z
< a
E0ε(ω)−1
ε(ω)+2 a3/ 3(2 cosθˆ
+ sin θˆ
θ) > a
,
(1.44)
wi h
ˆ
and
ˆ
θ
he uni ec o s along he adial and angen ial di ec ions, espec i ely.
No e ha , as expec ed,
Eind
(
, ω
)possesses a esonance a he DP equency,
ωDP =ωp/√3.
We show in Figu e 1.4c he absolu e alue o he elec ic ield
Eind
(
, ω
)induced
a he DP equency,
ωDP
= 3
.
4
eV
, in he p oximi y o a MNP o adius
a
= 5
nm
.
The MNP is cha ac e ized by a D ude dielec ic unc ion [Eq.
(1.10)
] using he
pa ame e s
ωp
= 5
.
89
eV
and
γp
= 0
.
21
eV
o desc ibe sodium. Resul s a e
no malized o he ampli ude o he incoming ield,
E0
. C ucially, he induced ield
is much s onge han he inciden ield, and i is localized in a space egion much
smalle (
∼
1
−
5
nm
) han he wa eleng h o he inciden ield (
∼
400
nm
). Thus,
his elec omagne ic- ield localiza ion su passes he di ac ion limi .
Mo eo e , we no e ha he elec ic ield
Eind
(
, ω
)induced ou side he MNP
bounda y (
> a
) co esponds o he elec os a ic ield c ea ed by a poin dipole
placed a he cen e o he MNP [
6
,
22
]. Indeed, he ex e nal ield
Eex
induces
a dipole momen
p
(
ω
) =
α
(
ω
)
Eex
a he sphe ical MNP ha depends on he
quasi-s a ic pola izabili y [22]
α(ω) = a3ε(ω)−1
ε(ω)+2.(1.45)
This induced dipole
p
(
ω
)emi s ligh in o he a ield. The MNP hus ac s
as an op ical nanoan enna ha can ex emely localize inciden elec omagne ic
adia ion in he nea ield and adia e i in o he a ield [
146
]. The powe
Psca
sca e ed in o he a ield is ela ed o p(ω)by [6]
Psca =ω4
3c3|p(ω)|2.(1.46)
21
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
Typically, he sca e ed powe
Psca
=
σscaI0
is no malized o he in ensi y
I0
=
c|Eex |2/
(8
π
)o he inciden plane wa e, leading o he sca e ing c oss
sec ion [7],
σsca(ω) = 8π
3ω
c4|α(ω)|2.(1.47)
Mo eo e , he o al powe p o ided by he incoming ligh o a MNP,
Pex
, is
gi en by he ex inc ion c oss sec ion
σex
h ough he ela ionship
Pex
=
σex I0
.
σex (ω)is de ined as [7]
σex (ω) = 4πω
cIm{α(ω)}=σabs(ω) + σsca(ω),(1.48)
and co esponds o he sum o bo h sca e ing
σsca
and abso p ion
σabs
c oss
sec ions, he la e de e mining he powe o ligh abso bed by he MNP,
Pabs =σabsI0.
Figu e 1.4d displays he ex inc ion
σex
(
ω
)(blue line) and sca e ing
σsca
(
ω
)
( ed line) c oss-sec ion spec a o he same sphe ical sodium MNP conside ed in
Figu e 1.4c (
a
= 5
nm
). Fo such small MNP,
σsca
(
ω
)
∝a6ω
c4
(wi h
ωa
c<<
1) is
abou wo-o de s o magni ude smalle han
σex
(
ω
)
∝a3ω
c
. As a consequence,
he ex inc ion o he MNP is comple ely go e ned in his case by i s abso p ion.
Thus, o he small size o he MNPs conside ed in his hesis,
σabs
(
ω
)and
σex
(
ω
)
a e almos equi alen . We he e o e use he ollowing exp ession o compu e he
abso p ion c oss sec ion in all he hesis [see Eq. (1.48)]:
σabs(ω) = 4πω
cIm{α(ω)}.(1.49)
LSPPs in nanopa icle dime s
When wo sphe ical MNPs a e placed in close p oximi y, hey o m a dime
sus aining coupled LSPPs ha can be exci ed by plane-wa e illumina ion [
23
].
The s udy o LSPPs in he dime con igu a ion is pa icula ly in e es ing because
hey induce much s onge ield enhancemen s han indi idual MNPs [
147
,
148
].
Mo eo e , he analysis o he op ical p ope ies o a sphe ical dime can help o
unde s and he physics behind he op ical esponse o mo e complex plasmonic
nanos uc u es.
The esonan LSPP equencies in a me allic dime di e om hose o he
indi idual MNPs because he elec omagne ic coupling be ween he wo MNPs
modi ies he esul ing plasmonic esonances [
149
]. The LSPP equencies in his
dime con igu a ion depend on he ma e ial and size o he MNPs o ming he
dime , as well as he su ace- o-su ace gap sepa a ion
D
. As an example, we
show in Figu e 1.5a he wa e all spec a o he abso p ion c oss sec ion,
σabs
(
ω
)
[Eq.
(1.49)
], o a MNP dime wi h gap sepa a ion anging om
D
= 1
.
05
nm
(bo om) o
D
= 3
.
6
nm
( op). We conside he same sodium MNPs as in Figu es
1.4c-d, and he ex e nal illumina ion is pola ized along he dime axis. Resul s a e
22
1.2. Plasmonics
0
500
1000
1500
2000
2500
3000
Figu e 1.5:(a) Wa e all spec a o he abso p ion c oss sec ion
σabs
(
ω
)o a dime composed
by wo sphe ical sodium MNPs o adius
a
= 5
nm
cha ac e ized by a D ude dielec ic unc ion
wi h
ωp
= 5
.
89
eV
and
γp
= 0
.
21
eV
(same as in Figu es 1.4c-d). The gap sepa a ion is a ied
om
D
= 1
.
05
nm
(bo om) o
D
= 3
.
6
nm
( op) in s eps o
∼
0
.
21
nm
.(b) Ske ch o he
op ical hyb idiza ion be ween LSPPs o indi idual MNPs leading o he bonding dipola plasmon
(BDP) and he bonding quad upola plasmon (BQP). (c) Colo map o he induced elec ic- ield
enhancemen
|E|ind/E0
in he middle o he gap o med by wo MNPs wi h adius
a
= 5
nm
as a
unc ion o he equency o he ex e nal plane-wa e illumina ion,
ω
, and he su ace- o-su ace
gap dis ance,
D
. Resul s a e ob ained unde he non e a ded app oxima ion conside ing an
ex e nal elec omagne ic ield, Eex =E0ˆ
z, pola ized along he dime axis (z-axis).
ob ained wi hin he non e a ded app oxima ion ollowing he p ocedu e explained
in Subsec ion 1.1.3.
Two dis inc esonances can be obse ed in he abso p ion spec a o Figu e
1.5a o he whole ange o gap sepa a ion
D
conside ed: an in ense bonding dipola
plasmon (BDP) shi ing om
ωBDP ∼
3
.
2
eV
o
∼
2
.
8
eV
as
D
is educed, and a
weake bonding quad upola plasmon (BQP) a
ωBQP ∼
3
.
5
eV
, mo e p onounced
o na ow gaps. The BQP also edshi s wi h dec easing
D
. The o igin o hese
plasmon esonances can be unde s ood using a hyb idiza ion pic u e ha conside s
he coupling be ween he mul ipole modes
ℓ
o he wo indi idual MNPs [Eq.
(1.42)
]
simila ly o he hyb idiza ion be ween a omic o bi als in dia omic molecules [
150
].
Acco ding o his hyb idiza ion pic u e (see ske ch in Figu e 1.5b), he BDP is
mainly c ea ed due o he elec omagne ic coupling be ween he DP esonances
(
ℓ
= 1) o he indi idual MNPs, while he BQP is mainly a consequence o he
mix u e be ween he DP mode o one MNP and he quad upola mode (
ℓ
= 2) o
23
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
+
−
Figu e 1.7:(a) Abso p ion c oss-sec ion spec a
σabs
(
ω
)o an indi idual sphe ical MNP (dashed
black line) and ha o a QE–MNP coupled sys em (blue line). The MNP has a adius
a
= 5
nm
,
and i is cha ac e ized by a D ude dielec ic unc ion wi h
ωp
= 5
.
89
eV
and
γp
= 0
.
21
eV
. The
QE, loca ed a 1
.
5
nm
om he su ace o he MNP, is cha ac e ized by an oscilla o s eng h
α0
QE
= 2
au
, an in insic loss a e
γQE
= 10
meV
, and a ansi ion equency
ωQE
=
ωDP
= 3
.
4
eV
in esonance wi h he DP o he MNP. (b) Same as in panel (a) bu o a MNP dime cha ac e ized
by a gap sepa a ion
D
= 3
nm
. The QE emi e is a esonance wi h he BDP mode o he dime ,
ωQE =ωBDP = 3.15 eV. The esul s a e ob ained wi hin he non e a ded app oxima ion.
by he co esponding componen s [145],
G⊥( QE, QE, ω) = ∞
X
ℓ=1
ε(ω)−1
ε(ω) + ℓ+1
ℓ
a2ℓ+1
R2ℓ+4 (ℓ+ 1)2,(1.58a)
G∥( QE, QE, ω) = ∞
X
ℓ=1
ε(ω)−1
ε(ω) + ℓ+1
ℓ
a2ℓ+1
R2ℓ+4
1
2ℓ(ℓ+ 1),(1.58b)
wi h
R
he dis ance be ween he QE posi ion
QE
and he cen e o he sphe ical
MNP. Mo eo e ,
Eind
(
=
QE, ω
)can be ob ained om Eq.
(1.44)
, and
ˆαMNP
(
ω
)
≡
α(ω)is gi en by Eq. (1.45).
On he o he hand, o ob ain he op ical esponse o a QE coupled o a
sphe ical dime , we nume ically sol e he quan i ies o Eq.
(1.57)
by ollowing
he me hodology explained in Subsec ion 1.1.3. Thus, he op ical esponse o
he QE–MNPs sys em is ob ained in his case in h ee s eps: i s , we calcula e
Eind
(
=
QE, ω
)and
ˆαMNP
(
ω
)by sol ing Eq.
(1.35)
o he ex e nal po en ial
ϕex
(
, ω
)gi en by Eq.
(1.43)
co esponding o plane-wa e illumina ion. Second,
ˆ
G
(
QE, QE, ω
)and
ˆαQE
MNP
(
ω
)a e ob ained by sol ing he same equa ions bu
conside ing he ex e nal po en ial
ϕex
(
, ω
)
≡ϕQE
(
)o a uni a y poin dipole
gi en by Eq.
(1.53)
. Finally, we use Eq.
(1.57)
o calcula e he o al dipole momen
induced a he coupled sys em, p(ω) = pMNP(ω) + pQE(ω).
As an example, we show in Figu e 1.7 he abso p ion c oss-sec ion spec a
σabs
(
ω
)o an indi idual MNP (panel a) and a MNP dime (panel b) in e ac ing
wi h a QE. The MNPs ha e a adius
a
= 5
nm
and a e cha ac e ized using he
same D ude pa ame e s as abo e (sodium). The QE is cha ac e ized [Eq.
(1.45)
]
30

1.4. Plexci onics: Quan um emi e exci ons coupled o plasmons
by an oscilla o s eng h
α0
QE
= 2
au
, an in insic loss a e
γQE
= 10
meV
, and
a ansi ion equency
ωQE
esonan wi h he main plasmon esonance o he
nanos uc u es. In bo h si ua ions, he p esence o he QE s ongly a ec s he
op ical abso p ion o he coupled sys em (blue lines) as compa ed o ha o he
isola ed MNPs (dashed black lines). In pa icula , o he indi idual MNP Figu e
1.7a shows a spec ally na ow Fano-like esonance a
ω∼ωDP
= 3
.
4
eV
due o
he des uc i e in e e ence be ween he exci a ion o he plasmon and ha o
he exci on [
173
,
174
]. The e ec o he QE on he op ical esponse is s onge
in he dime con igu a ion shown in Figu e 1.7b due o he ield enhancemen in
he gap and hus la ge elec omagne ic coupling be ween he QE exci on and he
plasmonic esonances (see he ollowing subsec ion o u he de ails). In Chap e s
3and 5, we s udy he impac o se e al quan um-mechanical phenomena in he
op ical esponse o QE–MNPs coupled sys ems.
1.4.3 Coupled ha monic-oscilla o model
The sel -in e ac ion G een’s unc ion o malism explained in he p e ious subsec ion
is gene ic and, in p inciple, can be applied o ob ain he op ical esponse o a
QE in e ac ing wi h any plasmonic nanos uc u e o a bi a y shape. Howe e ,
using Eq.
(1.57)
equi es he compu a ion o he op ical esponse o he plasmonic
nanos uc u e (e.g. using nume ical me hods [
112
–
115
]), which can be di icul in
many si ua ions. Mo eo e , he exac nume ical solu ion o he elec omagne ic
in e ac ion be ween QEs and MNPs is usually di icul o in e p e . In con as , a
simple model based on coupled ha monic oscilla o s, explained in his subsec ion,
can p o ide physical insigh in o such exci on–plasmon elec omagne ic in e ac ion
[42,175–178].
In his model, he LSPP in he MNP and he exci on in he QE a e each
desc ibed as damped ha monic oscilla o s [
42
]. These oscilla o s ep esen , o
example, he dipole momen induced a he MNP and a he QE, espec i ely, and
a e coupled h ough he elec ic nea ield induced by each s uc u e. The equa ions
o mo ion o he QE induced dipole momen ,
pQE
(
), and ha o he MNP,
pMNP
(
),
a e gi en unde ex e nal plane-wa e exci a ion
Eex
(
) =
ReE0e−iω 
in ime
domain by [42,178]
¨
pQE( ) + γQE ˙
pQE( ) + ω2
QEpQE( ) = α0
QEEex ( ) + βMNPpMNP( )
¨
pMNP( ) + γMNP ˙
pMNP( ) + ω2
MNPpMNP( ) = α0
MNPEex ( ) + βQEpQE( ),
(1.59)
whe e
ωMNP
,
γMNP
and
α0
MNP
a e he esonan equency, in insic loss a e and
oscilla o s eng h associa ed wi h he single LSPP mode o he MNP, espec i ely.
Only a single LSPP mode is conside ed in he MNP, so ha a gene aliza ion o
he model would be equi ed o simul aneously accoun o he elec omagne ic
coupling be ween he exci on and mul iple LSPPs, such as he BDP and BQP
esonances suppo ed by he me allic dime shown in Figu e 1.7. In Eq.
(1.59)
, he
exci on–plasmon coupling is in oduced ia he elec ic nea ield induced by he
31
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
MNP (QE), βMNPpMNP (βQEpQE), a he posi ion o he QE (MNP).
The di ec exci a ion o he QE by he plane wa e is usually much weake han
he exci a ion by he elec ic nea ield o he MNP, and hus one can conside
α0
QEEex
(
) = 0 in Eq.
(1.59)
[
42
]. Mo eo e ,
pMNP
(
)is expec ed o be much
s onge han
pQE
(
). Unde hese assump ions, he exp ession o he app oxima ed
pola izabili y o he coupled sys em can be ob ained in he equency domain (using
pMNP( ) = Re pMNP(ω)e−iω and pQE( ) = Re pQE(ω)e−iω )
α(ω)≈α0
MNPω2
QE −ω2−iωγQE
ω2
MNP −ω2−iωγMNPω2
QE −ω2−iωγQE−4g2ω2,(1.60)
whe e we de ine he coupling s eng h
g
such ha 4
g2ω2
=
α0
QEα0
MNPβQEβMNP
.
The ac o 4 is in oduced so ha
g
can be di ec ly compa ed o he coupling
s eng h used in ca i y quan um-elec odynamics models [
179
]. This de ini ion
implies a dependence o
g
on
ω
,
g∝
1
/ω
. Howe e , in p ac ice we conside
g
o
be a cons an (see below), consis en wi h he p ocedu e adop ed in he li e a u e
[
42
]. We expec ha an al e na i e assump ion o he de ini ion o
g
would no
subs an ially modi y he esul s.
The exp ession gi en by Eq.
(1.60)
can be use ul e.g. o es ima e he alue o
he coupling s eng h
g
be ween a QE exci on and a speci ic LSPP in he MNP.
This can be done by i ing he exac pola izabili y
α
(
ω
)o he QE–MNPs sys em
igo ously ob ained by using e.g. he me hodology explained in Subsec ion 1.4.2
o he app oxima ed esul ob ained om Eq.
(1.60)
. We ollow his p ocedu e
in Chap e 5, which allows us o iden i y whe he he s udied QE–MNPs sys em
is in he s ong-coupling egime. In ac , he c i e ia adop ed in he li e a u e o
iden i y he s ong-coupling egime a e in ima ely ela ed o he alue o
g
[
180
].
In b ie , he less demanding c i e ion s a es ha
g > |γMNP −γQE|/
4has o be
ul illed. O he c i e ia o en used a e
g >
(
γMNP
+
γQE
)
/
4and he mo e es ic i e
g > (γMNP +γQE)/2.
Figu e 1.8 shows he abso p ion spec a
σabs
(
ω
)[Eq.
(1.49)
] o a QE–MNPs
sys em ob ained by sol ing Eq.
(1.57)
wi hin he sel -in e ac ion G een’s unc ion
o malism (blue line) and he app oxima ed alue o
σabs
(
ω
)ob ained om
Eq.
(1.60)
using he coupled ha monic-oscilla o model ( ed do ed line). The
pa ame e s cha ac e izing he sys em a e he same as in Figu e 1.7. In he case o
he indi idual MNP (Figu e 1.8a), he coupling s eng h
g
be ween he DP and he
exci on is
g
= 18
meV <
(
γMNP −γQE
)
/
4
<
(
γMNP
+
γQE
)
/
4
<
(
γMNP
+
γQE
)
/
2,
which indica es ha he sys em is in he weak-coupling egime (he e
γMNP
=
γp
).
Fo he me allic dime (Figu e 1.8b), he coupling s eng h
g
be ween he BDP
and he exci on is
g
= 45
meV ≈
(
γMNP −γQE
)
/
4
≈
(
γMNP
+
γQE
)
/
4, hus he
sys em is nea ly in he s ong-coupling egime.
Fo si ua ions o s ong coupling, he coupled ha monic-oscilla o model depic ed
in his subsec ion can also be used o calcula e he equencies o he wo pola i ons
esul ing om he elec omagne ic in e ac ion be ween he exci on and he LSPP.
Once he alues o he coupling s eng h
g
is de e mined, as explained abo e, he
32
1.4. Plexci onics: Quan um emi e exci ons coupled o plasmons
+
Figu e 1.8:(a) Abso p ion c oss-sec ion spec a
σabs
(
ω
)o a QE–MNP coupled sys em ob ained
by using he sel -in e ac ion G een’s unc ion o malism explained in Subsec ion 1.4.2 (blue
line) and a coupled ha monic-oscilla o model (Eq.
(1.60)
, ed do ed line). The MNP has a
adius
a
= 5
nm
, and i is cha ac e ized by a D ude dielec ic unc ion wi h
ωp
= 5
.
89
eV
and
γp
= 0
.
21
eV
. Wi hin he coupled-oscilla o model
ωMNP
=
ωDP
= 3
.
4
eV
is used. The
QE, loca ed a 1
.
5
nm
om he su ace o he MNP, is cha ac e ized by an oscilla o s eng h
α0
QE
= 2
au
, an in insic loss a e
γQE
= 10
meV
, and a ansi ion equency
ωQE
=
ωDP
in
esonance wi h he DP o he MNP. The coupling s eng h is
g
= 18
meV
, and i is ob ained by
i ing he exac esul ob ained om Eq.
(1.57)
o he exp ession gi en by Eq.
(1.60)
.(b) Same
as in panel (a) bu o a MNP dime cha ac e ized by a gap sepa a ion
D
= 3
nm
. Wi hin he
coupled-oscilla o model
ωMNP
=
ωBDP
= 3
.
15
eV
is used. The QE emi e is a esonance wi h
he BDP mode o he dime ,
ωQE
=
ωBDP
, and he coupling s eng h is
g
= 45
meV
. In bo h
panels, γMNP =γpis used wi hin he coupled-oscilla o model.
equencies o he uppe (
ω+
) and he lowe (
ω−
) pola i ons can be ob ained by
sol ing he eigenmodes o Eq. (1.59) [179]
ω±=1
2(ωMNP +ωQE)±1
2Re

s4g2+ωMNP −ωQE +iγQE −γMNP
22


.
(1.61)
Acco ding o Eqs.
(1.61)
, in a esonan exci on–plasmon sys em (
ωMNP ≈ωQE
),
ω+
and
ω−
a e sepa a ed by a ac o 2
qg2−(γQE −γMNP)2/16
as long as
g > |γQE −γMNP|/
4, which jus i ies he less demanding c i e ion o s ong coupling
men ioned abo e. Mo eo e , o si ua ions whe e
g >> |γQE −γMNP|/
4, he
sepa a ion is 2
g
, leading o he so-called Rabi spli ing be ween he uppe and
lowe pola i ons in s ongly coupled sys ems [
181
–
183
]. In con as , in he weak-
coupling egime, whe e
g < |γQE −γMNP|/
4, he e is no spli ing be ween
ω+
and
ω−
, indica ing ha no pola i onic (hyb id) modes a e c ea ed as a consequence o
he elec omagne ic in e ac ion be ween he exci on and he LSPP.
33
Chap e 1. Classical desc ip ion o ligh –ma e in e ac ion
1.5 Summa y
In summa y, we ha e p esen ed in his chap e he classical iewpoin o plasmonic
exci a ions suppo ed by MNPs and hei elec omagne ic coupling o QE exci ons.
Fi s , we ha e e iewed Maxwell’s equa ions in he linea - esponse egime, and
ocused on he local- esponse app oxima ion (LRA) ha neglec s he spa ial
nonlocali y o he me al esponse. Nex , we ha e in oduced he non e a ded
app oxima ion adop ed in his hesis and p esen ed a me hod o ob ain he
op ical esponse o sphe ical indi idual nanopa icles and dime s. Then, a e
b ie ly desc ibing bulk and su ace plasmons, we ha e paid special a en ion o
he exci a ion o LSPPs in plasmonic nanos uc u es o ini e geome y such
as sphe ical MNPs and dime s. We ha e also in oduced a simple model o
unde s and he nonlinea op ical esponse om MNPs. Finally, we ha e p esen ed
he heo e ical app oaches used in his hesis o desc ibe he elec omagne ic
in e ac ion be ween QEs and MNPs wi hin a classical linea - esponse amewo k.
The concep s explained in his chap e hus p o ide he g ounds o classical ligh –
ma e in e ac ion and se e as a e e ence o s udy quan um phenomena ha a e
ou o he each o classical desc ip ions. In his hesis, quan um e ec s a ising
in he ligh –ma e in e ac ion a e s udied wi hin TDDFT as explained in he
ollowing Chap e 2.
34
Chap e
2
QUANTUM MANY-BODY
DESCRIPTION OF LIGHT–MATTER
INTERACTION
The classical heo e ical amewo k in oduced in Chap e 1can be used o
accu a ely desc ibe ligh –ma e in e ac ion when he cha ac e is ic dimensions o
he sys em, such as he size o he MNP o he gap sepa a ion in MNP ensembles,
a e ela i ely la ge. Howe e , when small MNPs (
≲
10
nm
) o ul a-na ow
gaps (
≲
1
nm
) a e conside ed, he quan um na u e o he elec ons dynamics
becomes impo an and classical desc ip ions a e no longe alid. In his con ex , a
me hodology capable o desc ibing he elec onic s uc u e om a quan um many-
body pe spec i e is equi ed, which is a conside able challenge [
184
]. Fo example,
o desc ibe he g ound s a e o a small sodium MNP con aining 1000 a oms (wi h
11 elec ons pe a om), one would ha e o sol e he Sch ödinge equa ion o a wa e
unc ion depending on 33000 spa ial a iables ( h ee spa ial a iables pe elec on
wi hou conside ing spin degene acy and neglec ing he deg ees o eedom o he
nuclei). Sol ing his emendously complex p oblem is ou o cu en compu a ional
capabili ies [185].
To educe he compu a ional complexi y o his many-body p oblem, one can
use densi y- unc ional heo y (DFT), a igo ous o malism ha deals wi h he
g ound-s a e elec on densi y a he han wi h he many-elec on wa e unc ion.
DFT can be applied only o s udy g ound-s a e elec onic p ope ies, so ha o
de e mine elec onic exci a ions in me als (and o he ma e ials) esul ing om ligh –
ma e in e ac ion i is necessa y o adop he ime-dependen ex ension o DFT,
he so-called ime-dependen densi y unc ional heo y (TDDFT). This app oach
add esses he ime e olu ion o he elec on densi y when he sys em is subjec ed o
a ime-dependen ex e nal po en ial. On he o he hand, a semiclassical app oach
35

Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
e e ed o as he su ace- esponse o malism (SRF) i s in oduced by Pe e
Feibelman in he 1980s [
98
] which inco po a es he pa ame e s
d⊥
(
ω
)and
d∥
(
ω
),
has p omp ed g ea in e es and p ac ical use in he Nanopho onics communi y
o e he las ew yea s o accoun o ce ain quan um many-body e ec s in he
op ical esponse o me als. The ad an age o using he SRF is ha i is much
simple compu a ionally as compa ed o TDDFT, and hus allows o s udying
quan um e ec s in la ge plasmonic nanos uc u es. Howe e , he SRF is a c ude
app oxima ion han TDDFT, and canno accoun o all quan um many-body
phenomena.
In Sec ions 2.1 and 2.2, we b ie ly ecall he undamen als o DFT and TDDFT,
and desc ibe he co esponding quan um many-body algo i hms based on he
wa e-packe p opaga ion (WPP) me hod employed in his hesis o add ess he
dynamics o he elec on densi y in ime domain. We apply his TDDFT amewo k
in Sec ions 2.3 and 2.4 o in oduce some o he main quan um many-body
phenomena mani es ed in he linea and nonlinea op ical esponse o canonical
plasmonic sys ems such as indi idual sphe ical and cylind ical MNPs and hei
dime s. Finally, we explain in Sec ion 2.5 he undamen als o he SRF used in
his hesis (complemen a y o TDDFT) o p o ide addi ional insigh s on he s udy
o pa icula quan um su ace e ec s on he op ical esponse.
2.1 Fundamen als o densi y unc ional heo y
(DFT)
Densi y- unc ional heo y (DFT) allows us o de e mine he g ound-s a e elec onic
p ope ies o an in e ac ing many-elec on sys em by only calcula ing he g ound-
s a e (o equilib ium) elec on densi y
n0
(
), i.e., wi hou he need o he exac
wa e unc ion sa is ying he many-elec on Sch ödinge equa ion. Acco ding o he
wo k by Hohenbe g and Kohn [
186
],
n0
(
)comple ely de e mines he g ound-s a e
ene gy,
E0
, and all o he elec onic p ope ies o he many-elec on sys em subjec ed
o an ex e nal ime-independen po en ial. This po en ial can be, o example, he
a ac i e Coulomb po en ial
Vion
(
)c ea ed by he posi i ely cha ged ions in a
me al, as we conside he e. In his case, he g ound-s a e ene gy
E0
o a me al can
be exp essed as a unique unc ional o n0( )(deno ed by he squa e b acke s),
E0[n0( )] = Zd n0( )Vion( ) + 1
2Z Z d d ′n0( )n0( ′)
| − ′|+G[n0( )],(2.1)
whe e he i s and second e ms on he igh -hand side (RHS) a e he ene gy
due o elec on–ion and elec on–elec on Coulomb in e ac ion in a me al, and
G
[
n0
(
)] is a uni e sal unc ional o he densi y
n0
(
) alid o any numbe o
elec ons and any po en ial
Vion
(
).
G
[
n0
(
)] accoun s o he kine ic ene gy and
exchange–co ela ion ene gy (associa ed e.g. wi h he Pauli exclusion p inciple)
o he in e ac ing many-elec on sys em. I
G
[
n0
(
)] we e known, de e mining he
g ound-s a e ene gy
E0
and elec on densi y
n0
(
)o a many-elec on sys em could
36
2.1. Fundamen als o densi y unc ional heo y (DFT)
be di ec ly ob ained by minimizing
E0
[
n0
(
)] ela i e o
n0
(
)acco ding o he
a ia ional p inciple [
186
]. Thus, he wo k by Hohenbe g and Kohn comple ely
changes he pa adigm o he elec onic many-body p oblem, since dealing wi h he
elec on densi y
n0
(
)as he undamen al quan i y ins ead o he he many-elec on
wa e unc ion allows he s udy o elec onic p ope ies o many-elec on sys ems
in ac able in he pas . No ice ha Eq.
(2.1)
is o mally exac . Un o una ely, he
uni e sal unc ional
G
[
n0
(
)] is gene ally unknown and, he e o e, DFT becomes in
p ac ice an app oxima ion. Conside able e o has been de o ed o e he yea s o
ind sui able app oxima ions o G[n0( )] [187–189].
Kohn and Sham [
190
] p oposed o exp ess
G
[
n0
(
)] as a sum o wo unc ionals,
G[n0( )] = Ts[n0( )] + Exc[n0( )],(2.2)
whe e
Ts[n0( )] = X
j∈occ Zd Ψ0
j( )∗ˆ
TΨ0
j( )(2.3)
is he kine ic ene gy o an auxilia y sys em o non-in e ac ing elec ons
6
and
Exc
[
n0
(
)] is he exchange–co ela ion ene gy ha con ains all emaining many-
body in e ac ions. In Eq.
(2.3)
,
ˆ
T
=
−1
2∇2
is he kine ic-ene gy ope a o , and
he summa ion uns o e he occupied (
j∈occ
) ime-independen Kohn-Sham
(KS) o bi als Ψ
0
j
(
) ha de e mine he equilib ium elec on densi y
n0
(
)o he
many-body sys em,
n0( ) = X
j∈occ
χj|Ψ0
j( )|2,(2.4)
wi h he s a is ical ac o s
χj
accoun ing o bo h spin and symme y degene acy.
No e ha , acco ding o he de ini ion o Eq.
(2.4)
,
n0
(
)is conside ed o be posi i e.
The applica ion o he a ia ional p inciple o Eq.
(2.1)
using he unc ionals
gi en by Eqs. (2.2) and (2.3) esul s in he ollowing equa ion [191]:
ˆ
T+Vion( ) + Zd ′n0( ′)
| − ′|+δExc[n0( )]
δn0( )
| {z }
ˆ
H0[n0( )]
Ψ0
j( ) = ϵjΨ0
j( ),(2.5)
whe e Ψ0
j( )and ϵja e he ime-independen KS o bi als and ene gies7.
Thus, he equilib ium elec onic densi y
n0
(
)[Eq.
(2.4)
] o he ue many-body
sys em can be ob ained wi hin he KS scheme om he solu ions o he Sch ödinge
equa ion [Eq.
(2.5)
] o an auxilia y sys em o non-in e ac ing elec ons using an
6Ts
is hus an explici unc ional o he Kohn–Sham o bi als Ψ
0
j
(
), bu an implici unc ional
o n0( )acco ding o Eq. (2.4) [72].
7
No e ha Ψ
0
j
(
)and
ϵj
in Eq.
(2.5)
a e he eigen unc ions and eigen alues o he auxilia y
non-in e ac ing elec on sys em used o cons uc he exac elec on densi y
n0
(
), and he e o e,
ϵj
and Ψ
0
j
(
)ha e no di ec physical in e p e a ion [
192
,
193
]. An excep ion is made o he
ene gy o he highes occupied KS o bi al, which can be used e.g. o es ima e he wo k unc ion
o a me al [194].
37
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
e ec i e ime-independen Hamil onian
ˆ
H0[n0( )] = ˆ
T+Ve [n0( )],(2.6)
whe e Ve [n0]is he e ec i e one-elec on po en ial gi en by
Ve [n0( )] = Vion( ) + Zd ′n0( ′)
| − ′|+Vxc[n0( )].(2.7)
In Eq.
(2.7)
, he second e m on he RHS is he elec on–elec on po en ial due o
Coulomb in e ac ion in he single-elec on pic u e, and
Vxc
[
n0
(
)] is he exchange–
co ela ion po en ial ob ained om Exc[n0( )] h ough he ela ionship
Vxc[n0( )] = δExc[n0( )]
δn0( ),(2.8)
which accoun s o all many-body in e ac ions ha a e no p esen in
ˆ
T
and in he
second e m o Eq.
(2.7)
, bu exis in he ue in e ac ing many-elec on sys em.
Acco ding o he p e ious discussion abou
G
[
n0
(
)], he exchange–co ela ion
ene gy unc ional
Exc
[
n0
]is no known exac ly. We explain in Subsec ion 2.1.1 he
app oxima ion adop ed in his hesis o calcula e Exc[n0( )].
Equa ions
(2.4)
and
(2.5)
a e e e ed o as he ime-independen KS equa ions,
and ha e o be sol ed sel -consis en ly: one can s a wi h an assumed elec on
densi y
n0
(
), hen cons uc
Ve [n0( )]
om Eq.
(2.7)
, and inally ob ain a
new alue o
n0
(
)using Eqs.
(2.5)
and
(2.4)
. This p ocedu e is epea ed un il
con e gence o he esul s is achie ed. In he ollowing Subsec ions 2.1.1 and
2.1.2, we desc ibe he app oxima ions used in his hesis o compu e he po en ials
Vxc[n0( )] and Vion( ).
2.1.1 The local-densi y app oxima ion (LDA)
The simples way o de e mine he exchange–co ela ion ene gy unc ional
Exc
[
n0
(
)]
in Eq.
(2.8)
is o adop he local-densi y app oxima ion (LDA). I we assume ha
n0
(
)does no change apidly, he a ia ion o
Exc
[
n0
(
)] wi h espec o he
g adien o n0( )can be neglec ed, and Exc[n0( )] can be exp essed as [62]
Exc[n0( )] = Zd n0( )ϵxc(n=n0( )),(2.9)
whe e
ϵxc
(
n
)is he exchange and co ela ion ene gy pe elec on o a homogeneous
elec on gas wi h a e age elec on densi y
n
[
190
]. Thus, wi hin he LDA, he
exchange–co ela ion ene gy pe pa icle loca ed a posi ion
in an inhomogeneous
sys em wi h densi y
n0
(
)is app oxima ed by he exchange–co ela ion ene gy pe
pa icle o a uni o m elec on gas wi h he same densi y,
n
=
n0
(
). The LDA has
been widely used o de e mine he elec onic p ope ies o many sys ems, including
a oms and solids whe e he densi y does no a y slowly. Fo me allic su aces,
38
2.1. Fundamen als o densi y unc ional heo y (DFT)
he LDA is ound o p o ide accu a e esul s, which migh be su p ising since he
equilib ium densi y n0( ) a ies apidly nea he me al su ace [62].
Using Eq.
(2.8)
, we ob ain he exchange–co ela ion po en ial wi hin he LDA
8
,
Vxc[n0( )] = ∂n0( )ϵxc(n=n0( ))
∂n0( )=ϵxc(n=n0( )) + n0( )∂ϵxc(¯n)
∂n n=n0( )
,
(2.10)
which equi es an analy ical exp ession o
ϵxc
(
n
). Se e al app oxima ions ha e
been p oposed o ha end [
191
,
195
–
197
], and h oughou his hesis, we use he
exchange–co ela ion ene gy-densi y unc ional
ϵxc
(
n
)gi en by Gunna sson and
Lundquis [198]9.
ϵxc(n) = −1
20.916/ s+ 0.0666(1 + x3)ln(1 + 1/x)−x2+x
2−1/3),(2.11)
wi h
s
=
3
4π¯n1/3
he Wigne –Sei z adius [Eq.
(1.11)
], and
x
=
s/
11
.
9. Finally,
acco ding o Eq.
(2.10)
, he exchange–co ela ion po en ial ha we use in his
hesis is gi en by [199]
Vxc[n0( )] = −1
21.222/ s+ 0.0666 ln(1 + 11.4/ s)n=n0( )
.(2.12)
2.1.2 The jellium model o ee-elec on me als
To ob ain he po en ial
Vion
(
)in Eq.
(2.7)
, we adop in his hesis he jellium
model o ee-elec on me als [
194
,
200
,
201
], whe e he ions a he la ice si es a e
modeled as a uni o m posi i e backg ound cha ge wi h densi y
n+( ) = 


ninside he me al
0ou side he me al
.(2.13)
Thus, wi hin he jellium model,
n
(o , equi alen ly,
s
[Eq.
(1.11)
]) is he only
pa ame e needed o cha ac e ize he me al. Fo example, Al is modeled wi h
s= 2.07 a0, Na wi h s= 4 a0, and K wi h s= 4.96 a0[62].
This posi i e cha ge densi y n+( )c ea es an a ac i e Coulomb po en ial
Vion( ) = −Zd ′n+( ′)
| − ′|.(2.14)
8
I can be shown ha , o a unc ional
F
[
ρ
(
)] o he o m
F
[
ρ
(
)] =
Rd  , ρ
(
)

,
he unc ional de i a i e is gi en by
δF
δρ( )
=
∂
∂ρ
. A demons a ion can be ound in
h ps://www.you ube.com/wa ch? =_n UQ_WBp0U.
9
We conside he spin-unpola ized case wi h
ξ
= 0. Mo eo e , no e ha in e .
198
he
exp ession o
ϵxc
(
n
)is gi en in Rydbe g a omic uni s (while we use Ha ee a omic uni s) and
hus i di e s om ou Eq. (2.11) by a ac o 1
2.
39
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
om he ime- o- equency Fou ie ans o m F[Eq. (1.3)],
p(ω) = ZT
0
d p( )eiω F( ),(2.36a)
Eind( , ω) = ZT
0
d Eind( , )eiω F( ),(2.36b)
j( , ω) = ZT
0
d j( , )eiω F( ),(2.36c)
whe e
T
is he o al p opaga ion ime used in ou simula ions, which mus be long
enough o achie e con e gence.
Impo an ly, he ime-dependen unc ion
F
(
)(o il e )inEq.
(2.36)
is
in oduced o a enua e he collec i e ime-dependen cha ge-densi y oscilla ions
(losses), since he ALDA-TDDFT scheme adop ed in his hesis does no accoun o
decay and dephasing p ocesses ela ed o he sca e ing o elec ons wi h phonons,
no o he in insic losses due o inelas ic elec on–elec on in e ac ions. This is
a well-known ailu e o he ALDA used o he exchange–co ela ion po en ials
Vxc
[
n
(
,
)] [
72
,
220
,
241
,
242
], and hus a il e
F
(
)has o be applied o mimic such
in insic losses in he sys em. In p ac ice, in his hesis we employ wo di e en
il e s
F
(
)depending on he ex e nal exci a ion, as desc ibed in Sec ions 2.3 and
2.4.
Calcula ion o he p ojec ed densi y o elec onic s a es (PDOS) wi hin
he WPP me hod
Gi en an equilib ium elec on densi y
n0
(
)and a g ound-s a e e ec i e po en ial
Ve
(
)o a sys em [Eq.
(2.17)
], i is possible o access he ene gies o bo h he
occupied and unoccupied KS one-elec on s a es o he sys em by analyzing he
p ojec ed densi y o elec onic s a es (PDOS), Σ(
ω
). Impo an ly, he PDOS can
be ob ained wi hin he WPP me hod desc ibed he e by p opaga ing an ini ial
wa e packe Φ(
,
= 0) = Φ
0
(
)o a pa icula symme y acco ding o he
ime-dependen Sch ödinge equa ion unde he ( ime-independen ) one-elec on
Hamil onian
ˆ
H
=
ˆ
T
+
Ve
(
). This Hamil onian co esponds o he Hamil onian
o a single elec on subjec ed o he e ec i e g ound-s a e po en ial
Ve
(
)o he
nanos uc u e unde s udy. In his case, using he WPP me hod o ob ain he
PDOS we do no apply any ex e nal po en ial
Vex
(
,
), and
Ve
(
)does no a y in
ime. Howe e , he PDOS p o ides in o ma ion abou he occupied and unoccupied
elec onic s a es ha will be in ol ed in elec onic ansi ions exci ed op ically
when an ex e nal ime-dependen po en ial is applied.
The PDOS Σ(ω)p ojec ed on o he ini ial wa e packe Φ0( )is gi en by
Σ(ω) = ∞
X
j=1 |cj|2δ(ω−ϵj),(2.37)
wi h
δ
(
ω−ϵj
) he Di ac del a,
ϵj
he eigenene gies o he one-elec on Hamil onian
46

2.2. Fundamen als o ime-dependen densi y unc ional heo y (TDDFT)
ˆ
H=ˆ
T+Ve ( ), and
cj=⟨ϕj( )|Φ0( )⟩(2.38)
he complex coe icien s co esponding o he expansion o he ini ial wa e packe
Φ0( )in o he eigen unc ions ϕj( )o he Hamil onian ˆ
H,
Φ0( ) = ∞
X
j=1
cjϕj( ).(2.39)
The eigen unc ions
ϕj
(
)a e no known a p io i, and we apply he WPP me hod
o ob ain Σ(ω)[Eq. (2.37)] wi hou he need o calcula e ϕj( ).
The PDOS Σ(
ω
)gi en by Eq.
(2.37)
ep esen s he numbe o one-elec on s a es
o a pa icula spa ial symme y ha he nanos uc u e sus ains a a gi en ene gy
le el
ω
and spa ial egion. The PDOS includes he con ibu ion o bo h he occupied
and he unoccupied s a es, and i gi es in o ma ion abou he deg ee o localiza ion
o a pa icula elec onic s a e a a ce ain spa ial egion (de e mined by he ini ial
wa e packe Φ
0
(
)used in he p opaga ion). No ice ha , since he PDOS shows
he one-elec on ene gy s a es a ailable in a ( ic i ious) non-in e ac ing KS elec on
sys em (see Sec ion 2.1), he esonan ene gies op ically exci ed a he in e ac ing
many-elec on sys em will be eno malized wi h espec o he non-in e ac ing one
ia Ha ee and exchange–co ela ion po en ials [
192
,
193
]. Howe e , he PDOS
s ill p o ides use ul insigh s in o he p ope ies o he elec onic s uc u e o he
sys em. Fo example, we s udy in Chap e 5 he PDOS in he elec onically coupled
QE–MNPs sys em o quan i y he deg ee o elec onic hyb idiza ion o he occupied
and unoccupied elec onic s a es.
In o de o calcula e Σ(
ω
)[
229
,
243
], we choose an ini ial wa e packe
Φ(
,
=0)=Φ
0
(
)o a pa icula symme y and p opaga e i acco ding o
he ime-dependen Sch ödinge equa ion co esponding o ˆ
H=ˆ
T+Ve ( ),
Φ( , ) = e−i(ˆ
T+Ve ( ))Φ0( ),(2.40)
by using he WPP algo i hm desc ibed abo e [Eqs.
(2.31)
and
(2.32)
]. No e ha no
ex e nal pe u ba ion is used in he WPP calcula ions o he PDOS,
Vex
(
,
) = 0,
and ha he sel -consis en p ocedu e is no applied so ha
Ve
(
) emains cons an
in ime.
Once he ime e olu ion o he ini ial wa e packe Φ(
,
)is ob ained by sol ing
Eq.
(2.40)
, o calcula e Σ(
ω
)we i s apply he ime- o- equency Laplace ans o m
ˆ
Lω o Φ( , ),
ˆ
LωΦ( , ) = Z∞
0
d ei(ω+iζ) Φ( , ) = Z∞
0
d ei(ω−ˆ
H+iζ) Φ0( ) = i
(ω−ˆ
H+iζ)Φ0( ),
(2.41)
whe e
ζ→
0
+
is a small posi i e numbe . Using Eq.
(2.39)
,Eq.
(2.41)
can be
47
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
w i en as
ˆ
LωΦ( , ) = i∞
X
j=1
cjϕj( )
(ω−ϵj+iζ).(2.42)
Then, p ojec ing Eq. (2.42) on o he ini ial wa e packe Φ0( )one ob ains
⟨Φ0( )|ˆ
LωΦ( , )⟩=i∞
X
j=1
|cj|2
(ω−ϵj+iζ).(2.43)
Finally, applying he Sokho ski–Plemelj heo em o Eq.
(2.43)
, he PDOS Σ(
ω
)
gi en by Eq. (2.37) can be exp essed as
Σ(ω) = 1
πlim
ζ→0+Re{⟨Φ0( )|ˆ
LωΦ( , )⟩},
=1
πlim
ζ→0+Re{ˆ
Lω⟨Φ0( )|Φ( , )⟩}
| {z }
A( )
,(2.44)
whe e
A
(
) =
⟨
Φ
0
(
)
|
Φ(
,
)
⟩
is he au oco ela ion unc ion ha can be calcula ed
in ime domain using he WPP me hod. No ice ha
cj
a e he coe icien s o
Φ
0
(
)decomposed in o he eigen unc ions
ϕj
(
)o he Hamil onian
ˆ
H
[Eq.
(2.38)
],
so ha choosing di e en ini ial wa e packe s Φ
0
(
)leads o a di e en PDOS
Σ(
ω
)[
244
]. This can be use ul o ocus on speci ic elec onic s a es wi h a special
symme y o spa ial dis ibu ion (as we do in Chap e 5), since only hese s a es o
he sys em ha a e no o hogonal o he ini ial wa e packe Φ
0
(
)can be accessed
by he WPP me hod used o calcula e he PDOS.
2.3 Linea op ical esponse o canonical
plasmonic nanos uc u es add essed wi hin
TDDFT
In his sec ion, we analyze TDDFT esul s o canonical sys ems o illus a e he
gene al p ope ies o he linea op ical esponse o plasmonic nanos uc u es. In
pa icula , we discuss in Subsec ion 2.3.1 he in luence o he nanopa icle size in
he op ical esponse o sphe ical MNPs desc ibed wi hin he jellium model, and
in oduce he quan um-mechanical concep s o F iedel oscilla ions, elec on spill
ou , and su ace-enabled Landau damping. In Subsec ion 2.3.2, we analyze he
e ec o he size and he gap sepa a ion in he op oelec onic esponse o sphe ical
MNP dime s, and in oduce he concep o elec on unneling. Finally, we s udy
in Subsec ion 2.3.3 he linea op ical esponse o indi idual cylind ical me allic
nanowi es and hei dime s, which allows us o s udy la ge nanos uc u es han
he sphe ical MNPs because he symme y o he sys em educes he compu a ional
demands.
48
2.3. Linea op ical esponse o canonical plasmonic nanos uc u es add essed
wi hin TDDFT
2.3.1 Indi idual sphe ical me allic nanopa icles
In his subsec ion, we conside indi idual sphe ical MNPs su ounded by acuum.
The elec onic s uc u e o he sys em is desc ibed wi hin he jellium model o
ee-elec on me als (see Subsec ion 2.1.2) using a Wigne –Sei z adius
s
= 4
a0
ha co esponds o sodium. We conside closed-shell MNPs, and he numbe o
conduc ion elec ons
Ne
is a ied o s udy he in luence o he MNP size in he
op ical esponse. The adius o he MNPs, a, is de e mined om
a=N1/3
e s.(2.45)
Be o e showing he esul s, we p o ide he main nume ical de ails o ca y ou he
TDDFT simula ions.
Nume ical implemen a ion
To ob ain he g ound-s a e KS o bi als o indi idual sphe ical MNPs con aining
Ne
conduc ion elec ons using DFT, we ake ad an age o he sphe ical symme y
o he p oblem and sol e he ime-independen KS equa ions gi en by Eqs.
(2.4)
and
(2.5)
in a sphe ical coo dina e sys em
=
{ , θ, φ}
(Figu e 1.2a). We w i e
he ime-independen KS o bi als as
Ψ0
j ,ℓ,m( )≡Ψ0
j ,ℓ,m( , θ, φ) = 1
ψ0
j ,ℓ( )Ym
ℓ(θ, φ),(2.46)
whe e he sequence o he adial (
j
= 1
,
2
, . . .
)and angula (
ℓ
= 0
,
1
, . . .
)quan um
numbe s is limi ed by he condi ion
ϵj ,ℓ ≤EF
, whe e
ϵj ,ℓ
is he ene gy o he
g ound-s a e KS o bi al (see below) and
EF
is he Fe mi ene gy. The magne ic
quan um numbe can ake he alues
m
= (
−ℓ, . . . ,
0
, . . . , ℓ
). In Eq.
(2.46)
, he
adial pa
ψ0
j ,ℓ
(
)sa is ies he ollowing one-dimensional Sch ödinge -like equa ion
[see Eqs. (2.5) and (1.24)]:
−1
2
d2
d 2+ℓ(ℓ+ 1)
2 2+Ve [n0( )]
| {z }
ˆ
H0[n0( )]
ψ0
j ,ℓ( ) = ϵj ,ℓψ0
j ,ℓ( ),(2.47)
wi h
Ve
[
n0
(
)] gi en by Eq.
(2.17)
, and he equilib ium elec on densi y
n0
(
)is
gi en by
n0( )=2 X
j ,ℓ∈occ
1
4π
2ℓ+ 1
2|ψ0
j ,ℓ( )|2.(2.48)
In Eq.
(2.48)
, he ac o 2is due o he spin, and
(2ℓ+ 1)
due o he degene acy o
he KS o bi al wi h he same angula quan um numbe ,
ℓ
, and di e en magne ic
quan um numbe ,
m
. Because o he sphe ical symme y o he p oblem,
n0
(
)
and Ve [n0( )] depend only on he adial coo dina e .
The Ha ee po en ial
VH
[
n0
(
)] con ained in
Ve
[
n0
(
)] [Eq.
(2.17)
] is calcula ed
om Poisson’s equa ion [Eq.
(2.16)
] by de ining
˜
VH
[
n0
(
)] =
VH
[
n0
(
)]. This
49
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
de ini ion allows us o ob ain he Ha ee po en ial om
d2
d 2˜
VH[n0( )] = − 4π(n0( )−n+( )),(2.49)
which can be sol ed by di ec ly applying he in e se ma ix o ope a o
d2
d 2
o he
RHS o Eq. (2.49) using space- o-momen um Fou ie sine ans o m.
As desc ibed in Sec ion 2.1, he ime-independen KS equa ions gi en by
Eqs.
(2.47)
and
(2.48)
a e sol ed sel -consis en ly using an i e a i e p ocedu e
by diagonaliza ion o he Fou ie g id Hamil onian (FGH) [
237
] ob ained om
H0
[
n0
(
)] [Eq.
(2.47)
], whe e he KS o bi als
ψ0
j ,ℓ
(
)a e ep esen ed in a eal-space
mesh o equidis an poin s in he coo dina e
, and he space- o-momen um Fou ie
sine ans o m is used o compu e he ope a o
d2
d 2
[
230
,
232
]. A he i s i e a ion,
he e ec i e po en ial is ini ialized o
Ve ( ) = V0
e
e−( −a−1)
1 + e−( −a−1) ,(2.50)
whe e
V0
e
is a (nega i e- alued) pa ame e ha de e mines he dep h o he
po en ial. Mo e de ails on he sel -consis en p ocedu e can be ound in e . 245.
Once he g ound-s a e KS o bi als a e calcula ed using DFT, we ob ain he
linea op ical esponse o he sphe ical MNP wi hin TDDFT by applying he
WPP me hod desc ibed in Subsec ion 2.2.1. We conside he e he ollowing ime-
dependen ex e nal po en ial,
Vex ( , ) = δ( )E0∆ cos θ, (2.51)
which ep esen s a pe u ba ion a he ini ial ime
= 0 wi hin he dipole
app oxima ion co esponding o a plane-wa e elec ic ield pola ized along he
z
-axis. In Eq.
(2.51)
,
δ
(
)is he Di ac del a unc ion,
E0
is he ampli ude o he
ex e nal pe u ba ion (we ypically use
E0∼
10
−5au
, weak enough so ha he
linea - esponse app oxima ion holds), and ∆
is he p opaga ion ime s ep used in
ou simula ions, ypically ∆
∼
0
.
25
−
0
.
1
au
(1
au ≈
2
.
419
×
10
−2 s
, see Appendix
A).
The ex e nal po en ial gi en by Eq.
(2.51)
b eaks he sphe ical symme y o he
sys em bu p ese es he o a ional symme y wi h espec o he
z
-axis (azimu hal
symme y), and he e o e he magne ic quan um numbe
m
is s ill a good quan um
numbe . I is hus con enien o exp ess he ime-dependen KS o bi als as15
Ψj,m( , θ, φ, ) = 1
ψj,m( , θ, )1
√2πeimφ,(2.52)
whe e, he e, he quan um numbe
j
eplaces he pai
{j , ℓ}
de ining he g ound-
15
In his hesis, all he conside ed ex e nal po en ials
Vex
(
,
)ac ing on sphe ical MNPs
p ese e he o a ional symme y o he sys em and hus we can always w i e he KS o bi als o
sphe ical MNPs using Eq. (2.52) and ollow he p ocedu e desc ibed in his subsec ion.
50
2.3. Linea op ical esponse o canonical plasmonic nanos uc u es add essed
wi hin TDDFT
s a e KS o bi al. The o bi als
ψj,m
(
, θ,
)in Eq.
(2.52)
a e ob ained om he
ime-dependen KS equa ions [Eq.
(2.19)
], wi h he kine ic-ene gy ope a o
ˆ
T
aking he o m [Eq. (1.24)]
ˆ
T=−1
2∂2
∂ 2+1
21
sinθ
∂
∂θ sin θ∂
∂θ−m2
sin2θ.(2.53)
The ime e olu ion o
ψj,m
(
, θ,
)and
ψj,−m
(
, θ,
)is iden ical, and he e o e we
only p opaga e
ψj,m
(
, θ,
) o
m≥
0. We can hus exp ess he ime-dependen
elec on densi y as
n( , θ, )X
{j,m≥0}∈occ
1
2πχm
1
2|ψj,m( , θ, )|2,(2.54)
wi h
χm=


2 o m= 0
4 o m > 0
(2.55)
accoun ing o bo h he spin and
±m
degene acy. The o bi als
ψj,m
(
, θ,
)a e
p opaga ed in ime by adap ing he WPP scheme gi en by Eq.
(2.32)
o he use
o sphe ical coo dina es. We ep esen he KS o bi als
ψj,m
(
, θ,
)in a meshg id
ex ending in adial di ec ion up o 25
−
35
a0
om he jellium edge, using a cons an
adial spacing o ∆
∼
0
.
35
−
0
.
5
a0
. The angula a iable
θ
is disc e ized om 0
o πusing 60 −120 poin s. In pa icula , he ope a o
e−i∆ −1
2
∂2
∂ 2
linked o he i s e m on he RHS o Eq.
(2.53)
is applied o
ψj,m
(
, θ,
)in he
ecip ocal space using space- o-momen um Fou ie sine ans o ms. On he o he
hand, he ope a o
e−i∆ −1
2 21
sin θ
∂
∂θ (sin θ∂
∂θ )−m2
sin2θ
linked o he second and hi d e ms on he RHS o Eq.
(2.53)
is applied in eal
space by expanding
ψj,m
(
, θ,
)in a basis o associa ed Legend e polynomials
Pm
ℓ
(
cosθ
), aking ad an age o he ac ha
Pm
ℓ
(
cosθ
)a e he eigen unc ions o
he ope a o
n1
sin θ
∂
∂θ sin θ∂
∂θ −m2
sin2θo
. This p ocedu e is desc ibed in de ail in
e . 75.
The linea op ical esponse o he MNP is analyzed by calcula ing i s abso p ion
c oss-sec ion spec um,
σabs
(
ω
) =
4π
cIm{α
(
ω
)
}
[Eq.
(1.49)
], wi h
α
(
ω
) he
pola izabili y o he sys em calcula ed om α(ω) = 1
E0∆ p(ω)16. p(ω)is ob ained
using he ime- o- equency Fou ie ans o m gi en by Eq.
(2.36a)
by conside ing
16 p
(
ω
)is he equency-dependen dipole momen induced along he
z
-axis, ob ained acco ding
o Eqs. (2.33) and (2.36a).
51

Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
2.5 3 3.5 4
Figu e 2.1:(a) Time e olu ion o he dipole momen
p
(
)induced a a sphe ical sodium MNP
(
s
= 4
a0
) con aining
Ne
= 1074
elec ons
in esponse o he ex e nal exci a ion gi en by
Eq.
(2.51)
. The esul s di ec ly ob ained om TDDFT wi hin he WPP me hod (g ay line) a e
damped by using di e en alues o he a enua ion ac o
η
[Eq.
(2.56)
], as indica ed in he
legend. (b) Abso p ion c oss-sec ion spec a
σabs
(
ω
)o he same MNP as in panel (a) ob ained
o di e en alues o he a enua ion ac o
η
. Using di e en alues o he ac o
η
(wi h
η < κ
)
leads o e y simila wid hs o he DP esonance,
κ
. The esul s co esponding o
η
= 0 a e
di ided by 2 o cla i y.
in his case he il e
F( ) = e−η/2 ,(2.56)
wi h an a enua ion ac o
η∼
0
.
05
−
0
.
2
eV
smalle han he ypical plasmon
esonance wid h (as de e mined e.g. h ough he classical calcula ions in Chap e
1,γpin Eq. (1.10)).
We show in Figu e 2.1a he e ec o he il e
F
(
)gi en by Eq.
(2.56)
in
he esul o he ime-dependen dipole momen
p
(
)induced a a sphe ical MNP
con aining
Ne
= 1074 conduc ion elec ons. Wi hou applying he il e (
η
= 0, g ay
line),
p
(
)oscilla es in ime showing e i als because, as discussed in Subsec ion
2.2.1, he ALDA-TDDFT model adop ed in his hesis does no include in insic
dissipa ion p ocesses in he ee-elec on gas. When an a enua ion ac o
η >
0
is applied,
p
(
)is exponen ially damped, hus accoun ing o he losses o he
me al phenomenologically. This a enua ion ac o
η
also a ec s d ama ically he
abso p ion c oss-sec ion spec um
σabs
(
ω
)o he sys em as illus a ed in Figu e
2.1b. Indeed, o
η
= 0 he plasmon peak in
σabs
(
ω
)a a ound
ω
= 3
.
2
eV
is
agmen ed in o a se o disc e e lines associa ed wi h he single elec on–hole
exci a ions ha build up he plasmon
17
(see below) [
206
,
246
]. The inclusion o
η
b oadens hese single elec on–hole exci a ion peaks and allows us o e ie e he
b oad Lo en zian-like p o ile o he plasmon esonance simila ly o he classical
17
The wid h o he peaks associa ed o single elec on–hole exci a ions in Figu e 2.1b o
η
= 0
is due o he ini e calcula ion ime
T
, whe e in his case we use
T ∼
13000
au
. Howe e , we
ypically use
T ∼
3000
−
4000
au
in his hesis o linea - esponse calcula ions, enough o achie e
con e gence when using η= 0.05 −0.2eV.
52
2.3. Linea op ical esponse o canonical plasmonic nanos uc u es add essed
wi hin TDDFT
p edic ion shown in Figu e 1.4d. We employ he il e gi en by Eq.
(2.56)
in
Chap e 3, Chap e 4, and Chap e 5, whe e we s udy he linea op ical esponse
o di e en plasmonic sys ems.
Figu e 2.2:(a) Equilib ium elec on densi y
n0
(
)(solid lines) and backg ound jellium denis y
n+
(
)(dashed illed lines) as a unc ion o he adial coo dina e,
, o sphe ical MNPs
cha ac e ized by a Wigne –Sei z adius
s
= 4
a0
(sodium) and con aining di e en numbe s o
conduc ion elec ons:
Ne
= 338 elec ons (pu ple), 638 elec ons (b own), 1074 elec ons (blue),
2260 elec ons (g een), and 4458 elec ons ( ed). Resul s a e no malized o he a e age elec on
densi y
¯n
=
4
3π 3
s−1
[Eq.
(1.11)
]. (b) Induced elec on densi y
δn
(
, ω
)(mul iplied by
2
)
along he adial axis
(
θ
= 0) a he DP equency
ω
=
ωDP
= 3
.
25
eV
o a sodium MNP
con aining 4458 elec ons in esponse o he ex e nal plane-wa e exci a ion gi en by Eq.
(2.51)
.
The dashed ed line ep esen s he posi ion o he jellium edge o he MNP. An a enua ion ac o
η= 0.2eV [Eq. (2.56)] is used o pe o m he Fou ie ans o m.
In luence o he size o small sphe ical nanopa icles on hei op ical
esponse
We show in Figu e 2.2a he equilib ium elec on densi y p o ile
n0
(
)ob ained o
sphe ical MNPs con aining di e en numbe o conduc ion elec ons wi hin he ange
Ne
= 338
−
4458, esul ing in a adius
a∼
1
.
5
−
3
.
5
nm
[Eq.
(2.45)
]. Quan um-
mechanical phenomena such as elec on spill-ou [
247
] and F iedel oscilla ions
[
248
,
249
] o he equilib ium elec on densi y
n0
(
)a e obse ed. The elec on spill
ou o
n0
(
)is a consequence o he ini e po en ial ba ie a he MNP su ace ha
allows elec ons o sp ead ou side he backg ound jellium edge loca ed a
=
a
.
Mo eo e , F iedel oscilla ions a e due o elec on e lec ion a
= 0 and a he MNP
bounda y, and can be unde s ood as a mani es a ion o he Gibbs phenomenon
occu ing in he Fou ie se ies o s ep-like unc ions: he elec on densi y
n0
(
)is
gi en by a summa ion o a ini e numbe o
ψ0
j ,ℓ
(
)o bi als [Eq.
(2.48)
] and hus
exhibi s oscilla ions along he adial coo dina e
. In addi ion,
n0
(
)in Figu e 2.2a
ea u es ei he a peak o a dip a
= 0 depending on he numbe o conduc ion
elec ons. While o
Ne
= 338 elec ons (pu ple), 638 elec ons (b own) and 1074
53
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
elec ons (blue) he e is a dip a
= 0, o
Ne
= 2260 elec ons (g een) and 4458
elec ons ( ed) he e is a peak. This beha io is ela ed o he ela i e con ibu ion
o he
ψ0
j ,ℓ
(
)o bi als wi h
ℓ
= 0 (which depends upon he numbe o occupied
closed shells), since only hose wi h
ℓ
= 0 can con ibu e o he elec on densi y
n0
(
)p ecisely a
= 0 and only low-
ℓ
o bi als
ψ0
j ,ℓ
(
)con ibu e close o he
cen e o he MNP because o he cen i ugal po en ial
ℓ
(
ℓ
+ 1)
/ 2
[Eq.
(2.47)
] ha
o bids elec ons wi h high angula quan um numbe
ℓ
o app oach he cen e o
he MNP.
Figu e 2.2b shows he elec on densi y
δn
(
, ω
)induced a he dipola plasmon
(DP) equency
ωDP
= 3
.
25
eV
in a sphe ical MNP (
Ne
= 4458 is conside ed
as an example) in esponse o an ex e nal plane-wa e exci a ion. The induced
densi y
δn
(
, ω
)is p edominan ly loca ed nea he su ace o he MNP as expec ed
om classical desc ip ions. Howe e , due o he nonlocal dynamic sc eening o
conduc ion elec ons desc ibed wi hin he jellium model [
58
,
126
],
δn
(
, ω
)sp eads
beyond he limi s o he classical sha p edge a
=
a
(no o be con used wi h he
spill ou o he equilib ium elec on densi y n0( )shown in Figu e 2.2a).
0
100
200
300
400
500
2.5 3 3.5 4 2.5 3 3.5 4
Figu e 2.3:(a) TDDFT esul s o he abso p ion c oss-sec ion spec a
σabs
(
ω
)o sphe ical
MNPs wi h di e en numbe o conduc ion elec ons,
Ne
=338
(pu ple),
638
(b own),
1074
(blue),
2260
(g een),
and 4458
( ed)
. The elec onic s uc u e o he MNPs is desc ibed
wi hin he jellium model using a Wigne –Sei z adius
s
= 4
a0
ha co esponds o sodium.
An a enua ion pa ame e
η
= 0
.
07
eV
is used [Eq.
(2.56)
]. (b) Classical LRA esul s
o
σabs
(
ω
) o sphe ical MNPs wi h di e en adius
a
, as ob ained om Eq.
(1.45)
.
a
=
27
.
9
a0(pu ple),
34
.
4
a0(b own),
40
.
96
a0(blue),
52
.
49
a0(g een),
and 65
.
83
a0( ed)
. The alue
o he adius
a
is de e mined acco ding o Eq.
(2.45)
. A D ude dielec ic unc ion [Eq.
(1.10)
]
wi h
ωp
= 5
.
63
eV
and
γp
= 0
.
175
eV
is used o all MNP sizes, which ep oduces he TDDFT
da a o he la ges MNP ( ed cu e).
We nex show in Figu e 2.3 he abso p ion c oss-sec ion spec a
σabs
(
ω
)o he
sphe ical sodium MNPs. The TDDFT esul s (panel a) illus a e ha quan um
54
2.3. Linea op ical esponse o canonical plasmonic nanos uc u es add essed
wi hin TDDFT
ini e-size e ec s [
250
–
252
] b eak he in a iance wi h pa icle size o he spec al
p o ile o
σabs
(
ω
)p edic ed by classical (non e a ded) LRA calcula ions (panel
b). Fi s , dec easing he size o he MNP wi hin TDDFT p oduces a edshi o
he DP esonance
18
om
ωDP ∼
3
.
25
eV
( ed, lowe spec um) o
ωDP ∼
3
eV
(pu ple, uppe spec um), in con as o he size-in a ian
ωDP
=
ωp/√3≈
3
.
25
eV
classical LRA alue
19
(see Subsec ion 1.2.3). This edshi is a consequence o
he spill ou o he induced cha ges due o nonlocal dynamic sc eening shown in
Figu e 2.2b. One can in e p e in an in ui i e pic u e ha he size o he MNP
is e ec i ely inc eased [
194
], which o he same numbe o elec ons educes he
elec on densi y and, hus, also he classical alue o
ωp
[Eq.
(1.12)
] and
ωDP
. The
impac o his spill ou , and hus he edshi , is mo e signi ican he smalle he
MNP is [96,256,257].
Fu he , he wid h o he DP esonances,
κ
, ob ained wi h TDDFT inc eases
wi h dec easing he size o he MNP [
258
]. Fo small-sized MNPs, as conside ed
in Figu e 2.3, he wid h o he plasmon esonance
κ
ob ained wi hin TDDFT is
de e mined by su ace-enabled Landau damping [
64
,
66
,
194
,
259
–
261
], consis ing
in he plasmon decay in o single elec on–hole exci a ions caused by he sca e ing
a he MNP su ace (see also Figu e 2.1b). Indeed, he MNP su ace p o ides
he momen um equi ed o elec ons o exci e an in aband ansi ion wi hin he
conduc ion band and c ea e an elec on–hole pai [
262
,
263
], which is o bidden in
he bulk due o momen um conse a ion. This quan um su ace e ec gains mo e
impo ance wi h dec easing he adius
a
o he MNP. Indeed, he plamon esonance
wid h associa ed wi h Landau damping scales as
∼a−1
[
264
–
267
] (see Subsec ion
2.5.1). Finally, he agmen ed shape o
σabs
(
ω
)ob ained o he smalles MNP
conside ed in Figu e 2.3a (pu ple,
Ne
= 338 elec ons) is also a consequence o
su ace-enabled Landau damping. Fo such small MNPs he ene gy di e ence
be ween di e en single elec on–hole ansi ions is la ge han he b oadening
η
= 0
.
07
eV
[Eq.
(2.56)
] accoun ing o dissipa ion p ocesses in he sys em [
268
].
These single-elec on ea u es g adually disappea wi h inc easing he numbe o
conduc ion elec ons, because he spec ally close elec on–hole ansi ions me ge
wi h each o he . In his si ua ion, a Lo en zian esonance p o ile o
σabs
is ob ained,
and hus he sys em is said o exhibi a “be e -de eloped” plasmonic beha io
wi h inc easing size [203,269].
2.3.2 Dime s o sphe ical me allic nanopa icles
We nex conside a dime composed by wo iden ical sphe ical me allic nanopa icles
(MNPs) o adius
a
, as schema ically shown in Figu e 2.4. The elec onic s uc u e
o he MNPs is desc ibed wi hin he jellium model by conside ing a closed-shell
con igu a ion as in oduced in Subsec ion 2.3.1. The gap sepa a ion be ween he
18
This is no he case o noble me als such as Au o Ag, whe e in e band ansi ions in ol ing
d
-band elec ons gi e ise o a blueshi o
ωDP
ins ead o a edshi wi h dec easing size o he
MNP [80,253–255].
19
In Figu e 2.3b we use
ωp
= 5
.
63
eV
so ha he alue o
ωDP
ob ained classically coincides
wi h ha ob ained in Figu e 2.3a wi hin TDDFT o he la ges MNP.
55
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
wi h
j, m, q
he quan um (in ege ) numbe s ha de ine a KS s a e. The ime-
independen KS equa ion is gi en in his coo dina es by [Eq. (2.5)]
ˆ
H[n0(ρ)] −1
2
∂2
∂z2Ψ0
j,m,q(ρ, z, φ) = ϵj,m +1
22π
Lq2!Ψ0
j,m,q(ρ, z, φ),
(2.70)
whe e ϵj,m a e he eigen alues o he adial pa o he Hamil onian ˆ
H[n0(ρ)],
ˆ
T+VH[n0(ρ)] + Vxc[n0(ρ)]
| {z }
ˆ
H[n0(ρ)]
ψ0
j,m(ρ) = ϵj,mψ0
j,m(ρ),(2.71)
and he ρ-space kine ic-ene gy ope a o ˆ
Tis exp essed as
ˆ
T=−1
21
ρ
∂
∂ρ ρ∂
∂ρ−m2
ρ2.(2.72)
The equilib ium elec on densi y n0(ρ)is gi en by
n0(ρ)=2 X
{j,m,q}∈occ |Ψ0
j,m,q(ρ, z, φ)|2
= 2 X
{j,m,q}∈occ
1
2π|ψ0
j,m(ρ)|21
LX
q
Θ EF− ϵj,m +1
22π
Lq2!!,
(2.73)
whe e
EF
is he Fe mi ene gy, Θ ep esen s he Hea iside s ep unc ion, and he
ac o 2 is due o he spin degene acy. Because o he cylind ical symme y, he
±ms a es a e degene a e so ha we only sol e Eqs. (2.70) and (2.71) o m≥0.
The case o he in ini e cylind ical nanowi e can be ob ained by aking
L→ ∞
in Eqs.
(2.69)
-
(2.73)
. In such a si ua ion, he summa ion o e he quan um numbe
q
in Eq.
(2.73)
can be ans o med in o an in eg al o e he con inuous a iable
kz=2π
Lq, and n0(ρ)can be exp essed as
n0(ρ) = X
{j,m≥0}∈occ
χj,m|ψ0
j,m(ρ)|2,(2.74)
wi h he s a is ical ac o
χj,m
accoun ing o spin and
±m
degene acy (see
Subsec ion 2.3.1), as well as o he degene acy due o he elec on mo ion along
he z-axis,
χj,m =


1
π2p2(EF−ϵj,m) o m= 0
2
π2p2(EF−ϵj,m) o m>0
.(2.75)
Simila ly o he equilib ium densi y o sphe ical MNPs in Subsec ion 2.3.1
[Eq.
(2.48)
],
n0
(
ρ
)o he cylind ical nanowi e, gi en by Eq.
(2.74)
, only depends
upon he adial coo dina e
ρ
. The o bi als
ψ0
j,m
(
ρ
)o he indi idual nanowi e a e
he e o e ob ained om Eqs.
(2.71)
and
(2.74)
using a sel -consis en p ocedu e
62

2.3. Linea op ical esponse o canonical plasmonic nanos uc u es add essed
wi hin TDDFT
based on he KS scheme desc ibed in Subsec ion 2.3.1, whe e he Hamil onian
ˆ
H
[
n0
(
ρ
)] in Eq.
(2.71)
is diagonalized by exp essing
ˆ
T
[Eq.
(2.72)
] in eal space
wi h ini e di e ences [75,130].
Once he ime-independen KS o bi als
ψ0
j,m
(
ρ
)o he indi idual nanowi e
a e ob ained om Eqs.
(2.71)
and
(2.74)
, we use he WPP me hod desc ibed
in Subsec ion 2.2.1 and p opaga e in eal ime he KS o bi als
ψp
subjec ed o
an ex e nal ime-dependen po en ial. The ime-dependen po en ial
Vex
(
x, y,
)
depends on he (
x, y
)-coo dina es so ha he ansla ional in a iance o he sys em
along he
z
-axis is p ese ed. Fo he ime p opaga ion, we disc e ize
ψp
on
an equidis an mesh in Ca esian coo dina es,
ψp≡ψp
(
x, y,
), whe e he ini ial
condi ions
ψp
(
x, y,
= 0)
≡ψ0
j,m
(
ρ, φ
)a e gi en by he KS o bi als o he g ound
s a e. No e ha when exp essing he ime-dependen KS o bi als in Ca esian
coo dina es, he quan um numbe
p
eplaces he pai
{j, m}
used in cylind ical
coo dina es o he g ound s a e. One o he ad an ages o using Ca esian
coo dina es
ψp≡ψp
(
x, y,
) o he ime p opaga ion is ha i is possible o
di ec ly apply he same algo i hm as o he indi idual nanowi e o s udy e.g.
he op ical esponse o a pai o pa allel nanowi es (nanowi e dime , Figu e 2.6a)
unde he in luence o any ex e nal po en ial
Vex
(
x, y,
)depending on (
x, y
).
This algo i hm basically consis s in applying he pseudospec al FGH me hod
[
230
,
236
,
237
] o calcula e he kine ic-ene gy ope a o
ˆ
T
as well as he Ha ee
po en ial VH[n(x, y, )] [Eq. (2.23)].
We ob ain he op ical esponse o nanowi es o a spa ially-cons an
x
-pola ized
ex e nal elec ic ield o ampli ude
E0
, co esponding o he ollowing ex e nal
po en ial [see Eq. (2.51)]:
Vex (x, y, ) = δ( )E0∆ x. (2.76)
Mesh s eps o he o de o ∆
x
= ∆
y∼
0
.
5
a0
and a ime s ep o ∆
∼
0
.
1
au
a e
ypically used in his hesis o he cylind ical geome y. The KS o bi als
ψp
(
x, y,
)
e ol e in ime acco ding o he ime-dependen KS equa ions [Eq.
(2.19)
], wi h he
ime-dependen elec on densi y n(x, y, )exp essed in Ca esian coo dina es as
n(x, y, ) = X
k∈occ
χp|ψp(x, y, )|2.(2.77)
In Eq. (2.77), he s a is ical ac o s χpa e now gi en by
χp=2
πq2(EF−ϵp),(2.78)
whe e ϵpis he eigenene gy o he g ound-s a e KS o bi al ψp(x, y, = 0).
Op ical esponse o indi idual cylind ical me allic nanowi es and dime s
We show he e he TDDFT esul s o he linea op ical esponse o an indi idual
me allic nanowi e and nanowi e dime s [
59
,
119
,
273
–
276
] consis ing o
˜
N
= 240
a−1
0
63
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
0
200
400
600
800
1000
1200
1400
1600
1800
Figu e 2.6:(a) Geome y o he indi idual cylind ical nanowi e (le ) and he dime composed
by wo pa allel nanowi es ( igh ). The adius o he nanowi es is
Rc
, which is de e mined by
he numbe o conduc ion elec ons
˜
N
pe uni leng h in he
z
-di ec ion acco ding o Eq.
(2.68)
.
The cylinde s a e in ini e along he
z
-axis ( ansla ionally in a ian ), and a e sepa a ed by a gap
dis ance
D
along he
x
-axis in he dime con igu a ion. (b) Abso p ion c oss-sec ion spec um
σabs
(
ω
)
/L
pe uni leng h
L
in he
z
-di ec ion o an indi idual nanowi e cha ac e ized by a
Wigne –Sei z adius
s
= 3
.
02
a0
and
˜
N
= 240
a−1
0
(
Rc≈
94
.
3
a0
). (c) Same as in (b) bu o
he dime con igu a ion whe e he gap sepa a ion is a ied om
D
= 14
a0
o
D
= 40
a0
. An
a enua ion pa ame e η= 0.07 eV [Eq. (2.56)] is used o pe o m he Fou ie ans o m.
elec ons pe uni leng h in he
z
-di ec ion ( adius
Rc≈
94
.
3
a0
). We use a jellium
model wi h a Wigne –Sei z adius
s
= 3
.
02
a0
(
ωp≈
8
.
98
eV
) cha ac e is ic o
he conduc ion elec on densi y o gold. In addi ion, we in oduce a s abilizing
po en ial
21
[
216
] inside he me al so ha he wo k unc ion
WF
= 5
.
5
eV
o gold
is e ie ed [
277
]. No e ha , since he jellium model does no accoun o op ical
ansi ions in ol ing localized
d
-band elec ons (see Subsec ion 2.1.2), he plasmonic
esponse ob ained he e p esen s impo an di e ences wi h espec o he esul s
ha would be ob ained o a mo e exac model o gold. Howe e , his desc ip ion
o he me allic nanowi es s ill allows one o p edic elec on anspo p ope ies o
gold junc ions in o - esonan exci a ion condi ions, as implemen ed in ecen wo ks
[
77
,
210
,
278
] when s udying he elec on-cu en s dynamics induced by ul a as
21
Applying a s abilizing po en ial simply consis s in in oducing a cons an po en ial inside
he me al in he e ec i e po en ial gi en by Eq.
(2.17)
bo h in he g ound-s a e and in he
ime-dependen calcula ions. This p ocedu e does no in oduce any addi ional compu a ional
di icul y.
64
2.4. Nonlinea e ec s in he op ical esponse o sphe ical plasmonic nanopa icles
add essed wi hin TDDFT
elec omagne ic ields in plasmonic gaps. He e we a e in e es ed in he gene al
ends o he op ical esponse o hese sys ems, whe e simila physical e ec s as
desc ibed in Subsec ions 2.3.1 and 2.3.2 o sphe ical MNPs a e also expec ed o
be p esen .
Figu e 2.6b shows he abso p ion c oss-sec ion spec um
σabs
(
ω
)
/L
pe uni
leng h
L
in he
z
-di ec ion o an indi idual me allic nanowi e o adius
Rc≈
94
.
3
a0
(
≈
5
nm
). Due o he ela i ely la ge size o he nanowi e,
σabs
(
ω
)
/L
does
no exhibi single elec on–hole ansi ion ea u es and he plasmonic esponse
is well de eloped. A single peak associa ed wi h he DP esonance (
m
= 1) o
he nanowi e eme ges a
ωDP
= 6
.
24
eV
, sligh ly below he classical non e a ded
p edic ion,
ωDP
=
ωSP
=
ωp/√2
= 6
.
35
eV22
[
22
], as a consequence o he spill ou
o he induced cha ges (see Subsec ion 2.3.1).
On he o he hand, Figu e 2.6c displays he abso p ion spec um
σabs
(
ω
)
/L
o
a dime o med by wo pa allel nanowi es, iden ical o ha discussed in Figu e 2.6b.
The gap sepa a ion,
D
, anges om
D
= 14
a0
o
D
= 40
a0
(
D∼
0
.
75
−
2
.
1
nm
).
The o e all quali a i e beha io o
σabs
(
ω
)
/L
wi h educing gap dis ance
D
is almos
iden ical o ei he he classical LRA (Figu e 1.5a) o TDDFT (Figu e 2.5c) esul s
o he sphe ical MNP dime s, and can be unde s ood using he elec omagne ic
hyb idiza ion pic u e in oduced in Subsec ion 1.2.3 [
150
,
151
]. No e ha , due o
he azimu hal symme y o he indi idual nanowi es, he e he magne ic quan um
numbe
m
plays a simila ole as he mul ipole o de
ℓ
in sphe ical MNPs. Fo he
la ges gap sepa a ion,
D
= 40
a0
(
D∼
2
.
1
nm
), wo dis inc modes eme ges: a
BDP a
ωDP ∼
5
.
35
eV
o med om he hyb idiza ion o he DP modes (
m
= 1)
o he indi idual nanowi es, and a b oad pseudomode a
ωPSM ∼
6
.
2
eV
o med
by he hyb idiza ion o nea ly-degene a ed highe -o de modes (
m >
1). This
pseudomode is sligh ly edshi ed wi h espec o he su ace plasmon equency.
Fo smalle gap sepa a ions,
D∼
28
a0
,
σabs
(
ω
)
/L
exhibi s h ee well-de ined peaks
because ano he dis inc esonance co esponding o he BQP a
ωBQP ∼
5
.
9
eV
eme ges. The BDP and he BQP esonances edshi wi h educing
D
because o
he inc easing a ac i e in e ac ion be ween he cha ges o opposi e sign ac oss
he junc ion [126].
2.4 Nonlinea e ec s in he op ical esponse o
sphe ical plasmonic nanopa icles add essed
wi hin TDDFT
In p e ious sec ions, we ocused on he linea op ical esponse o MNPs, which
a e shown o s ongly enhance he s eng h o he inciden elec ic ield a
op ical equencies because o he exci a ion o plasmon esonances. Howe e ,
plasmonic nanos uc u es also exhibi an e icien nonlinea op ical esponse o
22
While he DP esonance in sphe ical MNPs (
ℓ
= 1) is classically a
ωDP
=
ωℓ=1
=
ωp/√3
wi hin he non e a ded app oxima ion, he DP esonance in cylind ical nanowi es (
m
= 1) is a
he su ace plasmon equency, ωDP =ωm=1 =ωSP =ωp/√2[22].
65
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
s ong illumina ion in ensi ies, hus being good candida es o he ab ica ion o
nanode ices based on nonlinea op ics [
36
]. One manne o heo e ically s udy he
nonlinea op ical esponse o MNPs is o adop he classical scheme in oduced
in Sec ion 1.3 wi h he use o he nonlinea hype pola izabili ies
α(n)
,ad hoc
pa ame e s ha cha ac e ize he nonlinea op ical esponse o he sys em. In his
sec ion, we show ha he eal- ime TDDFT app oach based on he WPP me hod
employed in p e ious sec ions o s udy he linea op ical esponse o plasmonic
s uc u es can also be used o analyze nonlinea e ec s in he op ical esponse
wi hou using ad hoc pa ame e s.
As an example, we conside he nonlinea op ical esponse o an indi idual
sphe ical MNP. We desc ibe he elec onic s uc u e o he MNP wi hin he jellium
model in oduced in Subsec ion 2.1.2 using a Wigne –Sei z adius
s
= 4
a0
o
sodium. We add ess a sphe ical MNP ha con ains 1074 conduc ion elec ons,
wi h a adius
a
= 40
.
96
a0
(
≈
2
.
2nm). The nume ical implemen a ion is almos
iden ical o he one used in Subsec ion 2.3.1. The only di e ence is ha o his
s udy o nonlinea e ec s, ins ead o using he ex e nal po en ial
Vex
(
,
)gi en by
Eq. (2.51), he e we use he ollowing one:
Vex ( , ) = E0 cos θcos(ω( − 0)) e−( − 0
σ)2
,(2.79)
which co esponds o he po en ial expe ienced by an elec on in e ac ing wi h an
inciden Gaussian lase pulse pola ized along he
z
-axis. The undamen al equency
o he ex e nal illumina ion
ω
= 1
.
585
eV
in his sec ion is hal o he DP equency
ωDP
= 3
.
17
eV
(see Figu e 2.3a), he du a ion o he pulse is
σ
= 5
×
2
π/ω
, and
he a i al ime o he pulse 0is 0= 5σ.
In o de o analyze he nonlinea op ical esponse o he indi idual MNP, we
calcula e wi hin TDDFT he ime e olu ion o he induced elec on densi y
δn
(
,
)
and ob ain he ime-dependen induced dipole momen ,
p
(
)[Eq.
(2.33)
], as well as
he elec ic nea ield (induced ield),
Eind
(
,
)[Eq.
(2.34)
], c ea ed by he MNP
in esponse o an inciden elec omagne ic pulse. The equency- esol ed quan i ies
a e hen ob ained om he ime- o- equency Fou ie ans o m gi en by Eq.
(2.36)
,
δn( ,Ω) = Zd δn( , )eiΩ e−( − 0
σ)2
,
p(Ω) = Zd p( )eiΩ e−( − 0
σ)2
,
Eind( ,Ω) = Zd Eind( , )eiΩ e−( − 0
σ)2
,
(2.80)
whe e in his case he il e F( )is gi en by
F( ) = e−( − 0
σ)2
.(2.81)
The Gaussian il e
F
(
)in oduced in Eq.
(2.81)
pa ially accoun s o decay and
dephasing p ocesses o he collec i e densi y oscilla ions ha a e no included in
he p esen ALDA-TDDFT app oach [
72
,
241
,
242
] (see Sec ion 2.2), and allows us
66
2.4. Nonlinea e ec s in he op ical esponse o sphe ical plasmonic nanopa icles
add essed wi hin TDDFT
Figu e 2.7: Nonlinea op ical esponse o he indi idual sphe ical MNP as calcula ed wi hin
TDDFT o an inciden
z
-pola ized Gaussian elec omagne ic pulse wi h undamen al equency
ω
= 1
.
585
eV
(hal o he equency o he MNP dipola plasmon
ωDP
= 3
.
17
eV
).The in ensi y
is
I0
= 10
8
W cm
−2
(dashed blue line) o
I0
= 10
10
W cm
−2
( ed line). Panel (a) shows he
squa e o he induced dipole momen
|p
(Ω)
|2
, and panel (b) he absolu e alue o he spec um
o he elec ic nea ield
|Eind
(
,
Ω)
|
induced a he
z
-axis a 18
a0
(
≈
0
.
95
nm
) om he MNP
su ace. The symbol Ω ep esen s he equency o he induced elec omagne ic ields in esponse
o he inciden illumina ion wi h undamen al equency ω.
o each con e gen spec al esponse a high-ha monic equencies. This app oach
is jus i ied because he undamen al equency is s ongly de uned om he DP
esonance o he MNP, so ha no elec on-densi y oscilla ion and high-ha monic
gene a ion is expec ed when he lase is swi ched o . Consis en ly, we apply a
Gaussian il e gi en by he en elope o he inciden pulse. Mo eo e , in Eq.
(2.80)
we use he symbol Ω(compa e Eq.
(2.80)
wi h Eq.
(2.36)
) o e e o he oscilla ion
equency o he elec omagne ic ields (and dipole momen s) induced by he MNP
due o he nonlinea op ical esponse o he ex e nal exci a ion oscilla ing a ω.
The nonlinea op ical esponse o he indi idual sphe ical MNP is displayed
in Figu e 2.7. In panel (a), we show he in ensi y spec um o he induced dipole
momen
|p
(Ω)
|2
, which is p opo ional o he powe o ligh emi ed o he a ield
[Eq.
(1.46)
]. In panel (b), we show he spec um o he induced nea ield
|Eind
(
,
Ω)
|
a he
z
-axis, a 18
a0
(
≈
0
.
95
nm
) om he MNP su ace. Resul s a e ob ained
o an inciden Gaussian elec omagne ic pulse wi h in ensi y
I0
= 10
8
W cm
−2
(
E0
= 4
.
8
×
10
−5au
, dashed blue line) and
I0
= 10
10
W cm
−2
(
E0
= 4
.
8
×
10
−4au
,
ed line), a e aged o e he du a ion o he pulse
σ
. The co esponding ene gy pe
inciden pulse is well below he documen ed damage h eshold o small MNPs [
279
–
281
]. The induced dipole momen
|p
(Ω)
|2
in Figu e 2.7a exhibi s only odd ha monics
n
= 1
,
3
,
5
. . .
. Thus, only odd mul iples Ω =
ω,
3
ω,
5
ω, . . .
o he incoming
equency
ω
a e emi ed by he sys em in o he a ield, consis en ly wi h he
in e sion symme y o he MNP ha p e en s e en-ha monic gene a ion [
152
,
282
]
(see Sec ion 1.3). O e all, he nonlinea esponse
|p
(Ω)
|2
o
I0
= 10
10
W cm
−2
is se e al o de s o magni ude la ge han ha o
I0
= 10
8
W cm
−2
. This la ge
inc ease is in acco dance wi h he
In
0
dependence o
|p
(Ω =
nω
)
|2
, expec ed om
he s anda d heo y o nonlinea op ics as desc ibed in Sec ion 1.3 [152].
In con as o he a - ield esponse, bo h odd and e en ha monics a e p esen in
67

Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
Figu e 2.8: Colo maps o he eal pa o he induced elec on densi y
δn
(
,
Ω) (le ), o he
adial componen o he elec ic nea ield
Eind
(
,
Ω) (cen e ), and o he angen ial componen
o
Eind
(
,
Ω) ( igh ) induced a he undamen al, second, hi d, and ou h-ha monic equency
by a
z
-pola ized Gaussian elec omagne ic pulse wi h undamen al equency
ω
= 1
.
585
eV
and
in ensi y
I0
= 10
10
W cm
−2
inciden a he indi idual sphe ical MNP. Resul s a e o a ionally
symme ic wi h espec o he
z
-axis, and hey a e shown in he (
x, z
)-plane no malized o uni y.
he spec um o he elec ic nea ield induced by he indi idual MNP (Figu e 2.7b).
Indeed, a he me al– acuum in e ace he in e sion symme y is locally b oken,
and sho - ange e en-ha monic elec ic ields can be induced close o he MNP
su ace [
283
–
287
]. As expec ed, he
|Eind
(
,
Ω)
|
is o de s o magni ude la ge o
I0= 1010 W cm−2( ed line) han o I0= 108W cm−2(blue).
The colo maps o he induced elec on densi y
δn
(
,
Ω) and o he elec ic
nea ield
Eind
(
,
Ω) induced by he inciden
z
-pola ized Gaussian elec omagne ic
pulse a e shown in Figu e 2.8 o he undamen al, second, hi d, and ou h
ha monics. The induced cha ge densi y
δn
(
,
Ω =
nω
)o he
n
- h ha monic
and he co esponding nea ield
Eind
(
,
Ω =
nω
)a e shown in he (
x, z
)-plane.
Because o he symme y o he con igu a ion, he calcula ed colo maps a e
68
2.5. Semiclassical su ace- esponse o malism (SRF)
independen o a o a ion a ound he
z
-axis. A odd ha monics (
n
= 1
,
3),
he induced cha ge densi ies a e an isymme ic wi h espec o he (
x, y
)-plane,
δn
(
x, y, z, nω
) =
−δn
(
x, y, −z, nω
), which esul s in a ne dipole momen (see
Figu e 2.7a). In con as , a quad upola -like nea ield and symme ic cha ge-
densi y,
δn
(
, nω
) =
δn
(
- , nω
), a e induced a e en ha monics (
n
= 2
,
4) [
288
–
294
].
The dipole momen
p
(Ω) is ze o in his case, and hus he e is no emission in o he
a ield a e en ha monics. Thus, despi e he second ha monic being a esonance
wi h he dipola plasmon o he MNP, he la e can no be exci ed because o he
symme y selec ion ules. Ano he consequence o he symme y selec ion ules
is ha , o any poin loca ed in he (
x, y
)-plane, he e en-ha monic nea ield in
ha poin is o ien ed pe pendicula ly o he
z
-pola ized inciden pulse. We use in
Chap e 6 he insigh s ob ained in his sec ion o s udy nonlinea e ec s when he
sphe ical MNP is coupled o a QE loca ed nea by, demons a ing ha he p esence
o he QE enables he emission in o he a ield a he second-ha monic equency,
o he wise o bidden because o he in e sion symme y o he MNP.
2.5 Semiclassical su ace- esponse o malism
(SRF)
In his sec ion, we in oduce he semiclassical su ace- esponse o malism (SRF)
employed in his hesis, which allows one o accoun o quan um su ace e ec s using
much less compu a ionally-demanding calcula ions as compa ed o he TDDFT
me hodology. In b ie , he SRF is an ex ension o he classical LRA (Chap e
1) ha inco po a es su ace- esponse co ec ions a he me al–dielec ic in e ace
in he solu ion o Maxwell’s equa ions (Sec ion 1.1) by means o he so-called
Feibelman pa ame e s ob ained om quan um-mechanical calcula ions [
95
–
98
].
These pa ame e s, commonly deno ed as
d⊥
and
d∥
, we e i s in oduced by Pe e
Feibelman in he 1980s [
98
], and ha e ecei ed enewed a en ion du ing he las
yea s [
97
,
124
,
129
,
276
,
295
–
302
] due o hei use ulness o s udy sys ems ha
ha e ecen ly become expe imen ally easible. Indeed, he semiclassical SRF based
on he Feibelman pa ame e s allows us o accoun o he g adual a ia ion o he
induced elec on densi y ac oss he me al– acuum in e ace (see Figu e 2.2b), in
con as o he classical LRA desc ibed in Chap e 1 ha conside s he pola iza ion
cha ges o be loca ed s ic ly a he me al bounda y o in ini esimal wid h.
The Feibelman pa ame e s
d⊥
and
d∥
a e usually de ined by conside ing a
semi-in ini e me al su ace [
54
,
55
,
62
,
98
], al hough he exp essions o
d⊥
and
d∥
ha e been also p oposed o o he geome ies such as sphe ical MNPs [
95
]
23
.
To ou knowledge, he Feibelman pa ame e s ha e been so a compu ed in he
li e a u e wi hin he long-wa eleng h app oxima ion, which consis s in neglec ing
he nonlocali y o he op ical esponse in he di ec ion pa allel o he me al–
dielec ic in e ace. Thus, wi hin he long-wa eleng h app oxima ion,
d⊥≡d⊥
(
ω
)
and
d∥≡d∥
(
ω
)solely depend on he exci a ion equency,
ω
, and no on he
23 In Chap e 4we analyze he Feibelman pa ame e d⊥ o a cylind ical me allic nanowi e.
69
Chap e 2. Quan um many-body desc ip ion o ligh –ma e in e ac ion
wa enumbe pa allel o he me al su ace,
k∥
. In his hesis we hus e e o he
k∥-independen d⊥(ω)and d∥(ω)as he nondispe si e Feibelman pa ame e s. We
adop he long-wa eleng h app oxima ion in his sec ion and in Chap e 3, whe e
we iden i y si ua ions whe e his app oxima ion becomes inaccu a e. In Chap e
4, we calcula e he Feibelman pa ame e s as a unc ion o bo h
ω
and
k∥
and
p opose a dispe si e SRF ha o e comes he sho comings o he long-wa eleng h
app oxima ion.
The nondispe si e Feibelman pa ama e s de ined o a semi-in ini e plana me al–
acuum in e ace a
z
= 0 (Figu e 1.3a) a e usually exp essed as [
62
,
81
,
124
,
296
]:
d⊥(ω) = Rdz z δn(z, ω)
Rdz δn(z, ω),(2.82a)
d∥(ω) = Rdz z ∂
∂z j∥(z, ω)
Rdz ∂
∂z j∥(z, ω),(2.82b)
wi h
δn
(
z, ω
) he quan um-mechanical (complex- alued) elec on densi y induced
in esponse o he ex e nal exci a ion, and
j∥
(
z, ω
) he pa allel- o- he-su ace
componen o he associa ed induced elec on cu en densi y. O he de ini ions
equi alen o Eq.
(2.82)
ha exp ess he pa ame e s in e ms o he elec omagne ic
ields and nonlocal dielec ic unc ions ha e been also used [
62
,
98
,
296
,
303
]. In
Eq.
(2.82)
, he eal pa o
d⊥
(
ω
)(
Re{d⊥
(
ω
)
}
) co esponds o he posi ion o he
cen oid o he induced cha ge densi y wi h espec o he posi i e backg ound edge
o he me als (see schema ic ep esen a ion in Figu e 2.9a), while he imagina y
pa (
Im{d⊥
(
ω
)
}
) is ela ed o su ace-enabled Landau damping [
62
] (see below).
On he o he hand,
Re{d∥
(
ω
)
}
ep esen s he posi ion o he cen oid o he no mal
de i a i e o he elec on cu en pa allel o he me al su ace.
The Feibelman pa ame e s
d⊥
(
ω
)and
d∥
(
ω
)a e su ace- esponse unc ions
inhe en o a speci ic me al, bu also dependen on he su ounding ma e ial
[
62
]. Impo an ly,
d∥
(
ω
)gi en by Eq.
(2.82b)
anishes o cha ge-neu al plana
su aces [
62
,
304
], and i is also expec ed o be much less impo an han
d⊥
(
ω
)
o cu ed su aces [
95
]. In his hesis we hus conside
d∥
(
ω
) = 0, consis en wi h
he app oxima ion adop ed in o he s udies [
81
,
124
,
296
,
300
]. We show in Figu e
2.9b he nondispe si e Feibelman pa ame e
d⊥
(
ω
)used in his sec ion and in
Chap e 3, ob ained in e .
296
o a sodium plana su ace (
s
= 4
a0
) su ounded
by acuum
24
. The Feibelman pa ame e
d⊥
(
ω
)in Figu e 2.9b shows a esonance
a
ω∼
4
.
7
eV
, associa ed wi h he exci a ion o he Benne plasmon a
ω∼
0
.
8
ωp
[
79
,
305
], also e e ed o as he mul ipole su ace plasmon [
62
,
306
] (no o be
con used wi h localized mul ipole plasmons suppo ed e.g. by sphe ical MNPs).
K ame s-K onig ela ions connec he eal (blue line) and imagina y ( ed line)
pa s o d⊥(ω)[307].
The ad an age o he SRF is ha , once he Feibelman pa ame e s a e ob ained
24
Re .
124
p o ide he pa ame iza ion o exp ess he da a o
d⊥
(
ω
)ob ained in e .
296
wi hin TDDFT as a sum o Lo en zian unc ions.
70
2.5. Semiclassical su ace- esponse o malism (SRF)
-6
-4
-2
0
2
4
6
8
10
Figu e 2.9:(a) Schema ic ep esen a ion o he nondispe si e Feibelman pa ame e
d⊥
(
ω
)in a
plana me al– acuum in e ace.
Re{d⊥
(
ω
)
}
co esponds o he posi ion o he cen oid o he
induced cha ge
δn
(
z, ω
)wi h espec o he classical me al su ace loca ed a
z
= 0.(b) Real
(blue) and imagina y ( ed) pa s o he nondispe si e Feibelman pa ame e
d⊥
(
ω
)ob ained in
e .
296
o a plana sodium– acuum in e ace. The pa ame e shown in panel (b) is used in his
sec ion and in Chap e 3, whe e we s udy quan um su ace phenomena in he in e ac ion be ween
QEs and MNPs.
o a plana su ace o a gi en ma e ial using quan um-mechanical me hods, hese
pa ame e s can in p inciple be applied o accoun semiclassically o quan um
su ace e ec s in he op ical esponse o a bi a y-shaped MNPs. Wi hin he
semiclassical SRF, he elec omagne ic p oblem is add essed by sol ing Maxwell’s
equa ions [Eq.
(1.5)
] wi h he use o local dielec ic unc ions (e.g. a D ude dielec ic
unc ion, Eq.
(1.10)
), and in oducing a se o
modi ied
bounda y condi ions a
he me al–dielec ic in e aces ha di e om hose used wi hin he classical LRA
[Eqs.
(1.16)
and
(1.18)
].
d⊥
(
ω
)and
d∥
(
ω
)can be ela ed, espec i ely, o a su ace
pola iza ion o ien ed pe pendicula ly o he in e ace and o a pa allel su ace
cu en [
97
,
296
] leading o he ollowing modi ied bounda y condi ions (see also
e . 124):
ˆ
n×ESRF
ou −ESRF
in =−d⊥(ω)ˆ
n×∇hˆ
n·ESRF
ou −ESRF
in i,(2.83a)
ˆ
n·DSRF
ou −DSRF
in =d∥(ω)∇·hˆ
n×DSRF
ou −DSRF
in ׈
ni,(2.83b)
ˆ
n×HSRF
ou −HSRF
in =iωd∥(ω)hˆ
n×DSRF
ou −DSRF
in ׈
ni,(2.83c)
ˆ
n·BSRF
ou −BSRF
in = 0,(2.83d)
whe e
ˆ
n
is he no mal uni ec o poin ing ou wa ds om he me al bounda y, and
he supe sc ip “SRF” deno es ha he ields a e calcula ed wi hin he semiclassical
SRF. No e ha , as men ioned abo e, we conside
d∥
(
ω
)=0, so ha in p ac ice
we only conside he modi ica ions in oduced by Eq.
(2.83a)
in o he bounda y
condi ions. The es o he bounda y condi ions emain he same as in he classical
71
Chap e 3. Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles
which in he weak-coupling egime de e mines he Pu cell ac o [
101
,
102
] and
Lamb shi [
103
,
104
] in he emission by he QE p oduced by he plasmonic
en i onmen . Typically, a local dielec ic unc ion ob ained expe imen ally o om
simple heo e ical app oaches, such as he D ude model (Subsec ion 1.1.1), can be
used o cha ac e ize he MNPs. Howe e , such a dielec ic unc ion does no accoun
o quan um phenomena ele an in MNPs o small cha ac e is ic dimensions such as
elec on spill-ou , su ace-enabled Landau damping, o nonlocal dynamical sc eening
[
10
,
61
,
64
,
66
,
96
,
126
,
209
,
212
,
219
], in oduced in Sec ion 2.3 when analyzing
he linea op ical esponse o MNPs. These nonclassical phenomena, inhe en
o he quan um na u e o elec ons in me als, a e also expec ed o in luence he
QE–MNPs elec omagne ic in e ac ion when small MNPs and QE–MNPs dis ances
a e conside ed [65,124,300,330–336].
In his chap e , we use he TDDFT app oach [
58
,
79
,
80
,
202
,
210
,
214
,
337
–
341
]
based on he WPP me hod in oduced in Chap e 2 o p o ide a undamen al
desc ip ion o he elec omagne ic coupling be ween QEs and canonical MNPs. The
QE–MNP sepa a ions a e se su icien ly la ge so ha elec on unneling is negligible
and hus he elec omagne ic in e ac ion mainly de e mines he op ical p ope ies
o he sys em. Howe e , we conside QE–MNP sepa a ions small enough o he
a o emen ioned quan um su ace e ec s o be impo an . We i s use TDDFT
o calcula e he sel -in e ac ion G een’s unc ion
ˆ
G
(
QE, QE, ω
) ha go e ns he
QE–MNPs coupling. A compa ison wi h classical LRA esul s (Subsec ion 1.1.1)
e eals he impo ance o quan um e ec s. Fu he , we compa e TDDFT esul s
and he semiclassical nondispe si e SRF (Sec ion 2.5) ha inco po a es quan um
su ace- esponse co ec ions ia he Feibelman pa ame e
d⊥
(
ω
)ob ained in he
long-wa eleng h app oxima ion, which allows us o iden i y su ace-enabled Landau
damping and spill-ou o he induced elec on densi y as he dominan quan um
mechanisms d ama ically in luencing he elec omagne ic QE–MNPs in e ac ion.
TDDFT also p o ides a benchma k o es ablish he alidi y ange o he
(s anda d) nondispe si e SRF o adequa ely accoun o he dominan quan um
phenomena a ising in he elec omagne ic in e ac ion be ween QEs and MNPs. The
nondispe si e SRF, as used in his chap e , neglec s he nonlocal op ical esponse in
he di ec ion pa allel o he me al su ace (long-wa eleng h limi ), enabling a e y
e icien implemen a ion o nonlocali y in nanoscale geome ies [
97
]. In his chap e ,
we iden i y si ua ions o e y small QE–MNP dis ances whe e he nondispe si e
implemen a ion o he SRF e en ually ails, indica ing ha he dispe sion o he
Feibelman pa ame e s wi h espec o he wa enumbe pa allel o he su ace needs
o be conside ed (see Chap e 4).
3.1 Sys em and me hods
We analyze he elec omagne ic coupling be ween a poin -like QE and wo di e en
canonical plasmonic nanos uc u es. Fi s , we conside in Subsec ions 3.2.1,3.2.2,
and 3.2.3 he case o an indi idual sphe ical MNP, whe e he QE is placed a a
dis ance
d
om he MNP su ace (Figu e 3.1a). Then, in Subsec ion 3.2.4, we
78

3.1. Sys em and me hods
Figu e 3.1: Ske ch o he sys ems s udied in his chap e , consis ing o a QE modeled as a poin
dipole
pd
placed (a) a a dis ance
d
om he su ace o an indi idual sphe ical MNP, and (b)
a he cen e o a gap o size
D
o med by wo iden ical sphe ical MNPs. The poin dipole is
o ien ed along he
z
-axis, which is also he axis o he dime . Each MNP is ep esen ed wi hin
he ee-elec on jellium model using a Wigne –Sei z adius
s
= 4
a0
ha co esponds o sodium,
and con ains Ne= 4458 conduc ion elec ons esul ing in a adius a= 65.83 a0(≈3.5nm).
analyze a dime composed by wo iden ical sphe ical MNPs sepa a ed by a gap
dis ance
D
(Figu e 3.1b), wi h he QE si ua ed a he cen e o he gap (
z
= 0).
We de ine he coo dina es in he same way as in Subsec ions 2.3.1 and 2.3.2, such
ha he cen e o he nanopa icle(s) and he QE is a he
z
-axis. The en i e
sys em is su ounded by acuum.
We conside closed-shell jellium MNPs o adius
a
= 65
.
83
a0
(
≈
3
.
5
nm
), which
esul s in a well-de eloped plasmonic esponse [
203
] in he TDDFT simula ions.
The su ace- o-emi e dis ance is su icien ly la ge o ensu e ha he elec on
densi ies o he MNPs a he posi ion o he QE a e negligible, and he e o e
he e is no elec on unneling [
43
,
67
,
68
,
83
,
86
,
342
]. Speci ically, we conside
su ace- o-emi e dis ances
d
in he ange o
d
= 10
−
42
a0
(
≈
0
.
5
−
2
.
2
nm
)
o he case o he indi idual MNP, and gap sepa a ions o
D
= 2
d
= 20
−
45
a0
(
≈
1
.
1
−
2
.
4
nm
) o he dime s uc u e. We use he poin -dipole app oxima ion
desc ibed in Subsec ion 1.4.1 o model he QE and hus neglec i s spa ial ex en .
We ocus on he s udy o he sel -in e ac ion G een’s unc ion
ˆ
G
(
QE, QE, ω
)
de ined acco ding o Eq.
(1.56)
, which p o ides he elec ic ield
Esel
(
QE, ω
)
c ea ed by he me allic nanos uc u e a a posi ion
QE
in esponse o a poin
dipole
pd
loca ed a he same posi ion and oscilla ing a a equency
ω
. Impo an ly,
ˆ
G
(
QE, QE, ω
)de e mines he o al decay a e (Γ) and he Lamb shi (∆
ωQE
) o
a QE ha in e ac s weakly wi h he plasmonic nanos uc u e [104,170–172],
Γ = γ0+γn
QE + 2|µQE|2Im{ˆ
k·ˆ
G( QE, QE, ω =ωQE)·ˆ
k},(3.1a)
∆ωQE =−|µQE|2Re{ˆ
k·ˆ
G( QE, QE, ω =ωQE)·ˆ
k},(3.1b)
wi h
γ0
,
γn
QE
and
µQE
he spon aneous decay a e in acuum, he non- adia i e
79
Chap e 3. Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles
in insic loss a e, and he ansi ion dipole momen o he QE along he
ˆ
k
-di ec ion,
espec i ely [
22
]. The sel -in e ac ion G een’s unc ion
ˆ
G
(
QE, QE, ω
)in Eq.
(3.1a)
and Eq.
(3.1b)
is e alua ed a he ansi ion equency o he QE,
ω
=
ωQE
. The
uni ec o
ˆ
k
de ines he o ien a ion o
pd
. We no e ha he enhancemen o he
decay a e due o he QE–MNPs coupling is o en no malized by
γ0
, which gi es
he Pu cell ac o
FP=Γ−γn
QE
γ0
= 1 + 3c3
2ω3
QE
Im{ˆ
k·ˆ
G( QE, QE, ω =ωQE)·ˆ
k},(3.2)
cbeing he speed o ligh in acuum [102,343].
We es ic ou analysis o he case o a poin dipole o ien ed along he
z
-axis,
pd
=
pdˆ
z
, whe e
ˆ
z
is he uni ec o along he
z
-axis. Mo eo e ,
we use
QE
=
QE ˆ
z
, so ha
ˆ
G
(
QE, QE, ω
)can be conside ed as a scala ,
ˆ
G
(
QE, QE, ω
)
≡G
(
QE, QE, ω
), because o he symme y o ou sys em. In
he upcoming subsec ions, we b ie ly summa ize he key aspec s o he h ee
di e en models used in his chap e o ob ain
G
(
QE, QE, ω
), namely he TDDFT,
he classical LRA, and he semiclassical SRF based on he nondispe si e Feibelman
pa ame e s. Fu he de ails on he me hodologies adop ed in his chap e a e
p o ided in Chap e s 1and 2.
3.1.1 Time-dependen densi y unc ional heo y (TDDFT)
In he TDDFT calcula ions we desc ibe he MNPs wi hin he jellium model o
ee-elec on me als [
194
,
200
] as in oduced in Subsec ion 2.1.2. We use a Wigne –
Sei z adius equal o ha o sodium,
s
= 4
a0
, which allows us o use he alues
o he nondispe si e Feibelman pa ame e s ob ained in e .
296
(Figu e 2.9b) o
calcula e
G
(
QE, QE, ω
)wi hin he SRF and make he compa ison be ween he
TDDFT and SRF esul s. No e also ha he DP esonance o sodium MNPs lies a
op ical equencies (
ωDP ∼
3
eV
, see Figu e 2.3), e y close o ha o gold MNPs,
hus placing he esul s wi hin he equency ange ele an o ac ual applica ions
in plasmonics. The closed-shell MNPs con ain
Ne
= 4458 conduc ion elec ons,
which se s he adius o he backg ound jellium edge o a= 65.83 a0(≈3.5nm).
In o de o ob ain he sel -in e ac ion G een’s unc ion wi hin TDDFT,
GTDDFT
(
QE, QE, ω
), we use he WPP me hod explained in Subsec ion 2.2.1
o sol e he ime-dependen KS equa ions gi en by Eqs.
(2.19)
and
(2.20)
in ime
domain, and o calcula e he ime e olu ion o he elec on densi y
n
(
,
). Ins ead
o Eq.
(2.51)
used o model plane-wa e illumina ion in Chap e 2, he ex e nal
po en ial
Vex
(
,
) ha d i es he sys em [Eq.
(2.21)
] is gi en in his chap e by
an impulsi e po en ial c ea ed by a poin dipole,
Vex ( , ) = −pd∆ ˆ
z· − QE
| − QE|3δ( ),(3.3)
wi h
δ
(
) he Di ac del a unc ion and he ampli ude
pd
su icien ly small o ensu e
a linea esponse.
Vex
(
,
)appea s wi h a minus sign in Eq.
(3.3)
because i is
80
3.1. Sys em and me hods
ac ing on elec ons wi h cha ge
qe
=
−
1. Fo he indi idual MNP, we employ
he me hod desc ibed in Subsec ion 2.3.1 and ep esen he occupied KS o bi als
Ψ
j
(
,
)using a meshg id in sphe ical coo dina es, whe eas o he dime we adop
cylind ical coo dina es as in oduced in Subsec ion 2.3.2. The p opaga ion ime-s ep
is ∆ ∼0.05 au.
The Ha ee po en ial
VH
[
n
(
,
)] [Eq.
(2.23)
] is calcula ed o he indi idual
sphe ical MNP by exp essing
VH
[
n
(
,
)] as a sum o Legend e polynomials
Pℓ
(
cosθ
),
VH[n( , )] = ∞
X
ℓ=0
1
Vℓ( , )Pℓ(cosθ),(3.4)
since he sys em subjec ed o he ex e nal po en ial
Vex
(
,
)gi en by Eq.
(3.3)
possesses o a ional symme y wi h espec o he
z
-axis, and he e o e he esponse
is independen o he azimu hal angle
φ
. Using he Laplace ope a o
∇2
in sphe ical
coo dina es [Eq.
(1.24)
], Poisson’s equa ion gi en by Eq.
(2.23)
can be exp essed
as:
∞
X
ℓ=0
1
d2
d 2−ℓ(ℓ+ 1)
2
| {z }
Aℓ
Vℓ( , )Pℓ(cosθ) = −4π∞
X
ℓ=0
nℓ( , )Pℓ(cosθ),(3.5)
whe e
nℓ( , ) = 2
2ℓ+ 1 Z1
−1
(n( , )−n+)Pℓ(cosθ)d(cos θ),(3.6)
and we exp ess he ope a o
Aℓ
=
d2
d 2−ℓ(ℓ+1)
2
in ma ix o m using he me hod
o he Fou ie g id Hamil onian wi h sine basis unc ions [
344
]. F om Eq.
(3.5)
,
Vℓ( , )can hen be di ec ly ob ained om
Vℓ( , ) = −4πA−1
ℓ nℓ( , ).(3.7)
Fo he dime geome y we ob ain
VH
[
n
(
,
)] ollowing he p ocedu e desc ibed in
Subsec ion 2.3.2 [75].
Finally, we calcula e he ime-dependen elec ic ield
Eind
(
=
QE,
)induced
by he me allic nanos uc u e a he posi ion
QE
o he QE using
VH
[
n
(
,
)]
acco ding o Eq.
(2.34)
. The ime- o- equency Fou ie ans o m [Eq.
(2.36)
]
inally leads o he equency- esol ed sel -in e ac ion G een’s unc ion [compa e o
Eq. (1.56)],
GTDDFT( QE, QE, ω) = 1
pd∆ ZT
0
d Eind( QE, )ei(ω+iη/2)
| {z }
Esel ( QE,ω)
,(3.8)
whe e
T
= 3500
au
is he o al p opaga ion ime used in ou simula ions (enough
o achie e con e gence), and
η
= 0
.
07 eV accoun s o elaxa ion p ocesses beyond
81
Chap e 3. Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles
he ALDA-TDDFT desc ip ion [
241
,
242
] o he many-body dynamics such as he
in e ac ion o exci ed elec ons wi h phonons and many-body inelas ic elec on–
elec on sca e ing e en s (see Sec ions 2.2 and 2.3).
3.1.2 Classical local- esponse app oxima ion (LRA)
The
classical (non e a ded) local sel -in e ac ion G een’s unc ion,
GLRA
(
QE, QE, ω
),
is ob ained in he equency domain using he me hodology desc ibed in Subsec ion
1.1.3 o he ex e nal po en ial gi en by Eq.
(1.53)
,
ϕQE
(
) =
pQEˆ
z· − QE
| − QE|3
. The
speci ic de ails o ob ain
GLRA
(
QE, QE, ω
) o bo h he indi idual sphe ical MNP
and he dime a e gi en in Subsec ion 1.4.2. In his chap e , we use a D ude- ype
local dielec ic unc ion
ε
(
ω
) o cha ac e ize he me al [Eq.
(1.10)
], wi h a plasma
equency
ωp
=
q3
3
s
= 5
.
89
eV
and in insic damping pa ame e
γp
= 0
.
1
eV
. The
alue o pa ame e
γp
is ob ained om he compa ison o he abso p ion spec um
σabs
(
ω
)o he indi idual sphe ical MNP calcula ed wi h TDDFT and wi h he
SRF, as de ailed in he ollowing Subsec ion 3.1.3.
In he case o a poin -like QE a posi ion
QE
o ien ed in he adial di ec ion
and exci ing he indi idual MNP o adius
a
, he sel -in e ac ion G een’s unc ion
is gi en by Eq. (1.58a) [145]
GLRA( QE, QE, ω) = ∞
X
ℓ=1
(ℓ+ 1)2a2ℓ+1
R2ℓ+4
ε(ω)−1
ε(ω) + ℓ+1
ℓ
,(3.9)
wi h
R
he dis ance be ween he posi ion
QE
o he QE and he cen e o he
MNP, and ℓ he mul ipole o de o he plasmonic esonance.
3.1.3 Semiclassical su ace- esponse o malism (SRF)
The semiclassical SRF employed in his chap e is based on he nondispe si e
Feibelman pa ame e
d⊥
(
ω
)gi en by Eq.
(2.82a)
[
95
,
98
], a ( equency-dependen )
complex- alued unc ion ha allows o inco po a ing quan um su ace e ec s in o
an o he wise classical desc ip ion. Indeed,
Re{d⊥
(
ω
)
}
de e mines he posi ion o
he cen oid o he induced cha ge densi y wi h espec o he posi i e jellium edge
o he me al, and
Im{d⊥
(
ω
)
}
accoun s o su ace-enabled Landau damping [
62
]. As
explained in Sec ion 2.5, in his chap e we conside
d∥
(
ω
) = 0 [Eq.
(2.82b)
] and use
he nondispe si e pa ame e
d⊥
(
ω
)ob ained by Ch is ensen e al. [
296
] wi hin he
jellium model (Figu e 2.9b) o a semi-in ini e plana me al su ace wi h
s
= 4
a0
(see also e .
124
). Using he pa ame e
d⊥
(
ω
)ob ained o plana su aces is a
easonable app oxima ion when he adius o cu a u e o he nanos uc u e o
he ypical leng h o he a ia ion o he ex e nal po en ial along he su ace is
much la ge han he Feibelman pa ame e . One o he objec i es o his chap e is
indeed o es he alidi y o his app oxima ion o si ua ions whe e he adius o
cu a u e o he sys em is small and he ex e nal po en ial along he me al su ace
82
3.2. Resul s and discussion
(he e c ea ed by he poin -dipole QE) a ies apidly.
Simila ly o he classical LRA, wi hin he semiclassical SRF one can ob ain
an analy ical solu ion o he non e a ded sel -in e ac ion G een’s unc ion in he
p esen sys em,
GSRF
(
QE, QE, ω
). The exp ession o
GSRF
(
QE, QE, ω
)can
be ob ained om Eq.
(2.86)
and Eq.
(2.87)
conside ing he ex e nal po en ial
ϕex
(
, ω
)
≡ϕQE
(
)o a poin dipole gi en by Eq.
(1.53)
, which esul s in ( o
d∥(ω)=0) [95,124]:
GSRF( QE, QE, ω) = ∞
X
ℓ=1
(ℓ+ 1)2a2ℓ+1
R2ℓ+4
(ε(ω)−1)1 + ℓ
ad⊥(ω)
ε(ω) + ℓ+1
ℓ−(ε(ω)−1)ℓ+1
ad⊥(ω).(3.10)
Fo
d⊥
(
ω
)
/a →
0, i.e., o si ua ions whe e he adius o he MNP
a
is much la ge
han he su ace- esponse co ec ion
d⊥
(
ω
),Eq.
(3.10)
educes o he classical LRA
exp ession o GLRA( QE, QE, ω)gi en by Eq. (3.9).
The alue o he damping pa ame e
γp
used in he D ude- ype dielec ic
unc ion
ϵ
(
ω
)[Eq.
(1.10)
] o he classical LRA as well as o he SRF esul s
is ob ained om he i ing o he abso p ion c oss-sec ion spec um
σabs
(
ω
)o
he indi idual MNP calcula ed wi hin TDDFT o he esul ob ained wi hin
he nondispe si e SRF. The abso p ion c oss sec ion
σabs
(
ω
)wi hin TDDFT is
calcula ed using he me hodology desc ibed in Subsec ion 2.3.1 o plane-wa e
illumina ion (gi en by Eq.
(2.51)
), and he SRF alue is ob ained using he
pola izabili y
αSRF
(
ω
)gi en by Eq.
(2.88)
. Figu e 3.2 shows e y good ag eemen
be ween he TDDFT esul s (solid blue line) and he nondispe si e SRF (dashed
ed line) when using
γp
= 0
.
1
eV
, hus jus i ying he alue o
γp
used in his chap e .
Fo simplici y, in he ollowing we gene ically use
G
(
QE, QE, ω
) o e e o
any o
GTDDFT
(
QE, QE, ω
),
GLRA
(
QE, QE, ω
), and
GSRF
(
QE, QE, ω
). The
me hodology used o calcula e
G
(
QE, QE, ω
)will be clea in he con ex o each
subsec ion.
3.2 Resul s and discussion
In his sec ion, we p esen he esul s o he elec omagne ic coupling be ween a
QE and sphe ical me allic nanos uc u es. Fi s , in Subsec ion 3.2.1, we ocus on
he ole o quan um phenomena by compa ing quan um TDDFT and classical LRA
esul s o a QE in p oximi y o an indi idual sphe ical MNP. Then, in Subsec ion
3.2.2, we compa e esul s o TDDFT and he semiclassical nondispe si e SRF o
he same sys em. In Subsec ion 3.2.3, we use he nondispe si e SRF o analyze
he o igin o he obse ed quan um e ec s, and discuss he alidi y ange and
sho comings o he long-wa eleng h Feibelman pa ama e
d⊥
(
ω
) ha neglec s he
nonlocali y o he op ical esponse in he di ec ion pa allel o he me al su ace.
Finally, in Subsec ion 3.2.4, we ex end he analysis o he case o a dime o wo
iden ical sphe ical MNPs, showing ha he quan um su ace e ec s obse ed o
he indi idual MNP a e also mani es ed in he dime con igu a ion and ha he
83

Chap e 3. Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles
0
20
40
60
80
100
120
140
160
180
1 1.5 2 2.5 3 3.5 4 4.5 5
TDDFT
Figu e 3.2: Compa ison be ween he abso p ion c oss-sec ion spec a
σabs
ob ained om TDDFT
calcula ions wi hin he jellium model using a Wigne –Sei z adius
s
= 4
a0
(solid blue line),
and om he nondispe si e SRF using a plasma equency
ωp
= 5
.
89
eV
and in insic damping
pa ame e
γp
= 0
.
1
eV
in he D ude dielec ic unc ion gi en by Eq.
(1.10)
(dashed ed line). The
nondispe si e SRF esul s a e ob ained om Eq.
(2.88)
, and he TDDFT esul s a e ob ained
ollowing he p ocedu e desc ibed in Subsec ion 2.3.1. An a enua ion ac o
η
= 0
.
07
eV
[Eq. (2.56)] is used o pe o m he Fou ie ans o m wi hin TDDFT.
nondispe si e SRF does no desc ibe accu a ely si ua ions o gap dis ances na owe
han D∼1.5nm o he p esen dime geome y.
3.2.1 Quan um TDDFT s. classical LRA
We i s analyze he quan um e ec s ha in luence he sel -in e ac ion G een’s
unc ion
G
(
QE, QE, ω
)ob ained o a QE placed in on o an indi idual sphe ical
MNP. The QE is o ien ed in he adial di ec ion pe pendicula o he MNP su ace
(see ske ch in Figu e 3.1a). To iden i y he quan um e ec s, we i s compa e in
Figu e 3.3 he classical LRA (panels a,b) and he TDDFT (panels c,d) esul s.
We plo bo h he imagina y (a,c) and eal (b,d) pa s o
G
(
QE, QE, ω
)which a e
ela ed o he Pu cell ac o and Lamb shi , espec i ely [Eqs.
(3.1a)
and
(3.1b)
].
Resul s a e shown as a unc ion o he oscilla ion equency o he QE,
ω
, and he
dis ance dbe ween he QE and he su ace o he sphe ical MNP, d=R−a.
The classical LRA calcula ions p edic a dependence o
G
(
QE, QE, ω
)on
equency de e mined by a ious mul ipola plasmon modes exci ed by he QE,
esul ing in se e al peaks in he spec a o
Im{G
(
QE, QE, ω
)
}
(Figu e 3.3a). The
h ee lowe - equency sha p esonances a e associa ed wi h he dipola (DP,
ℓ
= 1),
quad upola (QP,
ℓ
= 2), and oc upola (OP,
ℓ
= 3) plasmons o he sphe ical MNP.
Thei equencies
ωℓ
a e gi en by he poles o Eq.
(3.9)
,
Reϵ(ωℓ) + ℓ+1
ℓ
= 0. As
84
3.2. Resul s and discussion
Figu e 3.3:(a) Classical LRA esul o he (a) imagina y pa (
Im{G
(
QE, QE, ω
)
}
), and (b) eal
pa (
Re{G
(
QE, QE, ω
)
}
) o he sel -in e ac ion G een’s unc ion
G
(
QE, QE, ω
)ob ained o a
poin -like QE placed in on o an indi idual sphe ical MNP o adius
a
= 65
.
83
a0
(
≈
3
.
5
nm
).
Resul s a e shown as a unc ion o he equency
ω
o he oscilla ing QE and he su ace- o-emi e
dis ance,
d
. Panels (c) and (d) co espond o he esul s ob ained wi h TDDFT simula ions. In
(a) and (b), he uppe and lowe ange o alues in he colo ba deno e sa u a ion.
we conside a me al desc ibed wi h a D ude dielec ic unc ion, his esul s in
ωℓ=ωp ℓ
2ℓ+ 1.(3.11)
F om Eq.
(3.11)
i ollows ha he equencies o he DP, QP and OP a e
espec i ely
ωDP ≈
3
.
4
eV
,
ωQP ≈
3
.
7
eV
, and
ωOP ≈
3
.
85
eV
. The high-
equency b oad peak a
ωPSM ∼
4
eV
(i.e., close o he su ace plasmon equency
ωSP
=
ωp/√2≈
4
.
16
eV
) co esponds o he so-called pseudomode [
143
], which
is composed by a pilling up o se e al o e lapping high-o de plasmonic modes
(ℓ= 4,5,6, . . . ) wi h closely-spaced esonan equencies.
85
Chap e 3. Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles
A small
d≈
0
.
53
−
1
nm
,
Im{G
(
QE, QE, ω
)
}
, as calcula ed wi hin he
classical LRA, is domina ed by he pseudomode exci a ion. As he dis ance
be ween he QE and he MNP inc eases,
Im{G
(
QE, QE, ω
)
}
dec eases, and he
ela i e con ibu ion o di e en plasmon modes changes in a o o he low
ℓ
esonances ( his beha io can be seen mo e clea ly in Figu e 3.4). Thus, o la ge
d∼
1
.
6
−
1
.
8
nm
, he alues o
Im{G
(
QE, QE, ω
)
}
a ained wi hin he pseudomode
equency ange become compa able o hose a he sha p DP and QP esonances.
The as e dec ease o he esonances associa ed wi h high-o de plasmon modes
wi h inc easing
d
can be in e ed om Eq.
(3.9)
, whe e
R
=
a
+
d
. We also no e ha
he alue o
Im{G
(
QE, QE, ω
)
}
ob ained o a QE esonan wi h he pseudomode
a
ωPSM
= 4
.
05
eV
and loca ed a
d
= 0
.
58
nm
co esponds o a Pu cell ac o
FP≈
5
.
2
×
10
6
[Eq.
(3.2)
]. This e y la ge alue is explained by he small olume
o he MNP (and he esul ing s ong ield localiza ion).
In con as o he classical esul s,
Im{G
(
QE, QE, ω
)
}
as calcula ed wi h
TDDFT (Figu e 3.3c) mainly e eals a single b oad ea u e [
345
] o he ange o
dis ances conside ed in his chap e . A small sepa a ion
d≈
0
.
53
−
1
nm
, he
maximum alue o
Im{G
(
QE, QE, ω
)
}
is eached wi hin he equency in e al
ω∼
3
.
6
−
3
.
7
eV
, i.e., i is edshi ed wi h espec o he classical pseudomode
peak. As
d
inc eases, he esonan ea u e sligh ly shi s o lowe equencies.
Mo eo e , he o e all p o ile somewha sha pens, albei , in shee con as wi h he
classical heo y, he con ibu ions o di e en plasmon modes emain spec ally
b oade and a e ba ely esol ed. Consis en wi h he s ong b oadening o he
plasmon esonances due o quan um e ec s, he TDDFT esul s show smalle
alues o
Im{G
(
QE, QE, ω
)
}
a esonance, and hus lowe QE decay a es, as
compa ed o he classical LRA p edic ion. Fo example, o a QE placed a a
dis ance
d
= 0
.
58
nm
, he esonan Pu cell ac o
FP
calcula ed wi hin TDDFT
is
FP≈
1
.
5
×
10
6
. This is mo e han h ee imes smalle han he maximum
LRA alue. On he o he hand, he b oadening o he spec a leads o a la ge
o - esonan
Im{G
(
QE, QE, ω
)
}
ob ained wi h TDDFT as compa ed o classical
LRA p edic ions.
We nex compa e he classical LRA and quan um TDDFT esul s o he eal
pa o he sel -in e ac ion G een’s unc ion,
Re{G
(
QE, QE, ω
)
}
, which de e mines
he Lamb shi ∆ωQE o he QE ansi ion equency [Eq. (3.1b)]. As depic ed in
Figu e 3.3b, and consis en wi h he esul s ob ained o he imagina y pa o he
G een’s unc ion (Figu e 3.3a), he equency dependence o
Re{G
(
QE, QE, ω
)
}
ob ained om classical LRA calcula ions ea u es a ich esonance p o ile. Fo an
indi idual plasmonic mode, he K ame s-K onig ela ions would lead o a sign
change o
Re{G
(
QE, QE, ω
)
}
a he esonance equency. In he ull calcula ions,
Re{G
(
QE, QE, ω
)
}
does no show he sign change a esonance o low
ℓ
modes
and small dis ances, because o he o - esonan con ibu ion associa ed wi h
neighbo ing plasmon modes wi h la ge
ℓ
. I is only a he pseudomode equency
ha he con ibu ion o he nea ly degene a e esonances leads o a change o sign
o
Re{G
(
QE, QE, ω
)
}
om posi i e alues a equencies below
ωPSM ∼
4
eV
o
nega i e alues abo e his equency. When
d
inc eases, he con ibu ion om
o - esonan neighbo ing modes is educed so ha , in addi ion o he pseudomode
86
3.2. Resul s and discussion
esonance, he sign change o
Re{G
(
QE, QE, ω
)
}
can be obse ed a he DP and
QP esonances. This appea s pa icula ly clea in Figu e 3.4 discussed below,
whe e we show he equency dependence o he G een’s unc ion calcula ed o a
se o ixed sepa a ions, d, be ween he QE and he MNP su ace.
Simila ly o he esul s ob ained o he imagina y pa , he TDDFT calcula ions
in Figu e 3.3d show smalle absolu e alues o
Re{G
(
QE, QE, ω
)
}
(Lamb shi )
and a b oade s uc u e a he esonan plasmon equency
ω∼
3
.
3
−
3
.
7
eV
as compa ed o he classical LRA. This holds o he en i e dis ance
d
ange
conside ed in his chap e . No ably, a single b oad esonance is app ecia ed in he
TDDFT esul s, and
Re{G
(
QE, QE, ω
)
}
changes i s sign om posi i e o nega i e
only a he esonan equency
ω∼
3
.
3
−
3
.
7
eV
, i.e. a lowe equency han
wi hin he classical LRA. As a consequence, o QEs wi h ansi ion equencies
wi hin he ange o
ωQE ∼
3
.
7
−
4
eV
, each model p edic s a pho onic Lamb shi
∆
ωQE ∝Re{G
(
QE, QE, ω
=
ωQE
)
}
o opposi e sign [Eq.
(3.1b)
]. Fo example,
acco ding o LRA, a QE loca ed a
d
= 0
.
58
nm
and cha ac e ized by a ansi ion
dipole momen
µQE
= 0
.
1
enm
and esonan equency
ωQE
= 4
eV
expe iences
a edshi o ∆
ωQE ≈ −
130
meV
. In con as , TDDFT p edic s a blueshi o
∆ωQE ≈34 meV unde he same condi ions.
3.2.2 Quan um TDDFT s. semiclassical nondispe si e
SRF
A e iden i ying he main quan um-mechanical e ec s in Figu e 3.3, we can use
he nondispe si e SRF o dissec he ole o nonlocali y and he spill-ou o he
induced cha ges ha can be behind he di e ences be ween he classical LRA and
TDDFT esul s o
G
(
QE, QE, ω
)discussed abo e. To his end, we compa e in
Figu e 3.4 he eal pa (uppe panels a-e) and he imagina y pa (lowe panels -j)
o
G
(
QE, QE, ω
)as calcula ed using he h ee di e en app oaches (TDDFT, LRA,
and SRF). Resul s a e shown as a unc ion o he oscilla ion equency o he QE,
o selec ed alues o he su ace- o-emi e dis ance
d
. The solid and dashed blue
lines show he e e ence TDDFT esul s and hose ob ained using he nondispe si e
SRF, espec i ely. The classical LRA esul s a e plo ed by g ay-do ed lines. The
o e all good ag eemen be ween he TDDFT and he nondispe si e SRF in Figu e
3.4 es ablishes he alidi y o he la e and allows us o use he amewo k o he
SRF o analyze he ole o he quan um phenomena mani es ed in
G
(
QE, QE, ω
),
as we discuss below.
We i s ocus on he esul s a ela i ely la ge dis ance
d
= 1
.
38
−
2
.
22
nm
(panels a-c and -h), whe e he ag eemen be ween he nondispe si e SRF and
TDDFT is pa icula ly good. The semiclassical SRF accu a ely ep oduces he
TDDFT esul s o he spec al posi ion and esonance p o ile o
G
(
QE, QE, ω
),
hus co ec ly accoun ing o he edshi and la ge b oadening o he peaks as
compa ed o he classical LRA. On he o he hand, o dis ances below
d≤
0
.
95
nm
(panels d-e and i-j), he nondispe si e SRF esul s a e edshi ed wi h espec
o hose o TDDFT, i.e., he semiclassical model based on he long-wa eleng h
app oxima ion o
d⊥
(
ω
)o e es ima es he edshi o he plasmonic modes om
87
Chap e 3. Quan um su ace e ec s in he elec omagne ic coupling be ween
quan um emi e s and me allic nanopa icles
a cu o Figu e 3.6. Al hough he e a e quan i a i e disc epancies be ween he
TDDFT and he SRF esul s o na ow gaps
D∼
1
−
1
.
5
nm
(panels d,e,i,j in
Figu e 3.7), he o e all good ag eemen be ween he wo app oaches indica es
ha he spill-ou o he induced cha ges and su ace-enabled Landau damping,
al eady discussed in he indi idual MNP esul s, a e also he main quan um
mechanisms in luencing
G
(
QE, QE, ω
)in he dime con igu a ion. Mo eo e , hese
esul s co obo a e he alidi y o he long-wa eleng h limi implemen a ion o he
SRF o adequa ely desc ibe he elec omagne ic in e ac ion be ween a QE and
a plasmonic gap nanos uc u e o si ua ions whe e he gap sepa a ion is la ge
han
D∼
1
.
5
nm
(panels a,b,c, ,g,h in Figu e 3.7). Howe e , simila o he case
o he indi idual MNP in Subsec ions 3.2.2 and 3.2.3, o smalle gap sepa a ions
in a dime ,
D <
1
.
5
nm
, he limi a ions o he app oxima ion used o ob ain he
Feibelman pa ame e d⊥(ω)(neglec ing he nonlocali y o op ical esponse in he
di ec ion pa allel o he me al su ace) a ec s he accu acy o he esul s, since in
such a case, high-o de plasmonic
ℓ
-modes a e also ele an in he esponse o he
sys em.
3.3 Summa y
In his chap e , we s udy he in luence o quan um phenomena on he
elec omagne ic in e ac ion be ween a poin -like quan um emi e (QE) and
canonical me allic nanos uc u es. We ocus on he s udy o he sel -in e ac ion
dyadic G een’s unc ion
ˆ
G
(
QE, QE, ω
)ob ained o an indi idual sphe ical MNP
and a dime comp ising wo iden ical sphe ical MNPs, wi h he QE o ien ed
pe pendicula o he me al su aces. In he case o he dime , he QE is loca ed in
he middle o he gap. We conside su icien ly la ge QE–MNP sepa a ions so ha
cha ge- ans e p ocesses ela ed o elec on unneling do no play a ole.
We i s calcula e
ˆ
G
(
QE, QE, ω
)in he p esence o an indi idual sodium
MNP using ime-dependen densi y unc ional heo y (TDDFT), and hen employ
analy ical exp essions de i ed om a semiclassical model ( e e ed o as he
nondispe si e SRF) in o de o iden i y he o igin o he quan um e ec s ha
in luence he QE–MNP coupling. This nondispe si e SRF inco po a es su ace
quan um- esponse co ec ions by means o he Feibelman
d⊥
(
ω
)pa ame e ob ained
unde he long-wa eleng h app oxima ion. The o e all good ag eemen be ween
TDDFT and he nondispe si e SRF o bo h he indi idual and he dime
con igu a ions con i ms ha su ace-enabled Landau damping and he spill-ou
o he induced elec on densi y d as ically a ec he elec omagne ic QE–MNPs
in e ac ion. These mechanisms explain why he esonances o
ˆ
G
(
QE, QE, ω
)
ob ained om TDDFT a e edshi ed and b oade as compa ed o hose ob ained
om a classical calcula ion using he local- esponse app oxima ion (LRA) o he
op ical esponse o he me als.
We ind ha hese quan um e ec s become mo e signi ican wi h inc easing
o de
ℓ
o he plasmonic esonance. The analysis o he TDDFT calcula ions
indica es ha he highe he alue o
ℓ
, he la ge he b oadening
κℓ
p oduced by
94

3.3. Summa y
su ace-enabled Landau damping as well as he edshi p oduced by he spill-ou
o he dynamical sc eening cha ges. Thus, he quan um phenomena explo ed in
his chap e show a conside able in luence in he op ical esponse o e y small
dis ance be ween a QE and a me allic su ace, and when he QE is coupled o
high-o de plasmonic modes o he nanos uc u e.
We also ind ha he nondispe si e SRF calcula ions based on he long-
wa eleng h limi o he Feibelman pa ame e
d⊥
(
ω
)desc ibe mo e accu a ely
he G een’s unc ion o a dipola emi e when conside ing QEs coupled o low-
o de plasmonic modes. Howe e , hese calcula ions a e no accu a e when he
con ibu ion om high-o de modes (
ℓ≳
5) o he nanos uc u e is la ge, as occu s
when he dis ance
d
be ween he QE and he MNP is e y small,
d∼
0
.
6
nm
.
As a consequence, in he dime con igu a ion he nondispe si e SRF calcula ions
unde es ima e he b oadening o he pseudomode and become inaccu a e o gap
sepa a ions o he o de o
D∼
1
−
1
.
5
nm
. The sho comings o he nondispe si e
SRF a e due o he limi a ions o add ess he pa allel nonlocali y o he esponse
when using a
d⊥
(
ω
)pa ame e whe e he dependence on he wa enumbe
k∥
pa allel
o he me al su ace (o , equi alen ly, he angula momen um
ℓ
o sphe ical MNPs)
is neglec ed. Fo mo e accu a e esul s, i is necessa y o go beyond he long-
wa eleng h limi o
d⊥
(
ω
) o p ope ly accoun o he nonlocali y o he su ace
esponse in he di ec ion pa allel o he su ace, as his impac s he esul s in
si ua ions whe e he QE–MNP dis ance is small (high-o de
ℓ
modes in ol ed).
The ex ension o he SRF o a dispe si e model ha akes in o accoun he
k∥-dependence o d⊥is p oposed in Chap e 4.
This chap e hus p o ides a undamen al desc ip ion o he quan um
phenomena in luencing he elec omagne ic in e ac ion be ween a QE and plasmonic
nanos uc u es o su ace- o-emi e dis ances as small as
≈
0
.
5
nm
. Fo e en
smalle sepa a ion, cha ge- ans e p ocesses be ween he QE and he MNPs can
in luence he op oelec onic esponse o he sys em, so ha a many-body ea men
based on TDDFT o bo h he QE and he MNPs is necessa ily equi ed o na u ally
accoun o any quan um e ec in hose ex eme si ua ions, including also cha ge-
ans e p ocesses. We adop his s a egy in Chap e 5.
95
Chap e
4
DISPERSIVE SURFACE-RESPONSE
FORMALISM TO ADDRESS OPTICAL
NONLOCALITY IN SITUATIONS OF
EXTREME PLASMONIC FIELD
CONFINEMENT
As shown in Chap e 3, he su ace- esponse o malism (SRF) based on he
Feibelman pa ame e
d⊥
allows us o cap u e quan um e ec s such as he spill
in/ou o he induced cha ges, su ace-enabled Landau damping, and nonlocal
dynamical sc eening in a compu a ionally simple manne . To he bes o ou
knowledge,
d⊥
has been so a compu ed using he long-wa eleng h app oxima ion
[
98
], which consis s in neglec ing he nonlocali y o he op ical esponse in he
di ec ion pa allel o he me al–dielec ic in e ace. The nondispe si e Feibelman
pa ame e is hen a unc ion o he exci a ion equency,
d⊥≡d⊥
(
ω
), and i does
no depend on he wa enumbe
k∥
pa allel o he me al su ace. Conside ing
he long-wa eleng h limi educes he compu a ional e o o ob ain
d⊥
(
ω
) om
quan um calcula ions. Mo eo e , i simpli ies he implemen a ion o he SRF
in exis ing nume ical ools ha sol e Maxwell´s equa ions in Nanopho onics as
employed in a numbe o ecen s udies [97,124,129,296,299,300].
Using he long-wa eleng h limi o
d⊥
(i.e., conside ing
k∥∼
0) is a easonable
app oxima ion when he nonlocali y o he op ical esponse in he di ec ion
pe pendicula o he su ace domina es, i.e., when he cha ac e is ic scale ∆
s
o he op ical ield a ia ion along he su ace is la ge as compa ed o ha o he
su ace- esponse co ec ion, ∆
s>> d⊥
. This is he case o e.g. ypical indi idual
me allic nanopa icles (MNPs) subjec ed o plane-wa e illumina ion. Howe e , in
97
Chap e 4. Dispe si e su ace- esponse o malism o add ess op ical nonlocali y in
si ua ions o ex eme plasmonic ield con inemen
Chap e 3we demons a e ha using he long-wa eleng h app oxima ion o
d⊥
(
ω
)
wi hin he nondispe si e SRF is no accu a e when desc ibing si ua ions in ol ing
mul ipole plasmon modes cha ac e ized by localized su ace cha ges ha apidly
a y along he MNPs su ace. This is analogous o exci ing plasmons wi h la ge
ans e se wa enumbe
k∥
, and he e o e equi es o go beyond he long-wa eleng h
limi o d⊥.
In his chap e , we demons a e ha accoun ing explici ly o he nonlocali y
o he op ical esponse in he di ec ion pa allel o he su ace li s he sho comings
o he nondispe si e SRF. Using a dispe si e Feibelman pa ame e
d⊥≡d⊥
(
ω, k∥
)
ha is a unc ion o
ω
and
k∥
allows o co ec ly p edic ing he op ical esponse
o plasmonic s uc u es in ex eme si ua ions whe e plasmon-induced cha ges
cha ac e ized by high
k∥
can be exci ed, such as in s uc u es wi h small adius o
small gaps, o ins ance. We i s s udy in Subsec ion 4.2.1 he ene gy dispe sion o
localized mul ipola plasmon esonances sus ained by a cylind ical me allic nanowi e
using TDDFT, and show ha he ene gies o mul ipola plasmon modes in such a
nanowi e a e go e ned by a uni e sal pa ame e
m/Rc
equi alen o he wa enumbe
k∥
o su ace plasmons a plana in e aces (he e
m
is he magne ic quan um numbe ,
and
Rc
is he nanowi e adius). Consis en wi h his s a emen , we demons a e
ha he ene gy dispe sion o localized mul ipola plasmon modes ela i e o
m/Rc
in cylind ical nanowi es ollows he p e iously-s udied
k∥
-dispe sion o non e a ded
su ace plasmons a plana me al su aces [
62
,
306
,
346
]. Thus, TDDFT calcula ions
o he cylind ical nanowi e a e used in Subsec ion 4.2.2 o ob ain he Feibelman
pa ame e
d⊥≡d⊥
(
ω, k∥
=
m/Rc
), which e eals a s ong dependence on
k∥
.
We show in Subsec ion 4.2.3 ha he dispe si e (
k∥
-dependen ) SRF accu a ely
ep oduces he ene gy dispe sion o plasmon esonances ob ained om TDDFT
o he cylind ical nanowi e. Fu he mo e, we demons a e ha he same se o
d⊥
(
ω, k∥
)can be used o add ess op ical nonlocali y in he pa allel di ec ion in
me allic nanos uc u es o di e en shapes. To his end, we apply he dispe si e
SRF o small sphe ical MNPs as well as o nanome e -gap sphe ical dime s coupled
o quan um emi e s (QEs), ob aining a good ag eemen be ween he dispe si e
SRF and TDDFT esul s. This chap e hus p o ides a signi ican ad ance owa d
he implemen a ion o a SRF ha adequa ely accoun s o quan um e ec s in he
op ical esponse o plasmonic sys ems exhibi ing ex eme op ical nonlocali y.
4.1 Sys em and me hods
Th ee di e en plasmonic nanos uc u es a e conside ed in his chap e o show
he gene ali y o he dispe si e SRF: (i) an in ini e cylind ical nanowi e o adius
Rc
ex ended along he
z
-axis, as desc ibed in Subsec ion 2.3.3, which is used o
calcula e
d⊥
(
ω, k∥
), (ii) an indi idual sphe ical MNP o adius
a
, as desc ibed
in Subsec ion 2.3.1, and (iii) a dime consis ing o wo iden ical sphe ical MNPs
wi h a QE loca ed in he middle o he gap. The dime conside ed he e is he
same as he one conside ed in Chap e 3. All he nanos uc u es conside ed in
his chap e a e desc ibed wi hin he jellium model o ee-elec on me als (see
98
4.1. Sys em and me hods
Figu e 4.1: Ske ch o he sys ems s udied in his chap e . (a) Cylind ical Na nanowi e wi h
adius
Rc
, in ini e along he
z
-axis.
ρ
is he adial coo dina e, and
φ
he azimu hal angle. The
sys em possesses o a ional symme y wi h espec o he
z
-axis. We conside h ee di e en
alues o he adius:
Rc
= 75
a0,
100
a0,and
150
a0
.(b) Sphe ical Na nanopa icle wi h adius
a
= 65
.
83
a0
.(c) A poin -like quan um emi e (QE) wi h a dipole momen
pd
placed a he
middle o he gap o med by wo iden ical sphe ical Na nanopa icles. The QE is ep esen ed
by he g een a ow. The gap sepa a ion dis ance is deno ed as
D
. In all he s uc u es, Na is
cha ac e ized by a Wigne –Sei z adius s= 4 a0.
Subsec ion 2.1.2) using a Wigne –Sei z adius
s
= 4
a0
ha co esponds o sodium.
The classical bulk plasma equency is he e o e
ωp
=
q3
3
s
= 5
.
89 eV, he su ace
plasmon equency
ωSP
=
ωp/√2
= 4
.
16 eV, and he localized dipola plasmon
(DP) equency
ωDP
=
ωp/√3
= 3
.
4eV. We choose his ma e ial because i allows
o a di ec compa ison be ween ou esul s and hose ob ained in Chap e 3
using he long-wa eleng h app oxima ion. The schema ic ep esen a ion o he
nanos uc u es s udied in his chap e is depic ed in Figu e 4.1.
The op ical esponse o indi idual sphe ical MNPs and dime s has been
s udied in de ail in p e ious chap e s, and hus he e we ocus on he nume ical
p ocedu e used o ob ain he ene gy dispe sion o localized mul ipola plasmon
esonances sus ained by he cylind ical nanowi e. We add ess cylind ical nanowi es
wi h la ge adii
Rc
(as compa ed o ha o sphe ical MNPs) wi hin he ange
Rc
= 75
−
150
a0
(
≈
4
−
8
nm
)cha ac e ized by well-de eloped mul ipola plasmons,
hus allowing us o span a la ge ange o pe iods 2
πRc/m
o he spa ial a ia ion o
plasmon-induced cha ges along he nanowi e su ace. He e, owing o he cylind ical
symme y o he sys em, he mul ipola plasmon modes can be cha ac e ized by
he magne ic numbe
m
ela ed o he
eimφ
dependence on he azimu hal angle
φ
o he po en ials, ields and induced cha ges.
In he equency domain, he po en ial
Vind
(
ρ, φ, ω
)induced a
ρ > Rc
in
esponse o an ex e nal exci a ion Vex (x, y, )can be exp essed as
99

Chap e 4. Dispe si e su ace- esponse o malism o add ess op ical nonlocali y in
si ua ions o ex eme plasmonic ield con inemen
Vind(ρ, φ, ω)=2
mmax
X
m=1 Rc
ρm
Qm(ω)cos(mφ),(4.1)
whe e
Qm
(
ω
)is he equency- esol ed mul ipole momen o o de
m
pe uni
leng h along he
z
-axis. Ou aim is o de e mine he equency and wid h o hese
mul ipola esonances.
Wi h his pu pose, we use he Kohn–Sham scheme (KS) o TDDFT in oduced
in Sec ion 2.2. We employ he same nume ical implemen a ion as desc ibed in
Subsec ion 2.3.3 bu , ins ead o conside ing a
x
-pola ized plane-wa e exci a ion
gi en by Eq.
(2.76)
, he e we conside he ollowing impulsi e ex e nal po en ial
Vex (x, y, ):
Vex (x, y, ) = ξ δ( )
mmax
X
m=1 ρ
Rcm
cos(mφ),(4.2)
whe e
x
=
ρcosφ
and
y
=
ρsinφ
,
δ
(
)is he Di ac del a unc ion, he ampli ude
ξ
is su icien ly weak o ensu e a linea esponse, and
mmax
= 30 is he highes
mul ipole o de conside ed in he simula ion. In Eq.
(4.2)
, we w i e explici ly
he dependence o
Vex
(
x, y,
)on he spa ial a iables (
x, y
) o s ess ha we use
Ca esian coo dina es as desc ibed in Subsec ion 2.3.3. Due o he exci a ion used
he e, mul ipola plasmon exci a ions a e localized a he (
x, y
)-plane, and do no
p opaga e along he
z
-axis. The e m localized e e ing o plasmons exci ed a
he nanowi e is o en omi ed bu implici ly assumed. Because o he cylind ical
symme y o he sys em,
m
is a good quan um numbe , i.e., an ex e nal po en ial
wi h angula dependence
cos
(
mφ
)exci es localized mul ipola plasmon modes a
he nanowi e cha ac e ized by induced cha ges wi h he same angula dependence
cos(mφ).
Using he eal- ime ALDA-TDDFT me hodology in Ca esian coo dina es as
in oduced in Subsec ion 2.3.3, we calcula e he ime-dependen mul ipole momen
Qm
(
)induced a he nanos uc u e pe uni leng h along he
z
-axis, de ined as
25
Qm( ) = −1
mZZ dx dy ρ
Rcm
cos(mφ)δn(x, y, ),(4.3)
whe e he elec on densi y
δn
(
x, y,
)induced by
Vex
(
x, y,
)in Eq.
(4.2)
is gi en
by [Eq. (2.77)]
δn(x, y, ) = n(x, y, )−n(x, y, = 0).(4.4)
The equency- esol ed spec um o he mul ipole momen
Qm
(
ω
)is inally
ob ained om he ime- o- equency Fou ie ans o m,
Qm(ω) = Zd Qm( )e(iω−η/2) ,(4.5)
25
The 1
/(Rc)m
ac o o he de ini ion o he cylind ical mul ipole momen in Eq.
(4.3)
(and
consis en ly he ac o (
Rc
)
m
in Eq.
(4.1)
) simpli ies he compa ison be ween he esul s o he
calcula ions pe o med o nanowi es o di e en adius Rc.
100
4.2. Resul s and discussion
whe e an a enua ion ac o
η
= 0
.
15
eV
is used [Eq.
(2.56)
] o mimic dissipa ion
p ocesses beyond he each o he ALDA-TDDFT scheme adop ed he e (see de ails
in Sec ion 2.2).
I is wo h men ioning ha , o
mmax
= 1,Eq.
(4.2)
exp esses he non e a ded
po en ial co esponding o a plane-wa e illumina ion pola ized along he
x
-axis.
Such illumina ion is ypically used in linea - esponse TDDFT calcula ions o ob ain
he dipola pola izabili y o he sys em, as desc ibed in Subsec ion 2.3.3.
4.2 Resul s and discussion
4.2.1 TDDFT s udy o he ene gy dispe sion o mul ipola
plasmon esonances in a me allic nanowi e
Figu e 4.2 shows he in ensi y spec um o he mul ipole momen s,
|Qm
(
ω
)
|2
,
ob ained o a cylind ical nanowi e o adius
Rc
= 150
a0
(panel a) and
Rc
= 100
a0
(panel b) in esponse o he ex e nal po en ial gi en by Eq.
(4.2)
. The esul s
o di e en alues o
m
a e plo ed, anging om
m
= 1 ( op) o
m
=
mmax
(bo om). Fo he la ges nanowi e
mmax
= 30 is conside ed, whe eas o he
smalles one
mmax
= 23. The gene al beha io o mul ipola plasmon esonances
|Qm
(
ω
)
|2
is independen o he size o he nanos uc u e: i s , o a gi en
m
, a
well-de ined esonance cen e ed a a equency
ωm
is ob ained associa ed wi h he
exci a ion o he mul ipola plasmon mode o o de
m
. The wid h o he mul ipola
plasmon esonance o he wo nanowi es inc eases wi h inc easing
m
because o
he enhancemen o su ace-enabled Landau damping, whe e he plasmon decays
in o elec on–hole pai exci a ions a he su ace egion, as discussed in Chap e 3
o sphe ical MNPs o inc easing
ℓ
. Mo eo e , ega dless o he speci ic alue o
he adius
Rc
,
ωm
i s edshi s wi h inc easing
m
and, a e eaching a minimum,
i con inuously blueshi s. Howe e , he mul ipola plasmon esonances o a gi en
o de
m
a e b oade o he smalle nanowi e, and do no eme ge a he same
equency
ωm
o he wo sizes. Fo example, he dipola plasmon esonance
(
m
= 1) eme ges a
ω1∼
4
.
1
eV
o
Rc
= 150
a0
(ex emely close o he classical
long-wa eleng h limi o he su ace plasmon equency,
ωSP
= 4
.
16
eV
), while
o
Rc
= 100
a0
i appea s a
ω1∼
4
.
06
eV
. Mo eo e , o he la ges nanowi e
he minimum alue o
ωm
is eached a
m
= 10 (
ω10 ∼
3
.
8
eV
), whe eas o he
smalles nanowi e he minimum is ob ained a
m
= 8. The di e ences in he esul s
ob ained o he wo nanowi es a e pa icula ly appa en when compa ing
ωm
o
m= 20: in his case, he e is a misma ch in ωmo he o de o ∼0.5eV.
The esul s shown in Figu e 4.2a,b a e closely ela ed o he dispe sion ela ion
o su ace plasmon esonances suppo ed by plana me al– acuum in e aces as a
unc ion o he pa allel wa enumbe
k∥
, as we discuss below. Indeed, he induced
ields and su ace cha ge densi ies o localized mul ipola plasmons conside ed
in his chap e ha e a dependence o
exp
(
imφ
)on he azimu hal angle
φ
. By
in oducing he coo dina e
∥
along he su ace o he nanowi e c oss-sec ion,
∥
=
Rcφ
, he angula dependence ans o ms in o
exp
(
imφ
)
→exp
(
im
Rc ∥
).
101
Chap e 4. Dispe si e su ace- esponse o malism o add ess op ical nonlocali y in
si ua ions o ex eme plasmonic ield con inemen
Figu e 4.2:(a) In ensi y spec um o he mul ipole momen
|Qm
(
ω
)
|2
induced a an in ini ely
long cylind ical Na nanowi e (
s
= 4
a0
) o adius
Rc
= 150
a0
(
≈
8
nm
). Resul s a e shown
as a unc ion o he equency o he ex e nal exci a ion,
ω
, o di e en alues o he magne ic
quan um numbe
m
, anging om
m
= 1 ( op) o
m
= 30 (bo om), as indica ed in he inse s.
All
|Qm
(
ω
)
|2
a e no malized o hei co esponding maximum alue. (b) Same as in (a) bu o
a smalle nanowi e wi h adius
Rc
= 100
a0
(
≈
5
.
3
nm
)and magne ic numbe wi hin he ange
m= 1 −23. Resul s a e displaced in he e ical axis o cla i y.
Fo
Rc→ ∞
, he cylind ical geome y ends o he plana -su ace geome y
wi h
∥
being he coo dina e pa allel o he su ace, so ha se ing
m
Rc→k∥
eco e s he s anda d dependence
exp
(
ik∥ ∥
)o a su ace plasmon p opaga ing
along he su ace [see Eq.
(1.38)
]. Thus, we in e p e
m
Rc
in he ollowing as
an “e ec i e” wa enumbe
k∥
by conside ing ha localized mul ipola plasmons
in he nanowi e co espond o con ined su ace plasmons wi h a quan ized
wa eleng h
λ∥
= 2
πRc/m
[
347
]. To suppo his co espondence, we compa e
in Figu e 4.3 he in ensi y spec a
|Qm
(
ω
)
|2
ob ained o nanowi es wi h adius
Rc
= 75
a0
(
dahsed lines
)
,
100
a0
(
do ed lines
)
,and
150
a0
(
solid lines
).
Resul s a e shown o selec ed alues o
m
such ha
k∥
=
m/Rc
=
0
.
013
,
0
.
02
,
0
.
04
,
0
.
067
,and
0
.
1
a−1
0
. Whene e he a io
m/Rc
is ixed, he
in ensi y spec um o he mul ipole momen
|Qm
(
ω
)
|2
ob ained o nanowi es o
di e en size shows nea pe ec ma ch wi h each o he , hus con i ming ha an
e ec i e wa enumbe k∥=m/Rcde e mines he op ical esponse o he sys em.
102
4.2. Resul s and discussion
Figu e 4.3: In ensi y spec um o he mul ipole momen s
|Qm
(
ω
)
|2
ob ained o cylind ical
nanowi es o di e en adii
Rc
and selec ed alues o he magne ic numbe
m
. Solid lines:
Rc
= 150
a0
. Do ed lines:
Rc
= 100
a0
. Dashed lines:
Rc
= 75
a0
. The selec ed alues o
m
a e such ha
k∥
=
m/Rc
= 0
.
013
a−1
0
( ed),
k∥
= 0
.
02
a−1
0
(blue),
k∥
= 0
.
04
a−1
0
(b own),
k∥
= 0
.
067
a−1
0
(o ange), and
k∥
= 0
.
1
a−1
0
(pu ple) o he h ee alues o
Rc
. The spec a
|Qm
(
ω
)
|2
co esponding o a speci ic wa enumbe
k∥
=
m/Rc
a e no malized o he maximum
alue ob ained o he case o Rc= 150 a0. Resul s a e displaced in he e ical axis o cla i y.
4.2.2 Calcula ion o he dispe si e Feibelman pa ame e
d⊥(ω, k∥)
The esul s p esen ed in he p e ious sec ion allow us o calcula e
d⊥≡d⊥
(
ω, k∥
)as
a unc ion o bo h he exci a ion equency
ω
and he e ec i e wa enumbe pa allel
o he su ace,
k∥
=
m/Rc
, using he cylind ical geome y. In his subsec ion, we
i s ob ain he exp ession o he Feibelman pa ame e
d⊥
o a cylind ical me allic
nanowi e wi hin he SRF, and hen discuss he TDDFT esul s o
d⊥≡d⊥
(
ω, k∥
)
calcula ed o he p esen sys em ollowing he me hodology employed in he
p e ious subsec ion.
Exp ession o he Feibelman pa ame e d⊥ o a cylind ical nanowi e
To ob ain he exp ession o he Feibelman pa ame e
d⊥
o a cylind ical nanowi e
in ini e along he
z
-axis, we ocus on he SRF solu ion o he elec os a ic po en ial
ϕind
(
ρ, φ, ω
)induced a he nanowi e (in he non e a ded app oxima ion). Due
o he ansla ional in a iance o he sys em wi h espec o he
z
-axis,
ϕind
only
depends upon he spa ial a iables (
ρ, φ
). The induced po en ial
ϕind
(
ρ, φ, ω
)can
hus be exp essed as:
ϕind(ρ, φ, ω) =
m=∞
X
m=−∞
ϕm(ρ, ω)eimφ,(4.6)
whe e o mally he sum ex ends om m=−∞ o m=∞.
The adial pa o he induced po en ial,
ϕm
(
ρ, ω
), is gi en by he solu ion o
103
Chap e 4. Dispe si e su ace- esponse o malism o add ess op ical nonlocali y in
si ua ions o ex eme plasmonic ield con inemen
D=2.33 nm D=2.33 nm
D=1.06 nm D=1.06 nm
Figu e 4.6: Compa ison be ween he esul s ob ained using TDDFT (solid lines), dispe si e
SRF (long-dashed lines), and nondispe si e SRF (sho -dashed lines). (a) Imagina y pa o
he i s en mul ipola pola izabili ies
αℓ
(
ω
)(
ℓ
= 1
−
10). The le -hand side panel p esen s
he compa ison be ween TDDFT and nondispe si e SRF esul s, whe eas he igh -hand side
panel p esen s he compa ison be ween TDDFT and dispe si e SRF esul s. Each spec a is
no malized o he co esponding maximum alue ob ained wi hin TDDFT o each alue o
ℓ
. The
spec a co esponding o di e en
ℓ
a e e ically displaced o isibili y. The TDDFT esul s a e
ep esen ed by solid lines wi h ha ched a ea. (b,c) Lamb shi ∆
ωQE
(le -hand side panels) and
Pu cell ac o
FP
( igh -hand side panels) ob ained wi hin he h ee me hods o a poin -dipole
quan um emi e (QE) a he cen e o a sphe ical MNP dime o adius
a
= 65
.
83
a0
(
≈
3
.
5
nm
).
The dipole is o ien ed along he dime axis, and i s ansi ion dipole momen is
µ
= 0
.
1
enm
(wi h e he elec on cha ge). In (b), he gap sepa a ion is D= 2.33 nm. In (c),D= 1.06 nm.
110

4.2. Resul s and discussion
plasmon mode appea s in each spec um o
Im{αℓ}
, which allows o he discussion
on hei ene gies and wid hs.
The TDDFT esul s o
Im{αℓ
(
ω
)
}
in Figu e 4.6a show ha he mul ipole
plasmon esonances o o de
ℓ
con inuously blueshi wi h inc easing
ℓ
in he
conside ed ange
ℓ
= 1
−
10. The esonance b oadens as
ℓ
inc eases due o he
enhancemen o su ace-enabled Landau damping [
64
,
66
,
194
,
259
–
261
]. As al eady
discussed in Chap e 3, he nondispe si e SRF accu a ely ep oduces he TDDFT
da a o
Im{αℓ
(
ω
)
}
o low alues o
ℓ∼
1
−
4, bu ails o make co ec p edic ions
o
ℓ≥
5. Indeed, o hese la ge alues o e ec i e
k∥
=
ℓ/a
, he plasmonic
esonances wi hin he nondispe si e SRF s a o edshi wi h inc easing
ℓ
in
con as o he con inuous blueshi ob ained om TDDFT calcula ions. Thus, he
nondispe si e SRF p edic s mul ipole plasmon equencies ha de ia e signi ican ly
om he TDDFT alues. In shee con as , by accoun ing o he dependence
o he Feibelman pa ame e
d⊥
on
k∥
, he dispe si e SRF co ec ly cap u es he
ene gy blueshi and b oadening o mul ipole plasmon esonances in
Im{αℓ
(
ω
)
}
(see igh -hand side panel in Figu e 4.6a). Al hough some quan i a i e di e ences
eme ge o la ge mul ipole o de
ℓ
= 7
−
10, one can obse e an o e all good
ag eemen be ween TDDFT and he dispe si e SRF esul s o e he en i e ange
o
ℓ
alues conside ed he e. Thus, he dispe si e SRF is use ul o desc ibe localized
mul ipole plasmon esonances o la ge o de ℓsus ained by small MNPs.
Finally, we add ess ano he canonical plasmonic sys em: a dime o sphe ical
MNPs. Speci ically, we s udy he case o a poin -dipole quan um emi e (QE)
loca ed a he cen e o he gap o med by wo iden ical sphe ical MNPs wi h
adius
a
= 65
.
83
a0
(
≈
3
.
5
nm
), as ske ched in Figu e 4.1c. This sys em is iden ical
o he one conside ed in Chap e 3. The QE is o ien ed along he axis o he MNP
dime ( he
z
-axis). The gap sepa a ion dis ance,
D
, is in he nanome e scale, and
hus nonlocali y s ongly in luences he op ical esponse o he sys em, as shown in
Chap e 3. We ocus on he enhancemen o he QE o al decay a e gi en by he
Pu cell ac o
FP
and he change o esonan equency ∆
ωQE
(Lamb shi ) due o
he sel -in e ac ion o he QE wi h he MNP dime . The Lamb shi is calcula ed
conside ing a ansi ion dipole momen
µ
= 0
.
1
enm
(wi h
e
he elec on cha ge).
The TDDFT and SRF esul s a e ob ained wi hin he non e a ded app oxima ion
ollowing he p ocedu e desc ibed in Chap e 3, whe e o he dispe si e SRF we
use he Feibelman pa ame e d⊥(ω, k∥)ob ained in his chap e .
Figu e 4.6b shows he Lamb shi ∆
ωQE
(le -hand side panel) and Pu cell ac o
FP
( igh -hand side panel) ob ained o a gap sepa a ion
D
= 2
.
33
nm
, as calcula ed
wi h he h ee models employed in his chap e (TDDFT, dispe si e SRF, and
nondispe si e SRF). The h ee app oxima ions show quali a i ely good ag eemen ,
al hough he nondispe si e SRF esul s sligh ly de ia e om he TDDFT and
dispe si e SRF p edic ions. Fo his ela i ely la ge gap, he exci a ion o low-
ℓ
mul ipole plasmon esonances domina es he esponse o he MNP dime o he
ield c ea ed by he poin -dipole QE (see Chap e 3), which alida es he long-
wa eleng h app oxima ion behind he nondispe si e SRF esul s. None heless, he
esul s ob ained wi hin he dispe si e SRF a e mo e accu a e when compa ed o
TDDFT.
111
Chap e 4. Dispe si e su ace- esponse o malism o add ess op ical nonlocali y in
si ua ions o ex eme plasmonic ield con inemen
The be e pe o mance o he dispe si e SRF o desc ibe he elec omagne ic
QE–MNPs in e ac ion is mo e e iden when conside ing a smalle gap, which
na u ally in ol es la ge alues o
k∥
in he esponse. Figu e 4.6c shows he Lamb
shi ∆
ωQE
and Pu cell ac o
FP
o a gap sepa a ion
D
= 1
.
06
nm
. In his
si ua ion, because o he highe spa ial con inemen o he induced cha ges a he
me al su aces ac oss he gap, plasmon modes wi h la ge mul ipola o de
ℓ
become
impo an . These la ge-
ℓ
modes ha e o e lapping esonan equencies and hus
con ibu e o a single b oad peak ( e e ed o as he pseudomode, see Chap e 3)
a
ω∼
3
.
4
eV
, as e ealed by he TDDFT calcula ions. Since he nondispe si e
model does no accu a ely desc ibe he ene gy o la ge-
ℓ
mul ipola modes o he
indi idual MNP (Figu e 4.6a), i also ails o p edic he ene gy and he wid h
o he plasmon pseudomode ob ained wi hin TDDFT o he dime o small gap.
Mo eo e , he nondispe si e SRF s ongly o e es ima es he Pu cell ac o and he
Lamb shi close o he bonding dipola plasmon (BDP) esonance a
ω∼
2
.
75
eV
because o he con ibu ion o high-
ℓ
mul ipola modes nea he BDP equency
wi hin he nondispe si e model (see Figu e 4.6a). In con as , he dispe si e SRF
p o ides accu a e esul s e en o his small gap sepa a ion, hus indica ing ha he
dispe si e SRF is well sui ed o co ec ly accoun o nonlocali y in si ua ions whe e
plasmon-induced cha ges a e cha ac e ized by a apid a ia ion in he di ec ion
pa allel o he me al su ace.
4.3 Summa y
In summa y, in his chap e we ha e p oposed a dispe si e SRF ha explici ly
accoun s o he dependence o he Feibelman pa ame e
d⊥
on he wa enumbe
pa allel o he me al su ace,
k∥
. Using TDDFT calcula ions as a e e ence,
we ha e demons a ed ha he dispe si e SRF is much mo e accu a e han he
nondispe si e SRF, usually implemen ed in he li e a u e, in desc ibing plasmonic
sys ems cha ac e ized by ex emely con ined induced ields. The dispe si e SRF
p oposed he e hus o e comes he limi a ions o he nondispe si e SRF iden i ied
in Chap e 3.
Using he analogy be ween localized mul ipola plasmons in in ini e cylind ical
nanowi es o adius
Rc
and p opaga ing su ace plasmons a plana me al–
acuum in e aces, we ha e demons a ed ha
m/Rc
can be in e p e ed as a
wa enumbe pa allel o he su ace,
k∥
=
m/Rc
(he e
m
is he magne ic quan um
numbe ). This s udy has allowed us o ob ain he dispe si e Feibelman pa ame e
d⊥
(
ω, k∥
=
m/Rc
)using cylind ical nanowi es, which is inco po a ed in o he SRF
o comple e he desc ip ion based on he long-wa eleng h alue
d⊥
(
ω, k∥
= 0), used
in Chap e 3and in o he ecen wo ks [124,129,296,299,300].
Suppo ed by he examples o cylind ical and sphe ical me allic nanos uc u es,
we ha e demons a ed ha , in con as o he nondispe si e model, he dispe si e
SRF accu a ely desc ibes he nonlocal op ical esponse in ex eme si ua ions whe e
he induced cha ges a e cha ac e ized by a apid a ia ion in he di ec ion pa allel
112
4.3. Summa y
o he me al su ace (la ge
k∥
). The esul s shown in his chap e hus con ibu e o
he de elopmen o a heo e ical model ha cap u es quan um nonlocal e ec s in
ex eme si ua ions, while keeping he nume ical e iciency and easy implemen a ion
in o he amewo k o classical elec omagne ic heo ies [
348
]. We hus belie e ha
he dispe si e SRF p oposed in his chap e can be use ul o co ec ly accoun
o op ical nonlocali y in nanos uc u ed sys ems wi h ex eme plasmonic ield
con inemen , as i can be he case o me allic nanos uc u es in e ac ing wi h
as elec ons, MNPs coupled o QEs in close p oximi y, o MNPs ensembles
wi h ex emely na ow junc ions. The dispe si e SRF subs an ially imp o es
he pe omance o he nondispe si e SRF o desc ibe nonlocali y in he op ical
esponse o na ow junc ions, howe e i s ill lacks he desc ip ion o cha ge- ans e
p ocesses. To accoun o such e ec s by using semiclassical models, i would be
necessa y o u he de elop he p esen amewo k combining he SRF wi h e.g. a
quan um-co ec ed model [83–87].
113
Chap e
5
ELECTRONIC EXCITON–PLASMON
COUPLING IN A NANOCAVITY
BEYOND THE ELECTROMAGNETIC
INTERACTION PICTURE
The main mechanism ha con ols he in e ac ion be ween quan um emi e s (QEs)
and me allic nanopa icles (MNPs) is he exci a ion o he QE exci on by he local
elec ic ield associa ed o he MNP plasmon. In p e ious chap e s, he plasmonic
esponse o MNPs is desc ibed ei he classically (Chap e 1) o by using TDDFT
simula ions ha cap u e nonlocal and quan um su ace e ec s (Chap e 3), while
he exci on dynamics o he QE is modeled wi hin he poin -dipole app oxima ion.
The success o he me hodologies used in p e ious chap e s o explain he main
ea u es o he op ical esponse in plasmonic nanoca i ies is due o he dominance
o he elec omagne ic in e ac ion in he QE–MNP coupling o sepa a ions as small
as one nanome e [
145
,
349
]. Howe e , a e en smalle sepa a ions be ween emi e s
and me al su aces, o he o de o Ångs oms, ano he quan um e ec becomes
impo an : elec onic s a es localized a he QE and a he MNPs hyb idize in o
"supe molecula " s a es which modi y op ical ansi ions, allowing o elec on
ans e be ween he QE and he MNP.
Despi e i s impo ance [
166
,
349
–
352
], he e ec o hyb idiza ion be ween he
QE and he MNP elec onic s a es as well as he co esponding elec on- ans e
p ocesses emain la gely unexplo ed in Nanopho onics, as he quan um heo e ical
ea men o he p oblem is challenging. I is only ecen ly ha such s udies ha e
become wi hin he each o heo e ical e o s [
107
–
109
,
323
,
329
] enabling e.g. a
be e unde s anding o ligh emission in unneling junc ions [
309
–
311
]. No ably,
i has been shown ha a QE b idging wo MNPs can igge elec on conduc ance
115

Chap e 5. Elec onic exci on–plasmon coupling in a nanoca i y beyond he
elec omagne ic in e ac ion pic u e
ac oss subnanome ic junc ions, which s ongly in luences he op ic and elec onic
(op oelec onic) esponse o he coupled sys em [107–109,329,353].
In his chap e , we apply a ully quan um many-body app oach based on
TDDFT o s udy he op oelec onic esponse and exci on dynamics in a QE–MNPs
sys em whe e he QE is loca ed a subnanome ic sepa a ion om he me allic
in e aces. In con as wi h he me hodologies employed in p e ious chap e s, he e
we use a TDDFT ea men o desc ibe he elec onic s uc u e o bo h he QE
and he MNPs. We place pa icula emphasis on he ole o elec onic coupling
and elec on ans e be ween he QE and he MNPs o un eil he mani es a ion
o hese quan um e ec s in he op ical esponse o he en i e coupled sys em.
Impo an ly, we demons a e ha he modi ica ion o he elec onic s uc u e o
he hyb id QE–MNPs sys em as well as he b oadening o he elec onic s a es o
he QE due o cha ge ans e lead o a b eakdown o he classical elec omagne ic
desc ip ion o plasmon–exci on in e ac ion. We e eal impo an quan i a i e and
quali a i e di e ences be ween quan um TDDFT and classical LRA esul s o he
linewid hs and equencies o he ele an op ical modes. Mo eo e , we also obse e
he o ma ion o a no el cha ge- ans e plasmon mode a low equencies media ed
by he emi e elec onic s uc u e.
5.1 Sys em and me hods
We conside a QE in e ac ing wi h a plasmonic dime o med by wo sphe ical
MNPs. As ske ched in Figu e 5.1a, he QE placed in he middle o a plasmonic
nanogap is illumina ed by a plane wa e pola ized along he dime axis (
z
-axis). In
his chap e , he gap sepa a ion
D
is a ied o explo e di e en egimes o elec onic
QE–MNPs coupling, anging om elec onically decoupled QE–MNPs (la ge
D
) o
elec onically coupled ones (small
D
). The calcula ion o he op ical esponse is
pe o med wi hin he Kohn–Sham (KS) scheme o ime-dependen densi y unc ional
heo y [
70
–
72
,
189
] (TDDFT) as in oduced in Sec ion 2.2, which success ully
inco po a es quan um phenomena such as many-body and single elec on–hole
pai exci a ions, elec onic spill-ou , nonlocal sc eening o elec on unneling in
(sub)-nanome ic me allic ca i ies [79,125,202,211,212,214,337,338,342].
The elec onic s uc u e o he MNPs is desc ibed wi hin he jellium model
o ee-elec on me als [
194
,
200
] in oduced in Subsec ion 2.1.2, using a Wigne –
Sei z adius o
s
= 4
a0
ha co esponds o sodium. Each MNP con ains 638
conduc ion elec ons ( adius
a
= 34
.
4
a0≈
1
.
8
nm
), and he Fe mi le el o he
MNPs s ands a
EF
=
−
2
.
86
eV
below he acuum le el. In con as o he
poin -dipole app oxima ion employed in p e ious chap e s o model he QE (e.g.
in Chap e 3), he e we conside a “mo e ealis ic” QE ha has a ini e spa ial
ex ension. The elec onic s uc u e o he QE is desc ibed as a wo-le el sys em
using a model po en ial
VQE
(
)(see below). The op ical esponse o he coupled
QE–MNPs is add essed using he wa e-packe p opaga ion (WPP) me hod in
cylind ical coo dina e sys em, as in oduced in Subsec ion 2.3.2. To exci e he
sys em, we apply an ex e nal po en ial
Vex
(
,
) =
E0
∆
z δ
(
)[Eq.
(2.67)
], which
116
5.1. Sys em and me hods
0
0.04
0.08
0.12
0.16
0=1 eV
V0=3 eV
V0=5 eV
V
Figu e 5.1:(a) Ske ch o he sys em s udied in his chap e . A QE wi h a single op ically-allowed
HOMO–LUMO ansi ion is loca ed in he middle o a gap o size
D
o med by wo sphe ical
MNPs. The QE and each MNP con ain 2 and 638 conduc ion elec ons, espec i ely. (b) E ec i e
one-elec on po en ial
Ve
(
)( op) and equilib ium elec on densi y
n0
(
)(bo om) along he
symme y
z
-axis o he coupled QE–MNPs sys em wi h gap size
D
= 26
a0
. The HOMO and
LUMO ene gy le els o he isola ed QE a e ep esen ed by ed and g een lines, espec i ely. The
Fe mi le el
EF
=
−
2
.
86
eV
o he MNPs is shown by he black dashed line. (c) Abso p ion
c oss-sec ion spec a
σabs
(
ω
)o he isola ed MNP dime o
D
= 26
a0
(blue line) and
D
= 38
a0
(g een). Dashed and solid lines co espond o he esul s ob ained wi h classical LRA and TDDFT
simula ions, espec i ely. (d) Abso p ion c oss-sec ion spec a
σabs
(
ω
)o he isola ed QE o
di e en alues o he pa ame e
V0
[Eq.
(5.3)
] used o con ol op ical and elec onic p ope ies
o he QE.
co esponds o plane-wa e illumina ion pola ized along he z-axis.
P io o s udying he coupled QE–MNPs sys em, we summa ize in Figu e 5.1c
he TDDFT esul s o he abso p ion spec a o he isola ed MNP dime o adius
a
= 34
.
4
a0≈
1
.
8
nm
conside ed in his chap e (solid lines). Gap sepa a ion
dis ance o
D
= 38
a0
(g een) and
D
= 26
a0
(blue) a e conside ed he e. The
op ical esponse o he MNP dime is cha ac e ized by a bonding dipola plasmon
(BDP) esonance a
ωBDP ∼
3
eV
. As expec ed om he esul s shown in Chap e 1
and Chap e 2, his BDP mode edshi s when educing he gap sepa a ion because
o he inc eased capaci i e coupling be ween he wo MNPs [
354
]. As a e e ence, in
Figu e 5.1c we also show he esul s om classical (non e a ded) LRA calcula ions
(dashed lines) in oduced in Sec ion 1.1. The MNPs a e desc ibed in his case
wi h a D ude dielec ic unc ion [Eq.
(1.10)
] using an “e ec i e” plasma equency
ωp
= 5
.
43
eV
and in insic damping pa ame e
γp
= 0
.
15
eV
. These pa ame e s a e
chosen o p o ide easonably good ag eemen be ween he TDDFT and classical
117
Chap e 5. Elec onic exci on–plasmon coupling in a nanoca i y beyond he
elec omagne ic in e ac ion pic u e
LRA esul s o he abso p ion spec um o he indi idual MNP (no shown) [
125
].
Using his alue o
ωp
= 5
.
43
eV
(i.e., sligh ly smalle han he nominal bulk plasma
equency
ωp
= 5
.
89
eV
o sodium o
s
= 4
a0
,Eq.
(1.12)
) allows us o accoun
o he edshi o he dipola plasmon (DP) equency o he small indi idual MNP
because o he elec on spill-ou and dynamical sc eening in oduced in Subec ion
2.3.1. In addi ion, he damping pa ame e
γp
= 0
.
15
eV
used in LRA in his
chap e accoun s o all he decay channels o he plasmon exci a ion including
he con ibu ion o su ace-enabled Landau damping. Fu he , in all classical LRA
calcula ions we also in oduce a gap scaling o ∆ = 3
.
4
a0
o (pa ially) accoun
o he spill-ou o he induced elec on densi y wi h espec o he geome ical
su ace o he MNPs in he dime con igu a ion (co ec ly cap u ed by he TDDFT
simula ions, as shown in Subsec ion 2.3.2). In oducing he gap scaling ∆=3
.
4
a0
is simila o conside ing he Feibelman pa ame e
d⊥
(
ω
)employed in Chap e 3and
Chap e 4, as epo ed in e .
58
. In his hesis we ound ha his p ocedu e is alid
o ep oduce he edshi o he BDP o he MNP dime , howe e i o e s ima es
he s eng h o he highe -o de plasmon modes such as he BQP. Thus, nonlocal
and quan um e ec s a e pa ially in oduced in he classical LRA calcula ions o
his chap e in an e ec i e manne , which allows us o co ec ly ep oduce he
TDDFT spec a o he isola ed MNP dime .
We nex in oduce in Subsec ion 5.1.1 he TDDFT desc ip ion adop ed in his
chap e o model he QE, which allows o s udying he e ec o he elec onic
in e ac ion be ween he QE and MNPs s a es.
5.1.1
Cha ac e iza ion o he model quan um emi e (QE)
The elec onic s uc u e o he QE is desc ibed wi hin he ee-elec on jellium
model (Subsec ion 2.1.2), in a simila way as we model he MNPs. We conside a
sphe ical QE o adius
RQE
= 5
a0
(
≈
0
.
26
nm
)con aining wo alence elec ons.
The spin- es ic ed case is conside ed [
355
]. In he g ound-s a e con igu a ion o
he QE, he o al spin is ze o, and he 2 elec ons wi h opposi e spins occupy he
same Kohn–Sham (KS) alence o bi al. F om he cha ge neu ali y condi ion, he
posi i e backg ound densi y
n+
[Eq.
(2.13)
] ep esen ing he a omic co es o he
QE and sp eading o e i s spa ial ex en sa is ies
4
3πR3
QEn+= 2.(5.1)
The occupied (
j
= 1) and unoccupied (
j
= 2
,
3
,
4
, . . .
) one-elec on KS o bi als
Ψ
0
QE,j
(
)o he QE and hei ene gies
ϵQE,j
a e ob ained om he ime-independen
KS equa ion o DFT [Eq. (2.5)],
ˆ
H[n0
QE( )]Ψ0
QE,j( ) = ϵQE,jΨ0
QE,j( ),(5.2)
whe e
ϵQE,j
a e he one-elec on ene gy le els o he QE, and we use sphe ical
coo dina es as desc ibed in Subsec ion 2.3.1. The equilib ium elec on densi y o
he isola ed QE,
n0
QE
(
), is gi en in his case by
n0
QE
(
)=2
|
Ψ
0
QE,1
(
)
|2
, wi h he
118
5.1. Sys em and me hods
ac o 2 accoun ing o spin degene acy.
Con olling he alues o he one-elec on ene gy le els
ϵQE,j
o he QE s a es
allows us o s udy di e en si ua ions o he QE–MNPs coupling, whe e he QE
exci on can be ei he in esonance o ou o esonance wi h he main BDP esonance
o he MNP dime a
ωBDP ∼
3
eV
(see Figu e 5.1c). To his end, we in oduce in
he Hamil onian
ˆ
H
[
n0
QE
(
)] [Eq.
(5.2)
] an addi ional a ac i e po en ial
VQE
(
)
[
216
]. This a ac i e po en ial
VQE
(
)can be hough o as a pseudopo en ial due
o he a omic co es, i is localized in he spa ial egion o he QE, and i is gi en
by
VQE( ) = −V0e−4| |2/R2
QE ,(5.3)
The one-elec on ene gy le els
ϵQE,j
o he QE s a es can be hus modi ied by
changing he pa ame e V0.
In his chap e , we e e o Ψ
0
QE,1
(
)as he highes occupied molecula o bi al
(HOMO), which is a 1
s
(
ℓ
= 0
, m
= 0) o bi al wi h ze o o bi al momen um (
ℓ
) and
magne ic quan um numbe (
m
). Thus, he elec onic con igu a ion o he QE is 1
s2
.
The ene gy le el o he HOMO is
EHOMO
=
ϵQE,1
. Fo he alues o
V0
[Eq.
(5.3)
]
conside ed in his chap e , we ind only h ee ene gy-degene a e unoccupied KS
o bi als accessible o op ical ansi ions om he g ound s a e. These o bi als
co espond o he 2
p
-shell and a e cha ac e ized by he o bi al momen um
ℓ
= 1
and magne ic quan um numbe s
m
= 0
,±
1. Wi h
ψQE,2p
(
) he adial pa o he
KS o bi al o he isola ed QE, we can de ine
Ψ0
QE,2( ) = Y0
1(θ, φ)ψQE,2p( ),
Ψ0
QE,3( ) = Y−1
1(θ, φ)ψQE,2p( ),
Ψ0
QE,4( ) = Y1
1(θ, φ)ψQE,2p( ),
(5.4)
whe e
Ym
ℓ
(
θ, φ
)a e he sphe ical ha monics. Because o he symme y o he sys em
conside ed in his chap e (see Figu e 5.1a), wi h an inciden elec omagne ic wa e
pola ized along he
z
-axis, he elec onic ansi ions p ese e he magne ic quan um
numbe
m
, and hus a e e ec i e be ween he g ound-s a e 1
s
KS o bi al and he
2
p
(
ℓ
= 1
, m
= 0) KS o bi al Ψ
0
QE,2
(
). Thus, he op ical abso p ion o he QE is
de e mined by he 1
s→
2
p
ansi ion. Fo he sake o simplici y, in his chap e
we e e o Ψ0
QE,2( )as he lowes unoccupied molecula o bi al (LUMO).
The ee pa ame e
V0
o he po en ial
VQE
(
)[Eq.
(5.3)
] is used o con ol he
ene gy le els o he HOMO (
EHOMO
=
ϵQE,1
) and he LUMO (
ELUMO
=
ϵQE,2
)
o he QE, as schema ically depic ed in Figu e 5.1a by he ed and g een lines,
espec i ely. As a consequence, he pa ame e
V0
also de e mines he oscilla o
s eng h
α0
QE
and ansi ion equency
ωQE
o he QE exci on [Eq.
(1.54)
] ele an
in he op ical esponse o he coupled QE–MNPs sys em. We show in Table
5.1 he ene gy le els
EHOMO
and
ELUMO
, as well as he oscilla o s eng h
α0
QE
and ansi ion equency
ωQE
o he QE exci on, ob ained o he h ee
di e en alues o he backg ound po en ial
V0
= 1
eV,
3
eV,and
5
eV
[Eq.
(5.3)
]
conside ed in his chap e . The alues o
EHOMO
and
ELUMO
a e di ec ly ob ained
om ime-independen DFT calcula ions ollowing he p ocedu e desc ibed in
119
Chap e 5. Elec onic exci on–plasmon coupling in a nanoca i y beyond he
elec omagne ic in e ac ion pic u e
The classical LRA esul s o
σabs
(
ω
)in Figu e 5.5a show a spli ing be ween
he LR (blue do s) and he UR (g een do s) wi h espec o he esonan equency
ωQE
o he isola ed QE al eady o la ge gap sepa a ion dis ance
D
= 40
a0
. This
LR–UR spli ing is a signa u e o he s ong coupling be ween he QE exci on and
he BDP esonance o he MNP dime , as we u he con i m by analyzing in Figu e
5.5b he coupling s eng h
g
. The coupling s eng h
g
is ob ained by i ing he
classical LRA esul s o
σabs
(
ω
) o he spec a ob ained om he coupled ha monic-
oscilla o model in oduced in Sec ion 1.4.3 [Eq.
(1.60)
]. Figu e 5.5b shows ha ,
o he la ges dis ance conside ed (
D
= 40
a0
), he c i e ion
g >
(
γMNP
+
γQE
)
/
4
o en used o iden i y s ong coupling is sa is ied. Wi h dec easing
D
, he LR–UR
equency di e ence in Figu e 5.5a s ongly inc eases wi hin he classical LRA
model owing o he s onge elec omagne ic coupling be ween he exci on and
he plasmon. This is consis en wi h he inc ease o
g
, obse ed in Figu e 5.5b,
which sa is ies he mo e es ic i e c i e ion
g >
(
γMNP
+
γQE
)
/
2 o smalle gaps
D≤
32
a0
. Mo eo e , wi hin he classical LRA amewo k, he highe -o de
plasmonic modes o he MNPs con ibu e o he elec omagne ic in e ac ions
be ween he dime and he QE o small gap sepa a ions. The e ec o hese
highe -o de modes, well documen ed o isola ed dime an ennas [
23
], esul s he e
in an addi ional edshi o bo h he LR and UR b anches, which explains why he
UR b anch appea s a lowe equencies han he exci on equency
ωQE
= 2
.
95
eV
o he isola ed QE. Resul s in Figu e 5.5a,b hus demons a e ha he esonan
QE–MNPs sys em would be in he s ong-coupling egime acco ding o he classical
LRA desc ip ion.
The TDDFT esul s o
σabs
(
ω
)in Figu e 5.5c also show a spli ing be ween he
LR and he UR wi h espec o
ωQE
o la ge gap sepa a ion dis ance
D
= 40
a0
,
al hough he s eng h o he LR is weake han he one p edic ed by he classical
model. This di e ence is a consequence o he nonlocali y and ini e-size e ec s
(in oduced in Sec ion 2.3) ha a ec he op ical esponse o he isola ed MNP
dime , as con i med by he analysis shown in Subsec ion 5.2.2 below. The gene al
simila i y be ween classical and TDDFT esul s o la ge sepa a ion occu s because
he e is no QE exci on quenching p oduced by elec onic hyb idiza ion o such
la ge sepa a ion. A hose dis ances he elec onic QE–MNPs coupling does no
play a ole. Howe e , upon educing he gap size D, he elec onic o bi als o he
QE hyb idize wi h hose o he MNPs. As shown in Figu e 5.5d, he LUMO e ol es
in o a b oad s uc u e e lec ing he as ans e o he exci ed elec on be ween
he LUMO o he QE and he conduc ion-band s a es o he MNPs quan ized by
he ini e-size e ec . This elec onic in e ac ion has an immedia e consequence on
he op ical esponse o he s ongly coupled QE–MNPs sys em since i hinde s
he ene gy ans e be ween he QE and he MNPs, hus a enua ing he UR–LR
spli ing in exci on-plasmon pola i on sys ems as well as p oducing a p og essi e
me ging o he LR and UR b anches in o a b oad spec al ea u e when dec easing
gap sepa a ion
D
. In his si ua ion, s ong elec omagne ic coupling is, he e o e,
us a ed due o elec onic QE–MNPs coupling.
126

5.2. Resul s and discussion
34 a0
38 a0
30 a0
26 a0
22 a0
18 a0
34 a0
38 a0
30 a0
26 a0
22 a0
18 a0
Figu e 5.6: Abso p ion c oss-sec ion spec a
σabs
(
ω
)o he s udied QE–MNPs sys em, as ob ained
om he semiclassical model employed in his sec ion. This semiclassical app oach is based on
he sel -in e ac ion G een’s unc ion [Eq.
(1.57)
] and conside s ha he QE is a classical poin
dipole, howe e
ˆ
G
(
QE, QE, ω
),
Eind
(
=
QE, ω
),
ˆαMNP
(
ω
)and
ˆαQE
MNP
(
ω
)a e ob ained om
TDDFT simula ions o he isola ed MNP dime . Resul s a e shown as a unc ion o he equency
ω
o gap size anging om
D
= 16
a0
o
D
= 40
a0
in s eps o 2
a0
. Panel (a) co esponds o
he esul s ob ained o ωQE = 2.58 eV, and panel (b) o ωQE = 2.95 eV.
5.2.2 Quan um ini e-size e ec s s. elec onic QE–MNPs
coupling
In o de o gain a be e unde s anding o he ole played by he elec onic QE–
MNPs coupling in he op ical esponse, and o disca d nonlocal and ini e-size
e ec s as s udied in Chap e 3in connec ion wi h QE exci on quenching, we
apply he e a semiclassical
27
app oach o he cu en QE–MNPs sys em. This
semiclassical app oach adop s he sel -in e ac ion G een’s unc ion o malism (see
Subsec ion 1.4.2), whe e he QE is in oduced as a classical poin dipole. The dipole
momen s induced a he MNPs and a he QE a e hen ob ained om Eq. (1.57).
Howe e , he quan i ies
ˆ
G
(
QE, QE, ω
),
Eind
(
=
QE, ω
),
ˆαMNP
(
ω
)and
ˆαQE
MNP
(
ω
)
a e ob ained om he TDDFT simula ions o he MNP dime , as desc ibed in
Subsec ion 2.3.2 and Chap e 3. This semiclassical app oach na u ally includes
ini e-size e ec s on he esponse o he MNP dime such as elec on spill-ou ,
nonlocali y, su ace-enabled Landau damping, and single elec on–hole ansi ions.
Howe e , since he QE is in oduced as a classical poin dipole, he elec onic
coupling be ween he MNPs and he QE, as well as he ac ual elec onic s uc u e
o he QE and he ini e-size ex ension o i s ansi ion densi y a e no accoun ed
o . Thus, we expec ha he di e ences be ween TDDFT and he semiclassical
app oach e eal he e ec o elec onic hyb idiza ion, only accoun ed o wi hin
he ully quan um TDDFT model.
We show in Figu e 5.6 he abso p ion c oss-sec ion
σabs
(
ω
)ob ained wi hin
he semiclassical model employed in his subsec ion o
ωQE
= 2
.
58
eV
(panel a)
27
The semiclassical app oach employed in his chap e should no be con used wi h he
semiclassical SRF employed in Chap e 3and Chap e 4.
127
Chap e 5. Elec onic exci on–plasmon coupling in a nanoca i y beyond he
elec omagne ic in e ac ion pic u e
and
ωQE
= 2
.
95
eV
(panel b). Fo la ge gap sepa a ion dis ances,
D∼
30
−
40
a0
,
he semiclassical esul s shown in Figu e 5.6 coincide wi h he esul s o TDDFT
calcula ions displayed in Figu e 5.2b (
ωQE
= 2
.
58
eV
) and Figu e 5.5c (
ωQE
=
2
.
95
eV
). In pa icula , consis en wi h he discussion o he p e ious subsec ion,
he LR calcula ed o he esonan case
ωQE
= 2
.
95
eV
wi hin bo h he semiclassical
model and TDDFT is conside ably weake han he one p edic ed by he classical
LRA app oach (Figu e 5.5a). Thus, he o igin o he di e ence be ween he TDDFT
and he classical LRA abso p ion spec a esides on quan um su ace e ec s ha
a e impo an o such small MNPs. Indeed, a la ge
D
he e is no hyb idiza ion
be ween he elec onic s a es localized a he QE and a he MNPs.
As al eady discussed, he elec onic QE–MNPs coupling s ongly a ec s he
abso p ion spec a o he sys em o
D
below
D∼
26
a0
, which is now u he
co obo a ed om he compa ison be ween he semiclassical esul s in Figu e 5.6,
he TDDFT esul s in Figu e 5.2b and Figu e 5.5c, and he classical esul s in
Figu e 5.2a and Figu e 5.5a. Fi s , TDDFT shows subs an ial b oadening and
educ ion o he ampli ude o he UR e ol ing om he BDP o he MNP dime
as compa ed o bo h semiclassical and classical esul s. These e ec s, no cap u ed
by he semiclassical model, a e a ibu ed o he cha ge- ans e p ocesses be ween
he MNPs. In ou sys em, elec on anspo can occu a la ge gap sepa a ions as
compa ed o ypical acuum junc ions (see Subsec ion 2.3.2) because i is assis ed
by pho oexci ed elec on ans e h ough he LUMO o he QE [356].
On he o he hand, he semiclassical model in Figu e 5.6 p edic s a con inuous
edshi o he LR o he o - esonan case (panels a), and inc easing LR–UR
spli ing o he esonan case (panel b) wi h dec easing gap size
D
, consis en wi h
he classical LRA p edic ions. As compa ed o hese classical esul s, he main
di e ence is ha he semiclassical model shows a weakening and a b oadening o
he LR upon dec easing
D
, which poin s owa d he ole o nonlocal op ical e ec s
ha can also a ec he elec omagne ic esponse o such a small sys em [
336
] as
discussed in de ail in Chap e 3. The e o e, he compa ison o he esul s ob ained
wi hin he classical LRA, he semiclassical model, and he TDDFT app oach as
employed in his chap e allows us o conclude ha he blueshi o he LR o
dec easing D below
D≤
26
a0
o he o - esonan QE–MNPs sys em, as well as he
weakening o he LR–UR spli ing o he esonan case, a e only obse ed when
he hyb idiza ion be ween he MNPs and he QE elec onic o bi als is possible,
i.e., when he (exci ed) elec on can unnel ac oss he sys em.
5.2.3 Cha ge- ans e esonances a low equencies
Finally, we discuss in Figu e 5.7 he ole o he QE in igge ing elec on anspo
be ween he wo MNPs ac oss he junc ion in esponse o ex e nal illumina ion.
In he las yea s, se e al wo ks ha e iden i ied he eme gence o cha ge- ans e
plasmons (CTP) suppo ed by me allic acuum junc ions o gap sepa a ions
ypically below
∼
0
.
4
nm
and esonan equencies o he o de o a ew elec on ol s
[
43
,
68
,
83
,
84
,
86
,
342
,
357
]. CTP a e plasmonic esonances whe e a ne elec on
anspo occu be ween he MNPs ha o m he nanogap (see Subsec ion 2.3.2).
128
5.2. Resul s and discussion
12 a0
14 a0
16 a0
18 a0
isola ed dime
0=1 eV
V
V0=5 eV
Figu e 5.7:(a) Abso p ion spec a o he hyb id QE–MNPs sys em o low illumina ion equencies
ω
= 0
−
1eV. Resul s a e shown o a gap size
D
anging om
D
= 12
a0
o
D
= 18
a0
, as indica ed
in he inse . The e e ence abso p ion spec um o he isola ed MNP dime o
D
= 12
a0
is shown
by he dashed black line. The si ua ions o
V0
= 1
eV
( op) and
V0
= 5
eV
(bo om) in Eq.
(5.3)
a e conside ed. (c) Colo maps o he induced elec on densi y (le ) and he elec on-cu en
densi y along he
z
-di ec ion ( igh ) o an inciden
z
-pola ized elec omagne ic plane wa e o
equency
ωCT
= 0
.
11 eV. The gap dis ance is
D
= 16
a0
, and
V0
= 1
eV
. On he igh -hand
side panel, he bounda ies o he jellium edges o he MNPs a e indica ed by dashed lines. The
snapsho s a e aken a he ins an s o ime when he absolu e alue o he o al dipole momen
(le -hand side panel) and o he elec on-cu en densi y in he middle o he junc ion ( igh -hand
side panel) a e maximum. (c) G ound-s a e po en ial
Ve
along he symme y
z
-axis o he
hyb id QE–MNPs sys em o D= 16 a0.
Mo eo e , as poin ed ou in p e ious wo ks [
107
–
109
,
329
,
353
], he p esence o a QE
b idging a me allic nanogap subs an ially modi ies he cha ge- ans e p ope ies o
he sys em and igge s ou he eme gence o low- equency esonances associa ed
wi h elec on anspo be ween he MNPs.
In ou s udy, he abso p ion c oss-sec ion
σabs
(
ω
)shown in Figu e 5.7a o
wo di e en QEs cha ac e ized by
V0
= 1
eV
( op) and
V0
= 5
eV
(bo om)
e eals ha a cha ge- ans e esonance eme ges in he low- equency egion,
ωCT ∼
0
.
1
−
0
.
2
eV
, o gap sizes
D
= 12
a0−
18
a0
(
D≈
0
.
6
−
0
.
95
nm
, hus
la ge han ypical unneling dis ances in me al– acuum–me al junc ions s udied
in Subsec ion 2.3.2). Ou esul s a e consis en wi h he indings epo ed in he
li e a u e [
74
,
107
,
108
,
353
]. This new esonance is only ac i a ed due o he
p esence o he QE (see he esponse o he isola ed dime depic ed by he dashed
line), and i blueshi s and s eng hens conside ably when dec easing in e pa icle
129
Chap e 5. Elec onic exci on–plasmon coupling in a nanoca i y beyond he
elec omagne ic in e ac ion pic u e
dis ance. The cha ge- ans e cha ac e o he mode is clea ly e ealed by he
induced elec on densi y shown in Figu e 5.7b (le -hand side panel), wi h each
MNP exhibi ing a monopola elec on densi y pa e n o opposi e sign, and i is
u he co obo a ed by he elec on-cu en densi y along he
z
-di ec ion ( igh -
hand side panel), which clea ly shows ha elec ons shu le om one MNP o
ano he . In he s udied QE–MNPs sys em, he obse ed cha ge- ans e esonances
a
ωCT ∼
0
.
1
−
0
.
2
eV
eme ge because o gap sepa a ions o
D∼
18
a0
and below,
he QE gi es ise o a dec ease o he po en ial ba ie close o he dime axis
below he Fe mi le el o he sys em (Figu e 5.7c), so ha e en a classically-allowed
o e - he-ba ie elec on anspo be ween he MNPs becomes possible. Thus,
he low- equency cha ge- ans e plasmon epo ed he e can be unde s ood as a
consequence o he ballis ic elec on anspo and does no equi e he unneling
mechanism aid by a localized s a e a he QE [109].
5.3 Summa y
In summa y, in his chap e we ha e iden i ied he ole played by elec onic
coupling in he op ical esponse o a canonical hyb id sys em consis ing in a wo-
le el quan um emi e (QE) placed in a nanogap o med by wo sphe ical me al
nanopa icles (MNPs). Using a ully TDDFT model o bo h he QE and he
MNPs, we ha e demons a ed he quenching o he QE exci on o igina ed by he
hyb idiza ion o he exci ed s a es localized a he QE and he elec onic s a es o
he MNPs. This exci on quenching d as ically a ec s he op oelec onic esponse
o he hyb id QE–MNPs sys em o small gap sepa a ions. Fo example, i gi es
ise o a blueshi o he lowe esonance (LR) wi h dec easing gap sepa a ion
dis ance below
D≲
26
a0
o si ua ions whe e he QE ansi ion equency
ωQE
is
ou o esonance wi h he main plasmonic mode o he MNP dime , in con as o
he classical LRA calcula ions ha p edic a con inuous edshi . Fu he , exci on
quenching p oduced by he elec onic in e ac ion also leads o a d as ic a enua ion
o he LR–UR spli ing in esonan QE–MNPs sys ems, hus us a ing he s ong
coupling p edic ed by classical LRA simula ions. On he o he hand, deple ion o
he po en ial ba ie wi hin sub-nanome ic gaps due o he p esence o he QE
gi es ise o a low- equency elec on- ans e esonance a
ωCT ∼
0
.
2
eV
, e en o
si ua ions whe e he elec onic s a es o he QE do no ac as a ga eway o elec on
anspo be ween he MNPs. Ou indings a e expec ed o quali a i ely apply
o plasmon–exci on sys ems i espec i e o he speci ic elec onic s uc u e o he
nanocons i uen s, since hey a e based on gene al and obus quan um-mechanical
phenomena such as elec on unneling and elec on ans e be ween he MNPs
and he QE. Thus, he esul s ob ained in his chap e s ess he need o conside
he QE–MNPs elec onic coupling, in addi ion o he s anda d elec omagne ic
in e ac ion, in o de o un eil undamen al quan um e ec s ela ed o cha ge
ans e , o en a ec ing p ac ical implemen a ion o nanoscale sou ces o pho on
emission and op oelec onic nanode ices.
130
Chap e
6
SECOND-HARMONIC GENERATION
FROM A QUANTUM EMITTER
COUPLED TO A METALLIC
NANOPARTICLE
In p e ious chap e s, we ocused on he analysis o quan um e ec s eme ging in he
op ical and elec onic esponse o plasmonic sys ems in si ua ions whe e he in ensi y
o he ex e nal illumina ion is weak and hus he esponse is linea . Howe e , when
he in ensi y o he ex e nal illumina ion is s ong, he exci a ion o plasmonic
esonances in me allic nanopa icles (MNPs) can also lead o nonlinea e ec s ha
can be use ul o (bio-)imaging [
358
–
360
] o o gene a ion o ex eme-ul a iole
a osecond lase pulses [
361
], among o he s [
36
,
37
]. In pa icula , second-ha monic
gene a ion (SHG), whe eby wo pho ons a he undamen al equency a e abso bed
o emi one pho on a he second-ha monic equency, is a he ocus o e y ac i e
esea ch owing o i s p ac ical and undamen al in e es [
362
–
371
]. In his con ex ,
i has been shown ha plasmonic nanos uc u es esonan a he undamen al o a
he second-ha monic equency (o a bo h equencies) can gi e ise o conside able
enhancemen o SHG [
294
,
367
,
368
,
372
–
383
]. Recen expe imen s ha e also shown
he pola iza ion- esol ed p obing o he nonlinea nea - ield dis ibu ion o me allic
nanos uc u es by using doubly esonan plasmonic an ennas [
384
]. To achie e
SHG, howe e , he symme y o he sys em needs o be conside ed. Fo example,
we show in Sec ion 1.3 and Sec ion 2.4 ha , o ypical plane-wa e incidence,
SHG is o bidden om nanos uc u es ha a e cen osymme ic. This nonlinea
esponse is hus e y sensi i e o he geome y o he sys em and o su ace e ec s
ha may e en ually b eak he symme y cons ain s and lead o he emission o
ligh a he second-ha monic equency [37,68,154,385–388].
131

Chap e 6. Second-ha monic gene a ion om a quan um emi e coupled o a
me allic nanopa icle
In his chap e , we s udy SHG om a coupled sys em consis ing o a quan um
emi e (QE) placed in he icini y o a sphe ical MNP [
111
,
389
,
390
], as scke ched
in Figu e 6.1a. The small indi idual cen osymme ic MNP does no allow o
second-ha monic emission in o he a ield, bu i c ea es second-ha monic nea
ields in he p oximi y o he MNP su ace. The p esence o he QE li s he
symme y cons ain s and allows o SHG. When he elec onic ansi ion equency
o he QE,
ωQE
, is esonan wi h he second ha monic o he inciden equency, he
QE plays he ole o an op ical esona o , which e icien ly couples o he nonlinea
elec ic nea ield induced close o he MNP (see Sec ion 2.4), ansduces his nea
ield in o he a ield, and hus p oduces SHG [
384
]. This QE–MNP sys em hus
enables equency con e sion and allows o i s con ol. To calcula e he nonlinea
esponse o he coupled sys em and o e eal he physical mechanisms behind
SHG in his si ua ion, we use TDDFT calcula ions [
70
,
71
] based on he wa e-
packe p opaga ion (WPP) me hod in oduced in Sec ion 2.2. Wi h he insigh s
ob ained om he TDDFT simula ions, we de elop a semi-analy ical model ha
accu a ely ep oduces he TDDFT esul s. This semi-analy ical model also allows
o add essing mo e gene al and complex si ua ions beyond he each o TDDFT,
making possible a de ailed s udy o he sensi i i y o SHG o di e en pa ame e s
ha cha ac e ize he sys em. In pa icula , we demons a e he pola iza ion
con e sion o he nonlinea signal, as well as he exis ence o a ious egimes o
SHG de e mined by he in insic losses o he QE. The me hodology and esul s
p esen ed in his chap e can pa e he concep ual oad o enhancing and op imizing
SHG media ed by QEs coupled o plasmonic sys ems [391,392].
6.1 Sys em and me hods
We conside a QE loca ed in he p oximi y o a sphe ical sodium MNP. The MNP
is cha ac e ized as in Sec ion 2.4 (Wigne –Sei z adius
s
= 4
a0
,
Ne
= 1074
conduc ion elec ons, and adius
a
= 40
.
96
a0
). The dipola plasmon (DP)
esonance o he indi idual MNP is a
ωDP
= 3
.
17
eV
, and he quad upola plasmon
(QP) esonance a
ωQP
= 3
.
4
eV
. A Gaussian-like ex e nal exci a ion
Vex
(
,
)
gi en by Eq.
(2.79)
wi h undamen al equency
ω
, du a ion
σ
= 5
×
2
π/ω
, and
in ensi y
I0
= 10
10
W cm
−2
(ampli ude
E0
= 4
.
8
×
10
−4au
) is used wi hin TDDFT
in his chap e . Impo an ly, as discussed in Sec ion 2.4, such sphe ical MNP
canno emi second-ha monic ligh in o he a ield due o symme y cons ain s,
howe e second-ha monic ields wi h a quad upola pa e n a e induced in he
p oximi y o he MNP because he in e sion symme y is locally b oken a he
su ace ( igh -hand side panel in Figu e 6.1b). The ansi ion equency o he QE
is se o be esonan wi h he second ha monic o he undamen al equency o he
ex e nal illumina ion,
ωQE
= 2
ω
, so ha a a ia ion o
ω
in ou calcula ions implies
simul aneous a ia ion o
ωQE
. The QE plays he ole o an op ical esona o ,
sensi i e o he second-ha monic elec ic nea ield [
384
]. We model he QE as
a poin -like dipole as desc ibed in Subsec ion 1.4.1, using an oscilla o s eng h
α0
QE
= 1
au
in Eq.
(1.54)
. The alue o he in insic damping pa ame e
γQE
is
132
6.1. Sys em and me hods
Figu e 6.1:(a) Ske ch o he sys em s udied in his chap e : he adius o he sphe ical sodium
MNP is
a
= 40
.
96
a0
(
≈
2
.
2nm), and he poin -like QE is loca ed a posi ion
QE
, a a dis ance
d
om he MNP su ace. A Wigne –Sei z adius
s
= 4
a0
is used o cha ac e ize he MNP wi hin
he jellium model. (b) Colo maps o he eal pa o he adial componen o he elec ic nea
ield
Eind
(
, ω
)induced a he undamen al (
ω
, le -hand side panel) and a he second-ha monic
equency (2
ω
, igh -hand side panel) by a
z
-pola ized Gaussian elec omagne ic pulse wi h
undamen al equency
ω
= 1
.
585
eV
and in ensi y
I0
= 10
10
W cm
−2
inciden a he indi idual
sphe ical MNP in he absence o he QE (same esul s a e also shown in Figu e 2.8). Resul s
a e o a ionally symme ic wi h espec o he
z
-axis, and hey a e shown in he (
x, z
)-plane
no malized o uni y. Red and blue colo s a e used o posi i e and nega i e alues, espec i ely
(whi e o ze o).
a ied in his chap e wi hin he ange γQE = 0.1eV −10−7eV.
The expec a ion alue o he QE dipole momen ,
pQE
(
), e ol es in ime
acco ding o [22] [see Eq. (1.54)]:
¨
pQE( ) + γQE ˙
pQE( ) + ω2
QEpQE( ) = α0
QEE o ( QE, ),(6.1)
whe e he o al elec ic ield
E o
(
QE,
)ac ing on he QE posi ion
QE
is gi en
by he sum o he inciden lase pulse (wi h ampli ude
E0
, du a ion
σ
= 5
×
2
π/ω
,
and a i al ime 0= 5σ),
Eex ( ) = ˆ
zE0cos(ω( − 0)) e−( − 0
σ)2
,(6.2)
and he ield
Eind
(
QE,
)induced by he MNP [Eq.
(2.34)
] a he posi ion
QE
o
he QE,
E o
(
QE,
) =
Eex
(
) +
Eind
(
QE,
). No e ha
Eind
(
QE,
)includes he
eac ion o he MNP no only o he inciden pulse, bu also o he elec ic ield
induced by he QE. I hus also accoun s o he QE sel -in e ac ion (see Subsec ion
1.4.2). We desc ibe in Appendix Chow we sol e Eq. (6.1) in his hesis.
The QE dipole
pQE
(
)ac s as a adia ion sou ce emi ing in o he a ield as
well as a ec ing he dynamics o he conduc ion elec ons o he MNP. Because o
he small size o he sys em, e a da ion e ec s can be neglec ed, so ha he QE
placed a a posi ion
QE
nea he MNP c ea es an elec os a ic po en ial gi en by
[Eq. (3.3)]:
VQE( , ) = −pQE( )· − QE
| − QE|3.(6.3)
133
Chap e 6. Second-ha monic gene a ion om a quan um emi e coupled o a
me allic nanopa icle
Thus, he Kohn–Sham Hamil onian
ˆ
H
[
n
(
,
)] wi hin TDDFT [Eq.
(2.19)
], and on
he MNP elec ons, is gi en by
ˆ
H[n( , )] = ˆ
T+VH[n( , )] + Vxc[n( , )] + Vex ( , ) + VQE( , ),(6.4)
whe e, as discussed in Sec ion 2.1 and Sec ion 2.2,
n
(
,
)is he ime-dependen
elec on densi y [Eq.
(2.20)
],
ˆ
T
=
−1
2∇2
is he kine ic-ene gy ope a o ,
VH
[
n
(
,
)] is
he Ha ee po en ial [Eq.
(2.23)
],
Vxc
[
n
(
,
)] is he exchange–co ela ion po en ial
[Eq.
(2.12)
] calcula ed using he ke nel o Gunna sson and Lundquis [
198
], and
Vex
(
,
)is he ex e nal po en ial gi en by Eq.
(2.79)
ha d i es he QE–MNP
sys em. We employ he WPP algo i hm in sphe ical coo dina es as desc ibed in
Subsec ion 2.3.1. In his case, he ime-dependen Kohn–Sham equa ions gi en
by Eq.
(2.19)
and Eq.
(2.20)
a e sol ed sel -consis en ly oge he wi h Eq.
(6.1)
,
Eq.
(6.3)
, and Eq.
(6.4)
. These equa ions a e sol ed in ime domain, and he
ime- o- equency Fou ie ans o m gi en by Eq.
(2.80)
is used o ob ain he
equency- esol ed quan i ies o in e es such as he nonlinea dipole momen
induced a he MNP
pMNP
(Ω), and a he QE
pQE
(Ω). The o al dipole momen
p
(Ω) is gi en by he sum o bo h,
p
(Ω) =
pMNP
(Ω) +
pQE
(Ω). He e, he symbol
Ωis used o deno e he equency o he induced dipole momen s, since we a e
conside ing he nonlinea egime whe e
ω
= Ω in gene al. A Gaussian il e
F
(
)
gi en by Eq.
(2.81)
is used in he Fou ie ans o ms o pa ially accoun o decay
and dephasing p ocesses o he collec i e densi y oscilla ions ha a e no included
in he p esen ALDA-TDDFT app oach (see Sec ion 2.2).
6.2 Resul s and discussion
6.2.1 TDDFT esul s
We ini ially place he QE a he
z
-axis, co esponding o he di ec ion o pola iza ion
o he inciden lase pulse, a a dis ance
d
= 18
a0
(
≈
0
.
95
nm
) om he MNP
su ace. Fo his geome y, only
z
-pola ized dipole momen s a e induced in he
QE and in he MNP. The sys em hen possesses cylind ical symme y wi h espec
o he
z
-axis, which g ea ly educes he compu a ional demands o he TDDFT
calcula ions. The equency o he ex e nal exci a ion
ω
= 1
.
585
eV
is i s
conside ed such ha i s second ha monic ma ches he DP equency o he MNP,
2
ω
=
ωDP
= 3
.
17
eV
, and he in insic damping pa ame e o he QE is se o
γQE = 0.1eV.
Figu e 6.2 shows ha he coupled QE–MNP sys em ea u es s ong emission a
bo h odd and e en ha monics, in con as o he indi idual MNP ha only emi s
a odd-ha monics due o he in e sion symme y o he sys em (see Sec ion 2.4).
In his igu e, he in ensi y spec um o he o al induced dipole momen
|p
(Ω)
|2
calcula ed wi hin TDDFT o he coupled QE–MNP s uc u e is shown by he blue
line, e ealing clea peaks a e en ha monics Ω = 2
ω,
4
ω
and 6
ω
. The e e ence
esul s ob ained o he nonlinea esponse o he indi idual MNP (wi hou QE)
a e shown by he dashed ed line (only ha monics a Ω =
ω,
3
ω,
5
ω
and 7
ω
a e
134
6.2. Resul s and discussion
Figu e 6.2: In ensi y spec um o he o al dipole momen
|p
(Ω)
|2
o he coupled QE–MNP
sys em (solid blue line) and o he indi idual MNP (dashed ed line). Resul s a e ob ained o an
inciden
z
-pola ized Gaussian elec omagne ic pulse wi h undamen al equency
ω
= 1
.
585
eV
and in ensi y
I0
= 10
10
W cm
−2
. The QE is loca ed a he
z
-axis, a a dis ance
d
= 18
a0
(≈0.95 nm) om he MNP su ace, and i is cha ac e ized by a ansi ion equency ωQE = 2ω,
an in insic damping pa ame e
γQE
= 0
.
1
eV
, and oscilla o s eng h
α0
QE
= 1
au
, ollowing
Eq.
(6.1)
. Resul s a e shown as a unc ion o he equency measu ed in uni s o he undamen al
equency
ω
. In he inse , he solid blue line co esponds o he same esul as in he main
igu e (wi h QE), and he dashed black line co esponds o he esul s ob ained o he ansi ion
equency o he QE esonan wi h he ou h ha monic o he inciden ligh ,
ωQE
= 4
ω
, wi h
ω= 0.79 eV.
obse ed, as discussed in Figu e 2.7). The e en ha monics in he a ield om he
coupled QE–MNP sys em eme ge because he QE b eaks he e lec ion symme y
wi h espec o he (
x, y
)-plane, and hus he o al in e sion symme y o he
sys em [
388
]. No e ha he spec a in Figu e 6.2 a e a i icially b oadened by
applica ion o he Gaussian il e gi en in Eq.
(2.81)
ha allows o in oducing
losses in he sys em. We show in Figu e 6.3 he e ec o he Gaussian il e gi en
by Eq.
(2.81)
. While he non il e ed spec um o he hyb id MNP-QE sys em (blue
line) appea s qui e noisy, he il e ed one p esen s e y well-de ined high-ha monic
peaks ( ed line). All he peaks a e b oadened by he il e , which esul s in an
a enua ion o he maximum alue o he peaks as compa ed o he non il e ed
signal. Figu e 6.3 hus illus a es how his il e ing p ocedu e allows us o each
con e gen spec al esponse a high-ha monic equencies. In Subsec ion 6.2.2, we
de elop a semi-analy ical me hod ha allows us o o e come he di icul ies o he
ALDA-TDDFT calcula ions o inco po a e losses [72,241,242].
The esonance be ween he ansi ion equency o he QE,
ωQE
, and he second
ha monic o he inciden pulse s ongly enhances he in ensi y emi ed by he sys em
a 2
ω
. To illus a e his esonance e ec , we show in he inse o Figu e 6.2 he
esul s ob ained o a di e en si ua ion. The QE ansi ion equency
ωQE
in his
case is se o be esonan wi h 4
ω
(dashed black line), and he sys em is illumina ed
by a Gaussian pulse wi h undamen al equency
ω
such ha he ou h ha monic
ma ches he equency o he MNP dipola plasmon, 4
ω
=
ωQE
=
ωDP
= 3
.
17 eV.
135