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Universidade Federal do Rio de Janeiro
Centro de Ciências Matemáticas e da Natureza
Instituto de Física
Di�erent strategies to harness quantum
electrodynamics phenomena at low energies
Patrícia Pinto Abrantes
Ph.D. Thesis presented to the Graduate Program
in Physics of the Institute of Physics of the Federal
University of Rio de Janeiro � UFRJ, as part of
the requirements for obtaining the title of Doctor
in Science (Physics).
Advisor: Carlos Farina de Souza
Co-advisor: Felipe Siqueira de Souza da Rosa
Rio de Janeiro
December, 2021
Di�erent strategies to harness quantum electrodynamics phenomena
at low energies
Patrícia Pinto Abrantes
Carlos Farina de Souza e Felipe Siqueira de Souza da Rosa
Tese de Doutorado submetida ao Programa de Pós-Graduação em Física, Instituto de Física, da
Universidade Federal do Rio de Janeiro � UFRJ, como parte dos requisitos necessários à obtenção
do título de Doutor em Ciências (Física).
Aprovada por:
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Prof. Dr. Carlos Farina de Souza, IF-UFRJ
(Presidente e Orientador)
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Prof. Dr. Felipe Siqueira de Souza da Rosa, IF-UFRJ
(Co-orientador)
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Prof. Dr. Danilo Teixeira Alves, UFPA
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Prof. Dr. Diney Soares Ether Junior, IF-UFRJ
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Prof. Dr. Leonardo de Souza Menezes, UFPE
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Prof. Dr. Marcelo Paleólogo Elefteriadis de França Santos, IF-UFRJ
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Prof. Dr. Nuno Miguel Machado Reis Peres, Universidade do Minho
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Prof. Dr. Paulo Américo Maia Neto, IF-UFRJ
Rio de Janeiro, RJ � Brasil
Dezembro de 2021
CIP - Catalogação na Publicação
Elaborado pelo Sistema de Geração Automática da UFRJ com os dados fornecidos
pelo(a) autor(a), sob a responsabilidade de Miguel Romeu Amorim Neto - CRB-7/6283.
AA161d
Abrantes, Patrícia Pinto
 Different strategies to harness quantum
electrodynamics phenomena at low energies / Patrícia
Pinto Abrantes. -- Rio de Janeiro, 2021.
 212 f.
 Orientador: Carlos Farina de Souza.
 Coorientador: Felipe Siqueira de Souza da Rosa.
 Tese (doutorado) - Universidade Federal do Rio
de Janeiro, Instituto de Física, Programa de Pós
Graduação em Física, 2021. 
 1. Dispersive interaction. 2. Quantum
reflection. 3. Resonance energy transfer. 4.
Magneto-optical materials. 5. Topological phase
transitions. I. Souza, Carlos Farina de, orient.
II. Rosa, Felipe Siqueira de Souza da, coorient.
III. Título.
Abstract
Di�erent strategies to harness quantum
electrodynamics phenomena at low energies
Patrícia Pinto Abrantes
Advisor: Carlos Farina de Souza
Co-advisor: Felipe Siqueira de Souza da Rosa
Abstract of the Ph.D. Thesis presented to the Graduate Program in Physics of
the Institute of Physics of the Federal University of Rio de Janeiro � UFRJ, as
part of the requirements for obtaining the title of Doctor in Sciences (Physics).
In this thesis, we investigate some phenomena of quantum electrodynamics at low en-
ergies, focusing on the discussion of novel approaches to tune radiation-matter interactions
down to the quantum level.
In the �rst part of this thesis, we discuss dispersive interactions. They typically present
an attractive character, which has given rise to an intense search for con�gurations that
may show repulsion. In this regard, we investigate the non-retarded dispersive force
exerted on a polarizable atom by a conducting toroid. Depending on the various length
scales of this problem (the two parameters that characterize the toroid and the distance
from its center to the atom), we demonstrate that the force on the atom can indeed be
repulsive.
We also address the possibility of applying an external electrostatic �eld to counter-
act the dispersive interaction between an atom and a sphere. By varying the intensity
and orientation of the electric �eld, our results illustrate that one can interchange the
iv
repulsive/attractive character of the resultant force between them, even for quite realistic
values of the electric �eld. Furthermore, we show that our results are robust concern-
ing changes in the size and materials constituting the sphere, as well as the atom under
consideration.
To conclude this �rst part of the thesis, we study the quantum re�ection (QR) of atoms
by surfaces, taking advantage of its strong dependency upon distance of the dispersive
interaction between the incident particle and a given re�ecting wall. We investigate the
QR of di�erent atomic species of experimental relevance by graphene family materials.
The two-dimensional sheets are subjected to an external electric �eld and a circularly po-
larized light beam, which induces several topological phase transitions in these materials.
We verify that the QR probability can be signi�cantly modi�ed by varying the intensities
of the external agents. Our results also show that the topological phase transitions leave
a characteristic signature on the QR probability, unveiling a simple optical mechanism to
probe these transitions experimentally at the nanoscale.
In the second part of this thesis, we study the resonance energy transfer (RET). Mo-
tivated by recent discussions concerning the dependence of the RET rate on the local
density of optical states (LDOS), we consider this rate in a system constituted by a cou-
ple of quantum emitters near a host dielectric matrix in which a dielectric-metal phase
transition of a percolative character may occur. This is an appropriate scenario for in-
vestigating this dependence since it is known in the literature that the LDOS exhibits
a peak close to percolation. We show that the RET rate does not present such a peak,
suggesting that RET does not strongly correlate with the LDOS.
Finally, we demonstrate that an external magnetic �eld acting on a graphene sheet
provides an extremely e�cient platform for continuously tuning the RET rate between
two emitters, both at low and room temperatures. We show how the RET rate can be
dramatically altered � up to six orders of magnitude with respect to the free space value
� even for tiny variations of the magnetic �eld. Moreover, we reveal the role played by
v
the magnetoplasmon polaritons supported by the graphene monolayer as the dominant
channel for the RET within a certain distance range. In addition, these remarkable e�ects
show up for quite realistic values of the magnetic �eld.
Keywords: 1. Dispersive interaction. 2. Quantum re�ection. 3. Resonance energy
transfer. 4. Percolation. 5. Magneto-optical materials. 6. Topological phase transitions.
vi
Resumo
Diferentes estratégias para explorar fenômenos da
eletrodinâmica quântica em baixas energias
Patrícia Pinto Abrantes
Orientador: Carlos Farina de Souza
Coorientador: Felipe Siqueira de Souza da Rosa
Resumo da Tese de Doutorado apresentada ao Programa de Pós-Graduação
em Física do Instituto de Física da Universidade Federal do Rio de Janeiro �
UFRJ, como parte dos requisitos necessários à obtenção do título de Doutor
em Ciências (Física).
Esta tese versa sobre eletrodinâmicaquântica em baixas energias, com foco em novas
abordagens para controlar diferentes formas de interação entre radiação e matéria em um
nível quântico.
Na primeira parte desta tese, discutimos as interações dispersivas. Tipicamente, elas
apresentam um caráter atrativo, o que deu origem a uma intensa busca por con�gu-
rações nas quais pode haver repulsão. Nesse sentido, investigamos a força dispersiva
não-retardada exercida sobre um átomo polarizável por um toroide condutor. Depen-
dendo dos valores das várias escalas de distâncias deste problema (os dois parâmetros que
caracterizam o toroide e a distância do seu centro ao átomo), mostramos que a força sobre
o átomo pode de fato ser repulsiva.
Também abordamos a possibilidade de aplicar um campo eletrostático externo para
contrabalançar a interação dispersiva entre um átomo e uma esfera. Ao variar a intensi-
dade e a orientação do campo elétrico, nossos resultados a�rmam que é possível alternar
vii
o caráter repulsivo/atrativo da força resultante entre eles, mesmo para valores bastante
realistas de campo elétrico. Além disso, mostramos que nossos resultados são robustos
com respeito às mudanças no tamanho da esfera e nos materiais que a constituem, assim
como em relação ao átomo em consideração.
Para fechar esta primeira parte da tese, estudamos a re�exão quântica (QR) de átomos
por superfícies, aproveitando a forte dependência na interação dispersiva com a distância
entre a partícula incidente e uma parede re�etora. Investigamos a QR de diferentes espé-
cies atômicas de particular relevância experimental por materiais da família do grafeno.
As folhas bidimensionais são submetidas a um campo elétrico externo e à incidência de
luz circularmente polarizada, induzindo várias transições de fase topológicas nestes mate-
riais. Veri�camos que a probabilidade de QR pode ser signi�cativamente modi�cada pela
variação das intensidades dos agentes externos. Nossos resultados também mostram que
as transições de fase topológicas deixam uma assinatura característica na probabilidade
de QR, revelando um mecanismo óptico simples para sondar estas transições experimen-
talmente.
Na segunda parte desta tese, estudamos a transferência de energia de ressonância
(RET). Motivados por discussões recentes sobre a dependência da taxa de RET na den-
sidade local de estados ópticos (LDOS), calculamos esta taxa em um sistema constituído
por um par de emissores quânticos próximos a uma matriz dielétrica na qual há a possibili-
dade de transição de fase dielétrico-metal. Esse é um cenário adequado para a investigação
desta dependência, uma vez que é conhecido na literatura que a LDOS apresenta um pico
próximo à chamada percolação, que caracteriza essa transição de fase. Mostramos que a
taxa de RET não apresenta tal pico, sugerindo que RET e LDOS não se correlacionam
fortemente.
Por �m, demonstramos que um campo magnético externo atuando sobre uma folha de
grafeno pode ser um agente extremamente e�ciente para o controle contínuo da taxa de
RET entre dois emissores, tanto em temperatura baixa quanto em temperatura ambiente.
viii
Mostramos como a taxa de RET pode ser dramaticamente alterada � até seis ordens
de magnitude em relação ao valor do espaço livre � mesmo para pequenas variações do
campo magnético. Além disso, revelamos o papel desempenhado pelos magnetoplasmon-
poláritons possibilitados pela monocamada de grafeno como o canal dominante para a
RET dentro de uma determinada faixa de distância. Além disso, esses efeitos notáveis
aparecem para valores bastante realistas do campo magnético.
Palavras-chave: 1. Interação dispersiva. 2. Re�exão quântica. 3. Transferência de
energia de ressonância. 4. Percolação. 5. Materiais magneto-óticos. 6. Transições de fase
topológica.
ix
Acknowledgments
Eu não tenho a menor dúvida de que todos estes meus anos no Instituto de Física da
UFRJ e a minha trajetória acadêmica lá teriam sido muito diferentes e certamente mais
difíceis se não fosse o apoio de tantas pessoas. Em particular, praticamente metade do
meu doutorado foi construído durante todas as incertezas e inseguranças que a pandemia
nos trouxe. Mas, felizmente, desde que o início da minha graduação até o �nal deste
doutorado, tive a sorte de conhecer e me aproximar de pessoas incríveis que tornaram
tudo isso possível.
Não posso deixar de iniciar meus agradecimentos sem expressar minha gratidão aos
meus orientadores Carlos Farina e Felipe Rosa. Se hoje tenho dez anos de vivência no
Instituto de Física, nove deles foram trabalhando ao lado destes dois. Farina e Felipe me
orientam desde que iniciei o terceiro período da graduação e, boa parte da pro�ssional que
tenho me tornado e do que entendo por fazer ciência, eu devo ao quanto eles investiram
em mim e na minha formação. Farina é possivelmente a pessoa mais apaixonada por física
e pelo seu ensino que já vi na minha vida. Peço licença para o uso do inglês, mas, para
usar uma palavra que ele adora, tudo em física para ele é earth-shaking. Um professor
extremamente querido por seus alunos e que tenho a sorte de poder dizer que é também
meu amigo, uma vez que seus ensinamentos diários comigo vão para muito além de apenas
física. Também tive o imenso prazer de desfrutar dos conhecimentos do Felipe. Além de
ter tido a oportunidade de fazer dois ótimos cursos de Eletromagnetismo com ele durante
a graduação, continuo aprendendo muito sobre diferentes aspectos da física e da vida
com ele. Tenho certeza de que o seu futuro pro�ssional será brilhante e espero poder me
manter sempre próxima, tanto pro�ssionalmente quanto pessoalmente.
x
Não posso deixar de expressar minha mais profunda gratidão ao meu companheiro de
vida Davi. Agradeço sempre por sua presença, paciência, carinho e amor diários. Foi ele
quem viveu intensamente comigo este último (e especialmente complicado) ano. Sem a
sua crença em mim, o seu apoio contínuo para que eu seguisse em frente e seus incentivos
para que buscasse fazer sempre o meu melhor, tenho certeza de que esses meses todos
teriam sido muito diferentes. Muito obrigada por dividir essa conquista comigo!
Agradeço também aos meus pais Elaine e Edilson. Reconheço o quanto eles se es-
forçaram ao longo da vida para oferecer para mim e para o meu irmão a melhor educação
que pudéssemos ter e, sem isso, eu provavelmente não teria chegado tão longe. Mesmo
sem entender perfeitamente o que faz uma física, nunca deixaram de me apoiar durante
todos estes anos nas escolhas que �z. Aproveito para agradecer também à minha avó Olga
por me ensinar a importância de me dedicar aos estudos desde muito pequena. Ao meu
irmão Felipe e à minha cunhada Vanessa, por todo o apoio e por sempre me receberem
tão bem em sua casa para umas semanas de férias.
Além dos meus orientadores, tive a oportunidade de ter contato com ótimos profes-
sores e colaboradores. Agradeço aos professores Felipe Pinheiro, Reinaldo de Melo e
Souza e Carlos Zarro por todas as interações pro�ssionais e conselhos pessoais. Agradeço
também aos professores Felipe Pinheiro, Reinaldo de Melo e Souza e Paulo Américo por
aceitarem escrever cartas de recomendação para mim. Aos professores Luca Moriconi,
Miguel Quartin, Paulo Américo e Rodrigo Capaz, por excelentes cursos durante a minha
pós-graduação.
Agradeço ao Grupo de Flutuações Quânticas pelo ambiente bastante agradável e leve
para se trabalhar. Eu acho admirável como alunos, mesmo os mais jovens, conseguem
se sentir tão à vontade neste grupo e contentes por fazerem parte dele. Em particular,
agradeço à Daniela por estar presente em momentos tão importantes da minha vida
pessoal e pro�ssional. Agradeço também ao Yuri França e ao Victor por me darem a
chance de co-orientar seus projetos de iniciação cientí�ca (e uma vez mais ao Farina por
xi
con�ar a mim esta tarefa).
Durante dois dentre os meus anos de doutorado, tive a oportunidade de atuar como
Professora Substituta no Instituto de Física. Esta experiênciafoi extremamente positiva
para mim, encorajando-me ainda mais na busca por uma carreira acadêmica, e não poderia
deixar de agradecer à professora Simone Coutinho por me guiar no início deste processo.
Agradeço aos alunos com os quais convivi ao longo desses dois anos. Certamente eles não
sabem disso, mas toda a experiência de estar com eles em sala de aula me trouxe muito
mais con�ança em mim mesma e nos passos que escolhi seguir.
A física de fato me trouxe muitos amigos e pessoas queridas. Eu não poderia deixar de
agradecer em particular ao Claudio, à Yara e ao Arouca. Estes três tiveram e continuam
tendo um papel singular ao longo da minha trajetória, apoiando-me desde os nossos
tempos de jovens graduandos. Nossas conversas, tanto as mais bobas e sem sentido até
os questionamentos mais profundos (não necessariamente sobre física), �zeram os meus
dias bem melhores. Agradeço também àqueles que dividiram sala comigo. Foster, Larissa
e Yuri Muniz me aturaram em momentos diferentes da minha pós-graduação e tornaram
aquela sala um cubículo muito mais divertido. Agradeço também ao Carlos, ao Henrique,
ao Gabriel e ao Tiago por partilharem seus dias comigo e deixarem esses anos mais
agradáveis. Agradeço ao Tarik por toda ajuda no trabalho desenvolvido no capítulo 4.
Agradeço também aos meus amigos de fora da física por tanto apoio e carinho e por
sempre estarem ao meu lado. Em particular, reforço meus agradecimentos aos amigos
Ana Carolina, Iasmim, Juliana, Marcela e Yuri Lira. Às minhas amigas de carona e de
rodízio de pizza (!), Nicole e Lanuza, e aos outros participantes dos grupos de caronas,
obrigada por tornarem as várias e várias horas de engarrafamento de ida e de volta do
Fundão mais engraçadas.
Gostaria de agradecer também aos funcionários do Instituto de Física, especialmente
ao secretariado da pós-graduação Igor e Khrisna por resolverem tão e�cientemente todas
as questões que precisei levar a eles ao longo desses anos.
xii
Agradeço, por �m, às agências de �nanciamento CNPq (Conselho Nacional de De-
senvolvimento Cientí�co e Tecnológico) e FAPERJ (Fundação de Amparo à Pesquisa do
Estado do Rio de Janeiro). Este suporte �nanceiro colaborou para que pudesse me dedicar
a este trabalho de doutorado e ajudar a desenvolver um pouco mais da ciência brasileira.
Tal incentivo é fundamental, ainda mais em tempos tão críticos nos quais as nossas Uni-
versidades Públicas e o trabalho excelente desenvolvido por todos nós pesquisadores têm
sofrido diversos ataques e são constantemente difamados por todos os lados, sobretudo
por aqueles que mais deveriam nos apoiar.
xiii
Contents
List of Figures xviii
List of Tables xxvii
Introduction 1
I Quantum theory of dispersive interactions 8
I.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I.2 The van der Waals forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I.3 The Casimir e�ect and the resigni�cation of the zero-point energy . . . . . 16
I.4 Scattering theory for dispersive interactions . . . . . . . . . . . . . . . . . 20
I.4.1 The scattering and transfer matrices . . . . . . . . . . . . . . . . . 21
I.4.2 Modi�cation of the zero-point energy by an optical element . . . . . 26
I.4.3 Lifshitz formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
I.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
II Repulsive dispersive forces with non-trivial geometries 38
II.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
xiv
CONTENTS
II.2 Non-retarded dispersive interaction between an atom and an arbitrary con-
ducting surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
II.2.1 Eberlein-Zietal method . . . . . . . . . . . . . . . . . . . . . . . . . 42
II.2.2 An atom close to a grounded conducting plane . . . . . . . . . . . . 50
II.2.3 An atom close to a conducting sphere . . . . . . . . . . . . . . . . . 51
II.3 Dispersive van der Waals interaction between an atom and a conducting
toroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
II.3.1 Toroidal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 55
II.3.2 Point charge near a grounded conducting toroid . . . . . . . . . . . 57
II.3.3 Repulsion in the atom-toroid system . . . . . . . . . . . . . . . . . 63
II.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
IIIOutdoing dispersive forces upon external agents 71
III.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
III.2 Atom-sphere interaction in the presence of an external electric �eld . . . . 75
III.2.1 Dispersive force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
III.2.2 Electrostatic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
III.3 Controlling the atom-sphere interaction . . . . . . . . . . . . . . . . . . . . 81
III.3.1 Hydrogen atom near a metallic sphere . . . . . . . . . . . . . . . . 82
III.3.2 Interactions of di�erent atoms . . . . . . . . . . . . . . . . . . . . . 88
III.3.3 Atoms near a dielectric sphere . . . . . . . . . . . . . . . . . . . . 90
III.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
IVQuantum re�ection of atoms by surfaces 94
IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
IV.2 Calculation of the quantum re�ection probability . . . . . . . . . . . . . . 97
IV.3 E�ects of topological phase transitions in graphene family materials on
quantum re�ection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xv
CONTENTS
IV.3.1 Mathematical description of graphene family materials . . . . . . . 104
IV.3.2 Re�ection coe�cients and optical conductivities of graphene family
materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
IV.3.3 Probing topological phase transitions via quantum re�ection . . . . 110
IV.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
V Quantum theory of resonance energy transfer 121
V.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
V.2 Mathematical derivation of the resonance energy transfer rate in free space 124
V.3 Green's function approach for the resonance energy transfer . . . . . . . . 132
V.3.1 Electromagnetic �elds in dispersive and absorptive material media . 133
V.3.2 Resonance energy transfer and the Green's dyadic . . . . . . . . . . 135
V.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
VITailoring the resonance energy transfer 143
VI.1 Resonance energy transfer rate close to percolation transition . . . . . . . . 144
VI.1.1 Bruggeman e�ective medium theory . . . . . . . . . . . . . . . . . . 145
VI.1.2 Resonance energy transfer near composite media . . . . . . . . . . . 147
VI.2 Tuning resonance energy transfer exploring the magneto-optical properties
of graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
VI.2.1 Re�ection coe�cients and optical conductivities of graphene in the
presence of a magnetic �eld . . . . . . . . . . . . . . . . . . . . . . 152
VI.2.2 Resonance energy transfer close to a graphene sheet in the presence
of a magnetic �eld . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
VI.2.3 A comparison between resonance energy transfer and spontaneous
emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
VI.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
xvi
CONTENTS
Final remarks and conclusions 167
Bibliography 171
A List of publications 207
B Atom data used in chapters III and IV 209
C Dielectric functions of the materials used in chapter III 211
D Parameters of graphene family materials used in chapter IV 212
xvii
List of FiguresI.1 The one-dimensional scattering problem by a single object characterized
by the scattering matrix S. ψ+L (ψ
−
L ) and ψ
+
R (ψ
−
R) are scalar �elds on the
left and right sides of the object, respectively, that propagate along the
positive (negative) x direction. . . . . . . . . . . . . . . . . . . . . . . . . . 21
I.2 The same one-dimensional scattering problem by a single object, but now
from the viewpoint of the transfer matrix T . ψ+L (ψ
−
L ) and ψ
+
R (ψ
−
R) re-
main to denote the scalar �elds on the left and right sides of the object,
respectively, that propagate along the positive (negative) x direction. . . . 23
I.3 Electromagnetic wave incidence on a composition of several optical ele-
ments i = 1, 2, 3, ..., N , each characterized by its own transfer matrix TLi . . 24
I.4 Electromagnetic wave incidence on an empty space of length L, with scat-
tering and transfer matrices SL and TL, respectively. . . . . . . . . . . . . . 25
I.5 Optical element described by the scattering matrix S (and transfer matrix
T ) placed in empty space. A quantization box of length L0 = L1 + L2 is
considered to quantify the change in the vacuum energy, containing the ob-
ject and two stretches of empty space, whose transfer matrices are denoted
by TL1 and TL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
xviii
LIST OF FIGURES
I.6 Optical element formed by two partially re�ective mirrors 1 and 2, sepa-
rated by a distance L, inside a quantization box of length L0 = L1 + L+ L2. 29
I.7 Closed contour C = R + Γ(R) + I of radius R for deriving the one-
dimensional Lifshitz formula at zero temperature. . . . . . . . . . . . . . . 32
II.1 Correspondence between the quantum and classical problems established
by the Eberlein-Zietal method. The con�guration of an atom, at position
r0, close to any conducting surface S is mapped into another one composed
of a point charge q at the same position r0 and close to the same conducting
surface S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
II.2 Point charge q close to a grounded conducting plane at a given position r′ =
(x′, y′, z′) and its corresponding image qi = −q at position r′i = (x′, y′,−z′). 50
II.3 Point charge q close to a grounded conducting sphere at a given position
r′ and its corresponding image qi = −qR/r′ at position r′i = r′R2/r′2. . . . 52
II.4 An atom close to a grounded conducting toroid, placed at an arbitrary
position of the symmetry axis Oz. . . . . . . . . . . . . . . . . . . . . . . . 55
II.5 Cross-section of the toroid to show its two radii a and b. The radius a
denotes the distance from the geometrical center of the toroid to any point
located at the center of the torus tube and b stands for the radius of the
torus tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
II.6 Toroidal surfaces characterized by constant values of ξ (dashed blue lines)
and spherical calottes characterized by constant values of η (solid red lines). 57
II.7 (a) A generic point of space P at position r and a point charge q at position
r′ near a perfectly conducting toroid of radii a and b. (b) The vertical plane
de�ned by ϕ = π/2 containing the point charge q and the point P . . . . . . 58
II.8 Electrostatic potential along the Oz axis, normalized by the absolute value
of the electrostatic potential at z = 0, created by the super�cial charges on
xix
LIST OF FIGURES
the toroidal surface induced by a point charge located at the origin (blue
solid line) and located at a point on the positive semiaxis Oz (green dashed
line). In this �gure, we chose b/a = 0.6, with a = 5 nm. . . . . . . . . . . . 60
II.9 Electrostatic potential VH(x, y) created by the super�cial charges on the
toroidal surface induced by a point charge located at the origin, normalized
by the absolute value of the electrostatic potential at x = y = 0. We are
considering points on the Oxy plane outside the toroid, but with 0 ≤ r <
a− b (or η = π). Here, we chose b/a = 0.75, with a = 4 nm. . . . . . . . . 61
II.10 Electrostatic interaction energy U(z′), normalized by |U(0)|, between the
point charge q at position (0, 0, z′) and the induced charges on the toroidal
surface. In this �gure, we chose b/a = 0.6, with a = 5 nm. . . . . . . . . . 62
II.11 The van der Waals interaction energy U (AT)NR as a function of the atom's
position z0, for a �xed value of the radius a = 5 nm. Di�erent colors refer
to di�erent values of the radius b. . . . . . . . . . . . . . . . . . . . . . . . 65
II.12 The van der Waals dispersive force F (AT)NR acting on the atom as a function
of its position z0, for a �xed value of radius a = 5 nm and di�erent values
of radius b. The interval of distances analyzed is the same as the one used
in the previous �gure in order to facilitate the comparison of both results. . 65
II.13 Non-retarded dispersive force F (AT)NR as a function of the ratio a/b. In this
plot, we set b = 1 nm and each color corresponds to a distance z0 from the
atom to the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
II.14 Contourplot depicting the magnitude of the van der Waals force F (AT)NR
acting on the atom as a function of its distance z0 to the origin and the
radius a, both divided by the radius b = 1 nm. . . . . . . . . . . . . . . . . 67
II.15 (a) Schematic representation of a needle-like object near an in�nite con-
ducting plane with a circular hole of diameter W . (b) The �eld lines of a
point dipole at the center of the plane. (c) Pro�le of the interaction energy
xx
LIST OF FIGURES
of the system shown in (a). This �gure was extracted from Ref. [92]. . . . 68
III.1 A neutral and isolated sphere of radius R and a polarizable atom in its
ground state at a distance a from the surface of the sphere. The Oz axis is
chosen parallel to the line connecting the atom to the center of the sphere.
An external uniform electrostatic �eld E0 is applied at an angle θ0 with
respect to the Oz axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
III.2 Ratio Γ as a function of the angle θ0 between the applied electric �eld and
the Oz axis. Di�erent colors refer to di�erent �eld intensities. We set
R = 60 nm and a = 700 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
III.3 Ratio Γ as a function of the �eld intensity E0. Di�erent colors denote
di�erent distances a from the atom to the sphere's surface. We set R =
60 nm and θ0 = π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
III.4 Ratio Γ as a function of the distance a between the atom and the sphere's
surface for di�erent �eld intensities. We set R = 60 nm and θ0 = π/2. . . . 84
III.5 Ratio Γ as a function of the distance a from the atom to the sphere's
surface for di�erent �eld intensities in the short-distance regime. We set
R = 60 nm and θ0 = π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
III.6 (a) Ratio Γ as a function of the distance a between a given atomic species
and the sphere's surface for E0 = 1× 105 V/m. (b) Ratio Γ as a function
of the �eld intensity E0 with a = 500 nm. In all panels, di�erent colors
refer to di�erent atomic species and we set R = 60 nm and θ0 = π/2. . . . 88
III.7 Contour plot of the ratio Γ for the cesium atom varying θ0 and E0. The
dashed black line indicates combinations of θ0 and E0 for which Γ = 0.
Here, we chose R = 60 nm and a = 700 nm. . . . . . . . . . . . . . . . . . 89
III.8 Contour plot of the ratio Γ for the cesium atom varying a and E0. The
dashed white line indicates combinations of a and E0 for which Γ = 0.
xxi
LIST OF FIGURES
Here, we chose R = 60 nm and θ0 = π/2. . . . . . . . . . . . . . . . . . . . 90
III.9 Ratio Γ as a function of the distance a between a cesium atom and the
sphere's surface for (a) E0 = 7 × 104 V/m, using Eq. (III.4), and (b)
E0 = 9× 106 V/m, using Eq. (III.9). In all panels, di�erent colors refer to
spheres made of di�erent materials,and we set R = 60 nm and θ0 = π/2.
The insets show the resultant force on the atom as a function of a. . . . . . 91
IV.1 (a) Atomic specimen being re�ected by a suspended 2D material of the
graphene family. The system is under the in�uence of a static electric �eld
Ez and it is shined on by a circularly polarized light, characterized by the
parameter Λ. (b) Topological phase diagram of the 2D graphene-family
materials. Horizontal dashed gray lines show the paths used in this work
to explore this diagram. The acronyms of each topological phase are as
follows: band insulator (BI; C = 0), quantum spin Hall insulator (QSHI;
C = 0, indexed by the non-trivial Z2-index [217]), polarized-spin quantum
Hall insulator (PS-QHI; C = − 1), and anomalous quantum Hall insulator
(AQHI; C = −2), with C standing for the corresponding Chern numbers. . 102
IV.2 Pro�le of the electronic spectrum of the hamiltonian (IV.17), along path I of
Fig. IV.1(b), i.e., Λ/λSO = 0, and (a) eℓEz/λSO = 0, (b) eℓEz/λSO = 0.5,
(c) eℓEz/λSO = 1, (d) eℓEz/λSO = 1.5, and (e) eℓEz/λSO = 2. The left
panels show the spectrum for the valley K (η = +1) and the right panels
show the spectrum for the valley K ′ (η = −1). The black dashed line
depicts the spectrum for the spin-↑ sector (s = +1), and the red dotted
line depicts the spectrum for the spin-↓ sector (s = −1). The topological
critical point is represented in panel (c), where the electronic spectrum
becomes gapless, and it separates the QSHI phase [panels (a) and (b)] and
the BI phase [panels (d) and (e)]. . . . . . . . . . . . . . . . . . . . . . . . 106
xxii
LIST OF FIGURES
IV.3 Panels (a) and (b): Real (solid) and imaginary (dashed) parts of σxx(ω)
and σxy(ω) for some combinations of parameters {eℓEz/λSO,Λ/λSO}. A:
{0, 0}, B: {0, 1} and C: {0, 2}, as indicated in the topological phase di-
agram of panel (f). Panels (c) and (d): The same as before but for A:
{0, 0}, D: {1, 0.5} and E: {2, 0}, as in panel (f). We set Γ = 10−4λSO/ℏ
and µ = 0 in all cases. Panels (e) and (f): Topological phase diagram
of the graphene family materials described by the Hamiltonian (IV.17).
Panel (e) shows the phase diagram with the acronyms and their respec-
tive Chern numbers, while panel (f) sketches the set of parameters used in
panels (a)�(d). A detailed discussion of this phase diagram and the optical
conductivities can be found in Refs. [82, 216,226]. . . . . . . . . . . . . . . 109
IV.4 Results for QR of a Rb atom with energy E = 10−4 neV by a suspended
sheet of pristine stanene (λSO = 50 meV, chemical potential µ = 0, and
inverse of the scattering time Γ = 10−4λSO/ℏ). (a) QR probability as a
function of the electric �eld for di�erent values of the laser parameter. (b)
Percentage modi�cation in the QR probability caused by the electric �eld. 111
IV.5 (a) The probability R of a Rb atom su�ering QR from a stanene sheet as
a function of its incident energy E. (b) Relative modi�cation in the CP
energy between the Rb atom and the 2D surface caused by the applied
electric �eld as a function of the distance z. (c) The plot of Q(z) as a
function of the distance z for the incident energy E = 10−4 neV chosen
in the intermediate energy regime between the classical and the quantum
limit of QR [see panel (a)]. In all plots, Γ = 10−4λSO/ℏ, µ = 0, and Λ = 0. 113
IV.6 (a) The probability R of a Rb atom su�ering QR from a stanene sheet as
a function of its incident energy E. (b) Relative modi�cation in the CP
energy between the Rb atom and the 2D surface caused by the applied
electric �eld as a function of the distance z. (c) The plot of Q(z) as a
xxiii
LIST OF FIGURES
function of the distance z for the incident energy E = 10−4 neV chosen
in the intermediate energy regime between the classical and the quantum
limit of QR [see panel (a)]. In all plots, Γ = 10−4λSO/ℏ, µ = 0, and Λ = 0. 114
IV.7 Results for QR of a Rb atom with energy E = 10−4 neV by a suspended
sheet of pristine germanene (λSO = 20 meV, µ = 0, and Γ = 10−4λSO/ℏ).
(a) QR probability as a function of the electric �eld for di�erent values of
the laser parameter. (b) Percentage modi�cation in the QR probability
caused by the electric �eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
IV.8 Results for QR of a Rb atom with energy E = 10−4 neV by a suspended
sheet of pristine silicene (λSO = 2 meV, µ = 0, and Γ = 10−4λSO/ℏ). (a)
QR probability as a function of the electric �eld for di�erent values of the
laser parameter. (b) Percentage modi�cation in the QR probability caused
by the electric �eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
IV.9 Results for a Na atom with incident energy E = 10−3 neV. QR probability
as a function of the applied electric �eld by a suspended sheet of pristine
(a) stanene, (c) germanene and (e) silicene. Percentage modi�cation in
the QR probability due to the electric �eld by a suspended sheet of pristine
(b) stanene, (d) germanene and (f) silicene. In all cases di�erent curves
refer to di�erent values of the laser parameter. . . . . . . . . . . . . . . . . 118
V.1 System under study: a couple of two-level identical atoms A and B sepa-
rated by a distance r and located in free space. . . . . . . . . . . . . . . . 125
V.2 Feynman diagrams evidencing the possible intermediate states |0, 0; 1kp⟩
[panel (a)] and |1, 1; 1kp⟩ [panel (b)] permitted during the time evolution
of the system under study. In each panel, time runs from the bottom to the
top along the vertical black lines. Also, the left and right lines correspond
to the time evolution of atoms A and B, the part in bold indicating excited
xxiv
LIST OF FIGURES
states. The exchange of virtual photons is shown by the green wavy lines. . 127
V.3 RET rate Γ(0) between atoms A and B in free space as a function of r. Panel
(a) shows that the atomic transition dipole moments are only parallel to
each other, whereas panel (b) illustrates the con�gurations in which the
atomic transition dipole moments are parallel and aligned with each other.
Here, we set |d01A | = |d10B | = ea0 and λ0 = 2π/k0 = 0.13 µm. . . . . . . . . . 132
VI.1 A pair of two-level emitters separated by r and at a distance z from a
semi-in�nite medium composed of metallic (gold) inclusions immersed in a
dielectric (polystyrene) host matrix. . . . . . . . . . . . . . . . . . . . . . . 144
VI.2 Results for quantum emitters close to a medium made of gold. (a) Nor-
malized RET rate as a function of the distance r between the emitters for
three distances z from them to the medium. (b) Normalized RET rate as
a function of z for three values of r. . . . . . . . . . . . . . . . . . . . . . . 147
VI.3 Results for quantum emitters close to a medium made of polystyrene. (a)
Normalized RET rate as a function of the distance r between the emitters
for three distances z from them to the medium. (b) Normalized RET rate
as a function of z for three values of r. . . . . . . . . . . . . . . . . . . . . 148
VI.4 Normalized RET rate as a function of the �lling factor f . Results were
evaluated for �ve distances r between the emitters, both at z = 0.01λC
from the material medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
VI.5 A pair of two-level emitters separated by a distance r, both at a distance
z from a suspended graphene sheet. An external magnetic �eld B = Bẑ is
applied perpendicularly to the sheet. . . . . . . . . . . . . . . . . . . . . . 151
VI.6 Real and imaginary parts of the longitudinal and transverse conductivities
of graphene as functions of the external magnetic �eld for ω0 = 6π ×
1013 rad/s, vF = 106 m/s and τ = 1 ps. The �rst, second and third rows
xxv
LIST OF FIGURES
were obtained using µc = 0 eV, µc = 0.1 eV and µc = 0.2 eV, respectively.
Also, the �rst column was evaluated with T = 4 K while the second one,
with T = 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
VI.7 Normalized RET rate as functions of the externalmagnetic �eld. Each color
represents a separation r between the emitters with dominant transition
wavelength λ0 = 10 µm, both at a distance z = 50 nm from the graphene
sheet. The �rst, second and third row panels were obtained using µc = 0 eV,
µc = 0.1 eV and µc = 0.2 eV, respectively. Also, the �rst column was
evaluated with T = 4 K while the second shows results for T = 300 K. . . . 157
VI.8 MPP propagation length as a function of the magnetic �eld for di�erent
values of the chemical potential and (a) T = 4 K and (b) T = 300 K.
The same parameters used in the analysis of the conductivities were also
employed here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
VI.9 Normalized RET rate as a function of the magnetic �eld for the case pre-
viously shown with T = 300 K, µc = 0.1 eV and r = 0.02λ0. The plots
are comparisons between results obtained with G(S)zz calculated using Eqs.
(VI.18) (blue curve) and (VI.19) (red curve). . . . . . . . . . . . . . . . . . 162
VI.10Ratio RRET/SE as a function of the external magnetic �eld. Each color
refers to a separation r between the emitters with dominant transition
wavelength λ0 = 10 µm, both at a distance z = 50 nm from the graphene
sheet. In addition, we set µc = 0.1 eV and (a) T = 4 K and (b) T = 300 K.164
xxvi
List of Tables
B.1 Data for Na, K, Fe, Rb, and Cs atoms. This table contains parameters of
the two-oscillator model to be used in Eq. (B.2) [146] (1 a.u. = 1.648×10−41
C2m2J−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
B.2 Data for Na and Rb atoms. This table contains parameters of the single
Lorentz-oscillator model needed in Eq. (B.1) [146] and their masses (1 a.u.
= 1.648× 10−41 C2m2J−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
C.1 Data for gold (Au) and silicon dioxide (SiO2) spheres. This table contains
parameters for each model in Eqs. (C.1) [289] and (C.2) [144]. . . . . . . . 211
D.1 Spin-orbit coupling (λSO), buckling (ℓ), and Fermi velocity (vF ) of each
graphene family material used in chapter IV. Parameters were taken from
Refs. [209,213�215]. Note: There exists some discordance in the literature
for λSO in the case of stanene. . . . . . . . . . . . . . . . . . . . . . . . . . 212
xxvii
LIST OF TABLES
xxviii
Introduction
As far as we know, there are four fundamental interactions in nature: gravitational,
weak, electromagnetic, and strong. Electromagnetic interaction, which dictates the mo-
tion of interacting charged particles, is the most important at the length scales that are
most familiar to us, being responsible for most everyday phenomena. The �nal develop-
ment of the classical theory of electromagnetism took place at the beginning of the second
half of the 19th century with de�nitive contributions from Maxwell, which led us to the
uni�cation of electromagnetism and optics by showing that light is an electromagnetic
wave. This theory allows us to understand several phenomena such as the dispersion of
light and mirages, in addition to continuously enabling new technological applications.
Even today, it revolutionizes human life in di�erent spheres, especially given the broad
frequency spectrum of electromagnetic waves (106 ∼ 1020 Hz), ranging from radio waves
and microwaves to x-rays and gamma rays, which opens the door to a consequent broad
spectrum of routes to take advantage of them.
However, this does not mean that all electromagnetic phenomena can be explained
within the scope of classical physics. Particularly at atomic scales, the need for a new
treatment was inevitable. As magni�cent as Maxwell's equations are and despite the huge
amount of physics they encompass, some problems cannot be covered if quantum physics
1
Introduction
is not considered. For example, we know that a stable classical atomic model based on a
negative charge rotating around a positive nucleus is not feasible. In fact, since accelerated
charges radiate, the electron would lose energy during its orbital motion and spiral toward
the nucleus. This contradiction to what was already known, and other open questions,
such as the lack of an explanation for the blackbody's emission spectrum, were issues that
took a long time to be �gured out. The new understanding had to wait for concepts born
only with quantum physics, which eventually culminated in the development of quantum
electrodynamics (QED). This required the e�ort and intellect of several notable physicists
from Planck, Einstein, and Bohr, to Schrödinger, Heisenberg, Pauli, and Dirac, and �nally
to Dyson, Feynman, Schwinger, and Tomonaga, the last three being awarded the Nobel
Prize in 1965 for their valuable contributions that helped to establish the QED in its �nal
form. In this thesis, we focus on QED at low energies, in the sense that the discussions to
be presented revolve around processes in which the energy of the photon involved is much
smaller than the rest energy of the quantum emitters, in contrast to the high-energy QED
studied in particle accelerators. For example, a photon emitted by a hydrogen atom in
the visible range has an energy of many orders of magnitude smaller than the rest energy
of hydrogen.
QED is the theory responsible for describing light-matter interactions with unprece-
dented success, presenting an excellent agreement between theoretical predictions and
experimental results. This theory explains several phenomena that we shall elaborate on
shortly, which are somehow correlated with quantum vacuum �uctuations, and may occur
even in our daily lives. In quantum electromagnetism, the electromagnetic �elds become
operators acting on a given Hilbert space, de�ned in such a way that, although their ex-
pectation values may be zero, their variance never vanishes, even in vacuum. Physically,
this implies that, even in the absence of any sources, the vacuum is not simply an empty
space as in a classical picture.
The very concept of vacuum has undergone some modi�cations throughout history,
2
Introduction
alternating between two antagonistic ideas. There were times when it was believed that
vacuum was an entirely empty space and also times when prevailed the idea that it was
a space �lled with some very subtle matter, and therefore, with some inherent structure.
The �rst one describes the vacuum as what would supposedly result from a given volume
of space if we removed all the particles and radiation from it. In this de�nition, two
vacua are indistinguishable, being merely inert media that do not respond to any external
stimuli, and nothing interesting ever happens.
Nevertheless, this is a naive view. As we will see throughout this thesis, measurable
e�ects emerge from what looks like nothing. To learn what vacuum actually is, it was
necessary to come to terms with the intriguing notions born within a new theory. Only
after formulating special relativity, quantum mechanics, and the proper combination of
these theories, generating the quantum �eld theory, some concepts such as the creation
and annihilation of particles, the Heisenberg uncertainty principle, and the zero-point
energy could arise. In this theory, there is the vacuum state, which is de�ned as the state
of the system under consideration in which there are no real particles. It is the lowest-
lying state, and its energy is known as zero-point energy. Although this state contains
no real particles, we shall see that, in a very anti-intuitive way, a sea of virtual particles
lives in this state, �ickering in and out of existence even at zero temperature, as long as
such events are conditioned not to violate the uncertainty principle. During their brief
lifetimes, these particles are susceptible to external agents, such as electromagnetic �elds
or the presence of surrounding objects. Strictly speaking, we can portray the quantum
vacuum as having its own macroscopic properties, similar to material media. Far from
being just a classicalempty space, it can polarize, magnetize, or even emit real particles
when exposed to extremely intense external �elds.
The quantum vacuum is at the root of a variety of surprising physical e�ects, currently
designated as �uctuation-induced phenomena, that can all be understood through QED.
Amid so many examples, we highlight the Lamb shift, the electron anomalous magnetic
3
Introduction
moment, the spontaneous emission, the Casimir e�ect, the dispersive forces, and the
resonance energy transfer. In this thesis, we will take a closer look at the last two examples.
Classical physics predicts that neutral bodies without permanent electric or magnetic
multipole moments, even if polarizable, should not interact. However, noble gas atoms,
for instance, can undergo condensation, which is a phase transition that occurs because of
the interaction between them. An intuitive understanding of the physics involved in this
interaction was only possible with the Heisenberg uncertainty principle, which posits the
atomic electric dipole to be null only on the average, allowing for �uctuations. These forces
arising from quantum �uctuations, called dispersive forces, are responsible for several
interesting phenomena, from �eld theory to colloid physics. It also �nds applications
in research areas beyond physics, with a fascinating daily example being that dispersive
forces are the dominant mechanism by which geckos stick to walls.
Another macroscopic manifestation of dispersive interactions is the well-known Casimir
e�ect, predicted by H. B. G. Casimir in 1948. It is one of the e�ects most commonly as-
sociated with the zero-point energy of the electromagnetic �eld, a quantity that for a long
time was considered by many in the physical community to be a pending error in QED.
In its original version, the Casimir e�ect predicts an attractive force between two neutral,
parallel, and perfectly conducting plates located in vacuum.
In addition to interactions involving con�gurations such as those mentioned above,
considering only atoms or macroscopic objects, dispersive forces are equally important
when dealing with interactions between an atom and a macroscopic object. We may de-
scribe a con�guration of particular interest. Imagine an atom moving towards any neutral
�at surface. The dispersive potential generated by this surface on the atom is well-known
and decreases monotonically upon decreasing the distance. Consequently, the atom is con-
tinually attracted to this interface, as there are no turning points. In a classical picture,
we would expect the atom to accelerate more and more towards the surface until it collides
with the plate. From the viewpoint of quantum mechanics, however, di�erent results can
4
Introduction
happen. Due to the wave nature inherent to quantum particles, there is a non-vanishing
probability that an atom with very low energy can be bounced o� before reaching the
surface. This highly counter-intuitive phenomenon is called quantum re�ection (QR),
presenting applications ranging from atom optics to high-precision measurements of the
short-range regimes of gravitational forces. As one can already guess, this interaction may
vary substantially as we change both the surface (or only its properties) and the atom
since, ultimately, everything can be traced back to the dispersive interaction that governs
the atom-surface interaction.
All situations mentioned above, such as the adhesion of geckos to walls, the conden-
sation of gases, the Casimir e�ect, and the atom-plane system used in QR studies, are
examples of situations exhibiting attractive forces. Actually, with some general arguments,
we can infer that dispersive interactions usually present an attractive character, which is
not always welcome. For example, these kinds of interactions impose fundamental restric-
tions on the miniaturization of technical devices. Since the in�uence of quantum features
becomes unavoidable when going down in size, the attraction due to dispersive forces can
be quite detrimental to the functioning of nano- and microelectromechanical structures.
As we shall discuss later, this has produced a signi�cant e�ort among researchers to �nd
arrangements for which dispersive forces are repulsive.
Another notable phenomenon of QED that we will explore in detail is called resonance
energy transfer (RET). This interaction is characterized by the energy transfer from an
excited quantum emitter to another emitter in the ground state and is the dominant
energy transfer mechanism for short distances. It plays a key role in photosynthesis: the
chlorophyll molecules present in the leaves absorb light energy and transfer it to other
neighboring chlorophyll molecules until this excitation energy reaches the reaction centers
of the leaf, where the chemical reactions take place.
Nowadays, besides seeking a deeper understanding of the basic concepts surround-
ing all QED e�ects, we can go further. As new properties regarding each of them are
5
Introduction
discovered, we can envision novel applications, manipulate them at fundamental levels,
and fuel an ever-accelerating advance and development of science. The idea of tuning all
these phenomena has received much attention since the pioneering work of E. M. Purcell
in 1946. Purcell was concerned with the spontaneous emission, a process characterized
by the inexorable decay of an initially excited atom to its ground state, emitting radia-
tion. He showed that the presence of media close to the quantum emitter could a�ect its
spontaneous emission rate. This phenomenon received the name of Purcell e�ect and is
essentially correlated with the local density of states at the position and evaluated at the
transition frequency of the emitter, i.e., the number of available electromagnetic modes,
per unit of frequency and volume, with which the emitter can interact. Depending on
the chosen neighborhood, the spontaneous emission rate may be enhanced, attenuated,
or even completely suppressed.
Interestingly, Purcell's idea can be applied for di�erent �uctuation-induced phenom-
ena, and, presently, we are witnessing exciting times for improving the physical aspects
of QED. In a few decades, the �elds of nanophotonics and plasmonics have experienced a
revolution of sorts, mainly due to the invention of new experimental fabrication techniques
combined with advances in computer simulations. Such progress has naturally enriched
the variety of strategies with which we can harness these phenomena. In this thesis,
we intend to shed light on this fascinating research �eld, discussing in detail theoretical
proposals of tunable material platforms for dispersive forces and RET.
This thesis is organized as follows. In chapter I, we discuss the physics underlying
dispersive forces. In chapter II, we investigate the non-retarded dispersive force between
an atom and a conducting grounded toroid in vacuum. In chapter III, we study the
e�ects on the interaction between an atom and a sphere when an external electrostatic
�eld is applied in this system. Chapter IV addresses the QR of atoms by surfaces. We
present the mathematical derivation of the QR probability and analyze the QR of di�erent
atomic species by graphene family materials exposed to static electric �elds and circularly
6
Introduction
polarized light beams. In chapter V, we show the mathematical formalism to evaluate
the RET rate between two quantum emitters surrounded by an arbitrary environment.
These expressions are applied in chapter VI in order to investigate the e�ects on the RET
rate when the emitters are close to di�erent neighborhoods: (i) a host dielectric matrix
in which a dielectric-metal phase transition of a percolative character may occur, and (ii)
a suspended graphene sheet in vacuum under the action of a magnetic �eld.
7
I. Quantum theory of dispersive in-
teractions
Dispersive forces are probably the most intriguing kind of intermolecularforces and
have been satisfactorily explained only after the advent of quantum mechanics. They
arise as a result of the electromagnetic interaction between neutral and nonpolar, albeit
polarizable, atoms or molecules that occurs due to quantum �uctuations in their atomic
charge and current distributions. Naturally, they are also present between polar molecules,
but they are generally not the dominant interaction in these cases. These interactions
are responsible for several interesting phenomena, from �eld theory to colloid physics,
in addition to multidisciplinary applications. In this chapter, we begin our discussions
with a brief history of intermolecular forces, with particular emphasis on dispersive forces
and some important aspects of this type of interaction. Then, we discuss the correlation
between these forces and the so-called Casimir e�ect, elucidating the fundamental role
played by the zero-point energy of the electromagnetic �elds in understanding this phe-
nomenon. Exploring this concept, we also derive the Lifshitz formula within the scattering
approach and describe the calculation of dispersive interactions in more generic systems
than the idealized one considered by Casimir.
8
I. Quantum theory of dispersive interactions
I.1 Introduction
Since the ancient times of the scientist and philosopher Aristotle (384 - 322 B.C.), the
concept of intermolecular forces was already a maturing idea, as the behavior of gases
indicated that such forces must exist. Speci�cally, a gas may undergo a phase transition
to a liquid � condensation � when subjected to appropriate changes in temperature and
pressure, which can only be explained by an attractive interaction of its atomic/molecular
components. Although more intriguing and not understood at the time, the existence of
these intermolecular interactions was accepted even for gases composed of neutral atoms
or molecules and with no permanent multipole moments, which, from a classical point of
view, should not respond to each other's presence.
Among so many famous names that were somehow involved in attempts to understand
intermolecular forces, such as Newton, Laplace, Gauss, Maxwell, and Boltzmann [1], a
historically important step forward was taken by the Dutch physicist Johannes Diderik
van der Waals in 1873. Moved by the desire to explain why real gases do not obey the
ideal gas law (PV = RT per mole of gas, where P is the pressure, V is the molar volume,
R is the gas constant, and T is the temperature), he proposed the �rst quantitative, albeit
indirect, characterization of such attractive forces. In his dissertation [2], van der Waals
wrote the well-known phenomenological state equation for real gases that today bears his
name, to wit, (
P +
a
V 2
)
(V − b) = RT . (I.1)
This equation is quite simple, but already encompasses the occurrence of phase transitions
through the parameters a and b, known as the van der Waals constants, whose values
depend on the gas and can be obtained by adjusting this equation with experimental
data. It is clear that, for vanishing a and b, we return to the equation of state for ideal
gases. In Eq. (I.1), van der Waals interpreted the term b as the volume that accounts for
the �nite size of molecules and associated the term a/V 2 with an attractive intermolecular
9
I. Quantum theory of dispersive interactions
force between two molecules.
These attractive forces came to be known by the generic name of van der Waals
forces. Since they depend strongly on the polarity of the molecules under study, there are
three types of intermolecular interactions, called orientation, induction, and dispersion.
In Sec. I.2, we describe each of them, with a special focus on the dispersive case, the
main topic of the �rst part of this thesis. Section I.3 is dedicated to discussions of the
Casimir e�ect between neutral, parallel, and conducting plates, obtained by means of the
variation of the zero-point energy of the electromagnetic �eld. We also develop in Sec. I.4
the scattering formalism, presenting the so-called scattering formula, which serves as a
basis for some analytical and numerical calculations in the subsequent chapters. Finally,
Sec. I.5 comprises our �nal comments.
I.2 The van der Waals forces
When we consider two polar molecules (i.e., endowed with permanent electric dipole
moments, such as water molecules), the dominant type of interaction between them is via
orientation van der Waals forces. This type of interaction was originally calculated by
W. H. Keesom [3, 4], who considered the thermal average of the electrostatic interaction
energy between two molecules with dipole moments d1 and d2 that rotate randomly. In
his calculations, he obtained
Uor = −
2d21d
2
2
3kBT (4πε0)2r6
(I.2)
at the high-temperature limit (kBT ≫ d1d2/(4πε0r3)), where kB is the Boltzmann con-
stant, ε0 is the electric permittivity of vacuum, and r is the separation between the
molecules. The negative sign appears because orientations that give rise to the attrac-
tion are statistically favored over repulsive ones. Indeed, even though the number of
attractive and repulsive instantaneous con�gurations is equal, they are all multiplied by
the Boltzmann weight e−U/kBT , which favors the less energetic, attractive, con�gurations.
10
I. Quantum theory of dispersive interactions
Also, the previous equation reveals that the interaction weakens as T increases because all
con�gurations become more and more equally likely, e�ectively washing out the thermal
average.
Around the same time, P. Debye [5, 6] and others recognized that other attractive
forces must exist between molecules since gases composed of nonpolar molecules have
nonvanishing values of the van der Waals constant a. They investigated the so-called
induction forces occurring between a polar molecule and a nonpolar one, provided that
the latter is polarizable. In this situation, the polar particle with a permanent dipole
moment d1 induces a dipole moment d2 in the nonpolar one. Assuming an isotropic
molecule, as well as that the �elds are weak enough so that the linear approximation is
valid, then d2 = α2E1, with E1 being the electric �eld generated by the dipole d1. The
interaction energy can be estimated as
Uind(r) ∼ −d2 ·E1 ∼ −α2E21 ∼ −
α2d
2
1
(4πε0)2r6
, (I.3)
which also generates an attractive force. For further details on these forces, see Ref. [7].
Finally, there are the interactions between apolar molecules, with no permanent dipole
moment (nor higher-order multipoles) but polarizable, called dispersive forces. Although
the existence of an interaction between such types of molecules is a little strange if we
think in terms of classical electrodynamics, this interaction was also postulated by van der
Waals to explain the liquefaction of noble gases, even though he had no means of knowing
the physical origin of this attraction. A fundamental di�erence is that, while the previous
forces can be understood by classical arguments, the explanation for the attraction be-
tween neutral molecules had to wait for the development of quantum mechanics � which
came up after the passing of van der Waals. It was only after the advent of this theory
that the necessary tools became available for a conceptual understanding. A few years
later, in 1930, London [8], and Eisenschitz and London [9] analyzed this problem and,
assuming identical atoms with dominant transition frequency ω0 and static polarizability
11
I. Quantum theory of dispersive interactions
α0, obtained
Udisp(r) = −
3ℏω0α20
4 (4πε0)
2 r6
, (I.4)
using stationary perturbation theory. The presence of ℏ in this expression, the Planck
constant divided by 2π, is the clear signature that endorses the quantum nature of dis-
persive forces. This interaction is genuinely quantum because it arises precisely from
the zero-point quantum �uctuations of the particles' charge and current distributions,
which generate instantaneous dipolesin each molecule. As they are brought closer to-
gether, these �uctuating dipoles become correlated because each one generates an electric
�eld that interacts with the other, becoming more intense as the distance between them
decreases.
This result was considered a triumph of quantum mechanics in explaining the at-
tractive universal force between two molecules, despite neither having a permanent dipole
moment. In other words, it is only necessary that a dipole can be induced in the molecules,
meaning that each one can be polarized (α0 ̸= 0). Actually, the terminology �dispersive�
was inspired by the fact that this interaction involves atomic polarizabilities, which is
in turn related to the refractive index and the dispersion of electromagnetic waves in a
medium composed of these atoms.
Despite its fundamental importance, the previous results have some practical limita-
tions. One of them is related to the characteristic distances for which these calculations
are valid. In all of them, it was implicitly assumed that the atoms and molecules were
close enough so all interactions could be taken as instantaneous, but far enough away to
avoid the overlapping of wave functions. However, when the distances involved become
greater � and soon we will elaborate on these scales, de�ning how much is a short or long
distance �, we must take into account that the information regarding the state of a given
atom takes a certain time to arrive at the other atoms or molecules around it.
In fact, Verwey and Overbeek were facing a problem of incompatibility between the
1/r6 power-law obtained by London and some experimental results concerning colloid
12
I. Quantum theory of dispersive interactions
physics, investigated at the Philips laboratories in the 1940s [10]. Colloidal systems, such
as blood, wine, and ink, are systems composed of two phases, with a solid phase dispersed
in another liquid one in the aforementioned examples [11]. Despite the gravitational e�ect,
the solid particles are too light and do not precipitate quickly. Instead, due to thermal
�uctuations, they execute a Brownian movement because of collisions with particles of
the liquid phase, remaining in suspension for a long time. At the time, the equilibrium
theory for colloidal suspensions was based on the competition of two forces: electrostatic
(repulsive) and dispersive (the attractive London result). This theory was called DLVO
(Derjaguin, Landau, Verwey, and Overbeek) [10]. Due to this competition, colloids can
become unstable. The attractive interaction can cause microscopic particles to stick
together, forming a macroscopic aggregate that then precipitates by the gravitational
e�ect and destabilizes the colloid. The controversy was that experimental results were
revealing that the dispersive attraction was less intense than predicted by London [11].
Overbeek had already suspected the solution could be linked to the fact that the
near-�eld approximation is no longer valid for larger distances, being crucial to take into
account the electromagnetic retardation [12]. This means that the dipole �eld of the �rst
molecule will reach the second one after a time interval r/c, and the reaction �eld of
the second molecule will only reach back the �rst one after 2r/c. In this case, however,
there is an extra mathematical complication, as it becomes mandatory to quantize not
only the degrees of freedom of the atomic system but also those of the electromagnetic
�eld. In 1942, Hendrik B. G. Casimir became a researcher at Philips and was eventually
introduced to the problem of colloids. Employing a fourth-order stationary perturbation
theory, Casimir and Polder [13, 14] corrected London's result in the 1940s, re-obtaining
the short-distance limit and showing that the dispersive interaction energy between two
atoms in the long-distance regime is given by
U
(R)
disp(r) = −
23ℏcαAαB
4π(4πϵ0)2r7
, (I.5)
13
I. Quantum theory of dispersive interactions
being αA and αB the static polarizabilities of each atom. Basically, retardation e�ects
change the behavior from 1/r6 to 1/r7, resulting in a weaker attractive interaction than
predicted by London in the case of large distances. Note that, in addition to ℏ, a marked
symbol of quantum mechanics, the light velocity in vacuum c also appears explicitly as
the hallmark of electromagnetic retardation.
In general, dispersive forces are commonly classi�ed in the literature into two regimes:
non-retarded and retarded. The �rst one is equivalent to the regime discussed in Lon-
don's work. His result was obtained by considering the light velocity as in�nite, which
implies treating the interaction as occurring instantaneously. Thus, this regime is valid
for a0 ≪ r ≪ λ0 = 2πc/ω0, where a0 is the Bohr radius and λ0 is the dominant transition
wavelength, holding as long as the separations remain smaller than the typical transi-
tion wavelengths of the system, but still much larger than the Bohr radius, making the
repulsion from the superposition of the atomic electron clouds negligible. Also, there is
no need to quantize the electromagnetic �eld. The second regime takes into account the
�niteness of the light velocity, requiring the quantization of the electromagnetic �eld, for
it is the mediator of the interaction between atoms. It is also common for some authors to
link this retardation to the progressive loss of correlation between the �uctuating dipoles
and, as a consequence, the retarded force falls faster with distance than the non-retarded
one [15]. Then, retardation e�ects become relevant for distances r ≳ λ0.
It is worth mentioning that such forces are often designated by di�erent names ac-
cording to the geometry and separation of the objects involved, which can cause a bit
of confusion for those who are not so familiar. For instance, it is frequently named �dis-
persive van der Waals force� when retardation e�ects can be ignored, �Casimir forces�
when dealing with macroscopic bodies at length scales where electromagnetic retardation
becomes relevant, and the nomenclature �Casimir-Polder force� usually concerns atom-
surface con�gurations, also taking into account retardation e�ects.
A notable feature regarding dispersive interactions is that they are non-additive. It
14
I. Quantum theory of dispersive interactions
implies that the resultant dispersive force acting on a given atom due to the presence
of others cannot be reliably calculated by simply adding the individual forces that each
of them would exert if all the others were not present. This e�ect was �rst noticed by
Axilrod and Teller in 1943 [16], and, in a system composed of three atoms (denoted by
A, B, and C), the total non-retarded dispersive energy can be written as
Udisp = UAB + UAC + UBC + UABC , (I.6)
where Uij (i, j = A,B,C) represents the so-called pairwise terms, standing for the energy
between atoms i and j calculated in the absence of the third one, and
UABC ∝ ℏω0
αAαBαC
r3ABr
3
ACr
3
BC
, (I.7)
where rij is the distance between the atoms i and j, and αi is the polarizability of the
i-th atom. This 3-body non-additive term mixes properties of the three atoms, capturing
the modi�cation on the interaction between two of them due to the presence of the third
one. Non-additivity e�ects can be positive or negative and quite important in some cases,
depending on the density of the medium. Interesting examples can be found in Ref. [1].
It is possible to understand this property of non-additivity in a very intuitive way. We
saw that each atom has its �uctuating dipole and that the dispersive interaction between
A and B comes from the correlation of these dipoles arising from �uctuations in the
electromagnetic �eld. If we bring another atom C close to them, the dipole �uctuations
in A and B will be in�uenced by the �eld generated by C. But why is this true? To address
this question, we must realize that the �uctuating charges in the atoms are notprescribed.
For instance, this is di�erent from the Coulomb interaction, which is additive and obeys
the superposition principle. The Coulomb force between two point charges is not altered
by the presence of another one because this new one cannot modify the charge of the other
electrons, as the latter is a fundamental quantity. In dispersive interactions, however, there
are �uctuating dipoles that depend directly on the electromagnetic �elds generated by all
15
I. Quantum theory of dispersive interactions
surrounding entities. Therefore, the analysis is di�erent since the interaction between A
and B is entangled with the presence of C.
Lastly, we point out that these ubiquitous forces play an important role not only in dif-
ferent areas of physics, such as atomic and molecular physics, condensed matter physics,
quantum �eld theory, astrophysics, and cosmology [17], but also in engineering, chem-
istry, and biology [18]. Some situations include the stability of colloids [1], as previously
mentioned, the drug binding in proteins and the double-helix stability in DNA [19], the ad-
hesion of geckos to walls [20,21], the generation of electric potentials in thunderstorms [22],
and the cohesion in the �rst steps of accretion of small rotating asteroids [23,24]. Another
important example comprises the van der Waals forces in graphene-based materials. Due
to the chemical inertness of graphene and its reduced dimensionality, these forces must
be considered in order to describe their properties. Similar considerations apply to other
systems, such as 2D dichalcogenides, silicene, germanene, and stanene [25]. With increas-
ing advances in the re�nement of experimental techniques [26�29], dispersive interactions
remain an active topic of study even today. For a detailed overview of theoretical and
experimental e�orts regarding dispersive forces, see Refs. [25,30�33].
I.3 The Casimir e�ect and the resigni�cation of the
zero-point energy
In the last section, we have discussed how Casimir and Polder corrected the result
obtained by London in the long-distance regime. Their result was obtained using a quite
cumbersome procedure, employing fourth-order perturbation theory and quantizing the
entire system (atoms and electromagnetic �eld). Nevertheless, the result obtained by
them, given by Eq. (I.5), was quite simple, which naturally surprised Casimir and Polder,
and left them pondering about the possibility of obtaining such a simple result based on
some shorter calculation. This discomfort is clear in the conclusion of their paper when
they state
16
I. Quantum theory of dispersive interactions
�The very simple form of Eq. (56) and the analogous formula (25) suggest
that it might be possible to derive these expressions, perhaps apart from the
numerical factors, by more elementary considerations. This would be desirable
since it would also give a more physical background to our result, a result which
in our opinion is rather remarkable. So far we have not been able to �nd such
a simple argument.�
During a walk in the fall of 1947, Casimir brought his disquiet to Niels Bohr, who
suggested that the solution could be linked to the vacuum energy of the electromagnetic
�eld, an in�nite quantity that most of the community of theoretical physicists at the time
thought was something that should be discarded or some unidenti�ed error within the
quantum �eld theory [11]. In fact, this divergent quantity can be found by making the
usual canonical quantization of the electromagnetic �eld in the Coulomb gauge. In such
a case, it can be shown that the Hamiltonian operator for the free radiation �eld is given
by [34,35]
Ĥrad =
∑
k,p
ℏωk
(
â†kpâkp +
1
2
)
, (I.8)
where ωk = |k|c, while â†kp and âkp are, respectively, the creation and annihilation opera-
tors of a photon with linear moment ℏk and polarization p. The electromagnetic vacuum
state |{0kp}⟩ is de�ned as the state in which there are no real photons in any mode (k, p)
of the �eld. Its energy, which we will refer to hereafter simply as the vacuum energy (or
zero-point energy), is then given by
E0 = ⟨{0kp}|Ĥrad|{0kp}⟩ =
∑
k,p
1
2
ℏωk . (I.9)
From this equation, we can see that even if there is no real photon in a given �eld mode,
each pair of (k, p) still contributes to the vacuum energy with the value ℏωk/2. The
total vacuum energy is, therefore, a divergent quantity, given by the in�nite sum of the
contributions of all modes.
17
I. Quantum theory of dispersive interactions
Even so, Casimir pursued Bohr's suggestion and came out with a new method, using
it to re-derive the results he had obtained with Polder in a much simpler way, essentially
calculating the variation of the zero-point energy of the quantized electromagnetic �eld
caused by the presence of the atoms [36]. More explicitly, his idea relied on the fact
that the absolute vacuum energy does not have direct physical meaning, but it exists
regardless of whether there are objects in the vacuum or not. However, the vacuum energy
changes when we introduce bodies into this medium, even though it remains in�nite. It is
precisely from this variation in the zero-point energy that physical results can be extracted,
providing measurable e�ects, and that eventually ended up with the name Casimir energy.
To test the concept of vacuum energy and how physical results can be obtained from
it, Casimir analyzes two neutral, parallel, and perfectly conducting plates localized in
vacuum in thermal equilibrium at zero temperature � a much simpler system than the
atomic one, as it is similar to the problem of the vibrating string with �xed ends, whose
normal modes were already known. As both in the presence and absence of the plates the
vacuum energy is a divergent quantity, it is necessary to adopt a regularization prescription
to give physical meaning to this di�erence. Thus, the precise de�nition for the Casimir
energy is given by
ECas = lim
s→0
(∑
k,p
1
2
ℏωk
)
I
−
(∑
k,p
1
2
ℏωk
)
II
 , (I.10)
each of these terms must be calculated in exactly the same region of space. In this equa-
tion, I means that the sum is made considering boundary conditions and regularization,
II implies that the sum is made without boundary conditions but keeping the regular-
ization, and s is the regularization parameter. In the case of two perfectly conducting
plates, the boundary conditions on the electric E and magnetic B �elds are
E × n̂
∣∣∣∣
plates
= 0 , (I.11)
B · n̂
∣∣∣∣
plates
= 0 . (I.12)
18
I. Quantum theory of dispersive interactions
As a result, Casimir obtained an attractive interaction energy between the plates
emerging from a quantum e�ect, namely [37],
ECas = −
π2ℏcA
720L3
, (I.13)
leading to the Casimir force
FCas = −
dECas
dL
= −π
2ℏcA
240L4
, (I.14)
where L is the separation between the plates and A is their area (supposedly very large,
such that A ≫ L2 and edge e�ects are negligible). As can be seen in Eq. (I.13), the
Casimir energy (i) is proportional to ℏ since it is a quantum e�ect, (ii) decays with the
distance, which is consistent with the fact that interaction energies do not have in�nite
range, and (iii) exhibits a negative sign, indicating that the interaction is attractive.
This attractive interaction between neutral, parallel, and perfectly conducting plates in
a vacuum received the name of the Casimir e�ect, one of the results most associated
with the zero-point energy of the electromagnetic �eld. For instance, considering a dis-
tance L = 100 nm, we obtain FCas/A = 12 N/m2, which represents a pressure that, by
today's standards, is not di�cult to measure experimentally. However, the �nite conduc-
tivity e�ects of real materials decrease this value considerably, although they remain very
important.
As a �nal comment in this section, we remark that, while the original version of the
terminology �Casimir force� consists of the attraction between the perfectly conducting
plates