EUSKAL HERRIKO UNIBERTSITATEA
THE UNIVERSITY OF THE BASQUE COUNTRY
Depa men o Elec ici y and Elec onics
CAMPUS OF
INTERNATIONAL
EXCELLENCE
Theo e ical desc ip ion o ligh emission in
he p esence o nanoscale esona o s: om
classical sca e ing o pho on s a es
en anglemen and s a is ics
Thesis by
Ál a o Noda Villa
Supe ised by
P o . Ja ie Aizpu ua I iazabal
and
D . Rubén Es eban Llo en e
Donos ia-San Sebas ián, Feb ua y 2023
(cc)2023 ALVARO NODAR VILLA (cc by-sa 4.0)
ACKNOWLEDGMENTS
Quisie a da las g acias / I would like o exp ess my g a i ude:
A Ja ie y a Rubén, po habe me dado la opo unidad de ealiza es a esis y
po habe me di igido du an e odos es os años. Ha sido un iaje la go y ha habido
e apas du as, pe o posiblemen e sin esos e os no es a ía an segu o ni an o gulloso
como lo es oy aho a de cada una de las páginas que siguen en es a esis. G acias
po o ece me ues o apoyo y ues a paciencia. He c ecido mucho bajo ues a
u ela.
A Gab iel, quien ambién me ha supe isado du an e una la ga e apa de es a
esis. G acias, Gab iel, po emanga e y echa e las cuen as conmigo. G acias po
u dedicación, me aleg a mucho habe podido con a con igo du an e es a e apa.
To Mike and Mikołaj, o welcoming me wi h open a ms in Sydney. The p ojec
on he co ela ions eally mo i a ed me h ough he las s ages o my hesis. I
wan o especially hank Mikołaj o showing me a di e en way o doing science.
Thank you o being so humble and suppo i e. The las ime we wen o a pub,
you in i ed me. I hope o ha e many chances o epaying his deb .
A mi gen e del segundo piso (en o den al abé ico, e i emos a o i ismos):
Ad ián, Albe o, An on, B uno, Ca li os, Ike , Jon, Jona han, Ma io, Ma ín,
Mikel, Mi iam, Robe o, Txemikel. ¿Qué deci ? Hemos compa ido el día a día
du an e años, ya sabéis que os conside o muy buenos amigos y que sois una pa e
impo an e de mi ida. Un cachi o de es a esis es de cada uno de oso os. Ya me
in i a éis a un ca é en el E na, ¿no?
I would like o hank he suppo I ecei ed om all he iends I ha e made
along he way in he Donos i scien i ic communi y. To he old gang: And ea, Be ni,
Bo inaga, Donaldi, Fede, Mo i z, Thomas, Tomáš... hank you o showing me
he opes! To he new gang: Alex, Augus e, Ca mela, C is (Sanz), Edu ne, Fe ,
Gab iele, Jo ge (JOT), Jo ge (Melillo), Josa, Joseba, Josu, Ma ia, Ma ina, Ma hias,
Ma eo, Miguel Angel, Mikel, Paul, Raulillo, S e an, Unai, Xabi... es a esis no
hab ía sido posible sin ues o apoyo y amis ad, g acias po es a a mi lado en es a
e apa.
A mis pad es y a oda mi amilia po odo el apoyo incondicional que me habéis
b indado du an e mi la ga e apa de es udian e. Sé que es a éis an o gullosos de
es a esis como lo es oy yo hoy.
iii
A C is, du an e es os años he sido (y sigo siendo) muy eliz a u lado. Es oy
muy ag adecido po habe compa ido cada día de es a e apa con igo. G acias po
apoya me en los momen os du os y celeb a conmigo cuando las cosas salían bien.
Te quie o mucho.
Po úl imo, y mucho más impo an e que el es o, quie o ag adece el apoyo
económico b indado po el p oyec o PID 2019-107432GB-I00P del Minis e io de
Ciencia e Inno ación y de la Agencia Es a al de In es igación, así como po el
p oyec o IT 1526-22 del Gobie no Vasco. Ha sido una in e sión muy ace ada, os
los asegu o.
i
RESUMEN
Al pasa un dedo po el bo de de una copa, su cue po ib a c eando un sonido.
Es o ocu e po que la copa sopo a una esonancia mecánica, que se exci a al pasa
el dedo con una elocidad y p esión especí icas. Algo simila ocu e con la luz y las
nanoes uc u as o ónicas. Al ilumina algunas nanoes uc u as con una longi ud
de onda o ecuencia (colo ) especí icas, podemos aumen a la in ensidad del campo
dispe sado po la nanoes uc u a. Po ejemplo, las nanoes uc u as me álicas
sopo an esonancias plasmónicas que consis en en una oscilación colec i a de su
densidad de ca ga elec ónica y que pueden se exci adas con luz a ecuencias
óp icas. La exci ación esonan e de las co ien es de pola ización en el in e io
de nanoes uc u as dieléc icas, es o o ejemplo de cómo la luz puede exci a
esonancias óp icas en o os ma e iales. Cada ipo de nanoes uc u a o ece cie as
en ajas pa a con ola la luz en la nanoescala. Po ejemplo, las nanoes uc u as
me álicas pueden con ina la luz en olúmenes mucho más pequeños, mien as que
las nanoes uc u as dieléc icas ienen pé didas po abso ción mucho meno es.
G an pa e de es e ipo de enómenos se pueden explica den o del ma co
de la eo ía de elec omagne ismo clásico p opues a po James Cle k Maxwell en
1865 [1]. Desde en onces, se han desa ollado modelos analí icos (y semianalí icos)
así cómo mé odos numé icos que pe mi en analiza la espues a de nanoes uc u as
o ónicas en condiciones muy gene ales. Po ejemplo, en es a esis empleamos la
llamada eo ía de Mie pa a ob ene una solución semianalí ica a las ecuaciones de
Maxwell de los campos dispe sados po nanopa ículas es é icas bajo di e sas
iluminaciones [2]. Las p edicciones que se ob ienen esol iendo la espues a
elec omagné ica clásica de nanoes uc u as han sido de g an u ilidad pa a el
desa ollo de di e sas aplicaciones o ónicas basadas en el aumen o y la localización
del campo elec omagné ico. Den o de es as aplicaciones, cabe des aca las écnicas
de espec oscopía aumen ada po supe icies [3
–
5], las écnicas de mic oscopía con
esolución submolecula [6
–
8], los a amien os de cánce [9,10], ó las mejo as en
la cap ación y almacenamien o de ene gía sola [11,12].
Po o a pa e, los a ances ecien es en la ab icación y ca ac e ización
de nanoes uc u as o ónicas han pe mi ido alcanza un ni el de p ecisión lo
su icien emen e al o como pa a demos a di e sos e ec os cuán icos, lo cual ha
gene ado un c ecien e in e és en el campo de la nano o ónica cuán ica du an e las
úl imas décadas [13,14].
La nano o ónica cuán ica pe mi e desc ibi enómenos muy a iados. Po
ejemplo, el campo elec omagné ico aumen ado que gene a una nanoes uc u a
o ónica puede da luga a una ue e in e acción óp ica con un emiso cuán ico
(po ejemplo, un pun o cuán ico o una molécula) si uado en su en o no. Si es a
in e acción es muy ue e se pueden da enómenos no lineales en la espues a del
sis ema emiso -nanoes uc u a que solo pueden se desc i os con un o malismo
cuán ico. Al mismo iempo, la in e acción de nanoes uc u as con es ados de luz
cuán icos es pa icula men e in e esan e po su po encial aplicación en di e sas
ecnologías de in o mación cuán ica. Los es ados cuán icos de luz son muy
esis en es a pe de la in o mación codi icada en ellos al man ene su p opagación,
cons i uyéndose en excelen es candida os pa a la ansmisión de in o mación
cuán ica. Sin emba go, las posibilidades de modi ica la in o mación codi icada en
es ados de luz cuán icos se en limi adas debido a la débil in e acción de la luz
con la ma e ia. Una solución p ome edo a a es e e o es ap o echa la in e acción
ampli icada en e la luz y ma e ia que se da al exci a las esonancias óp icas de
las nanoes uc u as.
Es a esis es á dedicada a es udia la in e acción en e es ados de luz clásicos y
cuán icos con nanoes uc u as o ónicas aisladas o in e accionando con emiso es
cuán icos. A con inuación se p esen a una discusión esumida de los con enidos
p incipales de cada capí ulo:
En el capí ulo 1 e isamos algunos aspec os de los undamen os del
elec omagne ismo clásico que emplea emos en es a esis. En conc e o mos amos el
a amien o eó ico pa a en ende la espues a a la luz de una nanopa ícula es é ica,
una nanoes uc u a canónica en el campo de la nano o ónica que nos pe mi e
ilus a di e sos enómenos de la in e acción de la luz con nanoes uc u as o ónicas
esonan es. Pa a el es udio de es e sis ema conside amos a ias desc ipciones.
P ime o in oducimos la ap oximación cuasies á ica, álida pa a desc ibi la
espues a de nanopa ículas con un diáme o mucho más pequeño que la longi ud
de onda de la luz inciden e. La ap oximación cuasies á ica nos pe mi e explica
de una mane a in ui i a los aspec os p incipales de la ísica de es os sis emas.
Po ejemplo, desc ibimos cómo la luz puede exci a las esonancias plasmónicas.
También discu imos cómo se puede mejo a es e modelo ap oximado pa a inclui
la co ección adia i a que iene en cuen a las pé didas de la nanoes uc u a al
dispe sa la luz inciden e. Es os modelos ap oximados son u ilizados en el capí ulo
3pa a analiza los di e en es mecanismos ísicos que in e ienen en la in e acción
en e una nanoes uc u a y un emiso cuán ico.
Jun o con es as desc ipciones ap oximadas, en el capí ulo 1 ambién
in oducimos o malmen e la eo ía semianali ica de Mie. En conc e o, la solución
de Mie pe mi e exp esa los campos dispe sados po una nanopa ícula es é ica
como una suma de dis in as con ibuciones, cada una de ellas co espondiendose
con la exci ación de di e en es modos esonan es de la nanopa ícula. A lo la go de
es a esis u ilizamos el modelo de Mie pa a analiza la in e acción de la luz con
nanopa ículas es é icas.
Finalmen e, en el capí ulo 1desc ibimos las p opiedades del momen o angula
de la luz (espín, helicidad, momen o angula o bi al y momen o angula o al), y
e isamos la in e acción en e haces de luz con p opiedades de momen o angula bien
i
de inidas y nanopa ículas es é icas. Pa a ello ex endemos la solución semianali ica
de la eo ía de Mie pa a desc ibi la dispe sión de es e ipo de haces. Es e ipo de
p oblemas de dispe sión son muy in e esan es en el campo de la nano o ónica, ya
que el momen o angula de la luz in oduce nue os g ados de libe ad que pueden
se con olados con nanoes uc u as y que p esen an un g an po encial en di e sas
aplicaciones ecnológicas, po ejemplo, pa a aumen a la in o mación codi icada en
un haz de luz [15–18] ó pa a de ec a p opiedades de subs ancias químicas (como
la qui alidad de las moléculas) [19,20].
En el capí ulo 2 e isamos algunos de los undamen os de la nano o ónica
cuán ica que emplea emos en el es o de es a esis. En pa icula , nos cen amos
en dos p oblemas canónicos. P ime o es udiamos cómo se puede a a la
ans o mación de la luz inciden e con un di iso de haz. Es a ans o mación es la
base o mal de nues o es udio p esen ado en el capi ulo 5. De mane a más gene al,
el di iso de haz es el elemen o p incipal en muchos in e e óme os, po ejemplo,
el in e e óme o Hanbu y-B own-Twiss (HBT) que empleamos en el capí ulo 4.
El in e e óme o HBT pe mi e la ca ac e ización de la es adís ica del núme o
de o ones emi idos po una uen e, y en el capi ulo 2discu imos de alladamen e
cuál es la base de su uncionamien o y cómo dicha ca ac e ización nos pe mi e
clasi ica di e en es ipos de luz dependiendo de si los o ones ienden a llega
indi idualmen e o en g upos.
El segundo p oblema es udiado en es e capí ulo es la desc ipción cuán ica de
la in e acción en e una ca idad óp ica ( al cómo una nanoes uc u a esonan e)
y un emiso cuán ico basada en el o malismo de elec odinámica cuán ica de
ca idades (ca i y-quan um elec odynamics, en inglés). En conc e o p esen amos
la o mulación del Hamil oniano que desc ibe es a in e acción según el modelo
cuán ico de Rabi (QRM, quan um Rabi model en inglés), el cuál es álido pa a
cualquie alo de la ene gía de in e acción en e la ca idad y el emiso . También
in oducimos el o malismo de la ecuación maes a (mas e equa ion en inglés) que
desc ibe la dinámica de un sis ema cuán ico in e accionando con su en o no, y
que nos pe mi e in oduci las pé didas de la ca idad y la iluminación incohe en e
del emiso . El Hamil oniano del QRM y la ecuación maes a son las p incipales
he amien as que u ilizamos en el capí ulo 4pa a desc ibi la emisión de sis emas
o mados po un emiso cuán ico in e accionando con una nanoes uc u a.
El capí ulo 3 es á dedicado al es udio de la asime ía en las esonancias
Fano que eme gen en el espec o de ex inción de sis emas o mados po una
nanoes uc u a me álica in e accionando débilmen e con un emiso cuán ico. Es e
ipo de sis emas emiso -nanoes uc u a han sido es udiados ex ensi amen e en
el con ex o de las écnicas de espec oscopía de campo aumen ado (su ace-
enhanced spec oscopy en inglés), dónde el campo aumen ado gene ado al exci a
las esonancias plasmónicas de la nanoes uc u a me álica se usa pa a mejo a la
señal espec al de una molécula en la ce canía de la nanoes uc u a, lo que pe mi e
de ec a y ca ac e iza can idades muy pequeñas de moléculas.
Cuándo la ue za de acoplamien o en e el emiso y la nanoes uc u a es débil,
el espec o del sis ema emiso -nanoes uc u a se ca ac e iza po la apa ición de una
esonancia ipo Fano, la cual es o iginada po la in e e encia en e una esonancia
ii
espec almen e es echa co espondien e al emiso y o a mucho más ancha (que
se compo a como un con inuo de modos) co espondien e a la nanoes uc u a
me álica.
Las esonancias de ipo Fano se iden i ican po la apa ición de un cambio de la
señal ab up o en una egión espec al es echa, dando luga a lo que se denomina
un pe il Fano. La o ma de es e pe il a ía con la na u aleza de la in e acción
emiso -nanoes uc u a, y puede p esen a a ios g ados de asime ía, pe o un
modelo sencillo p edice que la esonancia Fano es pe ec amen e simé ica si el
sis ema es esonan e, es deci , cuándo la esonancia del exci ón del emiso y de
la esonancia óp ica de la nanoes uc u a ienen la misma ecuencia cen al. Sin
emba go, abajos expe imen ales ecien es han demos ado que la esonancia Fano
puede se asimé ica incluso en sis emas esonan es [8].
Pa a en ende mejo es a obse ación expe imen al, analizamos en de alle
el o igen de la asime ía en las esonancias de ipo Fano en sis emas emiso -
nanoes uc u a esonan es. Pa a ello empleamos simulaciones numé icas de la
espues a óp ica del sis ema híb ido en es ipos de nanoes uc u as di e en es
(una nanopa ícula es é ica de pla a, una nanopa ícula es é ica de o o, y un díme o
compues o po dos nanopa ículas es é icas de o o, odas ellas iluminadas po una
onda plana), así como una se ie de modelos analí icos basados en sis emas de
dos oscilado es a mónicos acoplados. De es a mane a podemos iden i ica cinco
e ec os que p oducen la asime ía en la señal Fano: (i) la ase que adquie en los
campos inducidos po la nanoes uc u a al p opaga se has a el emiso cuán ico (y
ice e sa), (ii) la exci ación di ec a del emiso cuán ico po la luz inciden e sob e el
sis ema, así como la emisión di ec a del emiso cuán ico al de ec o , (iii) las pé didas
adia i as de la nanoes uc u a, (i ) la con ibución de los elec ones de alencia a
la cons an e dieléc ica de la nanoes uc u a me álica, y ( ) la con ibución de las
esonancias de o den al o que sopo a la nanoes uc u a.
Los dos p ime os e ec os (la ase de p opagación y la exci ación y emisión di ec a)
son cla es pa a explica el o igen de la asime ía en odos los sis emas conside ados.
El impac o de la con ibución de los elec ones de alencia depende del ma e ial
conside ado. Po ejemplo, encon amos que es a con ibución a ec a ue emen e a
la asime ía en las nanoes uc u as de o o, mien as que en las nanoes uc u as de
pla a es mucho meno . Po o a pa e, la con ibución de las esonancias de o den
al o es pequeña en los sis emas conside ados, pe o puede adqui i más impo ancia
en o o ipo de sis emas e iluminaciones, cómo la iluminación po la co ien e únel
de un mic oscopio de e ec o únel (scanning unneling mic oscope, en inglés). Po
úl imo, es necesa io conside a las pé didas adia i as de la nanoes uc u a pa a
ob ene una co ec a desc ipción de la in e acción emiso -nanoes uc u a, y po
an o pa a cap u a co ec amen e la in luencia de odos los o os e ec os en la
asime ía.
En el capí ulo 4 seguimos conside ando un sis ema o mado po un emiso
cuán ico in e accionando con una nanoes uc u a me álica en condiciones esonan es.
Sin emba go, a di e encia del capí ulo an e io , en el capí ulo 4conside amos un
amplio ango de ene gías de acoplamien o en e la nanoes uc u a y el emiso ,
pasando del égimen de acoplamien o débil (dónde el in e cambio de exci aciones
iii
en e la nanoes uc u a y el emiso es más len o que la disipación de la ene gía
inciden e po el sis ema híb ido), al égimen de acoplamien o ue e (dónde el
in e cambio de exci aciones es más ápido que la disipación, de o ma que apa ecen
nue os es ados híb idos), y po úl imo al égimen de acoplamien o ul a ue e
(ca ac e izado po la apa ición de enómenos no lineales asociados con é minos que
no conse an el núme o de exci aciones). Además, en es e capí ulo conside amos
si uaciones donde la iluminación es muy in ensa, y pa a desc ibi co ec amen e
la espues a del emiso conside amos que es e ac úa como un sis ema de dos
ni eles ( wo le el sys em, en inglés), lo cual puede in oduci enómenos no lineales
adicionales (po ejemplo, el denominado bloqueo de o ones, pho on blockade, en
inglés).
En es e capí ulo es udiamos las co elaciones de in ensidad de la luz emi ida
po es e sis ema híb ido bajo una exci ación incohe en e del emiso , lo cual
equie e i más allá de la desc ipción clásica. Pa a ello u ilizamos dos modelos
cuán icos di e en es. P ime o in oducimos una o mulación del QRM desa ollada
ecien emen e que es álida pa a cualquie égimen de acoplamien o. En segundo
luga conside amos el modelo Jaynes-Cummings (JCM, Jaynes-Cummings model,
en inglés), el cual puede de i a se a pa i del QRM ás aplica la ap oximación
de onda o an e (RWA, o a ing wa e app oxima ion, en inglés) que desp ecia
los é minos que no conse an el núme o de exci aciones en el Hamil oniano del
QRM. Es a ap oximación no es álida pa a desc ibi el égimen de acoplamien o
ul a ue e, donde los é minos que no conse an el núme o de exci aciones se
uel en más impo an es, sin emba go ha sido u ilizada con éxi o pa a simpli ica
el análisis de sis emas acoplados débilmen e [21,22].
Al compa a las co elaciones de in ensidad calculadas con el QRM y con el JCM
obse amos que, en el égimen de acoplamien o ul a ue e, el QRM p edice una
emisión amon onada (bunched, en inglés), mien as que el JCM p edice una emisión
an i-amon onada (an ibunched, en inglés). So p enden emen e, bajo iluminaciones
débiles, es a di e encia no solo se da en el égimen de acoplamien o ul a ue e,
sino que ambién ocu e en el égimen ue e y débil, dónde se espe a ía que el
QRM y el JCM coincidie an.
A con inuación, analizamos en de alle la in luencia de cada au oes ado del
sis ema en la emisión, y encon amos que la emisión amon onada en el QRM se puede
a ibui al decaimien o de un sólo au oes ado, el pola i on
|3−⟩R
(co espondien e
al quin o es ado exci ado del sis ema bajo ene gías de acoplamien o pequeñas).
Debido a la p esencia de é minos que no conse an el núme o de exci aciones en
el Hamil oniano del QRM, el es ado
|3−⟩R
puede se exci ado de mane a di ec a
desde el es ado base (g ound s a e, en inglés), así cómo decae emi iendo múl iples
o ones de mane a simul ánea, lo que p oduce la emisión amon onada del sis ema.
Po el con a io, en el ma co del JCM el es ado análogo
|3−⟩
sólo puede exci a se
de mane a secuencial, median e es p ocesos de abso ción, un mecanismo mucho
menos e icien e pa a in ensidades y ue zas de acoplamien o su icien emen e bajas.
Recalcamos que es a di e encia en e el JCM y el QRM se puede ex ende al égimen
de acoplamien o débil, donde no malmen e se espe a que los esul ados del JCM
y el QRM coincidan, lo que indica que los é minos que no conse an el núme o
ix
In oduc ion
emi e (QE, e.g. a quan um do o a molecule) loca ed in he p oximi y o he
nanopa icle. I his in e ac ion is e y s ong, a a ie y o nonlinea phenomena
can occu in he esponse o he QE-nanos uc u e sys em ha can be be e
desc ibed h ough a quan um amewo k. A he same ime, he in e ac ion o
nanos uc u es wi h quan um s a es o ligh is pa icula ly in e es ing o a ious
quan um in o ma ion echnologies. Quan um s a es o ligh a e e y esilien o he
loss o hei in o ma ion h ough p opaga ion, es ablishing hemsel es as excellen
candida es o he ansmission o quan um in o ma ion. Howe e , he possibili ies
o con olling he in o ma ion encoded in quan um ligh s a es a e limi ed by he
weak in e ac ion o ligh wi h ma e . A p omising solu ion o his challenge is o
exploi he ampli ied in e ac ion be ween ligh and ma e ha occu s upon he
exci a ion o op ical esonances in nanos uc u es.
This hesis is de o ed o s udying he in e ac ion be ween classical and quan um
s a es o ligh wi h pho onic nanos uc u es and QE-nanos uc u e sys ems. In he
ollowing, we in oduce a summa y o he main con en s o each chap e .
In chap e 1 we summa ize some o he undamen als o classical
elec omagne ism ha we use in his hesis. In pa icula , we e iew he esponse
o ligh o a sphe ical nanopa icle, a canonical nanos uc u e in he ield o
nanopho onics ha allows us o illus a e a ious phenomena o he in e ac ion
o ligh wi h esonan pho onic nanos uc u es. We use di e en app oaches o
desc ibe he op ical esponse. Fi s , we in oduce he quasi-s a ic app oxima ion,
alid o desc ibing he esponse o nanopa icles wi h a size much smalle han he
inciden wa eleng h o ligh . The quasi-s a ic app oxima ion allows o explaining,
in an in ui i e way, he main aspec s o ligh -ma e in e ac ion. Fo example, we
desc ibe how ligh can exci e he plasmonic esonances in me allic nanopa icles. We
also discuss how his app oxima ed model can be imp o ed o include he adia i e
co ec ion ha akes in o accoun he inc ease in he losses o he nanos uc u e
due o he sca e ing o ligh . We use hese app oxima ed models in chap e 3
o analyze he di e en aspec s o he in e ac ion be ween a QE and a me allic
nanos uc u e.
Along wi h hese app oxima e desc ip ions, in chap e 1we o mally in oduce
he semianaly ical Mie heo y ha desc ibes he exac solu ion o Maxwell’s
equa ions o he ield sca e ed by a sphe ical nanopa icle. Speci ically, Mie’s
solu ion allows o exp essing he ield sca e ed by a nanopa icle as a sum o
di e en con ibu ions, each co esponding o he exci a ion o di e en esonan
modes o he nanopa icle. Th oughou his hesis, we use Mie heo y o analyze
he in e ac ion o ligh wi h sphe ical nanopa icles.
Finally, in chap e 1we desc ibe he angula momen um p ope ies o ligh (spin,
helici y, o bi al angula momen um, and o al angula momen um), and e iew he
in e ac ion be ween ligh beams wi h well-de ined angula momen um p ope ies
and nanopa icles. Fo his pu pose, we ex end he semi-analy ic solu ion o Mie
heo y o desc ibe he sca e ing o such beams by a sphe ical nanopa icle. Such
sca e ing p oblem is e y in e es ing in he ield o nanopho onics, since he angula
momen um o ligh in oduces new deg ees o eedom ha can be con olled wi h
nanos uc u es and ha e g ea po en ial in a a ie y o applica ions, o example,
2
In oduc ion
o inc ease he in o ma ion encoded in a ligh beam [15
–
18] o o de ec addi ional
p ope ies o chemical subs ances (such as he chi ali y o molecules) [19,20].
In chap e 2 we e iew he undamen als o quan um nanopho onics ha we use
in his hesis. In pa icula , we ocus on wo canonical p oblems. Fi s , we desc ibe
he ans o ma ion o a s a e o ligh by a beam spli e . This ans o ma ion is he
o mal basis o he s udy p esen ed in chap e 5. Mo eo e , he beam spli e is he
main elemen in many in e e ome e s, o example, in he Hanbu y-B own-Twiss
(HBT) in e e ome e ha we use in chap e 4. The HBT in e e ome e enables
he cha ac e iza ion o he s a is ics o he numbe o pho ons emi ed by a ligh
sou ce. In chap e 2, we discuss in de ail wha is he basis o he HBT ope a ion
and how i allows us o cha ac e ize di e en ypes o ligh depending on whe he
he pho ons a e emi ed mos ly sepa a ely (an ibunched) o in g oups (bunched).
The second p oblem s udied in his chap e is he quan um desc ip ion wi hin
ca i y Quan um Elec odynamics (ca i y-QED) o he in e ac ion be ween an
op ical ca i y (such as a esonan nanos uc u e) and a QE. In pa icula , we
p esen he o mula ion o he Hamil onian ha desc ibes his in e ac ion acco ding
o he quan um Rabi model (QRM), which is alid o any alue o he in e ac ion
s eng h be ween he ca i y and he QE. We also in oduce he mas e equa ion
o malism ha desc ibes he dynamics o a quan um sys em in e ac ing wi h i s
en i onmen and allows us o desc ibe he losses and he incohe en illumina ion
o he sys em. The QRM Hamil onian and he mas e equa ion a e he wo pilla s
we use in chap e 4 o desc ibe he ligh emission om sys ems consis ing o a QE
in e ac ing wi h a nanos uc u e.
Chap e 3 is de o ed o he s udy o he asymme y in he Fano esonances ha
eme ges in he ex inc ion spec um o sys ems o med by a me allic nanos uc u e
in e ac ing weakly wi h a QE. Such QE-nanos uc u e sys ems ha e been ex ensi ely
s udied in he con ex o enhanced ield spec oscopy echniques, whe e he enhanced
ield gene a ed by exci ing he plasmon esonances o he me allic nanos uc u e is
used o inc ease he spec al signal o he molecule. These echniques allows o
he de ec ion and cha ac e iza ion o e y small quan i ies o molecules.
When he coupling s eng h be ween he QE and he nanos uc u e is weak,
he ex inc ion spec um o he hyb id QE-nanos uc u e sys em is cha ac e ized
by he appea ance o a Fano- ype esonance, caused by he in e e ence be ween a
spec ally na ow esonance co esponding o he QE and a much wide one (which
beha es as a con inuum o modes) co esponding o he me allic nanos uc u e.
Fano esonances a e iden i ied by he appea ance o a sha p spec al ea u e,
which is he so-called Fano lineshapes. These lineshapes depend on he na u e o he
QE-nanos uc u e in e ac ion and can exhibi di e en deg ees o asymme y. In
pa icula , a simple model indica es ha a Fano esonance is pe ec ly symme ic i
he sys em is esonan , i.e. when he cen al equency o he (exci onic) esonance
o he QE is uned o he cen al equency o he op ical esonance o he
nanos uc u e. Howe e , ecen expe imen al wo k has shown ha he Fano
esonance can be asymme ic e en in esonan sys ems.
To be e unde s and his expe imen al obse a ion, we analyze in de ail he
o igin o he asymme y in he Fano esonances o ze o-de uned QE-nanos uc u e
3
In oduc ion
sys ems. Fo his pu pose, we analyze he op ical esponse in h ee di e en ypes
o nanos uc u es (a sphe ical sil e nanopa icle, a sphe ical gold nanopa icle,
and a dime composed o wo sphe ical gold nanopa icles, all illumina ed by a
plane wa e) using nume ical simula ions and a se ies o analy ical models based
on coupled ha monic oscilla o s. In his way, we iden i y he di e en physical
mechanisms ha o igina e he Fano asymme y in ze o-de uned QE-nanos uc u e
sys ems.
In chap e 4 we also conside a QE in e ac ing wi h a nanos uc u e unde
esonan condi ions. Howe e , unlike he p e ious chap e , in chap e 4we conside
a wide ange o coupling s eng hs be ween he nanos uc u e and he QE, anging
om he weak coupling egime (whe e he exchange o exci a ions be ween he
nanos uc u e and he QE is slowe han he dissipa ion o he inciden ene gy by
he sys em), o he s ong coupling egime (whe e he exchange o exci a ions is
as e han he dissipa ion so ha new hyb id s a es appea ), and inally o he
ul a-s ong coupling egime ( he coupling s eng h becomes so la ge ha nonlinea
phenomena associa ed wi h e ms ha do no conse e he numbe o exci a ions
becomes possible). Fu he mo e, in his chap e , we ake in o accoun ha an
(exci onic) ansi ion o a QE ac s as a wo-le el-sys em, which in oduces addi ional
nonlinea phenomena unde s ong illumina ion ( o example, he so-called pho on
blockade).
We s udy he in ensi y co ela ions o he ligh emi ed by his hyb id QE-
nanos uc u e sys em unde incohe en exci a ion o he QE. This s udy equi es
going beyond he classical desc ip ion. In pa icula , we use wo di e en quan um
models. Fi s , we in oduce a ecen ly-de eloped o mula ion o he QRM ha is
alid o any coupling egime. Second, we conside he Jaynes-Cummings model
(JCM), which can be de i ed om he QRM a e applying he o a ing wa e
app oxima ion (RWA) ha neglec s he e ms ha do no p ese e he numbe o
exci a ions in he Hamil onian o he QRM. This app oxima ion is known o ail in
he ul a-s ong coupling egime, whe e he ole o he e ms ha do no conse e
he numbe o exci a ions becomes mo e impo an , bu i has been success ully
used o simpli y he analysis o many sys ems in he weak and s ong coupling
egime [21,22].
We compa e he in ensi y co ela ions calcula ed wi h he QRM and he JCM,
and obse e ha , in he ul a-s ong coupling egime, he QRM p edic s a bunched
emission while he JCM p edic s an an ibunched emission. Su p isingly, unde
weak illumina ions, his di e ence does no only occu s in he ul a-s ong coupling
egime bu also in he s ong and weak egimes, whe e he QRM and JCM a e
expec ed o ag ee. In his chap e , we analyze in de ail he o igin o he bunched
emission in he QRM and he de ia ion be ween he QRM and he JCM, i.e. he
b eakdown o he RWA.
In chap e s 3and 4we ocus on he esponse o nanos uc u es illumina ed
wi h classical beams. In con as , in chap e 5 we conside a nanos uc u e
illumina ed by a quan um s a e o ligh . Speci ically, we s udy he esponse o
o a ionally symme ic nanos uc u es illumina ed by a quan um s a e composed
o wo en angled pho ons wi h in o ma ion encoded in hei angula momen um
4
In oduc ion
p ope ies. On he one hand, such s a es a e pa icula ly in e es ing o quan um
in o ma ion p ocessing applica ions, since he angula momen um o ligh is a
p ope y ha is no limi ed o wo alues like he spin o a apped ion qubi . The
o bi al angula momen um o ligh opens an (in p inciple) in ini e Hilbe space
o encode in o ma ion. On he o he hand, o a ionally symme ic nanopa icles
p ese e he o al angula momen um o he s a e a e sca e ing, hus o e ing he
possibili y o manipula ing he quan um s a e in a con olled manne . This con ol
in he manipula ion o a quan um s a e o ligh is e y in e es ing o p ocessing i s
quan um in o ma ion, bu equi es ha he quan um pu i y o he inciden s a e is
espec ed o a high deg ee.
We de elop a gene al heo e ical o malism o model he sca e ing p ocess
in hese sys ems, which is based on he ans o ma ion o a quan um s a e by a
lossy beam spli e . Using his o malism we calcula e he ou pu s a e sca e ed
by he nanos uc u e and ind ha he pu i y o he inciden s a e can be los in
he in e ac ion wi h he nanos uc u e. We hen de elop a semi-analy ical model
based on ea ing he quasi-monoch oma ic inpu and ou pu modes ha allows us
o iden i y he physical mechanism ha causes he loss o pu i y.
In summa y, he esea ch p esen ed in his hesis ad ances ou unde s anding o
undamen al physical aspec s o he in e ac ion o bo h classical ligh and quan um
s a es o ligh wi h nanos uc u es.
5
Chap e
1
CLASSICAL DESCRIPTION OF THE
INTERACTION BETWEEN LIGHT
AND MATTER AT THE NANOSCALE
The main opic o his hesis is he in e ac ion be ween ligh and ma e a he
nanoscale. In his chap e , we in oduce he heo e ical amewo k ha we use o
s udy he in e ac ion be ween classical s a es o ligh and a nanos uc u e.
Nanos uc u es can be used o con ol ligh a he nanoscale, allowing o
concen a ing inciden elec omagne ic ields in e y small egions [3,23
–
26], change
he pola iza ion p ope ies o inciden ligh [27
–
29], o gene a e ligh o a equency
di e en han ha o he illumina ion [30
–
32]. Mos o hese e ec s can be desc ibed
wi hin he heo y o classical elec omagne ism, whe e ligh -ma e in e ac ion
can be cap u ed Maxwell’s equa ions, a se o di e en ial equa ions ha desc ibe
how elec omagne ic ields e ol e in ime and space in a pa icula dielec ic
con igu a ion. By applying he app op ia e bounda y condi ions, Maxwell’s
equa ions can be sol ed o ob ain he esponse o an a bi a y nanos uc u e
unde speci ic illumina ion.
In sec ion 1.1 o his chap e , we e iew he o mula ion o Maxwell’s equa ions.
In sec ions 1.2 and 1.3 we e iew he analy ical and semi-analy ical solu ions o
Maxwell’s equa ions o a canonical nanos uc u e: a sphe ical nanopa icle. The
analy ical o semi-analy ical solu ions allow us o discuss he gene al p ope ies
o he op ical esponse. Speci ically, in sec ion 1.2 we ea he nanopa icle as
a pola izable objec ha beha es as an elec ic poin -like dipole, a commonly
used app oxima ion in nanopho onics. We also discuss how a simila app oach
can be used o desc ibe he op ical esponse o molecules and o he quan um
emi e s (QEs). On he o he hand, in sec ion 1.3 we in oduce a semi-analy ical
solu ion o Maxwell’s equa ions (ob ained wi hou any app oxima ion) o he
7
Chap e 1. Fundamen als o classical nanopho onics
ields sca e ed by a sphe ical nanopa icle. We e alua e hese amewo ks in a
canonical con igu a ion, he sca e ing o linea ly pola ized ligh by a me allic
sphe ical nanopa icle. Finally, in sec ion 1.4 we expand he o malism in oduced
in sec ion 1.3 o desc ibe he sca e ing o a sphe ical nanopa icle illumina ed by
complex beams o ligh , in pa icula , beams o ligh wi h well-de ined angula
momen um p ope ies, as hose used in chap e 5.
1.1 Maxwell’s equa ions
In 1865 he Sco ish ma hema ician James Cle k Maxwell published “A Dynamical
Theo y o he Elec omagne ic Field” [1], a pape con aining he o iginal o mula ion
o his amous equa ions showing he in e ela ionship be ween elec ic ields
E
(
,
)
and magne ic ields
B
(
,
)in a dielec ic medium (
E
(
,
)and
B
(
,
)a e e alua ed
a a posi ion
and ime
). Nine een yea s la e , in 1884, he English ma hema ician
Oli e Hea iside used his de elopmen s in ec o ial and complex numbe calculus o
e o mula e Maxwell’s equa ions in o he o m ha has been known e e since [33],
∇×E( , ) = −∂B( , )
∂ ,
∇×H( , ) = ∂D( , )
∂ +Jex ( , ),
∇·D( , ) = ρex ( , ),
∇·B( , )=0,(1.1)
whe e
Jex
is he ex e nal cu en densi y, and
ρex
is he ex e nal cha ge densi y.
The elec ic ield displacemen is
D
(
,
) =
ε0E
(
,
) +
PD
(
,
), whe e
PD
is he
pola iza ion ield o he medium and
ε0
is he elec ic pe mi i i y in a acuum.
Simila ly, he magne izing ield is
H
(
,
) =
B
(
,
)
/µ0−MB
(
,
), wi h
MB
he
magne iza ion ield o he medium and
µ0
he magne ic pe meabili y in a acuum.
In his hesis, we assume linea ligh -ma e in e ac ion, wi h
PD
(
,
)
∝E
(
,
)
and
MB
(
,
)
∝B
(
,
)and we assume ha he ma e ials we a e ea ing a e
iso opic. Thus, we in oduce he cons i u i e ela ionship [34
–
36] o elec ic ield
displacemen and he magne izing ield,
D( , ) = ε0εE( , ),(1.2)
and
B( , ) = µ0µH( , ),(1.3)
whe e
ε
and
µ
a e he ela i e dielec ic and ela i e magne ic pe mi i i y o he
medium, espec i ely.
Fo con enience, in his hesis, we ea he ields in he equency domain, wi h
E
(
, ω
) =
´d E
(
,
)
eiω /
(2
π
)and
B
(
, ω
) =
´d B
(
,
)
eiω /
(2
π
), espec i ely,
whe e
ω
is he (angula ) equency o ligh . Maxwell’s equa ions
(1.1)
can hen be
8
1.1. Maxwell’s equa ions
w i en as:
∇×E( , ω) = iωB( , ω),
∇×H( , ω) = Jex ( , ω) + iωD( , ω),
∇·D( , ω) = ρex ( , ω),
∇·B( , ω) = 0.(1.4)
Fu he mo e, o all he sys ems s udied in his hesis, we conside he case
whe e he e a e no ex e nal cu en s o cha ges [37], i.e.,
Jex
= 0 and
ρex
= 0, so
ha Eq. (1.4) simpli ies o,
∇×E( , ω) = iωB( , ω),
∇×B( , ω) = iω ε0ε
µ0µE( , ω),
∇·D( , ω)=0,
∇·B( , ω)=0,(1.5)
whe e we ha e also accoun ed o he cons i u i e ela ionship in Eqs.
(1.2)
and
(1.3).
In 1901 he Ge man ma hema ician Hein ich Webe e o mula ed Eq.
(1.5)
in
acuum (ε= 1 and µ= 1) as [37]
i∂F±( , )
∂ =c0∇×F±( , ),
∇·F±( , )=0,(1.6)
whe e c0= 1/√ε0µ0is he speed o ligh in acuum and
F±( , ) = E( , )±ic0B( , ),(1.7)
a e he Riemann–Silbe s ein ec o s named a e he Ge man ma hema ician
Be nha d Riemann, who inspi ed Hein ich Webe o publish Eq.
(1.6)
, and a e
he Polish-Ame ican physicis Ludwik Silbe s ein, who also published his equa ion
independen ly o Webe ’s wo k in 1907 [38,39].
Equa ion (1.6) can be easily w i en in he equency domain,
∇×
kF±( , ω) = ±F±( , ω),(1.8)
whe e
F±( , ω) = E( , ω)±ic0B( , ω),(1.9)
and k=ω/c0is he wa e ec o o ligh .
In his hesis, we mos ly use he s anda d o mula ion in Eq.
(1.5)
ha
conside s he elec ic and magne ic ields
E
and
B
. Howe e , we also discuss he
ad an ages o he Riemann–Silbe s ein o malism (Eq.
(1.8)
) when ea ing ligh
9
Chap e 1. Fundamen als o classical nanopho onics
k
E0
R
x
z
εou
εin
Figu e 1.1: Scheme o he p oblem s udied in subsec ions 1.2 and 1.3. A
x
-pola ized plane wa e
ha p opaga es along he
z
-axis in e ac s wi h a sphe ical nanopa icle o adius
R
. We ha e
explici ly indica ed he ela i e dielec ic pe mi i i y inside and ou side he nanopa icle,
εin
and εou , espec i ely.
wi h well-de ined angula momen um in sec ion 1.4.
1.2 Elec omagne ic esponse o e y small
objec s
Th oughou his hesis, we o en calcula e he elec omagne ic esponse o e y
small objec s, such as a small nanopa icle o a quan um emi e (QE, e.g., a
molecule, a quan um do , o a ni ogen- acancy cen e in diamond). This sec ion
discusses a e y common app oach o sol ing his p oblem: conside ing ha he
small objec is exci ed (o pola ized) by he illumina ion and beha es as a poin -like
elec ic dipole [35]. Du ing his sec ion we ocus on desc ibing he sepa a e esponse
o a nanopa icle and a QE. In chap e 3, we use he same amewo k in oduced
he e o desc ibe he esponse o a nanopa icle in e ac ing wi h a QE.
This sec ion is s uc u ed as ollows: we i s in oduce, in subsec ion 1.2.1,
he ields induced (o sca e ed) by a sphe ical nanopa icle unde plane-wa e
illumina ion ea ed wi hin he quasis a ic app oxima ion. Nex , in subsec ion
1.2.2, we b ie ly o mula e he esponse beyond his quasis a ic app oxima ion. In
subsec ion 1.2.3, we use he amewo k de eloped in he p e ious subsec ions o
illus a e he a - ield spec al esponse o he nanopa icle. Finally, in subsec ion
1.2.4 we use a simila o malism o ea he esponse o a QE.
1.2.1 Response o a e y small sphe ical nanopa icle
wi hin he quasis a ic app oxima ion
Nea - ield esponse
We i s e iew he solu ion o he nea ields induced (sca e ed) by a e y small
sphe ical nanopa icle o adius
R
much smalle han he wa eleng h o ligh
λ
.
The ma e ial o he nanopa icle has a ela i e dielec ic pe mi i i y
εin
and he
10
1.2. Elec omagne ic esponse o e y small objec s
medium ou side he nanopa icle has a ela i e dielec ic pe mi i i y
εou
. We
conside he pa icle o be illumina ed by an inciden plane wa e o ampli ude
E0
,
pola ized along he
x
-axis, and p opaga ing along he
z
-axis (Fig. 1.1). Fi s , we
conside ha he pa icle is e y small compa ed o he wa eleng h, and hus we
can assume he elec ic ield o be cons an along he space. This is he so-called
elec os a ic app oxima ion, wi hin which he elec omagne ic ields ul ill
∇×E( , ω)=0,
∇×H( , ω)=0,
∇·D( , ω)=0,
∇·B( , ω)=0.(1.10)
in he equency domain, whe e we ha e al eady accoun ed o he absence o
ex e nal cu en s o cha ges, i.e.,Jex = 0 and ρex = 0.
To sol e
E
in Eq.
(1.10)
, we no ice i s ha he o a ional o he elec ic ield
is ze o. Thus, we can w i e Eas he g adien o a scala unc ion [35,36,40],
E( , ω) = −∇VE( , ω)(1.11)
whe e he scala unc ion,
VE
, is he elec ic o elec os a ic po en ial. Using
he s anda d cons i u i e ela ions [34
–
36] o he displacemen ec o (
D
(
, ω
)
∝
E
(
, ω
)) in Eq.
(1.2)
, and
∇·D
(
, ω
)=0(in Eq.
(1.10)
) we ind ha
VE
mus
sa is y Laplace equa ion,
∇2VE( , ω)=0.(1.12)
The ields ou side he nanopa icle include he inciden plane wa e and he ields
sca e ed by he nanopa icle. Howe e , we conside ha he ields induced by he
nanopa icle decay wi h he dis ance, and hus, a om he pa icle
Eou
educes
o he inciden plane wa e,
lim
→∞ Eou ( , ω) = E0ux,(1.13)
wi h
E0ux
he elec ic ield o he inciden plane wa e (
ux
is he uni y ec o along
he
x
-axis). The bounda y condi ions a he su ace o he sphe ical nanopa icle,
R, a e gi en by [35,36],
n×[Eou (R, ω)−Ein(R, ω)] = 0,(1.14)
and
n·[Dou (R, ω)−Din(R, ω)] = 0.(1.15)
The solu ion o Eqs. (1.12)-(1.15) is [36],
Ein( , ω) = 4πε0
3εou (ω)
εou (ω)+2εin(ω)E0uz,(1.16)
11
Chap e 1. Fundamen als o classical nanopho onics
λ = 350 nm
R = 70 nm
z
x
Figu e 1.5: Fa - ield di ec i i y o a
R
= 70 nm adius sil e nanopa icle illumina ed by a
x
-pola ized plane wa e p opaga ing along he
z
-di ec ion wi h
λ
= 350 nm. The esul s a e
ob ained wi hin he quasis a ic app oxima ion (using Eqs.
(1.20)
,
(1.35)
, and
(1.36)
). The
di ec i i y is shown o di e en θs-angles along he xz-plane.
owa ds λ≈380 nm.
Las , o he la ges adius conside ed,
R
= 70 nm, he maximum enhancemen
ac o spec a ob ained wi h he quasis a ic app oxima ion (Fig. 1.2b) is almos
an o de o magni ude la ge han he esul in Fig. 1.4a. The main cause o
his e ec is he adia i e co ec ion, which acco ding o Eqs.
(1.29)
and
(1.36)
scales wi h
R3
. Simila o he
R
= 30 nm case, he adia i e co ec ion no only
causes a dec ease in he maximum enhancemen , bu i b oadens he enhancemen
peak, and induces a ed-shi in he esponse. Wi hou he adia i e co ec ion,
he peak o he
R
= 70 nm nanopa icle was cen e ed a
λ≈
350 nm, and wi h
he adia i e co ec ion, i is cen e ed a
λ≈
450 nm. We show in sec ion 1.3
ha he adia i e-co ec ed model cap u es he main changes o he lowes -ene gy
esonance wi h inc easing adius bu becomes inaccu a e as
R
inc eases and does
no cap u e some phenomena as he exci a ion o highe o de esonances.
Figu e 1.4b shows he enhancemen ac o dis ibu ion o a sil e sphe ical
nanopa icle o adius
R
= 70 nm illumina ed a
λ
= 350 nm (we only e alua e
he sca e ed ields ou side he nanopa icle). The enhancemen is signi ican ly
educed wi hin he adia i e co ec ed model, bu he egion o s onge ields is
s ill localized nea he nanopa icle in a e y simila way o he quasis a ic solu ion
(Fig. 1.2a).
Fa - ield emission
Nex , we e alua e he ields sca e ed by he nanopa icle in he a - ield egion
(
≫λ
) wi hin he adia i e-co ec ed model. Figu e 1.5 shows he a - ield
di ec i i y o a
R
= 70 nm sil e sphe ical nanopa icle illumina ed a
λ
= 350
nm by he same
x
-pola ized,
z
-p opaga ing plane wa e. The di ec i i y is de ined
18
1.2. Elec omagne ic esponse o e y small objec s
as [42]
Da(ϕs, θs) = 4π
‚|EF F
sca ( s, ϕs, θs, ω)|2dΩ|EF F
sca ( s, ϕs, θs, ω)|2,(1.37)
wi h
s≫λ
(in he a - ield).
Da
desc ibes he emission o he ields in a (
θs, φs
)
di ec ion (in his case, we only show he dependence on
θs
because he emission
is
φs
-symme ic). The di ec i i y pa e n in Fig. 1.5 shows wo lobes o ien ed
along he
z
-axis, whe e he a - ield emission o he nanopa icle is maximized.
In e es ingly, hese lobes a e o ien ed along he di ec ion o p opaga ion o he
inciden plane wa e, con a y o he nea - ields gene a ed by he nanopa icle,
which a e concen a ed along he x-axis (pola iza ion o he inciden plane wa e).
1.2.3 Op ical c oss-sec ions o a small pa icle
The adia i e co ec ed model in oduced in his sec ion allows us o ob ain he
a - ield esponse o a sphe ical nanopa icle exci ed by a linea ly pola ized plane
wa e o ampli ude
E0
. The a - ield spec al esponse o nanos uc u es is usually
cha ac e ized by h ee di e en quan i ies: he abso p ion c oss-sec ion
σabs
, he
sca e ing c oss-sec ion
σsca
, and he ex inc ion c oss-sec ion
σex
. The abso p ion
c oss-sec ion can be ela ed o he powe dissipa ed by he nanopa icle,
Pabs
,
as
Pabs
(
ω
) =
σabs
(
ω
)
I0
, whe e
I0
is he in ensi y o he inciden ligh . On he
o he hand, he sca e ing c oss sec ion ela es o he powe o he ligh elas ically
sca e ed in all di ec ions by he nanopa icle as
Psca
(
ω
) =
σsca
(
ω
)
I0
. Finally, he
ex inc ion c oss-sec ion co esponds o he sum o he abso p ion and sca e ing
c oss-sec ions,
σex
(
ω
) =
σsca
(
ω
)+
σabs
(
ω
), and desc ibes he o al powe ex inc ed:
Pex
(
ω
) =
σex
(
ω
)
I0
. In his sec ion we in oduce all he op ical c oss sec ions o a
pa icle in a acuum, and he ex ension o o he non-dissipa ing and linea media
is s aigh o wa d.
The powe sca e ed by a nanopa icle can be ob ained acco ding o Poyn ing’s
heo em [35,36,43] as,
Psca(ω) = ¨A
¯
Ssca( , ω)·nAdA,(1.38)
whe e
A
is an a bi a y closed a ea su ounding he dipole,
nA
is he no mal ec o
o his su ace a each su ace poin , and
¯
Ssca( , ω) = 1
2Re{Esca( , ω)×Hsca( , ω)∗}(1.39)
is he ime-a e aged Poyn ing ec o o he sca e ed elec omagne ic ields. Using
σsca
(
ω
) =
Psca
(
ω
)
/I0
, he exp essions o
Esca
and
Hsca
in Eqs.
(1.20)
-
(1.25)
, and
pa(ω) = αRC
a(ω)E0, we ob ain
σsca(ω) = k4
6πε0|αRC
a(ω)|2.(1.40)
19
Chap e 1. Fundamen als o classical nanopho onics
The ex inc ion c oss-sec ion,
σex
, can be di ec ly ob ained by using he op ical
heo em o a sys em d i en by a linea ly
x
-pola ized plane wa e p opaga ing along
z
. The op ical heo em ela es
σex
wi h he
x
-componen o he ield sca e ed by
he nanopa icle a some poin zd≫λ( a - ield) along he z-axis [36,40,44],
σex (ω)=2λIm zd
Esca(zd, ω)·ux
E0(zd, ω)·ux,(1.41)
whe e
ux
is he uni y ec o along he
x
-axis. By using Eqs.
(1.20)
,
(1.35)
,
(1.36)
,
and (1.41) we ob ain,
σex (ω) = k
ε0
Im{αRC
a(ω)}.(1.42)
No e ha we ha e w i en σsca and σex in Eqs. (1.40) and (1.42) in e ms o he
adia i e-co ec ed pola izabili y, bu an equi alen exp ession can be ound o
he quasis a ic app oxima ed model, by conside ing he quasis a ic pola izabili y
αagi en in Eq. (1.29) ins ead o αRC
a.
Las , he abso p ion c oss-sec ion can be ob ained om he ex inc ion c oss-
sec ion as
σabs(ω) = σex (ω)−σsca(ω).(1.43)
Figu e 1.6 shows he e alua ion o he abso p ion ( ed lines), ex inc ion
(black lines), and sca e ing (blue lines) c oss-sec ions spec a o sil e sphe ical
nanopa icles o di e en adius. All c oss-sec ions spec a a e no malized o
πR2
,
he a ea o he geome ical c oss-sec ion o he sphe ical nanopa icle (
σ
(
ω
)
/
(
πR2
)is
also called he e iciency ac o . Figu es 1.6(a), (c), and (e) a e ob ained conside ing
he adia i e co ec ed pola izabili y, and Figs. 1.6 (b), (d), and ( ) a e ob ained
using he quasis a ic pola izabili y, i.e., subs i u ing
αRC
a
(Eq.
(1.36)
) by
αa
(Eq.
(1.29)) in Eqs. (1.40) and (1.42).
Simila ly o he nea - ield enhancemen spec a in Figs. 1.2b and 1.4a, he
ex inc ion c oss-sec ion spec um o he nanopa icles is domina ed by a single peak
co esponding o he exci a ion o a dipola plasmonic esonance o he nanopa icle.
This peak edshi s and b oadens o inc easing adius, simila ly o he nea - ield
enhancemen spec a (Fig. 1.4a).
Fo he smalles adius conside ed,
R
= 5 nm, in Fig. 1.6(a), he o al ex inc ion
c oss-sec ion (black line) is mos ly domina ed by he abso p ion ( ed line), and
he sca e ing c oss-sec ion (blue line) is almos negligible. The weak con ibu ion
om he sca e ing can be unde s ood om he quasis a ic exp essions. In his
app oxima ion, he pola izabili y o he nanopa icle ollows Eq.
(1.29)
, and hus,
scales as
∝R3
. As a consequence,
σsca
in Eq.
(1.40)
scales as
∝R6
and
σex
in Eq.
(1.42)
as
∝R3
. Fo small alues o
R
,
σex
is hus much la ge han
σsca
(
R3≫R6
).
On he o he hand,
σsca
g ows much as e han
σabs
wi h inc easing
R
(e en i he
R6
scaling is no alid once he adia i e co ec ion becomes la ge). In ac , o
he in e media e
R
= 30 nm adius (Fig. 1.6c), he alues o he sca e ing (blue
line) become compa able o he abso p ion ( ed line) c oss-sec ion. Fu he , he
ex inc ion c oss-sec ion spec um o he
R
= 70 nm case in Fig. 1.6e is domina ed
20
1.2. Elec omagne ic esponse o e y small objec s
abs
ex
sca
abs
ex
sca
abs
ex
sca
Quasi-s a ic
R = 30 nm
Radia i e-co ec ed
R = 30 nm
R = 70 nmR = 70 nm
R = 5 nm
(b)
R = 5 nm
(a)
(d)(c)
( )(e)
abs
ex
sca
abs
ex
sca
abs
ex
sca
Figu e 1.6: Abso p ion ( ed lines), ex inc ion (black lines), and sca e ing (blue lines) c oss
sec ion spec a o sil e sphe ical nanopa icles wi h di e en adius. The c oss-sec ion spec a
a e calcula ed using he model p esen ed in sec ion 1.2. Figu es (a), (c), and (e) a e ob ained
conside ing he adia i e co ec ion o he pola izabili y o he nanopa icle, and igu es (b), (d),
and ( ) igno e his adia i e co ec ion. Figu es (a)-(b), (c)-(d), and (e)-( ) show he c oss-sec ion
spec a o a sil e sphe ical nanopa icle o adius
R
= 5,30, and 70 nm, espec i ely. The
c oss-sec ions a e no malized o
πR2
, he a ea o a ci cle wi h he same adius o he sphe ical
nanopa icle. The dielec ic pe mi i i y o sil e used o hese calcula ions is ob ained om
e e ence [41].
by he sca e ing con ibu ion.
By compa ing hese esul s wi h he ones ob ained wi h he quasis a ic
app oxima ion, we ind ha he e is a e y good ag eemen o he
R
= 5 nm
nanopa icle (compa e Figs. 1.6 a and b). Howe e , o he la ge
R
= 30 nm
and
R
= 70 nm he quasis a ic app oxima ion b eaks, simila o he esul s o
he enhancemen spec a in Figs. 1.2b and 1.4a. Mos no ably, he quasi a ic
esul s o
R
= 30 nm and
R
= 70 nm (in Figs. 1.6 d and , espec i ely) show
unphysical nega i e alues o he abso p ion c oss-sec ion spec a, which iola es
21
Chap e 1. Fundamen als o classical nanopho onics
he conse a ion o ene gy.
1.2.4 Response o a quan um emi e
In he p e ious sec ions we ha e discussed he esponse o a e y small sphe ical
nanopa icle, bu a simila app oach can also be alid o desc ibe he elec omagne ic
esponse o a quan um emi e (QE), such as a molecule, a quan um do , o a
ni ogen- acancy cen e in a diamond. The elec omagne ic esponse o he QE can
also be desc ibed as an oscilla ing elec ic poin -like dipole,
pe
. In his si ua ion,
pe
ep esen s he e ec o exci ing he ansi ion be ween he wo lowes ene ge ic
le el sys ems o he QE ( he g ound s a e, and he i s exci ed s a e), ene ge ically
sepa a ed by
ℏωσi
. I an ex e nal ield
Eex
d i es he QE wi h a equency close o
ωσ
, he i s exci ed s a e becomes popula ed (o exci ed). Then, a e an a e age
elaxa ion ime
τ
, he exci a ion s o ed in he exci ed s a e decays o he g ound
s a e by emi ing a pho on wi h a e age equency
ωσ
. Thus, he esponse o he
QE can be modeled by ea ing i s elec ic dipole momen as a damped ha monic
oscilla o wi h damping a e γ,
∂2
∂ 2pe( ) + γ∂
∂ pe( ) + ω2
σpe( ) = AeEex ( ),(1.44)
whe e Ae= 2ωσ 2
0/ℏ, and 0is he oscilla o s eng h [35].
Equa ion
(1.44)
can be ew i en in he equency domain (conside ing ha
pe( )∝e−iω , see discussion o Eq. (1.4)) as
pe(ω) = αe(ω)Eex (ω)(1.45)
wi h
αe(ω) = Ae
ω2
σ−ω2−iγω ,(1.46)
he pola izabili y o he QE.
The decay a e
γ
o he QE in his equa ion can be decomposed as
γ
=
γ0
+
γin
,
which accoun o he adia i e o spon aneous decay a e o he QE,
γ0
, (due o
he emission o pho ons) and o in insic non- adia i e losses,
γin
(due o he
decay o he QE wi hou emi ing pho ons).
γ0
can be ob ained om a quan um
elec odynamic desc ip ion o he coupling o he QE wi h he acuum ields [35],
esul ing in
γ0=ω3
σ 2
0
3ℏπc3
0
(1.47)
i
To app oxima e he esponse o he QE as he esponse o an elec ic poin -like dipole we a e
also conside ing ou addi ional condi ions sa is ied in common se -ups: (i) The ene gy di e ence
be ween he highe -ene ge ic s a es (beyond he i s exci ed s a e) and he g ound s a e is much
la ge han he ene gy di e ence be ween he g ound s a e and he i s exci ed s a e. (ii) Ligh
d i ing he QE has a simila equency o he wo-le el ansi ion
ωσ
, and hus, i does no d i e
highe -ene ge ic s a es. (iii) Ligh d i ing he QE is no con ined in e ec i e olumes smalle
han he size o he QE. (i ) The in ensi y o he ield d i ing he QE is small enough o a oid
non-linea e ec s.
22
1.3. Full elec omagne ic esponse o sphe ical nanopa icles
In e es ingly, his alue can also be ob ained by conside ing a QE wi hou losses and
applying he adia i e co ec ion discussed in subsec ion 1.2.2. The pola izabili y
o a lossless QE co esponds o conside ing γ= 0 in Eq. (1.46),
α0
e(ω) = Ae
ω2
σ−ω2.(1.48)
Using he adia i e-co ec ion o mula (Eq. (1.36)) we ob ain,
α0−RC
e(ω) =
=α0
e(ω)1
1−iω3
6πε0c3
0
α0
e(ω)=Ae
ω2
σ−ω2−iω3
6πε0c3
0
Ae≈Ae
ω2
σ−ω2−iω ω3
σ 2
0
3ℏπc3
0
,(1.49)
whe e in he las s ep we ha e app oxima ed
ω3≈ωω2
σ
, alid when he esponse o
he QE is negligible a om esonance. F om Eq.
(1.46)
and he las iden i y in
Eq.
(1.49)
we can iden i y
γ0
=
ω3
σ 2
0/
(3
ℏπc3
0
), which is he same alue as in Eq.
(1.47).
The equa ions desc ibing he c oss-sec ion spec a o he small sphe ical
nanopa icle can also be ex ended o he QE, whe e Eqs.
(1.40)
,
(1.42)
, and
(1.43) a e e alua ed wi h he use o αeins ead o αRC
a.
1.3 Full elec omagne ic esponse o sphe ical
nanopa icles
In he p e ious sec ion we ha e in oduced a desc ip ion o he esponse o small
nanopa icles by ea ing hem as elec ic dipoles. Nex , we desc ibe Mie’s o Lo enz-
Mie heo y, a semi-analy ical solu ion o Maxwell’s equa ions o he elec omagne ic
ield sca e ed by an a bi a ily la ge sphe ical pa icle, which is used in chap e s 3
and 5. As we discuss in his sec ion, his comple e solu ion o Maxwell’s equa ions
e eals ha , al hough nanopa icles can indeed be ea ed as elec ic dipoles,
la ge nanopa icles canno . In pa icula , we show ha la ge nanopa icles ha e a
complex beha io ha can be exp essed as he sum o di e en elec ic and magne ic
mul ipoles, whe e each mul ipole co esponds o a di e en ield dis ibu ion. In
subsec ion 1.3.1, we i s e iew he Mie’s o malism. Then, in subsec ion 1.3.2
we use his o malism o calcula e he op ical esponse in a canonical scena io
as s udied in his hesis: he ields sca e ed by a sphe ical nanopa icle when
illumina ed by a linea ly pola ized plane wa e. Fu he , Mie’s o malism is he
s a ing poin o desc ibe he op ical esponse o nanopa icles unde he exci a ion
by mo e complex beams in he las sec ion o his chap e .
23
Chap e 1. Fundamen als o classical nanopho onics
1.3.1 Mie heo y o mula ion
In his subsec ion, we e iew he o mula ion o Mie heo y as gi en by chap e 4 o
e e ence [40], which is e y simila o he one ollowed by Gus a Mie in his o iginal
pape w i en in 1908 [2]. This o mula ion depa om Helmhol z equa ions (o
elec omagne ic wa e equa ion). The Helmhol z equa ions a e a e o mula ion o
Maxwell’s equa ions in he absence o ex e nal cu en s and cha ges (
Jex
= 0 and
ρex
= 0 in Eq.
(1.1)
). By using he cons i u i e ela ions in Eqs.
(1.2)
and
(1.3)
in
Maxwell’s equa ions
(1.1)
, a e some algeb aic manipula ion, we ob ain [35,36,40]
1
c2
0
∂2E( , )
∂ 2=∇2E( , ),
1
c2
0
∂2B( , )
∂ 2=∇2B( , ).(1.50)
In he linea egime, we can w i e hese equa ions in he equency domain using
E
(
, ω
) =
´d E
(
,
)
eiω /
(2
π
)and
B
(
, ω
) =
´d B
(
,
)
eiω /
(2
π
), which leads
o,
−iω2εµ
c2
0
E( , ω) = ∇2E( , ω),
−iω2εµ
c2
0
B( , ω) = ∇2B( , ω),(1.51)
whe e
ε
and
µ
a e he ela i e dielec ic and magne ic pe mi i i y o he medium,
espec i ely.
Equa ion
(1.51)
is sa is ied by he ollowing amily o equa ions, so-called
mul ipoles:
Mb,n,m,e( , ω) = m
sin(θs)sin(−mφs)Pm
n(cos(θs))B(b)
n(ζ)uθs−
−cos(mφs)dPm
n(cos(θs))
dθs
B(b)
n(ζ)uφs,
Mb,n,m,o( , ω) = m
sin(θs)cos(−mφs)Pm
n(cos(θs))B(b)
n(ζ)uθs−
−sin(mφs)dPm
n(cos(θs))
dθs
B(b)
n(ζ)uφs,
(1.52a)
24
1.3. Full elec omagne ic esponse o sphe ical nanopa icles
Nb,n,m,e( , ω) = B(b)
n(ζ)
ζcos(mφs)n(n+ 1)Pm
n(cos(θs))u s+
+cos(mφs)dPm
n(cos(θs))
dθs
1
ζ
d[ζB(b)
n(ζ)]
dζ uθs+
+msin(−mφs)Pm
n(cos(θs))
sin(θs)
1
ζ
d[ζB(b)
n(ζ)]
dζ uφs,
Nb,n,m,o( , ω) = B(b)
n(ζ)
ζsin(mφs)n(n+ 1)Pm
n(cos(θs))u s+
+sin(mφs)dPm
n(cos(θs))
dθs
1
ζ
d[ζB(b)
n(ζ)]
dζ uθs+
+mcos(−mφs)Pm
n(cos(θs))
sin(θs)
1
ζ
d[ζB(b)
n(ζ)]
dζ uφs,(1.52b)
whe e
Pm
n
a e he associa ed Legend e unc ions o he (
m, n
)-o de ,
ζ
=
√εµk0 s
is he op ical dis ance (wi h
k0
he wa e numbe in acuum and
s
he adial
sphe ical coo dina e),
φs
and
θs
a e he azimu hal and pola sphe ical coo dina es
ii
,
espec i ely.
u s
,
uφs
, and
uθs
a e he uni y ec o s in sphe ical coo dina es
(co esponding o he adial, azimu hal, and pola di ec ions, espec i ely).
The subindex
n
indica es he o de o he mul ipole, and he subindex
m
sa is ies
m∈
[
−n, n
]. The
e
and
o
labels indica e ha he unc ions a e e en o odd wi h
espec o
φs
, espec i ely. Fo con enience, we subs i u e he
e
and
o
labels o
Eqs.
(1.52a)
and
(1.52b)
wi h a gene ic
σ∈ {o, e}
, so, o he es o his hesis,
we e e o
Mb,n,m,e
and
Mb,n,m,o
as
Mb,n,m,σ
, and o
Nb,n,m,e
, and,
Nb,n,m,o
as
Nb,n,m,σ.
Las , he label subindex
b
in Eqs.
(1.52a)
and
(1.52b)
indica es ha he
mul ipoles depend on he sphe ical Bessel unc ion o he
b
-kind (and
n
-o de ),
B(b)
n
(
ζ
), whe e we only need o conside
b
= 1
,
3( o he scena ios s udied in his
hesis). On he one hand, we use he
b
= 1, i s kind unc ions, o desc ibe inciden
beams and he ield inside a nanopa icle because only he i s kind unc ions,
B(1)
n
(
ζ
)
≡jn
(
ζ
), a e ini e a he o igin. On he o he hand, only he
b
= 3 hi d
kind unc ions (equi alen o he i s o de Hankel unc ions,
B(3)
n
(
ζ
)
≡h(I)
n
(
ζ
)),
beha e asymp o ically as an ou going sphe ical wa e,
lim
ζ→∞ B(3)
n(ζ)=(−i)n+1 eiζ
ζ.(1.53)
Thus, hey a e he only app op ia e unc ions o desc ibe he expec ed beha io o
ligh sca e ed by a nanopa icle (in con as wi h he asymp o ical beha io o
b= 1,2,and 4 unc ions, which become non-physical o la ge dis ances [40,45]).
ii
As in s anda d no a ion,
θs
is he angle wi h espec o he
z
-axis o he Ca esian coo dina es.
25
Chap e 1. Fundamen als o classical nanopho onics
(b)(a)
(d)(c)
x
z
y
x
z
y
x
z
y
x
z
y
Figu e 1.7: Di ec ion o he angen ial componen s o
M1,n,1,o
and
N1,n,1,e
(in Eqs.
(1.52a)
and
(1.52b)
) wi h
n
= 1
,
2o e a 3D sphe ical cap wi h
k
= 10,
φs∈
[0
, π
], and
θs∈
[0
, π
], ollowing
he plo ing con en ion in e e ence [2] ( he pe spec i e o he igu e shows he
xz
-plane in he
backg ound, see axis labels o e e ence). All he a ows ha e he same leng h in he 3D space,
and hey a e colo ed only o help he 3D isualiza ion. The do s ep esen he posi ion a which
he unc ions become ze o.
Elec ic and magne ic mul ipoles
Figu e 1.7a-d ep oduces some o he o iginal esul s om Mie [2] by e alua ing he
angen ial componen s o
M1,n,1,o
and
N1,n,1,e
wi h
n
= 1
,
2o e a 3D sphe ical
cap wi h
ζ
= 10,
φs∈
[0
, π
], and
θs∈
[0
, π
]( he pe spec i e o he igu e shows he
xz
-plane). We can indeed obse e how he
N1,1,1,e
and
N1,2,1,e
unc ions in Figs.
1.7a and 1.7c, espec i ely, show a ec o ial pa e n o ields going om one o wo
poin s (“di e gen ” poles) o he same numbe o “con e gen ” poles (indica ed
wi h g ey spo s in he igu e). These ec o ial pa e ns a e analogous o hose
ob ained i we placed posi i e cha ges in he di e gen and nega i e cha ges in he
con e gen poin s, o ming an elec ic dipole in he case o
N1,1,1,e
(Fig. 1.7a) and
an elec ic quad upole o
N1,2,1,e
(Fig. 1.7c). As a consequence, he
N
- unc ions
a e o en called elec ic o elec ic- ype mul ipoles, as al eady poin ed ou by Mie
in 1908 [2]. Simila ly, he
M1,1,1,o
and
M1,2,1,o
unc ions in Figs. 1.7b and 1.7d,
espec i ely, show a ec o ial pa e n o ields ha su ound he poles (g ey spo s
26
1.3. Full elec omagne ic esponse o sphe ical nanopa icles
in he igu e), which esembles he elec ic ield p oduced by a magne ic dipole
and a magne ic quad upole, espec i ely. Hence, he
M
- unc ions a e o en called
magne ic o magne ic- ype mul ipoles.
Al hough he ields gene a ed by a single
N
- unc ion do indeed co espond o
he ields gene a ed by an elec ic mul ipole, i is con enien o poin ou ha he
ep esen a ion o he
N
- unc ions ollowed by Mie and ep oduced in Figs. 1.7a and
1.7c can be misleading [40]. In Figs. 1.7b and 1.7d, i migh seem ha he e a e
cha ges placed a he poles, bu he e a e no ; he indica ed poles (g ey spo s in he
igu e) co espond o a ze o o he ans e se componen o he
N
- unc ions, no o
a cha ge exis ing a he pole. In hese posi ions he ec o ial ields become pu ely
adial, which canno be shown in his ep esen a ion. Despi e his ca ea , he
iden i ica ion o he ec o ial pa e ns o he
N
- unc ions wi h he ields gene a ed
by di e en elec ic mul iples (e.g., an elec ic dipole o an elec ic quad upole) is
qui e use ul o analyze he esponse o a nanopa icle.
Expansion o ields in he elec ic and magne ic mul ipoles
The mul ipola unc ions in Eqs.
(1.52a)
and
(1.52b)
,
Mb,n,m,σ
and
Nb,n,m,σ
(wi h
σ∈ {e, o}
) o m an o hogonal basis hemsel es, i.e.,
´d
Ω
Xb,n,m,σ ·X′
n′,m′,σ′∝
δK
X,X′δK
b,b′δK
n,n′δK
m,m′δK
σ,σ′
wi h
d
Ω he solid angle di e en ial,
δK
he K onecke
del a, and
X∈ {M,N}
(we di ec he eade o he ull o hogonali y ela ions
o he mul ipoles in chap e 7 o e e ence [46]). Thus, any a bi a y elec ic (o
magne ic) ield
Ea b
can be expanded on o he unc ions in Eqs.
(1.52a)
and
(1.52b) as:
Ea b( , ω) =
n
X
m=−n
∞
X
n=0 X
σ=e,o CM
b,n,m,σMb,n,m,σ( , ω) + CN
n,m,σNb,n,m,σ( , ω)
(1.54)
whe e
CM
b,n,m,σ
and
CN
b,n,m,σ
a e he no malized p ojec ions o
Ea b
on o he basis
{Mb,n,m,σ,Nb,n,m,σ}:
CM
b,n,m,σ =´dΩEa b( , ω)·Mb,n,m,σ( , ω)
´dΩ|Mb,n,m,σ( , ω)|2,
CN
n,m,σ =´dΩEa b( , ω)·Nb,n,m,σ( , ω)
´dΩ|Nb,n,m,σ( , ω)|2,(1.55)
Simila ly, he same me hod can be applied o decompose he magne ic ield on he
same basis. Thus, we can use his o malism o desc ibe any a bi a y elec ic and
magne ic ield.
1.3.2 Fields sca e ed by a sphe ical nanopa icle unde
plane wa e illumina ion
27
Chap e 1. Fundamen als o classical nanopho onics
Elec ic
dipole
Magne ic
dipole
Magne ic
quad upole
ex
Figu e 1.10: Ex inc ion c oss sec ion spec um o a
R
= 230 nm silicon sphe ical nanopa icle
ob ained wi h Mie heo y and accoun ing o he con ibu ion o all mul ipoles. The dielec ic
pe mi i i y o silicon used o hese calcula ions is ob ained om e e ence [48].
nanopa icle con ibu e signi ican ly o he c oss-sec ion spec a.
1.4 Angula momen um o ligh
So a , we ha e ocused on desc ibing he in e ac ion be ween a linea ly pola ized
plane wa e and a sphe ical nanopa icle. Howe e , i is also in e es ing o explo e
he in e ac ion be ween a sphe ical nanopa icle wi h mo e sophis ica ed ligh
beams, such as ligh beams wi h well-de ined angula momen um p ope ies, as
hese ype o beams will be used in chap e 5. The angula momen um p ope ies
in oduce new deg ees o eedom ha can be exploi ed in di e en applica ions,
such as sensing [19,28,54] o in o ma ion p ocessing [15–18].
The e a e ou p ope ies associa ed wi h he angula momen um o ligh ,
namely he spin angula momen um, he o bi al angula momen um, he helici y,
and he o al angula momen um. In he simple case o a weakly- ocused beam,
so-called a pa axial beam, he spin alue is di ec ly gi en by i s ci cula pola iza ion;
o example, a plane wa e wi h ci cula ly le pola iza ion has spin
s
= +1. On
he o he hand, he o bi al angula momen um is connec ed wi h changes in he
spa ial dis ibu ion o he phase on an indi idual wa e on (how many imes he
phase changes om 0 o 2
π
when mo ing along a ci cula ajec o y a ound he
axis o p opaga ion). In subsec ion 1.4.1, we discuss he spin and o bi al angula
momen um in mo e de ail o weakly ocused beams wi h a wa e on p opaga ing
in a single de ined di ec ion (pa axial beams).
On he o he hand, we also conside in his sec ion he angula momen um
p ope ies o non-pa axial beams, o example, wi h a sphe ical wa e on . Fo hese
ype o ields, he spin and o bi al angula momen um o ligh a e ill-de ined [55].
Thus, in ha case, we in oduce wo auxilia y p ope ies: he o al angula
momen um and he helici y. The helici y is an in insic p ope y o he pho ons [56],
de ined as he spin p ojec ed in he di ec ion o p opaga ion, and i de e mines he
o que ha ligh can exe on ma e [57
–
59]. On he o he hand, in simple pa axial
34
1.4. Angula momen um o ligh
beams, he o al angula momen um is he sum o he spin and he o bi al angula
momen um. The o al angula momen um o ligh de e mines he o al angula
momen um ha can be ans e ed o ma e , o example, o d i e ci cula mo ion
o nanopa icles [58
–
60]. In subsec ion 1.4.2, we adap Mie heo y o mula ion
in sec ion 1.3.2 o o mally desc ibe he helici y and o al angula momen um o
non-pa axial beams. Finally, in subsec ion 1.4.3, we de i e he exp essions o ields
sca e ed by a sphe ical nanopa icle when illumina ed by a beam wi h well-de ined
angula momen um ocused by a lens, a ypical expe imen al se up ha we conside
in chap e 5.
1.4.1 Pa axial beams wi h non-ze o o bi al angula
momen um and spin
Lague e-Gauss beams
Le us i s conside he simpli ied case o pa axial beams o in oduce he angula
momen um p ope ies o ligh . A pa axial ligh beam is an elec omagne ic
wa e ha p opaga es owa ds a ixed di ec ion and ha has a ela i ely small
di e gence o expansion while p opaga ing ( o example, he beam emi ed by
a lase ). Fo mally, he de ining cha ac e is ic o a pa axial beam is ha one
componen o he wa e ec o , k= (kx, ky, kz), domina es he es ; o example, a
z
-p opaga ing pa axial beam sa is ies
kz≫
(
kx
+
ky
). I we conside a pa axial
beam p opaga ing in ee space along he z-di ec ion wi h an elec ic ield,
E( , ω) = E0( )ei(kz−ω ) E,(1.71)
whe e
E0
(
)is he spa ial dis ibu ion ha a ies wi h
z
only slowly (compa ed
wi h he wa eleng h), and
E
is he pola iza ion ec o , pe pendicula o
z
[35,36].
We can app oxima e he Helmhol z equa ion (Eq.
(1.51)
) in he pa axial egime as
∇2
⊥E0( )+2ik ∂E0( )
∂z = 0.(1.72)
He e, we ha e app oxima ed
∂2E0
(
)
/∂z2≈ −k2E0
(
) + 2
ik∂E0
(
)
/∂zeikz
(i.e.,
we neglec
∂2E0
(
)
/∂z2
o e he es o he con ibu ions due o he slow spa ial
a ia ion o he ields along
z
), whe e
∇2
⊥
is he Laplacian in he coo dina es
pe pendicula o he
z
–axis. The s anda d solu ion o
E0
(
)in Eq.
(1.72)
ob ained
in cylind ical coo dina es ( adial
c
, pola
φc
, and axial
zc
coo dina es) is [61,62],
E0( )≡LGl
q( c, φc, zc) = s2q!
π(q+|l|)!
w0
w(zc) c√2
w(zc)|l|
exp− 2
c
w(zc)2
L|l|
q2 2
c
w(zc)2exp−ik 2
c
2RC(zc)exp(ilφc) exp(iψG(zc)),(1.73)
35
Chap e 1. Fundamen als o classical nanopho onics
whe e
LGl
q
a e he Lague e-Gauss beams o (
l, q
)–o de ,
ψG
(
zc
) =
a c an
(
zc/zR
)is
he Gouy phase,
RC
(
zc
) = (
z2
c
+
z2
E
)
/zc
is he posi ion-dependen adius o cu a u e
o he beam, and
zR
=
w2
0k/
2is he Rayleigh ange.
w
(
zc
) =
w0p1+(zc/zR)2
in
Eq.
(1.73)
is he posi ion-dependen wais o he beam, which shows a minimum
wais
w0
a he ocal plane
zc
= 0. In Eq.
(1.73)
,
Lq
|l|
a e he gene alized Lague e
polynomials o (|l|, q)–o de .
O bi al angula momen um
The ield in Eqs.
(1.71)
and
(1.73)
is said o ha e well-de ined o bi al angula
momen um. This means ha
LGl
q
is an eigen unc ion o a pa icula p ojec ion o
he o bi al angula momen um ope a o ,
L
=
−i
(
×∇
). In ou case,
LGl
q
is an
eigen unc ion o Lz, he p ojec ion along z, he di ec ion o p opaga ion,
Lz=L·uz=−i∂
∂φc
.(1.74)
The eigen alue o LGl
qwi h espec o Lzis l,
LzLGl
q( ) = lLGl
q( ),(1.75)
which is de e mined by he
exp(ilφc)
e m in Eq.
(1.73)
,i.e., he o a ion o
he phase a ound he
z
axis di ec ly esul s in he alue o he o bi al angula
momen um ca ied by he beam. Fo illus a ion, we show in Fig. 1.11 he spa ial
dis ibu ion o he ampli ude and phase o he
LG0
0
,
LG1
0
, and
LG2
0
Lague e-Gauss
beams in he
zc
= 0 plane. The phase o he
LGl
0
beams cycles
l
- imes om
−π
o
π
as
φc
a ies o e [
−π, π
]. Fo example, he phase o
LG1
0
in he igu e changes
(wi h
φc
) om
−π
o
π
one ime, while he phase o
LG2
0
goes om
−π
o
π
wice.
On he o he hand, he
LG0
0
beam (le column in he igu e) has a cons an phase,
which shows i s l= 0 o bi al angula momen um.
Spin angula momen um
In ee space o in a homogeneous medium, Maxwell’s equa ions impose ha any
elec omagne ic wa e has a pola iza ion ans e se o i s p opaga ion [63]. Hence,
Ein Eq. (1.71) mus ha e a o m (in Ca esian coo dina es) as:
E=
x
y
0z
,(1.76)
i.e., a ze o con ibu ion in he di ec ion o p opaga ion, which in his case is
z
.
The
x
- and
y
-componen s,
x
and
y
, espec i ely can ha e any a bi a y complex
alue. Howe e , speci ic alues o
x
and
y
make he
E
ec o an eigen ec o o
he spin ope a o p ojec ed in he di ec ion o p opaga ion,
Sz
. The spin ope a o
36
1.4. Angula momen um o ligh
max
min
π
—
π
Ampli udePhase
Figu e 1.11: Rep esen a ion o di e en Lague e-Gauss ields in he
zc
= 0 ocal plane. The
h ee columns in he igu e co espond om le o igh o
LG0
0
,
LG1
0
, and
LG2
0
beams. In he
uppe ow, we plo he ampli ude o he ields and in he ow below we plo he phase o he
ields. These beams a e ob ained o a z-p opaga ing beam.
in he Ca esian coo dina es is [64]
←→
S=
0 0 0
0 0 −i
0i0
ux+
0 0 i
0 0 0
−i0 0
uy+
0−i0
i0 0
000
uz.(1.77)
The eigen ec o s o Sz=←→
S·uza e,
+=1
√2
1x
iy
0z
,(1.78)
and
−=1
√2
1x
−iy
0z
,(1.79)
These eigen ec o s (no malized in Eqs.
(1.78)
and
(1.79)
) co espond o ci cula ly
le pola ized ligh and ci cula ly igh pola ized ligh , espec i ely.
+
has a
spin
s
= +1 eigen alue (
Sz +
= +
+
), and
−
has a spin
s
=
−
1eigen alue
(Sz −=− −).
As an example o a pa axial beam wi hou a well-de ined spin, we can conside
a beam in Eq. (1.71) wi h a linea pola iza ion ec o
x=
1x
0y
0z
.(1.80)
The spin associa ed wi h his ec o is ill-de ined because
x
is no an eigen alue
37
Chap e 1. Fundamen als o classical nanopho onics
o
Sz
. In ac ,
x
can be w i en as he sum o wo di e en con ibu ions wi h
opposi e spin, x∝ ++ −.
To al angula momen um
F om he discussion abo e, a pa axial beam wi h ield
ELG( , ω) = LGq
l( )ei(kz−ω ) ±(1.81)
has a well-de ined spin and o bi al angula momen um. Thus i we de ine he o al
angula momen um as ←→
J=←→
S+L,(1.82)
ELG is an eigen unc ion o Jz=←→
J·uzwi h eigen alue
m=l+s. (1.83)
1.4.2 Beyond he pa axial app oxima ion
Helici y and o al angula momen um
We conside a ield ha is non-pa axial, such as, a s ongly ocused beam o he ield
sca e ed by a nanopa icle. In his case, he o bi al and he spin angula momen um
become ill-de ined because he pola iza ion and he di ec ion o p opaga ion become
posi ion-dependen , and he o bi al and spin angula momen um alues a e de ined
as he eigen alues wi h espec o he o bi al and spin angula momen um ope a o s
p ojec ed on o he di ec ion o p opaga ion. To a oid hese issues, we can use
wo al e na i e quan i ies o desc ibe he angula momen um p ope ies o he
ield. On he one hand, i is con enien o wo k di ec ly wi h he o al angula
momen um [55], as he eigen alues om
Jz
a e well de ined o some solu ions o
Maxwell’s equa ions, such as he Mie’s mul ipoles (see below). On he o he hand,
we can add ess he pola iza ion p ope ies o ligh by de ining he helici y as he
spin p ojec ed in he di ec ion o p opaga ion. Fo mally, he helici y ope a o is
←→
Λ=←→
S·uk=∇×
k,(1.84)
whe e in he las iden i y, we ha e used Eq.
(1.77)
and he ac ha
uk
, he
uni y ec o along he di ec ion o p opaga ion can be ound using he ope a o
←→
uk
=
−i∇/k
[36,64]. No ably, he eigen ec o s o hese ope a o s a e he Riemann-
Silbe s ein ec o s as de ined in Eq.
(1.8)
. Thus, any elec omagne ic ield ha can
be w i en as a F+o F−Riemann-Silbe s ein ec o s has a well-de ined helici y.
Mul ipoles wi h well-de ined o al angula momen um
We can ede ine he s anda d Mie heo y mul ipoles desc ibed in sec ion 1.3 o
ob ain mul ipoles wi h a well-de ined o al angula momen um ope a o . Acco ding
38
1.4. Angula momen um o ligh
o e e ence [65], hese mul ipoles a e
A(M)
b,n,m( , ω) =
=i(−1)m
(sign(m))mpn(n+ 1)s2n+ 1
4π
(n−|m|)!
(n+|m|)!(Mb,n,|m|,e +isign(m)Mb,n,|m|,e)
(1.85)
A(E)
b,n,m( , ω) =
=(−1)m
(sign(m))mpn(n+ 1)s2n+ 1
4π
(n−|m|)!
(n+|m|)!(Nb,n,|m|,e +isign(m)Nb,n,|m|,o)
(1.86)
whe e
sign
(
m
) = 1 o
m≥
0and
sign
(
m
) =
−
1 o
m <
0.
A(E)
b,n,m
and
A(M)
b,n,m
a e a combina ion o elec ic (
Nb,n,m,σ
) and magne ic (
Mb,n,|m|,σ
) mul ipoles,
espec i ely and hey, hence, main ain hei espec i e elec ic o magne ic cha ac e .
Because o he o hogonal ela ionships be ween
Mb,n,|m|,σ
and
Nb,n,|m|,σ
,
A(M)
b,n,m
a e A(E)
b,n,m also o hogonal, i.e.,
˚d dΩA(X)
b,n,m( , ω)A(X′)
b′,n′,m′( , ω)=0,(1.87)
o
b
=
b′
,
n
=
n′
,
m
=
m′
, o
X
=
X′
, wi h
X
=
M
o
E
. The new
A(M)
b,n,m
and
A(E)
b,n,m
elec ic mul ipoles a e eigen alues o
Jz
, he
z
-componen o he o al
angula momen um ope a o ,
JzA(M)
b,n,m( , ω) = mA(M)
b,n,m( , ω),(1.88)
and
JzA(E)
b,n,m( , ω) = mA(E)
b,n,m( , ω).(1.89)
Fu he analysis o he angula momen um p ope ies o
A(M)
b,n,m
and
A(E)
b,n,m
can
be ound in e e ences [65,66].
Mul ipoles wi h well-de ined helici y
The
A(M)
b,n,m
and
A(E)
b,n,m
mul ipoles ha e well-de ined o al angula momen um,
bu no well-de ined helici y. Nex , we de ine a se o combina ions o
A(M)
b,n,m
and
A(E)
b,n,m
ha esul s in a new basis o mul ipoles wi h well-de ined helici y.
To ind hese combina ions, we use he exp essions o he Riemann-Silbe s ein
ec o s, which a e he eigen unc ions o he helici y ope a o (Eq.
(1.8)
), and a e
combina ions o elec ic and magne ic ields (Eq.
(1.7)
). Thus, we i s w i e he
39
Chap e 1. Fundamen als o classical nanopho onics
elec ic and magne ic ields using he A(M)
b,n,m and A(E)
b,n,m mul ipolesi ,
E( , ω) = X
n,m
[iC(E)
n,mA(E)
b,n,m( , ω) + C(M)
n,m A(M)
b,n,m( , ω)].(1.90)
By using his exp ession o he elec ic ield and applying Eqs.
(1.63)
,
(1.85)
and
(1.86)
on Maxwell’s equa ions (Eq.
(1.5)
), we can ob ain he exp ession o he
magne ic ield,
B( , ω) = 1
c0X
n,m
[C(M)
n,m A(E)
b,n,m( , ω)−iC(E)
n,mA(E)
b,n,m( , ω)].(1.91)
Using hese exp essions o
E
and
B
we can w i e he Riemann-Silbe s ein ec o s
in Eq. (1.9) as,
F±( , ω) = X
n,m
(C(E)
l,m +C(M)
l,m )[A(M)
b,n,m( , ω)±iA(E)
b,n,m( , ω)].(1.92)
F±
a e eigen unc ions o he helici y ope a o wi h eigen alue Λ. I can be
p o ed [65–67] ha each e m in he sum o Eq. (1.92),
A(Λ)
b,n,m( , ω) = A(M)
b,n,m( , ω)+ΛiA(E)
b,n,m( , ω),(1.93)
a e also eigen unc ions o he helici y ope a o ,
←→
Λ
, wi h he same eigen alue
Λ =
±
1,i.e.
A(+)
b,n,m
sa is ies
←→
Λ A(+1)
b,n,m
(
, ω
) =
A(+1)
b,n,m
(
, ω
), and
A(−1)
b,n,m
sa is ies
←→
Λ A(−1)
b,n,m
(
, ω
) =
−A(−1)
b,n,m
(
, ω
). Fu he , he
A(Λ)
b,n,m
mul ipoles a e also he
z
-
componen s o o al angula momen um,
Jz
=
←→
J·uz
. Thus,
A(Λ)
b,n,m
cons i u es he
new mul ipoles we a e looking o , which allows o decomposing any elec omagne ic
ield on a basis well-sui ed o di ec ly add ess he helici y and o al angula
momen um p ope ies.
1.4.3 Sca e ing o a beam wi h well-de ined angula
momen um by a sphe ical nanopa icle
In chap e 5we s udy he sca e ing o a ocused ligh beam wi h well-de ined
angula momen um p ope ies by a sphe ical nanopa icle. In his sec ion, we show
ha he Mie’s o malism desc ibed in sec ion 1.4.2 allows o ackling his sca e ing
p ocess. We show in Fig. 1.12 a scheme o his sca e ing p oblem, an inciden
ci cula ly-pola ized Lague e-Gauss beam ocused by a high-nume ical-ape u e
lens a he cen e o a sphe ical nanopa icle. Be o e ocusing, he Lague e-Gauss
beam is a pa axial beam and hus has a well-de ined alue o he spin and he
o bi al angula momen um (see subsec ion 1.4.1). Fo example, we can conside
i
No e ha o con enience, in equa ions
(1.90)
and
(1.91)
we chose o add a
π/
2phase (i.e.,
ai ac o ) be ween he elec ic and magne ic mul ipoles.
40
1.4. Angula momen um o ligh
Inciden ci cula ly
pola ized LG beam
High NA lens
Focusing
Nanopa icle
(a) Collima ion
(b)
Figu e 1.12: Scheme o he sca e ing p ocess s udied in sec ion 1.4.3. (a) Scheme o he
illumina ion: an inciden ci cula ly pola ized Lague e-Gauss (LG) beam is ocused on he cen e
o a small sphe ical nanopa icle by using a high nume ical ape u e lens. In he scheme we plo
he ampli ude dis ibu ion o a
LG0
l
beam wi h
l
= 1 and
s
=
−
1(co esponding o ci cula
igh -pola iza ion). These alues esul in a o al angula momen um m= 0. (b) Scheme o he
collima ion p ocess: he ligh back-sca e ed by he nanopa icle is collec ed and collima ed by
he same lens used o ocusing. In he ocusing p ocess he ield (in (a)) inciden on he ape u e
o he lens ( e ical a ow in he shaded blue a ea) is mapped on o a sphe ical su ace o adius
(dashed line) and hen o a ed as desc ibed in sec ion 1.4.3. The collima ion (in (b)) co esponds
o he in e se p ocess, so ha he ield is e alua ed in he same sphe ical su ace, o a ed, and
hen mapped on o he ape u e o he lens.
a ci cula ly igh -pola ized beam wi h a spa ial dis ibu ion ollowing a
LG0
1
(see
Eq.
(1.73)
) Lague e-Gauss, esul ing in
s
=
−
1,
l
= +1, and
m
=
l
+
s
= 0. We
conside ha he beam p opaga es in he
z
-di ec ion so ha
←→
Λ
=
Sz
(Eq.
(1.84)
),
and hus s=Λ=−1.
The sca e ed beam in he backwa d di ec ion (i.e., opposi e o he p opaga ion
o he inciden beam) is collima ed wi h he same lens used o ocus he inciden
beam. A e he collima ion we sepa a e he ield in o wo con ibu ions, one wi h
he same helici y as he inciden beam, and ano he con ibu ion wi h opposi e
helici y. The collima ed beams a e pa axial, and hus, we can also add ess he
s
,
l
,
o he sca e ed ields. We nex desc ibe he equa ions ha we use o implemen
all hese s eps o he sca e ing p ocess.
The inciden Lague e-Gauss beam ollows Eq.
(1.81)
. This beam is ocused
by he lens a he plane
zc
= 0. The i s s ep is o w i e he ocused ield as an
expansion o mul ipoles wi h well-de ined helici y
A(Λ)
b,n,m
o di e en o de
n
. The
mul ipoles
A(Λ)
b,n,m
a e de ined in Eq.
(1.93)
. In pa icula , we use mul ipoles wi h
b= 1 (A(Λ)
1,n,m) (see sec ion 1.3). The ocused elec ic ield is:
E oc( , ω) =
∞
X
n=0
√2Cn(ω)A(Λ)
1,n,m( , ω),(1.94)
whe e,
ω
is he angula equency o he ligh , Λis he helici y o he inciden beam
41
Chap e 1. Fundamen als o classical nanopho onics
(co esponding o i s ci cula pola iza ion, see subsec ion 1.4.1), and we only need
o sum o e mul ipoles wi h
m
equal o he angula momen um o he beam. Fo
con enience, he
coo dina es in Eq.
(1.94)
a e chosen o be cen e ed a he ocal
poin o he lens (no o be con used wi h he
c
,
ϕc,
and
θc
coo dina es o he
LG
beam in Eq. (1.73)). The coe icien s Cna e,
Cn=i(n−1)k√2π√2n+ 1ˆθmax
0
dn
m,Λ(θ)sin(θ)LGl
q( sin(θ),0,0)pcos(θ) e−ik dθ,
(1.95)
whe e
dn
m,Λ
is he small Wigne
d
- unc ion [68] and
θmax
is he maximal hal -angle
o he lens o nume ical ape u e
NA
=
nsin
(
θmax
)(
n
is he e ac i e index o he
medium a e he lens). These coe icien s a e de i ed in e e ence [67] using he
aplana ic lens model [35] (see scheme on Fig. 1.12a). In b ie , he modeling o he
ocusing p ocess can be sepa a ed in o h ee s eps. Fi s , he inciden elec ic ield
in he ape u e o he lens is mapped on o a e e ence su ace wi h coo dina es
=
,
φs∈
[0
,
2
π
], and
θ∈
[
π−θmax, π
](i.e. a sphe ical cap si ua ed a he ocal
dis ance,
, om he nanopa icle). Second, he mapped ( ec o ial) elec ic ield is
o a ed such ha om each poin o he e e ence su ace eme ges a plane wa e
ha p opaga es owa d he ocal poin . This o a ion esul s in he
dn
m,Λ
unc ion
in Eq.
(1.95)
. Thi d, we ob ain he ield a he ocal poin as he sum o all hese
plane wa es. The sum o hese plane wa es leads o he in eg a ion in Eq. (1.95).
We nex calcula e he ields sca e ed by he sphe ical nanopa icle unde
he illumina ion o he s ongly ocused Lague e-Gauss beams. Equa ion
(1.61)
desc ibes he esponse o a sphe ical nanopa icle unde plane wa e illumina ion.
The ex ension o his solu ion o ou case is [67],
Esca
LG( , ω) =
∞
X
n=0
Cn(ω)Vn(ω)A(Λ)
3,n,m( , ω) + Wn(ω)A(−Λ)
3,n,m( , ω),(1.96)
whe e
Vn
and
Wn
co espond o combina ions o he
an
and
bn
coe icien s (in Eqs.
(1.62a) and (1.62b), espec i ely),
Vn(ω) = −an(ω) + bn(ω)
2,(1.97) Wn=an(ω)−bn(ω)
2.(1.98)
No e ha due o he angula momen um p ope ies o he
A(Λ)
b,n,m
mul ipoles
(see Eqs.
(1.88)
,
(1.89)
, and
(1.93)
) he sca e ed ield in Eq.
(1.96)
p ese es he
o al angula momen um o he incoming and ocused beam. This p ese a ion is a
consequence o he o a ional symme y o he nanopa icle and will be exploi ed
in chap e 5.
The backsca e ed ield in Eq.
(1.96)
is collima ed h ough he same lens ha
ocuses he inciden beam. To model his collima ion using he aplana ic lens model,
we ollow he in e se p ocess o he ocusing. The backsca e ed ield is e alua ed
a he same sphe ical e e ence su ace as o he ocus (dashed line in Fig. 1.12b).
We hen pe o m he in e se o a ion compa ed o he ocusing p ocess so ha he
Poyn ing ec o s o he sca e ed ield become pe pendicula o he ape u e o he
42
1.4. Angula momen um o ligh
lens a all poin s. Finally, we map he o a ed ield o he e e ence ape u e on o
he su ace o he lens. This collima ion p ocess co esponds ma hema ically o:
ECol
LG ( c, φc, zc= 0, ω) = ˆ
R( , φs, θs)·Esca
LG( , φs, θs, ω)cos(θs)−1,(1.99)
whe e
ECol
LG
is he collima ed ield, (
c
,
φc
,
zc
) a e he cylind ical coo dian es in
he ape u e o he lens (wi h
zc
= 0 and
c
=
sin
(
θs
)), and
ˆ
R
(
, φs, θs
)is he
posi ion-dependen Eule o a ion ma ix:
ˆ
R( , φs, θs) =
sin(φs)−cos(φs) 0
cos(φs) sin(φs) 0
0 0 1
·
1 0 0
0 cos(θs)−sin(θs)
0 sin(θs) cos(θs)
·
sin(φs) cos(φs) 0
−cos(φs) sin(φs) 0
0 0 1
.(1.100)
Las , he
cos
(
θs
)
−1
ac o in Eq.
(1.99)
accoun s o he di e ences be ween he
di e en ial a ea a he e e ence sphe ical su ace,
dAS
, and he di e en ial a ea a
he ape u e o he lens,
dAL
(
dAS
=
dAL/cos
(
θs
), see chap e 3 o e e ence [35]).
43
Chap e 2. Fundamen als o quan um nanopho onics
Figu e 2.1: Ske ch o he classical and quan um ans o ma ion o ligh induced by a s anda d
beam spli e . In he classical desc ip ion, he inpu
Ei
1
and
Ei
2
modes a e ans o med on o he
ou pu
Eo
1
and
Eo
2
modes. In he quan um o malism, he inpu quan um
ˆai
1
and
ˆai
2
ope a o s
a e ans o med on o he ou pu ˆao
1and ˆao
2ope a o s.
spli e . We hen ex end his o malism o desc ibe he ans o ma ion by a lossy
beam spli e , i.e., a beam spli e ha can dissipa e he inciden pho ons and
hus making i lose hei co esponding ene gy. In his hesis we use he quan um
ans o ma ion o beam spli e s o desc ibe he p ocess occu ing in a Hanbu y-
B own Twiss in e e ome e in sec ion 2.4, and o s udy he in e ac ion be ween
quan um s a es o ligh and a nanos uc u e in chap e 5.
Classical ans o ma ion by a beam spli e
We i s conside he s anda d desc ip ion o he classical ans o ma ion o ligh
by a s anda d (lossless) beam spli e . This ans o ma ion can be unde s ood as a
change o basis be ween wo inpu modes a he beam spli e ,
Ei
1
(
ω
)and
Ei
2
(
ω
),
and he wo ou pu modes o he beam spli e
Eo
1
(
ω
)and
Eo
2
(
ω
). The inpu and
ou pu channels wi h he co esponding ields a e shown in he scheme o Fig. 2.1.
The inpu
Ei
1
(
ω
)and
Ei
2
(
ω
)and ou pu
Eo
1
(
ω
)and
Eo
2
(
ω
)modes a e ela ed by a
uni a y ans o ma ion [75],
Eo
1(ω) = 1(ω)Ei
1(ω) + 1(ω)Ei
2(ω),
Eo
2(ω) = 2(ω)Ei
1(ω) + 2(ω)Ei
2(ω),(2.24)
whe e
1
and
2
a e he ansmi ance coe icien s, and
1
and
2
a e he e lec ance
coe icien s o he beam spli e [76]. Fo a lossy beam spli e ,
1
,
2
,
1
, and
2
can ake any a bi a y alue as long he ene gy o he ields a he ou pu o he
beam spli e is lowe han he alue o he ene gy o he ields a he inpu o he
beam spli e , i.e.,|Eo
1(ω)|2+|Eo
2(ω)|2<|Ei
1(ω)|2+|Ei
2(ω)|2.
50
2.3. Quan um ans o ma ions by a lossless and a lossy beam spli e
On he o he hand, o a lossless beam spli e , he ene gy o he inpu ields is
he same as he ou pu ields, i.e.,
|Eo
1
(
ω
)
|2
+
|Eo
2
(
ω
)
|2
=
|Ei
1
(
ω
)
|2
+
|Ei
2
(
ω
)
|2
. The
conse a ion o ene gy imposes he ansmi ance
-coe icien s and he e lec ance
-coe icien s o sa is y [75,76]
| 1(ω)|2+| 1(ω)|2=| 2(ω)|2+| 2(ω)|2= 1,(2.25)
and
1(ω) 2(ω)∗+ 1(ω) 2(ω)∗= 0.(2.26)
These cons ain s on
1
,
2
,
1
, and
2
allow us o w i e Eq.
(2.24)
in e ms o
a single ansmi ance, and a single e lec ance coe icien [75,77,78],
Eo
1(ω) = (ω)Ei
1(ω) + (ω)Ei
2(ω),
Eo
2(ω) = (ω)Ei
1(ω) + (ω)Ei
2(ω),(2.27)
whe e and mus sa is y | (ω)|2+| (ω)|2= 1 and (ω) (ω) = −( (ω) (ω))∗.
Quan um ans o ma ion by an ene gy-conse ing beam spli e
In quan um op ics, he beam spli e ans o ma ion also co esponds o a change
o basis be ween he ou pu and he inpu s a es. Impo an ly, in quan um op ics,
his change o basis is de ined by he annihila ion ope a o s and no di ec ly
by he quan um s a es [70,77,78]. Following he p ocedu e o quan iza ion o
elec omagne ic ields used in sec ion 2.2, we in oduce a se o
ˆai
1
(
ω
)and
ˆai
2
(
ω
)
annihila ion ope a o s ha de ine he o hogonal inpu modes o he beam spli e
and ano he se o
ˆao
1
(
ω
)and
ˆao
2
(
ω
)annihila ion ope a o s ha de ine he o hogonal
ou pu modes (Fig. 2.1). The ela ionship be ween he inpu and ou pu quan um
ope a o s o a lossless beam spli e is analogous o he classical ans o ma ion
in oduced in Eq. (2.27) [70,77,78],
ˆao
1(ω) = (ω)ˆai
1(ω) + (ω)ˆai
2(ω),
ˆao
2(ω) = (ω)ˆai
1(ω) + (ω)ˆai
2(ω),(2.28)
whe e
and
a e he same ansmi ance and e lec ance coe icien s as in he
classical desc ip ion. The close connec ion be ween he classical and quan um
ans o ma ion is due o he ac ha Maxwell’s equa ions de e mine he e olu ion
o elec omagne ic modes bo h in he classical and quan um egimes [64].
Quan um ans o ma ion by a lossy beam spli e
We nex conside he quan um ans o ma ion o a lossy beam spli e . In his case,
he ou pu s a e o he lossy beam spli e does no conse e he ene gy o he
inciden quan um s a e, which means ha he ou pu s a e can ha e he same o
ewe pho ons han he inciden s a e. We model he “dissipa ion” modes (also called
“ancilla” modes [79]) by in oducing he Lange in
ˆ
L1
and
ˆ
L2
ope a o s associa ed
51
Chap e 2. Fundamen als o quan um nanopho onics
wi h losses due, o example, o luc ua ing cu en s in he beam spli e [78]. These
ope a o s a e added di ec ly o he quan um ans o ma ion in Eq.
(2.28)
, leading
o he quan um ans o ma ion o a lossy beam spli e [78],
ˆao
1(ω) = 1(ω)ˆai
1(ω) + 1(ω)ˆai
2(ω) + ˆ
L1(ω),
ˆao
2(ω) = 2(ω)ˆai
1(ω) + 2(ω)ˆai
2(ω) + ˆ
L2(ω).(2.29)
In he lossy beam spli e , he e is no imposi ion o
1
=
2
o
1
=
2
. Howe e ,
1
,
2
,
1
, and
2
s ill co espond o he classical alues o he ansmi ance and
e lec ance coe icien s in Eq.
(2.24)
. On he o he hand, he Lange in ope a o s
in Eq.
(2.29)
mus sa is y h ee equi emen s [78,80,81]: (i) hei expec ed alue
mus anish,
⟨ˆ
L1(ω)⟩=⟨ˆ
L†
1(ω)⟩=⟨ˆ
L2(ω)⟩=⟨ˆ
L†
2(ω)⟩= 0,(2.30)
(ii) he Lange in ope a o s mus commu e wi h he inpu ope a o s (e.g.,
[
ˆ
L1
(
ω
)
,ˆai
1
(
ω
)] = 0), as he inpu ields and noise sou ces inside he beam spli e
mus be independen , and (iii) he Lange in ope a o s do no change he canonical
commu a ion ela ionships be ween he bosonic ope a o s in Eq.
(2.21)
. Addi ional
p ope ies o Lange in ope a o s and hei explici o m o a one-dimensional beam
spli e (a e y hin one-laye beam spli e ) can be ound in e e ences [78,80,81].
2.4 The Hanbu y-B own and Twiss
in e e ome e
The Hanbu y-B own and Twiss (HBT) in e e ome e is a simple de ice ypically
composed o a beam spli e and wo de ec o s. The HBT has played a c ucial ole
in he his o y o quan um op ics because i enables o cha ac e ize he di e en
s a es o ligh and i has shown how pho ons composing a s a e o ligh can in e e e
be ween hem [69,70,82,83]. In his sec ion, we b ie ly e iew he measu emen s
ha can be pe o med wi h an HBT in e e ome e and how hey ela e o he
s a is ics o he numbe o pho ons emi ed by a ligh sou ce. We i s de i e a
ma hema ical o malism ha desc ibes he ans o ma ions p oduced in he HBT
measu emen s (subsec ion 2.4.1). We hen illus a e how o in e p e he esul s
om he HBT in e e ome y by applying his o malism o analyze he emission
om a a ie y o canonical ligh sou ces (subsec ions 2.4.2-2.4.5).
2.4.1
Gene al desc ip ion o he Hanbu y-B own and Twiss
in e e ome e
Figu e 2.2 shows he schema ics o a s anda d HBT in e e ome e : ligh inciden
along one inpu pa h o a beam spli e is di ided in o wo pa hs. A he end o
bo h pa hs, a de ec o ,
D1
and
D2
, is loca ed o he ho izon al and e ical pa hs,
espec i ely. The HBT measu es in ensi y co ela ions o ligh : he de ec ion o
52
2.4. The Hanbu y-B own and Twiss in e e ome e
Figu e 2.2: Ske ch o a s anda d HBT in e e ome e . The inpu ligh , which can be a classical o
quan um s a e o ligh , is spli ed in o wo ou pu pa hs by a beam spli e . In his sec ion, we
choose a “50-50” beam spli e , whe e ligh is e lec ed o ansmi ed wi h equal p obabili y. We
indica e his ans o ma ion wi h he inpu
ˆai
1
and ou pu
ˆao
1
and
ˆao
2
ope a o s in oduced in Fig.
2.1. Ligh lea ing he beam spli e is con e ed in o an elec ic cu en a wo pho ode ec o s
D1
and
D2
, each o hem a he end o he ou pu pa hs. Bo h de ec o s a e placed a he
same dis ance om he beam spli e . The elec ic signal o bo h de ec o s is manipula ed in an
addi ional de ice o ob ain he co ela ion measu emen , g(2)(0).
ligh simul aneously a bo h de ec o s no malized o he indi idual de ec ion a
each de ec o . The esul ing measu emen o he HBT is he in ensi y co ela ion,
g(2)
(
τ
), whe e
τ
indica es he ime delay be ween ligh a eling om he beam-
spli e o each de ec o . This hesis conside s only he ze o-delayed case wi h
τ
= 0, co esponding o a si ua ion whe e bo h de ec o s a e se a he same
dis ance om he beam spli e .
In a ypical expe imen based on HBT in e e ome y,
g(2)
(0) is ob ained by
aking he a e age o he in ensi y measu ed a each de ec o (
⟨ˆ
I1⟩
and
⟨ˆ
I2⟩
a
D1
and
D2
, espec i ely) and he a e age o he mul iplica ion o he in ensi y a bo h
de ec o s (⟨ˆ
I1ˆ
I2⟩). The esul ing in ensi ies co ela ion co esponds o
g(2)(0)(ω1, ω2) = ⟨ˆ
I1(ω1)ˆ
I2(ω2)⟩
⟨ˆ
I1(ω1)⟩⟨ˆ
I2(ω2)⟩,(2.31)
whe e
ω1
and
ω2
a e he equencies o de ec ion o each de ec o ( hey co espond,
o example, o he cen al equency o a il e placed a he ape u e o each
de ec o ) [83]. In his hesis we conside ha he de ec o s a e colo -blind, i.e.,
all pho ons a e de ec ed independen ly o hei equency. Thus we w i e he
“colo -blind” in ensi y co ela ions as
g(2)(0) = ⟨ˆ
I1ˆ
I2⟩
⟨ˆ
I1⟩⟨ˆ
I2⟩,(2.32)
whe e we a e acing ou o e he equency deg ee o eedom.
The in ensi y ope a o s a e p opo ional o he co esponding numbe o pho ons
ope a o :
ˆ
I1∝ˆao
1†ˆao
1
and
ˆ
I2∝ˆao
2†ˆao
2
. Nex , we conside ha he beam spli e
is a lossless, equency independen , “50-50” beam spli e , which co esponds o
se ing
(
ω
)
→
= 1
/√2
and
(
ω
)
→
=
i/√2
[77,78]. Assuming monoch oma ic
53
Chap e 2. Fundamen als o quan um nanopho onics
illumina ion, using he beam spli e ans o ma ion in oduced in Eq.
(2.28)
, and
conside ing ha he e is no inpu s a e on he e ical pa h, we can w i e Eq.
(2.32) in e ms o he inpu ope a o s as
g(2)(0) = ⟨ˆai
1†ˆai
1†ˆai
1ˆai
1⟩
⟨ˆai
1†ˆai
1⟩2=⟨ˆa†ˆa†ˆaˆa⟩
⟨ˆa†ˆa⟩2,(2.33)
whe e in he las equali y, we ha e simpli ied ou no a ion,
ˆai
1≡ˆa
. I can be shown
(see, o example, chap e 5 in e e ence [70]) ha
g(2)
(0) can be di ec ly connec ed
wi h he s a is ics o emission om a ligh sou ce
g(2)(0) = 1 + ⟨(∆ˆn)2⟩−⟨ˆn⟩
⟨ˆn⟩2,(2.34)
whe e
⟨ˆn⟩
is he mean numbe o pho ons emi ed by he sou ce in a ime in e al.
The ime in e al ha de ines he numbe o pho ons in each “packe ” o pho ons
emi ed by he sou ce is called cohe ence ime,
τC
, and i is in e sely ela ed o he
na u al line wid h o he spec al lines o he sou ce. In ou heo e ical desc ip ion,
we a e conside ing ha he de ec o s in he HBT de ec he numbe o pho ons
a i ing in ime windows o
τC ii
[70].
⟨(∆ˆn)2⟩
=
|⟨ˆn2⟩−⟨ˆn⟩2|
in Eq.
(2.34)
a e he
luc ua ions in he a e age numbe o emi ed pho ons. In he ollowing subsec ions
(subsec ions 2.4.2 o 2.4.5), we discuss he ela ionship be ween
g(2)
(0) and he
s a is ical na u e o he emission om ou canonical ligh sou ces: a cohe en
sou ce, a he mal sou ce, a quan um single-pho on sou ce, and a quan um ealis ic
wo-pho on sou ce.
2.4.2 S a is ics o ligh emi ed by a cohe en sou ce
Cohe en sou ces o ligh , such as a lase , emi pho ons wi h he same equency
and same wa e on s. The s a is ics o he numbe o pho ons emi ed om a
cohe en ligh sou ce ollows a cha ac e is ic Poisson’s dis ibu ion wi h luc ua ions
⟨(∆ˆn)2⟩
=
⟨ˆn⟩
[70]. Thus, using Eq.
(2.34)
, we ind ha he HBT in e e ome e
esul s in
g(2)(0) = 1.(2.35)
To illus a e his esul , we analyze in Fig. 2.3 he s a is ics and in ensi y co ela ions
o a cohe en ligh sou ce wi h an a e age emission o
⟨ˆn⟩
= 5 pho ons. Figu e 2.3a
shows he dis ibu ion o he numbe o pho ons emi ed by his sou ce (a Poisson
dis ibu ion wi h a e age
⟨ˆn⟩
= 5 and luc ua ions
⟨(∆ˆn)2⟩
= 5) a a ime in e al.
Figu e 2.3b shows a ske ch illus a ing he esponse o he HBT in e e ome e o
his inpu sou ce. In his ske ch, he packe s o pho ons a i e a he beam spli e
a ela i ely egula in e als and a e spli owa ds he wo de ec o s. No e ha he
beam spli e di ides he inciden s a es in o a supe posi ion o s a es wi h di e en
ii
Fo la ge ime windows, coun ing he pho ons in each packe can esul in a s a is ical
Poisson dis ibu ion, and a sho e ime windows, he in o ma ion in he pho on in ensi y
co ela ions can be los [77,83].
54
2.4. The Hanbu y-B own and Twiss in e e ome e
Cohe en sou ce
0 1 2 3 4 5 6 7 n
P(n)
8
4
2
1 4
6 9
25
2
3
3
3
4
3
18
6
5
3
4 6
2 4
3 6
2
2
2
3
6
8
4
9
2 4
68
2
2
4
12
3.1 2.4 7.4
Numbe o
pho ons
A e age
Measu emen
numbe
1
2
3
4
5
6
7
8
9
10
(a) (b)
(c)
Figu e 2.3: Example o he measu emen o he in ensi y co ela ions
g(2)
(0) o a cohe en sou ce.
(a) Poissonian dis ibu ion wi h mean occupa ion numbe
⟨n⟩
= 5 ep esen ing he s a is ics o he
numbe o pho ons emi ed by a cohe en sou ce. The g aph shows he p obabili y dis ibu ion
e alua ed up o
n
= 8 pho ons. (b) Ske ch o he esponse o he HBT in e e ome e unde
cohe en illumina ion, whe e each o ange ci cle ep esen s an indi idual pho on. In he igu e, we
ep esen one o he possible ou comes om he beam spli e . (c) Simula ed measu emen by a
HBT in e e ome e . In he i s en ows we show ( om le o igh columns): he measu emen
numbe , he in ensi y measu ed by he de ec o in he ho izon al pa h,
¯
I1
, he in ensi y measu ed
by he de ec o in he e ical pa h,
¯
I2
, he mul iplica ion o he in ensi y measu ed by bo h
de ec o s,
¯
I1¯
I2
, and he numbe o pho ons emi ed by he cohe en sou ce. These alues a e
ob ained wi h andom numbe gene a o s based on he numbe o pho ons s a is ically emi ed
by he cohe en sou ce and on he esponse o he beam spli e (see discussion in he ex ). In
he las ow o he able, we show he a e age alue o
¯
I1
,
¯
I2
, and
¯
I1¯
I2
. Below he able we show
he alue o g(2)(0) ob ained om he a e age alues.
55
Chap e 2. Fundamen als o quan um nanopho onics
numbe s o pho ons a each ou pu b anch o he beam spli e . Fo example, in
he igu e, an inciden s a e wi h
N
= 5 pho ons is spli in o a supe posi ion o
all possible
|n1, n2⟩
s a es wi h
n1
+
n2
=
N
= 5, whe e
n1
and
n2
indica e he
numbe o pho ons a each ou pu pa h o he beam spli e . When measu ed, his
supe posi ion o s a es collapses in o a single s a e; o example, in he igu e, we
chose i o be he |3,2⟩s a e.
In Fig. 2.3c, we show a able ha simula es he beha io o he HBT
in e e ome e . The i s en ows co espond o en simula ed measu emen s
unde cohe en illumina ion (measu emen numbe in he i s column). Fo his
able and he ollowing examples in Figs. 2.4-2.6, we i s use a andom numbe
gene a o o ob ain
N
, he numbe o pho ons emi ed by he sou ce in a gi en
ime in e al (co esponding o each indi idual simula ed measu emen ). This
andom numbe gene a o ollows he s a is ic dis ibu ion o he sou ce, i.e., a
Poisson dis ibu ion in he case o Fig. 2.3. Nex , we ob ain he in ensi y o an
indi idual measu emen by each de ec o ,
¯
I1
and
¯
I2
, shown in he second and
hi d columns, espec i ely ( he line o e a a iable indica es simula ed alues o a
single measu emen ). We assign
¯
I1
o
¯n1
, a andomly gene a ed numbe o pho ons
ha a i e a he
D1
de ec o . To ob ain
¯n1
we ake in o accoun ha , using he
beam spli e ans o ma ion in Eq.
(2.28)
, he ou pu s a e a each de ec o o he
HBT is
|Ψo
HBT⟩
= [(
ˆa†
1
+
iˆa†
2
)
/√2
]
N|0⟩
, wi h
|0⟩
being he acuum s a e (wi h ze o
pho ons). We can hen ob ain he p obabili y o measu ing
n1
pho ons in he
D1
de ec o as
|⟨Ψo
HBT|n1, n2⟩|2
, whe e he
|n1, n2⟩
desc ibe ha ing
n1
and
n2
pho ons
a he ho izon al and e ical ou pu pa hs o he beam spli e , espec i ely. The
alue o
¯n1
esul s om using a andom numbe gene a o ollowing he s a is ical
dis ibu ion gi en by he
|⟨Ψo
HBT|n1, n2⟩|2
p obabili ies. Then we assign
¯
I2
o
¯n2
=
N−¯n1
. Using hese alues o
¯
I1
and
¯
I2
we show he esul ing
¯
I1¯
I2
p oduc
in he ou h column. Finally, in he las ow, we gi e he mean alues
⟨¯
I1⟩
,
⟨¯
I2⟩
, and
⟨¯
I1¯
I2⟩
, which co esponds o he a e age o he indi idual simula ed
measu emen s,
¯
I1
,
¯
I2
, and
¯
I1¯
I2
, espec i ely, in he ows abo e. The esul o he
HBT in e e ome e co esponds o
g(2)(0) = ⟨¯
I1¯
I2⟩
⟨¯
I1⟩⟨¯
I2⟩.(2.36)
Fo he able in Fig. 2.3 we ob ain
g(2)(0) ≈
0
.
99 (indica ed a he bo om o
he igu e), which is e y close o he heo e ical alue
g(2)
(0) = 1 expec ed om
Eq.
(2.35)
(la ge sampling esul s in
g(2)(0) ≈
1, no shown he e). This simple
example illus a es no only he pe o mance o a s anda d HBT bu also how he
in ensi y co ela ions measu ed by an HBT in e e ome e a e di ec ly connec ed
o he s a is ics o he emission o a sou ce.
2.4.3 S a is ics o ligh emi ed by a he mal sou ce
We nex conside an HBT in e e ome e unde he mal illumina ion, such as ligh
emi ed by a gas discha ge lamp. The numbe o pho ons emi ed by a he mal
56
2.4. The Hanbu y-B own and Twiss in e e ome e
10
25
0 2
1 1
1
3
2
0
0
6
0
0
1
0
0 1
0 0
33
1
1
0
0
0
0
0
0
6 13
47
7
3
42
13
1.6 1.8 6.0
Numbe o
pho ons
A e age
Measu emen
numbe
1
2
3
4
5
6
7
8
9
10
The mal sou ce (bunching)
01 2 3 4 5 6 7 n
P(n)
8
(a) (b)
(c)
Figu e 2.4: Example o he measu emen o he in ensi y co ela ions
g(2)
(0) o a he mal sou ce.
(a) Bose-Eins ein dis ibu ion wi h mean occupa ion numbe
⟨n⟩
= 5 ep esen ing he s a is ics o
he numbe o pho ons emi ed by a cohe en sou ce. The g aph shows he p obabili y dis ibu ion
e alua ed up o
n
= 8 pho ons. (b) Ske ch o he esponse o he HBT in e e ome e unde
cohe en illumina ion, whe e each ed ci cle ep esen s an indi idual pho on. In he igu e, we
ep esen one o he possible ou comes om he beam spli e . (c) Simula ed measu emen by a
HBT in e e ome e . In he i s en ows we show ( om le o igh columns): he measu emen
numbe , he in ensi y measu ed by he de ec o in he ho izon al pa h,
¯
I1
, he in ensi y measu ed
by he de ec o in he e ical pa h,
¯
I2
, he mul iplica ion o he in ensi y measu ed by bo h
de ec o s,
¯
I1¯
I2
, and he numbe o pho ons emi ed by he he mal sou ce. These alues a e
ob ained wi h andom numbe gene a o s based on he numbe o pho ons s a is ically emi ed
by he he mal sou ce and on he esponse o he beam spli e (see discussion in he ex ). In
he las ow o he able, we show he a e age alue o
¯
I1
,
¯
I2
, and
¯
I1¯
I2
. Below he able we show
he alue o g(2)(0) ob ained om he a e age alues.
57
Chap e 2. Fundamen als o quan um nanopho onics
ligh sou ce ollows a Bose-Eins ein, posi i e-de ini e, hal -no mal dis ibu ion (a
no mal dis ibu ion,
P
(
n
), bu only de ined o posi i e alues,
n≥
0) [70]. Figu e
2.4a shows he e alua ion o such dis ibu ion wi h a
⟨(∆ˆn)2⟩
= 5 a iance. A
hal -no mal dis ibu ion sa is ies
⟨(∆ˆn)2⟩
=
⟨ˆn⟩2
+
⟨ˆn⟩
[70], and,
g(2)
(0) in Eq.
(2.34) becomes,
g(2)(0) = 2.(2.37)
Figu e 2.4b shows a ske ch o he emission o a he mal sou ce on o an HBT
in e e ome e . The ske ch emphasizes how his sou ce o en emi s pho on packages
con aining a signi ican numbe o pho ons, wi h a ela i ely long pe iod o ime
be ween hese g oups wi h no pho on emission. This beha io con as s wi h he
mo e egula emission o a cohe en sou ce (Fig. 2.3b). S a is ically, i he inciden
pho on package o he beam spli e con ains many pho ons, he ou pu de ec ed
s a e would con ain a simila numbe o pho ons a each de ec o .
Figu e 2.4c shows a simula ion o he HBT in e e ome e measu emen s o
a he mal sou ce. The alues shown in he igu e a e ob ained using he same
me hodology p esen ed when discussing Fig. 2.3c, excep o he use o he Bose-
Eins ein s a is ics o he pho ons emi ed by he he mal sou ce (Fig. 2.4a).
Consis en wi h he p e ious discussion, he numbe o pho ons emi ed by he
he mal sou ce in a ime in e al (las column o Fig. 2.4c) is e y i egula , wi h
some in e als con aining many pho ons and o he s e y ew o none. This ype o
beha io , whe e pho ons a e emi ed in “bunches” (packe s wi h la ge numbe s o
pho ons), co esponds o a bunched emission and esul s in
g(2)
(0)
>
1.
iii
. The
las ow o he able shows he a e age alue o he in ensi ies measu ed a each
de ec o o he HBT a e en simula ed measu emen s in he ows abo e. Using
hese a e age alues in Eq.
(2.36)
we ob ain
g(2)(0) ≈
2
.
08 (indica ed a he bo om
o he igu e), a alue e y close o he heo e ical
g(2)(0)
= 2 esul in Eq.
(2.37)
.
2.4.4 S a is ics o ligh emi ed by a single pho on sou ce
We s udy in his sec ion he in ensi y co ela ions and s a is ics o ligh showing a
e y di e en beha io as compa ed o he bunched he mal emission: he emission
om a single pho on sou ce. In Fig. 2.5a, we show he dis ibu ion o he numbe
o pho ons emi ed by a ealis ic single-pho on sou ce, which can emi one pho on
a each ime in e al, bu can also emi ze o pho ons. As illus a ed by he ske ch
in Fig. 2.5b, he ou pu s a e om he beam spli e can only be de ec ed in a
single ou pu pa h. Thus, he in ensi y measu ed by one o he de ec o s is ze o,
so ha ⟨ˆ
Ihˆ
I ⟩= 0, and acco ding o Eq. (2.32), in his case,
g(2)(0) = 0.(2.38)
The able in Fig. 2.5c shows he esul s o a simula ed measu emen (pe o med
as in p e ious sec ions) o a single pho on sou ce. In he las ow o he able, we
iii
Mo e echnically, “bunching” occu s when
g(2)
(
τ
)
< g(2)
(0). In any case, he s a is ical
beha io o he emission o pho ons in bunches esul s in
g(2)
(0)
>
1, and o he es o his
hesis, we e e o g(2)(0) >1as “bunching”.
58
2.4. The Hanbu y-B own and Twiss in e e ome e
00
01
0 1
0 1
0
1
1
1
0
0
0
0
0
0
0 0
0 0
00
0
0
0
0
0
0
0
0
1 1
00
0
0
0
0
0.3 0.1 0
Numbe o
pho ons
A e age
Measu emen
numbe
1
2
3
4
5
6
7
8
9
10
Single pho on sou ce (an ibunching)
0 1 2 3 4 5 6 7 n
P(n)
8
(a) (b)
(c)
Figu e 2.5: Example o he measu emen o he in ensi y co ela ions
g(2)
(0) o a single pho on
sou ce. (a) S a is ic dis ibu ion o he numbe o pho ons emi ed by a single pho on sou ce.
(b) Ske ch o he esponse o he HBT in e e ome e unde single pho on illumina ion, whe e
each blue ci cle ep esen s an indi idual pho on. In he igu e, we ep esen one o he possible
ou comes om he beam spli e . (c) Simula ed measu emen by a HBT in e e ome e . In he
i s en ows we show ( om le o igh columns): he measu emen numbe , he in ensi y
measu ed by he de ec o in he ho izon al pa h,
¯
I1
, he in ensi y measu ed by he de ec o in he
e ical pa h,
¯
I2
, he mul iplica ion o he in ensi y measu ed by bo h de ec o s,
¯
I1¯
I2
, and he
numbe o pho ons emi ed by he single-pho on sou ce. These alues a e ob ained wi h andom
numbe gene a o s based on he numbe o pho ons s a is ically emi ed by he single pho on
sou ce and on he esponse o he beam spli e (see discussion in he ex ). In he las ow o he
able, we show he a e age alue o
¯
I1
,
¯
I2
, and
¯
I1¯
I2
. Below he able we show he alue o
g(2)
(0)
ob ained om he a e age alues.
59
Chap e 2. Fundamen als o quan um nanopho onics
whe e ˆ
H(0)
C=ℏωcˆc†ˆc(2.53)
is he Hamil onian o he ca i y unpe u bed by he emi e , and
ˆ
Hσ=(ˆp−qˆ
A)2
m+ˆ
V , (2.54)
co esponds, a p io i, o he Hamil onian o he emi e unpe u bed by he ca i y,
whe e he second e m (
ˆ
V
) is he po en ial ene gy expe ienced by he cha ges o
he emi e , and he i s e m ((
ˆp−qˆ
A
)
2/m
) is equi alen o he kine ic ene gy o
he emi e (see discussion o he (
p−qA
) e m in Eq.
(2.46)
). This means ha in
his Hamil onian, de i ed in he Coulomb-gauge,
ˆ
HC
, we can no di ec ly iden i y
an independen e m desc ibing he in e ac ion be ween he emi e and he ca i y
( he ca i y-emi e in e ac ion e m is hidden in he ene gy o he “unpe u bed”
emi e ).
So a , we ha e in oduced he c ea ion and annihila ion ope a o s o he ca i y,
and he nex na u al s ep in he quan iza ion scheme equi es in oducing he
analogous “c ea ion” and “annihila ion” ope a o s o he emi e . The s anda d
p ocedu e o in oduce hese ope a o s equi es unca ing he Hilbe space o he
emi e , and, in o de o do so,
ˆ
Hσ
(Eq.
(2.54)
) should desc ibe he ene gy o he
emi e unpe u bed by he p esence o he ca i y. Howe e , he
ˆ
Hσ
Hamil onian
includes he ac ion o he ca i y wi h e ms
∝ˆ
A
. The la e
∝ˆ
A
e ms appea s
om he desc ip ion o he classical momen um
p
in Eq.
(2.43)
, and i does no
allow us o de i e u he he quan iza ion o
ˆ
Hσ
[84]. Mo eo e , i we di ec ly
unca e he
ˆ
V
e m in
ˆ
Hσ
(Eq.
(2.54)
),
ˆ
V
will also become dependen on he ield
o he ca i y. Be o e en e ing in o de ails on how o a oid his issue, le us conside
wha would happen i we di ec ly unca e he Hilbe space o he emi e .
The unca ion o he Hilbe space o he emi e is a e y common
app oxima ion in which he esponse o an emi e , such as a molecule o a
quan um do , is desc ibed by only conside ing he ansi ion be ween wo elec onic
le els wi h he lowes ene gy o he emi e . A e educing he esponse o he
emi e o only i s wo lowes elec onic le els, he emi e is e e ed o as a
“ wo-le el-sys em”. The wo lowes ene ge ic s a es o he unpe u bed emi e a e
e med as he g ound
|g⟩
s a e and he exci ed
|e⟩
s a e, and he ene gy s uc u e
o he emi e can be app oxima ed by hese wo s a es i wo main condi ions
a e me : (i) ha he wo s a es a e well sepa a ed om he highe ene gy le els,
and (ii) ha he ene gy di e ence ∆
ECT S
=
ℏ|ωc−ωσ|
gi en by he ansi ion
de ined by hese wo le els,
ℏωσ
, and he ene gy o he elec omagne ic mode o he
ca i y,
ℏωc
, is much smalle han he ene gy di e ence be ween
ℏωc
and he ene gy
o he o he elec ic ansi ions o he emi e . In his hesis, we always conside
ha hese wo condi ions a e me . Then, he Hamil onian o he “unpe u bed”
emi e in Eq.
(2.52)
is e ec i ely app oxima ed as
ˆ
Hσ
=
ℏωσ
(
|e⟩⟨e|−|g⟩⟨g|
)
/
2,
implying ha he ene gy o he unpe u bed TLS becomes
ℏωσ/
2i exci ed o
−ℏωσ/
2o he wise. I is s aigh o wa d o o esee ha eplacing his exp ession o
ˆ
Hσ
in Eq.
(2.52)
comple ely neglec s he ca i y-TLS in e ac ion, and hus, i is no
66
2.5. The Hamil onian o ca i y-QED sys ems
co ec . On he o he hand, e e ences [84,97] poin ou ano he common e ec i e
app oach o desc ibe he “unpe u bed” Hamil onian o he emi e , which consis s
in subs i u ing ˆp2
m+ˆ
V→ℏωσ
2(|e⟩⟨e|−|g⟩⟨g|)(2.55)
in
ˆ
Hσ
. Howe e , his app oxima ion in oduces wo c ucial sou ces o e o . Fi s ,
he
ˆp
ope a o in Eq.
(2.55)
depends on he ac ion o he ca i y ia a
qˆ
A
e m
(Eq.
(2.43)
). Second, in he Coulomb gauge, he Hilbe space unca ion o he
emi e (on o he
|g⟩
and
|e⟩
s a es) causes he po en ial
ˆ
V
o gain dependence
also on he ield
ˆ
A
(see he beginning o he nex subsec ion). Thus, no he
kine ic
ˆp2
e m, no he po en ial
ˆ
V
e m in Eq.
(2.55)
a e sui ed o desc ibe he
ene gy o he unpe u bed emi e . In he ollowing subsec ions, we b ie ly e iew
a solu ion o success ully unca e he Hilbe space o he emi e and ob ain a
co ec exp ession o he QRM Hamil onian in he Coulomb gauge.
2.5.2
T unca ion o he Hilbe space o he emi e , loss o
locali y, and quan um Rabi Hamil onian in he dipole
gauge
Loss o locali y o he emi e po en ial
Nex we b ie ly e iew he gene al discussion in e e ence [84] o explain how he
po en ial o he unpe u bed emi e becomes dependen on
ˆ
A
due o he unca ion
o he Hilbe space o he emi e in he Coulomb gauge. A e Eq.
(2.48)
we
discussed ha he po en ial ope a o
ˆ
V
could be w i en in he dis ance basis in
e ms o he unc ion
V
(
dσ, d′
σ
). We expec ha
V
(
dσ, d′
σ
) =
V
(
dσ
)
δ
(
dσ−d′
σ
)wi h
V
(
dσ
) he alue o he classical po en ial ha we in oduced in he beginning o
he p e ious subsec ion (Eq.
(2.41)
). This is because we ini ially conside ha
V
(
dσ, d′
σ
)is a local po en ial in he quan um con ex
xi
. Howe e , as we show nex ,
when unca ing he esponse o he QE o only i s wo lowes ene gy le els, he
locali y o his po en ial can be los .
We i s conside a comple e basis o he eigens a es o he emi e ,
|nσ⟩ ∈ {|g⟩,|e⟩,|e2⟩,|e3⟩, ...},(2.56)
whe e
|g⟩
is he g ound s a e,
|e⟩
is he i s exci ed s a e, and
|e2⟩,|e3⟩, ...
a e he
highe exci ed s a es. Then we w i e V(dσ, d′
σ)in Eq. (2.48) in his |nσ⟩basis,
V(dσ, d′
σ) = V(dσ)δ(dσ−d′
σ) =
∞
X
nσ=n′
σ
V(dσ)⟨dσ|nσ⟩⟨d′
σ|n′
σ⟩,(2.57)
whe e in a ealis ic emi e , he
⟨dσ|nσ⟩
and
⟨d′
σ|n′
σ⟩
e ms co espond o smoo h
xi
No e ha he e we discuss locali y and non-locali y in a quan um con ex , whe e quan um
non-locali y desc ibes he ins an aneous p opaga ion o co ela ions be ween en angled sys ems,
o in Albe Eins ein’s wo ds “spooky ac ion-a -a-dis ance” [98–100].
67
Chap e 2. Fundamen als o quan um nanopho onics
wa e unc ions ep esen ing he spa ial dis ibu ion o he eigens a es o he emi e
(no e ha he
dσ
dis ance is a con inuous a iable). Nex , we unca e he sum on
he |nσ⟩s a es o he i s wo eigens a es, |nσ⟩ ∈ {|g⟩,|e⟩}, and hus,
V(dσ, d′
σ)≈V(dσ)(⟨dσ|g⟩⟨d′
σ|e⟩+⟨dσ|e⟩⟨d′
σ|g⟩).(2.58)
The wo e ms added inside he pa en hesis in Eq.
(2.58)
canno esul in
δ
(
dσ−d′
σ
),
because he Di ac del a is he sum o all he elemen s o he basis; in o he wo ds, he
Di ac del a canno simply be ac o ized as he p oduc o wo smoo h wa e unc ions.
Thus, Eq.
(2.58)
esul s in a loss o locali y o he po en ial. Fu he mo e, any
non-local po en ial
V
(
dσ, d′
σ
)can be exp essed as a momen um-dependen po en ial,
V
(
dσ, d′
σ
)
→V
(
dσ, p
)[84,101
–
103], and his dependence on he momen um is a
signi ican issue in he Coulomb gauge because, acco ding o Eq.
(2.43)
,
p
depends
on he po en ial ec o
A
o he elec omagne ic ca i y mode. As we discussed
abo e, i he po en ial ope a o
ˆ
V
has a dependence on he ield o he ca i y, i is
no sui ed o desc ibe he po en ial ene gy o he unpe u bed emi e .
The me hod o a oid in oducing he dependence o
ˆ
V
on
ˆ
A
consis s in changing
om he Coulomb o he dipole gauge. In he dipole gauge,
p
only desc ibes he
p ope ies o he unpe u bed emi e , allowing one o unca e he Hamil onian
o he unpe u bed emi e . A e his change, i becomes possible o apply a
ans o ma ion on o he Hamil onian o he emi e in he dipole gauge and e u n
o he Coulomb gauge.
F om he Coulomb gauge Lag angian o he dipole gauge Hamil onian
The Lag angian in oduced in Eq.
(2.42)
is w i en in he Coulomb gauge. To
change om he Coulomb o he dipole gauge wi hou a ec ing he equa ions
o mo ion o he sys em, we mus pe o m a ans o ma ion o he ype
LD
=
L
+
dGL/d
, whe e
GL
is a unc ion o he posi ion a iables (
dσ
), he ield
a iables (
A
and Π), and ime (
). In pa icula , he
GL
unc ion ha desc ibes
he ans o ma ion om he Coulomb gauge o he dipole gauge is
GL
=
−qdσA
,
esul ing in
LD=L−q(˙
dσA+dσ˙
A) = 1
4m˙
d2
σ+V(dσ) + 1
2ε0VE ˙
A2−1
2ε0VE ω2
cA2−qdσ˙
A.
(2.59)
This change o gauge a ec s he choice o he canonical momen a om
p
and Πin
Eqs. (2.43) and (2.44) o
pD=∂L
∂˙
dσ
=1
2m˙
dσ,(2.60)
ΠD=∂L
∂˙
A=ε0VE ˙
A−qdσ.(2.61)
Impo an ly, he new canonical momen um
pD
in he dipole gauge does no depend
on any p ope y o he ex e nal ield. Using Eqs.
(2.59)
-
(2.61)
we can ob ain he
68
2.5. The Hamil onian o ca i y-QED sys ems
classical Hamil onian in he dipole gauge:
HD-Class. =p2
D
m+V(dσ) + (ΠD+qdσ)2
2ε0VE
+1
2ε0VE ω2
cA. (2.62)
Nex we use he co espondence p inciple and con e
pD
,Π
D
,
dσ
, and
A
in Eq.
(2.62) in o quan um ope a o s,
ˆ
H(N)
D=ˆp2
D
m+ˆ
V+(ˆ
ΠD+qˆ
dσ)2
2ε0VE
+1
2ε0VE ω2
cˆ
A. (2.63)
In a simila manne o he Coulomb gauge (Eqs.
(2.49)
and
(2.50)
) we can in oduce
he second-quan iza ion c ea ion
ˆc†
and annihila ion
ˆc
ope a o s associa ed o he
elec omagne ic mode o he ca i y,
ˆ
A= ℏ
2ε0VE ωc
(ˆc+ ˆc†),(2.64)
and
ˆ
ΠD=−i ℏε0VE ωc
2(ˆc−ˆc†).(2.65)
By subs i u ing hese exp essions on ˆ
HDwe ob ain,
ˆ
H(N)
D=ˆ
H(D)
σ+ℏωcˆc†ˆc+(qˆ
dσ)2
2εVE −iqˆ
dσ√ℏωc
√2ε0VE
(ˆc−ˆc†) + ℏωc
2,(2.66)
whe e we ha e de ined
H(D)
σ
=
ˆp2
D/m
+
ˆ
V
, as he Hamil onian o he unpe u bed
emi e .
Nex , we unca e he Hilbe space o he emi e . We in oduce his unca ion
on he Hamil onian o he sys em
ˆ
HD
in ou s eps [84,97]: (i) We i s conside
he comple e basis o he eigens a es o he emi e in Eq.
(2.56)
, and unca e i
o he i s wo s a es, he g ound
|g⟩
and he exci ed
|e⟩
s a e. (ii) We in oduce
he annihila ion (o lowe ing) ope a o
ˆσ=|g⟩⟨e|,(2.67)
and he c ea ion (o ising) ope a o
ˆσ†=|e⟩⟨g|.(2.68)
ˆσ
desc ibes he decay om he exci ed
|e⟩
s a e o he g ound
|g⟩
s a e o he emi e ,
and
ˆσ†
desc ibes he exci a ion om
|g⟩
o
|e⟩
. (iii) We w i e he Hamil onian o
he unpe u bed emi e ,
H(D)
σ
, as he ene gy di e ence be ween he exci ed and
g ound s a e,
ˆ
H(D)
σ=ℏωσ
2ˆσz,(2.69)
69
Chap e 2. Fundamen als o quan um nanopho onics
whe e ˆσz= [ˆσ†,ˆσ]. (i ) We w i e ˆ
dσand ˆpDin e ms o ˆσand ˆσ†,
ˆ
dσ= ℏ
2mωσ
(ˆσ+ ˆσ†),(2.70)
ˆpD=−i ℏmωσ
2(ˆσ−ˆσ†).(2.71)
Then, he Hamil onian in Eq. (2.66) can be w i en as,
ˆ
H(N)
D=ℏωσ
2ˆσz+ℏωcˆc†ˆc−iℏg ωc
ωσ
(ˆc−ˆc†)(ˆσ+ ˆσ†) + ℏq2
4mωσε0VE
+ℏωc
2,(2.72)
wi h [84]
g=q
2√mε0VE
.(2.73)
A e e-no malizing (neglec ing he cons an e ms in he Hamil onian), we ind
he well-known o mula o he Rabi Hamil onian in he dipole gauge,
ˆ
HD=ˆ
H(N)
D−ℏq2
4mωσε0VE
+ℏωc
2=ℏωσ
2ˆσz+ℏωcˆc†ˆc−iℏg ωc
ωσ
(ˆc−ˆc†)(ˆσ+ˆσ†).
(2.74)
2.5.3 F om he dipole gauge o he Coulomb gauge
So a we ha e shown he de i a ion o he QRM Hamil onian in he dipole gauge.
Ou nex and inal goal is o ob ain his Hamil onian in he Coulomb gauge.
In Eq.
(2.59)
we in oduced he change o gauge as a ans o ma ion on he
Lag angian ope a o . Howe e , we can also di ec ly change be ween he dipole and
coulomb gauges by pe o ming a uni a y ans o ma ion on he Hamil onian o he
sys em [84,85,87,92,104],
ˆ
HC=ˆ
Uˆ
HDˆ
U†+i˙
ˆ
Uˆ
U†,(2.75)
wi h ˆ
U= exp(i(−ˆ
GL)),(2.76)
being a uni a y ma ix, and
ˆ
GL
=
−qˆ
dσˆ
A
co esponds o applying he quan um
co espondence p inciple o he same
GL
unc ion ha we use o ans o m he
Lag angian om he Coulomb gauge o he dipole gauge. Using he exp essions o
ˆ
Aand ˆ
dσin Eqs. (2.64) and (2.70), ˆ
U esul s in
ˆ
U= expig
√ωσωc
(ˆσ+ ˆσ†)(ˆc+ ˆc†).(2.77)
70
2.6. Quan um dynamics in open quan um sys ems: he quan um mas e equa ion
No e ha he ˆ
U ans o ma ion does no depend explici ly on ime, and hus he
i˙
ˆ
Uˆ
U† e m in Eq. (2.75) anish, and we ob ain
ˆ
HC=ℏωcˆc†ˆc+ˆ
Uℏωσ
2ˆσzˆσˆ
U†(2.78)
Fo simplici y, we ocus on he esonan case wi h
ω0≡ωσ
=
ωc
, whe e Eq.
(2.78)
becomes
ˆ
HC=ℏω0ˆc†ˆc+ℏω0
2ˆσzcos[2η(ˆc+ ˆc†)] + ˆσysin[2η(ˆc+ ˆc†)],(2.79)
wi h
ˆσy
=
i
(
ˆσ†−ˆσ
), and
η
=
g/ω0
. This is he exp ession o he QRM Hamil onian
in he Coulomb gauge.
F om his de i a ion, i migh seem easie o wo k in he dipole gauge, whe e
he unca ion o he Hilbe space o he basis o he emi e is s aigh o wa d,
and hence, he de i a ion o he Hamil onian is di ec . Howe e , we no e ha
wo king in he dipole gauge also p esen s some disad an ages. Fo ins ance,
whe eas in he Coulomb gauge he ope a o o he elec ic ield gene a ed by he
ca i y is simply p opo ional o
ˆc
, in he dipole gauge i becomes p opo ional o
ˆc
+
iη
(
ˆσ
+
ˆσ†
)[85,87]. This change in he elec ic ield ope a o a ec s, o example,
he ope a o s desc ibing he emission and he dissipa ion o he sys em, which is
e y incon enien o analyzing he dynamics in ca i y-QED sys ems.
2.5.4 Jaynes-Cummings model Hamil onian
Many ca i y-QED se ups show coupling s eng hs ha a e much smalle han he
na u al equency o he ca i y, i.e.,
g≪ω0
. In his case, he Hamil onian in Eq.
(2.79)
can be simpli ied in he limi o e y small coupling
η
=
g/ω0→
0. In his
limi , we can expand he sine and cosine unc ions o he i s o de , and we ob ain
ˆ
HC≈ℏω0ˆc†ˆc+ℏω0
2ˆσz+ℏgˆσy(ˆc+ ˆc†).(2.80)
Fu he , we can pe o m he o a ing wa e app oxima ion (RWA), which consis s in
neglec ing he double o a ing, no-numbe conse ing e ms, i.e.,
ˆσˆc
and
ˆσ†ˆc†
. This
app oxima ion esul s in he widely-used Jaynes-Cummings Hamil onian [21,105]:
ˆ
HJC =ℏω0ˆc†ˆc+ℏω0
2ˆσz+iℏg(ˆσ†ˆc−ˆc†ˆσ).(2.81)
2.6 Quan um dynamics in open quan um
sys ems: he quan um mas e equa ion
E e y quan um sys em in e ac s wi h i s en i onmen , o example wi h phonons
a a speci ic empe a u e o simply wi h acuum luc ua ions. O en we know e y
71
Chap e 2. Fundamen als o quan um nanopho onics
li le abou his en i onmen , and we app oxima e i as a Ma ko ian ese oi (a
Ma ko ian ese oi has no memo y o i s p e ious in e ac ions wi h he quan um
sys em, no is i able o c ea e cohe ences due o i s in e ac ion wi h he sys em).
In his sec ion, we ocus on desc ibing how he in e ac ion o a quan um sys em
wi h i s en i onmen a ec s i s dynamics, i.e., he ime e olu ion o he s a e
o he quan um sys em. Fo ha pu pose, we in oduce he quan um mas e
equa ion o malism, which allows us o desc ibe he dynamics o he sys em
wi hou gi ing a comple e desc ip ion o he en i onmen (we jus app oxima e
he en i onmen as a Ma ko ian ese oi ). In he ollowing subsec ions 2.6.1 and
2.6.2, we e iew he c i ical s eps in de i ing he mas e equa ion ( o a comple e
and pedagogical de i a ion o he mas e equa ion, including a de ailed discussion
o he app oxima ions in ol ed, e e ences [72,73,106] a e good op ions).
2.6.1 F om he Von Neumann equa ion o he Ma ko ian
mas e equa ion
In quan um mechanics, he e a e h ee pic u es o unde s and he ime e olu ion
o a sys em: he Sch ödinge , he In e ac ion, and he Heisenbe g pic u es. The
s udies p esen ed in his hesis (chap e s 5and 4) a e gi en in he Sch ödinge
pic u e, whe e he s a es desc ibe he e olu ion o a sys em. In he Sch ödinge
pic u e, he e olu ion o a s a e (desc ibed by i s densi y ma ix,
ˆρ
) ollows he
Von Neumann equa ion [72],
d
d ˆρ( ) = −i
ℏ[ˆ
H,ˆρ( )],(2.82)
being
ˆ
H
he Hamil onian desc ibing he ene gy o he quan um sys em, he ene gy
o he en i onmen , and he sys em-en i onmen in e ac ion. Ou aim in his
sec ion is o ope a e he Von Neumann equa ion (Eq.
(2.82)
) o ob ain he so-
called quan um mas e equa ion, an exp ession ha ocuses only on he e olu ion
o he s a e o he sys em, desc ibed by he densi y ma ix
ˆρS
, and he in e ac ion
be ween he sys em and he en i onmen is con enien ly simpli ied.
ˆρS
is con ained
in he o al densi y ma ix
ˆρ
(in Eq.
(2.82)
), which desc ibes bo h he s a e o he
sys em (independen ly o he en i onmen ) and o he s a e o he en i onmen
(independen ly o he sys em). We can ex ac ˆρS om ˆρas,
ˆρS( ) = T R{ˆρ( )},(2.83)
whe e
T R
deno es he ace o e he Hilbe space o he en i onmen (o
ese oi ), [72–74,106].
Be o e s a ing he de i a ion o he mas e equa ion, we need o in oduce
ˆ
H
(in Eq.
(2.82)
), which is he Hamil onian o he sys em and he en i onmen . This
Hamil onian can be spli in o h ee e ms,
ˆ
H=ˆ
HS+ˆ
HR+ˆ
HSR,(2.84)
72
2.6. Quan um dynamics in open quan um sys ems: he quan um mas e equa ion
whe e
ˆ
HS
desc ibes he ene gy o he sys em ha we a e in e es ed in s udying.
Fo con enience, ou de i a ion o he mas e equa ion is ca ied ou on he basis
o he |ν⟩eigens a es o ˆ
HS, such ha
ˆ
HS=X
ν
ℏων|ν⟩⟨ν|,(2.85)
whe e
ℏων
is he eigen alue o he
|ν⟩
eigens a e.
ˆ
HR
in Eq.
(2.84)
desc ibes he
ene gy o he en i onmen o he sys em, and
ˆ
HSR
desc ibes he in e ac ion ene gy
be ween he sys em and he en i onmen . The mos gene al o m o he
ˆ
HSR
Hamil onian desc ibing he in e ac ion be ween he en i onmen and he sys em
is [72], ˆ
HSR =ℏX
α
ˆ
Sα⊗ˆ
Rα,(2.86)
whe e
ˆ
Sα
=
ˆ
S†
α
and
ˆ
Rα
=
ˆ
R†
α
a e he mi ian ope a o s o he sys em and o he
en i onmen , espec i ely. The subindex
α
unso e he di e en componen s o
he Hamil onian o he sys em. Fo ins ance, in he ca i y-QED sys ems s udied
in his hesis, we conside wo di e en
α
e ms, one o he in e ac ion be ween
he ca i y wi h he en i onmen , and ano he e m accoun ing o he in e ac ion
be ween he TLS and he en i onmen .
We now p oceed o ope a e he Von Neumann equa ion (Eq.
(2.82)
) o ob ain
he mas e equa ion. Fo his de i a ion, i is mos con enien o ope a e Eq.
(2.82)
in he in e ac ion pic u e. In he in e ac ion pic u e, bo h he s a es and
he Hamil onian e ol e in ime, and he Von Neumann equa ion in he in e ac ion
pic u e eads as [72],
d
d ˆρ(i)( ) = −i
ℏ[ˆ
H(i)( ),ˆρ(i)( )].(2.87)
Along his sec ion we use he label “(
i
)” o indica e ha he Hamil onian and
he densi y ma ix o he sys em,
ˆ
H(i)
and
ˆρ(i)
(
), espec i ely, a e w i en in
he in e ac ion pic u e.
ˆ
H(i)
and
ˆρ(i)
(
)a e connec ed wi h he ope a o s in he
Sch ödinge pic u e by a uni a y ans o ma ion [72,73,106]:
ˆ
H(i)( ) = ˆ
U†
(s−i)( )ˆ
HSR ˆ
U(s−i)( ),(2.88)
and
ˆρ(i)( ) = ˆ
U†
(s−i)( )ˆρˆ
U(s−i)( ).(2.89)
wi h he uni a y ans o ma ion being
ˆ
U(s−i)( ) = exp[i(ˆ
HS+ˆ
HR) /ℏ].(2.90)
We nex e-exp ess Eq. (2.87) by in eg a ing i ,
ˆρ(i)( ) = ˆρ(i)(0) −i
ℏˆ
0
[ˆ
H(i)( ′),ˆρ(i)( ′)]d ′,(2.91)
73
Chap e 2. Fundamen als o quan um nanopho onics
and hen subs i u ing
ˆρ(i)
(
)in Eq.
(2.91)
back in o he igh -hand side o Eq.
(2.87),
d
d ˆρ(i)( ) = −i
ℏ[ˆ
H(i)( ),ˆρ(i)(0)] −1
ℏ2ˆ
H(i)( ),ˆ
0
[ˆ
H(i)( ′),ˆρ(i)( ′)]d ′.(2.92)
As s a ed abo e, we a e in e es ed only in he ime e olu ion o he s a e o he
sys em, which we can ex ac om Eq.
(2.92)
using he pa ial ace o e he
Hilbe space o he en i onmen (Eq. (2.83)),
d
d ˆρ(i)
S( ) = −T Ri
ℏ[ˆ
H(i)( ),ˆρ(i)(0)] + 1
ℏ2ˆ
H(i)( ),ˆ
0
[ˆ
H(i)( ′),ˆρ(i)( ′)]d ′.
(2.93)
To ope a e Eq.
(2.93)
we need o in oduce ou i s app oxima ion in his
de i a ion: we assume ha he en i onmen co esponds o a Ma ko ian ese oi ,
whe e we assume ha he dissipa ion o he co ela ions o he ese oi is much
as e han any a ia ion o he sys em. Ma hema ically his implies ha [72,73,106]:
⟨ˆ
R(i)
α( )⟩=T {ˆ
R(i)
α( )ˆρ(i)(0)} ≈ 0,(2.94)
o all α, whe e ˆ
R(i)
α( ) = ˆ
U†
(s−i)( )ˆ
Rαˆ
U(s−i)( )(2.95)
a e he ope a o s o he ese oi in he in e ac ion pic u e. This is he main
app oxima ion ha we a e going o use o he de i a ion o he mas e equa ion.
I can be p o en ha he assump ion in Eq. (2.94) implies [72],
T R{[ˆ
H(i)( ),ˆρ(i)(0)]} ≈ 0.(2.96)
Thus, Eq. (2.93) simpli ies o
d
d ˆρ(i)
S( )≈ −T R1
ℏ2ˆ
H(i)( ),ˆ
0
[ˆ
H(i)( ′),ˆρ(i)( ′)]d ′.(2.97)
Nex , we in oduce wo u he app oxima ions in Eq.
(2.97)
: we i s assume
ha he s a e o he sys em a a ime
′
does no depend on he his o y o he
in e ac ion wi h he ese oi a p e ious imes ′, and hus,
[ˆ
H(i)( ′),ˆρ(i)( ′)] ≈[ˆ
H(i)( ′),ˆρ(i)( )].(2.98)
This app oxima ion simpli ies Eq. (2.97) o
d
d ˆρ(i)
S( )≈ −T R1
ℏ2ˆ
H(i)( ),ˆ
0
[ˆ
H(i)( ′),ˆρ(i)( )]d ′.(2.99)
No e ha in his las equa ion
ˆρ(i)
depends on
ins ead o
′
as in Eq.
(2.97)
. Second,
we in oduce he Bo n app oxima ion, which s a es ha o a weak in e ac ion
74
2.6. Quan um dynamics in open quan um sys ems: he quan um mas e equa ion
be ween he sys em and he ese oi , we can ac o ize he o al densi y ma ix
ˆρ(i)( )≈ˆρ(i)
S( )⊗ˆρ(i)
R,(2.100)
whe e he s a e o he ese oi , desc ibed by he densi y ma ix
ˆρ(i)
R
, is conside ed
o be cons an in ime because we assume ha he ime scales a which cohe ences
o he ese oi a e dissipa ed a e much as e han he ime scales a which he
sys em a ies, and hus,
ˆρR
is no a ec ed by he dynamics o he sys em (same
app oxima ion as o Eq.
(2.94)
). Using he Bo n app oxima ion (Eq.
(2.100)
) in
Eq. (2.99), we can a i e a he so-called Red ield equa ion [107],
d
d ˆρ(i)
S( ) = −T R1
ℏ2ˆ
H(i)( ),ˆ
0
[ˆ
H(i)( ′),ˆρ(i)
S( )⊗ˆρ(i)
R]d ′.(2.101)
I we again assume ha he ese oi co ela ions disappea e y as in compa ison
wi h he e olu ion o he sys em, we can conside ha he [
ˆ
H(i)
(
′
)
,ˆρS
(
)
(i)⊗ˆρ(i)
R
]
e m in Eq.
(2.101)
is only de e mined by he e alua ion o
′
close o
, and hus,
we can app oxima e he lowe limi o he in eg al in ime by
−∞
[72,73,106]. This
esul s in he Bo n-Ma ko o Ma ko ian mas e equa ion,
d
d ˆρ(i)
S( ) = −T R1
ℏ2ˆ
H(i)( ),ˆ∞
0
[ˆ
H(i)( −s),ˆρ(i)
S( )⊗ˆρ(i)
R]ds,(2.102)
whe e we ha e subs i u ed ′→ −s o con enience.
2.6.2 F om he Ma ko ian mas e equa ion o he
Lindbladian mas e equa ion
The in eg al in Eq.
(2.102)
does no ensu e ha
ˆρ(i)
S
is a posi i e-de ini e ma ix,
which is a equisi e o any quan um densi y ma ix. To ensu e ha
ˆρ(i)
S
is
posi i e-de ini e, we need o pe o m one las app oxima ion, he o a ing wa e
app oxima ion on he in e ac ion be ween he sys em and he ese oi (RRWA
x
).
To be e explain he RRWA, le us i s ind he exp ession o
ˆ
H(i)
, he Hamil onian
in he in e ac ion pic u e. We i s in oduced
ˆ
H(i)
in Eq.
(2.88)
, whe e i is shown
o explici ly depend on
ˆ
HSR
, he Hamil onian desc ibing he sys em- ese oi
in e ac ion in he Sc h ödinge pic u e.
ˆ
HSR
is w i en di ec ly in e ms o he
ˆ
Sα
ope a o s o he sys em and he
ˆ
Rα
ope a o s o he ese oi (Eq.
(2.86)
). In he
in e ac ion pic u e,
ˆ
H(i)
can also be di ec ly exp essed in e ms o he ope a o s o
he sys em and he ese oi [72],
ˆ
H(i)( ) = X
α
ˆ
S(i)
α( )⊗ˆ
R(i)
α( ),(2.103)
x
Do no con use wi h he RWA o he Jaynes-Cummings model in oduced in subsec ion 2.5.4.
75
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
om he QE o he nanoan enna, i.e.,
Gqs
x,x
(
e
)
≡Gqs
x,x
(
e, a
) =
Gqs
x,x
(
a, e
)(see
Eq.
(1.27)
).
αa
and
αe
in Eqs.
(3.1a)
and
(3.1b)
a e he pola izabili y o he
nanoan enna and o he QE, espec i ely. Wi hin a D ude model desc ip ion o
he pe mi i i y o he me al, and using a D ude-Lo en z model o he op ical
esponse o he QE, he αeand αapola izabili ies become:
αe(ω) = Ae
ω2
0−ω2−iγ0ω,(3.2a)
αa(ω) = Aa
ω2
0−ω2−iκω ,(3.2b)
whe e he s eng h o he coupling be ween he wo dipoles is de e mined by he
pola izabili y ampli ude o he QE,
Ae
, and ha o he plasmonic nanoan enna,
Aa
. The equency
ωR
=
ωp/√3
is he equency o he dipola plasmon esonance
in he nanoan enna (wi h
ωp
he D ude plasma equency). He e we chose
ωR
o
ma ch
ω0
(
ωR
=
ω0
), he esonan exci a ion equency o he QE. Las ,
γ0
and
κ
a e he spon aneous decay a e o he QE and he plasmonic in insic decay
a e, espec i ely. Th oughou his chap e we conside ha he dipole momen
o he exci onic ansi ion is
0
= 0
.
05
e·
nm (
e
is he elec on cha ge), which se s
he pola izabili y ampli ude o he QE
Ae
= 2
ω0 2
0/ℏ
and he spon aneous decay
a e
γ0
=
ω3
0 2
0/
(3
πε0ℏc3
0
)(see sec ion 1.2.4). We do no conside o he in insic
molecula losses beyond
γ0
. Subs i u ing Eqs.
(3.2a)
and
(3.2b)
in o Eqs.
(3.1a)
and (3.1b) one ob ains:
(ω2
0−ω2)pe(ω)−iωγ0pe(ω) = AeGqs
x,x( e)pa(ω),(3.3a)
(ω2
0−ω2)pa(ω)−iωκpa(ω) = AaGqs
x,x( e)pe(ω) + AaE0.(3.3b)
He e, he pola izabili y ampli ude o he nanoan enna de e mines how e icien ly
he sys em is exci ed ( ia he e m
AaE0
). Eqs.
(3.1a)
-
(3.3b)
and he exp ession
used o ob ain Aaa e discussed in mo e de ail in sec ion 3.4.2.
We no e ha Eqs.
(3.3a)
and
(3.3b)
a e e y simila o hose ob ained wi h
phenomenological models ha assume ha he QE and he plasmonic nanopa icle
can be ea ed as wo coupled ha monic oscilla o s. This coupled-ha monic-
oscilla o s model has been equen ly used o desc ibe he in e ac ion be ween
quan um emi e s and nanoan ennas [8,125,126].
We show in Fig. 3.1b he ex inc ion c oss-sec ion spec um o he hyb id
QE-nanoan enna sys em ob ained using Eqs.
(3.3a)
and
(3.3b)
o di e en alues
o he sepa a ion dis ance
d
be ween he QE and he su ace o he nanoan enna
( he alue o
d
a ec s he quasis a ic G een’s unc ion,
Gqs
x,x
, and, hus, he QE-
nanoan enna in e ac ion). The ex inc ion c oss sec ion
σex
is no malized o he
co esponding alue o he ba e nanoan enna
σ(0)
ex
and i is ob ained assuming ha
he di ec emission o he QE is negligible (
pa≫pe
), which is a ypical si ua ion
82
3.2. Fano asymme y unde esonan condi ions
in plasmonic sys ems. In his case, he op ical heo em (Eq. (1.41)) [40] gi es
σsimp
ex (ω) = 2π
λε0
Im{pa(ω)/E0},(3.4)
whe e he supe index “
simp
” emphasizes ha his is a simpli ied exp ession ha
only conside s he emission om he nanoan enna.
The no malized ex inc ion c oss-sec ion in Fig. 3.1b ea u es an almos cons an
backg ound and he eme gence o a spec ally na ow dip a he esonan wa eleng h
λ0
o all he sepa a ion dis ances
d
conside ed. The backg ound co esponds o
he e y b oad plasmonic esponse and he dip o he Fano ea u e. Impo an ly,
he Fano dip ob ained wi hin his simple dipole-dipole in e ac ion model (Eqs.
(3.3a)-(3.4)) is always pe ec ly symme ic.
We nex ob ain he exac elec omagne ic esponse o he hyb id sys em, which
we calcula e om he op ical heo em (Eq.
(1.41)
) by ob aining he o al sca e ed
ield wi hin he igo ous Mie heo y o malism in oduced in sec ion 1.3 (see also
e e ences [40,45,127]). In pa icula , we use Mie heo y o calcula e he esponse
o he sphe ical nanoan enna illumina ed by he incoming plane wa e (sec ion 1.3.2)
and by he induced dipole momen o he QE (sec ion 1.2.4). Fo all Mie heo y
calcula ions shown in his chap e , we ha e used an expansion o 60 mul ipoles,
which we e i ied ha ensu es con e gence. The de ails o he calcula ion o he
ex inc ion c oss-sec ion a e gi en in sec ion 3.3.1.
Figu e 3.1c shows he esul ing ex inc ion c oss sec ion ob ained o he same
sys em and dis ances
d
as in Fig. 3.1b. These exac calcula ions exhibi again a
b oad backg ound, due o he esponse o he ba e plasmonic nanoan enna, and
a na ow Fano ea u e caused by he coupling be ween he QE exci on and he
plasmonic esonance o he nanoan enna. Howe e , he Fano lineshapes show clea
di e ences compa ed o hose ob ained wi h he simple dipole-dipole in e ac ion
model (Eqs.
(3.3a)
-
(3.4)
and Fig. 3.1b). O e all, he Fano ea u es a e b oade
o he exac calcula ions han o he simple dipole model. Fu he , he exac
calcula ion also esul s in a shi o he Fano ea u es, induced by he pho onic
lamb shi , no included in he simple model. The shi and la ge b oadening a e
clea e o small nanoan enna-QE sepa a ion dis ances (
d <
20 nm) and a e mainly
a consequence o he coupling be ween he QE exci on and he highe -o de modes
o he nanoan enna [35,95,96,128
–
136]. C ucially, he Fano ea u e ob ained wi hin
he exac calcula ions is no necessa ily a pe ec ly symme ic dip, bu i can ake
di e en lineshapes. This shape e ol es om a b oad and almos symme ic dip a
sho sepa a ion dis ances (
d <
15 nm) owa ds a na ow and almos symme ic
peak a la ge sepa a ion dis ances (
d >
60 nm). In he ange be ween hese wo
ex emes, he Fano ea u e becomes clea ly asymme ic.
Thus, we ha e shown ha he p edic ion o a symme ic Fano dip ob ained
wi h a simple dipole-dipole in e ac ion model can s ongly di e om he esul s
o he exac calcula ions, whe e signi ican ly asymme ic lineshapes eme ge. We
emphasize ha his asymme y is no due o plasmon-exci on de uning as he
esonance condi ion o ze o de uning is p ese ed in all cases. In he ollowing, we
83
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
analyze in de ail he di e en physical mechanisms ha lead o asymme ic Fano
lineshapes o QE-nanoan enna sys ems unde esonan condi ions.
3.3 Fano lineshape
3.3.1 Analy ical de i a ion o he Fano lineshape in he
ex inc ion c oss-sec ion
Fi s , we show ha he ex inc ion c oss-sec ion spec um
σex
o a QE in e ac ing
wi h a nanoan enna ollows he modi ied Fano lineshape discussed in e e ences
[118,119],
σex (ω)
σ(0)
ex ≈(Ω(ω) + q)2+B
Ω(ω)2+ 1 ,(3.5)
whe e
q
is he Fano asymme y ac o ( he key pa ame e ha we analyze in his
chap e ), Bis he ze o-dip pa ame e , and
Ω(ω) = ω′
0
2−ω2
ωγ′, ω′
0=ω0+ ∆ω, (3.6)
wi h ∆
ω
he Lamb shi .
γ′
is he enhanced decay a e o he QE in he p esence
o he nanoan enna, which can be w i en in e ms o he Pu cell Fac o PFas
γ′= (PF+ 1)γ0+γNR
i(3.7)
whe e
γNR
i
a e o he in insic (non- adia i e) losses o he QE (in his chap e
we conside
γNR
i
= 0).
q
,
B
,∆
ω
, and
PF
a e he main pa ame e s de ining he
Fano lineshape. In he i s pa o his subsec ion, we ocus on de i ing simple
analy ical exp essions o calcula e hem in an a bi a y QE-nanoan enna sys em.
In he second pa o his subsec ion, we e alua e hem o a QE in e ac ing wi h
a sil e sphe ical nanopa icle and discuss in mo e dep h hei physical o igin.
To ob ain he ex inc ion c oss-sec ion o he QE–nanoan enna sys em using he
op ical heo em [40], e alua ed o ou sys em and illumina ion
σex (ω) = 4π
k2Re(−ikzd)e−ikzdEFF
x( d, ω)
E0,(3.8)
whe e
EFF
x
is he
x
-componen o he sca e ed elec ic ield ha he QE-
nanoan enna sys em induces on a poin -like de ec o placed in he a - ield egion,
a he (Ca esian) coo dina es
d
= (0
x,
0
y,
(
zd
)
z
).
σex
is independen o he
chosen alue o
zd
since
EFF
x
(
d, ω
)
∝eikzd/zd
[40]. We no e ha h oughou his
de i a ion, i is only necessa y o calcula e he
x
-componen o all o he conside ed
elec ic ields because o he geome y conside ed [35,40]. Thus, o a oid epe i ion,
we do no always s a e explici ly in he discussion below ha we a e e e ing o
he
x
-componen o he ields, o he (
x, x
)-componen o he G een’s unc ions,
84
3.3. Fano lineshape
bu we indica e his by an x(o (x, x)) subindex.
EFF
xcan be exp essed as he sum o wo con ibu ions,
EFF
x( d, ω) = EA
x( d, ω) + EE-To
x( d, ω),(3.9)
whe e
EA
x
is he ield di ec ly sca e ed by he nanoan enna in he absence o
he QE and
EE-To
x
is he o al elec ic ield ha he QE induces a he de ec o
conside ing he p esence o he nanoan enna. By de ining he a - ield enhancemen
ac o a he posi ion o he de ec o as
KFF
(
d, ω
) =
EA
x
(
d, ω
)
/E0
we can w i e
EA
xas
EA
x(ω) = KFF(ω)E0.(3.10)
On he o he hand,
EE-To
x
can be w i en using he G een’s unc ion o malism
as
EE-To
x( d, ω) = GFF
x,x( d, e, ω)pe(ω),(3.11)
whe e
pe
is he induced dipole momen o he QE (o ien ed along he
x
-axis) and
GFF
x,x
is he G een’s unc ion ha desc ibes he emission o he QE owa ds he
de ec o in he p esence o he nanoan enna.
We decompose GFF
x,x as a sum o wo con ibu ions
GFF
x,x( d, e, ω) = GFF
0x,x ( d, e, ω) + SFF
x,x( d, e, ω),(3.12)
whe e
GFF
0x,x
(al eady in oduced in Eq.
(1.24)
) is he acuum G een’s unc ion ha
desc ibes he ield
EE
x
ha he QE induces in he de ec o when no nanoan enna
is p esen ,
EE
x( d, ω) = GFF
0x,x( d, e, ω)pe(ω),(3.13)
whe e we can app oxima e
e≈
0in he e alua ion o
GFF
0x,x
(i.e., he e ec o
his change becomes negligible small in he de ec o si ua ed in he a - ield), so
ha
GFF
0x,x ( d, e, ω)≈GFF
0x,x ( d,0, ω) = k2
4πε0zd
eikzd,(3.14)
On he o he hand,
SFF
x,x
is he dyadic unc ion ha desc ibes he elec ic ields
induced by he QE on he de ec o ia he nanoan enna
EEA
x
[137] (i.e. he elec ic
ields ha he nanoan enna sca e s owa ds he de ec o when i is illumina ed
only by he QE),
EEA
x( d, ω) = SFF
x,x( d, e, ω)pe(ω).(3.15)
Fo he exac esul s ob ained wi h Mie heo y, we calcula e
SFF
x,x
using Mie heo y
as discussed in e e ence [127]. To e alua e Eqs.
(3.11)
,
(3.13)
, and
(3.15)
we need
i s o ob ain he alue o he induced dipole momen
pe
.
pe
is gi en by (see
85
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
sec ion 1.2.4),
pe(ω) = αe(ω)ENF
x( e, ω),(3.16)
whe e αeis gi en in Eq. (3.2a). ENF
xis he ield ha exci es he QE,
ENF
x( e, ω) = E0+EAE
x( e, ω) + EEAE
x( e, ω),(3.17)
which we w i e as he sum o he ield
E0
due o he di ec illumina ion o he plane
wa e, and wo addi ional con ibu ions induced by he p esence o he nanoan enna,
EAE
x
and
EEAE
x
.
EAE
x
co esponds o he elec ic ield ha he nanoan enna induces
a he QE posi ion in he absence o he QE due o he illumina ion by he inciden
plane wa e, i.e.
EAE
x
depends only on he esponse o he isola ed nanoan enna. I
we use a ypical de ini ion o he nea - ield enhancemen ac o a he posi ion o
he QE, K( e, ω) = EAE
x( e, ω)/E0, we can w i e
EAE
x( e, ω) = K( e, ω)E0.(3.18)
On he o he hand,
EEAE
x
is he elec ic ield ha he QE induces a i s own
posi ion ia he nanoan enna, ha is, he ield sca e ed by he nanoan enna a
he QE posi ion when he only sou ce is he QE. We w i e his con ibu ion as a
unc ion o
pe
by using again he G een’s unc ion o malism. Fo doing so, we
in oduce he dyadic unc ion
SNF
x,x
ha desc ibes he emission o he QE on o i sel
ia he nanoan enna [137],
EEAE
x( e, ω) = SNF
x,x( e, ω)pe(ω).(3.19)
Using Eqs. (3.17), (3.18), and (3.19) we ind he exp ession o pe,
pe(ω) = Ae
ω′
0(ω)2−ω2−iγ′(ω)ω[K( e, ω) + 1]E0,(3.20)
wi h
ω′
0(ω)2=ω2
0−AeRe{SNF
x,x( e, ω)}, γ′(ω) = γ0+γNR
i+1
ωAeIm{SNF
x,x( e, ω)},
(3.21)
whe e
γNR
i
is he non- adia i e decay a e o he QE (in his chap e , we conside
γNR
i
= 0). We obse e ha he
Ae/[ω′
0(ω)2−ω2−iγ′(ω)ω]
p e ac o in Eq.
(3.20)
ollows a simila exp ession o he o iginal
αe
pola izabili y o he QE (Eq.
(3.2a)
)
bu o he cen al equency, which is shi ed om
ω0
o
ω′
0
in Eq.
(3.21)
(see Eq.
(3.6)
) and o he decay a e o he QE, which is augmen ed om
γ0
o
γ′
in Eq.
(3.21) (see Eq. (3.7)).
By using Eqs. (3.10), (3.11), and (3.20) we can w i e EFF
xas
86
3.3. Fano lineshape
EFF
x( d, ω) =
=nKFF( d, ω)+[GFF
0x,x( d, e, ω) + SFF
x,x( d, e, ω)]
| {z }
GFF
x,x( d, e,ω)
Ae[K( e, ω) + 1]
ω′
0(ω)2−ω2−iγ′(ω)ωoE0,
(3.22)
and by subs i u ing Eq. (3.22) in o Eq. (3.8) we ob ain
σex (ω) =
=4π
k2Re(−ikzd)e−ikzdKFF( d, ω) + GFF
x,x( d, e, ω)Ae(K( e, ω) + 1)
ω′
0(ω)2−ω2−iγ′(ω)ω.
(3.23)
We no malize his exp ession by he ex inc ion c oss-sec ion o he ba e
nanoan enna
σ(0)
ex
( ha can be calcula ed using he op ical heo em as
σ(0)
ex
(
ω
) =
(4
π/k2
)
Re{
(
−ikzd
)
e−ikzdKFF
(
d, ω
)
}
), and ob ain he no malized ex inc ion c oss-
sec ion o he hyb id sys em as:
σex (ω)
σ(0)
ex (ω)=
=Ren1 + 1
k2
4πσ(0)
ex (ω)(−ikzd)e−ikzdGFF
x,x( d, e, ω)Ae(K( e, ω) + 1)
ω′
0(ω)2−ω2−iγ′(ω)ωo.
(3.24)
Nex , we de ine
Ω(ω) = ω′
0(ω)2−ω2
γ′(ω)ω,(3.25)
q(ω) = 1
2
1
k2
4πσ(0)
ex (ω)
Ae
γ′(ω)ωRe(−ikzd)e−ikzdGFF
x,x( d, e, ω)(K( e, ω) + 1),
(3.26)
B(ω) =
= 1 −q(ω)2−1
k2
4πσ(0)
ex (ω)
Ae
γ′(ω)ωIm(−ikzd)e−ikzdGFF
x,x( d, e, ω)(K( e, ω) + 1),
(3.27)
87
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
and, aking in o accoun ha
Ae
,
γ0
,
ω
,
ω0
, and
σ(0)
ex
(
ω
)a e eal numbe s, we can
simpli y Eq. (3.24) as
σex (ω)
σ(0)
ex (ω)=1 + 2q(ω)Ω
Ω2+ 1 +−1 + q(ω)2+B(ω)
Ω2+ 1 =[Ω + q(ω)]2+B(ω)2
Ω2+ 1 .(3.28)
Fo gene ali y, we ha e kep in his de i a ion he explici equency dependence
o he pa ame e s
σ(0)
ex
(
ω
),
γ′
(
ω
),
ω′
0(ω)2
,
q
(
ω
), and
B
(
ω
), so ha Eq.
(3.28)
is exac .
I we conside ha he spec al wid h o he Fano ea u e (de e mined by
γ′
) is
much smalle han he spec al wid h o he plasmon esonance o he nanoan enna
we can e alua e all hese pa ame e s a he exci onic esonan equency ω0,
σ(0)
ex ≈σ(0)
ex (ω0), γ′≈γ′(ω0), ω′
0
2≈ω′
0(ω0)2, q ≈q(ω0), B ≈B(ω0),
(3.29)
and app oxima ex i Eq. (3.28) as he modi ied Fano lineshape [8,118,119],
σex (ω)
σ(0)
ex ≈(Ω(ω) + q)2+B2
Ω(ω)2+ 1 ,(3.30)
wi h Ω(
ω
) = (
ω′
0
2−ω2
)
/
(
ωγ′
). To ensu e ha he app oxima ion o Eq.
(3.29)
is jus i ied, we conside in his chap e a dipola oscilla o s eng h
0
o he QE
su icien ly small so ha he Fano ea u es emain e y na ow [138,139] (in ou
calcula ions we use 0= 0.05e·nm, ebeing he elec on cha ge).
Nex we summa ize he exp essions o he pa ame e s p esen in Eq. (3.30),
σ(0)
ex =4π
k2
0
Re(−ik0zd)e−ik0zdKFF( d, ω0),(3.31a)
∆ω=ω′
0−ω0=qω0−AeRe{SNF
x,x( e, ω0)}−ω0,(3.31b)
PF=γ′−γNR
i
γ0−1 = 1
ωAeIm{SNF
x,x( e, ω0)},(3.31c)
q=1
2
1
k2
0
4πσ(0)
ex
Ae
γ′ω0
Re(−ik0zd)e−ik0zdGFF
x,x( d, e, ω0)(K( e, ω0) + 1),(3.31d)
B= 1 −q2−1
k2
0
4πσ(0)
ex
Ae
γ′ω0
Im(−ik0zd)e−ik0zdGFF
x,x( d, e, ω0)(K( e, ω0) + 1),
(3.31e)
whe e
k0
=
ω0c0
is he wa e ec o a he esonan equency
ω0
. We ha e e i ied
ha Eqs.
(3.30)
-
(3.31e)
desc ibe all he spec a ha we p esen in his chap e
e y accu a ely (i.e. he app oxima ion gi en in Eq.
(3.29)
is alid). We no e ha
all he pa ame e s in Eqs.
(3.31a)
-
(3.31e)
can be ob ained om s anda d classical
x i
In he sys ems s udied in his chap e
ω0≈ω′
0
, and hus, e alua ing he pa ame e s
σ(0)
ex
,
γ′,q, and Bpa ame e s in Eq. (3.29) a ω0o ω′
0gi es e y simila esul s.
88
3.3. Fano lineshape
elec omagne ic calcula ions.
Finally, we simpli y Eq.
(3.31d)
using he ecip oci y heo em [140
–
142].
Acco ding o his heo em, nea - ield enhancemen ac o
K
(
e, ω
) =
EA
x
(
e, ω
)
/E0
is connec ed wi h he ields ha he QE induces a he de ec o di ec ly (
EE
x
) and
ia he nanoan enna (EEA
x) by he ollowing equa ion,
K( e, ω) = EEA
x( e, ω)
EE
x( e, ω).(3.32)
Using Eqs. (3.13), (3.15), and (3.32) we w i e he ecip oci y heo em as
SFF
x,x( d, e, ω)
GFF
0x,x( d, e, ω)=K( e, ω).(3.33)
In p inciple, he ecip oci y heo em [140
–
142] ela es he ield enhancemen
induced by a plane-wa e wi h he emission o he QE in he backwa d di ec ion
(i.e. in he nega i e
z
-di ec ion o he conside ed sys ems), while
GFF
0x,x
and
SFF
x,x
in Eq.
(3.31d)
desc ibe he emission o he QE in he o wa d di ec ion (i.e. in he
posi i e
z
-di ec ion). Howe e , he symme y o ou sys em implies ha
GFF
0x,x
and
SFF
x,x
a e iden ical in he o wa d and backwa d di ec ion, and Eqs.
(3.32)
and
(3.33) a e alid. By using Eq. (3.12) we can w i e Eq. (3.33) as
GFF
x,x( d, e, ω) = GFF
0x,x( d, e, ω)(K( e, ω) + 1).(3.34)
Las , using Eqs. (3.14), (3.31d), and (3.34) we ob ain
q=Ae
2σ(0)
ex γ′c0ε0
(Im{K( e, ω0)}+Re{K( e, ω0)}Im{K( e, ω0)}),(3.35)
3.3.2 E alua ion o he Fano lineshape
In he subsec ion abo e, we ha e shown ha he ex inc ion c oss-sec ion o a
QE-nanoan enna sys em exci ed by a plane wa e can be desc ibed by a modi ied
Fano lineshape [8,118,119] (assuming ha he spec al wid h o he QE emission
is much smalle han he wid h o he spec al esponse o he nanoan enna). Fo
con enience, we epea he e he o mula o he modi ied Fano lineshape in Eq.
(3.30)
σex (ω)
σ(0)
ex ≈(Ω(ω) + q)2+B
Ω(ω)2+ 1 ,
whe e Ω(ω)=(ω′
0
2−ω2)/(ωγ′)(Eq. (3.25)).
The modi ied Fano lineshape in Eq.
(3.30)
depends on h ee pa ame e s,
q
,
Ω(
ω
), and
B
(calcula ed om Eqs.
(3.31a)
-
(3.31e)
and
(3.35)
). As in oduced
abo e,
q
is he o al asymme y ac o (also called Fano-pa ame e ) ha cap u es
he asymme y o he Fano lineshape o he ex inc ion c oss-sec ion spec um, and
i is he main ocus o his chap e . Ωis a no malized equency gi en in Eq.
(3.6)
.
89
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
1.0
1.1
1.2
1.3
-0.8 -0.4 0 0.4 0.8
(nm)
1.0
1.2
1.4
1.6
ex /(0)
ex
ex /(0)
ex
3
4
4
0.4
0.6
0.8
1.0
ex /(0)
ex
2
0.5
1.0
1.5
ex /(0)
ex
1
(a)
R = 50 nm
d = 7 nm
R = 20 nm
d = 50 nm
R = 50 nm
d = 100 nm
R = 70 nm
d = 100 nm
2
1
3
4
(c)
2
1
3
4
(b)
2
1
3
4
(d)
2
1
3
(e)
RR
Figu e 3.2: (a-d) Con ou plo s o he pa ame e s de ining he Fano lineshape, ob ained wi hin
Mie heo y. (a) Pu cell Fac o ,
PF
, (b) Lamb Shi , ∆
ω
, (c) con as ,
C
, and (d) o al asymme y
ac o ,
q
, as a unc ion o he dis ance,
d
( he minimum dis ance in he panels is
d
= 2 nm), om
he QE o he su ace o a sil e sphe ical an enna wi h di e en adius,
R
. The esonance o he
QE is chosen o ma ch he equency o esonance o he nanoan enna o all sizes o pa icles,
R
.
(e) No malized ex inc ion spec a o he hyb id sys em a e e alua ed a poin s ma ked as 1,2,3,
and 4in panels (a)-(d). The alues o
R
and
d
o each poin a e indica ed in he labels o panel
(e).
∆
ω
is he pho onic Lamb Shi (Eq.
(3.6)
) ha co esponds o a sligh shi in he
esonan equency om
ω0
o
ω′
0
, and
γ′
= (
PF
+ 1)
γ0
a e he enhanced losses
o he QE (Eq.
(3.7)
), which desc ibes he b oadening o he Fano dip. Bo h
e ec s can be clea ly obse ed in he spec a o Fig. 3.1c. The exp ession o
γ′
assumes no in insic losses beyond
γ0
, bu i can be modi ied in a s aigh o wa d
manne o include addi ional in insic losses. Las ,
B
in Eq.
(3.30)
is he ze o-dip
pa ame e (Eq.
(3.31e)
) ha can be ela ed o he ac o
q
and he con as
C
=
p2(B+ 1)q2+ (B−1)2+q4
. He e, we de ine he con as
C
o he Fano
ea u e as he di e ence be ween he maximum and he minimum o he Fano
ea u e in he no malized ex inc ion c oss-sec ion spec um (see inse in panel 2 o
Fig. 3.2e).
Thus, changes on he Fano spec al lineshape can be unde s ood by analyzing
he pa ame e s,
q
,
C
,
PF
, and ∆
ω
( he las wo de e mining Ω). No e ha hese
pa ame e s can be ob ained om he classical G een’s unc ion and he ield
enhancemen o he plasmonic an enna a he posi ion o he emi e acco ding o
Eqs.
(3.31a)
-
(3.31e)
. In Fig. 3.2a-d, we sys ema ically s udy he dependence o
hese pa ame e s wi h he adius o he sil e sphe ical nanopa icle,
R
, and he
dis ance be ween he an enna and he emi e ,
d
. All alues a e ob ained om exac
Mie heo y calcula ions using he expe imen al alues o he sil e pe mi i i y [41]
and assuming esonan condi ions; i.e., o each adius, we ind he equency o he
dipola plasmonic esonance o he an enna (lowes -ene gy peak in he ex inc ion
90
3.3. Fano lineshape
c oss-sec ion spec um o he ba e nanoan enna), and we modi y he ene gy o he
QE ansi ion acco dingly. To illus a e he esul ing Fano lineshape desc ibed by
he pa ame e s in Figs. 3.2a-b, we also show, in Fig. 3.2e, he ex inc ion c oss
sec ion spec um o ou di e en poin s indica ed in Figs. 3.2a-d ( he alues o
R
and d o each poin a e gi en in he labels o Fig. 3.2e).
Fo all he adii conside ed, he Pu cell Fac o ,
PF
, (Fig. 3.2a), and he
pho onic Lamb shi ∆
ω
(Fig. 3.2b) s ongly inc ease when he QE app oaches
he an enna, as shown in p e ious s udies [35,95,96,128
–
133]. This inc ease is
due o he mo e e icien coupling o he QE wi h he plasmonic modes o he
nanoan enna, pa icula ly wi h high-o de modes. On he o he hand, he con as ,
C
, shows a mo e complex dependence wi h he adius,
R
, and he dis ance,
d
(Fig.
3.2c). We can dis inguish h ee di e en dis ance egimes in his igu e. Fo sho
dis ances (
d≲
10 nm), he QE couples e y e icien ly o he highe -o de modes o
he sphe ical nanopa icle, and he esul ing quenching o he emission [134
–
136]
leads o he disappea ance o he Fano dip (small con as ). Fo in e media e
dis ances (compa ed o he adius, i.e. 10 nm
≲d≲
3
R
), he quenching becomes
less signi ican , and he Fano ea u e eme ges wi h a easonably big con as
(1
≳C≳
0
.
5, pu ple- eddish egion in Fig. 3.2c). In his egime o dis ances, he
con as is smalle o sphe es wi h
R≳
40 nm, which is mainly a consequence o
wo des uc i e in e e ence e ec s, he i s be ween he exci a ion o he QE by
he illumina ion plane–wa e and by he an enna-induced nea ields, and he second
be ween he ligh emi ed by he QE di ec ly and ia he nanoan enna. Las , as
he sepa a ion dis ance is made signi ican ly la ge han he adius (
d≳
3
R
), he
QE p og essi ely decouples om he nanoan enna, and he ex inc ion c oss-sec ion
o he whole sys em e ol es owa d he supe posi ion o he peak o he ex inc ion
c oss-sec ion o he QE in a acuum on op o he b oad backg ound spec um
o he ba e sphe ical nanopa icle. We can hen exp ess he ex inc ion c oss–
sec ion o he whole sys em a e y long sepa a ion dis ances as (
σ(0)
ex
+
σQE
ex
)
/σ(0)
ex
,
whe e
σQE
ex
is he ex inc ion c oss-sec ion o he QE in a acuum. As we ha e
conside ed ha he QE only has adia i e losses due o he spon aneous decay,
σQE
ex
is la ge han he ex inc ion c oss-sec ion o he ba e sphe ical nanopa icle
σ(0)
ex
[36] (
σQE
ex
= 6
π
(
ω0/c0
)
2> σ(0)
ex
), and he con as becomes highe han one,
C > 1.
Las , Fig. 3.2d shows he ela i ely complex dependence o he o al asymme y
ac o
q
wi h adius
R
and dis ance
d
. The Fano asymme y is small (
|q|<
0
.
2)
o
d≲
10 nm (co esponding o a symme ic dip) and o la ge dis ances,
d≳
150
nm, (co esponding o an almos symme ic peak). A in e media e dis ances (10
nm
≲d≲
150 nm) he asymme y is signi ican ly la ge , wi h a maximum alue
a a dis ance
d∼
50 nm, which is s ongly dependen on he adius. We also ind
ha a la ge dis ances (
d≳
150 nm) he asymme y can ake nega i e alues and
show a damped oscilla o y beha io o
q
wi h
d
( he dependence o
q
o a la ge
ange o dis ance is s udied in sec ion 3.4.3). Unde s anding his complex beha io
is he main objec i e o his chap e and i is analyzed in de ail nex .
91
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
We no e ha , once he adia i e co ec ion is in oduced, he maximum o he
ex inc ion c oss-sec ion σ(0)
ex o he ba e nanoan enna dipola esonance ed-shi s
spec ally wi h inc easing size and is ound a a sligh ly sho e wa eleng h han
he co esponding maximum o he nea – ield enhancemen
|K|
[144,145]. When
no s a ed o he wise, we conside by de aul ha he esonance equency o he
QE,
ω0
, ma ches he equency a which
σ(0)
ex
is maximum. Below, in sec ion 3.4.3,
we show ha he e ec o se ing ω0 o he maximum o |K|is weak.
We show in Fig. 3.4b (second column o he igu e) he esul ing o al asymme y
ac o
q
(and i s con ibu ions,
qE
and
qR
) ob ained a e subs i u ing
αMD
a
by
αRC
a
in he simples dipole-dipole in e ac ion model
(3.44a)
-
(3.44c)
. This change mainly
a ec s he alue o
K
, which acqui es a la ge eal pa a esonance as compa ed
o he p e ious model ( he phase o
K
de ia es u he om
π/
2) and, hus,
q
=
qR
inc eases (Eq.
(3.37)
), wi h
qE
emaining equal o ze o. In pa icula ,
qR
becomes
la ge wi h inc easing
R
, as he e ec o he adia i e co ec ion inc eases wi h
he size o he nanoan enna.
qR
also inc eases o sho e
d
due o he s onge
enhancemen
|K|
. Howe e , he asymme y emains small (ha dly no iceable in
Fig. 3.4b, wi h
max
(
qR
)
≈
0
.
14). Las , we no e ha his low alue o
q
may lead
o hink ha he adia i e co ec ion is o li le impo ance in he desc ip ion o
he o al asymme y ac o . Howe e , we emphasize in he las model p esen ed in
his sec ion ha , once we go beyond he quasis a ic app oxima ion, i is c i ical o
conside a co ec desc ip ion o he adia i e co ec ion.
Di ec exci a ion and emission o he QE
In he nex s ep we in oduce he di ec exci a ion o he QE by he plane wa e
and he di ec emission o he QE o he a ield ( hi d column, Fig. 3.4c). These
wo e ec s a e in oduced a he same ime because, due o ecip oci y [140
–
142],
hei con ibu ion o he asymme y is iden ical (demons a ion in appendix A).
A e all hese changes he esponse o he sys em is gi en by he ollowing modi ied
equa ions:
pa(ω) = αRC
a(ω)(E0+Gqs
x,x( e)pe(ω)),(3.45a)
pe(ω) = αe(ω)(1 + K( e, ω))E0,(3.45b)
K( e, ω)E0=Gqs
x,x( e)pa(ω),(3.45c)
γ′=1 + ImAeαRC
a(ω)
γ0ω0
(Gqs
x,x( e, ω))2γ0,(3.45d)
σex (ω) = 2π
λε0
Impa(ω) + pe(ω)
E0.(3.45e)
The di ec exci a ion o he QE by he inciden plane wa e o ampli ude
E0
is
desc ibed by he e m
αeE0
in Eq.
(3.45b)
. Simila ly, he di ec con ibu ion
om he QE o he ex inc ion c oss sec ion is gi en by he e m
∝Im{pe}
in Eq.
(3.45e)
. In his scena io, he
qE
con ibu ion o he asymme y is no longe ze o,
and he ull exp ession o he o al asymme y ac o ,
q
=
qE
+
qR
, needs o be
98
3.4. Dissec ion o he asymme y
conside ed (Eq.
(3.36)
). On he o he hand,
qR
emains unchanged as compa ed
o he p e ious model.
As shown in Fig. 3.4c he esul ing
qE
domina es he o al asymme y ac o
q
,
and ollows simila ends wi h dis ance as hose desc ibed when discussing he
esul s o he exac calcula ion in Fig. 3.3.
qE
is small a long sepa a ion dis ances
(
d >
100 nm) because he QE and he nanoan enna s a o beha e independen ly,
and also a sho dis ances (
d≲
20 nm) because he di ec exci a ion o he QE is
e y small compa ed o he exci a ion ia he nanoan enna.
qE
is hus maximum
a in e media e dis ances wi hin his model (20 nm
≲d≲
80 nm) whe e he
exci a ion o he QE ia he nanoan enna is o he same o de o magni ude han
he di ec exci a ion by he inciden plane wa e. The dis ance ha maximizes
qE
ollows a linea dependence wi h inc easing adius
R
. Mo e p ecisely, he maxima
a e ound o an app oxima ely cons an dis ance be ween he QE and he cen e
o he nanoan enna (d+R). This beha io occu s because in his desc ip ion he
nea ields a e e alua ed using he quasis a ic G een’s unc ion, which only depends
on (
d
+
R
)
3
. Fu he , despi e he simila beha io o
qE
ob ained wi h his model,
and ha ob ained wi h he igo ous calcula ion (compa e Figs. 3.4c and 3.3b),
some di e ences s ill emain. In pa icula he la e decays mo e slowly wi h
dis ance, i changes i s sign as he dis ance inc eases, and he maximum o
qE
is
ound a a simila dis ance
d
o all adii. Mo eo e , he cu en model, gi en by
Eqs.
(3.45b)
-
(3.45e)
, is clea ly insu icien o ep oduce he exac
qR
con ibu ion
(compa e Figs. 3.4c and 3.3a).
Full e a ded G een’s unc ion
In o de o u he app oach he exac esponse o he in e ac ing sys em, we
eplace he quasis a ic nea - ield G een’s unc ion in Eqs.
(3.45a)
-
(3.45e)
wi h
he ull exp ession o he G een’s unc ion
Gx,x
in Eq.
(1.21)
[35].
Gqs
x,x
(
e
) =
1
/
[(2
π
)
ε0
(
R
+
d
)
3
](Eq.
(3.43)
) is always a eal numbe bu
Gx,x
is complex,
wi h a phase ha changes wi h dis ance
d
la gely due o he e a da ion phase
associa ed wi h he p opaga ion o he ields. Fu he mo e,
Gx,x
decays mo e slowly
han Gqs
x,x wi h dbecause i includes e ms decaying as 1/(R+d)and i/(R+d)2
(co esponding o he a - and in e media e- ield con ibu ions, espec i ely). These
changes di ec ly a ec he phase and he modulus o he enhancemen ac o
(
K
(
e, ω
) =
Gx,x
(
e, ω
)
pa
(
ω
)
/E0
) and hus bo h
qE
and
qR
(Eq.
(3.36)
), as shown
in Fig. 3.4d ( ou h column).
We i s obse e ha he dis ance-dependence o he ampli ude and phase o
K
induces he change o sign o
qE
o
d≈
150 nm (change om ed o blue colo ),
and also he o e all slowe decay o i s absolu e alue (
|qE|
) wi h
d
discussed abo e.
We show in sec ion 3.4.3 ha
qE
oscilla es o la ge sepa a ion dis ances. The
maxima alues o
qE
a e la ge han hose in he p e ious model, mainly due o
he a - and in e media e ield con ibu ions.
Fu he , we ob ain clea ly la ge alues o
|qR|
han in he p e ious model, wi h
alues o up o
|qR| ≈
0
.
45, as compa ed o
|qR|≲
0
.
14 in Fig. 3.4c. Acco ding
o Eq.
(3.37)
we can di ec ly ela e hese high alues o
qR
o changes o phase
99
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
o he ield enhancemen ,
φA
. When he ull G een’s unc ion
Gx,x
is used,
φA
can conside ably di e om
π/
2 o mode a e and la ge
d
, which explains he
ela i ely la ge alues o
|qR|
.
|qR|
is maximum o (
R
+
d
)
≈
100 nm and i decays
o la ge dis ances because he ield enhancemen becomes e y small and, hus,
|qR|∝|K( e, ω)|2/γ′(Eq. (3.37)) p og essi ely app oaches ze o.
The asymme y con ibu ions
qE
and
qR
ake simila absolu e alues o opposi e
signs a in e media e dis ances (20 nm
< d <
100 nm). As a consequence, he o al
asymme y
q
=
qE
+
qR
pa ially cancels in his egime, specially o adius 25 nm
≲R≲
70 nm. Thus,
q
p esen s a saddle poin cen e ed a
R≈
40 nm and
d≈
80
nm. The quali a i e dependence o
q
wi h adius and dis ance wi hin his model
is in good ag eemen wi h he igo ous Mie heo y esul s (Fig. 3.2d). We hus
conclude ha his imp o ed model con ains he undamen al elemen s o cap u e
he main ea u es o he beha io o he o al asymme y ac o .
Expe imen al pe mi i i y
We can u he inc ease he ag eemen wi h he esul s ob ained wi h he Mie
heo y calcula ions by using he same expe imen al alues
εExp
a
o he pe mi i i y
o sil e used in ha Mie calcula ions (ins ead o he modi ied D ude model).
The pola izabili y o he nanoan enna hen becomes
αRC-Exp
a
(
ω
) =
αExp
a
(
ω
)[1
/
(1
−
iαExp
a
(
ω
)
k3/
(6
πε0
))] wi h
αExp
a
(
ω
) = 4
πR3
(
εExp
a
(
ω
)
−
1)
/
(
εExp
a
(
ω
)+2). Figu e 3.4e
( i h column) shows he asymme y con ibu ions calcula ed wi h his assump ion.
The changes as compa ed wi h he p e ious model (Fig. 3.4d) a e ela i ely small
and a e mos ly ound o small sphe es (
R <
25 nm), whe e we ind an inc ease o
|qE|
and a dec ease o
|qR|
. Indeed, smalle sphe es esona e a sho e wa eleng hs,
o which he con ibu ion o he
d
-elec ons o he expe imen al pe mi i i y
signi ican ly modi ies he plasmonic esponse. The changes on he asymme y due
o he in luence o he
d
-elec ons can be la ge in o he ma e ials, such as gold. Fo
example, we show in sec ion 3.4.3 ha including his e ec is c ucial o accu a ely
desc ibe he asymme y ac o o a QE in e ac ing wi h a gold nanoan enna.
Imp o ed desc ip ion o he adia i e co ec ion
Las we in oduce a mo e accu a e desc ip ion o he adia i e–co ec ed
pola izabili y ollowing e e ence [47]:
αIRC
a(ω) = αExp
a(ω)
1−3
5ζ2εExp
a(ω)−2
εExp
a(ω)+2 −iαExp
a(ω)(2π/λ)3
6πε0−3ζ4
350
(εExp
a(ω))2−24εExp
a(ω)+16
εExp
a(ω)+2 (3.46)
wi h ζ= 2πR/λ.
We implemen his imp o emen o he dipole-dipole in e ac ion model, which
can be summa ized in a se o equa ions as:
pa(ω) = αIRC
a(ω)(E0+Gx,x( e, ω)pe(ω)),(3.47a)
100
3.4. Dissec ion o he asymme y
pe(ω) = αe(ω)(E0+K( e, ω)),(3.47b)
K( e, ω)E0=Gx,x( e, ω)pa(ω),(3.47c)
γ′=1 + ImAeαIRC
a(ω)
γ0ω0
(Gx,x( e, ω))2γ0,(3.47d)
σex (ω) = 2π
λε0
Impa(ω) + pe(ω)
E0.(3.47e)
Fo ease o e e ence, we summa ize all he aspec s ha a e included in Eqs.
(3.47a)
-
(3.47e)
bu no in Eqs.
(3.44a)
-
(3.44e)
( he la e co esponding o he
simples model conside ed in his sec ion): (i) he di ec exci a ion and emission o
he QE a e included in Eqs.
(3.47b)
and
(3.47e)
, espec i ely, (ii) he p opaga ion
o he ields beyond he quasis a ic app oxima ion is included by he ull G een’s
unc ion in Eqs.
(3.47a)
,
(3.47c)
, and
(3.47d)
, and (iii) we use a modi ied e sion
o he pola izabili y o he sphe ical nanopa icle
αIRC
a
in Eqs.
(3.47a)
and
(3.47d)
ha inco po a es he e ec o he adia ion damping o he nanoan enna and
conside s he in luence o d-elec ons on he pe mi i i y o he ma e ial.
Figu e 3.4 (six h column) shows ha by in oducing he imp o ed adia i e
co ec ion (Eq.
(3.46)
) he alues o he asymme y change e y li le, wi h he
la ges changes occu ing o
R >
50 nm (as compa ed o he esul s o he p e ious
model in Fig. 3.4e). In pa icula he maxima o
|qR|
and
qE
o
R >
50 nm ha e
been displaced in Fig. 3.4 owa ds sligh ly la ge dis ances
d
. The eason o his
displacemen is ha he new adia i e co ec ion edshi s he esonan wa eleng h
o la ge pa icles, which changes he a io be ween he QE-nanoan enna dis ance
and he wa eleng h, (R+d)/λ (a ec ing he ull G een’s unc ion Gx,x).
The esul ing alues o he o al asymme y ac o
q
and he
qR
and
qE
con ibu ions ha a e ob ained wi hin his imp o ed model (Fig. 3.4 ) a e in
e y good ag eemen wi h he exac esul s shown in Figs. 3.2d and 3.3a-b o
he adius and dis ances conside ed. The main di e ence occu s in he sho es
ange o dis ances,
d <
10 nm. Fo such dis ances he Mie heo y calcula ion
esul s in a e y la ge inc ease o he decay a e
γ′
due o he coupling wi h he
high-o de modes o he plasmonic esponse [134], which is no included in he
dipole-dipole desc ip ion analyzed he e. The la ge inc ease o
γ′
s ongly educes
he asymme y by inc easing he denomina o in Eq.
(3.36)
. Howe e , his dec ease
is ha d o app ecia e in he igu es, as he alue o
q
p edic ed by he mos e ined
dipole-dipole in e ac ion model (Eqs.
(3.47b)
-
(3.47e)
) is al eady small o hese
sho dis ances.
Equa ions
(3.47b)
-
(3.47e)
allow o a simple quan i a i e desc ip ion o he
o al asymme y ac o ha enables o iden i y he di e en e ec s ha in luence
he alue o
q
. Howe e , i is ins uc i e o u he analyze he impo ance o
he adia i e co ec ion. In he discussion abo e, he in oduc ion o he simple
adia i e co ec ion only led o a e y small change o he asymme y (compa e
Fig. 3.4a and Fig. 3.4b), bu his e ec is small only when conside ing e y simple
dipole-dipole in e ac ion models. I he adia i e co ec ion is neglec ed in he
inal exp essions (Eqs.
(3.47b)
-
(3.47e)
) we ob ain a comple ely inaccu a e esponse
101
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
o he o al asymme y ac o (and i s con ibu ions). This can be obse ed by
compa ing Fig. 3.4g (se en h column) wi h Fig. 3.4 , whe e he only di e ence
be ween he wo is ha Fig. 3.4g igno es he adia i e co ec ion. We ha e e i ied
ha including he adia i e co ec ion is necessa y o all he models ha use he
ull G een’s unc ion.
We ha e hus shown, in his subsec ion, how each app oxima ion in he dipole-
dipole in e ac ion a ec s he desc ip ion o he Fano asymme y o ze o-de uning.
This has allowed us o associa e he di e en aspec s o he asymme y wi h ele an
physical e ec s.
3.4.3 The Fano asymme y ac o in addi ional scena ios
In sec ions 3.2,3.3, and 3.4 we s udy he asymme y ac o o he Fano ea u e
p esen in he ex inc ion c oss-sec ion spec um o a QE placed a a dis ance
d∈
[2
,
200] nm om he su ace o a single sphe ical nanopa icle. In pa icula , we
s udied he si ua ion whe e he plasmonic mode o he nanoan enna and he exci on
o he QE a e esonan a e he same wa eleng h
λ0
, wi h
λ0
he wa eleng h ha
maximizes he ex inc ion c oss-sec ion o he ba e nanoan enna (in he spec al
egion whe e he dipola plasmonic esonance domina es he esponse). Howe e ,
some o he dependencies o he Fano asymme y p edic ed (such as he oscilla o y
beha io o
q
wi h he sepa a ion dis ance
d
o he in luence o he
d
-elec ons) a e
ha d o app ecia e in he si ua ions conside ed so a . The e o e, we explo e in his
subsec ion h ee addi ional cla i ying scena ios. Fi s , we conside a la ge ange
o dis ances
d∈
[2
,
1200] nm. Second, we change he ma e ial o he sphe ical
nanopa icle om sil e o gold. Las , o assess he obus ness o ou esul s,
we analyze he e ec o uning he esonan wa eleng h o he QE o ma ch he
maximum alue o he nea - ield enhancemen ac o
|K|
associa ed wi h he dipola
mode ins ead o he maximum o he c oss-sec ion.
The analysis p esen ed he e ollows he same p ocedu e as in subsec ion 3.4.2,
i.e. we calcula e he asymme y ac o
q
and i s con ibu ions
qR
and
qE
(Eq.
(3.36)
) wi h a se ies o simple models based on he dipole-dipole app oxima ion.
The simples model always p edic s an almos negligible asymme y ac o , and
we p og essi ely inco po a e di e en physical e ec s ha inc ease he accu acy
o he desc ip ion. Columns a- o Figs. 3.5,3.7, and 3.6 co espond o he same
models as hose de eloped o Fig. 3.4a- . We no e ha all he models in his
subsec ion do no include he quenching e ec due o he coupling wi h highe
o de plasmonic modes [134] ha s ongly dec ease q o d < 10 nm.
Longe ange o dis ances
So a we ha e ocused on he beha io o he asymme y o a ange o dis ances
d∈
[2
,
200] nm o he di e en dipole-dipole in e ac ion models. We show in
Fig. 3.5a- he alues o he asymme y ac o
q
and i s con ibu ions
qR
and
qE
calcula ed o a la ge ange o dis ances,
d∈
[2
,
1200] nm. Figu e 3.5 shows he
alues o
q
,
qR
, and
qE
using he mos accu a e dipole-dipole in e ac ion model
102
3.4. Dissec ion o he asymme y
No RC RC IRCRC RC
(a) (b) (c) (d) (e) ( )
RC
R R R R R R
Figu e 3.5: Values o he asymme y ac o calcula ed o a QE in he p oximi y o a sphe ical
sil e nanopa icle o di e en adius
R
. The sys em is iden ical as in Fig. 3.4, excep ha
he asymme y is calcula ed o a la ge ange o dis ances be ween he QE and he su ace o
he nanopa icle
d∈
[2
,
1200] nm. (a)-( ) Asymme y ac o
q
(panels in he hi d ow) and
i s con ibu ions
qE
( i s ow) and
qR
(second ow) calcula ed as a unc ion o
R
and
d
. Each
column co esponds o a di e en model, as indica ed by he labels a he bo om, ollowing he
same scheme as used in Fig. 3.4a- (see cap ion o ha igu e o u he de ails).
using he imp o ed desc ip ion o he adia i e co ec ion (Eqs.
(3.47a)
-
(3.47e)
).
Consis en ly wi h he discussion in subsec ion 3.4.2, as we inc ease
d
we can obse e
a clea oscilla o y beha io (supe imposed o a gene al endency o dec ease) o
q
,
qR
, and
qE
. The oscilla o y beha io causes changes o he sign o hese h ee
ac o s. Compa ing he esul s o he di e en models in Fig. 3.5 we ind ha he
oscilla ions appea when we include he ull G een’s unc ion
Gx,x
in ou model
(compa e Figs. 3.5c and d).
Las , we no e ha Fig. 3.5a (co esponding o he simples dipole-dipole
in e ac ion model) also shows a small bu clea ly non-ze o alue o asymme y,
q
= 0 o
d≲
100 nm, which was ha de o app ecia e in Fig. 3.4a because o he
chosen colo scheme.
Gold sphe ical nanopa icle
Figu e 3.6a- shows he analysis o he asymme y ac o
q
o he Fano ea u e in
he ex inc ion c oss-sec ion spec um and i s con ibu ions
qR
and
qE
when he
single sphe ical nanopa icle is made o gold. The QE esonan equency is again
se o he alue ha maximizes he ex inc ion c oss-sec ion spec um nea he
dipola esonance. Figu e 3.6 , co esponding o he mos p ecise dipole-dipole
in e ac ion model conside , shows ha
q
is ela i ely big o a la ge ange o
103
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
No RC RC IRCRC RC
(a) (b) (c) (d) (e) ( )
RC
R R R R R R
Figu e 3.6: Values o he asymme y ac o calcula ed o a QE in he p oximi y o a sphe ical
gold nanopa icle o di e en adius
R
o he nanopa icle and di e en dis ances
d
be ween he
QE and he su ace o he nanopa icle. (a)-( ) Asymme y ac o
q
(panels in he hi d ow) and
i s con ibu ions
qE
( i s ow) and
qR
(second ow) calcula ed as a unc ion o
R
and
d
. Each
column co esponds o a di e en model, as indica ed by he labels a he bo om, ollowing he
same scheme as used in Fig. 3.4a- (see cap ion o ha igu e o u he de ails).
dis ances,
q >
0
.
25 o
d≲
200 nm (whe eas in he sil e case
q >
0
.
25 o 20 nm
≲d≲
100 nm, see Fig. 3.4d and 3.4 ). Fu he , he dependence o
q
wi h
d
o he
gold nanopa icles shows a b oad single maximum and does no change i s sign in
he ange o dis ances conside ed. This is in con as wi h he sil e esul s, whe e
he e is a change o sign o
q
ollowing an oscilla o y pa e n. Addi ionally, o
he gold sphe ical nanopa icle (Fig. 3.6 )
|qE|
is o e all much la ge han
|qR|
, so
ha i domina es he dependence o he o al asymme y ac o
q
wi h
R
and
d
,
while in he sil e nanopa icles
|qE|
and
|qR|
a e o he same o de o magni ude
and bo h con ibu ions s ongly in luence he alues o
q
. Las , Fig. 3.6 shows
ha he
qR
con ibu ions akes la ge posi i e alues (
q >
0
.
25) o
d≲
50 nm
when conside ing gold as he plasmonic ma e ial, whe eas o sil e ,
qR
is o e all
nega i e o he same ange o dis ances (see Fig. 3.4 ).
The di e ences be ween he calcula ions o sil e and gold nanopa icles a e
mos ly due o he con ibu ion o he
d
-elec ons o he ma e ial o he dielec ic
pe mi i i y, which is much la ge o gold. This can be con i med by looking a Fig.
3.6d, which was ob ained using he modi ied D ude model o gold. Speci ically, we
use
εMD
a=ε∞−ω2
p
ω(ω+iκ),(3.48)
104
3.4. Dissec ion o he asymme y
IRC
No RC RC RC RC
(a) (b) (c) (d) (e) ( )
RC
R R R R R R
Figu e 3.7: Values o he asymme y ac o calcula ed o a QE in he p oximi y o a sphe ical
sil e nanopa icle o di e en adius
R
o he nanopa icle and di e en dis ances
d
be ween he
QE and he su ace o he nanopa icle. The asymme y ac o is calcula ed by uning he QE
esonance o ma ch he equency a which he nanoan enna nea - ield enhancemen is maximized
due o he dipola esonance o each pa icula se o
R
and
d
(in con as , in Fig. 3.4 he QE
esonance was ma ched o he ex inc ion maximum o he plasmonic dipola esonance). (a)-( )
Asymme y ac o
q
(panels in he hi d ow) and i s con ibu ions
qE
( i s ow) and
qR
(second
ow) calcula ed as a unc ion o
R
and
d
. Each column co esponds o a di e en model, as
indica ed by he labels a he bo om, ollowing he same scheme as used in Fig. 3.4a- (see
cap ion o ha igu e o u he de ails).
wi h
ε∞
= 9,
ℏωp
= 9
.
07eV, and
ℏκ
= 71 meV o gold. These alues a e
ob ained om i ing he expe imen al da a [40] o la ge
ω
. The dependence o
he asymme y ac o wi h
R
and
d
shown in Fig. 3.6d (modi ied D ude model)
a e e y simila o he esul s o he sil e nanosphe e (Fig. 3.4 ).
QE uned o he equency o he maximum ield enhancemen
All he esul s p esen ed in his chap e excep o Fig. 3.7 a e ob ained conside ing
ha he esonan equency o he QE ma ches he maximum o he ex inc ion
c oss-sec ion o he ba e nanoan enna. In Fig. 3.7a- we show he analysis o he
Fano asymme y in he ex inc ion c oss-sec ion spec um o he case whe e he
esonan equency o he QE ma ches he maximum o he ield enhancemen
induced by he nanoan enna a he posi ion o he QE (in bo h cases we conside
he maximum ha is mos ly de e mined by he dipola plasmonic mode).
O e all, he alues o
|q|
,
|qR|
, and
|qE|
a e sligh ly highe in Fig. 3.7a- han in
Fig. 3.4a- . Howe e , hese di e ences a e small and he ends o he dependence
o
q
,
qR
, and
qE
wi h he dis ance
d
and he adius o he nanopa icle
R
a e he
105
Chap e 3. Fano asymme y in ze o–de uned exci on–plasmon sys ems
same in Fig. 3.7a- and in Fig. 3.4a- .
3.5 Fano esonance in dime s
In he p e ious sec ions we ha e analyzed in de ail he asymme y o he Fano
lineshape ha is e ealed in he ex inc ion c oss sec ion spec um o a QE placed
nea a sphe ical me allic nanopa icle (Fig. 3.1a), chosen as an example o a
canonical nanoan enna. To demons a e ha a simila analysis can be applied
o mo e gene al nanos uc u es, we conside nex he Fano asymme y o a
QE si ua ed in a junc ion be ween wo sphe ical gold nanopa icles (a dime
nanoan enna). This dime con igu a ion has been in ensely s udied because i
induces a much la ge nea – ield enhancemen han he single sphe ical nanopa icle,
as sough , o example, in su ace-enhanced spec oscopy [25,26,146–151].
We show in Fig. 3.8a a scheme o he dime con igu a ion. The sys em is d i en
by an inciden plane wa e o ampli ude
E0
ha p opaga es along he
z
-axis, and
pola ized along he
x
-axis pa allel o he o ien a ion o he poin -like dipole ha
ep esen s he QE and o he axis o symme y o he wo sphe ical nanopa icles.
We conside gold ins ead o sil e nanopa icles in his sec ion. Despi e ha ing la ge
abso p ion losses, gold is widely used in su ace-enhanced spec oscopy because
i does no oxidize and i is mo e handleable in expe imen s. The pe mi i i y o
gold is aken om e e ence [41], he wo sphe ical nanopa icles ha e a adius o
R
= 40 nm, and we a y hei su ace- o-su ace dis ance 2
d
. The emi e is placed
in he middle o he gap be ween he wo nanopa icles (a dis ance
d
om he
su ace o each o hem), and i s p ope ies a e he same as in he p e ious sec ions
(s eng h
0
= 0
.
05
e·
nm, in insic decay a e co esponding o he spon aneous
adia i e decay, and esonance equency uned as a unc ion o
d
o always ma ch
he dipola esonance o he nanoan enna [152], as gi en by he maximum o he
ex inc ion c oss-sec ion), i.e., we keep he condi ion o ze o-de uning in all cases
analyzed and shown he e.
Fig. 3.8b shows he ex inc ion c oss-sec ion spec um o his hyb id sys em
calcula ed o di e en alues o
d
, as ob ained om Eq.
(3.8)
. In his sec ion, he
alue o all he necessa y inpu elec omagne ic pa ame e s (such as he nea - ield
enhancemen and he sel -in e ac ion G een’s unc ion) a e ob ained om he
solu ion o Maxwell’s equa ions unde plane-wa e o dipola illumina ion as gi en
by he Ma lab package MNPBEM17 [153
–
155] ( he de ails o hese calcula ions
a e gi en in appendix C). A clea Fano ea u e is obse ed in all spec a, showing a
quali a i ely simila dependence wi h dis ance as he esul s o he single sphe ical
nanopa icle (Fig. 3.1c). In bo h si ua ions he Fano lineshape ob ained a small
dis ances
d≲
10 nm co esponds o a b oadened and almos symme ic dip, while
a much la ge sepa a ion dis ances,
d≳
200 nm, we obse e an almos symme ic
na ow peak. Thus,
q≈
0in hese wo si ua ions. Fo alues o
d
be ween hese
wo ex emes, he Fano spec um shows a ious deg ees o asymme y.
Despi e hese quali a i e simila i ies, he esul s ob ained o he dime
nanoan enna (Fig. 3.8b) and a single sil e nanopa icle (Fig. 3.1c) show some
106
3.5. Fano esonance in dime s
Figu e 3.8: Cha ac e iza ion o he Fano asymme y in he ex inc ion c oss–sec ion o a QE
coupled o a me allic dime ob ained a ze o de uning ( esonan condi ions). (a) Scheme o he
dime nanos uc u e. A QE wi h dipole momen um pola ized along he
x
-axis is placed be ween
wo gold sphe ical nanopa icles o adius
R
= 40 nm a a dis ance
d
om he su ace o each o
hem ( he sepa a ion be ween he cen e o he wo nanopa icles is 2(
d
+
R
)). The dime axis is
pa allel o he
x
-axis. The sys em is illumina ed by a plane wa e p opaga ing along he
z
-axis
and wi h he elec ic ield pola ized along he
x
-axis. (b) No malized ex inc ion c oss-sec ion
spec a
σex /σ(0)
ex
o he coupled emi e -dime nanoan enna sys em. The spec a a e e ically
displaced by 1
.
5 o cla i y. Each Fano lineshape is e alua ed o di e en alues o
d
ha ange
om
d
= 2
.
5nm o
d
= 250 nm (see labels in he igu e). The spec a a e g ouped in h ee
sepa a e panels, each o hem plo ed o e a di e en spec al ange, ∆
λ
. (c) Dependence wi h
dis ance
d
o he Fano o al asymme y ac o
q
(blue line) oge he wi h i s con ibu ions
qE
(o ange line) and
qR
(g een line). Fo each sepa a ion dis ance
d
o he calcula ions in (b) and (c)
we ha e se he esonance o he QE o ma ch he equency o esonance o he nanoan enna.
clea quan i a i e di e ences. Fo ins ance, he Pu cell ac o
PF
and he pho onic
Lamb shi ∆
ω
expe ienced by he emi e , which desc ibe he b oadening and shi
o he Fano ea u e, espec i ely, a e much la ge in he case o he dime due o he
s onge ield con inemen [25,146,151] (
PF≈
1
.
4
×
10
4
and ∆
ω/γ ≈
3
.
2
×
10
5
o he
dime and
d
= 2
.
5nm, o be compa ed wi h
PF≈
4
.
3
×
10
2
and ∆
ω/γ ≈
4
.
4
×
10
3
o he single sil e sphe ical nanopa icle sys em o he same adius
R
and dis ance
d
). We also obse e ha he e is a clea asymme y o a la ge ange o dis ances
in he dime as compa ed o he single sil e nanoan enna o he same adius
(compa e he h ee panels in Fig. 3.8b and Fig. 3.1c)
To s udy he Fano asymme y o he dime sys em in mo e de ail, we show in
Fig. 3.8c he dependence wi h
d
o he o al asymme y ac o
q
(blue line) and i s
wo componen s
qE
(o ange line) and
qR
(g een line), as ob ained om Eq.
(3.36)
(wi h
q
=
qE
+
qR
). Fo sepa a ion dis ances
d≳
30 nm he o al asymme y ac o
q
is mainly in luenced by he
qE
con ibu ion, i.e. i is mos ly due o he di ec
exci a ion and emission o he QE. In a simila way as o he single sphe ical
nanopa icle,
qE
is la ge a in e media e dis ances (20 nm
≲d≲
200 nm o he
107
Chap e 4. Unbounded s ong bunching and b eakdown o he RWA in he QRM
se en elemen s o cha ac e ize he dynamics o he sys em in he mas e equa ion
in oduced in sec ion 2.6 (Eq.
(2.120)
): (i) he Hamil onian desc ibing he ene gy
o he CTS, (ii) he ope a o s desc ibing he losses o he ca i y, (iii) he decay a e
o he ca i y, (i ) he ope a o s desc ibing he losses o he TLS, ( ) he decay a e
o he TLS, ( i) he ope a o s desc ibing he incohe en exci a ion o he TLS, and
( ii) he pumping a e o he TLS. In subsec ions 4.2.1 and 4.2.2 we in oduce hese
se en elemen s in he comple e QRM and in he app oxima ed JCM, espec i ely.
4.2.1 Quan um Rabi model
In e ac ion and dynamics in he quan um Rabi model
In sec ion 2.5, we de i ed he QRM Hamil onian
ˆ
HC
desc ibing he ene gy in he
CTS o any a bi a y coupling s eng h
η
=
g/ω0
be ween he ca i y and he TLS.
Fo con enience, we epea he e he o mula o ˆ
HC, (Eq. (2.79)),
ˆ
HC=ℏω0ˆc†ˆc+ℏω0
2ˆσzcos[2η(ˆc+ ˆc†)] + ˆσysin[2η(ˆc+ ˆc†)],
whe e, again,
ˆc
is he annihila ion ope a o o he ca i y mode,
ω0
is he esonan
equency o he ca i y and o he TLS,
ˆσz
= [
σ†, σ
], and
ˆσy
=
i
(
ˆσ†−ˆσ
), being
ˆσ
he lowe ing ope a o o he TLS (see Eq.
(2.67)
). No e ha he
ˆ
HC
Quan um
Rabi Hamil onian does no conse e he exci a ion numbe , bu
ˆ
HC
conse es he
pa i y o exci a ions.
As men ioned abo e, he Hamil onian is he i s elemen ha we need o
add ess he dynamics o he s a e o ou sys em, desc ibed by he ime e olu ion o
he densi y ma ix o he CTS,
ˆρ
. Nex , we cha ac e ize he dissipa ion o he ca i y.
To do so, we use he d essed ope a o s o malism in oduced in subsec ion 2.6.2,
whe e he d essed ope a o s appea ed in he Lindblad e ms o he mas e equa ion
(Eqs.
(2.117)
and
(2.120)
). We use his same o malism applied o he
ˆ
Sˆc
=
i
(
ˆc†−ˆc
)
ope a o s desc ibing he in e ac ion be ween he ca i y and he en i onmen [85,86].
As in he de i a ion o he mas e equa ion in sec ion 2.6.2, i is con enien o
decompose
ˆ
Sˆc
in he basis o he eigens a es o he QRM Hamil onian (see also Eq.
(2.106)),
ˆ
Sˆc= ˆx(0)
ˆc+X
ω
[ˆxˆc(ω) + ˆx†
ˆc(ω)],(4.1)
whe e
ˆx(0)
ˆc=X
µ|µ⟩R R⟨µ|i(ˆc†−ˆc)|µ⟩R R⟨µ|,(4.2)
and
ˆxˆc(ω) = |ν⟩R R⟨ν|i(ˆc†−ˆc)|µ⟩R R⟨µ|,(4.3)
whe e
ω
=
ωµ−ων>
0, and he ke s
|ν⟩R
and
|µ⟩R
a e he eigen ec o s o he
QRM Hamil onian, and
ℏων>ℏωµ
a e hei espec i e eigen alues. We plo
∆
E
=
ℏ
(
ων−ωG
)(being
ℏωG
he eigenene gy o he CTS g ound s a e) as a
114
4.2. Ca i y- wo-le el-sys em Hamil onian models
(b) JCMQRM
(a)
R
R
R
R
RR
R
Figu e 4.2: Eigen alues o he CTS ob ained as unc ion o he coupling pa ame e
η
=
g/ω0
wi hin (a) he JCM and (b) he QRM. Fo each
|ν⟩
( o he JCM) o
|ν⟩R
( o he QRM)
eigens a e, we show ∆
E
: he di e ence be ween i s eigenene gy (
ℏων
) and he g ound s a e ene gy
(ℏωG).
unc ion o
η
in Fig. 4.2a. The no a ion o QRM eigens a es used h oughou
his chap e ,
|ν⟩R∈ {|0⟩R,|1−⟩R,|1+⟩R,|2−⟩R,|2+⟩R, . . . }
, is chosen o ecall
he JCM pola i ons (
|n±⟩
= (
|n, g⟩±|n−1, e⟩
)
/√2
), as he wo ma ch in he limi
o anishing coupling
g
. In Fig. 4.2a, we can obse e how he eigen alues o
|n+⟩R
(
|n−⟩R
) inc ease (dec ease) linea ly wi h
η
o
η≲
0
.
1. This same beha io is
ollowed by he JCM pola i ons (
|n±⟩
). Howe e , whe eas he JCM pola i ons keep
inc easing o dec easing linea ly wi h
η
o
η≳
0
.
1, he eigen alues o he QRM
show a mo e complex dependence wi h
η
, p esen ing c ossings and an ic ossings
wi h he eigen alues o o he eigens a es. No e ha h oughou his chap e we
keep he labeling o he eigens a es e en a e he eigen alues c ossing poin s (see
colo code in he igu e).
Each e m on he igh hand side o Eq.
(4.1)
ep esen s a di e en p ocess in
he in e ac ion o he sys em wi h i s en i onmen ;
ˆxˆc
desc ibes he dissipa ion ( he
ca i y gi es ene gy o he en i onmen ),
ˆx†
ˆc
desc ibes he incohe en pumping ( he
en i onmen gi es ene gy o he ca i y), and
ˆx(0)
ˆc
desc ibes he so-called dephasing
( he e is no di ec ene gy ans e be ween he ca i y and he en i onmen ) [21,72,86].
In he sys ems s udied in his hesis, we a e in e es ed in he dissipa ion o he
ca i y, desc ibed by he
ˆxˆc
e ms. These ope a o s a e hen used o build he
Dˆxˆc(ω)
dissipa ion Lindblad supe ope a o s acco ding o Eq.
(2.117)
, whe e each
dissipa ion supe ope a o has associa ed a equency-dependen dissipa ion a e,
γˆc
in he inal mas e equa ion (Eq.
(2.120)
). In gene al,
γˆc
can be s ongly dependen
on he ene gy
ℏω
o he ansi ions be ween he eigens a es. Howe e , in his
hesis, we conside simple dissipa ion mechanisms, whe e
γˆc
becomes cons an , i.e.,
γˆc
(
ω
)
→κ
, wi h
κ
being he classical plasmonic losses in oduced in sec ion 1.2,
when he ca i y is a plasmonic esona o . In he plasmonic CTS conside ed in his
chap e , he ca i y is cha ac e ized by esonan equency ω0(se o ℏω0= 1 eV),
and a low quali y ac o
Q
=
ω0/κ
, de ined by he dissipa ion a e
κ
, and se o
Q
= 20 ( hus
κ
= 50 meV). Thus, we only need a single Lindblad e m o desc ibe
he dissipa ion o he ca i y,
κDˆxˆc[ˆρ( )] = κ
22ˆxˆcˆρ( )ˆx†
ˆc−{ˆx†
ˆcˆxˆc,ˆρ( )}(4.4)
115
Chap e 4. Unbounded s ong bunching and b eakdown o he RWA in he QRM
wi h
ˆxˆc=X
ωµ>ων|ν⟩R R⟨ν|i(ˆc†−ˆc)|µ⟩R R⟨µ|.(4.5)
Nex , we use his same app oach o add ess he dissipa ion and incohe en
pumping o he TLS. We i s conside he ope a o
ˆ
Sˆσ
= (
ˆσ†
+
ˆσ
)desc ibing he
in e ac ion be ween he TLS and he en i onmen [85,86]. Then we decompose
ˆ
Sˆσ
in he basis o he QRM eigen alues:
ˆ
Sˆσ= ˆx(0)
ˆσ+X
ω
[ˆxˆσ(ω) + ˆx†
ˆσ(ω)],(4.6)
whe e,
ˆx(0)
ˆσ=X
µ|µ⟩R R⟨µ|(ˆσ†+ ˆσ)|µ⟩R R⟨µ|,(4.7)
and
ˆxˆσ(ω) = |ν⟩R R⟨ν|(ˆσ†+ ˆσ)|µ⟩R R⟨µ|,(4.8)
wi h
ω
=
ωµ−ων>
0(as in Eq.
(4.3)
).
ˆxˆσ
and
ˆx†
ˆσ
in Eq.
(4.6)
a e used in
Lindblad supe ope a o s o add ess he dissipa ion, and he incohe en pumping,
espec i ely [85,86]. Fo he CTS conside in his chap e , we conside equency-
independen dissipa ion and pumping a es o he TLS, so we use a single Lindblad
e m o desc ibe he losses o he TLS,
γDˆxˆσ[ˆρ( )] = γ
22ˆxˆσˆρ( )ˆx†
ˆσ−{ˆx†
ˆσˆxˆσ,ˆρ( )},(4.9)
and a single Lindblad e m o desc ibe he incohe en pumping o he TLS,
ΓDˆx†
ˆσ
[ˆρ( )] = Γ
22ˆx†
ˆσˆρ( )ˆxˆσ−{ˆxˆσˆx†
ˆσ,ˆρ( )}.(4.10)
He e
γ
and Γa e he decay and incohe en pumping a es o he TLS, espec i ely,
and
ˆxˆσ=X
ωµ>ων|ν⟩R R⟨ν|(ˆσ†+ ˆσ)|µ⟩R R⟨µ|.(4.11)
In pa icula , we chose a decay a e o he TLS
γ/ω0
= 10
−3
, negligible compa ed
o he decay a e o he ca i y
κ
. Thus, we can es ablish he uppe limi o he
WC egime as η=g/ω0≲κ/(2ω0)=0.025.
We now ha e all he elemen s o desc ibe he dynamical e olu ion o
ˆρ
as ollows
om he mas e equa ion in Eq. (2.120),
d
d ˆρ( ) = −i
ℏ[ˆ
HC,ˆρ( )] + κDˆxˆc[ˆρ( )] + γDˆxˆσ[ˆρ( )] + ΓDˆx†
ˆσ
[ˆρ( )].(4.12)
116
4.2. Ca i y- wo-le el-sys em Hamil onian models
Co ela ions in he quan um Rabi model
To cha ac e ize he s a is ical p ope ies o he emission o he CTS, we s udy
he in ensi y co ela ions o he emi ed pho ons,
g(2)
(
τ
= 0), as measu ed by
an HBT in e e ome e , in oduced in sec ion 2.4. The measu emen o a HBT
in e e ome e co esponds o,
g(2)(τ) = ⟨I1( +τ)I2( )⟩
⟨I1( +τ)⟩⟨I2( )⟩,(4.13)
whe e
I1
(
+
τ
)and
I2
(
)a e he pho ocu en s egis e ed by he wo de ec o s
o he HBT in e e ome e , and
τ
is he ime delay be ween he de ec ion e en s
(see Fig. 4.1a).
Fo ze o ime delay (
τ
= 0) and su icien ly la ge
(so he sys em eaches he
s eady s a e), his quan i y is ela ed o he s a is ics o pho ons inside he ca i y
as [21,85]:
g(2)(0) = ⟨ˆx†
ˆcˆx†
ˆcˆxˆcˆxˆc⟩ss
⟨ˆx†
ˆcˆxˆc⟩2
ss
.(4.14)
C ucially, wi hin he QRM,
g(2)
(0) depends on he
ˆxˆc
d essed ope a o s in oduced
in Eq.
(4.5)
, ensu ing Gauge-in a iance. On he o he hand,
⟨ˆ
O⟩ss
in he exp ession
o
g(2)
(0) deno es he expec a ion alue o ope a o
ˆ
O
in he s eady s a e (
ss
).
Then, by using Eq.
(4.12)
we can ob ain he s eady s a e o he sys em ia he
densi y ope a o ˆρss such ha ∂ ˆρss = 0.
This amewo k is used o calcula e he dependence o
g(2)
(0) (Eq.
(4.14)
) on
he coupling pa ame e
η
=
g/ω0
, plo ed in Fig. 4.1b as solid o ange and blue lines,
o he case o s ong (Γ
/γ
= 10), and weak (Γ
/γ
= 10
−6
) pumping, espec i ely.
All calcula ions o he e olu ion o he s eady s a e and expec a ion alues in he
sys em ha we ob ain in his chap e ha e been ca ied ou using he Py hon
package QuTiP [177,178]. We ha e conside ed in all o ou QuTiP calcula ions an
expansion o he Fock s a es o he ca i y up o
Nc
= 10, which we e i ied ensu es
con e gence.
4.2.2 Jaynes-Cummings model
In sec ion 2.5.4 we de i ed he JCM Hamil onian by aking wo app oxima ions.
Fi s , we expand he in e ac ion e m in he QRM Hamil onian as
ˆσzcos 2η(ˆc+ ˆc†)+ ˆσysin 2η(ˆc+ ˆc†)= ˆσz+ 2ηˆσy(ˆc+ ˆc†) + O(η2),(4.15)
and d op he nonlinea e ms in
η
. Nex , we in oduce he o a ing wa e
app oxima ion (RWA) by emo ing he so-called non-numbe -conse ing e ms
ˆσˆc
+
ˆσ†ˆc†
, o ob ain he JCM Hamil onian in Eq.
(2.81)
ha we ep oduce he e
o con enience,
ˆ
HJC =ℏω0ˆc†ˆc+ℏω0
2ˆσz+iℏg(ˆσ†ˆc−ˆc†ˆσ).
117
Chap e 4. Unbounded s ong bunching and b eakdown o he RWA in he QRM
Fu he , he JCM in oduces addi ional app oxima ions ega ding he emission,
dissipa ion, and abso p ion o he sys em.
Dynamics in he Jaynes Cummings model
In he JCM, he in e ac ion be ween he sys em and he en i onmen is simpli ied
by ea ing he dissipa ion o he ca i y and o he TLS as sepa a e elemen s.
Le us i s explo e he dissipa ion o he ca i y in he absence o he TLS. The
Hamil onian o he single ca i y mode is (see Eq. (2.53)),
ˆ
HCa i y =ℏω0ˆc†ˆc. (4.16)
The eigens a es o hese Hamil onians a e he Fock numbe s a es
|n⟩
(see sec ion
2.2), wi h eigen alues
ℏnω0
, (
n≥
0any in ege numbe ). The dissipa ion o he
ca i y is desc ibed using he same d essed ope a o s o malism in oduced in he
sec ion abo e (Eqs.
(4.1)
-
(4.5)
). Howe e , he d essed ope a o s a e buil using
he
|n⟩
eigens a es o he Hamil onian o he ba e ca i y. Thus, he
ˆxˆc
d essed
ope a o s in Eq. (4.5) now esul in
ˆxˆc→X
ωm>ωn|n⟩⟨n|i(ˆc†−ˆc)|m⟩⟨m|=−iX
n
√n|n−1⟩⟨n|=−iˆc, (4.17)
whe e, in he las iden i y, we ha e simply used he de ini ion o he
ˆc
annihila ion
ope a o s in Eq.
(2.18)
. Hence, he Lindblad dissipa ion e m o he ca i y becomes,
κDˆxˆc[ˆρ(JC)( )] →κD(−iˆc)[ˆρ(JC)( )] = κDˆc[ˆρ(JC)( )] =
=κ
22ˆcˆρ(JC)( )ˆc†−{ˆc†ˆc, ˆρ(JC)( )}.(4.18)
No e ha he e we use he label “(
JC
)” o di e en ia e he densi y ma ix desc ibing
he s a e in he QRM and ha in he JCM.
Simila ly, he dissipa ion o he TLS is desc ibed in he absence o he ca i y.
The Hamil onian o he ba e TLS is (see Eq. (2.69)),
ˆ
HTLS =ℏω0
2ˆσz,(4.19)
wi h eigens a es
|g⟩
and
|e⟩
, and eigen alues
−ω0/
2 o
|g⟩
and
ω0/
2 o
|e⟩
. As
o he case o he ca i y, he dissipa ion and incohe en exci a ion o he TLS is
desc ibed using he same d essed ope a o o malism in oduced o he QRM (Eqs.
(4.6)
-
(4.11)
). Howe e , he d essed ope a o s a e buil using only he decomposi ion
on he
|g⟩
and
|e⟩
eigens a es. Thus, he
ˆxˆσ
d essed ope a o s in Eq.
(4.11)
become,
ˆxˆσ→X
ωe>ωg|g⟩⟨g|(ˆσ†+ ˆσ)|e⟩⟨e|= ˆσ. (4.20)
Subs i u ing his exp ession o
ˆxˆσ
in o he dissipa ion and incohe en pumping
118
4.2. Ca i y- wo-le el-sys em Hamil onian models
e ms in Eqs. (4.9) and (4.10), we ob ain
γDˆxˆσ[ˆρ(JC)( )] →γDˆσ[ˆρ(JC)( )] = γ
22ˆσˆρ(JC)( )ˆσ†−{ˆσ†ˆσ, ˆρ(JC)( )},(4.21)
o he TLS dissipa ion e m, and
ΓDˆx†
ˆσ
[ˆρ(JC)( )] →ΓDˆσ†[ˆρ(JC)( )] = Γ
22ˆσ†ˆρ(JC)( )ˆσ−{ˆσˆσ†,ˆρ(JC)( )},(4.22)
o he TLS incohe en pumping e m.
In summa y, we can w i e he mas e equa ion wi h all he con ibu ions o he
sys em dynamics as in oduced he e, as:
d
d ˆρJC( ) = −i
ℏ[ˆ
HJC,ˆρ(JC)( )] + κDˆc[ˆρ(JC)( )] + γDˆσ[ˆρ(JC)( )] + ΓDˆσ†[ˆρ(JC)( )].
(4.23)
In ensi y co ela ions in he Jaynes Cummings model
As o he QRM (in Eq.
(4.14)
), he co ela ions a ising in he JCM a e also
ela ed o he s a is ics o pho ons inside o he ca i y and, can be exp essed in
e ms o he d essed ca i y ope a o s [21]. Howe e , he ope a o ha desc ibes
he pho ons o he ca i y co esponding o he d essed ope a o s in Eq.
(4.17)
educe o ˆxˆc→ −iˆc, and hus, in he JCM he in ensi y co ela ions become
g(2)
JC (0) = ⟨ˆc†ˆc†ˆcˆc⟩ss
⟨ˆc†ˆc⟩2
ss
.(4.24)
The expec ed alues in Eq.
(4.24)
a e ob ained wi h he s eady-s a e
ˆρ(JC)
ss
, which
we calcula e om he solu ion o
∂ ˆρ(JC)
ss
= 0 in he s anda d mas e equa ion
in oduced in Eq. (4.23).
Figu e 4.1b (dashed lines) shows he in ensi y co ela ions
g(2)
JC
(0) ob ained
applying Eq.
(4.24)
in he JCM o s ong (dashed o ange line) and weak (dashed
blue line) pumping. In he o me case, he JCM co ec ly ep oduces he esul s
o he exac QRM below he USC h eshold
η≲
0
.
1, bu comple ely ails o la ge
η
, whe e he
g(2)
JC
(0) ob ained wi h he JCM sa u a es,
g(2)
JC
(0)
≲
2
/
3, while, in he
exac QRM,
g(2)
(0) keeps inc easing s ongly wi h
η
. Fu he mo e, o he weak
incohe en pumping (blue lines), we ind signi ican di e ences be ween he JCM
and QRM e en in he WC egime ( he JCM and QRM di e o
η≳
5
×
10
−3
).
This unexpec ed b eakdown o he JCM o small
η
is discussed in de ail in sec ion
4.4.
119
Chap e 4. Unbounded s ong bunching and b eakdown o he RWA in he QRM
Analy ical limi o he co ela ions in he Jaynes-Cummings
Hamil onian
The JCM has been ex ensi ely used o desc ibe he p ope ies o weakly-coupled
CTSs [21]. In his chap e , i es ablishes a poin o compa ison o iden i y new
ea u es ha can a ise in he QRM. Fu he , we show nex ha he simpli ica ions
in oduced by he JCM allow us o ob ain an app oxima e analy ical exp ession o
g(2)
(0) in he weak-illumina ion case, whe e he incohe en pumping o he TLS
has a a e much lowe han he TLS losses, Γ≪γ.
As a i s s ep, we iden i y he minimum se o ope a o s o
which he mas e equa ions o m an almos closed sys em: =
ˆc†ˆc, ˆσ†ˆσ, ˆc†ˆσ, ˆcˆσ†,ˆc†ˆcˆσ†ˆσ, ˆc†ˆcˆcˆσ†,ˆc†ˆc†ˆcˆσ, ˆc†ˆc†ˆcˆcT
. The equa ions o mo ion o
he expec a ion alues o hese ope a o s can be app oxima ely exp essed as
d
d ⟨ ⟩=M⟨ ⟩+b,(4.25)
wi h b= (0,Γ,0,0,0,0,0,0)T, and
M=
−κ0−g−g0 0 0 0
0−γ g g 0 0 0 0
g−g−M10−g0 0 0
g−g0−M1−g0 0 0
Γ 0 0 0 −(γ+κ)g g 0
0 0 0 0 −2g−M20g
0 0 0 0 −2g0−M2g
0 0 0 0 0 −2g−2g−2κ
,(4.26)
whe e
M1
= (Γ +
κ
+
γ
)
/
2and
M2
= (Γ +
γ
+ 3
κ
)
/
2. To de i e Eqs.
(4.25)
-
(4.26)
we unca ed he se o equa ions by conside ing ha he pumping a e o he
sys em, Γ, is e y small. Thus, in he s eady-s a e, he TLS is mos ly in he g ound
s a e, and we app oxima e [179] (i)
⟨ˆσˆσ†⟩ ≈
1, (ii)
⟨ˆc†ˆc†ˆcˆcˆσˆσ†⟩≈⟨ˆc†ˆc†ˆcˆc⟩
, and (iii)
⟨ˆc†ˆc†ˆcˆcˆσ†ˆσ⟩ ≈ 0.
In he s eady s a e
∂ ⟨ ⟩ss
= 0, and we can de i e closed exp essions o
⟨ˆc†ˆc⟩ss
and ⟨ˆc†ˆc†ˆcˆc⟩ss.
⟨ˆc†ˆc⟩ss ≈ −4g2Γ(4g2γ+ 12g2κ+ Γγκ +γ2κ+ Γκ2+ 4γκ2+ 3κ3)×
×16g4Γγ−16g4γ2−64g4γκ + 4g2Γ2γκ −4g2Γγ2κ−8g2γ3κ−48g4κ2−
−8g2Γγκ2−36g2γ2κ2−Γ2γ2κ2−2Γγ3κ2−γ4κ2−4g2Γκ3−40g2γκ3−
−Γ2γκ3−6Γγ2κ3−5γ3κ3−12g2κ4−4Γγκ4−7γ2κ4−3γκ5−1,(4.27)
120
4.3. O igin o he bunching in ul as ongly coupled sys ems
⟨ˆc†ˆc†ˆcˆc⟩ss ≈32g4Γ2−16g4Γγ+ 16g4γ2+ 64g4γκ −4g2Γ2γκ + 4g2Γγ2κ+
+ 8g2γ3κ+ 48g4κ2+ 8g2Γγκ2+ 36g2γ2κ2+ Γ2γ2κ2+ 2Γγ3κ2+γ4κ2+
+ 4g2Γκ3+ 40g2γκ3+ Γ2γκ3+ 6Γγ2κ3+ 5γ3κ3+
+12g2κ4+ 4Γγκ4+ 7γ2κ4+ 3γκ5−1.(4.28)
In pa icula , we a e in e es ed only in he alues o
⟨ˆc†ˆc⟩ss
and
⟨ˆc†ˆc†ˆcˆc⟩ss
in
he wo opposi e limi s o
g≪
Γ( anishing coupling), and
g≫κ
(beyond he
SC egime). We ob ain an exp ession ha is alid in bo h limi s by e aining he
e ms dependen on he leading powe s o he wo ee pa ame e s, Γand
g
, in Eqs.
(4.27) and (4.28):
⟨ˆc†ˆc⟩ss ≈4g2(4g2+κ2)Γ
κ3(4g2+γκ)= Γ C
C+ 1
4g2+κ2
κ3,(4.29)
whe e C= 4g2/(κγ)is he coope a i i y, and
⟨ˆc†ˆc†ˆcˆc⟩ss ≈32g4Γ2
3κ2(16g4+γκ3).(4.30)
The in ensi y co ela ions a e ound, in he wo limi s o in e es , as
g(2)
JC (0) κ≫γ≫Γ≫g
−−−−−−−→ 2
3
γ
κ,(4.31)
g(2)
JC (0) g≫κ≫γ≫Γ
−−−−−−−→ 2
3.(4.32)
No e ha
g(2)
JC
(0) is de i ed he e by neglec ing he di ec emission om he TLS,
which is an in alid app oxima ion o e y small
g
in equa ion
(4.31)
(unless he
emission o he TLS is il e ed-ou ). I di ec emission o he TLS is included
(4.31)
should be modi ied, as i is clea es in he limi o
g
= 0. In his case, he emission
o he sys em is only gi en by he TLS, which emi s one pho on a a ime esul ing
in g(2)
JC = 0.
Impo an ly, bo h limi s in Eqs.
(4.31)
and
(4.32)
a e an ibunched, i.e., o all
coupling s eng hs, we expec he JCM o esul in an an ibunching signal. This is
con i med by he esul s in Fig. 4.1b (dashed blue line), which shows he in ensi y
co ela ions
g(2)
JC
(0) ob ained applying Eq.
(4.24)
o he JCM o weak incohe en
pumping. The esul s o his igu e also alida es he limi s in Eqs.
(4.31)
and
(4.32)
, wi h
g(2)
(0) = 1
.
33
×
10
−2
= (2
/
3)(
γ/κ
) o
η
= 10
−3
and
g(2)
(0) = 2
/
3 o
η= 1.
121
Chap e 4. Unbounded s ong bunching and b eakdown o he RWA in he QRM
exac
calcula ion
diagonal
app ox. R R
Figu e 4.3: In ensi y co ela ions as a unc ion o he coupling pa ame e
η
=
g/ω0
ob ained
wi hin se e al app oxima ions: exac alues o
g(2)
(0) (solid blue line); he diagonal app oxima ion
unca ed o he s a es
{|0⟩R,|1−⟩R,|1+⟩R,|2−⟩R,|2+⟩R,|3−⟩R}
in Eq.
(4.33)
(solid o ange
line); he diagonal app oxima ion conside ing only he R⟨3−| ˆxˆcˆxˆc|1−⟩R e m in he nume a o
o
g(2)
(0) in Eq.
(4.37)
(
|3−⟩R→ |1−⟩R
, solid g een line). The calcula ions shown in his igu e
a e ob ained wi hin he QRM o Γ/γ = 10−3.
4.3 O igin o he bunching in ul as ongly
coupled sys ems
In his sec ion, we demons a e ha he s ong bunching iden i ied in he USC
egime in Fig. 4.1b can be ela ed o he cha ac e is ics (decay pa hways and
popula ion) o he single
|3−⟩R
eigens a e. To jus i y he ocus on ha pa icula
eigens a e o he QRM Hamil onian, we plo in Fig. 4.3 he exac alues o
g(2)
(0)
(solid blue line) ob ained o an in e media e pumping Γ
/γ
= 10
−3
oge he wi h
app oxima ed esul s. We s a by app oxima ing he s eady-s a e densi y ma ix
as being diagonal in he basis o he |ν⟩Reigens a es o he QRM Hamil onian,
ˆρss ≈X
ν
Rν|ν⟩R R⟨ν|,(4.33)
whe e
Rν
is he popula ion o he
|ν⟩R
eigens a e. Thus, we can w i e he expec ed
alue o any ope a o ˆ
Oin he s eady s a e as:
⟨ˆ
O⟩ss =T {ˆ
Oˆρss} ≈ T (ˆ
O X
ν
Rν|ν⟩R R⟨ν|!)=X
ν
RνR⟨ν|ˆ
O|ν⟩R.(4.34)
We hen apply his o mula o he expec ed alue o
⟨(ˆx†
ˆc)n(ˆxˆc)n⟩ss
(
n
= 1 and
n= 2 o he nume a o and denomina o o g(2)(0), espec i ely), esul ing in:
⟨(ˆx†
ˆc)n(ˆxˆc)n⟩ss ≈X
ν
RνR⟨ν|(ˆx†
ˆc)n(ˆxˆc)n|ν⟩R=X
ν
RνR⟨ν|(ˆx†
ˆc)nˆ
I(ˆxˆc)n|ν⟩R,
(4.35)
122
4.3. O igin o he bunching in ul as ongly coupled sys ems
R0
R1R1 +
R2R2 +
R3
Figu e 4.4: Popula ions o he pola i onic s a es
Rν
(see labels) o a CTS calcula ed wi h he
QRM (solid lines) and JCM (dashed lines) as a unc ion o he coupling pa ame e
η
=
g/ω0
, o
he in e media e pumping Γ/γ = 10−3.
whe e we ha e included in he las s ep he iden i y ma ix
ˆ
I
. Because he eigens a es
o he QRM Hamil onian a e o hono mal we can w i e he iden i y ma ix as
ˆ
I=Pµ|µ⟩R R⟨µ|, and hus:
X
ν
RνR⟨ν|(ˆx†
ˆc)nˆ
I(ˆxˆc)n|ν⟩R=
=X
µ,ν
RνR⟨ν|(ˆx†
ˆc)n|µ⟩R R⟨µ|(ˆxˆc)n|ν⟩R=X
µ,ν
Rν|R⟨µ|(ˆxˆc)n|ν⟩R|2,(4.36)
whe e we ha e used he p ope y
⟨b|ˆ
O†|a⟩
= (
⟨a|ˆ
O|b⟩
)
∗
. Applying Eq.
(4.36)
o
n
= 1 (denomina o o
g(2)
(0) in Eq.
(4.14)
) and
n
= 2 (nume a o o
g(2)
(0)),
esul s in
g(2)(0) ≈Pν,µ Rν|R⟨µ|ˆxˆcˆxˆc|ν⟩R|2
Pν,µ Rν|R⟨µ|ˆxˆc|ν⟩R|22,(4.37)
The in ensi y co ela ions calcula ed by unca ing he double sum in he
nume a o up o
|3−⟩R
a e shown wi h he solid o ange line in Fig. 4.3 —
his app oxima ion gi es a e y good ag eemen wi h he exac calcula ion o
η≳
2
.
5
×
10
−2
; as we ha e nume ically e i ied, he de ia ion obse ed in he WC
egime
η≲
2
.
5
×
10
−2
is no due o he unca ion o he basis, bu a he due o
he e ec o he o -diagonal e ms o ˆρss.
We can u he app oxima e
g(2)
(0) by limi ing he double sum in Eq. (4.37)
o e νand µ o he |ν⟩=|3−⟩Rand |µ⟩=|1−⟩R e m:
g(2)(0) ≈R3−|R⟨1−|ˆxˆcˆxˆc|3−⟩R|2
Pν,µ Rν|R⟨µ|ˆxˆc|ν⟩R|22.(4.38)
This app oxima ion explo es he ole o he co ela ed wo-pho on emission om
|3−⟩R
o he
|1−⟩R
s a e. We show in Fig. 4.3 (g een line) he esul ing
g(2)
(0)
123
Chap e 4. Unbounded s ong bunching and b eakdown o he RWA in he QRM
in bo h conside ed pumping a es conside ed, bu hese di e ences would be likely
di icul o iden i y in expe imen al se ings. On he o he hand, as shown in Figs.
4.7 and 4.8b, he
g(2)
(0) ob ained o Γ
/γ
= 10
−3
and Γ
/γ
= 10
−6
show s ong
quali a i e di e ences o
η≳
2
.
5
×
10
−2
due o he di ec exci a ion pa hway o
he s a e
|3−⟩R
in oduced in sec ion 4.3. The lack o sensi i i y o
S(1)
(
ω
) o
his di ec exci a ion pa hway is due o he ac ha he emission om he lowe
|1−⟩R
and
|1+⟩R
s a es p edominan ly go e n he emission spec a o
η≲
0
.
3.
The exci a ion and emission om hese eigens a es a e no impac ed by he di ec
exci a ion mechanism discussed abo e, and do no lead o he b eakdown o he
RWA. We hus conclude ha he cha ac e iza ion o he co ela ions is a mo e
powe ul ool han measu ing he one-pho on emission spec a o he iden i ica ion
o phenomena caused by he non-numbe -conse ing e ms o he QRM Hamil onian
o coupling below he adi ional USC h eshold η≈0.1.
4.5 Conclusions
In his chap e , we analyze he s a is ics o he emission om a gene ic quan um
sys em comp ising an incohe en ly d i en wo-le el emi e in e ac ing wi h a ca i y.
We iden i y he eme gence o unbounded bunching as he sys em app oaches he
USC egime. By exp essing he dynamics o he sys em in he basis o he pola i onic
eigens a es o he quan um Rabi Hamil onian, we can a ibu e he bunching o
he singula beha io o he indi idual eigens a e
|3−⟩R
, which (i) decays h ough
a co ela ed wo-pho on emission, and (ii) is e y s ongly popula ed by a new,
di ec exci a ion mechanism om he g ound s a e.
We show ha in ensi y co ela ions
g(2)
(0) a e a much mo e sensi i e ool o
obse ing he phenomena induced by he non-numbe -conse ing e ms in he
QRM, han he one-pho on emission spec a. Indeed, we ind ha he in ensi y
co ela ions can iden i y a b eakdown o he o a ing wa e app oxima ion a below
he con en ional limi o he USC, wi h he exac limi de e mined by he a e o
incohe en pumping.
These indings call o expe imen al e i ica ion, and u he heo e ical s udies,
o e i y he obus ness o he iden i ied exci a ion and emission mechanisms. Ou
model can be ex ended o accoun o mo e complex decay dynamics and ene gy
s uc u e o he quan um emi e , in ol ing da k exci onic s a es, o pu e dephasing,
as well as he in e ac ion wi h a s uc u ed ese oi .
130
Chap e
5
PRESERVATION AND DESTRUCTION
OF THE PURITY OF TWO-PHOTON
STATES IN THE INTERACTION WITH
A NANOSCATTERER
5.1 In oduc ion
Classical ligh beams can ca y well-de ined angula momen um, as desc ibed in
chap e 1. Simila ly, quan um s a es o ligh (in oduced in chap e 2) can also
ca y well-de ined angula momen um. This aspec opens up a a ie y o po en ial
bene i s o quan um echnologies. Fo ins ance, he angula momen um o quan um
s a es o ligh can be used o encode in o ma ion [15
–
18]. This is pa icula ly
use ul o quan um in o ma ion applica ions as quan um s a es o ligh a e e y
esilien o lose hei en anglemen and pu i y du ing p opaga ion [180
–
182]. Thus,
he s a es o ligh wi h well-de ined angula momen um a e ideal candida es as
quan um in o ma ion ca ie s.
On he o he hand, pho ons do no in e ac s ongly wi h ma e ial pa icles and
s uc u es, limi ing he possibili ies o p ocessing pho onic quan um in o ma ion
[183,184]. Se e al echniques a e being de eloped in o de o enhance pho on
in e ac ions, such as he use o quan um op omechanical in e ac ion, op ical
me ama e ials, high-densi y gases, slow-ligh ma e ials and se e al o he s [185
–
189].
Enginee ing nanopho onic nanos uc u es o quan um in o ma ion p ocessing
o e s he possibili y o, no only enhancing ligh -ma e in e ac ions, bu also
manipula ing ligh in de ices wi h a oo p in o he o de o he wa eleng h. While
he use o nanos uc u es and he s udy o hei op ical esonances o enhance
131
Chap e 5. Loss o pu i y in he sca e ing o wo-pho on en angled s a es
he classical in e ac ion o ligh and ma e has a long adi ion [49,111,190], a
o mal s udy o he e ec o he in e ac ion o quan um s a es o ligh wi h such
nanos uc u es is s ill needed.
In his chap e , we p o ide a amewo k o s udy he in e ac ion be ween
quan um s a es o ligh wi h well de ined angula momen um and a nanos uc u e.
Ou app oach is gene al, bu we ocus on an expe imen ally ele an si ua ion: he
sca e ing o wo-pho on s a es o ligh by a o a ionally symme ic nanos uc u e.
C ucially, he o a ional symme y imposes he conse a ion o he o al angula
momen um
m
=
l
+
s
, whe e
l
and
s
ep esen he o bi al and spin angula
momen um, espec i ely. Thus, hese nanos uc u es allow o he manipula ion o
s a es o ligh wi h a ixed and well-de ined o al angula momen um in a con olled
manne [20,191].
5.2 Inpu and ou pu s a es
The heo e ical amewo k used o desc ibe he quan um sca e ing p ocess is based
on an inpu /ou pu gene al o malism [78,79,191,192]. We conside ha he inpu
and ou pu s a es o he sys em a e quan um s a es o ligh composed by wo
en angled pho ons, whe e he wo pho ons ha e o al angula momen um
m
= 0,
and he in o ma ion is encoded in hei helici y Λ(de ined as he p ojec ion o
he spin ope a o
←→
S
on he di ec ion o p opaga ion, see Eq.
(1.84)
), which akes
Λ = +1 o Λ =
−
1 alues (see sec ion 1.4) [191]. We conside a basis o ou inpu
wo-pho on modes ha comple ely desc ibes any inpu monoch oma ic wo-pho on
s a es:
|ψi
±(ω1, ω2)⟩=1
2nˆa†
i(ω1)ˆa†
i(ω2)±ˆ
b†
i(ω1)ˆ
b†
i(ω2)o|0⟩,(5.1a)
|χi
±(ω1, ω2)⟩=1
2nˆa†
i(ω1)ˆ
b†
i(ω2)±ˆ
b†
i(ω1)ˆa†
i(ω2)o|0⟩,(5.1b)
whe e
|0⟩
is he acuum s a e, and
ω1
and
ω2
a e he equencies o he wo-pho ons.
The basis o he wo-pho on ou pu monoch oma ic modes also has ou elemen s,
|ψo
±(ω1, ω2)⟩
and
|χo
±(ω1, ω2)⟩
, whcih ollow Eqs.
(5.1a)
and
(5.1b)
, espec i ely,
bu he inpu “i” labels a e subs i u ed by he ou pu “o” labels,
|ψo
±(ω1, ω2)⟩=1
2nˆa†
o(ω1)ˆa†
o(ω2)±ˆ
b†
o(ω1)ˆ
b†
o(ω2)o|0⟩,(5.2a)
|χo
±(ω1, ω2)⟩=1
2nˆa†
o(ω1)ˆ
b†
o(ω2)±ˆ
b†
o(ω1)ˆa†
o(ω2)o|0⟩.(5.2b)
The modes o ligh a e desc ibed by he inpu (ou pu )
ˆa†
i(o)
(
ω
)and
ˆ
b†
i(o)
(
ω
)ope a o s
ha indica e he c ea ion o an inpu (ou pu ) pho on wi h helici y Λ = +1 o
Λ =
−
1, espec i ely. The p ope ies o he c ea ion and annihila ion ope a o s
desc ibing he quan iza ion o ligh we e discussed in sec ion 2.2. No ably, he
ˆa†
i(o)
(
ω
)and
ˆ
b†
i(o)
(
ω
)bosonic ope a o s sa is y he canonical commu a ion ela ions
(Eq. (2.21)) and ope a e on a single equency ω.
132
5.2. Inpu and ou pu s a es
Recen expe imen s ha e measu ed a deg ada ion o inpu quan um s a es
(quan i ied below by means o he loss o pu i y) a e sca e ing o a nanos uc u e
[191]. In pa icula , in hese (and o he simila ) expe imen s, he inciden pho on
pai s a e gene a ed in a supe posi ion o s a es o di e en equencies, ypically by a
s anda d spon aneous pa ame ic down-con e sion (SPDC) p ocess. In his chap e ,
we analyze he mechanism by which he sca e ing o hese non-monoch oma ic
s a es can esul in a loss o pu i y. Wi h his pu pose, we i s ocus on he
sca e ing o s a es ha a e equency supe posi ions o he monoch oma ic wo-
pho on s a e
|ψi
+(ω1, ω2)⟩
(in sec ion 5.3.3 we s udy he sca e ing o he es o
he elemen s in he basis),
|Ψi
+⟩=¨dω1dω2ϕ(ω1, ω2)|ψi
+(ω1, ω2)⟩.(5.3)
whe e
ϕ
(
ω1, ω2
)is he wo-pho on spec al unc ion ha we app oxima e as he
p oduc o wo Gaussian unc ions bo h cen e ed a he cen al equency
ωin
(o
cen al wa eleng h
λin
= 2
πc/ωin
) and a iance
σ
= 3 THz ( his alue is chosen o
be simila o he one used in ecen expe imen s [193,194]),
ϕ(ω1, ω2) = 1
σ√πexp−(ω1−ωin)2
2σ2exp−(ω2−ωin)2
2σ2.(5.4)
No e ha his exp ession o
ϕ
(
ω1, ω2
)implies ha he wo pho ons a e
indis inguishable, since ϕ(ω1, ω2) = ϕ(ω2, ω1).
Expe imen ally-accesible densi y ma ix
In o de o app oach a ealis ic expe imen al cha ac e iza ion o quan um s a es,
we conside ha he sca e ed s a es a e measu ed h ough a pos -selec ion o wo-
pho on s a es, whe e he sca e ed s a es wi h less han wo pho ons a e igno ed.
We also conside in he ollowing ha he de ec o s a e “blind” o he equency
deg ee o eedom. Then, he inpu and ou pu quan um s a es a e bes desc ibed
wi h he expe imen ally-accessible pos -selec ed densi y ma ix,
ˆϱ
, esul ing om
acing ou he equency deg ee o eedom. Thus, he elemen s
⟨ξ|ˆϱ|ξ′⟩
o he
densi y ma ix co esponds o he esul s o s anda d quan um s a e omog aphy
measu emen s [74,195],
⟨ξ|ˆϱi(o)|ξ′⟩=K¨dω1dω2⟨ξ(ω1, ω2)|Ψi(o)
+⟩⟨Ψi(o)
+|ξ′(ω1, ω2)⟩,(5.5)
whe e
K
is a no maliza ion cons an ha ensu es
T {ˆϱi(o)}
= 1.
|ξ(ω1, ω2)⟩
and
|ξ′(ω1, ω2)⟩
can be any o he
|ψi(o)
±(ω1, ω2)⟩
and
|χi(o)
±(ω1, ω2)⟩
s a es gi en in
Eqs.
(5.1a)
and
(5.1b)
, espec i ely. Fo example, Fig. 5.1 shows he densi y
ma ix
ˆϱi
o he inciden s a e
|Ψi
+⟩
(Eq.
(5.3)
) calcula ed using Eq.
(5.5)
.
ˆϱi
is cha ac e ized by a single non-ze o elemen co esponding o
⟨ψ+|ˆϱi|ψ+⟩
, and
he e is no con ibu ion om he o he elemen s o he basis (Eqs.
(5.1a)
and
133
Chap e 5. Loss o pu i y in he sca e ing o wo-pho on en angled s a es
ψ+χ+ψ−χ−
ψ+
χ+
ψ−
χ−
0.00
0.25
0.50
0.75
1.00
Re{ i}
ψ+χ+ψ−χ−
ψ+
χ+
ψ−
χ−
0.00
0.25
0.50
0.75
Im{ i}
Inpu s a e
Figu e 5.1: Real (le ) and imagina y ( igh ) componen s o
ˆϱi
, he densi y ma ix associa ed o
he inpu s a e
|Ψi
+⟩
. The wo-pho on spec al unc ion is cen e ed a
ωin
= 17
.
5
×
10
14
ad/s
and has a a iance o σ= 3 THz. Li= 0 indica es ha he inpu s a e is pu e.
(5.1b)
). The loss o pu i y o such a quan um s a e can hen be quan i ied using
Li(o)
= 1
−T {
(
ˆϱi(o)
)
2}
(in oduced in Eq.
(2.3)
), whe e
Li(o)
= 0 indica es a pu e
s a e and
Li(o)>
0a mixed s a e. The inpu densi y ma ix
ˆϱi
in Fig. 5.1 sa is ies
Li= 1 −T {(ˆϱi)2}= 0, which con i ms ha ˆϱiis a pu e s a e [74].
5.3 Quan um ans o ma ion
We nex discuss how o analyze he loss o pu i y due o he sca e ing o
m
= 0
pho ons by a o a ionally symme ic nanos uc u e. As discussed in sec ion 1.4.3,
o a ionally symme ic s uc u es conse e he o al angula momen um o he
inciden ligh . Howe e , he conse a ion o he o al angula momen um does
no imply he conse a ion o he ec o ial deg ee o eedom o ligh , which is
de e mined by he helici y Λ. Since he s a es o ligh s udied in his chap e
a e de e mined by
m
and Λ(see sec ion 1.4.2), he inpu elec omagne ic modes
wi h
m
= 0 and Λ = +1 (o Λ =
−
1) can only be sca e ed in o wo di e en
ou pu elec omagne ic modes wi h
m
= 0 (due o
m
conse a ion) and Λ = +1
o Λ =
−
1. Fu he , we conside ha pho ons can be los o dissipa ed in he
sca e ing p ocess. This si ua ion whe e wo inpu elec omagne ic modes a e ei he
los o ans o med in o wo o he ou pu elec omagne ic modes is analogous o
he si ua ion p oduced in a lossy beam spli e [78,79,192]. Thus, we can di ec ly
adap he ans o ma ion o lossy beam spli e s ha was in oduced in sec ion
2.3 (Eq.
(2.29)
) o ou sys em, esul ing in he ollowing equa ions connec ing he
ou pu and inpu annihila ion ope a o s:
ˆao(ω) = α+1(ω)ˆai(ω) + β+1(ω)ˆ
bi(ω) + ˆ
L+1(ω),
ˆ
bo(ω) = α−1(ω)ˆ
bi(ω) + β−1(ω)ˆai(ω) + ˆ
L−1(ω),(5.6)
whe e
ˆ
L+1
and
ˆ
L−1
a e he Lange in ope a o s accoun ing o he losses in he
sca e ing p ocess (see sec ion 2.3) [78,80,81], and
α+1
,
α−1
,
β+1
, and
β−1
a e he
helici y-spli ing coe icien s ha a e ully desc ibed in he nex subsec ion.
134
5.3. Quan um ans o ma ion
5.3.1 Helici y-spli ing coe icien s
Equa ion
(5.6)
desc ibes he sca e ing o quan um s a es o ligh , bu he
α+1
,
α−1
,
β+1
, and
β−1
coe icien s can be calcula ed om he classical esponse o
he sys em as ob ained om Maxwell’s equa ions because Maxwell’s equa ions
de e mine how he elec omagne ic modes ge ans o med, bo h in he classical
and quan um egimes [64,196]. Thus, o ob ain he helici y-spli ing coe icien s,
we conside a classical sca e ing p oblem whe e inciden ligh beam wi h
m
= 0
is sca e ed by he nanos uc u e. The sca e ed ligh is sepa a ed by i s helici y
con ibu ions, and each helici y con ibu ion is de ec ed sepa a ely. Fo example,
in his chap e , we conside ha he de ec ion is done by coupling each helici y
con ibu ion o he sca e ed ield o a single-mode ibe connec ed o a de ec o .
In pa icula , we conside wo classical inpu beams wi h a helici y o ei he
Λ = +1 o Λ =
−
1and o al angula momen um
m
= 0. The elec ic ields o hese
inpu beams a e ep esen ed by
Ei
+1
and
Ei
−1
, espec i ely. We hen calcula e he
sca e ed ields,
Esca
+1
(
Esca
−1
), when he nanos uc u e is illumina ed by he inpu
beam
Ei
+1
(
Ei
−1
). The helici y-spli ing coe icien s a e de e mined by p ojec ing
he sca e ed ield in o wo classical ou pu beams,
Eo
+1
and
Eo
−1
, which also ha e
an angula momen um o
m
= 0 and a helici y o Λ = +1 and Λ =
−
1, espec i ely:
α+1(ω) = ¨A
dA[Eo
+1( , ω)]∗·Esca
+1 ( , ω),
α−1(ω) = ¨A
dA[Eo
+1( , ω)]∗·Esca
−1( , ω),
β+1(ω) = ¨A
dA[Eo
−1( , ω)]∗·Esca
+1 ( , ω),
β−1(ω) = ¨A
dA[Eo
−1( , ω)]∗·Esca
−1( , ω).
(5.7)
This ope a ion co esponds o calcula ing he coupling be ween he sca e ed ields
and a single-mode ibe connec ed o he de ec o , and Ais he a ea o he ibe .
In his chap e , we conside a simple de ec ion scheme (discussed in sec ion 5.4)
whe e
α+1
(
ω
) =
α−1
(
ω
) =
α
(
ω
)and
β+1
(
ω
) =
β−1
(
ω
) =
β
(
ω
). In his case, Eq.
(5.6) simpli ies o
ˆao(ω) = α(ω)ˆai(ω) + β(ω)ˆ
bi(ω) + ˆ
L+1(ω),
ˆ
bo(ω) = α(ω)ˆ
bi(ω) + β(ω)ˆai(ω) + ˆ
L−1(ω).(5.8)
5.3.2 Ou pu |Ψo
+⟩s a e
By conside ing he simpli ied ans o ma ion in Eq.
(5.8)
we can ob ain a he ou pu
s a e
|Ψo
+⟩
om he p ojec ion o he inpu s a e,
|Ψi
+⟩
, on all he wo-pho on
135
Chap e 5. Loss o pu i y in he sca e ing o wo-pho on en angled s a es
nanos uc u e
Ro a ionally
symme ic
s a e
Inpu
s a e
Ou pu
Figu e 5.2: Scheme o he sca e ing p ocess. The
|Ψi
+⟩
inpu s a e is sca e ed as a supe posi ion
o |ψo
+⟩and |χo
+⟩wi h ampli udes gi en by Cψϕand Cχϕ, espec i ely
s a es o he ou pu basis:
|Ψo
+⟩=h|ψo
+(ω3, ω4)⟩⟨ψo
+(ω3, ω4)|+
+|ψo
−(ω3, ω4)⟩⟨ψo
−(ω3, ω4)|+|χo
+(ω3, ω4)⟩⟨χo
+(ω3, ω4)|+
+|χo
−(ω3, ω4)⟩⟨χo
−(ω3, ω4)|i·¨dω1dω2ϕ(ω1, ω2)|ψi
+(ω1, ω2)⟩.(5.9)
To e alua e Eq.
(5.9)
, we i s subs i u e he exp essions o he
ˆao
(
ω
)and
ˆ
bo
(
ω
)ope a o s in Eq.
(5.8)
in o he exp essions o
⟨ψo
+(ω1, ω2)|
,
⟨ψo
−(ω1, ω2)|
,
⟨χo
+(ω1, ω2)|
, and
⟨χo
−(ω1, ω2)|
in Eqs.
(5.2a)
and
(5.2b)
,i.e., we exp ess he
ou pu basis b a s a es in e ms o he inpu ope a o s. Then we pe o m he
p ojec ion o each ou pu b a s a e on o he
|ψi
+(ω1, ω2)⟩
s a e. A e some algeb aic
manipula ion, Eq. (5.9) becomes:
|Ψo
+⟩=¨dω1dω2ϕ(ω1, ω2)Cψ(ω1, ω2)|ψo
+(ω1, ω2)⟩+Cχ(ω1, ω2)|χo
+(ω1, ω2)⟩,
(5.10)
wi h Cψ(ω1, ω2)and Cχ(ω1, ω2)de ined as:
Cψ(ω1, ω2) = α(ω1)α(ω2) + β(ω1)β(ω2),(5.11a)
Cχ(ω1, ω2) = α(ω1)β(ω2) + β(ω1)α(ω2).(5.11b)
Equa ion
(5.10)
shows ha , in gene al, he ou pu s a e is a supe posi ion
o wo di e en en angled pho on modes,
|ψo
+(ω1, ω2)⟩
and
|χo
+(ω1, ω2)⟩
. Each
ou pu mode has a di e en ampli ude,
Cψ
(
ω1, ω2
)
ϕ
(
ω1, ω2
) o
|ψo
+(ω1, ω2)⟩
and
Cχ
(
ω1, ω2
)
ϕ
(
ω1, ω2
) o
|χo
+(ω1, ω2)⟩
, as we show in he scheme o igu e 5.2. The
Cψ
(
ω1, ω2
)and
Cχ
(
ω1, ω2
)coe icien s in Eqs.
(5.11a)
and
(5.11b)
may ha e
a s ong equency dependence due o apid spec al changes o he
α
and
β
coe icien s, g ea ly a ec ing he pu i y o he ou pu s a e. To illus a e he
o malism, we a i icially se
β
(
ω
)=0
.
2and
α
(
ω
) =
ωLγ/
[2(
ω2
L−ω2
+
iγω
)] a
Lo en zian unc ion ha mimics he esonan beha io o a nanos uc u e. Figu e
136
5.3. Quan um ans o ma ion
ψ+χ+ψ−χ−
ψ+
χ+
ψ−
χ−
0.2
0.0
0.2
0.4
0.6
Re{ o}
ψ+χ+ψ−χ−
ψ+
χ+
ψ−
χ−
0.2
0.0
0.2
0.4
0.6
Im{ o}
Ou pu s a e
Figu e 5.3: Real (le ) and imagina y ( igh ) componen s o
ˆϱo
, he pos -selec ed densi y ma ix
o he ou pu s a e
|Ψo
+⟩
ha esul s om he sca e ing o he inciden inpu s a e
|Ψi
+⟩
in Fig.
5.1. Fo he calcula ion, we chose ha he helici y spli ing coe icien
β
(
ω
)=0
.
2, and
α
(
ω
)is a
Lo en zian unc ion wi h
ωL
= 17
.
5
×
10
14
ad/s and
γ
= 1 THz.
Lo≈
0
.
4
>
0indica es ha
he ou pu s a e is no pu e
5.3 shows he ou pu densi y ma ix
ˆϱo
calcula ed using Eqs.
(5.5)
and
(5.10)
-
(5.11b)
, o hese
α
and
β
helici y-spli ing coe icien s. The
ˆϱo
ob ained o he
ou pu s a e ep esen s a pa ially cohe en supe posi ion o
|ψo
+(ω1, ω2)⟩
and
|χo
+(ω1, ω2)⟩
(as indica ed by Eq.
(5.10)
). The pu i y o his s a e is
Lo≈
0
.
4
>
0
(i.e., he ou pu s a e is mixed). Thus, his simple example demons a es ha
he pu i y o he inciden quan um s a e can be los in he in e ac ion wi h a
nanos uc u e.
O igin o he loss o pu i y unde he quasi–monoch oma ic
app oxima ion
To iden i y he o igin o his loss o pu i y, we conside ha he inpu wo-pho on
spec al unc ion,
ϕ
(
ω1, ω2
)is quasi-monoch oma ic (i.e., i s spec al a iance
σ
is signi ican ly smalle han he cen al equency o he pulse,
ωin
). Unde his
app oxima ion, we i s expand he
α
and
β
coe icien s o i s o de a ound he
cen al equency o he wo-pho on spec al unc ion ωin:
α(ω)≈A1 + A′
A∆ω,(5.12)
β(ω)≈B1 + B′
B∆ω,(5.13)
wi h
A
=
α
(
ωin
),
B
=
β
(
ωin
),
A′
=
dα
(
ω
)
/dω|ωin
,
B′
=
dβ
(
ω
)
/dω|ωin
, and
∆ω=ω−ωin.
Using Eqs. (5.12) and (5.13) we can w i e Eq. (5.11a) and (5.11b) as:
Cψ(ω1, ω2)≈(A2+B2)[1 + (∆ω1+ ∆ω2)(Fψ+iτψ)],(5.14)
Cχ(ω1, ω2)≈2AB[1 + (∆ω1+ ∆ω2)(Fχ+iτχ)],(5.15)
137
Chap e 5. Loss o pu i y in he sca e ing o wo-pho on en angled s a es
wi h
Fψ=1
|A|4+|B|4+ 2|A|2|B|2cos(2δ)|A|′
|A||A|4+|B|′
|B||B|4+
|A|2|B|2cos(2δ)|A|′
A+|B|′
B+ sin(2δ)(a g{A}′−a g{B}′),(5.16)
τψ=1
|A|4+|B|4+ 2|A|2|B|2cos(2δ)a g{A}′|A|4+ a g{B}′|B|4+
|A|2|B|2cos(2δ)(a g{A}′+ a g{B}′) + sin(2δ)|B|′
|B|−|A|′
|A|,(5.17)
Fχ=1
2|A|′
|A|+|B|′
|B|,(5.18)
τχ=1
2(a g{A}′+ a g{B}′).(5.19)
In Eqs.
(5.16)
-
(5.19)
we ha e in oduced
a g{A}′
=
da g{α
(
ω
)
}/dω|ωin
,
a g{B}′
=
da g{β
(
ω
)
}/dω|ωin
,
δ
=
a g{B} − a g{A}
,
|A|′
=
d|α
(
ω
)
|/dω|ωin
, and
|B|′
=
d|β
(
ω
)
|/dω|ωin
. We u he make he app oxima ion 1 +
x
∆
ω≈ex∆ω
in Eqs.
(5.14) and (5.15), which gi es
Cψ(ω1, ω2)≈(A2+B2)exp[(∆ω1+ ∆ω2)(Fψ+iτψ)],(5.20)
Cχ(ω1, ω2)≈2AB exp[(∆ω1+ ∆ω2)(Fχ+iτχ)].(5.21)
Then, he
ϕ
(
ω1, ω2
)
Cψ
(
ω1, ω2
)and
ϕ
(
ω1, ω2
)
Cχ
(
ω1, ω2
) unc ions (Eq.
(5.10)
)
esul in:
ϕ(ω1, ω2)Cψ(ω1, ω2)≈
≈Aψexp−[ω1−(ωin +σ2FΨ)]2
2σ2+iω1τψexp−[ω2−(ωin +σ2FΨ)]2
2σ2+iω2τψ,
(5.22)
ϕ(ω1, ω2)Cχ(ω1, ω2)≈
≈Aχexp−[ω1−(ωin +σ2Fχ)]2
2σ2+iω1τχexp−[ω2−(ωin +σ2Fχ)]2
2σ2+iω2τχ.
(5.23)
No e ha
AΨ
,
FΨ
,
τΨ
,
Aχ
,
Fχ
, and
τχ
depend only on he classical esponse o
he sys em e alua ed a he cen al equency o he inciden pulse, ωin.
Equa ions
(5.22)
and
(5.23)
show ha he ou pu wo-pho on modes can be
ep esen ed as wo dis inc wo-pho on pulses. Each wo-pho on pulse can be
ac o ized as wo Gaussian pulses, one o each pho on. Howe e , he pulses
138
5.3. Quan um ans o ma ion
associa ed wi h he
|ψo
+(ω1, ω2)⟩
and
|χo
+(ω1, ω2)⟩
s a es ha e di e en p ope ies.
The pulse in Eq.
(5.22)
has an ampli ude
Aψ
, a common cen al equency o bo h
pho ons (
ωin
+
σ2Fψ
), and a cen al ime
τψ
(iden i ied om he
iω1τψ
and
iω2τψ
e ms in Eq.
(5.22)
). Simila ly, he pulse in Eq.
(5.23)
has an ampli ude
Aχ
, a
cen al equency (
ωin
+
σ2Fχ
), and a cen al ime
τχ
. This implies ha a e he
in e ac ion wi h he nanos uc u e, he esul ing quan um s a e is a supe posi ion
o wo di e en quan um s a es wi h di e en ime- equency p ope ies. As we
show nex , he loss o pu i y in he ou pu s a e can be a ibu ed o he ime
delay and equency shi be ween he ou pu pulses.
Analy ical exp ession o he loss o pu i y
Using Eqs.
(5.5)
,
(5.10)
,
(5.22)
, and
(5.23)
we ob ain an analy ical exp ession o
he pu i y o he ou pu s a e in he quasi-monoch oma ic app oxima ion,
Lo=2|AΨ|2|Aχ|2eσ2(F2
χ+F2
Ψ)
(|AΨ|2e2σ2F2
Ψ+|Aχ|2e2σ2F2
χ)2[eσ2∆F2−e−σ2∆τ2],(5.24)
wi h ∆
F
=
FΨ−Fχ
and ∆
τ
=
τΨ−τχ
. This equa ion indica es ha he ou pu
s a e is pu e unde any o he ollowing condi ions:
•
I
σ
= 0, co esponding o a pu ely monoch oma ic inciden s a e. In his
case, he ou pu s a e is pu e and consis s o a supe posi ion o wo di e en
s a es wi h ampli udes AΨand Aχ(Eqs. (5.10)-(5.23)).
•
I
β
(
ωin
) = 0, co esponding o he condi ion o helici y p ese a ion [27].
In his case, he ou pu s a e becomes a equency supe posi ion o only
|ψo
+(ω1, ω2)⟩s a es.
•
I
α
(
ωin
) = 0, co esponding o he condi ion o o al con e sion o helici y [27].
In his case, he ou pu s a e also becomes a equency supe posi ion o only
|ψo
+(ω1, ω2)⟩s a es.
•
I
α
(
ωin
) =
±β
(
ωin
), co esponding o he si ua ion whe e he inciden ligh
only exci es ei he magne ic o elec ic modes o he nanos uc u e [27,29,197].
In his case, he ou pu s a e is a supe posi ion o ou pu
|ψo
+(ω1, ω2)⟩
and
|χo
+(ω1, ω2)⟩
s a es wi h he same ampli ude, and he ime delays and
equency shi s be ween he ou pu pulses become ze o (∆
τ
= 0 and ∆
F
= 0).
This si ua ion is discussed in mo e de ail a he end o sec ion 5.4.2.
•
I
α
(
ω
)and
β
(
ω
)a e almos cons an in a spec al ange gi en by
σ
. F om
Eqs. (5.16)-(5.19) we ind ha his case leads o ∆F≈0and ∆τ≈0.
F om he condi ions abo e, we can expec a subs an ial loss o pu i y i an
inciden non-monoch oma ic pulse (
σ
= 0) exci es bo h elec ic and magne ic
esonances o he nanos uc u e ( hus,
α
(
ωin
)
=
±β
(
ωin
)
= 0), while
α
(
ω
)and
β(ω)change ab up ly nea he illumina ion equency.
139
