NOVEL REGIMES OF QUANTUM OPTOMECHANICS
lukas neumeie
PhD Thesis
Thesis supe iso : P o . Da ick E. Chang
ICFO-The Ins i u e o Pho onic Sciences
Uni e s i a Poli ècnica de Ca alunya
Ap il 2018 – Ba celona
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Lukas Neumeie : No el egimes o quan um op omechanics, PhD
Thesis, © Ap il 2018
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Fü Valen in
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ABSTRACT
In e e yday li e he impac o ligh on he mo ion o mechani-
cal objec s is negligible. Howe e , mode n expe imen s making
use o high quali y op ical esona o s a e able o obse e signi i-
can e ec s o igina ing om he o ces associa ed wi h pho ons
on small mechanical sys ems. The common ea u e o hese sys-
ems is he dependence o he op ical esonance equency on
he posi ion o he mechanical objec , laying he amewo k o
op omechanics. Many in e es ing egimes ha e been explo ed
which allow o pho on-ligh en anglemen , lase cooling o mo-
ion, gene a ion o squeezed s a es o ligh , and e en he de ec-
ion o g a i a ional wa es. In e es ingly, he op omechanical in-
e ac ion is so gene ic ha i s unde lying concep s and de i ed
insigh s can be gene ally applied o a la ge a ie y o sys ems,
as we will see in his hesis.
In Chap e 1, we p o ide a b ie o e iew o key concep s
and esul s om he ield o op omechanics, be o e going on o
discuss he no el egimes and applica ions ha we ha e iden i-
ied and p oposed.
In Chap e 2, we heo e ically in es iga e esul s om a cou-
ple o expe imen s, ha we e p e iously no well-unde s ood.
These expe imen s ap dielec ic nano-pa icles h ough an op-
ical esona o mode and obse e ha he in ensi ies expe i-
enced by he pa icles a e s ongly educed compa ed o a con-
en ional op ical weeze ap. We ind ha hese sys ems can
be ully desc ibed by a simple op omechanical oy model and
de i e ha he op ical po en ial inside esona o s can app oach
a nea ly pe ec squa e well. This po en ial can be dynamically
eshaped by changing he d i ing lase equency and we ind
a d ama ic educ ion o in ensi ies seen by he apped pa i-
cle, which could signi ican ly inc ease he ange o sys ems o
which op ical apping can be applied. These esul s a e qui e
ema kable and should ha e impo an implica ions o u u e
apping echnologies.
In Chap e 3, we ecognize ha a majo end wi hin he ield
o ca i y QED is o a ain he s ong coupling egime. Addi-
ional ich dynamics can occu by conside ing he a omic mo-
ional deg ee o eedom. In pa icula , we show ha such a
sys em is a na u al candida e o explo e he single-pho on op-
omechanical s ong coupling egime o quan um op omechan-
ics, bu whe e he mo ional equency canno be esol ed by
he ca i y. We show ha his egime can esul in a numbe o
ema kable phenomena, such as s ong en anglemen be ween
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he a omic wa e- unc ion and he sca e ing p ope ies o sin-
gle inciden pho ons, o an anomalous hea ing mechanism o
a omic mo ion.
In Chap e 4we show ha an a om apped in and cou-
pled o a ca i y cons i u es an a ac i e pla o m o ealiz-
ing he op omechanical single-pho on s ong coupling egime
wi h esol ed mechanical sidebands. Realizing his egime is
a majo goal wi hin he ield o op omechanics, as i would
enable he de e minis ic gene a ion o non-classical s a es o
ligh . Howe e , his egime is di icul o achie e wi h con en-
ional mechanical sys ems due o hei small ze o-poin mo-
ions. As an example, we show ha op omechanically-induced
pho on blockade can be ealized in ealis ic se ups, whe ein
non-classical ligh is gene a ed due o he in e ac ion o pho-
ons wi h he a omic mo ion alone.
RESUMEN
En la ida co idiana, el impac o de la luz sob e el mo imien-
o de los obje os mecánicos es insigni ican e. Sin emba go, los
expe imen os mode nos que usan esonado es óp icos de al a
calidad son capaces de obse a e ec os signi ica i os que se
o iginan de las ue zas asociadas con los o ones en pequeños
sis emas mecánicos. La ca ac e ís ica común de es os sis emas
es la dependencia de la ecuencia de esonancia óp ica en la
posición del obje o mecánico, que es ablece el campo de la op-
omecánica. Se han explo ado muchos egímenes in e esan es
que pe mi en el en elazamien o de o ones, el en iamien o del
mo imien o po láse , la gene ación de es ados de luz comp i-
midos e incluso la de ección de ondas g a i acionales. Cu io-
samen e, la in e acción op omecánica es an gené ica que sus
concep os subyacen es y sus p o undas consecuencias pueden
aplica se gene almen e a una g an a iedad de sis emas, como
e emos en es a esis.
En el Capí ulo 1, p opo cionamos una b e e desc ipción de
los p incipales concep os y esul ados del campo de la op ome-
cánica, an es de pasa a analiza los nue os egímenes y aplica-
ciones que hemos iden i icado y p opues o.
En el Capí ulo 2, in es igamos eó icamen e los esul ados de
un pa de expe imen os que an es no se en endían bien. Es os
expe imen os a apan nanopa ículas dieléc icas a a és de un
modo de un esonado óp ico y obse an que las in ensidades
i
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expe imen adas po las pa ículas se educen conside ablemen-
e en compa ación con una ampa de pinzas óp icas con encio-
nal. Encon amos que es os sis emas se pueden desc ibi com-
ple amen e median e un modelo op omecánico de jugue e sim-
ple y demos amos que el po encial óp ico den o de los eso-
nado es puede ap oxima se a un pozo cuad ado casi pe ec o.
Es e po encial se puede modi ica dinámicamen e cambiando
la ecuencia de en ada del láse y encon amos una educción
d ás ica de las in ensidades is as po la pa ícula a apada, lo
que pod ía aumen a signi ica i amen e el ango de sis emas
a los que se puede aplica el a apamien o óp ico. Es os esul-
ados son bas an e no ables y debe ían ene implicaciones im-
po an es pa a las u u as ecnologías de a apamien o.
En el Capí ulo 3, econocemos que una endencia impo an e
en el campo de la elec odinámica cuán ica de ca idades (del
inglés, ça i y QED") es log a un égimen de acoplamien o ue -
e. Se pueden p oduci dinámicas adicionales al conside a el
g ado de libe ad de mo imien o a ómico. En pa icula , mos-
amos que dicho sis ema es un candida o na u al pa a explo a
el égimen de acoplamien o ue e op omecánico de un único
o ón en op omecánica cuán ica, pe o donde la ecuencia de
mo imien o no puede se esuel a po la ca idad. Mos amos
que es e égimen puede da luga a una se ie de enómenos
no ables, como un ue e en elazamien o en e la unción de
onda a ómica y las p opiedades de dispe sión de los o ones
inciden es indi iduales, o un mecanismo de calen amien o anó-
malo del mo imien o a ómico.
En el Capí ulo 4mos amos que un á omo a apado y aco-
plado a una ca idad cons i uye una pla a o ma a ac i a pa a
ob ene el égimen de acoplamien o ue e op omecánico con
un único o ón y con bandas la e ales mecánicas esuel as. La
ob ención de es e égimen es un obje i o p incipal en el campo
de la op omecánica, ya que pe mi i ía la gene ación de e mi-
nis a de es ados de luz no clásicos. Sin emba go, es e égimen
es di ícil de log a con los sis emas mecánicos con encionales
debido a sus pequeños mo imien os de pun o ce o. Como ejem-
plo, mos amos que el bloqueo de o ones inducido de o ma
mecánica puede ealiza se en con igu aciones ealis as, donde
la luz no clásica se gene a solamen e debido a la in e acción de
o ones con el mo imien o a ómico.
ii
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PUBLICATIONS
1. Neumeie , L. Quidan , R & Chang, D. E. Sel -induced
back-ac ion op ical apping in nanopho onic sys ems, New
J. Phys. 17,123008 (2015).
2. Neumeie & Chang, D. E. Explo ing un esol ed sideband,
op omechanical s ong coupling using a single a om cou-
pled o a ca i y: soon
3. Neumeie , L, No hup T. E & Chang, D. E. Reaching he
op omechanical s ong coupling egime wi h a single a om
in a ca i y, a Xi :1711.09619 (2017)
The esul s o he i s publica ion a e included in Chap e 2
and hose o he second and hi d publica ion a e included in
Chap e 3and 4, espec i ely.
ix
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1
INTRODUCTION
1.1 o ces o ligh
A pho on walks in o a ho el and he ecep ionis asks "Hi! Can
we help you wi h you luggage?" And he pho on esponds:
"No hanks, I am a eling ligh !"
So wha can we lea n om his joke? Pa icles o ligh a e
called pho ons, which we can coun . O he han being coun -
able hey also change he eloci y o hings hey hi . The o ce
a ising om con inuous hi ing is known as he adia ion p es-
su e o ce [1]. Due o he la ge mass o mac oscopic mechanical
objec s, he e ec o his o ce applied by single pho ons is in-
c edibly weak. As an example, when a single pho on e lec s o
a sma phone a es , he eloci y o ha sma phone a e he
in e ac ion is abou one a om-size pe age o he uni e se. How-
e e , i we e lec a pho on on a single a om a es , he a om has
a eloci y o oughly one sma phone leng h pe second, which
is a decen e ec a he single pho on le el and some hing o
keep in mind. Op ical o ces ha e many applica ions anging
om physics o li e sciences. Fo example hey a e exploi ed
o sola sails [2], o cooling a oms [3] and o op ical weeze s
ha can ap and mo e small pa icles a ound [4].
The e ec s o op ical o ces a e mos easily seen wi h la ge
lase in ensi ies, due o he small e ec ha a single pho on
ypically has. One way o inc ease he e ec o op ical o ces,
wi hou inc easing he inciden in ensi y, is o u ilize an op ical
esona o (ca i y). An example o an op ical ca i y wi h leng h
Lis shown in Figu e 1.1a). We assume ha he ca i y is d i en
by a cohe en lase d i e wi h equency ωLand numbe lux
E2
0 h ough he le mi o . The ca i y suppo s op ical modes
wi h equencies ωc=2πc/λ wi h possible wa e eleng h obey-
ing m·λ/2 =L,mbeing any posi i e in ege numbe . Fo
mos se ups i is su icien o only conside a single op ical
mode, ha o which he equency ωcis closes o he lase
equency. He e, o simplici y we assume equal mi o s wi h
a decay a e o κ/2 each, and igno e in insic losses. Resonan
pho ons (ωL=ωc) bounce back and o h be ween he mi o s
many imes be o e hey decay wi h a e κ. Thus, he numbe o
esonan pho ons inside he ca i y is p opo ional o nc∝E2
0/κ,
as shown in Fig. 1.1b), whe e we plo he numbe o in a-ca i y
pho ons ncas a unc ion o lase equency. I becomes ob ious
3
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4 in oduc ion
Figu e 1.1:a) Illus a ion o an op ical ca i y wi h leng h Lc ea ing
a s anding wa e op ical mode wi h wa eleng h λco e-
sponding o a esonance equency ωc=2πc/λ. The ca -
i y consis s o wo equal mi o s each ha ing a decay a e
o κ/2 and is cohe en ly d i en wi h a lase o equency
ωL h ough he le mi o .
b) The numbe o pho ons ncinside he ca i y (qual-
i a i ely) as a unc ion o lase equency ωL o ms a
Lo en zian cen e ed a ound he ca i y equency (ωL=
ωc) wi h wid h κand a maximum alue o nc∝1/κ.
ha good mi o s (small κ) can lead o a huge build up o ligh
in ensi y inside he ca i y. This allows enhanced op ical o ces
Fop ∝ncon objec s apped inside he ca i y (and on he ca i y
mi o s hemsel es).
The idea o using ca i ies o enhance op ical o ces (o many
o he e ec s in ol ing ligh ) is qui e old. Howe e , in he pas
en yea s, he ield o “op omechanics” has seen explosi e g ow h.
A a b oad le el, his ield aims o obse e and exploi in e es -
ing dynamical e ec s ha can occu , when op ical ca i y o ces
and he mo ion hey induce modi y he p ope ies o he ca i y
i sel . A simple model whe e such e ec s can be unde s ood is
illus a ed in Fig. 1.2a), whe e now one o he ca i y mi o s is
moun ed on a sp ing and allowed o mo e.
Fo an emp y ca i y, he mi o has an equilib ium posi ion
x0and he ca i y a leng h Lde e mining i s esonance equency
ωc(x0). In Fig. 1.2b) we u n on an ex e nal lase d i e popula -
ing he ca i y wi h pho ons. Fo la ge nc, he balance o op ical
o ces and he es o ing o ce o he sp ing esul s in a new equi-
lib ium posi ion ¯x0, which inc eases he leng h o he ca i y and
esul s in a lowe esonance equency ωc(¯x0). This enables an
in e es ing dynamic: The esonance equency o he ca i y de-
pends on he posi ion o he mi o , he posi ion o he mi o
depends on he numbe o pho ons inside he ca i y and he
numbe o pho ons depends again on he esonance equency
o he ca i y. The dynamics a ising om his in e play can gi e
ise o ema kable e ec s. Pe haps mos no ably, i enables an
incoming lase o ex ac ene gy om a mo ional deg ee o ee-
dom, he eby educing i s e ec i e empe a u e [5,6].
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1.2 s anda d egimes o op omechanics 5
Figu e 1.2:Illus a ion o he s anda d op omechanical se up.
a) An emp y ca i y wi h leng h L, whe e he igh mi o
is a ached o a sp ing ep esen ing a ib a ional mode.
The ca i y equency ωc(x0)depends on i s equilib ium
posi ion x0.
b) A cohe en ly d i en ca i y. The adia ion p essu e o ce
on he mi o s is p opo ional o he numbe o pho ons
ncinside he ca i y. Thus, many pho ons inside he ca -
i y push he igh mi o o a new equilib ium posi ion
¯x0, which inc eases he ca i y leng h > L and as a conse-
quence educes i s esonance equency o ωc(¯x0).
I u ns ou ha a simple and “s anda d” physical model un-
de lying he sys em illus a ed in Fig. 1.2can equally apply o
a b oad class o sys ems ha con ain coupled op ical and me-
chanical esonances. This p o ides a la ge numbe o ways in
which op omechanical e ec s can be obse ed and exploi ed
(see Sec. 1.2). A he same ime, in all sys ems explo ed hus
a , a single pho on s ill has a e y weak op omechanical e -
ec , which necessi a es ha a la ge numbe o pho ons a e sen
in. Wi hin his con ex , he b oad ques ions his hesis aims o
answe can be summa ized in wo bulle poin s:
• Can one, inspi ed by he concep s o op omechanics, ind
new applica ions o iden i y new phenomena in sys ems,
which go beyond he “s anda d” op omechanical model?
• Can we ind new sys ems, whe e he in e ac ion be ween
indi idual pho ons and mo ion becomes e y s ong, c e-
a ing a new playg ound o explo e op omechanical phe-
nomena in he quan um egime?
Be o e we answe hese ques ions in Chap e s 2-4, we will p o-
ide a basic in oduc ion in o he heo y o ca i y op omechan-
ics.
1.2 s anda d egimes o op omechanics
He e, we will in oduce he s anda d egimes o op omechan-
ics, which ha e been bo h heo e ically analyzed and expe i-
men ally obse ed. As hin ed by Fig. 1.2, a minimal model o
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6 in oduc ion
op omechanical in e ac ions in ol es a single op ical and me-
chanical deg ee o eedom, and whe e he op ical esonance
depends on he posi ion o he mechanical sys em. A co e-
sponding Hamil onian hus eads [7]:
Hom =ωmb†b+ωc(x)a†a. (1.1)
We use aand bas he annihila ion ope a o s o pho ons and
phonons in he op ical and mechanical modes, espec i ely, and
ωmis he equency o he mechanical mode. Fo simplici y
we neglec mechanical damping. ωc(x)desc ibes he posi ion-
dependen ca i y esonance equency. Fo mally, we can ex-
pand he esonance equency in powe s o he displacemen
a ound some equilib ium posi ion x0,
ωc(x) = ωc(x0) + ω0
c(x0)(x−x0) + .... (1.2)
Gi en he na u ally weak o ce associa ed wi h ligh , he cou-
pled mechanical deg ee o eedom is displaced by hese o ces
by ypically in ini esimal dis ances. This mo i a es expanding
he esonance equency o he ca i y only up o linea displace-
men s in Eq. (1.2) which de ines he op omechanical in e ac ion
as gi en by
HI=ω0
c(x0)(x−x0)a†a=gm(b+b†)a†a. (1.3)
He e, we ha e e-w i en he displacemen in e ms o he un-
damen al c ea ion and annihila ion ope a o s, x−x0=xzp(b†+
b), whe e xzp =p
h/(2mωm)is he quan um mechanical un-
ce ain y associa ed wi h mo ion, which dec eases wi h he e -
ec i e mass mo he mi o . The single-pho on, single-phonon
op omechanical coupling s eng h is de ined by
gm≡ω0
c(x0)xzp. (1.4)
E en hough we ha e linea ized he displacemen o he me-
chanical mo ion, he op omechanical in e ac ion Eq. (1.3) s ill
gi es ise o non-linea equa ions o mo ion due o he p od-
uc o h ee ope a o s in he Hamil onian. Wi hou losses and
he mal e ec s, s a ing om a classical (e.g., cohe en ) s a e,
he in e ac ion could e en ually cause he s a e o become non-
classical. This is in e es ing o a numbe o easons; o exam-
ple, i migh be ha op omechanical sys ems could be used o
gene a e and manipula e non-classical s a es o ligh o quan-
um in o ma ion p ocessing. Howe e , as he bes demons a ed
a io o coupling s eng h o ca i y linewid h hus a is gm/κ ∼
10−2[8,9], such quan um e ec s a e oo small o be obse ed.
To in ui i ely mo i a e he op omechanical Hamil onian (1.1)
we conside ed he simple pic u e o a mo ing mi o a ached
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1.2 s anda d egimes o op omechanics 7
o a sp ing. Real op omechanical se ups seldom look ha way
as he op omechanical desc ip ion o a posi ion-dependen op i-
cal esonance equency is qui e gene ic and success ully mod-
els a wide ange o di e en sys ems. Examples o ecen ly de-
eloped op omechanical geome ies a e shown in Fig. 1.3, wi h
mechanical equencies anging om Hz o GHz, and masses
anging om kilog ams o sub-picog ams. The la ges op ome-
chanical s uc u e and also he mos sensi i e o mechanical
displacemen o da e is he g a i a ional wa e de ec o (LIGO),
which can esol e a change in leng h o less han 1/10000 he
size o a p o on. The leng h o i s in e e ome e a ms (4km) is
a ec ed by dis o ions o space i sel . O he (a bi smalle ) ap-
p oaches o include a mechanical deg ee o eedom a e g am-
scale mi o s, which a e op ically apped a mechanical e-
quencies o ωm∼2π ×200 Hz [10] and coa ing o can ile e s
[11–13] which cons i u e mo able ca i y mi o s. Op omechan-
ical dynamics can also be achie ed by placing memb anes [14]
inside a ca i y mode. The esonances o whispe ing galle y
mode mic o-ca i ies [15,16] can be a ec ed by he elas ic de-
o ma ions o he dielec ic s uc u e i sel , whe eas in supe -
conduc ing mic owa e esona o s he capaci i e coupling o a
nanomechanical beam gi es ise o he op omechanical in e -
ac ion [17]. The smalles op omechanical sys ems a e pho onic
c ys al ca i ies whe e suspended memb anes unc ion as a me-
chanical oscilla o [18] and pho onic c ys al nanobeam ca i ies,
whe e he de o ma ions o he beam i sel suppo s ib a ional
equencies in he GHz ange [5] by ha ing an e ec i e mass o
a ac ion o a picog am.
While mos sys ems ely on he de o ma ion o displacemen
o esona o bounda ies in o de o achie e a posi ion depen-
den esonance equency, he apping o nano-sphe es inside
a ca i y mode cons i u es an op omechanical pla o m as well
[19–24].
As men ioned, all hose expe imen s emain in he so-called
op omechanical weak coupling egime, gmκ, whe e many
pho ons inside he op ical mode a e equi ed o see an app e-
ciable e ec on he ib a ional mode. In he ollowing we will
demons a e how o model his egime and gi e some in ui ion
abou i s consequences.
1.2.1Weak op omechanical coupling
Since gmκ, a la ge inciden ield mus be sen in o d i e he
sys em. This enables one o de elop a linea ized heo y o quan-
um luc ua ions a ound he classical s eady-s a e solu ion. The
op omechanical Hamil onian including a cohe en lase d i e
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8 in oduc ion
Figu e 1.3:Examples o ecen op omechanical sys ems
(Top o Bo om) G a i a ional wa e de ec o s [pho o c edi
LIGO Labo a o y], ha monically suspended g amscale
mi o s [10], coa ed a omic o ce mic oscopy can ile e s
[11], coa ed mic omi o s [12,13], SiN3memb anes dis-
pe si ely coupled o an op ical ca i y [14], op ical mic o-
ca i ies [15,16], supe conduc ing mic owa e esona o s
coupled o a nanomechanical beam [17], suspended mem-
b anes in pho onic c ys al ca i ies [18] and SI nanobeam
ca i ies [5]. Pa s o he igu e and cap ion a e aken om
a e iew on op omechanics [25].
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1.2 s anda d egimes o op omechanics 9
and w i en in a ame o a ing wi h lase equency ωLis gi en
by
Ho= −δca†a+ωmb†b+gm(b†+b)a†a+√κexE0(a†+a). (1.5)
He e, δc=ωL−ωc(x0)is he de uning o he lase om he
ca i y equency and κex deno es he decay a e o he ca i y
in o some pa icula ex e nal channel, which also se es as he
sou ce o injec ion o pho ons. Wi h he s anda d Heisenbe g-
Lange in equa ions [26], one can ind he s eady-s a e equilib-
ium posi ion ¯x0and he s eady-s a e expec a ion alue o he
ampli ude hai=¯α o his Hamil onian. The op omechanical
in e ac ion can be linea ized by spli ing he op ical mode in o
his s eady-s a e solu ion and quan um luc ua ions δa a ound
i :
a=¯α+δa. (1.6)
The e ec o he lase d i e is hen abso bed in o he s eady
s a e solu ion ¯α∝E0and he las e m o Eq. 1.5can be omi ed.
The in e ac ion Hamil onian u ns in o
HI=gm(¯α∗+δa†)( ¯α+δa)(b†+b). (1.7)
The i s e m gm|¯α|2(b+b†)jus desc ibes an a e age adia ion
p essu e o ce. In ui i ely, such a cons an o ce pushing on he
mi o jus esul s in a new equilib ium posi ion ¯x0, and a co e-
sponding s a ic shi in he ca i y esonance equency ωc(¯x0).
We also omi he e m p opo ional o δa†δa as i is smalle by
a ac o o ¯α han he e m we a e in e es ed in. Then, we end
up wi h he “s anda d model” o (linea ized) op omechanical
in e ac ions, which desc ibes essen ially e e y op omechanical
expe imen o da e:
HL≈gm√¯nc(δa†+δa)(b†+b). (1.8)
He e, ¯nc=|¯α|2is he mean in a-ca i y pho on numbe .
To ge some in ui ion, we no e ha HLenables a p ocess
whe e a pho on is c ea ed (δa†), along wi h he c ea ion o an-
nihila ion o a phonon. I he pho on exi s he ca i y (due o he
ini e ca i y losses κ), hen his cycle has esul ed in he hea -
ing o cooling o he mechanical ene gy, wi h he ene gy di e -
ence ca ied away by he ou going pho on (so-called S okes o
an i-S okes sidebands). In pa icula , when ωm> κ (sideband
esol ed) , one can use he na ow op ical esonance o signi -
ican ly enhance he cooling p ocess o e hea ing, by choosing
he lase equency ωL≈ωc(x0) − ωm o be ed de uned. This
in p inciple p o ides a ou e o cool he mechanical mo ion o
i s quan um g ound s a e.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
10 in oduc ion
Fo ωm< κ (un esol ed sidebands) he adiaba ic esponse o
he ca i y ield o he mo ion gi es ise o he op ical sp ing e -
ec , whe e he ib a ional equency can be op ically enhanced
(sp ing) o educed (an i-sp ing). All hese e ec s can be com-
ple ely unde s ood om a classical pe spec i e. The e o e we
will now de i e he classical (and linea ) esponse o he op-
omechnical sys em, which is also called he suscep ibili y.
1.2.2Linea esponse: suscep ibili y
Many in e es ing e ec s a ising om ca i y op omechanics can
be explained classically. When he equa ions o mo ions can be
linea ized, he expec a ion alues o he quan um Heisenbe g-
Lange in equa ions coincide wi h he classical obse ables. The
linea esponse o a mechanical sys em o an ex e nal d i e
wi h equency ωgi es in o ma ion abou i s esonance e-
quencies and damping o ampli ica ion a es. The o al linea ized
Hamil onian o he op omechanical sys em, w i en in a ame
o a ing wi h an ex e nal lase equency ωLis gi en by
HLO =p2
2m +1
2mω2
m(x−¯x0)2−¯
δca†a+(g/xzp)(a†+a)x−xFex ( ).
(1.9)
The i s e m desc ibes he kine ic ene gy o he mechanical
mo ion wi h momen um p. He e, ¯
δc=ωL−ωc(¯x0)is he de-
uning o he lase om he s eady s a e ca i y equency. Fo
simplici y we choose ¯x0=0and changed he no a ion om δa
o a. We de ine he in a-ca i y ield enhanced op omechanical
coupling s eng h g=gm√¯nc. We added an ex e nal d i ing
o ce Fex ( )shaking he mechanical sys em wi h equency ω.
The sys em dynamics unde his Hamil onian is desc ibed by
s anda d Heisenbe g-Lange in equa ions [26]. A e aking he
classical expec a ions alues o he posi ion x=hx( )iand he
ca i y ampli ude luc ua ions α=hai, he equa ions o mo ion
a e gi en by:
m¨x= −mω2
mx−mΓm˙x+ (g/xzp)(α∗+α) + Fex ( )(1.10)
˙α= (i¯
δc−κ
2)α+i(g/xzp)x. (1.11)
In addi ion o he uni a y dynamics unde HLO, we ha e added
he ca i y decay a e κand a mechanical damping a e Γm.
These a e linea coupled equa ions, which can be s aigh o -
wa dly sol ed o x(ω)in equency space, whe e we eplace all
a iables ( )wi h hei Fou ie ans o ms ( ) = Rdωe−iω (ω).
The suscep ibili y χ(ω)is hen de ined as he a io be ween he
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
1.2 s anda d egimes o op omechanics 11
mo ional ampli ude and he ex e nal o ce, x(ω)≡χ(ω)Fex (ω).
I is gi en by
χ(ω) = 1
m(ω2
m−ω2−iΓmω) + Σ(ω). (1.12)
All e ec s o igina ing om he op omechanical in e ac ion a e
con ained in
Σ(ω) = g2
x2
zp 1
(¯
δc+ω) + iκ/2 +1
(¯
δc−ω) − iκ/2(1.13)
which could be called he “op omechanical sel -ene gy” [27]
summa izing he e ec s o he op omechanical in e ac ion.
1.2.3The op ical sp ing e ec and cooling/hea ing
Fo Γmgκ, all dynamics ake place in he icini y o
ω≈ωmand we can app oxima e ω2
m−ω2≈2ωm(ωm−ω)
and e alua e Σ(ωm)a he ba e mechanical equency. Then he
suscep ibili y (Eq. 1.12) akes a Lo en zian shape:
χ(ω) = 1
2mωm
1
(ωm+δωop ) − ω−i(Γm+Γop )/2. (1.14)
Thus, we a e able o iden i y he op omechanically induced
damping a e Γop = −Im[Σ(ωm)]/(mωm)and mechanical e-
quency shi δωop =Re[Σ(ωm)]/(2mωm):
Γop =g2κ
(¯
δc+ωm)2+κ2/4 −κ
(¯
δc−ωm)2+κ2/4(1.15)
δωop =g2¯
δc+ωm
(¯
δc+ωm)2+κ2/4 +¯
δc−ωm
(¯
δc−ωm)2+κ2/4. (1.16)
No e ha he e ec o he op omechanical in e ac ion is, as
g2∝¯nc, inc easing linea ly wi h lase powe . The i s and sec-
ond e ms in Γop can be iden i ied wi h an i-S okes and S okes
sca e ing. In pa icula , he an i-S okes p ocess leads o cool-
ing (Γop > 0)and i s a e is maximized when ¯
δc≈−ωm,
such ha he equency o he sca e ed pho on (which akes
away a phonon o ene gy) aligns wi h he ca i y esonance.
Likewise, he S okes p ocess leads o hea ing (Γop < 0) and
is maximized when ¯
δc≈ωm. No e ha o subs an ial hea -
ing/cooling o ake place, he mechanical sidebands ha e o be
esol ed ωm> κ; o he wise bo h p ocesses ake place wi h an
almos equal a e. The maximal cooling a e is Γm
op =4g2/κ.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
18 in oduc ion
|
0
,
0
〉
|
1
0
〉
|
2
,
0
〉
|
1
,
1
〉
,
Figu e 1.7:Op omechanical pho on blockade. Spec um o he op-
omechanical Hamil onian (1.19). |nc,mideno es he s a e
wi h ncpho ons and mphonons. In his diag am, we o-
cus on ansi ions in ol ing s a es wi h m=0phonons
(black lines), while o he s a es (m=1shown he e) a e
deno ed by g ay lines. A lase wi h equency ωL, which
is esonan wi h he ansi ion |0c,0i → |1c,0i( he ze o-
phonon line), canno esonan ly exci e a second pho on
|2c,0ias op omechanical in e ac ions shi he ela i e en-
e gy o his s a e by an amoun 2g2
m/ωm.
olu ion, ωm> κ, o p e en he nea - esonan ansmission o
he second pho on ia he exci a ion o phonon s a es |2c,m6=0i.
In Chap e 4we demons a e ha an a om apped in and
dispe si ely coupled o a high inesse ca i y cons i u es an a -
ac i e pla o m o ealizing his egime. In pa icula , we
show ha cu en expe imen s should be al eady able o ob-
ain s ong op omechanical coupling and esol ing mechanical
sidebands. This can hen be expe imen ally e i ied by he an i-
bunched s a is ics o he ansmi ed ligh .
Consequences o he s ong op omechanical coupling egime
wi h un esol ed sidebands κωma e hus a unexplo ed in
li e a u e. This is pe haps because wi h con en ional op ome-
chanical sys ems one canno cool o he g ound s a e in he i s
place in his egime, and hus any quan um e ec s ha a ise
would need o be obse ed on op o a he mal backg ound
o phonons. In Chap e 3, we will p o ide o he i s ime a
heo e ical analysis o his egime, which is na u ally eached
by dispe si ely coupling a oms o ca i ies wi h small mode ol-
umes. The use o a oms o explo e his egime is no el, as he
a omic mo ion can be sepa a ely cooled o he g ound s a e by
s anda d echniques, and is highly decoupled om any he mal
o decohe en en i onmen . This allows no el quan um e ec s
o eme ge.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

1.4 o e iew o he hesis esul s 19
1.4 o e iew o he hesis esul s
In he p e ious sec ions we ha e e iewed he s anda d egimes
o op omechanics whe e he mo ion and he ield ha e been lin-
ea ized leading o he “op ical sp ing” e ec and cooling and
hea ing o mechanical mo ion. These e ec s ha e been bo h he-
o e ically analyzed and also expe imen ally obse ed. Then we
ocused on non-s anda d egimes o op omechanics, whe e ei-
he he mo ion o he ield canno be linea ized. Those e ec s
ha e no been obse ed ye . Now we will gi e an o e iew
o he hesis esul s, which on one hand analyze addi ional
no el egimes wi hin op omechanics o he i s ime, and on
he o he hand p opose speci ic sys ems in which non-s anda d
egimes o op omechanics could be ealized expe imen ally.
1.4.1Sel -induced back-ac ion (SIBA) op ical apping in nanopho-
onic sys ems
The beginning o ou scien i ic jou ney was mo i a ed by a cou-
ple o expe imen s [47,48], which obse ed a quali a i ely new
apping beha io in nanopho onic ca i ies leading o s ongly
educed local in ensi ies expe ienced by he apped pa icles.
Fo example, by apping nano-pa icles inside a cohe en ly
d i en pho onic c ys al ca i y, long apping imes (up o 20min)
wi hou appa en pho o- he mal damage o pho o-bleaching o
he pa icles ha e been obse ed [48]. In pa icula , as p esen ed
in Fig. 1.8a), hey measu ed he pa icle induced esonance shi
in ime, which shows he op ical esponse o he ca i y in he
p esence o absence o a pa icle using a scanning a e o 1
Hz. Addi ionally, in Fig. 1.8b), hey p o ide a snapsho , which
shows he ca i y ansmission wi h and wi hou a apped pa -
icle as a unc ion o lase wa eleng h. One can see an a e -
age esonance shi o 1.8nm, which is la ge han he ca i y
linewid h. This clea ly shows he capabili y o he pa icle o
shi he ca i y in and ou o esonance and as a consequence
u ning i s own apping ield on and o . Thus, one can con-
clude ha he pa icle plays an ac i e ole in he apping mech-
anism by ac ing back on i s own apping ield, hus coining he
e m “sel -induced back-ac ion apping" (SIBA).
Fu he mo e, he g oup o Romain Quidan a ICFO was
unning an expe imen o apping gold nano-pa icles in plas-
monic ca i ies du ing ha ime and was seeing simila e ec s.
In Fig. 1.8c) we see hei measu ed ansmission as a unc ion
o ime changing a ound 50% o an emp y ca i y compa ed
o a ca i y con aining a pa icle. Howe e , bo h expe imen s
we e lacking a simple heo e ical model desc ibing SIBA, which
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
20 in oduc ion
Figu e 1.8:Expe imen al demons a ion o a pa icle-induced eso-
nance equency shi .
(a) Reco d o he ime e olu ion o he ca i y spec um
while a pa icle is apped in a pho onic c ys al ca i y and
a e i is eleased.
(b) Snapsho s om (a), displaying an a e age esonance
shi o 1.8nm om he unloaded ca i y esonance, which
is la ge han he ba e ca i y linewid h (wid h o he
blue peak). c) T ansmission o an emp y (g ey) o wi h
a apped pa icle (o ange) plasmonic ca i y as a unc ion
o ime, no malized o he ansmission o an emp y ca -
i y.
Plo s a) and b) a e aken om [48] and plo c) is aken
om [49].
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
1.4 o e iew o he hesis esul s 21
could possibly be used o boos i s pe o mance and unde -
s and he ad an ages o SIBA apping o e con en ional op i-
cal weeze s. To ind such a heo y became hen ou ask. I wan
o men ion ha s imula ing discussions wi h Romain Quidan
and lea ning expe imen al de ails om Pau Mes es and Jo-
hann Be helo made his p ocess e y enjoyable. In Chap e 2
we will p esen such a simple heo e ical model which ully cap-
u es he physics o SIBA and p o ides a clea p esc ip ion o
how o op imize back-ac ion e ec s. Some o he insigh s om
his heo y helped Pau Mes es e al. o nicely demons a e he
back-ac ion o he pa icle on i s own apping po en ial wi h
hei o iginal expe imen al se up [49]. We will now gi e a sho
o e iew o ou app oach o model SIBA and he ema kable
esul s.
As has been expe imen ally obse ed, he key physics is ha
he posi ion o he apped pa icle al e s he esonance e-
quency o he ca i y, which esul s in a s ong in e play be-
ween he in a-ca i y ield in ensi y and he o ces exe ed. As
his sounds a lo like s anda d op omechanics, we will apply i s
o malism o his p oblem. Howe e , he equency shi ωc(x)
needs o be ea ed globally as he pa icle is allowed o eely
di use h ough he ca i y, and in con as o s anda d op ome-
chanics (whe e he e usually exis s a na u al es o ing o ce o
he mechanics), he pa icle only expe iences o ces om he ex-
e nally d i en ca i y mode i sel . As he SIBA e ec has been
obse ed by apping pa icles in wa e a oom empe a u e,
we conclude ha his e ec is pu ely classical.
In Chap e 2we will show ha SIBA apping exhibi s se -
e al su p ising ea u es, when compa ed o con en ional op-
ical weeze aps. Fi s , he pa icle is e ec i ely apped in
an in ensi y minimum, e en i i is nominally high-in ensi y
seeking, which explains he s ong educ ion o pho o- he mal
damage seen by expe imen s. Fu he mo e, we show ha back-
ac ion can be exploi ed o c ea e aps wi h s ongly sub-wa eleng h
spa ial ea u es, e en i he ca i y mode i sel obeys he di ac-
ion limi , e en allowing a squa e well po en ial o a la ge
enough ”back-ac ion pa ame e “. The spa ial ea u es o his
ap can also be dynamically shaped using only changes in
lase equency. We belie e ha hese p ope ies o SIBA will
ha e impo an implica ions o u u e apping echnologies.
1.4.2Quan um SIBA wi h a single a om in a nano/mic o-ca i y
(un esol ed sidebands)
In he p e ious sec ion, we sol ed o he classical expec a ion
alues as ou sys em o in e es was a om any quan um be-
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
22 in oduc ion
ha io . Howe e , mo i a ed by he obse a ion ha he SIBA
e ec p o ides us wi h a squa e well po en ial, we we e e y
cu ious abou whe he such an analysis holds up in he quan-
um domain. In pa icula , a squa e well is in e es ing due o
i s highly anha monic spec um, and migh be use ul o c ea e,
e.g., a wo-le el phonon. Fo his eason, he nex s ep o ou
jou ney is o de i e a ull quan um mas e equa ion capable o
desc ibing he SIBA e ec in he quan um egime, which we
will do in Chap e 3.
E en assuming ha he squa e-well po en ial holds in he
quan um domain, a quick analysis shows ha he ene gy scales
associa ed wi h ealis ic dielec ic pa icles a e much oo low o
make p epa a ion o he g ound s a e ealis ic. We hus u n
o a single a om in a ca i y, wi h he goal o aking ad an age
o i s ligh mass and decoupling o i s mo ion om a he mal
ba h.
The e a e al eady many expe imen s coupling single neu al
a oms [50–53] o ions [54–60] o high- inesse ca i ies. In pa ic-
ula , expe imen s [50,51,61,62] now ou inely each he s ong
coupling egime o ca i y QED, whe ein an a om maximally
coupled o he ca i y (in an an i-node) shi s he ba e ca i y
equency by mo e han a linewid h. Mo ing he a om by a
qua e -wa eleng h o a node elimina es his shi . Thus, a ze o-
poin mo ion on he o de o a ac ional wa eleng h is su i-
cien o a ain op omechanical s ong coupling, which is easily
achie able gi en he ligh single-a om mass.
Thus, we de i e a ull mas e equa ion o a single a om in
a cohe en ly d i en nano-ca i y. Al hough he quan um calcu-
la ion yields an expec a ion alue o he o ce, whose in eg al
co esponds o he classical squa e well po en ial, he squa e
well po en ial does no appea as he He mi ian Hamil onian
o he pa icle i sel . In o he wo ds, a signi ican pa o he
classical o ce comes om ca i y dissipa ion, and he en angle-
men ha builds up be ween sca e ed pho ons and he posi ion
o he a om.
We decided o shi ou a en ion u he on o he en angle-
men be ween ligh and mo ion. To in ui i ely unde s and how
s ong posi ion-pho on en anglemen a ises, we no e ha when
gm> κ, he unce ain y o he ze o-poin mo ion i sel ans-
la es in o an unce ain y o he ca i y esonance equency ha
is much la ge han he linewid h. Thus, obse ing he e lec-
ion o ansmission o a single inciden pho on ( e ealing an
o - o on- esonance ca i y) is consis en only wi h he a om
being loca ed o no in a spa ial egion much smalle han he
ze o-poin unce ain y. This egime na u ally eme ges in he
s ong op omechanical coupling egime wi h un esol ed me-
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
1.4 o e iew o he hesis esul s 23
chanical sidebands κ > ωm. We ealize ha he e a e al eady
many expe imen s ocusing on achie ing s ong coupling be-
ween a pho on and he a omic in e nal deg ee o eedom
wi hin he ield o ca i y QED and we disco e ed ha his same
esou ce also na u ally enables one o each his op omechani-
cal s ong coupling egime. We hough ha one expe imen al
candida e sys em o ou heo y could be a new ibe -ca i y
se up o T acy No hup in Innsb uck.
Funny side s o y: Be o e we ealized ha s ong op omechan-
ical coupling could be achie ed wi hin ca i y QED I me T acy
on a con e ence in Benasque whe e she explained me he ibe
ca i y expe imen in which she s ongly couples single 40Ca+
ions o a ca i y mode. We also discussed how she could model
op omechanics wi h he sys em and wha she could measu e
and hings looked eally in e es ing. Howe e , in he end she
acciden ally ga e me he pa ame e s om an olde expe imen
o a high inesse Fab y-Pé o ca i y (well, mos likely I con-
used i ). Looking a hose pa ame e s a home I was i s disap-
poin ed, as in ha pa icula expe imen mechanical sidebands
we e esol ed ωm> κ and hus ou heo y was no alid. How-
e e , a quick calcula ion wi h hose pa ame e s showed ha
she would be able o each he s ong op omechanical coupling
egime wi h esol ed sidebands ( he holy g ail o op omechan-
ics!) wi h he olde se up. Fu he mo e, she would be able o
show ha she eached i by demons a ing op omechanically
induced pho on blockade o he i s ime. So, ou o his mis-
unde s anding, a collabo a ion and a new esa ch p ojec was
bo n which we will discuss now.
1.4.3Reaching he op omechanical s ong coupling egime wi h a
single a om in a ca i y ( esol ed sidebands)
As men ioned in Sec. 1.3.2, in o de o each he op omechan-
ical s ong coupling egime, a ze o-poin mechanical displace-
men should shi he equency o he op ical esona o by an
amoun compa able o i s linewid h, which is di icul due o
he la ge mass o con en ional mechanical elemen s and he
implied small ze o-poin mo ion. Finding a pla o m whe e his
single-pho on s ong coupling egime o op omechanics can be
explo ed cons i u es a e y impo an goal o he ield.
As a speci ic example, we show heo e ically ha one can
obse e op omechanically induced pho on blockade in ealis ic
ca i y QED se ups, whe e a non-classical an i-bunched ield is
p oduced as he sys em is unable o ansmi mo e han a sin-
gle pho on a a ime. We also desc ibe how his op omechanical
beha io can be clea ly dis inguished om, and domina e o e ,
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

24 in oduc ion
he usual an i-bunching associa ed wi h he wo-le el na u e o
he a om. Expe imen ally showing pho on blockade induced by
mo ion p o es ha one eally has eached he s ong coupling
egime o op omechanics and we an icipa e ha he p oposed
pla o m o single a oms coupled o a ca i y will also enable
many o he exo ic new egimes o op omechanics o be iden i-
ied and explo ed.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
Pa II
RESULTS
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
2
SELF-INDUCED BACK-ACTION OPTICAL
TRAPPING IN NANOPHOTONIC SYSTEMS
2.1 in oduc ion
Op ical apping is one o he mos impo an expe imen al
ools in physics and li e sciences because i enables p ecise con-
ol o e small dielec ic pa icles [4]. Famous examples o i s
use a e op ical le i a ion and cooling o nanoscale pa icles [23,
63–66], apping o bac e ia [67] and cells [68], op ical so ing
in mic o luidic channels [69], he manipula ion and s e ching
o DNA [70], and ecen ly, e en apping o indi idual HIV-1
i uses [71]. Howe e , he di icul y o apping a pa icle gen-
e ally inc eases wi h dec easing size, due o he dec eased op-
ical esponse o he pa icle. This equi es a commensu a e in-
c ease in ield in ensi y o main ain ap s abili y, and leads o
associa ed p oblems such as he mal o ma e ial damage. An-
o he limi ing ac o is he di ac ion limi , which cons ains
he leng h scale o e which ields can a y, and hus he s i -
ness o possible spa ial ea u es ha a ap can possess.
A numbe o expe imen s in ecen yea s ha e mig a ed om
apping in ee-space beams o he ields gene a ed in nano-
op ical esona o s [47,48,72–75] as illus a ed in Fig. 2.1. Such
a pa adigm can enable some echnical ad an ages. Fo example,
he esona o allows one o build up a highe in ensi y seen by
he pa icle wi hin he s uc u e compa ed o he inpu , hus
elaxing inpu powe equi emen s. Enginee ing he nanopho-
onic s uc u e also p o ides some lexibili y o e he ield p o-
ile, and hus he apping po en ial. Howe e , i is clea ha
simply eplacing he inpu ield wi h he enhanced one does no
elax any equi emen s om he s andpoin o in ensi y seen
by he pa icle. The e o e i emains an open ques ion whe he
one can ci cum en hese seemingly undamen al ade o s be-
ween pa icle size and he in ensi ies equi ed o achie e gi en
ap dep hs, equencies, and spa ial con inemen . A he same
ime, doing so would ha e signi ican implica ions o op ical
manipula ion as a ool in physics, chemis y and biology.
In his con ex , a numbe o expe imen s ha e obse ed qual-
i a i ely new apping beha io in nanopho onic ca i ies [47,
48]. The key physics is ha he posi ion o he apped pa icle
al e s he esonance equency. This esul s in a “sel -induced
back-ac ion" (SIBA) e ec in which he mo ion dynamically a -
27
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
34 siba classic
ec i ely apped in a dynamical in ensi y minimum, despi e
he ac ha i has posi i e pola izabili y and is hus nomi-
nally high-in ensi y seeking. This would ha e emendous con-
sequences in he educ ion o he mal damage due o op ical
abso p ion by he pa icle. Mo i a ed by his obse a ion, we
seek o quan i y how much he ime-a e aged in ensi y seen by
he pa icle can be educed.
We de ine he ime-a e aged expe ienced in ensi y hIexpi as
he local in ensi y expe ienced by he pa icle a i s posi ion,
a e aged o e one mo ional pe iod T. I is hus gi en by
hIexpi =c
hωL
2VmTZT
0
n(xp( )) (xp( ))d (2.9)
whe e xp( )is a solu ion o he di e en ial Eq. (2.4) oge he
wi h Eq. (2.7). In o de o p oceed u he , we conside a simple
case o he undamen al mode o a 1D Fab y-Pe o ca i y, (x) =
cos2(kx)wi h k=π/L, whe e L=λ/2 is he ca i y leng h.
Al hough we ha e swi ched o a speci ic model o illus a e
he back-ac ion mechanism, we belie e he o e all conclusions
a e gene ally alid. A ini e empe a u e o he en i onmen
can be aken in o accoun by a e aging he esul s o di e en
maximal kine ic ene gies Ekin (kine ic ene gy o he pa icle in
he ap minimum) acco ding o a Bol zmann dis ibu ion.
We ha e e alua ed Eq. (2.9) by nume ically sol ing he equa-
ions o mo ion (2.3)-(2.5). In Fig. 2.3, we plo he ime-a e aged
expe ienced in ensi y hIexp(η)i no malized by he alue in he
op ical weeze egime hIexp,Ti , as a unc ion o back-ac ion pa-
ame e η. As seen be o e, he op ical weeze egime is eached
by aking η1. To make a ai compa ison, we en o ce ha he
ap dep hs in he wo cases a e equal, δU(η) = δUT. Fo a ixed
x , he igu e shows a signi ican educ ion in ime-a e aged in-
ensi y o high back-ac ion pa ame e , which also depends on
he a io o kine ic ene gy Ekin o ap dep h δU. In he high
back-ac ion egime, i is possible o de i e an analy ic exp es-
sion (see Appendix A.3):
lim
η→∞hIexp(η)i =2c0
α(ω)
(x )
| 0(x )|
Ekin
x
(2.10)
A new ea u e o he back-ac ion ap is he g adual decoupling
be ween ap dep h and he spa ial egion δx =|x 2−x 1|(x 1
and x 2a e he classical u ning poin s) o which he pa icle is
con ined. Fo la ge enough η hey decouple comple ely since
he classical u ning poin s con e ge o he esonan posi ions
(i.e., he edges o he squa e well) and hus δx →d. In his
egime, con inemen only depends on lase equency, whe eas
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

2.5 wo mode back-ac ion 35
ap dep h only depends on lase powe . This independen con-
ol again highligh s he abili y o dynamically eshape he ap.
In con as , in he op ical weeze egime, he ap dep h, kine ic
ene gy and con inemen a e ine i ably connec ed.
Ins ead o compa ing he expe ienced in ensi y a ixed ap
dep h, we can also in es iga e he ade-o be ween in ensi y
and con inemen δx =din he la ge back-ac ion limi . The lo-
ca ions o he apping wells a e always cen e ed a ound he
mode p o ile maximum x0=0. Fo small x , an asymp o ic ex-
pansion yields (x )
| 0(x )|≈1
2k2x . Thus, o high back-ac ion and
s ong con inemen , we ob ain hIexpi ≈4c0
α(ω)
1
(kδx)2Ekin. In e es -
ingly, expanding Eq. (2.1) o he op ical weeze a ound he
bo om o a s anding wa e po en ial also p oduces hIexp,Ti ≈
I(x0)≈4c0
α(ω)
1
(kδx)2Ekin, which seems o indica e ha no imp o e-
men is gained in in ensi y s. con inemen wi h back-ac ion.
Looking a Eq. (2.10), in he s ong back-ac ion egime, one o
he ac o s o 1
δx o igina es simply om he ime T∝δx ha
he pa icle akes o a el be ween he walls o he squa e well.
This pa o he scaling seems undamen al and canno be im-
p o ed wi hin his model. On he o he hand, he second ac-
o o 1
kδx clea ly o igina es om he anishing o back-ac ion
e ec s a ound he maximum o he mode p o ile, as he e-
quency shi becomes insensi i e o i s -o de changes in he
pa icle displacemen , 0(x0) = 0. We show ha his ac o is
no undamen al, and can be elimina ed by p ope ly d i ing a
second op ical mode o he sys em.
2.5 wo mode back-ac ion
In his Sec. we show how he scaling be ween expe ienced in-
ensi y and con inemen can be imp o ed o hIexpi ∝1
kδx by
using wo di e en ca i y modes o apping. In o de o ob-
ain conc e e esul s, we conside he simple geome y whe e
he wo modes consis o he i s and second ha monics o a
Fab y-Pe o (see Fig. 2.4), al hough we belie e ha he conclu-
sions hold qui e gene ally. We assume ha each mode can be
d i en wi h i s own lase , wi h ampli ude E0iand equency
ωLi. As he equa ion o he in a-ca i y ields βi(gene alized
om Eq. (2.5)) o each mode a e decoupled om one ano he ,
hey can be sepa a ely in eg a ed as in he single-mode case.
Thus, he o al po en ial U o (x) = Pi=1,2Ui(x)is he incohe en
sum o he po en ials in Eq. (2.8) o each mode. To unde s and
he ele an physics, i is su icien o assume ha he mode
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
36 siba classic
0.1 1 10 100 1000
10- 2
0.05
0.10
0.50
1
Figu e 2.3:Time-a e aged expe ienced in ensi y o a apped pa i-
cle. We plo he ime-a e aged expe ienced in ensi y as a
unc ion o back-ac ion pa ame e hIexp(η)i , no malized
wi h he alue in he op ical weeze egime η1. The
wo cases a e se o ha e equal ap dep h. The plo is nu-
me ically calcula ed o he case o apping in he unda-
men al mode o a Fab y-Pe o ca i y (x) = cos2(kx)wi h
esonan posi ions kx =π/4. The back-ac ion egime can
enable much lowe a e age local in ensi ies han in he op-
ical weeze egime.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
2.5 wo mode back-ac ion 37
d i ing ampli udes E0i, decay a es κex,κin, and back-ac ion pa-
ame e s a e iden ical, al hough he concep s can be easily gen-
e alized.
The in e es ing egime will be when he esonan posi ions o
each mode a e uned by hei espec i e d i ing lase equen-
cies such ha each mode is esponsible o p o iding one ap-
ping wall. This is illus a ed in Fig. 2.4b), whe e he le and
igh walls x 1and x 2o igina e om he i s and second ca -
i y modes, espec i ely. Signi ican ly, he well can be loca ed
a om he nodes/an inodes 0
i(x) = 0whe e he e ec s o
back-ac ion would anish o ei he mode. In he ollowing we
will dis inguish h ee di e en egimes conce ning he a io be-
ween he dis ance d=|x 1−x 2|and he wid h ∼2
kη o hese
in ensi y peaks illus a ed in Fig. 2.4.
We s a by examining he high back-ac ion egime, when he
dis ance o he in ensi y peaks is much la ge han hei wid h,
kd =k|x 1−x 2|2
η, such ha we encoun e an almos pe -
ec squa e-well po en ial as shown in Fig. 2.4b) and Fig. 2.5.
I is s aigh o wa d o gene alize he high back-ac ion limi o
Eq. (2.10) in he single mode case. As he pa icle is apped a
om poin s whe e back-ac ion e ec s anish ( 0
i(x) = 0), we e-
co e he imp o ed scaling be ween expe ienced in ensi y and
con inemen , hIexpi ∝1
kδx as al eady an icipa ed.
Quali a i ely, he condi ions needed o each his scaling a e
ha kd 2
η, so ha he po en ial esembles a squa e well, bu
also ha he a io o kine ic ene gy o po en ial dep h is su i-
cien ly la ge ha he pa icle ac ually app oaches he edges o
he well. This is schema ically illus a ed in Fig. 2.5. Assuming
ha he la e condi ion is ini ially sa is ied o a la ge alue
o d(subplo 1), i con inues o be sa is ied by dec easing d
(subplo 2). On he o he hand, i can be seen ha d ama ically
inc easing he ap dep h p e en s he pa icle om eaching
he edge (subplo 3), which esul s in a less a o able scaling
o in ensi y e sus con inemen . In Fig. 2.5, we ha e plo ed he
esul s o expe ienced in ensi y s. con inemen om ull nu-
me ical simula ions o equa ions (2.3)-(2.5) (gene alized o wo
modes). The di e en poin s o a ixed back-ac ion pa ame e
ηa e ob ained by a ia ion o he inpu powe s and esonan
posi ions x ( ia he lase equencies). Tuning he esonan po-
si ions o educe d=|x 1−x 2|indeed enables one o sa u a e
he scaling o hIexpi ∝1
kδx as long as kδx &2
η, as illus a ed in
Fig. 2.6a).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
38 siba classic
Figu e 2.4:SIBA wi h wo op ical modes, illus a ed he e o he i s
wo modes o a Fab y-Pe o ca i y. Top: he mode p o-
iles a e gi en by 1(x) = cos2(kx), 2(x) = sin2(2kx).
G een: he esul ing op ical apping po en ial U(x). Bo -
om: in a-ca i y in ensi ies I(x)as a unc ion o pa icle
posi ion. a)In he ha monic back-ac ion egime, he dis-
ance be ween he esonan poin s is compa able o he
wid h o he in ensi y peaks, kd ∼2
η.b)In he high back-
ac ion egime, he dis ance signi ican ly exceeds he wid h,
kd 2
η.
Fo kδx .2
η, he op imal scaling seen in he nume ics goes
like hIexp,hbi ∝1
η(kδx)2. The scaling wi h δx−2 esembles he
op ical weeze case, bu he in ensi y is supp essed by a ac-
o o η. We call his he “ha monic back-ac ion egime” (see
Fig. 2.4a)). To unde s and his case, we i s no e ha he pa -
icle mo es by a small enough amoun a ound he ap mini-
mum ha he o ces om each mode can be linea ized a ound
small displacemen s o yield a ha monic ap. Fu he mo e, o
small displacemen s, he o al ime a e aged expe ienced in en-
si y hIexpi =PihIexp,ii ≈PiIi(x0)is jus he sum o he in en-
si ies o he espec i e mode a he ap minimum xp=x0. The
associa ed sp ing cons an is:
kop = −F0(x0) = X
i
n0
i(x0)ω0
c,i(x0) + ni(x0)ω00
c,i(x0)(2.11)
whe e he sum goes o e all apping modes. The i s e m
n0
i(x0)is a new con ibu ion o he op ical sp ing cons an kop
o igina ing om he change in pho on numbe wi h pa icle
posi ion a ound he ap minimum. In ui i ely, his back-ac ion
con ibu ion o he sp ing cons an is maximized by ensu ing
he pho on numbe o each mode maximally changes a ound
x0. This is oughly op imized by se ing kd ∼2
η, such ha x0
co esponds o si ing hal a ca i y linewid h away om he es-
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
2.5 wo mode back-ac ion 39
onan posi ion x . Such an op imiza ion yields (see Appendix
A.4):
kop ,i=α(ω)
c0hIexp,ii
1
i(x0)hηi 0
i(x i)2− 00
i(x0)i. (2.12)
The i s e m in he b acke s o igina es om he change in pho-
on numbe wi h pa icle posi ion, whe eas he second e m e-
duces o he op ical weeze sp ing cons an gi en by Eq. (2.1):
kT=U00(x0). Since 00(x0)∼ 0(x0)2∼k2, i can be seen ha he
back-ac ion con ibu ion is a ac o o ηla ge . We can equi -
alen ly in e p e his con ibu ion as a ising om an e ec i e
educed wa eleng h λe ∼λ
√η, which enables he gene a ion o
ap ea u es a below he di ac ion limi . We emphasize ha
his e ec o igina es om he apid change in in a-ca i y pho-
on numbe wi h pa icle displacemen a he han a change
in he spa ial mode i sel (see Eq. (2.11)), and hus he e is no
b eakdown o he dipole app oxima ion in which all o hese ex-
p essions a e de i ed. This is analogous o he “op ical sp ing"
e ec desc ibed in Sec. 1.2.3and Fig. 1.4, whe e an op ical ca -
i y can exe la ge es o ing o ces o small displacemen s o
a mechanical sys em. In he con en ional op ical sp ing e ec
he s i ness o he mechanical mode i sel plays he ole o ou
second op ical mode, and se es o keep he equilib ium posi-
ion a a poin o non- anishing back-ac ion ( 0(x0)6=0) [78,79].
Exploi ing he no ion o a educed wa eleng h, in he ha monic
back-ac ion egime one can immedia ely conclude ha he scal-
ing o a e age expe ienced in ensi y imp o es om hIexp,Ti ∝
1
(kδx)2 o an op ical weeze o hIexp,hbi ∝1
η(kδx)2. A mo e de-
ailed op imiza ion o he sys em shown in Fig. 2.4 e eals ha
(see Appendix A.4):
hIexp,hbi
hIexp,Ti
=4
η(2.13)
o equal con inemen and kine ic ene gy. We wan o empha-
size ha o each his op imal scaling, one should ix kd =
k|x 1−x 2| ∼ 2
η. In o he wo ds, o achie e he bes con inemen
o a gi en in ensi y, one should inc ease he lase in ensi y (see
Fig. 2.5, subplo 4and 5). This p ocedu e enables one o s ay
along he do ed line o he in ensi y e sus con inemen plo
illus a ed in Fig. 2.6. In con as , Fig. 2.6also shows ha by de-
c easing he dis ance be ween he esonan posi ions, kd 2
η,
he scaling de ia es back owa ds he op ical weeze limi and
he bene i s o back-ac ion anish.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

40 siba classic
2.6 conclusion
The e ha e al eady been wo ypes o sys ems, plasmonic ca i-
ies [80] and pho onic c ys al ca i ies [76], whe e SIBA has been
obse ed, and we now discuss he po en ial igu es o me i as-
socia ed wi h each. As he plasmon esonances associa ed wi h
small me allic sys ems do no obey a di ac ion limi , hey a e
able o achie e s ongly sub-wa eleng h mode olumes. On he
o he hand, ealis ic quali y ac o s a e limi ed o Q.10 −102.
A he same ime, an uppe bound on he alidi y o ou calcula-
ion is ha he pa icle size V.Vmdoes no exceed he mode
olume, and hus we an icipa e maximum possible alues o
η∼10 −102 o such sys ems. In pho onic c ys al ca i ies, he
mode olume is limi ed by he di ac ion limi o Vm&(λ
2)3,
while ex emely high quali y ac o s o Q∼106a e possible
[76]. This yields η∼10,100,400 o a dielec ic sphe e wi h a-
dius ∼6.5nm,15nm, 28nm (Appendix A.2). The e has been
signi ican ac i i y in ecen yea s o de elop design p inciples
in o de o ailo he spa ial modes o plasmonic [80] and pho-
onic c ys al s uc u es [81] o apping. Combined wi h he po-
en ially la ge back-ac ion pa ame e s achie able, we an icipa e
ha ou wo k will open up signi ican new oppo uni ies o
op ical apping. I would also be in e es ing o explo e he use
o la ge back-ac ion pa ame e s in o he unc ionali ies, such as
pa icle de ec ion and eedback cooling.
Finally, i is in iguing o ask whe he back-ac ion apping,
and he esul ing squa e wells, could be applicable o a oms.
Combined wi h he long cohe ence imes o he a om, a squa e
well could po en ially be used o gene a e a " wo-le el" phonon.
This would build upon he al eady ich ield o mechanical e -
ec s o ligh on a oms in ca i ies [82–86] and ecen success ul
e o s o in e ace cold a oms wi h nanopho onic sys ems [87,
88]. We hus u n o his ques ion in he nex chap e .
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
2.6 conclusion 41
Figu e 2.5:Po en ial wi h wo op ical modes. He e we show how he
apping po en ial ans o ms as one ei he dec eases he
dis ance d=|x 1−x 2|be ween he esonan posi ions ( ia
changes in he lase de uning), o inc eases he lase in en-
si y. T ans o ma ions ia changes in dand in ensi y a e
depic ed by he g een and ed a ows, espec i ely. Fo a
gi en kine ic ene gy Ekin inc easing he lase powe lowe s
he a io be ween kine ic ene gy and ap dep h. This a io
hen de e mines he egion o he po en ial he pa icle is
allowed o explo e. In po en ial 1) we a e in he high back-
ac ion egime (kd =k|x 2−x 1|2
η) whe e dand δU
(lase powe ) decouple and he po en ial o ms a squa e
well ha he pa icle has su icien ene gy o explo e. De-
c easing dun il he condi ion kd ∼2
ηis eached enables
one o s ay in he high back-ac ion egime, as shown in
2). On he o he hand, a signi ican inc ease in powe , il-
lus a ed in 3), p e en s he pa icle om coming in o con-
ac wi h he edges o he well, and one loses he a o able
scaling o in ensi y e sus con inemen . A kd ∼2
η, one
eaches he ha monic back-ac ion egime, whe e he pa i-
cle expe iences an app oxima e ha monic po en ial ega d-
less o lase powe . Op imum in ensi y e sus con inemen
is achie ed by hen inc easing lase powe , as opposed o
u he educ ion in d, as illus a ed in 4) and 5). The sub-
plo numbe s 1), 2), 3), 4), 5) co espond o he same num-
be s as indica ed in Fig. 2.6b).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
42 siba classic
- 4 - -
1000
10
105
107
109
10- 4 10- 3 10- 2 10-1
Figu e 2.6:Time-a e aged expe ienced in ensi y s. con inemen o
wo op ical modes. a) Time-a e aged expe ienced in en-
si y in uni s o cEkin
α(ω)as a unc ion o con inemen kδx =
k|x 2−x 1|. The indi idual poin s o igina e om di e -
en combina ions o back-ac ion pa ame e , lase powe
and de unings. The solid lines indica e he scalings in
he op ical weeze egime and high back-ac ion egime
(kd =k|x 2−x 1|2
η). The dashed line shows he op-
imized ha monic back-ac ion egime, whe e kd ∼2
η. b)
Illus a ion o igu e 2.6a) o a ixed ηand Ekin, and a
schema ic o he p o ocol o sa u a e he scaling bounds.
The g een a ows deno e a dec ease in dis ance be ween
he esonan posi ions d=|x 1−x 2|, while he ed a -
ows deno e an inc ease in lase powe . The numbe s 1),
2), 3), 4), 5) co espond o he same numbe s as indica ed
in Fig. 2.5.6) dec easing kd < 2
ηsupp esses back-ac ion
as he pa icle mo ion no longe shi s he ca i y mode
equencies.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
3
EXPLORING UNRESOLVED SIDEBAND,
OPTOMECHANICAL STRONG COUPLING
USING A SINGLE ATOM COUPLED TO A
CAVITY
3.1 in oduc ion
In op omechanics much p og ess has been made imp o ing he
con ol o e he in e ac ion be ween pho ons and phonons a
he quan um le el [89]. La ely he e ha e been many impo -
an expe imen al successes, which include he gene a ion o
slow ligh wi h op omechanics [90], he en anglemen o mo ion
wi h mic owa e ields [91], and e y ecen ly emo e en angle-
men be ween wo mic omechanical oscilla o s [92]. Fo mos o
he quan um phenomena obse ed hus a o en isioned, side-
band esolu ion, whe e he mechanical equency ωmexceeds
he ca i y linewid h κ, is equi ed. Fo example, his enables
cooling o he quan um g ound s a e [93,94], which ep esen s
a iducial pu e s a e p epa a ion. In one ema kable heo e i-
cal wo k [46], i has been p edic ed ha he combina ion o
sideband esolu ion and single-pho on op omechanical s ong
coupling – whe e he ze o-poin mo ional unce ain y induces
a shi in he op ical esonance equency la ge han he ca i y
linewid h – would enable he gene a ion o non-classical, an i-
bunched ligh .
He e, we s udy he complemen a y egime o single-pho on
op omechanical s ong coupling, bu wi h un esol ed sidebands
[95,96]. We show ha in e es ing quan um e ec s bo h in he
ligh and mo ion can be obse ed, a leas when he mechanical
sys em is well-isola ed and can be sepa a ely p epa ed in he
g ound s a e. A na u al candida e sys em consis s o a single
a om [50–53,97,98] o ion [54–59] in ca i y QED, whose elec-
onic ansi ion is s ongly coupled o a nea - esonan op ical
mode. To p o ide an in ui i e pic u e, s ong coupling wi hin
ca i y QED [99,100] implies ha a poin -like a om p oduces a
shi in he ca i y esonance equency ha is la ge han he
ca i y linewid h, when he a om is si ua ed a a ca i y an i-
node. I he a om is displaced by a qua e wa eleng h o a
node, his shi anishes. Gi en he ligh mass, i is s aigh -
o wa d o a apped a om o ha e a ze o-poin mo ion on
ha scale, hus ealizing single-pho on op omechanical s ong
coupling. Fu he mo e, ealis ic ap equencies o a oms a e
43
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
50 siba quan um
single phonons is gi en by Homs =gm(b†+b)a†a, whe e gm=
∆0
c(x0)xzp ∼gomηLD. Thus, in o de o achie e s ong op ome-
chanical coupling on he single-pho on, single-phonon le el (gm&
κ), addi ionally a su icien ly la ge Lamb-Dicke pa ame e ηLD
is equi ed. Gi en he abo e conside a ions, we nex de i e an
e ec i e mas e equa ion o he a omic mo ion alone ha is
alid o s ong and nonlinea op omechanical coupling, which
can be iewed as a gene aliza ion o he ypical op ically-induced
cooling and hea ing a es ob ained o linea ized op omechan-
ical coupling [93,94,106]. Ou mas e equa ion also comple-
men s p e ious wo k in es iga ing in a-ca i y op ical o ces
on a oms in he semi-classical limi [82,83,110–112].
3.3.2E ec i e Mas e Equa ion o Mo ion
S a ing wi h Eq. (3.7) we can use he Nakajima-Zwanzig ech-
nique o e ec i ely elimina e he ca i y deg ees o eedom
(Appendix A.5.2). He e, o simplici y we assume ha spon a-
neous emission can be igno ed. The esul ing mas e equa ion
o a omic mo ion in con en ional Lindblad- o m is hen gi en
by:
˙ρ= −i[Hm,ρ] − 1
2J†Jρ +ρJ†J+JρJ†. (3.13)
The He mi ian Hamil onian and jump ope a o s a e gi en e-
spec i ely by
Hm=ωmb†b+κ E2
0∆c(x)
∆2
c(x) + κ2
4
(3.14)
and
J=i√κκ E0
∆c(x) + iκ
2
. (3.15)
We will p o ide an in ui i e pic u e o his mas e equa ion
in Sec. 3.5. Now we ocus on he e ec i e mechanical po en-
ial which a ises in he Hamil onian. We can always ew i e a
mas e equa ion in e ms o an e ec i e non-He mi ian Hamil-
onian Hcwhich hen con ains a complex po en ial:
˙ρ= −i(Hcρ−ρH†
c) + JρJ†(3.16)
Hc=ωmb†b+V(x)(3.17)
wi h
V(x) = κ E2
0∆c(x)
∆2
c(x) + κ2
4
−i
2J†J. (3.18)
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

3.3 ca i y qed wi h mo ion 51
Figu e 3.3:Quan um and classical mechanical po en ial a ising
om a cohe en ly d i en ca i y mode
Real pa Re[V(x)] (blue) and imagina y pa Im[V(x)]
( ed) o he quan um po en ial Eq. (3.18) as a unc ion
o posi ion. Also plo ed is he classical po en ial U(x)
(dashed, g een) de i ed by in eg a ing he expec a ion
alue o he o ce ac ing on he a om. One can obse e
ha he eal pa o he quan um po en ial is signi ican ly
di e en om he classical expec a ion alue. He e, we
choose a lase equency ωLsuch ha he esonan po-
si ion kcx =π/4, and Jaynes-Cummings pa ame e s o
g0/κ ∼20 and δ0= −2g0(yielding an e ec i e op ome-
chanical coupling s eng h o gom ∼10κ). The po en ials
a e plo ed in uni s o
h(κ /κ)E2
0.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
52 siba quan um
The eal and imagina y pa s o he complex po en ial V(x)
a e illus a ed in Fig. 3.3. As he esonance equency o he
ca i y depends on he posi ion o he a om, he e can be a omic
posi ions o which he ca i y is esonan wi h he cohe en
d i e. These posi ions x a e called esonan posi ions and a e
de ined by ∆c(x ) = 0. A ound hese posi ions, he eal pa
o he po en ial changes sign and he imagina y pa has sinks
indica ing inc eased hea ing a ound hose posi ions.
I is also in e es ing o compa e he “cohe en ” po en ial, Re[V(x)],
wi h he classical po en ial U(x)as de i ed om he a e age
o ce F(x) = dhpi/d =T (pρ)on he a om, and de ined ia
dU/dx = −F(x). The esul is gi en by
U(x)=−2κ
κE2
0a c an 2∆c(x)
κ, (3.19)
which ag ees wi h ou p e ious, comple ely classical analysis o
a dielec ic objec apped in a ca i y in Chap e 2(see Eq. (2.8)).
The po en ial is illus a ed in Fig. 3.3. Fo la ge gom/κ,U(x)is
seen o app oach a squa e well, wi h he walls o he well align-
ing wi h he esonan posi ions ∼x whe e he la ge in aca i y
ield esul s in a la ge classical es o ing o ce. By compa ing
V(x)and U(x), i is clea ha a signi ican con ibu ion o he
a e age o ce mus a ise om he s ochas ic p ocess associa ed
wi h he quan um jumps J. As one consequence, al hough i
would be highly in e es ing o ealize a squa e well o a oms
(leading, e.g., o a highly anha monic phonon spec um), he
di ec quan iza ion o U(x)in his case is no meaning ul.
3.4 single-pho on sca e ing heo y:op omechan-
ical s ong coupling wi h un esol ed sidebands
A complemen a y physical pic u e o he op omechanical cou-
pling be ween an a om and ca i y can be gained by conside ing
no a cohe en ex e nal d i e, bu single inciden pho ons. F om
Eq. (3.7), he e ec i e non-He mi ian Hamil onian associa ed
wi h an und i en sys em is
He =ωmb†b− (∆c(x) + iκ
2)a†a(3.20)
whe e ∆c(x) = ωL−ωc(x)is he posi ion-dependen de uning
be ween pho on equency ωLand ca i y equency ωc(x) =
ωc−gomu2(x). To be speci ic, we will conside single pho ons
inciden h ough he le mi o (see Fig. 3.2), which has a decay
a e back in o he e lec ion channel o κ . The igh mi o is
coupled o he con olled ansmission channel wi h κ . The
o al ca i y linewid h is hus κ=κ +κ . Fo simplici y we
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
3.4 single-pho on sca e ing 53
igno e he e an in insic loss a e, al hough i is s aigh o wa d
o include la e on.
A connec ion can be made be ween he eigens a es o He
and he p ope ies o single-pho on sca e ing ia he S-ma ix
o malism. Fo mally, he S-ma ix desc ibes a cohe en e olu-
ion mapping an inpu s a e ( = −∞) o an ou pu s a e ( =
+∞):
|Ψou (ωL)i=S|Ψin(ωL)i. (3.21)
He e, we assume a single monoch oma ic pho on wi h equency
ωLinciden on he le ca i y mi o
|Ψin(ωL)i=|(ωL)le ,0i, (3.22)
whe eas he op omechanical sys em ini ially is in i s g ound
s a e ep esen ed by he second en y in he ke s a e. Gene i-
cally he ou pu s a e will consis o a supe posi ion o nphonons
in he mechanical s a e, which we e exci ed by he incoming
pho on, and an ou going pho on o ene gy ωL−nωmin ei he
he e lec ion po ( ) o ansmission po ( ):
|Ψou (ωL)i=X
n
S ,n(ωL)|(ωL−nωm) ,ni
+X
n
S ,n(ωL)|(ωL−nωm) ,ni. (3.23)
Due o a connec ion be ween he sca e ing ma ix and he
Heisenbe g inpu -ou pu ope a o s [113] one can exp ess he
S-ma ix elemen s in e ms o he eigen alues λβand eigen-
s a es |βio he e ec i e Hamil onian He [114]. We p o ide a
de ailed de i a ion o he S-ma ix elemen s in Appendix A.6.
In e lec ion, he ou pu consis s o a supe posi ion be ween a
non-in e ac ing p opaga ing pho on (δn,0) and pho on emission
om he exci ed op omechanical sys em:
S ,n(ωL) = δn,0+iκ X
βh1c,n|βi1
λβhβ|1c,0i. (3.24)
He e, h1c,n|βiis he p ojec ion o he eigens a es |βion o he
basis s a es h1c,n|wi h 1c e e ing o a single pho on inside he
ca i y mode. Simila ly, he ma ix elemen s o pho on ans-
mission a e gi en by
S ,n(ωL) = i√κ κ X
βh1c,n|βi1
λβhβ|1c,0i. (3.25)
The ma ix elemen o pho on ansmission lacks he con ibu-
ion om he non-in e ac ing p opaga ing pho on as he inpu
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
54 siba quan um
channel on he ansmi ing side o he ca i y is in he acuum
s a e. To p oceed u he , we assume in he ollowing ha a de-
ec o canno e ec i ely esol e he equency o he ou going
pho on. Then, we can e ec i ely w i e he ou going s a e as
|Ψou (ωL)i=S (ωL,x)Ψ0(x)|1 i+S (ωL,x)Ψ0(x)|1 i, (3.26)
whe e |1 / iindica es an ou going e lec ed/ ansmi ed pho-
on, espec i ely, and Ψ0(x)is he ini ial mo ional wa e unc-
ion o he a om. The en anglemen be ween he pho on e-
quency and he mo ional s a e has been supp essed, as we ha e
assumed ha any p ojec i e measu emen o a pho on in ei he
po is no equency- esol ing. Fu he mo e, we now assume
ha we ope a e in he sideband-un esol ed limi κωm. The
Hamil onian He is app oxima ely diagonal in he posi ion ba-
sis, as he op omechanical in e ac ion domina es o e he ee
Hamilon ian ωmb†bin He (Eq. 3.20). Thus, he eigen alues
o He a e app oxima ely λ≈−∆c(x) − iκ
2and he sca e ing
ma ix elemen s can be simply w i en as
S (ωL,x) = 1−iκ
∆c(x) + iκ
2
(3.27)
and
S (ωL,x)=− i√κ κ
∆c(x) + iκ
2
. (3.28)
As he shape o he mechanical wa e unc ion a e he decay o
a single pho on in o one speci ic channel is he p oduc be ween
he co esponding S-ma ix elemen and he ini ial wa e unc-
ion Ψ0(x), we obse e ha he shape o he mechanical wa e
unc ion a e one such sca e ing e en is s ongly en angled
wi h whe he he decaying pho on is e lec ed o ansmi ed.
3.5 connec ion be ween sca e ing heo y and mas-
e equa ion
Mo i a ed by he obse a ion ha he sca e ing ma ices S
and S o Eqs. (3.27) and (3.28) a e e y simila o he jump
ope a o s J(Eq. 3.15), we exp ess he mas e equa ion (3.13) in
a way ha i s jump ope a o s co espond o he single pho on
sca e ing ma ices:
˙ρ= −i(Hsρ−ρH†
s) + E2
0(S ρS†
+S ρS†
)(3.29)
wi h he Hamil onian
Hs=ωmb†b−i
2E2
0. (3.30)
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
3.6 quan um e ec s due o ze o-poin mo ion 55
Figu e 3.4:Re lec ion spec um p as a unc ion o lase equency
ωL.He e, we ake c i ical coupling (κ =κ/2) and a ap
equilib ium posi ion o kcx0=π/4. We assume he ini ial
a omic wa e unc ion is in he mo ional g ound s a e.
a) I he ze o-poin mo ion is un esol ed, he e lec ion
spec um (blue) jus beha es like he e lec ion spec um
o an emp y ca i y (g een, dashed) bu is shi ed o a
new esonance ωc(x0). He e we choose gom =κand
ηLD =0.01, implying zp =0.02.
b) I he ze o-poin mo ion is esol ed, he e lec ion spec-
um is b oadened by oughly gmand becomes shallowe .
He e we choose gom =5κ and ηLD =0.2, implying zp =2.
W i en in his o m he connec ion be ween sca e ing heo y
and jump o malism becomes clea . The non-He mi ian e m
in Hsdesc ibes he a e ha quan um jumps a e applied o
he mo ional wa e unc ion, which co esponds o he a e E2
0
o inciden pho ons on he ca i y. The jump ope a o s hem-
sel es, J / =E0S / , wi h (J†
J +J†
J =E2
0), a e p opo ional o
he single-pho on sca e ing ma ix elemen s in e lec ion and
ansmission, encoding he wo p ocesses by which he o iginal
wa e unc ion can change by becoming en angled wi h a sca -
e ed pho on. In e es ingly, he cohe en pa o he po en ial,
Re[V(x)] in Eq. (3.18), is seen o a ise om he e m S ρS†
in
Eq. (3.29), and speci ically om he in e e ence be ween he in-
ciden and sca e ed componen s ( i s and second e ms on he
igh o Eq. (3.27), espec i ely).
3.6 quan um e ec s due o ze o-poin mo ion
We ha e al eady seen ha he sca e ing o a single pho on on
a ca i y con aining an a om leads o an en angled ou pu s a e
(3.26). This ou pu s a e desc ibes he coexis ence o he possi-
bili ies o pho on e lec ion and pho on ansmission and how
he wa e unc ion o he a om ge s modi ied o each o hose
e en s. We now p oceed o desc ibe some o he ele an obse -
a ional consequences.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

56 siba quan um
We can expand he posi ion-dependen ca i y de uning a ound
a esonan posi ion x (de ined by ∆c(x ) = 0) un il linea o de :
∆c(x)≈δc+gomu2(x ) − gom sin(2kcx )kc(x−x ). (3.31)
This is a good app oxima ion in he Lamb-Dicke egime ηLD 
1. In o de o p edic obse ables, linea izing displacemen is
also a good app oxima ion o gom κ, e en i ηLD ∼1, since
hen he ca i y equency shi s ou o esonance o displace-
men s kcδx 1. The e m sin(2kcx )indica es ha he ca i y
equency is mos sensi i e o displacemen s i kcx =±π/4,
hal way be ween a ca i y node and an i-node. Then i can be
seen ha i he a omic wa e unc ion is cen e ed a ound kcx0=
kcx =π/4, he ca i y equency shi s by a linewid h κ, i he
a om mo es a dis ance o kcR=κ/gom. As he ansmission/ e-
lec ion o a single, nea - esonan pho on changes signi ican ly
as i s equency a ies o e a ca i y linewid h, Rcan be iewed
as he spa ial esolu ion o e which he single pho on "lea ns"
abou he a omic posi ion ia i s sca e ing di ec ion. We will
now de ine he ze o-poin esolu ion
zp ≡(2xzp)/R = (2gm)/κ, (3.32)
wi h gm=gomηLD being he single-pho on, single phonon cou-
pling s eng h as de ined in Sec. 3.3.1. The ze o-poin esolu ion
ells us how much ine he esolu ion o an inciden pho on is
compa ed o he wid h o he a omic wa e unc ion. I dis in-
guishes wo egimes: un esol ed ze o-poin mo ion zp 1,
which co esponds o he usual egime o weak op omechan-
ical in e ac ions, and he esol ed ze o-poin mo ion egime
zp 1, whe e he esolu ion o he sys em becomes smalle
hen he ze o-poin mo ion, which is un il now unexplo ed and
which gi es ise o no el e ec s as we will demons a e in he
ollowing.
3.6.1In luence o he ze o-poin mo ion on he e lec ion spec um
He e, we assume he a om o be ini ially in i s mo ional g ound
s a e Ψ0(x)∝e−1
4(x−x0)2/x2
zp wi h a ap equilib ium kcx0=π/4
and κ =κ/2 (c i ical coupling). The spec um o e lec ion, as
a unc ion o he inciden pho on equency ωL, is hen gi en
by
p (ωL) = Zdx|S (ωL,x)|2|Ψ0(x)|2. (3.33)
Fig. 3.4a) shows p as a unc ion o ca i y de uning δc=ωL−
ωc o zp 1(un esol ed ze o-poin mo ion). The g een dashed
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
3.6 quan um e ec s due o ze o-poin mo ion 57
0.1 0.5 1 5 10 50
0.0
0.2
0.4
0.6
0.8
1.0
Figu e 3.5:Resolu ion beyond ze o-poin unce ain y
a) Fo zp =2, he spa ial wid h ∼2xzp o he a omic p oba-
bili y densi y |Ψ0(x)|2(blue) exceeds he spa ial esolu ion
R, which co esponds o he wid h o he absolu e alue
o he sca e ing ma ix |S (x)|2( ed dashed). As he ca i y
is only esonan wi h an incoming pho on i he a om is
loca ed wi hin R, he e is a la ge p obabili y ha he ca i y
is o - esonan , e en hough ωL=ωc(x0). The p obabil-
i y o e lec ion is calcula ed by he o e lap o bo h plo ed
unc ions.
b) Same as a), bu wi h he absolu e alue o he S-ma ix
o ansmission ( ed, dashed).
c) P obabili y o pho on e lec ion p ( ed) and ansmis-
sion p (g een) as a unc ion o ze o-poin esolu ion zp,
o an inciden pho on ha is esonan wi h he ca i y
in he limi ha a omic mo ion luc ua ions a e igno ed
(i.e., δc= −gomu2(x0)). One sees ha o la ge zp, he
p obabili y o ansmission becomes negligible, because
he p obabili y o inding he a om wi hin R(which would
imply a esonan sys em and consequen ansmission) ap-
p oaches ze o o zp 1.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
58 siba quan um
line is he e lec ion spec um o an emp y ca i y wi h decay
a e κ. The blue solid line is calcula ed wi h Eq. (3.33) o zp =
0.02, whe e p ≈|S (ωL,x0)|2. One can see ha i exhibi s he
same Lo en zian esponse as an emp y ca i y, bu wi h a eso-
nance equency shi ed by −gomu2(x0). Fig. 3.4b) shows he e-
lec ion spec um p o esol ed ze o-poin mo ion zp =2. We
obse e ha he p obabili y o e lec ion is s ongly inc eased
o δc= −gomu2(x0), compa ed o he case o small zp. This be-
ha io can be unde s ood om Eq. (3.31). In pa icula , he es-
onance equency o he coupled a om-ca i y sys em depends
on he posi ion o he a om, and δc= −gomu2(x0)co esponds
o he esonance o he mos likely a omic posi ion. Howe e ,
he la ge sp ead o he a omic wa e unc ion esul s in a la ge
unce ain y o he esonance equency, which inc eases he e-
lec ion p obabili y. Con e sely, an inciden pho on wi h e-
quency a om δc= −gomu2(x0)sees a dec eased e lec ion
p obabili y ( hus he b oadening o he spec um), as he e is
some chance ha he sp ead in a omic posi ion allows he cou-
pled sys em o be on esonance wi h he pho on. This is illus-
a ed in Fig. 3.5a), whe e we plo he a omic p obabili y den-
si y |Ψ0(x)|2(blue) and he absolu e alue o he e lec ion S-
ma ix |S (x)|2( ed dashed) (Eq. 3.27) as a unc ion o posi ion
xand o zp =2. One can see, ha he wid h o he a omic
wa e unc ion ∼2xzp exceeds he spa ial esolu ion R, wi hin
which he ca i y is esonan . Fo comple eness, we also p o ide
a plo o he absolu e alue o he ansmission S-ma ix |S (x)|2
( ed dashed) in Fig. 3.5b). Fig. 3.5c) shows he p obabili y o e-
lec ion and ansmission o δc= −gomu2(x0)as a unc ion o
zp. Fo zp 1 he p obabili y o e lec ion anishes and he
ansmission app oaches uni y as i would o an emp y eso-
nan ca i y. Howe e , wi h inc easing zp i becomes less likely
o ind he a om wi hin he spa ial esolu ion Rwi hin which
he ca i y is esonan , leading o an inc ease o p . Finally, he
e lec ion p obabili y p app oaches uni y o zp 1.
Mos o his plo is al eady expe imen ally accessible wi h
cu en echnology. Fo example a neu al a om apped in i s
g ound s a e inside pho onic c ys al ca i ies can each zp ∼10
(Appendix A.8.1) whe eas a cu en ibe ca i y expe imen
eaches zp ∼1(Appendix A.8.2). While measu ing p , he
ze o-poin esolu ion zp can hen be g adually dec eased by
inc easing he a om-ca i y de uning ω0−ωc, inc easing ap
equency ωmo by mo ing he ap equilib ium x0away om
he posi ion o maximal op omechanical coupling kcx0=±π/4.
This p ocedu e would expe imen ally ep oduce pa s o Fig. 3.5c).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
3.6 quan um e ec s due o ze o-poin mo ion 59
Figu e 3.6:Illus a ion o a single-pho on sca e ing e en o e-
sol ed ze o-poin mo ion
a) Inpu s a e: An inciden pho on (g een) wi h a e-
quency ensu ing x0=x is lying owa ds a ca i y con-
aining a apped a om wi h p obabili iy densi y |Ψ0(x)|2
(black). Due o i s ze o-poin unce ain y, he sys em is in
an e ec i e supe posi ion o esonance equencies. This
inpu s a e is gi en by Eq. (3.22).
b) Ou pu s a e: Illus a ion o he en angled ou pu s a e
gi en by Eq. (3.26), which is a supe posi ion o he pho-
on being e lec ed, which implies an o - esonan sys em
and a pho on being ansmi ed, which implies a esonan
sys em. The plo ed p obabili y densi ies |Ψ / (x)|2a e he
no malized p oduc o |Ψ0(x)|2and he espec i e sca e -
ing ma ix |S / (x)|2o Fig. 3.4a) and 3.4b), whe e zp =2.
Fo his alue, he p obabili y o e lec ion is p ≈0.56.
3.6.2En anglemen and condi ional p ojec ion o he a omic wa e
unc ion
Ha ing p e iously in es iga ed he uncondi ional e lec ion spec-
um o an inciden pho on, we now s udy mo e ca e ully he
co ela ions ha build up be ween he a omic mo ion and pho-
on e lec ion o ansmission o he case when he ap equi-
lib ium alls a he esonan posi ion (x0=x ). As he a om is in
a cohe en supe posi ion o being wi hin he spa ial esolu ion
Rand no , and an incoming pho on ge s ansmi ed i he a om
is wi hin ha spa ial esolu ion and e lec ed i o he wise, he
esul ing s a e (Eq. 3.26) is en angled. Gi en ha he pho on has
been ansmi ed, he no malized condi ional wa e unc ion is
gi en by
Ψ (x) = p−1/2
S (x)Ψ0(x). (3.34)
I s p obabili y densi y is p opo ional o he p oduc o |Ψ0(x)|2
and |S (x)|2as indi idually d awn in Fig. 3.5b). Thus, o zp 
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
66 pho on blockade
posi ion x0, he ca i y equency is gi en by ωc(x)≈ωc(x0) +
ω0
c(x0)(x−x0). The o al Hamil onian o he sys em, including
a cohe en ex e nal d i ing ield, is gi en in a ame o a ing
wi h he lase equency ωLby
Hop =ωmb†b− (ωL−ωc(x0) + iκ
2)a†a
+gm(b+b†)a†a+ κ
2E0(a†+a). (4.1)
He e, ωmis he equency o he ib a ional mode, and aand
bdeno e he pho on and phonon annihila ion ope a o s, e-
spec i ely. The quan i y ωL−ωc(x0)is he de uning be ween
lase equency ωLand he ca i y equency ωc(x0)when he
mechanical sys em lies a i s equilib ium posi ion. Each ca i y
mi o has a decay a e o κ/2 in o ou going adia ion, while
he le side also se es as he sou ce o injec ion o a cohe en
s a e in o he ca i y wi h pho on numbe lux E2
0. The posi ion-
dependen ca i y shi desc ibed p e iously has been e-w i en
in e ms o phonon ope a o s as ω0
c(x0)(x−x0) = gm(b+b†)
whe e gm=ω0
c(x0)xzp is he single pho on-phonon coupling
s eng h and xzp =p
h/(2me ωm)is he ze o-poin mo ional
unce ain y (me being he e ec i e mass). The cubic in e ac-
ion e m (b+b†)a†agi es ise o nonlinea equa ions o mo-
ion, bu quan um signa u es ha e no been obse ed, as he
bes a io o coupling s eng h o linewid h so a is gm/κ ∼
10−2[8,9]. Thus, cu en expe imen s emain in he so-called
op omechanical weak coupling egime, whe e many pho ons
inside he op ical mode a e equi ed o see an app eciable in-
e ac ion, and allowing o linea iza ion a ound he s ong clas-
sical ca i y ield. Howe e , he e we will ocus on he egime
whe e his linea iza ion b eaks down and he nonlinea na u e
o he op omechanical coupling mani es s i sel ia pho on co-
incidence measu emen s [46].
To quan i y he op omechanical nonlinea i y we change in o
a displaced oscilla o ep esen a ion, which diagonalizes Hop
in he limi o weak d i ing [46]. The eigen alues as E0→0
can hen be w i en as En,m =mωm+nωc(x0) − g2
m
ωmn2and co -
espond o he (displaced) eigens a es |n,mi. The spec um is
shown in Fig. 4.1b). I he lase equency is esonan wi h he
ansi ion |0c,0i → |1c,0i(ze o phonon line ≡ZPL) hen he
ansi ion o he second pho on is o esonan om he an-
si ion |1c,0i → |2c,0iby an amoun E2,0−2E1,0= −2g2
m/ωm.
In o de o ha e a subs an ial e ec , his anha monici y should
be esol able, g2
m/ωm&κ, and u he mo e, one should op-
e a e in he sideband esol ed egime ωm&κso ha ansi-
ions o o he mo ional s a es, e.g., he i s phonon sideband
|0c,0i → |1c,1ia e supp essed. These equi emen s o an i-
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

4.2 op omechanical pho on blockade 67
bunching can also be obse ed in Fig. 4.1c), whe e we ha e plo -
ed he second-o de co ela ion unc ion g(2)(0) o he ansmi -
ed ield gi en a weak cohe en s a e inpu o di e en alues
o κand gm, aking he lase equency ωLas being esonan
wi h he ZPL.
Fo mally, he quan um p ope ies o he ansmi ed ield
a e encoded in he inpu -ou pu ela ion aou ( ) = ain( ) +
pκ/2a( ). As he ex e nal d i ing ield is injec ed h ough he
o he mi o , he inpu ield in he ansmi ed po is he ac-
uum s a e, and hus he second-o de co ela ion unc ion g(2)(0) =
h(a†
ou )2a2
ou i/ha†
ou aou i2=h(a†)2a2i/ha†ai2depends only on
he in a-ca i y ield. We nume ically calcula e he necessa y ex-
pec a ion alues om he sys em wa e unc ion |Ψ( )i=Pn,mcn,m( )|n,mi
(whe e ndeno es he pho on numbe and m he phonon num-
be ), which we unca e o nmax > 2 (gi en ha a su icien ly
weak inpu s a e is unlikely o gene a e mo e han wo ca -
i y pho ons), and mmax depending on con e gence. In he case
o he pu e op omechanical Hamil onian Hop, we sol e o he
s eady-s a e ampli udes cn,m om he e ec i e Sch oedinge
equa ion i |˙
Ψ( )i=Hop |Ψ( )i. Then, ha†ai=Pm|c1,m|2+2|c2,m|2
and h(a†)2a2i=Pm2|c2,m|2. No e ha we neglec mechanical
damping as ou ue subspace o in e es consis s o apped
a oms. Fo mally, he inclusion o ca i y dissipa ion in he e ec-
i e wa e unc ion e olu ion mus be supplemen ed wi h s ochas-
ic quan um jumps [26]. Howe e , in he weak d i ing limi
E0→0 ha we conside he e, he e ec o jumps on obse ables
becomes anishingly small and hus we do no need o explic-
i ly accoun o hem. While we ha e explici ly discussed he
op omechanical Hamil onian Hop he e, he cases o he Jaynes-
Cummings model wi hou mo ion o Jaynes-Cummings model
including mo ion a e sol ed in an immedia ely simila ashion
in he ollowing.
A alue o g(2)(0)< 1 indica es non-classical an ibunching,
and a minimum alue occu s a ound a ound gm≈0.5ωm,
which o well- esol ed sidebands dec eases as g(2)(0)≈20(κ/ωm)2.
One also sees ha inc easing he a io gm/ωm u he does
no imp o e he amoun o an ibunching, due o he possibil-
i y o esonan ly coupling o o he exci ed s a es. Fo exam-
ple, a gm/ωm≈1/√2, he educed an ibunching a ises as
a second pho on can esonan ly exci e he s a e |2c,1i, since
E2,0−2E1,0= −ωm.
While ma hema ically he deg ee o an ibunching is de e -
mined by he pa ame e s gm,ωm,κ, i will also be help ul o
“ isualize” how he an ibunching changes as he equilib ium
posi ion x0is scanned om a ca i y an i-node o node, o p o-
ide a use ul compa ison wi h a oms la e . Fo a weak dielec ic
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
68 pho on blockade
|
0
,
0
〉
|
1
0
〉
|
2
,
0
〉
|
1
,
1
〉
,
Figu e 4.1: Op omechanical pho on blockade. a) A memb ane wi h
equilib ium posi ion x0inside a ca i y wi h in ensi y
mode p o ile u2(x), which is d i en wi h numbe lux E2
0
om he le . Each mi o has a decay a e o κ/2. The
pho ons a e measu ed on he ansmi ing side o he ca -
i y ( igh ). b) Spec um o he op omechanical Hamil o-
nian Hop o E0→0. He e, |n,mideno es he s a e wi h
npho ons and mphonons. In his diag am, we ocus on
ansi ions in ol ing s a es wi h m=0phonons (black
lines), while o he s a es (m=1shown he e) a e deno ed
by g ay lines. A lase wi h equency ωL, which is eso-
nan wi h he ansi ion |0c,0i → |1c,0i( he ze o-phonon
line), canno esonan ly exci e a second pho on |2c,0ias
op omechanical in e ac ions shi he ela i e ene gy o
his s a e by an amoun 2g2
m/ωm.c) No malized second-
o de co ela ion unc ion o he ansmi ed ield, g(2)(0),
as a unc ion o gm/ωmand κ/ωm.d) Top: g(2)(0)as
a unc ion o equilib ium posi ion x0and de uning om
he emp y ca i y δc=ωL−ωc, no malized by he ap
equency ωm. The mechanical sys em is coupled o an
in ensi y mode p o ile u2(x) = cos2(kcx), whe e kcis
he wa e ec o o he ca i y mode. The dashed ed/black
lines deno e a de uning whe e he ca i y is esonan ly
d i en on he ze o phonon line (ZPL)/ i s phonon side-
band, espec i ely. Bo om: g(2)(0)along he ZPL. The pa-
ame e s chosen o Fig. 4.1d) a e gm0=2π ×0.16 MHz,
κ=2π ×0.02 MHz, ωm=2π ×0.2MHz.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
4.3 ca i y qed wi hou mo ion 69
pe u ba ion such as a hin memb ane, in ui i ely one expec s
ha he a ia ion in he ca i y equency ollows he in ensi y
p o ile o he s anding wa e i sel , δωc(x)∝−cos2(kcx)[22,
120]. I ollows hen ha gm(x0) = gm0sin(2kcx0). In pa icula ,
gm(x0) anishes a a node o an i-node, and eaches he maxi-
mum possible alue o gm0hal way be ween. In Fig. 4.1d) we
plo g(2)(0)as a unc ion o apping posi ion x0and de uning
om he emp y ca i y δc=ωL−ωc o a mechanical sys em
ini ially in i s g ound s a e. The dashed ed line co esponds o
a d i ing lase esonan wi h he ZPL, which equi es he lase
equency o be uned ollowing he ene gy eigen alue E1,0. In
addi ion o he ea u es along he ZPL, an ibunching can also
be obse ed when a mo ional sideband |1c,miis esonan ly
d i en, ollowing he equa ion |ωL=E1,mi(see black dashed
cu e o m=1). Below, we plo g(2)(0) ollowing he ZPL ( ed,
dashed). The oscilla ions in g(2)(0)along he ZPL e sus x0oc-
cu as gm(x0) sweeps in o and away om he op imal alues
o an ibunching (compa e wi h Fig. 4.1c)). He e, we ha e cho-
sen pa ame e s o gm0 =2π ×0.16 MHz, κ=2π ×0.02 MHz
and ωm=2π ×0.2MHz. These do no necessa ily co espond
o a physically ealizable op omechanical sys em, bu allow he
in e es ing ea u es o be obse ed.
4.3 ca i y qed wi hou mo ion
We now conside an a om coupled o a ca i y mode wi h ampli-
ude u(x) = cos(kcx)(see Fig. 4.2a)), which is desc ibed by he
Jaynes-Cummings (J-C) Hamil onian [105]. Due o he wo-le el
na u e o he a om, he spec um o he J-C Hamil onian is non-
linea . We hus s udy he e ec o his nonlinea i y on g(2)(0)
i s wi hou mo ion (i.e., he a om is in ini ely igh ly apped),
so ha we can la e clea ly dis inguish mo ional e ec s. The J-C
Hamil onian, in an in e ac ion pic u e o a ing a ωL, is gi en
by
HJC =−(δ0+iγ
2)σee − (δc+iκ
2)a†a
+ κ
2E0(a+a†) + g0u(x0)(a†σge +h.c.). (4.2)
The lase -a om de uning is δ0=ωL−ω0wi h ω0being he
esonance equency o he a om, while σ¸˛ =|αihβ|, whe e
α,β=g,eco espond o combina ions o he a omic g ound
and exci ed s a es. As be o e, δc=ωL−ωcis he de uning
ela i e o he ba e ca i y esonance. The a om-ca i y coupling
s eng h g0u(x0)depends on he apping posi ion x0, whe e g0
is he magni ude o he acuum Rabi spli ing a he an i-node
a he ca i y wais . The emission a e o an exci ed a om in o
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
70 pho on blockade
Figu e 4.2: Ca i y QED wi hou mo ion. a) Schema ic o an a om in-
ini ely igh ly apped inside a ca i y mode a posi ion x0.
The ca i y and a omic exci ed s a e decay a es a e κand γ,
espec i ely. b) Second-o de co ela ion unc ion g(2)(0)
o he ansmi ed ield, as a unc ion o apping posi ion
x0and de uning om he emp y ca i y δc=ωL−ωc,
no malized by he ca i y linewid h κ. He e, we es ic
ou sel es o d i ing equencies nea he esonance o he
pho on-like d essed s a e o he Jaynes-Cummings model.
To gene a e his plo , we ake idealized pa ame e s such
ha an ibunching a ising om s ong a om-ca i y cou-
pling can be easily seen: ∆=3g0,g0=2π ×2MHz,
κ=γ=2π ×0.02 MHz.
ee space is gi en by γ.
Igno ing dissipa i e p ocesses o he momen , he sys em
is block diagonal o n o al exci a ions in he sys em, wi h
possible s a es |g,ni,|e,n−1i. The ene gy eigen alues in each
block a e gi en by E±
n=nωc+ (±q4g2
0u2(x0)n+∆2+∆)/2,
whe e ∆=ω0−ωc. In he ollowing we conside he dis-
pe si e egime ∆g0,κ,γ, whe e he single-exci a ion eigen-
s a es o he J-C Hamil onian a e ei he mos ly a omic (|ψ+i ≈
|e,0i) o pho onic (|ψ−i ≈ |g,1i). These eigens a es ha e co e-
sponding eigenene gies E+
1≈ω0+g2
0
∆u2(x0)and E−
1≈ωc−
g2
0
∆u2(x0), espec i ely. He e, we ocus on he case when he sys-
em is d i en nea esonan ly wi h he pho onic eigens a e. In
ha limi , he a om can app oxima ely be iewed as a classi-
cal dielec ic ha p o ides a posi ion-dependen ca i y shi
∝g2
0
∆. Howe e , he wo-le el na u e o he a om p o ides a
esidual nonlinea i y o exci e a second pho on, o magni ude
E−
2−2E−
1≈2(g4
0/∆3)u4(x0). Such a nonlinea i y esul s in an
an i-bunched ansmi ed ield i i is compa able o he ca -
i y linewid h κ. In Fig. 4.2b) we plo g(2)(0) o ∆=3g0, as
a unc ion o a om posi ion x0and de uning δc, o equen-
cies a ound he pho onic eigenene gy E−
1(do ed line). He e,
we ha e chosen idealized pa ame e s g0=2π ×2MHz, κ=
γ=2π ×0.02 MHz, which enable he an ibunching ea u es
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
4.4 ull model:ca i y qed wi h mo ion 71
o be clea ly seen. Wi hou mo ion, he la ges deg ee o an i-
bunching na u ally occu s a ound he an i-node (x0=0)and
mono onically dec eases as one app oaches he nodes.
4.4 ull model:ca i y qed wi h mo ion
We now include a omic mo ion in o he Jaynes-Cummings Hamil-
onian H=ωmb†b+HJC by ea ing x0→xas a dynamical
a iable. We assume ha he a om sees an in e nal-s a e inde-
penden and ha monic apping po en ial, which occu s na -
u ally o apped ions o using magic wa eleng h aps o
neu al a oms [121]. In Fig. 4.3a), we plo g(2)(0)as a unc ion
o lase -ca i y de uning δcand he cen al posi ion x0o he
ap, o pa ame e s g0=2π ×10 MHz, κ=γ=2π ×0.02 MHz,
∆=5g0,ωm=2π ×0.5MHz, and an a omic ecoil equency
ω ec =2π ×6.8kHz co esponding o a 40Ca+ion. I can be
seen ha his igu e cap u es a combina ion o he pu e J-C
plo (Fig. 4.2b) and pu e op omechanical plo (Fig. 4.1d), whe e
he la ges deg ee o an ibunching occu s a ound he an i-node
(x0=0)o in be ween he node and an i-node, espec i ely.
In pa icula , he p esence o sideband ea u es, and he ex-
ended an ibunching away om he an i-node a e quali a i e
signa u es o mo ional e ec s. Below we plo g(2)(0) ollowing
he ZPL ( ed, dashed). The egion o negligible an ibunching,
g(2)(0)≈1, a kcx0≈ ±π/8 o igina es om an exac cancella-
ion o he nonlinea i ies induced by mo ion and he wo-le el
na u e.
To be e unde s and he con ibu ion om mo ion, unde
ce ain condi ions one can e ec i ely map he J-C model o he
op omechanical Hamil onian. In pa icula , o la ge lase -a om
de unings δ0g0, he a omic g ound-s a e popula ion is ap-
p oxima ely one which allows o an e ec i e elimina ion o
he a omic exci ed s a e [107,122] using he Nakajima-Zwanzig
p ojec ion ope a o o malism [108,109]. In he Lamb-Dicke
egime ηLD =pω ec/ωm=kcxzp 1 he e ec i e op ome-
chanical Hamil onian (4.1) is ep oduced by eplacing gm→
ge wi h he e ec i e op omechanical coupling s eng h ge =
g2
0δ0/(δ2
0+γ2/4)ηLD sin(2kcx0)and κ→κe wi h he e ec i e
ca i y linewid h κe =κ+γg2
0/(δ2
0+γ2/4)u2(x0), b oadened
by a omic spon aneous emission (see Sec. 3.3.1and Appendix
A.5.1 o he de i a ion o he e ec i e op omechanical model:
ge =∆0
c(x0)xzp wi h Eq. (3.9) and κe o igina es om a e ag-
ing κ(x)wi h he a omic wa e unc ion in Eq. (3.11)). No e ha
δ0≈−∆ o ∆g0and when he sys em is d i en esonan ly
on he ZPL. Fo small ηLD, he nonlinea i y a ising om mo-
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

72 pho on blockade
Figu e 4.3: J-C model including mo ion. a) Top: g(2)(0)o he ans-
mi ed ield e sus apping posi ion x0and de uning
om he emp y ca i y δc=ωL−ωc, o de unings nea
he pho onic eigens a e and o a om-ca i y de uning ∆=
5g0. He e, we use idealized pa ame e s g0=2π ×10 MHz,
κ=γ=2π ×0.02 MHz , and ωm=2π ×0.5MHz so
ha all o he key ea u es can be clea ly obse ed. Below:
g(2)(0) ollowing he ZPL ( ed, dashed). b) We plo he
same as in Fig. 4.3a), bu using he pa ame e s o a e-
alis ic ca i y QED expe imen gi en below. In his igu e,
we choose ∆=12g0and ωm=2π ×0.1MHz. c) g(2)(0)
as a unc ion o a om-ca i y de uning ∆and apping e-
quency ωm.d) g(2)(0)as a unc ion o apping posi ion
x0and apping equency ωm o ∆=12g0. Fo Fig. 4.3b),
4.3c) and 4.3d) we choose pa ame e s g0=2π ×1.4MHz,
κ=2π ×0.05 MHz, γ=2π ×11 MHz and ecoil equency
ω ec =2π ×6.8kHz.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
4.5 mo ional pho on blockade in an exis ing expe imen 73
ion simply adds o ha a ising om he wo-le el na u e o
he a om, and he ene gy spec um eads
En,m ≈mωm+ ωc−g2
0
∆u2(x0)!n+ g4
0
∆3u4(x0) − g2
e
ωm!n2.
(4.3)
He e, ndeno es he numbe o exci a ions in he pho on-like
eigens a e o he J-C model. Thus, he essen ial ing edien s needed
o obse e a quan um nonlinea i y associa ed wi h he mo ion
a e g2
e /ωm&κe and ωm&κe (along wi h ηLD < 1, such
ha he a omic mo ion can be linea ized, see Appendix A.9
o second o de co ec ions). As he wo-le el and mo ional
anha monici ies scale wi h ∆−3and ∆−2, espec i ely, inc eas-
ing ∆se es as a way o make wo-le el an ibunching anish
while nonlinea mo ional e ec s pe sis . Fu he mo e, as he
maximum allowed alue o ge o e ain alidi y o he e ec-
i e model is ge ∼g0, one can see ha he ca i y QED s ong
coupling condi ion g0&κna u ally enables op omechanical
s ong coupling. Ac ually, he mo e con en ional c i e ion o
ca i y QED s ong coupling, g0> κ,γ, is no equi ed, as we
illus a e nex .
4.5 mo ional pho on blockade in an exis ing ex-
pe imen
To p esen he ealis ic possibili ies o obse ing op omechan-
ical blockade, we conside an exis ing ca i y QED se up wi h
apped 40Ca+-ions [123] wi h g0=2π ×1.4MHz, κ=2π ×
0.05 MHz and γ=2π ×11 MHz. No e ha wi hou mo ion, he
la ge spon aneous emission a e γg0in his pa icula se up
p e en s one om obse ing blockade a ising om he Jaynes-
Cummings ladde when he a om and ca i y a e on esonance.
Blockade canno be obse ed by wo king o esonance ei he ,
as he nonlinea i y in he spec um dec eases as e (∝∆−3)
han he a omic con ibu ion o he decay a e o he ca i y
(∝∆−2). Howe e , op omechanical blockade can be obse ed
as i s nonlinea i y dec eases also as ∆−2. In Fig. 4.3b) we plo
g(2)(0)as a unc ion o a om posi ion x0and de uning δc, o
∆=12g0and ωm=2π ×0.1MHz, and also o a de uning
δc ollowing he ZPL ( ed, dashed). As he maximal wo-le el
anha monici y 2(g4
0/∆3)u4(0)≈2π ×1.6kHz κe is a om
being esol ed, no pho on blockade occu s due o he wo-le el
na u e and hus no an ibunching can be seen a he an i-nodes.
Howe e , he mo ional nonlinea i y 2g2
e /ωm≈2π ×15 kHz is
almos an o de o magni ude la ge and allows a minimum
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
74 pho on blockade
alue o g(2)(0)≈0.83 d i ing he ZPL ( ed do ed line) a ound
kcx0≈π/3.
This alue ac ually ep esen s he op imum ha can be ob-
se ed a his posi ion, scanning o e he pa ame e s ωmand
∆/g0as we illus a e in Fig. 4.3c). Fo lowe alues o ∆, he side-
band esolu ion is los owing o he la ge alue o he a omic
spon aneous emission a e γand i s con ibu ion o he e ec-
i e ca i y linewid h κe (κe ≈2π ×84 kHz a he op imized
poin ). On he o he hand, o inc easing ωm, he magni ude o
he mo ional nonlinea i y 2g2
e /ωmbecomes educed, while o
dec easing ωmagain sideband esolu ion is los . No e as well
ha he an i-bunching is negligible o any de uning, when he
mo ion is ozen ou (ωm→∞). This dependence o g(2)(0)on
ωm e eals he pu e mo ional o igin o an ibunching. Fig. 4.3d)
shows g(2)(0)as a unc ion o a om posi ion x0and ap e-
quency ωm, o ∆=12g0and esonan ly d i ing he ZPL. He e
one again sees ha he an ibunching occu s only be ween he
nodes and an i-nodes, and he adeo in ωm.
4.6 conclusion
In conclusion, we ha e shown ha ca i y QED expe imen s ap-
p oaching he s ong coupling egime a e na u al pla o ms
o explo e he single-pho on, single-phonon s ong coupling
egime o op omechanics, in he limi ha he mo ional side-
bands can be esol ed. Since many o hose expe imen s, which
allow o he ealiza ion o mo ional nonlinea e ec s, al eady
exis , we an icipa e ha such pla o ms will s imula e much
heo e ical and expe imen al wo k o u he explo e he gene -
a ion o non-classical ligh om mo ion and i s consequences.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
Pa III
APPENDIX
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
82 appendix
Su p isingly his equa ion is alid o all ηas long as he pa i-
cle is con ined su icien ly close o he an inode. Fig. A.2shows
he excellen ag eemen be ween he nume ical simula ion and
he analy ic solu ion ob ained by Eq. (A.18). Taking he limi
η→∞o Eq. (A.18) implies x →x which ep oduces Eq. (2.10)
o he main ex :
lim
η→∞hIexp(η)i =2c0
α(ω)
(x )
| 0(x )|
Ekin
x
(A.19)
a.4 op imiza ion o he ha monic back-ac ion egime
He e we wan o maximize he sp ing cons an kop =khb +kT
gi en by Eq. (2.11) in he main ex . khb =Pin0
i(x0)ω0
c,i(x0)de-
sc ibes he i s e m in Eq. (2.11) and o igina es om changes
o pho on numbe wi h pa icle posi ion, whe eas kTis he a-
milia e m known om op ical weeze s. The op imiza ion is
done o a ixed expe ienced in ensi y hIexpi i we conside wo
apping modes o a ca i y. As a esul we will de i e Eq. (2.12)
and Eq. (2.13) in he main ex and conclude how o op imally
choose he lase de unings o he apping modes.
We ocus on he egime whe e he ap minimum x0is loca ed
oughly a a dis ance ∼1
kη away om bo h esonan posi ions,
whe e he pho on numbe n(xp)can be linea ized a ound he
ap minimum x0 o each apping mode i:ni(x)≈ni(x0) +
n0
i(x0)(x−x0). A linea change in pho on numbe wi h displace-
men implies a ha monic ap, because he o ce is p opo ional
o he pho on numbe (see Eq. (2.4) and Eq. (2.6) in he main
ex and no e ha 0
i(x)≈ 0
i(x0)≈ 0
i(x i) o kδx 1). Using
Eq. (2.9) in he main ex , he e m p opo ional o n0
i(x0)does
no con ibu e o he ime-a e aged in ensi y due o he ha -
monic mo ion. Unde hese ci cums ances Eq. (A.16) is alid in
he ha monic back-ac ion egime as well and he pa icle expe i-
ences he ollowing ime a e aged in ensi y om each apping
mode i:
hIexp,ii ≈2E2
0κexc
hωL
κ2Vm
i(x0)
1+ (η 0
i(x i))2(x0−x i)2, (A.20)
whe e we linea ized he mode p o iles a ound hei esonan
posi ions in Eq. (2.9) in he main ex . This is a good app oxi-
ma ion i he he wid h o he in ensi y peaks is smalle han
he spa ial a ia ions o he mode p o iles, which is he case o
η1. Now can w i e he con ibu ions o he i s e m khb o
Eq. (2.11) in he main ex as:
khb,i≈4E2
0
κex
κη 0
i(x i)2η| 0
i(x i)(x0−x i)|
1+ (η 0
i(x i))2(x0−x i)22. (A.21)
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

A.4 nakajima-zwanzig 83
Exp essing he op ical weeze e m kTin he same way, we can
w i e kop ,iin e ms o hIexp,ii :
kop ,i=α(ω)
c0hIexp,ii
1
i(x0)"2 i
1+ 2
i
ηi 0
i(x i)2− 00
i(x0)#. (A.22)
i=|ηi 0
i(x i)(x i−x0)|physically desc ibes he a io be ween
hal o he wid h o an in ensi y peak 1
ηi 0
i(x i)and he dis ance
o he espec i e esonan posi ion o mode i om he ap min-
imum |x i−x0|. The sp ing cons an is maximized o i=1
o which Eq. (A.22) educes back o Eq. (2.12) in he main ex .
Fo ηi1and i=1 he con ibu ion o he sp ing cons an
p opo ional o 00
i(x0)can be neglec ed and he sp ing cons an
pu ely a ises om changes o pho on numbe s wi h pa icle
posi ion. In con as , o η1we can neglec he con ibu-
ion p opo ional o 0
i(x0)2 eaching again he op ical weeze
egime.
Eq. (2.13) in he main ex is de i ed by o ming he a io o
hese wo con ibu ions o he sp ing cons an khb
kTand compa -
ing he wo expe ienced in ensi ies necessa y o c ea e he same
sp ing cons an in each egime. To de i e his, we also assume
ha he apping modes consis o he i s and second modes
o a Fab y-Pe o ca i y, which ha e equal back-ac ion pa am-
e e s ηi. We also use ha hIexp,1i 2(x0)≈ hIexp,2i 1(x0)using
Eq. (2.4) in he main ex wi h Eq. (A.16) and | 0
1(x0)|≈| 0
2(x0)|
close o he ap minimum.
a.5 om he jaynes-cummings model including mo-
ion o an e ec i e model o mo ion only
Eq. (3.2) o he main ex desc ibes he ull mas e equa ion o a
mo ing wo-le el a om in e ac ing wi h a ca i y, in he p esence
o ca i y losses and a omic spon aneous emission. In he limi
whe e he ca i y is d i en nea esonan ly and he a om is a -
de uned, he a omic exci ed s a e can be elimina ed o yield
an e ec i e op omechanical sys em in ol ing jus he a omic
mo ion and he ca i y mode. One can go a s ep u he and
elimina e he ca i y mode, o yield he educed dynamics o jus
he a omic mo ion. The p ocedu e by which a ce ain deg ee o
eedom can be elimina ed om an open sys em is known as
he Nakajima-Zwanzig p ojec ion ope a o o malism [106,108,
109], which we now desc ibe he e.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
84 appendix
Figu e A.3: The comple e Hilbe space o he in e nal deg ees o ee-
dom o he a om. Pρ is he pa we a e in e es ed in and
he emainde is cha ac e ized by he p ojec ion ope a o
Q.
a.5.1P ojec ing ou he a omic exci ed s a e
Fi s , we wan o elimina e he a omic exci ed s a e om he ull
dynamics o Eq. 3.2. I is con enien o de ine a se o ope a o s
P,Q, which p ojec he en i e sys em densi y ma ix
ρ=|gihg|ρgg +|gihe|ρge +|eihg|ρeg +|eihe|ρee, (A.23)
in o he subspace spanned by |gihg|(which we wan o p ojec
he dynamics in o), and i s o hogonal 1−|gihg|. He e ρij =
hi|ρ|jia e he educed densi y ma ices o he educed Hilbe
space, which s ill con ain all o he exis ing deg ees o eedom.
Thus, we de ine a p ojec ion ope a o P:
Pρ =|gihg|ρgg (A.24)
and i s complemen a y
Qρ =|gihe|ρge +|eihg|ρeg +|eihe|ρee. (A.25)
I is s aigh o wa d o show P2=P,Q2=Q,QP =0,P+Q=1.
In Fig. A.3we d aw a simple pic u e o he ull Hilbe space o
he in e nal deg ees o eedom o he a om in o de o isualize
he pa o he Hilbe space we a e in e es ed in (desc ibed by
Pρ) and he pa we a e no (desc ibed by Qρ). We will now
di ide he supe -ope a o Lup in pa s acco ding o he way
hey ac on he Hilbe space desc ibing he in e nal deg ees o
eedom o he a om:
L=Lo+La+LI+J. (A.26)
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
A.5 nakajima-zwanzig 85
He e, Lo=Lm+Lcis composed o e ms ha do no ac on he
in e nal deg ees o eedom, wi h Lmand Lcdesc ibing espec-
i ely he apped a omic mo ion and he ba e dynamics o he
d i en ca i y mode:
Lmρ= −i[ωmb†b,ρ](A.27)
Lcρ=iδc[a†a,ρ]−i√κ E0[(a+a†),ρ]− κ
2a†aρ +ρa†a−2aρa†.
(A.28)
The supe -ope a o
Laρ=iδ[σee,ρ] − γ
2{σee,ρ}(A.29)
ac s on |eihg|,|gihe|,|eihe|( he subspace spanned by Q) and jus
mul iplies hose e ms by a c-numbe . I desc ibes e olu ion
and damping o he exci ed in e nal s a e o he a om.
LIρ= −i[g(x)(σega+σgea†),ρ](A.30)
ac s on all he s a es and all Hilbe spaces, desc ibing he in e -
ac ion o he a om wi h he ca i y ield and
Jρ =γσgee−ikcxρeikcxσeg (A.31)
desc ibes he spon aneous jump o he exci ed s a e o he a om
in o i s g ound s a e accompanied by a momen um ecoil. In
Fig. A.4we d aw a ows showing how hese supe -ope a o s
ac on di e en pa s o he Hilbe space o a omic in e nal
deg ees o eedom. We a e in e es ed in he dynamics o he
subspace Pρ, while accoun ing o luc ua ions in o Qρ. Thus,
only closed loops which s a and end in Pρ con ibu e o he
e olu ion o he educed densi y ma ix Pρ. To see how his
wo ks, we de ine =Pρ and w=Qρ and inse P+Q=1in o
Eq. (3.2):
˙ =P˙ρ=PLρ =PLPρ +PLQρ. (A.32)
Le us i s look a PLP:
PLPρ =P(Lo+La+LI+J)Pρ. (A.33)
To quickly iden i y anishing e ms we ake ad an age o Fig. A.4
by ollowing he pa h he supe -ope a o s ake us h ough he
Hilbe space applying hem om he igh o he le . He e a e
some examples:
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
86 appendix
Figu e A.4: The Hilbe space o he in e nal deg ees o eedom o
he a om. The no a ion is as ollows: The label o an a ow
co esponds o a Liou illian, while he di ec ion o he
a ow indica es he possible beginning and ending sub-
spaces o he Liou illian. Fo example, he ed a ow in-
dica es ha he Liou illian Jac ing on he subspace |eihe|
akes his subspace o |gihg|. Since we assume δ0o γ o
be much la ge han κand ωm, we can neglec he ac ion
o Lo=Lm+Lcdu ing a luc ua ion ou o Pρ, which we
indica e by c ossing hem ou in he igh - op co ne and
neglec ing hem in Eq. (A.35).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
A.5 nakajima-zwanzig 87
1. The e m PLIP:Pp ojec s in o he subspace |gihg|, while
LImaps a s a e om P o Q. Thus, ac ing again wi h P
causes his e m o anish.
2.PLaP:Pp ojec s in o |gihg|and we immedia ely see ha
Ladoes no ac on i , so his e m anishes.
3.PJP =0because Jdoes no ac on |gihg|.
A e iden i ying all anishing e ms, we ob ain:
˙ =Lo +P(J+LI)w(A.34)
and
˙w=QLI +Q(Lo+La+LI)w. (A.35)
No e ha wdesc ibes he e olu ion o he luc ua ions ou o
he subspace o in e es . As he imescale o hese luc ua ions
is se by δ0and γand we assume ha ei he δ0o γis much
la ge han bo h ωmand κ, we can neglec he ee e olu ion
o he ca i y o mo ion du ing one o hese luc ua ions and
app oxima e Low≈0in Eq. (A.35), as also indica ed in Fig. A.4.
Then he gene al solu ion o his equa ion eads:
w( ) = Z
0
dτeQ(Lo+La)( −τ)QLIw(τ) + Z
0
dτeQ(Lo+La)( −τ)QLI (τ)
(A.36)
whe e we se w(0) = 0as he ini ial condi ion. Now we plug
his equa ion wice in o Eq. (A.34) (i e a i ely) in o de o ca ch
a e m o he o de JL2
I:
˙ ( ) = Lo +P(J+LI)Z
0
dτeQ(Lo+La)( −τ)QLI (τ)
+P(J+LI)Z
0
dτeQ(Lo+La)( −τ)QLIZτ
0
dτ0eQ(Lo+La)( −τ0)QLI (τ0).
(A.37)
He e we neglec ed he e m p opo ional o w(τ0)since i p o-
duces only e ms ∝L3
Io highe . Again by ollowing he pa h
o how hese supe -ope a o s ac wi h Fig. A.4, we can quickly
iden i y which e ms anish since all con ibu ing e ms need
o ha e closed loops s a ing and ending in |gihg|. So we a e le
wi h:
˙ ( ) = Lo +PLIZ
0
dτe(Lo+La)( −τ)LI (τ)
+PJ Z
0
dτe(Lo+La)( −τ)LIZτ
0
dτ0e(Lo+La)( −τ0)LI (τ0).
(A.38)
A e ex ending he lowe in eg al bo de s o −∞(Ma ko ap-
p oxima ion), we ob ain Eq. (3.7) o he main ex .
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

88 appendix
a.5.2P ojec ing ou he ca i y ield
The nex s ep is o ind a mas e equa ion only con aining mo-
ional deg ees o eedom (pand x) o he a om as ope a o s.
In o de o ind his equa ion we need o use he Nakajima-
Zwanzig echnique o p ojec ou he ca i y mode om Eq. (3.7).
Fo he sake o simplici y we assume δ0γ(and hus g2
0
δ2
0+γ2
4≈
g2
0
δ2
0
) and κγin he ollowing, so we can igno e he a omic de-
cay channel o his de i a ion by app oxima ing Lom ≈Lκ. Fo
weak d i ing, we can es ic ou sel es o he pho on subspace
de ined by |0i,|1i. Subsequen ly, we can adop ou p ojec ion
ope a o o malism om abo e and w i e he densi y ope a o
as ollows:
ρ=|0ih0|ρ00 +|0ih1|ρ01 +|1ih0|ρ10 +|1ih1|ρ11 (A.39)
wi h ρij =hi|ρ|jibeing he educed densi y ma ix desc ibing
a omic mo ion. As we a e in e es ed in he subspace spanned
by |0ih0|we de ine an p ojec ion ope a o P:
Pρ =|0ih0|ρ00 (A.40)
and
Qρ =|0ih1|ρ01 +|1ih0|ρ10 +|1ih1|ρ11. (A.41)
We again decompose he o al Liou illian in pa s acco ding o
he way hey ac :
L=Lm+Lca +LD+J(A.42)
wi h Lmde ined in Eq. (A.27),
Lca ≈−i[−∆(x)a†a,ρ] − κ
2{a†a,ρ}(A.43)
and LDρ= −i√κ E0[a+a†,ρ], which desc ibes he in e ac ion
o he ca i y mode wi h an ex e nal cohe en lase d i e. Jρ =
κaρa†desc ibes he spon aneous decay o he ca i y mode. Now
we d aw in Fig. A.5a pic u e o he Hilbe space o he deg ees
o eedom o he ca i y, including he a ows which illus a e
how hese de ined supe -ope a o s ac . A simila p odecu e as
in Appendix A.5.1leads o he quan um mas e equa ion (3.13)
o he main ex desc ibing a omic mo ion.
a.6 single pho on sca e ing heo y
He e we p o ide de ails o he de i a ion o Eqs. (3.24) and
(3.25) in he main ex . Inse ing Eqs. (3.22) and (3.23) in o
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
A.6 single pho on sca e ing heo y 89
Figu e A.5: The Hilbe space o he single exci a ion subspace o he
ca i y. The label o an a ow co esponds o a Liou il-
lian, while he di ec ion o he a ow indica es he pos-
sible beginning and ending subspaces o he Liou illian.
Fo example, he ed a ow indica es ha he Liou illian J
ac ing on he subspace |1ih1| akes his subspace o |0ih0|.
As we assume κωm, we can neglec he ime e olu ion
due o he supe -ope a o Lmdu ing a luc ua ion ou o
Pρ.
Eq. (3.21) and mul iplying wi h h(ω0) / ,m| om he le gi es
us an equa ion o he S-ma ix elemen s:
S / ,n(ωL)δ(ωL−ω0−nωm) = h(ω0) / ,n|S|(ωL)le ,0i(A.44)
whe e ω0 e e s o he equency o he e lec ed o ansmi -
ed pho on. In he ollowing, we will es ablish a connec ion
be ween he S-ma ix elemen s, and he s anda d inpu -ou pu
o malism o ca i y QED [114]. Con enien ly, his connec ion
enables one o calcula e S-ma ix elemen s based upon knowl-
edge o he eigen alues and eigens a es o he sys em Hamil-
onian He . The inpu -ou pu equa ion s a es ha he ou pu
ield in each decay channel ( e lec ion/ ansmission) is he sum
o he inpu ield and he ield emi ed by he sca e ing cen e .
Fo example he inpu -ou pu equa ion o pho on e lec ion is
gi en by
aou ( ) = ain( ) − i√κ a( )(A.45)
whe e o no a ional con enience we lea e ou he subsc ip
“ ” in he inpu and ou pu po s. The sca e ing ope a o s
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
90 appendix
ain/ou (ω)a e connec ed o he inpu -ou pu Heisenbe g-Lange in
ope a o s ain/ou ( )by a simple Fou ie ans o m [113]
ain/ou (ω) = 1
√2π Zd eiω ain/ou ( ). (A.46)
Now we ocus on he S-ma ix o he p ocess o pho on e lec-
ion
S ,n(ωL)δ(ωL−ω0−nωm) = h0c,n|aou (ω0)a†
in(ωL)|0c,0i
(A.47)
whe e we exp essed he S-ma ix in e ms o sca e ing op-
e a o s a†
in(ωL)and aou (ω0)which c ea e in- and ou -going
monoch oma ic sca e ing s a es [125]. Using he inpu -ou pu
equa ion, one can e-w i e aou in e ms o he ca i y ield and
inpu ield, yielding
S ,n(ωL)δ(ωL−ω0−nωm) = δ(ωL−ω0)δn,0
−i√κ h0c,n|a(ω0)a†
in(ωL)|0c,0i.
(A.48)
Now we eplace he sca e ing ope a o s wi h he Fou ie ans-
o m o he co esponding inpu -ou pu ope a o s. The ma ix
elemen h0c,n|a( 0)a†
in( L)|0c,0i anishes o L> 0since
[a( 0),a†
in( L)] = 0 o L> 0and h0c|a†
in( L) = 0. Thus, we in-
oduce he ime o de ing ope a o Tmaking su e ha 0> L.
Then we ha e
h0c,n|T[a( 0)a†
in( L)] |0c,0i= −i√κ h0c,n|T[a( 0)a†( L)] |0c,0i,
(A.49)
whe e we eplaced ain( L)wi h a( L)using he inpu -ou pu
equa ion. The e m con aining he ou pu ope a o anishes as
[a( 0),a†
ou ( L)] = 0 o 0> L(which is al eady ensu ed by T)
and h0c|a†
ou ( L) = 0. Finally, we a i e a
S ,n(ωL)δ(ωL−ω0−nωm) = δ(ωL−ω0)δn,0−κ τn(ωL)(A.50)
wi h
τn(ωL) = 1
2π Zd Ld 0ei(ω0 0−ωL L)h0c,n|Ta( 0)a†( L)|0c,0i.
(A.51)
Fo he S-ma ix desc ibing he p ocess o pho on ansmission
we ob ain
S ,n(ωL)δ(ωL−ω0−nωm)=−√κ κ τn(ωL). (A.52)
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
A.7 he ull e ec i e heo y and i s alidi y 91
No e ha he S-ma ix o e lec ion S includes he e m δ(ωL−
ω0)δn,0desc ibing in e ac ion- ee e lec ion o pho ons. In con-
as , in he S-ma ix o ansmission S he e is no such e m,
since he inpu ield on he ansmi ing side o he ca i y is in
he acuum s a e and hus he ansmi ed ield is buil exclu-
si ely om he emission o pho ons by he sca e ing cen e . We
can w i e
ha( 0)a†( L)i=T haeLs( 0− L)aρ(0)i, (A.53)
whe e ρ(0) = |0c,0ih0c,0|and Lsρ= −i[He ,ρ] + κaρa†wi h
He desc ibed by Eq. (1.3) om he main ex . Since he e m
κaρa† educes he numbe o pho ons, i s con ibu ion anishes
as he co ela o conse es he numbe o pho ons. Thus, he
e olu ion o a( )is go e ned by He alone and o e alua ing
he S-ma ix we can e ec i ely use
a( ) = eiHe a†e−iHe . (A.54)
We u he exp ess
h0c,n|Ta( 0)a†( L)|0c,0i=Θ( L− 0)eiωnn Lh1c,n|e−iHe ( L− 0)|1c,0i
(A.55)
whe e eiωnn Lcoun s he ene gy o he c ea ed phonons du ing
he sca e ing p ocess and he s ep unc ion Θ( L− 0)which
anishes o L< 0ensu es ime o de ing. In o de o exp ess
he S-ma ix ully in e ms o eigen alues λβand eigens a es
|βio He wi h He |βi=λβ|βiwe inse a uni y ope a o 1=
Pβ|βihβ| igh be o e |1c,0i. The e o e we w i e
h1c,n|e−iHe ( L− 0)|1c,0i=X
βh1c,n|βie−iλβ( L− 0)hβ|1c,0i
(A.56)
whe e h1c,n|βiis he p ojec ion o he eigens a es |βiin o he
basis s a es h1c,n|. A e e alua ing he Fou ie ans o m in
Eq. (A.51) we a e le wi h
τn(ωL)=−iδ(ωL−ω0−nωm)X
βh1c,n|βi1
λβhβ|1c,0i. (A.57)
which oge he wi h Eq. (A.50) and (A.52) ep oduces Eq. (3.24)
and (3.25) in he main ex .
a.7 he ull e ec i e heo y and i s alidi y
He e we begin by gene alizing ou e ec i e heo y p esen ed in
he main ex (sec ions 3.3and 3.4) by including spon aneous
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
98 appendix
To ensu e ha he esul s a e no signi ican ly a ec ed by
his ela i ely la ge Lamb-Dicke pa ame e , we will now in-
clude he nex o de e m u(x)≈u(x0) + u0(x0)kc(x−x0) +
(1/2)u00(x0)(x−x0)2. In Fig. A.9a), we plo he adjus ed g(2)(0)
as a unc ion o a om posi ion x0and de uning δc. He e we
choose ∆=10g0and ωm=2π ×0.09 MHz in o de o minimize
g(2)(0)including quad a ic o de co ec ions. Fig. A.9b) shows
g(2)(0)as a unc ion o a om posi ion x0 ollowing he ZPL o
a) (blue). In ed we plo g(2)(0), whe e u(x)has only been ex-
panded un il linea o de o he same pa ame e s. We obse e
a easonable ma ch and conclude ha linea izing mo ion on he
Hamil onian le el a leas quali a i ely ully cap u es he ele-
an physics e en o ela i ely la ge ηLD. Fo comple eness, we
plo g(2)(0)as a unc ion o ωmand ∆in Fig. A.9c) o a ixed
a omic posi ion kcx0=1.15, and in Fig. A.9d) we plo g(2)(0)as
a unc ion o apping posi ion x0and ap equency ωm o
∆=10g0.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]

A.9 nex o de co ec ion 99
Figu e A.9: J-C model wi h mo ion expanding u(x)un il quad a ic
o de . a) g(2)(0)o he ansmi ed ield e sus apping
posi ion x0and de uning om he emp y ca i y δc=
ωL−ωc, o de unings nea he pho onic eigens a e and
by using he pa ame e s o a ealis ic ca i y QED expe -
imen gi en below. In his igu e, we choose an a om-
ca i y de uning ∆=10g0and a omic ap equency
ωm=2π ×0.09 MHz, which p oduces he minimum pos-
sible g(2)(0)including quad a ic o de co ec ions. b) Fol-
lowing he ZPL o a) ( ed, dashed). W compa e g(2)(0)
calcula ed wi h only linea displacemen s ( ed) in Hamil-
onian Eq. (4.2) o he main ex wi h g(2)(0)calcula ed by
also including e ms o quad a ic o de (blue). c) g(2)(0)
as a unc ion o a om-ca i y de uning ∆and apping
equency ωmincluding e ms o quad a ic o de . o -
de . He e, he a omic posi ion is ixed a kcx0=1.15.d)
g(2)(0)as a unc ion o apping posi ion x0and apping
equency ωm o ∆=10g0including e ms o quad a ic
o de . As in he main ex , we choose pa ame e s o an ex-
is ing ca i y QED expe imen wi h apped 40Ca+-ions:
g0=2π ×1.4MHz, κ=2π ×0.05 MHz, γ=2π ×11 MHz
and ecoil equency ω ec =2π ×6.8kHz.
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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