Novel regimes of quantum optomechanics
Abstract
Institutional repository that preserves and disseminates the academic and scientific output of the institution.
Full text
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
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
Lukas Neumeie : No el egimes o quan um op omechanics, PhD
Thesis, © Ap il 2018
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
Fü Valen in
[ 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]
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
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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
[5de ab il de 2018 a 10:42 –classic hesis e sion 4.2]
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
[ 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]
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
[ 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]
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
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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].
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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, gmκ, 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 gmκ, 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
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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].
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
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 Γmgκ, 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 =g2κ
(¯
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 κωma 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 η1. 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 =2c0
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 ≈4c0
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)≈4c0
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 η1. 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)
c0hIexp,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
2J†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 κωm. 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 δ0g0, 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 γg0in 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 =2c0
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
η1. 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
κη 0
i(x i)2η| 0
i(x i)(x0−x i)|
1+ (η 0
i(x i))2(x0−x i)22. (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)
c0hIexp,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 ηi1and 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 η1we 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
2a†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 δ0γ(and hus g2
0
δ2
0+γ2
4≈
g2
0
δ2
0
) and κγin 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 κωm, 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]
BIBLIOGRAPHY
1. Ashkin, A. Accele a ion and apping o pa icles by a-
dia ion p essu e. Physical e iew le e s 24,156 (1970).
2. Tsuda, Y e al. Fligh s a us o IKAROS deep space sola
sail demons a o . Ac a As onau ica 69,833–840 (2011).
3. Hänsch, T. W. & Schawlow, A. L. Cooling o gases by
lase adia ion. Op ics Communica ions 13,68–69 (1975).
4. Ashkin, A. Op ical apping and manipula ion o neu al
pa icles using lase s. P oceedings o he Na ional Academy
o Sciences 94,4853–4860 (1997).
5. Chan, J. J. Chan, TPM Aleg e, AH Sa a i-Naeini, JT Hill,
A. K ause, S. G öblache , M. Aspelmeye , and O. Pain e ,
Na u e (London) 478,89 (2011). Na u e (London) 478,89
(2011).
6. Teu el, J., Ha low, J., Regal, C. & Lehne , K. Dynamical
backac ion o mic owa e ields on a nanomechanical os-
cilla o . Physical e iew le e s 101,197203 (2008).
7. Law, C. In e ac ion be ween a mo ing mi o and adi-
a ion p essu e: A Hamil onian o mula ion. Physical Re-
iew A 51,2537 (1995).
8. Meenehan, S. M. e al. Silicon op omechanical c ys al es-
ona o a millikel in empe a u es. Physical Re iew A 90,
011803 (2014).
9. Pi kkalainen, J.-M. e al. Ca i y op omechanics media ed
by a quan um wo-le el sys em. Na u e communica ions 6,
6981 (2015).
10. Co bi , T. e al. An all-op ical ap o a g am-scale mi o .
Physical e iew le e s 98,150802 (2007).
11. Kleckne , D. & Bouwmees e , D. Sub-kel in op ical cool-
ing o a mic omechanical esona o . Na u e 444,75 (2006).
12. A cize , O., Cohadon, P.-F., B ian , T., Pina d, M. & Hei-
dmann, A. Radia ion-p essu e cooling and op omechani-
cal ins abili y o a mic omi o . Na u e 444,71 (2006).
13. Gigan, S. e al. Sel -cooling o a mic omi o by adia ion
p essu e. Na u e 444,67 (2006).
14. Thompson, J., Zwickl, B., Jayich, A., Ma qua d , F., Gi in,
S. & Ha is, J. S ong dispe si e coupling o a high- inesse
ca i y o a mic omechanical memb ane. Na u e 452,72
(2008).
101
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
102 bibliog aphy
15. Kippenbe g, T., Rokhsa i, H, Ca mon, T, Sche e , A & Va-
hala, K. Analysis o adia ion-p essu e induced mechan-
ical oscilla ion o an op ical mic oca i y. Physical Re iew
Le e s 95,033901 (2005).
16. Schliesse , A., Del’Haye, P., Nooshi, N., Vahala, K. & Kip-
penbe g, T. Radia ion p essu e cooling o a mic omechan-
ical oscilla o using dynamical backac ion. Physical Re-
iew Le e s 97,243905 (2006).
17. Regal, C., Teu el, J. & Lehne , K. Measu ing nanome-
chanical mo ion wi h a mic owa e ca i y in e e ome e .
Na u e Physics 4,555 (2008).
18. Ga a in, E. e al. Op omechanical coupling in a wo-dimensional
pho onic c ys al de ec ca i y. Physical e iew le e s 106,
203902 (2011).
19. Kiesel, N., Blase , F., Deli´c, U., G ass, D., Kal enbaek, R.
& Aspelmeye , M. Ca i y cooling o an op ically le i a ed
submic on pa icle. P oceedings o he Na ional Academy o
Sciences 110,14180–14185 (2013).
20. Kuhn, S. e al. Ca i y-assis ed manipula ion o eely o-
a ing silicon nano ods in high acuum. Nano le e s 15,
5604–5608 (2015).
21. Genoni, M. G., Zhang, J., Millen, J., Ba ke , P. F. & Se -
a ini, A. Quan um cooling and squeezing o a le i a -
ing nanosphe e ia ime-con inuous measu emen s. New
Jou nal o Physics 17,073019 (2015).
22. Chang, D. E. e al. Ca i y op o-mechanics using an op-
ically le i a ed nanosphe e. P oceedings o he Na ional
Academy o Sciences 107,1005–1010 (2010).
23. Giesele , J., Deu sch, B., Quidan , R. & No o ny, L. Sub-
kel in pa ame ic eedback cooling o a lase - apped nanopa -
icle. Physical e iew le e s 109,103603 (2012).
24. Mes es, P., Be helo , J., Spaseno i´c, M., Giesele , J., No o ny,
L. & Quidan , R. Cooling and manipula ion o a le i a ed
nanopa icle wi h an op ical ibe ap. Applied Physics Le -
e s 107,151102 (2015).
25. Kippenbe g, T. J. & Vahala, K. J. Ca i y op omechanics:
back-ac ion a he mesoscale. science 321,1172–1176 (2008).
26. Meys e, P. & Sa gen , M. Elemen s o quan um op ics (Sp inge
Science & Business Media, 2007).
27. Ma qua d , F., Chen, J. P., Cle k, A. & Gi in, S. Quan um
heo y o ca i y-assis ed sideband cooling o mechanical
mo ion. Physical Re iew Le e s 99,093902 (2007).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
bibliog aphy 103
28. Teu el, J. e al. Sideband cooling o mic omechanical mo-
ion o he quan um g ound s a e. Na u e 475,359 (2011).
29. Shea d, B. S., G ay, M. B., Mow-Low y, C. M., McClelland,
D. E. & Whi comb, S. E. Obse a ion and cha ac e iza ion
o an op ical sp ing. Physical Re iew A 69,051801 (2004).
30. Vogel, M, Moose , C, Ka ai, K & Wa bu on, R. Op ically
unable mechanics o mic ole e s. Applied Physics Le e s
83,1337–1339 (2003).
31. Schumake , B. L. Quan um mechanical pu e s a es wi h
Gaussian wa e unc ions. Physics Repo s 135,317–408 (1986).
32. Pa kins, A. & Kimble, H. Quan um s a e ans e be ween
mo ion and ligh . Jou nal o Op ics B: Quan um and Semi-
classical Op ics 1,496 (1999).
33. Zhang, J., Peng, K. & B auns ein, S. L. Quan um-s a e
ans e om ligh o mac oscopic oscilla o s. Physical Re-
iew A 68,013808 (2003).
34. Vi ali, D. e al. Op omechanical en anglemen be ween a
mo able mi o and a ca i y ield. Physical e iew le e s
98,030405 (2007).
35. Genes, C, Ma i, A, Tombesi, P & Vi ali, D. Robus en-
anglemen o a mic omechanical esona o wi h ou pu
op ical ields. Physical Re iew A 78,032316 (2008).
36. Pa e nos o, M. e al. C ea ing and p obing mul ipa i e
mac oscopic en anglemen wi h ligh . Physical Re iew Le -
e s 99,250401 (2007).
37. Chang, D., Sa a i-Naeini, A. H., Ha ezi, M. & Pain e ,
O. Slowing and s opping ligh using an op omechanical
c ys al a ay. New Jou nal o Physics 13,023003 (2011).
38. And ews, R. W. e al. Bidi ec ional and e icien con e -
sion be ween mic owa e and op ical ligh . Na u e Physics
10,321 (2014).
39. Galland, C., Sangoua d, N., Pi o, N., Gisin, N. & Kippen-
be g, T. J. He alded single-phonon p epa a ion, s o age,
and eadou in ca i y op omechanics. Physical e iew le -
e s 112,143602 (2014).
40. Nunnenkamp, A, Bø kje, K, Ha is, J. & Gi in, S. Cool-
ing and squeezing ia quad a ic op omechanical coupling.
Physical Re iew A 82,021806 (2010).
41. Ganga , A. A., S ace, T. M. & Milbu n, G. J. Phonon num-
be quan um jumps in an op omechanical sys em. New
Jou nal o Physics 13,043024 (2011).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
104 bibliog aphy
42. Sankey, J. C., Yang, C., Zwickl, B. M., Jayich, A. M. &
Ha is, J. G. S ong and unable nonlinea op omechan-
ical coupling in a low-loss sys em. Na u e Physics 6,707
(2010).
43. Thompson, J., Zwickl, B., Jayich, A., Ma qua d , F., Gi in,
S. & Ha is, J. S ong dispe si e coupling o a high- inesse
ca i y o a mic omechanical memb ane. Na u e 452,72
(2008).
44. Miao, H., Danilishin, S., Co bi , T. & Chen, Y. S anda d
quan um limi o p obing mechanical ene gy quan iza-
ion. Physical e iew le e s 103,100402 (2009).
45. Hu ne , B. & Ba ne , S. M. Quan iza ion o he elec o-
magne ic ield in dielec ics. Physical Re iew A 46,4306
(1992).
46. Rabl, P. Pho on blockade e ec in op omechanical sys-
ems. Physical e iew le e s 107,063601 (2011).
47. Juan, M. L., Go don, R., Pang, Y., E ekha i, F. & Quidan ,
R. Sel -induced back-ac ion op ical apping o dielec ic
nanopa icles. Na u e Physics 5,915 (2009).
48. Descha mes, N., Dha anipa hy, U. P., Diao, Z., Tonin, M.
& Houd é, R. Obse a ion o backac ion and sel -induced
apping in a plana hollow pho onic c ys al ca i y. Phys-
ical e iew le e s 110,123601 (2013).
49. Mes es, P., Be helo , J., A´cimo i´c, S. S. & Quidan , R.
Un a eling he op omechanical na u e o plasmonic ap-
ping. Ligh : Science & Applica ions 5,e16092 (2016).
50. Bi nbaum, K. KM Bi nbaum, A. Boca, R. Mille , AD Booze ,
TE No hup, and HJ Kimble, Na u e (London) 436,87
(2005). Na u e (London) 436,87 (2005).
51. Reise e , A. & Rempe, G. Ca i y-based quan um ne wo ks
wi h single a oms and op ical pho ons. Re iews o Mode n
Physics 87,1379 (2015).
52. Volz, J., Scheuche , M., Junge, C. & Rauschenbeu el, A.
Nonlinea πphase shi o single ib e-guided pho ons
in e ac ing wi h a single esona o -enhanced a om. Na-
u e Pho onics 8,965 (2014).
53. Shom oni, I., Rosenblum, S., Lo sky, Y., Bechle , O., Guen-
delman, G. & Dayan, B. All-op ical ou ing o single pho-
ons by a one-a om swi ch con olled by a single pho on.
Science 345,903–906 (2014).
54. Gu höh lein, G., Kelle , M, Hayasaka, K, Lange, W &
Wal he , H. A single ion as a nanoscopic p obe o an op-
ical ield. Na u e 414,49 (2001).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
bibliog aphy 105
55. Mund , A. e al. Coupling a single a omic quan um bi
o a high inesse op ical ca i y. Physical e iew le e s 89,
103001 (2002).
56. Russo, C e al. Raman spec oscopy o a single ion cou-
pled o a high- inesse ca i y. Applied Physics B 95,205–
212 (2009).
57. Leib and , D. R., Labaziewicz, J., Vule i´c, V. & Chuang,
I. L. Ca i y sideband cooling o a single apped ion. Phys-
ical e iew le e s 103,103001 (2009).
58. S e k, J., Luo, L, Manning, T., Maunz, P & Mon oe, C.
Pho on collec ion om a apped ion-ca i y sys em. Phys-
ical Re iew A 85,062308 (2012).
59. S eine , M., Meye , H. M., Deu sch, C., Reichel, J. & Köhl,
M. Single ion coupled o an op ical ibe ca i y. Physical
e iew le e s 110,043003 (2013).
60. Takahashi, H., Kassa, E., Ch is o o ou, C. & Kelle , M.
Ca i y-induced an ico ela ed pho on-emission a es o
a single ion. Physical Re iew A 96,023824 (2017).
61. Black, A. T., Chan, H. W. & Vule i´c, V. Obse a ion o col-
lec i e ic ion o ces due o spa ial sel -o ganiza ion o
a oms: om Rayleigh o B agg sca e ing. Physical e iew
le e s 91,203001 (2003).
62. Pu dy, T. P., B ooks, D., Bo e , T., B ahms, N., Ma, Z.-Y. &
S ampe -Ku n, D. M. Tunable ca i y op omechanics wi h
ul acold a oms. Physical e iew le e s 105,133602 (2010).
63. Li, T. in Fundamen al Tes s o Physics wi h Op ically T apped
Mic osphe es 81–110 (Sp inge , 2013).
64. Kiesel, N., Blase , F., Deli´c, U., G ass, D., Kal enbaek, R.
& Aspelmeye , M. Ca i y cooling o an op ically le i a ed
submic on pa icle. P oceedings o he Na ional Academy o
Sciences 110,14180–14185 (2013).
65. Millen, J, Fonseca, P, Ma ogo da os, T, Mon ei o, T &
Ba ke , P. Op omechanical cooling o a le i a ed nanosphe e
in a hyb id elec o-op ical ap. P ep in a h p://a xi . o g/abs/1407.3595
(2014).
66. Ranji , G., A he on, D. P., S u z, J. H., Cunningham, M.
& Ge aci, A. A. A onew on o ce de ec ion using mic o-
sphe es in a dual-beam op ical ap in high acuum. Phys-
ical Re iew A 91,051805 (2015).
67. Ashkin, A. & Dziedzic, J. M. Op ical apping and ma-
nipula ion o i uses and bac e ia. Science 235,1517–1520
(1987).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
106 bibliog aphy
68. Liu, Y, Sonek, G., Be ns, M. & T ombe g, B. Physiological
moni o ing o op ically apped cells: assessing he e ec s
o con inemen by 1064-nm lase weeze s using mic o lu-
o ome y. Biophysical Jou nal 71,2158–2167 (1996).
69. MacDonald, M., Spalding, G. & Dholakia, K. Mic o luidic
so ing in an op ical la ice. Na u e 426,421 (2003).
70. Wang, M. D., Yin, H., Landick, R., Gelles, J. & Block, S. M.
S e ching DNA wi h op ical weeze s. Biophysical jou nal
72,1335–1346 (1997).
71. Pang, Y., Song, H., Kim, J. H., Hou, X. & Cheng, W. Op i-
cal apping o indi idual human immunode iciency i uses
in cul u e luid e eals he e ogenei y wi h single-molecule
esolu ion. Na u e nano echnology 9,624 (2014).
72. Volpe, G., Quidan , R., Badenes, G. & Pe o , D. Su ace
plasmon adia ion o ces. Physical e iew le e s 96,238101
(2006).
73. Nie o-Vespe inas, M, Chaume , P., Rahmani, A., e al. Nea -
ield pho onic o ces. Philosophical T ansac ions-Royal Soci-
e y o London Se ies A Ma hema ical Physical and Enginee -
ing Sciences, 719–738 (2004).
74. G igo enko, A., Robe s, N., Dickinson, M. & Zhang, Y.
Nanome ic op ical weeze s based on nanos uc u ed sub-
s a es. Na u e Pho onics 2,365 (2008).
75. Righini, M., Zelenina, A. S., Gi a d, C. & Quidan , R.
Pa allel and selec i e apping in a pa e ned plasmonic
landscape. Na u e Physics 3,477 (2007).
76. Asano, T., Song, B.-S. & Noda, S. Analysis o he expe i-
men al Q ac o s (˜ 1million) o pho onic c ys al nanoca -
i ies. Op ics exp ess 14,1996–2002 (2006).
77. Boh en, C. F. & Hu man, D. R. Abso p ion and sca e ing
o ligh by small pa icles (John Wiley & Sons, 2008).
78. Co bi , T. e al. An all-op ical ap o a g am-scale mi o .
Physical e iew le e s 98,150802 (2007).
79. Hamme e , K. e al. S ong coupling o a mechanical oscil-
la o and a single a om. Physical e iew le e s 103,063005
(2009).
80. Juan, M. L., Righini, M. & Quidan , R. Plasmon nano-
op ical weeze s. Na u e Pho onics 5,349 (2011).
81. Hung, C., Meenehan, S., Chang, D., Pain e , O & Kimble,
H. T apped a oms in one-dimensional pho onic c ys als.
New Jou nal o Physics 15,083026 (2013).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]
bibliog aphy 107
82. Niedenzu, W., Schü z, S., Habibian, H., Mo igi, G. & Ri sch,
H. Seeding pa e ns o sel -o ganiza ion o pho ons and
a oms. Physical Re iew A 88,033830 (2013).
83. Domokos, P. & Ri sch, H. Collec i e cooling and sel -o ganiza ion
o a oms in a ca i y. Physical e iew le e s 89,253003 (2002).
84. Black, A. T., Chan, H. W. & Vule i´c, V. Obse a ion o col-
lec i e ic ion o ces due o spa ial sel -o ganiza ion o
a oms: om Rayleigh o B agg sca e ing. Physical e iew
le e s 91,203001 (2003).
85. Baumann, K., Gue lin, C., B ennecke, F. & Esslinge , T.
Dicke quan um phase ansi ion wi h a supe luid gas in
an op ical ca i y. Na u e 464,1301 (2010).
86. Mu ch, K. W., Moo e, K. L., Gup a, S. & S ampe -Ku n,
D. M. Obse a ion o quan um-measu emen backac ion
wi h an ul acold a omic gas. Na u e Physics 4,561 (2008).
87. Thompson, J. D. e al. Coupling a single apped a om o
a nanoscale op ical ca i y. Science 340,1202–1205 (2013).
88. Goban, A e al. A om–ligh in e ac ions in pho onic c ys-
als. Na u e communica ions 5,3808 (2014).
89. Aspelmeye , M., Kippenbe g, T. J. & Ma qua d , F. Ca i y
op omechanics. Re iews o Mode n Physics 86,1391 (2014).
90. Sa a i-Naeini, A. H. e al. Elec omagne ically induced
anspa ency and slow ligh wi h op omechanics. Na u e
472,69 (2011).
91. Palomaki, T., Teu el, J., Simmonds, R. & Lehne , K. En-
angling mechanical mo ion wi h mic owa e ields. Sci-
ence, 1244563 (2013).
92. Riedinge , R. e al. Remo e quan um en anglemen be-
ween wo mic omechanical oscilla o s. a Xi p ep in a Xi :1710.11147
(2017).
93. Genes, C., Vi ali, D., Tombesi, P., Gigan, S. & Aspelmeye ,
M. G ound-s a e cooling o a mic omechanical oscilla-
o : Compa ing cold damping and ca i y-assis ed cooling
schemes. Physical Re iew A 77,033804 (2008).
94. Ma qua d , F., Chen, J. P., Cle k, A. & Gi in, S. Quan um
heo y o ca i y-assis ed sideband cooling o mechanical
mo ion. Physical Re iew Le e s 99,093902 (2007).
95. Benne , J. S., Khosla, K., Madsen, L. S., Vanne , M. R.,
Rubinsz ein-Dunlop, H. & Bowen, W. P. A quan um op-
omechanical in e ace beyond he esol ed sideband limi .
New Jou nal o Physics 18,053030 (2016).
[ Ap il 5,2018 a 10:42 –classic hesis e sion 4.2]