npj | quan um in o ma ion A icle
Published in pa ne ship wi h The Uni e si y o New Sou h Wales
h ps://doi.o g/10.1038/s41534-025-01058-1
Cohe en enhancemen o collec ion o
ligh om linea ion c ys als
Check o upda es
Thuy Dung T an1, Daniel Babjak1, A em Ko alenko1,K a ee Singh
1,TuanMinhPham
2,Pe Obšil1,
Adam Lešundák1,Ondřej Číp2&LukášSlodička1
The e ficien de ec ion o ligh om apped ions in ee space is pa amoun o mos o hei
applica ions. We p opose a scheme o enhance he pho on collec ion om linea ion s ings. I
employs he cons uc i e in e e ence o ligh sca e ed om ions along he axial di ec ion in linea Paul
aps. The cohe en enhancemen o pho on collec ion is nume ically op imized o a ange o easible
spa ial angles and ealis ic ion posi ions in a single ha monic Coulomb po en ial. Despi e he la ge
mu ual dis ance o sca e e s on he o de o many wa eleng hs o sca e ed ligh , he expe imen al
es s p esen ed confi m he easibili y o enhancemen s by a ac o o 3.05 ± 0.09 wi h a c ys al o nine
40Ca+ions. The p oposed collec ion geome y is in insic o di e se linea ion ap designs, and he
me hodology can be di ec ly applied o an obse a ion o sca e ing om ion c ys als p epa ed in
collec i e elec onic exci a ions.
Realizing an e ficien in e ace be ween ligh and a oms ep esen s one o he
mos ac i e esea ch di ec ions in expe imen al quan um op ics. While col-
lec i e in e ac ions be ween ligh and a oms can p o ide a easible solu ion in
la ge a omic ensembles1–3, small apped ensembles o indi idual a oms o
ions ypically u ilize a complemen a y app oach based on high nume ical
ape u e (NA) collec ion op ics. Di e se op ical and apping designs o
e ficien co e age o la ge solid angles wi h bulk op ics 4–18 o ap-embedded
op ical mic o-de ices19–22 ha e been de eloped and es ed. The imple-
men a ions o such couplings ha e ypically been significan ly mo e chal-
lenging o apped ions due o he high sensi i i y o hei mo ion o he
p oximi y o dielec ic su aces, which limi s he accep able wo king dis ances
o collec ion op ics. A he same ime, howe e , he igh and ex emely s able
apping po en ials o Paul aps make his pla o m supe io in di e se
undamen al es s and a b oad ange o applica ions23–31. The a ailabili y o an
e ficien in e ace be ween in e nal elec onic s a es and well-defined pho-
onic modes plays a c i ical ole in a majo i y o hei implemen a ions. Fo
example, he limi ed collec ion e ficiency o ligh om many apped ions o a
single mode se e ely hinde s he ex endabili y o s udies o hei in e ac ion
wi h ligh o measu emen o highe pho on co ela ions32,33. I s enhancemen
would di ec ly p o ide a equi ed speedup o gene a ion and dis ibu ion o
en anglemen be ween quan um epea e nodes based on apped ions34–36.
Recen p og ess o in e aces based on op ical ca i ies demons a ed a pos-
sible iable solu ion31,37,38, howe e , he selec ion o a pa icula spa ial mode
s uc u e and slow ca i y decay a es impose limi s on applica ions which ely
on inhe en ly as dynamics39,40. On he o he hand, se ups employing high-
NA collec ion op ics in ee space a e mos ly limi ed o absolu e pho on
de ec ion e ficiencies on he o de o a ew pe cen , e en in pa icula ly
op imized apping and imaging geome ies12,13,15,18,20–22.Theseo en esul in
se e e es ic ions o o he c i ical implemen a ion pa ame e s, including he
apping po en ial s uc u e, mo ional hea ing a es, o es ic ed field o iew
co esponding o only single o ew apped ions.
He e, we de ise and demons a e a scheme o enhancing he collec ion
e ficiency o ligh om linea ion s ings by ailo ing hei a -field sca e ing
pa e ns in he limi o small collec ion spa ial angles. We conside he ealis ic
spa ial s uc u e o apped ion s ings in a single ha monic po en ial o med
along he axial di ec ion o a linea Paul ap. The ela i e collec ion
enhancemen s in a gi en solid angle along he linea ap axis a e op imized by
scanning he leng h o he ion s ing, which can be expe imen ally con olled
by he applied s a ic elec ic po en ial. The expe imen al es s a e conside ed in
he elas ic sca e ing limi and build on he demons a ed easibili y o scalable
in e e ence om la ge ion s ings41,42. The pe o med simula ions p edic ha
wi hin he ypical mo ional egimes and co esponding posi ion unce ain ies
o ions, he e is no significan disad an age om he non-equidis an c ys al
s uc u e compa ed o idealized egula ly spaced chains o a omic sca e e s.
Following he expe imen al implemen a ion limi ed by fini e he mal posi ion
unce ain ies o 40Ca+ions, we analyze he p ospec s o u he enhancemen s
by employing 138Ba+, which p o ides a mo e sui able configu a ion o longe
sca e ing wa eleng h and highe spa ial localiza ion a compa able apping
equencies and he Dopple cooling limi . In addi ion, an analogous collec-
ion app oach o spon aneously emi ed pho ons om ion s ings p epa ed
in an en angled s a e o elec onic exci a ions can be en isaged, ollowing he
me hodology demons a ed p e iously wi h wo-ion c ys als43.
1Depa men o Op ics, Palacký Uni e si y, 17. lis opadu 12, 771 46 Olomouc, Czech Republic. 2Ins i u e o Scien ific Ins umen s o he Czech Academy o
Sciences, K álo opolská 147, 612 64 B no, Czech Republic. e-mail: [email p o ec ed];slodicka@op ics.upol.cz
npj Quan um In o ma ion | (2025) 11:117 1
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We illus a e he capabili y o he p oposed scheme on a gene ic
expe imen al scena io wi h b oadly easible ion apping pa ame e s. The
pho ons sca e ed om he linea ion s ing a e collec ed wi hin a spa ial
angle pa ame e ized by he nume ical ape u e in a acuum NA ¼sin θ,
whe e θis hal o he ape u e angle. The ele ance o he axial op ical access
along he z-di ec ion eme ges om he a ailabili y o he smalles angula
g adien o he in ensi y pa e ns gene a ed by sca e ing ligh om linea
ion s ings along hei symme y axis. The maximiza ion o he coupling in
he pa icula solid angle ac ion cen e ed along he axial di ec ion o he
ap using a -field in e e ence c i ically depends on he collec i e con-
s uc i e con ibu ion o he coupled mode gi en by he ela i e posi ions o
ions in he same linea ha monic po en ial. The equilib ium posi ion o i- h
ion in he lase -cooled s ing can be ound by minimizing he po en ial
ene gy o he whole n-ion c ys al. Since he spa ial configu a ion in a
common ha monic po en ial is e ec i ely de e mined by a single pa ame e ,
ions posi ions can be con enien ly pa ame e ized using a common spa ial
leng h scale l¼ðq2=ð4πϵ0mω2
zÞÞð1=3Þ,whe eϵ
0
is pe mi i i y o ee space,
such ha z
i
=l
i
,and
i
is dimensionless equilib ium posi ion o i- h ion
ound nume ically by minimizing he o al po en ial ene gy o he ion
s ing44. The uppe limi on he axial apping equency is ul ima ely gi en
by he equi emen on he geome ically linea configu a ion o ions in he
ap, which se s he a io A¼ðωz=ω Þ2 o be lowe han A
c i
=cnB,
whe e pa ame e s c=2.94andB=−1.845–47. Howe e , we no e ha in he
expe imen , o he ac o s a ec ing he s abili y o he linea c ys al and he
easibili y o he op imal lase cooling will u he es ic he con enien
ange o axial mo ional equencies. The opposi e - maximal spa ial leng h
scale is se o he equency o he secula mo ion in he axial di ec ion
ωmin
z¼ð
λ
4Þ2ℏ
2m, such ha he co esponding posi ion unce ain y a he
Dopple cooling limi e ec i ely s ill allows o esol ing op ical in e e ence
om wo sca e e s.
Resul s
Simula ions
We access he easible pho on collec ion enhancemen s by e alua ing he
a -field spa ial angula dependence o op ical in ensi y esul ing om he
in e e ence o ligh sca e ed elas ically om a linea ion s ing. We con-
side a sca e ing o he beam wi h a plane wa e on and di ec ion gi en by
he exci a ion angle α. The e alua ion o an in ensi y pa e n on a sphe ical
sc een ep esen s a gene ic scena io, which can be applied o compa ing he
achie able ela i e collec ion enhancemen s i espec i e o he pa icula
collec ion op ics and pho on de ec ion sys em. Any linea op ical imaging
wi h a co esponding inpu nume ical ape u e will me ely a ec he
esul ing spa ial in ensi y pa e ns a he de ec o . The complex ampli ude o
ligh field sca e ed om he fi s ion in he a -field can be w i en as
U1¼ε0eiðk φ0Þ,whe eε
0
is he ampli ude, kis he wa enumbe , is he
dis ance a eled by he wa e om he fi s ion o he sc een, and φ
0
is he
phaseo heligh a heposi iono hefi s ion. He e, we omi he esidual β-
dependen phase o se , as he ollowing e alua ion o in e e ence pa e ns
depends only on he ela i e phase delay o he fields sca e ed om di e en
ions. The complex ampli ude o ligh sca e ed om he j- h ion can
hen be exp essed as Uj¼εjeiðk φ0ÞeikΔj;1
d, whe e he pa h di e ence
Δj;1
d¼lð j 1Þðcos αcos βÞis aken wi h espec o he fi s ion. The
a -field ampli ude esul ing om he sca e ing o he monoch oma ic ligh
wi h a plane wa e on and a wa enumbe k om a s ing o niden ical wo-
le el sca e e s is gi en by he sum o complex ampli udes om di e en
ions U o ¼Pn
j¼1Uj.Fo simplifica ion, we assume iden ical ampli udes ε
j
= 1, which is an expe imen ally plausible app oxima ion in he conside ed
axial c ys al spa ial leng hs and easible exci a ion beam wid hs41,48.The
spa ial in ensi y pa e n I(β)=∣U
o
(β)∣2can hen be e alua ed as
IðβÞ¼ P
n
j¼1
UjðβÞ
2
¼P
n
a;b¼1
UaðβÞUbðβÞ
¼
¼P
n
a;b¼1
cosðkΔa;b
dÞ;ð1Þ
whe e Δa;b
dis a pa h di e ence be ween fields sca e ed om he a- h ion and
he b- h ion p opo ional o hei mu ual dis ance l(
a
−
b
). We no e ha
his simplified model neglec s pola iza ion e ec s and co esponding
o ien a ions o a omic dipoles, as hey p o ide on a e age he same
enhancemen o he conside ed sca e ing om mul i-ion c ys als and o
he e e ence a e age emission om a single emi e .
The op imiza ion o collec ion enhancemen s can be quan ified by
he e alua ion o he maximal pho on flux Φ
NA
(n)=∫
Ω
IdΩwi hin a
gi en solid angle Ω¼2πð1cos θÞo e expe imen ally easible spa ial
leng h scales l. Equi alen ly, pa ame iza ion by he co esponding
nume ical ape u e θ¼a csinðNAÞcan be employed. Due o he ci cula
symme y o he sca e ing pa e ns a ound he z-axis, he pho on flux
can be e alua ed in a pa axial limi , e alua ed in sphe ical coo dina es as
ΦNAðnÞ¼R2π
ϕ¼0Rθ
β¼0Isin βdβdϕ¼2πRθ
0IðβÞsin βdβ. Using he o e all
flux om an nion s ing Φ
4π
(n)=n4π, he collec ion e ficiency can be
defined as PD¼ΦNA
Φ4π,whe eΦ
NA
deno es he con en ional pa a-
me iza ion by he nume ical ape u e NA. Fo accessing a mo e di ec
es ima ion o he enhancemen , he ela i e enhancemen ac o P
D, el
can be defined as he collec ion e ficiency o he nion s ing no malized
by he collec ion e ficiency o a single ion
PD; elðnÞ¼ PDðnÞ
PDðn¼1Þ¼ΦNAðnÞ
ΦNAðn¼1Þn:ð2Þ
The enhancemen o pho on collec ion e ficiency by he p esen ed
me hodology depends on he easibili y o geome ical a angemen s p o-
iding a close o cons uc i e in e e ence o he sca e ed ligh om many
ions wi hin he selec ed solid angle. The p ac ical limi s on such config-
u a ions in ion aps a e imposed by he s abili y o linea ion c ys als. We
p esen he simula ions o easible enhancemen s o up o n=10ions
conside ing expe imen ally b oadly easible adial secula apping e-
quencies ω
=(2π) × 5 MHz o e alua ion o he uppe limi on ωmin
z.Fo
highe ion numbe s, insu ficien axial comp ession o he s ing in he
ha monic ap p e en s u he enhancemen in he p ac ically use ul limi
o NA > 0.05. In addi ion, he fini e he mal mo ion is expec ed o sig-
nifican ly educe he p ac ical applicabili y o he employmen o high ion
numbe s, which equi e inc easingly lowe axial apping equencies
necessa y o achie ing close o a quasi-pe iodic s uc u e. The exci a ion
angle α=45
∘is a ailable in many expe imen al se ups, as his configu a ion
allows o simul aneous e ficien cooling o he axial and adial mo ion o
ions in a linea Paul ap. Employmen o smalle exci a ion angles would
allow u he dec ease in he sensi i i y o he mal mo ion, and he op imal
se ing could co espond o an exci a ion angle ha is sligh ly la ge han he
maximal collec ion angle θ, such ha i would closely a oid he con ibu ion
o he exci a ion lase ligh o he de ec ed pho on signal. The p esen ed
simula ion cons ain s esul in e alua ed alues o minimal spa ial leng h
scales anging om lmin ¼1:61 μm, o lmin ¼4:23 μm o n= 2 and 10 ions,
espec i ely. The alue o he maximal spa ial leng h scale lmax ¼81:18 μm
co esponding o ωmin
zis kep he same o all simula ions. The pho on
collec ion enhancemen s ha e been nume ically op imized o e he spa ial
pa ame e l.
Figu e 2a illus a es he easibili y o significan gainsinab oad ange
o nume ical ape u es. In he limi o NA << 0.1 and smallion numbe s, he
enhancemen P
D, el
becomes close o linea wi h n.Thisisaconsequenceo
app oaching he scaling o ideal cons uc i e in e e ence om a quasi-
pe iodic a ay o poin sca e e s, whe e Imax /n2,as heangula in e -
e ence pa e n does no app oach he fi s des uc i e node wi hin he
co esponding analyzed small spa ial angles. The di e gence om his
idealized scaling o la ge nisgi enonlyby esidualde ia ions om he
quasi-pe iodic ion posi ions wi hin he a ailable ange o axial apping
equencies. Fo highe nume ical ape u es NA >> 0.1, he ad an age due
o he cohe en con ibu ions g adually dec eases. The phase o he in e -
e ence pa e ns oscilla es mo e apidly o la ge sca e ing angles, and he
o e all gain anishes due o he a e aging o cons uc i e and des uc i e
in e e ence con ibu ions. Mo e comple e unde s anding o he eme gence
h ps://doi.o g/10.1038/s41534-025-01058-1 A icle
npj Quan um In o ma ion | (2025) 11:117 2
o enhancemen scaling hus mus conside he ac ual beha io o I(β) o
di e en leng h scales l, as he ela i e heigh s o he esul ing cons uc i e
peaks depend on he in eg a ed spa ial angle gi en by he chosen NA. This
esul s in he sha p ansi ions be ween di e en in e e ence peaks o
di e en NA co esponding o kinks in he cu es shown in Fig. 2a. A mo e
de ailed analysis o he co esponding beha io o P
D, el
can be ound in he
Supplemen a y In o ma ion.
As becomes eme gen om he simula ions in Fig. 2a, he conside ed
maximal nume ical ape u e NA = 0.3 app oaches field dis ibu ions wi h
la ge sca e ing angles β, which esul in fini e angula dis ibu ions o
cons uc i e in e e ence lobes. These limi he achie able enhancemen s o
highe numbe s o ions nwi h he co esponding longe spa ial leng hs o
he ion c ys al. Fo n> 3, he dependence o P
D, el
on NA displays kinks,
which co espond o poin s o ansi ions be ween di e en op imal max-
imal cons uc i e lobes, which esul in maximal P
D, el
a di e en nume ical
ape u es NA. Gene ally, lowe NA allows o he employmen o a na -
owe angula dis ibu ion o he con ibu ing cons uc i e lobe o he
in e e ence pa e n.
The ideal quasi-pe iodic ion posi ions canno be gua an eed on he axis
o he case o ou o mo e ions in a single ha monic apping po en ial. The
op imized nume ical solu ions p esen ed in Fig. 2a include many config-
u a ions, whe e mu ual ion posi ions p o ide significan collec ion
enhancemen s in a gi en solid angle despi e esidual de ia ions om he ully
cons uc i e in e e ence configu a ions. We quan i y he co esponding
ine ficiency o ela i e enhancemen by analysis o he case wi h equal
mu ual dis ances be ween neighbo ing ions. To allow o a physically ele-
an compa ison, he ange o he mu ual dis ances was in his case limi ed o
he a e age dis ance o he same numbe o ions in he ha monic ap gi en
by he p esen ed limi s on hei mo ional equencies ωmin
zand ωmax
z.The
example o simula ion in Fig. 2b o di e en ion numbe s and NA ~0.07 and
α=45
∘co esponding o ou expe imen al se ing p o ides a clea illus a-
ion o he eme gence o a close- o-linea scaling wi h n o small ion
numbe s. The simula ion sugges s ha he ideal equidis an spacing could
s ill allow o u he enhancemen in his limi . The addi ional ela i e
enhancemen o collec ion e ficiency achie able by employing an equidis an
s ing o a omic sca e e s can be quan ified as Peq
D; el=Pha m
D; el ¼1:48 wi h 9
ions in an ideal pe iodic la ice. In compa ison, he easible imp o emen by
implemen a ion o exci a ion a α=90
∘, while keeping he ha monic
po en ial, eaches P90
D; el=P45
D; el ¼1:26 o he same 9 ion s ing. We no e ha
such pe pendicula exci a ion can be addi ionally ad an ageous due o he
eflec ion symme y o he in e e ence pa e ns, whe e simul aneous
enhancemen o coupling o bo h opposi e axial di ec ions can be achie ed.
The pa icula ly high spa ial localiza ion, long- e m s abili y, and la ge
mu ualdis ancesbe weenionsinPaul apsallow o e ficien op ical
add essing schemes49,50. Exci a ion o he ion s ing wi h an a ay o igh ly
ocused lase beams wi h indi idually con ollable n−1 ela i e phases allows
o maximiza ion o he coupling o sca e ed ligh in he gi en solid angle.
Al e na i ely, such a scena io could also be ealized in he inelas ic sca e ing
egime by implemen ing local co ec ions o phases o ligh sca e ed by
di e en ions o he collec i ely o med a -field in e e ence pa e n using he
add essable AC-S a k shi s on a s ing o ions p epa ed in a collec i e single-
exci a ion Dicke s a e o in e nal elec onic le els33,43.Anexampleo asimu-
la ion o he op imal phase di e ences o he gi en exci a ion geome y and
o he case o n= 5 ions in he single ha monic po en ial conside s n−1=4
independen ly op imized phases. Eq. (1)can henbemodified o
IðβÞX
n
a;b¼1
cos½Δa;b
φþðφ0
aφ0
bÞ;ð3Þ
whe e φ0
aand φ0
ba e phase shi s o di e en ions wi h espec o he ini ial
phase o he fi s ion in he s ing. Thus, φ0
1¼0andφ0
2φ0
na e a ying
om 0 o 2π. Thei independen op imiza ion in p inciple allows o
employing he sho es easible alue o he spa ial leng h scales l!lmin o
he gi en adial confinemen . Figu e 2c p esen s an example compa ison o
he h ee coupling me hods o a linea s ing o fi e ions. The g een da a
poin s depic he esul s o he app oach wi h nume ically op imized
independen phase ac o s in sca e ing om di e en con ibu ing ions in a
single ha monic ap. The esul depic ed as he solid line co esponds o he
equal ini ial phase ac o s o all ions and op imiza ion o e a single pa a-
me e — he spa ial leng h scale l, and is iden ical o he esul p esen ed in
Fig. 2a. The dashed line ep esen s he op imiza ion o e he ange o dis-
ances be ween ions in a pe iodic ion s ing. In he limi o low nume ical
ape u es NA < 0.1, he h ee app oaches p o ide nea ly iden ical collec ion
enhancemen s, and he ha monic apping po en ial does no p esen any
limi a ion. Fo highe solid angles, he possibili y o op imiza ion o ini ial
phases allows o significan imp o emen o enhancemen s, which can
p o ide P
D, el
e en beyond he equidis an case. This esul s om he ea-
sibili y o he op imal cons uc i e in e e ence o e gi en solid angles
achie able a he smalles dis ance allowed by he ap lmin.Theemploymen
o he op imiza ion o phases becomes e en mo e beneficial in he limi o a
la ge numbe o ions, whe e a close- o-op imal pa e n is no easible o any
ealis ic anges o spa ial leng h pa ame e l.
Expe imen al es
The p ac ical easibili y o he enhancemen o collec ion e ficiency has been
es ed o a c ys al o 40Ca+ions apped in he linea Paul ap, as illus a ed
in he example p esen ed in Fig. 1a. The equency o he adio- equency
(RF) ol age applied o adial elec odes was ω
=(2π)×29.9MHz.The wo
adial mo ional modes had close o degene a e secula equencies o abou
(2π) × 2.2 MHz. The axial secula equencies ω
z
we e expe imen ally
op imized by scanning he applied s a ic ol age U
ip
o confi m he p edic ed
collec ion enhancemen s wi hin anges ω
z
≈(2π)×(0.60−1.22) MHz o 2
ions o ω
z
≈(2π)×(0.30−0.77) MHz o 9 ions c ys al. They co espond o l
pa ame e anging om 3.9 o 6.2 μmand om5.3 o9.6μm o 2and9
ions, espec i ely. The linea s uc u e o he s ing was moni o ed by he
came a in he adial apping di ec ion du ing he whole expe imen .
The a -field collec ion o pho ons sca e ed by ion s ings in he axial
di ec ion has been based on a single plan-con ex lens wi h a ocal leng h o
100 mm. The limi on he e ec i e obse a ion spa ial angle was se by he
ape u e o he axially posi ioned apping elec ode, which has been es i-
ma ed om he ap design o NA ≈0.07. The collec ed ligh passed a
pola iza ion beam-spli e , which ansmi ed he linea pola iza ion
o ien ed pe pendicula o he di ec ion o he applied magne ic field, such
ha i maximizes he po ion o he elas ically sca e ed field om he σ-
ansi ions in he de ec ed signal. The p ecise measu emen s o he coun
a e om a single ion and he backg ound coun a e we e implemen ed o
allow o he exac e alua ion o he ela i e enhancemen s o he coupling o
pho ons om ions. A coun a e om a single apped ion o he lase
exci a ion se ings allowing o he s able con inuous measu emen and
op imal Dopple cooling o all ion c ys als was es ima ed o C(n= 1) = 270
± 17 coun s/s. I co esponds o a sa u a ion pa ame e o he sca e ed
397 nm lase o abou s
397
≈0.65 and de uning Δ
397
=−21 MHz, op imized
o he maximal c ys al s abili y and isibili y o he in e e ence on he
la ges employed ion s ing wi h n= 9. To allow o he eliable expe imen al
es ima ion o scaling wi h he numbe o ions, he lase pa ame e s and
adial apping equencies we e kep iden ical in all measu emen s. The
exci a ion lase equencies we e s abilized o an op ical equency comb
which p o ides a long- e m s abili y51. The measu ed backg ound o C
bg
=
24 ± 5 coun s/s o igina ed dominan ly om he esidual sca e ing o he
397 nm lase o he adial apping elec odes.
An example o an expe imen al es in Fig. 1b, illus a es he unabili y
o he collec ion enhancemen on he sca e ing om n= 5 ions. The
op imiza ion o e axial po en ial s eng h was pe o med o all p esen ed
expe imen al alues wi hin he easible anges o he spa ial leng h scales l.
The no malized enhancemen o he pho on collec ion e ficiency was hen
e alua ed om he measu ed pho on a es as
Pexp
D; elðnÞ¼ CðnÞCbg
ðCðn¼1ÞCbgÞn:ð4Þ
h ps://doi.o g/10.1038/s41534-025-01058-1 A icle
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The measu ed da a we e fi ed using he model Pcal
D; el conside ing he ideal
in e e ence o he cohe en ly sca e ed pa acco ding o he Eq. (1)wi h he
p obabili y
coh
and he complemen a y incohe en pa o he sca e ed ligh
incoh
=1−
coh
,
Pexp
D; el ¼ incoh þ cohPcal
D; el:ð5Þ
The pa ame e
coh
was e alua ed om he local fi s o he model (5) o he
expe imen al da a a ound he maximal measu ed Pexp
D; el o supp ess he
e ec o measu emen noise and he esidual de ia ions o he simplified
heo e ical model om he expe imen al eali y. The esul ing local max-
imum is depic ed as a g een diamond, and he e o ba shows a s a is ically
e alua ed single s anda d de ia ion co esponding o pho on numbe
fluc ua ions o measu ed coun a e close o he maximal P
D, el.
Figu e 3p esen s he summa y o he measu ed maximal alues o Pexp
D; el
( ed diamonds) and he es ima ed ac ions o cohe en ly sca e ed ligh
(black c osses). The alues o cohe en ac ion
coh
es ima ed om he fi
using he Eq. (5)a esimila ac oss hewhole angeo ionnumbe s,which
suppo s he ele ance o he obse ed scaling o P
D, el
wi h n.Weno e ha i s
ela i e dec ease by abou 20 % o he ini ial alue o n>5ionscanbemos ly
a ibu ed o inc eased spa ial posi ion unce ain ies, as he op imal wo king
poin s co espond o la ge spa ial leng h scales l o hisnumbe o ions.The
aw collec ion enhancemen s ha e been measu ed om abou 1.51 ± 0.01 o
wo ions and mono onously inc eased o 3.05 ± 0.09 o a nine-ion c ys al.
The measu ed wo-ion enhancemen alue is close o wha has been
p esen ed p e iously43. Despi e wo king he e in a much smalle NA limi ,
hese alues can be di ec ly compa ed because P
D, el
is independen o NA
o a e y small numbe o ions in he co esponding e ec i e NA egimes,
as illus a ed in Fig. 2a. The in insic egula i y o ion spacing o wo and
h ee-ion s ings enhances hei obus ness o a ious geome ical ac o s
and enables compa ison among di e en expe imen al egimes and ion
apping pla o ms. The esidual di e ences can be a ibu ed mos ly o
he mal mo ional e ec s specific o di e en a omic species. The unique
easibili y o es ing wi h a highe numbe o ions p esen ed in his wo k
allows o he fi s obse a ion o he scaling o expe imen al gains o n>2
apped ions. We no e ha he absolu e o e all de ec ion e ficiency o
pho on sca e ed om a single apped ion es ima ed o P
D,abs
=1.7×10
−2%
emains ela i ely small mos ly due o he employmen o ap wi h he axial
access co esponding o he op ical solid angle o Ω≈0.015 and he fini e
de ec ion e ficiency o employed single-pho on coun ing module o η
SPCM
≈
50%. Howe e , he ela i e enhancemen o 9 ions gi es final
P
D,abs
= 0.051 ± 0.001%, which is close o he op imized alue o he o e all
de ec ion e ficiency in adial di ec ion P
D,abs
= 0.06% measu ed wi h he
collec ion objec i e wi h much highe NA o ≈0.3 co e ing abou 2% o he
ull solid angle in he same appa a us and in he analogous pho on de ec ion
se ings52. These compa isons p o ide an expe imen al alida ion o he
applicabili y o he p oposed app oach o di e se applica ions equi ing
e ficien pho on collec ion om ions, including he implemen a ion o
pa icula ly challenging expe imen s equi ing mul i-pho on de ec ion
e en s om many ions43,53.
The fini e sa u a ion pa ame e and he mul ile el elec onic le el
s uc u e o 40Ca+ educe he obse ed enhancemen s, which amoun s o he
esidual di e ence be ween he measu ed and simula ed da a poin s. The
con ibu ion o he inelas ically sca e ed ligh has been es ima ed o be
abou 13% using da k esonance spec oscopy. In addi ion, he s eady-s a e
popula ion o he me as able 32D
3/2
mani old o abou 13% e ec i ely
co esponds o a andom swi ching o he con ibu ion o di e en ions o
he in e e ing signal. Howe e , we emind ha hese e ec s a e in insic o
he expe imen al es s elying on he elas ic sca e ing and should no a ec
he collec ion e ficiency o pho ons sca e ed om he ion c ys alp epa ed in
en angled jWis a es wi h cohe en ly sha ed elec onic exci a ion33,43,54,55.
The educ ion o he in e e ence isibili y due o he posi ion unce -
ain y o ions has been es ima ed o be he mos significan de e io a ion
p ocess, simila o p e ious demons a ions o in e e ence om s ings o
apped ions41–43,56. The impac o mo ion inc eases o low apping e-
quencies and co esponding la ge spa ial leng h scales ldue o he g ea e
mean posi ion unce ain y o ions, which also becomes eme gen om he
measu ed maximal enhancemen s. Conside ing he posi ion unce ain ies
o he mal mo ional s a es, equa ion (1)canbemodified o
IX
n
a;b¼1
e1
2k2
e σ2
a;bcos Δa;b
φ:ð6Þ
He e ~
ke ¼~
kou ~
kin,~
kou and ~
kin a e he sca e ed and inciden wa e
ec o s, espec i ely, and σ
a,b
is he s anda d de ia ion o he mu ual dis-
ance be ween a- h and b- h ions conside ing ull decomposi ion o n
Fig. 1 | Illus a ion o he cohe en enhancemen o he collec ion o ligh om
linea ion c ys als in a single ha monic apping po en ial. a depic s he p inciple
o employmen o a -field angula in ensi y dis ibu ions I(β) o ligh sca e ed om
he exci a ion beam wi h a wa e ec o ~
kin o maximizing he gain in he collec ion
e ficiency P
D, el
in a ailable axial op ical collec ion nume ical ape u e NA ¼sin θ
along he ion c ys al axis. The op imiza ion o e he spa ial leng h scale lo he ion
c ys al can be ailo ed by con olling he axial apping po en ial, which allows o
di ec expe imen al unabili y and e alua ion o he enhancemen . The p esen ed
measu emen example in (b) wi h n= 5 apped 40Ca+ions illus a es he easible
modifica ion o ela i e de ec ion e ficiency o exci a ion angle α=45
∘. The
co esponding simula ed no malized sca e ing spa ial pa e ns I(β) depic se e al
no able cases, including des uc i e emission along bo h axial di ec ions, con-
s uc i e in he di ec ion o he de ec o , and cons uc i e in he opposi e di ec ion,
om le o igh , espec i ely. They illus a e he basic p inciple o maximiza ion o
he ela i e collec ion e ficiency in a gi en solid angle along he axial apping
di ec ion by con olling he spa ial leng h scale lo he ion c ys al. The no mal-
iza ion o he de ec ed pho on a e C(n) o he a e om he same numbe o ions ina
ully incohe en sca e ing egime nC(n= 1) allows o a di ec e alua ion o he
ela i e collec ion enhancemen P
D, el
.
h ps://doi.o g/10.1038/s41534-025-01058-1 A icle
npj Quan um In o ma ion | (2025) 11:117 4
no mal axial mo ional modes. The co esponding simula ions o collec ion
enhancemen s a ec ed by mo ional dephasing a e shown in Fig. 3as g ay
squa es. The a e age educ ion o he ela i e enhancemen o n=2 o6ion
c ys als due o he he mal mo ion a he Dopple cooling limi has been
e alua ed o a ound 25 %, while he es ima ed supp ession becomes clea ly
mo e significan wi h he inc easing spa ial leng h scale o highe ion
numbe s. The enhancemen o he gi en apping pa ame e s is hus
expec ed o be u he significan ly imp o ed when employing hea ie ion
species, as he a iance o he mu ual ion posi ions is gi en by σ2
a;bℏ
2mωz,o
by employing a highe sca e ing wa eleng h. Figu e 3includes examples o
simula ions wi h 138Ba+conside ing sca e ing a he 62S
1/2
↔62P
1/2
dipole
ansi ion wi h λ≈493 nm, shown as g ay ci cles. The wo p esen ed da a
poin s ep esen no able cases o he la ges easible ela i e enhancemen in
compa ison wi h he measu emen s on 40Ca+ o n=5 and he highes
measu ed Pexp
D; el o n=9. Fo n= 5, he simula ion p edic s addi ional
imp o emen PBa
D; el=PCa
D; el 1:45, i.e., a o al enhancemen o abou
PBa
D; el 3:93.Thep edic edop imum o n= 5 is he consequence o he
enhanced sensi i i y o mo ion o la ge n, which equi es longe spa ial
leng h scales o app oach quasi-pe iodic c ys al s uc u e. A he same ime,
hese longe c ys als esul in spa ial in e e ence pa e ns wi h a much
smalle angula wid h o he fi s cons uc i e lobe. These e ec s join ly
con ibu e o he dec easing in maximal gain Pexp
D; el=n o high n.
Discussion
The p esen ed scheme o he enhancemen o collec ion e ficiency om
linea ion s ings employs a sca e ing geome y ha is in insic o di e se
mode n linea ion aps. I s simula ions confi m he easibili y o he close o
an ideal linea gain by a ac o o nin he collec ion e ficiency o ligh
sca e ed om up o n= 5 ions o small nume ical ape u es NA < 0.1, and
p edic s ill significan enhancemen s e en o highe ion numbe s and
nume ical ape u es, s ill wi hin he expe imen ally easible posi ion
unce ain ies and axial mo ional equencies. The scheme p o ides he mos
a ac i e enhancemen s wi hin he ange o nume ical ape u es 0.05 <
NA < 0.2, whe e he maximum gain can be in he ideal case o ully cohe en
sca e ing app oached al eady o ela i ely small ion numbe s n≈5. They
allow o he employmen o small spa ial leng h scales lmin o he gi en
adial secula equency and hus na u ally p o ide op imal enhancemen
configu a ions due o he esul ing in e e ence pa e ns wi h smalle
angula g adien s. Fo highe ion numbe s, i becomes un easible o achie e
asu ficien axial comp ession o he s ing in a single linea ha monic
apping po en ial, and he co esponding minimal spa ial leng h scales lead
o he lowe maximal collec ion enhancemen s. Al hough he op imiza ion
in he case o equidis an emi e s p edic s he easibili y o u he
enhancemen s, hese gains can be, in p inciple, ully elimina ed by con ol o
(a)
(b)
(c)
Fig. 2 | Simula ions o he ela i e enhancemen o he collec ion e ficiency P
D, el
o ligh om linea ion c ys als in a single ha monic apping po en ial. a shows
he op imized P
D, el
o di e en numbe s o 40Ca+ions wi hin he p ac ically ele-
an ange o nume ical ape u es NA o he collec ion op ical mode. The sca e ing
wa eleng h λ= 397 nm and α=45
∘.bdepic s easible P
D, el
o di e en numbe s o
ions and he nume ical ape u e NA = 0.07 co esponding o he employed
expe imen al es . Resul s o op imiza ion in he ha monic po en ial o he wo
angles o incidence α=45
∘,α=90
∘, and o he case wi h egula ion spacing and α=
45∘(ci cles) a e shown as ull squa es, iangles, and ci cles, espec i ely. cillus a es
he example o di e en op imiza ion p o ocols o he s ing o n= 5 ions. The
op imiza ion solely o e he single apping pa ame e — he spa ial leng h scale lin
he ha monic apping po en ial is shown as a solid cu e. The add essable unabili y
o indi idual sca e ing phases in he single ha monic po en ial allows o max-
imiza ion o P
D, el
o he lowes easible mu ual dis ance o ions gi en by l¼lmin,
wi h esul ing enhancemen s shown as g een poin s. The dashed line depic s he
example o op imiza ion o equidis an sca e e s.
Fig. 3 | Summa y o he measu ed enhancemen s o collec ion e ficiency o a
linea chain o 40Ca+ions o he NA ≈0.07. Values e alua ed om he measu e-
men s shown as ed diamonds can be compa ed wi h simula ions shown as squa es,
which conside he he mal mo ion o ions a he Dopple cooling limi . The
cohe en ac ion
coh
shown in he bo om g aph was e alua ed om he measu ed
in e e ence dependencies on he spa ial leng h scale lusing he Eq. (5). The e o
ba s co espond o a single s anda d de ia ion and a e smalle han he displayed
da a poin symbols, whe e no shown. The examples o p edic ions o 138Ba+
illus a e he expec ed enhancemen s o he equi alen exci a ion configu a ion.
h ps://doi.o g/10.1038/s41534-025-01058-1 A icle
npj Quan um In o ma ion | (2025) 11:117 5
indi idual sca e ing phases in s ings o ions p epa ed in s a es possessing a
collec i e cohe en spin exci a ion43.
The ealized expe imen demons a ed enhancemen s anging om
≈1.51 o he wo-ion c ys al o ≈3.05 o he nine-ion c ys al, limi ed
dominan ly by he esidual he mal mo ion, fini e sa u a ion pa ame e s,
and mul ile el e ec s. While he de imen al impac o he inelas ically
sca e ed ligh and mul ile el e ec s would be p ac ically absen in he
schemes employing he collec i ely sha ed spin exci a ion33,43, heposi ion
unce ain y can be u he imp o ed by employing ions wi h la ge a omic
mass, employmen o ansi ions wi h longe wa eleng hs, u he dec ease
o he inpu sca e ing angle, o hei combina ion.
The p esen ed app oach enables a b oadly applicable, e ficien pho on
collec ion om ions o small spa ial angles. When compa ed o he ee space
high-NA collec ion se ups, he conside ed small solid angle limi p o ides an
inhe en ad an age o he educ ion o wa e on abe a ions o he collec ed
ligh wi h simple pa axial collec ion op ics, which p omises pe spec i es in
di e se in e e ome ic applica ions using collec i ely coupled apped ion
c ys als57–59. Demons a ed me hodology wi h apped ion s ings can be
di ec ly ex ended o implemen a ionso p o ocolsin ol ing he enhanced
di ec ionali y o iginally based on o wa d-sca e ing o ligh in cold a omic
clouds60–63. The scaling o he co esponding collec i e enhancemen s in a
single-mode de ec ion limi is expec ed o be compa able o he cohe en
enhancemen s o he collec ion e ficiency p esen ed he e. As he la ge ocus
dep h o he small-NA fluo escence collec ion allows o simul aneous
obse a ion o ligh om many ions, he a ailable con ol o la ge apped ion
quan um egis e s64,65 can allow o mapping o he collec i e in e nal s a es o
he ion s ing on he di ec ion o he sca e ed ligh 33,43.
Me hods
Expe imen al scheme
The sca e ing wa eleng h and exci a ion angle a e selec ed acco ding o he
p esen ed expe imen al es s wi h 40Ca+, howe e , he esul s can be easily
adap ed o any o he combina ion o a omic ion species and exci a ion
geome ies. The expe imen al demons a ion employs a 3D linea Paul ap
and collec ion o he 397 nm pho ons. This co esponds o se e al ecen
expe imen s on cohe en elas ic sca e ing om ion s ings41,42,66.The
apping o linea ion s ings is achie ed by he RF ol age U cosðω Þ
applied o adial elec odes and a s a ic ol age U
ip
a wo axial cone-shaped
hollow ip elec odes, which oge he p o ide a h ee-dimensional apping
po en ial. The equency o he secula mo ion o ions along he axial
di ec ion is gi en by ωz¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2qU ipκ=ðmz2
0Þ
q,whe emis he ion mass, qis
he ion cha ge, 2z
0
is he dis ance be ween he wo ip elec odes, U
ip
is he
ol age applied o he ip elec odes, and κis he geome ical ac o . The adial
mo ional equencies ω ¼qU =ðm 2
0ω ffiffiffi
2
pÞdepend on he ampli ude o
he RF po en ial U
and i s angula equency ω
.Figu e1a illus a es he
exci a ion and de ec ion geome y, which is simila o ha p e iously
employed in bo h neu al a om and apped ion in e e ence expe imen s41,67.
The sca e ing o he 397 nm field wi h a ed-de uned equency ω
l
om he
42S
1/2
↔42P
1/2
elec ic dipole ansi ions p o ides a simul aneous Dopple
cooling o ions. The lase sca e s o he ion s ing a an angle α≈45 ∘wi h
espec o he axial apping di ec ion in he z−xplane. The 866 nm lase
beam wi h he same di ec ion is used o eshu fling he popula ion o he
me as able 32D
3/2
mani old back o he cooling ansi ion. The degene acy o
Zeeman s a es is li ed by applying a s a ic magne ic field wi h a magni ude
j~
Bj¼3:3o Gaussalong hey-di ec ion. Pola iza ions o bo h exci a ion
lase s a e conside ed o be linea wi h he ec o pe pendicula o he applied
magne ic field.
Da a a ailabili y
The expe imen al and heo e ical da a ha suppo he findings o his
a icle can be ound a h ps://doi.o g/10.5281/zenodo.15721196.
Recei ed: 27 Janua y 2025; Accep ed: 9 June 2025;
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Acknowledgemen s
D.B. and T.D.T. a e g a e ul o na ional unding om he MEYS unde g an
ag eemen No. 731473 and om he QUANTERA ERA-NET co und in
quan um echnologies implemen ed wi hin he Eu opean Union’s Ho izon
2020 P og am (p ojec PACE-IN, 8C20004). A.K. acknowledges he suppo
o he Czech Science Founda ion unde he p ojec GA21-13265X. T.D.T.
and A.L. we e suppo ed by na ional unding om he MEYS unde he
p ojec CZ.02.01.01/00/22 008/0004649. O.Č., T.M.P., and L.S. we e sup-
po ed by he Minis y o In e io o he Czech Republic wi hin he P og am
OPSEC unde he p ojec VK01030193. K.S. is g a e ul o he Palacký
Uni e si y g an IGA-P F-2024-008. We hank Romain Bachela d and Phi-
lipp Schindle o ui ul discussions.
Au ho con ibu ions
T.D.T. pe o med he simula ions and analyzed he expe imen al da a. A.K.
and K.S. ca ied ou he p esen ed measu emen s. L.S. concei ed and
supe ised he p ojec . T.D.T. and L.S. w o e he manusc ip . All au ho s
con ibu ed o he ealiza ion o he employed ion apping appa a us,
e iewed he esul s, and app o ed he final e sion o he manusc ip .
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