SIMULATION STUDIES OF LASER COOLING FOR THE GAMMA FACTORY PROOF-OF-PRINCIPLE EXPERIMENT AT THE CERN SPS

SIMULATION STUDIES OF LASER COOLING FOR THE GAMMA
FACTORY PROOF-OF-PRINCIPLE EXPERIMENT AT THE CERN SPS
P. K uy ∗1, D. Gamba, G. F anche i1,2,3
CERN, Gene a, Swi ze land
1also a Goe he Uni e si y, F ank u , Ge many, 2also a GSI, Da ms ad , Ge many,
3also a HFHF, F ank u am Main, Ge many
Abs ac
The Gamma Fac o y p oo -o -p inciple (GF PoP) expe i-
men a he Supe P o on Synch o on (SPS) a CERN aims
a demons a ing lase cooling o high ene gy Li-like
Pb79+
in a synch o on. Fi s , we p esen lase cooling simula-
ions wi h ealis ic lase and beam pa ame e s o he GF PoP
expe imen a he SPS. Fu he mo e, we in es iga e he ex-
pec ed cooling pe o mance o a Fou ie -limi ed lase pulse
and compu e he pe o mance me ics, such as emi ance
educ ion. These me ics a e hen compa ed o in a-beam
sca e ing (IBS) g ow h a es o de e mine he cooling ea-
sibili y. Las ly, he angula sp ead o he ou going gamma
ays is in es iga ed.
INTRODUCTION
The GF PoP expe imen a he SPS in ends o demons a e
he abili y o con ol he exci a ion o pa ially s ipped ions
in a high-ene gy synch o on h ough head-on in e ac ion
wi h a lase beam. The GF PoP expe imen migh open he
doo o possible implemen a ion in o a machine like he
LHC o p oduce high ene gy pho ons [1, 2]. A omic physics
s udies ha can be made possible by using hese high-ene gy
pho on beams can be ound in Re . [3, 4]. Addi ionally,
his echnique educes he beam emi ances and, he e o e,
inc eases a collide ’s luminosi y. Mo e de ailed in o ma ion
abou lase cooling can be ound in Re . [5].
The GF PoP expe imen will use he
2𝑠→2𝑝1/2
ansi-
ion o Li-like Pb ions. The e ha e al eady been simula ion
s udies on he easibili y o he GF PoP expe imen , mos o
which we e pe o med by A. Pe enko, e.g. see Re . [6]. The
simula ions in his pape build on he p e ious wo k using
a new implemen a ion in he Xsui e amewo k [7]. This
beam- acking code is being ac i ely de eloped a CERN
wi h mul i-pu pose accele a o physics in mind. This en-
ables he s udy o cooling along wi h hea ing e ec s such as
In a-Beam Sca e ing (IBS) and Space Cha ge e ec s (SC).
METHOD AND PARAMETERS
The GF PoP expe imen was simula ed in Xsui e assum-
ing a linea ized ep esen a ion o he SPS op ics [8], i.e.,
only de ined by a one- u n anspo ma ix ha embeds he
ans e se and longi udinal unes and a newly implemen ed
lase cooling elemen [9] which models he u n-by- u n in-
e ac ion o he ion beam wi h a pulsed lase . The exci a ion
p obabili y o lase cooling wi h a Fou ie -limi ed pulsed
∗pie e .ma in.k uy @ce n.ch
lase in Xsui e is calcula ed using he op ical Bloch equa-
ions wi h damping, which p o ide he s eady-s a e solu ion
o he popula ion o he exci ed s a e as de ailed in [10,
11]. This app oach does no conside any pola iza ion o
he lase ligh . In he case o a success ul exci a ion, he
ion ene gy is educed due o he spon aneous emission o a
pho on. The amoun o ene gy educ ion is de e mined by
he Dopple -boos ed exci a ion ene gy o he ion and he an-
dom emission angle. Any ans e se hea ing due o emission
in he ho izon al o e ical di ec ion was no conside ed
because he ans e se kicks a e small (
≈12 p ad
). Cooling
a es we e assessed by moni o ing he emi ance and ene gy
sp ead while acking. This acking did no in ol e hea ing
e ec s like SC o IBS, which will be discussed sepa a ely.
Table 1 summa izes he lase pa ame e s necessa y o simu-
la ions and he beam and Twiss pa ame e s a he in e ac ion
poin [6]. These pa ame e s es ablish he expe imen al se up
and condi ions o he GF PoP expe imen a he SPS, se ing
as he basis o subsequen simula ions o he lase cooling
pe o mance. Fu he de ails ega ding he op ical sys em
can be ound in Re . [12].
COOLING RATES
The cooling a e is ound in acking simula ions by i -
ing he i s
50 ms
o he emi ance (o ela i e momen um
sp ead) ime e olu ion o an exponen ial unc ion
𝜖(𝑡)=
𝜖0exp(−𝑐𝑟𝑡)
, whe e
𝜖(𝑡)
is he emi ance a ime
𝑡
,
𝜖0
is he
ini ial emi ance, and
𝑐𝑟
ep esen s he cooling a e. Lase
cooling ac s p ima ily in he longi udinal plane by educing
he ene gy sp ead o he beam. Howe e , i he e is dispe -
sion a he in e ac ion poin , he cooling can also be coupled
o he ans e se plane by dispe si e cooling [15]. Reducing
he momen um o a pa icle in a dispe si e egion will shi
i s o bi , which a ec s i s be a on ampli ude. To educe he
be a on ampli ude in a egion wi h posi i e dispe sion, he
momen um mus be educed a a ho izon al posi ion
𝑥 < 0
.
Fo lase cooling, his can be done by displacing he beam
ela i e o he cen e o he lase beam. A pa ame ic sweep
was pe o med in he simula ion o de e mine he op imal
displacemen o he ion beam ela i e o he lase beam o
cool ho izon ally. The esul s a e shown in Fig. 1, which
shows ha lase cooling is a e sa ile ool ha p o ides he
lexibili y o selec he amoun o longi udinal and ho izon al
cooling by adjus ing he ho izon al o bi o he beam ela i e
o he lase beam. The SPS’s op imal ho izon al displace-
men o ho izon al cooling is
−1.24 mm
. The e o e, his
alue was chosen in he simula ions o he ho izon al cooling
Table 1: Pa ame e s o he GF PoP Expe imen
Pa ame e Value
Lase Pa ame e s
Wa eleng h (𝜆) 1031 nm (1.2 eV)
RMS ela i e band sp ead (𝜎𝜆/𝜆)2×10−4
Single pulse ene gy a IP (𝐸) 5 mJ
Lase wais adius a IP 1.3 mm
Collision angle (𝜃𝐿) 2.6 deg ees
Fou ie -limi ed pulse wid h (𝜎𝑡) 2.74 ps
Twiss Pa ame e s a he In e ac ion Poin
𝛽𝑥, 𝛽𝑦54.47 m, 44.40 m
𝛼𝑥, 𝛼𝑦-1.55, 1.32
𝐷𝑥2.4 m
𝐷𝑝𝑥0.09
Machine Pa ame e s
T ans e se unes (𝑄𝑥, 𝑄𝑦) 26.299, 26.249
Synch o on une (𝑄𝑠) 0.00628
Ion Beam Pa ame e s
Ion species 208Pb79+
Ion mass (𝑚) 193.687 GeV/𝑐2
Mean ene gy (𝐸) 18.652 TeV
Mean Lo en z ac o (𝛾) 96.3
Numbe o ions pe bunch (𝑁)0.9×108
RMS ela i e ene gy sp ead (𝜎𝐸/𝐸)2×10−4
No malized ans e se emi ances (𝜖𝑛) 1.5 mm m ad
RMS bunch leng h (𝜎𝑧) 6.3 cm
Exci ed s a e li e ime 76.6 ps
Ion exci a ion ene gy ℎ𝜔0[13, 14] 231 eV
o he GF PoP expe imen , shown in Fig. 2. Op imal lon-
gi udinal cooling isn’ a ze o o se due o dispe sion. The
lase is uned o in e ac wi h high-ene gy pa icles, which
a e mo e p e alen o posi i e
𝑥
due o he posi i e dispe -
sion. Ve ical cooling, which is no conside ed he e, can be
achie ed by in oducing coupling be ween ho izon al and
e ical planes. Addi ionally, by changing he beam’s ene gy,
he lase can cause longi udinal blow-up, which can mi iga e
IBS.
INTRA-BEAM SCATTERING
The IBS g ow h a es can be ansla ed o an exponen ial
inc ease o he emi ance o oo -mean-squa e dp/p [16, 17],
which can be compa ed o he exponen ial decay due o
cooling. The ac ha o sho ime in e als, he emi ance
e olu ion due o IBS is desc ibed by an exponen ial allows
o a di ec compa ison wi h he exponen ial educ ion due o
cooling, hus de e mining whe he he beam is in a cooling-
domina ed o IBS-domina ed egime. The IBS g ow h a es
ha e been compu ed using he Bjo ken-M ingwa model [18]
and he pa ame e s om Table 1 using xibs [17], which is
an IBS module wi hin Xsui e [7]. The ho izon al emi ance
Figu e 1: Cooling a e as a unc ion o lase -ion ho izon al
o se o
Pb79+
ions. The blue line ep esen s he ho izon al
cooling a e, while he o ange line indica es he longi udinal
cooling a e.
Figu e 2: Time e olu ion o no malized ho izon al emi ance
(blue line) and ela i e momen um sp ead dp/p (o ange line)
o op imized ho izon al cooling in he SPS.
and longi udinal IBS g ow h a es ha e been compu ed o a
ange o no malized emi ance and momen um sp eads and
hen compa ed agains he cooling a es, which a e assumed
o be cons an . Figu e 3 shows he poin s whe e he g ow h
a es om IBS a e equal o he cooling a es o ho izon al
emi ance, solid blue line, o momen um sp ead, dashed blue.
The plo also shows he ime e olu ion o he lase cooling
p ocess wi h he ed do indica ing he assumed ini ial beam
pa ame e s, and each black a ow indica es a ime s ep o
10 s
. Following he a ows, one can see he lase coole can
educe he emi ance and momen um sp ead by almos
50%
.
Ini ially, he cooling a e is la ge enough o o e come he
IBS g ow h a es, and cooling is expec ed o s op when he
beam eaches he cooling-IBS equilib ium a e abou 90 s.
PHOTON PRODUCTION
Fo maximum pho on gene a ion, he lase -ion o se was
se o 0 acco ding o Fig 1. The es ima ed numbe o pho ons
p oduced om a Li-like
Pb79+
beam using he lase om
he GF PoP expe imen is shown in Table 2, which shows
ha he lase can exci e
14.1%
o he ions. This co esponds
Figu e 3: Time e olu ion o he oo -mean-squa e momen-
um sp ead and no malized emi ance o he op imized ho -
izon al cooling in he GF PoP expe imen . The ed do
indica es he beam be o e cooling. Each a ow ep esen s
10 seconds o cooling. The solid blue line indica es he equi-
lib ium be ween ho izon al cooling and IBS g ow h a es,
and he dashed line o he longi udinal plane.
o a pho on in ensi y o
1.27 ×107
pho ons pe bunch o a
single passing by he lase . The Dopple -shi ed lase pho on
ene gy mus ma ch he ion’s exci a ion ene gy. The Dopple
equency shi is compu ed using he Lo en z ans o ma ion
be ween he wo e e ence ames, acco ding o
𝜔′=(1+𝛽cos 𝜃)𝛾𝐿𝜔≈2𝛾𝐿𝜔, (1)
whe e
𝜔′
is he pho on equency in he ion- es ame,
𝜃
is he pho on-ion angle,
𝛾𝐿
he Lo en z ac o ,
𝜔
is he e-
quency in he lab ame, and he small angle app oxima ion
was used. The Eq.
(1)
shows ha he pho on equency in
he ion- es ame
𝜔′
is
2𝛾𝐿
imes la ge han he equency
𝜔
in he lab ame. The Lo en z ans o ma ion also a ec s
he angula sp ead o he pho ons ha a e emi ed by he
exci ed ions. In he como ing ame o he ion, he emis-
sion is equally p obable in e e y di ec ion. Howe e , in he
lab ame, he pho ons will be emi ed wi h angula sp ead
𝜃𝑒∼1/𝛾𝐿
, which means ha he small angle app oxima ion
can also be applied o he Dopple equency shi o he
emi ed pho ons. To de e mine he ene gy o he pho ons a -
e spon aneous emission, he Dopple shi is applied o he
exci a ion ene gy o he ion, which will gi e ano he ac o
o
2𝛾𝐿
. Consequen ly, he ene gy o he emi ed pho on is
enhanced by a ac o
4𝛾2
𝐿
compa ed o he pho ons p oduced
by he lase . Fo example, his ac o would be
≈108
o
LHC beams [19].
The dis ibu ion o pho ons p oduced om he GF PoP
expe imen is shown in Fig. 4 o wo cases: he case whe e
he di e gence o he ion beam is neglec ed, in o ange, and
he case ha also conside s he ion om which he pho on
was emi ed and i s a eling angle. As expec ed, aking his
in o accoun he o al dis ibu ion o pho on angles is wide .
In he case o a cooled ion beam, he angula sp ead will be
smalle , and he angula sp ead o he ions will con ibu e
less o he angula sp ead o he ou going pho ons.
Table 2: Pho on p oduc ion pa ame e s
Pa ame e Value
Numbe o ions pe bunch [20] 0.90 ×108
F ac ion o exci ed pa icles 14.1%
Numbe o emi ed pho ons pe bunch 1.27 ×107
Lase wa eleng h 1.2 eV (1031 nm)
Ion exci a ion ene gy ℎ𝜔0231 eV
Maximum emi ed pho on ene gy 44 keV
Figu e 4: Ene gy dis ibu ion o pho ons p oduced by he
GF as a unc ion o sca e ing angle 𝜃 o Pb79+ions.
CONCLUSION
The simula ions in his s udy ocus on he lase cooling o
Li-like
Pb79+
in he GF PoP expe imen . In pa icula , ho i-
zon al cooling is maximized by inding he op imal lase -ion
beam displacemen o p o i om he la ge ho izon al dispe -
sion a he in e ac ion poin . This op imal ho izon al cooling
se up is used o a long simula ion o he lase cooling pe -
o mance wi hou any hea ing e ec s. The esul ing cooling
a e is compa ed agains he compu ed IBS g ow h a es o
de e mine he equilib ium be ween he lase cooling and he
emi ance blow-up om IBS. The esul s show ha he GF
PoP will be able o achie e an app eciable dec ease in emi -
ance be o e he cooling eaches an equilib ium wi h IBS.
Las ly, simula ions we e pe o med o es ima e he in ensi y
o he p oduced pho on beams and hei ene gy dis ibu ion.
ACKNOWLEDGMENTS
Fi s ly, he au ho s exp ess hei g a i ude o A. Pe enko
o pionee ing and sha ing his ools o simula ing lase
cooling wi h a Fou ie -limi ed lase pulse. Secondly, he au-
ho s would like o hank all membe s o he Gamma-Fac o y
wo king g oup o sha ing hei insigh s and expe ise. Fu -
he mo e, his wo k is suppo ed by he Physics Beyond
Collide s S udy G oup. Las ly, his wo k is pa ially sup-
po ed by he Eu opean Union’s Ho izon 2020 Resea ch and
Inno a ion p og am unde G an Ag eemen No. 101004730
(iFAST).
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