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Mul i-scale molecula dynamics simula ions o enhanced ene gy ans e in o ganic
molecules unde s ong coupling
© The Au ho (s) 2023
Published e sion
Sokolo skii, Ilia; Tichaue , Ru h H.; Mo ozo , Dmi y; Feis , Johannes; G oenho ,
Ge i
Sokolo skii, I., Tichaue , R. H., Mo ozo , D., Feis , J., & G oenho , G. (2023). Mul i-scale
molecula dynamics simula ions o enhanced ene gy ans e in o ganic molecules unde s ong
coupling. Na u e Communica ions, 14, A icle 6613. h ps://doi.o g/10.1038/s41467-023-
42067-y
2023
A icle h ps://doi.o g/10.1038/s41467-023-42067-y
Mul i-scale molecula dynamics simula ions
o enhanced ene gy ans e in o ganic
molecules unde s ong coupling
Ilia Sokolo skii
1,3
,Ru hH.Tichaue
1,2,3
, Dmi y Mo ozo
1
, Johannes Feis
2
&
Ge i G oenho
1
Exci on anspo can be enhanced in he s ong coupling egime whe e
exci ons hyb idize wi h confined ligh modes o o m pola i ons. Because
pola i ons ha e g oup eloci y, hei p opaga ion should be ballis ic and long-
anged. Howe e , expe imen s indica e ha o ganic pola i ons p opaga e in a
di usi e manne and mo e slowly han hei g oup eloci y. He e, we esol e
his con o e sy by means o molecula dynamics simula ions o Rhodamine
molecules in a Fab y-Pé o ca i y. Ou esul s sugges ha pola i on p opa-
ga ion is limi ed by he ca i y li e ime and appea s di usi e due o e e sible
popula ion ans e s be ween pola i onic s a es ha p opaga e ballis ically a
hei g oup eloci y, and da k s a es ha a e s a iona y. Fu he mo e, because
long-li ed da k s a es ansien ly ap he exci a ion, p opaga ion is obse ed
on imescales beyond he in insic pola i on li e ime. These insigh s no only
help o be e unde s and and in e p e expe imen al obse a ions, bu also
pa e he way owa ds a ional design o molecule-ca i y sys ems o cohe en
exci on anspo .
Sola cells based on o ganic molecules a e p omising al e na i es o
he silicon-based echnologies ha domina e oday’s ma ke , mos ly
because o ganic pho o ol aics (OPV) a e cheape o mass-p oduce,
ligh e , mo e flexible and easie o dispose o . A key s ep in ligh ha -
es ing is anspo o exci ons om whe e pho ons a e abso bed o
whe e his ene gy is needed o ini ia ing a pho ochemical p ocess1,
usually deepe inside he ma e ial o he sola cell. Because exci ons in
o ganic ma e ials a e p edominan ly localized on o single molecules,
exci on anspo p oceeds ia incohe en hops2. Such andom-walk
di usion is, howe e , oo slow o compe e wi h ul a- as deac i a ion
p ocesses o single exci ons, such as adia i e and non- adia i e
decay. As exci on di usion is u he mo e hinde ed by he mal dis-
o de , p opaga ion dis ances in o ganic ma e ials ypically emain
below 10 nm2. Such sho di usion leng hs limi he e ficiency o sola
ene gy ha es ing and equi e complex mo phologies o ac i e laye s
in o nanome e sized domains, e.g., bulk he e ojunc ions in OPVs,
which no only complica es de ice ab ica ion, bu also educes de ice
s abili y3,4.
Dis ances o hund eds o nanome e s ha e been obse ed o
he di usion o longe -li ed iple s a es5, bu because no all
o ganic ma e ials can unde go e ficien in e sys em c ossing o
single fission, i may be di ficul o exploi iple di usion in gene al.
Exci on mobili y can also be inc eased h ough ansien exci on
delocaliza ion6–8, bu as he di ec exci onic in e ac ions a e weak in
mos o ganic ma e ials, molecules need o be o de ed o each his
enhanced anspo egime.
Al e na i ely, pe manen delocaliza ion o e la ge numbe s o
molecules can be achie ed by s ongly coupling exci ons in he
ma e ial o he confined ligh modes o op ical ca i ies, such as Fab y-
Pé o esona o s (Fig. 1a) o nano-s uc u ed de ices9–11.In hiss ong
ligh -ma e coupling egime he a e o ene gy exchange be ween
molecula exci ons and confined ligh modes exceeds he in insic
Recei ed: 6 Feb ua y 2023
Accep ed: 21 Sep embe 2023
Check o upda es
1
Nanoscience Cen e and Depa men o Chemis y, Uni e si y o Jy äskylä, P.O. Box 35, Jy äskylä 40014, Finland.
2
Depa amen o de Física Teó ica de la
Ma e ia Condensada and Condensed Ma e Physics Cen e (IFIMAC), Uni e sidad Au ónoma de Mad id, Mad id, Spain.
3
These au ho s con ibu ed equally:
Ilia Sokolo skii, Ru h H. Tichaue . e-mail: ge i .x.g oenho @jyu.fi
Na u e Communica ions | (2023) 14:6613 1
1234567890():,;
1234567890():,;
decay a es o bo h he exci ons and he confined modes leading o he
o ma ion o new cohe en ligh -ma e s a es, called pola i ons12–20.
The majo i y o hyb id s a es in ealis ic molecule-ca i y sys ems
a e da k21–23, meaning ha hey ha e negligible con ibu ions om he
ca i y pho ons. In con as , he ew s a es wi h such con ibu ions a e
he b igh pola i onic s a es ha ha e dispe sion and hence g oup
eloci y, defined as he de i a i e o he pola i on ene gy wi h espec
o in-plane momen um (i.e., k
z
in Fig. 1b). In he ou -o -plane ca i y
di ec ion (i.e., pe pendicula o he mi o s), hese s a es a e deloca-
lized o e he molecules inside he mode olume, while in he in-plane
di ec ion (i.e., pa allel o he mi o s) hey beha e as quasi-pa icles
wi h a low e ec i e mass and la ge g oup eloci y (i.e., ac ions o he
speed o ligh ). These pola i on p ope ies can be exploi ed o bo h
ou -o -plane9–11,24–33, and in-plane ene gy anspo 34–54.
Indeed, a c yogenic empe a u es, in-plane ballis ic p opaga ion
a he g oup eloci y o pola i ons was obse ed o pola i on wa e-
packe s in a Fab y-Pé o mic oca i y con aining an In
0.05
Ga
0.95
As
quan um well34. Ballis ic p opaga ion was also obse ed o pola i ons
o med be ween o ganic molecules and Bloch su ace wa es38,42,51,
while a combina ion o ballis ic anspo on an ul asho imescale
(sub-50 s) and di usi e mo ion on longe imescales was obse ed o
ca i y- ee pola i ons43, o whichs ongcouplingwasachie ed
h ough a misma ch o he e ac i e indices be ween hin laye s o
densely-packed o ganic molecules and a hos ma e ial55.Incon as ,
expe imen s on s ongly coupled o ganic J-agg ega es in me allic
mic o-ca i ies sugges ha molecula pola i ons p opaga e in a di u-
si e manne and much mo e slowly han hei g oup eloci ies40.
Fu he mo e, despi e a low ca i y li e ime on he o de o ens o
em oseconds in hese expe imen s, p opaga ion was obse ed o e
se e al picoseconds, which was a ibu ed o a long li e ime o he
lowe pola i on (LP)17,40. He e, we add ess hese con o e sies by
p o iding a omis ic insigh s in o pola i on p opaga ion wi h mul i-
scale molecula dynamics (MD) simula ions56,57 o sol a ed Rhodamine
molecules s ongly coupled o he confined ligh modes o a one-
dimensional (1D) Fab y-Pé o mic oca i y58, shown schema ically
in Fig. 1a.
Resul s and discussion
Resonan exci a ion
Fi s , we explo e how pola i ons p opaga e a e esonan exci a ion o
a Gaussian wa epacke o LP s a es wi h a b oad-band lase pulse. In
Fig. 2, we show he ime e olu ion o he p obabili y densi y o he
pola i onic wa e unc ion, ∣Ψ( )∣2a e such exci a ion in bo h a pe ec
lossless ca i y wi h an infini e Q- ac o (γ
ca
=0ps
−1, oppanels)anda
lossy ca i y wi h a low Q- ac o (γ
ca
=66.7ps
−1, bo om panels) con-
aining 1024 Rhodamine molecules. Plo s o wa epacke p opaga ion
in sys ems wi h 256 and 512 molecules a e p o ided as Supplemen a y
In o ma ion (SI, Figs. S5–S6), as well as anima ions o he wa epacke s
o all sys em sizes (Supplemen a y Mo ies 1–9and13–21).
Lossless ca i y. In he pe ec lossless ca i y, he o al wa epacke
∣Ψ( )∣2ini ially p opaga es ballis ically close o he maximum g oup
eloci y o he LP b anch ( LP,max
g =68μmps−1, Fig. 1c), un il a ound 100
s (see anima ions in he SI), when i slows down as e idenced by a
dec ease in he slope o he expec a ion alue o he posi ion o he
wa epacke 〈z〉in Fig. 3a. The change om a quad a ic o a linea ime-
dependence o he Mean Squa ed Displacemen (Fig. 3c) a =100 s
u he mo e sugges s a ansi ion om ballis ic o di usi e mo ion.
Du ing p opaga ion, he wa epacke b oadens and sha p ea u es
appea , isible as e ical lines in bo h he o al and molecula wa e-
packe s in Fig. 2a, b and as peaks in he wa epacke anima ions p o-
ided as SI. These peaks coincide wi h he zposi ions o molecules ha
con ibu e o he wa epacke wi h hei exci a ions du ing p opaga-
ion. Such peaks a e no obse ed i he e is no diso de and he
molecula deg ees o eedom a e ozen (Fig. S16), bu appea al eady
a he s a o he simula ion when he ini ial configu a ions o he
molecules a e all di e en (Fig. S22). Simila obse a ions we e made
by Ag ano ich and Ga s ein35 who a ibu ed hese peaks o ene ge ic
diso de among he molecula exci ons. We he e o e also assign hese
peaks o a pa ial localiza ion o he wa epacke a he molecules due
o s uc u al diso de ha al e s hei con ibu ion o he wa epacke .
In con as , because he ca i y modes a e delocalized in space, he
Fig. 1 | Rhodamine-ca i y sys em. a Schema ic illus a ion o an op ical Fab y-
Pé o mic oca i y filled wi h Rhodamine ch omopho es (no o scale). The quan-
um mechanical (QM) subsys em, shown in ball-and-s ick ep esen a ion in he
inse , is desc ibed a he Ha ee-Fock (HF/3-21G) le el o heo y in he elec onic
g ound s a e (S
0
), and a he Configu a ion In e ac ion le el o heo y, unca ed a
single elec on exci a ions (CIS/3-21G), in he fi s single exci ed s a e (S
1
). The
molecula mechanical (MM) subsys em, consis ing o he a oms shown in s ick
ep esen a ion and he wa e molecules (no shown), is modeled wi h he Ambe 03
o ce field. bNo malized angle- esol ed abso p ion spec um (P
abs
)o heca i y,
showing Rabi spli ing be ween lowe pola i on (LP, ed line) and uppe pola i on
(UP, blue line) b anches. The ca i y dispe sion and exci a ion ene gy o he mole-
cules (4.18 eV a he CIS/3-21G//Ambe 03 le el o heo y) a e plo ed wi h do -
dashed and dashed whi e lines, espec i ely. cG oup eloci y o he LP ( ed) and UP
(blue), defined as he de i a i e o he equency o pola i ons ω(k
z
) wi h espec o
he in-plane wa e- ec o k
z
,i.e.,∂ω(k
z
)/∂k
z
.
A icle h ps://doi.o g/10.1038/s41467-023-42067-y
Na u e Communica ions | (2023) 14:6613 2
Fig. 2 | Pola i on p opaga ion a e esonan exci a ion o a wa epacke in he
lowe pola i on (LP) b anch cen e ed a z=5μm. a,b,andc: o al p obabili y
densi y ∣Ψ(z, )∣2, p obabili y densi y o he molecula exci ons ∣Ψ
exc
(z, )∣2and o he
ca i y mode exci a ions ∣Ψ
pho
(z, )∣2, espec i ely, as a unc ion o dis ance (z,ho -
izon al axis) and ime ( e ical axis), in a ca i y wi h pe ec mi o s (i.e., γ
ca
=0ps
−1).
The ed dashed line indica es p opaga ion a he maximum g oup eloci y o he LP
(68 μmps−1). dCon ibu ions o molecula exci ons (black) and ca i y mode
exci a ions ( ed) o ∣Ψ(z, )∣2as a unc ion o ime in he pe ec ca i y. Wi hou ca i y
losses, no g ound s a e popula ion (blue) can build up. e–g∣Ψ(z, )∣2,∣Ψ
exc
(z, )∣2and
∣Ψ
pho
(z, )∣2, espec i ely, as a unc ion o dis ance (z, ho izon al axis) and ime
( e ical axis), in a lossy ca i y (γ
ca
= 66.7 ps−1). hCon ibu ions o he molecula
exci ons (black) and ca i y mode exci a ions ( ed) o ∣Ψ(z, )∣2as a unc ion o ime.
The popula ion in he g ound s a e, c ea ed by adia i e decay h ough he impe -
ec mi o s, is plo ed in blue. Sou ce da a a e p o ided as a Sou ce Da a file.
Fig. 3 | P opaga ion o he pola i onic wa epacke a e on- esonan exci a ion.
Top panels: Expec a ion alue o he posi ion o he o al ime-dependen wa e-
unc ion, hzi=Ψðz, Þ

^
zΨðz, Þ
=Ψðz, ÞjΨðz, Þ

, in an ideal ca i y (a,γ
ca
=0ps
−1)
and a lossy ca i y (b,γ
ca
= 66.7 ps−1). The black lines ep esen 〈z〉while he shaded
a ea a ound he lines ep esen s he oo mean squa ed de ia ion (RMSD, i.e.,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hðzð Þhzð ÞiÞ2i
q). Bo om panels: Mean squa ed displacemen (MSD, i.e., hðzð Þ
hzð0ÞiÞ2i)in heideal(c) and he lossy (d) ca i y. Magen a lines a e quad a ic fi s o
he MSD and cyan lines a e linea fi s. Sou ce da a a e p o ided as a Sou ce Da a file.
A icle h ps://doi.o g/10.1038/s41467-023-42067-y
Na u e Communica ions | (2023) 14:6613 3
pho onic wa epacke emains smoo h h oughou he p opaga-
ion (Fig. 2c).
The ansi ion om ballis ic p opaga ion o di usion a ound 100
s coincides wi h he onse o he molecula exci ons domina ing he
pola i onic wa epacke , as shown in Fig. 2d, in which we plo he
con ibu ions o he molecula exci ons (black line) and ca i y mode
exci a ions ( ed line) o he o al wa e unc ion (see Me hods o
de ails o his analysis). Because in he pe ec ca i y, pho on leakage
h ough he mi o s is absen (i.e., γ
ca
=0ps
−1), he dec ease o ca i y
mode exci a ions is due o popula ion ans e om b igh LP s a es
in o he da k s a e mani old (Fig. S20b)59–61. Thus, while esonan
exci a ion o LP s a es ini ially leads o ballis ic mo ion wi h he cen al
g oup eloci y o he wa epacke , as e idenced by he quad a ic
dependence o he Mean Squa ed Displacemen on ime (Fig. 3c),
popula ion ans e in o da k s a es u ns he p opaga ion in o a di -
usion p ocess, as e idenced by a linea ime-dependence o he Mean
Squa ed Displacemen a e ~100 s.
Since da k s a es lack g oup eloci y, and a e he e o e s a iona y,
while exci onic couplings be ween molecules a e neglec ed in ou
model (see SI), p opaga ion in he di usi e egime mus s ill in ol e
b igh pola i on s a es. Ou simula ions he e o e sugges ha while,
ini ially, molecula ib a ions d i e popula ion ans e om he p o-
paga ing b igh s a es in o he s a iona y da k s a es62, hisp ocessis
e e sible, causing new wa epacke s o o m con inuously wi hin he
ull ange o LP g oup eloci ies. Likewise, he p opaga ion o an-
sien ly occupied b igh s a es is con inuously in e up ed by ans e s
in o da k s a es, and e-s a ed wi h di e en g oup eloci ies. This e-
spawning p ocess leads o he di usi e p opaga ion o he exci a ion
obse ed in Fig. 2, wi h an inc easing wa epacke wid h (Fig. 3a), in line
wi h expe imen al obse a ions40,42,43,51.
Lossy ca i y. Including a compe ing adia i e decay channel by adding
pho on losses h ough he ca i y mi o s a a a e o γ
ca
=66.7ps
−1,
leads o a apid deple ion o he pola i on popula ion (Fig. 2h), bu
does no a ec he o e all anspo mechanism: he wa epacke s ill
p opaga es in wo phases, wi h a as ballis ic egime ollowed by
slowe di usion. Howe e , in con as o he p opaga ion in he ideal
lossless ca i y, we obse e ha he wa epacke empo a ily con ac s.
This con ac ion is isible as a educ ion o bo h he expec a ion alue
o 〈z〉and he Mean Squa ed Displacemen be ween 60 o 130 s in he
igh panels o Fig. 3.
Ini ially, he p opaga ion o he wa epacke is domina ed by bal-
lis ic mo ion o he popula ion in he b igh pola i onic s a es mo ing
a he maximum g oup eloci y o he LP b anch. Howe e , due o non-
adiaba ic coupling62, some o ha popula ion is ans e ed in o da k
s a es ha a e s a iona y. Because non-adiaba ic popula ion ans e is
e e sible, he wa epacke p opaga ion unde goes a ansi ion in o a
di usion egime, which is significan ly slowe , as also obse ed in he
ideal ca i y (Fig. 3c).
In addi ion o hese non-adiaba ic ansi ions, adia i e decay
u he deple es popula ion om he p opaga ing b igh pola i onic
s a es. Because be o e decay, his popula ion has mo ed much u he
han he popula ion ha go apped in he da k s a es, he expec a-
ion alue o 〈z〉, as well as he Mean Squa ed Displacemen , which
we e domina ed ini ially by he as -mo ing popula ion, dec ease un il
he slowe di usion p ocess ca ches up and eaches he same dis ance
a ound 130 s ( igh panels in Fig. 3). Such con ac ion o he wa e-
packe in a lossy ca i y is consis en wi h he measu emen s o Musse
and co-wo ke s, who also obse e such con ac ion a e on- esonan
exci a ion o UP s a es47.
Because o he con ac ion, i is di ficul o see whe e he ansi-
ion be ween he ballis ic and di usion egimes occu s in Fig. 3d. We
he e o e ex apola ed he linea egime ins ead, and es ima e he
u n-o e a 30 s, whe e he quad a ic fi o he ballis ic egime
in e sec s he ex apola ed fi o he di usion egime. As in he pe ec
lossless ca i y, he ansi ion be ween ballis ic and di usion egimes
occu s when he popula ion o molecula exci ons exceeds he
popula ion o ca i y mode exci a ions (Fig. 2h). Howe e , due o he
adia i e decay o he la e , his u no e al eady happens a ound 30
s in he lossy ca i y simula ions.
Owing o he sho ca i y mode li e ime (15 s), mos o he exci-
a ion has al eady decayed in o he g ound s a e a 100 s, wi h a small
emainde su i ing in da k s a es (Fig. 2h) ha lack mobili y. Because
ca i y losses es ic he li e ime o b igh LP s a es, he dis ance a
wa epacke can each is limi ed due o (i) a sho ening o he ballis ic
phase, and (ii) a educ ion o he di usion coe ficien (i.e., he slope o
〈z〉, Fig. 3b) in he second phase. The e o e, he o e all eloci y is
significan ly lowe han in he pe ec ca i y, sugges ing a connec ion
be ween ca i y Q- ac o and p opaga ion eloci y47, while also he
b oadening o he wa epacke is educed (Fig. 3b). Fu he mo e,
because he a e o popula ion ans e is in e sely p opo ional o he
ene gy gap62, and hence highes when he LP and da k s a es o e lap60,
we specula e ha he u n-o e be ween he ballis ic and di usion
egimes depends on he o e lap be ween he abso p ion line wid h o
he molecules and he pola i onic b anches, and can hence be con-
olled by uning he exci a ion ene gy o mo e he cen e o he ini ial
pola i onic wa epacke along he LP b anch. In addi ion, he di ec ion
o ballis ic p opaga ion can be con olled by a ying he incidence
angle o he on- esonan exci a ion pulse.
Compa ison o expe imen s. Ou obse a ions a e in line wi h an-
sien mic oscopy expe imen s, in which b oad-band exci a ion pulses
we e used o ini ia e pola i on p opaga ion. A low empe a u es
F eixane e al. obse ed ballis ic wa epacke p opaga ion o a
s ongly coupled quan um do 34. I we supp ess ib a ions ha d i e
popula ion ans e by eezing he nuclea deg ees o eedom, we
also obse e such pu ely ballis ic mo ion (Figs. S16–S17). In con as , in
oom empe a u e expe imen s on ca i y- ee molecula pola i ons,
Pandya e al.43 iden ified wo anspo egimes: a sho ballis ic phase
ollowed by di usion. Based on he esul s o ou simula ions, we
a ibu e he fi s phase o pu ely ballis ic wa epacke p opaga ion o
pho o-exci ed LP s a es. The slow-down o anspo in he second
phase is a ibu ed o e e sible apping o popula ion inside he
s a iona y da k s a e mani old. Owing o he e e sible ans e o
popula ion be ween hese da k s a es and he LP s a es, p opaga ion
con inues di usi ely a ime scales exceeding he pola i on li e ime, in
line wi h expe imen 40,43.
O - esonan exci a ion
Nex , we in es iga e pola i on p opaga ion a e an o - esonan exci-
a ion o he molecule-ca i y sys em. Expe imen ally such o - esonan
exci a ion condi ions a e achie ed by op ically pumping a highe -
ene gy elec onic s a e o he molecules38,40,42,51, which hen apidly
elaxes in o he lowes ene gy exci ed s a e (S
1
) acco ding o Kasha’s
ule63. We he e o e modeled o - esonan pho o-exci a ion by s a ing
he simula ions di ec ly in he S
1
s a e o a single molecule, loca ed a
z=5μm in he ca i y (SI). In Fig. 4, we show he ime e olu ion o he
p obabili y densi y o he o al pola i onic wa e unc ion, ∣Ψ( )∣2,a e
such exci a ion in bo h a pe ec lossless ca i y wi h an infini e Q- ac o
(γ
ca
=0ps
−1, op panels) and a lossy ca i y wi h a low Q- ac o
(γ
ca
=66.7ps
−1, bo om panels) con aining 1024 Rhodamine mole-
cules. Plo s o he wa epacke p opaga ion in sys ems wi h 256 and 512
molecules a e p o ided as SI (Figs. S9–S10),aswellasanima ionso he
wa epacke s o all sys em sizes (Supplemen a y Mo ies 25–33
and 37–45).
Lossless ca i y. In he lossless ca i y wi h pe ec mi o s, he exci-
a ion, ini ially localized a a single molecule, apidly sp eads o o he
molecules (see anima ion in he SI). In con as o he ballis ic mo e-
men obse ed o on- esonan exci a ion, he wa epacke sp eads ou
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ins ead, wi h he on o he wa epacke p opaga ing a a eloci y ha
closely ma ches he maximum g oup eloci y o he LP b anch
(68 μmps−1, Fig. 1c), while he expec a ion alue o he wa epacke
posi ion (〈z〉,Fig.5a) mo es a a lowe pace (~10 μmps−1).
Because we do no include nega i e k
z
- ec o s in ou ca i y
model, p opaga ion can only occu in he posi i e zdi ec ion. Wi h
nega i e k
z
- ec o s, p opaga ion in he opposi e di ec ion cancels such
mo ion leading o 〈z〉≈0 (Fig. S15a). Ne e heless, since he Mean
Squa ed Displacemen is no a ec ed by b eaking he symme y o he
1D ca i y, and inc eases linea ly wi h ime in bo h uni- and bi-
di ec ional ca i ies (Figs. 5c and S15b), we conside i easonable o
assume ha he mechanism unde lying he p opaga ion p ocess is
iden ical.
Because he popula ion o da k s a es domina es h oughou
hese simula ions (Fig. 4d), and di ec exci onic couplings a e no
accoun ed o in ou model (SI), he obse ed p opaga ion mus again
in ol e b igh pola i on s a es. Since he ini ial s a e, wi h one mole-
cule exci ed, is no an eigens a e o he molecule-ca i y sys em,
popula ion exchange om his s a e in o he p opaga ing b igh s a es
is no only due o displacemen s along ib a ional modes ha a e
o e lapping wi h he non-adiaba ic coupling ec o 62, bu also due o
Rabi oscilla ions, in pa icula a he s a o he simula ion.
To quan i y o wha ex en he o e all p opaga ion is d i en by
popula ion ans e s due o he molecula displacemen s, we pe -
o med addi ional simula ions a 0 K wi h all nuclea deg ees o
eedom ozen. As shown in Fig. S18, he p opaga ion is educed a
0 K, and he wa epacke emains mo e localized on he molecule ha
was ini ially exci ed, han a 300 K. A quad a ic ime-dependence o
he Mean Squa ed Displacemen o he ca i y mode con ibu ions o
he wa epacke (Fig. S19 ) u he mo e sugges ha he mobili y a
0 K is d i en by he cons uc i e and des uc i e in e e ences o he
b igh pola i onic s a es, which e ol e wi h di e en phases
(i.e., eiEm =_).
The educed mobili y o he wa epacke a 0 K compa ed o
300 K (Fig. S19) confi ms ha he mally ac i a ed displacemen s o
nuclea coo dina es, which a e absen a 0 K, a e essen ial o d i e
popula ion in o he b igh s a es and sus ain he p opaga ion o he
pola i on wa epacke . Thus, as du ing he di usion phase obse ed
o on- esonan exci a ion, ballis ic mo ion o b igh s a es is con-
inuously in e up ed and es a ed wi h di e en g oup eloci ies,
which makes he o e all p opaga ion appea di usi e wi h a Mean
Squa ed Displacemen ha depends linea ly on ime (Fig. 5c), in line
wi h expe imen al obse a ions40,42.
In he pe ec ca i y, p opaga ion and b oadening con inue
indefini ely due o he long- ange ballis ic mo ion o s a es wi h highe
g oup eloci ies. Indeed, a small ac ion a he on o he wa e-
packe , which mo es e en as e han he maximum g oup eloci y o
he LP (indica ed by a yellow dashed line in Fig. 4c), is mos ly com-
posed o highe -ene gy UP s a es. These s a es no only ha e he
highes in-plane momen a, bu also elax mos slowly in o he da k
s a e mani old o he pe ec ca i y due o he in e se dependence o
he non-adiaba ic coupling on he ene gy gap57. Momen um- esol ed
pho o-luminenscence spec a a wo dis ances om he ini ial exci a-
ion spo (Fig. S24) confi m ha he on o he wa epacke is indeed
composed o UP s a es: a sho dis ances (z=10μm) om he exci-
a ion spo (z=5μm), he emission spec um, accumula ed o e 100 s
simula ion ime, closely ma ches he ull pola i on dispe sion o Fig. 1b,
displaying bo h he LP and UP b anches. In con as , u he away om
he exci a ion spo (z=20μm), he emission exclusi ely o igina es
om he highe ene gy UP s a es, sugges ing ha only hese s a es can
each he longe dis ance wi hin 100 s.
Lossy ca i y. Adding a adia i e decay channel o he ca i y mode
exci a ions (γ
ca
=66.7ps
−1) es ic s he dis ance o e which pola -
i ons p opaga e (Fig. 4e–g), bu does no a ec he o e all anspo
mechanism, as we also obse e a linea inc ease o he Mean Squa ed
Fig. 4 | Pola i on p opaga ion a e o - esonan exci a ion in o he S
1
s a e o a
single molecule loca ed a z=5μm. a–c: To al p obabili y densi y ∣Ψ(z, )∣2,
p obabili y densi y o he molecula exci ons ∣Ψ
exc
(z, )∣2and o he ca i y mode
exci a ions ∣Ψ
pho
(z, )∣2, espec i ely, as a unc ion o dis ance (z, ho izon al axis)
and ime ( e ical axis), in a ca i y wi h pe ec mi o s (i.e., γ
ca
=0ps
−1). The ed
and yellow dashed lines indica e p opaga ion a he maximum g oup eloci y o he
lowe pola i ons (68 μmps−1) and uppe pola i ons (212 μmps−1), espec i ely.
dCon ibu ions o he molecula exci ons (black) and ca i y mode exci a ions ( ed)
o ∣Ψ(z, )∣2as a unc ion o ime in he pe ec ca i y. Wi hou ca i y decay, he e is
no build-up o g ound s a e popula ion (blue). e–g∣Ψ(z, )∣2,∣Ψ
exc
(z, )∣2and
∣Ψ
pho
(z, )∣2, espec i ely, as a unc ion o dis ance (z, ho izon al axis) and ime
( e ical axis), in a lossy ca i y (i.e., γ
ca
= 66.7 ps−1). hCon ibu ions o he mole-
cula exci ons (black), and ca i y mode exci a ions ( ed) o ∣Ψ(z, )∣2as a unc ion o
ime in he lossy ca i y. The popula ion in he g ound s a e, c ea ed by adia i e
decay h ough he impe ec mi o s, is plo ed inblue. Sou ce da a a e p o idedas
a Sou ce Da a file.
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Displacemen wi h ime (Fig. 5d). While he p opaga ion in he lossy
ca i y ini ially is e y simila o ha in he ideal lossless ca i y, adia i e
decay selec i ely deple es popula ion om he p opaga ing b igh
s a es and he wa epacke slows down, as e idenced by he expec a-
ion alue o he displacemen , 〈z〉, le elling o in Fig. 5b. In addi ion,
since he maximum dis ance a wa epacke can a el in a lossy ca i y is
de e mined by he ca i y li e ime in combina ion wi h he g oup
eloci y38, he b oadening o he wa epacke is also mo e limi ed when
ca i y losses a e included (Fig. 5b). Fu he mo e, e en i da k s a es do
no ha e a significan con ibu ion om he ca i y mode exci a ions,
he e e sible ans e o popula ion be ween he da k s a e mani old
and he decaying b igh pola i onic s a es, also leads o a significan
educ ion o da k s a e popula ion in he lossy ca i y as compa ed o
he ideal lossless ca i y (Fig. 4d, h). Ne e heless, da k s a es s ill
p o ide p o ec ion om ca i y losses as he o e all li e ime o he
pho o-exci ed molecule-ca i y sys em (>150 s) significan ly exceeds
ha o he ca i y modes (15 s).
Compa ison o expe imen s. In mic oscopy expe imen s elying on
o - esonan op ical pumping, pola i on emission is ypically obse ed
be ween he exci a ion spo and a poin se e al mic ons u he
away38,40,42,46,51. While such a b oad emission pa e n is eminiscen o a
di usion p ocess, he ma ch be ween he o al dis ance o e which
ha emission is de ec ed on he one hand, and he p oduc o he
maximum LP g oup eloci y and ca i y li e ime on he o he hand,
sugges s ballis ic p opaga ion. The esul s o ou simula ions a e hus
in quali a i e ag eemen wi h such obse a ions as also ou esul s
sugges ha , while pola i on p opaga ion appea s di usi e unde o -
esonan exci a ion condi ions, he on o he wa e packe p opa-
ga es close o he maximum g oup eloci y o he LP b anch.
Based on he analysis o ou MD ajec o ies we p opose ha on
he expe imen ally accessible imescales, pola i on p opaga ion
appea s di usi e due o e e sible popula ion ans e s be ween s a-
iona y da k s a es and p opaga ing b igh s a es. Fo lossy ca i ies,
adia i e decay o he ca i y modes u he slows down pola i on
anspo such ha he exci a ion eaches a maximum dis ance be o e
decaying comple ely. Because a la ge ac ion o he popula ion
esides in he non-decaying da k s a es, he li e ime o he molecule-
ca i y sys em is ex ended60, and pola i on p opaga ion can be
obse ed on imescales a beyond he ca i y li e ime, in line wi h
expe imen 40.
No e ha in ou simula ions we only couple exci ons o he modes
o he Fab y-Pé o ca i y, whe eas in expe imen s wi h mic o-ca i ies
cons i u ed by me allic mi o s, exci ons can in p inciple also couple o
su ace plasmon pola i ons (SPPs) below he ligh line ha a e sup-
po ed by hese me al su aces. While hei ole will depend on he
de ails o he se -up (e.g., he ma e ials used, ene gy o he ele an
molecula exci a ions, e c.), we canno ule ou ha e e sible popu-
la ion ans e be ween da k s a es and SPP-exci on pola i ons also
con ibu es o he e ec i e di usion cons an obse ed in hose
expe imen s40,48. Howe e , because SPPs decay exponen ially away
om he me al su ace, and SPP-exci on pola i ons also ha e g oup
eloci y, he quali a i e beha io is no expec ed o change.
Size dependence
The Rabi spli ing depends on he acuum field s eng h, E
y
,and he
numbe o molecules, N, ia_ΩRabi ≈2μTDM Eyffiffiffiffi
N
p,wi hμTDM he
molecula ansi ion dipole momen ,which o o ganic molecules is on
he o de o a ew Debye. Because he acuum field s eng h o a ca i y
is in e sely p opo ional o he squa e oo o he mode olume (Eq. 3
Fig. 5 | P opaga ion o he pola i onic wa epacke a e o - esonan exci a-
ion. Top panels: Expec a ion alue o he posi ion o he o al ime-dependen
wa e unc ion, hzi=Ψðz, Þ

^
zΨðz, Þ
=Ψðz, ÞjΨðz, Þ

, in an ideal ca i y (a,
γ
ca
=0ps
−1) and a lossy ca i y (b,γ
ca
= 66.7 ps−1).The black lines ep esen 〈z〉while
he shaded a ea a ound he lines ep esen s he oo mean squa ed de ia ion
(RMSD, i.e., ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hðzð Þhzð ÞiÞ2i
q). Bo om panels: Mean squa ed displacemen (MSD,
i.e., (hðzð Þzð0ÞÞ2i)) in he ideal lossless ca i y (c) and he lossy ca i y (d). Cyan
lines a e linea fi s o he MSD. Sou ce da a a e p o ided as a Sou ce Da a file.
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in SI), he Rabi spli ing scales wi h he molecula concen a ion in he
mode olume, V
ca
, o he ca i y, i.e., _ΩRabi /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N=Vca
p. Reaching he
s ong coupling egime o o m pola i ons wi h o ganic molecules in
Fab y-Pé o ca i ies wi h mode olumes on he o de o Vca /
ðλca =nÞ3(whe e λ
ca
is he wa eleng h o he ca i y mode and n he
e ac i e index), hus equi es collec i e coupling o many molecules
(i.e., 105-108)64–66. Because he numbe o molecules we can include in
ou simula ions is much smalle due o limi a ions on ha d- and so -
wa e, we in es iga ed how ha numbe a ec s he p opaga ion by
epea ing simula ions o di e en N. To keep he Rabi spli ing con-
s an , and hence he pola i on dispe sion he same, we scaled he
mode olume wi h N,i.e.,V
ca
=NV
ca ,0
,whe eV
ca ,0
is he mode
olume equi ed o achie e a Rabi spli ing o 325 meV wi h a single
Rhodamine molecule in he ca i y.
Wi h he excep ion o he smalles ensemble ha lacks da k s a es,
we obse e o all o he ensemble sizes ha he p opaga ion
mechanism in ol es e e sible popula ion exchange be ween he
s a iona y da k s a e mani old and p opaga ing pola i ons
(Figs. S5–S13, SI). These addi ional simula ions he e o e unde sco e
he ole o da k s a es in he p opaga ion p ocess and sugges ha he
mechanism does no s ongly depend on N.Incon as o he
mechanism, howe e , he a es a which hese popula ion exchanges
occu , depend on he numbe o molecules. Indeed, hese a es a e
in e sely p opo ional o N23,62,67. Because he numbe o da k s a es
scales wi h N, whe eas he numbe o pola i onic s a es is cons an
(Fig. S4), we obse e ha o he la ge ensembles, he ac ion o
popula ion esiding wi hin he da k s a e mani old is highe han o
he smalle ensembles. Such di e ences a ec (i) he p opaga ion
eloci y (e.g., Fig. S14); (ii) he li e ime (e.g., igh columns in Figs. 4and
S11) and he e o e also (iii) he dis ance o e which he exci on-
pola i ons a e ans e ed (e.g., Figs. 5,andS11–S13).
Because he eloci y is in e sely p opo ional o N(Fig. S14), he
p opaga ion eloci y in expe imen s, wi h 105-108molecules inside he
mode olume64–66, is much lowe han in ou simula ions. Ne e heless,
because o he 1/Nscaling, he e ec i e pola i on p opaga ion eloci y
app oaches he lowe expe imen al limi o 105coupled molecules64
al eady a ound 1000 molecules. We he e o e conside he esul s o
he simula ions wi h 1024 Rhodamines su ficien ly ep esen a i e o
expe imen and o p o iding quali a i e insigh s in o pola i on p o-
paga ion. Indeed, a p opaga ion speed o 9.6 μmps−1in he ca i y
con aining 1024 molecules is abou an o de o magni ude below he
maximum g oup eloci y o he LP (68 μmps−1)inlinewi hexpe i-
men s on o ganic mic oca i ies40, and ca i y- ee pola i ons43.
Summa y and ou look
To conclude, we ha e in es iga ed exci on anspo in ca i ies filled
wi h Rhodamine molecules by means o a omis ic MD simula ions ha
no only include he de ails o he ca i y mode s uc u e57,58,bu also
he chemical de ails o he ma e ial56. The esul s o ou simula ions
sugges ha anspo is d i en by an in e play be ween p opaga ing
b igh pola i onic s a es and s a iona y da k s a es. Re e sible popu-
la ion exchanges be ween hese s a es in e up ballis ic mo ion in
b igh s a es and make he o e all p opaga ion p ocess appea di u-
si e. While o o - esonan exci a ion o he molecule-ca i y sys em,
hese exchanges a e essen ial o ans e popula ion om he ini ially
exci ed molecule in o he b igh pola i onic b anches and s a he
p opaga ion p ocess, he exchanges limi he du a ion o he ini ial
ballis ic phase o on- esonan exci a ion. As adia i e decay o he
ca i y modes selec i ely deple es he popula ion in b igh s a es, bal-
lis ic p opaga ion is es ic ed e en u he i he ca i y is lossy.
Because da k s a es lack in-plane momen um, he e e sible popula-
ion exchange be ween da k and b igh s a es causes di usion in all
di ec ions. The e o e, unde o - esonan exci a ion condi ions,
he p opaga ion di ec ion canno be con olled. In con as ,
because b igh s a es ca y momen um, he p opaga ion di ec ion in
he ballis ic phase can be con olled p ecisely by uning he incidence
angle and exci a ion wa eleng h unde on- esonan exci a ion
condi ions.
The a e a which popula ion ans e s be ween b igh and da k
s a es depends on he non-adiaba ic coupling ec o , whose di ec ion
and magni ude a e de e mined by he Huang-Rhys ac o in combi-
na ion wi h he equency o he F anck-Condon ac i e ib a ions62,
bo h o which a e ela ed o he molecula S okes shi 68.Inaddi ion,
because he non-adiaba ic coupling is in e sely p opo ional o he
ene gy gap62, he S okes shi in combina ion wi h he Rabi spli ing,
also de e mines he egion on he LP b anch in o which popula ion
ans e s a e o - esonan exci a ion o a single molecule69–72.We
he e o e specula e ha he S okes shi can be an impo an con ol
knob o uning he cohe en p opaga ion o pola i ons.
Because ou Rhodamine model ea u es he key pho ophysical
cha ac e is ics o an o ganic dye molecule, we specula e ha he
p opaga ion mechanism obse ed in ou simula ions is gene ally alid
o exci on anspo in s ongly-coupled o ganic mic o-ca i ies, in
which he abso p ion line wid h o he ma e ial exceeds he Rabi
spli ing and he e is a significan o e lap be ween b igh and da k
s a es. To confi m his, we ha e also pe o med simula ions o exci on
anspo in ca i ies con aining Te acene and Me hylene Blue and
obse ed ha he p opaga ion mechanism emains he same
(Figs. S25–S30, Supplemen a y Mo ies 10–12 and 46–48). Fu u e wo k
will be aimed a in es iga ing how he p opaga ion can be con olled
by uning molecula pa ame e s, empe a u e, Rabi spli ing (Fig. S23),
o ca i y Q- ac o 73. Because we include he s uc u al de ails o bo h
ca i y and molecules, ou simula ions, which a e in quali a i e ag ee-
men wi h expe imen s, could be used o sys ema ically op imize
molecule-ca i y sys ems o enhancing ene gy ans e .
Me hods
Mul iscale Ta is-Cummings simula ion model
We used he mul i-scale Ta is-Cummings model, in oduced by Luk
e al.56, and ex ended o he mul iple modes o a one-dimensional (1D)
Fab y-Pé o mic o-ca i y58 by Tichaue e al.57, o pe o m molecula
dynamics (MD) simula ions o 1024 sol a ed Rhodamine molecules
s ongly coupled o he confined ligh modes o a 1D Fab y-Pé o
mic o-ca i y, shown in Fig. S158. In his model, we apply he Bo n-
Oppenheime app oxima ion o sepa a e he nuclea deg ees o ee-
dom, which we ea classically, om he elec onic deg ees o ee-
dom and he ca i y modes. Wi hin he single-exci a ion subspace,
p obed expe imen ally unde weak d i ing condi ions, and employing
he o a ing wa e app oxima ion (RWA), alid o ligh -ma e cou-
pling s eng hs below 10% o he ma e ial exci a ion ene gy74,we
model he elec onic plus ca i y mode deg ees o eedom wi h he
Ta is-Cummings model o Quan um Op ics75,76. In he long-wa eleng h
app oxima ion, he in e ac ion be ween he molecula exci ons and
he ca i y modes a e modeled as he inne p oduc s be ween he
ansi ion dipole momen s and he acuum field associa ed wi h an
exci a ion o he Fab y-Pé o ca i y modes. The mul i-scale Ta is-
Cummings model is desc ibed in ou p e ious wo ks56,57,60,andwe
p o ide a concise summa y o he de ails ele an o his wo k in
Sec ion 1 o he SI.
Rhodamine model
The elec onic g ound s a e (S
0
) o he Rhodamine molecules was
modeled a he hyb id Quan um Mechanics / Molecula Mechanics
(QM/MM) le el77,78, using he es ic ed Ha ee-Fock (HF) me hod in
combina ion wi h he 3-21G basis se 79 o he QM subsys em, which
con ains he used ings o he molecule. The MM subsys em, con-
sis ing o he es o he molecule and 3,684 TIP3P wa e s80,was
modeled wi h he Ambe 03 o ce field81.Thefi s elec onic exci ed
s a e (S
1
) o he QM egion was modeled wi h Configu a ion In e ac-
ion, unca ed a single elec on exci a ions (CIS/3-21G//Ambe 03). A
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Na u e Communica ions | (2023) 14:6613 7
his le el o heo y, he exci a ion ene gy o Rhodamine is 4.18 eV,
which is significan ly o e es ima ed wi h espec o expe imen s. This
disc epancy is due o he limi ed size o he basis se and he neglec o
elec on-elec on co ela ion in he ab ini io me hods we used. While
including elec on-elec on co ela ion in o he desc ip ion o he QM
egion imp o es he e ical exci a ion ene gy, we show in he SI ha
his does no significan ly change he opology o he ele an
po en ial ene gy su aces (Fig. S3), which de e mines he molecula
dynamics. Fu he de ails o he Rhodamine simula ion se up, as well
as he ullde ails o addi ional simula ions o Te acene incyclohexane
and o Me hylene Blue in wa e , a e p o ided in he SI.
Molecula dynamics o Rhodamine-ca i y sys ems
Ca i y model. A e a 200 ns equilib a ion a he o ce field le el, and a
u he 100 ps equilib a ion a he QM/MM le el, he 1024 Rhodamine
molecules we e placed wi h equal in e -molecula dis ances on he z-
axis o a pe iodic 1D ca i y35,58 o leng h L
z
=50μm, whe e zindica es
he in-plane di ec ion (i.e., pa allel o he mi o s). Wi h a dis ance o
L
x
= 163 nm be ween he mi o s (ca i y wid h), whe e xindica es he
ou -o -plane di ec ion (i.e., pe pendicula o he mi o s), he unda-
men al mode o he ca i y has an ene gy o ℏω
0
= 3.81 eV a no mal
incidence (i.e., k
z
= 0) and hence i s dispe sion is ed-de uned wi h
espec o he molecula exci a ion ene gy a 4.18 eV (ho izon al
dashed whi e line in Fig. 1b). The dispe sion, ωca ðkzÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω2
0+c2k2
z=n2
q
(do -dashed whi e line in Fig. 1b), was modeled wi h 160 modes
(0 ≤p≤159 o k
z
=2πp/L
z
,wi hc he speed o ligh and n he e ac i e
index)35.Wi haca i y acuumfield s eng h o 0.26 MVcm−1, heRabi
spli ing, defined as he ene gy di e ence be ween he b igh lowe
(LP) and uppe pola i on (UP) b anches a he wa e- ec o k es
zwhe e
he ca i y dispe sion ma ches he molecula exci a ion ene gy (Fig. 1b),
was~ 325 meV. While he choice o a 1D ca i y model wi h only posi-
i e k
z
ec o s was mo i a ed by he necessi y o keep ou simula ions
compu a ionally ac able, i p ecludes he obse a ion o elas ic
sca e ing e en s ha would change he di ec ion (i.e., in-plane
momen um, ℏk) o p opaga ion. Fu he mo e, wi h only posi i e k
z
ec o s, pola i on mo ion is es ic ed o he + zdi ec ion, bu we show
in he SI (Fig. S15) ha his assump ion does no a ec ou conclusions
abou he anspo mechanism. To maximize he collec i e ligh -
ma e coupling s eng h, he ansi ion dipole momen s o he Rho-
damine molecules we e aligned o he acuum field a he s a o he
simula ion. The same s a ing coo dina es we e used o all Rhoda-
mines, bu di e en ini ial eloci ies we e selec ed andomly om a
Maxwell-Bol zmann dis ibu ion a 300 K. We checked ha adding
diso de by andomly selec ing configu a ions om he equilib ium
QM/MM ajec o y, o by andomly placing molecules on he z-axis,
does no a ec he conclusions o ou wo k (Figs. S21 and 22).
Mean-field molecula dynamics. Eh en es MD ajec o ies we e
compu ed by nume ically in eg a ing New on’s equa ions o mo ion
using a leap- og algo i hm wi h a 0.1 s imes ep82. The mul i-mode
Ta is-Cummings Hamil onian (Eq. 4 in SI) was diagonalized a each
ime-s ep o ob ain he (adiaba ic) pola i onic eigens a es75,76:
ψm
=X
N
j
βm
j
^
σ+
j+X
nmode
p
αm
p
^
ay
p
!

S1
0S2
0::SN1
0SN
00ð1Þ
wi h eigenene gies E
m
. He e, jS1
0S2
0::SN1
0SN
0ij0i ep esen s he wa e
unc ion o he molecule-ca i y sys em in he g ound s a e, in which
nei he he molecules, no he ca i y modes a e exci ed. The c ea ion
ope a o s ^
σ+
j=jSj
1ihSj
0jand ^
ay
p=j1pih0pjexci e molecule jand ca i y
mode pwi h in-plane momen um k
z
=2pπ/L
z
, espec i ely. The βm
jand
αm
pexpansion coe ficien s hus eflec he con ibu ion o he
molecula exci ons (jSj
1i) and o he ca i y mode exci a ions (j1pi) o
pola i on jψmi.
The o al wa e unc ion, Ψð Þ
, was cohe en ly p opaga ed along
wi h he classical deg ees o eedom o he molecules as a ime-
dependen supe posi ion o he pola i onic eigens a es:
Ψð Þ
=X
m
cmð Þψm
ð2Þ
whe e c
m
( ) a e he ime-dependen expansion coe ficien s o he ime-
independen eigens a es jψmi. A uni a y p opaga o in he local
diaba ic basis was used o in eg a e hese coe ficien s83,while he
nuclea deg ees o eedom o he molecules we e e ol ed on he
mean-field po en ial ene gy su ace.
Resul s epo edin hiswo kwe eob ainedasa e ageso e a
leas wo ajec o ies. Fo all simula ions we used G omacs 4.5.384,in
which he mul i-mode Ta is-Cummings QM/MM model was
implemen ed57, in combina ion wi h Gaussian1685.Fu he de ailso
he simula ions, including o he ensemble sizes o he Rhodamine-
ca i y sys ems, and di e en molecules, i.e., Te acene and Me hylene
Blue (Fig. S2), a e p o ided in he SI.
Exci a ion condi ions. Resonan exci a ion in o he LP b anch by a
sho b oad-band lase pulse, as o en used in ime- esol ed
expe imen s34,43,47, was modeled by p epa ing a Gaussian wa epacke
o LP s a es cen e ed a ℏω= 3.94 eV whe e he g oup eloci y o he LP
b anch is highes , and wi h a bandwid h o σ=0.707μm−135.Thus, he
expansion coe ficien s o he wa e unc ion, Ψð =0Þ
(Eq. 2), we e
ini ia ed as
cmð =0Þ=ζ
2π3

1
4
exp½ζðkm
zkcÞ2ð3Þ
wi h ζ=10
−12m2cha ac e izing he wid h o he wa epacke and km
z he
expec a ion alue o he in-plane momen um o pola i on ψm

(i.e., hkm
zi=Pnmode
pjαm
pj2kz,p=Pnmode
pjαm
pj2).
Expe imen ally, an o - esonan exci a ion in a molecule-ca i y
sys em is achie ed by op ically pumping a highe -ene gy elec onic
s a e o he molecules38,40,42,51, which hen apidly elaxes in o he
lowes ene gy exci ed s a e (S
1
) acco ding o Kasha’s ule
63.We
he e o e modeled o - esonan pho o-exci a ion by s a ing he
simula ions di ec ly in he S
1
s a e o a single molecule, loca ed a
z=5μm in he ca i y. This was achie ed by ini ia ing he expansion
coe ficien s o he wa e unc ion (Eq. 2)ascmð =0Þ=βm
j.Amo e
de ailed de i a ion o hese ini ial condi ions is p o ided in he SI.
We assume ha he in ensi y o he exci a ion pulse in bo h cases
is su ficien ly weak o he sys em o emain wi hin he single-
exci a ion subspace in ou simula ions. We hus exclude mul i-pho on
abso p ion and model he in e ac ion wi h he pump pulse as an
ins an aneous abso p ion o a single pho on.
Ca i y li e ime. Because he ligh -confining s uc u es used in p e-
ious expe imen s (e.g., Fab y-Pé o ca i ies34,40,47,48, Bloch su ace
wa es38,42,51, o plasmonic la ices41,46,54) span a wide ange o quali y
ac o s (Q- ac o s), we also in es iga ed he e ec o he ca i y mode
li e ime on he anspo by pe o ming simula ions in an ideal
lossless ca i y wi h no pho on decay (i.e., γ
ca
=0ps
−1), and a lossy
ca i y wi h decay a e o 66.7 ps−1. This decay a e co esponds o a
li e ime o 15 s, which is in he same o de o magni ude as he 2–15
s li e imes epo ed o me allic Fab y-Pé o ca i ies in
expe imen s40,86–88. Ca i y losses we e modeled as a fi s -o de decay
o popula ion om eigens a es wi h con ibu ions om ca i y mode
exci a ions. Assuming ha he in insic decay a es γ
ca
a e he same
o all modes, he o al loss a e o an eigens a e, ψm
, is calcula ed
as he p oduc o γ
ca
and he o al pho onic weigh , Ppjαm
pj2, o ha
eigens a e.
A icle h ps://doi.o g/10.1038/s41467-023-42067-y
Na u e Communica ions | (2023) 14:6613 8