Dipola d ople s o s ongly in e ac ing molecules
Tim Langen,1, 2 Jo di Bo ona ,3Juan Sánchez-Baena,3Raúl Bombín,4Tijs Ka man,5and Fe an Mazzan i3
15. Physikalisches Ins i u and Cen e o In eg a ed Quan um Science and Technology,
Uni e si ä S u ga , P a enwald ing 57, 70569 S u ga , Ge many
2Vienna Cen e o Quan um Science and Technology,
A omins i u , TU Wien, S adionallee 2, 1020 Vienna, Aus ia∗
3Depa amen de Física, Uni e si a Poli ecnica de Ca alunya, Campus No d B4-B5, 08034 Ba celona, Spain
4Ins i u des Sciences Moléculai es (ISM), Uni e si é de Bo deaux,
351 Cou s de la Libé a ion, 33405 Talence, F ance
5Ins i u e o Molecules and Ma e ials, Radboud Uni e si y, Nijmegen, The Ne he lands
(Da ed: July 16, 2024)
We simula e a molecula Bose-Eins ein condensa e in he s ongly dipola egime, obse ing he
exis ence o sel -bound d ople s, as well as hei spli ing in o mul iple d ople s by con inemen -
induced us a ion. Ou quan um Mon e Ca lo app oach goes beyond he limi s o he es ablished
e ec i e mean- ield heo ies o dipola quan um gases, e ealing small d ople s p oduced by s ong
dipola in e ac ions ou side known s able egimes. The simula ions include ealis ic molecula in-
e ac ions and he e o e ha e di ec ele ance o cu en and u u e expe imen s.
In oduc ion. Ul acold dipola Bose-Eins ein con-
densa es (BECs) a e known o exhibi many ex-
o ic quan um phases [1,2], anging om sel -bound
d ople s [3,4] o d ople supe solids [5–7] and exo ic
supe solid pa e ns [8–10]. A key ole in he o ma ion,
s abili y, and dynamics o hese phases is played by e-
pulsi e quan um luc ua ions, which can s abilize he
dipola sys ems agains a mean- ield collapse [11,12].
In he heo e ical modelling, hese luc ua ions a e usu-
ally aken in o accoun using an e ec i e mean- ield de-
sc ip ion based on he ex ended G oss-Pi ae skii equa-
ion (eGPE) [13–15].
So a , he expe imen al s udy o dipola gases has
mos ly been based on magne ic BECs, o med by
weakly dipola lan hanide a oms [2]. In hese cases,
he e ec i e desc ip ion yields — excep o excep-
ional cases [16–18] — a good desc ip ion o he physics
obse ed. Howe e , wi h he ecen d ama ic ad-
ances [19] in p epa ing s ongly dipola molecula gases
close o o in he quan um egime [20–27] i can be an-
icipa ed ha his desc ip ion will soon each i s lim-
i s [10,28]. Mo eo e , on a mo e undamen al le el,
i is expec ed ha s ongly in e ac ing molecula gases
will sha e impo an simila i ies wi h solid helium [10],
whe e acancy-induced o ms o supe solidi y ha e been
in es iga ed o decades [29]. I is hus essen ial o ex-
plo e he limi s o he cu en e ec i e mean- ield ap-
p oaches and ex end he heo e ical desc ip ion o dipo-
la quan um ma e in o he s ongly in e ac ing limi .
He e, we use quan um Mon e Ca lo simula ions
o in es iga e he exis ence o dipola d ople s in
a molecula BEC. Using ealis ic in e ac ion po en-
ials o mic owa e-shielded molecules, we obse e ha
∗ e an.mazzan [email protected]; im.langen@ uwien.ac.a
(103
a0)
( 0)
V( ,θ)/E0
( 0)
(103
a0)
( 0)
(103
a0)
δ = 0.1 δ = 0.5 δ = 1
θ = 0
θ = π/4
θ = 54.7º
θ = π/3
θ = 2π/5
θ = π/2
Figu e 1. Molecula in e ac ion po en ials o single e-
quency mic owa e d essing o NaK molecules wi h a Rabi
equency o Ω/ℏ= 2π×10 MHz and a ying ela i e de-
uning δ [30]. The d essing es ablishes a shielding co e lo-
ca ed a a ound 1000 a0, which p e en s molecula wo-body
losses. He e, a0deno es he Boh adius and 0 he dipo-
la leng h, as de ined in he ex . By changing he d essing
pa ame e s, he po en ials can be uned and a ious in e -
ac ion pa ame e s can be ealized, po en ially esul ing in
di e en many-body s a es o he ul acold molecula gas.
sel -bound d ople s pe sis in he s ongly in e ac ing
egime — well ou side he egime o alidi y o mean-
ield desc ip ions — o ealis ic in e ac ion pa ame-
e s achie able in expe imen s. Mo eo e , we obse e
he o ma ion o me as able d ople assemblies upon
con inemen -induced us a ion, ou lining a pa h o-
wa ds molecula supe solids. Taken oge he , his high-
ligh s he p ospec s o molecula BECs o explo e no el
egimes o dipola quan um ma e .
Molecula in e ac ions. Ul acold molecula gases a e
known o be s ongly a ec ed by wo- and h ee-body
losses a ul acold empe a u es. These losses can, o
example, occu h ough chemical eac ions ha a e al-
a Xi :2407.09391 2 [cond-ma .quan -gas] 15 Jul 2024
2
δ C6(10−3)de (D) as(a0)add (a0)ϵdd Nc i
(I) 0.10 0.9080 0.7813 -9558.3 3579.9 -0.375 7
(II) 0.50 1.5471 0.7023 -3644.6 2892.5 -0.794 12
(III) 0.75 2.7026 0.6282 -1666.1 2314.0 -1.389 31
(IV) 1.00 5.0097 0.5552 -596.6 1807.8 -3.030 135
(V) 1.25 9.3089 0.4905 -110.1 1411.0 -12.820 400
Table I. Summa y o he molecula in e ac ion pa ame e s conside ed in his wo k. We use NaK molecules and a ixed Rabi
equency o Ω/ℏ= 2π×10 MHz. By changing he ela i e de uning δ =|δ|/Ω, he C6coe icien o he in e ac ion, e ec i e
dipole momen de ,s-wa e sca e ing leng h asand mean- ield dipola leng h add, and dipola pa ame e ϵdd =add/ascan
be uned in expe imen s. The c i ical pa icle numbe Nc i is de e mined h ough ou Mon e Ca lo simula ions and ma ks
he ansi ion be ween sel -bound d ople s and egula gases. No ice he c i ical alue o C6 equi ed o a wo-body bound
s a e o appea is Cc i
6∼0.531 ×10−3. Fo C6> Cc i
6no wo-body bound s a e is o med, while d ople s s ill exis .
lowed o some species [31], o h ough he o ma ion o
sho -li ed complexes ha can be los om he ap-
ping po en ial con ining he molecules [32]. In con-
as o ul acold a oms i is hus impe a i e o shield
molecules om hese losses a sho ange. A e y
e sa ile me hod o achie e his is mic owa e d ess-
ing [33–37], which has been used o ealize bo h de-
gene a e Fe mi gases [21] and, e y ecen ly, also a
BEC o dipola molecules [27]. An added bene i o
such shielding is he unabili y o molecula in e ac-
ions h ough a simple change o mic owa e powe and
equency. Mo eo e , as he esul ing in e ac ion po-
en ials a e o med om mic owa e-d essed o a ional
s a es o he molecules, hey a e known wi h high p e-
cision and hus ideally sui ed o nume ical simula ions,
such as he ones p esen ed he e. This is in s a k con as
o magne ic a oms [2], whe e he in e -a omic sca e -
ing is chao ic and he shape o he in e ac ion po en ial
is hus highly unce ain [16].
In his Le e , we use a ecen ly p oposed analy ical
app oxima ion o model he mic owa e-d essed molec-
ula in e ac ions [30]. This app oxima ion has been
shown o ag ee e y well wi h he ull molecula in e -
ac ion po en ials o e he pa ame e ange ele an in
ypical expe imen s. We no e ha ou simula ions can
easily be adap ed o desc ibe mo e complex shielding
po en ials such as, o example, he ecen ly in oduced
double mic owa e d essing [27], DC shielding [38], o
combina ions o MW and DC ields [10,39].
Exp essed in sphe ical coo dina es and o ci cula ly
pola ized mic owa es, he in e ac ion po en ial o wo
molecules sepa a ed by a dis ance ec o is gi en by
V( ) = C6
6sin2θ(1 + cos2θ) + C3
3(3 cos2θ−1) ,(1)
wi h θ he declina ion angle. In his exp ession, C3=
d2/[48 πϵ0(1 + δ2
)] and C6=d4/[128 π2ϵ2
0Ω(1 + δ2
)3/2]
a e unc ions o he ela i e de uning δ =|δ|/Ω. He e,
δand Ωa e he de uning and Rabi equency o he
mic owa e ield, espec i ely. Fu he mo e, ddeno es
he pe manen dipole momen in he molecula ame
and ϵ0is he acuum pe meabili y. The esul ing e -
ec i e dipole momen in he lab ame is gi en by
de =d/p12(1 + δ2
), and o ien ed in he z-di ec ion,
pe pendicula o he mic owa e pola iza ion. No ice
ha he dipola con ibu ion ∼C3/ 3has a e e sed
—an i-dipola — o m [2,40,41]. The co esponding
change in sign has a di ec in luence on he inal shape
o he sys em as we discuss below.
As a conc e e example, in his wo k we use pa ame-
e s o bosonic NaK molecules [42], wi h a dipole mo-
men o 2.72 D, mass m= 62 amu and o a ional con-
s an B/ℏ= 2π×2.089 GHz. Fo he Rabi equency,
we use Ω/ℏ= 2π×10 MHz, which is well-achie able in
ypical expe imen s [21]. These pa ame e s desc ibe an
in e ac ion po en ial whe e bound s a es a e loca ed a
om h eshold, which is expec ed o minimize losses in
expe imen s [24]. Example plo s o he in e ac ion po-
en ial a e depic ed in Fig. 1 o di e en declina ion
angles θand di e en in e ac ion s eng hs. The plo s
highligh , in pa icula , he la ge size o he epulsi e
shielding co e ∼1000 a0, which is in s a k con as o
he usual con ac in e ac ion po en ial used o desc ibe
ul acold a oms.
Besides he e ec i e dipole momen de , changes in
in e ac ion s eng h a e e lec ed in he s-wa e sca e -
ing leng h as, which is expec ed o be he only ele an
sca e ing pa ame e in a dilu e bosonic gas a ul acold
empe a u es. Rela ing his las quan i y o he po-
en ial pa ame e s equi es sol ing he low-momen um
limi o he sca e ing T-ma ix as in Re . [43], o equi -
alen ly he long- ange beha io o he s-wa e compo-
nen o he wa e unc ion o he E= 0 wo-body p ob-
lem. Fo he gi en po en ial, hese calcula ions a e mo e
in ol ed han usual, due o he coupling be ween modes
l, l±2and l±4coming om he pu ely epulsi e pa o
he in e ac ion. The esul ing alues o as, and ela ed
o his, dipola pa ame e s ϵdd =add/as, a e gi en in
Table I o ypical pa ame e s used in his wo k. He e,
add =m d2
e /[12πϵ0ℏ2]deno es he dipola leng h.
3
In iew o he la ge shielding co e and s ong dipola
in e ac ions o he molecules, i is an in e es ing ques-
ion whe he he sca e ing leng h emains he ele an
low-ene gy pa ame e ha ully cha ac e izes he in e -
ac ions, whe he eno maliza ion o i s s eng h is e-
qui ed as in dysp osium a oms [16,44], o whe he a
new app oach is necessa y al oge he [45–47]. In his
wo k, we he e o e p o ide bo h pu ely expe imen al
pa ame e s such as de uning and Rabi equency, in
pa allel wi h he con en ional sca e ing leng h.
Simula ions. Based on he gi en ealis ic in e ac ion
po en ial, in he ollowing we analyze a sys em o N
indis inguishable dipola bosonic molecules in h ee di-
mensions desc ibed by he Hamil onian
H=−ℏ2
2m
N
X
j=1
∇2
j+X
i<j
V( ij),(2)
whe e V( )is he wo-body in e ac ion po en ial gi en
in Eq. (1).
We use he Pa h In eg al G ound S a e (PIGS) algo-
i hm [48,49] o simula e he g ound s a e o he sys-
em. I s a s om a a ia ional wa e unc ion ha is
p opaga ed in imagina y ime o elimina e all he com-
ponen s ha a e o hogonal o he ue g ound s a e,
p o ided he p opaga ion is ca ied ou o e a la ge
enough imagina y ime, and a good sho - ime app ox-
ima ion o he ime-e olu ion ope a o is used. No ice
ha his is an ab ini io me hod ha leads o he exac
g ound-s a e solu ion o a gi en Hamil onian. Rema k-
ably and in con as o p e ious wo ks [16,50–52], in
he cu en case an accu a e desc ip ion o he Hamil-
onian is known and gi en by he exp essions abo e,
hus allowing o a obus simula ion o he molecu-
la BEC’s beha io . The ac ha he Hamil onian is
known, makes quan um Mon e Ca lo me hods, such as
PIGS, an op imal choice o s udy his sys em.
In ou simula ions, we use dimensionless quan i ies
exp essed in e ms o he dipola leng h 0=mC3/ℏ2
and ene gy E0=ℏ2/m 2
0. No e he di e ence o 1/3in
he nume ical p e- ac o o 0when compa ing o he al-
e na i e dipola leng h add, which is used in he mean-
ield desc ip ion o dipola gases. In ou uni s con en-
ion, C3is always 1, while he changes in he in e ac-
ion pa ame e s modi y he C6pa ame e . Two-body
bound s a es could appea when he epulsi e compo-
nen o he in e ac ion is educed below a c i ical alue
C6≈0.531 ×10−3, which is, howe e , ou side he se
o pa ame e s conside ed he e.
In p ac ice, we s a he simula ion by se ing up he
sys em in a igh ly con ining ap in o de o gua an-
ee all pa icles a e close enough, and subsequen ly le
he sys em equilib a e. Once his has been achie ed,
he ap is emo ed and he sys em is le o e ol e
eely. When p esen in he simula ions, d ople s a e
2
1 0
1
2
1 2 3
Posi ion ( 0)
0
Column Densi y ( 0 )
-1
1500
1000
500
2
1
0
-1
-2
2
1 0 1 2
x ( 0)y ( 0)
z ( 0)
----
(a)
Sel -bound
d ople
Gas
C i ical pa icle numbe Nc i
In e ac ion s eng h
(b)
0.12
ela i e de uning δ
0.21 0.37 0.61 0.91 1.29
(I) (II) (III)
(IV)
(V)
Figu e 2. (a) Single sel -bound dipola d ople in he
s ongly in e ac ing egime o δ = 0.75 and N= 300
molecules. Due o he an i-dipola in e ac ions be ween he
molecules, he densi y dis ibu ion is s ongly p ola e. The
column densi ies ha e been ob ained om he a e age o
a la ge numbe o con igu a ions, no malized o he o al
numbe o molecules. The inse shows a single snapsho
o a d ople . (b) Phase diag am o he ansi ion om
sel -bound d ople s o a no mal gas phase, as a unc ion
o ela i e de uning δ . Fo e e ence, we also show he in-
e ac ion s eng h pa ame e ized by he dipola pa ame e
ϵ−1
dd =as/add, as i is commonly used o cha ac e ize he
mean- ield limi [16]. Pa ame e s o he whi e da a poin s
(I) o (V) a e summa ized in Table I.
ecognized by bo h hei ene ge ic and s uc u al p op-
e ies: being sel -bound s a es, he d ople s o al en-
e gy emains nega i e, while a he same ime pa icles
s ay close o each o he and do no sp ead o e he
whole simula ion box. These wo a e clea signa u es
o d ople o ma ion and allow o a good es ima ion
o he c i ical a om numbe equi ed o one o mo e
d ople s o o m.
Resul s. In Fig. 2a we show example esul s o a sim-
ula ion wi h δ = 0.75 and N= 300 pa icles, leading
4
o a s able dipola d ople . Due o he speci ic o m o
he in e ac ion, which has a ully epulsi e sho - anged
pa combined wi h he sign- e e sed dipola e m, he
sys em ends o a ange in he plane pe pendicula o
he pola iza ion o he dipoles. The obse ed d ople
hus shows a p ola e shape, wi h a adial size ha is
signi ican ly la ge han i s mino axis. This shape is
eminiscen o o he , p e iously p edic ed an i-dipola
d ople s in he mean- ield egime [41,53,54]. No e ha
as he in e ac ion po en ials do no suppo a wo-body
bound s a e o he pa ame e s conside ed, he d ople s
o med a e he ue mani es a ion o a many-body sel -
bound s a e [28].
A adial in eg a ion o he o al densi y shows ha
he local gas pa ame e a he cen e o he sys em is
n|as|3≈4.36, well beyond he each o mean- ield he-
o y [55–57]. Rema kably, his alue yields a mean in e -
pa icle spacing o n−1/3∼50 nm, which is compa a-
ble o he size o he shielding co e (c . Fig. 1). This
sugges s ha he sho ange de ails o he in e ac ion,
which a e neglec ed in a mean- ield app oach, play an
impo an ole in he desc ip ion o he d ople .
The ac ha bo h leng h scales — he size o he
shielding co e and he mean in e pa icle spacing —
a e compa able could also be expec ed o lead o h ee-
dimensional s uc u e o ma ion, whe e he epulsi ely
in e ac ing molecules would spon aneously a ange in a
c ys al wi h he same la ice spacing [39]. Howe e , we
do no ind any e idence o his in ou simula ions o
he pa ame e s in es iga ed.
Finally, he obse ed densi y ∼6×1021 m−3in he
d ople is highe han he densi y in he molecula BEC
and compa able o he densi ies obse ed in d ople s o
magne ic a oms [1]. This emphasizes he need o educe
h ee-body losses in expe imen al implemen a ions o a
alue ha is also compa able o ha in a oms, whe e
loss a e coe icien s L3= 1.3×10−41 m6s−1ha e been
de e mined [3].
Nex , we sys ema ically map ou he beha io o he
sys em o di e en numbe s o pa icles Nand in e -
ac ion pa ame e s. Ou esul s a e shown in Fig. 2b.
The pa ame e s o pa icula alues o Nc i a e also
summa ized in Table I.
Simila o he beha io in magne ic a oms, whe e a
sub le balance be ween dipola in e ac ions, con ac in-
e ac ions, and quan um luc ua ions is equi ed o s a-
ble d ople s o o m [1], we obse e a sha p ansi ion
be ween a pa ame e egime whe e sel -bound d ople s
a e p e alen , and a gaseous BEC.
Ou esul s con i m sugges ions [10] indica ing ha
o inc easing dipola in e ac ion s eng h he sys em
suppo s d ople s o dec easing pa icle numbe . Speci -
ically, we ind c i ical pa icle numbe s as low as Nc i =
7 o δ = 0.10. The co esponding d ople s a e cha ac-
e ized by s ongly a ac i e sca e ing leng hs, which
is in s a k con as o he usual mean- ield desc ip ion,
2
1
0
-1
-2
-2 -1 0
1
2 3
1.5
1.0
0.5
0
-0.5
-1.0
-2.5
x (add)
-3 -2 -1 0 1 2 3
(a) (b)
y (add)
x (add)y (add)
z (add)
z (add)
-3 -2 -1 0 1 2
-2 -1 0 1
Figu e 3. Snapsho s o d ople s o med om simula ions
con aining (a) N= 90 and (b) N= 300 pa icles o
δ = 0.75, equilib a ed in igh con ining aps ha us-
a e he single d ople g ound s a e. The d ople s emain
in a me as able s a e a e he ap is emo ed.
whe e such pa ame e s lead o a collapse o he sys-
em. No ably, he low pa icle numbe s equi ed a e
well wi hin he ange o he mode a e-sized condensa es
ha ha e been achie ed in expe imen s o da e [27].
Fo la ge de unings o he mic owa e ield, co e-
sponding o weake dipoles, he c i ical pa icle num-
be inc eases, e en ually app oaching alues o se e al
hund ed pa icles, which is simila o alues p e iously
obse ed in weakly-dipola dysp osium gases [10,16].
Nex we compa e he esul s o he quan um Mon e
Ca lo simula ions o he p edic ions o he s anda d
eGPE desc ip ion, which equi es he inclusion o a Lee-
Huang-Yang (LHY) e m o desc ibe he quan um luc-
ua ions ha s abilize he d ople s. Due o he e e sed
sign o he dipola e m in he in e ac ion, howe e ,
he LHY co ec ion e m has o be ede i ed (see Ap-
pendix). The esul ing c i ical a om numbe s, ob ained
as he minimum numbe o pa icles equi ed o p o-
duce a nega i e ene gy, a e a below he ones ob ained
om he PIGS simula ions. Fo δ = 1.25,1.5,1.75 and
2we ind N(eGPE)
c i = 9,31,69 and 193, espec i ely. The
la ge disc epancies he e a e no su p ising, conside ing
he la ge cen al densi y o he d ople s, oge he wi h
he ac ha wi h he in e ac ion o Eq. (1) and he
condi ions conside ed, he LHY co ec ion acqui es an
imagina y pa ha can be as la ge as a 50%. This
again indica es ha he sys em is well beyond a mean-
ield egime, e en when he leading quan um luc ua ion
e ms a e aken in o accoun .
Despi e he ac ha in i s g ound s a e he sys em
o ms a single sel -bound d ople , di e en me as able
con igu a ions can be ealized by placing i in a mo e
s ongly con ining ap. When he oscilla o leng hs as-
socia ed o he adial equencies a e smalle han he
adial size o he g ound s a e d ople , us a ion oc-
cu s and he sys em spli s, p oducing wo o mo e nucle-
a ion cen e s depending on he o al numbe o pa icles
and he ap equencies employed in he ini ial equili-
b a ion [58]. We obse e a c i ical maximum size ha
each o hese d ople s can a o d, a ou ing he spli ing
5
in o mo e d ople s when he o al numbe o pa icles
inc eases. An example o ha is shown in Fig. 3, whe e
wo snapsho s o equilib a ed con igu a ions con aining
N= 90 and N= 300 pa icles a e p esen ed. The
d ople s o med in his way a e well isola ed, wi h li le
o no exchange o pa icles be ween hem. Fu he mo e,
despi e no being he ue g ound s a e, hey s ill ha e
a e y la ge and nega i e ene gy, hus indica ing ha
he elaxa ion imes equi ed o hem o e ol e in o
he ue g ound s a e d ople g ea ly exceed he ime
scales accessible in he simula ion. These me as able
con igu a ions a e he e o e expec ed o be easily e-
alized and obse ed in expe imen s, when ini ial igh
con ining aps a e used o he malize he sys em.
Conclusion. We ha e obse ed sel -bound dipola
d ople s in a ealis ic quan um Mon e Ca lo simula ion
o a molecula BEC. In pa icula , he esul s sugges
ha single d ople s and d ople assemblies, which a e
well known om magne ic quan um gases [1], exis in
molecula BECs o e a wide ange o in e ac ions pa-
ame e s and s eng hs. In ce ain aspec s, s ongly
dipola sys ems can hus beha e simila ly o weakly
dipola sys ems in he mean- ield limi , as well as o
o he sys ems wi h compe ing in e ac ions [50,59–61].
Howe e , he pa ame e s and s abili y egimes o he
o ma ion o many-body s a es can be undamen ally
di e en , as highligh ed by he obse a ion o d ople s
e en o s ongly a ac i e sho ange in e ac ions.
We no e ha , wi hin he scope o he p esen in es-
iga ion, he sys em’s beha io s ill shows a s ong de-
pendence on he alue o he sca e ing leng h, despi e
being well ou side he uni e sali y egime, so ha o he
sca e ing pa ame e s mus also be ele an . Mo e wo k
along he lines o Re s. [44–47] is needed o explo e his
ques ion in u he de ail.
While he d ople s ealized in his wo k do no o e -
lap, i will be in e es ing o in es iga e whe he su-
pe solid s a es also pe sis in he s ongly in e ac ing
egime [10]. Ce ainly, he educed numbe o pa icles
pe d ople makes PIGS me hod app op ia e o s udy
he possible BEC-supe solid ansi ion, as i has been
done ecen ly o dysp osium sys em con aining hun-
d eds o a oms [62]. Fu he mo e, we expec ha by
sui ably enginee ing he con inemen , a ciga shaped
molecula BEC could be u ned in o a sel -o ganized
s ack o s ongly in e ac ing laye s wi h s ong in a-
and in e laye dipola couplings [53], which is known o
gi e ise o addi ional many-body phenomena [63–65].
Ou wo k es ablishes quan um Mon e Ca lo simula-
ions as a ool o ealis ically model cu en and u-
u e expe imen s wi h ul acold molecula gases. We
an icipa e ha sys ema ic compa isons o such simula-
ions, mean- ield models, and expe imen s will g ea ly
ad ance ou unde s anding o dipola quan um ma e
in he nea u u e.
No e added.— Du ing he p epa a ion o his
manusc ip , we ha e become awa e o he i s
expe imen al obse a ion o d ople s in a molecula
BEC, including ones ha a e s able a nega i e sca -
e ing leng hs [66].
We hank Jens He ko n, Thomas Pohl, Ma eo Cia-
di, and Sebas ian Will o discussions, as well as
Phillip G oß and Lukas Leczek o a ca e ul eading
o he manusc ip . T.L. acknowledges unding om
he Eu opean Resea ch Council (ERC) unde he Eu o-
pean Union’s Ho izon 2020 esea ch and inno a ion p o-
g amme (G an ag eemen No. 949431), om Ca l Zeiss
Founda ion, and he Aus ian Science Fund (FWF)
10.55776/COE1. R.B., J.S-B., F.M. and J.B. acknowl-
edge inancial suppo om Minis e io de Ciencia e In-
no ación MCIN/AEI/10.13039/501100011033 (Spain)
unde G an No. PID2020-113565GB-C21 and om
AGAUR-Gene ali a de Ca alunya G an No. 2021-
SGR-01411. R.B. acknowledges unding om ADA-
GIO (Ad anced Manu ac u Ing Resea ch Fellowship
P og amme in he Basque – New Aqui aine Region)
MSCA COFUND Pos -Doc o al ellowship p og amme
(G an ag eemen No. 101034379).
[1] F. Bö che , J.-N. Schmid , J. He ko n, K. S. H. Ng,
S. D. G aham, M. Guo, T. Langen, and T. P au, Re-
po s on P og ess in Physics 84, 012403 (2020).
[2] L. Chomaz, I. Fe ie -Ba bu , F. Fe laino, B. Labu he-
Tol a, B. L. Le , and T. P au, Repo s on P og ess in
Physics 86, 026401 (2022).
[3] M. Schmi , M. Wenzel, F. Bö che , I. Fe ie -Ba bu ,
and T. P au, Na u e 539, 259 (2016).
[4] L. Chomaz, S. Baie , D. Pe e , M. J. Ma k,
F. Wäch le , L. San os, and F. Fe laino, Phys. Re .
X6, 041039 (2016).
[5] F. Bö che , J.-N. Schmid , M. Wenzel, J. He ko n,
M. Guo, T. Langen, and T. P au, Phys. Re . X 9,
011051 (2019).
[6] L. Tanzi, E. Lucioni, F. Famà, J. Ca ani, A. Fio e i,
C. Gabbanini, R. N. Bisse , L. San os, and G. Mod-
ugno, Phys. Re . Le . 122, 130405 (2019).
[7] L. Chomaz, D. Pe e , P. Ilzhö e , G. Na ale, A. T au -
mann, C. Poli i, G. Du as an e, R. M. W. an Bijnen,
A. Pa scheide , M. Sohmen, M. J. Ma k, and F. Fe -
laino, Phys. Re . X 9, 021012 (2019).
[8] Y.-C. Zhang, T. Pohl, and F. Mauche , Phys. Re . A
104, 013310 (2021).
[9] J. He ko n, J.-N. Schmid , M. Guo, F. Bö che ,
K. S. H. Ng, S. D. G aham, P. Ue lings, T. Langen,
M. Zwie lein, and T. P au, Phys. Re . Res. 3, 033125
6
(2021).
[10] M. Schmid , L. Lassabliè e, G. Quéméne , and T. Lan-
gen, Phys. Re . Res. 4, 013235 (2022).
[11] A. R. P. Lima and A. Pels e , Phys. Re . A 84, 041604
(2011).
[12] I. Fe ie -Ba bu , H. Kadau, M. Schmi , M. Wenzel,
and T. P au, Phys. Re . Le . 116, 215301 (2016).
[13] F. Wäch le and L. San os, Phys. Re . A 93, 061603
(2016).
[14] D. Baillie, R. M. Wilson, R. N. Bisse , and P. B. Blakie,
Phys. Re . A 94, 021602 (2016).
[15] H. Sai o, Jou nal o he Physical Socie y o Japan 85,
053001 (2016).
[16] F. Bö che , M. Wenzel, J.-N. Schmid , M. Guo,
T. Langen, I. Fe ie -Ba bu , T. P au, R. Bombín,
J. Sánchez-Baena, J. Bo ona , and F. Mazzan i, Phys.
Re . Res. 1, 033088 (2019).
[17] D. Pe e , G. Na ale, R. M. W. an Bijnen, A. Pa schei-
de , M. J. Ma k, L. Chomaz, and F. Fe laino, Phys.
Re . Le . 122, 183401 (2019).
[18] R. Bombín, V. Cikoje ić, F. Mazzan i, and J. Bo ona ,
Phys. Re . A 109, 033312 (2024).
[19] T. Langen, G. Val olina, D. Wang, and J. Ye, Na u e
Physics 20, 702 (2024).
[20] L. De Ma co, G. Val olina, K. Ma suda, W. G. Tobias,
J. P. Co ey, and J. Ye, Science 363, 853 (2019).
[21] A. Schindewol , R. Bause, X.-Y. Chen, M. Duda,
T. Ka man, I. Bloch, and X.-Y. Luo, Na u e 607, 677
(2022).
[22] M. Duda, X.-Y. Chen, A. Schindewol , R. Bause, J. on
Milczewski, R. Schmid , I. Bloch, and X.-Y. Luo, Na-
u e Physics 19, 720 (2023).
[23] J. Cao, H. Yang, Z. Su, X.-Y. Wang, J. Rui, B. Zhao,
and J.-W. Pan, Phys. Re . A 107, 013307 (2023).
[24] N. Bigagli, C. Wa ne , W. Yuan, S. Zhang, I. S e en-
son, T. Ka man, and S. Will, Na u e Physics 19, 1579
(2023).
[25] J. J. Pa k, Y.-K. Lu, A. O. Jamison, and W. Ke e le,
Na u e Physics 19, 1567 (2023).
[26] J. Lin, G. Chen, M. Jin, Z. Shi, F. Deng, W. Zhang,
G. Quéméne , T. Shi, S. Yi, and D. Wang, Phys. Re .
X13, 031032 (2023).
[27] N. Bigagli, W. Yuan, S. Zhang, B. Bula o ic, T. Ka -
man, I. S e enson, and S. Will, Na u e (2024),
10.1038/s41586-024-07492-z.
[28] W.-J. Jin, F. Deng, S. Yi, and T. Shi, “Bose-eins ein
condensa es o mic owa e-shielded pola molecules,”
(2024), a Xi :2406.06412 [cond-ma .quan -gas].
[29] S. Baliba , Na u e 464, 176 (2010).
[30] F. Deng, X.-Y. Chen, X.-Y. Luo, W. Zhang, S. Yi, and
T. Shi, Phys. Re . Le . 130, 183001 (2023).
[31] P. S. Żuchowski and J. M. Hu son, Phys. Re . A 81,
060703 (2010).
[32] R. Bause, A. Ch is ianen, A. Schindewol , I. Bloch, and
X.-Y. Luo, The Jou nal o Physical Chemis y A 127,
729 (2023).
[33] L. Lassabliè e and G. Quéméne , Phys. Re . Le . 121,
163402 (2018).
[34] T. Ka man and J. M. Hu son, Phys. Re . Le . 121,
163401 (2018).
[35] T. Ka man and J. M. Hu son, Phys. Re . A 100, 052704
(2019).
[36] T. Ka man, Phys. Re . A 101, 042702 (2020).
[37] L. Ande egg, S. Bu chesky, Y. Bao, S. S. Yu, T. Ka -
man, E. Chae, K.-K. Ni, W. Ke e le, and J. M. Doyle,
Science 373, 779 (2021).
[38] K. Ma suda, L. De Ma co, J.-R. Li, W. G. Tobias,
G. Val olina, G. Quéméne , and J. Ye, Science 370,
1324 (2020).
[39] A. V. Go shko , P. Rabl, G. Pupillo, A. Micheli,
P. Zolle , M. D. Lukin, and H. P. Büchle , Phys. Re .
Le . 101, 073201 (2008).
[40] T. Lahaye, C. Meno i, L. San os, M. Lewens ein, and
T. P au, Repo s on P og ess in Physics 72, 126401
(2009).
[41] S. B. P asad, T. Bland, B. C. Mulke in, N. G. Pa ke ,
and A. M. Ma in, Phys. Re . Le . 122, 050401 (2019).
[42] K. K. Voges, P. Ge sema, M. Meye zum Al en Bo -
gloh, T. A. Schulze, T. Ha mann, A. Zenesini, and
S. Ospelkaus, Phys. Re . Le . 125, 083401 (2020).
[43] A. Macia, J. Sánchez-Baena, J. Bo ona , and F. Maz-
zan i, Phys. Re . Le . 117, 205301 (2016).
[44] R. Ołdziejewski and K. Jachymski, Phys. Re . A 94,
063638 (2016).
[45] S. Yi and L. You, Phys. Re . A 63, 053607 (2001).
[46] C. Tickno , Phys. Re . Le . 100, 133202 (2008).
[47] D.-W. Wang, New Jou nal o Physics 10, 053005
(2008).
[48] A. Sa sa, K. Schmid , and W. Mag o, The Jou nal o
Chemical Physics 113, 1366 (2000).
[49] R. Ro a, J. Casulle as, F. Mazzan i, and J. Bo ona ,
Phys. Re . E 81, 016707 (2010).
[50] F. Cin i and M. Boninsegni, Phys. Re . A 96, 013627
(2017).
[51] Y. Ko a and M. Boninsegni, Jou nal o Low Tempe a-
u e Physics 197, 337 (2019).
[52] M. Boninsegni, Resul s in Physics 31, 104935 (2021).
[53] K. Mukhe jee, M. N. Tengs and, T. A. Ca dinale, and
S. M. Reimann, Phys. Re . A 108, 023302 (2023).
[54] W. Ki kby, T. Bland, F. Fe laino, and R. N. Bisse ,
SciPos Phys. Co e 6, 084 (2023).
[55] S. Gio gini, J. Bo ona , and J. Casulle as, Phys. Re .
A60, 5129 (1999).
[56] F. Mazzan i, A. Polls, and A. Fab ocini, Phys. Re . A
67, 063615 (2003).
[57] A. Macia, F. Mazzan i, J. Bo ona , and R. E. Zillich,
Phys. Re . A 84, 033625 (2011).
[58] M. Wenzel, F. Bö che , T. Langen, I. Fe ie -Ba bu ,
and T. P au, Phys. Re . A 96, 053630 (2017).
[59] Y. H. Liu, L. Y. Chew, and M. Y. Yu, Phys. Re . E
78, 066405 (2008).
[60] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesano sky,
and P. Zolle , Phys. Re . Le . 104, 223002 (2010).
[61] C. R. Cab e a, L. Tanzi, J. Sanz, B. Naylo , P. Thomas,
P. Cheiney, and L. Ta uell, Science 359, 301 (2018).
[62] R. Bombín, F. Mazzan i, and J. Bo ona , New Jou nal
o Physics 26, 013052 (2024).
[63] C. T e zge , C. Meno i, and M. Lewens ein, Phys. Re .
Le . 103, 035304 (2009).
[64] C.-C. Huang and W.-C. Wu, Phys. Re . A 82, 053612
(2010).
[65] L. Du, P. Ba al, M. Can a a, J. de Hond, Y.-K. Lu,
7
and W. Ke e le, Science 384, 546 (2024).
[66] S. Zhang, N. Bigagli, W. Yuan, B. Bula o ic, H. Kwak,
T. Ka man, I. S e enson, and S. Will, “Explo a ion o
dipola quan um phases wi h a degene a e gas o NaCs
molecules.” Talk a DAMOP 2024.
[67] S. Yi and L. You, Phys. Re . A 61, 041604 (2000).
[68] R. Schü zhold, M. Uhlmann, Y. Xu, and U. R. Fische ,
In . J. Mod. Phys. B 20, 3555 (2006).
[69] A. R. P. Lima and A. Pels e , Phys. Re . A 86, 063609
(2012).
[70] S. Ronen, D. C. E. Bo olo i, and J. L. Bohn, Phys.
Re . A 74, 013623 (2006).
[71] F. Mazzan i, A. Polls, and A. Fab ocini, Phys. Re . A
71, 033615 (2005).
8
Appendix: The ex ended G oss-Pi ae skii equa ion
o he molecula po en ial
The minimiza ion o he ene gy unc ional wi h e-
spec o he single-pa icle wa e unc ion wi hin a mean
ield app oxima ion yields he non-linea wa e equa ion
µψ( ) = −ℏ2∇2
2m+Zd ′VGPE( − ′)|ψ( ′)|2+Hqu( )ψ( ),(A.1)
which de e mines he equilib ium s a e o he conden-
sa e o a gi en chemical po en ial µ. The pseudopo-
en ial VGPE( )is gi en by
VGPE( ) = gδ( ) + C3
3(3 cos2θ−1) ,(A.2)
whe e g=4πℏ2a
mis he usual con ac copuling cons an ,
wi h a he s-wa e sca e ing leng h. This pseudopo en-
ial is buil such ha i s sca e ing p ope ies compu ed
in he i s -o de Bo n app oxima ion ep oduce hose
o he ull po en ial o Eq. (1) [45,67]. On he o he
hand, he e m
Hqu( ) = 32
3g a3
πQ5− 0
3a|ψ( )|3(A.3)
accoun s o he e ec o quan um luc ua ions, which
p o ide a c ucial s abiliza ion mechanism o he o h-
e wise collapsing sys em. The ac o 3di iding he
a gumen o he Q5 unc ion s ems om ou de ini-
ion o he dipole leng h, which is h ee imes la ge
han he usual de ini ion add =mC3/(3ℏ2). No ice
also ha , compa ed o he beyond-mean- ield co ec-
ion o a s anda d dipole-dipole in e ac ion, which we
e e o as HDDI
qu ( ), he exp ession in Eq. (A.3) ea-
u es an ex a minus sign in he a gumen o he Q5
unc ion o accoun o he e e sed sign o he dipola
e m o Eq. (1) [11,68,69]. Rema kably, his e e sed
sign signi ican ly changes he eal and imagina y con-
ibu ions o he Q5 e m. We illus a e his in Fig. 4,
whe e we show he eal and imagina y pa s o bo h Hqu
and HDDI
qu compu ed o he homogeneous sys em wi h
a/ 0= 0.11 (i.e. δ = 1.75). As i can be seen om he
igu e, he LHY co ec ion o he e e sed DDI shows
a smalle eal pa , oge he wi h a signi ican ly la ge
and unphysical imagina y pa , which ende s he ap-
plica ion o Eq. (A.1) un eliable o desc ibe he p esen
sys em in he egime o pa ame e s conside ed.
In o de o compu e he c i ical numbe s p o ided
in he main ex , we sol e Eq. (A.1), and subsequen ly
compu e nume ically he Hankel ans o m o he wa e
unc ion [70,71] o ake ad an age o he cylind ical
symme y o he p oblem. The c i ical numbe is de-
e mined by ob aining he pa icle numbe o which
he ene gy o he g ound s a e con igu a ion becomes
nega i e, indica ing he p esence o a sel -bound s able
solu ion.
0 5 10 15 20 25
n 3
0
0
100
200
300
400
500
Hqu/E0
Re{Hqu/E0}
Im{Hqu/E0}
Re{HDDI
qu /E0}
Im{HDDI
qu /E0}
Figu e 4. Real (dashed) and imagina y (do -dashed) pa s
o he quan um luc ua ions co ec ion o he chemical po-
en ial Hqu o he po en ial o Eq. A.2 (blue) and he DDI
pseudopo en ial ( ed) as a unc ion o he homogeneous den-
si y n. The pa ame e s a e a/ 0= 0.11 (δ = 1.75)
