PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
Accessing elusi e wo-dimensional phases o dipola Bose-Eins ein condensa es
by ini e empe a u e
Liang-Jun He,1Juan Sánchez-Baena ,2Fabian Mauche ,3and Yong-Chang Zhang 1,*
1MOE Key Labo a o y o Nonequilib ium Syn hesis and Modula ion o Condensed Ma e , Shaanxi Key Labo a o y o Quan um In o ma ion
and Quan um Op oelec onic De ices, School o Physics, Xi’an Jiao ong Uni e si y, Xi’an 710049, People’s Republic o China
2Depa amen de Física, Uni e si a Poli ècnica de Ca alunya, Campus No d B4-B5, 08034 Ba celona, Spain
3Facul y o Mechanical Enginee ing; Depa men o P ecision and Mic osys ems Enginee ing, Del Uni e si y o Technology,
2628 CD Del , The Ne he lands
(Recei ed 29 Oc obe 2024; accep ed 18 Ma ch 2025; published 7 Ap il 2025)
I has been shown ha dipola Bose-Eins ein condensa es ha a e igh ly apped along he pola iza ion
di ec ion can ea u e a ich phase diag am. In his pape we show ha ini e empe a u e can assis in accessing
pa s o he phase diag am ha o he wise appea ha d o ealize due o excessi ely la ge densi ies and numbe
o a oms being equi ed. These include honeycomb and s ipe phases bo h uncon ined and wi h a ini e ex en .
To map ou a phase-diag am, we employ bo h a ia ional analysis and ull nume ical calcula ions sol ing he
ini e- empe a u e ex ended G oss-Pi ae skii equa ion (TeGPE). Fu he mo e, we exhibi eal- ime e olu ion
simula ions leading o such s a es. We accoun o he e ec o he mal luc ua ions by means o Bogoliubo
heo y, employing he local densi y app oxima ion. We ind ha ini e empe a u es can lead o a signi ican
dec ease in he necessa y pa icle numbe and densi y ha migh ul ima ely pa e a ou e o u u e expe imen al
ealiza ions.
DOI: 10.1103/PhysRe Resea ch.7.023019
I. INTRODUCTION
Dipola Bose-Eins ein condensa es (dBECs) ep esen an
ou s anding pla o m o explo ing he in e play be ween
long- anged dipole-dipole in e ac ion, con ac in e ac ion, and
quan um luc ua ions [1–3]. Quan um luc ua ions [4,5] can
play a c ucial ole in s abilizing he condensa e agains col-
lapse [6–9], p o iding access o pa ame e domains whe e
exo ic physics occu . This mainly includes sel -o ganized
pa e n o ma ion akin o classical e o luids [10] and he
eme gence o ul adilu e liquid d ople s [11–14].
Supe solidi y is a s a e o ma e ha simul aneously
ea u es bo h disc e e ansla ional symme y and a la ge
supe luid ac ion [15–18]. Such phase-cohe en densi y-
modula ed s a es we e ealized using addi ional ex e nal ields
[19,20]. Wi h he pionee ing expe imen o Re . [11], dBECs
eme ged as a sel -o ganizing al e na i e. Since hen, a ange
o exci ing expe imen s ha e explo ed pa e n o ma ion and
supe solidi y in dBECs, including hei exci a ion spec a
[21–25], nuclea ion o o ices [26], and he eme gence o
pa e ns in elonga ed ciga -shaped aps wi h one-dimensional
symme y b eaking [21,22,27–32] and in a pancake apping
geome y, whe e he condensa e is igh ly con ined along he
*Con ac au ho : [email p o ec ed]
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI.
pola iza ion di ec ion leading o a wo-dimensionally b oken
symme y [11,33]. This in ense expe imen al ac i i y has been
complemen ed wi h a ange o heo e ical wo ks explo ing he
physics o dBECs in ciga -shaped [34–43] and pancake aps
[44–50].
Theo e ical wo ks explo ing such pancake geome ies a
ze o empe a u e ha e e ealed a ich phase diag am in his
sys em [44,47,49,50] wi h in e es ing supe solid p ope ies
[44,51]. I has been shown ha he di e en coexis ing phases
con e ge o a single poin , a which he phase- ansi ion is
o second o de and a ound which supe solidi y occu s [44].
This second-o de poin un o una ely equi es la ge densi ies
ha appea expe imen ally un easible, and he new phases,
namely, s ipe and honeycomb s a es, equi e e en highe den-
si ies. Thus, inding pa ame e s ha pe mi he expe imen al
ealiza ion o hese phases ep esen s a signi ican challenge
[47,49].
Ye , he e appea s an al e na i e way o p omo e o on-
so ening and, subsequen ly, d i e he quan um phase-
ansi ion apa om only inc easing densi y and changing
apping pa ame e s. Gi en ha dBECs a e s ongly suscep-
ible o quan um luc ua ions, i migh seem plausible ha
he mal luc ua ions ha e a s ong e ec as well. In ac ,
ecen expe imen s ha e explo ed he e ec o ini e empe a-
u es on he dBEC [32]. La e , heo e ical conside a ions ha
ea ed he mal luc ua ions by means o Bogoliubo heo y
employing local densi y app oxima ion showed ha inc eas-
ing empe a u e can indeed p omo e pa e n o ma ion and
possibly supe solidi y [42,43].
He e, we pu sue he idea whe he a ini e empe a-
u e migh lowe he densi y equi ed o accessing he
2643-1564/2025/7(2)/023019(9) 023019-1 Published by he Ame ican Physical Socie y
HE, SÁNCHEZ-BAENA, MAUCHER, AND ZHANG PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
second-o de poin and he high-densi y phases o expe imen-
ally mo e accessible alues o a dBEC in a pancake ap.
The shi in densi y due o a ini e empe a u e has al eady
been explo ed in a ciga -shaped ap [43]. The mo i a ion
o explo ing he la e again in a pancake ap is due o he
ac ha dimensionali y and con inemen play a c ucial ole
o he densi y and he pa icle numbe a he second-o de
poin [44,47]. Thus, we add ess hese ques ions in his pa-
pe , which is o ganized as ollows. In Sec. II,we e iew he
model we employ o desc ibe he ini e- empe a u e e ec s. In
Sec. III, we p esen he ini e- empe a u e phase diag am. In
Sec. IV, we explo e whe he a eal- ime e olu ion accoun ing
o h ee-body losses can ac ually d i e he ansi ion o he
high-densi y phases. Finally, in Sec. Vwe p esen he main
conclusions o ou wo k.
II. FINITE-TEMPERATURE THEORY
To iden i y g ound s a es o he condensa e we use bo h
nume ical and a ia ional me hods. We employ Bogoliubo
heo y and he local densi y app oxima ion (LDA) o accoun
o he mal luc ua ions on he condensa e [42,52,53]. Wi hin
he LDA, his leads o he empe a u e-dependen ex ended
G oss-Pi ae skii equa ion (TeGPE) o he condensa e wa e
unc ion ψ( ) gi en by
μψ( )=−¯h2∇2
2m+U( )+d Vdd( − )|ψ( )|2
+4π¯h2as
m|ψ( )|2+Hqu( )+H h( )ψ( ).(1)
He e, μis he chemical po en ial, mis he a omic mass,
Vdd deno es he dipole-dipole in e ac ion, and asis he s-wa e
sca e ing leng h. Udesc ibes he apping po en ial, which
in he he modynamic limi eads U( )=1
2mω2
zz2and o he
ully apped sys em is gi en by U( )=1
2m(ω2
xx2+ω2
yy2+
ω2
zz2), wi h ωzωx,ωy. He e, by he he modynamic limi ,
we e e o he si ua ion whe e he dBEC is uncon ined in one
o mo e di ec ions, such ha he pa icle numbe and he ol-
ume di e ge whils he densi y emains cons an . In ou case,
his is he si ua ion when he condensa e is solely con ined in
he zdi ec ion by he po en ial U( )=1
2mω2
zz2, whils being
in ini ely ex ended in he x-yplane, such ha he o al pa icle
numbe N→∞and he a ea S→∞di e ge, whe eas he
a e age wo-dimensional (2D) densi y ρ2D =N/S emains
ini e. The e ms Hqu and H h accoun o he e ec o quan um
and he mal luc ua ions, espec i ely. Wi hin he LDA, hey
a e gi en by [5,42,52]
Hqu( )=32
3√πga3
sQ5(add/as)|ψ( )|3,(2)
H h( )=dk
(2π)3
1
(eβεk−1)˜
V(k)τk
εk( ),(3)
whe e g=4π¯h2as
m,εk( )=τk[τk+2|ψ( )|2˜
V(k)] is he
Bogoliubo exci a ion spec um o a gi en local densi y
|ψ( )|2o he dBEC, τk=¯h2k2
2m,β=1/kBT, and Tdeno es
empe a u e. ˜
V(k) co esponds o he Fou ie ans o m o
he sum o he dipole-dipole in e ac ion and he con ac
in e ac ion, gi en by
˜
V(k)=4π¯h2as
m+4π¯h2add
m3k2
z
k2−1,(4)
whe e dipoles a e assumed o be pola ized along he zaxis.
The pa ame e add =mCdd/(12π¯h2) co esponds o he dipole
leng h, Cdd desc ibes he s eng h o he dipola in e ac ion,
and he auxilia y unc ion Q5(add/as) is gi en by [5]
Q5(add/as)=1
0
du1−add
as+3add
asu25/2
.(5)
Equa ion (2) desc ibes he pa o he TeGPE ha esul s
om he impac o he quan um luc ua ions on he mean
ield o he a oms and is esponsible o a es ing collapse
[9] o he condensa e ha would o he wise occu [6,7]. Ca e
mus be aken in he e alua ion o Eq. (3), since imagi-
na y exci a ion ene gies a ise o as<add a low momen a.
The applica ion o he apping po en ial in all h ee spa ial
dimensions implies a ini e size o ou sys em in a gi en
apping di ec ion which p o ides a lowe bound o he pos-
sible momen a o he exci a ions en e ing Eq. (3). Due o
he symme y o he dipole-dipole in e ac ion, he con i-
bu ion o he luc ua ion ene gies depends only on kzand
kρ=k2
x+k2
y. Thus, we only e ain exci a ions ha ul ill
kz>2π/lzand kρ>(2π/lx)2+(2π/ly)2, wi h lx,y,z ep-
esen ing he size o ou sys em along he x,y, and zaxes,
espec i ely. Fo he homogeneous sys em, we se kz>2π/lz,
kρ>0. In ou calcula ions, we assume ha he condensa e
exhibi s a Thomas-Fe mi p o ile wi h a ypical wid h o σz=
(ρ2D(as/add+2)
2ω2
z)1/3in he zdi ec ion [44,47,54] (also see he
subsequen discussion on a ia ional app oxima ion), and we
app oxima ely se lz=2σz. Fo he ans e se size in he ully
apped si ua ion, we i s ob ain a s able solu ion o Eq. (1)
using imagina y- ime e olu ion wi hou a ans e se cu o
(i.e., kρ>0). Subsequen ly, we i he ans e se densi y p o-
ile wi h a Gaussian unc ion cha ac e ized by a wid h σ⊥,
which allows us o de e mine lx=ly=2σ⊥and hus es ablish
he ans e se cu o kρ>π/σ
⊥. Using his ini e cu o , we
ecalcula e he g ound s a e o Eq. (1). No e ha he cu o
may sligh ly al e he exac posi ion o he pa ame e do-
mains [55]. Fo he a ia ional app oxima ion, we conside he
ene gy di e ence E=E(ρ)−E(ρ0) be ween he unmod-
ula ed s a e ρ0and a modula ed s a e ρcon aining pe iodic
densi y pe u ba ions as below [44,47,54],
ρ( )=ρ0(z)[1+P( ⊥)],(6)
whe e he unmodula ed s a e is app oxima ed by a Thomas-
Fe mi p o ile ρ0(z)=3ρ2D
4σz(1 −z2
σ2
z) in he con ined zdi ec ion,
ρ2D ep esen s he a e age 2D densi y in he ans e se
di ec ion, and P( ⊥) desc ibes he pe iodic pe u ba ion
in he ans e se x-yplane. Fo he modula ion exhibi -
ing h ee old o a ional symme y, we de ine he densi y as
P( ⊥)=A3
j=1cos(pj· ⊥), whe e A ep esen s he mod-
ula ion ampli ude and he h ee wa e ec o s pj o m
an equila e al iangle in he ans e se di ec ion, sa is y-
ing 3
j=1pj=0 and |pj|=p. In his scena io, Eq. (6)
e eals wo dis inc densi y dis ibu ions depending on
023019-2
ACCESSING ELUSIVE TWO-DIMENSIONAL PHASES … PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
FIG. 1. Va ia ional phase diag am o T=0 (blue dashed lines)
and o kBT/dd =2 (solid ed line) o a pancake dipola BEC in he
he modynamic limi . The ed ma ke s ep esen he co esponding
c i ical poin s be ween di e en phases ob ained ia nume ically
sol ing he TeGPE. The densi y p o iles o he modula ed s a es (i.e.,
iangula , s ipe, and honeycomb) a e displayed in hei co espond-
ing domains. The densi y alue is indica ed by he colo dep h, whe e
blue (whi e) co esponds o a high (low) densi y. No e, ha he e
is a di e ence in scale o he 2D condensed densi y ρ2D shown
a he bo om (i.e., blue axis) o he ze o- empe a u e case and a
he op (i.e., ed axis) o he ini e- empe a u e case. The densi y is
exp essed in uni s o 1/ 2
0, wi h 0=12πadd.
he sign o he modula ion ampli ude: a iangula s a e
o posi i e Aand a honeycomb s a e o nega i e A,
as discussed in subsequen sec ions. Simila ly, o he
modula ed s a e wi h wo old o a ional symme y, such
as he s ipe phase, he densi y modula ion can be ex-
p essed as P( ⊥)=Acos(p· ⊥), which in ol es only
one wa e- ec o componen . By subs i u ing his ansa z in o
he ene gy di e ence equa ion, we ob ain he ene gy di e -
ence E(A,p) as a unc ion o Aand p. By nume ically
minimizing Ewi h espec o he wo a ia ional pa am-
e e s, one can de e mine he g ound s a e wi h he lowes
ene gy. A non-nega i e E o a bi a y Aand pindica es an
unmodula ed supe luid g ound s a e, while a nega i e Ea
ini e Aand pindica es he ansi ion o a modula ed s a e. By
compa ing he ene gy shi s o di e en ypes o modula ed
s a es, we can iden i y he bounda ies be ween he iangula ,
s ipe, and honeycomb s a es.
To p esen he esul s o ou wo k, we choose he cha ac-
e is ic leng h and ene gy scales gi en by 0=12πadd and
dd =¯h2/(m 2
0). The e o e, and i no speci ied o he wise, all
subsequen leng h and ene gy scales a e exp essed in e ms o
hese cha ac e is ic quan i ies.
III. FINITE-TEMPERATURE PHASE DIAGRAM
We s a by e alua ing he e ec o empe a u e on he
phase diag am in he he modynamic limi , o which he ap-
ping po en ial eads U( )=1
2mω2
zz2. The apping s eng h
is ixed o ¯hωz/dd =0.08. We show in Fig. 1 he phase
diag am o a dipola condensa e wi h pancake geome y o
kBT=0 (blue dashed lines) and kBT/dd =2 [ ed solid lines
( a ia ional esul ) and ma ke s (TeGPE solu ion)]. The la e
co esponds o T=87 nK o a sys em o 164Dy a oms. The
2D condensa e densi y is de ined as ρ2D =dz|ψ( )|2, whe e
ψ( ) co esponds o he solu ion o Eq. (1) no malized o
he pa icle numbe N=d |ψ( )|2. Fo he simula ion in
he he modynamic limi we ha e employed pe iodic bound-
a y condi ions in he x-yplane, whe e he a e age 2D densi y
ρ2D is ixed while he pa icle numbe Ncan a y wi h he size
o he nume ical box.
Compa ing he a ia ional and nume ical esul s o he
ini e- empe a u e case in Fig. 1, we no e ha he a ia ional
analysis cap u es he quali a i e physics easonably well and
hus ep esen s a compa ably inexpensi e ool o i s ex-
plo a ion. The ull nume ical solu ion o Eq. (1) essen ially
amoun s o a shi in bo h sca e ing leng h and densi y.
We can add ess he consis ency o employing he LDA in
ou heo y by inspec ing one o he modula ed g ound-s a e
solu ions o he phase diag am o Fig. 1. Acco ding o Re . [5],
i s applicabili y is p ima ily go e ned by he Thomas-Fe mi
pa ame e Nas/aho, whe e aho is he ha monic oscilla o
leng h. The LDA is a alid app oxima ion in he egime whe e
Nas/aho 1. To examine his Thomas-Fe mi pa ame e o
ou se ing, we conside he size o a uni cell o , e.g., he
iangula s a e a ρ2D =80 and as/add =0.797 (i.e., nea he
second-o de c i ical poin ), which con ains a single d ople
wi h Nuc ≈2.5×104a oms. The densi y p o ile o such a sin-
gle d ople can be i ed by a Gaussian ∝e−(x2+y2)/a2
⊥,ho−z2/a2
z,ho ,
yielding a⊥,ho =1.42 µm and az,ho =5.3 µm. He e, we ha e
used he ele an pa ame e s (i.e., mass and dipole leng h)
o 164Dy. Consequen ly, he Thomas-Fe mi pa ame e s a e
Nucas/a⊥,ho =98 and Nucas/az,ho =26. As Nas/aho 1is
ul illed, he applica ion o he LDA appea s easonable.
A ze o empe a u e (blue lines), he dashed lines indica e
a i s -o de phase ansi ion be ween he luid-solid, luid-
honeycomb, and honeycomb-s ipe phases. They con e ge o
a poin a which he phase ansi ion is o second o de [44].
This quali a i e phenomenology and quali a i e shape o he
phase diag am emains ue a ini e empe a u es ( ed lines)
as well.
We ocus he discussion now on he second-o de poin . We
no e ha a sligh shi o he second-o de poin owa ds la ge
alues o he sca e ing leng h occu s in he ini e- empe a u e
case. This shi co esponds o as≃2.4a0 o 164Dy a oms,
wi h a0deno ing he Boh adius. The mos s iking poin
when compa ing he ze o- and ini e- empe a u e phase dia-
g ams is he signi ican shi in condensa e densi y. Fo he
ini e- empe a u e case, he densi y o he second-o de poin
is mo e han hal ed, ρc
2D(87 nK)/ρc
2D(0) =0.49 (no e he
di e en scales o he ze o- and ini e- empe a u e cases in
Fig. 1). This educ ion in densi y in he he modynamic limi
is p omising o he ealiza ion o hese phases in an expe i-
men , since acco ding o he es ima e in Re . [56] he li e ime
due o h ee-body losses scales like 3∼1/ρ2. Thus, in ou
case, he li e ime could be expec ed o inc ease oughly by a
ac o o ≃4.2. We explo e he e ec o ini e empe a u es in
he dynamical o ma ion o hese phases u he in Sec. IV.
These obse a ions a e consis en wi h p e ious esul s
[42,43]. Ye , in his case, he shi in densi y is subs an ially
la ge , highligh ing ha dimensionali y plays an impo an
ole. The inc ease o empe a u e o a gi en condensed
023019-3
HE, SÁNCHEZ-BAENA, MAUCHER, AND ZHANG PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
s ipe
honeycomb
unmodula ed
FIG. 2. Tempe a u e-d i en supe solidi y, isualized by showing
he con as o he g ound-s a e wa e unc ion [see Eq. (7)] as a
unc ion o empe a u e o ρ2D 2
0=105 and as/add =0.807.
densi y can p omo e a phase ansi ion om he luid phase
o a modula ed s a e. Figu e 2p o ides an example o
he g ound-s a e phase ansi ions d i en by empe a u e o
ρ2D 2
0=105 and as/add =0.807. I depic s he con as o he
g ound s a e,
C=|ψ(z=0)|2
max −|ψ(z=0)|2
min
|ψ(z=0)|2
max +|ψ(z=0)|2
min
,(7)
a a ixed densi y. We see ha , when he empe a u e su passes
∼83 nK, he honeycomb eme ges as g ound s a e wi h a ini e
con as unde going a i s -o de phase ansi ion. I we u he
inc ease he empe a u e beyond ∼88 nK, he honeycomb
becomes ene ge ically less a o able as compa ed o he s ipe
phase.
Le us now explo e how a ying he apping equency
ωzquan i a i ely changes he shi in densi y o he c i ical
poin . P e ious wo ks a ze o empe a u e show he e is a
compe i ion be ween he peak densi ies and he pa icle num-
be in he ully apped sys em [47]. Employing pa ame e s
ha yield he honeycomb, laby in h, and s ipe s uc u es a
densi ies o which he h ee-body losses a e mode a e o
expe imen s in ol e p ohibi i ely la ge condensed pa icle
numbe s (N∼106) and, ice e sa, employing lowe pa i-
cle numbe s (N∼105) leads o peak densi ies la ge han
1015 cm−3.
Figu e 3depic s he c i ical densi y ρc
2D as a unc ion o
ωz o ze o and ini e empe a u e wi h kBT/dd =2. The
igu e shows ha he di e ence in he c i ical densi y o
he second-o de poin o a ini e empe a u e dec eases
as he equency inc eases. This beha io s ems om he
densi y dependence o he quan um and he mal luc ua ion
e mso Eqs.(2) and (3). I has al eady been es ab-
lished ha he mal luc ua ions dec ease upon inc easing
densi y while quan um luc ua ions ollow he opposi e end
[42,43].
To u he elucida e his poin , Fig. 4shows he depen-
dence o bo h he 3D c i ical densi y ρc=3ρc
2D/(4σz) and
he numbe o condensed pa icles pe uni cell a he c i ical
poin , Nc=2ρc
2Dλ2/√3, as a unc ion o he apping s eng h
FIG. 3. Dependence o he c i ical 2D densi y as a unc ion o he
apping equency o T=0 (blue solid line) and kBT/dd =2( ed
solid line). The densi y and he ha monic equency a e exp essed in
uni s o 1/ 2
0and dd/¯h, espec i ely.
ωz o bo h kBT/dd =0 and 2. He e, he leng h λ=2π/p
is gi en by he wa e ec o o he modula ed densi y a he
c i ical poin . Using Ncwe can es ima e how many pa icles
a e equi ed o a gi en numbe o uni cells o a densi y
modula ed s a e in he ully apped sys em. Again, we no e
ha empe a u e educes Ncsigni ican ly o small apping
equencies.
In iew o he ecen p og ess in he con ol and educ ion
o eac i e losses o ul acold dipola molecules [57–62] and
he ealiza ion o he i s molecula dBEC [62], i is in e -
es ing o pu he p e ious esul s in he con ex o molecules.
Due o hei conside ably la ge dipole momen as compa ed
o dBECs o a single species, he educed empe a u e o
kBT/dd =2 co esponds o T=1 nK o a gas o NaCs
molecules. Fo his ex emely low empe a u e, he alues o
Ncand ρca e in he ange Nc∈(0.5,2.5) ×104and ρc∈
(1011,1012)cm
−3 o apping equencies ωz∈(3,40) Hz.
While he alues o he c i ical densi y a e less o equal
compa ed o hose in he ecen expe imen o Re . [62],
××
FIG. 4. Th ee-dimensional c i ical densi y (dashed lines) and
numbe o condensed pa icles pe uni cell a he c i ical poin (solid
lines) as a unc ion o he apping equency o T=0 nK (blue)
and T=87 nK ( ed) o a gas o 164Dy a oms.
023019-4
ACCESSING ELUSIVE TWO-DIMENSIONAL PHASES … PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
×
FIG. 5. Subsequen dynamics a e an in e ac ion quench om as/add =0.8 o 0.757 a T=0 nK (a) and T=87 nK (b). The apping
equencies a e ¯hωz/dd =0.11 and ¯hω⊥/dd =0.052. Ini ially, he numbe o condensed pa icles co esponds o N=3.5×105. Panel
(c) shows he dec ease o he a om numbe due o he h ee-body losses.
he numbe o condensed pa icles pe uni cell g ea ly
exceeds ha o he expe imen al condensa e, which consis s
o a ew hund ed molecules. Tha being said, he p oduc ion
o molecula dBECs is s ill a an ea ly s age and u u e de el-
opmen s migh lead o molecula dBECs wi h highe pa icle
numbe s.
IV. REAL-TIME EXCITATION OF A HONEYCOMB
STATE IN A DIPOLAR BEC
Thus a we ha e es ic ed ou discussion o phases in he
he modynamic limi . We ocus ou a en ion now on he ully
apped sys em. Fo ha ma e , we un eal- ime simula ions
023019-5
HE, SÁNCHEZ-BAENA, MAUCHER, AND ZHANG PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
×
FIG. 6. Dynamics ollowing an in e ac ion quench om as/add =0.65 o 0.595 a T=0 nK (a) and T=87 nK (b). The apping
equencies a e ¯hωz/dd =0.75 and ¯hω⊥/dd =0.32. The ini ial numbe o condensed pa icles is se o N=105. Panel (c) shows he dec ease
o he a om numbe due o he h ee-body losses.
o he TeGPE o model he expe imen al ealiza ion o he
honeycomb s a e o a sys em o 164Dy a oms ollowing a
quench o he sca e ing leng h a T=0 nK and T=87 nK.
The ime-dependen TeGPE is ob ained by eplacing he
le -hand side o Eq. (1)byi¯h∂
∂ ψ. The ime-e ol ed conden-
sa e wa e unc ion is ob ained h ough an i e a i e p ocess,
whe e, o each i e a ion, he inal s a e ψ( , + )is
ob ained om he ini ial one ψ( , ) by e alua ing he ime-
dependen TeGPE. The quan um and he mal luc ua ion
e ms a e ob ained by inse ing ψ( , )inEqs.(2) and (3),
analogously o he p ocedu e ollowed a ze o empe a u e
[26,63–65]. We ha e included h ee-body losses in he same
023019-6
ACCESSING ELUSIVE TWO-DIMENSIONAL PHASES … PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
way as in Re s. [47,66]. The ull ime-e olu ion equa ion
eads
i¯h∂
∂ ψ=−¯h2∇2
2m+U( )+d Vdd( − )|ψ( )|2
+4π¯h2as
m|ψ( )|2+Hqu( )+H h( )
−i¯hL3
2|ψ( )|4ψ( ),(8)
wi h L3=1.5×10−41 m6/s[66]. The esul s o he ime
e olu ion a e shown in Figs. 5and 6.
Fo a apping s eng h o ¯hωz/dd =0.11 and N=3.5×
105condensed a oms, a honeycomb s a e o a la ge li e ime
o ∼40 ms wi h a mode a e peak densi y o ρpeak ∼6×
1014 cm−3is displayed in Fig. 5. We can see om he
esul s ha empe a u e a o s he o ma ion o he honey-
comb s a e, while he calcula ion a ze o empe a u e does
no lead o a modula ed s a e. Un o una ely, dec easing he
pa icle numbe unde hese condi ions u he esul s in he
disappea ance o he honeycomb s uc u e. To dec ease he
condensed pa icle numbe while e aining he honeycomb
s a e, we ha e o se a igh e con inemen along he zaxis,
which will inc ease he densi y. We obse e a s uc u e wi h
a much sho e li e ime o ∼7ms o ¯hωz/dd =0.75 and
N=105condensed a oms (Fig. 6), howe e , a he cos o
a conside ably la ge peak densi y ρpeak ∼3×1015 cm−3.
The mal e ec s he e ha e a smalle impac han in he p io
case, as he mal luc ua ions domina e a smalle densi ies as
al eady discussed [42,43].
In summa y, he wo cases we p esen ed ha e he pu pose
o po ay wo ex eme pa ame e domains, one case wi h a
la ge pa icle numbe and a compa ably small densi y ha is
s ongly a ec ed by empe a u e and one case wi h a compa-
ably small pa icle numbe and a la ge densi y ha is less
a ec ed. In he i s case, he pa icle numbe is signi ican ly
educed as compa ed o he ze o- empe a u e si ua ion [47].
V. CONCLUSIONS
In his pape we ha e explo ed whe he he mal luc ua-
ions migh assis in p omo ing pa e n o ma ion o such an
ex en ha he high-densi y physics o a dBEC wi h pancake
symme y becomes expe imen ally accessible. This includes
access o no el phases such as honeycomb and s ipe phases
as well as he second-o de poin o he phase diag am. We
ha e ound ha an inc ease in empe a u e indeed can lead o a
signi ican dec ease in he necessa y densi y o p obe he high-
densi y physics o he la ened dBEC. We ha e also shown
eal- ime simula ions wi h ealis ic in e ac ion quenches ha
ga e ise o he o ma ion o a honeycomb. Thus, we conclude
ha empe a u e indeed migh p esen a p omising ou e o-
wa ds he po en ial ealiza ion o hese high-densi y phases.
Beyond p obing he high-densi y physics o dBECs and
pa e n o ma ion, his wo k migh pa e a u he pa hway o-
wa ds explo ing ini e- empe a u e e ec s in dBECs due o he
clea signa u e o he eme ging pa e ns. Fu he mo e, highe -
o de heo ies beyond wha has been p esen ed he e could,
o ins ance, quan i a i ely s udy he e ec o empe a u e on
he supe luid p ope ies o he densi y-modula ed s uc u es,
like he honeycomb o he s ipe [49–51], as well as he phase
ansi ion be ween no mal gas and he supe luid s a e in he
s ong local in e ac ion egime [67]. Fo his pu pose, ab ini io
me hods, like Mon e Ca lo algo i hms, ep esen an excellen
op ion. Such me hods would be able o s udy he egime o
empe a u es e en highe han hose conside ed he e, whe e
he sys em is mos ly noncondensed.
ACKNOWLEDGMENTS
This wo k was suppo ed by he Na ional Key
Resea ch and De elopmen P og am o China (G an
No. 2021YFA1401700), he Na ional Na u e Science
Founda ion o China (G an No. 12104359), and he
Shaanxi Academy o Fundamen al Sciences (Ma hema ics,
Physics) (G an No. 22JSY036). J.S.-B. acknowledges
suppo by he Spanish Minis e io de Ciencia e Inno ación
(MCIN/AEI/10.13039/501100011033, G an s No.
PID2020-113565GB-C21 and No. PID2023-147469NB-C21)
and by he Gene ali a de Ca alunya (G an No. 2021 SGR
01411). Y.-C.Z. acknowledges he suppo o he Xiaomi
Young Talen s p og am, Xi’an Jiao ong Uni e si y h ough
he “Young Top Talen s Suppo Plan” and Basic Resea ch
Funding as well as he High-pe o mance Compu ing
Pla o m o Xi’an Jiao ong Uni e si y o he compu ing
acili ies.
[1] T. Lahaye, C. Meno i, L. San os, M. Lewens ein, and T. P au,
The physics o dipola bosonic quan um gases, Rep. P og. Phys.
72, 126401 (2009).
[2] V. I. Yukalo , Dipola and spino bosonic sys ems, Lase Phys.
28, 053001 (2018).
[3] L. Chomaz, I. Fe ie -Ba bu , F. Fe laino, B. Labu he-Tol a,
B. L. Le , and T. P au, Dipola physics: A e iew o expe -
imen s wi h magne ic quan um gases, Rep. P og. Phys. 86,
026401 (2023).
[4] A. R. P. Lima and A. Pels e , Quan um luc ua ions in dipola
Bose gases, Phys. Re . A 84, 041604(R) (2011).
[5] A. R. P. Lima and A. Pels e , Beyond mean- ield low-lying
exci a ions o dipola Bose gases, Phys. Re . A 86, 063609
(2012).
[6] T. Koch, T. Lahaye, J. Me z, B. F öhlich, A. G iesmaie , and
T. P au, S abiliza ion o a pu ely dipola quan um gas agains
collapse, Na . Phys. 4, 218 (2008).
[7] T. Lahaye, J. Me z, B. F öhlich, T. Koch, M. Meis e , A.
G iesmaie , T. P au, H. Sai o, Y. Kawaguchi, and M. Ueda,
d-Wa e collapse and explosion o a dipola Bose-Eins ein con-
densa e Phys.Re .Le .101, 080401 (2008).
[8] D. S. Pe o , Quan um mechanical s abiliza ion o a col-
lapsing Bose-Bose mix u e, Phys.Re .Le .115, 155302
(2015).
[9] F. Wäch le and L. San os, Quan um ilamen s in dipola Bose-
Eins ein condensa es, Phys. Re . A 93, 061603(R) (2016).
[10] R. E. Rosensweig, Fe ohyd odynamics (Camb idge Uni e si y,
New Yo k, 1985).
023019-7
HE, SÁNCHEZ-BAENA, MAUCHER, AND ZHANG PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
[11] H. Kadau, M. Schmi , M. Wenzel, C. Wink, T. Maie ,
I. Fe ie -Ba bu , and T. P au, Obse ing he Rosensweig
ins abili y o a quan um e o luid, Na u e (London) 530, 194
(2016).
[12] M. Schmi , M. Wenzel, F. Bö che , I. Fe ie -Ba bu , and T.
P au, Sel -bound d ople s o a dilu e magne ic quan um liquid,
Na u e (London) 539, 259 (2016).
[13] I. Fe ie -Ba bu , H. Kadau, M. Schmi , M. Wenzel, and T.
P au, Obse a ion o quan um d ople s in a s ongly dipola
Bose gas, Phys. Re . Le . 116, 215301 (2016).
[14] L. Chomaz, S. Baie , D. Pe e , M. J. Ma k, F. Wäch le , L.
San os, and F. Fe laino, Quan um- luc ua ion-d i en c osso e
om a dilu e Bose-Eins ein condensa e o a mac od ople in a
dipola quan um luid, Phys. Re . X 6, 041039 (2016).
[15] E. P. G oss, Uni ied heo y o in e ac ing bosons, Phys. Re .
106, 161 (1957).
[16] A. F. And ee and I. M. Li shi z, Quan um heo y o de ec s in
c ys als, So . Phys. Usp. 13, 670 (1971).
[17] G. V. Ches e , Specula ions on Bose-Eins ein condensa ion and
quan um c ys als, Phys. Re . A 2, 256 (1970).
[18] A. J. Legge , Can a solid be supe luid? Phys. Re . Le . 25,
1543 (1970).
[19] J. Léona d, A. Mo ales, P. Zupancic, T. Esslinge , and T.
Donne , Supe solid o ma ion in a quan um gas b eaking a
con inuous ansla ional symme y, Na u e (London) 543,87
(2017).
[20] J.R. Li, J. Lee, W. Huang, S. Bu chesky, B. Sh eynas, F. C.
Top, A. O. Jamison, and W. Ke e le, A s ipe phase wi h
supe solid p ope ies in spin-o bi -coupled Bose-Eins ein con-
densa es, Na u e (London) 543, 91 (2017).
[21] M. Guo, F. Bö che , J. He ko n, J.-N. Schmid , M. Wenzel,
H. P. Büchle , T. Langen, and T. P au, The low-ene gy golds one
mode in a apped dipola supe solid, Na u e (London) 574, 386
(2019).
[22] L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Famà, A. Fio e i, C.
Gabbanini, G. Modugno, A. Reca i, and S. S inga i, Supe solid
symme y b eaking om comp essional oscilla ions in a dipola
quan um gas, Na u e (London) 574, 382 (2019).
[23] G. Na ale, R. M. W. an Bijnen, A. Pa scheide , D. Pe e , M. J.
Ma k, L. Chomaz, and F. Fe laino, Exci a ion spec um o a
apped dipola supe solid and i s expe imen al e idence, Phys.
Re . Le . 123, 050402 (2019).
[24] E. Poli, D. Baillie, F. Fe laino, and P. Blai Blakie, Exci a ions
o a wo-dimensional supe solid, Phys.Re .A110, 053301
(2024).
[25] P. B. Blakie, Di ac poin s and shea ins abili y induced c ys al
ansi ions in honeycomb supe solids, Phys. Re . Le . 134,
013401 (2025).
[26] E. Caso i, E. Poli, L. Klaus, A. Li ino , C. Ulm, C. Poli i,
M. J. Ma k, T. Bland, and F. Fe laino, Obse a ion o o -
ices in a dipola supe solid, Na u e (London) 635, 327
(2024).
[27] L. Tanzi, E. Lucioni, F. Famà, J. Ca ani, A. Fio e i, C.
Gabbanini, R. N. Bisse , L. San os, and G. Modugno, Obse -
a ion o a dipola quan um gas wi h me as able supe solid
p ope ies, Phys.Re .Le .122, 130405 (2019).
[28] F. Bö che , J.-N. Schmid , M. Wenzel, J. He ko n, M. Guo,
T. Langen, and T. P au, T ansien supe solid p ope ies in an
a ay o dipola quan um d ople s, Phys.Re .X9, 011051
(2019).
[29] 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, Long-li ed and ansien
supe solid beha io s in dipola quan um gases, Phys. Re . X 9,
021012 (2019).
[30] L. Tanzi, J. G. Malobe i, G. Biagioni, A. Fio e i, C. Gabbanini,
and G. Modugno, E idence o supe luidi y in a dipola su-
pe solid om nonclassical o a ional ine ia, Science 371, 1162
(2021).
[31] G. Biagioni, N. An olini, A. Alaña, M. Modugno, A. Fio e i, C.
Gabbanini, L. Tanzi, and G. Modugno, Dimensional c osso e
in he supe luid-supe solid quan um phase ansi ion, Phys.
Re . X 12, 021019 (2022).
[32] M. Sohmen, C. Poli i, L. Klaus, L. Chomaz, M. J. Ma k, M. A.
No cia, and F. Fe laino, Bi h, li e, and dea h o a dipola supe -
solid, Phys.Re .Le .126, 233401 (2021).
[33] M. A. No cia, C. Poli i, L. Klaus, E. Poli, M. Sohmen, M. J.
Ma k, R. N. Bisse , L. San os, and F. Fe laino, Two-dimensional
supe solidi y in a dipola quan um gas, Na u e (London) 596,
357 (2021).
[34] D. Edle , C. Mish a, F. Wäch le , R. Na h, S. Sinha, and L.
San os, Quan um luc ua ions in quasi-one-dimensional dipo-
la Bose-Eins ein condensa es, Phys. Re . Le . 119, 050403
(2017).
[35] S. M. Roccuzzo and F. Ancilo o, Supe solid beha io o a
dipola Bose-Eins ein condensa e con ined in a ube, Phys. Re .
A99, 041601(R) (2019).
[36] P. B. Blakie, D. Baillie, and S. Pal, Va ia ional heo y
o he g ound s a e and collec i e exci a ions o an elon-
ga ed dipola condensa e, Commun. Theo . Phys. 72, 085501
(2020).
[37] P. B. Blakie, D. Baillie, L. Chomaz, and F. Fe laino, Supe so-
lidi y in an elonga ed dipola condensa e, Phys.Re .Res.2,
043318 (2020).
[38] S. I. Ma eenko, M. S. Baho adino , M. A. Ba ano , and
G. V. Shlyapniko , Ro ons and hei damping in elonga ed
dipola Bose-Eins ein condensa es, Phys. Re . A 106, 013319
(2022).
[39] J. C. Smi h, D. Baillie, and P. B. Blakie, Supe solidi y and
c ys alliza ion o a dipola Bose gas in an in ini e ube, Phys.
Re . A 107, 033301 (2023).
[40] P. B. Blakie, L. Chomaz, D. Baillie, and F. Fe laino, Comp ess-
ibili y and speeds o sound ac oss he supe luid- o-supe solid
phase ansi ion o an elonga ed dipola gas, Phys. Re . Res. 5,
033161 (2023).
[41] A. Alaña, Supe solid- o ma ion- ime sho cu and exci a ion
educ ion by manipula ing he dynamical ins abili y, Phys. Re .
A109, 023308 (2024).
[42] J. Sánchez-Baena, C. Poli i, F. Mauche , F. Fe laino, and
T. Pohl, Hea ing a dipola quan um luid in o a solid, Na .
Commun. 14, 1868 (2023).
[43] J. Sánchez-Baena, T. Pohl, and F. Mauche , Supe luid-
supe solid phase ansi ion o elonga ed dipola Bose-Eins ein
condensa es a ini e empe a u es, Phys. Re . Res. 6, 023183
(2024).
[44] Y.-C. Zhang, F. Mauche , and T. Pohl, Supe solidi y a ound a
c i ical poin in dipola Bose-Eins ein condensa es, Phys. Re .
Le . 123, 015301 (2019).
[45] T. Bland, E. Poli, C. Poli i, L. Klaus, M. A. No cia, F. Fe laino,
L. San os, and R. N. Bisse , Two-dimensional supe solid
023019-8
ACCESSING ELUSIVE TWO-DIMENSIONAL PHASES … PHYSICAL REVIEW RESEARCH 7, 023019 (2025)
o ma ion in dipola condensa es, Phys.Re .Le .128, 195302
(2022).
[46] L. E. Young-S. and S. K. Adhika i, Mini d ople , mega
d ople and s ipe o ma ion in a dipola condensa e, Physica
D (Ams e dam, Ne h.) 455, 133910 (2023).
[47] Y.-C. Zhang, T. Pohl, and F. Mauche , Phases o supe solids in
con ined dipola Bose-Eins ein condensa es, Phys.Re .A104,
013310 (2021).
[48] M. A. No cia, E. Poli, C. Poli i, L. Klaus, T. Bland, M. J. Ma k,
L. San os, R. N. Bisse , and F. Fe laino, Can angula oscilla ions
p obe supe luidi y in dipola supe solids? Phys. Re . Le . 129,
040403 (2022).
[49] 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, Pa e n o ma ion in quan um e o luids: F om
supe solids o supe glasses, Phys. Re . Res. 3, 033125
(2021).
[50] Y.-C. Zhang, T. Pohl, and F. Mauche , Me as able pa e ns in
one- and wo-componen dipola Bose-Eins ein condensa es,
Phys. Re . Res. 6, 023023 (2024).
[51] A. Gallemí and L. San os, Supe luid p ope ies o a honeycomb
dipola supe solid, Phys.Re .A106, 063301 (2022).
[52] E. Ayba and M. Ö. Ok el, Tempe a u e-dependen den-
si y p o iles o dipola d ople s, Phys. Re . A 99, 013620
(2019).
[53] S. F. Öz ü k, E. Ayba , and M. Ö. Ok el, Tempe a u e depen-
dence o he densi y and exci a ions o dipola d ople s, Phys.
Re . A 102, 033329 (2020).
[54] Y.-C. Zhang and F. Mauche , Va ia ional app oaches o wo-
dimensionally symme y-b oken dipola Bose-Eins ein conden-
sa es, A oms 11, 102 (2023).
[55] L.-J. He, F. Mauche , and Y.-C. Zhang, In a ed cu o o
dipola d ople s, Phys.Re .A110, 053316 (2024).
[56] M. K s aji´
c, P. Juhász, J. Kuˇ
ce a, L. R. Ho e , G. Lamb,
A. L. Ma chan , and R. P. Smi h, Cha ac e iza ion o
h ee-body loss in 166E and op imized p oduc ion o
la ge Bose-Eins ein condensa es, Phys. Re . A 108, 063301
(2023).
[57] A. V. Go shko , P. Rabl, G. Pupillo, A. Micheli, P. Zolle , M. D.
Lukin, and H. P. Büchle , Supp ession o inelas ic collisions
be ween pola molecules wi h a epulsi e shield, Phys. Re .
Le . 101, 073201 (2008).
[58] J. Lin, G. Chen, M. Jin, Z. Shi, F. Deng, W. Zhang, G.
Quéméne , T. Shi, S. Yi, and D. Wang, Mic owa e shielding
o bosonic NaRb molecules, Phys.Re .X13, 031032 (2023).
[59] C. Ka am, R. Vexiau, N. Boulou a-Maa a, O. Dulieu, M.
Lepe s, M. M. z. A. Bo gloh, S. Ospelkaus, and L. Ka pa,
Two-pho on op ical shielding o collisions be ween ul acold
pola molecules, Phys.Re .Res.5, 033074 (2023).
[60] N. Bigagli, C. Wa ne , W. Yuan, S. Zhang, I. S e enson, T.
Ka man, and S. Will, Collisionally s able gas o bosonic dipola
g ound-s a e molecules, Na . Phys. 19, 1579 (2023).
[61] B. Mukhe jee and J. M. Hu son, Con olling collisional loss
and sca e ing leng hs o ul acold dipola molecules wi h s a ic
elec ic ields, Phys. Re . Res. 6, 013145 (2024).
[62] N. Bigagli, W. Yuan, S. Zhang, B. Bula o ic, T. Ka man, I.
S e enson, and S. Will, Obse a ion o Bose–Eins ein conden-
sa ion o dipola molecules, Na u e (London) 631, 289 (2024).
[63] L. Klaus, T. Bland, E. Poli, C. Poli i, G. Lampo esi, E. Caso i,
R. N. Bisse , M. J. Ma k, and F. Fe laino, Obse a ion o o -
ices and o ex s ipes in a dipola condensa e, Na . Phys. 18,
1453 (2022).
[64] E. Poli, T. Bland, S. J. M. Whi e, M. J. Ma k, F. Fe laino, S.
T abucco, and M. Manna elli, Gli ches in o a ing supe solids,
Phys. Re . Le . 131, 223401 (2023).
[65] S. Das and V. W. Sca ola, Un eiling supe solid o de ia o ex
ajec o y co ela ions, a Xi :2407.02481 [Phys. Re . Le . ( o
be published)].
[66] 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, Dilu e dipola quan um d ople s be-
yond he ex ended G oss-Pi ae skii equa ion, Phys.Re .Res.1,
033088 (2019).
[67] Y. He, Z. Chen, H. Zhen, M. Huang, M. K. Pa i , and G.-B.
Jo, Explo ing he Be ezinskii-Kos e li z-Thouless ansi ion in
a wo-dimensional dipola Bose gas, a Xi :2403.18683.
023019-9
