PHYSICAL REVIEW A 108, 033316 (2023)
Supe solid o ma ion in a dipola condensa e by o on ins abili y
Ai o Alaña ,1,2Iñigo L. Egusquiza ,1,2and Michele Modugno 1,2,3
1Depa men o Physics, Uni e si y o he Basque Coun y UPV/EHU, 48080 Bilbao, Spain
2EHU Quan um Cen e , Uni e si y o he Basque Coun y UPV/EHU, 48940 Leioa, Biscay, Spain
3IKERBASQUE, Basque Founda ion o Science, 48013 Bilbao, Spain
(Recei ed 31 May 2023; accep ed 8 Sep embe 2023; published 26 Sep embe 2023)
We cha ac e ize he ole o o on ins abili y in he o ma ion o a supe solid s a e o an elonga ed dipola
condensa e, ollowing a quench o he con ac in e ac ions ac oss he supe luid-supe solid ansi ion, as obse ed
in ecen expe imen s. We pe o m dynamical simula ions by means o he ex ended G oss-Pi ae skii equa-
ion including quan um co ec ions, o di e en inal alues o he s-wa e sca e ing leng h. The co esponding
exci a ion spec um is compu ed using an e ec i e one-dimensional desc ip ion, which p o ides a easonably
accu a e p edic ion o he g ow h a e o he mos uns able mode obse ed in he simula ions. To analyze he
beha io o he sys em, we employ he in e se pa icipa ion a io, which con enien ly cha ac e izes he di e en
deg ee o localiza ion in he supe luid and supe solid phases. By means o a sui able e ec i e ansa z o he
densi y, we de i e a simple ye e ec i e exp ession o he o ma ion ime o he supe solid. This exp ession
p o ides aluable insigh s ega ding i s scaling beha io wi h espec o he s-wa e sca e ing leng h.
DOI: 10.1103/PhysRe A.108.033316
I. INTRODUCTION
Ul acold dipola gases, cha ac e ized by signi ican mag-
ne ic o elec ic dipole momen s, ha e eme ged as a ascina -
ing a ea o esea ch in he ield o cold a oms and quan um
gases. Unlike adi ional Bose-Eins ein condensa es (BECs),
wi h only sho - ange con ac in e ac ions, dipola conden-
sa es exhibi also long- ange, aniso opic in e ac ions (see,
e.g., Re s. [1–3]). No ably, hei exci a ion spec um is cha ac-
e ized by a o onic mode [4–9], which may become uns able
[10,11] and lead o exo ic and complex beha io , including
he o ma ion o d ople s [12] and supe solids [13–16], among
o he s [17–19].
In pa icula , in ecen yea s an in ense esea ch ac i i y
has been di ec ed owa ds un a eling he p ope ies o he
supe solid phase o dipola gases, bo h om he expe imen al
[20–33] and heo e ical poin o iew [34–47]. P oposed in
he pas cen u y [48,49], supe solids a e an exci ing phase o
ma e combining a supe luid na u e [50,51] wi h he ans-
la ional symme y-b eaking cha ac e is ic o solid s uc u es
[52–55]. No ably, he supe solid (SS) phase o ma e can
be achie ed h ough bo h a classical ansi ion om a gas
o a supe solid [30] and a quan um ansi ion om an un-
modula ed supe luid (SF) o a supe solid. Expe imen ally and
heo e ically bo h discon inuous [13,14,22,56–58] and con in-
uous [27,59] ea u es ha e been obse ed, eminiscen o he
i s -o de and second-o de ansi ions p edic ed in he he -
modynamic limi in wo dimensions (2D) and one dimension,
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.
espec i ely. In e es ingly, he e ec i e dimensionali y o he
sys em can be con olled by uning he ans e se con inemen
and he a om numbe , as expe imen ally obse ed by Biagioni
e al. [31] and heo e ically discussed in Re . [44].
A ema kable ea u e o his ansi ion is ha whe eas one
can elax an ini ial SS s a e on o a SF s a e by c ossing he
ansi ion almos adiaba ically, in he opposi e di ec ion he
supe solid equi es a ini e ime o o m [14,26,31,44]. This
co esponds o he ime equi ed o in insic luc ua ions—
associa ed o he o on ins abili y— o g ow up and b eak he
ansla ional in a iance o he sys em. This dis inc i e o onic
spec um o a dipola supe luid was ecen ly obse ed and
cha ac e ized in he g oundb eaking expe imen by Chomaz
e al. in Re . [7].
He e, we p o ide a heo e ical cha ac e iza ion o he
ole o his ins abili y in he dynamics o o ma ion o
a supe solid in a quasi-one-dimensional (quasi-1D) dipola
condensa e, by conside ing a ypical con igu a ion o he
expe imen in Re . [31]. In pa icula , we ocus on he sce-
na io in which he sys em c osses he SF-SS phase ansi ion
by unde going a quench o he con ac in e ac ions which,
o quasi-1D sys ems, can s ill p o ide a smoo h connec ion
be ween an unmodula ed BEC and a supe solid a ay [44]
(unlike wha occu s in 2D [32]). We pe o m a sys ema ic
analysis by means o nume ical simula ions o he ex ended
G oss-Pi ae skii equa ion, conside ing a ious ampli udes o
he in e ac ion quench. The co esponding o on spec um
is compu ed by means o he one-dimensional heo y p o-
posed by Blakie e al. [36], which p o ides a simpli ied ye
accu a e cha ac e iza ion o he simula ion ou comes. Speci i-
cally, we in es iga e he e olu ion o he sys em by analyzing
he in e se pa icipa ion a io (IPR), a aluable measu e o
cha ac e izing he a ying deg ees o localiza ion in he supe -
luid and supe solid phases. By employing an e ec i e ansa z
2469-9926/2023/108(3)/033316(7) 033316-1 Published by he Ame ican Physical Socie y
ALAÑA, EGUSQUIZA, AND MODUGNO PHYSICAL REVIEW A 108, 033316 (2023)
o he densi y, we de i e a simple exp ession con aining a
minimal se o i ing pa ame e s, ha p o ides a eliable
desc ip ion o he IPR beha io in he egime domina ed by
he g ow h o he o on mode. These esul s a e hen used o
discuss he gene al ea u es ha de e mine he o ma ion ime
o he supe solid and i s scaling beha io wi h espec o he
s-wa e sca e ing leng h.
The pape is o ganized as ollows. In Sec. II, we p o ide
an o e iew o he sys em unde conside a ion and b ie ly
summa ize he ele an o mulas de ining he ex ended G oss-
Pi ae skii heo y o dipola condensa es. In Sec. III,we
discuss he gene al p o ocol employed o induce he in e ac-
ion quench ac oss he SF-SS phase ansi ion, p esen ing he
gene al phenomenology obse ed h ough nume ical simula-
ions. In Sec. IV we b ie ly e iew he 1D e ec i e heo y o
Re . [36] used o s udy he exci a ion spec um o an elonga ed
dipola condensa e. The la e is compu ed o ou speci ic
case in Sec. V, whe e i is used o cha ac e ize sys ema i-
cally he ole o he o on ins abili y in he o ma ion o he
supe solid. The e, we also explo e he implica ions o his
ins abili y on he scaling beha io o he supe solid o ma ion
ime wi h espec o he s-wa e sca e ing leng h. Finally, we
p esen a summa y o ou indings and concluding ema ks in
Sec. VI.
II. SYSTEM
In he ollowing analysis, we conside he quasi-1D con-
igu a ion in es iga ed in he expe imen o Re . [31] and
heo e ically analyzed in Re . [44]. I consis s o a dipola
condensa e, a ze o empe a u e, composed by N=3×104
magne ic a oms o 162Dy—wi h unable s-wa e sca e ing
leng h asand dipola leng h add =130a0(a0being he Boh
adius)— apped by a ha monic po en ial wi h equencies
(ωx,ω
y,ω
z)=2π×(15,101,94) Hz. While he choice o
hese pa ame e s is mo i a ed by hei expe imen al easibili y
in a speci ic case, i is wo h no ing ha he ollowing analysis
is concep ually gene al, which enables i s ex ension o o he
scena ios.
This sys em can be desc ibed in e ms o an ex ended
G oss-Pi ae skii (GP) heo y including dipola in e ac ions
[60] and he Lee-Huang-Yang (LHY) co ec ion accoun ing
o quan um luc ua ions, wi hin he local densi y app ox-
ima ion [61–63]. The ene gy unc ional can be w i en as
E=EGP +Edd +ELHY wi h
EGP =¯h2
2m|∇ψ( )|2+V( )n( )+g
2n2( )d ,
Edd =Cdd
2 n( )Vdd( − )n( )d d ,(1)
ELHY =2
5γLHY n5/2( )d ,
whe e EGP =Ek+Eho +Ein is he s anda d GP ene gy
unc ional including he kine ic, po en ial, and con ac in-
e ac ion e ms, V( )=(m/2)α=x,y,zω2
α 2
αis he ha monic
apping po en ial, n( )=|ψ( )|2 ep esen s he conden-
sa e densi y (no malized o he o al numbe o a oms
N), g=4π¯h2as/mis he con ac in e ac ion s eng h, Vdd
( )=(1 −3 cos2θ)/(4π 3) he in e pa icle dipole-dipole
po en ial, Cdd ≡μ0μ2i s s eng h, μ he modulus o he
dipole momen μ, he dis ance be ween he dipoles, and θ
he angle be ween he ec o and he dipole axis, cos θ=μ·
/(μ ). As in Re s. [31,44], we conside he magne ic dipoles
o be aligned along he zdi ec ion by a magne ic ield B.The
LHY coe icien is γLHY =128√π¯h2a5/2
s/(3m)(1 +32
dd/2),
wi h dd =μ0μ2N/(3g).
As discussed in Re s. [31,44], he equilib ium con igu a-
ion o he sys em co esponds o ei he a con en ional SF
s a e o a SS s a e. The ansi ion om one phase o he o he
can be induced by uning he s-wa e sca e ing leng h as.
Fo he p esen alues o he numbe o a oms and apping
equencies, he c i ical poin is loca ed a ac
s≃94.4a0and
he ansi ion has a con inuous cha ac e [31,44].
III. PROTOCOL
In o de o s udy he o ma ion o a supe solid, we adop
he ollowing app oach. The sys em is ini ially p epa ed in
an equilib ium con igu a ion wi hin he SF phase, a a(in)
s=
ac
s+1.5a0. Fo he sake o simplici y, his alue is kep ixed
h oughou his wo k. Then, he condensa e is quenched in he
SS phase by a sudden change o he con ac sca e ing leng h,
o a inal alue a( i)
s=ac
s−δas. In he p esen s udy, δasis
a ied in he ange [1.0,6.5]a0.
The dynamics o he sys em ollowing he quench is ob-
ained by sol ing he GP equa ion [51]
i¯h∂ ψ=δE[ψ,ψ∗]/δψ∗,(2)
whe e he ene gy unc ional E[ψ,ψ∗] is he one in Eq. (1)
[64]. To unco e he physics in ol ed in he o ma ion
p ocess, we simpli y he analysis by excluding dissipa ion
mechanisms and pa icle losses, which a e no essen ial o
he p esen discussion.
A use ul quan i y o cha ac e izing he o ma ion p ocess
o he supe solid is ep esen ed by he IPR, which measu es
he deg ee o localiza ion (high IPR) o sp ead (low IPR) o
a ce ain quan um s a e. Fo a con inuous sys em, i can be
de ined as IPR =|ψ|4d . In he p esen case, i u ns ou
o be p opo ional o he con ac in e ac ion ene gy Ein [see
Eq. (1)],
IPR ∝Ein ( )=g
2n2( , )d .(3)
In o de o acili a e he ongoing discussion, i is con enien
o no malize he abo e exp ession as
¯
Ein ( )≡Ein ( )/Ein (0),(4)
which is also equi alen o no malizing he in e se pa ici-
pa ion a io o i s ini ial alue a =0. The beha io o his
quan i y as a unc ion o he ime elapsed a e he quench is
shown in Fig. 1(a), o di e en alues o δas. I is no ewo hy
ha all he cases p esen ed exhibi he same quali a i e beha -
io , which is exempli ied in Fig. 1(b) o he case δas=2.0a0.
Jus a e he quench, he sys em e ains i s ini ial cha ac-
e o a SF condensa e o a ce ain pe iod o ime, du ing
which i unde goes mos ly b ea hing-like oscilla ions. These
oscilla ions exhibi a consis en pa e n among he di e en
alues o δas, as indica ed by he blue a ows in Fig. 1(a).
033316-2
SUPERSOLID FORMATION IN A DIPOLAR CONDENSATE … PHYSICAL REVIEW A 108, 033316 (2023)
(a)
(b)
FIG. 1. (a) Beha io o ¯
Ein ( ) du ing he e olu ion a e he
quench, o di e en alues o δas anging om 1 a0 o 3 a0.The
ed do s co espond o he alues ¯
E∗
in a which he exponen ial
g ow h sa u a es; he blue a ows he oscilla ions maxima be o e
he o ma ion o he SS pa e n. The ho izon al lines indica e he
equilib ium alue o ¯
ESS
in o he SS g ound s a e a he same alue o
δas(same line ype). (b) The case wi h δas=2a0in de ail. Densi y
plo s (colo scale weigh ed by each densi y dis ibu ion) show he SF
con igu a ion and he SS con igu a ion co esponding o he blue and
ed do s, espec i ely.
A e some ime, he supe solid s uc u e eme ges, ollowing
a sudden inc ease o ¯
Ein . As we shall see la e on, his is a
ypical signa u e o an unde lying o ma ion p ocess d i en
by he o on ins abili y, namely, he exponen ial g ow h o he
co esponding mode, which becomes uns able. In Sec. IV,we
will p o ide a de ailed quan i a i e analysis and cha ac e iza-
ion o his beha io .
I should be emphasized ha he SS s a e gene a ed by
his mechanism can display signi ican de ia ions om he
supe solid g ound s a e a he same alue o as, due o
he highly ou -o -equilib ium na u e o bo h he quench
and he o ma ion p ocess (see also he discussion in
Re s. [7,16,17]). In addi ion, since dissipa ion p ocesses a e
no included in he p esen analysis, he sys em emains in
such a highly exci ed s a e e en long a e he o ma ion o
he supe solid. Howe e , i is wo h no icing ha o su -
icien ly low alues o δas[up o δas=2a0, o he cases
shown in Fig. 1(a)], he alue ¯
E∗
in a which he ins abili y
sa u a es almos coincides wi h he equilib ium alue ¯
ESS
in o
he g ound s a e, indica ed by ho izon al lines in he igu e.
This is no longe he case a highe alues o δas, o which
IPR sa u a ion akes place a lowe alues han IPR o he SS
g ound s a e, ¯
E∗
in <¯
ESS
in . This e ec becomes mo e p ominen
by inc easing δas, owing o he inc ease o nonlinea e ec s by
mo ing deepe in o he SS phase (ha ing s a ed in all cases
in he same ini ial con igu a ion).
In Fig. 1(b), he ime ame du ing which he sys em ex-
hibi s ins abili y is illus a ed as a (blue) shaded a ea be ween
wo consecu i e oscilla ion maxima ep esen ed by blue and
ed do s. The densi y dis ibu ions o he co esponding s a es
a e shown in he inse s. The i s one co esponds o a s a e ha
can be s ill associa ed wi h he SF phase, despi e displaying
a weak densi y modula ion. The o he clea ly co esponds o
a well- o med SS s a e. Conside ing he abo e scena io, he
ime i akes o he supe solid o o m om he ins an o
he quench, deno ed as τSS and named o ma ion ime in wha
ollows, can be he e o e con enien ly de ined by e e ing o
he posi ion o he ed do (s), as illus a ed in he same igu e.
Figu e 1(a) shows ha his o ma ion ime ge s educed by
inc easing δas.
IV. EFFECTIVE 1D DESCRIPTION
To gain a quan i a i e unde s anding o he supe luid o -
ma ion p ocess, we will employ he e ec i e 1D model o an
elonga ed dipola condensa e desc ibed in Re . [36], simpli-
ying he subsequen analysis. Owing o he s ong ans e se
con inemen o he p esen con igu a ion, ωx/ωy,z≈0.15, and
he small asymme y be ween he ans e se apping equen-
cies, ωy/ωz≈1 (see Sec. II), his app oxima ion is expec ed
o be easonably accu a e.
In summa y, his app oach consis s o ac o izing he
condensa e wa e unc ion as ψ( )=ϕ(x)χ(y,z), whe e he
ans e se wa e unc ion χ(y,z) is con enien ly aken as a
Gaussian,
χ(y,z)=(√πl)−1e−(ηy2+z2)/2ηl2,(5)
wi h l=lylz,ly(lz) being he 1/ehal wid h o he ans-
e se densi y along he yaxis (zaxis), and η=ly/lz he
ans e se aniso opy o he densi y dis ibu ion. F om a
Gaussian i o he ans e se p o ile o he ini ial SF densi y
dis ibu ion we ob ain ly≃0.82 μm and lz≃2.19 μm[65].
In eg a ing ou he ans e se di ec ions, one ob ains
an e ec i e 1D GP model whe e he con ac in e ac-
ion s eng h and he LHY coe icien ge eno malized
as g1D =g/(2πlylz), γ(1D)
LHY =γ⊥γLHY, wi h γ⊥=2/5π3/2l3.
The dipole-dipole e ec i e in e ac ion po en ial can be con-
enien ly app oxima ed in momen um space by
˜
V(1D)
dd (q)=1
1+η
μ0μ2
Dy
2πlylzq2eq2Ei(−q2)+2−η
3,(6)
wi h q≡η1/4lk/√2 and k ep esen ing he momen um com-
ponen ela i e o he xaxis.
The desc ip ion can be u he simpli ied by conside -
ing small de ia ions om uni o mi y, wi h linea densi y
n0. The longi udinal wa e unc ion can be exp essed as
ϕ(x, )=√n0+δϕ(x, ), wi h he la e e m ep esen ing a
small pe u ba ion o e he ini ial s a e ϕ0=√n0. The col-
lec i e exci a ions o he condensa e a e hen desc ibed by
he associa ed Bogoliubo –de Gennes equa ions, see, e.g.,
Re s. [7,11,36,60]. By looking o solu ions o he o m
δϕ(x, )∝u(x)e−iω + ∗(x)eiω and conside ing ha o a
uni o m sys em he exci a ions a e plane wa es o momen um
¯hk, namely, uk(x)=Uke−ikx and k(x)=Vke−ikx, one inally
ob ains he ollowing exp ession o he exci a ion ene gies
[36]:
¯hωk=±
¯h2k2
2m¯h2k2
2m+2n0˜
V(k)+3γ(1D)
LHY n3/2
0,(7)
033316-3
ALAÑA, EGUSQUIZA, AND MODUGNO PHYSICAL REVIEW A 108, 033316 (2023)
FIG. 2. Imagina y pa o ωk, as a unc ion o he s-wa e sca -
e ing leng h asand o he exci a ion momen a k. The e ical ( ed)
dashed line indica es he c i ical alue o he sca e ing leng h, ac
s=
94.4a0. The black a ea co esponds o Im[ωk(as)] =0. The (yellow)
do -dashed line ep esen s he posi ion o he o on mode, which
disappea s a ac
s, whe e i is eplaced by he mos uns able mode
(black do -dashed line). The whole line is indica ed as kR(as).
whe e ˜
V(k) deno es he Fou ie ans o m o he in e ac ion
po en ial, see Eq. (6),
˜
V(k)=g1D +˜
V(1D)
dd (η1/4lk/√2).(8)
The abo e spec um is cha ac e ized by a o on exci a ion
( ha is, a local minimum in he exci a ion dispe sion ela ion)
ha so ens o ze o ene gy and becomes dynamically uns able
when he s-wa e sca e ing leng h is uned below a ce ain
c i ical alue ac
s[7,36].
V. STABILITY ANALYSIS
The quasi-1D e ec i e o mula ion p esen ed abo e allows
a s aigh o wa d s abili y analysis o ou sys em a e he
quench. In o de o do so, we need o es ablish a c i e ion o
accoun o he nonuni o mi y o he sys em. In pa icula , we
use he ac ha o a con inuous ansi ion, he c i ical poin
o he SF-SS ansi ion is expec ed o coincide wi h he o on
ins abili y, see Re s. [39,47]. Speci ically, by de ining n(x)≡
n( )dydz,wese n0=cn(0), whe e cis chosen o ep oduce
he c i ical alue o he s-wa e sca e ing leng h ac
s=94.4a0
ob ained om nume ical simula ions (see Sec. III). We ind
c≃0.5.
In Fig. 2we show he imagina y pa o he posi i e b anch
o he equency ωkas a unc ion o he s-wa e sca e ing
leng h asand o he exci a ion momen a k. We ecall ha
he p esence o imagina y equencies in he spec um is
associa ed wi h exponen ially g owing modes ha make he
sys em modula ionally uns able, i hey a e ini ially popula ed.
In he igu e, he e ical ( ed) dashed line co esponds o he
c i ical alue o he sca e ing leng h, ac
s=94.4a0. Abo e ac
s
he spec um is pu ely eal, co esponding o a s able SF s a e.
Below ac
s he equency o some modes becomes imagina y, as
indica ed by he colo ed a ea. This uns able egion b oadens
by dec easing he alue o as. The do -dashed line co esponds
o he posi ion o he mos uns able mode, o as<ac
s, which
connec s o ha o he o on mode, o as>ac
s. We indica e
i s posi ion as kR(as). I is also wo h no ing ha he LHY
co ec ion in Eq. (7) has a negligible con ibu ion o he spec-
um in Fig. 2 o he pa ame e alues a hand.
Supe solid o ma ion
Now ha we ha e de e mined he p ope ies o he ex-
ci a ion spec um, we can e isi he dynamical beha io o
he condensa e a e he quench, discussed in Sec. III, and
examine he o ma ion o he supe solid in e ms o he eme -
gence o exponen ially g owing modes. In pa icula , in o de
o cap u e he essen ial ea u es o he exponen ial g ow h o
he o on mode, we make use o he ollowing ansa z o he
densi y, n(x, )≃n0+nR( ) cos(kRx), wi h nR( )=ηRe2 /τR,
whe e ηR ep esen s he ini ial popula ion o he o on mode,
and τ−1
R≡Im[ωkR]. Such ansa z is phenomenological, p o-
iding a i s app oxima ion owa ds he nonhomogeneous
densi y dis ibu ion. Then, he no malized in e se pa icipa-
ion a io in Eqs. (3) and (4) can be app oxima ed as
¯
Ein ( )≃1
Ln2
0n(x, )2dx
=1+β1
ηR
n0
e2 /τR+β2
η2
R
n2
0
e4 /τR,(9)
whe e Lis he o al leng h o he sys em and he pa ame e s
β1and β2a e nume ical ac o s ha depend on he geome y
o he sys em. Typically, in a uni o m sys em, hei de e mi-
na ion is s aigh o wa d. Howe e , in he p esen case, we
ea hem as ee pa ame e s o accommoda e he inhe en
nonuni o mi y o he sys em. Based on Eq. (9), we in oduce
he ollowing exp ession o i ing he nume ical da a,
( )≡1+2γAe2 /τ +A2e4 /τ ,(10)
whe e we ha e con enien ly eabso bed he e m ηR/n0in o
he de ini ion o A, namely, A≡√β2ηR/n0,γ=β1/(2√β2).
The abo e exp ession will be used o discussing he scaling
beha io o he ins abili y. I depends on he h ee independen
pa ame e s: A,γ, and τ. In his espec , i is impo an o
men ion ha while τis an in insic p ope y o he sys em,
de e mined by Eq. (7), Adepends h ough ηR/n0on he ini ial
popula ion o he uns able modes, namely, on he p epa a ion
o he ini ial s a e, whe eas γis jus a nume ical ac o . To
illus a e he ins abili y beha io , in Fig. 3we conside he
case δas=5a0, as an example. In Fig. 3(a), he e olu ion o
¯
Ein ( ) (see also Fig. 1) is compa ed wi h he i ing unc ion
( )inEq.(10). The i is es ic ed o he shaded a ea, whe e
he ins abili y becomes mani es [66]. Ob iously, he simpli-
ied model in Eq. (10) canno desc ibe he ini ial luc ua ions
in he SF phase, ha s em om he nonuni o m na u e o he
ac ual sys em, no he collec i e oscilla ions o he SS s a e,
which clea ly all ou side he egime o linea exci a ion o
he ini ial s a e. Howe e , i p o ides a clea explana ion o
he wo di e en egimes obse ed a e he quench, be o e
he o ma ion o he supe solid. The ini ial egime, in which
he condensa e e ains i s SF cha ac e , co esponds o he
one in which he popula ion o he uns able modes emains
negligible, namely, ( )≃1. Then, an e iden exponen ial
beha io eme ges, leading o he o ma ion o he supe solid,
a =τSS. A his poin , he sys em has al eady le he linea
033316-4
SUPERSOLID FORMATION IN A DIPOLAR CONDENSATE … PHYSICAL REVIEW A 108, 033316 (2023)
FIG. 3. Time e olu ion o he sys em a e a quench wi h δas=
5a0. (a) Plo o ¯
Ein ( ) ( hick line) along wi h he i ing unc ion in
Eq. (10) ( hin ed line). The g ay a ea indica es he ange in which
he i has been applied (see ex ). (b) Densi y plo o he momen um
dis ibu ion along he xdi ec ion as a unc ion o ime. The ho izon al
(g een) dashed line co esponds o he nominal mos uns able modes
a ±kR,seeFig.2. Side peaks a in ege mul iples o he o me a e
also isible (no ice ha he plo is in loga i hmic scale).
exci a ion egime, and Eq. (10) no longe applies. F om τSS
on, he dynamics o he sys em once again become p edomi-
nan ly d i en by nonlinea e ec s, wi h quan um luc ua ions
being ins umen al o s abilizing he sys em.
The g ow h o uns able modes is clea ly isible in Fig. 3(b),
whe e we show a densi y plo ep esen ing he ime e olu ion
o he momen um dis ibu ion along he xdi ec ion (see also
Re . [7]). The ho izon al (g een) dashed lines co espond o
he nominally mos uns able mode in Fig. 2, namely, he o on
exci a ions a ±kR. No ably, i is in good ag eemen wi h he
esul o he nume ical simula ion (see also Re . [7,13]). The
same analysis can be epea ed o he o he alues o δas
conside ed in Sec. III.InFig.4we compa e he in e se g ow h
a e ex ac ed om he i , τ i , wi h he alue τR=Im[ωkR]−1
co esponding o he mos uns able mode in Fig. 2.Wealso
show, as a ligh shaded a ea, how τRchanges by a ying he
linea densi y n0. As one may expec , he ins abili y mecha-
nism is mo e e ec i e o la ge densi ies. This obse a ion is
consis en wi h he esul s o he GP simula ions, which show
ha he o ma ion o he SS pa e n i s occu s a he cen e o
he condensa e, whe e he densi y is highes , and subsequen ly
sp eads ou wa d owa d he lowe densi y egion in he ail.
O e all, he ag eemen be ween he model p edic ions and
he nume ical da a is ema kably good, indica ing ha he
model cap u es he essen ial physics o he sys em despi e
he inhe en simpli ica ions leading o Eq. (7), such as he
assump ion o uni o mi y.
In Fig. 4we also show he o al o ma ion ime τSS,as
ed squa es. In e es ingly, τSS displays a scaling beha io as
a unc ion o δas ha closely esembles ha o τR, namely,
τSS ≃ατR(δas) (no ice he log scale o he e ical axis), wi h
αbeing a p opo ionali y ac o . In he p esen case we ind
α≃6.5, see he black do ed line in he igu e. Ac ually, he
FIG. 4. Plo o he di e en cha ac e is ic imes τen e ing he
o ma ion o he supe solid. τ i is he in e se g ow h a e o he
ins abili y ex ac ed om a i o ¯
Ein ( ) as in Fig. 3, o be compa ed
wi h τRob ained om he s abili y analysis in Fig. 2. The ligh shaded
a ea ep esen s he a ia ion o he la e upon a a ia ion o ±5% o
he linea densi y n0.τSS is he o ma ion ime o he SS s a e de ined
in Fig. 1. The black do ed line co esponds o ατR(δas), wi h α≃6.5
(see ex ).
igu e shows ha he e mus also be a sligh co ec ion o he
scaling, namely, α=αδas. This can be explained as ollows.
F om he simpli ied model in Eq. (10) i is s aigh o wa d o
exp ess he supe solid o ma ion ime as
τSS =τR
2ln ¯
E∗
in −1+γ2−γ+ln n0
√βηR.(11)
This esul con i ms ha he scaling o τSS as a unc ion o
δasis indeed mos ly de e mined by he beha io o τR(δas),
wi h he o he pa ame e s p o iding loga i hmic co ec ions.
Rega ding he ini ial popula ion ηRo he uns able modes, i
is wo h no ing ha —gi en a ce ain ealiza ion o he ini ial
s a e (and o i s momen um dis ibu ion)—i may sligh ly de-
pend on asdue o he a ying posi ion o he o on momen um
kR(as) (see Fig. 3). Al hough we do no include empe a u e
in ou conside a ions, we signal ha i he e is an inc ease in
he ini ial popula ion a he o on mode because o he mal
exci a ion, hen he o ma ion ime o he supe solid would be
sho ened wi h espec o he T=0 p edic ion, as has been
obse ed in expe imen s [14,31,44]. I is also impo an o
ema k ha he loga i hmic e ms, hough hey only p o ide
a co ec ion o he scaling, a e essen ial o ixing he alue
o he p opo ionali y cons an αδas. In pa icula , we ind ha
he majo con ibu ion comes om he e m ln(n0/ηR)in he
ange o alues o δasconside ed he e.
VI. CONCLUSIONS
We ha e cha ac e ized he ole o he o on ins abili y in
he o ma ion o a supe solid s a e o an elonga ed dipola
condensa e ac oss a con inuous supe luid-supe solid ansi-
ion, as expe imen ally epo ed by Biagioni e al. [31] and
heo e ically in es iga ed in Re . [44]. By means o nume ical
simula ions based on he ex ended G oss-Pi ae skii equa ion,
we ha e sys ema ically analyzed he dynamic beha io o he
sys em a e a quench o he con ac in e ac ions conside ing
a ious ampli udes δaso he in e ac ion quench. We ha e
033316-5
ALAÑA, EGUSQUIZA, AND MODUGNO PHYSICAL REVIEW A 108, 033316 (2023)
also compu ed he co esponding exci a ion spec um by using
he e ec i e one-dimensional app oach o Blakie e al. [36],
inding ha he calcula ed g ow h a es o he uns able o on
modes p o ide an accu a e cha ac e iza ion o he beha io
obse ed in he simula ions.
Ou main inding is ha he scaling beha io o he o -
ma ion ime τSS o a supe solid s a e as a unc ion o δasis
mos ly de e mined by he scaling o he in e se g ow h a e
o he o on mode, namely, τSS(δas)≃αδasτR(δas), wi h αδas
being a p opo ionali y ac o ixed by he ini ial popula ion o
he (mos ) uns able mode. Rema kably, his implies ha he
supe solid o ma ion ime τSS is de e mined by he p ope ies
o he ini ial s a e a he han by he ene gy di e ence be ween
he ini ial supe luid s a e and he inal supe solid s a e, as i
was specula ed in Re s. [14,44]. The p esen analysis—which
admi s a s aigh o wa d ex ension also o o he cases besides
a quench (e.g., by means o a ime escaling as discussed in
Re . [7])—o e s a deepe insigh in o he o ma ion dynamics
o he supe solid s uc u es in dipola condensa es, he eby
p o iding a aluable amewo k o u u e expe imen s and
heo e ical s udies.
ACKNOWLEDGMENTS
This wo k was suppo ed by G an No. PID2021-
126273NB-I00 unded by MCIN/AEI/10.13039/5011000
11033 by “ERDF A way o making Eu ope,” by he Basque
Go e nmen h ough G an No. IT1470-22, and by he
Eu opean Resea ch Council h ough he Ad anced G an “Su-
pe solids” (G an No. 101055319).
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033316-7
