PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
Dynamical o ma ion o quan um d ople s in a 39Kmix u e
G. Fe ioli,1,2G. Semeghini ,3,*S. Te adas-B iansó ,4L. Masi,1,2M. Fa o i,1,2and M. Modugno 5,6
1LENS and Dipa imen o di Fisica e As onomia, Uni e si á di Fi enze, 50019 Ses o Fio en ino, I aly
2CNR Is i u o Nazionale O ica, 50019 Ses o Fio en ino, I aly
3Depa men o Physics, Ha a d Uni e si y, Camb idge, Massachuse s 02138, USA
4Ins i u o de Ciencia de Ma e iales de A agón and Depa amen o de Física de la Ma e ia Condensada,
CSIC–Uni e sidad de Za agoza, 50009 Za agoza, Spain
5Depa amen o de Física Teó ica e His o ia de la Ciencia, Uni e sidad del Pais Vasco UPV/EHU, 48080 Bilbao, Spain
6IKERBASQUE, Basque Founda ion o Science, 48013 Bilbao, Spain
(Recei ed 19 Decembe 2019; accep ed 14 Feb ua y 2020; published 12 Ma ch 2020)
We epo on he dynamical o ma ion o sel -bound quan um d ople s in a ac i e mix u es o 39Ka oms.
Conside ing he expe imen al obse a ions o Semeghini e al. [Phys.Re .Le .120,235301 (2018)], we pe o m
nume ical simula ions o unde s and he ele an p ocesses in ol ed in he o ma ion o a me as able d ople om
an ou -o -equilib ium mix u e. We i s analyze he so-called sel -e apo a ion mechanism, whe e he d ople
dissipa es ene gy by eleasing a oms, and hen we conside he e ec s o losses due o h ee-body ecombina ions
and o he balancing o popula ions in he mix u e. We discuss he impo ance o hese h ee mechanisms in he
obse ed d ople dynamics and hei implica ions o u u e expe imen s.
DOI: 10.1103/PhysRe Resea ch.2.013269
I. INTRODUCTION
Sel -bound d ople s o ul acold a oms we e ecen ly dis-
co e ed as a new exo ic quan um phase [1–4]. Al hough hey
o m in dilu e a omic gases, hey display p ope ies which a e
unique in ha con ex bu common o di e en sys ems such
as classical liquids, helium nanod ople s, o a omic nuclei.
A e hei exis ence in a ac i e bosonic mix u es was he-
o e ically p edic ed in [1], hey we e expe imen ally obse ed
in dipola condensa es [5–8], homonuclea mix u es o 39K
[9,10], and ecen ly in a he e onuclea mix u e o 41Kand
87Rb [11].
The i s pionee ing expe imen s pe o med wi h homonu-
clea mix u es a ICFO (Ins i u de Ciencies Fo oniques)
[9,12] and LENS ( he Eu opean Labo a o y o Non-Linea
Spec oscopy) [10,13] we e able o demons a e he exis ence
o sel -bound d ople s in hese sys ems and o p o ide a i s
cha ac e iza ion o hei peculia ea u es. While hese wo ks
we e mainly de o ed o s udy he d ople s’ equilib ium p op-
e ies, he expe imen epo ed in Re . [10] also showed ha
du ing he o ma ion o he d ople an in e es ing and complex
dynamics akes place. The nonadiaba ic p epa a ion o he
mix u e leads o an ini ial comp ession o he a omic cloud
and o ollowing oscilla ions o i s size, while he p esence
o s ong h ee-body losses (3BL) con inuously d i es he
sys em ou o equilib ium. Du ing his e olu ion, he d ople
*Co esponding au ho : semeghini@ as.ha a d.edu
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.
e en ually eaches a me as able s a e, which was he main
ocus o he in es iga ion pe o med in [10]. In his wo k
we concen a e ins ead on he dynamical e olu ion obse ed
du ing he d ople o ma ion, ying o unde s and he di e en
mechanisms in ol ed and hei ela i e impo ance. While
h ee-body ecombina ions a e a well-known phenomenon in
he ield o ul acold a oms, he o he wo mechanisms playing
a ole in his e olu ion a e speci ic o quan um d ople s and
hus equi e u he a en ion. The i s is ela ed o he need
o adjus he popula ions in he componen s o he mix u e
o balance i s in e ac ion ene gy. The second, he so-called
sel -e apo a ion, is a mo e peculia dissipa ion mechanism
p edic ed in Re . [1]. Calcula ing he exci a ion spec um o
he d ople , Pe o no iced ha , in some speci ic condi ions,
he d ople canno hos any disc e e exci a ion, since all he
exci ed s a es a e highe in ene gy han he pa icle emission
h eshold. This sugges ed he idea ha he d ople could be
able o dissipa e any excess o ene gy by expelling a oms,
om which he e m sel -e apo a ion o igina ed.
In his pape , we conside he speci ic expe imen al case
o Re . [10] and use nume ical simula ions o unde s and how
hese di e en mechanisms come in o play in he e olu ion
o he d ople . In Sec. II we summa ize he condi ions o
he exis ence o sel -bound s a es in a 39Ka omic cloud a
ze o empe a u e. In Sec. III we analyze he phenomenon
o sel -e apo a ion, conside ing he ideal case o a mix u e
wi hou 3BL. We i s s udy he linea egime, whe e he
d ople is p epa ed wi h a small ini ial exci a ion, compa ible
wi h he assump ions o Re . [1], and hen we in es iga e how
he concep o sel -e apo a ion ex ends o he mo e ealis ic
case whe e he mix u e is p epa ed a om equilib ium. In
Sec. IV we in oduce 3BL, hus ully eco e ing he expe i-
men al condi ions o Re . [10], and we analyze he dynamical
e olu ion o he d ople , iden i ying which mechanisms play
2643-1564/2020/2(1)/013269(8) 013269-1 Published by he Ame ican Physical Socie y
G. FERIOLI e al. PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
a key ole in he d ople dynamics. Finally, conclusions a e
d awn in Sec. V.
II. SELF-BOUND DROPLETS IN 39K
The key ing edien o he o ma ion o sel -bound a omic
clouds is he compe i ion be ween a ac i e and epulsi e
o ces which, scaling di e en ly wi h he a omic densi y,
gene a e a binding po en ial [1]. In a mix u e o 39Ka oms
in he hype ine s a es |1,0≡|1and |1,−1≡|2, his
si ua ion occu s in a speci ic ange o magne ic ields B, whe e
he in aspecies sca e ing leng hs a11 and a22 a e posi i e,
while he in e species a12 is nega i e. Fo B<56.85 G, he
quan i yδa≡−|a12|+√a11a22 becomes nega i e, so ha he
global mean- ield (MF) in e ac ion is a ac i e. In his egime,
a epulsi e e ec is p o ided by quan um luc ua ions, co -
esponding o he so-called Lee-Huang-Yang (LHY) ene gy
e m [14], ha s abilizes he sys em agains collapse and gi es
ise o a sel -bound a omic d ople . As de i ed in Re . [1], he
compe i ion be ween he LHY ene gy and he a ac i e MF
e m locks he a io be ween he equilib ium densi ies in he
wo species o
n(0)
1n(0)
2=a22/a11,(1)
wi h
n(0)
i=25π
1024
1
√aii
δa2
a11a22(√a11 +√a22)5.(2)
The expe imen s epo ed in Re s. [9,10] ha e e i ied he
exis ence o hese sel -bound a omic clouds in he p edic ed
in e ac ion egime a he nominal popula ion a io o Eq. (1).
III. SELF-EVAPORATION
As discussed in Re . [1], o small a om numbe s ( ead u -
he o a mo e p ecise de ini ion) he d ople has no collec i e
modes wi h ene gy lowe han he pa icle emission h eshold.
The e o e, no bound collec i e exci a ion can be sus ained
in ha egime, so ha any pe u ba ion o he equilib ium
s a e would esul in a elease o pa icles o in a b eakup
o he d ople in o smalle pieces. In his sense, a quan um
d ople ep esen s a sel -e apo a ing objec . In his sec ion we
use nume ical simula ions o cha ac e ize his phenomenon,
looking a he dynamical e olu ion o an exci ed d ople . We
i s conside he linea egime, whe e he sys em is p epa ed
wi h a small ini ial exci a ion, and hen we ex end ou analysis
o a egime close o he expe imen al condi ions, whe e he
d ople is p epa ed in a highly exci ed s a e.
A. Linea egime
Fo small-ampli ude exci a ions, we can desc ibe he sys-
em wi h a single wa e unc ion φ( , ), neglec ing any pos-
sible ela i e mo ion o he mix u e componen s. We use he
same o malism as [1], which we b ie ly summa ize he e. We
in oduce he escaled spa ial coo dina e ρ= /ξ, wi h
ξ≡384
25π2
a11a22(√a11 +√a22)6
|δa|31/2
,(3)
and he escaled ime τ≡¯h /mξ2. The d ople wa e unc-
ion φ(ρ,τ) e ol es acco ding o he ime-dependen G oss-
Pi ae skii equa ion
i∂τφ=−1
2∇2
ρ−3|φ|2+5
2|φ|3φ, (4)
whe e |φ|2d3ρ=
Nde ines he chemical po en ial ˜μand 
N
is ela ed o he numbe o a oms in he wo a omic species Ni
by
Ni=n(0)
iξ3
N,i=1,2.(5)
I was p edic ed in Re . [1] and con i med by he expe imen s
in Re s. [9,10] ha s able d ople s exis only o 
N>
Nc≈
18.65. The egime o sel -e apo a ion co esponds o 20.1<

N<94.2[1].
In he ollowing, we will es ic ou analysis o he case
o a sphe ically symme ic sys em, which educes Eq. (4)
o an e ec i ely one-dimensional equa ion which depends
on he adial coo dina e ρonly. This assump ion will be
main ained h oughou his wo k, o ease o calcula ions and
concep ual cla i y. No e ha in his app oxima ion he only
possible exci a ion is ha wi h angula momen um =0,
i.e., he monopole mode. Highe angula momen um modes
a e no accoun ed o , so ha he sys em is sel -e apo a ing
up o 
N≃934, whe e he monopole mode een e s in o he
spec um (see Re . [1]).
To s udy he dynamics o an exci ed d ople , we p epa e he
sys em sligh ly ou o equilib ium, by sol ing he s a iona y
e sion o Eq. (4) wi h a ini e ole ance,1so ha he ini ial
wa e unc ion φ(i)
0(ρ) co esponds o an exci ed s a e wi h a
sligh ly la ge ene gy han he ac ual g ound s a e φ0(ρ).2We
hen calcula e he e olu ion o he sys em by sol ing Eq. (4)
wi h he abo e ini ial condi ion. To dis inguish he d ople
om unbound expanding componen s ha may o m du ing
he e olu ion, we de ine a d ople olume as ha con ained
wi hin a ce ain bulk adius Rd(τ), de ined as he posi ion
o he minimum o ¯n(ρ,τ)≡ρ2n(ρ,τ) (see Appendix A o
de ails).3
The e olu ion o he d ople size σ(τ), de ined as he ms
size o he densi y dis ibu ion wi hin he d ople olume, is
shown in Fig. 1. We ind wo di e en beha io s depending on
he alue o 
N. Fo low alues o 
N[in panel (a)], we obse e
a damped oscilla ion wi h dec easing equency ω0(τ). In his
egime, whe e all he d ople exci a ions lie in he con inuum,
he d ople canno hos a bound monopole mode. This means
1The s a iona y solu ions o Eqs. (4)and(9) a e ob ained by means
o imagina y ime e olu ion, using a spli -s ep algo i hm which
includes a C ank-Nicholson p opaga o o he adial coo dina e. A
simila app oach is employed o compu ing he eal ime e olu ion.
See, e.g., [15].
2The g ound s a e o he sys em is compu ed by p opaga ing a ial
wa e unc ion in imagina y ime, un il he a ia ion o he chemical
po en ial in he uni s ep is below a gi en ole ance. An exci ed s a e
can be hen ob ained simply by inc easing he alue o ha ole ance,
he e o e s opping be o ehand he imagina y ime e olu ion.
3We ema k ha he nume ical simula ions o he G oss-Pi ae skii
equa ion a e pe o med on a compu a ional box ha is a leas one
o de o magni ude la ge han Rd.
013269-2
DYNAMICAL FORMATION OF QUANTUM DROPLETS IN A … PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
-2.00
0.00
2.00
4.00 (a)
-4.00
-2.00
0.00
2.00
4.00
0 20 40 60 80 100 120 140 160 180 200
(b)
[σ(τ)−σ0]/σ0(%)
-2.00
0.00
2.00
4.00
τ
-4.00
-2.00
0.00
2.00
4.00
0 20 40 60 80 100 120 140 160 180 200
FIG. 1. E olu ion o he d ople size σ(τ), plo ed in e ms o i s
ela i e a ia ion wi h espec o he equilib ium alue σ0, o small
ini ial exci a ions, o (
N−˜
Nc)1/4≃3.5 (a) and (
N−˜
Nc)1/4≃7
(b) [1]. Pu ple do s co espond o he alues ex ac ed om he
nume ical simula ion, while he g een solid lines a e he esul o
damped sinusoidal i s on sepa a e ime in e als (a) and o a pu e
sinusoidal i (b).
ha when he cloud s a s oscilla ing, pa o he a oms mo e
away om i , hus educing he d ople ene gy by E=
−μN+Ekin, whe e Nis he numbe o a oms lea ing he
d ople and Ekin is hei a e age kine ic ene gy. Fo la ge 
N
ins ead, when he monopole has een e ed in o he spec um
[panel (b)], he size σpe o ms a sinusoidal oscilla ion wi h a
well de ined equency ω0.
To cha ac e ize he la e egime, we i σ(τ) o 
N>940
wi h Acos(ω0τ) o ex ac he monopole equency. Fo lowe

N, ins ead, we i he oscilla ion wi h sepa a e damped cosine
unc ions o he o m Acos(ω0τ)exp(−τ/˜τ) on di e en
in e als o leng h τ =25, and we ex ac he co esponding
equency ω0and damping a e ˜τ.InFig.2(a) we epo
he i ed alues o ω0 o di e en alues o 
Nand we
compa e hem wi h he heo e ical p edic ions o Re . [1] o
ω0(
N) and o he emission h eshold −˜μ.Fo 
N>940 we
ind a e y good ag eemen be ween ou nume ical esul s
and he p edic ion o ω0. In he sel -e apo a ion egime we
see ha he oscilla ion equency dec eases in ime un il i
eaches i s lowe alue, se by −˜μ.InFig.2(b) we plo he
ex ac ed alues o ω0(τ) and ˜τ(τ) om he i s o Fig. 1(a).
In Figs. 2(c) and 2(d) we epo he a ia ion o he pa ame e

Nand o he d ople ene gy Ein he co esponding ime in e -
als, namely 
N(τi)≡
N(τi−τ/2)−
N(τi+τ/2) and
E(τi)≡E(τi−τ/2)−E(τi+τ/2). We obse e ha
he cloud ini ially oscilla es wi h la ge equencies and as
damping a es, associa ed wi h a signi ican elease o a oms
om he d ople in o an unbound cloud. As he d ople ene gy
dec eases due o a om losses, he oscilla ion slows down and
ω0e en ually sa u a es a −˜μ. Close o his h eshold he
a oms ha lea e he d ople ca y away ≃−˜μand hus ha e
negligible kine ic ene gy. The damping o hese small inal
oscilla ions is hen ex emely slow and he d ople eaches
he s a iona y g ound s a e only asymp o ically. Howe e ,
hese esidual exci a ions a e ex emely small and one can
e ec i ely conside ha he d ople has eached i s equi-
0.0
0.1
0.2
0.3
0.4
0.5
012345678
(a)
ω0
(˜
N−˜
Nc)1/4
0.0
0.1
0.2
0.3
0.4
0.5
012345678
0.0
0.2
0.4
(b)
(c)
(d)
0.0
0.4
0.8
0.0
0.1
0 50 100 150
ω0,τ−1
0.0
0.2
0.4
ΔN
0.0
0.4
0.8
ΔE
τ
0.0
0.1
0 50 100 150
FIG. 2. (a) Plo o he monopole equency ω0(ob ained om
he nume ical simula ion) as a unc ion o (
N−
Nc)1/4[1]. In he
sel -e apo a ion egion (shaded a ea) he ange o a ia ion o ω0is
shown by (pu ple) a ows (see Fig. 1). Ou o he sel -e apo a ion
egime, whe e he d ople size pe o ms a sinusoidal oscilla ion
as in Fig. 1(b), he alue o ω0is ep esen ed by solid ci cles.
The lines co espond o he heo e ical p edic ions o Re . [1] o
−˜μ(
N) (g een dashed line) and ˜ω0(
N) ( ed do ed line). (b) Values
o ω0(pu ple ci cles) and 1/˜τ(g een squa es) i ed on consecu i e
ime in e als o leng h τ =25 (indica ed as e o ba s), ex ac ed
om he damped oscilla ion o Fig. 1(a). The black dashed line
co esponds o −˜μ. (c),(d) Va ia ion o he pa ame e 
Nand o he
d ople ene gy Ein each ime in e al (see ex ), espec i ely.
lib ium con igu a ion well be o e he asymp o ic egime is
eached.
B. Nonlinea egime
In he expe imen o Re . [10], he mix u e is p epa ed
a om he equilib ium p o ile φ0o a d ople . Then, o
unde s and he dynamics which akes place du ing he d ople
o ma ion, we ex end he p e ious analysis beyond he linea
case. Since we canno exclude ap io ia di e en beha io o
he wo componen s o he mix u e, he e we eplace Eq. (4)
wi h a sys em o wo coupled gene alized G oss-Pi ae skii
(GP) equa ions [16] including LHY co ec ions [10,11](we
keep he assump ion o sphe ical symme y o simpli y he
discussion):
i¯h∂ ψ1=−¯h2∇2
2m+μ1(n1,n2)ψ1,
i¯h∂ ψ2=−¯h2∇2
2m+μ2(n1,n2)ψ2,
013269-3
G. FERIOLI e al. PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
whe e ψi( ) a e he wa e unc ions o he wo species and he
co esponding chemical po en ials a e gi en by
μi=giini+g12nj+δELHY
δni,i= j(6)
wi h gij =4π¯h2aij/m. The LHY ene gy is [1]
ELHY =8m3/2
15π2¯h3V(g11n1+g22n2)5/2,(7)
so ha
δELHY
δni=32
3√πgii(a11n1+a22n2)3/2.(8)
No e ha a simila model, ob ained by means o a local
densi y app oxima ion, has been used also o he desc ip ion
o dipola quan um d ople s [17–19].
In ou nume ical simula ions we conside a p epa a ion
o he mix u e simila o he one implemen ed in he expe -
imen o Re . [10]. A single species Bose-Eins ein condensa e
(BEC) o Na oms o 39Kin he hype ine le el |2is p epa ed
in he g ound s a e o a ha monic ap wi h apping equency
ω.A =0, N1a oms a e ins an aneously ans e ed o |1,
wi h N2=N−N1a oms emaining in he ini ial s a e. The
ha monic po en ial is hen u ned o o s udy he e olu ion o
he mix u e in ee space. He e, we conside a ypical se o
expe imen al pa ame e s, wi h N=N1+N2=4×105and
sca e ing leng hs a11 ≃69.99a0,a12 ≃−53.37a0, and a22 ≃
34.11a0, which co espond o a Feshbach magne ic ield B=
56.45 G. Fo his se o pa ame e s, we ha e 
N=200, which
lies in he sel -e apo a ion egime iden i ied abo e.
The g ound s a e wa e unc ion ψ0o he ini ial condensa e
is ob ained om he ollowing s a iona y GP equa ion [16]
−¯h2
2m∇2
+1
2mω2 2+g22N|ψ0|2ψ0=μψ0(9)
wi h |ψ0|2d3 ≡1. Conside ing he p epa a ion sequence
desc ibed abo e, we simula e he ins an aneous ans e o
a oms in |1by assuming ha he (no malized) wa e unc ions
o he wo componen s a =0 a e equal o ψ0, wi h he
co esponding densi ies being ni=Ni|ψ0|2. He e we keep he
a io N1/N2 ixed o he nominal a io in Eq. (1). The ollowing
e olu ion is hen ob ained by sol ing Eq. (6).
One can easily guess ha he d ople is minimally exci ed
when he apping equency ωis such ha he ini ial a omic
dis ibu ion is as close as possible o ha o he d ople g ound
s a e o ha alue o 
N, namely |ψ0( )|2≈ξ−3|φ0( /ξ )|2.
To quan i y he di e ence be ween he ini ial p o ile o he
condensa e and ha o he d ople , one can conside , e.g.,
he ela i e de ia ion be ween he co esponding ene gies,
[
N,ω]≡|E(ψ0)−E(φ0)|/E(φ0). Fo he p esen case, we
ind ha [
N=200,ω] is minimized o a apping e-
quency ω/2π≃600 Hz.
We hen s udy he dynamics o he sys em as a unc ion
o he dis ance o he ini ial s a e wi h espec o φ0,by
a ying he ap equency ω. Simila ly o he case discussed
in he p e ious sec ion we obse e ha , a e he mix u e
is o med and eleased in o ee space, he a omic clouds
unde go damped oscilla ions. Fo ω/2π=600 Hz he densi y
o he bina y mix u e smoo hly adap s o a d ople p o ile wi h
036
=0ms
036
=4ms
036
=12ms
2.0
3.0
4.0
5.0
-50
-25
0.4
0.6
0.8
1.0
120
150
180
210
02468101214161820
densi y (a b. uni s)
(μm)
036
(μm)
036
(μm)
036
σ(μm)
2.0
3.0
4.0
5.0
(a)
(b)
(c)
(d)
Ed(a b. uni s)
-50
-25
Nd
1/Nd
2
0.4
0.6
0.8
1.0
NR
(ms)
ω/2π= 200Hz 600Hz
120
150
180
210
02468101214161820
FIG. 3. E olu ion o he d ople wid h σ( ) (a), i s ene gy Ed( )
(b), he a io Nd
1( )/Nd
2( ) (c), and he unning alue o 
NR( ) [(d),
see ex ], o ω/2π=200 Hz (pu ple) and ω/2π=600 Hz (g een).
The inse s in he op ow show he o al densi y o he bina y
mix u e, n( , )=2
i=1Ni|ψi( , )|2, a di e en e olu ion imes,
co esponding o he ed ci cles in (a). As a e e ence, he p o iles
o a d ople wi h 
N=200 (do -dashed line, =0,4 ms), 
N=193
and 
N=159 (do ed and dashed line espec i ely, o =12 ms),
a e also shown in he inse s. In (a), (b) he dashed (do ed) ho izon al
lines ep esen he equilib ium alues o he ms size [in (a)] and he
ene gy o a d ople wi h 
N=193 (159) [in (b)]. The ho izon al line
in (c) ep esen he equilib ium alue N1/N2=√a22/a11 ≃0.698,
and he dashed a ea he co esponding ole ance (see ex ).

N=193 in a ew milliseconds a e he elease om he ap
(see op inse s in Fig. 3). In his case, he ini ial exci a ion en-
e gy is e y small and indeed we obse e a limi ed dynamics.
Rema kably, a d ople is quickly o med e en i he apping
equency di e s signi ican ly om he op imal alue, as in
he case o ω/2π=200 Hz (he e wi h 
N=159), which
highligh s he e iciency o he sel -e apo a ion mechanism o
dissipa e he ini ial exci a ion ene gy.
To cha ac e ize he elaxa ion p ocess owa ds a s a iona y
d ople , he e we indica e wi h Nd
i( )(i=1,2) he numbe o
013269-4
DYNAMICAL FORMATION OF QUANTUM DROPLETS IN A … PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
a oms emaining wi hin he d ople olume and we de ine a
unning alue o 
N om Eq. (5)as

NR( )≡5π2
3√6|δa|5/2
(√a11 +√a22)5Nd
1( )+Nd
2( )
≃0.5×10−32

i=1
Nd
i( ),(10)
whe e he second (app oxima e) equali y holds o he cu en
alues o he sca e ing leng hs. We indica e wi h σ( ) he
d ople size (e alua ed as he a e age ms o he densi y
dis ibu ions o he wo a omic componen s de ined as in he
p e ious sec ion; see Appendix A), and wi h Ed( ) he co e-
sponding ene gy. The e olu ion o σ( ), Ed( ), Nd
1( )/Nd
2( )
and o he unning alue o 
NR( ) a e plo ed in Fig. 3, o
ω/2π=200 Hz and 600 Hz. We ind ha he elaxa ion
p ocess occu s ia a sudden expulsion o a oms (a abou =
2–4 ms, depending on ω), which allows dissipa ion o mos o
he ini ial exci a ion ene gy. F om Fig. 3(b) we can es ima e
an a e age dissipa ion a e, by e alua ing he dec ease in
ene gy occu ing in he i s pa o he e olu ion, un il he
de ia ion om he equilib ium d ople ene gy becomes e y
small. We ind ha i a ies om 0.3 o 2.5 MHz/ms, as ω/2π
goes om 600 o 200 Hz. Figu e 3(c) shows ha he a io
be ween he a om numbe s in he wo componen s always
emains e y close o he nominal equilib ium alue. We ecall
ha a d ople can sus ain an excess o pa icles in one o
he wo componen s δNi/Niup o a c i ical alue ∼|δa|/aii,
beyond which he pa icles in excess a e expelled [1]. He e
he de ia ions o Nd
1( )/Nd
2( ) a e always wi hin his ole ance
[shaded a ea in Fig. 3(c)].
We can conclude ha —in he absence o 3BL— he elax-
a ion dynamics o a bina y mix u e p oduced by means o he
nonadiaba ic expe imen al p o ocol o [10] is domina ed by
he sel -e apo a ion mechanism, consis ing in he dissipa ion
o he ini ial exci a ion ia he elease o pa icles wa e pack-
e s emi ed om he d ople (see also Fig. 5in Appendix A).
IV. DYNAMICS OF THE DROPLET FORMATION
IN THE PRESENCE OF 3BL
Ha ing now a clea idea o how sel -e apo a ion wo ks,
we can discuss he dynamical o ma ion o he d ople in
he ealis ic expe imen al condi ions o Re . [10], whe e a
signi ican ole is played by 3BL. To do his, we use he
same model as in Sec. IIIB, adding o each equa ion in (6)
a nonuni a y e m −i¯hKiii
2n2
iψi,(11)
whe e Kiii a e he in aspecies 3BL a es.4Thei alues
a e K
111/3! =9×10−40 m6/s, K
222/3! =1×10−41 m6/s,
whe e he p imed alues co espond o he loss a es o he -
mal a oms and he ac o 1/3! accoun s o he Bose s a is ics
o condensed a oms [20]. Following Re . [10], we assume ha
4We neglec he e he e ec o in e species 3BL, ollowing he
expe imen al obse a ions discussed in Re s. [9,10].
he wo hype ine le els a e equally popula ed ini ially, N1=
N2=2×105, and we ix he apping equency in Eq. (9)
o a alue o he same o de as he geome ic a e age o he
expe imen al equencies, namely ω=2π×200 Hz.
In he p esen case, owing o he complex dynamics ha
o igina es om he p esence o 3BL, especially in he i s
pa o he e olu ion, we use a di e en s a egy o dis inguish
he d ople om unbound expanding a oms, simila o he
p o ocol implemen ed in [10] o analyze he expe imen al
images. Fi s we ans o m he sphe ically symme ic adial
da a in o a ull h ee-dimensional (3D) da a dis ibu ion, ha is
hen in eg a ed along one di ec ion ( he “imaging di ec ion”)
in o de o ob ain a 2D column densi y dis ibu ion (see
Appendix B). The la e is i ed wi h a double Gaussian unc-
ion (R)=Aexp[−R2/(2σ2)] +Bexp[−R2/(2σ2
exp)], whe e
he i s Gaussian co esponds o he cen al d ople and he
second one o he unbound expanding cloud. He e R ep e-
sen s he adius in he 2D plane.
In Fig. 4we show he e olu ion o he d ople size σ( ), o
he a om numbe in he d ople and in he expanding cloud
su ounding i , o he d ople popula ion a io N1/N2, and
inally o he d ople ene gy. In Fig. 4(e) we compa e he
measu ed d ople ene gy wi h he ene gy o he g ound s a e
o he same a om numbe Eeq(N( )). In he i s s ages o he
e olu ion, he dynamics is domina ed by losses o a oms om
s a e |1, which b ing he a io N1/N2close o he nominal
alue o Eq. (1) and hus signi ican ly educe he d ople
ene gy. These losses come bo h om 3B ecombina ions and
om he elease o a oms om he d ople in o he unbound
componen , due o he imbalance in he in e ac ion ene gies.
In his s age, he sys em dissipa es he excess ene gy p e y
quickly: We measu e an ene gy loss a e o abou 20 MHz/ms.
No ice ha his is almos 10 imes la ge han he sel -
e apo a ion cooling a e measu ed in he p e ious sec ion o
he same ini ial ω. When he a io N1/N2 eaches i s nominal
alue, he d ople keeps losing a oms in |1due o 3BL. To
compensa e o ha , he d ople s a s eleasing a oms in |2.
This popula ion dynamics, oge he wi h he comp ession o
he a omic cloud, causes he wo appa en bumps in Fig. 4(e).
A his poin we ind he sys em close o i s equilib ium
con igu a ion, wi h he cen al cloud o ming a me as able
d ople wi h an a om numbe signi ican ly smalle han he
ini ial one. Since he sys em keeps losing a oms, due o 3BL in
|1and o popula ion eequilib a ion in |2, we see ha a ound
=12 ms he d ople popula ion d ops below he c i ical
alue 
Nc[1,9,10], so ha he binding mechanism b eaks and
he cloud s a s expanding.
F om his analysis we can conclude ha he ene gy a i-
a ions d i en by 3BL and he co esponding popula ion bal-
ancing occu on imescales much sho e han hose ypical o
he sel -e apo a ion mechanism desc ibed abo e, so ha i is
ha d o de e mine i he la e occu s a all. E en i ha we e
he case, i s e ec would be negligible wi h espec o he wo
leading loss mechanisms.
We ha e also in es iga ed whe he a di e en choice o
he ini ial pa ame e s, such as he apping equency o he
popula ion a io N1/N2, would lead o a as e equilib a ion
ime and/o o longe d ople li e imes. We ind ha he alues
o ωand N1/N2discussed abo e, which co espond o he
alues chosen in he expe imen [10] by op imizing he d ople
013269-5

G. FERIOLI e al. PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0
50
100
150
200
250
0
40
80
120
160
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-60
-40
-20
0
20
02468101214161820
σ(μm)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
(a)
(b)
(c)
(d)
(e)
N(103)
0
50
100
150
200
250
N3BL (103)
0
40
80
120
160
N1/N2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
E(MHz)
(ms)
-60
-40
-20
0
20
02468101214161820
FIG. 4. Nume ical esul s o he dynamical o ma ion o he
d ople in p esence o s ong 3BL. (a) E olu ion o he d ople size
σ( ), (b) o he numbe o a oms in he d ople (solid) and in he
unbound cloud (dashed), (c) o he numbe o a oms los due o
3BL, (d) o he popula ion a io in he d ople N1/N2,and(e)o he
d ople ene gy. In (b),(c) we plo he a om numbe s sepa a ely o he
wo a omic species |1(black) and |2(b own). In (d) we compa e
he popula ion a io wi h he nominal ange whe e he in e ac ions
ene gies in he d ople a e p ope ly compensa ed, N1/N2≃√a22/a11
(shaded a ea) [1]. In (e) he d ople ene gy is shown along wi h he
g ound s a e ene gy o he co esponding a om numbe Eeq(N( ))
(black dashed line).
li e ime, ep esen indeed a con enien choice. In ac , we
obse e ha a la ge alue o he ini ial apping equency ω
leads o s onge 3BL in he i s s ages o he e olu ion, hus
educing e en u he he d ople li e ime. On he o he hand, a
a io N1/N2close o i s equilib ium alue √a22/a11 does no
help ei he , because he s onge losses in |1quickly d i e
he sys em away om his equilib ium condi ion, igge ing
an ea lie elease o a oms om s a e |2.
To s udy expe imen ally he e ec o sel -e apo a ion, one
would need o ind a sui able a ac i e mix u e wi h signi i-
can ly lowe 3BL. This migh be he case o he 41K-87Rb
mix u e o he ecen expe imen in Re . [11], whe e hanks o
he di e en alues o he sca e ing leng hs he d ople o ms
a lowe densi ies, so ha 3BL a e less e ec i e. Indeed, he e
he li e ime is expec ed o be one o de o magni ude la ge
han ha o he 39Kmix u e conside ed he e, a leas . In his
case, as discussed in Sec. IIIB, i is p e e able o p epa e
he ini ial s a e a he nominal equilib ium a io N1/N2and as
close as possible o he d ople densi y p o ile, o ha e sho
equilib a ion imes and limi ed o e all dynamics (see Fig. 2).
Rema kably, i he d ople is p epa ed in he sel -e apo a ion
egime, any unwan ed exci a ion would be hen e icien ly
dissipa ed ia ha mechanism.
V. CONCLUSIONS
We ha e cha ac e ized, by means o nume ical simula-
ions, he dynamics o sel -e apo a ion o an a omic quan um
d ople , conside ing he cases o bo h small and la ge ini ial
exci a ions. We ha e e i ied ha he egime o i s occu ence
co esponds o ha p edic ed in Re . [1], and we ha e dis-
cussed how and on which imescales his mechanism allows
he dissipa ion o ene gy in he d ople . We ha e simula ed
he same p epa a ion o he d ople ealized in Re . [10],
including he e ec o 3BL. This has allowed us o iden i y
he ele an mechanisms in ol ed in he dynamical o ma ion
o a sel -bound d ople o 39K. In his case, he e olu ion o
he sys em is domina ed by he p esence o 3BL and by a
con inuous elease o a oms o es o e he p ope popula ion
a io, whe eas he sel -e apo a ion mechanism plays only a
negligible ole, i any. Finally, we ha e discussed he op imal
s a egies o he p epa a ion o d ople s close o hei g ound
s a e, bo h o he expe imen al case conside ed he e and o
possible u u e expe imen s wi h educed 3BL. The esul s
epo ed he e, which p o ide an accu a e cha ac e iza ion o
he dynamical e olu ion o exci ed d ople s, a e bo h use ul
o a deepe unde s anding o he ecen expe imen al obse a-
ions o quan um d ople s and ele an o u u e expe imen al
s udies, especially ega ding he e ec o sel -e apo a ion.
ACKNOWLEDGMENTS
We acknowledge insigh ul discussions wi h D. Pe o
and E. She man. This wo k was suppo ed by he Spanish
Minis y o Science, Inno a ion and Uni e si ies and he
Eu opean Regional De elopmen Fund FEDER h ough G an
No. PGC2018-101355-B-I00 (MCIU/AEI/FEDER, UE), by
he Basque Go e nmen h ough G an No. IT986-16, by
he EC-H2020 G an QUIC No. 641122, and by he p ojec
TAIOL o Quan ERA ERA-NET Co und in Quan um Tech-
nologies (G an Ag eemen No. 731473) implemen ed wi hin
he Eu opean Union’s Ho izon 2020 P og amme.
APPENDIX A: DEFINING THE DROPLET RADIUS
The dynamical o ma ion o s a iona y (o me as able)
d ople s occu s ia a elaxa ion p ocess du ing which some
o he a oms a e expelled om he cen al olume. The e o e,
a c i e ion is needed o dis inguishing be ween he d ople
and he ou going densi y wa es. He e we exploi he adial
symme y o he p esen p oblem and we conside —along
wi h he p ope densi y dis ibu ion n( , ) in adial coo di-
na es [Figs. 5(a) and 5(b)]— he quan i y ¯n( , )≡ 2n( , ),
which includes he ac o 2co esponding o he Jacobian
de e minan in sphe ical coo dina es [Figs. 5(c) and 5(d)].
The e o e, a linea in eg a ion o ¯n( , ) co esponds o a
olume in eg a ion o he densi y n( , ). This ep esen a ion
also p o ides a con enien way o isualize he poin a which
he ou going densi y wa es “de ach” om he inne olume,
013269-6
DYNAMICAL FORMATION OF QUANTUM DROPLETS IN A … PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
(a) (b)
(c) (d)
=4ms =12ms
0 5 10 15 20 25 30
=4ms
0 5 10 15 20 25 30
=12ms
densi y (a b. uni s)
(µm)
051015202530
(µm)
0 5 10 15 20 25 30
FIG. 5. Plo o he o al densi y n( , )=2
i=1Ni|ψi( , )|2
(a),(c) and o he quan i y ¯n( , )≡ 2n( , ), a di e en e olu ion
imes ( =4,12 ms; he e o he case ω/2π=200 Hz o he
simula ions discussed in Sec. IIIB). The abscissa o he ed do (see
ex ) ep esen s he alue o he d ople adius Rd( ).
which we iden i y wi h he i s minimum o ¯n( , )a he
igh -hand side o he bulk, ma ked by a ed do in Figs. 5(c)
and 5(d). The abscissa o his poin is he alue ha we iden-
i y wi h he d ople adius Rd( ). In he nume ical simula ions
o Sec. IIIB, o example, he d ople adius is ini ialized
a Rd(0) =10 μm and i hen e ol es—bo h con inuously
o ia sudden jumps— ollowing he posi ion o he le mos
minima (ou side o he o igin). The la e is ypically ound
be ween 5 and 10 μm du ing he whole e olu ion conside ed
he e (up o 20 ms), co esponding o he ac ha he bulk
hos s a b ea hing mode and i does no expand. Con a ily,
he densi y wa es which a e expelled om he d ople mo e
ou wa d while expanding, as shown in Figs. 5(c) and 5(d).
APPENDIX B: EXTRACTING THE 2D COLUMN DENSITY
In he p esence o 3BL he dynamics o he sys em is
complica ed by he ac ha he amoun o unbound expanding
a oms is o he same o de as hose emaining in he cen al
d ople . This beha io makes i di icul o iden i y he d ople
adius di ec ly om he densi y dis ibu ion in adial coo di-
na es n( ). The e o e, we in eg a e n( ) along one di ec ion
(z) in o de o ob ain a 2D densi y dis ibu ion,
n2D(R)=+∞
−∞
n(x2+y2+z2)dz,(B1)
whe e R=x2+y2is he adial coo dina e in he (x,y)
plane. In o de o sol e he in eg al, i is ad an ageous o make
he change o a iables z2= 2−R2, ha allows one o w i e
Eq. (B1)as
n2D(R)=2+∞
Rn( ) d
√ 2−R2,(B2)
whe e /√ 2−R2co esponds o he Jacobian de e minan o
he ans o ma ion and he ac o 2 akes in o accoun he ac
ha he in eg al o e zcan be w i en as wo imes he in eg al
om ze o o in ini y. The in eg al (B2) exhibi s a singula i y
wi h a loga i hmic di e gence a =R. In o de o emo e
his singula i y we di ide Eq. (B2) in wo pa s:
n2D(R)=2R+
R
n( ) d
√ 2−R2++∞
R+
n( ) d
√ 2−R2,(B3)
wi h  being he uni s ep o he (nume ical) adial g id.
Fo su icien ly small  , we can assume n( ) cons an in he
i s e m in Eq. (B3) and i can be mo ed ou o he in e-
g a ion sign. The e o e, he emaining in eg al can be sol ed
analy ically and, inally, i is possible o ob ain he 2D column
densi y dis ibu ion n2D(R) s a ing om he 3D sphe ically
symme ic adial da a n( ), exploi ing he exp ession
n2D(R)=2n(R)(2R + 2)++∞
R+ n( ) d
√ 2−R2.
(B4)
[1]D.S.Pe o ,Phys. Re . Le . 115,155302 (2015).
[2]D.S.Pe o ,Na . Phys. 14,211 (2018).
[3] I. Fe ie -Ba bu and T. P au, Science 359,274 (2018).
[4] I. Fe ie -Ba bu , Phys. Today 72(4), 46 (2019).
[5] I. Fe ie -Ba bu , H. Kadau, M. Schmi , M. Wenzel, and T.
P au, Phys.Re .Le .116,215301 (2016).
[6] M. Schmi , M. Wenzel, F. Bö che , I. Fe ie -Ba bu , and T.
P au, Na u e (London) 539,259 (2016).
[7] L. Chomaz, S. Baie , D. Pe e , M. J. Ma k, F. Wäch le ,
L. San os, and F. Fe laino, Phys. Re . X 6,041039
(2016).
[8] L. Tanzi, E. Lucioni, F. Famà, J. Ca ani, A. Fio e i, C.
Gabbanini, R. N. Bisse , L. San os, and G. Modugno, Phys. Re .
Le . 122,130405 (2019).
[9] C. Cab e a, L. Tanzi, J. Sanz, B. Naylo , P. Thomas, P. Cheiney,
and L. Ta uell, Science 359,301 (2018).
[10] G. Semeghini, G. Fe ioli, L. Masi, C. Mazzinghi, L. Wolswijk,
F. Mina di, M. Modugno, G. Modugno, M. Inguscio, and M.
Fa o i, Phys. Re . Le . 120,235301 (2018).
[11] C. D’E ico, A. Bu chian i, M. P e edelli, L. Salasnich, F.
Ancilo o, M. Modugno, F. Mina di, and C. Fo , Phys. Re .
Resea ch 1,033155 (2019).
[12] P. Cheiney, C. R. Cab e a, J. Sanz, B. Naylo , L. Tanzi, and
L. Ta uell, Phys. Re . Le . 120,135301 (2018).
[13] G. Fe ioli, G. Semeghini, L. Masi, G. Gius i, G. Modugno, M.
Inguscio, A. Gallemi, A. Reca i, and M. Fa o i, Phys.Re .Le .
122,090401 (2019).
[14] T. D. Lee, K. Huang, and C. N. Yang, Phys. Re . 106,1135
(1957).
[15] W. H. P ess, S. A. Teukolsky, W. T. Ve e ling, and B. P.
Flanne y, Nume ical Recipes: The A o Scien i ic Compu ing,
2nd ed. (Camb idge Uni e si y P ess, New Yo k, 1996).
013269-7
G. FERIOLI e al. PHYSICAL REVIEW RESEARCH 2, 013269 (2020)
[16] F. Dal o o, S. Gio gini, and S. S inga i, Re . Mod. Phys. 71,
463 (1999).
[17] F. Wäch le and L. San os, Phys. Re . A 93,061603(R)
(2016).
[18] F. Wäch le and L. San os, Phys. Re . A 94,043618 (2016).
[19] D. Baillie, R. M. Wilson, R. N. Bisse , and P. B. Blakie, Phys.
Re . A 94,021602(R) (2016).
[20] P. A. Al in, G. R. Dennis, G. D. McDonald, D. Dö ing, J. E.
Debs, J. D. Close, C. M. Sa age, and N. P. Robins, Phys. Re .
A84,033632 (2011).
013269-8