Dynamical formation of quantum droplets in a K-39 mixture

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. 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