Beating dark-dark solitons in Bose-Einstein condensates

Bea ing da k-da k soli ons in Bose-Eins ein condensa es D. Yan,1J.J. Chang,2C. Hamne ,2M. Hoe e ,3P.G. Ke ekidis,1P. Engels,2V. Achilleos,4D.J. F an zeskakis,4and J. Cue as5 1Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , Massachuse s 01003-4515, USA 2Washing on S a e Uni e si y, Depa men o Physics & As onomy, Pullman, Washing on 99164, USA 3No h Ca olina S a e Uni e si y, Depa men o Ma hema ics and S a is ics, Raleigh, No h Ca olina 27695, USA 4Depa men o Physics, Uni e si y o A hens, Panepis imiopolis, Zog a os, A hens 157 84, G eece 5Nonlinea Physics G oup. Escuela Poli ´ ecnica Supe io . Depa amen o de F´ ısica Aplicada I. Uni e sidad de Se illa. C/ Vi gen de ´ A ica, 7. 41011-Se illa (Spain) Mo i a ed by ecen expe imen al esul s, we s udy bea ing da k-da k soli ons as a p o o ypical cohe en s uc u e ha eme ges in wo-componen Bose-Eins ein condensa es. We showcase hei connec ion o da k- b igh soli ons ia SO(2) o a ion, and in e om i bo h hei in insic bea ing equency and hei equency o oscilla ion inside a pa abolic ap. We iden i y hem as exac pe iodic o bi s in he Manako limi o equal in e - and in a-species nonlinea i y s eng hs wi h and wi hou he ap and showcase he pe sis ence o such s a es upon weak de ia ions om his limi . We also conside la ge de ia ions om he Manako limi illus a ing ha his b ea hing s a e may be b oken apa in o da k-an ida k soli on s a es. Finally, we conside he dynamics and in e ac ions o wo bea ing da k-da k soli ons in he absence and in he p esence o he ap, in e ing hei ypically epulsi e in e ac ion. I. INTRODUCTION One o he p incipal hemes o s udy in he eme ging ield o a omic Bose-Eins ein condensa es (BECs) is he examina ion o he cohe en s uc u es ha a ise in hem [1–4]. When such explo a ions s a ed o e a decade ago [5–9], hey we e conside ably hinde ed by ei he geome ic o he mal e ec s, which we e de imen al owa ds he li e ime o da k soli ons and o ices ha can be o med in epulsi e BECs. Ye , he newe gene a ions o expe imen s ha e enabled conside able s ides owa ds he obse a ion o dynamics and in e ac ions o such nonlinea wa e o ms [10–15]. In addi ion o he abo e con ex o single-componen BECs, soli on and o ex s a es may also a ise in mul i-componen condensa es, such as he wo-componen pseudo-spino BECs, o he h ee and highe componen spino BECs [1, 2, 4]. One DB DB DD DB DB DD DD Densi y [a.u.] 0x[mic ons] 100 200 DB DB DD DB 2,2 1,1 2,2 1,1 2,2 - 1,1 - DB DB DD DB DB DD DD Densi y [a.u.] 0x[mic ons] 100 200 DB DB DD DB 2,2 1,1 2,2 1,1 2,2 - 1,1 - FIG. 1: (Colo online) P o o ypical expe imen al images o da k-b igh and da k-da k soli ons in a wo-componen BEC. The wo componen s a e e ically o se o sepa a e imaging. All dynamics occu wi h e ically o e lapped componen s be o e he imaging p ocedu e. Clea examples o da k-b igh and da k-da k soli ons a e ma ked as DB and DD espec i ely. In he ou h panel, he ed ( hick) line shows a adially in eg a ed c oss sec ion o he uppe componen in he boxed egion o he hi d panel, while he black ( hin) line shows he c oss sec ion o he lowe componen . The |F, mF⟩hype ine s a es used o hese images a e gi en o he igh o each componen . 2 o he p o o ypical examples o a soli on s a e in hese se ings is he so-called da k-b igh (DB) soli on [16, 17]. Expe imen al images o DB soli ons in a wo-componen BEC a e p esen ed in Fig. 1. The BEC in his igu e is comp ised o wo di e en hype ine s a es o 87Rb, and he soli ons a e gene a ed by subjec ing he BEC o in e -componen coun e low; de ails o his echnique a e desc ibed in Re s. [18, 19]. In each panel, he a om clouds o he wo componen s a e e ically o se o imaging only, while all he dynamics leading o he soli on o ma ion occu s in o e lapped clouds. Clea examples o da k- b igh soli ons a e ma ked as “DB” in he igu e and hey consis o a da k soli on in one componen ha is coupled o a b igh soli on in he second componen . These s uc u es can be hough o as “symbio ic” (o e en pa asi ic) s a es because hei b igh componen canno be suppo ed alone in he case o epulsi e in e ac ions [3]; in ac , he b igh soli on is only sus ained because o he p esence o i s da k-coun e pa , which ope a es as an ex e nal apping po en ial. Al hough da k-b igh soli ons (and e en a p o o ypical in e ac ion he eo ) we e i s obse ed some ime ago in he con ex o nonlinea op ics [20, 21], hei obse a ion in ecen a omic BECs expe imen s [10] igge ed a sizeable bu s o esea ch ac i i y cen e ed a ound hem. Topics o s udy included (bu we e no limi ed o) mul i-DB soli on solu ions om he iewpoin o in eg able sys ems [22], nume ical s udy o DB soli on in e ac ions [23], disc e e DB soli ons [24], expe imen al ealiza ions o DB soli on ains [18], DB soli on oscilla ions and in e ac ions [25, 26], as well as in e ac ion o DB soli ons wi h localized impu i ies [27]. Recen ly, a “cousin” o hese DB soli ons, namely he da k-da k (DD) soli on — which in ol es wo da k soli ons bu wi h po en ially a b ea hing oscilla ion be ween hei densi ies was also expe imen ally obse ed [19]. Pe inen examples a e ma ked as “DD” in Fig. 1. These soli ons show in e es ing dynamics in which hey pe iodically change hei o m, om he one shown in he i s panel o he one shown in he second panel, and back (no e he o de o he hump/no ch ea u es in each o he DD’s componen ; see also Fig. 5 below). Such “bea ing da k-da k soli ons” a e expec ed o eme ge in he in eg able wo-componen (so-called Manako ) limi o he ele an mean- ield heo e ic models [28] and we e, in ac , ea lie obse ed in nume ical expe imen s in ol ing he d agging o de ec s h ough he bina y condensa es [29]. The cu en expe imen al ad ances, such as he ones leading o he soli on images o Fig. 1, mo i a e he p esen heo e ical s udy, in which we e isi DD soli on s a es a he in eg able Manako limi and ex ac in o ma ion om hei connec ion o he DB soli ons (sec ion II). These esul s a e co obo a ed by he iden i ica ion o such single DD soli on s a es, as genuine pe iodic o bi s o he Manako case (wi h and wi hou a ap) and he s udy o hei s abili y, in e nal modes and associa ed nea -equilib ium dynamics (sec ion III). In addi ion, we examine he dynamics o indi idual such soli ons, upon depa u e om he in eg able limi (sec ion IV). Expe imen ally i has also become possible o gene a e se e al soli ons, and e en soli ons o di e en ypes, in a single BEC – see, e.g., he hi d panel o Fig. 1 which demons a es he coexis ence o da k-b igh and da k-da k soli ons. Al hough he expe imen ally exploi ed coun e low be ween he wo componen s is beyond he scope o ou cu en analysis, hese expe imen al indings mo i a e ou in es iga ion o he in e ac ions be ween wo da k-da k soli ons (sec ion V). Finally, conclusions o ou s udy, as well as a numbe o in e es ing pe spec i es o u u e wo k, a e also p esen ed (sec ion VI). II. DARK-BRIGHT AND DARK-DARK SOLITONS: THEORETICAL BACKGROUND We conside a wo-componen elonga ed (along he x-di ec ion) epulsi e BEC, composed o wo di e en hype ine s a es o he same alkali iso ope. In he case o a highly aniso opic ap (i.e., i he longi udinal and ans e se apping equencies a e such ha ωx≪ω⊥), his sys em can be desc ibed by wo coupled G oss-Pi ae skii equa ions (GPEs) o he o m [1]: i~∂ ψj=(−~2 2m∂2 xψj+V(x)−µj+ 2 ∑ k=1 gjk|ψk|2)ψj.(1) He e, ψj(x, )(j= 1,2) deno e he mean- ield wa e unc ions o he wo componen s (no malized o he numbe s o a oms Nj=∫+∞ −∞ |ψj|2dx), mis he a omic mass, and µja e he chemical po en ials; u he mo e, gjk = 2~ω⊥ajk a e he e ec i e one-dimensional (1D) coupling cons an s, ajk deno e he h ee s-wa e sca e ing leng hs (no e ha a12 =a21) which accoun o collisions be ween a oms belonging o he same (ajj) o di e en (ajk, j =k) species, while V(x) = (1/2)mω2 xx2is he ex e nal apping po en ial. Le us now assume ha he wo-componen BEC unde conside a ion consis s o wo di e en hype ine s a es o 87Rb, such as he s a es |1,−1⟩and |2,1⟩used in he expe imen o Re . [30], o he s a es |1,−1⟩and |2,−2⟩used in he expe imen s o Re s. [18, 19, 25]. In he i s case he sca e ing leng hs ake he alues a11 = 100.4a0,a12 = 97.66a0and a22 = 95.00a0, while in he second case he espec i e alues a e a11 = 100.4a0,a12 = 98.98a0and a22 = 98.98a0(whe e a0is he Boh adius). In ei he case, i is clea ha he sca e ing leng hs ake app oxima ely he same alues, say aij ≈a. This way, measu ing he densi ies |ψj|2, leng h, ime and ene gy in uni s o 2a,a⊥=√~/ω⊥,ω−1 ⊥and ~ω⊥, espec i ely, we may cas Eqs. (1) in o 3 he ollowing dimensionless o m, i∂ u1=−1 2∂2 xu1+V(x)u1+ (|u1|2+|u2|2−µ)u1,(2) i∂ u2=−1 2∂2 xu2+V(x)u2+ (|u1|2+|u2|2−µ)u2,(3) whe e we ha e also assumed ha he chemical po en ials cha ac e izing each componen a e equal. No e ha he po en ial in Eqs. (2)-(3) is now gi en by V(x) = (1/2)Ω2x2, whe e Ω = ωx/ω⊥is a na u al small pa ame e o he sys em. The abo e sys em o Eqs. (2)-(3) is in a ian unde SU(2) o a ions [28]. In pa icula , le us i s ecall ha a gene al ma ix elemen o SU(2) akes he o m U=(α−β∗ β α∗), whe e αand βa e complex cons an s such ha |α|2+|β|2= 1. Then, i can be shown ha i (u1, u2)Ta e solu ions o Eqs. (2)-(3), hen, (u′ 1 u′ 2)≡U(u1 u2)=(αu1−β∗u2 βu1+α∗u2), a e also solu ions o Eqs. (2)-(3). This sugges s ha we may s a om he exac da k-b igh (DB) soli on solu ion (which exis s in he absence o he po en ial) and, de i e he bea ing da k-da k (DD) soli on solu ion. Mo e speci ically, in he absence o he ex e nal po en ial (V(x) = 0), and o he bounda y condi ions |u1|2→µand |u2|2→0as |x| → ∞, Eqs. (2)-(3) possess an exac analy ical single DB soli on solu ion o he ollowing o m: u1(x, ) = √µ{cos ϕ anh ξ+isin ϕ},(4) u2(x, ) = ηsech ξexp{ikx +iθ( )},(5) whe e ξ=D(x−x0( )),ϕis he da k soli on’s phase angle, cos ϕand η ep esen he ampli ude o he da k and b igh soli ons, and Dand x0( )a e associa ed wi h he in e se wid h and he cen e posi ion o he DB soli on. Fu he mo e, k=D an ϕand θ( )a e he (cons an ) wa enumbe and phase o he b igh soli on, espec i ely. The abo e pa ame e s o he DB soli on a e connec ed h ough he ollowing equa ions: D2=µcos2ϕ−η2,(6) ˙x0=k=D an ϕ, (7) ˙ θ=1 2(D2−k2),(8) wi h ˙x0and ˙ θdeno ing he DB soli on eloci y and angula equency, espec i ely (o e do s deno e ime de i a i es). Thus, he DB soli on (4), (5) is cha ac e ized by h ee ee pa ame e s (se en pa ame e s µ,ϕ,η,k,D,˙x0,˙ θand ou cons ain s (6)-(8)). No ice ha he ampli ude ηo he b igh soli on, he chemical po en ial µo he da k soli on, as well as he (in e se) wid h pa ame e Do he DB soli on a e connec ed o he numbe o a oms NBo he b igh soli on by means o he ollowing equa ion: NB≡∫R|u2|2dx=2√µη2 D.(9) Acco ding o he abo e a gumen s, one may s a om he DB soli on and cons uc SU(2) o a ed solu ions, in he ollowing o m: u1(x, ) = α√µ{cos ϕ anh ξ+isin ϕ}−β∗ηsech ξexp{ikx +iθ( )},(10) u2(x, ) = β√µ{cos ϕ anh ξ+isin ϕ}+α∗ηsech ξexp{ikx +iθ( )}.(11) Wi h he addi ional ou pa ame e s α,β∈Cand he cons ain |α|2+|β|2= 1, he solu ion (10)-(11) is cha ac e ized by six ee pa ame e s. In oducing a new pa ame e c, he eloci y o he backg ound luid, ano he solu ion can be cons uc ed om Eqs. (10)-(11) ia a Galilean boos : exp[i(cx −c2 /2)]u1,2(x−c , ). Thus, in he mos gene al case, his DD soli on solu ion is cha ac e ized by se en ee pa ame e s. One na u al se o pa ame e s can be ound om he a - ield, |x| → ∞ beha io consis ing o wo densi ies, an o e all luid eloci y, and ou phases. 4 Due o Galilean in a iance and phase in a iance, u′ j(x, ) = eiφjuj(x, ), we will assume, wi hou loss o gene ali y, ha he backg ound is a es (c= 0) and ocus, mo e speci ically, on he case o he SO(2) o a ed DB soli on. In his case, he co esponding o hogonal ma ix is gi en by: U=(cos(χ)−sin(χ) sin(χ) cos(χ)),(12) whe e χis an a bi a y angle. This way, he ele an SO(2) o a ed soli on solu ion akes he o m: u1(x, ) = cos(χ)√µ{cos ϕ anh(D(x−x0( ))) + isin ϕ}−sin(χ)ηsech(D(x−x0( ))) exp{ikx +iθ( )},(13) u2(x, ) = sin(χ)√µ{cos ϕ anh(D(x−x0( ))) + isin ϕ}+ cos(χ)ηsech(D(x−x0( ))) exp{ikx +iθ( )},(14) The solu ion (13)-(14) is a DD soli on solu ion cha ac e ized by 4 ee pa ame e s. The asymp o ics o his solu ion a e |u1|2→ µcos2(χ)and |u2|2→µsin2(χ)as |x| → ∞. The densi ies o he abo e da k soli ons ead: n1≡ |u1|2=µcos2(χ)−(µcos2(χ) cos2ϕ−η2sin2(χ))sech2ξ −√µη sin(2χ){sin ϕsin[kx +θ( )] + cos ϕcos[kx +θ( )] anh ξ}sechξ, (15) n2≡ |u2|2=µsin2(χ)−(µsin2(χ) cos2ϕ−η2cos2(χ))sech2ξ +√µη sin(2χ){sin ϕsin[kx +θ( )] + cos ϕcos[kx +θ( )] anh ξ}sechξ, (16) while he o al densi y n o o he DD soli on is gi en by: n o =n1+n2=µ−D2sech2ξ. (17) No ice ha he o al densi y o he DD soli on is ime-independen and has he o m o a da k soli on densi y o dep h D2on op o a backg ound densi y µ. The abo e densi y is, in ac , iden ical o he densi y o he DB soli on; his is due o he ac ha unde SO(2) o a ion he o al densi y, as well as all o he conse ed quan i ies o he sys em, emain unchanged. This will be pa icula ly impo an when conside ing he mo ion o he DD soli on in a ap — see below. On he o he hand, one may conside he indi idual da k soli on densi ies, n1and n2, ac oss he ajec o y o he DD soli on, i.e., o ξ= 0: in such a case, x=x0( ) = k and he densi ies ead: n1(ξ= 0) = µcos2(χ) sin2ϕ+η2sin2(χ) −√µη sin(2χ) sin ϕsin [1 2(k2+D2) ],(18) n2(ξ= 0) = µsin2(χ) sin2ϕ+η2cos2(χ) +√µη sin(2χ) sin ϕsin [1 2(k2+D2) ].(19) I is clea ha n1,2(ξ= 0) a e pe iodic unc ions o ime; he ele an angula equency (which cons i u es he in e nal bea ing equency o he DD soli on) is gi en by: ω0=1 2(k2+D2) = 1 2(µ−η2sec2ϕ),(20) whe e we ha e also used Eq. (6). The equency ω0is bounded by wo limi ing alues. Fi s , in he case η→0, he bea ing DD soli on becomes a egula DD soli on, cha ac e ized by a wid h D=√µcos ϕand a eloci y k=√µsin ϕ; in his case, ω0→(1/2)µ. Second, in he limi ing case D→0, he bea ing DD soli on is educed o a plane wa e; in his case, ω0→(1/2)k2. In o he wo ds, he in insic oscilla ion equency ake alues in he ange: 1 2k2< ω0<1 2µ. (21) III. DARK-DARK SOLITONS AS PERIODIC ORBITS IN THE MANAKOV MODEL In his sec ion, we analyze he exis ence, s abili y and dynamics o single bea ing DD soli ons in a ap o he o m V(x) = 1 2Ω2x2, conside ing hem as pe iodic o bi s. In he p esence o he ap, he dynamics o he cen e o mass x0( )o he bea ing DD soli on is s ill desc ibed by he dynamics o he o iginal (un o a ed) DB soli on cen e x0. This is due o he ac ha he GPEs (2)-(3) a e in a ian unde SO(2) o a ions e en in he p esence o V(x), and so a e all conse ed quan i ies o he sys em, 5 −20 −15 −10 −5 0 5 10 15 20 −1 −0.5 0 0.5 1 x u1(x), u2(x) −20 −15 −10 −5 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 x |u1(x)|2, |u2(x)|2 −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Im(λ) Re(λ) FIG. 2: Le panel: P o iles and densi ies o he bea ing da k-da k soli on solu ion wi h Ω=0,ω0= 0.5and µ= 1.5a = 0. Righ panel: Floque mul iplie s spec um o he da k-da k soli on displayed in he le panel. such as he o al ene gy. Since he de i a ion o he equa ion o mo ion o he DB soli on cen e x0in Re . [16] was elying on he change o ene gy (due o he p esence o he ap), i is clea ha he e olu ion o he bea ing DD soli on cen e ollows he same dynamics: i pe o ms a ha monic oscilla ion in he ap acco ding o he equa ion ¨x0+ω2 oscx0= 0, whe e he oscilla ion equency ωosc is gi en by [16]: ω2 osc = Ω2(1 2− 8√1 + ( 4)2)(22) whe e =NB √µis a measu e o he a io o he numbe o a oms in he b igh and da k soli on componen . In o de o compu e he soli on p o ile and de e mine i s s abili y, we conside he solu ion o Eqs. (2)-(3), wi h g11 =g22 =g12 = 1, as a Fou ie se ies expansion o pe iod ω0[32], namely, u1(x, ) = ∞ ∑ k=−∞ zk(x)eikω0 , u2(x, ) = ∞ ∑ k=−∞ yk(x)eikω0 ,(23) wi h {zk},{yk} ∈ R. Then, he dynamical equa ions a e educed o a se o coupled equa ions: [µ−kω0−V(x)]zk+1 2∂2 xzk=∑ p∑ q (zpz∗ q+ypy∗ q)zk−p+q(24) [µ−kω0−V(x)]yk+1 2∂2 xyk=∑ p∑ q (zpz∗ q+ypy∗ q)yk−p+q(25) whe e we ha e used he no a ion zk≡zk(x),yk≡yk(x). I he ap is absen , i is s aigh o wa d o see ha z0(x) = √µ 2 anh(√2ω0x) = y0(x),(26) z1(x) = −√µ 2−ω0sech(√2ω0x) = −y1(x),(27) zj(x) = yj(x)=0,|j|>1o j =−1,(28) is ac ually he solu ion (13)-(14) o χ=π/4,ϕ=k= 0, and ω0=D2/2. In o de o nume ically ind a DD soli on solu ion in he sys em wi h he ap, he p e ious solu ion (wi h he da k componen {zk}mul iplied by he Thomas-Fe mi cloud wi h uT F 1=√max(µ−V(x),0)) is in oduced as a seed o a ixed-poin me hod in he sys em o Eqs. (24)-(25). Th oughou his sec ion, we ha e conside ed — o con enience— a ap s eng h Ω=0.2in o de o consume less ime in he nume ical calcula ions, as will be explained below. Figu es 2 and 3 show he pe iodic o bi o = 0 wi hou and wi h a ap po en ial, espec i ely. I is wo h ema king ha solu ions in he ap exis o µ > 2ω0, as p edic ed in he end o sec ion II. Once a pe iodic solu ion is ound, i s (linea ) o bi al s abili y can be analyzed by means o Floque analysis. To his end, he ime e olu ion o a small pe u ba ion ξ1(x, ), ξ2(x, ) o a pe iodic solu ion {u1,0(x, ), u2,0(x, )}mus be aced. Fo he double indices o ui,j,i ep esen s he componen index, i.e., i= 1,2is he i s and second componen o he da k-da k soli on 6 −20 −15 −10 −5 0 5 10 15 20 −1 −0.5 0 0.5 1 x u1(x), u2(x) −20 −15 −10 −5 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 x |u1(x)|2, |u2(x)|2 −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Im(λ) Re(λ) FIG. 3: Same as Fig. 2 bu in he apped case wi h Ω = 0.2. solu ion espec i ely. The index j= 0 deno es ha his is he (nume ically) exac pe iodic solu ions o u1and u2, a ound which we linea ize in ou Floque analysis. The pe u ba ions a e in oduced in he dynamical equa ions (2)-(3) as: u1(x, ) = [u1,0(x, ) + δξ1(x, )] , u2(x, ) = [u2,0(x, ) + δξ2(x, )] ,(29) and he esul ing equa ion a o de O(δ) eads: i∂ ξ1= [−1 2∂2 x+ 2|u1,0|2+|u2,0|2−µ+V(x)]ξ1+u2 1,0ξ∗ 1+u∗ 2,0u1,0ξ2+u2,0u1,0ξ∗ 2,(30) i∂ ξ2= [−1 2∂2 x+|u1,0|2+ 2|u2,0|2−µ+V(x)]ξ2+u2 2,0ξ∗ 2+u∗ 1,0u2,0ξ1+u1,0u2,0ξ∗ 1.(31) Then, in he amewo k o Floque analysis, he s abili y p ope ies o pe iodic o bi s a e esol ed by diagonalizing he mon- od omy ma ix M, which is de ined as:    Re(ξ1(x, T)) Im(ξ1(x, T)) Re(ξ2(x, T)) Im(ξ2(x, T))   =M   Re(ξ1(x, 0)) Im(ξ1(x, 0)) Re(ξ2(x, 0)) Im(ξ2(x, 0))   .(32) wi h T= 2π/ω0. As he sys em is symplec ic and Hamil onian, he linea s abili y o he solu ions equi es ha he monod omy eigen alues, λ(also called Floque mul iplie s) mus lie on he uni ci cle (see, e.g., [35, 36] o de ails). The Floque mul iplie s can also be w i en as λ= exp(iΘ), wi h Θbeing deno ed as he Floque exponen . An in e nal mode o he soli on co esponds o a spa ially localized solu ion o Eqs. (30)- (31), wi h i s oscilla ion equency ela ed o he Floque exponen s as ωm= Θω0/(2π). Figu es 2 and 3 show a ypical Floque mul iplie spec a, indica ing s abili y o he pe iodic o bi s. All he analyzed solu ions (i.e. wi h Ω = 0 and Ω = 0.2) a e s able. The choice o a ap s eng h Ω = 0.2 o s udying he s abili y o pe iodic o bi s ins ead o he alue Ω = 0.01 as in he es o he pape is wo old. On he one hand, as indica ed by Eq. (22), he oscilla ion pe iod scales wi h Ω−1; consequen ly, dec easing 20 imes he ap s eng h implies an in eg a ion ime 20 imes la ge ; on he o he hand, he Thomas-Fe mi adius (RT F =√2µ/Ω), which measu es he da k soli on size, would also inc ease 20 imes, so he numbe o equa ions o in eg a e also inc eases a his a e. Wi h Ω=0.2, ob aining he monod omy ma ix akes a ound 30 minu es. Consequen ly, he compu a ion o he monod omy ma ix o Ω = 0.01 would ake abou 200 hou s. Some in e es ing esul s can be ex ac ed by he analysis o he in e nal modes o he pe iodic o bi s. Figu e 4(le ) shows he dependence o h ee in e nal modes o he Floque spec um wi h espec o µ o ω0= 0.5. The blue line is close o he equency p edic ed by Eq. (22) [depic ed as a dashed ed line]. Indeed, pe u bing he bea ing DD soli on wi h he co esponding eigenmode, we ha e con i med ha his pe u ba ion leads o an oscilla ion o he soli on in he ap wi h a equency equal o ha o he eigenmode (c . le panel o Fig. 5). I can be obse ed ha he ag eemen be ween he nume ical eigen equency and ha p edic ed by Eq. (22) imp o es when µinc eases, as expec ed. This is because he assump ion ha he soli a y wa e is a pa icle inside a Thomas-Fe mi cloud is one o inc easing alidi y he deepe one is wi hin he Thomas-Fe mi limi o la ge µ. The igh panel o Figu e 4 shows he dependence o he equency o he in e nal mode co esponding o he oscilla ion o he ap wi h espec o ω0 o ixed µ= 5 and compa es i wi h he equency p edic ed by Eq. (22). We no e he e, as an aside in he case Ω = 0, ha he in e nal soli on modes a e neu al modes loca ed a (1,0) on he uni ci cle. In pa icula , he mode associa ed wi h he oscilla ion o he DD soli on in he ap becomes in his case a neu al mode associa ed wi h he ansla ion o he soli on. The algeb aic mul iplici y o he mul iplie a (1,0) in he case o Ω = 0 is 8, while in he apped case (due o he li ing o ansla ional in a iance) i is 6. 7 1.5 2 2.5 3 3.5 4 4.5 5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 µ ωm 0.5 1 1.5 2 2.5 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ωo ωm FIG. 4: (Le panel) Dependence o he eigen equencies o some in e nal modes (solid lines) wi h espec o µwi h ixed ω0= 0.5(The black, blue, g een and ed lines a e he uppe solid, middle solid, lowe solid and he dashed espec i ely). The igh panel shows ha dependence wi h espec o ω0wi h ixed µ= 5 o he mode in blue in he le panel. In bo h panels, he ed dashed line co esponds o he oscilla ion equency (22) in a ap wi h Ω = 0.2. x 0 50 100 150 −10 −8 −6 −4 −2 0 2 4 6 8 10 x 0 50 100 150 −10 −8 −6 −4 −2 0 2 4 6 8 10 x 0 50 100 150 −10 −8 −6 −4 −2 0 2 4 6 8 10 x 0 50 100 150 −10 −8 −6 −4 −2 0 2 4 6 8 10 x 0 50 100 150 −10 −8 −6 −4 −2 0 2 4 6 8 10 x 0 50 100 150 −10 −8 −6 −4 −2 0 2 4 6 8 10 FIG. 5: Con ou plo s showing he e olu ion o he densi ies o he DD soli on componen s ( i s and second ows depic espec i e componen s) when pe u bed by h ee di e en eigenmodes: he le panel co esponds o a soli on pe u bed along he blue mode and leads he soli on o ha monically oscilla e nea he cen e o he ap; in he cen al panel, he pe u ba ion (along he black mode) leads o a b ea hing o he soli on wid h, whe eas in he igh panel (pe u ba ion along he g een mode), he ou come co esponds o an oscilla ion o he whole condensa e. In bo h cases, µ= 3,ω0= 0.5and Ω = 0.2. In o de o obse e he p ope ies o o he in e nal modes, we ha e pe u bed he bea ing DD soli on wi h he co esponding eigenmodes. In pa icula , a pe u ba ion along he di ec ion o he localized mode depic ed in black in he le panel o Fig. 4, leads o a b ea hing in he wid h o he soli on — see cen al panel o Fig. 5. On he o he hand, a pe u ba ion along he di ec ion o he mode depic ed in g een in he le panel o Fig. 4, leads o an oscilla ion o he condensa e along he ap, lea ing he bea ing DD soli on una ec ed (i.e., he soli on s ays a he ap cen e ) — see igh panel o Fig. 5. Fo p og essi ely weake aps, he modes o he backg ound condensa e and o he da k-b igh soli a y wa e essen ially decouple and in ac wo o he equencies shown in Fig. 4 (g een and blue) end o 0, as he co esponding mo ions (o he soli a y wa e h ough he backg ound o o he backg ound h ough he soli a y wa e) become neu al. Finally, we make a ema k abou he way we ha e calcula ed he alue NB ha mus be in oduced in Eq. (22). The p ocedu e consis s in pe o ming an SO(2) o a ion wi h χ=−π/4 o he pe iodic DD soli on a = 0. This solu ion is shown in he le panel o Fig. 6, whe eas he o a ed solu ion is depic ed in he igh panel o he same igu e. Thus, NBis he no m o he b igh componen o he o a ed solu ion. I can also be in e ed om he Fou ie coe icien s o he pe iodic o bi : NB=∫R|u2|2dx=1 2∫R [|∑ k zk|2+|∑ k yk|2−2Re((∑ k z∗ k)(∑ k yk))]dx. (33) 8 −20 −15 −10 −5 0 5 10 15 20 −2 −1 0 1 2 x u1(x), u2(x) −20 −15 −10 −5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 x |u1(x)|2, |u2(x)|2 −20 −15 −10 −5 0 5 10 15 20 −2 −1 0 1 2 x u1(x), u2(x) −20 −15 −10 −5 0 5 10 15 20 0 1 2 3 x |u1(x)|2, |u2(x)|2 FIG. 6: (Le panel) P o iles and densi ies o a pe iodic o bi a = 0 wi h µ= 3,ω0= 0.5and Ω = 0.2. The igh panel shows he da k-b igh soli on a ising by o a ing wi h χ=−π/4 he da k-da k soli on o he le panel. x g11:g12:g22=1:1:1 100 200 300 400 −5 0 5 x 100 200 300 400 −5 0 5 g11:g12:g22=1.03:1:0.97 100 200 300 400 −5 0 5 100 200 300 400 −5 0 5 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 FIG. 7: The compa ison be ween he in eg able case g11 :g12 :g22 =1:1:1(le column) and he non-in eg able case g11 :g12 :g22 = 1.03 : 1 : 0.97 ( igh column), is demons a ed. The uppe panels show he densi ies o he i s da k soli on componen while he lowe ones show he second da k componen . He e η= 0.6, χ =π/4, θ = 0, k = 0, µ = 1. Based on he simila i y o he ele an dynamics, we will ocus on he case o uni nonlinea coe icien s. IV. SINGLE BEATING DARK-DARK SOLITONS NEAR AND FAR FROM THE MANAKOV LIMIT We now u n o a nume ical s udy o he p ope ies o he bea ing DD soli on s a es. Fi s ly, in he absence o a ap, we a e going o compa e he in eg able case wi h equal sca e ing leng hs g11 :g12 :g22 = 1 : 1 : 1 o he non-in eg able case g11 :g12 :g22 = 1.03 : 1 : 0.97 (see Re . [30]). F om Fig. 7, we obse e ha bo h o he da k componen s a e oscilla ing wi h ixed equencies and hese wo cases a e e y simila [38] All o he uns epo ed below o one o hese pa ame e se s ha e been epea ed wi h he second one and in all cases we ha e obse ed a close simila i y be ween he dynamical phenomenology o hese wo cases. To highligh he ac ha subs an ial a ia ions o he sca e ing leng h–which can be imposed by i ue o a Feshbach esonance–may ha e a signi ican impac on he obus ness o he bea ing DD soli ons, we conside sca e ing leng hs in he se wi h a ios g11 :g12 :g22 =g: 1 : 1. In pa icula , we ake g= 1.1,1.2,1.6in Fig. 8. When gis no so la ge, i.e., g= 1.1,1.2, he bea ing DD soli on oscilla es and, as inc eases, he change in he sca e ing leng h esul s in mobili y o he cohe en s uc u e. Howe e , mo e d ama ic e en s can a ise when gis ela i ely la ge, e.g., o g= 1.6. The e, we can see ha he soli on is inally spli in o wo agmen s (upon g ow h o he in insic bea ing oscilla ion which e en ually induces he spli - ing) and esul s in wo s a es ha esemble da k-an ida k soli ons [31] (see also Re . [29]). In pa icula , each o he componen s acqui es a da k soli on coupled o an an i-da k soli on, i.e., a densi y hump (ins ead o a dip) on op o a ini e backg ound, in he second componen . In Fig. 9, we show a pa icula example o he DD soli on in he ap, which oscilla es a ound i s cen e ; he pa ame e alues a e µ= 1,η= 0.6, ini ial soli on posi ion x0( =0)=2.5, and ap s eng h Ω = 0.05. No e ha o hese uns, he ini ial p o ile o he bea ing DD soli on in he ap is app oxima ed by he nume ically ound (in ap) g ound s a e — i.e., he Thomas- 9 g=1.1 200 400 600 800 −5 0 5 200 400 600 800 −5 0 5 g=1.2 200 400 600 800 −5 0 5 200 400 600 800 −5 0 5 g=1.6 200 400 600 800 −20 0 20 200 400 600 800 −20 0 20 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 FIG. 8: The dynamics o he densi ies o a DD soli on in he absence o a ap bu o g= 1.1, g = 1.2, g = 1.6 espec i ely (g≡g11); uppe and lowe panels show he densi ies o he i s and second da k soli on componen , espec i ely. When g= 1.1o g= 1.2, he soli on is se in o ansla ional mo ion. Howe e , o g= 1.6 he cohe en s uc u e execu es a g owing oscilla ion which e en ually esul s in i s spli ing in o a pai o da k-an ida k soli ons (i.e., a da k soli on in he one componen coupled o a lump in he o he ). The pa ame e s used he e a e he same as in Fig. 7. x 200 400 600 800 −20 0 20 x 200 400 600 800 −20 0 20 0 500 −2 0 2 x0 2 4 6 8 10 12 14 0 0.02 0.04 ωosc 0 0.5 0 0.5 FIG. 9: Mo ion o a da k-da k soli on in a ap o s eng h Ω = 0.05. O he pa ame e s a e he same as in Fig. 7. The uppe wo panels show he oscilla ion o he soli on ini ially cen e ed a x0= 2.5( he chemical po en ial is µ= 1). The lowe le panel demons a es he cen e o mass o he DD in he uppe panels. The analy ical oscilla ion equency, gi en by Eq. (22), is 0.03123, while he nume ical equency, calcula ed by Fou ie ans o m, is 0.03238. The lowe igh panel yields he compa ison be ween he analy ically calcula ed equencies ( ed line) e sus he nume ical ob ained ones ( he blue iangles), as a ies be ween 0.1 and 14. Fe mi cloud — mul iplied by he bea ing DD solu ion (wi hou a ap) o Eqs. (13)-(14). Then ia a ime-s epping algo i hm (a ou h-o de Runge-Ku a scheme), we ob ain he ime e olu ion o he densi ies o he oscilla ing soli ons in he uppe wo panels. Mo eo e , he le lowe panel shows he cen e o mass o he bea ing DD soli on in he ap. Using Fou ie analysis, we can in e he nume ical equency o in- ap oscilla ion, which can, in u n, be compa ed o he analy ical one, c . Eq. (22). As shown in he bo om igh panel o he igu e, he e is e y good ag eemen be ween he wo. Nex , we conside he in- ap dynamics o a single bea ing DD soli on bu o he non-in eg able cases. Again, when g11 : g12 :g22 = 1.03 : 1 : 0.97, we obse e a nea ly iden ical phenomenology o ha o uni gij’s. Fo he mo e signi ican de ia ions om ha case o he o m g11 :g12 :g22 =g: 1 : 1 whe e g= 1.1,1.2,1.6, he esul s a e epo ed in Fig. 10. Fo lowe alues o g= 1.1,1.2, he beha io o he DD is simila o he case wi h g= 1, howe e , we p og essi ely obse e mo e signi ican adia i e emissions which also a ec he oscilla ion equency. Howe e , once again he modi ica ions o he phenomenology a e mos d ama ic in he case o g= 1.6o he bo om panels. The e, he adia ion emission is accompanied by g owing in insic oscilla ions which e en ually esul in he b eakup and o ma ion o a single da k-an ida k soli a y wa e.