Dynamics of matter wave solitons

MSc in Pho onics PHOTONICSBCN Uni e si a Poli ècnica de Ca alunya (UPC) Uni e si a Au ònoma de Ba celona (UAB) Uni e si a de Ba celona (UB) Ins i u de Ciències Fo òniques (ICFO) h p://www.pho onicsbcn.eu Mas e in Pho onics MASTER THESIS WORK TOWARDS INTERFEROMETRY WITH MATTER WAVE BRIGHT SOLITONS Juan Polo Gomez Supe ised by D . Ve ònica Ahu inge B e o (UAB) P esen ed on da e 4 h o Sep embe , 2012 Regis e ed a Towa ds in e e ome y wi h ma e wa e b igh soli ons Juan Polo Gomez G up d’ `Op ica, Dep . de F´ısica, Uni e si a Au `onoma de Ba celona, E-08193 Bella e a, Spain E-mail: zes zes zes gmail.com Abs ac . We s udy he implemen a ion o a ma e wa e b igh soli on in e e ome e o med by a Gaussian po en ial ba ie placed a he cen e o a ha monic ap po en ial. A e nume ically e alua ing he ansmission coe icien o he po en ial ba ie as a unc ion o he a io be ween he kine ic ene gy o an inciden ma e wa e b igh soli on and he heigh o he ba ie , we ocus on he case o which he ansmi ed and he e lec ed spli soli ons ha e he same numbe o a oms. This beam spli e beha io is ully cha ac e ized in e ms o he phase and he ene gies o he wo ou come soli ons. The ecombina ion p ocess h ough he ba ie is also in es iga ed in e ms o he phase di e ence be ween he wo a ms o he in e e ome e . Keywo ds: ma e wa e soli ons, in e ac ion wi h po en ial ba ie s, a om in e e ome e . 1. In oduc ion In e e ome y has been o many yea s an impo an scien i ic ool wi h a my iad o applica ions, anging om p ecision measu emen s o undamen al s udies. A omic in e e ome e s ha exploi he wa e cha ac e o a oms allow o e y high sensi i i ies due o he associa ed small wa eleng h. Mo eo e , he inhe en a omic p ope ies like mass o pola izabili y o e new possibili ies o high-p ecision measu emen s o , o ins ance, undamen al cons an s, in e nal o ces o o a ions [1–4]. Speci ically, he use o ul acold a oms, ha can be p ecisely manipula ed a he quan um le el and can be held o long imes in aps [5], p o ides e y high p ecision and sensi i i y [6]. Ne e heless, in linea in e e ome y, he minimum unce ain y in de e mining he phase di e ence is gi en by he s anda d quan um limi [7], i.e., he expec ed o a classical measu emen . I has been shown ha his limi can be su passed, app oaching he Heisenbe g limi , using nonclassical inpu s a es [8–12] o en angled s a es wi hin he in e e ome e in he egime o la ge a om numbe wi h con olled in e ac ions [7]. One po en ial p oblem ha a omic in e e ome e s ace is based on he ac ha he wa e unc ion dispe ses du ing i s e olu ion. A possible solu ion is o use ma e wa e b igh soli ons in Bose-Eins ein condensa es (BEC). Ma e wa e b igh soli ons a e Towa ds in e e ome y wi h ma e wa e b igh soli ons 3 soli a y wa es wi h a ac i e in e a omic in e ac ions, ha p o e o be a he obus when colliding wi h each o he o when in e ac ing in an ex e nal po en ial and p esen a e y impo an cha ac e is ic: hey can p opaga e wi hou dispe sion [13,14]. Wi hin he mean ield app oxima ion, hei dynamics is well desc ibed by he G oss Pi ae skii equa ion (GPE), al hough e ec i e pa icle models also exis and ha e been used o cha ac e ize he cen e o mass ajec o ies o he soli ons [15–17]. No e ha he use o soli ons o in e e ome y [18–26] opens new pe spec i es and allows o add ess undamen al ques ions due o he dual na u e (pa icle and wa e) o soli ons in BEC. Wi hin he quasi 1-D limi , collisions o ma e wa e b igh soli ons wi h ba ie s ha e been la gely s udied in o de o cohe en ly spli he soli on [25] and he ecombina ion o iden ical soli ons h ough a po en ial ba ie has also been add essed [19,20]. In his wo k, we will in es iga e he implemen a ion o a ull ma e wa e b igh soli on in e e ome e o med by a ha monic po en ial ap and a Gaussian ba ie a i s cen e in which he inciden ma e wa e soli on spli s and he wo ou pu spli ones ecombine a e some ime. This se up is cu en ly unde expe imen al s udy in he g oup o P o . R. Hule [27]. In sec ion 2, we in oduce he physical sys em ha we a e conside ing. In sec ion 3, we analyze in de ail he ole o he ba ie as a beam spli e cha ac e izing he wo ou pu soli ons as a unc ion o he sys em pa ame e s. Sec ion 4 is de o ed o he ecombina ion p ocess and in sec ion 5 we p esen he conclusions. 2. Physical sys em We conside a Bose-Eins ein condensa e o 7Li a T= 0 whose dynamics wi hin he mean ield app oach is desc ibed by he 3D ime dependen GPE [28]: i¯h∂ ∂ ϕ( , ) = −¯h2∇2 2m+V( ) + g3D|ϕ( , )|2!ϕ( , ) ; (1) whe e he i s e m o he igh hand side o Eq. (1) co esponds o he kine ic con ibu ion, he second e m is he ex e nal po en ial ha eads V( ) = m[ω2 (x2+ y2) + ω2 zz2]/2, and he las e m is he nonlinea e m, whe e g3D= 4π¯h2asN/m, and N,asand ma e he a om numbe , s-wa e sca e ing leng h and a omic mass, espec i ely. The wa e unc ion is no malized o 1, and he sca e ing leng h is nega i e, as<0, which means ha he in e ac ions a e a ac i e. Fo su icien ly igh adial con inemen (ω ωz), bu allowing ha in e ac ions emain 3D, as(¯h/mω )1/2, i is possible o sepa a e he adial and he axial dynamics using he ansa z ϕ( ) = Ψ(z)(mω /π¯h)1/2exp (−mω [x2+y2]/2¯h), ob aining he educed 1D GPE [26]: i¯h∂ ∂ Ψ(z, ) = −¯h2 2m ∂2 ∂z2+V(z) + g1D|Ψ(z, )|2!Ψ(z, ),(2) whe e g1D= 2N¯hω as[29] and he ex e nal po en ial educes o V(z) = 1 2mω2 zz2. By using C ank-Nicolson me hod in imagina y ime e olu ion we calcula e he g ound s a e o he sys em o s a e o he a pa ame e alues [13]: ω = 2π×710 Hz, ωz= 2π×78 Hz, as=−0.21 nm, m= 1.165 ×10−26 kg and N= 15000 a oms, ob aining a ma e Towa ds in e e ome y wi h ma e wa e b igh soli ons 4 wa e b igh soli on. Once a s able soli on is c ea ed, we displace he cen e o he ha monic po en ial a dis ance z0and in oduce a Gaussian po en ial ba ie cen e ed a z= 0 on which he soli on will collide. The ex e nal po en ial eads now: V(z) = 1 2mω2 z(z−z0)2+Vbez2 2σ2,(3) whe e Vband σa e he s eng h and he wid h o he ba ie , espec i ely. Wi h his con igu a ion we implemen an in e e ome e o ma e wa e b igh soli ons ha can be sepa a ed in o h ee s eps: a) he soli on is spli in wo by colliding wi h he ba ie ; b) he wo spli soli ons e ol e in he ex e nal ha monic po en ial educing hei kine ic ene gy and inc easing he po en ial one un il hey s op and go back in opposi e di ec ions; and c) a ecollision o he soli ons wi h he ba ie occu s a he cen e o he ap. Fig. 1 shows he p ocess schema ically. In o de o cha ac e ize he dynamics o he spli ing o he ini ial soli on and he Figu e 1. Schema ics o he in e e ome e sequence: (a) ini ial si ua ion in which he soli on is displaced wi h espec o he cen e o he ha monic po en ial ap by a dis ance z0; (b) he wo spli soli ons ob ained a e he collision wi h he ba ie sepa a e om each o he (c) he wo soli ons e u n o he posi ion o he ba ie and ecollide. e olu ion o he sys em a e wa ds, we calcula e he expec ed alue o he posi ion and momen um o he soli on be o e i s collision wi h he ba ie and o he wo soli ons ob ained a e he spli ing: hzii=K−1 iZCi dz|Ψ(z, )|2z, (4) hpii=−i¯hK−1 iZCi dzΨ∗(z, )∂ ∂zΨ(z, ),(5) whe e Ki=ZCi dz|Ψ(z, )|2and Cideno es he egion con aining he soli ons: being i=I, R, T o Inciden , Re lec ed and T ansmi ed egions, espec i ely. We also de ine he posi ion and momen um unce ain y: ∆zi= (hz2 ii−hzii2)1/2,(6) ∆pi= (hp2 ii−hpii2)1/2,(7) whe e: hz2 ii=K−1 iZCi dz|Ψ(z, )|2z2,(8) Towa ds in e e ome y wi h ma e wa e b igh soli ons 5 hp2 ii=−¯h2K−1 iZCi dzΨ∗(z, )∂2 ∂z2Ψ(z, ).(9) The numbe o pa icles o each soli on is gi en by Ni=NKiand he ene gy pe pa icle eads: ei=Ei N=ZCi dz   ¯h2 2m ∂ ∂zΨ(z, ) 2 +V(z)|Ψ(z, )|2+g1D|Ψ(z, )|4 ,(10) whe e he i s e m o he igh hand side co esponds o he kine ic ene gy Ek, he second one is he ex e nal po en ial ene gy, Eho, and he las one is he in e nal ene gy due o he a om-a om in e ac ion, Ein . No e also ha by di ec in eg a ion o he GPE unde he s a iona y ansa z Ψ(z, ) = Ψ(z) exp (−iµ /¯h), whe e µis he chemical po en ial, we ob ain µ=Ek+Eho + 2Ein [28]. 3. Spli ing p ocess In his sec ion we analyze he main aspec s ha a ec he dynamics o he spli ing p ocess o a ma e wa e b igh soli on unde he collision wi h a Gaussian po en ial ba ie . 3.1. T ansmission By an ins an aneous displacemen o he apping po en ial, he cen e o he soli on is placed a −z0and he soli on acqui es a po en ial ene gy ha , as p edic ed by pa icle models [30,31], is ully con e ed in o kine ic ene gy once he soli on eaches he cen e o he po en ial. The e o e, he kine ic ene gy o he soli on when in e ac ing wi h he ba ie is Eho =1 2mω2 z(z−z0)2. We ha e checked ha his esul is in ull ag eemen wi h Eq. (10). Fig. 2(a) shows he nume ically e alua ed ansmission coe icien as a unc ion o he a io be ween he kine ic ene gy o he soli on and he heigh o he po en ial ba ie o a ixed po en ial ba ie heigh o Vb= 17.14¯hωz(squa es), and o a ixed kine ic ene gy o Ek= 10.83¯hωz(ci cles); in bo h cases σ= 0.5µm. The di e en beha io s ob ained close o T= 1 can be unde s ood by aking in o accoun ha he changes in he s eng h o he po en ial o in he kine ic ene gy o he inciden soli on do no p oduce exac ly he same e ec due o he di e en a ea ha he soli on has o pene a e in o de o pass h ough he ba ie . Ne e heless, a common ea u e is he high slope in he egion whe e he ansmission coe icien is 0.5, i.e. whe e he ini ial soli on spli s in o wo soli ons wi h he same numbe o a oms, which makes his poin e y sensi i e o he ini ial condi ions. Fig. 2(b) shows he ansmission as a unc ion o he wid h o he ba ie (ci cles) o a ixed a io Ek/Vb= 0.63. As expec ed, he inc ease o he wid h o he ba ie co esponds o a dec ease in ansmission. In Fig. 2, we also plo he analy ical ansmission coe icien o a ec angula ba ie (do ed-dashed line), which Towa ds in e e ome y wi h ma e wa e b igh soli ons 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 T ansmission T Ek/Vb (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 T ansmission T σ(µm) (b) Figu e 2. T ansmission as a unc ion o (a) he a io be ween Ek/Vband (b) he wid h o he ba ie ob ained by ixing Ek= 17.14¯hωz(ci cles) o Vb= 10.83¯hωz(squa es). The analy ic esul o a ec angula ba ie (do ed-dashed line) and o he WKB app oxima ion (dashed line) in he linea case a e also plo ed. The pa ame e s o he sys em a e N= 15000, ωz= 2π×78 Hz, ω = 2π×710 Hz, as=−0.21 nm. In (a), σ= 0.5µm and in (b) Ek/Vb= 0.63. eads [32]: TuE<Vb=1 1 + V2 bsinh2(k1(2σ)) 4E(Vb−E) TuE>Vb=1 1 + V2 bsin2(k1(2σ)) 4E(E−Vb) ; (11) whe e k1=q2m|Vb−Ek|/¯h2; and he Wen zel−K ame s−B illouin (WKB) app oxima ion (dashed line). The WKB app oxima ion es ima es he ansmission coe icien , o small alues o Tand o po en ials ha do no change ab up ly, aking in o accoun he shape o he po en ial and conside ing he ac ha he ampli ude and he phase o he wa e unc ion a y slowly. The de ailed analy ical de i a ion o he ansmission coe icien can be ound in [32]: TWKB ≃e1 ¯hRb adz√2m(V(z)−Ek)+1 4e −1 ¯hRb adz√2m(V(z)−Ek)−2 ; (12) whe e aand ba e he z-axis poin s whe e he kine ic ene gy and he ba ie coincide, which o he Gaussian shape ba ie a e b=−a=q2σ2ln (Vb/Ek). No e ha hese analy ical ansmission unc ions (Eq. (11) and (12)) a e only alid in he linea egime. The e o e, in o de o be alid o ou nume ical simula ion Ekhas o be la ge enough o he wid h o he ba ie small enough o minimize he in e ac ion ime o he soli on wi h he ba ie . Ne e heless, ou speci ic se up limi s hese wo pa ame e s: he minimum wid h o he ba ie is ixed by he di ac ion limi , since he Gaussian po en ial is expe imen ally c ea ed by using a ocalized lase beam wi h a beam wais equal o σand he kine ic ene gy is limi ed by he ini e po en ial ap displacemen , z0. Mo eo e , he e is an addi ional aspec ha makes he compa ison wi h he analy ical unc ions di icul : he soli on is a wa e packe desc ibed by a mac oscopic wa e unc ion, whe eas he analy ical cases conside plane wa es. The e o e, while plane wa es ha e a Towa ds in e e ome y wi h ma e wa e b igh soli ons 7 well de ined momen um, he soli on has a e y well de ined posi ion and he momen um unce ain y is e y la ge, ul illing he unce ain y ela ion ∆xI∆pI≥¯h/2 close o he sa u a ion alue. Fo ins ance, o an inciden soli on wi h N= 15000 he unce ain ies using Eq. (6) and (7) a e ∆xI= 0.583 nm and ∆pI/¯h= 0.898 1/nm, gi ing ∆xI∆pI/¯h= 0.524. In spi e o he discussed limi a ions, he ec angula po en ial ba ie i s a he well wi h he nume ical esul s (see Fig. 2). Howe e he slope o he ansmission unc ion o he ec angula ba ie nea T= 0.5, egion o in e es o in e e ome ic applica ions, is lowe han he one ob ained nume ically wi h he Gaussian ba ie . The WKB app oxima ion ag ees wi h he nume ical ansmission coe icien when T→0. Howe e his egion is ou o he egime o in e e ome ic applica ions whe e he aim is o spli he soli on in wo iden ical soli ons (T= 0.5). Fig. 3 shows he po en ial s eng h Vbas a unc ion o Ek o which T= 0.5 and o wo di e en alues o he numbe o pa icles N. We can obse e ha he dependence is app oxima ely linea and independen o N o la ge kine ic ene gies. Ne e heless, o small kine ic ene gies he nonlinea i y s a s o play an impo an ole and b eaks he linea dependence (see inse o Fig. 3). In he ollowing we conside a soli on wi h N= 15000. This alue has been chosen in o de o ob ain wo soli ons in he spli ing p ocess i.e, he nonlinea i y is su icien o compensa e he dispe sion a e he spli ing, whe e he numbe o a oms o each o he ou come soli ons is N/2. 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 Vb/¯hωz Ek/¯hωz 0.3 0.45 0.6 0.75 0.9 0 20 40 60 Ek/Vb Ek/¯hωz Figu e 3. Po en ial ba ie heigh as a unc ion o he kine ic ene gy o he inciden soli on o a ixed ansmission coe icien T= 0.5, o an inciden soli on o N= 15000 (squa es) and N= 7500 (ci cles). The inse shows he a io Ek/Vbas a unc ion o he kine ic ene gy. The es o he pa ame e s o he sys em a e ωz= 2π×78 Hz, ω = 2π×710 Hz, as=−0.21 nm and σ= 0.5µm. 3.2. Ene gy and Phase The sensi i i y o an in e e ome e has a s ong dependence on how he phase di e - ence is measu ed and o how long he spli pa hs can accumula e phase wi hou losing cohe ence [6]. The e o e, in sec ion 4, we will analyze he phase e olu ion o he inciden Towa ds in e e ome y wi h ma e wa e b igh soli ons 8 soli on and o he wo spli soli ons p oduced by he in e ac ion wi h he ba ie . Ne e heless, in his sec ion we s udy i s he phase di e ence be ween he ansmi ed and e lec ed soli ons, in oduced by he in e ac ion wi h he ba ie du ing he spli - ing p ocess, by se ing he ha monic ex e nal po en ial equal o ze o. Fig. 4(a) shows 0.5 0.75 1 1.25 1.5 1.75 2 2.25 20 30 40 50 60 70 80 ∆φ( ad) Ek/¯hωz (a) 58 60 62 Ek/¯hωz (b) -17 -15 -13 3 3.5 4 4.5 5 5.5 6 Ein /¯hωz (ms) Figu e 4. (a) Phase di e ence be ween he ansmi ed and he e lec ed soli on as a unc ion o he kine ic ene gy o he inciden soli on o T= 0.5. (b) Kine ic and in e nal ene gy o a o al e lec ed soli on wi h ini ial kine ic ene gy Ek= 58.85¯hωz. The solid (dashed) line shows he ime a e aged ene gy be o e (a e ) he collision wi h he ba ie o bo h Ekand Ein . The in e al o ime in which he collision wi h he ba ie occu s is ma ked wi h he black egion. The pa ame e s used a e: N= 15000, ω = 2π×710 Hz, ωz= 0 Hz, σ= 0.5µm and as=−0.21 nm. he linea dependence o he phase di e ence be ween he ansmi ed and he e lec ed soli on, in oduced by he e lec ion p ocess, wi h he kine ic ene gy o he inciden soli on. We conside N= 15000, po en ial wid h σ= 0.5µm and a heigh such ha he T= 0.5 condi ion is ul illed, i.e., he wo esul ing soli ons ha e he same numbe o a oms. By using exp ession (10), one can also calcula e he e olu ion o he di e en con ibu ions o he ene gy du ing he spli ing p ocess. We ob ain ha he e lec ed soli on has less kine ic ene gy han he ansmi ed one, and mo eo e , in he e lec ion p ocess some kine ic ene gy is ans e ed o in e nal ene gy. The e o e, he ansmi ed and e lec ed soli ons a e no iden ical ene ge ically. As an example, an inciden soli on wi h an ini ial ene gy Ek= 20.08¯hωzand Ein =−14.84¯hωzspli s in o wo soli ons wi h he same numbe o pa icles when in e ac ing wi h a ba ie wi h Vb= 20.57¯hω and σ= 0.5µm. The kine ic (in e nal) ene gy o he e lec ed and he ansmi ed soli on is EkT= 5.20¯hωz(Ein T=−1.73¯hωz) and EkR= 3.57¯hωz(Ein R=−1.80¯hωz), espec- i ely. Al hough he o al ene gy is conse ed, EkR+Ein R+EkT+Ein T=Ek+Ein , i is clea ha he e lec ed soli on has less ene gy han he ansmi ed one. An ex eme case occu s o he o al e lec ion o a soli on o N= 15000 whe e he ans e o kine ic o in e nal ene gy is clea ly obse ed (Fig. 4(b)): he solid (dashed) line shows he ime a e aged ene gy be o e (a e ) he collision wi h he ba ie o bo h Ekand Ein . I is wo h no icing ha he e lec ion p ocess can also induce b ea hing exci a ions in which some ac ion o he ene gy is exchanged pe iodically be ween Ekand Ein as shown in Towa ds in e e ome y wi h ma e wa e b igh soli ons 9 Fig. 4(b), o he o al e lec ion case (T= 0). 4. Recombina ion p ocess In his sec ion we add ess he las s ep o he p oposed implemen a ion o a ma e wa e b igh soli on in e e ome e , i.e., he ee e olu ion o he spli soli ons and hei ecombina ion a he cen e o he ha monic po en ial (Fig. 1(c)). We conside he case desc ibed in sec ion 3.2 in which one soli on collides wi h a Gaussian ba ie and spli s in o wo soli ons wi h he same Nand a ela i e phase induced by he po en ial ba ie . Du ing hei e olu ion in he ha monic po en ial, he phases o he spli soli ons e ol e di e en ly due o he di e en eloci y acqui ed du ing he spli ing p ocess (see sec ion 3.2) as shown in Fig. 5. The inciden soli on inc eases i s phase as i p opaga es owa ds he ba ie . A e he collision wi h he ba ie a = 4.34 ms, wo soli ons appea wi h a phase di e ence (see inse Fig.5) as desc ibed in sec ion 3.2. A e hei e olu ion in he ha monic po en ial pe o ming a dipole oscilla ion, he wo spli soli ons collide a he posi ion o he ba ie a = 10.40 ms. The ou come o his collision depends on he ela i e phase be ween he wo soli ons ha ecombine. In gene al, one ob ains wo soli ons wi h di e en numbe o pa icles a e he ecombina ion. The e o e, he accumula ed ela i e phase in he wo a ms o he in e e ome e can be de e mined by measu ing he ela i e pa icle numbe o he wo ou going soli ons [25]. Fig. 6 shows di e en ou pu s o he s udied a omic in e e ome e co esponding o 42 44 46 48 50 52 54 56 58 60 62 4 5 6 7 8 9 10 φ( ad) (ms) 45 46 47 48 4.2 4.4 4.6 φ( ad) (ms) Figu e 5. Time e olu ion o he phase o he inciden soli on (black line) be o e he in e ac ion wi h he ba ie and o he e lec ed (blue line) and ansmi ed ( ed line) soli on c ea ed in he spli ing p ocess du ing i s dipole oscilla ion in he ha monic ap. The pa ame e s used a e N= 15000, ωz= 2π×78 Hz, ω = 2π×710 Hz, as=−0.21 nm, Ek= 10.74¯hωz,σ= 0.5µm, he po en ial used o spli ing in o wo soli ons wi h he same numbe o pa icles is Vb= 17.70¯hωz. The ime in e al in which he collision wi h he ba ie occu s is ma ked wi h he black egion. The inse shows an enla ged iew o he spli ing p ocess. di e en phase di e ences be ween he ecombined soli ons. The phase di e ence in his