Some mathematical aspects in determining the 3D controlled solutions of the Gross-Pitaevskii equation

Some mathematical asp ects in determining the 3D cont rolled solutions of the Gross-Pitaevskii equation R. F edele, 1 , ∗ D. Jo v anovi ´ c, 2 , † S. De Nicola, 3, 1 , ‡ B. Eliasson, 4, 5 , § and P . K. Sh ukla 4, 6 , ¶ 1 Dip artimento di Scienze Fisiche, Universit` a F e deric o I I and INFN Se zione di Nap oli, Complesso Universi tario di M.S. Angelo, via Cintia, I-80126 Nap oli, Italy 2 Institute of Physics, P. O. Box 57, 11001 Belg r a d e, Serbia 3 Istituto di Cib ernetic a “Eduar do Caiani el lo” del CNR Compr ensorio “ A. Olivetti” F abbr. 70, Via Camp i Fle gr ei, 34, I-80078 Pozzuoli (NA), Italy 4 Institut f¨ ur The o r e tisch e Physik IV, Ru hr–Universit¨ at Bo chum, D-44 7 80 B o chum, Germa ny 5 Dep artment of Physics, Ume ˚ a University, SE-90 187 Ume ˚ a, Swe den 6 SUP A, Dep a rtment of Physics, Unive rsity of Str athclyde, Glasgow G4 ONG, Unite d Kin g d om (Dated: Octob er 22, 2021) 1 Abstract The p ossibilit y of the decomp osition of the three d imensional (3D) Gross-Pitae vskii equation (GPE) in to a pair of coupled Sc hr ¨ odinger-t yp e equations, is inv estigated. It is sho wn that, und er suitable mathematic al conditions, solutions o f the 3D con trolled GPE can b e constructed fr om the solutions of a 2D linear Sc hr¨ odinger equation (transv erse comp onent of the GPE) coupled with a 1D nonlinear Sc hr¨ odinger equ ation (longitudinal comp onent of the GPE). Such a decomp osition, called the ’con trolling p oten tial metho d’ (CPM), allo ws one to cast the ab o ve solutions in the form of the pro duct of the s olutions of the transv erse and the longitudin al comp onents of the GPE. Th e coupling b et w een these t w o equations is the fun ctional of b oth the trans verse and the longitudinal profiles. The analysis sho ws that the C PM is based on the v ariational principle that sets up a condition on the con trolling p otentia l w ell, and whose physical in terpr etatio n is give n in terms of the m inimization of the (energy) effects introdu ced by the control op eration. P ACS nu mbers: 03.75.-b Matter wa ves; 67.85.-d Ultra c old ga ses, tra ppe d gas e s; 52.35.Mw Nonlinear phe- nomena: w av es , wav e pro pa gation, and o ther interactions ∗ Electronic address: renato.fedele@na.infn.it † Electronic addr ess: djo v anov@ph y .bg.ac.yu ‡ Electronic addr ess: s.denicola@cib.na.cnr .it § Electronic addr ess: bengt@ tp4 .rub.de ¶ Electronic address: ps@tp4.rub.de 2 I. INTR ODUCTION Since its exp erimen ta l observ ation [1], the study of the three dimensional (3D) collectiv e and nonlinear dynamics of the Bose Einstein condensate (BEC) [2] in an external p oten tial trap [3] has receiv ed a great deal of attention b y a v ery wide scien tific comm unity and in t he in ves tigations concerning fundamental phy sics, by mathematical ph ysics and sophisticated tec hnological applications [4]. Although rapid scien tific and tec hnological adv ances ha v e b een ac hiev ed in this area, finding the exact analytical 3D solutions of the Gross-Pitaevskii equation (G PE) [3], that corresp ond to the coheren t state of a BEC in a suitable external p oten tial w ell (suc h as soliton-lik e structures), still remains a c hallenging task fo r ph ysicists and mathematicians. A n umber of v aluable approximativ e analytical [5] and n umerical ev aluations [6] hav e b een presen ted in the literatur e and hav e b een adequately compared with a v ery wide sp ectrum of exp erimen tal observ ations. The exp erience gained from these in v estigations ma y suggest the idea that a BEC’s dynamics exhibits the features of a nonlinear non-auto nomous system [7] f or whic h it seems to b e necessary to include some con trol op erations in o r der to allo w the exis tence of coheren t structures. In particular, to retain the 3D coheren t stationary structures of the BEC fo r a long time, suitable ”ad ho c” time-dep enden t external p oten tia ls and con trol op erations are k no wn to be necessary [8]. F urthermore, in the presence of an inhomogeneous time-dep enden t external potential one encoun ters some difficulties to find exact soliton solutions in one or more dimensions, altho ugh sev eral kind of solitons ha v e b een found in certain approxim ations [9]. Consequen tly , one easily arriv es to t he conclusion that, in order to get exact soliton structures, some sort of the ’control of the system ’ seems to b e necessary . This implies that the cor r ect analysis of the system should include a con tr o l p oten tial term in the GPE whic h is to b e determined dynamically by the system itself. In principle, this pro cedure may be extended to an arbitra ry ’con tro lled solution’ with the appropriate choic e of the external p oten tial (so-called ’controlling p otential’ [10] ). In fact, a controlling p oten tial metho d (CPM) has b een recen t ly prop osed in t he literatur e and used to find m ulti-dimensional con trolled lo calized solutions of the GPE. In the preliminary in ves tigations [11], this metho d has established reasonable experimen tal control op erations that ensure the stabilit y of the solution against relativ ely small errors in the exp erimen tal realization of the prescrib ed con trolling p otential. The main goa l of the CPM is to fix the 3 t ype of the desired con trolled solution and to find the appropriate family of the con tro lling p oten tials. Then, the set of suitable mathematical conditions ha s to b e found allo wing us to select the desired solution, with the adopted con tr o lling p otential. In this man uscript, w e dev elop a n analytical pro cedure to construct exact three dimen- sional solutions of a con trolled Gro ss-Pitaevskii equation, b y improving the CPM. T o this end, w e dev elop the theory of the BEC con trol based on t w o de c omp os i tion the or ems leading to suitable ph ysical conditions to express the BEC w av e function as the pr o duct of a 2D w av e function and a 1D wa v e function, ta king into accoun t the ’transv erse’ and ’longitudinal’ BEC profiles, resp ectiv ely . Such a f a ctorization allo ws us to decompo se the 3D controlled GPE in to a set of coupled equations, comprising a 2D linear Schr¨ odinger equation (gov erning the ev olution of the ’transv erse’ w av e function), a 1 D nonlinear Schr¨ odinger equation ( g o v erning the ev olutio n of the ’longitudinal’ w a ve f unction) and a v ariational condition in v olving the con tr o lling p otential. The requireme n t fo r the minimization o f the effects intro duced b y the con tr o l op erations (i. e. the requiremen t that the av erage of the controlling p otential ov er the transv erse plane is equal to zero) allo ws us to dete rmine explicitly the self-consisten t con tr o lling p oten tial whic h also play s the role of the coupling term b et w een transv erse and longitudinal BEC dynamics. I I. CON TR O LLED GR OSS-PIT AEVSKI I EQUA TION It is w ell kno wn that the spatio- temp oral ev olutio n of the ultra cold system of iden tical atoms fo rming a BEC in the presence of the external p oten tial U ext ( r , t ), within the mean field approximation, is gov erned by the three dimensional Gross-Pitaevskii equation [3], viz., i ¯ h ∂ Ψ ∂ t = − ¯ h 2 m a ∇ 2 Ψ + N Q | Ψ | 2 Ψ + U ext ( r , t ) Ψ , (1) where Ψ( r , t ) is the w a vefunc tion describing the BEC state, m a is the atom mass and Q is a coupling coefficien t r elat ed to the short range scattering (s-w av e) length a represen ting the in teractions b et ween atomic particles, viz., Q = 4 π ¯ h 2 a/m a , and N is the n um b er of atoms. Note that the short range scattering length can b e either p ositiv e o r negativ e. W e assume that U ext is the sum of the 3D trapping p otential well, U trap , that is used t o confine the particles of a BEC, and the controlling p otential U contr whic h will b e determined self- consisten tly . W e conv enien tly introduce the v ariable s = ct ( c b eing the sp eed of ligh t) and 4 divide b oth sides of Eq.(1) b y m a c 2 , and we use the notation U ext ( r , t ) m a c 2 = U trap ( r , t ) m a c 2 + U contr ( r , t ) m a c 2 ≡ V trap ( r , s ) + V contr ( r , s ) , (2) Eq. (1) can b e cast in the form i λ c ∂ ψ ∂ s = − λ 2 c 2 ∇ 2 ψ +  V trap ( r , s ) + V contr ( r , s ) + q | ψ | 2  ψ , (3) where ψ ( r , s ) ≡ Ψ( r , t = s/c ), λ c ≡ ¯ h/m a c 2 is the Compton wa v elength of the single atom of BEC and q ≡ N Q/mc 2 . In this pap er, w e will in v estigate the pr o p erties of Eq. (3) and V contr that enable the existence of the controlled 3D solutions in the facto r ized form ψ ( r , s ) = ψ ⊥ ( r ⊥ , s ) ψ z ( r ⊥ , z , s ) , (4) pro vided that V trap can b e split in t o tw o parts, a s V trap ( r , s ) = V ⊥ ( r ⊥ , s ) + V z ( z , s ) (5) where, in Cartesian co ordinates, r ≡ ( x, y , z ) and r ⊥ ≡ ( x, y ) denotes, b y definition, the ’transv erse’ part of the part icle’s v ector p osition r . W e also refer to z as to the ’longitudinal’ co ordinate. By substituting Eqs. (4) and (5) in Eq. (3), w e easily get: ψ ⊥ λ 2 c 2  ∇ 2 ⊥ ψ z + 2 ∇ ⊥ ψ ⊥ ψ ⊥ · ∇ ⊥ ψ z  + ψ ⊥  i λ c ∂ ψ z ∂ s + λ 2 c 2 ∂ 2 ψ z ∂ z 2 −  V z + V contr + q | ψ ⊥ | 2 | ψ z | 2  ψ z  + ψ z  i λ c ∂ ψ ⊥ ∂ s + λ 2 c 2 ∇ 2 ⊥ ψ ⊥ − V ⊥ ( r ⊥ , s ) ψ ⊥  = 0 , (6) where, in Cartesian co ordinates, ∇ ⊥ ≡ ˆ x ∂ /∂ x + ˆ y ∂ /∂ y . Let us define as ’controlled parameter’ t he follow ing time-dep enden t quantit y: q 1 D ( s ) = q Z d 2 ~ r ⊥ | ψ ⊥ | 4 ; (7) and the fo llo wing linear and nonlinear op erators, resp ectiv ely: b H ⊥ = − λ 2 c 2 ∇ 2 ⊥ + V ⊥ ( r ⊥ , s ) (8) b H z = − λ 2 c 2 ∂ 2 ∂ z 2 + V z ( z , s ) + q 1 D ( s ) | ψ z ( r ⊥ , z , s ) | 2 + V 0 (9) where V 0 is an arbitrary real constan t. Then, Eq. (6) can b e rewritten as: ψ z  i λ c ∂ ∂ s − b H ⊥  ψ ⊥ + ψ ⊥  i λ c ∂ ∂ s − b H z  ψ z +  q 1 D ( s ) − q | ψ ⊥ | 2  | ψ z | 2 ψ z + ( V 0 − V contr ) ψ z  + ψ ⊥ λ 2 c 2  ∇ 2 ⊥ ψ z + 2 ∇ ⊥ ψ ⊥ ψ ⊥ · ∇ ⊥ ψ z  = 0 . (10) 5 I I I . THE DECOMPOSITION PR OPER TIES OF THE CONTROLLED GR OSS- PIT AEVSKI I EQUA TION By the definition of the con trolling p ot ential, V contr dep ends b oth on ψ ⊥ and ψ z . In particular, w e assume here that the space and time dep endence o f V contr is give n a lso thro ugh ρ ⊥ ( r ⊥ , s ) ≡ | ψ ⊥ ( r ⊥ , s ) | 2 , viz., V contr = V contr ( ρ ⊥ ( r ⊥ , s ) , z , s ) . (11) Moreo ver, defining a lso the follow ing functional of ρ ⊥ : V [ ρ ⊥ ; z , s ] = Z ρ ⊥ ( r ⊥ , s ) V contr ( ρ ⊥ ( r ⊥ , s ) , z , s ) d 2 r ⊥ , (12) the followin g theorem ho lds: DECOMPOSITI ON THEOREM 1. If ψ z ( r ⊥ , z , s ) = ψ z ( z , s ) , (13) and ψ ⊥ ( r ⊥ , s ) is the solution of the fol l o wing 2D line ar Schr¨ o d i nger e quation  i λ c ∂ ∂ s − b H ⊥  ψ ⊥ = 0 , (14) and V is a stationary functional (with r esp e ct to variations δ ρ ⊥ of ρ ⊥ ) , assuming the value V = V 0 , c onditione d by the c onstr aints Z ρ ⊥ d 2 r ⊥ = 1 , (15) (normalization c ondition for ψ ⊥ ) , and Z ρ 2 ⊥ d 2 r ⊥ = q 1 D ( s ) q = gi v e n func tion , (16) then ψ z is the solution of the fol lowing 1D non line ar Schr¨ odinger e quation  i λ c ∂ ∂ s − b H z  ψ z = 0 , (17) and V contr is given by V contr ( r ⊥ , z , s ) =  q 1 D ( s ) − q | ψ ⊥ ( r ⊥ , s ) | 2  | ψ z ( z , s ) | 2 + V 0 . (18) 6 T o pro v e this theorem, first of all, w e note that the assumptions (13) a nd (14) allow us to reduce Eq. (10) to  i λ c ∂ ∂ s − b H z  ψ z +  q 1 D ( s ) − q | ψ ⊥ | 2  | ψ z | 2 ψ z + ( V 0 − V contr ) ψ z = 0 . (19) Secondly , the required stationarit y of V with resp ect to v aria tions δ ρ ⊥ of ρ ⊥ implies that δ V + α ( z , s ) δ Z ρ ⊥ d 2 r ⊥ + β ( z , s ) δ Z ρ 2 ⊥ d 2 r ⊥ = 0 , (20) where α ( z , s ) and β ( z , s ) are Lagra ngian m ult ipliers. T aking in to accoun t Eq. (12), conditio n (20) allows us to solv e the corr espo nding ordinary inhomogeneous first-order differential equation for V contr where ρ ⊥ pla ys the role of the independen t v ariable and z and s are parameters, yielding the follow ing general solution V contr ( r ⊥ , z , s ) = h ( z , s ) ρ ⊥ ( r ⊥ , s ) − α ( z , s ) − β ( z , s ) ρ ⊥ ( r ⊥ , s ) , (21) where h ( z , s ) is an arbitrary function. Actually , to ensure the con v ergence of the in tegral in the definition of the functional V , see Eq. (12), it is easy to see that we m ust ha v e h ( z , s ) = 0. Consequen tly , the appropriat e V contr satisfying the stationar ity condition V = V 0 is giv en b y V contr ( r ⊥ , z , s ) =  q 1 D ( s ) q − ρ ⊥ ( r ⊥ , s )  β ( z , s ) + V 0 , (22) whic h after the substitution in Eq. (19) giv es  i λ c ∂ ∂ s − b H z  ψ z +  q 1 D ( s ) − q | ψ ⊥ | 2   | ψ z | 2 − β /q  ψ z = 0 . (23) No w, according to the h ypo thesis (13), to preserv e the r ⊥ -indep endence of ψ z , Eq. (23) can b e satisfied o nly when β ( z , s ) = q | ψ z ( z , s ) | 2 , (24) whic h immediately implies that Eqs. (17) and (18 ) are satisfied. DECOMPOSITI ON THEOREM 2. L et us supp ose that ψ z = ψ z ( z , s ) is the solution of the 1D nonline ar S c h r¨ odinger e quation (17). Th e n, the functional V give n by (12)and c onditione d by the c o n str aints (15), an d (16 ) , is s tationa ry (with r esp e ct to varia tion s δ ρ ⊥ of ρ ⊥ ), V = V 0 if, a nd o n ly if, ψ ⊥ = ψ ⊥ ( r ⊥ , s ) is the sol ution of the 2D line ar Schr¨ odinger e quation (14) . 7 T o pro v e this prop osition, w e o bserve that since ψ z ( z , s ) satisfies Eq. (17), Eq. (10) b ecomes  i λ c ∂ ∂ s − b H ⊥  ψ ⊥ +  q 1 D ( s ) − q | ψ ⊥ | 2  | ψ z | 2 + ( V 0 − V contr )  ψ ⊥ = 0 . (25) By m ultiplying the latter on t he left by ψ ∗ ⊥ and in t egrating o ver all the transv erse plane, w e easily obtain Z ψ ∗ ⊥  i λ c ∂ ∂ s − b H ⊥  ψ ⊥ d 2 r ⊥ + V 0 − V [ ρ ⊥ ; z , s ] = 0 , (26) where constrain ts (15) a nd (16) hav e b een used. Conse quen tly , if ψ ⊥ satisfies Eq. (14), then V is a stationary functional with the v alue V = V 0 , conditioned b y (15) and (16). Con v ersely , the assumed stationarity of V implies that the functional form of V contr with resp ect to r ⊥ , z and s is giv en b y Eq. (22), whic h substituted in Eq. (25) give s  i λ c ∂ ∂ s − b H ⊥  ψ ⊥ +  q 1 D ( s ) − q | ψ ⊥ | 2   | ψ z | 2 − β ( z , s ) /q  = 0 . (27) Ho w ev er, if β ( z , s ) /q 6 = | ψ z ( z , s ) | 2 , then ψ z w ould b e also function of r ⊥ whic h would con- tradict the assumption ψ z = ψ z ( z , s ). It follo ws that β ( z , s ) /q = | ψ z ( z , s ) | 2 and, in turn, that Eq. (14) is satisfied. The results presen ted ab o v e a llow us to draw the follo wing conclusion. If ψ ⊥ ( r ⊥ , s ) and ψ z ( z , s ), ar e two c om plex functions which ar e exact solutions of the 2D lin e ar Schr¨ odinger e quation (14) and the 1D nonline a r Schr¨ od inger e quation (17), r esp e ctively, pr ovide d that V contr is given by Eq. (18), the function ψ ( r , s ) = ψ ⊥ ( r ⊥ , s ) ψ z ( z , s ) is the exact solution of the c ontr ol le d 3D Gr oss-Pitaevski i e q uation (3) . Of course, the inv erse is not necessarily true. In fa ct, it is easy to see that, in principle, it is not true that an arbitrary solution of Eq. (3) can b e express ed as the pro duct of tw o w av e f unctions ψ ⊥ ( r ⊥ , s ) and ψ z ( z , s ) t ha t ob ey the Eqs. (14) and (17), resp ectiv ely . In other w ords, w e can decomp ose the controlled 3D G PE (3) in to the sys tem of equations (14), (17) and (18) only for the subset of its solutions of the t yp e (4). How ev er, using suc h a decomp osition w e are able to solv e Eq. (3) and to obtain a wide sp ectrum of exact solutions of the ty p e (4). IV. CONCLUSIO NS AND REMARKS In t his pap er, w e ha v e presen ted some mathematical prop erties of the controlled 3D GPE (3). After formulating and proving t wo decomp osition theorems , w e ha v e found the 8 mathematical conditions that mak e p ossible the construction of the solution in a factor- ized form, i.e. ψ ( r , s ) = ψ ⊥ ( r ⊥ , s ) ψ z ( z , s ), where ψ ⊥ ( r ⊥ , s ) and ψ z ( z , s ) satisfy the 2D linear Sc hr¨ odinger equation ( i λ c ∂ ψ ⊥ /∂ s = b H ⊥ ψ ⊥ ) and the nonlinear con trolled nonlinear Sc hr¨ odinger equation ( i λ c ∂ ψ z /∂ s = b H z ψ z ), r esp ectiv ely . The results presen ted here improv e the formulation of the recen tly prop o sed Con tro lling Poten tial Metho d [10, 11]. It is worth y observing that the set of equations (14), (17) and (18) op ens up the p ossibility to find the controlled solutions of the type (4) which exhibit the quan tum c haracter in the tra nsve rse part (sup erp osition principle with conse quen t interferen ce effec ts) and the classical c haracter in the longit udinal part (due to the nonlinearit y of the 1D no nlinear Sc hr¨ odinger equation), although the en tire solution of the con trolled 3D GPE is nonlinear and, therefore, has a classical c haracter. By means of suitable con trolling and trapping p oten tials, this p ossibilit y w ould allow, for instance, for a v ery stable soliton-like longitudinal profile of the BEC whose transv erse pro file w o uld ha ve a quan tum character as a result of the quan tum interferen ce at the macroscopic lev el. Note tha t, when ψ ⊥ satisfies Eq. (14), according to definition (12), V represen ts t he a v erage of V contr in the transv erse plane. The v alue o f this av erage corr esp onds to the arbitrary constant V 0 . Without loss of generality , we put V 0 = 0 , viz. Z d 2 ~ r ⊥ ψ ∗ ⊥ V contr ψ ⊥ = 0 . (28) This w a y , among all p o ssible c ho ices of V contr , w e adopt the one whic h do es not c hange the mean energy of the system (note that the a v erage of the Hamiltonian op erator in Eq. (3) is the same with or without V contr ) a nd thus minimizes the effects in tro duced b y our con t rol op eration. In our fo rthcoming pap ers, we will use the metho d dev elop ed in the presen t pap er t o solv e exactly the 3D controlled GPE with a 3D parab olic p otential tra p. W e find the controlled en ve lop e solutions in the form of lo calized as well a s p erio dic structures for whic h suitable stabilit y analysis is p erformed. 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