Spontaneous symmetry breaking of gap solitons in double-well traps
M. Trippenbach, E. Infeld, J. Gocalek, Michal Matuszewski, M. Oberthaler, B. A. Malomed
aa r X i v : . [ c ond - m a t . o t h e r] F e b Spontaneous symmetry breaking of gap solitons in double-well traps
M. Trippenbach, E. Infeld, J. Goca lek, Micha l Matuszewski, M. Oberthaler, and B. A. Malomed. Institute of Theoretical Physics, Physics Department,Warsaw University, Ho˙za 69, PL-00-681 Warsaw, Poland Soltan Institute for Nuclear Studies, Ho˙za 69, PL-00-681 Warsaw, Poland Institute of Physics, Polish Academy of Sciences, Al. Lotnikw 32/46, Warsaw, Poland Nonlinear Physics Center and ARC Center of Excellence for Quantum Atom Optics,Research School of Physical Sciences and Engineering,Australian National University, Canberra ACT 0200, Australia Kirchhoff-Institut f¨ur Physik, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany Department of Interdisciplinary Sciences, School of Electrical Engineering,Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
We introduce a two-dimensional model for the Bose-Einstein condensate with both attractive andrepulsive nonlinearities. We assume a combination of a double-well potential in one direction, andan optical-lattice along the perpendicular coordinate. We look for dual-core solitons in this model,focusing on their symmetry-breaking bifurcations. The analysis employs a variational approxima-tion, which is verified by numerical results. The bifurcation which transforms antisymmetric gapsolitons into asymmetric ones is of supercritical type in the case of repulsion; in the attraction model,increase of the optical latttice strength leads to a gradual transition from subcritical bifurcation (forsymmetric solitons) to a supercritical one.
PACS numbers: 03.75.Lm, 05.45.Yv, 42.65.Tg
I. INTRODUCTION
The Gross-Pitaevskii equation (GPE) provides a pow-erful model for studying the mean-field dynamics of Bose-Einstein condensates (BECs) [1]. Important examplesare the prediction of 1D gap solitons (GSs) in a self-repulsive condensate trapped in a periodic optical-lattice(OL) potential [2]. This was realized experimentally inan ultracold gas of Rb atoms confined in a cigar-shapedtrap [3], and the prediction of the Josephson effect in aBEC [4].It was subsequently observed in a condensatetrapped in a macroscopic double-well potential [5]. Incontrast to hitherto realized Josephson systems in super-conductors and superfluids, interactions between tunnel-ing particles play a crucial role in a bosonic junction. Theeffective nonlinearity induced by the interactions givesrise to new effects in the tunneling. In particular, an-harmonic Josephson oscillations were predicted [6, 7, 8],provided that the initial population imbalance in the twopotential wells falls below a critical value [9, 10]. This dy-namic regime can be well explained by means of a simplemodel derived from the GPE, which amounts to a sys-tem of equations for the inter-well phase difference andpopulation imbalance. The nonlinearity specific to theBEC also gives rise to a self-trapping effect in the formof a self-maintained population imbalance.One-dimensional dynamics of a BEC in potentials com-posed of two rectangular potential wells were studied inseveral papers [11]. Stationary states with different pop-ulations in the two wells are generated by symmetry-breaking bifurcations from symmetric and antisymmetricstates, for attractive and repulsive nonlinearity, respec-tively [9, 10]. A natural 2D extension of the double-wellconfiguration is a dual-channel one, with the potential featuring the two wells in the direction of x , which areextended into parallel troughs along the y axis [12, 13].In the case of an attractive nonlinearity, this settingmay naturally give rise to dual-core solitons, which areself-trapped in the y direction (similar to the ordinarymatter-wave solitons created in a single-core trap [14]),and are supported by a double-well structure in the per-pendicular direction. Furthermore, if the nonlinearity isstrong enough, or else the tunnel coupling between thetroughs is weak, the obvious symmetric dual-core solitonmay bifurcate into an asymmetric one. This was demon-strated both in the full 2D model [12], and in its 1Dcounterpart, which replaces the 2D equation by a pairof one-dimensional GPEs with coordinate y , while thetunneling in the x direction is approximated by a linearcoupling between the equations [13]. In fact, the lat-ter model resembles the standard one widely accepted innonlinear optics to describe dual-core nonlinear opticalfibers and asymmetric solitons [15, 16]. In a similar way,the double-well potential may be uniformly extended intwo transverse directions, giving rise to a 3D structurebased on a pair of parallel “pancakes”.If the dual-channel potential in 2D geometry is com-bined with an axial optical lattice, which runs along bothpotential troughs, it is natural to consider a dual-core gapsoliton in the self-repulsive BEC filling this structure.In Ref. [13], this was done using the above-mentionedapproximation which replaced the corresponding two-dimensional GPE by a pair of linearly-coupled 1D equa-tions. It was demonstrated that a symmetric gap soli-tons may be stable in this case, and never bifurcate,while asymmetric solitons are generated by a symmetry-breaking bifurcation from antisymmetric ones. Similarresults (including the emergence of asymmetric gap soli-tons carrying intrinsic vorticity) where obtained in the2D extension of the model. This model pertains tothe above-mentioned “dual-pancake” structure [17]. Innonlinear optics, asymmetric gap solitons were studiedin models of dual-core fiber Bragg gratings, which alsoamount to systems of linearly coupled 1D equations [18].The prediction of symmetry breaking for matter-wavesolitons in a setting combining the transverse double-wellpotential and a longitudinal optical lattice in experimen-tally relevant conditions makes it necessary to study thefull 2D model (especially for the stability of the emerg-ing asymmetric solitons) for both repulsive and attrac-tive condensates, which is the purpose of the presentwork. Parameter regions admitting asymmetric solitonswill be predicted by means of the variational approxima-tion (VA) [16]. These results will be verified by numerics.The character of the symmetry-breaking bifurcations forthe dual-core solitons will also be identified (we obtaina gradual transition from a subcritical bifurcation to asupercritical one with increase of the OL strength).The paper is organized as follows. The model and theVA are introduced in Sec. II. In Sec. III we analyze thesymmetry-breaking bifurcations in both attraction andrepulsion models, and Sec. V concludes the paper. II. THE MODEL AND VARIATIONALAPPROXIMATION
The normalized form of the GPE for the mean-fieldwave functions Ψ in 2D geometry is i Ψ t = − (1 /
2) (Ψ xx + Ψ yy )+ (cid:2) U ( x ) + σ | Ψ | + ρ cos (2 y ) (cid:3) Ψ , (1)where σ = +1 and − ρ cos (2 y ) represents the longitudi-nal optical lattice potential. The transverse double-wellstructure is taken as U ( x ) = (cid:26) , | x | < L/ | x | > L/ D, − U , L/ < | x | < L/ D, (2)with D , U and L being, respectively, the width anddepth of each well, and the width of the barrier betweenthem, see Fig. 1 below.Stationary solutions to Eq. (1) are assumed in the formΨ( x, y, t ) = e − iµt Φ( x, y ), where the real function Φ( x, y )satisfies the equation µ Φ + (1 /
2) (Φ xx + Φ yy ) − U ( x )Φ − σ Φ + ρ cos(2 y )Φ = 0 . (3)It can be derived from the Lagrangian, L stat = Z Z dxdy (cid:2) µ Φ − (1 / (cid:0) Φ x + Φ y (cid:1) −− U ( x )Φ − ( σ/
2) Φ + ρ cos (2 y ) Φ (cid:3) . (4)To apply the VA, we follow Ref. [12] and adopt an ansatz consisting of two distinct parts. First, inside each po-tential trough, i.e., at | x ∓ ( L + D ) / | < D/
2, the trial y x
U L D FIG. 1: (Color online) The shape of the quasi-one-dimensionaldouble-well potential, U ( x, y ). The wiggles indicate quasi-1Dlattice along y . function isΦ ± ( x, y ) = A ± cos (cid:18) π x ∓ ( L + D ) / D (cid:19) exp (cid:18) − y W (cid:19) , (5)where A ± and W are three variational parameters. Thisexpression implies different amplitudes and a commonlongitudinal width, W , of the wave-function patterns inboth troughs. In the x direction, the ansatz (5) emulatesthe ground-state wave function in an infinitely deep po-tential box, which vanishes at the edges of the trough,see Fig. 1. In the y direction, the ansatz approximatesthe self-trapped soliton by a Gaussian profile. Outsidethe troughs (at | x | > L/ D and | x | < L/ x, y ) = X + , − A ± exp (cid:18) − p − µ (cid:12)(cid:12)(cid:12)(cid:12) x ∓ L + D (cid:12)(cid:12)(cid:12)(cid:12) − y W (cid:19) , (6)with the same amplitudes A ± and width W as in Eq. (5).The ansatz is not continuous at the edges of the troughs;however, comparison with numerical findings (see Fig. 2below) clearly suggest that the VA can be used despitethis local discrepancy.Substitution of expressions (5) and (6) into Eq. (4) andintegration produce the following simplified Lagrangian,in which contributions from the exponentially decayingfunctions in the outer region, | x | > L/ D , are ne-glected, the contribution from the optical lattice poten-tial is taken into account only inside the troughs, andthe Thomas-Fermi approximation in the x direction isadopted, i.e., term − (1 / x in the Lagrangian densityis omitted: 2 D √ π L eff = 12 ρW e − W (cid:0) A + A − (cid:1) (7)+ X + , − (cid:18) µ + U A ± W − A ± W − σ / A ± W (cid:19) + 4 √− µD e −√− µ ( L + D ) A + A − W. (8)We now define N ± ≡ (cid:0) / √ (cid:1) A ± W , and λ ≡ (2 /D ) p − µ exp (cid:16) − p − µ ( L + D ) (cid:17) , (9) N ≡ N + + N − √ λ , ν ≡ N + − N − √ λ , ǫ ≡ µ + U . (10)The numbers of atoms trapped in the two troughs areproportional to the respective partial norms of the wavefunction, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z + ∞−∞ dy Z ± ( D + L/ ± L/ dx (Φ( x, y )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 √ π DN ± , (11)hence ν , defined in Eq. (10), measures the populationimbalance. In this notation, the Lagrangian (8) simplifiesto 38 √ πλD L eff = (12) ≡ ǫN − N W − σ √ λ N + ν W − sλ p N − v + 12 ρN e − W , (13)with s = +1 and − A + A − < A + A − >
0, respectively).Our Lagrangian gives rise to variational equations ∂L/∂W = ∂L/∂ν = ∂L/∂N = 0: N + 2 σ √ λ (cid:0) N + ν (cid:1) W − ρN W e − W = 0 , (14) ν − σW + s r λN − ν ! = 0 , (15)14 W + σ √ λNW + 2 sλN √ N − ν − ρe − W = ǫ. (16)Equation (15) has two solutions: ν = 0, which corre-sponds to symmetric or antisymmetric solitons, and ν = N − λW , (17)for asymmetric ones. Comparison of typical asymmetricand symmetric solitons, found from a numerical solutionof Eq. (3), with their counterparts predicted by the VA,is presented in Fig. 2.For symmetric and antisymmetric solitons, Eqs. (14)and (16), with ν = 0, are tantamount to equations thatwere derived, by means of the VA, for solitons in 1Dmodels with a periodic sinusoidal potential and attractiveor repulsive nonlinearity [19, 20]. In particular, in the -5 0 5 x F ( x ) -5 0 5 x F ( x ) FIG. 2: (Color online) The top and bottom panels demon-strate examples of cross-section profiles, along y = 0, of sta-ble asymmetric and symmetric gap solitons in the model withrepulsion, as obtained from a numerical solution to Eq. (3)and predicted by the variational approximation (dashed andcontinuous lines, respectively). Parameters of the double-wellpotential are L = D = 1, U = − . ρ = 1. Norms of the asymmetric and symmetric solitons are,respectively, N = 0 .
52 and 0 .
34. The asymmetry parameterfor the former soliton, see Eqs. (10), is ν = 0 . latter case (for σ = +1) a known fact is that solutionsexist only for ρ > ρ (0) ≡ e / ≈ .
462 (in fact, thisconstraint predicts, with high accuracy, the edge of thefirst finite bandgap in the linear spectrum induced by theOL [20]). Results for asymmetric solitons are presentedin the next section.
III. ASYMMETRIC SOLUTIONSA. Equations for the bifurcation point
According to Eq. (15), asymmetric solutions exist intwo cases: σ = s = +1 (repulsion, with the asymmet-ric branch bifurcating from the antisymmetric one), or σ = s = − ν in Eqs. (14) and(16) by means of Eq. (17) yields a system of equationsfor N and W : N + 2 σ √ λW (cid:0) N − λW (cid:1) = 4 ρN W e − W , (18)14 W + σ √ λNW + 2 s √ λNW − ρe − W = ǫ. Taking into account definitions (9) and (10), solutions toEqs. (18) depend on parameters
L, D, U , and ρ .At the bifurcation point, ν = 0, Eq. (17) yields N = √ λW , hence Eqs. (17) generate a system of twoequations for two coordinates of the bifurcation point, µ [via relations (10) and (9)] and W :1 + 2 σλW = 4 ρW e − W , (19)14 W + 2 ( σ + s ) λ − ρe − W = ǫ. Without the OL, i.e., for ρ = 0 (the case considered inRef. [12]), the first equation in (19) gives the bifurcationpoint at N = 1 / √
2. To obtain explicit results in themodel with ρ = 0, one can start with an obvious solutionto Eqs. (19), at λ = µ = N = 0, ρ = ρ (0) (recall ρ (0) ≡ e / U = U (0)0 ≡ /
16, and W = W (0) ≡ √
2. Thissolution, which has N = 0 is, by itself, trivial, but anontrivial one can be obtained as an expansion aroundit. B. The model with self-attraction
Consider the attraction model corresponding to σ = s = −
1. Then, straightforward analysis of Eqs. (19) forsmall δρ = ρ − ρ (0) and δU = U − U (0)0 demonstrates thatthe bifurcation of symmetric solitons (which pertain to s = −
1, see above) may occur at two values of the norm, N = 12 √ p − ( e − δρ + δU ) (cid:20) − (cid:0) e − δρ + δU (cid:1) ± p e − δρ − δU (cid:21) , (20)the respective value of the width being W ≈√ (cid:2) − (cid:0) e − δρ + δU (cid:1) / (cid:3) . Note that the second termin the square brackets in Eq. (20) is a small correctionto 2, the main correction given by the last term, whichdemonstrates that theoretically there may be two differ-ent bifurcation points. Obviously, expressions (20) aremeaningful, i.e., the bifurcation takes place, if (cid:0) e / (cid:1) δU < δρ < − e δU (21)(in other words, δU must be negative, while δρ may haveeither sign). Numerical calculations imply that only thelower value of N is valid. n r=0.0r=0.2r=0.4r=0.8 U=-0.7, l=1, d=1
FIG. 4: (Color online)A set of numerically found bifurcationdiagrams in the model with attraction, showing degree of aasymmetry of dual-core soliton, ν , as a function of the soli-ton’s total norm, N , see Eqs. (10). The diagrams pertain tofixed values of parameters of the transverse double-well con-figuration, L = D = 1, U = − . ν = 0 and ± N/ √
3. One can clearlysee that the supercritical bifurcation will turn into subcriticalbifurcation with increase of the optical latttice strength.
A set of bifurcation diagrams in the attraction model,in the form of ν ( N ), i.e., curves showing the asymme-try of the dual-core solitons versus the total norm, wasgenerated by a numerical solution of the full system ofEqs. (18). The set is displayed in Fig. 4, where anoteworthy feature is the transition from the subcritical shape (backward-directed one), which is a characteristicof the attraction model without the longitudinal OL [12](as well as to the model of dual-core optical fibers [15]),to the simpler supercritical (forward-directed) shape atsufficiently large values of OL strength ρ . Note thatthe symmetry-breaking bifurcations of dual-core solitons,studied in systems of linearly-coupled GPEs including theattractive nonlinearity and OL potential [13, 17], as wellas in the system of linearly-coupled fiber Bragg gratings[18], are of supercritical type too. The physical signifi-cance of the subcritical bifurcation is that it allows bista-bility of the solitons (the coexistence of stable symmetricand asymmetric ones) in a limited interval of values of N . C. The model with self-repulsive nonlinearity
In the case of the self-repulsion, i.e., σ = s = +1, theexpansion of Eqs. (19) predicts the following values ofthe norm at which asymmetric gap solitons may bifur-cate from the antisymmetric ones (recall antisymmetric n r=0.6r=0.8r=1.0 U=-4, l=1, d=1
FIG. 5: (Color online) A set of bifurcation diagrams for gapsolitons in the model with repulsive nonlinearity, for L = D =1, U = 4, and a set of different values of the OL strength, ρ . solitons corresponds to s = +1): N = 12 √ p e − δρ + δU (cid:20) − (cid:0) e − δρ + δU (cid:1) ± p e − δρ − δU (cid:21) , (22)where the notation is the same as in Eq. (20) for the attractive model. This expression predicts the bifurcationin the following region [cf. Eq. (21) in the attractionmodel]: − e − δρ < δU < e − δρ, which implies δρ > δU may be both positive and negative, in contrastwith the case of the attraction model, that demanded δU <
0, while allowing δρ to take either sign. Onceagain numerical calculations imply that only the lowervalue of N is valid.A typical set of bifurcation diagrams in the repulsivemodel is displayed in Fig. 5. It is seen that the bifur-cation generating asymmetric gap solitons from the an-tisymmetric ones is always of supercritical type, in com- pliance with results obtained for the models based onlinearly coupled GPEs with the optical lattice potentialand repulsive nonlinearity [13, 17]. These bifurcation di-agrams exist only for ρ > ρ (0) ≡ e /
16, because, as saidabove, at smaller values of the optical lattice strength theVA does not predict antisymmetric GSs that might giverise to a bifurcation.
IV. CONCLUSIONS
We have introduced a 2D model for self-attractive andself-repulsive BECs, which combines a double-well po-tential in the transverse direction, and a periodic poten-tial along the longitudinal coordinate. The analysis in-volved symmetry-breaking bifurcations for dual-core soli-tons. Systematic results were obtained by means of thevariational approximation, which was verified by numer-ical results. In the case of a repulsive nonlinearity, thebifurcation is of supercritical type, while in the modelwith attraction an increase of the optical lattice strengthleads to a gradual transition from subcritical bifurcationto a supercritical one. This is an important result.
V. ACKNOWLEDGEMENTS
M.T. acknowledges the support of the Polish Govern-ment Research Grant for 2006-2009. E.I and M.M. ac-knowledges the support of the Polish Government Re-search Grant for 2007-2010 and 2007+2009. The workof B.A.M. was partially supported by the Israel Sci-ence Foundation through Excellence-Center grant No.8006/03. He would like to thank Soltan Institute for Nu-clear Studies, Warsaw, for an invitation in 2007. B.A.M.and M.O. acknowledge the support bz German-IsraelFoundation through grant No. 149/2006. [1] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999).[2] F. Kh. Abdullaev et al ., Phys. Rev. A , 043606 (2001);I. Carusotto, D. Embriaco, and G. C. La Rocca, ibid . ,053611 (2002); B. B. Baizakov, V. V. Konotop and M.Salerno, J. Phys. B , 5105 (2002); E. A. Ostrovskayaand Y. S. Kivshar, Phys. Rev. Lett. , 160407 (2003);Opt. Exp. , 19 (2004).[3] B. Eiermann et al. ., Phys. Rev. Lett. , 230401 (2004).[4] J. Javanainen, Phys. Rev. Lett. , 3164 (1986); A.Smerzi et al ., Phys. Rev. Lett. et al ., Phys. Rev. A , 620 (1999); S. Giovanazzi, A.Smerzi, and S. Fantoni, Phys. Rev. Lett. , 4521 (2000);E. A. Ostrovskaya et al ., Phys. Rev. A 61, 031601(R)(2000); K. W. Mahmud, J. N. Kutz, and W. P. Rein-hardt; Phys. Rev. A , 063607 (2002).[5] M. Albiez et al ., Phys. Rev. Lett. ,010402, (2005).[6] J. Javanainen, Phys. Rev. Lett. , 3164 (1986). [7] M. W. Jack, M. J. Collett, and D. F. Walls, Phys. Rev.A , R4625 (1996).[8] I. Zapata, F. Sols, and A. J. Leggett, Phys. Rev. A ,R28 (1998).[9] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls,Phys. Rev. A 55, 4318 (1997).[10] A. Smerzi et al ., Phys. Rev. Lett. , 4950 (1997); S.Raghavan et al ., Phys. Rev. A , 620 (1999).[11] P. Zin et al ., Phys. Rev. A , 022105 (2006), E. Infeld et al ., Phys. Rev. E , 026610 (2006); for a review, seeR. Gati and M. Oberthaler, J. Phys. B. , R61 (2007).[12] M. Matuszewski, B. A. Malomed, and M. Trippenbach,Phys. Rev. A , 063621 (2007).[13] A. Gubeskys and B. A. Malomed, Phys. Rev. A ,063602 (2007).[14] K. E. Strecker et al ., Nature , 150 (2002); L.Khaykovich et al ., Science , 1290 (2002).[15] E. M. Wright, G. I. Stegeman, and S. Wabnitz, Phys. Rev. A , 4455 (1989); N. Akhmediev and A. Ankiewicz,Phys. Rev. Lett. , 2395 (1993); P. L. Chu, B. A. Mal-omed, and G. D. Peng, J. Opt. Soc. Am. B , 1379(1993).[16] V. M. P´erez-Garc´ıa et al ., Phys. Rev. A , 1424 (1997);B. A. Malomed, in: Progress in Optics , vol. , p. 71 (ed.by E. Wolf: North Holland, Amsterdam, 2002).[17] A. Gubeskys and B. A. Malomed, Phys. Rev. A ,043623 (2007). [18] W. Mak, B. A. Malomed, and P. L. Chu, J. Opt. Soc.Am. B , 1685 (1998); Y. J. Tsofe and B. A. Malomed,Phys. Rev. E , 1197 (1999).[20] S. Adhikari and B. A. Malomed, Europhys. Lett. ,50003 (2007); Phys. Rev. A76