Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons on one-dimensional 2 1 optical lattice 0 2 n S. K. Adhikari1 , and P. Muruganandam1,2 a ‡ § J 1Instituto de F´ısica Te´orica, UNESP - Universidade Estadual Paulista, 01.140-070 9 S˜ao Paulo, S˜ao Paulo, Brazil 2School of Physics, Bharathidasan University, Palkalaiperur Campus, Tiruchirappalli ] s 620024, Tamilnadu, India a g - Abstract. t n By solving the three-dimensional Gross-Pitaevskii equation we generate two- a dimensionalaxially-symmetricandanisotropicdipolarBose-Einsteincondensatebright u q solitons, for repulsive atomic interaction, stabilized by only a weak one-dimensional . optical lattice (OL) aligned along and perpendicular, respectively, to the dipole t a polarization direction. In the former case vortex solitons can also be created. We m show that it is possible to make a stable array of small interacting axially-symmetric - d dipolar solitons put on alternate OL sites. Further, we demonstrate the elastic nature n of the collision of two such solitons. o c [ 2 PACS numbers: 03.75.Lm,03.75.Nt,05.30.Jp v 9 3 5 3 . 2 1 1 1 : v i X r a Email: [email protected]; URL: http://www.ift.unesp.br/users/adhikari/ ‡ [email protected] § Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 2 A bright soliton is a self-reinforcing solitary wave that maintains its shape, while traveling at constant speed, due to a cancellation of nonlinear attraction and dispersive effects. Integrable solitons without any external trap or intervention for cubic nonlinearity exist only in one dimension (1D). Experimentally, bright matter-wave solitonsandsolitontrainswerecreatedinaquasi-1DBose-Einsteincondensate(BEC)of 7Li [1, 2] and 85Rb atoms [3] by turning the atomic interaction attractive from repulsive using a Feshbach resonance (FR) [4] and employing a transverse trap. Although, the normal three-dimensional (3D) BEC soliton [5] is of great interest, such a BEC has only short-range attraction which makes it vulnerable against collapse. Physicalsystemsarestableduetoapeculiarnatureofinteractionamongitsconstituents (atoms, molecules and nuclei), e.g., short-range repulsion and long-range attraction. Lately, BEC of 52Cr [6, 7] and 164Dy [8, 9] atoms with a large long-range dipolar interaction has been observed. Also, experimental tuning of the long-range dipolar interaction by means of rapidly rotating orienting fields [10] as well as of the short- range atomic interaction using a FR [4] are completely under control. This engineering of the atomic and dipolar interactions makes the dipolar BEC (DBEC) an interesting system for the formation of soliton [11, 12, 13]. The long-range anisotropic dipolar interaction is attractive in some directions and repulsive in others. If it were attractive inalldirections, stablerobust3DDBECsolitons, correspondingtoaminimuminenergy functional, would naturally be formed for repulsive short-range atomic and attractive long-range dipolar interactions [14]. Normaldipolarinteractionleadstoattractionalongthepolarizationz directionand repulsion along transverse directions. It is possible to have the opposite by tuning the dipole interaction to “negative” values by orienting fields [10] and this set-up was used in some studies [13, 15]. For normal dipolar interaction, an anisotropic two-dimensional (2D) soliton can be obtained for repulsive short-range atomic interaction if a weak OL is placed along y axis, perpendicular to the polarization direction z, to overcome the dipolar repulsion in transverse directions. For dipolar interaction tuned to negative values[10], axially-symmetric2Dbrightandvortexsolitonscanbeobtainedforrepulsive short-range atomic interaction if a weak OL is placed along z axis to overcome the dipolar repulsion in that direction in this sign-changed setting. In all cases, the dipolar repulsion is weak and we do not need any trap in other directions to stabilize a soliton. Such 2D solitons cannot be stabilized without the dipolar interaction [13]. We present a linear stability analysis for the axially-symmetric soliton [16]. We study the 2D DBEC solitons using the numerical and Lagrangian variational analysis of the 3D Gross-Pitaevskii (GP) equation. The effective Lagrangian of the variational analysis has the same structure as that of a generalized classical dynamical system with two degrees of freedom. We find that stable (referred to as “center” as it corresponds to a stable periodic orbit around a center in phase space) and unstable stationary (called “saddle” as it corresponds to a saddle point in energy) states appear and disappear through the mechanism of saddle-center bifurcation [17]. There have been studies of 2D DBEC solitons with strong harmonic traps along Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 3 y [11] or z [13] axis and of 1D DBEC solitons under transverse harmonic trap [12]. The present solitons confined by only a weak OL along y or z axis, respectively, are distinct. The previous studies [11, 13, 15] will essentially have an approximate Gaussian density distribution along the infinite trap direction, whereas the present solitons will have an exponential density distribution due to weak finite traps in these directions. More interestingly, an OL simulates the periodic electron-atom potential in a solid and the study of solitons in an OL is also of interest in condensed-matter physics [18, 19]. We show that a new type of stable interacting 1D array of solitons can be formed in 3D space when tiny axially-symmetric interacting DBEC bright solitons are placed on alternate sites of the OL. However, if the solitons are placed on all sites of the OL, the array is destroyed due to strong long-range dipolar interaction among its constituents. Statics and dynamics of such periodic array of tiny droplets of dipolar matter are of concern in condensed matter physics [19], as they simulate many problems of general interest, such as, a periodic linear array of tiny magnets. Polarized droplets of 52Cr and 164Dy have permanent magnetic dipole moment. In a repulsive BEC on 3D OL, gap solitons having negative effective mass responsible for attraction, with the chemical potential lying in the band-gap, can be made [20, 21]. The present solitons on 1D OL, free to move in the transverse plane are bright, and not gap, solitons. We consider a DBEC of N atoms, each of mass m, using the GP equation: [6] ∂φ(r,t) (cid:34) 2 (cid:35) i = ∇ +V1D +g φ 2 +F φ(r,t) (1) OL ∂t − 2 | | with g = 4πaN, F = (cid:82) U (r r(cid:48)) φ(r(cid:48),t) 2dr(cid:48), r x,y,z ρ,z , V1D dd OL − | | ≡ { } ≡ { } ≡ V cos(2z) or V cos(2y) is the weak OL for stabilizing the soliton, U (R) = z y dd − − g (1 3cos2θ)/R3,g = 3a Nα, R = r r(cid:48), normalization (cid:82) φ(r)2dr = 1, a the dd dd dd − − scattering length, θ the angle between R and z, a = µ µ¯2m/(12πh¯2) the strength of dd 0 dipolarinteraction, µ¯ the(magnetic)dipolemomentofanatom, andµ thepermeability 0 of free space. The parameter α (1 > α > 1/2) can be tuned by a rapidly rotating − magnetic field allowing the change of the sign of dipole interaction. In (1), length is measured in units of l λ/(2π), time t in units of t = ml2/h¯, V ,V , and energy in 0 ≡ 0 0 y z units of 2E , where E = h2/(2mλ2) is recoil energy, with λ the OL wave length. R R First we consider the axially-symmetric soliton for V1D = V cos(2z) and α < 0. OL z − In this case the Lagrangian density of (1) is [22, 23] 1 1 = i(φφ(cid:63) φ(cid:63)φ )+ φ 2 +2πaN φ 4 +V1D φ 2 L 2 t − t 2|∇ | | | OL| | 1 (cid:90) + N φ 2 U (r r(cid:48)) φ(r(cid:48)) 2dr(cid:48). (2) dd 2 | | − | | For a variational study we use the Gaussian ansatz [22, 23]: φ(r,t) = exp( ρ2/2w2 − ρ − z2/2w2 +iγρ2+iβz2)/(w √w π3/4) where w and w are time-dependent widths and γ z ρ z ρ z and β are time-dependent chirps. The effective Lagrangian L (per particle) is (cid:90) (cid:18) 1 (cid:19) L dr = w2γ˙ + w2β˙ +2w2γ2 +w2β2 ≡ L ρ 2 z ρ z Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 4 t=0 t=70 (a) (b) (c) (d) Figure 1. (Color online) (a) Numerical 3D contour of an axially-symmetric soliton for g =50 and g = 15 on OL V1D = 2cos(2z). (b) The same for an anisotropic dd − OL − soliton with g = 50 and g = 20 on OL V1D = 2cos(2y) and that for a vortex dd OL − soliton with g = 5 and g = 9 on OL V1D = 2cos(2z) at t = (c) 0 and (d) 70. dd − OL − The density φ(r)2 on the contour is 0.001. | | + E +E +E , (3) kin trap int with kinetic, trap, and interaction energies given, respectively, by E = (1/2w2 + kin ρ 1/4w2),E = V exp( w2), E = N[a a f(κ)]/(√2πw2w ), where f(κ) = z trap − z − z int − dd ρ z [1+2κ2 3κ2 d(κ)]/(1 κ2),d(κ) = (atanh√1 κ2)/√1 κ2,κ = w /w . The Euler- ρ z − − − − Lagrange equations for parameters w ,w ,γ,β can be used to obtain the following ρ z equations of the widths for the dynamics of the DBEC state 1 1 N w¨ = + [2a a e(κ)], (4) ρ w3 √2πw3w − dd ρ ρ z 1 1 2N 4V w z z w¨ = + [a a h(κ)] , (5) z w3 √2πw2w2 − dd − exp(w2) z ρ z z withe(κ) = [2 7κ2 4κ4+9κ4d(κ)]/(1 κ2)2,h(κ) = [1+10κ2 2κ4 9κ2d(κ)]/(1 κ2)2. − − − − − − The widths of a stationary soliton of energy E E + E + E are obtained by kin trap int ≡ solving (4) and (5) for w¨ = w¨ = 0. ρ z To obtain a quantized vortex of unit angular momentum h¯; around z axis, we introduce a phase (equal to the azimuthal angle) in wave function [24]. This procedure introduces a centrifugal term 1/[2(x2 + y2)] in the GP equation for a vortex and we adopt this method to study an axially-symmetric vortex soliton on a 1D OL along z axis for the dipolar interaction tuned to negative values. For the anisotropic 2D soliton on OL V1D = V cos(2y) with α > 0, we consider a OL y − minimization of energy E for a soliton using the Gaussian ansatz φ(r) = exp( x2/2w2 − x− y2/2wy2 − z2/2wz2)/(√wxwywzπ3/4), with Ekin = 1/4wx2 + 1/4wy2 + 1/4wz2,Etrap = V exp( w2), E = [a+a s(k ,k ) a ]/[√2πw w w ] and − y − y int dd x y − dd x y z (cid:90) 1 3κ κ u2du x y s(k ,k ) = , (6) x y (cid:113) (cid:113) 0 1+(κ2 1)u2 1+(κ2 1)u2 x − y − where κ = w /w ,κ = w /w . x x z y y z We perform numerical simulation of the 3D GP equation (1) using the split-step Crank-Nicolson method [25]. The dipolar term is treated by fast Fourier transformation Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 5 Table 1. Numerical (n) and variational (v) energy and rms sizes E, x , y , z of (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) bright (br) and vortex (vor) solitons. Experimental parameters a (15a for 52Cr dd 0 and 130a for 164Dy), α, N for realizing these solitons are given for scattering length 0 a=5a and wavelength λ=10000 ˚A. 0 g g a (a ) α N E x y z dd dd 0 (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) n, br 50 15 15 0.418 2400 0.836 2.00 2.00 0.450 − − − v, br 50 15 130 0.0482 2400 0.814 2.086 2.086 0.423 − − − n, br 50 20 15 0.557 2400 0.752 0.533 1.38 3.65 − n, vor 5 9 130 0.289 240 0.744 4.02 4.02 0.663 − − − [22]. The error of the reported numerical results is less than 1 %. We present in figure 1 (a) the 3D contour of the axially-symmetric bright soliton for g = 50, g = 15 and dd − V1D = 2cos(2z). In figure 1 (b), we show the anisotropic bright soliton for g = 50, OL − g = 20 and V1D = 2cos(2y). For the anisotropic soliton, the numerical energy is dd OL − 0.752 in agreement with the energy 0.739 obtained from the minimization in (6). − − The anisotropy in the x z plane in figure 1 (b) is due to dipolar interaction. In figure 1 − (c) we show an axially-symmetric vortex soliton on 1D OL, V1D = 2cos(2z) for g = 5 OL − and g = 9. A relatively large g is needed to overcome the centrifugal barrier dd dd − | | and stabilize a vortex soliton. The bright solitons of figures 1 (a) and (b) are stable in real-time propagation. However, the vortex soliton with the parameters of figure 1 (c) suffers from transverse instability at large times (t > 70), which eventually leads to its destruction [26]. The snapshot of the vortex soliton after real-time propagation at t = 70 in figure 1 (d) does not, however, show any distortion or sign of instability. The numerical energy and root-mean-square (rms) sizes of the solitons of figures 1 are shown in Table I with variational results in the axially-symmetric case. In this table we also show the parameters a , α, and N for these solitons for 52Cr and 164Dy stabilized by a dd laser of wavelength λ = 10,000 ˚A, and atomic scattering length a = 5a obtained using 0 a Feshbach resonance. For the axially-symmetric bright soliton, we have a conservative system with two degrees of freedom w and w with Lagrangian (3). The stable state appears and ρ z disappears by saddle-center bifurcation [17] as V is increased as shown in figure 2 (a) z for g = 50 and g = 15, where the unstable stationary (M and M ) and stable (S) dd 1 2 − states are shown in the w versus V plot. For small V (< 0.4278) there exists only the ρ z z unstable stationary state M . At V = 0.4278 and w 10 a stable (S) and a unstable 1 z ρ ≈ stationary(M )stateappear“outofnothing”bysaddle-centerbifurcation. Withfurther 2 increase of V , the center S comes towards the saddle M and the two disappear “to z 1 nothing” by a reverse (sub-critical) saddle-center bifurcation at V = 12.2, whereas the z state M moves towards infinity [17]. We show the equal-energy variational contours in 2 the w versus w phase plot for different V in figures 2 (b) (h), where the positions of z ρ z − the stable and unstable stationary states are also shown. Figures 2 (d) (g) show close − ups of the appearance and disappearance of the state S by saddle-center bifurcations. Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 6 40 0.8 1 2 (a) M (b) .0 (c) 2 5 0.5 1. 30 0.6 5 S 1 g = 50 1. 0.7 2 -0.8 wρ20 gd = −15 wz 0. -0.7 0 0.4 10 M1 M1 M1 −1 0.2 S V =0 V =2 00 2 4 6Vz8 10 12 14 00..571.10.5 1 2wρ 3 z 4-0.8-0.7 1 2wρ 3 z 4 −2 1.6 0.4 2.0 (d) Vz =0.4278 (f) Vz =0.45 (h) 0.09 1.5 0.3 wz wz 1.8 0. 1.4 0.2 061 0.060 0.08 V =12.2 z 1.6 M2 11..36 00..14 wz 0.07 (e) S (g) 1.4 1.5 M2 0.3 0.06 wz1.4 Vz =0.429 wz0.2 V =11.8 SM1 1.2 S0.060 0.062 0.063 0.05 z 1.3 0.1 1.0 8.0 8.5 9.0 9.5 10.0 0.5 1.0 1.5 2.0 4 8 12 16 w w w ρ ρ ρ Figure 2. (Color online) (a) Bifurcation diagram showing saddle-center bifurcation involvingthestable(center,S)andunstablestationarystates(saddles,M andM )in 1 2 the w versus V plot. The lines correspond to solutions of the variational equations. ρ z The thresholds for saddle-center bifurcation are at V = 0.4278 and 12.2. (b) (h) z − The equal-energy contours in the w versus w phase plot for different V . A center ρ z z (open circle) S corresponds to a minimum of energy surrounded by closed loops in these contour plots. A saddle (solid square), M or M , corresponds to intersection 1 2 of two equal-energy lines. In (d) (g) the appearance of a saddle and a center for a − small change of V near the thresholds at V = 0.4278 and 12.2 is illustrated. In all z z cases g =50 and g = 15. dd − To perform a linear stability analysis of the axially-symmetric states, we rewrite (4) and (5) as [16] x˙ = x , x˙ = x , (7) 1 3 2 4 1 1 N (cid:20) (cid:18)x (cid:19)(cid:21) 1 x˙ = + 2a a e , (8) 3 x3 √2πx3x − dd x 1 1 2 2 1 4V x 1 2N (cid:20) (cid:18)x (cid:19)(cid:21) z 2 1 x˙ = + a a h , (9) 4 x3 − exp(x2) √2πx2x2 − dd x 2 2 1 2 2 where (x ,x ,x ,x ) (w ,w ,w˙ ,w˙ ). These equations for widths can be written 1 2 3 4 ρ z ρ z ≡ as x˙ f(x), where x x ,x ,x ,x . If x(0) denote the fixed points with x˙ = 0, 1 2 3 4 ≡ ∈ { } so that f(x(0)) = 0, then the linearization matrix is J ∂f(x)/∂x [16]. An x=x(0) ≡ | examination of eigenvalues of J reveals the nature of stability of the states. The eigenvalues come in pairs λ and lead to exponential growth unless all of them are ± Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 7 imaginary corresponding to a spectrally stable equilibrium, which is of interest in the present context. For the parameters of figure 1 (a), there exist the saddle M at 1 x(0) (0.9379,0.2147,0,0), and the center S at x(0) (2.9497,0.5976,0,0), and the ≡ ≡ saddle M at x(0) (22.8943,2.3485,0,0) of which S and M are shown in figure 2 2 2 ≡ (h). The eigenvalues are ( 4.8644, 19.9241i) for M , ( 4.4835i, 0.2274i) for S, 1 ± ± ± ± and ( 0.4751, 0.0018i) for M . The center with pairs of pure imaginary eigenvalues 2 ± ± confirm its stability. Using variational equations, we analyze the appearance of axially-symmetric bright solitons using the phase plots of g versus g for V = 2 in figure 3 (a). For g in a dd z dd | | | | window of critical values, stable bright solitons can be formed. For smaller g , there is dd | | too much repulsion and the system expands to infinity and for larger g , there is too dd | | much attraction leading to collapse allowing only unstable stationary states. In figure 3 (b) we plot the variational and numerical rms sizes and energies for g corresponding dd | | to the numerical points ((cid:63)) in figure 3 (a). Next, we consider a stable soliton array by mounting tiny DBEC bright solitons along the supporting OL sites. Such solitons will be interacting due to long-range dipolar interaction. We prepare a stable array of axially-symmetric solitons each with g = g = 5 by putting them on alternate sites of OL V1D = 2cos(2z) at dd OL − − x = y = 0,z = (2n+1)π,n = 0,1,2,3. Such an array of bright solitons with small g ± and g are stable, whereas those of large g and g , e.g. the one of figure 1 (a), are dd dd | | | | unstable. In figure 4 (a) and (b) we show the initial array and the the final profile after real-time propagation at t = 200. A similar array of anisotropic solitons has a finite life and is destroyed at large times. Finally, we investigate the collision of two axially-symmetric bright solitons with 200 10 10 × Unstable 8 150 x (v) xx ((nn)) (Collapse) E h i hh ii − 6 z (v) zz ((nn)) dd|100 S t a ble z,i h Ei (v) hh EEii ((nn)) g V = 2 h − 10 −− | z , 4 × i Unstable x 50 h (Expansion) 2 (a) (b) 0 0 0 200 400 600 800 1000 0 100 200 300 400 500 g g Figure 3. (Coloronline)(a)Thephaseplotof g versusg fromvariationalanalysis dd | | of the axially-symmetric solitons showing the region of stable solitons for V = 2. z The (cid:63)’s denote the numerical points showing the stable-unstable boundary. (b) The numerical(n)andvariational(v)rmssizesandenergyversusg for g corresponding dd | | to the (cid:63)’s in (a). Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 8 2 t = 0 0 2 t = 0 −2 3.2 0 2 −2 6.4 0 2 −2 9.6 0 (a) 2 −2 z 12.8 0 2 −2 16.0 0 2 t = 200 −2 19.2 0 2 −2 22.8 0 2 −2 (c)20 10 0x 10 2025.6 (b) 20 − − − 20 10 0 10 20 − − 0.00 0.01 0.02 0.03 Figure 4. (Color online) 3D contour of the stable array of eight axially-symmetric solitons each with g = g = 5 on OL V1D = 2cos(2z) at (a) t = 0 and at (b) − dd OL − t=200. The density at contour is 0.001. (c) Contour plot of density φ(x,y,0,t)2 of | | two colliding solitons each with g = 5 and g = 3 and velocity 1, before, during, dd − and after collision. g = 5 and g = 3 each, placed at x = 12.8, y = z = 0 at t = 0. Each soliton is dd − ± given a velocity of v = 1 towards center x = 0 by a phase factor exp( ix), respectively, ± in the initial wave functions. The collision dynamics is illustrated in figure 4 (c) where we show the snapshots of contour plots of density φ(x,0,z,t) 2 at different times. The | | solitons come towards each other, interact at x = 0 and t 12.8 and come out without ≈ deformation showing their robustness. In this simulation, not only the parameters of two individual solitons should lead to a stable state, the combined nonlinearities 2g and 2g should also correspond to a stable configuration in figure 3 (a) to avoid collapse dd during collision. It was demonstrated in [13] that, under harmonic confinement, after collision at very low velocities, two quasi-2D dipolar BEC solitons may merge together to form a single soliton molecule. However, in the present case the solitons appear in a narrow window of nonlinearities g and g , as can be seen from figure 3. If two equal dd 2D solitons, as in figure 4 (c), coalesce at low velocities, the nonlinearities of the merged soliton molecule will be outside the domain of stability in figure 3. Hence, the formation of soliton molecule is mostly not possible in the present case. At large velocities the solitons of [13] undergo quasi-elastic collision quite similar to the present collision shown in figure 4 (c). Also, both quasi-elastic collision at large velocities and merging at low velocitiesoftwoquasi-1Dsolitonsundertransverseharmonicconfinementwasillustrated in [12]. To summarize, we studied different types of 2D bright solitons in a DBEC with repulsive atomic interaction using the solution of the 3D GP equation. Anisotropic stable 2D bright solitons in DBEC are possible on a weak 1D OL perpendicular to the Two-dimensional dipolar Bose-Einstein condensate bright and vortex solitons 9 polarization direction. Axially-symmetric stable 2D bright and vortex solitons in DBEC can be generated on a weak 1D OL along the polarization direction when the dipolar interaction is tuned to negative values [10]. In this sign-changed dipolar-interaction configuration, bright and vortex solitons are stable due to the long-range attractive dipolar interaction in the quasi-2D shape. In the axially-symmetric case, an 1D stable array of tiny solitons, placed on alternate OL sites, can be made. Such 1D array with empty OL sites between solitons is of interest in condensed matter physics [18, 19] and bears some similarity with stable checkerboard pattern of DBEC on 2D OL with emptysitesinbetween[27], botharisingduetodipolarinteraction. Theelasticnatureof collisionoftwoaxially-symmetricsolitonsisalsodemonstrated. Withpresenttechnology these stable 2D solitons and their 1D arrays can be created and studied in laboratory. Acknowledgments We thank Prof. B. A. Malomed for a critical reading of the manuscript and FAPESP (Brazil), CNPq (Brazil), DST (India), and CSIR (India) for partial support. 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