Structure and stability of two-dimensional Bose- Einstein condensates under both harmonic and lattice confinement

Transcription

1 Universit of Massachusetts Amherst Amherst Mathematics and Statistics Department Facult Publication Series Mathematics and Statistics 28 Structure and stabilit of two-dimensional Bose- Einstein condensates under both harmonic and lattice confinement KJH Law PG Kevrekidis Universit of Massachusetts - Amherst, kevrekid@math.umass.edu BP Anderson R Carretero-Gonzalez DJ Frantzeskakis Follow this and additional works at: Part of the Phsical Sciences and Mathematics Commons Recommended Citation Law, KJH; Kevrekidis, PG; Anderson, BP; Carretero-Gonzalez, R; and Frantzeskakis, DJ, "Structure and stabilit of two-dimensional Bose-Einstein condensates under both harmonic and lattice confinement" (28). JOURNAL OF PHYSICS B-ATOMIC MOLECULAR AND OPTICAL PHYSICS. 38. Retrieved from This Article is brought to ou for free and open access b the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Facult Publication Series b an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact scholarworks@librar.umass.edu.

2 arxiv:83.324v2 [nlin.ps] 6 Ma 28 Structure and stabilit of two-dimensional Bose-Einstein condensates under both harmonic and lattice confinement K.J.H. Law and P.G. Kevrekidis, B.P. Anderson 2, R. Carretero-González 3, and D.J. Frantzeskakis 4 Department of Mathematics and Statistics, Universit of Massachusetts, Amherst, MA 3-4 USA law@math.umass.edu 2 College of Optical Sciences and Department of Phsics, Universit of Arizona, Tucson, AZ 872 USA 3 Nonlinear Dnamical Sstems Group, Department of Mathematics and Statistics, San Diego State Universit, San Diego, California USA 4 Department of Phsics, Universit of Athens, Panepistimiopolis, Zografos, Athens 784, Greece Abstract. In this work, we stud two-dimensional Bose-Einstein condensates confined b both a clindricall smmetric harmonic potential and an optical lattice with equal periodicit in two orthogonal directions. We first identif the spectrum of the underling two-dimensional linear problem through multiple-scale techniques. Then, we use the results obtained in the linear limit as a starting point for the eistence and stabilit analsis of the lowest energ states, emanating from the linear ones, in the nonlinear problem. Two-parameter continuations of these states are performed for increasing nonlinearit and optical lattice strengths, and their instabilities and temporal evolution are investigated. It is found that the ground state as well as some of the ecited states ma be stable or weakl unstable for both attractive and repulsive interatomic interactions. Higher ecited states are tpicall found to be increasingl more unstable. PACS numbers: 3.7.Lm, 3.7.Kk Submitted to: J. Phs. B: At. Mol. Phs. URL:

3 Structure and stabilit of 2D BECs under harmonic and lattice confinement 2. Introduction The last decade has witnessed a tremendous amount of research efforts in the phsics of atomic Bose-Einstein condensates (BECs) [, 2]. The stud of BECs has ielded a wide arra of interesting phenomena, not onl because of ver precise eperimental control that eists over the relevant eperimental procedures [3, 4], but also because of the intimate connections of the description of dilute-gas BECs with other areas of phsics, such as superfluidit, superconductivit, lasers and coherent optics, nonlinear optics, and nonlinear wave theor. Of particular emphasis in much eperimental and theoretical work is the setting of a BEC trapped in periodic potentials, usuall combined with an additional harmonic trapping potential. From the standpoint of nonlinear interactions, mathematical descriptions of BECs held in purel harmonic traps are now well known. Nevertheless, apart from studies focusing on the transition between superfluidit and an insulator state [], relativel little attention has been given to an understanding of the varieties of man-bod states that ma possibl eist with intermediate lattice strengths, where phase coherence is maintained across the sample. A more complete understanding of BEC behavior in such lattice potentials is relevant and important to current work with BECs, and to an even broader arra of topics, in particular discrete nonlinear optics and nonlinear wave theories. Such regimes of BEC phsics are eperimentall and theoreticall accessible, and comparisons between theoretical and eperimental results are certainl possible. Here, we present a theoretical stabilit eamination of BECs with either attractive or repulsive interatomic interactions in a combined harmonic and periodic potential. Man of the common elements between BECs and other areas of phsics, and in particular optics, originate in the eistence of macroscopic coherence in the man-bod state of the sstem. Mathematicall, BEC dnamics are therefore often accuratel described b a mean-field model, namel a partial differential equation of the nonlinear Schrödinger (NLS) tpe, the so-called Gross-Pitaevskii (GP) equation [, 2, 3]. The GP equation is particularl successful in drawing connections between BEC phsics, nonlinear optics and nonlinear wave theories, with vortices and solitar waves eamples of common elements between these areas. The GP equation is a classical nonlinear evolution equation (with the nonlinearit originating from the interatomic interactions) and, as such, it permits the stud of a variet of interesting nonlinear phenomena. These phenomena have primaril been studied b treating the condensate as a purel nonlinear coherent matter-wave, i.e., from the viewpoint of the nonlinear dnamics of solitar waves. Relevant studies have alread been summarized in various books (see, e.g., Ref. [4]) and reviews (see, e.g., Refs. [6] for bright matter-wave solitons, [7, 8] for vortices in BECs, [9] for dnamical instabilities in BECs, [, ] for nonlinear dnamics of BECs in optical lattices). On the other hand, man static and dnamic properties of BECs confined in various tpes of eternal potentials can be studied b starting from the non-interacting limit, where the nonlinearit is considered to be negligible. The basic idea of such an approach

4 Structure and stabilit of 2D BECs under harmonic and lattice confinement 3 is that in the absence of interactions the GP equation is reduced to a linear Schrödinger equation for a confined single-particle state; in this limit, and in the case of, e.g., a harmonic eternal potential, the linear problem becomes the equation for the quantum harmonic oscillator characterized b discrete energies and corresponding eigenmodes [2]. Eploiting this simple phsical picture, one ma then use analtical and/or numerical techniques for the continuation of these linear eigenmodes supported b the particular tpe of the eternal trapping potential into nonlinear states as the interactions become stronger. This idea has been eplored at the level of one-dimensional (D) [3] and higher-dimensional states [4] in the case of a harmonic trapping potential, where nonlinear stationar modes were found from a continuation of the (linear) states of the quantum harmonic oscillator. The same problem has been studied in the framework of the so-called Feshbach resonance management technique in Ref. [], where a linear temporal variation of the nonlinearit was considered. The continuation of the linear states to their nonlinear counterparts has also been eplored from the point of view of bifurcation and stabilit theor [6]. Finall, in the same spirit but in the twodimensional (2D) setting, radiall smmetric nonlinear states of harmonicall trapped pancake-shaped condensates were recentl investigated in Ref. [7]. Importantl, all of the above studies provide a clear phsical picture of how genuinel nonlinear states of harmonicall confined BECs (such as dark and bright matter-wave solitons in D or ring solitons and vortices in 2D), are connected to and emanate from the eigenmodes of the quantum harmonic oscillator. Similar considerations also hold for BECs confined in optical lattices. In this case, pertinent nonlinear stationar states (such as spatiall etended nonlinear Bloch waves, truncated nonlinear Bloch waves, matter-wave gap solitons in D, and gap vortices in 2D and 3D), can be understood b the structure of the band-gap spectrum of the linear Bloch waves supported in the non-interacting limit (see, e.g., Ref. [8] and references therein). However, there are onl few studies for condensates confined in both harmonic and optical lattice potentials, and these are basicall devoted to the dnamics of particular nonlinear structures (such as dark [9] and bright [2] solitons in D, and vortices in 2D [2]). Thus, the structure of condensates confined in such superpositions of harmonic and periodic potentials remains, to the best of our knowledge, largel uneplored. Nevertheless, such a stud is particularl relevant to current work with BECs, and even suggests new avenues for eploration. In particular, one might ask whether the addition of a weak optical lattice might increase the stabilit of ecited states that are known to be unstable in harmonic traps. Stabilit, if it is found, ma add new realistic options for the engineering of new quantum states of BECs. It ma also be interesting to investigate the Mott insulator transition b starting from a stable ecited state of a weak optical lattice. Also, the transport of ecited states (which is not discussed in this paper) through a lattice structure ma have application in future BEC interferometr eperiments. Finall, the advances of far-off-resonant optical trapping techniques allow for the creation of strongl pancake-shaped condensates that ma be confined b harmonic and spatiall periodic components, and we epect that

5 Structure and stabilit of 2D BECs under harmonic and lattice confinement 4 the theoretical considerations described here ma be directl eplored using current eperimental techniques. Our aim in the present work is to contribute to this direction and stud the structure and the stabilit of a pancake-shaped condensate confined b the combination of a harmonic trap and a periodic potential, with periodicit in two orthogonal directions. We will adopt the above mentioned approach of continuation of linear states to nonlinear ones, thus providing a host of interesting solutions that have not been eplored previousl and et should be tractable within presentl available eperimental settings. In particular, our analsis starts b first considering the non-interacting limit. In this regime, we emplo a multiscale perturbation method (which uses the harmonic trap strength as a formal small parameter) to find the discrete energies and the corresponding eigenmodes of the pertinent single-particle Schrödinger equation with the combined harmonic and periodic potential. We then use this linear limit as a starting point for initializing a 2D nonlinear solver that identifies the relevant stationar nonlinear eigenstates as a function of the chemical potential (i.e., the nonlinearit strength) and of the optical lattice depth. Once the basic structure of the condensate is found, we subsequentl perform a linear stabilit analsis of the nonlinear modes that can be initiated b the noninteracting ground and first few ecited states. When nonlinear states are found to be unstable, we use direct numerical simulations to stud their dnamics and monitor the evolution of the relevant instabilit. Essential results that will be presented below are the following: for a fied harmonic trap strength, there eist certain regions in the parameter plane defined b the chemical potential and the optical lattice depth, where not onl the ground state, but also ecited states are stable or onl weakl unstable. Particularl, an ecited state with a shape resembling an out-of-phase matter-wave soliton pair (for attractive interactions) is found to persist for long times, being stable (weakl unstable) for attractive (repulsive) interatomic interactions. Thus, the ground state and the aforementioned ecited state have a good chance to be observed in a real eperiment with either attractive or repulsive pancake BECs. Similar conclusions can be drawn for more comple states, such as a quadrupolar one which ma also be stable in the attractive case, however, higher ecited states are tpicall more prone to instabilities, as is shown in our detailed numerics below. The paper is organized as follows: In Section II we present the model and stud analticall the non-interacting regime. The continuation of the linear states to the nonlinear ones, as well as the stabilit properties of the nonlinear states are presented in Section III. Finall, in Section IV, we summarize our findings and present our conclusions. 2. The model and its analtical consideration At sufficientl low temperatures, and in the framework of the mean-field approach, the condensate dnamics can be described b the order parameter Ψ(r, t). We assume

6 Structure and stabilit of 2D BECs under harmonic and lattice confinement that the condensate is kept in a highl anisotropic trap, with the transverse (, ) and longitudinal (z) trapping frequencies chosen so that ω = ω ω ω z. In such a case, the condensate has a nearl planar, so-called pancake shape (see, e.g., Refs. [22] for relevant eperimental realizations), which allows us to assume a separable wave function, Ψ = Φ(z)ψ(, ), where Φ(z) is the ground state of the respective quantum harmonic oscillator. Then, averaging of the underling three-dimensional (3D) GP equation along the longitudinal direction z [23] leads to the following 2D GP equation for the transverse component of the wave function (see also Refs. [8, 9, 4]): i h t ψ = h2 2m ψ + g 2D ψ 2 ψ + V et (, )ψ. () Here, is the 2D Laplacian, m is the atomic mass, and g 2D = g 3D / 2πa z is an effective 2D coupling constant, where g 3D = 4π h 2 a/m (a being the scattering length) and a z = h/mω z is the longitudinal harmonic oscillator length. Finall, the potential V et (, ) in the GP Eq. () is assumed to consist of a harmonic component and a square 2D optical lattice (OL) created b two pairs of interfering laser beams of wavelength λ: V et (, ) = 2 mω2 r 2 + V [cos 2 (k) + cos 2 (k)]. V H (r) + V OL (, ) (2) In the above epression, r , while the optical lattice is characterized b two parameters, namel its depth V and its periodicit d = π/k = (λ/2)/ sin(θ/2), where θ is the angle between the two beams that create the -direction lattice, and between the two beams that create the -direction lattice. Measuring length in units of a L = d/π, time in units of ωl = h/e L, and energ in units of E L = 2E rec = h 2 /ma 2 L (where E rec is the lattice recoil energ), the GP Eq. () can be put into the following dimensionless form: i t ψ = 2 ψ + s ψ 2 ψ + V (, )ψ. (3) In the normalized GP Eq. (3), the wavefunction is rescaled as ψ g 2D /E L ψ ep [i(v /E L )t], the parameter s is given b s = sign(g 2D ) = ± (with s = + or s = corresponding, respectivel, to repulsive or attractive interatomic interactions), while the normalized trapping potential V (, ) is now given b: V (, ) = 2 Ω2 r 2 + V (cos(2) + cos(2)). (4) In the above equation, the normalized lattice depth V is measured in units of 4E rec, while the normalized harmonic trap strength is given b Ω = a2 L = ω, () a 2 ω L where a = h/mω is the transverse harmonic oscillator length. Note that using realistic parameter values (see, e.g., Ref. [24]), namel, a lattice periodicit.3 m, a recoil energ E rec /h 6KHz (assuming an atomic mass corresponding to 87 Rb), and

7 Structure and stabilit of 2D BECs under harmonic and lattice confinement 6 E/ Ω (a) E/ Ω (b) 2 n n 3 4 u().. (c) V() u().. (d) V() u().. (e) V() Figure. (Color Online) Panel (a) shows the energ spectra corresponding to a purel parabolic potential (pluses), a parabolic and lattice one found numericall (circles) and parabolic and lattice potential found analticall (stars). Shown are onl the first few eigenvalues for Ω =. and V =.3. A similar result is demonstrated in panel (b) but for a larger lattice depth, V =. (and the same value of Ω). For the latter case (V =.), panels (c), (d) and (e) show the first few eigenmodes for the parabolic potential (thick solid), parabolic and lattice potential computed numericall (thin solid), and the same ones given b Eq. (4) (dashed). The potential (rescaled for visibilit) is shown b the dash-dotted line. using a transverse trap frequenc ω = 2π Hz, the parameter Ω is of order of 4 ; thus, it is a natural small parameter of the problem. Our analsis starts b considering the non-interacting limit s, in which the GP equation becomes a linear Schrödinger equation. Then, seeking stationar localized solutions of the form ψ(,, t) = ep( ie m,n t)u m,n (, ) [where E m,n are discrete energies and u m,n (, ) are the corresponding linear eigenmodes], and rescaling spatial variables b Ω, we obtain the following equation: E m,n Ω u m,n = 2 u m,n + 2 (2 + 2 )u m,n (6) + V [ ( ) ( 2 2 cos + cos )]u m,n. Ω Ω Ω The net step is to separate variables through u(, ) = u m ()u n () and split the energ into E m,n = E m +E n, to obtain two D eigenvalue problems of the same tpe as above, namel, d 2 u m 2 d u m + V ( ) 2 Ω cos u m = E m Ω Ω u m, (7) and a similar one for (with replaced b and the subscript m replaced b n).

8 Structure and stabilit of 2D BECs under harmonic and lattice confinement 7 We now restrict ourselves to the phsicall relevant regime of < Ω as discussed above. In this case, we ma use ν Ω as a formal small parameter and develop methods of multiple scales and homogenization techniques [6] in order to obtain analtical predictions for the linear spectrum. In particular, introducing the fast (i.e., rapidl varing) variable X = /ν and rescaling the energ as ε m = E m /Ω, the eigenvalue problem of Eq. (7) is epressed as follows: ( ) where 2 ν 2 L H ν X + L OL 2 u m = ν 2 ε m u m, (8) L H = , (9) 2 L OL = 2 X + V 2 cos(2x) () (note that V /Ω was treated as an O() parameter). Additionall, we consider a formal series epansion (in ν) for u m and ε m [6], namel, u m = u + νu + ν 2 u , () ε m = ε 2 ν 2 + ε ν + ε + ε ν (2) To this end, substitution of this epansion in the eigenvalue problem of Eq. (8) and use of the solvabilit conditions for the first three orders of the epansion [i.e., O(), O(ν) and O(ν 2 )] ields the following results for the eigenvalue problem of the original operator. The energ of the m-th mode can be approimated b: E m = ( 4 V 2 + ) ( 4 V 2 Ω m + 2 while the corresponding eigenfunction is given b: u m () = c m H m V 2 [ V π 2 cos 4 ep ), (3) 2 2 V 2 2 ( )] 2, (4) Ω where c m = (2 m m! π) (/2) is the normalization factor and H m () = e 2 ( ) m (d m /d m )e 2 are the Hermite polnomials. Combining the results in the two orthogonal directions and ields a total energ eigenvalue E m,n = ( 2 V V 2 ) Ω (n + m + ), () and a corresponding eigenfunction which up to normalization factors can be written as: [ u m,n (, ) H m V ( )] 2 V 2 2 cos Ω /2 H n V [ V ( )] 2 2 cos Ω /2

9 Structure and stabilit of 2D BECs under harmonic and lattice confinement 8 r2 ep. (6) 2 V 2 2 The particularl appealing feature of this epression is that it allows us to combine various (ground or ecited) states in the direction with different ones along the direction. In this work we will focus on the simplest possible combinations of m, n {,, 2} and eamine the various states generated b the combinations of these, which we hereafter denote m, n. In particular, below we will focus on the ground state, and the ecited states,,, +,,,, and 2,. Our aim is to investigate which of these states persist in the nonlinear regime in both cases of attractive and repulsive interatomic interactions, and stud the stabilit of these states in detail. Notice that we will onl illustrate (b an appropriate curve in the numerical results that follow) the linear limit, E m,n of Eq. () above, for the case of attractive interactions. 3. Numerical Results 3.. The non-interacting limit We start b eamining the validit of the above analtical predictions concerning the linear limit of the problem, namel, Eqs. () and (6). The results are summarized in Fig.. Panel (a) shows the D harmonic oscillator energ spectrum (for Ω =.) and compares it with the energ spectrum obtained from numerical and approimate theoretical [see Eq. (3)] solutions of the combined harmonic and optical lattice potential for V =.3. Panel (b) offers a similar comparison but for a larger lattice depth, namel V =.. One can clearl see that the theoretical calculation approimates ver accuratel the numerical results for the first few states (i.e., n =,, 2), while deviations become more significant for higher-order ecited states. Panels (c), (d) and (e) show the zeroth, first and second eigenfunction of the purel harmonic potential (thick solid line), as well as of the harmonic trap and optical lattice potential as found numericall (thin solid line) and analticall [given b Eq. (4)]. The green dash-dotted line represents the form of the combined potential. We once again note the good agreement of our analtical results above in comparison with the full numerical computation The approach for the interacting case We now consider the full nonlinear problem of Eq. (3). In the following, we will monitor the two-dimensional, V, parameter space (where is the chemical potential of the relevant modes) for nonlinear ecitations that stem from the linear spectrum of the problem. We perform the relevant analsis first in the case of attractive interatomic interactions and then in the case of repulsive ones. Notice that the parameter Ω will be fied to a relativel large value, namel Ω =.; this is done for convenience in our numerical simulations (such large values of Ω correspond to smaller condensates that can be analzed numericall with relativel coarser spatial grids), but we have

10 Structure and stabilit of 2D BECs under harmonic and lattice confinement 9 checked that our results remain qualitativel similar for smaller values of Ω (results not shown here). Nevertheless, it should be noted that even such a value of Ω, together with the considered range of values of the other normalized parameters (chemical potential, number of atoms, lattice depth, etc see below) is still phsicall relevant. For eample, our choice ma realisticall correspond to a 87 Rb condensate containing, atoms, confined b a harmonic potential with frequencies ω z = 2ω = 2π 24Hz (so that ω = 2π 2 Hz) and an optical lattice potential with a periodicit d 3m. The recoil energ in this case is E rec /h = 6 Hz (so that ω L = 2π 2 Hz, giving Ω =.), and a lattice depth of V =.3 corresponds to.2 E rec. To further set the scale for the simulations described below, such a BEC with repulsive interactions in the purel harmonic trap (where the optical lattice is not applied) would have a chemical potential of. in our dimensionless units. In the following sections, stationar solutions of the full nonlinear problem are sought in the form ψ(,, t) = ep( i m,n t)u m,n (, ), (7) where m,n (which is the nonlinear analog of the energ E m,n found in the non-interacting limit) represents the chemical potential. Note that we will henceforth avoid using subscripts m and n when the meaning is clear, in the interest of avoiding notational clutter Attractive interatomic interactions Eistence and Stabilit We begin b looking at the case of attractive interatomic interactions. The most fundamental solutions are those belonging to the, branch, which represents the ground state of the sstem and is shown in Fig. 2. The top left panel of this figure shows the diagnostic that we will tpicall use to follow the V surface, namel the rescaled number of particles N(V, ) = u m,n 2 dd as a function of the chemical potential introduced above, and the optical lattice depth V. Essentiall, the grascale values in this plot correspond to the number of atoms needed to obtain a particular chemical potential with a particular lattice depth; lighter values correspond to more atoms. As N becomes smaller, through the appropriate variation of, we approach the linear limit so one epects the solution to degenerate to the corresponding linear eigenmode (for tending to the corresponding eigenvalue of the linear problem). Figure 2 shows our observations for this fundamental branch, which seems to disappear for m,n (V ) E m,n (V ) = V ( V ) 2 4 Ω (n + m + ). Naturall the surface degenerates to its linear limit for m,n (V ) E m,n (V ) and the number of particles is a decreasing function of, contrar to what is the case in the repulsive nonlinearit (see below). This is a well-known difference between the two cases that has been documented elsewhere (see, e.g., Refs. [, 6]). It is relevant to indicate here that although there is a direct correspondence between the atom number N and the chemical potential, we opt to illustrate our results as a function of (and V ), since the latter is the relevant parameter entering the mathematical setup of the problem and it is the one for which

11 Structure and stabilit of 2D BECs under harmonic and lattice confinement we developed an analtical prediction in the linear limit. Nevertheless, we also give N(V, ), so as to associate in each case the relevant chemical potential (and lattice strength) for a given configuration with the corresponding phsical quantit, i.e., the atom number. It is important to highlight here that that the numerical computations have been performed in a domain of size 2 2, with = =.. The size of the grid weakl affects the value of the respective eigenvalues. In particular, the energ eigenvalues in the non-interacting limit of the one-dimensional decoupled eigenvalue problem (with V =.) for the n = and n = mode that we report below are E.43 and E.7639 respectivel for this coarser domain, while for the a domain of size 3 with =. the become.48 and.7692 respectivel (computations show the discrepanc is uniforml smaller for smaller values of V ). This feature will weakl affect the quantitative aspects of the results that follow (in essence, providing an error bar in the estimates below of 4 and similar for the eigenvalues λ introduced below), but is essentiall necessar, given the limitations of standard eigenvalue solvers for such big Jacobian eigenvalue problems. The linear stabilit of the solutions is analzed b using the following standard ansatz for the perturbation, ψ = e it [ u(, ) + ( a(, )e λt + b (, )e λ t )], (8) where λ = λ r + iλ i is the generall comple eigenvalue (subscripts r and i denote, respectivel, the real and imaginar parts of λ) of the ensuing Bogoliubov-de Gennes equations [, 2, 3, 4], and (a, b) T is the corresponding eigenvector. The real part λ r of the eigenvalue then determines the growth rate of the potential instabilit of the solution, since for positive real values the perturbation will grow eponentiall in time. The imaginar part λ i denotes the oscillator part (frequenc) of the relevant eigenmode. The top right of Fig. 2 depicts the stabilit domain, denoted b S(V, ) = ma λ (λ r ), in terms of the maimum real part of all eigenvalues as a function of the lattice depth V and the chemical potential. This quantit S is a particularl important one from a phsical point of view since if a perturbation to the sstem has initiall a projection p() onto the most unstable eigenmode of the linearization, then this perturbation will grow in time according to p(t) = p() ep(st) while the solution is sufficientl close in space to the original profile. Hence, given the initial condition profile [which determines p()] and S, we can determine for an unstable configuration the time t required for the instabilit to manifest itself, i.e., for p(t) to become of the order of the solution amplitude. It is important to note, in connection to our numerical linear stabilit results, that the, branch can become unstable for < cr (V ) (see top right panel of Fig. 2) due to the appearance of a real pair of eigenvalues. This instabilit for large N is something that ma be epected in the case of attractive interactions under consideration, as the corresponding 2D GP equation for an a homogeneous BEC (i.e., without an eternal potential) is well-known to be subject to collapse [2]. However, it should also be

12 Structure and stabilit of 2D BECs under harmonic and lattice confinement V V λ i λ r Figure 2. (Color Online) The ground state in the case of attractive interatomic interactions. The top left panel shows the rescaled number of particles N(V, ) = u 2 dd as a function of the amplitude of the optical lattice V and the chemical potential ; the red line represents the approimation of the energ eigenvalue E(V ) of the linear problem given b Eq. (). For each V, the number of atoms, N V () approaches zero in the limit E(V ). The top right panel shows the stabilit domain S(V, ) = ma(λ r ); the red line here corresponds to the stabilit window S < 4. It is clear that for each V, there is a window of values of for which S V () < 4. This is epected, since the attractive nature of the interatomic interactions leads to blowup above a critical value of N. The bottom left and right panels depict, respectivel, a contour plot of an unstable solution u in the (, ) plane and its corresponding spectral plane (λ r, λ i ) [for (V, ) = (.2,.23) corresponding to the parameter value depicted b the red circle in the panels of the top row]. The bottom-left colorbar represents atomic densit. In the bottom-right plot, the presence of real eigenvalue pairs denotes instabilit (its growth rate is given b the magnitude of the real part), while the imaginar pairs indicate the frequencies of oscillator modes.

13 Structure and stabilit of 2D BECs under harmonic and lattice confinement V V.. λ i λ r Figure 3. (Color Online) Similar to Fig. 2 but for the case of the, state for attractive interactions. The results shown in the bottom row correspond to parameter values (V, ) = (.2,.8) (see red circle in top panels). epected that ver close to the linear limit the growth rate of the instabilit is essentiall zero (cf. with the top left panel of Fig. 2 of Ref. [7] for V = which is not shown here). Essentiall, the potential appears to stabilize the solitar wave against dispersion in this regime (i.e., close to the linear limit), but cannot stabilize it against the catastrophic collapse-tpe instabilit. Furthermore, in the presence of the optical lattice we can observe that there is alwas a range of chemical potentials for which the condensate is stabilized, in accordance with what was originall suggested in Ref. [26]. Furthermore, even in the 3D case it is in principle possible to arrest collapse b appropriate choices of the parameters [27]. Net, we consider real-valued solutions with m + n =. The, state (again in the case of attractive interactions) is shown in Fig. 3. This branch is alwas unstable, due to up to two real eigenvalue pairs and one comple quartet. A tpical eample of the branch in the bottom panels of the figure reveals this instabilit. The, +, configuration for the attractive interactions case is shown in

14 Structure and stabilit of 2D BECs under harmonic and lattice confinement V V λ i λ r.. λ i λ r Figure 4. (Color Online) The state, +, for attractive interatomic interactions. The top row is similar to Fig., the middle row is for parameter values (V, ) = (.3,.2) (see blue cross in top panels), and the bottom one is for (.3,.48) (see red circle in top panels).

15 Structure and stabilit of 2D BECs under harmonic and lattice confinement V V λ i λ r λ i λ r 3 Figure. (Color Online) The state, for attractive interatomic interactions. The laout of the figure is similar to the one used in the previous figures. The parameters for the solution depicted in the middle and bottom rows are (V, ) = (.2,.8) (see blue cross in top panels) and (.4,.39) respectivel (see red circle in top panels).

16 Structure and stabilit of 2D BECs under harmonic and lattice confinement V.2.4 V λ i λ r Figure 6. (Color Online) The state 2, for attractive interatomic interactions. The laout of the figure is similar to the one used in the previous figures. The parameters for the solution depicted in the bottom row are (V, ) = (.3,.) (see red circle in top panels). Fig. 4. This configuration turns out to be unstable in a large fraction of the regime of parameters considered due to a quartet of comple eigenvalues. However, remarkabl, as V and are increased and decreased respectivel, it is possible to actuall trap this state in a linearl stable form (eliminating the relevant oscillator instabilit). This indicates that it would be of particular interest to tr to identif such a state (which resembles an out-of-phase soliton pair) in a real eperiment. Also, as epected, the solution degenerates to its linear counterpart as E. Images of a tpical unstable solution and its comple quartet are shown along with a stable solution from the top right-hand region of the two-parameter diagram. We should also note in passing that states in the form of, +i, would produce a vorte waveform; however since such states have been studied in some detail earlier in Ref. [2] in a similar setting (i.e., in the presence of an eternal potential containing both harmonic and lattice components), we do not eamine them in more detail here.

17 Structure and stabilit of 2D BECs under harmonic and lattice confinement 6 (a) (b) (c) (d) (e) Figure 7. (Color Online) Dnamics of the unstable states (in the case of attractive interatomic interactions) that were shown in the previous figures. Shown are space-time evolution plots given b a characteristic densit isosurface D k = {,, t u(,, t) 2 = k}, where k = a(ma {,} { u(,, 2 }) and [a=.7 for (a),. for (b) and (d), and.3 for (c) and (e)]. (a) Ground state,, which collapses ver quickl. (b) First ecited state,, which collapses shortl after the ground state. (c) Degeneration of a, +, state into an eventual single-pulse structure that survives for a long time after the merger. The unstable, (d) and 2, (e) states deform, for ver short times as epected from the strong instabilities identified in their spectra, and subsequentl collapse. We now turn to solutions featuring m + n = 2. First, we consider the, branch for the attractive case in Fig.. In this case the solution ma possess between one and three comple eigenvalue quartets in its linearization (the middle panel of the figure shows a particular unstable case where there are two such quartets). However, once again, there eists a region in the right side of the relevant parameter space [i.e., for appropriate (V, )] where the solution is found to be linearl stable and all potential oscillator instabilities are suppressed. The bottom panel of Fig. shows such a linearl stable case of the quadrupolar configuration, which, again, should be eperimentall accessible. Finall, we consider the state 2,, as depicted in Fig. 6. This configuration is

18 Structure and stabilit of 2D BECs under harmonic and lattice confinement 7 highl unstable throughout our parameter space, with up to four real pairs and one comple quartet of eigenvalues. A tpical eample of the unstable configuration and its spectral plane of eigenvalues is shown in the bottom panel of Fig Dnamics Now we corroborate our eistence and stabilit results (for the attractive interactions case) with an investigation of the actual dnamics of tpical unstable solutions selected from the above families. For each case, the particular solution presented in the corresponding figures is perturbed with random noise distributed uniforml between. and. and integrated over time. It is important to note that although a random perturbation is used here to emulate the eperimental noise, the sstem is deterministic and the sole relevant feature of an (generic) random perturbation is its projection onto the most unstable eigendirection(s) of the perturbed solution profile. These projections, as indicated above, will grow (determinsticall) according to the corresponding growth rate. For the time propagation, we implement a standard 4th-order Runge-Kutta integrator scheme where we have numerical consistenc and stabilit for the conservative time step of t = 3. The results are compiled in Fig. 7. Panel (a) in Fig. 7 depicts the catastrophic instabilit of the ground state,,, which is subject to collapse, occurring at t 2. Panel (b) shows similar behavior for the, state, in which the two lobes appear to self-focus independentl, although one eventuall prevails and collapses for t 3. It is ver interesting to note that while the, state collapses, its more stable superposition with the, state survives for longer times, as epected, and also eventuall merges into a ground-state-like (single pulse) configuration, which was found to have a number of atoms just on the unstable side of the boundar of stabilit for such a structure. The resulting state actuall survives for ver long times, oscillating within one of the wells where it originall collected itself [see the panel (c)], apparentl stabilized b the ensuing oscillations. Panels (d) and (e) show, respectivel, the relativel rapid break up and subsequent collapse of the, and 2, states Repulsive interatomic interactions Eistence and Stabilit Now we will investigate the results pertaining to repulsive interatomic interactions for the same linear states eamined above. We once again start with the ground state, branch shown in Fig. 8. The top left panel of Fig. 8 shows the number of atoms N(V, ) as in the previous section. However, the linear stabilit S(V, ) for this case is omitted because, as ma be epected, this branch is stable throughout the parameter space, in contrast to its attractive counterpart (which is subject to collapse). We now turn to ecited states with m + n =. Figure 9 shows features similar to the previous one, but now for the state,. This branch is found to alwas be unstable due to the appearance of up to three real eigenvalue pairs. The top panels depict the

19 Structure and stabilit of 2D BECs under harmonic and lattice confinement V λ i λ r 3 Figure 8. (Color Online) The ground state state, solution in the case of repulsive interatomic interactions. The bottom panels correspond to (V, ) = (.3,.37). The stabilit of the ground state persists over the entire parameter space, and hence the stabilit surface is omitted from this set. dependence of the number of atoms N(V, ) (left), and instabilit growth rate S(V, ) (right) on the lattice depth V and the chemical potential. A sample profile and its eigenvalue spectrum are given in the bottom panels, indicating the presence, in this case, of three real eigenvalues pairs. The net state we consider is the, +, state which is presented in Fig.. This state alwas possesses a quartet of comple eigenvalues, and up to two additional pairs of real eigenvalues, and is thus unstable for all. It it worth noticing, however, that the instabilit weakens to relativel benign small magnitude comple quartets for intermediate values of the chemical potential, roughl (.4,.9), and large lattice depths, V >.3. This suggests that such a configuration should be long-lived enough that it could be observable in eperiments with repulsive condensates. Net, we consider the states with n + m = 2 (again for repulsive interatomic interactions), starting with the, branch in Fig.. The branch is also alwas

20 Structure and stabilit of 2D BECs under harmonic and lattice confinement V..2.3 V λ i λ r Figure 9. (Color Online) The state, in the case of repulsive interatomic interactions. The bottom row illustrates a sample profile (left) and the corresponding eigenvalue spectrum (right) for (V, ) = (.2,.47) displaing the strong instabilit arising from the three real eigenvalue pairs. unstable, possessing a comple quartet and then up to four additional real pairs for larger values of. The instabilit is shown in the right subplots of Fig., where the spectral plane of the bottom right panel corresponds to the solution of the bottom left one, for parameter values (V, ) = (.2,.6). It is interesting to note that such states are reminiscent of the domain-walls presented in Ref. [28] (here the domain-wall is imposed b the difference in phase), which however were found as potentiall stable structures in multi-component condensates. Net, the case of the 2, state is shown in Fig. 2. Here, there are up to three comple quartets along with three real pairs of eigenvalues. It is notable that for higher values of the lattice depth, these states are deformed as the lattice squeezes the central maimum separating the two minima (see middle left panel of the figure). The contour plot shown in the bottom right panel suggests that further increase of lattice depth ma lead to a new configuration altogether when the two local maima eventuall pinch off of

21 Structure and stabilit of 2D BECs under harmonic and lattice confinement V..2.3 V λ i λ r. λ i..... λ r Figure. (Color Online) Same as the previous figures but for the, +, state in the case of repulsive interatomic interactions. This state is alwas unstable due to at least an eigenvalue quartet and up to two other real pairs. Note that there eists a region of weak instabilit, where onl small magnitude quartets are present. The middle and bottom panels show the contour plots of this state and its linearization spectrum for (V, ) = (.2,.) (see blue cross in top panels) and (.37,.9) (see red circle in top panels), respectivel.

22 Structure and stabilit of 2D BECs under harmonic and lattice confinement V..2.3 V λ i λ r Figure. Same as in Fig. 9, but for the state, in the case of repulsive interatomic interactions for parameter values (V, ) = (.2,.6). each other. This deformation is a direct consequence of the presence of the (repulsive) nonlinearit, which results in drasticall different configurations as compared to the linear limit of the structure Dnamics We performed numerical simulations to investigate the evolution of tpical unstable states in the case of repulsive interatomic interactions, using similar time-stepping schemes as discussed above in the case of attractive interactions. Apart from the ground state, all ecited states presented in the previous section were predicted to be unstable, and this is confirmed in this section. In the particular case of the state, +, which was found to be weakl unstable (see bottom row of Fig. ), the instabilit takes a considerable time to manifest itself. The evolution of this state is depicted in panel (b) of Fig. 3; it is clearl seen that the state persists up to t 3. On the other hand, panel (a) shows the evolution of the state, which persists up to t 4, while the bottom row of panels shows the dnamics of the (c), state and of the (d) 2,, both persisting also up to t 4. All of these ecited states degenerate

23 Structure and stabilit of 2D BECs under harmonic and lattice confinement V V λ i λ r Figure 2. (Color Online) Same as in Fig. 9, but for the state 2, (in the case of repulsive interatomic interactions) with parameter values (V, ) = (.2,.7) (see red circle in top panels). The bottom row shows profiles for smaller, V =. (left, see blue cross in top panels), and larger, V =.3 (right, see green diamond in top panels), values of the optical lattice depth.

24 Structure and stabilit of 2D BECs under harmonic and lattice confinement 23 (a) (b) (c) (d) Figure 3. (Color Online) The dnamics of the unstable states in the case of repulsive interatomic interactions. Panels (a) and (b) show, respectivel, the evolution of the states, and, +,. It is clear that the state, +, is subject to a weaker oscillator instabilit for the parameter values mentioned in the bottom row of Fig. and, as a result, the original configuration persists for a long time. Panels (c) and (d) show the dnamics of the states with m + n = 2, namel (c),, and (d) 2,. All these solutions ultimatel degenerate into ground-state-like configurations. The densit isosurfaces are taken at a =. with the eception of (d) at a =.4. into ground-state-like configurations. Notice that transient vorte-like structures seem to appear during this process but the do not persist in the eventual dnamics and are hence not further discussed here. 4. Conclusions and discussion In summar, we have studied the structure and the stabilit of a pancake-shaped condensate (with either attractive or repulsive interatomic interactions) confined in a potential with both a harmonic and an optical lattice component. Starting from the non-interacting limit, and eploiting the smallness of the harmonic trap strength, we have emploed a multiscale perturbation method to find the discrete energies and the corresponding eigenmodes of the pertinent 2D linear Schrödinger equation. Then, we used the results found in this linear (non-interacting) limit in order to identif states persisting in the nonlinear (interacting) regime as well. This investigation revealed that the most fundamental states (emanating from combinations of the ground state and the first few ecited states in the two orthogonal directions of the optical lattice) can indeed be continued in the nonlinear regime. To demonstrate this continuation, we used two-parameter diagrams involving the effective strength of the nonlinearit (through the chemical potential) and the optical lattice depth. Ecited states were tpicall found to be unstable. The instabilit was found to result in either wavefunction collapse or a robust single-lobed structure in the case of attractive interactions; on the other hand, in the case of repulsive interactions, the

25 Structure and stabilit of 2D BECs under harmonic and lattice confinement 24 instabilit was alwas found to lead to the ground state of the sstem. Nevertheless, noteworth eceptions of stable or ver weakl unstable states were also revealed. These include the, +, and the, states in the case of attractive interatomic interactions. Moreover, in the case of repulsive interactions, the same state,, +,, was found (in certain parameter regimes) to be onl ver weakl unstable. Direct numerical simulations confirmed that the instabilit of this state is indeed weak, and it manifests itself at large times, an order of magnitude larger than the ones pertaining to the manifestation of instabilities of other ecited states. Thus, it is clear that the obtained results suggest that the state, +, has a good chance to be observed in eperiments with either an attractive or a repulsive pancake condensate. It is especiall important to highlight that these states are stabilized (or quasi-stabilized) onl in the presence of a sufficientl strong optical lattice; hence the latter potential plas a critical role in determining the stabilit of the states presented herein. As described in Section III B, the parameters used in our analsis have been chosen in order to facilitate convenience of the numerical computations, while also within range of eperimentall achievable limits of atom number, chemical potential, harmonic oscillator frequencies, and optical lattice depth and periodicit. Furthermore, we believe that the stabilit and spatial structure of the states eamined here can be eamined eperimentall. For eample, we imagine utilizing a BEC held in a pancakeshaped harmonic trap, created b an optical field. B using an optical trap rather than a magnetic trap, the scattering length of a BEC ma be adjusted using a Feshbach resonance. We envision that an optical lattice potential is ramped on and superimposed on a BEC with an interatomic scattering length tuned to be near zero. Once the lattice has reached the desired depth, the scattering length can be further adjusted with a magnetic field to be either positive or negative (the latter option would need to be within a region of stabilit that does not result in collapse of the BEC). Finall, phase imprinting techniques [29] can be used to generate the desired phase profile of the BEC. B opticall eamining the state of the BEC at various points in time after phase profile imprinting, the stabilit of the generated states can then be eamined and compared with our numerical results and stabilit analsis. For eample, with an optical lattice frequenc of ω L = 2π 2 Hz (as in the eample of Section III B), the time unit of our dnamical evolution plots is.3 ms. This implies that for the cases we have eamined, signatures of instabilit would be tpicall visible on the eperimentall feasible to ms timescale. We therefore believe that our predictions could be eamined with current eperimental techniques. There are various directions along which one can etend the present considerations. A natural one is to etend the analsis to full 3D condensates and eamine the persistence and stabilit of higher-dimensional variants of the presented states. A perhaps more subtle direction is to consider a different basis of linear eigenfunctions in the 2D problem, namel instead of the Hermite-Gauss basis used here, to focus on the Laguerre-Gauss basis of the underling linear problem with the parabolic potential. Under such a choice, it would be interesting to eamine how solutions of that tpe, in-

26 Structure and stabilit of 2D BECs under harmonic and lattice confinement 2 cluding one-node and multi-node ring-like structures (see, e.g., Ref. [7] and references therein), are deformed in the presence of the lattice and how their stabilit is correspondingl affected. Finall, as discussed above, it appears that the setting considered herein should be directl accessible to present eperiments with pancake-shaped BECs. In view of that, it would be particularl relevant to eamine which ones among the structures presented in this work can survive for evolution times that are of interest within the time scales of an eperiment. Acknowledgments P.G.K. and R.C.G. gratefull acknowledge the support of NSF-DMS-663, and P.G.K. additionall acknowledges support from NSF-DMS-69492, NSF-CAREER and the Aleander von Humboldt Foundation. B.P.A. acknowledges support from the Arm Research Office and NSF Grant No. MPS The work of D.J.F. was partiall supported b the Special Research Account of the Universit of Athens. [] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge Universit Press (Cambridge, 2). [2] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oford Universit Press (Oford, 23). [3] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phs. 7, 463 (999). [4] P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-González (eds.), Emergent nonlinear phenomena in Bose-Einstein condensates. Theor and eperiment (Springer-Verlag, Berlin, 28). [] M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature (London) 4, 39 (22). [6] F.Kh. Abdullaev, A. Gammal, A.M. Kamchatnov and L. Tomio, Int. J. Mod. Phs. B 9, 34 (2). [7] A.L. Fetter and A.A. Svidzinks, J. Phs.: Cond. Matter 3, R3 (2). [8] P.G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis and I. G. Kevrekidis, Mod. Phs. Lett. B 8, 48 (24). [9] P.G. Kevrekidis and D.J. Frantzeskakis, Mod. Phs. Lett. B 8, 73 (24). [] V. Brazhni and V.V. Konotop, Mod. Phs. Lett. B 8, 627 (24). [] O. Morsch and M.K. Oberthaler, Rev. Mod. Phs. 78, 79 (26). [2] L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon Press, Oford, 987). [3] Yu.S. Kivshar, T.J. Aleander, and S.K. Turitsn, Phs. Lett. A 278, 22 (2). [4] Yu.S. Kivshar and T.J. Aleander, in Proceeding of the APCTP-Nankai Smposium on Yang- Bater Sstems, Nonlinear Models and Their Applications, edited b Q-Han Park et al. (World Scientific, Singapore, 999). [] P.G. Kevrekidis, V.V. Konotop, A. Rodrigues and D.J. Frantzeskakis J. Phs. B: At. Mol. Opt. Phs. 38, 73 (2). [6] T. Kapitula and P.G. Kevrekidis, Chaos, 374 (2); T. Kapitula and P.G. Kevrekidis, Nonlinearit 8, 249 (2). [7] G. Herring, L.D. Carr, R. Carretero-González, P.G. Kevrekidis, and D.J. Frantzeskakis, Phs. Rev. A (28). [8] E.A. Ostrovskaa, M.K. Oberthaler, and Yu.S. Kivshar in Emergent nonlinear phenomena in