Pitchfork bifurcations in blood-cell shaped dipolar Bose-Einstein condensates arXiv:1001.1253v1 [cond-mat.quant-gas] 8 Jan 2010 Stefan Rau, J¨ org Main, Patrick K¨ oberle, and G¨ unter Wunner Institut f¨ ur Theoretische Physik 1, Universit¨ at Stuttgart, 70550 Stuttgart, Germany (Dated: November 15, 2018) We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation of condensates with electromagnetically induced attractive 1 /r interaction, or with dipole-dipole interaction. Moreover, Gaussian wave packets are superior in that they are capable of producing both stable and unstable stationary solutions, and thus of giving access to yet unexplored regions of the space of solutions of the Gross- Pitaevskii equation. We apply the method to clarify the theoretical nature of the collapse mechanism of blood-cell shaped dipolar condensates: On the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation. PACS numbers: 67.85.-d, 03.75.Hh, 05.30.Jp, 05.45.-a with ϑ ′ the angle between r − r ′ and the axis of an exter- nal magnetic field. For completeness we will also consider the case of an isotropic “gravity-like” attractive 1 /r long- range interaction, where for dipolar interaction we have The most obvious way of solving the Gross-Pitaevskii equation (1) is its direct numerical integration on multi- dimensional grids using, e.g., fast-Fourier techniques. The stationary ground state can be obtained by imagi- nary time evolution. These calculations, however, may turn out laborious, and physical insight can often be gained using approximate, in particular, variational solu- tions. A common approach employed for determining the dynamics and stability of condensates both with contact interaction only [10, 11] and with additional long-range interaction [8] is to assume a simple Gaussian form of the wave function, with time-dependent width parame- ters and phase, and to investigate the dynamics of these The GPE for ultracold gases with long-range interac- tions, described by the interatomic potential V$_{lr}$ ( r ), has the form According to O’Dell et al. [5] this interaction could be electromagnetically induced by exposing the condensate atoms to an appropriately arranged set of triads of laser beams. The appealing feature of such “monopolar” condensates is that they can be self-trapping, i.e. ex- ist without an external trapping potential. The equa- tions above have been brought into dimensionless form by introducing natural units, which for monopolar in- teraction ( V$_{mono}$ = − u/r ) are [6–8] the “Bohr radius” a$_{u}$ = ℏ $^{2}$/ ( mu ) for lengths, the “Rydberg energy” E$_{u}$ = ℏ $^{2}$/ (2 ma 2 $_{u}$) for energies and ω$_{u}$ = E$_{u}$/$_{ℏ}$ for frequen- cies. Natural units for dipolar atoms with magnetic mo- ment µ are [9] the dipole length a$_{d}$ = µ$_{0}$µm/ (2 π$_{ℏ}$ $^{2}$), the dipole energy E$_{d}$ = ℏ $^{2}$/ (2 ma 2 $_{d}$) and the dipole frequency ω$_{d}$ = E$_{d}$/$_{ℏ}$ . The quantities γ$_{x,y,z}$ in (1) denote the trap- ping frequencies in the three spatial directions measured in the respective frequency units, N is the number of bosons, and a the scattering length in units of a$_{u}$ and a$_{d}$ , respectively. Bose-Einstein condensates (BECs) with dipole-dipole interaction have become an active and exciting field of re- search because they offer the possibility of tuning the rel- ative strengths of the short-range isotropic contact inter- action and the anisotropic long-range dipole interaction by manipulating the s -wave scattering length via Fesh- bach resonances, and thus of studying a wealth of new phenomena that occur as one crosses the whole range from dominance of the contact interaction to that of the dipole interaction. The experimental realization of a BEC of chromium atoms [1–3], which possess a strong magnetic dipole moment, has given additional impetus to the field (for a comprehensive list of references see the recent review by Lahaye et al. [4]). In the dilute limit, the theoretical description of these condensates can be done in the framework of the Gross-Pitaevskii equa- tion (GPE). This nonlinear Schr¨ odinger equation has been solved in the literature so far by simple variational ansatzes, where the mean-field energy is minimized, e.g., with the conjugate gradient method or by imaginary time evolution. In this Letter we will show that the method of coupled Gaussian wave packets is an adequate alterna- tive to solving the GPE of BECs with long-range interac- tions. Moreover, we will show that the method is superior in that it also yields unstable stationary solutions, and thus opens access to regions of the space of solutions of the GPE unexplored heretofore. As an application of the method we will analyze in detail the theoretical nature of the collapse mechanism of dipolar BECs. V$_{lr}$ ( r ) = − 2 N ∫ d $^{3}$r ′ | ψ ( r $^{′}$, t ) | 2 | r − r $^{′}$| . (3) i d d t ψ ( r , t ) = [ − Δ + γ 2 $_{x}$x 2 + γ 2 $_{y}$y 2 + γ 2 $_{z}$z 2 +8 πNa | ψ ( r , t ) | 2 + V$_{lr}$ ( r ) $^{]}$ψ ( r , t ) , (1) V$_{lr}$ ( r ) = N ∫ d $^{3}$r $_{′}$1 − 3 cos $^{2}$ϑ ′ | r − r $^{′}$| 3 | ψ ( r $^{′}$, t ) | $^{2}$. (2)