Rotating quantum gases 38At low temperatures, we have found that the vortex density follows an activation law ofthe form exp(−Δ/k B T ) with an activation energy Δ weakly depending on temperaturein both the ideal and the interacting cases. Concerning the spatial correlations betweenthe positions of vortices, we have found that no qualitative difference appears betweenthe ideal and the interacting cases at high temperatures, while in the activation regimevortex pairs in the ideal gas have a much longer size.[1] A. Recati, F. Zambelli, and S. Stringari, Phys. Rev. Lett. 86, 377 (2001).K. W. Madison, F. Chevy, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 86, 4443(2001)[2] G. Tonini, F. Werner and Y. Castin, Eur. Phys. J. D 39, 283 (2006).[3] I. Bausmerth, A. Recati, and S. Stringari, Phys. Rev. Lett. 100, 070401 (2008).[4] J. Carlson et al., Phys. Rev. Lett. 91, 050401 (2003).[5] G.E. Astrakharchick et al., Phys. Rev. Lett. 93, 200404 (2004).[6] I. Bausmerth, A. Recati, and S. Stringari, Phys. Rev. A 78, 063603 (2008).[7] M. Antezza, M. Cozzini, and S. Stringari, Phys. Rev. A 75, 053609 (2007).[8] G. Watanabe, M. Cozzini, and S. Stringari, Phys. Rev. A 77, 021602(R) (2008).[9] S. Schwartz, M. Cozzini, C. Menotti, I. Carusotto, P. Bouyer, and S. Stringari, ,New J. Phys. 8, 162 (2006).[10] L. Giorgetti, I. Carusotto, Y. Castin, Phys. Rev. A 76, 013613 (2007).
Nonlinear dynamics and solitons 39NONLINEAR DYNAMICS AND SOLITONSSound waves, Bogoliubov quasiparticles, topological and nonlinear excitations, suchas vortices and solitons, in Bose-Einstein condensates have been the objects of a veryintense activity in the last decade. This is also a traditional field of research for thegroup in Trento. Here we briefly report on recent advances on solitons in both Boseand Fermi gases, as well as on the flow of a superfluid past an obstacle.Solitons in two-dimensional Bose-Einstein condensatesBose-Einstein condensates of ultracold atoms are ideal systems for exploring matterwave solitons. For most purposes, these condensates at zero temperature are welldescribed by the Gross-Pitaevskii (GP) equation which has the form of a NonlinearSchrödinger Equation, where nonlinearity comes from the interaction between atoms.In 1D, the GP equation for condensates with repulsive interaction admits solitonicsolutions corresponding to a local density depletion, namely gray and dark solitons.Such solitons have already been created and observed in elongated condensates withdiverse techniques. Also multidimensional solitons in condensates have attracted muchattention [1]. An interesting type of excitation in a 2D condensate is represented bya self-propelled vortex-antivortex pair which is a particular solitonic solution of theGP equation [2, 3]. In the low momentum limit, the relation between the energy Eand momentum P of the soliton approaches the dispersion law of Bogoliubov phonons,ɛ = cq, from below. In this limit, when the 2D soliton moves at a velocity V closeto the Bogoliubov sound speed c, the phase singularities of the vortex-antivortex pairdisappear and the soliton takes the form of a localized density depletion, also calledrarefaction pulse. Moreover, if V is close to c the GP equation can be rewritten in asimpler form, known as Kadomtsev-Petviashvili (KP) equation [4].In [5] we studied the excitations of the condensate in the presence of the soliton bylinearizing the KP equation around the stationary solution. By looking at the shapeof the eigenfunctions we found excitations localized near the soliton, having shape anddispersion law similar to those of the transverse oscillations of a 1D gray soliton ina 2D condensate. We also used a stabilization method in order to obtain a betterdetermination of the dispersion law of the localized states. The same method allowsus to visualize the coupling between the localized states and the free states, i.e., theBogoliubov phonons of the uniform condensate. The occurrence of a coupling betweenbound and unbound states, which is visible in the avoided crossings in the stabilizationdiagram, is worth stressing. The width of the avoided crossings is directly related to
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