We variationally determine the dynamics of bright soliton trains composed of harmonically trapped Bose-Einstein condensates with attractive interatomic interactions. In particular, we obtain the interaction potential between two solitons. We also discuss the formation of soliton trains due to the quantum mechanical phase fluctuations of a one-dimensional condensate.

Multi-component Bose±Einstein condensates1±3 provide opportunities to explore experimentally the wealth of physics associated with the spin degrees of freedom 4±7 . The ground-state properties 8±11 and line-like vortex excitations 8,12,13 of these quantum systems have been studied theoretically. In principle, nontrivial spin textures consisting of point-like topological excitations, or skyrmions 14,15 , could exist in a multi-component Bose±Einstein condensate, owing to the super¯uid nature of the gas. Although skyrmion excitations are already known in the context of nuclear physics and the quantum-Hall effect, creating these excitations in an atomic condensate would offer an opportunity to study their physical behaviour in much greater detail, while also enabling an ab initio comparison between theory and experiment. Here we investigate theoretically the stability of skyrmions in a ®ctitious spin-1/2 condensate of 87 Rb atoms. We ®nd that skyrmions can exist in such a gas only as a metastable state, but with a lifetime comparable to (or even longer than) the typical lifetime of the condensate itself.An essential feature of a spinor Bose±Einstein condensate is that two or more hyper®ne states of the atoms in the condensate have almost the same energy. As a result, this spin degree of freedom becomes a relevant dynamical variable, which gives rise to new excitations that are not present in the usual single-component Bose±Einstein condensates, where the spins are effectively frozen. One of these excitations is the skyrmion, which is a topological nontrivial spin texture. Roughly speaking, the skyrmion is a pointlike object that can be created out of the ground state, in which all the spins are aligned, by reversing the average spin in a ®nite region of space. Although topological considerations indeed allow for these excitations, which we note are fundamentally different from the topologically trivial coreless vortices discussed in ref. 8, we need to know whether such a con®guration is also energetically stable. However, when solving the appropriate Gross±Pitaevskii equation 8 , we ®nd that, in equilibrium, the energy is always minimized by collapsing the skyrmion to zero size. Fortunately, it turns out that for suf®ciently small sizes of the skyrmion a nonequilibrium stability mechanism starts to work. A number of atoms in the centre of the skyrmion, which we denote here as the core atoms, will be trapped by an effective three-dimensional potential barrier, that is, a repulsive shell with a ®nite radius that is induced by the gradients in the spin texture of the skyrmion itself. As the skyrmion shrinks in size, the barrier height of the repulsive shell increases and the radius decreases. This leads to a squeezing of the core atoms and thus to an increase in their energy, which ultimately stabilizes the skyrmion. As mentioned, this is not an equilibrium state of the condensate because the core atoms will tunnel over the barrier and give the skyrmion a ®nite lifetime. We note that our calculations are for a unifor...

We improve on the Popov theory for partially Bose-Einstein condensed atomic gases by treating the phase fluctuations exactly. As a result, the theory becomes valid in arbitrary dimensions and is able to describe the low-temperature crossover between three-, two-, and one-dimensional Bose gases, which is currently being explored experimentally. We consider both homogeneous and trapped Bose gases.

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We show that even in three dimensions an antiferromagnetic spin-1 Bose-Einstein condensate, which can, for instance, be created with 23 Na atoms in an optical trap, has not only singular linelike vortex excitations, but also allows for singular pointlike topological excitations, i.e., monopoles similar to the 't Hooft -Polyakov monopoles. We discuss the static and dynamic properties of these monopoles. DOI: 10.1103/PhysRevLett.87.120407 PACS numbers: 03.75.Fi, 14.80.Hv, 32.80.Pj, 67.40. -w Introduction.-Quantum magnetism plays an important role in such diverse areas of physics as high-temperature superconductivity, quantum phase transitions, and the quantum Hall effect. Moreover, it now appears that magnetic properties are also very important in another area, namely Bose-Einstein condensation in trapped atomic gases. This is due to two independent experimental developments. The first development is the realization of an optical trap for 23 Na atoms [1], whose operation no longer requires the gas to be doubly spin polarized and has given rise to the creation of a spin-1 Bose-Einstein antiferromagnet [2]. The second development is the creation of a two-component condensate of 87 Rb atoms [3], which by means of rf fields can be manipulated so as to make the two components essentially equivalent [4]. As a result also a spin-1͞2 Bose-Einstein ferromagnet can now be studied in detail experimentally.The spin structure of these condensates has recently been worked out by a number of authors [5 -8] and also the first studies of the linelike vortex excitations have appeared [5,9,10]. An immediate question that comes to mind, however, is whether the spin degrees of freedom lead also to other topological excitations that do not have an analogy in the case of a single component Bose-Einstein condensate. The answer to this question is in general affirmative. Indeed, we have recently shown that ferromagnetic Bose-Einstein condensates have long-lived Skyrmion excitations, which are nonsingular but topologically nontrivial pointlike spin textures [11]. Moreover, we show here that also spin-1 Bose-Einstein antiferromagnets have pointlike topological excitations. In particular, there exist singular pointlike spin textures, which are analogous to the magnetic monopoles in particle physics discovered by 't Hooft and Polyakov [12]. Having done so, we then turn to the investigation of the precise texture and the dynamics of these monopoles.As indicated above, Skyrmion and monopole excitations have already been studied in the context of nuclear and high-energy physics, respectively. However, in these areas of physics there does not exist a satisfactory microscopic theory for these topological excitations. For example, the Skyrme model gives only a rather rough description of a nucleon. Moreover, magnetic monopoles

We present an improved many-body T-matrix theory for partially Bose-Einstein condensed atomic gases by treating the phase fluctuations exactly. The resulting mean-field theory is valid in arbitrary dimensions and able to describe the low-temperature crossover between three-, two-, and one-dimensional Bose gases. When applied to a degenerate two-dimensional atomic hydrogen gas, we obtain a reduction of the three-body recombination rate, which compares favorably with experiment. Supplementing the mean-field theory with a renormalization-group approach to treat the critical fluctuations, we also incorporate into the theory the Kosterlitz-Thouless transition that occurs in a homogeneous Bose gas in two dimensions. In particular, we calculate the critical conditions for the Kosterlitz-Thouless phase transition as a function of the microscopic parameters of the theory. The proposed theory is further applied to a trapped one-dimensional Bose gas, where we find good agreement with exact numerical results obtained by solving a nonlinear Langevin field equation.

We propose a method of forming matter-wave soliton molecules that is inspired by the recent experiment of Dris et al. [1]. In the proposed set-up we show that if two solitons are initially prepared in phase and with a sufficiently small separation and relative velocity, a bound pair will always form. This is verified by direct numerical simulation of the Gross-Pitaevskii equation and by the derivation of the exact interaction energy of two solitons, which takes the form of a Morse potential. This interaction potential depends not only on the separation but also on the relative phase of the solitons and is essential for an analytical treatment of a host of other problems, such as the soliton gas and the Toda lattice of solitons.

We show that a ferromagnetic Bose-Einstein condensate has not only line-like vortex excitations, but in general, also allows for pointlike topological excitations, i.e., skyrmions. We discuss the thermodynamic stability and the dynamic properties of these skyrmions for both spin-1/2 and ferromagnetic spin-1 Bose gases.