Abstract
We analytically solve two problems that may be useful in the context of the recent observation of matter wave bright solitons in a one-dimensional attractive atomic Bose gas. The first problem is strictly beyond mean field: from the Bethe ansatz solution we extract the internal correlation function of the particle positions in the quantum soliton, that is for a fixed center of mass position. The second problem is solved in the limit of a large number of particles, where the mean field theory is asymptotically correct: it deals with the number of excitations created by the opening of the trap, starting from a pure soliton in a weakly curved harmonic potential.
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Let us assume that the wavefunction Φ(R) of the center of mass of the gas is a Gaussian wavepacket of width σ, centered in R=0. To obtain in this case a pair correlation ρ(x,y)-ρ(x)ρ(y) close to the one calculated here for a center of mass perfectly localized in R=0, we estimate that the condition \(\sigma< \xi/\sqrt{N}\) should be satisfied, from the expansion \(\rho(x)\simeq \rho(x|0) +\frac{1}{2} \langle R^2\rangle [\partial_R^2\rho(x|R)]_{R=0}\) and \(\rho(x,y)\simeq \rho(x,y|0) +\frac{1}{2} \langle R^2\rangle [\partial_R^2\rho(x,y|R)]_{R=0}\), where 〈R2〉 is the expectation value in the wavefunction Φ(R). Starting with a harmonically trapped gas in its internal plus center of mass ground state, as discussed in Weiss, one may change the scattering length after trap opening to adiabatically increase the soliton size, in order to decrease \(\sqrt{N}\sigma /\xi\)
C. Weiss, Y. Castin, arXiv:0806.3395 (2008)
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Castin, Y. Internal structure of a quantum soliton and classical excitations due to trap opening. Eur. Phys. J. B 68, 317–328 (2009). https://doi.org/10.1140/epjb/e2008-00407-3
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DOI: https://doi.org/10.1140/epjb/e2008-00407-3
PACS
- 03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations
- 03.75.Hh Static properties of condensates; thermodynamical, statistical, and structural properties
- 03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow