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Derivation of the 1d Gross–Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons

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Abstract

We consider the dynamics of N interacting bosons initially forming a Bose–Einstein condensate. Due to an external trapping potential, the bosons are strongly confined in two dimensions, where the transverse extension of the trap is of order \(\varepsilon \). The non-negative interaction potential is scaled such that its range and its scattering length are both of order \((N/\varepsilon ^2)^{-1}\), corresponding to the Gross–Pitaevskii scaling of a dilute Bose gas. We show that in the simultaneous limit \(N\rightarrow \infty \) and \(\varepsilon \rightarrow 0\), the dynamics preserve condensation and the time evolution is asymptotically described by a Gross–Pitaevskii equation in one dimension. The strength of the nonlinearity is given by the scattering length of the unscaled interaction, multiplied with a factor depending on the shape of the confining potential. For our analysis, we adapt a method by Pickl (Rev Math Phys 27(01):1550003, 2015) to the problem with dimensional reduction and rely on the derivation of the one-dimensional NLS equation for interactions with softer scaling behaviour in Boßmann (Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons. arXiv preprint, 2018. arXiv:1803.11011).

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Correspondence to Lea Boßmann.

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Communicated by Claude-Alain Pillet.

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Boßmann, L., Teufel, S. Derivation of the 1d Gross–Pitaevskii Equation from the 3d Quantum Many-Body Dynamics of Strongly Confined Bosons. Ann. Henri Poincaré 20, 1003–1049 (2019). https://doi.org/10.1007/s00023-018-0738-7

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  • DOI: https://doi.org/10.1007/s00023-018-0738-7

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