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Journal of Mathematical Physics

Improved stability for 2D attractive Bose gases

Phan Thành Nam, Nicolas Rougerie

Published Feb 1, 2020• 15 citations
Publisher

Abstract

We study the ground-state energy of N attractive bosons in the plane. The interaction is scaled for the gas to be dilute so that the corresponding mean-field problem is a local non-linear Schrödinger (NLS) equation. We improve the conditions under which one can prove that the many-body problem is stable (of the second kind). This implies, using previous results, that the many-body ground states and dynamics converge to the NLS ones for an extended range of diluteness parameters.

DOI
10.1063/1.5131320

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We study the ground-state energy of N attractive bosons in the plane. The interaction is scaled for the gas to be dilute so that the corresponding mean-field problem is a local non-linear Schrödinger (NLS) equation. We improve the conditions under which one can prove that the many-body problem is stable (of the second kind). This implies, using previous results, that the many-body ground states and dynamics converge to the NLS ones for an extended range of diluteness parameters.
We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.
We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state.
Abstract. We consider a one-dimensional, trapped, focusing Bose gas where [Formula: see text] bosons interact with each other via both a two-body interaction potential of the form [Formula: see text] and an attractive three-body interaction potential of the form [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The system is stable either for any [Formula: see text] as long as [Formula: see text] —the critical strength of the one-dimensional focusing quintic nonlinear Schrödinger (NLS) equation— or for [Formula: see text] when [Formula: see text]. In the former case, fixing [Formula: see text], we prove that in the mean-field limit the many-body system exhibits the Bose–Einstein condensation on the cubic-quintic NLS ground states. When assuming [Formula: see text] and [Formula: see text] as [Formula: see text], with the former convergence being slow enough and “not faster” than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case, [Formula: see text] fixed, we obtain the convergence of many-body energy for small [Formula: see text] when [Formula: see text] is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence [Formula: see text] is “faster” than the slow enough convergence [Formula: see text].
We consider a system of NNN interacting bosons in a rotating harmonic trap in R2\mathbb{R}^2R2, where the two-body interaction is attractive and scaled as N2αU(Nαx)N^{2α}U(N^αx)N2αU(Nαx) with 0<α<1/120<α<1/120<α<1/12, and the three-body interaction is repulsive and scaled as N4βW(Nβx,Nβy)N^{4β}W(N^βx,N^βy)N4βW(Nβx,Nβy) with 0<β<1/240<β<1/240<β<1/24. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling aaa approaches the critical value a∗a_*a∗​ and the three-body coupling bbb tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies ENLS=(1−ζ/4+o(1))Qpotℓn2E^{\mathrm{NLS}} = (1-ζ/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2ENLS=(1−ζ/4+o(1))Qpotℓn2​ with ζ⩾0ζ\geqslant0ζ⩾0 determined by the relative rates of an→a∗a_n\to a_*an​→a∗​ and bn↘0b_n\searrow0bn​↘0. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.
We study the ground-state properties of a one-dimensional Bose gas with NNN-body attractive contact interactions. By using the explicit form of the bright soliton solution of a generalized nonlinear Schr\"odinger equation, we compute the chemical potential and the ground-state energy. For N=3N=3N=3, a localized soliton wave function exists only for a critical value of the interaction strength: in this case the ground state has an infinite degeneracy that can be parametrized by the chemical potential. The stabilization of the bright soliton solution by an external harmonic trap is also discussed, and a comparison with the effect of NNN-body attractive contact interactions in higher dimensions is presented.