Abstract We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order
ε ε . The interaction is non-negative and scaled in such a way that its scattering length is of order
ε/N ε / N , while its range is proportional to
(ε/N)β ( ε / N ) β with scaling parameter
β∈(0,1] β ∈ ( 0 , 1 ] . We consider the simultaneous limit
(N,ε)→(∞,0) ( N , ε ) → ( ∞ , 0 ) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter
β β . For
β∈(0,1) β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice
β=1 β = 1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.
We consider the dynamics of N interacting bosons initially exhibiting Bose–Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap being of order ε. The non-negative interaction potential is scaled such that its scattering length is positive and of order (N/ε2)−1 and the range of the interaction scales as (N/ε2)−β for β ∈ (0, 1). We prove that in the simultaneous limit N → ∞ and ε → 0, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a cubic defocusing nonlinear Schrödinger equation in one dimension, where the strength of the nonlinearity depends on the interaction and on the confining potential. This is the first derivation of a lower-dimensional effective evolution equation for singular potentials scaling with β≥12 and lays the foundations for the derivation of the physically relevant one-dimensional Gross–Pitaevskii equation (β = 1). For our analysis, we adapt an approach by Pickl to the problem with strong confinement.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.
We present reduced-dimensional stochastic projected Gross-Pitaevskii equations describing regimes of confinement and temperature where a one-dimensional (1D) or 2D superfluid is immersed in a 3D thermal cloud. The projection formalism provides both a formally rigorous and physically natural way to effect the dimensional reduction. The 3D form of the number-damping (growth) terms is unchanged by the dimensional reduction. Projection of the energy-damping (scattering) terms leads to modified stochastic equations of motion describing energy exchange with the thermal reservoir. The regime of validity of the dimensional reduction is investigated via variational analysis. For the 1D case, we validate our variational treatment by comparing numerical simulations of a trapped prolate system in three dimensions with the 1D theory and establish a consistent choice of cutoff for the 1D theory. We briefly discuss the scenario involving two components with different degeneracy, suggesting that a wider regime of validity exists for systems in contact with a buffer-gas reservoir.
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by
a3β−1V(aβ⋅) where
∫V⩽0 and
a matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D
N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty $. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.
Using the F-expansion method, we present analytical matter-wave solutions to Bose-Einstein condensates with two- and three-body interactions through the generalized onedimensional Gross-Pitaevskii equation with time-dependent coefficients, for the experimentally relevant quadratic potential strength. Various shapes of the analytical matter-wave solutions which have important applications of physical interest are studied in detail. In particular, the matter-wave solitons are obtained in the quintic Gross-Pitaevskii equation model, which may be produced by tuning the cubic nonlinearity to zero via the Feschbach resonance technique. Direct numerical simulation has been performed to show the stable region of the matter-wave soliton.