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.
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.
Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by
Wβ(x)=N−1+2βW(Nβx) Wβ(x)=N-1+2βW(Nβx) , for any
β>0 β>0 , or to be given by
VN(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported
W,V∈L∞(R2,R) W,V∈L∞(R2,R) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential
VN VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
Abstract. 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 a 3β−1 V (a β ·) 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 ∞. 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
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.
The goal of this thesis is to provide a mathematical rigorous derivation of the Schrödinger-Klein-Gordon equations, the Maxwell-Schrödinger equations and the defocusing cubic nonlinear Schrödinger equation in two dimensions. We study the time evolution of the Nelson model (with ultraviolet cutoff) in a limit where the number N of charged particles gets large while the coupling of each particle to the radiation field is of order N^{−1/2}. At time zero it is assumed that almost all charges are in the same one-body state (a Bose-Einstein condensate) and that the radiation field is close to a coherent state. We show the persistence of condensation over time and prove that the time evolution is approximately described by the Schrödinger-Klein-Gordon system of equations in the large N limit. Subsequently, we consider the spinless Pauli-Fierz Hamiltonian which models the interaction between charged bosons and the quantized electromagnetic field. We discuss the limit previously described and prove that the time evolution is approximated by the Maxwell-Schrödinger equations. To our knowledge, this is the first rigorous result concerning a mean-field limit of the Pauli-Fierz Hamiltonian. We then turn to the evolution of Bose-Einstein condensates in two dimensions and consider N bosons which interact by a repulsive two-body potential. The interaction is given either by N^{−1+2β}V(N^{β}x) with β∈R^{+}_{0} or by e^{2N}V(e^{N}x), for some spherical symmetric, positive and compactly supported V∈L_{∞}(R^{2},R). We prove that the dynamics is approximated by the defocusing two-dimensional cubic nonlinear Schrödinger equation in the large N limit. In case of the exponential scaling, we show that a short-scale correlation structure affects the dynamics of the condensate. This is the first rigorous derivation that considers an exponential scaling of the interaction. All derivations rely on a method developed by Pickl in [Lett. Math. Phys. 97(2), 151–164 (2011)]. The first two results are obtained by an extension of the method to systems which interact with quantized radiation fields. The latter is derived by an appropriate adaption of the proof in three space dimensions [Rev. Math. Phys., 27, 1550005 (2015)]. The crucial insight to derive the Maxwell-Schrödinger equations is to restrict the class of many-body wave functions to a subspace of states whose energy per particle only fluctuates little around the energy functional of the Maxwell-Schrödinger system. To derive the two-dimensional Gross-Pitaevskii equation it is essential to define a measure of condensation which properly incorporates the correlations that arise from the exponential scaling of the interaction. This thesis is based on the preprints [54, 47].
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.