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arXiv:1907.04547 · Archive for Rational Mechanics and Analysis

Derivation of the 2d Gross–Pitaevskii Equation for Strongly Confined 3d Bosons

Lea Boßmann

Published Jul 18, 2020• 10 citations
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Abstract

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 $$\varepsilon $$ ε . The interaction is non-negative and scaled in such a way that its scattering length is of order $$\varepsilon /N$$ ε / N , while its range is proportional to $$(\varepsilon /N)^{\beta }$$ ( ε / N ) β with scaling parameter $$\beta \in (0,1]$$ β ∈ ( 0 , 1 ] . We consider the simultaneous limit $$(N,\varepsilon )\rightarrow (\infty ,0)$$ ( 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 $$\beta $$ β . For $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice $$\beta =1$$ 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.

DOI
10.1007/s00205-020-01548-w

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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\varepsilon ε ε . The interaction is non-negative and scaled in such a way that its scattering length is of order ε/N\varepsilon /Nε/N ε / N , while its range is proportional to (ε/N)β(\varepsilon /N)^{\beta }(ε/N)β ( ε / N ) β with scaling parameter β∈(0,1]\beta \in (0,1]β∈(0,1] β ∈ ( 0 , 1 ] . We consider the simultaneous limit (N,ε)→(∞,0)(N,\varepsilon )\rightarrow (\infty ,0)(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\beta β β . For β∈(0,1)\beta \in (0,1)β∈(0,1) β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β=1\beta =1β=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.
Consider a system of NNN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j))N2V(N(xi​−xj​)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HNH_NHN​ denote the Hamiltonian of the system and let ψN,tψ_{N,t}ψN,t​ be the solution to the Schrödinger equation. Suppose that the initial data ψN,0ψ_{N,0}ψN,0​ satisfies the energy condition <ψN,0,HNkψN,0>≤CkNk < ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k <ψN,0​,HNk​ψN,0​>≤CkNk for $k=1,2,... $. We also assume that the kkk-particle density matrices of the initial state are asymptotically factorized as N→∞N\to\inftyN→∞. We prove that the kkk-particle density matrices of ψN,tψ_{N,t}ψN,t​ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential VVV. We also prove the same conclusion if the energy condition holds only for k=1k=1k=1 but the factorization of ψN,0ψ_{N,0}ψN,0​ is assumed in a stronger sense.
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)W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)Wβ​(x)=N−1+2βW(Nβx) Wβ(x)=N-1+2βW(Nβx) , for any β>0\beta >0β>0 β>0 , or to be given by VN(x)=e2NV(eNx)V_N(x)=e^{2N} V(e^N x)VN​(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R)W,V \in L^\infty ({\mathbb {R}}^2,{\mathbb {R}})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 VNV_NVN​ VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
Consider a system of NNN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j))N2V(N(xi​−xj​)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HNH_NHN​ denote the Hamiltonian of the system and let ψN,t\psi_{N,t}ψN,t​ be the solution to the Schrodinger equation. Suppose that the initial data ψN,0\psi_{N,0}ψN,0​ satisfies the energy condition ≤CN>. \leq C N >. ≤CN>. and that the one-particle density matrix converges to a projection as N→∞N \to \inftyN→∞. Then, we prove that the kkk-particle density matrices of ψN,t\psi_{N,t}ψN,t​ factorize in the limit N→∞N \to \inftyN→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential VVV. In \cite{ESY}, we proved the same statement under the condition that the interaction potential VVV is sufficiently small; in the present work we extend this result to large potentials.
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.
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.