Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0808.1178

On the Time Dependent Gross Pitaevskii- and Hartree Equation

Peter Pickl

Published Aug 8, 2008• 4 citations
arXivHTMLPDF

Abstract

We are interested in solutions $Ψ_t$ of the Schrödinger equation of $N$ interacting bosons under the influence of a time dependent external field, where the range and the coupling constant of the interaction scale with $N$ in such a way, that the interaction energy per particle stays more or less constant. Let $\mathcal{N}^{ϕ_0}$ be the particle number operator with respect to some $ϕ_0\in L^2(\mathbb{R}^3\to\mathbb{C})$. Assume that the relative particle number of the initial wave function $N^{-1}< Ψ_0,\mathcal{N}^{ϕ_0}Ψ_0>$ converges to one as $N\to\infty$. We shall show that we can find a $ϕ_t\in L^2(\mathbb{R}^3\to\mathbb{C})$ such that $\lim_{N\to\infty}N^{-1}< Ψ_t,\mathcal{N}^{ϕ_t}Ψ_t>=1$ and that $ϕ_t$ is -- dependent of the scaling of the range of the interaction -- solution of the Gross-Pitaevskii or Hartree equation. We shall also show that under additional decay conditions of $ϕ_t$ the limit can be taken uniform in $t

DOI
10.48550/arxiv.0808.1178

Related papers

Rigorous Derivation of the Gross-Pitaevskii Equation: the Case of a Strong Potential

László Erdős, Benjamin Schlein, Horng‐Tzer Yau

Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schrodinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ \leq C N >. \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $\psi_{N,t}$ factorize in the limit $N \to \infty$. 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 $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we extend this result to large potentials.

•Feb 26, 2008•2 citations

Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields

Peter Pickl

Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.

arXiv:1001.4894•Jan 27, 2010•24 citations

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

László Erdős, Benjamin Schlein, Horng‐Tzer Yau

Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $ψ_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $ψ_{N,0}$ satisfies the energy condition \[ < ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k \] for $k=1,2,... $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $ψ_{N,t}$ 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 $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $ψ_{N,0}$ is assumed in a stronger sense.

arXiv:math-ph/0606017•Annals of Mathematics•Jun 27, 2010•328 citations

Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

Ofir E. Alon, Alexej I. Streltsov, L. S. Cederbaum

The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of $N$ interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or $\text{MCTDHB}(M)$, where $M$ specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number $M$ of orbitals is demonstrated in a specific example of four interacting bosons in a double well.

arXiv:cond-mat/0703237•Physical Review A•Mar 14, 2008•355 citations

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

Lea Boßmann

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.

arXiv:1907.04547•Archive for Rational Mechanics and Analysis•Jul 18, 2020•10 citations

Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas

Crispin W. Gardiner

The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number $N$ of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator $A$ for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the $N$ particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different $N$. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for $N$ particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed $N$ is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of $\sqrt{N}$ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.

arXiv:quant-ph/9703005•Physical Review A•Aug 1, 1997•256 citations
We are interested in solutions ΨtΨ_tΨt​ of the Schrödinger equation of NNN interacting bosons under the influence of a time dependent external field, where the range and the coupling constant of the interaction scale with NNN in such a way, that the interaction energy per particle stays more or less constant. Let Nϕ0\mathcal{N}^{ϕ_0}Nϕ0​ be the particle number operator with respect to some ϕ0∈L2(R3→C)ϕ_0\in L^2(\mathbb{R}^3\to\mathbb{C})ϕ0​∈L2(R3→C). Assume that the relative particle number of the initial wave function N−1<Ψ0,Nϕ0Ψ0>N^{-1}< Ψ_0,\mathcal{N}^{ϕ_0}Ψ_0>N−1<Ψ0​,Nϕ0​Ψ0​> converges to one as N→∞N\to\inftyN→∞. We shall show that we can find a ϕt∈L2(R3→C)ϕ_t\in L^2(\mathbb{R}^3\to\mathbb{C})ϕt​∈L2(R3→C) such that lim⁡N→∞N−1<Ψt,NϕtΨt>=1\lim_{N\to\infty}N^{-1}< Ψ_t,\mathcal{N}^{ϕ_t}Ψ_t>=1limN→∞​N−1<Ψt​,Nϕt​Ψt​>=1 and that ϕtϕ_tϕt​ is -- dependent of the scaling of the range of the interaction -- solution of the Gross-Pitaevskii or Hartree equation. We shall also show that under additional decay conditions of ϕtϕ_tϕt​ the limit can be taken uniform in $t
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.
Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.
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.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
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.
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number NNN of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator AAA for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the NNN particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N\sqrt{N}N​, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different NNN. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for NNN particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed NNN is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N\sqrt{N}N​ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.