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For the Proceedings of `Quantum Theory and Symmetries' (Goslar, 18-22 July 1999) (World Scientific, 2000), edited by H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke
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Bose Gas, Gross-Pitaevskii equation
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\begin{document}
\begin{center}
\vspace*{1.0cm}
{\LARGE{\bf The Ground State Energy and Density \\
of Interacting Bosons in a Trap}}
\vskip 1.5cm
{\large {\bf Elliott~H.~Lieb$^{1}$, Robert Seiringer$^2$ and Jakob
Yngvason$^2$}}
\vskip 0.5cm
$^1$Departments of Physics and Mathematics, Princeton University, \\
P.~O.~Box 708, Princeton, New Jersey 08544-0708, USA\\
$^2$ Institut f\"ur Theoretische Physik, Universtit\"at Wien, \\
Boltzmanngasse 5, A-1090 Wien, Austria
\end{center}
\vspace{1 cm}
\begin{abstract}
In the theoretical description of recent experiments with dilute Bose
gases confined in external potentials the Gross-Pitaevskii equation
plays an important role. Its status as an approximation for the
quantum mechanical many-body ground state problem has recently been
rigorously clarified. A summary of this work is presented here.
\end{abstract}
\footnotetext[0]{\copyright 1999 by the authors. Reproduction
of this work, in its entirety, by any means, is permitted for
non-commercial purposes.}
\vspace{1 cm}
\section{Introduction}
The Gross-Pitaevskii (GP) equation is a nonlinear Schr\"odinger equation
that was introduced in the early sixties \cite{G1961}--\cite{G1963} as a
phenomenological equation for the order parameter in superfluid
${\rm He}_{4}$. It has come into prominence again because of recent
experiments on Bose-Einstein condensation of dilute gases in magnetic traps.
The paper \cite{DGPS} brings an up to date review of these
developments.
One of the inputs needed for the justification of the GP equation
starting from the many body Hamiltonian is the ground state energy of
a a dilute, thermodynamically infinite, homogeneous Bose gas. The
formula for this quantity is older than the GP equation but it has
only very recently been derived rigorously for suitable interparticle
potentials. See \cite{LY1998} and \cite{LY1999}. The paper
\cite{LSY1999} goes one step further and derives the GP as
a limit of the full quantum mechanical description. This derivation
is summarized in the present contribution.
The starting point of the investigation is the
Hamiltonian for $N$ bosons that interact with each other via a
spherically symmetric pair-potential $v(|\x_i - \x_j|)$ and are
confined by an external potential $V(\x)$:
\begin{equation}
H = \sum_{i=1}^{N} \{- \Delta_i + V(\x_{i})\}+
\sum_{1 \leq i < j \leq N} v(|\x_i - \x_j|).
\end{equation}
The Hamiltonian acts on {\it symmetric} wave functions in
$L^2(\R^{3N},d^{3N}x)$. The pair interaction $v$ is assumed to be
{\it nonnegative} and of short range, more precisely, $v(r)\leq {\rm
(const.)}\,r^{-(3+\varepsilon)}$ as $r\to\infty$, for some
$\varepsilon>0$. The potential $V$ that represents the trap is
continuous and $V(\x)\to\infty$ as $|\x|\to\infty$. By shifting the
energy scale we can assume that $\min_{\x}V(\x)=0$.
Units are chosen so that $\hbar=2m=1$, where $m$ is the
particle mass. A natural energy unit is given by the ground state
energy $\hbar\omega$ of the one particle Hamiltonian $-(\hbar^2/2m)\Delta+V$.
The corresponding length unit, $\sqrt{\hbar/(m\omega)}$, measures the
effective extension of the trap.
The ground state wave function $\Psi_{0}(\x_{1},\dots,\x_{N})$
satisfies
$H\Psi_{0}=E^{\rm QM}\Psi_{0}$,
with the ground state energy $E^{\rm QM}=\inf{\rm \,spec\,}H$.
If $v=0$, then
%\begin{equation}
$\Psi_{0}(\x_{1},\dots,\x_{N})=
\prod_{i=1}^{N}\Phi_{0}(\x_{i}),$
%\end{equation}
with $\Phi_{0}$ the ground state wave function of $- \Delta + V(\x)$.
On the other hand, if
$v\neq 0$ the ground state
$\Psi_{0}(\x_{1},\dots,\x_{N})$ is, in general,
far from being a product state if
$N$ is large. In spite
of this fact, recent experiments on BE condensation are usually
interpreted in terms of a function $\Phi^{\rm GP}(\x)$ of a single
$\x\in{\mathbb
R}^3$, the solution of the
{\it Gross-Pitaevskii equation}
\begin{equation}\label{gpe}(- \Delta + V+8\pi
a|\Phi|^2)\Phi=\lambda\Phi, \end{equation}
together with the normalization condition
\begin{equation}\label{norm}
\int_{{\mathbb R}^3}|\Phi(\x)|^2=N.
\end{equation}
Here $a$ is the {\it scattering length} of the potential $v$:
\begin{equation}a=\lim_{r\to\infty}\left(r-\frac{u_{0}(r)}{
u_{0}'(r)}\right)\end{equation}
where $u_{0}$ satisfies the zero energy scattering equation,
\begin{equation}\label{scatteq}- u^{\prime\prime}(r)+\mfr1/2 v(r)
u(r)=0,\end{equation}
and $u_{0}(0)=0$. (The factor $1/2$ is due to the reduced mass of the
two-body problem.)
The GP equation (\ref{gpe}) is the variational equation for the
minimization of the GP {\it energy
functional}
\begin{equation}\label{gpf}\E^{\rm
GP}[\Phi]=\int\left(|\nabla\Phi|^2+V|\Phi|^2+4\pi
a|\Phi|^4\right)d^3\x\end{equation}
with the subsidiary condition (\ref{norm}). The corresponding energy is
\begin{equation}E^{\rm GP}(N,a)=\inf_{\int|\Phi|^2=N}\E^{\rm GP}[\Phi]=
\E^{\rm GP}[\Phi^{\rm GP}
],\end{equation}
with a unique, positive $\Phi^{\rm GP}$. The eigenvalue,
$\lambda$, in (\ref{gpe}) is related to $E^{\rm GP}$ by
\begin{equation}
\lambda=dE^{\rm GP}(N,a)/dN=E^{\rm GP}(N,a)/N+4\pi a\bar \rho,
\end{equation}
where
\begin{equation}\label{rhobar}
\bar\rho=\frac 1N\int|\Phi^{\rm GP}(\x)|^4 d^3\x
\end{equation}
is the {\it mean density}. The minimizer $\Phi^{\rm GP}$ of
(\ref{gpf}) with the condition (\ref{norm}) depends on $N$ and $a$, of
course, and when this is important we denote it by $\Phi^{\rm GP}_{N,a}$.
Mathematically, the GP equation is quite similar to the
Thomas-Fermi-von Weizs\-\"acker equation \cite{lieb81} and its basic
properties can be established by similar means. See \cite{LSY1999},
Sect.~2 and Appendix A.
The idea is now that for {\it dilute} gases one should have
\begin{equation}\label{approx}E^{\rm GP}
\approx E^{\rm QM}\quad{\rm and}\quad \rho^{\rm QM}(\x)\approx
\left|\Phi^{\rm GP}(\x)\right|^2\equiv \rho^{\rm GP}(\x),\end{equation}
where the quantum mechanical particle density in the ground state is
defined by \begin{equation} \rho^{\rm
QM}(\x)=N\int|\Psi_{0}(\x,\x_{2},\dots,\x_{N})|^2d\x_{2}\cdots
d\x_{N}. \end{equation} {\it Dilute} means here that
\begin{equation}\bar\rho a^3\ll 1.\end{equation} The task is to make
(\ref{approx}) precise and prove it!
The first remark is that by scaling
\begin{equation}E^{\rm GP}(N,a)=NE^{\rm GP}(1,Na)\quad
\mbox{and}\quad
\Phi^{\rm GP}_{N,a}=N^{1/2}\Phi^{\rm GP}_{1,Na}.
\label{scaling}
\end{equation}
Hence $Na$ is the natural parameter in GP theory. It should also be
noted that $E^{\rm QM}$ depends on $N$ and $v$ and not only on the
scattering length $a$. However, in the limit we are about to define,
it is really only $a$ that matters. To bring this out we write
\begin{equation}v(r)=(a_1/a)^2v_1(a_1r/a),
\end{equation}
where $v_{1}$ has scattering length $a_{1}$ and regard $v_{1}$ as {\it
fixed}. Then $E^{\rm QM}$ is a function $E^{\rm QM}(N,a)$
of $N$ and $a$ and we can
state our main result.
\begin{thm}[Dilute limit of the QM ground state energy and density]
\begin{equation}\label{econv}
\lim_{N\to\infty}\frac{{E^{\rm QM}(N,a)}}{ {E^{\rm
GP}(N,a)}}=1\end{equation}
and
\begin{equation}\label{dconv}
\lim_{N\to\infty}\frac{1}{ N}\rho^{\rm QM}_{N,a}(\x)=
\left |{\Phi^{\rm GP}_{1,Na}}(\x)\right|^2\end{equation}
if $Na$ is fixed. The convergence in (\ref{dconv}) is in the
weak $L_1$-sense.
\end{thm}
\noindent In particular, the limits depend only on the scattering
length $a_{1}=Na$ of $v_{1}$ and not on details of the potential.
\medskip
\noindent{\it Remark.} Since $\bar\rho\sim N$ the attribute `dilute'
may seem a bit strange for these limits. However, what matters is
that the scattering length $a$ is small compared to the mean particle
distance, $\sim\bar\rho^{1/3}$, and if $Na$ is kept constant, then
$\bar\rho a^3\sim N^{-2}$. It is also important to remember that the
unit of length, $\sqrt{\hbar/(m\omega)}\equiv a_{\rm trap}$, depends on
the external potential, and $a$ really stands for $a/a_{\rm trap}$.
If the external potential is scaled with $N$, the both $a_{\rm trap}$
and the energy unit $\hbar\omega$ depend on $N$. For instance, in
order to achieve a finite transition temperature for the BE
condensation of a noninteracting gas in a parabolic trap it is
necessary to keep $N\omega^3$ fixed in the thermodynamic limit, and
hence $a_{\rm trap}\sim N^{1/6}$ and $\hbar\omega\sim N^{-1/3}$.
(See \cite{DGPS}, Eq.\ (14).) However, since the energy unit cancels in
(\ref{econv}) a dependence of $V$ on $N$ does not affect the
validity of (\ref{econv}), and (\ref{dconv}) also remains valid taking
into account that both sides really contain the factor
$a_{\rm trap}^{-3}$.
\section{The dilute homogeneous Bose gas}
The motivation for the last term in the GP energy functional
(\ref{gpf}) is an asymptotic formula for the quantum mechanical ground
state energy $E_{0}(N,L)$ of $N$ bosons in a rectangular {\it box} of
side length $L$ (i.e., the {\it homogeneous} case), that was put
forward by several authors many decades ago. Consider the energy per
particle in the
thermodynamic limit with $\rho=N/L^3$ fixed:
\begin{equation}e_{0}(\rho)=
\lim_{L\to\infty}E_{0}(\rho L^3,L)/(\rho L^3).\end{equation}
According to the pioneering work of Bogoliubov \cite{BO} the leading
term in the
{\it low density asymptotics} of $e_{0}(\rho)$ is given by
\begin{equation}e_{0}(\rho)\approx4\pi\rho a\end{equation}
for $\rho a^3\ll 1$.
In the 50's and early 60's several derivations of this formula were
presented \cite{Lee-Huang-YangEtc}.% \cite{Lieb63}.
They all depended on some special
assumptions about the ground state that have never been proved or the
selection of some special terms from a perturbation expansion that
most likely diverges.
The only rigorous estimates of this period were obtained by Dyson \cite{dyson}
for hard spheres:
\begin{equation}\frac{1+2 Y^{1/3}}{ (1-Y^{1/3})^2}\geq
\frac{e_{0}(\rho)}{ 4\pi\rho a}\geq\frac{1}{ 10\sqrt 2} \end{equation}
with $Y\equiv 4\pi\rho a^3/3$. The upper bound has the right
asymptotic form, and it is not very difficult to generalize it to other
potentials than hard spheres.
The lower bound on the other hand is too low by a factor of about $1/14$
and a remedy for this situation was obtained only 40 years
after Dyson's paper:
\begin{thm}[Lower bound for a homogeneous gas] If $v$ is nonnegative and of
finite range, then
\begin{equation}\label{lbd}\frac{e_{0}(\rho)}{ 4\pi\rho a}\geq (1-C\,
Y^{1/17})\end{equation}
\medskip
with some constant $C$.
\end{thm}
The proof of this theorem is given in \cite{LY1998}, see also the
exposition in \cite{LY1999}. For the application to the proof of Theorem 1.1
we need a version for finite boxes that is implicitly contained in
\cite{LY1998} and explicitly stated in \cite{LY1999}:
\begin{thm}[Lower bound in a finite box] \label{lbthm2}
For a positive potential $v$ with finite range there is
a $\delta>0$ such that
\begin{equation}\label{lbd2}E_{0}(N,L)/N\geq 4\pi\mu\rho
a \left(1-C\,
Y^{1/17}\right)
\end{equation}
for all $N$ and $L$ with $Y<\delta$ and $L/a>C'Y^{-6/17}$. Here
$C$ and $C'$ are constants,
independent of $N$ and $L$. (Note that the condition on $L/a$
requires in particular that $N$ must be large enough,
$N>\hbox{\rm (const.)}Y^{-1/17}$.)
\end{thm}
These results are stated for interactions of finite range. An
extension to potentials $v$ of infinite range
decreasing faster than $1/r^3$ at infinity is
obtained by approximating $v$ by finite range potentials, controlling the
change of the scattering length as the cut-off is removed. In this
case the estimate holds also, but possibly with an exponent different
from $1/17$ and a different constant.
It should be noted, however, that the form of the error term in
(\ref{lbd}) is
dictated by the method of proof. The true error term
presumably
does not have a negative sign for sufficiently small $Y$.
\section{An upper bound for the QM energy}
We now turn to the inhomogeneous gas. In order to prove Eq.\ (\ref{econv}) one
has to establish upper and
lower bounds for $E^{\rm QM}$ in terms of $E^{\rm GP}$ with errors
that vanish in the limit considered. As usual, the upper bound
is easier. It is based on test wave
functions of the form
\begin{equation}\label{ansatz}\Psi(\x_{1},\dots,\x_{N})
=\prod_{i=1}^N\Phi^{\rm
GP}(\x_{i})F(\x_{1},\dots,\x_{N}).\end{equation}
where $F$ is constructed in the following way:
\begin{equation}F(\x_1,\dots,\x_N)=\prod_{i=1}^N
%F_i(\x_1,\dots,\x_i) \quad
%{\rm with} \quad
%F_i(\x_1,\dots,\x_i)=
f(t_i(\x_1,\dots,\x_i)),\end{equation}
where $t_i = \min\{|\x_i-\x_j|, 1\leq j\leq i-1\}$ is the distance
of $\x_{i}$ to its {\it nearest neighbor} among the points
$\x_1,\dots,\x_{i-1}$ and $f$ is a function of $t\geq 0$.
With $u_{0}$ the zero energy scattering solution and
$f_{0}(r)=u_{0}(r)/r$ the function $f$ can be taken as
\begin{equation}\label{eff}f(r)=f_{0}(r)/f_{0}(b)\end{equation}
for $r**0$ so that
\begin{equation}N_{0}>cN\end{equation}
for all $N$.
\medskip
This definition applies also if $\langle\cdot\rangle_0$ is replaced by
a thermal equilibrium state at nonzero temperature.
Note that the density in momentum space is
\begin{equation}\langle
\tilde a^*(\p)\tilde
a(\p)\rangle_0=\int\exp(i\p\cdot(\x-\x^\prime))\gamma{}_{N}(\x,\x^\prime)d\x
d\x^\prime \ , \end{equation}
and this differs from $\left|\int
\exp(i\p\cdot\x)\sqrt{\rho(\x)}d\x\right|^2$ unless $\gamma{}_{N}$ has rank 1. \medskip
It is often claimed that $\Phi^{\rm GP}$ is (approximately) the
eigenfunction to the highest eigenvalue of $\gamma{}_{N}(\x,\x^\prime)$
and hence that $|\tilde\Phi^{\rm GP}(\vec p)|^2$ gives the momentum
distribution of the condensate, but this is not proved yet. In fact,
so far the only cases with genuine interaction where BE condensation has been
rigorously established in the ground state are lattice gases at
precisely half filling. (The hard core lattice Bose gas corresponds to
the $XY$ spin $1/2$ model and BE condensation was proved in dimension
$\geq3$ \cite{DLS} and dimension $2$ \cite{LKS1988a}. The hard core
lattice gas with nearest neighbor repulsion corresponds to the
Heisenberg antiferromagnet and condensation was proved for high
dimension in \cite{DLS} and dimension $\geq3$ in \cite{LKS1988}.
Dimension $2$ is still open.)
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\end{document}
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