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Bose-Einstein condensation, Gross-Pitaevskii approximation
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\begin{document}
\twocolumn[
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\title{Proof of Bose-Einstein Condensation for Dilute Trapped Gases}
\author{Elliott H. Lieb and Robert Seiringer\cite{leave}}
\address{Department of Physics, Jadwin Hall, Princeton University,
P.O. Box 708, Princeton, New Jersey 08544, USA}
\date{December 16, 2001}
\maketitle
\begin{abstract}
The ground state of bosonic atoms in a trap has been shown
experimentally to display Bose-Einstein condensation (BEC). We
prove this fact theoretically for bosons with two-body repulsive
interaction potentials in the dilute limit, starting from the basic
Schr\"odinger equation; the condensation is 100\% into the state
that minimizes the Gross-Pitaevskii energy functional. This is
the first rigorous proof of BEC in a physically realistic,
continuum model.
\end{abstract}
\vfill \pacs{PACS numbers: 05.30.Jp, 03.75.Fi, 67.40-w} \twocolumn
\vskip.5pc ] \narrowtext
It is gratifying to see the experimental realization, in traps, of
the long-predicted Bose-Einstein condensation (BEC) of gases. From
the theoretical point of view, however, a rigorous demonstration
of this phenomenon -- starting from the many-body Hamiltonian of
interacting particles -- has not yet been achieved. In this letter
we provide such a rigorous justification for the ground state of
2D or 3D bosons in a trap with repulsive pair potentials, and in
the well-defined limit (described below) in which the Gross-Pitaevskii
(GP) formula is applicable. It is the first proof of BEC for interacting
particles in a continuum (as distinct from lattice) model and in a
physically realistic situation.
The difficulty of the problem comes from the fact that BEC is not
a consequence of energy considerations alone. The correctness
\cite{LY98} of Bogolubov's formula for the ground state energy
per particle, $e_0(\rho)$, of bosons at low density $\rho$, namely
$e_0(\rho) = 2\pi \hbar^2 \rho a /m$ (with $m=$ particle mass and $a=$
scattering length of the pair potential) shows only that
`condensation' exists on local length scales. The same is true
\cite{LY01} in 2D, with Schick's formula \cite{Schick} $e_0(\rho)
= 2\pi \hbar^2 \rho / (m |\ln (\rho a^2)|)$. Although it is convenient to
{\it assume} BEC in the derivation of $e_0(\rho)$, these formulas for
$e_0(\rho)$ do not {\it prove} BEC. Indeed, in 1D the assumption
of BEC leads to a correct formula \cite{LL63} for $e_0(\rho)$, but
there is, presumably, no BEC in 1D ground states \cite{PS91}.
The results just mentioned are for homogeneous gases in the thermodynamic
limit. For traps, the GP formula is exact \cite{LSY00,LSY01} in the
limit, and one expects BEC into the GP
function (instead of into the constant, or zero
momentum, function appropriate for the homogeneous gas). This is
proved in Theorem 1. In the homogeneous case the BEC is not
100\%, even in the ground state. There is always some `depletion'. In
contrast, BEC in the GP limit is 100\% because the $N\to \infty$
limit is different.
In the homogeneous case one fixes $a>0$ and takes $N\to \infty$
with $\rho = N/{\mathrm{volume}}$ fixed. For the GP limit one fixes
the external trap potential $V(\x)$ and fixes $Na$, the `effective
coupling constant', as $N\to \infty$. A particular, academic
example of the trap is $V(\x)=0$ for $\x$ inside a unit cube and
$V(\x) =\infty $ otherwise. By scaling, one can relate this
special case to the homogeneous case and thereby compare the two
limits; one sees that the homogeneous case corresponds,
mathematically, to the trap case with this special $V$, but with
$N a ^3 =\rho a^3$ fixed as $N\to \infty$. Thus, BEC in the trap case
is the easier of the two,
reflecting the incompleteness of BEC in the
homogeneous case.
The lack of depletion in the GP limit is
consistent with $\rho a^3 \to 0$ and with
Bogolubov theory.
We now describe the setting more precisely. We concentrate on the
3D case, and comment on the generalization to 2D at the end of
this letter. The Hamiltonian for $N$ identical bosons in a trap
potential $V$, interacting via a pair potential $v$, is
\begin{equation}\label{ham}
H=\sum_{i=1}^N \left(-\Delta_i+V(\x_i)\right)+\sum_{1\leq i0$ let $\K=\{\x\in\R^3,
|\x|\leq R\}$, and define
$$ \langle f_\X\rangle_\K=\frac
1{\int_\K |\pgp(\x)|^2 d^3\x} \int_\K |\pgp(\x)|^2 f_\X(\x)\, d^3\x \ .
$$
We shall use Lemma 2, with $m=3$,
$h(\x)=|\pgp(\x)|^2/\int_\K|\pgp|^2$, $\Omega=\Omega_\X\cap\K$ and
$f(\x)= f_\X(\x)-\langle f_\X \rangle_\K$ (see (\ref{defomega})
and (\ref{deff})). Since $\pgp$ is bounded on $\K$ above and below
by some positive constants, this Lemma also holds (with a
different constant $C'$) with $d^3\x$ replaced by $|\pgp(\x)|^2d^3\x$
in (\ref{poinc}). Therefore,
\begin{eqnarray}\nonumber
&&
\int d\X \int_\K d^3\x |\pgp(\x)|^2 \left[f_\X(\x)-\langle
f_\X\rangle_\K\right]^2
\\ \nonumber && \leq C'\int d\X\left[\int_{\Omega_\X\cap \K}
|\pgp(\x)|^2|\nabla_{\x} f_\X(\x)|^2 d^3\x\right. \\ &&\left.
\qquad\quad\qquad + \frac {N^{-8/51}}{R^2} \int_\K
|\pgp(\x)|^2|\nabla_{\x} f_\X(\x)|^2 d^3\x \right], \label{21}
\end{eqnarray}
where we used that $|\Omega_\X^c\cap\K|\leq (4\pi/3)
N^{-4/17}$. The first integral on the right side of (\ref{21})
tends to zero as $N\to\infty$ by Lemma 1, and the second is
bounded by (\ref{bound}). We conclude, since $\int_\K |\pgp(\x)|^2
f_\X(\x) d^3\x\leq \int_{\R^3} |\pgp(\x)|^2 f_\X(\x)d^3\x$, that
\begin{eqnarray}\nonumber &&\liminf_{N\to\infty} \frac 1N \langle
\pgp|\gamma|\pgp\rangle \geq \\ \nonumber &&\geq \int_\K
|\pgp(\x)|^2 d^3\x \, \lim_{N\to\infty}\int d\X \int_\K d^3\x
|\Psi(\x,\X)|^2 \ .
\end{eqnarray}
It follows from (\ref{part1}) that the right side of this inequality
equals $\left[\int_\K |\pgp(\x)|^2 d^3\x\right]^2$. Since the radius
of $\K$ was arbitrary, $\frac 1 N \langle\pgp|\gamma|\pgp\rangle\to 1$,
implying Theorem 1 (cf. \cite{S79}, Theorem 2.20). \hfill QED
We remark that the method presented here also works in the case of a 2D
Bose gas. The relevant parameter to be kept fixed in the GP limit is
$g=4\pi N/|\ln (a^2 N)|$, all other considerations carry over without
essential change, using the results in \cite{LSY01,LY01}. A minor
difference concerns the parameter $s$ in Theorem 2, which can be shown
to be always equal to $1$ in 2D, i.e., the interaction energy is purely
kinetic in the GP limit (see \cite{CS01b}). We also point out that our
method necessarily fails for the 1D Bose gas, where there is presumably
no BEC \cite{PS91}. An analogue of Lemma 1 cannot hold in the 1D case
since even a hard core potential with arbitrarily small range produces an
interaction energy that is not localized on scales smaller than the total
size of the system. There is also no GP limit for the one-dimensional
Bose gas in the above sense.
We are grateful to Jakob Yngvason for helpful discussions.
E.H.L. was partially supported by the U.S. National Science Foundation
grant PHY 98 20650. R.S. was supported by the Austrian Science Foundation
in the form of an Erwin Schr\"odinger Fellowship.
\begin{references}
\bibitem[*]{leave} Erwin Schr\"odinger Fellow.
On leave from Institut f\"ur
Theoretische Physik, Universit\"at Wien, Boltzmanngasse 5, 1090
Vienna, Austria
\bibitem{LY98} E.H. Lieb and J. Yngvason,
%{\it Ground State Energy of the Low Density Bose Gas},
Phys. Rev. Lett. \textbf{80}, 2504 (1998)
\bibitem{LY01} E.H. Lieb and J. Yngvason,
%{\it Ground State Energy of a Dilute Two-dimensional Bose Gas},
J. Stat. Phys. {\bf 103}, 509 (2001)
\bibitem{Schick}
M. Schick,
%{\it Two-Dimensional System of Hard Core Bosons},
Phys. Rev. A {\bf 3}, 1067 (1971)
\bibitem{LL63}
E.H. Lieb and W. Liniger,
%{\it Exact Analysis of an Interacting Bose Gas.
%I. The General Solution and the Ground State},
Phys. Rev. {\bf 130}, 1605 (1963)
\bibitem{PS91} L. Pitaevskii and S. Stringari, J. Low
Temp. Phys. {\bf 85}, 377 (1991)
\bibitem{LSY00} E.H. Lieb, R. Seiringer, and J. Yngvason,
%{\it Bosons in a Trap: A Rigorous Derivation of the
%Gross-Pitaevskii Energy Functional},
Phys. Rev. A {\bf 61}, 043602 (2000)
\bibitem{LSY01}
E.H. Lieb, R. Seiringer, and J. Yngvason,
%{\it A Rigorous Derivation of the Gross-Pitaevskii
%Energy Functional for a Two-Dimensional Bose Gas},
Commun. Math. Phys. {\bf 224}, 17 (2001)
\bibitem{CS01a} A.Y. Cherny and A.A. Shanenko, Preprint
%{\it The kinetic and interaction energies of a trapped
%Bose gas: Beyond the mean field},
arXiv:cond-mat/0105339
\bibitem{LY00} E.H. Lieb and J. Yngvason,
%{\it The Ground State Energy of a Dilute Bose Gas},
in: {\it Differential Equations and Mathematical Physics}, {\it
Proceedings of 1999 conference at the Univ. of Alabama}, R.
Weikard and G. Weinstein eds., p. 295, International Press (2000)
\bibitem{LL}
E.H. Lieb and M. Loss, {\it Analysis}, 2nd ed., Amer. Math.
Society, Providence, R.I. (2001)
\bibitem{S79} B. Simon, {\it Trace ideals and their application},
Cambridge University Press (1979)
\bibitem{CS01b} A.Y. Cherny and A.A. Shanenko,
%{\it Dilute Bose gas in two dimensions: Density
%expansions and the Gross-Pitaevskii equation},
Phys. Rev. E {\bf 64}, 027105 (2001)
\end{references}
\end{document}
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