text:
\chapter{BEC AND THE NEW WORLD OF COHERENT MATTER WAVES}
\chapterauthors{Allan Griffin}
\noindent\chapteraddress

\vskip 12pt

After decades of effort to produce an atomic Bose condensate, this was
finally achieved in June, 1995 in a laser-cooled magnetically trapped gas
of $^{87}$Rb atoms by Eric Cornell and Carl Wieman at JILA (University of
Colorado and N.I.S.T.).  As of July, 1999 (4 years later), there are now
over 20 experimental groups around the world who can routinely produce and
study such atomic Bose condensates.  In addition, over 1000 theoretical
papers have been published!

I will give three {\it introductory} lectures on this new {Quantum Phase
of Matter}, which I think will continue to be a growth point of
fundamental physics research for the next decade and will also be the
source of new technologies based on using this source of {\it coherent
matter waves}.
A brief sketch of my three lectures is as follows:
\begin{enumerate}
\item A BRIEF HISTORY OF BEC STUDIES.
\begin{itemize}
\item Before 1995, going back to pioneering work of Einstein (1925) and
Fritz London (1938).
\item Introduce the concept of a {\it Bose macroscopic wavefunction}
$\Phi({\bf r}, t)$ describing a Bose condensate and the key
Gross-Pitaevskii equation of motion for $\Phi({\bf r}, t).$
\item Review some of the results obtained over the last 4 years - and
indicate why trapped Bose gases are so interesting, possibly even more
than superfluid $^4$He and BCS superconductors, which also involve Bose
condensation.
\end{itemize}
\item DYNAMICS OF A PURE CONDENSATE $(T\ll T_{BEC})$.
\begin{itemize}
\item Time-dependent Gross-Pitaevskii (1961) equation of motion.
\item Crucial effect of weak interatomic interactions.
\item Collective oscillations of the condensate.
\item The Bose condensate as a {\it classical} quantum object!
\end{itemize}
\item DYNAMICS OF COUPLED CONDENSATE AND NON-CONDEN-SATE COMPONENTS
(TWO-FLUID HYDRODYNAMICS):
\begin{itemize}
\item Derivation of a {\it quantum} Boltzmann equation for the
non-conden-sate excited atoms and a generalized GP equation for the
condensate.
\item More complex behaviour than in superfluid $^4$He.
\item Comparison with the well-known Landau two-fluid theory (1941).
\end{itemize}
\end{enumerate}
Further references which are relevant for these lectures are:
\begin{enumerate}
\item BEC Homepage (maintained by a BEC theorist, Mark Edwards):
\newline
http://amo.phy.gasou.edu/bec.html/.  This widely-used homepage has played
a crucial role in BEC research.
\item A recent article \cite{Dalgiorpit} by Dalfovo, Giorgini, Pitaevskii
and Stringari is an authoritative review of recent theory on atomic Bose
condensates and expands on the material I cover in Sections
\ref{sec:Overview} and \ref{sec:Dynamics}.
\item  Articles in a book \cite{Ingstrwie}, {\bf Bose-Einstein
condensation in atomic gases},  {\it Proceedings of the International
School of Physics
``Enrico Fermi''}, ed by M. Inguscio, S. Stringari and C. Wieman.  This
contains many review articles on current BEC research.  In particular, I
call attention to (all these can be downloaded from the LANL website under
cond-mat):
\begin{itemize}
\item W. Ketterle, D.S. Durfee and D.M. Stamper-Kurn, ``Making, probing
and understanding Bose-Einstein condensates'' - a 100 page review of
recent experiments \cite{ketterle}.
\item A. Griffin ``A brief history of our understanding of BEC: From Bose
to Beliaev'' \cite{Agrif}.
\item A. Griffin, ``Theory of excitations of the condensate and
non-condensate at finite temperature'' \cite{Agrif}.
\item A.L. Fetter, ``Theory of a dilute low-temperature trapped Bose
condensate.''  A very detailed analysis at $T = 0$.
\end{itemize}
\item A long article by Zaremba, Nikuni and Griffin \cite{Zarnikgrif}  on
the non-equilibrium dynamics of trapped Bose gases at finite temperatures.
This expands on the material covered in Section~\ref{sec:Coupled} of these
lectures.
\end{enumerate}

\section{AN OVERVIEW OF PAST AND RECENT WORK}
\label{sec:Overview}

\subsection{Some history before 1980}

Einstein predicted that a non-interacting gas of atoms (Bosons) would
undergo a {\it phase transition} at low temperatures, when a {\it
macroscopic} $(0(N))$ number of atoms occupy the lowest energy level (in a
uniform ideal Bose gas, this is the the zero momentum single-particle
state ).  His work was inspired by a novel derivation of the Planck
distribution for photons by Bose in 1924.  The basic physics of this phase
transition is worked out in every text in statistical mechanics
\cite{Khuang}.  The simplest way of estimating $T_{BEC}$ is to note that
the transition occurs when

\begin{equation}
\lambda_T\, \gapx \ d = {\rm average \ distance\ between\ atoms} \sim {1
\over n^{1/3}}, \label{eqoverview1}
\end{equation}
where $\lambda_T$ is the thermal De Broglie wavelength of a gas of atoms
at temperature $T$,

\begin{equation}
\lambda_T \equiv \left({2\pi\hbar^2 \over mk_BT}\right)^{1\over
2}.\label{eqoverview2}
\end{equation}
The criterion in (\ref{eqoverview1}) is equivalent to
$n\lambda_T^3\,\gapx\, 1$, while a more careful analysis \cite{Khuang}
gives $n\lambda^3_T = \zeta(3/2) = 2.612.$  One sees that $\lambda_T
\rightarrow $ large as $T\rightarrow 0.$  When $\lambda_T\, \gapx \ d,\, $
all the atoms become {\it correlated} and the gas exhibits new collective
behaviour (even in {\it absence} of interactions).
Using (\ref{eqoverview1}),  one finds that
\begin{equation}
k_B T_{BEC} \sim {2\pi\hbar^2\over m}n^{2/3}.\label{eqoverview3}
\end{equation}
Below $T_{BEC}$, the number of atoms in the ${\bf p} = 0$ single-particle
state increases and is given by the well-known formula
\begin{equation}
{N_c(T)\over N} = \left[1 - \left({T\over
T_{BEC}}\right)^{3/2}\right].\label{eqoverview4}
\end{equation}
At $T=0$, all the atoms in an ideal gas are in this ${\bf p}=0$ state
(this state is the Bose condensate in non-interacting 3D gas).

Nothing much happened until 1938.  Then the neglected work of Einstein was
re-discovered and developed by Fritz London \cite{Flondon}, who suggested
that it might be the basis of an explanation for the strange effects
noticed in liquid $^4$He at $T_c\sim 2.17K.$  London's suggestion was
based on the fact that the $^4$He atom was a ``composite'' Boson $(S = 0)$
and the formula in (\ref{eqoverview3}) gives  $T_{BEC}\sim 3K$ if we used
the density for liquid $^4$He.
L. Tisza used London's idea and suggested (somehow!) that the condensate
atoms act in a coherent way - a {\it new collective degree of freedom
moving without friction}.  This ``picture'' led to a rudimentary two-fluid
model that could explain experiments showing superfluidity (especially by
Kapitza as well as by Allen and Meisner) as a counterflow of the
superfluid and the normal fluid.  In a dilute, weakly interacting Bose
gas, these two components are the condensate and non-condensate,
respectively.

Until the 1960's,  the theory of BEC in interacting systems was dominated
by efforts to use it to understand superfluid $^4$He.  The London-Tisza
scenario was essentially correct, but to formulate it properly needed
field-theoretic many body techniques and the concept of broken-symmetry,
which were only developed in the period 1957-1965.  In this period, a
large amount of work was done on a {\it toy} problem, a dilute weakly
interacting Bose gas, since a Bose liquid like superfluid $^4$He was too
difficult to deal with theoretically.  These early studies \cite{Grif2}
are the foundation of our current understanding of trapped atomic gases.

A phenomenological theory of superfluid $^4$He was introduced by Landau in
1941, based on the idea of quasiparticles (phonon-roton spectrum) and a
two-fluid superfluid hydrodynamics \cite{Land}.  This brilliant theory has
been very successful and is the basis of modern descriptions of superfluid
$^4$He \cite{Khalatnikov}.  However,  it made no explicit mention of BEC
or even the fact that $^4$He atoms obeyed Bose statistics.  Only in the
1960's did it become clear that Landau formulation had it's {\it
microscopic} basis in the existence of a condensate macroscopic
wavefunction,
\begin{equation}
\Phi({\bf r}, t) = <{\hat\psi}({\bf r})> = \sqrt{N_c({\bf r}, t)}
e^{i\theta({\bf r}, t)},\label{eqoverview5}
\end{equation}
where $\hat\psi({\bf r})$ is the quantum field operator (see
Section~\ref{sec:Dynamics}).
This concept was first formally introduced by Beliaev in 1957 \cite{Bel2},
extending the pioneering work of Bogoliubov in 1947 \cite{Nnbog}.  The
superfluid motion is associated with the gradient of the phase of this
two-component order parameter,
\begin{equation}
e^{i\theta} = e^{i(\theta_0 + {\bf
r}\cdot\mbox{\boldmath$\nabla$}\theta)}\ ; \ {\bf k}_s \equiv
{\mbox{\boldmath$\nabla$}}\theta\equiv{m{\bf v}_s\over\hbar}
\label{eqoverview6}
\end{equation}
The essential relation between superfluidity and $\Phi({\bf r}, t)$ is
simply and elegantly described in the classic monograph by Nozi\`eres and
Pines \cite{Gorkov}.
For further discussion of the development of our current understanding of
Bose condensates, see the review article by Griffin in Ref.
\cite{Ingstrwie}.

It might be useful to make a brief digression here on the BCS theory of
superconductors based on formation of Cooper pairs (total spin $S=0),$
which was developed in 1957.  In the BCS theory, superconductors exhibit
the same kind of {\it macroscopic} quantum behaviour as superfluid $^4$He,
as first argued in the late 1930's by F.London.  In the Gor'kov version of
the  BCS theory, the spin-singlet Cooper pair order parameter
\begin{equation}
\Phi({\bf r}, t)=<{\hat\psi}_\uparrow({\bf r}){\hat\psi}_\downarrow({\bf
r})>
\label{eqoverview7}\end{equation}
is the equivalent of
\begin{equation}
\Phi({\bf r}, t) = <{\hat\psi({\bf r})}>
\label{eqoverview8}\end{equation}
in Bose superfluids.
The essential equivalence of these two systems was obscured by the
complexity of the original many-particle BCS-wavefunction, due to the
large spatial size of overlapping Cooper pairs.  As a result, Cooper pairs
{\it form} and {\it condense} into a coherent state at {\it same}
temperature.  It was only in the 1980's that theorists (Leggett,
Nozi\`{e}res and others \cite{Randiera}) realized that for small Cooper
pairs (tightly bound and hence high $T_c$), the BCS theory smoothly goes
over to theory of a weakly interacting Bose gas of non-overlapping Cooper
pairs.
Recently a gas of $^{40}$K atoms (a composite fermion) have been
laser-cooled at JILA to slightly below the Fermi temperature
\cite{Demarco}.  There are ways to make the interaction attractive (see
later) and then one might look for the formation of Cooper pairs
\cite{HTCStoof} of ultra-cold Fermi atoms!

\subsection{More recent developments (1980-1995)}
Since the 1970's, there has been increasing interest by experimentalists
to find a ``pure'' form of BEC, namely, in a low temperature gas.  The two
early candidates for Bosons were excitons (electron-hole pairs) in
semiconductors and spin-polarized hydrogen atoms (see articles by Greytak
and by Wolfe et al. in the book mentioned in Ref. \cite{Randiera}). A gas
of $H_\uparrow$ atoms was predicted to be stable as a gas even at $T=0$.
This is because the atoms cannot combine since there is no bound state of
the interatomic potential between two spin-polarized H atoms.  Thus, one
cannot form liquid or solid phase.  Many people got interested in BEC in
$H_\uparrow$ gas, including theorists \cite{TJGreytak}.

Several  of the key ideas that led to success in alkali atoms in 1995 grew
out of the pioneering work on $H_\uparrow$ gas in the 1980's.
However 3-body interactions become increasingly important at higher
densities and these cause spin flips, allowing formation of H-molecules.
High densities were needed since cooling was by cryogenic methods, which
could reach $\sim 10^{-4}$K but no lower.  BEC in $H_\uparrow$ gas was
finally produced at MIT in June, 1998, after almost 20 years of work
\cite{Physicstoday}.  Unfortunately, the alkali atom condensates are much
easier to create and study, and appear to be more interesting gases.

Since the early 1990's, attention has focussed on the alkali atoms: Li,
Na, K,
Rb, Cs.  The atoms are Bosons, with an even number of neutrons.  The
strategy was to use laser-cooling to get to very low temperatures, where
the low density gas would Bose-condense.  The essential idea behind laser
cooling is that when an atom absorbs a photon, it slows down.  In the
summer of 1995, BEC was announced by three groups led by:

\begin{itemize}
\item C. Wieman and E. Cornell (JILA), using $^{87}$Rb atoms
\cite{Andensmatwiecor}.
\item W. Ketterle (MIT), using $^{23}$Na atoms \cite{Davmewand}.
\item R. Hulet (Rice University), using $^7$Li atoms \cite{Bradsach}.
\end{itemize}

\noindent Parenthetically, it is now the general feeling  that the
original Rice data, interesting as it was,  did not give an unambiguous
signature of a (very small, since the interaction is attractive)
condensate in $^7$Li gas \cite{Bradsachhul}.

Alkali atoms are perfect for BEC studies.  They have a magnetic moment,
and hence can be trapped by magnetic fields.  They essentially have a
``one-electron'' structure.  They are thus simple atoms, and have been
well studied by atomic physicists.  One can easily selectively flip the
``spin'' of higher energy trapped atoms.  These ``hot'' atoms are then
quickly ejected from the magnetic trap and the remaining atoms quickly
thermalize to a lower temperature.  This ``evaporative cooling'' is very
efficient and quickly brings one into the temperature region required for
BEC.

It is useful to mention a few experimental facts about the magnetic traps
currently in use.  As it turns out, these traps are well described as a
harmonic potential
\begin{eqnarray}
V_{ex}({\bf r})&=&{1\over 2}m\omega^2_0 r^2 \ ({\rm{isotropic}})
\label{eqoverview9}\\
&=&{1\over 2}m(\omega_{ox}^2 x^2+\omega_{oy}^2 y^2 +\omega_{oz}^2 z^2)\
({\rm anisotropic}).\label{eqoverview10}
\end{eqnarray}
Most current traps are either:
\begin{eqnarray}
{\rm{pancakes}} &,&  \omega_{oz} \gg \omega_{ox} (=\omega_{oy}) \nonumber\\
{\rm{cigars}} &,& \omega_{oz} \ll \omega_{ox}
(=\omega_{oy})\label{eqoverview11}
\end{eqnarray}
and the trap frequencies are of the order $\omega_0 \sim 2\pi\times
100$Hz.  In 1995, the first condensates were small $\sim 10^3$ atoms and
$T_{BEC}\sim$ 100nK.  However in 1999, the condensates can be quite large
$\sim 10^8$ atoms at $T_{BEC}\sim\mu$K.  These have a size $\sim$ many
microns, which can be easily seen optically .  When the condensates are
small, the trap is turned off and cloud allowed to expand, and then
measured by optical methods.  The results are simple to analyze if gas is
non-interacting.  However, more analysis is needed to include the effects
of interactions during expansion.

Early reports on atomic condensates discussed the system as an {\it ideal}
Bose gas.  It was soon realized that even in these very dilute gases, the
interactions played a crucial role.  Indeed,  since the 1960's, it has
been understood that an interacting Bose gas is quite different from an
ideal Bose gas.  In particular, interactions stabilize (or ``lock'') the
phase of the condensate and allow {\it coherent} properties to emerge.
(We recall that a free Bose gas has a condensate but is not a superfluid).
Of course, interactions are also crucial for cooling.  After  hot atoms
are removed by rf-induced spin-flips, it is important that remaining atoms
can quickly re-thermalize through collisions.

However, it is useful to first consider an {\it ideal} Bose gas in a trap,
to illustrate some characteristic features.  For atoms in an external
potential, we have
\begin{equation}
N = \sum_i f_0(\epsilon_i),\label{eqoverview12}
\end{equation}
where the Bose distribution is
\begin{equation}
f_0(\epsilon_i) = {1\over e^{\beta(\epsilon_i-\mu)}-1}.
\label{eqoverview13}
\end{equation}
In a harmonic trap, the energy levels are:
\begin{equation}
\epsilon_i=\epsilon_{{n_x}{n_y}{n_z}} = (n_x+n_y+n_z+{3\over
2})\hbar\omega_0,\label{eqoverview14}
\end{equation}
with $n_x, n_y, n_z = 0, 1, 2, 3\dots$.
The condensate is described by the ground state single-particle
wavefunction
\begin{equation}
\phi_0({\bf r})\sim e^{-r^2/2a^2_{HO}},\label{eqoverview15}\end{equation}
where the S.H. oscillator length is $a_{HO}\equiv\left({\hbar \over
m\omega_0}\right)^{1\over 2}
\sim 1 \mu m$ in current traps.  Clearly $a_{HO}$ gives the ``size'' of
condensate $n_{c0}({\bf r})= |\Phi_0({\bf r})|^2$ in a trapped gas. For
the non-condensate density ${\tilde n}_0({\bf r})$ (often called the
``thermal cloud''), we can use the {\it semiclassical limit}, since the
thermal energy $(k_B T)$ is much larger than the spacing between the S.H.
energy levels $(\hbar \omega_0).$  Then we have
\begin{eqnarray}
{\tilde n}_0({\bf r})&\sim& e^{-V_{ex}({\bf r})/k_B T} \nonumber\\
&=& e^{-r^2/2R_T^2},\label{eqoverview16}
\end{eqnarray}
where
\begin{equation}
R_T = {\sqrt{k_BT\over m\omega^2_0}} = a_{HO} \left({kT\over
\hbar\omega_0}\right)^{1\over 2} \gg a_{HO}.\label{eqoverview17}
\end{equation}
Thus we see that the size of the thermal cloud $R_T$ is much larger than
the condensate.
The signature for condensate is this {\it sharp} high density peak at the
centre of the trap, which suddenly starts to grow out of the broad thermal
distribution at the predicted transition temperature $T_{BEC}$.  As
$T\rightarrow 0$ (effectively $T\ \lapx \ 0.4 \ T_{BEC}),$ the thermal
cloud steadily disappears as all atoms go into the ground state
$\phi_0({\bf r})$ given by (\ref{eqoverview15}), which is the macroscopic
wavefunction for a non-interacting trapped Bose gas.  The temperature of
the gas is measured from the temperature dependence of the tail of the
thermal distribution given by (\ref{eqoverview16}).

It is easy to calculate the transition temperature for atoms in an
harmonic trap.  Separating out the condensate contribution in
(\ref{eqoverview15}), we have
\begin{equation}
N=N_c+\sum_{i\ne 0} f^0(\epsilon_i); \ N_c = \int d{\bf r}|\phi_0({\bf
r})|^2.\label{eqoverview18}\end{equation}
Making a change of variable $\beta \hbar\omega_0 n_x \equiv {\bar n}_x$,
and using the continuum approximation, we have
\begin{eqnarray}
N-N_c&\simeq& \left({k_BT\over\hbar\omega_0}\right)^3\int^\infty_0 d{\bar
n}_x
\int^\infty_0 d{\bar n}_y \int^\infty_0 d{\bar n}_z {1\over e^{({\bar n}_x
+{\bar n}_y +{\bar n}_z)}-1} \nonumber\\
&=&\zeta(3) \left({k_B T\over \hbar\omega_0}\right)^3.
\label{eqoverview19}
\end{eqnarray}
We note that the chemical potential $\mu_0 = {3\over 2}\hbar\omega_0,$ but
${\hbar\omega_0\over k_B T}\ll 1$ and hence the zero point energy has
been  neglected in (\ref{eqoverview19}).  Since $N_c = 0$ at $T_{BEC}$, we
have
\begin{equation}
k_BT_{BEC} \simeq 0.94 (N^{1\over 3})\hbar\omega_0, \ \mbox{with}\
{N_c(T)\over N} = \left[1-\left({T\over
T_{BEC}}\right)^3\right].\label{eqoverview20}
\end{equation}
One can improve on these simple estimates for $T_{BEC}$ and $N_c(T)$, but
the corrections are only a few percent at best \cite{Dalgiorpit}.

Interactions make a dilute, weakly-interacting Bose condensed gas into a
full non-trivial many body problem, even though the system is very dilute
and the interactions are weak.
In a dilute gas, we need only consider binary collisions.  The real
interatomic potential $v({\bf r})$ has a hard core with a radius of a few
Angstroms and a weak, long-range attractive tail. In a dilute, very cold
gas, we can approximate $v({\bf r})$ using the $s$-wave scattering length
approximation effectively replacing $v({\bf r})$ by a pseudopotential
\cite{Dalibard}
\begin{equation}
v({\bf r}) \Rightarrow {4\pi h^2\over m}a\delta({\bf r})\equiv
g\delta{(\bf r}).  \label{eqoverview21}
\end{equation}
We require $a\ll$ average distance between atoms, or $na^3\ll 1,$ which is
very well satisfied in these gases.  For alkali atoms, $v({\bf r})$ almost
has a bound state of two atoms.  This quasi-bound state is very sensitive
to the long range part of the potential, and thus the value of the
$s-$wave scattering length $a$ can be very large.  Current values for
atoms used in BEC experiments are:
\begin{eqnarray}
^{87}{\rm Rb} &:& a = 58 \stackrel{\circ}{\rm A} \nonumber \\
^{23}{\rm Na} &:& a=28 \stackrel{\circ}{\rm A} \nonumber\\
^7{\rm Li}&:& a=-14 \stackrel{\circ}{\rm A} \label{eqoverview22}
\end{eqnarray}

One can adjust the energy of the quasi-bound state and, as a result,
change the value of s-wave scattering length $a$ with a small magnetic
field.  Near a so-called {\it Feshbach resonance}, one can even change the
interaction sign, going from repulsive $(a>0)$ to attractive $(a<0).$
There is a lot of current work \cite{Phystodayart} trying to exploit this
ability to change the interaction strength and sign by ``turning a
knob'',  perhaps even making $a\rightarrow 0$!

The key equation for the macroscopic wavefunction for a $T=0$ condensate
was written down and discussed by Pitaevskii \cite{LPPit} and Gross
\cite{EPGross} in 1961:
\begin{equation}
i\hbar {\partial\Phi({\bf r}, t)\over\partial t} =
\left[-{\hbar^2\nabla^2\over 2m}+V_{ex}({\bf r})+gn_c({\bf r},
t)\right]\Phi({\bf r}, t),\label{eqoverview23}
\end{equation}
where $n_c = |\Phi|^2.$  This equation describes the condensate atoms
moving in dynamic self-consistent Hartree field produced by the
condensate,
\begin{equation} V_H({\bf r}, t) =
\int d{\bf r}^\prime v({\bf r}-{\bf r}^\prime)n_c({\bf r}^\prime, t) =
gn_c({\bf r}, t).\label{eqoverview24}\end{equation}
The non-linear GP equation (\ref{eqoverview23}) will be the subject of
Section~\ref{sec:Dynamics}.  Hundreds of papers have been written on it in
the last four years.  For $T \ \lapx \ 0.4 \ T_{BEC},$ it describes both
the static properties and the dynamic fluctuations (linear and non-linear)
very well, usually within a few percent \cite{Dalgiorpit, Ingstrwie}.

To complete this brief introduction, I mention several other research
topics that make trapped Bose-condensed gases so exciting:

\begin{enumerate}
\item Alkali atoms have several different atomic hyperfine states.  Apart
from Cs, the alkali atoms have a nuclear spin $I=3/2$ and an electron spin
$S = 1/2$.  Thus the total spin operator ${\bf F} = {\bf I}+{\bf S}$ has
values of 1 and 2, leading to 8 possible atomic states.  One usually works
with one of these atomic hyperfine states - trapped in a magnetic well.
However, purely optical traps (MIT) using the dipole force of a laser beam
can be used to trap low energy atoms in different hyperfine states. Thus
one can now also deal with gases with several different states, ie, a
multicomponent Bose gas. Moreover, one can induce transitions between
different hyperfine states.  One sees that spin is a new degree of freedom
in such multicomponent Bose-condensed gases \cite{Dalgiorpit, Ingstrwie}.

\item Independent of the special ``coherent'' features of a Bose
condensate, these trapped gases give us a source of high density, very
cold atoms.  Lene Hau \cite{Phystodayjul} has used this high density to
slow down the speed of light to that of slow car ($\sim$ 40 km/hour) using
self-induced transparency.  One can also switch on an optical lattice
(produced by intersecting laser beams) on a trapped Bose gas
\cite{Pzoller}.  Turning off  the magnetic trap, the low energy atoms will
occupy the potential minima of this periodic lattice. This could not be
done with high energy atoms since the dipole-induced potentials of the
optical lattice are very weak.  With ultra-cold trapped fermions, one may
also be able to produce a Hubbard model, of the kind extensively studied
in connection with the cuprate-oxide high temperature
superconductors.\end{enumerate}

We agree with the opinion of Pitaevskii \cite{LPPita} that the discovery
of BEC in alkali gases ``can be considered as one of the most beautiful
results of experimental physics in our century''.  The next two sections
will flesh out this qualitative overview with some theoretical
calculations on the collective oscillations at $T\ll T_{BEC}$ (pure
condensate) and at finite temperatures $T\sim T_{BEC}$ (mixture of
condensate and non-condensate) of these strange quantum ``wisps of
matter''.

\section{DYNAMICS OF THE PURE CONDENSATE}
\label{sec:Dynamics}
The theory of interacting Bose-condensed fluids is most usefully discussed
using quantum field operators.  This procedure was formalized by Beliaev
(1957) and developed by Bogoliubov \cite{Bog2}, Gavoret and Nozi\`{e}res
\cite{Gavnoz}, Martin and Hohenberg \cite{Hohmar}, and others in the
1960's \cite{review}.
We recall:
\begin{eqnarray}
& {\hat\psi}^+({\bf r})= \mbox{creates atom at} \ {\bf r} \nonumber \\
&{\hat\psi}({\bf r}) =\mbox{destroys atom at} \ {\bf
r}.\label{eqdynamics25}
\end{eqnarray}
These fields satisfy the usual Bose commutation relations, such as
\begin{equation}
\left[{\hat\psi}({\bf r}), {\hat\psi}^+({\bf r}^\prime)\right] =
\delta({\bf r} - {\bf r}^\prime). \label{eqdynamics26}
\end{equation}
All observables can be written in terms of these quantum field operators,
such as the interaction energy
\begin{eqnarray}
{\hat V}_{ext} &=& {1\over 2}\int d{\bf r} \int d{\bf r}^\prime
{\hat\psi}^+ ({\bf r}^\prime){\hat\psi}^+({\bf r}) v({\bf r}-{\bf
r}^\prime){\hat\psi}({\bf r}^\prime){\hat\psi}({\bf r})\nonumber \\
&=& {1\over 2}g \int d{\bf r} {\hat\psi}^+({\bf r}){\hat\psi}^+({\bf
r}){\hat\psi}({\bf r}){\hat\psi}({\bf r}).\label{eqdynamics27}
\end{eqnarray}

The crucial idea due to Bogoliubov (1947) and later generalized by Beliaev
is to separate out the condensate part
\begin{equation}
{\hat\psi}({\bf r}) = \left<{\hat\psi}({\bf r})\right> +{\tilde\psi}({\bf
r}), \label{eqdynamics28}
\end{equation}
where
\begin{equation}
\left<{\hat\psi}({\bf r}) \right> \equiv \Phi({\bf r}) = \mbox{Bose
macroscopic wavefunction}.\label{eqdynamics29}
\end{equation}
This quantity plays the role of the order parameter for the superfluid
phase transition:
\begin{eqnarray}
\Phi({\bf r}) &= 0 \ \ \ \ T > T_c \nonumber \\
&\ne 0 \ \ \ \ T < T_c. \label{eqdynamics30}
\end{eqnarray}
We note that $\Phi({\bf r}) \equiv\sqrt{N_c} e^{i\theta}$ is a 2-component
order parameter.  Clearly, $\Phi({\bf r})$ is not simply related to the
many-particle wavefunction $\Psi({\bf r}_1, {\bf r}_2, \dots {\bf r}_N).$
The thermal average in $<{\hat\psi}({\bf r})>$ involves a small
symmetry-breaking perturbation to allow $\Phi$ to be finite,
\begin{equation}
{\hat H}_{SB} = \int d{\bf r}\left[\eta ({\bf r}){\hat\psi}^+({\bf r}) +
\eta^\ast({\bf r}) {\hat\psi}({\bf r})\right].\label{eqdynamics31}
\end{equation}

It is useful to make a few comments on the physics behind $\Phi({\bf r},
t)$.  $\Phi ({\bf r}, t)$ is a {\it coherent} state, with a ``clamped''
value of phase - rather than a Fock-state of fixed $N$, with no
well-defined phase.  $\Phi({\bf r}, t)$ acts like a {\it classical} field,
since quantum fluctuations are negligible when $N_c$ is large.  Probably
P.W. Anderson deserves the greatest credit for understanding (in the
period 1958-1963) the new physics behind working with a broken-symmetry
state $\Phi({\bf r}, t)$, both in BCS superconductors and in superfluid
$^4$He \cite{PWand}.   It captures the physics of the new phase of matter
(such as the occurence of the Josephson effect) and the associated
superfluidity.  The symmetry-breaking perturbation allows $<{\hat\psi}>$
to be finite.  More precisely, it allows the system to internally set up
off-diagonal symmetry-breaking fields, which persist even when the
external symmetry-breaking perturbation in (\ref{eqdynamics31}) is set to
zero at the end $(\eta \rightarrow 0)$.  The same sort of physics is
behind the BCS theory of superconductors.

The exact Heisenberg equation of motion for the field operator is
\begin{eqnarray}
i\hbar{\partial{\hat\psi}({\bf r}, t)\over\partial t} &=&
\left[-{\hbar^2\nabla^2\over 2m}+V_{ex}({\bf r}) + \delta V({\bf r},
t)\right]
{\hat\psi}({\bf r}, t)\nonumber \\
&+&\eta({\bf r})+g{\hat\psi}^+ ({\bf r}, t){\hat\psi}({\bf r},
t){\hat\psi}({\bf r}, t),\label{eqdynamics32}
\end{eqnarray}
where $\delta V({\bf r}, t)$ is a small time-dependent driving potential.
This gives an exact equation of motion for $\Phi({\bf r}, t)
\equiv\left<{\hat\psi}({\bf r}, t)\right>,$
\begin{eqnarray}
i\hbar{\partial\Phi({\bf r}, t)\over\partial t}
&=&\left[-{\hbar^2\nabla^2\over 2m}+V_{ex}({\bf r})
+\delta V({\bf r}, t)\right]\Phi({\bf r}, t)\nonumber \\
&+&\eta({\bf r})+g\left<{\hat\psi}^+ ({\bf r}, t){\hat\psi}({\bf r},
t){\hat\psi}({\bf r}, t)\right>,\label{eqdynamics33}
\end{eqnarray}
with
\begin{equation}
{\hat\psi}^+{\hat\psi}{\hat\psi} = |\Phi|^2\Phi +
2|\Phi|^2{\tilde\psi}+\Phi^2{\tilde\psi}^+ +
\Phi^\ast{\tilde\psi}{\tilde\psi}+2\Phi{\tilde\psi}^+{\tilde\psi} +
{\tilde\psi}^+{\tilde\psi}{\tilde\psi}.\label{eqdynamics34}
\end{equation}
Taking the symmetry-breaking average, one finds
\begin{equation}
\left<{\hat\psi}^+{\hat\psi}{\hat\psi} \right> = n_c\Phi+{\tilde
m}\Phi^\ast+2{\tilde n}\Phi+\left<{\tilde\psi}^+
{\tilde\psi}{\tilde\psi}\right>,\label{eqdynamics35}
\end{equation}
where
\begin{eqnarray}
n_c ({\bf r}, t) &\equiv& |\Phi({\bf r}, t)|^2 = \mbox{condensate density}
\nonumber\\
{\tilde n}({\bf r}, t) &\equiv&  \left<{\tilde\psi}^+({\bf r},
t){\tilde\psi}({\bf r}, t)
\right> = \mbox{non-condensate density} \nonumber\\
{\tilde m}({\bf r}, t) &\equiv&\left<{\tilde\psi}({\bf r},
t){\tilde\psi}({\bf r}, t)
\right> = \mbox{off-diagonal (anomalous) density}.\nonumber
\end{eqnarray}
Here we have separated out the {\it condensate} and {\it non-condensate}
parts
\begin{equation}
{\hat\psi}\Rightarrow\left<{\hat\psi}\right>+{\tilde\psi}=\Phi
+{\tilde\psi}.
\label{eqdynamics36}
\end{equation}
In general, the equation (\ref{eqdynamics33}) for $\Phi({\bf r}, t)$ is
not closed - it is coupled to the dynamics of the non-condensate.
However, in this Section we limit ourselves to $T\ll T_{BEC}$, where we
can assume the non-condensate fraction is negligible, leaving
\begin{equation}
i\hbar{\partial\Phi({\bf r}, t)\over\partial t} =
\left[-{\hbar^2\nabla^2_r\over 2m}+V_{ex}({\bf r}) +g|\Phi({\bf r},
t)|^2\right]\Phi({\bf r}, t).\label{eqdynamics37}
\end{equation}
This is the famous time-dependent Gross-Pitaevskii equation for the
condensate macroscopic wavefunction.  It gives a complete description of
the dynamics of a coherent matter wave at $T=0.$

\subsection{Static condensate}
We first consider the time-dependent stationary GP equation, which has the
solution
\begin{equation}
\left<{\tilde\psi}({\bf r}, t)\right>\equiv\Phi({\bf r}, t)=\Phi_0({\bf
r})e^{-i\mu t/\hbar},
\label{eqdynamics38}
\end{equation}
where $\mu$ is the chemical potential.  The physics behind this can be
seen from
\begin{eqnarray}
 \left<N-1|{\hat\psi}({\bf r}, t)|N\right>
&=& e^{iE_{N-1} t/\hbar} \left<N-1|{\hat\psi}({\bf r})|N\right>e^{-iE_N
t/\hbar} \nonumber\\
&=&\left<N-1|\sqrt{N}|N-1\right>e^{-i(E_N-E_{N-1})t/\hbar}\nonumber\\
&=& \sqrt{N} e^{-i\mu t/\hbar}.
\label {eqdynamics39}\end{eqnarray}
Using (\ref{eqdynamics38}) in (\ref{eqdynamics37}) gives
\begin{equation}
i\hbar\left(-{i\mu\over\hbar}\right)\Phi_0({\bf
r})=\left[-{\hbar^2\nabla^2\over 2m} +V_{ex}({\bf r})+g|\Phi_0({\bf
r})|^2\right]\Phi_0({\bf r}).\label{eqdynamics40}
\end{equation}
The {\it static} GP equation for the static condensate wavefunction
$\Phi_0({\bf r})$ is thus
\begin{equation}
\left[-{\hbar^2\nabla^2\over 2m}-\mu+V_{ex}({\bf r}) +g|\Phi_0({\bf
r})|^2\right]\Phi_0({\bf r})= 0.\label{eqdynamics41}
\end{equation}

A simple approximation in solving (\ref{eqdynamics41}) is to ignore the
kinetic energy of the condensate, ie, neglect the $-{\hbar^2\nabla^2\over
2m}$ term.  This is called the ``Thomas-Fermi''  approximation (TF) in the
recent Bose gas literature \cite{Dalgiorpit}.  In this TF approximation,
the static GP (\ref{eqdynamics41}) equation for $\Phi_0({\bf r})$ reduces
to
\begin{equation}
\left[V_{ex}({\bf r}) + g|\Phi_0({\bf
r})|^2\right]=\mu,\label{eqdynamics42}
\end{equation}
which is easly inverted to give the condensate density profile
\begin{eqnarray}
n_{c0}({\bf r}) &=& {1\over g}\left[\mu - V_{ex}({\bf r})\right] \nonumber
\\
&=& {1\over g}\left[\mu - {1\over 2}
m\omega^2_0r^2\right]>0.\label{eqdynamics43}
\end{eqnarray}
Clearly in the TF approximation, the ``size'' of the condensate is
$R_{TF}$, where
\begin{equation}
\mu = {1\over 2} m\omega^2_0 R^2_{TF}. \label{eqdynamics44}
\end{equation}
One finds $\mu$ from the condition $\int d{\bf r}n_c({\bf r})= N_c = N$, or
\begin{equation}
N_c = 4\pi\int^{R_{TF}}_0 dr r^2{1\over g}\left[\mu- {1\over 2}
m\omega^2_0r^2\right].
\label{eqdynamics45}
\end{equation}
This gives
\begin{equation}
\mu = \hbar\omega_0 \left[15{Na\over a_{HO}}\right]^{2/5}; \ \ a_{HO}
\equiv (\hbar/m\omega_0)^{1/2}. \label{eqdynamics46}
\end{equation}
We note that SH oscillator length $a_{HO}$ is the size of the ground state
wavefunction (\ref{eqoverview15}) of an atom in a parabolic potential.
Combining (\ref{eqdynamics45}) and (\ref{eqdynamics44}) gives
\begin{eqnarray}
R_{TF} &=& a_{HO}\left(15 {Na\over a_{HO}}\right)^{1/5} \nonumber \\
& \gg & a_{HO}, \  \mbox{if} \ {Na\over a_{HO}}\gg 1. \label{eqdynamics47}
\end{eqnarray}
We thus find the surprising result that interactions (while weak) spread
out the ideal gas condensate $(R_{TF} \gg a_{HO})$ and decrease the
density of the condensate at centre of trap.  The TF approximation for
$n_{c0}({\bf r})$ is very good for large $N$, except for a small region
near the edge of condensate $(\simeq R_{TF}).$  Experimental data confirms
these GP predictions for the $n_{c0}({\bf r})$ condensate profile,
emphasizing that the condensate is {\it not} simply the ground state
wavefunction of the harmonic trap potential.

\subsection{Dynamics of the condensate (collective modes)}
If we linearize around the static equilibrium value of the condensate
\begin{equation}
\Phi({\bf r}, t) = e^{-i\mu t/\hbar}\left[\Phi_0({\bf r}) +\delta\Phi({\bf
r}, t)\right], \label{eqdynamics48}
\end{equation}
where $\delta\Phi\ll\Phi_0, $ we see that
\begin{equation}
i\hbar{\partial\Phi\over\partial t} = \left[-{\hbar\nabla^2\over
2m}+V_{ex}({\bf r}) + g
\left[|\Phi_0|^2+\Phi^\ast_0\delta\Phi+\Phi_0\delta\Phi^\ast\right]
\left[\Phi_0+\delta\Phi\right]e^{-{i\mu t\over
\hbar}}\right]\label{eqdynamics49}
\end{equation}
which gives
\begin{equation}
i\hbar{\partial\delta\Phi({\bf r}, t)\over\partial t} =
\left[-{\hbar\nabla^2\over 2m}+V_{ext}({\bf r}) + 2g
|\Phi_0|^2-\mu\right]\delta\Phi({\bf r}, t)+g\Phi^2_0\delta\Phi^\ast({\bf
r}, t).\label{eqdynamics50}
\end{equation}
We also have a similar equation of motion for $\delta\Phi^\ast({\bf r},
t).$  Solving these two coupled equations with the ansatz
\begin{equation}
\delta\Phi({\bf r}, t) = u({\bf r})e^{-i\omega t} +v({\bf r}) e^{i\omega
t},
\label{eqdynamics51}\end{equation}
we find two coupled ``Bogoliubov equations'' for the amplitudes $u$ and
$v$ \cite{Fetwalecka, Edrupbur}:
\begin{eqnarray}
&\left[-{\hbar^2\nabla^2\over 2m}+V_{ex}({\bf r})-\mu+2g n_{c0} ({\bf
r})\right]u({\bf r}) + gn_{c0} ({\bf r}) v({\bf r}) = E_i u({\bf r})
\nonumber \\
&\left[-{\hbar^2\nabla^2\over 2m}+V_{ex}({\bf r})-\mu+2g n_{c0} ({\bf
r})\right]v({\bf r}) + gn_{c0} ({\bf r}) u({\bf r}) = -E_i v({\bf
r})\label{eqdynamics52}
 \end{eqnarray}
Here $E_i\equiv \hbar\omega$ are the excitation energies of the
condensate.

The equations in (\ref{eqdynamics52}) have been solved numerically by
several groups and the observed oscillations are in good agreement with
these predictions \cite{Dalgiorpit, Edrupbur}. As an illustration of the
physics, it is useful to solve (\ref{eqdynamics52}) for a uniform Bose
gas.  In this case we have
\begin{eqnarray}
u({\bf r}) = ue^{i{\bf k}\cdot{\bf r}}\nonumber \\
v({\bf r}) = ve^{i{\bf k}\cdot{\bf r}},\label{eqdynamics53}
\end{eqnarray}
which gives
 \begin{eqnarray}
(\hbar\omega)^2 &=& \left[{\hbar^2k^2\over 2m}-\mu + 2g n_{c0}\right]^2 -
\left[gn_{c0}\right]^2 \nonumber \\
&=& \epsilon^2_k + 2gn_{c0}\epsilon_k. \label{eqdynamics54}\end{eqnarray}
This is the famous Bogoliubov spectrum at $T=0$ \cite{Nnbog, Fetwalecka}.
Here we have used $\mu_0 = gn_{c0}$ discussed earlier (see
(\ref{eqdynamics42})).  One finds
a phonon region at long wavelengths
\begin{equation}\hbar \omega_k = \hbar v_Bk \ ; \ v_B \equiv
\left({gn_{c0}\over m}\right
)^{1/2}.\label{eqdynamics55}
\end{equation}
The cross-over from particle-like to this collective phonon region occurs
at $k_c,$ where
\begin{equation}
{\hbar^2 k^2_c\over 2m}=2gn_{c0} \rightarrow k_c = \sqrt{4mn_{c0} g/\hbar}.
\label{eqdynamics56}\end{equation}
This shows how the interactions changes the qualitative nature of low
energy exciations in a Bose-condensed gas.  This feature can be shown to
stabilize superfluid motion against dissipation \cite{Gorkov}.

These oscillations of the condensate can be understood as excitations
involving the {\it non-condensate}.  Using (\ref{eqdynamics36}), the
Hamiltonian is given by \cite{AlFetter, Agriffin}
\begin{eqnarray}
{\hat H}-\mu{\hat N} &=& \int d{\bf r}{\tilde\psi}^+({\bf
r})\left[-{\hbar^2\nabla^2\over 2m}+V_{ex}({\bf
r})-\mu\right]{\tilde\psi}({\bf r}) \nonumber \\
&+&\int d{\bf r} \Phi^\ast_0 ({\bf r})\left[-{\hbar^2\nabla^2\over 2m}
+V_{ex}({\bf r})-\mu\right]\Phi_0({\bf r})\nonumber \\
&+&2g\int d{\bf r}|\Phi_0({\bf r})|^2{\tilde\psi}^+({\bf
r}){\tilde\psi}({\bf r})\nonumber\\
&+&{1\over 2}g\int d{\bf r}\Phi^2_0({\bf r}){\tilde\psi}^+({\bf
r}){\hat\psi}^+({\bf r})\nonumber \\
&+&{1\over 2}g\int d{\bf r}\Phi^{\ast 2}_0({\bf r}){\tilde\psi}({\bf
r}){\tilde\psi}({\bf r}).\label{eqdynamics57}
\end{eqnarray}
We can diagonalize the quadratic part of the Hamiltonian, using
\begin{equation}
{\tilde\psi}({\bf r}) = \sum_i\left[u_i({\bf r}){\hat\alpha}_i +
v^\ast_i({\bf r})\alpha_i^+\right],\label{eqdynamics58}
\end{equation}
where $\left[{\hat\alpha}_i, {\hat\alpha}_j^+\right] = \delta_{ij}$ (Boson
quasiparticles).  Thus one finds
\begin{equation}
{\hat H}-\mu{\hat N} = \mbox{const.} + \sum_i\hbar
\omega_i{\hat\alpha}_i^+ {\hat\alpha}_i.\label{eqdynamics59}\end{equation}

This transformation shows how the non-condensate part of Hamiltonian can
be reduced to a system of {\it non-interacting} quasiparticles with a
spectrum identical to the condensate fluctuations.  This equivalence is
easy to understand.  The condensate fluctuations
\begin{equation}
\delta\Phi \equiv \left<{\hat\psi}({\bf r})\right> - \Phi_0
\label{eqdynamics60}\end{equation}
can be calculated to first order in the symmetry-breaking perturbation
(\ref{eqdynamics31}),
\begin{equation}
H_{sb} = \int d{\bf r} \left[\eta{\hat\psi}^+ +
\eta^\ast{\hat\psi}\right].\label{eqdynamics61}\end{equation}
Then standard linear response theory \cite{Agrif} gives (schematically)
\begin{eqnarray}
\delta\Phi &\sim&  \int <[{\hat\psi}, H_{sb}]>\nonumber \\
&\sim& \int <[{\tilde\psi}, {\tilde\psi}^+]>\eta +<[{\tilde\psi},
{\tilde\psi}]>\eta^\ast.\label{eqdynamics62}\end{eqnarray}
This shows that the single-particle Green's functions of the
non-condensate fields have the same spectrum as $\delta\Phi$.  This
identity of the spectrum of density fluctuations and single-particle
excitations is a characteristic signature of all Bose-condensed systems
which persists at finite temperatures \cite{Agrif, Grif2}.

One interesting collective oscillation is the dipole mode corresponding to
{\it rigid} oscillation of the centre of mass of the static condensate
profile, and predicted to have the trap frequency $\omega_0$.  This mode
is described by
\begin{equation}
n_c({\bf r}, t) = n_{c0} ({\bf r} - \mbox{\boldmath$\eta$}(t)), \ \ \
{\dot{\mbox{\boldmath$\eta$}}}(t) = {\bf v}_c,
\label{eqdynamics63}\end{equation}
where the time-dependent centre of mass satisfies
\begin{equation}{\partial^2\mbox{\boldmath$\eta$} (t)\over \partial t^2} =
-\omega^2_0\mbox{\boldmath$\eta$}(t). \label{eqdynamics64}\end{equation}
This mode at frequency $\omega_0$ is special feature of a parabolic trap
and is called the Kohn mode for in the case of interacting fermions
\cite{Jfdobson}. This ``sloshing mode''  is used in BEC experiments to
measure the natural frequency $\omega_0$ of the trap and it exists at
finite temperatures as well (see Section \ref{sec:Coupled}).

\subsection{Quantum hydrodynamic formulation }
One often rewrites the time-dependent GP equation using the amplitude and
phase variables \cite{Dalgiorpit}
\begin{equation}\Phi({\bf r}, t) = \sqrt{n_c}
e^{i\theta}.\label{eqdynamics65}
\end{equation}
Inserting this into the GP equation (\ref{eqdynamics37}) and separating
out the real and imaginary parts of the equation gives:
\begin{eqnarray}&& {\partial n_c({\bf r}, t)\over\partial t} +
\mbox{\boldmath$\nabla$}\cdot n_c({\bf r}, t){\bf v}_c({\bf r}, t)= 0 , \
\mbox{continuity equation} \label{eqdynamics66}\\
&&\hbar{\partial \theta({\bf r}, t)\over\partial t} = -\left[\mu_c({\bf
r}, t)+{1\over 2}mv^2_c({\bf r}, t)\right], \mbox{Josephson equation}
\label{eqdynamics67}\end{eqnarray}
Here the gradient of the phase is related to the superfluid velocity by
\begin{equation}m{\bf v}_c ({\bf r}, t)
\equiv\hbar\mbox{\boldmath$\nabla$}\theta ({\bf r}, t)
\label{eqdynamics68}\end{equation}
and the condensate chemical potential is
\begin{equation}
\mu_c({\bf r}, t)\equiv-{\hbar^2\nabla^2\sqrt{n_c}\over 2m\sqrt{n_c}} +
V_{ex}({\bf r}) + gn_c({\bf r}, t).\label{eqdynamics69}\end{equation}
Taking the gradient of (\ref{eqdynamics67}) gives
\begin{equation}
m\left({\partial{\bf v}_c\over\partial t} +{1\over 2}
\mbox{\boldmath$\nabla$}{\bf v}_c^2\right) = -
\mbox{\boldmath$\nabla$}\mu_c.\label{eqdynamics70}\end{equation}
The equations in (\ref{eqdynamics66}) and (\ref{eqdynamics70}) ``look''
like those in classical hydrodynamic theories. They show that the
condensate can be described in terms of coherent motions involving two
variables:
\begin{equation}
n_c ({\bf r}, t), \ {\bf v}_c({\bf r}, t)\label{eqdynamics71}\end{equation}
The Landau 2-fluid equations \cite{Land, Khalatnikov} reduce to these same
equations at $T=0$  (where $\rho_s=\rho,\ \rho_n = 0),$ namely
\begin{eqnarray}
&{}&{\partial n\over\partial t} + \mbox{\boldmath$\nabla$}\cdot n{\bf v}_c
= 0 \nonumber \\
&{}&m\left({\partial{\bf v}_c\over\partial t}+{1\over 2}\nabla{\bf
v}_c^2\right) =
-\mbox{\boldmath$\nabla$}\mu.\label{eqdynamics72}\end{eqnarray}
We will find these equations useful in Section~\ref{sec:Coupled}, where
they complement the hydrodynamic equations describing the {\it
non-condensate} in the collision-dominated region.

This approach also allows a simple theory  developed by Stringari
\cite{Sstringari} for linearized collective modes when we use the TF
approximation.  Taking the time-derivative of (\ref{eqdynamics66}) gives
$({\bf v}_{c0} = 0)$
\begin{equation}
{\partial^2\delta n_c\over \partial t^2} = - \mbox{\boldmath$\nabla$}\cdot
\left[n_{c0} \left(
{\partial\delta{\bf v}_c\over \partial
t}\right)\right].\label{eqdynamics73}
\end{equation}
Using (\ref{eqdynamics70}), we have
\begin{eqnarray} {\partial\delta{\bf v}_c\over \partial t} &=& -{1\over m}
 \mbox{\boldmath$\nabla$}\left[V_{ex}({\bf r}) + gn_{c0}({\bf r}) +
g\delta n_c({\bf r}, t)\right]\nonumber \\
&=& -{g\over m} \mbox{\boldmath$\nabla$}\delta n_c ({\bf r}, t).
\label{eqdynamics74}\end{eqnarray}
Combining this last result with (\ref{eqdynamics73}), we obtain the very
useful Stringari equation of motion \cite{Sstringari}
\begin{equation}
{\partial^2\delta n_c\over \partial t^2} =
\mbox{\boldmath$\nabla$}\cdot\left\{\left[\mu-{1\over 2}m\omega^2_0
r^2\right]\mbox{\boldmath$\nabla$}\delta
n_c\right\}.\label{eqdynamics75}\end{equation}
This describes the collective oscillations of the condensate in terms of a
single differential equation.  As one example, the breathing mode of the
condensate has a frequency $\hbar\omega = \sqrt{5}\hbar \omega_0.$  This
example points out that in the TF limit (large $N_c$), the frequencies are
independent of the interaction strength and the size of the condensate
$N_c$.

We also note that using (\ref{eqdynamics74}), (\ref{eqdynamics75}) can be
equally well rewritten in terms of the superfluid velocity ${\bf v}_c
({\bf r}, t)$ defined in (\ref{eqdynamics68}).  This emphasizes that the
condensate fluctuations are directly related to the existence of phase
fluctuations.  Their existence may thus be viewed as ``evidence'' of
superfluidity, the latter being always a consequence of the phase
coherence of the macroscopic wavefunction given by (\ref{eqdynamics65})
\cite{Gorkov}.

The great thing about the collective oscillations of a condensate in a
trapped gas is you can ``see'' them. A beautiful example from MIT is shown
in Fig. 2 of Ref. \cite{Dalgiorpit}.  As Ketterle has remarked, these
condensates are {\it robust} - one can kick them, shake them and these
``wisps'' of Bose-condensed matter keep their integrity.

\subsection{Interference of coherent matter waves}
In the pioneering matter wave interference experiments done at MIT using a
de-tuned cigar-shaped trap \cite{Wketterle}, one first destroys the
condensate at the centre using laser beam.  Then the confining trap is
turned off and the two condensates are allowed to expand and interfere.
One observes nice interference fringes at the mid-point, as expected.
Using \begin{eqnarray} \Phi_{\mbox{system}}  &=& \Phi_A({\bf r}, t) +
\Phi_B({\bf r}, t)\nonumber \\
&=&\sqrt{N_A}e^{{i\theta}_A}+\sqrt{N_B}e^{{i\theta}_B},\label{eqdynamics76}
\end{eqnarray}
the density is given by
\begin{eqnarray} n({\bf r}, t) &=& |\Phi_{\mbox{system}}|^2 \nonumber \\
&=& N_A + N_B + 2\sqrt{N_AN_B}\cos
\Delta\theta,\label{eqdynamics77}\end{eqnarray}
where $\Delta\theta=\theta_A -\theta_B$.  In the region of interference,
the density is low and hence interaction effects are {\it small} (ie, the
$gn_c$ term is small in the GP equation).  Asymptotically, the solution of
GP equation gives $\theta({\bf r}, t) = {mr^2\over 2\hbar t}.$  This
implies \cite{Dalgiorpit}
\begin{equation}
\Delta\theta = {m\left(z+{d\over 2}\right)^2\over 2\hbar t} -
{m\left(z-{d\over 2}\right)^2\over 2\hbar t} = {mzd\over\hbar
t}\equiv{2\pi z\over\lambda(t)},\label{eqdynamics78}\end{equation}
where $\lambda(t)=2\pi\hbar t/md.$ This wavelength is in good agreement
with experimental observations (See Fig. 2 of Ref. \cite{Wketterle}).

A condensate described by $\Phi({\bf r}, t)$ may be viewed as a
``classical'' matter wave, as recently emphasized by Pitaevskii and
Stringari \cite{Lpitsstring}.  This is quite {\it different} from ordinary
quantum deBroglie waves, since one can ignore quantum fluctuations (large
$N_c$).  It is also quite different than ordinary (classical) macroscopic
objects and electromagnetic waves, since $\Phi({\bf r}, t)$ is described
by the GP equation (\ref{eqdynamics37}) which involves Planck's constant
$\hbar$.
Thus $\Phi({\bf r}, t)$ is a classical object which is described by a
quantum equation!!  This promises to be a challenge for the quantum theory
of measurement.

One can describe a two-component Bose gas (see Section~\ref{sec:Overview})
using coupled equations:
\begin{eqnarray}
i\hbar{\partial\Phi_1\over\partial t}&=& \left[-{\hbar^2\nabla^2\over 2m}
+V_1({\bf r})+g_1|\Phi_1|^2+g_{12}|\Phi_2|^2\right]\Phi_1 \nonumber \\
  i\hbar{\partial\Phi_2\over\partial t}&=& \left[-{\hbar^2\nabla^2\over
2m} +V_2({\bf r})+g_2|\Phi_2|^2+g_{12}|\Phi_1|^2\right]\Phi_2
\label{eqdynamics79}.\end{eqnarray}
There are two {\it coupled} GP equations for {\it two} macroscopic wave
functions.  Extensive studies \cite{Eacornhall} have been made at JILA
using the two atomic hyperfine states of $^{87}$Rb:
\begin{equation}
|F=1, m_F = -1 >, \ \ |F = 2, m_F = 1>. \label{eqdynamics80}
\end{equation}
In particular, one can study interesting interference effects between
these coupled wavefunctions.  Recent work at JILA has used such
two-component Bose fluids to produce the long sought-for vortex state in
one of the components \cite{Matand}.

\section{COUPLED DYNAMICS OF THE CONDENSATE AND NON-CONDENSATE}
\label{sec:Coupled}

In this Section, we switch our attention from $T=0$ (ie, $T\ \lapx\ 0.4\
T_{BEC})$ to finite temperatures, where $N_c$ and ${\tilde N}$ are
comparable in size.  We first consider how the GP equation of motion for
$\Phi({\bf r}, t)$ is modified.  As a first step, we could use
\cite{Agrif, Ezarnikgrif},
\begin{equation}
i\hbar{\partial\Phi({\bf r}, t)\over\partial t}=
\left[-{\hbar^2\nabla^2\over 2m} +V_{ex}({\bf r})+gn_c({\bf r}, t)+
2g{\tilde n}({\bf r}, t)\right]\Phi({\bf r},
t).\label{eqcoupled81}\end{equation}
The last term takes into account that the condensate moves in the dynamic
Hartree-Fock (HF) field produced by non-condensate atoms.  Immediately,
one sees this generalized GP equation is no longer closed.  It requires a
theory of the non-condensate fluctuations, as described by ${\tilde
n}({\bf r}, t)= {\tilde n}_0({\bf r})+\delta{\tilde n}({\bf r}, t).$

A simpler version of (\ref{eqcoupled81}) is to treat the effect of the
non-condensate as a {\it static} HF field \cite{Agriffin, Hutzargrif}:
\begin{equation}
2g{\tilde n}({\bf r}, t) \simeq 2g{\tilde n}_0({\bf r})
.\label{eqcoupled82}\end{equation}

\begin{enumerate}
\item This corresponds to treating the condensate moving in a static HF
field of the non-condensate.
\item ${\tilde n}_0({\bf r})$ can be calculated (self-consistently) using
the fluctuations of $\Phi$, as discussed in Section \ref{sec:Dynamics}.
One finds that the depletion of the condensate at $T=0$ is only a few
percent.
\item This procedure gives reasonable results for the thermodynamic
properties at finite temperatures, as discussed in the recent literature
\cite{Dalgiorpit}.
Within the Thomas-Fermi approximation (good for $N\ \gapx\ 10^4$ atoms),
the linearized version of (\ref{eqcoupled81}) using (\ref{eqcoupled82})
leads to the same Stringari equation at finite $T$ as the $T=0$ result in
(\ref{eqdynamics75}).  Since the solutions of (\ref{eqdynamics75}) do not
depend on the magnitude of the condensate, one concludes that the
collective modes of the condensate will show no temperature dependence,
even though the condensate is being thermally depleted.  This prediction
does not appear to agree with experimental results when $T \ \gapx \ 0.6\
T_{BEC}.$  This suggests that the {\it dynamics} of the non-condensate has
to be included.
\end{enumerate}

We now go on to determining ${\tilde n}({\bf r}, t)$ directly by deriving
a {\it quantum} Boltzmann equation for the single-particle distribution
function of excited atoms $ f({\bf p}, {\bf r}, t)$ and then use:
\begin{equation}{\tilde n}({\bf r}, t)\equiv\int{d{\bf p}\over (2\pi)^3}
f({\bf p}, {\bf r}, t).\label{eqcoupled83}\end{equation}
This procedure generalizes the approach of Boltzmann (1880's) for a
classical gas, including the effect of binary collisions.  Clearly one
must make some approximations!  One wants, initially, to find a {\it
useful} kinetic equation that builds in {\it just} enough physics.  Here I
will discuss such a quantum Boltzmann equation for a trapped
Bose-condensed gas at finite temperatures, which has been extensively
discussed by Zaremba, Nikuni and the author \cite{Zarnikgrif,
Tnikzargrif}.  It is only valid in the so-called semi-classical limit,
where it is sufficient to work with $f({\bf p}, {\bf r}, t)$.  The
conditions are
\begin{equation}
k_BT\gg gn\ , \ k_BT\gg \hbar\omega_0 . \label{eqcoupled84}\end{equation}
In this domain, one also can assume that the important thermal excitaitons
can be approximated by simple Hartree-Fock particle-like spectrum:
\begin{equation}
{\tilde\varepsilon}({\bf r}, t) = {p^2\over 2m}+ 2g \left[n_c({\bf r},
t)+{\tilde n}({\bf r}, t)\right] + V_{ex}({\bf r})\equiv {p^2\over 2m} +
U({\bf r}, t).\label{eqcoupled85}\end{equation}
Clearly the resulting kinetic equation is not valid at very low
temperatures, where the thermal excitations are described by a
Bogoliubov-type spectrum.

We simply write down our quantum kinetic equation \cite{Zarnikgrif},
\begin{eqnarray}
{\partial f({\bf p}, {\bf r}, t)\over\partial t} &+&{{\bf p}\over
m}\cdot\mbox{\boldmath$\nabla$}_r
f({\bf p}, {\bf r}, t) - \mbox{\boldmath$\nabla$}_r
U({\bf r}, t) \cdot\mbox{\boldmath$\nabla$}_p f({\bf p}, {\bf r}, t)
\nonumber \\
&=& C_{22}[f] + C_{12}[f].\label{eqcoupled86}\end{eqnarray}
The right hand side describes how binary collisions effect the value of
the single-particle distribution function $ f({\bf p}, {\bf r}, t)$.
The effect of collisions between excited atoms in the non-condensate is
described by:
\begin{eqnarray}
C_{22}[f] &=& {2g^2\over (2\pi)^5\hbar^7}\int d{\bf p}_2\int d{\bf
p}_3\int d{\bf p}_4\delta({\bf p}+{\bf p}_2 - {\bf p}_3 - {\bf
p}_4)\nonumber \\
&\times&\delta\left({\tilde\varepsilon}_p+{\tilde\varepsilon}_{p_2}-{\tilde\varepsilon}_{p_3}-{\tilde\varepsilon}_{p_4}\right)\nonumber
\\
&\times&\left[(1+f)(1+f_2)f_3f_4 -
ff_2(1+f_3)(1+f_4)\right].\label{eqcoupled87}\end{eqnarray}
This collision integral was discussed in detail in 1933 by Uehling and
Uhlenbeck for $T>T_{BEC}$ \cite{Uehlbeck}.
We recall that creating a Boson gives a factor $(1 + f)$ and destroying a
Boson gives $f.$  In the classical high temperature limit, $f\ll 1$ and
the collision integral $C_{22}$ considerably simplifies.

Where does $f({\bf p}, {\bf r}, t)$ come from in a microscopic derivation
of (\ref{eqcoupled86})?  Basically we calculate the non-equilibrium
real-time single-particle Green's functions of the non-condensate field
operators (using the Kadanoff-Baym formalism \cite{Kadbaym}).  This gives
(schematically)
\begin{equation}
g_1(1,
1^\prime)\sim\left<{\tilde\psi}^+(1){\tilde\psi}(1^\prime)\right>,\label{eqcoupled88}\end{equation}

where $1\equiv {\bf r}_1, t_1 ; \  1^\prime={\bf r}^\prime_1,
t^\prime_1.$  We then express this as $g_1({\bf r}, t; {\bf R}, T)$, where
the {\it relative} and {\it centre of mass} coordinates are
\begin{eqnarray}
&{}&{\bf r}={\bf r}_1-{\bf r}^\prime_1 ; \ {\bf R} = {1 \over 2}
\left({\bf r}_1+{\bf r}^\prime_1\right)\nonumber \\
&{}&t=t_1 - t_1^\prime \ \ \  T = {1\over
2}\left(t_1+t_1^\prime\right).\label{eqcoupled89}\end{eqnarray}
Finally we Fourier transform
$g_1({\bf r}, t; {\bf R}, T)$ to find $g_1({\bf p}, \omega; R, T)$,
which gives the number of atoms at ${\bf R}, T$ with momentum ${\bf p}$
and energy $\hbar\omega.$  The single-particle Wigner distribution
function is given by
\begin{equation}
f({\bf p}, {\bf R}, T) \equiv\int^\infty_{-\infty} d\omega g_1({\bf p},
\omega; {\bf R}, T).\label{eqcoupled90}\end{equation}
The Wigner distribution function  $f({\bf p}, {\bf R}, T)$ is the quantum
generalization the classical single-particle distribution function
\cite{Kadbaym}.  These remarks should indicate how we can go from an
equation of motion for the single-particle Green's function (within a
given self-energy approximation) to a kinetic equation for $f({\bf p},
{\bf R}, T)$.  We refer to the classic account given (for
non-Bose-condensed gases) in the book by Kadanoff and Baym \cite{Kadbaym}
for further details.  This powerful approach was generalized to uniform
Bose-condensed gases by Kane and Kadanoff \cite{Kanekad}, and has been
extended to trapped gases in recent work \cite{Tomgrif}.

In addition to $C_{22}$ collisions, we also have collisions which involve
{\it one} condensate atom:
\begin{eqnarray}
C_{12}[f] &=& {2g^2\over (2\pi)^2\hbar^4}\int d{\bf p}_1 \int d{\bf p}_2
\int d{\bf p}_3\delta (m{\bf v}_c +{\bf p}_1-{\bf p}_2-{\bf p}_3)
\nonumber\\
&\times& \delta\left(\varepsilon_c
+{\tilde\varepsilon}_{p1}-{\tilde\varepsilon}_{p2}-{\tilde\varepsilon}_{p3}\right)\times\left[\delta\left({\bf
p}-{\bf p}_1)-\delta({\bf p}-{\bf p}_2)-\delta({\bf p}-{\bf
p}_3\right)\right]\nonumber \\
&\times&\left[n_c(1+f_1)f_2f_3
-n_cf_1(1+f_2)(1+f_3)\right].\label{eqcoupled91}\end{eqnarray}
Here the condensate atom has
\begin{eqnarray}
&{}&\mbox{energy:} \  \varepsilon_c = \mu_c + {1\over 2} mv_c^2 \ ; \
\mu_c = V_{ex} + gn_c +2g{\tilde n}\nonumber\\
&{}&\mbox{momentum:} \  {\bf p}_c = m{\bf v}_c
\label{eqcoupled92}\end{eqnarray}
We note the {\it key} difference between $C_{12}$ and $C_{22}$ collisions:
\begin{itemize}
\item $C_{22}$ and $C_{12}$ conserve energy and momentum in collisions.
\item $C_{12}$ does {\it not} (but $C_{22}$ does) conserve the number of
condensate atoms. $C_{12}$ describes how atoms are ``kicked'' in and out
of condensate.
\end{itemize}

It turns out the generalized GP equation (\ref{eqcoupled81}) is also
modified by a term related to $C_{12}[f]$.  This makes sense, since the
$C_{12}$ collisions  modify the condensate wavefunction $\Phi({\bf r},
t).$  One finds the new GP equation is given by (see also
Ref.\cite{Hcstoof}):
\begin{equation}
i\hbar{\partial\Phi({\bf r}, t)\over\partial t} =
\left[-{\hbar^2\nabla^2\over 2m}+V_{ex} ({\bf r}) + gn_c({\bf r}, t) +
2g{\tilde n}({\bf r}, t) -iR({\bf r}, t)\right]\Phi({\bf r},
t),\label{eqcoupled93}
\end{equation}
where
\begin{equation}
R({\bf r}, t)\equiv\int {d{\bf p}\over(2\pi)^3} {C_{12}[f({\bf p}, {\bf
r}, t)]\over 2n_c({\bf r}, t)}.\label{eqcoupled94}\end{equation}
\vskip 2pt
\noindent More precisely, the dissipative {\it iR} term in
(\ref{eqcoupled93}) arises from a three field correlation function in the
exact equation of motion [see (\ref{eqdynamics33}) and
(\ref{eqdynamics35})], and is given by \cite{Zarnikgrif}
\begin{equation}
\int{d{\bf p}\over (2\pi)^3} C_{12}[f] = {2g\over\hbar}\sqrt{n_c}\
Im\left<{\tilde\psi}^+{\tilde\psi}{\tilde\psi}\right>.\label{eqcoupled95}
\end{equation}

We have to solve for $f({\bf p}, {\bf r}, t)$ and $\Phi({\bf r}, t),$
treating $C_{12}[f]$ very carefully. We see that there will be an exchange
of atoms between the ${\tilde n}({\bf r}, t)$ and $n_c({\bf r}, t)$
components through the $C_{12}$ collisions.  We can use these coupled
equations for a variety of problems.  In these lectures, we will consider
the collective oscillations of the combined system composed of condensate
and non-condensate.
It is useful to introduce {\it two} regimes to describe collective modes
in interacting systems  \cite{Agrif, chap1}:
\begin{enumerate}
\item[I.]Collisionless (produced by {\it mean fields})
\begin{displaymath}
\omega\tau_R\gg 1 \ \ \mbox{or}\ \ T\ll \tau_R \
\left(\omega\equiv{2\pi\over T}\right)
\end{displaymath}
\item[II.] Hydrodynamic (produced by {\it collisions})
\begin{displaymath}
\omega\tau_R\ll 1 \ \ \mbox{or}\ \ T\gg \tau_R,
\end{displaymath}
\end{enumerate}
where $\tau_R$ is some appropriate relaxation time.
What should we use for $\tau_R$?  For a classical gas, this is the
collision time \cite{Khuang}
\begin{equation}
{1\over\tau_c}={\tilde n}\sigma {\bar v}\label{eqcoupled96}\end{equation}
where
\begin{eqnarray*}
&{}&\sigma= 8\pi a^2 \ \mbox{(for Bose particles)}; \ a = s\mbox{-wave
scattering length}.\\
&{}&{\bar v} \simeq \ \mbox{average velocity of atoms}\
\sim\sqrt{k_BT\over m}.\\
&{}&{\tilde n}  =\  \mbox{density of excited atoms}.
\end{eqnarray*}
Even for a Bose-condensed gas, taking $\tau_R \sim\tau_c$ is a reasonable
first estimate \cite{Tnikzargrif}.
To get into the interesting hydrodynamic region $(\omega\tau_R\ll 1)$, we
need small $\tau_R $, ie, a large  density ${\tilde n}$ or a large
collision cross-section  $\sigma$ (perhaps using a Feshbach resonance, as
discussed in Section~\ref{sec:Overview}).

Let us look at the kinetic equation (\ref{eqcoupled86}), writing it in the
schematic form:
\begin{equation}{\hat{\cal L}} f = C_{22}[f]
+C_{12}[f].\label{eqcoupled97}\end{equation}
In the collisionless region, we need only solve ${\hat{\cal L}} f = 0.$
In contrast, in the hydrodynamic region, the collisions are so {\it
strong} they produce {\it local} equilibrium \cite{Khuang}.  That is, they
force $f$ to satisfy $C_{22}[f]=0.$ The unique solution ${\tilde f}$ of
this equation is well-known to be given by
\begin{equation}
{\tilde f}({\bf p}, {\bf r}, t) = {1\over e^{\beta[{({\bf p}-m{\bf
v}_n)^2\over 2m}+U({\bf r}, t) - {\tilde\mu}(r,
t)]}-1},\label{eqcoupled98}\end{equation}
where ${\bf v}_n$ is the average local velocity and ${\tilde\mu}$ is the
local chemical potential of the thermal atoms.
This local equilibrium Bose distribution involves the local variables
$\beta, {\bf v}_n, {\tilde\mu}$ and $U$, all of which depend on $({\bf r},
t)$.

Why must ${\tilde f}$ have the form in (\ref{eqcoupled98})?  To satisfy
$C_{22}[f_1] = 0,$ we must have [see (\ref{eqcoupled87})]
\begin{equation}
(1+f_1)(1+f_2)f_3f_4 -
f_1f_2(1+f_3)(1+f_4)=0,\label{eqcoupled99}\end{equation}
and this requires that
$f$ be given by the Bose distribution.
We have used the fact that
\begin{equation}
f(x)\equiv{1\over e^x-1}  = -[f(-x)+1]\label{eqcoupled100}\end{equation}
and that
\begin{equation}
\left.\begin{array}{cc}
&{\bf p}_1 + {\bf p}_2 = {\bf p}_3 + {\bf p}_4 \nonumber \\
&{}\nonumber\\
&
{\tilde\varepsilon}_{p_1}+{\tilde\varepsilon}_{p_2}={\tilde\varepsilon}_{p_3}+

{\tilde\varepsilon}_{p_4}\end{array}\right\}
\ \mbox{energy and momentum
conservation},\label{eqcoupled101}\end{equation}
where ${\tilde\varepsilon}_p={p^2\over 2m} + U({\bf r}, t).$ As an aside,
using a kinetic equation is the most physical way of deriving the
equilibrium Bose distribution.  The standard approach in statistical
mechanics texts based on calculating a partition function does not bring
out the reason {\it why}
\begin{equation}
f_{B,F} = {1\over e^{\beta(\epsilon-\mu)}\mp
1}.\label{eqcoupled102}\end{equation}

However, while the fact that ${\tilde f}$ is given by the local
equilibrium Bose distribution in $(\ref{eqcoupled98})$ ensures that
$C_{22}[{\tilde f}]=0, $ one finds that $C_{12}[{\tilde f}] \ne 0.$  More
precisely, we find from (\ref{eqcoupled91})

\begin{eqnarray}
&{}&\left[(1+{\tilde f}_1){\tilde f}_2{\tilde f}_3 - {\tilde
f}_1(1+{\tilde f_2})(1+{\tilde f_3})\right] \nonumber\\
&{}&\propto \left[e^{-\beta[{\tilde\mu}-\mu_c-{1\over 2}m({\bf v}_n-{\bf
v}_c)^2]}-1\right](1+{\tilde f}_1){\tilde f}_2{\tilde
f}_3.\label{eqcoupled103}\end{eqnarray}
The expression in the square bracket only vanishes if the condensate and
non-condensate are in {\it diffusive} equilibrium, which requires that
\begin{equation}
{\tilde\mu}=\mu_c+{1\over 2}m({\bf v}_n-{\bf
v}_c)^2.\label{eqcoupled104}\end{equation}
When we {\it perturb} the system, this may not be true, ie, the two
components may be out of diffusive equilibrium.

We can now derive hydrodynamic equations for non-condensate by taking
moments of Boltzmann equation, the standard procedure used in classical
gases \cite{Khuang}. The first moment gives a continuity equation with a
source term:
\begin{eqnarray}
\int d{\bf p}\left\{{\cal L}{\tilde f} = C_{12}[{\tilde
f}]\right\}\rightarrow {\partial{\tilde n}\over\partial t} =
-\mbox{\boldmath$\nabla$}\cdot({\tilde n}{\bf v}_n)+
\Gamma_{12}[{\tilde f}], \label{eqcoupled105}\end{eqnarray}
where
\begin{eqnarray}
{\tilde n} &\equiv& \int{d{\bf p}\over (2\pi)^3} {\tilde f}({\bf p}, {\bf
r}, t)\nonumber \\
{\tilde n}{\bf v}_n &\equiv& \int{d{\bf p}\over (2\pi)^3} {{\bf p}\over
m}{\tilde f}({\bf p}, {\bf r}, t)\nonumber\\
\Gamma_{12}[{\tilde f}]&\equiv&\int{d{\bf p}\over(2\pi)^3} C_{12}[{\tilde
f}].\label{eqcoupled106}\end{eqnarray}
More explicitly, we find
\begin{eqnarray}
\Gamma_{12}[{\tilde f}]&=& {2g^2n_c\over(2\pi)^5\hbar^7}
[e^{-\beta[{\tilde\mu}-\mu_c-{1\over 2}m({\bf v}_n-{\bf
v}_c)^2]}-1]\nonumber\\
&\times&\int d{\bf p}_1\int d{\bf p}_2\int d{\bf p}_3 \delta(m{\bf v}_c
+{\bf p}_1 - {\bf p}_2 - {\bf p}_3) \nonumber \\
&\times&\delta(\varepsilon_c+{\tilde\varepsilon}_1-{\tilde\varepsilon}_2-{\tilde\varepsilon}_3)(1+{\tilde
f}_1){\tilde f}_2{\tilde f}_3 \nonumber \\
&\equiv& \left[e^{-\beta[{\tilde\mu}-\mu_c-{1\over 2}m({\bf v}_n-{\bf
v}_c)^2-1]}-1\right]{n_c\over\tau_{12}}.\label{eqcoupled107}\end{eqnarray}
We note that $\tau_{12}$ is a collision time \cite{Tnikzargrif} which
describes the $C_{12}$ collisions between the C and N.C. atoms.  Combining
(\ref{eqcoupled105}) with the continuity equation which results from
(\ref{eqcoupled93}),
\begin{equation}
{\partial n_c\over\partial t} = -\mbox{\boldmath$\nabla$}\cdot(n_c{\bf
v}_c)-\Gamma_{12}[{\tilde f}], \label{eqcoupled108}\end{equation}
we see that the source term $\Gamma_{12}$ cancels out to give
\begin{equation}{\partial(n_c+{\tilde n})\over\partial
t}=-\mbox{\boldmath$\nabla$}\cdot(n_c{\bf v}_c+{\tilde n}{\bf
v}_n).\label{eqcoupled109}\end{equation}
Thus our theory gives the exact continuity equation for the total local
density $n=n_c+{\tilde n}.$

Similarly, one finds
\begin{eqnarray}
&&\int d{\bf p}{\bf p}\left\{{\hat{\cal L}}{\tilde f}\right.=
\left.C_{12}[{\tilde f}]\right\}\rightarrow m{\tilde n}\left({\partial{\bf
v}_n\over\partial t}+{1\over 2}\mbox{\boldmath$\nabla$}{\bf
v}^2_n\right)\nonumber\\
&& \ \ \ \ \  = -\mbox{\boldmath$\nabla$}{\tilde P}({\bf r}, t)-{\tilde
n}\nabla U({\bf r}, t) -m({\bf v}_n-{\bf v}_c)\Gamma_{12}[{\tilde f}],
\label{eqcoupled110}\end{eqnarray}
where the kinetic pressure is given  by
\begin{equation}
{\tilde P}({\bf r}, t) = {m\over 3}\int {d{\bf p}\over (2\pi)^3} ({\bf
p}-m{\bf v}_n)^2{\tilde f}({\bf p}, {\bf r},
t).\label{eqcoupled111}\end{equation}
The second moment gives
\begin{eqnarray}
&&\int  d{\bf p} p^2\left\{{\cal L}{\tilde f} = C_{12} [{\tilde
f}]\right\}\rightarrow{\partial{\tilde P}\over\partial
t}+\mbox{\boldmath$\nabla$}\cdot({\tilde P}{\bf v}_n)\nonumber \\
&&\  = -{2\over 3}{\tilde P}\mbox{\boldmath$\nabla$}\cdot{\bf v}_n+{2\over
3}\left[\mu_c +{1\over 2}m({\bf v}_n-{\bf
v}_c)^2-U\right]\Gamma_{12}[{\tilde f}].\label{eqcoupled112}\end{eqnarray}

The detailed derivation of these results is not important here
\cite{Zarnikgrif}.  The main thing is that the hydrodynamic equations
(\ref{eqcoupled105}), (\ref{eqcoupled110}) and (\ref{eqcoupled112}) can be
shown to describe the non-condensate in terms of three new
``coarse-grained'' variables:
\begin{displaymath}{\tilde n}({\bf r}, t), {\bf v}_n({\bf r}, t) \
\mbox{and}\ {\tilde P}({\bf r}, t).\end{displaymath}
These are coupled to the two additional variables which describe the
condensate:
\begin{displaymath}
n_c({\bf r}, t), \ {\bf v}_c({\bf r}, t). \end{displaymath}
We note that the two condensate equations of motion given by
(\ref{eqdynamics70}) and (\ref{eqcoupled108}) are always ``hydrodynamic''
in form.  In contrast, it is only in the collision-dominated region that
the non-condensate dynamics can be described in terms of a few collective
variables.
We thus have 5 variables and 5 equations, which form a closed system.
Both components exhibit {\it coupled}, {\it coherent} collective motions.
This is the essence of two-fluid superfluid behaviour \cite{Land,
Khalatnikov}, a new unexplored frontier in trapped Bose gases.

What is {\it new} about the two-fluid hydrodynamic equations derived above
is the role of the source term $\Gamma_{12}[{\tilde f}].$  In a linearized
theory expanded around the {\it static} equilibrium Bose distribution
${\tilde f}_0$
 (where $\Gamma_{12}[{\tilde f}_0]$ vanishes),  one finds
\cite{Zarnikgrif, Tnikzargrif}
\begin{equation}
\Gamma_{12}[{\tilde f}]=\delta\Gamma_{12}[{\tilde f}]=-{\beta_0
n_{0c}\over\tau^0_{12}}\delta\mu_{diff},\label{eqcoupled113}\end{equation}
where $\mu_{diff}\equiv{\tilde\mu}-\mu_c$.  We find an equation of motion
of the kind
\begin{equation}
{\partial\delta\mu_{diff}\over\partial t} =
-{\delta\mu_{diff}\over\tau_\mu}+\dots,\label{eqcoupled114}
\end{equation}
where (see Eq.(87) in Ref. \cite{Zarnikgrif}).

\begin{equation}
{1\over\tau_\mu}\equiv
\left({gn_{c0}\over
k_BT}\right){1\over\sigma}{1\over\tau_{12}^0}.\label{eqcoupled115}\end{equation}

Here $\sigma$ involves various static equilibrium thermodynamic
functions.  The new relaxation time $\tau_\mu$ (which we can calculate!)
determines {\it how fast} ${\tilde\mu}\rightarrow\mu_c$, ie, how fast we
reach diffusive equilibrium between the condensate and non-condensate.  We
can have
\begin{equation}
\left.\begin{array}{cc}
&\omega\tau_{22}\ll 1\nonumber \\
& \omega\tau_{12}\ll 1\end{array}\right\}
\ \mbox{required for hydrodynamics}\label{eqcoupled116}\end{equation}
but simultaneously
\begin{equation}
\omega\tau_\mu\gg 1\label{eqcoupled117}\end{equation}
near $T_{BEC}$, where $n_{c0}\rightarrow 0$.  Our hydrodynamic equations
predict the existence of a new relaxational mode \cite{Zarnikgrif,
Tnikzargrif}
\begin{equation}
\omega\simeq -i/\tau_\mu.\label{eqcoupled118}\end{equation}
This mode is not included in the standard Landau 2-fluid equations (where
$\rho_s$ and $\rho_n$ are assumed to be always in local equilibrium with
each other).
\vskip 4pt

In a uniform gas, the two-fluid hydrodynamic equations give two normal
mode solutions \cite{grifzar, Zarnikgrif}:
\begin{itemize}
\item First sound (oscillation of the non-condensate mainly)
\begin{equation}\omega = u_1k\ , \ u^2_1 \simeq{5\over 3}{{\tilde
P}_0\over m{\tilde n}_0}\sim{kT\over m}.\label{eqcoupled119}\end{equation}
\item Second sound (oscillation of the condensate mainly)
\begin{equation}
\omega=u_2k\ , \ u^2_2 \simeq {gn_{c0}\over
m}.\label{eqcoupled120}\end{equation}
\end{itemize}
We note the second sound mode is the hydrodynamic version of famous $T=0$
Bogoliubov phonon mode discussed in Section \ref{sec:Dynamics}.  It is the
``soft mode'' at $T_{BEC}$.  This second sound mode couples to the new
relaxational mode given in (\ref{eqcoupled118}) and is damped as a result,
the maximum damping occuring when $\omega\tau_\mu=1.$

In  a trapped gas, we can work out the spectrum of hydrodynamic
oscillations $(\sim e^{-i\omega t})$.  Both the condensate and
non-condensate components have the same frequency.  The most interesting
one is the dipole mode, described by
\begin{eqnarray}
&{}&{\tilde n}({\bf r}, t) = {\tilde n}_0 ({\bf
r}-\mbox{\boldmath$\eta$}_n(t)), \ \ \ {\dot{\mbox{\boldmath$\eta$}}}_n(t)
= {\bf v}_n \nonumber \\
&{}&n_c({\bf r}, t) = n_{c0} ({\bf r}-\mbox{\boldmath$\eta$}_c(t)), \ \ \
{\dot{\mbox{\boldmath$\eta$}}}_c(t) = {\bf v}_c.
\label{eqcoupled121}\end{eqnarray}
One finds there are two modes of this kind \cite{Ezarnikgrif, Zarnikgrif}:
\begin{itemize}
\item In-phase (or Kohn) mode, where $\mbox{\boldmath$\eta$}_n=
\mbox{\boldmath$\eta$}_c$ and $\omega=\omega_0$ (trap frequency). It is
the finite temperature version of the sloshing mode described by
(\ref{eqdynamics63}) and (\ref{eqdynamics64}).  We note that this mode is
generic (occuring in both the hydrodynamic and collisionless limit) and is
not damped \cite{Zarnikgrif}.
\item Out-of-phase dipole mode, with  $\mbox{\boldmath$\eta$}_n\ne
\mbox{\boldmath$\eta$}_c$ and in {\it opposite} directions.  The frequency
of this mode is different from the trap frequency.  This out-of-phase mode
is of special interest since it is the analogue of the out-of-phase second
sound mode in superfluid $^4$He. \end{itemize}

We conclude this Section with some remarks:
\begin{enumerate}\item The {\it specific calculation} sketched above is
also of interest in the general field of non-equilibrium statistical
physics.  It describes the detailed dynamics of a system with a
two-component order parameter self-consistently coupled to a gas of
excitations based on a fully microscopic theory.
\item More work is needed to extend our analysis to {\it low} but finite
temperatures and also into the critical region {\it very close} to
$T_{BEC}$.  In both cases, our simple Hartree-Fock particle-like thermal
excitation spectrum (\ref{eqcoupled85}) is no longer valid.

\item The {\it classical}
  kinetic theory of gases has been a rich subject in mathematical physics
in the twentieth century, with well-known contributions by people like
Boltzmann, Hilbert, Enskog, Chapman, Uhlenbeck and Burnett.  These new
equations of motion for a Bose-condensed gas promise to yield a lot of new
physics in the next century - and surprises, as our work in this Section
has already shown.
\end{enumerate}

\section*{Acknowledgements}

The work in Section \ref{sec:Coupled} is part of an on-going collaboration
with Eugene Zaremba (Queen's University) and Tetsuro Nukuni (Tokyo
Institute of Technology).
I would like to thank Yvan Saint-Aubin for his strong desire that BEC be
represented at this CRM Summer School and insisting that I could fit it
into my post-sabbatical schedule.  My work is supported by a research
grant from NSERC.  I would also like to thank JILA (University of
Colorado) for support during the time when I prepared these lectures.


text:
Editorial Board Guest Author
Ms. Ford
Executive Director of Member Services, Wireless Infrastructure Association
Tracy Ford is Executive Director of Member Services at the Wireless Infrastructure Association and guides HetNet Forum initiatives for the association. The Wireless Infrastructure Association represents the businesses that develop, build, own and operate the nation's wireless infrastructure. Members include wireless carriers, infrastructure providers, and professional services firms that collectively own and operate telecommunications facilities around the globe. Through public affairs and advocacy efforts — on the local, state and federal level — WIA works to support the widespread deployment of wireless infrastructure in order to enable wireless broadband everywhere.
The HetNet Forum is a membership group of WIA dedicated to the advancement of the heterogeneous network. The HetNet is a wireless ecosystem composed of a variety of mobile and wireless technologies and infrastructure, interoperable with the macrocellular network providing harmonious voice and data communications. Ms. Ford also leads WIA's Innovation & Technology Council, which produces thought-leadership items on topics including 5G, CBRS and in-building wireless solutions.
Ms. Ford has spent more than two decades covering the rapidly changing wireless industry, tracking its changes as it grew from a voice-centric marketplace to the dynamic data-intensive industry it is today. She started her technology journalism career at RCR Wireless News and has held a number of titles there, including associate publisher and executive editor. She is a winner of the American Society of Business Publication Editors Silver Award for both trade show and government coverage. She joined WIA in 2011. A graduate of the Minnesota State University-Moorhead, Ms. Ford holds a B.S. degree in Mass Communications with an emphasis on public relations.
Please visit http://www.wia.org for more information.
Ms. Ford can be contacted at +1 703-535-7459 or tracy.ford@wia.org
New Spectrum Brings Opportunities for More Back-Office Functions to Hospitality
By Tracy Ford, Executive Director of Member Services, Wireless Infrastructure Association

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Animating when filtering divs I can't quite wrap my head around how to do this, so any help would be appreciated.
I have a search field and as you type into it, it is looking at a data-attribute and hiding the divs that don't have the correct data. I would like to fade out the divs that don't have the correct data. I'd also like them to fade in as you clear out the search box. I can't quite figure out what I need to do, should I use .fadeIn/.fadeOut or should I be adding/removing classes?
This is my code for the filtering:
$("#filter").keyup(function(){
    var selectSize = $(this).val();
    filter(selectSize);
});

function filter(e) {
    if(e) {
        var regex = new RegExp('\\b\\w*' + e + '\\w*\\b', 'i');
        $('.oneStaff').hide().filter(function () {
            return regex.test($(this).data('regions'))
        }).show();
    } else {
        $('.oneStaff').show();
      }
}

redpajama_set_name: RedPajamaStackExchange


Laurie Mileur is one of my clients and a special lady. Her recent article about breast cancer written for our local paper helped me become better informed as I hope it will for you too!
October's arrival brings scarecrows and (cringe) a profusion of pink ribbons and products to remind us it is (teeth gritting) Breast Cancer Awareness Month. The effort is designed to provide community education about breast cancer risk factors, promote early detection and raise funds for research. These are laudable goals. So you may wonder why I, as a breast cancer survivor, cringe every October.
First, the goals are misleading.
Risk Factors that include being female, aging, family history, and lifestyle only predict 20 to 30 percent of breast cancers. Except for tobacco and hormone replacement therapy, toxins and chemicals linked to breast cancer are ignored. This omission suggests a serious conflict as many of these toxins and chemicals are produced by corporations that donate large sums to Breast Cancer Awareness Month. For example, Estée Lauder Cosmetics, which founded Breast Cancer Awareness Month and created the pink ribbon campaign, produce beauty products containing paraben which has been linked to breast cancer.
Many people don't realize that while early detection does benefit most women, it doesn't guarantee that breast cancer, when found, will be in an early stage. Unfortunately, even women at low risk who have regular mammograms can be diagnosed with metastatic (late stage) breast cancer. Metastatic means the breast cancer has spread to other organs (bone, lungs, brain, liver) with no treatment option other than palliative care.
Only a portion of the billions of dollars raised over the past two decades during Breast Cancer Awareness month are spent on research or patient services. Further, the focus of research is diagnosis and treatment instead of breast cancer prevention, which receives only three to five percent of all research dollars (including federal grants). Surprised? You shouldn't be.
Many corporations and pharmaceutical companies who give large donations to Breast Cancer Awareness Month also profit from breast cancer-related technology and drugs they themselves produce. An inconvenient truth is that breast cancer is a lucrative industry with annual medical costs predicted to exceed 20 billion dollars by 2020.
Second, Breast Cancer Awareness Month's packaging is just wrong.Branding breast cancer with the color pink, because it is comforting, non-threatening, and feminine, conveys the wrong message to women with breast cancer and the public.
Breast cancer is a terrible disease and its treatment is disfiguring, barbaric, and painful. Dressing cancer up in pink ribbons won't change that. Treatment, which includes surgery, radiation, and chemotherapy, can last months, years, or may even never end. Treatment side effects are nasty, with long-term consequences to physical, mental, emotional, and financial well-being There is nothing comforting, non-threatening or feminine about breast cancer.
This year more than 250,000 women will be diagnosed with breast cancer and over 40,000 will die. Today 3.1 million women in the United States have a history of breast cancer. This includes women who are getting or have completed treatment, and 150,000 women who are receiving palliative care for metastatic breast cancer. When I think of these women and women who risk cancer in their future including my daughter and granddaughters, I see and feel red.
So what can be done? The only way to eliminate breast cancer is to prevent it. More pressure needs to be applied to charities, pharmaceutical companies and government agencies to force them to focus on preventing breast cancer rather than continuing to profit from its treatment. Add your voice to activist groups such as "Think Before You Pink" that are trying hold these organization accountable.
Be informed and ask questions. Watch "Pink Ribbons, Inc.", a highly rated documentary on the dark side of the breast cancer industry.
If you donate, thank you, but make sure your dollars are spent wisely on areas that interest you: research, support services, or education and public awareness. Charity Navigator (www.charitynavigator.org) is a good resource.
Laurie. Mileur, PhD, RD. Is a registered dietitian who has worked in health care for over 35 years. She is a member of the Healthcare Advocacy Committee for the Cambria Community Healthcare District. Her mother and grandmother died from metastatic breast cancer; Laurie and her sister, Jan, are survivors (13 years and counting).


text:
<?php
/**
 * Hush Framework
 *
 * @category   Hush
 * @package    Hush_Bpm_Lang
 * @author     James.Huang <shagoo@gmail.com>
 * @license    http://www.apache.org/licenses/LICENSE-2.0
 * @version    $Id$
 */

require_once 'Hush/Bpm/Lang.php';
require_once 'Hush/Bpm/Lang/Exception.php';

/**
 * Php Bpm Execution Language
 * @package Hush_Bpm_Lang
 */
class Hush_Bpm_Lang_Pbel extends Hush_Bpm_Lang
{
	/**
	 * @var array
	 */
	public $methods = array(
		'model_field',
		'model_form_add',
		'model_form_edit',
		'audit_by_role',
		'audit_by_user',
		'audit_check',
		'forward',
	);

	/**
	 * @var array
	 */
	public $returns = null;

	/**
	 * @var array
	 */
	public $phpfile = null;

	/**
	 * Set file for $this->getDocs()
	 * @param string $file
	 */
	public function setFile ($file)
	{
		$this->phpfile = (string) $file;
		return $this;
	}

	/**
	 * Get all docs
	 */
	public function getDocs ()
	{
		if (!file_exists($this->phpfile)) {
			throw new Hush_Bpm_Lang_Exception("Pbel class '$file' does not exists");
		}

		$docs = array();
		foreach ((array) file($this->phpfile) as $line) {
			if (preg_match('/@pbel(.*)/i', $line, $lres)) {
				if (empty($lres[1])) continue;
				$docs[] = array(
					'method' => trim($lres[1]),
					'intros' => trim($lres[1]),
				);
			}
		}

		return $docs;
	}

	/**
	 * Set return data for executing code
	 * @param mixed $return
	 */
	protected function setReturn ($return)
	{
		$this->returns = $return;
		return $this->returns;
	}

	/**
	 * Get return data after executing code
	 */
	protected function getReturn ()
	{
		return $this->returns;
	}

	/**
	 * Get all going next nodes for saving into db
	 * @param string $code
	 */
	public function getNextNodes ($code)
	{
		$gotoNodes = array();
		@preg_match_all('/forward\(([0-9]+)\)/i', $code, $cres);
		if ($cres[1]) $gotoNodes = (array) $cres[1];
		return $gotoNodes;
	}

	/**
	 * Check syntax for pbel code before executing
	 * @param string $code
	 */
	public function prepare ($code)
	{
		return ($this->_safeCheckSyntax($code) === false) ? false : true;
	}

	/**
	 * Executing code
	 * @param string $code
	 */
	public function execute ($code)
	{
		return $this->_safeEvalCode($code);
	}

	/**
	 * Private checking syntax method
	 */
	private function _safeCheckSyntax ($code)
	{
		$code = trim(str_replace("pbel.", "", $code));
		$code = "return true;\n" . $code;
		if (!$code) return false;

		ob_start();
		$eval = @eval($code);
		if ($eval === false) {
			ob_end_clean();
			return false;
		}
		ob_end_clean();

		return true;
	}

	/**
	 * Private eval code method
	 */
	private function _safeEvalCode ($code)
	{
		$class = (string) get_class($this);
		$lang = new $class;

		$code = trim(str_replace("pbel.", "\$lang->", $code));
		if (!$code) return false;

//		error_log($code); // for code debug

		ob_start();
		$return = false;
		$eval = eval($code);
		if ($eval === false) {
			ob_end_clean();
			return false;
		}
		$oput = ob_get_contents();
		ob_end_clean();

		return $lang->getReturn();
	}
}

redpajama_set_name: RedPajamaGithub


text:
Jorge Posada apologizes for comments about his former Yankee teammates, A-Rod is quick to accept
May 14, 2015 | 11:48 PM
Alex Rodriguez (r.) says Thursday he appreciates the apology offered up by Jorge Posada. (Ron Antonelli/New York Daily News)
Consider it, A-pology accepted.
Alex Rodriguez, blasted by former teammate Jorge Posada on Wednesday for his link to PED use, said he's accepted the catcher's apology and is ready to move on.
"Apology accepted. Much appreciated," Rodriguez said upon arriving at Tropicana Field on Thursday.
RELATED: POSADA SAYS A-ROD, CLEMENS DON'T BELONG IN HALL OF FAME
Posada, while promoting his book on Wednesday, said A-Rod and Roger Clemens, another former teammate, should not be allowed into the Hall of Fame due to their ties to steroids.
"I don't think it's fair," Posada said on "CBS This Morning." "I really don't. I think the guys that need to be in the Hall of Fame need to be a player that played with no controversy."
When asked specifically about A-Rod and Clemens going into the Hall, Posada said, "no."
But on Thursday on the Boomer and Carton radio show on WFAN, Posada tried to take back the words, blaming the interviewer for pushing him toward the answer.
"I tried to text him yesterday," Posada said Thursday morning. "I feel like I was cornered into this answer. I know better. I was really caught off-guard. He knows how I feel about him. And I would like to — if I caused him anything, I would like to apologize. . . . I felt like the guy kept pushing towards that response, and I should have known better."
When asked Thursday if Posada had texted him, A-Rod said, "Yeah, I did receive a text."
When asked if Posada's apology was necessary, Rodriguez responded, "Guys, like I said, 2015. I'm not the right person to ask what's necessary or not."
As for possibly seeing him on Sunday, May 24 for Bernie Williams Day, Rodriguez said, "I'm fine."
Yankees star Aaron Judge happy to give back as coach of fan softball game with Didi Gregorius and Gleyber...


text:
RSR Award Detail
Award Detail
Awardee: NEVADA SYSTEM OF HIGHER EDUCATION
Doing Business As Name: Board of Regents, NSHE, obo University of Nevada, Reno
PD/PI:
George Bebis
bebis@cse.unr.edu
Award Date: 09/09/2009
Estimated Total Award Amount: $ 599,970
Funds Obligated to Date: $ 599,970
FY 2009=$599,970
Transaction Type: Grant
Agency: NSF
Awarding Agency Code: 4900
Funding Agency Code: 4900
CFDA Number: 47.070
Primary Program Source: 040101 RRA RECOVERY ACT
Award Title or Description: CPS: Small: Real-time, Simulation-based Planning and Asynchronous Coordination for Cyber-Physical Systems
Federal Award ID Number: 0932423
DUNS ID: 146515460
Parent DUNS ID: 067808063
Program: CPS-Cyber-Physical Systems
Program Officer:
David Corman
dcorman@nsf.gov
Awardee Location
Street: 1664 North Virginia Street
County: Reno
Awardee Cong. District: 02
Primary Place of Performance
Organization Name: Board of Regents, NSHE, obo University of Nevada, Reno
Cong. District: 02
Abstract at Time of Award
CPS:Small: Real-time, Simulation-based Planning and Asynchronous Coordination for Cyber-Physical Systems This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The objective of this research is to investigate how to replace human decision-making with computational intelligence at a scale not possible before and in applications such as manufacturing, transportation, power-systems and bio-sensors. The approach is to build upon recent contributions in algorithmic motion planning, sensor networks and other fields so as to identify general solutions for planning and coordination in networks of cyber-physical systems. The intellectual merit of the project lies in defining a planning framework, which integrates simulation to utilize its predictive capabilities, and focuses on safety issues in real-time planning problems. The framework is extended to asynchronous coordination by utilizing distributed constraint optimization protocols and dealing with inconsistent state estimates among networked agents. Thus, the project addresses the frequent lack of well-behaved mathematical models for complex systems, the challenges of dynamic and partially-observable environments, and the difficulties in synchronizing and maintaining a unified, global world state estimate for multiple devices over a large-scale network. The broader impact involves the development and dissemination of new algorithms and open-source software. Research outcomes will be integrated to teaching efforts and undergraduate students will be involved in research. Underrepresented groups will be encouraged to participate, along with students from the Davidson Academy of Nevada, a free public high school for gifted students. At a societal level, this project will contribute towards achieving flexible manufacturing floors, automating the transportation infrastructure, autonomously delivering drugs to patients and mitigating cascading failures of the power network. Collaboration with domain experts will assist in realizing this impact.
Publications Produced as a Result of this Research
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
Marble J, Bekris KE. "Asymptotically Near-Optimal Planning with Probabilistic Roadmap Spanners"	IEEE Transactions on Robotics,	v.29,	2013,	p.432.
For specific questions or comments about this information including the NSF Project Outcomes Report, contact us.


Multi purpose style defines this chic daybed, featuring a robust wooden construction, stylish curving sides, and decorative round finials. A pull-out trundle offers extra bedding space as needed.
Slat kits included, bunkie board suggested.
FL Comment: Nice solid wood, simple design. Completed bed save money. Assembly required. Nationwide shipping available for this item, the cost for this item range from $110 - $245 depending on where you live.


text:
Pure Poly Floral Square Neck Jumpsuit
Select Size 6-7 Years 7-8 Years 8-9 Years Out of Stock 9-10 Years 10-11 Years Out of Stock 11-12 Years 12-13 Years Out of Stock 13-14 Years 14-15 Years Out of Stock 15-16 Years Out of Stock
Our 'Standard' womenswear range has been designed for a woman whose height is 168cm (5ft 6") and who has an inside leg of 79cm. The fit of a garment may vary depending on the cut, style and fabric.
UK/AU/NZ US/CA EU DE Bust Waist Hips
78 82 87 92 97 102.5 108 114 120 126 30¾ 32¼ 34¼ 36¼ 38 40¼ 42½ 45 47¼ 49½
61 65 70 75 80.5 86 92 98 104 110 24 25½ 27½ 29½ 31¾ 34 36¼ 38½ 41 43¼
86 90 95 100 105 110 115 121 127 133 33¾ 35½ 37½¼ 39½ 41½ 43¼ 45¼ 47½ 50 52½
Size UK/AU/NZ US/CA EU DE Bust Waist Hips
4-6 8-10 12-14 16-18 20-22 22-24
82-87 92-97 102.5-108 114-120 126-132 138-144 32¼-34¼ 36¼-38 40¼-42½ 45-47¼ 49¼-52
65-70 75-80.5 86-92 98-104 110-116 122-128 25½-27½ 29½-31¾ 34-36¾ 38½-41 43¼-45½
90-95 100-105 110-115 121-127 133-139 145-151 35½-37½ 39½-41½ 43¼-45¼ 47½-50 52½-54¾
In bare feet take the measurement from the top of the inside leg and measure down the inside leg to the floor.
With bare feet together measure from the top of the head to the floor.
6. Skirt length
Measure from natural waist line to the knee.
Measure under the arms at the fullest part around the bust horizontally.
Measure around the natural waistline.
With the feet together measure around fullest part.
Product ID: T74/1727S
Let your little girl steal the limelight. This adorable pink jumpsuit from Marks and Spencer is perfect to add heaps of charm to her look for the next day out she attends. Made with 100% polyester, it has a zipper closure and is supremely comfortable.
Gentle machine warm wash, Do not bleach, Low tumble dry, Cool iron, do not dry clean, Wash Simillar Colours together, Wash and Tumble dry inside out
Name and address of the manufacturer(s): POOJA INTERNATIONAL A-20 SECTOR-64 GAUTAM BUDDHA NAGAR UP INDIA 201307
Date of Import (First date): NA
Date of Import (Latest date): NA
Name and address of importer: Marketed By Marks and Spencer Reliance India Pvt. Ltd-: Ground Floor Infinity Tower C,DLF Cyber City Phase II Gurgaon Haryana-122002
See all Dresses & Jumpsuits


text:
Varker Surname Ancestry Results
Our indexes 1000-1999 include entries for the spelling 'varker'. In the period you have requested, we have the following 6 records (displaying 1 to 6):
varian variar variava varicas varick varie varielles varienne varier varillat varin varing vario varion varior varipati varis varister variyar varkala varkevisser varkey varkie varkki varkvisser varle varlect varlee varler varlet varlet-maree varleth varlett varley varlien varlo varloe varlos varlow varlows
Sailors on board H. M. S. Swiftsure (1796-1798)
His Majesty's ship the Swiftsure took part in the destruction of the French fleet in Aboukir Bay at the mouth of the Nile ('the Battle of the Nile') on the evening of the 1st and morning of the 2nd August 1798. This is the muster book for 1 August to 30 September 1798: being a continuation book in a series covering wages and victualling from May 1796, it also includes the names of some men who had died, deserted or been discharged from the ship from then to August 1798. Of the ship's complement of 590, this index covers the sailors, volunteers, and boys, as well as the supernumeraries and the retinue of William Parker, Rear-Admiral of the Red: but not the marines, or the French prisoners taken after the battle. Usually each man's entry gives his birthplace, and also his age on entering the ship.
VARKER. Cost: £8.00.
Owners of Merchantmen (1822)
The Society for the Registry of Shipping was instituted in 1760, and published an annual register and supplement. The annual register consisted of an alphabetical list of ships surveyed for insurance in Britain and Ireland, together with an alphabetical supplement. The society maintained a Registry Office at which alterations and additions were notified, and members delivering their registers when called for had them updated and returned on the following or the ensuing day. Each ship was given a number within each letter of the alphabet: ships' names were not unique, so within each name a ship was identified by the name of the captain or master at the time of the last survey. Then abbreviations indicate the type of vessel (Bg, brig; Bk, barque; Cr, cutter; Dr, dogger; G, galliott; H, hoy; K, ketch; S, ship; Sk, smack; Sp, sloop; Sr, schooner; St, schoot; Sw, snow), and whether sheathed (s) and/or doubled (d) with copper (C) and iron bolts (I B) or over boards (W & C), patent felt (P F), copper fastened (c f), copper bolted (c b), or copper repaired (C rp) sometimes with a date, such as (18)18. The third column, reserved for masters' names, is not particularly wide; with short surnames, an initial will be given; but longer surnames omit the initials, and even longer surnames are abbreviated. It will be borne in mind that these are the names of the masters not (necessarily) in 1822, but at the time of the last survey. Often new masters had been appointed by the time of re-survey, and their names are added in slightly smaller type under the original master's names in the third column. In the fourth column is the tonnage: where there is a blank under the number this indicates that the ship had two decks; more often the letters S D (B) for single deck (with beams); D W for deep waist; S D W single deck with deep waist; B D W single deck with beams and deep waist. Underneath the entry may run references to recent repairs: Cl. clincher built; Drp. damages repaired; grp. good repair; len. lengthened; lrp. large repair; N. (new) B. bottom, D. deck, Kl. keel, Sds. sides or UW. upper-works; rb. rebuilt; rsd. raised; Srprs. some repairs; or trp. thorough repair. In italics, the timber of the ship is described - B. B., black birch; C., cedar; H., hazel; Hk., hackmetack; J., juniper; L. O., live oak; M., mahogany; P., pine; P. P., pitch pine; S., spruce; W. H., witch hazel; W. O., white oak. The fifth column gives the place that the ship was built. For foreign ships this may be as vague as 'Dutch' or 'French'; but nothing in this record specifically indicates the nationality of ship, master or owners, except that an A. under the owner's name indicates that the vessel was American property. The sixth column gives the year of the ship's age; a few were still sailing after 30 or 40 years. The seventh column gives the owner's name, abbreviated in the same way as the master's name. Where the master was the owner, the word Capt. will appear. With vessels owned abroad, the name in this column is sometimes that of the port of origin, not the surname of the owner. Where there has been a change of owner by the time of re-survey, the new name is put underneath in smaller type. The printer sought to avoid confusion by aligning names of ports to the left and surnames to the right, but that leaves longer names doubtful. The eighth column gives the feet of the draught of water when loaded. The ninth column shows the destined voyage for which the survey took place, with the port of survey abbreviated (Be., Belfast; Br., Bristol; Co., Cork; Cs, Cowes; Da., Dartmouth; Du., Dublin; Eh, Exmouth; Ex., Exeter; Fa., Falmouth; Gr., Greenock; Hl, Hull; Hn, Harrington; La., Lancaster; Lh, Leith; Li., Liverpool; Lo., London; Ly., Lynn; Mt., Maryport; Po., Poole; Ph, Portsmouth; Pl., Plymouth; Sc., Star-Cross; Tn., Teignmouth; Tp., Topsham; Wa., Waterford; Wn, Whitehaven; Wo., Workington; Ya., Yarmouth), and the letter C where the vessel was a constant trader between the two ports. The tenth column gives the classification of the vessel (A, first; E, second) and its stores (1, first; 2, second) and the year of survey, e. g. 09 for 1809, or, if surveyed during 1821, the month, e. g. 3 for March. Where the vessel has been re-surveyed, the classification letter and number will be repeated or revised in the final column. The sample scan is from the main list. This is the index to owners in the main list and the supplement.
Outstanding soldiers of the 28th regiment of Foot (1860-1870)
The 28th (The North Gloucestershire) Regiment of Foot embarked for Turkey in 1854, and took part in the Crimean War; then to Malta in 1856; and thence to India. It was stationed at Ahmednuggur in 1860. The regiment returned to England in 1865, and in 1867 to Ireland: the depot was at Fermoy. In 1868 the troops were sent to Gibraltar. Each year just a handful of outstanding soldiers of the regiment were chosen for good conduct medals and gratuities: these are listed here. There were two lists, one for men recommended for the Good Conduct Medal without a gratuity, and one for gratuities - £5 to a private, £10 to a corporal, and £15 to a serjeant. Both lists are indexed here, and each gives rank, name, regimental number, date of recommendation and date of issue. (The sample scan is from the 105th foot)
County Court Judgments: Cornwall (1890)
Extracts from the Registry of County Courts' Judgments. These judgments were not necessarily for debt. In some cases they were for damages on properly disputed causes of action, but no distinction was made on the Register. Judgments settled otherwise than through the Court may appear, unless 'Satisfaction' was entered up within the fourteen days allowed for that purpose. These printed extracts include occasional notes giving more detail about certain cases, and also list Satisfactions entered on the Register.
Civil Servants and Office Holders (1913)
The Imperial Calendar gives lists of officials and office-holders throughout England, Wales, Scotland and Ireland
Boys entering Truro School (1901-1954)
Truro College was founded 19 January 1880; the name was changed to Truro School in 1931. In 1954 this register of old boys was published, arranged alphabetically by surname and then christian name (or initials), in bold; then years of entering and leaving the school, and, in many cases, address as in 1954. Sometimes occupation is given.