text:
\section{Introduction} \label{sec:Intro}

An atom laser is a device which, in analogy to the optical laser,
would emit a coherent beam of bosonic atoms.  Various models of atom
lasers have been presented
\cite{Holland95,Wiseman95a,Wiseman95b,Guzman96,Spreeuw95,Olshanii95,Moy97,Wiseman97}.

Many of these schemes are based around a master equation description
for the dynamics of the
system\cite{Holland95,Wiseman95a,Wiseman95b,Guzman96,Wiseman97}.  In
these, the atom laser output is described by a Born-Markov master
equation.  Making the Born-Markov approximation involves assuming that
reservoir correlations decay rapidly compared with the system damping
time and that the reservoir does not change significantly with time
due to the effect of the system.  We discuss these approximations in
the context of atom output coupling.

Born-Markov master equations were initially derived for optical
systems\cite{QuantumOptics}.  There they are used to describe a system
(for instance an optical laser mode) coupled to a large, unchanging
reservoir (the external modes).  In optics the Born-Markov
approximations allow an equation containing only system variables to
be derived.  One of the fundamental assumptions involved in deriving
such a Born-Markov master equation is that the reservoir correlation
functions decay rapidly.  This allows the reservoir to be approximated
as unchanging in time.  While it is assumed the system does not affect
the reservoir, the reservoir does affect the system.

An equation in terms of system variables alone is also an important
goal for describing atom lasers.  However, atom and photon devices
work in substantially different parameter regimes.  Moreover, atoms
and photons have different dispersion relations, which affects the
behaviour of the reservoir correlation functions.  Furthermore,
typically the reservoir for an optical cavity is taken to be at
thermal equilibrium at a nonzero temperature.  For atoms, a vacuum
reservoir with all modes initially empty is often more appropriate.

In this paper we look at the validity of the Born-Markov master
equation for describing output coupling from a single mode atomic
system to a reservoir.  The reservoir is described as a continuum of
free-space modes.  Our description of the coupling will initially be
quite general, though our later discussion will focus on particular
schemes in which the atom becomes free of the system through a change
of state.  Such a change of state to an untrapped state can be
achieved using either a Raman transition \cite{Moy97} or an RF
transition\cite{MITExpt,MITExpt2,MIT3}.  We will also discuss the
effects of gravity on output coupling.  In the latter part of the
paper we will focus on broadband coupling.  This allows a comparison
with exact results\cite{Hope97,Moy97b}, however we also present finite
coupling results which more accurately describe output coupling
through a change of state.  We will discuss exact equations which can
be obtained in regimes where the Born-Markov approximations fail.

In section \ref{sec:Output} we present our model of output coupling
from a single mode trap.  This allows exact equations of motion, and
their solutions, to be obtained for system variables.  In section
\ref{sec:MasterEqn} we model this system using the master equation
formalism, emphasising the importance of the Born and Markov
approximations.  In section \ref{sec:Validity} we then consider the
validity of these approximations.  In the optical case, the adequacy
of the Born-Markov approximation means that the standard model we are
using leads to exponential decay of the number of photons in a cavity.
The equivalent model for atoms does not lead to an atom number which
tends to zero for sufficiently long time.  In section
\ref{sec:nonzero} we discuss the reasons for this and consider the
results obtained from models which include the effects of gravity.
Having noted that the Born-Markov approximations may fail, we discuss
in section \ref{sec:nonmarkov} a non-Markovian master equation and
show that for atoms the Born-approximation is not valid for our
parameters even if we avoid making the Markov approximation.  In the
regimes where these approximations fail, the output field becomes
correlated with the system and hence cannot be traced over to provide
a master equation.  Finally we discuss ways of proceeding when the
Born-Markov master equation fails.  The most straightforward method is
that used in section \ref{sec:Output} where exact equations for the
whole system and reservoir are obtained.

\section{Exact solutions} \label{sec:Output}

Dilute gas Bose Einstein Condensates (BEC) are now available in the
laboratory.  To produce a continuously running atom laser from a BEC
requires the addition of a suitable pumping mechanism, and an output
coupler.  A change of atomic state through RF transition has
been used to produce a ``pulsed atom laser''\cite{MITExpt,MITExpt2}.
The output coupling from a single mode to a large reservoir is
sometimes described by a Born-Markov master equation of the form %
\begin{eqnarray}
\frac{d \rho(t)}{dt} &=& C (2 a \rho(t) a^{\dag} -
a^{\dag} a \rho(t) -  \rho(t) a^{\dag} a),
\label{Eq.QuantumOpticalEqn}
\end{eqnarray}
where $C$ describes the strength of the coupling and $a$ ($a^{\dag}$) is
the annihilation (creation) operator for the single mode system.  Two
important approximations involved in such a description are that the
evolution is Markovian and that it depends only on the system operators.
The Markovian property means that the rate of change of the state depends
only on the state at that time.  There is no explicit dependence on the
state at previous times.  The equation is a function of system variables
only, due to tracing over the reservoir.  This is appropriate if the
reservoir remains uncorrelated with the system.  In fact, it is
approximated in deriving Eq.  (\ref{Eq.QuantumOpticalEqn}) that the
reservoir does not change with time.

We wish to investigate these two approximations, and their validity
for describing atomic output coupling.  In a full atom laser model it
is essential that a pumping term is also included.  However, in this
paper we consider a single mode coupled only through output coupling
to the outside world.  Experimentally this corresponds to a leaky BEC.
We focus on the output coupling term which would be present in a full
master equation model alongside other terms.

We begin by considering a generic output coupling mechanism which we
have previously analysed in the context of Heisenberg equations of
motion\cite{Hope97,Moy97b}.  Here we investigate how such a model can
be described by a density operator equation - in particular a master
equation.  When the output field is constrained by a waveguide such as
in a hollow optical fibre model\cite{Moy97}, the output field is
effectively one dimensional.  We consider a single mode system (the
lasing mode with creation operators $a^{\dag}$) coupled to a
one-dimensional continuum of free space modes.  The free space modes
are labelled by their momentum, $\hbar k$ (creation operators
$b_{k}^{\dag}$).  In a general three dimensional model $b_{k}$ would
also be labelled by indicies describing the transverse modes.  Here,
however, we suppress these indicies and assume that the output is
coupled into a waveguide with the atoms remaining in a single
transverse mode.  The Hamiltonian describing the system is given by %
\begin{eqnarray}
{ H} &=& { H_{\mbox{s}}} +
      { H_{\mbox{r}}}  + { H_{\mbox{sr}}}, \label{Eq.Heff}\\
{ H_{\mbox{s}}} &=& \hbar \omega_{0} ~ a^{\dag} a,
\label{Eq.Hsys}\\
{ H_{\mbox{r}}} &=& \int_{-\infty}^{\infty} \mbox{dk} ~ \hbar \omega_{k} ~
      b_{k}^{\dag} b_{k}, \label{Eq.Hext}\\
{ H_{\mbox{sr}}} &=& -i \hbar \int_{-\infty}^{\infty} \mbox{dk}~~
(\kappa (k) ~b_{k} a^{\dag} - \kappa^{*} (k)~ b_{k}^{\dag} a).
\label{Eq.Hint}
\end{eqnarray}
The function $\kappa(k)$ describes the shape of the coupling in $k$
space, and is left general here.  By appropriately choosing
$\kappa(k)$ we may simulate a wide range of practical output coupling
mechanisms, including position dependant coupling and the effect of
momentum kicks.  The latter may result from laser photons when an atom
undergoes a change of state.  Choosing $\kappa(k)$ as constant over a
broad region corresponds to broadband coupling.  This model is used
for optical input-output theory\cite{OpticalModel}, and in proposed
atom laser theories which result in Born-Markov master equations.  For
the atom case this model ignores potentially important effects such as
gravity and atom-atom interactions.  For now, we neglect these effects
in order to investigate the validity of the Born-Markov
approximations.  We use this model to understand the differences
between optical and atomic output coupling.  We will extend the model
in section \ref{sec:nonzero}.

We use the Hamiltonian presented above to write Heisenberg equations
of motion for the operators $a$, $b_{k}$.  We can also obtain
equations for combinations of these operators which may be of more
interest, such as the number of atoms in the system, $a^{\dag} a$.
These equations can be difficult to solve in general.  However, since
they include the output and system explicitly, they describe the
dynamics of the model exactly\cite{Hope97,Moy97b}.  Exact solutions
can be compared with Born-Markov master equations.

To facilitate this comparison we present exact equations for the
expectation values of the system operators, $a^{\dag}$ and $a^{\dag}
a$.  In these we assume (as we will do in the master equation
descriptions to follow) that the reservoir is initially empty,
$\langle b_{k}^{\dag}(0) b_{k}(0) \rangle = 0$.  We do not place any
further restrictions on $\langle b_{k}^{\dag}(t) b_{k}(t) \rangle$.
This is the first fundamental difference between the atom and photon
case.  An empty reservoir for photons is inapropriate at finite
temperatures\cite{Carmichael91}.  In experiments where a Bose-Einstein
condensate is allowed to leak out of a trap, however, the most
appropriate initial state for the outside atom modes is a vacuum.
Similarly, for an atom laser in a hollow fibre, which we discussed in
\cite{Moy97}, the initial output modes would be empty.

We obtain %
\begin{eqnarray}
\frac{d}{d t} \langle a^{\dag} (t)\rangle &=& i \omega_{0} \langle
a^{\dag} (t) \rangle - \nonumber\\
&&~  \int_{0}^{t} d \tau ~ f'^{*}(\tau)  \langle
a^{\dag}(t-\tau) \rangle e^{i \omega_{0} \tau}, \label{Eq.adag} \\
\frac{d}{d t} \langle a^{\dag}(t) a(t) \rangle &=& - \int_{0}^{t} d
\tau ~ f'(\tau) ~ \langle a^{\dag}(t) a(t-\tau) \rangle e^{-i
\omega_{0} \tau} ~ \nonumber \\
  && ~+ \mbox{h.c}. \label{Eq.adaga}
\end{eqnarray}
These equations have solutions given by \cite{Hope97,Moy97b}
\begin{eqnarray}
\langle a^{\dag} (t)\rangle  &=& \langle a^{\dag}(0) \rangle e^{i
\omega_{0} t} ~{\cal
L}^{-1} \left \{ \frac{1}{s + {\cal L} \left\{f'(t)^{*} \right\}(s) }
\right\} (t), \label{Eq.adagsoln}
\end{eqnarray}
\begin{eqnarray}
\langle a^{\dag}(t) a (t)\rangle &=& \langle a^{\dag}(0) a(0) \rangle
~ \times \nonumber\\
&& ~ \left| {\cal L}^{-1} \left \{ \frac{1}{s + {\cal L} \left\{f'(t)
\right\}(s) } \right\} (t) \right|^{2}, \label{Eq.adagasoln}
\end{eqnarray}
where ${\cal L}$ and ${\cal L}^{-1}$ denote Laplace and inverse
Laplace transforms respectively.  $f'(t)$ is the function defined by
\cite{Moy97b} %
\begin{eqnarray}
f'(t) &=& \int_{-\infty}^{\infty} dk |\kappa(k)|^{2} e^{-i
(\omega_{k}-\omega_{0}) t}.
\label{Eq.f't}
\end{eqnarray}
The function $f(t) = f'(t) e^{-i \omega_{0} t}$ is the ``reservoir
correlation function'' in the master equation picture which we
describe in the next section.  Here $\omega_{k} = \hbar k^{2}/(2 m)$
for atoms, in contrast to $\omega_{k} = c_{L} |k|$ for photons.  Here
$m$ is the mass of the atoms, and $c_{L}$ is the speed of light.

For a previously described output coupling from a condensate through state
change\cite{Moy97b}, $\kappa(k)$ is a Gaussian of width $\sigma_{k}$, %
\begin{eqnarray}
\kappa(k) &=& i \Gamma^{\frac{1}{2}} (2 \pi \sigma_{k}^{2})^{-1/4}
\exp{\left(-k^{2} / (4 \sigma_{k}^{2}) \right) }. \label{Eq.kappa}
\end{eqnarray}
Here, the strength of the coupling is given by the coupling constant,
$\Gamma$.  For this form of coupling $f'(t)$ may be evaluated as: %
\begin{equation}
f'(\tau) = \frac{e^{i \omega_{0} \tau} \Gamma}{\sqrt{1 + i \alpha \tau}},
\label{Eq.f'b}
\end{equation}
where we have defined
\begin{equation}
\alpha = \hbar \sigma_{k}^{2}/m. \label{Eq.alpha}
\end{equation}

For the broadband limit of Eq.~(\ref{Eq.alpha}) (discussed in section
\ref{sec:Validity}) we may use methods similar
to those given in reference \cite{Moy97b} to obtain an analytical form for
the inverse Laplace transform,
\begin{eqnarray}
{\cal L}^{-1} \left \{ \frac{1}{s + {\cal L} \left\{f'(t)
\right\}(s) } \right\} (t)  &=& e^{i \omega_{0} t} \nonumber\\
    &&\rule{-35mm}{0cm}\times ~(W(a,b,c) +
  W(b,a,c) +  W(c,b,a) ),  \label{Eq.LaplaceSoln} \\
W(a,b,c) &=& \frac{a^{2} e^{a^{2} t} ~ (1 + \mbox{erf}[a
\sqrt{t}])}{(a-b)(a-c)}. \label{Eq.Wabc}
\end{eqnarray}
The variables, a,b and c are the three solutions of the equation $s^{3} + i
\omega_{0} s + \Gamma c \sqrt{i} = 0$.  Note that Eq.~(\ref{Eq.adagsoln})
gives that $\langle a(t) \rangle$ will always remain zero if $\langle a(0)
\rangle = 0$.  For the case of damping of a BEC which we consider here, the
initial state corresponds to a BEC in an atom trap.  According to
spontaneous symmetry breaking arguments BECs are in coherent states with a
definite global phase \cite{StatMech}, so that $\langle a(0) \rangle \neq
0$.  This is a useful assumption.  Nevertheless, even if $\langle a \rangle
= 0$, the form of the equation for $\langle a(t)
\rangle$ must be as given.

The equations of motion given by Eqs.  (\ref{Eq.adag},\ref{Eq.adaga})
and the corresponding solutions, Eqs.
(\ref{Eq.adagsoln},\ref{Eq.adagasoln}) are exact for the system under
consideration.  In specific cases it is very difficult to solve for
these inverse Laplace transforms.  Moreover, the Heisenberg equations
for system operators depend on the external operators, $b_{k}(0)$ in
general.  We next investigate equations of motion based on the Born
and Markov approximations.  These are compared with the exact
solutions given above.

\section{The Born-Markov Master Equation} \label{sec:MasterEqn}

Derivations of the Born-Markov master equation for a general system
reservoir interaction are given in references
\cite{Carmichael91,Louisell90,Tannoudjii92}.  Here we present
a derivation for the specific model of Eq.  (\ref{Eq.Heff}) to
Eq.(\ref{Eq.Hint}).  We assume that the atom reservoir is initially in
a vacuum state - that is, there are initially no atoms outside the
system.  This assumption was also made in the exact solutions
presented in Sec.  \ref{sec:Output}.  We make the Born approximation.
It involves ignoring correlations which may arise between the system
and reservoir and ignoring any time evolution of the reservoir density
operator.  We use the interaction Hamiltonian in Eq.  (\ref{Eq.Hint}).
This leads to the non-Markovian master equation: %
\begin{eqnarray}
\frac{d{\widetilde{\rho}}}{dt} &=& - \int_{0}^{t}~ d\tau \left\{ a^{\dag}a
\widetilde{\rho}(t-\tau) - a \widetilde{\rho}(t-\tau) a^{\dag}
\right\} \times \nonumber\\
&& f'(\tau) ~ + \mbox{h.c}, \label{Eq.Born}
\end{eqnarray}
where, ${\widetilde{\rho}}$ is the density operator in the interaction
picture.  The function, $f'(\tau)$, defined as $f'(\tau) = f(\tau)~ e^{i
\omega_{0} \tau}$ is the same as defined in Eq.  (\ref{Eq.f't}) and
Eq.~(\ref{Eq.f'b}).  Here $f(\tau)$ is the reservoir correlation function.

Eq.  (\ref{Eq.Born}) is non-Markovian.  The second major approximation
required to produce a Born-Markov master equation is the Markov
approximation.  The Markov approximation is made on the assumption that the
reservoir correlation function, $f(\tau)$ goes to zero rapidly compared
with the time scale on which $\widetilde{\rho}(t)$ changes.  Making the
Markov approximation thus involves replacing the terms $\widetilde{\rho}(t-
\tau)$ in Eq.  (\ref{Eq.Born}) with $\widetilde{\rho}(t)$.  In the optics
case, this approximation is also usually accompanied with the extension of
the upper limit of the integral from $t$ to infinity.  Making these
approximations gives %
\begin{eqnarray}
\dot{\widetilde{\rho}} &=& \left\{ (c + c^{*}) a \widetilde{\rho}(t)
a^{\dag} - c~ a^{\dag}a \widetilde{\rho}(t) - c^{*}
\widetilde{\rho}(t) a^{\dag} a \right\} , \label{Eq.Markov}
\end{eqnarray}
where
\begin{eqnarray}
c &=&  \int_{0}^{t \rightarrow \infty} d \tau
\int_{-\infty}^{\infty} |\kappa(k)|^{2} e^{-i (\omega_{k}-\omega_{0})
\tau} dk \nonumber\\
&=& \int_{0}^{\infty} f'(\tau) d\tau \nonumber\\
 &=& \frac{\Gamma \sqrt{2 \pi} \exp{\left[- \omega_{0} / \alpha
\right]}}{\sqrt{2 \omega_{0} \alpha }} ~ \left(1 + \mbox{Erf} \left[ i
\sqrt{\frac{\omega_{0}}{ \alpha}} \right] \right). \label{Eq.c}
\end{eqnarray}

The upper integration limit $t$ has been extended to $\infty$, as is
done in the optical case, without affecting our subsequent conclusion
regarding the Born and Markov approximations.  This produces an
equation of the form which has been used previously to describe atom
lasers.  We further note that we could redefine $c$ to be real without
loss of generality by incorporating the imaginary part of $c$ in with
the free part of the system Hamiltonian.  This reduces the form of the
loss term to the familiar $C (2 a \rho a^{\dag} - a^{\dag}a \rho -
\rho a^{\dag} a) $, with $C = {\cal R}e[c]$ the (real) coupling
strength.

The value of the constant $c$ depends on the form of $f'(\tau)$.  In the
following, we consider $f'(t)$ to be either that resulting from the
Gaussian coupling presented in this section, or the broadband limit of this
function which we discuss in the next section.

\section{Timescale conditions} \label{sec:Validity}

The Born-Markov master equation, Eq.  (\ref{Eq.Markov}), is the form
used recently in various discussions of atom laser dynamics
\cite{Holland95,Wiseman95a,Wiseman95b,Guzman96,Wiseman97}.  This
Born-Markov master equation is used ubiquitously in quantum optics.
The validity of the Markov approximation depends on the interplay
between the reservoir correlation timescale, the system timescale, and
the timescale on which the system decays.

The condition for the validity of the Born-Markov approximations is
given in standard optics texts by the timescale separation condition
\cite{Carmichael91,Louisell90,Tannoudjii92}
\begin{equation}
t_{R} << \Delta t << t_{D}
\end{equation}
where $t_{R}$ is the reservoir correlation time, and $t_{D}$ is the
cavity decay time.  $\Delta t$ defines a coarse grained timescale on
which the equations of motion are valid.  Generally $t_{R}$ is defined
as the ``timescale on which the reservoir correlations are non-zero''.
$t_{D}$ is the decay timescale of the system which can be obtained by
solving the equations for system and reservoir self consistently.  In
the Markov approximation this timescale goes as $1/(c + c^{*})$ for
$c$ similar to that defined in Eq.  (\ref{Eq.c}) except with the
optics dispersion relation.

Both $t_{R}$ and $t_{D}$ depend on the function $f'(\tau)$, and thus
in turn on the form of $\omega_{k}$ as a function of $k$.  For atoms
$\omega_{k} = \hbar k^{2}/(2 m)$, whereas for photons $\omega_{k} =
c_{L} |k|$, where $c_{L}$ is the speed of light.  It also depends on
factors such as the nature of the reservoir and the parameter regime
in which atom lasers operate.

A second timescale condition for the Born-Markov approximation which
is discussed in some treatements \cite{Louisell90,Tannoudjii92} of the
optical Born-Markov approximation is that the system timescale,
defined as $t_{s} = 1/\omega_{0}$ must satisfy
\begin{equation}
t_{s} << \Delta t. \label{Eq.condition2}
\end{equation}
This is equivalent to requiring $\omega_{0}$ to be very large.  A large
$\omega_{0}$ condition is required in optics for a number of reasons.
First, the initial coupling Hamiltonian, of the form ($a b_{k}^{\dag} +
a^{\dag} b_{k})$, is in the rotating wave approximation and ignores terms
of the form $a b_{k}$ and $a^{\dag} b_{k}^{\dag}$.  This rotating wave
approximation in optics can only be made in the case of large $\omega_{0}$.
For the atom coupling, however, the correct form of the coupling does not
include such (non atom number conserving) terms, even for small
$\omega_{0}$.  The terms which are eliminated in the optics
case\cite{Louisell90,Tannoudjii92} through the assumption of large
$\omega_{0}$ are already zero for our model, due to the assumption of an
atom vacuum reservoir.  Thus, one may be led from these treatments to
suppose that the Born-Markov approximation is made independently of
$\omega_{0}$ for an atom-vacuum reservoir.  This however is not the case as
we will discuss later.

In optics, these timescale conditions are usually satisfied.  For a
coupling based on a mirror it is standard to assume that the coupling
is broadband\cite{QuantumOptics}.  That is, we assume $\kappa(k)$ is
flat in k-space.  In this case the reservoir correlation function,
$f'(\tau)$, given by Eq.  (\ref{Eq.f't}) becomes %
\begin{eqnarray}
f'(\tau) &\approx& |\kappa(k_{0})|^{2} e^{i \omega_{0}\tau}
~\int_{-\infty}^{\infty} ~ e^{-i \omega_{k} \tau}~ dk \nonumber\\
 &=& 2 |\kappa(k_{0})|^{2}
~\int_{-\omega_{0}/c_{L}}^{\infty} ~ e^{-i c_{L} k \tau}~ dk \nonumber\\
&\approx& 2 |\kappa(k_{0})|^{2}
~ \delta(\tau). \label{Eq.opticcompare1}
\end{eqnarray}
In the final equation the Dirac delta function, $\delta(\tau)$, is
obtained by extending the frequency integral into physically
unrealistic negative frequencies.  This is a standard technique in
optics \cite{QuantumOptics} where $\omega_{0}$ is typically large.

Typically, for a laser, $\omega_{0} \approx 10^{15} \mbox{s}^{-1}$ is
large and the assumption of a Dirac delta function decay is very good.
Atom traps work in rather different parameter regimes with $\omega_{0}
\approx 10^{3} \mbox{s}^{-1}$.  If we avoid using negative
frequencies, with the assumption of the empty reservoir considered
here, we obtain the sum of a Dirac delta function, and an imaginary
part corresponding to the Cauchy principal value of the integral in
Eq.  (\ref{Eq.opticcompare1}).
\begin{equation}
f'(\tau) = \frac{2 \pi}{c_{L}} |\kappa(k_{0})|^{2} e^{i \omega_{0}
\tau} \left( \delta(\tau) - \frac{i}{\pi \tau} \right).
\end{equation}
However, for an optical reservoir this estimate of correlation function
decay based on our reservoir model is not strictly valid.  This is because,
while we have considered here the photonic dispersion relation, a vacuum is
unrealistic for an optical reservoir at finite temperatures.  More
appropriate is a thermal reservoir, which leads to a decay time of order
$\hbar / k_{B} T
\approx 10^{-13} \mbox{s}$\cite{Carmichael91}.

These reservoir correlation times must be short compared with the
decay time, $t_{D}$.  The system timescale, $1/\omega_{0}$ must also
be short compared with $t_{D}$ for a standard optical reservoir.
$t_{D}$ is the timescale of exponential decay, given by $e^{-(c+c^{*})
t}$ in the Born-Markov limit.  From an equivalent derivation of the
optical master equation to that given in section \ref{sec:MasterEqn},
the decay timescale is given by $t_{D} = 1/(c+c^{*}) \propto
(\sigma_{k}/\Gamma)$.  That is $t_{D}$ is inversely proportional to
the strength of the coupling, given by $\Gamma/\sigma_{k}$ in the
broadband limit.  We will see later that for the atom coupling, the
different dispersion relation makes $t_{D}$ depend on $\omega_{0}$ and
other parameters, such as the mass of the atom.  For optical systems
this decay time is typically of the order $10^{-8} \mbox{s}$.  Thus,
in typical optical systems, the Born-Markov approximation holds for a
number of reasons.  The condition $t_{R} << t_{D}$ holds, because in
the large $\omega_{0}$ limit the reservoir correlations decay as a
Dirac delta function.  $t_{D}$ does not depend on $\omega_{0}$ and is
typically many orders of magnitude larger than the reservoir decay
times.  Similarly, the system timescale $t_{s}$ for realistic optical
cavities is very much shorter than the decay timescale, $t_{D}$.

In contrast, the large $\omega_{0}$ limit is not necessarily valid for
realistic atom traps.  Moreover, even in the limit of infinitely large
$\omega_{0}$, the atom correlation function $f'(\tau)$ does not tend
towards a Dirac delta function.  This is due to the atomic dispersion
relations, which lead to a dependence of $t_{D}$ on parameters other
than the coupling strength.  Furthermore, the assumption of an
initially empty reservoir is realistic for atoms.  For atoms, the
broadband limit of the function $f'(t)$ is given by %
\begin{eqnarray}
f'(\tau) &\approx& |\kappa(k_{0})|^{2} e^{i \omega_{0}\tau}
~\int_{-\infty}^{\infty} ~ e^{-i \omega_{k} \tau}~ dk \nonumber\\
&=& 2 |\kappa(k_{0})|^{2} e^{i \omega_{0}\tau}
~\int_{0}^{\infty} ~ e^{-i \hbar k^{2}/(2 m) \tau}~ dk
\nonumber\\
 &=&  |\kappa(k_{0})|^{2} ~e^{i \omega_{0}\tau} ~\frac{\sqrt{m
 \pi} \left(1 - i  \right)}{\sqrt{\hbar \tau}}.
 \label{Eq.opticcompare2}
\end{eqnarray}
This is the broadband limit of the general reservoir
correlation function, Eq.  (\ref{Eq.f'b}) given in Sec.
\ref{sec:MasterEqn}.  Both broadband and Gaussian coupling give forms for
$f'(\tau)$ which fall off as inverse $\sqrt{\tau}$.  The broadband limit of
Eq.  (\ref{Eq.f'b}) is obtained as both $\sigma_{k}$ and $\Gamma$ tend to
infinity, with $\Gamma/\sigma_{k} =
\mbox{const.} = \sqrt{2 \pi} |\kappa(k_{0})|^{2}$.  $\sigma_{k}$ and
$\Gamma$ are both defined in Sec.  \ref{sec:MasterEqn} with
$\sigma_{k}$ giving the width of the Gaussian coupling, $\kappa(k)$.
We note that the broadband limit of Eq.  (\ref{Eq.f'b}) is
not correctly obtained by taking $\sigma_{k} \rightarrow \infty$ while
keeping $\Gamma$ constant.  If $\sigma_{k} \rightarrow \infty$ then
the coupling function $\kappa (k)$ and the constant $c$ in the master
equation, Eq.  (\ref{Eq.Markov}), go to zero everywhere due to the
normalisation of our coupling, Eq.~(\ref{Eq.kappa}).

We may now highlight three significant differences between optical and
atomic output coupling.  Firstly, Eq.~(\ref{Eq.opticcompare2}) shows that
the atomic reservoir correlation function decays as $1/\sqrt{\tau}$.  This
is different from the optical case even for finite $\omega_{0}$, and will
lead to different behaviour of the exact equations.  Secondly, the system
timescale, $1/\omega_{0}$, is significantly larger for atom traps than for
optical cavities.  Thirdly, the timescale on which the system decays is
given by $t_{D} \approx 1 / (c + c^{*})$ where $c$ is related to the
integral of the correlation function, $f'$, given in Eq.~(\ref{Eq.c}).

The optical dispersion relation causes the system decay time to depend
only on the coupling strength, $\Gamma/\sigma_{k}$, and the speed of
light, $c_{L}$.  For the atom dispersion relation, $t_{D}$ also
depends on the system frequency, $\omega_{0}$ and the mass of the
atom, $m$, and is given by
\begin{equation}
t_{D} = 2 \sqrt{\frac{\omega_{0} \hbar}{m \pi}}
\left(\frac{\sigma_{k}}{\Gamma} \right) \label{Eq.tD}
\end{equation}

These timescale considerations based on the differing dispersion relations,
reservoir model, and parameter regimes for atoms compared with optics
suggest that the Born-Markov approximation may not be generally valid in
describing output from practical atom lasers.  In fact, the optical
Born-Markov theory is not universally valid for optics, either.  In a
photonic band gap material the dispersion relation for the photons and the
radiation reservoir may be modified.  Bay {\it et al.} \cite{Bay97} have
discussed fluorescence into a radiation continuum in which a band gap with
dispersion relation near the band edge, $\omega_{k} = \omega_{e} + A
(k-k_{0})^{2}$, similar to the atomic dispersion relation, is present.
They find behaviour which cannot be described using the Born-Markov theory.

In the next section, we discuss equations of motion obtained from the
Born-Markov master equation for $\langle a^{\dag}(t) \rangle$ and
$\langle a^{\dag}(t) a (t) \rangle$.  We will compare these with those
obtained from the exact equations, Eq.  (\ref{Eq.adag}) and Eq.
(\ref{Eq.adaga}) and hence examine the regimes of validity of the
Born-Markov approximation.

\section{The validity of the Born-Markov approximation.}
\label{sec:equations}

We consider first the Born-Markov master equation, Eq.
(\ref{Eq.Markov}).  Using the relation,
\begin{equation}
\frac{d \langle \tilde{o} \rangle}{dt} = \mbox{Tr} \left[\tilde{o}
\frac{d \tilde{\rho}}{d t}
\right], \nonumber
\end{equation}
where $\tilde{o}$ is any system operator in the interaction picture, we
obtain the following equations of motion for $\langle a^{\dag} \rangle$ and
$\langle a^{\dag} a \rangle$ from the Markovian master equation, Eq.
(\ref{Eq.Markov}).
\begin{eqnarray}
\frac{d \langle a^{\dag}(t) \rangle}{dt} &=& \left[i \omega_{0} -
c^{*} \right] \langle a^{\dag} \rangle (t),\label{Eq.dadagdt}\\
\frac{d \langle a^{\dag}(t) a(t) \rangle}{dt} &=& -(c + c^{*})
\langle a^{\dag}(t) a(t) \rangle. \label{Eq.dadagadt}
\end{eqnarray}

We compare the exact equations, Eq.  (\ref{Eq.adag}) and Eq.
(\ref{Eq.adaga}) with Eq.  (\ref{Eq.dadagdt}) and Eq.
(\ref{Eq.dadagadt}) respectively.  The equations derived from the
Born-Markov master equations are equivalent to the exact equations
under the approximation that the term $\langle a^{\dag}(t-\tau)
\rangle e^{i \omega_{0} \tau} = \langle a^{\dag}(t) \rangle$ and
$\langle a^{\dag}(t) a(t-\tau) \rangle e^{-i \omega_{0} \tau} =
\langle a^{\dag}(t) a(t) \rangle$.  That is, if we ignore the effect
of the interaction on the system evolution.  Alternatively, the exact
and Born-Markov equations will agree if $f'(\tau)$ can be approximated
by a Dirac delta function.  For atoms, however, there is no limit in
which $f'(\tau)$ tends to a Dirac delta function.

When $f'(\tau)$ is not given by a Dirac delta function, the
approximation to replace $\langle a^{\dag}(t-\tau) \rangle e^{i
\omega_{0} \tau}$ by $\langle a^{\dag}(t) \rangle$ in Eq.
(\ref{Eq.adag}) will still be valid in some parameter regimes.  In
particular, if we consider the exact equation, Eq. (\ref{Eq.adaga}),
and the solutions obtained from the Heisenberg equations of motion, we
can see that the exact equation can be rewritten as
\begin{eqnarray}
\frac{d}{d t} \langle a^{\dag}(t) a(t) \rangle = - \langle
a^{\dag}(t) a(t) \rangle \times \int_{0}^{t} ~ f'(\tau) ~ \nonumber \\
 \times \frac{{\cal L}^{-1} \left \{ \frac{1}{s
 + {\cal L} \left\{f'(t) \right\}(s) } \right\} (t-\tau)}{{\cal
 L}^{-1} \left \{ \frac{1}{s + {\cal L} \left\{f'(t) \right\}(s) }
 \right\} (t)} ~d \tau ~+ \mbox{h.c}.  \label{Eq.newadaga}
\end{eqnarray}
From this form of the exact equation, the Born-Markov equation is
obtained by the assumption that $f'(\tau)$ decays rapidly on the
timescale on which the other (inverse Laplace transform) terms in the
integral change with $\tau$.  For parameters in which the Born-Markov
approximation is valid, we know that this ratio, given exactly from
Eq.  (\ref{Eq.LaplaceSoln}), is approximately exponential with a
timescale of order $t_{D}$.  This fact can be motivated by considering
Eq~(\ref{Eq.adagasoln}).  This equation shows that the number of atoms
in the cavity as a function of time is given by the square of the
absolute value of the inverse Laplace transform term, identical to
that found in Eq.~(\ref{Eq.newadaga}) above.  In the Born-Markov
regime we know the number of atoms in the cavity must decay
approximately exponentially on the timescale $t_{D}$.  Thus, for the
Born-Markov approximation to hold we require that the the timescale on
which $f'(t)$ decays must be small compared to $t_{D}$.

For the non-broadband case, we can define a timescale on which $f'(t)$
decays by the width at half maximum of the absolute value of the
reservoir correlation function $|f(\tau)|$ %
\begin{eqnarray}
t_{R} &=& \frac{\sqrt{15} m}{\hbar \sigma_{k}^{2}} =
\frac{\sqrt{15}}{\alpha}. \label{Eq.tr}
\end{eqnarray}
Here $m$ is the mass of the atoms and $\sigma_{k}$ is the width of the
Gaussian lasing mode in momentum space.  This timescale must be small
compared with the decay timescale, $t_{D}$ discussed earlier.  This
condition, by itself, would suggest that as the coupling becomes
increasingly broadband the Born-Markov approximations become increasingly
good.  However, this is not the case.  Although the function $f'(\tau)$
becomes infinite at zero in the broadband limit and therefore must have a
zero half width, $t_{R}$, there are significant contributions to the
integral in Eq.~(\ref{Eq.newadaga}) away from $\tau=0$.  Instead of the
half width of the reservoir correlation function, $t_{R}$, we are more
interested in the half width of the integral of $f'(\tau)$.  This timescale
is defined in terms of the reservoir correlation function, $f(\tau)$, and
the system frequency, $\omega_{0}$.

We define $t_{rs}$ such that
\begin{equation}
	\int_{0}^{t_{rs}} f'(\tau) d\tau = 1/2
	\int_{0}^{\infty} f'(\tau) d\tau.
	\label{Eq.trs}
\end{equation}

For broadband coupling we find that $t_{rs} = 1/\omega_{0}$ is
equivalent to the system timescale, $t_{s}$, defined earlier.  The
atomic dispersion relations make $t_{D}$ also depend on $\omega_{0}$,
Eq.~(\ref{Eq.tD}) which means that the condition $t_{rs} << t_{D}$, can
be written as
\begin{eqnarray}
\omega_{0}^{3/2} \sqrt{\frac{\hbar}{m \pi}}
\left(\frac{\sigma_{k}}{\Gamma} \right) >> 1. \label{Eq.inequality}
\end{eqnarray}

For broadband coupling, this timescale condition determines the
parameter regimes in which the Born-Markov approximation is valid.
The dependence on $\omega_{0}^{3/2}$ and on the mass in this condition
comes from the dependence of $t_{D}$ on $\mbox{m}$ and $\omega_{0}$.
In the equivalent model with optical dispersion relations, $t_{D}$
only depends on the strength of the coupling, given by
$\Gamma/\sigma_{k}$.

We now compare the results of the exact and Born-Markov equation.  We
initially consider Gaussian coupling with similar parameters to those
discussed in \cite{Moy97b}.  For atoms, $m \approx 5 \times 10^{-26}
\mbox{kg}$.  Atom traps in which BEC has been achieved have
frequencies of $\omega_{0} \approx 2 \pi \times 123
\mbox{s}^{-1}$\cite{MIT3}.  Values for the coupling strength, $\Gamma$
depend on the method used.  For Raman coupling, for instance, $\Gamma$
depends on the intensity of the lasers\cite{Moy97b}, so a range of
values down to zero can be achieved.  The value $\Gamma \approx 10^{6}
\mbox{s}^{-2}$ can be achieved with laser intensities similar to those
discussed in \cite{Moy97b}.  The width of the Gaussian we assume to be
determined by $\sigma_{k} \approx 10^{6} \mbox{m}^{-1}$ corresponding
to a ground state wavefunction of size $\approx 2 \mu \mbox{m}$.

In Fig \ref{Fig.1}a the solution for the expectation value of the
number of atoms in the system at time $t$ is plotted for the
parameters quoted above.  The exact solution is given by Eq.
(\ref{Eq.adagasoln}).  The solution derived from the Born-Markov
master equation is also shown.  This is given by %
\begin{eqnarray}
\langle a^{\dag}(t) a(t) \rangle &=& \langle a^{\dag}(0) a(0) \rangle
~ e^{ -(c+c^{*})t}.  \label{Eq.dadagasolnbm}
\end{eqnarray}
Figure \ref{Fig.1} demonstrates that the results for the number of
atoms using the Born-Markov approximations disagree with the exact
solutions in the case of Gaussian output coupling.  For these
parameters, $t_{R} \approx 2.0 \times 10^{-3} \mbox{s}$, $t_{s}
\approx 1.4 \times 10^{-3} \mbox{s}$ and $t_{D} \approx 5.0 \times
10^{-4} \mbox{s}$,  so that both the inequalities $t_{R} < t_{D}$
and $t_{s} < t_{D}$ fail.

\begin{figure}
\begin{center}
\epsfxsize=\columnwidth
\epsfbox{Fig1.eps}
\end{center}
\caption{A comparison between $\langle a^{\dag}(t) a(t) \rangle $
found using the Born-Markov master equation (dashed line) and the
exact solution (solid line).  Atom numbers are normalized so that
$\langle a^{\dag}(0) a(0) \rangle = 1$.  Parameters are (a) $\Gamma =
1 \times 10^{6} \mbox{s}^{-2}$, $\sigma_{k} = 10^{6} \mbox{m}^{-1}$
for Gaussian coupling and (b) $|\kappa(k_{0})|^{2} = 1.0/\sqrt{2 \pi}
\mbox{m}\mbox{s}^{-2}$ for broadband coupling.  Other parameters are
$m = 5 \times 10^{-26} \mbox{kg}$, $\omega_{0} = 2 \pi \times 123~
\mbox{s}^{-1}$.}
\label{Fig.1}
\end{figure}

We have presented results in terms of the Gaussian coupling discussed in
Sec.  (\ref{sec:MasterEqn}).  This reflects a possible realistic output
coupling method.  However from the inequality $t_{s} < t_{D}$ we see that
even in the broadband limit, atoms do not couple out of the system in a
Born-Markov manner for parameters which correspond to the ones we discuss
above.  Figure \ref{Fig.1}b compares the exact and Born-Markov solutions to
our model in the broadband limit.  In this limit the timescale, $t_{R}$ as
defined above tends to zero, thus the first inequality is satisfied.
However, the inequality, Eq.  (\ref{Eq.inequality}), fails.  This figure
demonstrates that even for infinitely broad atom coupling the Born-Markov
approximation may fail.  The parameters chosen are the same as those for
the Gaussian coupling, with a strength of coupling in the exact solutions
given by
\begin{equation}
|\kappa(k_{0})|^{2} = \frac{\Gamma}{\sqrt{2 \pi}~\sigma_{k}} \approx
0.4 \mbox{m s}^{-2},
\label{Eq.kappa0}
\end{equation}
In this case the Born-Markov solution again gives exponential decay.
However, the decay constant is now given as the broadband limit of Eq.
(\ref{Eq.c}).

Figure \ref{Fig.1} demonstrates that for our model of output coupling
from an atom laser, results for the number of atoms using the
Born-Markov approximations disagree qualitatively with the exact
solutions.  One of the effects of not being able to ignore the
back-action from the reservoir is that for the model we are
considering the number of atoms in the cavity does not tend to zero
for long times (Figure \ref{Fig.1}).  We discuss reasons for this
behaviour in the next section. If effects such as gravity and repulsive
interactions are included the cavity number will tend to zero.
However, even with these effects included the loss is not exponential
and the conclusion that the Born-Markov approximation fails to
describe the output coupling remains.

Here, we have demonstrated the failure of the Born-Markov
approximation by the use of the particular system variable, $\langle
a^{\dag} a \rangle$.  However, the problems with using the Born-Markov
approximations are not confined to this particular example.  For
instance, if the output from a BEC is described using a Born-Markov
master equation, the resulting long time energy spectrum is
Lorentzian.  However, if we avoid making the Born-Markov
approximations for atoms the exact spectrum may be non-Lorentzian for
some values of coupling strength, $\Gamma$, and frequency,
$\omega_{0}$ \cite{Moy97b}.

\section{Effect of gravity on the model} \label{sec:nonzero}

In this section we present a quasi-single particle model which allows
us to consider the effects of gravity on our output coupling.  Such a
model cannot be extended to show interesting effects, such as noise
suppression due to gain saturation if pumping is added to the model as
it does not give information about the general quantum statistics of
the output atom field.  However, it shows that the inclusion of
gravity causes the atom number to asymptote to zero.  It does this in
a non-exponential way, however, and therefore cannot be modelled by a
damping term based on the Born-Markov approximation.  Moreover, we
demonstrate that the short time behaviour predicted by the models with
gravity agree with the exact solutions we have presented earlier which
ignore the effects of gravity.

The previous section demonstrated qualitative differences between the
exact solution of our model and the solution which uses the
Born-Markov approximation.  In fact, the exact solution of the model
has a stable, nondispersing state which means that not all of the
atoms leave the cavity, whereas the approximate solution shows an
exponential decay of atoms from the cavity.  The presence of the
stable state is due to a coherence between the atoms in the cavity and
the output modes.  The Born-Markov approximation ignores any coherence
between the cavity mode and the output modes, and therefore cannot
describe any model which produces such features.  We now show that
such features would be destroyed by gravity.

For coherent dynamics without atom-atom interactions, the
multiparticle evolution is identical to the evolution of a single
particle \cite{Kasevich}.  The gravitational term makes it impossible
to derive analytical results as in section ~\ref{sec:Output}, so we
solved the corresponding time-dependent Schr\"{o}dinger equation
numerically.  This was done in the position basis rather than the
momentum basis for convenience.  The Hamiltonian for our system with
the inclusion of a gravitational potential is
\begin{eqnarray}
H &=& H_{s} + H_{r} + H_{sr} + H_{g}, \label{Eq.H2} \\
H_{s} &=& \hbar \omega_{0} |\psi_{a} \rangle \langle \psi_{a} |
,\label{Eq.Hs2} \\
H_{r} &=& \frac{\hat{P}^{2}}{2 M}, \label{Eq.Hr2} \\
H_{sr} &=& \int ~ dx~ \left( g^{*}(x) |\psi_{a} \rangle \langle x | ~+~
g(x) |x \rangle \langle \psi_{a} | \right), \label{Eq.Hsr2} \\
H_{g} &=& M g \hat{x} \sin(\theta), \label{Eq.Hgrav}
\end{eqnarray}
where the coupling function, $g(x)$ is related to $\kappa(k)$ by a
Fourier transform,
\begin{eqnarray}
g(x) &=& \int~ dk \frac{i \hbar \kappa^{*}(k) ~ e^{i k x}}{\sqrt{2
\pi}}. \label{Eq.gx}
\end{eqnarray}

$\hat{P}$ is the momentum operator.  With the term $H_{g} = 0$, this
model is equivalent to a single particle version of our earlier many
particle description, Eq.  (\ref{Eq.Heff}), and produces a time
dependance for the probability of finding an atom in the cavity which
is identical to $\langle a^{\dag}(t) a(t) \rangle$, found previously.
These show a long time steady state in the number of atoms in the
cavity which does not tend towards zero.  With the inclusion of
gravity ($H_{g} \neq 0$), however, the number of atoms decays to zero
in a non-exponential manner.  This behaviour is shown in Figure
\ref{Fig.2}.
\begin{figure}
\begin{center}
\epsfxsize=\columnwidth
\epsfbox{Fig2.eps}
\end{center}
\caption{The effect of introducing gravity into the model on the
number of atoms in the cavity $\langle a^{\dag}(t) a(t) \rangle$.
Atom numbers are normalized so that $\langle a^{\dag}(0) a(0) \rangle
= 1$.  The solid line shows the figure with gravity turned off and
agrees with the results presented in Fig.  \ref{Fig.1}a.  The dashed
line includes gravity, $g ~\mbox{sin} (\theta)$, $g = 9.8 \mbox{m
s}^{-1}, \theta = \pi/20$.}
\label{Fig.2}
\end{figure}

In figure \ref{Fig.2} we can see collapses and revivals in the number
of atoms in the cavity.  This is due to the evolving phase
relationship between the atoms in the cavity and the atoms which have
been coupled into the output field modes.  There is no version of the
Born-Markov approximation that can describe behaviour such as this, as
such an approximation requires that there be no entanglement between
the lasing mode and the output modes.

The presence of gravity changes the long time behaviour of our model
so that the number of atoms asymptotes to zero, while the short time
behaviour remains the same.  Other effects may also lead to the long
time decay of atom number.  Repulsive interactions, for instance,
would be expected to destroy the thin stable structure in position
space which leads to the long time non-zero population of the cavity
mode.  The effect of repulsive interactions can be included in this
model by including in Eq.  (\ref{Eq.H2}) a nonlinear Hamiltonian term
given by
\begin{eqnarray}
H_{i} &=& N U_{0} |\psi(x)|^{2}. \label{Eq.Hrep}
\end{eqnarray}
Including such interactions into our model produces a Gross-Pitaevskii
type equation \cite{GPEqn}, where $N$ is the number of atoms in the
system and $U_{0}$ an interaction strength.  We find that introducing
such an interaction term does cause the atom number to go below the
nonzero steady state predicted in the interaction free model.

\section{A non Markovian master equation} \label{sec:nonmarkov}

In the previous two sections we have shown that the standard master
equation does not correctly describe the dynamics of our model for
particular parameters.  However a master equation, in terms of only
the system variables, would be an important tool for describing an
atom laser.  We now consider whether a non-Markovian master equation
can give a correct description of the atom laser.  To do this, we
continue to make the Born approximation, but do not make a Markov
approximation.

  The master equation with the Born approximation only is given in Eq.
  (\ref{Eq.Born}).  Again, we check the validity of the Born
  approximation by comparing the results obtained from this Born
  master equation with the exact solutions.  We begin by considering
  the resulting equation for the expectation value $\langle a^{\dag}
  \rangle$, %
\begin{eqnarray}
\frac{d \langle a^{\dag}(t) \rangle}{dt} &=& i \omega_{0} \langle a^{\dag}
\rangle (t) - \int_{0}^{t} dt' \langle a^{\dag}(t') \rangle \times
\nonumber\\
&& \int |\kappa(k)|^{2} e^{i \omega_{k}(t-t')}
dk,\label{Eq.dadagdtnm}
\end{eqnarray}

Eq.  (\ref{Eq.dadagdtnm}) is the same as that obtained through the
full system plus reservoir equations given in Eq.  (\ref{Eq.adag}).
This can be seen by making the transformation $\tau = t-t'$ in Eq.
(\ref{Eq.dadagdtnm}) to obtain the alternative form, Eq.
(\ref{Eq.adag}).  Despite this success, the density operator equation
with the Born approximation, Eq.  (\ref{Eq.Born}) is not correct.  In
particular, the Born approximation does not give correct values for
higher order expectation values, such as $\langle a^{\dag} a \rangle$.

The equation derived from the non-Markovian master equation, Eq.
(\ref{Eq.Born}) for the number operator expectation value is
\begin{eqnarray}
\frac{d \langle a^{\dag}(t) a(t) \rangle}{dt} &=& - \int_{0}^{t}
d\tau \langle a^{\dag} a \rangle (t-\tau) \times \nonumber\\
&& \int |\kappa (k)|^{2}
e^{i (\omega_{0} - \omega_{k}) \tau} dk + \mbox{h.c},
\label{Eq.dadagadtnm}
\end{eqnarray}
with solution
\begin{eqnarray}
\lefteqn{\langle a^{\dag}(t) a(t)\rangle = \langle a^{\dag}(0) a(0)
\rangle
\times } \hspace{1cm} \nonumber\\
& &  {\cal
L}^{-1} \left\{ \frac{1}{s + {\cal L} \left\{f'(t) \right\}(s) + {\cal
L} \left\{f'(t)^{*} \right\}(s)} \right\}(t).  \label{Eq.adagasolnnm}
\end{eqnarray}

These do not agree with the corresponding exact equations given in Eq.
(\ref{Eq.adaga}) and Eq.  (\ref{Eq.adagasoln}), as is shown in Fig.
\ref{Fig.3}.
\begin{figure}
\begin{center}
\epsfxsize=\columnwidth
\epsfbox{Fig3.eps}
\end{center}
\caption{A comparison between $\langle a^{\dag}(t) a(t) \rangle $
found using the Born (non-Markovian) master equation (dashed line) and
the exact solution (solid line) respectively in the broadband regime.
Atom numbers are normalized so that $\langle a^{\dag}(0) a(0) \rangle
= 1$.  Parameters are the same as in Fig.  \ref{Fig.1}b.}
\label{Fig.3}
\end{figure}
This figure compares the exact results for $\langle
a^{\dag}(t) a(t) \rangle$ with the solution in the Born approximation
only, Eq.  (\ref{Eq.adagasolnnm}).  The parameter values chosen are
the same as those discussed in section \ref{sec:equations} and the
comparison is made for broadband coupling.  The reason for the failure
of the non-Markovian master equation is that where the correct
equations lead to the term $\langle a^{\dag}(t) e^{-i \omega_{0} \tau}
a(t-\tau) \rangle$, the non-Markovian master equation with the Born
approximation leads to $\langle a^{\dag}(t-\tau) a(t-\tau) \rangle$.
These two terms are equal if we consider only the free evolution of
the system, and ignore the interaction term.  That is, to zeroth order
in $|\kappa(k)|^{2}$, $\langle a^{\dag} (t-\tau) a(t-\tau) \rangle
\approx \langle a^{\dag}(t) a(t-\tau) \rangle (t) e^{-i \omega_{0}t}$.
However, there are interaction terms between the system and reservoir.
Due to the interaction, terms obtained using the Born approximation
are not equal to the exact results.  To reproduce equations for the
correct system dynamics we cannot ignore the correlations which arise
between the system and reservoir.

In optics the Born-Markov approximation is a useful approximation in
almost all practical parameter regimes.  In fact, in optics the value
of $\omega_{0}$ is sufficiently large that, if one assumes a linear
dispersion relation the reservoir correlation can be well
approximatied by a Dirac delta function.  This is not the case for
atoms.

The non-Markovian master equation, Eq.  (\ref{Eq.Born}), gives correct
results for single operator expectation values.  Nevertheless, the
Born approximation neglects higher orders in the interaction term,
$|\kappa(k)|^{2}$, thus ignoring reservoir evolution.  This leads to
incorrect results for higher order expectation values, such as
$\langle a^{\dag}(t) a(t) \rangle$.

\section{Conclusion} \label{sec:Conclusion}

We have discussed the use of master equations and other density
operator equations for atom lasers.  We find that the Born-Markov
master equation, as is commonly used in optics, is not valid in
parameter regimes in which atom lasers may work.  The output modes
correlation function for photons is well approximated by a delta
function in the broadband limit, and for typical optical parameters.
In comparison, this function falls off as $1/\sqrt{\tau}$ for atoms
because of the atom dispersion relations.

The Born-Markov approximations must be made self consistently.  In
regimes where the Born-Markov approximations fail, the system can be
solved in the Heisenberg picture, treating the output modes fully
\cite{Hope97,Moy97b}.  For broadband coupling, the parameter regimes
in which the Born-Markov approximation is valid is given in Eq.
(\ref{Eq.inequality}).  Another possibility for describing such
systems involves the use of quantum trajectories.  These have been
found to be useful for other systems in which no Born-Markov master
equation exists\cite{Bay97}.

The breakdown of the Born and Markov approximations means that new
theoretical methods will be required to understand the atom laser in
some regimes.  However, it also opens up the possibility of finding
new properties of atom lasers significantly different from those found
in the optical laser.

redpajama_set_name: RedPajamaArXiv


The firm's primary areas of work are described below and comprise of company and commercial, property, employment, dispute resolution, consultancy and private client, supplemented by niche units specialising in areas such as construction law, planning and environmental and insolvency. All departments within the firm work closely to ensure a client receives expert advice to achieve commercial objectives.
While the firm is an independent Northern Ireland owned, and managed practice, it has an alliance with Matheson Solicitors in the Republic of Ireland to provide an all-island service; it is a member of LawNet, an association of independent UK law firms, and is a member of the International Lawyers Network. This network gives the firm's lawyers access to commercial law firms in 66 countries across six continents.
Planning and environment: Cleaver Fulton Rankin is at the forefront of this specialised area of law, with a wealth of experience in providing practical legal advice throughout the planning application process, appeals and inquiries. The firm also provides environmental and planning due diligence for property transactions and company acquisitions.
Above material supplied by Cleaver Fulton Rankin.


〔1〕Michel Oksenberg, "Congress, Executive-Legislative Relations, and American China Policy", in Edmund Muskie, et al, eds., The President, Congress and Foreign Policy (N. Y.: University Press of America, 1986), p.209.
〔2〕U. S. Congress. Senate.Committee on Foreign Relations.Hearings on the Taiwan Relations Act. 96th Congress, 1st Session, p.397.
〔3〕Senator Glenn, U. S. Congress. Senate. Committee on Foreign Relations. Subcommittee on East Asia and Pacific Affairs.Hearings on the Oversight of TRA. 96th Congress, 1st Session, p.2.
〔4〕Senator Church, note 2, p.1.
〔8〕Congressional Record, March 5, 1979, E842.
onEast Asia and Pacific Affairs. Hearings on the Oversight of TRA. 96th Congress, 2nd Session, p.2.
〔12〕"Implementation of the TRA: the First Year", A Staff Report to the Committee on Foreign Relations, U. S. Serate, p.3.
〔18〕U. S. Congress. Senate. Committee on Foreign Relations.Subcommittee on East Asia and Pacific Affairs. Hearings on US-China Relations. 98th Congress, 2nd Session, p.3.
〔19〕Wallop, Congressional Record, October 2, 1986, S 14555.
〔20〕David Maybew, Congress: The Electoral Connection (New Haven: Yale University Press, 1979), p.5.
〔22〕McDonald, Congressional Record, April 10, 1979, E1693.
〔23〕Ashbrook, Congressional Record, May 21, 1980, H3919.
〔24〕Rudd, Congressional Record, June 12, 1980, H4917.
〔25〕Congressional Quarterly's Weekly Report, 1986, p.968.
〔26〕Congressional Record, August 15, 1986, E2920.
〔27〕Congressional Record, October 2, 1986, H9008.
〔28〕Congressional Record, November 21, 1985, S16195.
〔29〕Congressional Record, November 13, 1985, S15260.
〔31〕China: U. S. Policy since 1945 (Washington, DC: Congressional Quarterly Inc., 1980), p.54.
〔32〕Congressional Record, July 23, 1980, S9575-9576.
〔33〕Congressional Record, August 20, 1980. H7547.


BAKU, July 17 (Reuters) - Azerbaijan's top gold producer, Anglo Asian Mining, said on Tuesday it had increased its gold production by 43 percent year-on-year in the first half of 2018 to 33,255 ounces from 23,218 ounces in the same period last year.
The London-listed firm said its total production of gold-equivalent ounces (GEO) rose by 22 percent to 37,349 GEO from 30,561 GEO.
Gold is produced at Gedabek and other Azeri mines under a production venture in which Anglo Asian holds 51 percent and the state the remainder.
The company said copper production declined to 587 tonnes from 1,322 tonnes in January-June 2017, and silver output declined to 84,785 ounces from 85,087 ounces.
Anglo Asian said in February its gold output would rise to 64,000-70,000 ounces in 2018, from 59,617 ounces last year, and it expected its total metal production to increase to 78,000-84,000 GEO in 2018, from 71,461 GEO last year.
Anglo Asian began production at Gedabek, the bigger of two mines it operates, in July 2009 with plans to extract 22 tonnes of gold. It eventually plans to develop seven mines in western Azerbaijan, including Gedabek, with estimated gold reserves of 430 tonnes in total.
The company started a significant exploration programme at the end of 2016 after making a new gold discovery at Ugur, 3 km (2 miles) from its Gedabek processing facilities.
Gold bullion sales in the first half of 2018 totalled 25,778 ounces at an average price of $1,319 per ounce, up from 15,689 ounces sold in January-June 2017 at an average of $1,238 per ounce.
Net debt totalled $2.9 million by June 30, down from $10.4 million at the end of the first quarter of 2018 and $18.3 million at the end of 2017.


text:
Home » The Partners » Dr Alexander Doney
Dr Alexander Doney
Senior Lecturer and Honorary Consultant
Dr Alexander Doney is a Senior Lecturer at the University of Dundee and Honorary Consultant Physician at Ninewells Hospital and Medical School, Dundee.
He is currently working within the Medicines Monitoring Unit (MEMO) with Professor Tom MacDonald and Dr Isla Mackenzie. He also directs the development of the eClinical Phenome within the Health Informatic Centre, Dundee which manages large bioresources in Tayside linked to electronic Medical Records (EMR).
He initially studied pure genetics and went on to a PhD in molecular genetic epidemiology of cardiovascular disease and dyslipidaemia but subsequently trained in clinical medicine specialising in Clinical Pharmacology and General Internal Medicine. His clinical interests led him towards Acute Stroke Medicine and neurovascular health and he led the Acute Stroke Programme in Ninewells Hospital for a number of years.
Research interests have been in genomics combined with health informatics, information technology and medical image processing as a means of providing a wide range of clinical phenotypes for high quality studies in investigating genetic determinants of disease risk and therapeutic response with the ultimate aim of translation of this knowledge into clinical health benefits.
He has played a central role in inaugurating and expanding large EMR-linked bioresources at the University of Dundee including GoDARTS, GoSHARE, BIOSTAT CHF and INSPIRE, which aims to deliver genomically informed Precision Medicine Globally.
ORCID page:- https://orcid.org/0000-0002-6210-5620

redpajama_set_name: RedPajamaCommonCrawl


text:
Sir Thomas Henry Hall Caine CH KBE (1853 – 1931) – British writer, poet, critic
"The Manxland Bard… a genuius … a powerful pen" ("Dundee Courrier, 25th December 1901)
Portrait of Hall Caine by R.E. Morrison
With thanks to Stuart Allen of The Runcorn & District Historical Society for telling me about Hall Caine. Although I knew about "King Albert's Book", I did not know that Hall Caine was the Editor.
Known later as Hall Caine, Thomas Henry Hall Caine was born on 14th May 1853 in Runcorn, Cheshire, UK. His parents were John Caine, from the Isle of Man, and his wife, Sarah, nee Hall (from Cumberland – now called Cumbria). Hall's siblings were John (b. 1856), William Ralph (b.1865), who also became a writer, and Elizabeth Ann (1869-1914), who became a famous actress using the name Lilly Hall Caine. Caine is a Celtic name and Hall, his mother's maiden name, is Norse.
Hall was educated at Hope Street School in Liverpool, where he was taught by George Gill, who was head of a school book publisher, and encouraged Hall's talent for writing. Hall spent time with his Father's family on the Isle of Man when he was growing up. Leaving school at the age of fifteen, Hall was apprenticed to an artichect and trained as an architectural draughtsman. He also contributed articles to local newspapers and magazines and gave lectures around Merseyside to various societies. The subject of one of those lectures - Dante Gabriel Rossetti - invited Hall to go to London. Hall lived with the old man right up until Rossetti died in 1882 and the two became great friends.
Under Rossetti's influence Hall developed his literary talents and eventually became a novelist, playwright, short story writer, poet and critic. Hall worked for the "Academy" (a review of literature and general topics published in London from 1869 to 1902) and also contributed atricles to the "Athenaum" (a literary magazine published in London, from 1828 to 1921). His novel "The Eternal City" was the first novel to sell over a million copies worldwide. His plays were performed to wide acclaim both in America and Britain and Hall became very famous indeed.
In 1886, Hall married Mary Chandler and their sons were Gordon Ralph (1884 – 1962), who became a writer, publisher and Conservative politician and Derwent (b. 1891).
Hall became a resident of The Isle of Man in 1895 and sat from 1901 to 1908 in the Manx House of Keys, the lower house of Tynwald, the parliament of the Isle of Man (the other branch being the Legislative Council). He was elected President of the Manx National Reform League in 1903 and became Chairman of the Keys' Committee that prepared the 1907 petition for constitutional reform. In 1929, Hall was granted the Freedom of the Borough of Douglas, Isle of Man. He visited Russia in 1892 on behalf of the persecuted Jews and in 1895, travelled in north America and Canada, where he represented the Society of Authors.
During the First World War (1914 – 1919), he wrote many patriotic articles and edited "King Albert's Book", the proceeds of which went to help Belgian refugees.
Cover of "King Albert's Book"
In 1917, Hall was created an Officer of the Order of Leopold by King Albert I of Belgium. He cancelled his literary engagements in America in order to devote his time and energy to the British war effort. At the recommendation of then Prime Minister Lloyd George, King George V made him a Knight of the British Empire (KBE) in 1918, for services as an Allied propagandist in the United States. In 1922, Hall Caine was made a Companion of Honour (CH).
Hall died on 311st August 1931 on the Isle of Man.
Sources: Find my Past, Free BMD, "Hall Caine: The Man and the Novelist" by C. Fred Kenyon (Greening & Co. Ltd., London, 1901)
http://www.isle-of-man.com/manxnotebook/people/writers/thc.htm
https://shakespeare.emory.edu/lily-hall-caine/
https://en.wikipedia.org/wiki/Gordon_Hall_Caine
http://www.victorianweb.org/authors/caine/bio1.html
Robert Underwood Johnson (1853-1937) - American po...
Sir Thomas Henry Hall Caine CH KBE (1853 – 1931) –...
Ralph Hale Mottram (1883–1971) – WW1 soldier poet...
Arthur Hugh Sidgwick (1882 - 1917) – British soldi...


text:
Memphis protest slams immigration policy
The gathering of many nationalities met at the historic church where other civil rights marches began decades ago.
Memphis protest slams immigration policy The gathering of many nationalities met at the historic church where other civil rights marches began decades ago. Check out this story on commercialappeal.com: http://memne.ws/2jZYAXI
Linda A. Moore, USA TODAY NETWORK – Tennessee Published 5:24 p.m. CT Feb. 1, 2017 | Updated 9:24 p.m. CT Feb. 1, 2017
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries — Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen — for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days.(Photo: Nikki Boertman, The Commercial Appeal)Buy Photo
A crowd estimated at close to 2,000 people marched from Clayborn Temple in Memphis under cloudy skies Wednesday evening in protest of President Trump's executive order banning immigrants from certain countries for as long as 120 days.
It took about 20 minutes for the exuberant marchers to reach the National Civil Rights Museum, where many lit candles as part of a prayer vigil, which opened with the Maghreb, one of the five Muslim evening prayers. Other prayers followed in English and in Spanish.
Stationed in front of the spot where Martin Luther King Jr. was assassinated, like the protesters, a diverse lineup of speaker derided Trump, to the delight of the cheering crowd, while championing the ideals upon which the country was founded.
Sarah Ahmed, center, and Ayat Ayesh, center left, join other concerned citizens as they participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries Ñ Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen Ñ for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Memphian Mohamed Aldafari, who immigrated to the United States 31 years ago from Yemen, participates in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries Ñ Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen Ñ for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Concerned citizens participate in a march organized by the Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant and Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. Nikki Boertman / The Commercial Appeal
Citizens participated in a Downtown march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition in response to Trump's executive order halting immigration from seven majority Muslim countries for 90 days. Nikki Boertman/The Commercial Appeal File
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries — Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen — for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began at ClaybornTemple and ended at theNational Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries — Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen — for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began atClaybornTemple and ended at theNational Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries — Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen — for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began atClaybornTemple and ended at theNational Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries Ñ Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen Ñ for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Concerned citizens participate in a march in Memphis last month organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The march was in response to President Donald Trump's controversial immigration executive order. Nikki Boertman/The Commercial Appeal
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. Nikki Boertman/The Commercial Appeal
Concerned citizens participated in a January march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. Nikki Boertman/The Commercial Appeal File
ClaybornTemple and ended at theNational Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries Ñ Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen Ñ for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. Nikki Boertman, The Commercial Appeal
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant and Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. Nikki Boertman/The Commercial Appeal
"America will not become great by spreading hatred or by spreading fear. America will not become great by cutting itself off from its neighbors. America will not raise the torch of liberty by breaking up families and denying treatment to the sick," said Yasir Qadhi, with the Islamic Center of Memphis. "On the day you leave office, we will make America great again."
Jasmin Magana, a native of Los Angeles, whose parents are from Mexico, was not surprised that Trump was elected.
"I am no longer accepting things I cannot change, I'm changing things I cannot accept," she said, quoting Angela Davis, an icon of the black power movement.
Born in Yemen, 12-year-old Abdallah Altareb, has been in the United States six years and lives here with his parents, eight sisters and other relatives.
Abdallah waved an American flag as he marched, and with family still in Yemen, he finds Trumps ban alarming.
"It's not America without diversity," he said. "You can't judge a book by its cover so you can't judge every Muslim by one group of Muslims."
Often thought of as a young people's movement, many in the crowd, like 59-year-old Rodney Jeffrey, were of another generation. He referenced Martin Niemoller's poem about the rise of Nazi's in Germany that begins with "First they came came for the Socialist" as his reason for participating.
"It's just one of those things I had to do," he said.
The protesters, many carrying United States flags of all sizes or protest signs, chanted "This is what democracy looks like," and some carried signs proclaiming "If you aren't a native then you're an immigrant."
"I appreciate the diversity of people who came here to show their support to refugees around the world,' said Souad Habibi, 22, who came to this country from Syria in 2011.
The city had first said the protest wouldn't be granted a permit because an application was not filed 14 days in advance of the march. But city leaders agreed Tuesday to grant an exception to the permitting process after receiving a written notice of the march.
Concerned citizens participate in a march organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. The event began at Clayborn Temple and ended at the National Civil Rights Museum. The march was in response to Trump's controversial order signed Friday halting immigration from seven majority-Muslim countries — Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen — for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days. (Photo: Nikki Boertman, The Commercial Appeal)
Protesters had already affirmed that, with our without a permit, they intended to march and would condemn the city on social media for denying their First Amendment rights over a technicality.
The march was organized by Refugee Empowerment Program, Comunidades Unidas en Una Voz and the Tennessee Immigrant & Refugee Rights Coalition. It was in response to Trump's controversial executive order signed Friday halting immigration from seven majority-Muslim countries — Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen — for 90 days, barring entrance to the U.S. from Syria indefinitely, and suspending all refugee applications for 120 days.
Reach Linda Moore at linda.moore@commercialappeal.com, on Facebook at Linda Anita Moore and on Twitter at @LindaAnita.
Read or Share this story: http://memne.ws/2jZYAXI
Miss Tennessee contest coming to Memphis


Adfactor and our publishers are very very flexible. Please do not hesitate to contact us for alternative options. Please submit your creative two working days in advance of the campaign starting date. Traffic e-mail: tr@adfactor.nl.
The specsheet about the Adfactor Autonative can be found here. Please provide assets, mock-ups are optionable.


text:
DaniMer Scientific Releases Breakthrough Biobased Label Adhesive for PET Industry
Myriant announced today that its high-performing bio-succinic acid was selected as key renewable building block chemical in a bio-based label adhesive announced today by DaniMer Scientific. The adhesive, designated DaniMer 92721, was created to eliminate the problems associated with polyethylene terephthalate (PET) container recycling. To develop the adhesive, DaniMer Scientific utilized renewable building blocks from two world-class suppliers: Myriant and DuPont Tate & Lyle. DaniMer combines Myriant's bio-succinic acid with Susterra® Propanediol from DuPont Tate & Lyle into a proprietary, patent pending resin that enables the breakthrough properties of the new 92721 label adhesive.
DaniMer's new label adhesive is designed to meet the demands of PET container packaging recycling streams and has a renewable content higher than 50%. The adhesive dissolves completely in PET flake caustic wash recycling operations without clumps or "stickies." This new bio-based formulation is competitively priced with block co-polymers, with pricing not linked to crude oil derivatives. In addition, rheology of DaniMer 92721 enables the adhesive to operate in existing adhesive systems at temperatures under 325F. Results from industry standard protocol testing show the product enables zero contamination in PET recycle streams.
"DaniMer is pleased to announce this advancement towards eliminating problems with PET recycling in streams," said Daniel Carraway, Founder and CEO of DaniMer Scientific. "We have been actively engaged with our target market in order to understand what our consumers need, and the issue of PET contamination was a recurring and significant matter that needed to be resolved. Of course, we chose Myriant and DuPont as our suppliers for this project due to their world-class capabilities, knowledge and extensive background with the development of innovative, renewable building blocks."
Succinic acid is traditionally produced from petroleum and is used in a wide variety of applications, including the production of polymers, fibers, surfactants, detergents and flavors. The global market for succinic acid is currently estimated at approximately $7.5 billion annually.
Myriant's proprietary technology platform reduces the environmental impact from petroleum-based chemical manufacturing by making chemicals, such as bio-succinic acid, from renewable, non-food feedstocks. Myriant's high purity bio-succinic acid is a drop-in replacement for petroleum-based succinic acid. Customer analysis of Myriant's bio-succinic acid validates that it is chemically identical to petroleum-based succinic acid, while also being more environmentally friendly and cost-competitive without government subsidies.
"The recycling industry has long been challenged to find ways to overcome the issues and complications associated with recycling PET containers," said Stephen J. Gatto, Chairman and CEO of Myriant. "Myriant is proud that its high-performing bio-succinic acid is part of this innovative solution to a decades-old challenge. DaniMer's new adhesive is a great example of the innovation and market opportunities available with bio-based chemicals."
"Susterra® propanediol is certified 100% bio-based by the US Department of Agriculture, making it attractive for companies seeking to add renewable content to their products. We are excited to be selected by DaniMer as a key ingredient for this new bio-based adhesive technology," commented Pete Colombo, account executive for DuPont Tate & Lyle Bio Products.
DaniMer Scientific will be providing details and announcing the commercial launch of their new label adhesive at the Nova-Pack 2012 Conference in Arlington, Virginia. The product exhibition will be on June 7-8 at West Arlington Gateway Hotel, just outside of Washington, D.C. The exhibition gives packaging suppliers and users a chance to display products they designed that overcome challenges faced with the PET recycling industry. For more information and to register for the Nova-Pack 2012, please visit http://novapackconference.com.
About Myriant Corporation
Myriant utilizes its proprietary technology platform to develop innovative, performance-based, renewable chemicals utilizing low-cost sugars. In December 2010, Myriant broke ground on its flagship 30 million pound commercial bio-succinic acid facility in Lake Providence, Louisiana and anticipates beginning commercial production in early 2013. The company's D(-) lactic acid started production at commercial scale in June 2008 for use in polylactic acid. Myriant has agreements with ThyssenKrupp Uhde GmbH for engineering, Davy Process Technology for the integration of Myriant's bio-succinic acid process with the Davy butanediol process for the production of bio-based butanediol, and PTT Chemical for the commercialization of Myriant's technology in Southeast Asia. Myriant is headquartered in Quincy, Massachusetts. Details are available at www.myriant.com.
About DuPont Tate & Lyle Bioproducts
DuPont Tate & Lyle Bio Products is a joint venture between DuPont, a global science company, and Tate & Lyle, a world-leading renewable food and industrial ingredients company. DuPont Tate & Lyle Bio Products provides natural and renewably sourced ingredients that do not compromise product performance. For more information on the company's products, visit www.duponttateandlyle.com.
About DaniMer Scientific
DaniMer Scientific, an international corporation, believes people's lives can be improved and the environment can be better protected by utilizing renewable resources to manufacture articles that are intended for short-term use. One of the company's goals is to reduce dependence on petroleum, enabling people and communities to benefit from environmentally-friendly products. For more information about DaniMer Scientific, please visit: www.danimer.com.
About Nova-Pack 2012
Nova-Pack 2012 Conference on PET Containers for Food and Beverages is mainly focused on helping packing suppliers design products that meet the demands of the PET container recycling stream. Exhibition participants have faced the challenge of avoiding contamination to the PET recycle stream when trying to break down the adhesive. Another issue contributing to the impact of the PET market is the development of a cost-effective route. The Nova-Pack 2012 Conference will be providing display tables for the major industry stakeholders to display their products that meet the extensive needs of the industry. http://novapackconference.com/.
Source: http://www.myriant.com
adhesives Company News
Etiquetas: adhesives Company News


text:
Line of Duty Act; under coverage individuals employed as fire company personnel for National Guard. (SB424)
Sen. Frank Ruff (R-Clarksville) with support from co-patrons Del. Tommy Wright (R-Victoria), Sen. John Edwards (D-Roanoke), and Sen. Louise Lucas (D-Portsmouth)
Line of Duty Act; fire company personnel of the Virginia National Guard and the Virginia Air National Guard. Includes under the coverage of the Line of Duty Act individuals employed as fire company personnel for the Virginia Air National Guard or the Virginia National Guard's Fort Pickett Reserve. Read the Bill »
Bill Has Passed
01/11/2012 Prefiled and ordered printed; offered 01/11/12
01/11/2012 Prefiled and ordered printed; offered 01/11/12 12101796D
01/11/2012 Referred to Committee for Courts of Justice
01/18/2012 Reported from Courts of Justice with substitute (15-Y 0-N) (see vote tally)
01/18/2012 Rereferred to Finance
01/18/2012 Committee substitute printed 12104363D-S1
01/20/2012 Impact statement from DPB (SB424S1)
01/20/2012 Incorporates SB99
01/24/2012 Reported from Finance (15-Y 0-N) (see vote tally)
01/25/2012 Constitutional reading dispensed (40-Y 0-N) (see vote tally)
01/26/2012 Reading of substitute waived
01/26/2012 Committee substitute agreed to 12104363D-S1
01/26/2012 Engrossed by Senate - committee substitute SB424S1
01/27/2012 Read third time and passed Senate (39-Y 0-N) (see vote tally)
02/13/2012 Referred to Committee on Appropriations
02/17/2012 Assigned App. sub: Compensation and Retirement
02/28/2012 Subcommittee recommends reporting (8-Y 0-N)
02/29/2012 Reported from Appropriations (22-Y 0-N) (see vote tally)
03/05/2012 Passed House BLOCK VOTE (96-Y 0-N)
03/05/2012 VOTE: BLOCK VOTE PASSAGE (96-Y 0-N) (see vote tally)
03/05/2012 Reconsideration of House passage agreed to by House
03/05/2012 VOTE: BLOCK VOTE PASSAGE #2 (97-Y 0-N) (see vote tally)
03/08/2012 Bill text as passed Senate and House (SB424ER)
03/08/2012 Impact statement from DPB (SB424ER)
04/04/2012 G Approved by Governor-Chapter 573 (effective 7/1/12)
04/04/2012 G Acts of Assembly Chapter text (CHAP0573)
This bill mentions Richmond.
This bill was discussed on the floor of the General Assembly. Below is all of the video that we have of that discussion, 1 clip in all, totaling 40 seconds.
Duplicate Bills
The following bills are identical to this one: SB10.
No Richmond Sunlight visitors voted on this bill while voting was open.
SB508: Tuition, in-state; members of Virginia National Guard.
HB644: In-state tuition; eligibility of members of Virginia National Guard residing in state.
HB820: National Guard; eligibility for service.
SB527: Virginia Personnel Act; hiring preference in state employment for members of National Guard.
HB384: Virginia Personnel Act; hiring preference in state employment for members of National Guard, etc.


text:
Blender Announces the Animation 2025 Project
The initiative aims to gradually improve the software's animation tools over a three-year period.
The Blender Foundation has recently unveiled Animation 2025 – a new project with the goal of gradually improving and enhancing Blender's rigging and animation tools over a three-year period. Aiming to provide animators with a robust toolset, the initiative is set to start on January 1, 2023, and run until the end of 2025, with the newly added features being merged into the software along the way.
During the first year of the initiative, the company plans to build prototypes of the new systems, prioritizing the Rig Explainer, 3D Onion Skinning, and Rigging/Constraint Nodes, which would then "be implemented in a way that compartmentalizes those aspects that depend on the current implementation of Armatures, Actions, etc."
In 2024, the team aims to overhaul Blender's Armatures, Actions, and Objects in accordance with the 2023 data, planning to allow multi-object animation, better data handling, layering of animation data, and more. As for the Animation 2025 project's final year, the developers didn't share any concrete plans regarding it, only intending to add "more usable cool stuff."
"These plans are BIG, and we are excited to have the opportunity to work on them," states the team. "Three years is likely not enough time to implement all of it, but with a team working on this for three years it's certain to have a big impact on Blender."
You can learn more about the initiative by clicking this link. Also, don't forget to join our Reddit page and our Telegram channel, follow us on Instagram and Twitter, where we share breakdowns, the latest news, awesome artworks, and more.


text:
Concert Review: Beach Slang
Posted on December 23, 2017 July 20, 2022 by Jonathan
God, I'm almost 30. I've been suffering from some sort of existential, crotchety, malaise lately, or at least more so than the average high school teacher. I could find any number of reasons for this (my students, our President, Taylor Swift) but I rather find a remedy to jumpstart my attitude before I collapse underneath the weight of all these mental burdens. Beach Slang's Drunk or Lust tour was the punch to the heart that I needed.
Firstly, it should be expected that any tour involving Beach Slang would be dripping with nostalgia (the band recently completed a run with Jimmy Eat World). The ticket from open to close did not disappoint in this department. See Through Dresses opened the night with their brand of 80's/90's dream pop and emo that's seen a resurgence with bands like Turnover, before switching to what I imagine is their older work, which was more 90's garage punk fare. Being an opener is no easy act, and the energy and passion of this young act was very impressive, especially considering people were still trickling in.
Dave Hause and the Mermaid kept the crowd going with their brand of revival rock, hearkening back to the stylings of Springsteen and Petty. As the world's biggest self proclaimed Brian Fallon fan, I initially went into this set skeptical, seeing as Fallon reigns supreme in that genre and especially since Hause himself is a contemporary and peer of Fallon (he starred in The Gaslight Anthem's video for "Bring It On"). Any doubts I had were quickly quelled by Hause's enthusiasm. From the opening guitar and piano chords of "Autism Vaccine Blues" through the somber acoustic number "Season's Greetings from Ferguson", Dave Hause proved himself to be a regular bandleader working the audience and getting everyone belting his sing-alongs.
By the time James Alex and Co. took the stage, the crowd was primed enough for the unrelenting barrage of feelings that is their live show. The band filled their set list with all of the hits from their punk rock arsenal plus a slew of covers including The Cure, Jawbreaker, The Pixies, The Gin Blossoms, and The Replacements (twice!). This might seem like cheating, but everything about this show was so intimate, that it didn't even matter. Due to the charisma oozing out of Alex during his audience banter and uninhibited stage antics (I've never seen a man bend backwards that far while wielding a classic Dark Cherry ES), there was never once a feeling of distance between them and us in the audience. The whole thing felt as if you were seeing an old friend's band in a jam packed basement, and they played all of your favorite songs of theirs, as well as covers of the songs you both loved. To be a band that sells that sort of romanticism is one thing, to so effortlessly rope the audience in to that feeling and community is another.
Needless to say, this group, their music, their ethos, has been EXACTLY what I've needed recently. They are a reminder that life doesn't have to be dominated by petty squabbles between celebrities or politicians or young, stupid kids, but by those enduring moments of when you were a young, stupid kid with nothing but a lust for life and a disregard for everyone who told you "no". What an invigorating reminder heading into this new year.
Bring it on, 2018.
Beach Slang will be supporting Dashboard Confessional on their 2018 "We Fight" Tour.
Follow Jon on Twitter at @anotherrahulJ
Tags: Beach Slang, Concert Review, Dave Hause, punk rock, See Through DressesCategories: Music
Jonathan has spent a majority of his life either in Toronto, Ontario or St. Louis, Missouri and as a result, like Cory in that one episode of Boy Meets World with the jelly beans, sometimes can't decide which he loves more. He contemplates this tension, his life as a teacher, as well as many other existential questions concerning the world (i.e. who is God? what happens after people die? is it really NOT butter?) in his car while listening to Spotify. Follow him on Twitter @anotherRahulJ.


You can be the lucky one with thick and healthy hair. But the catch is you always have to control the growth of your mane. Otherwise, it will grow to produce a mushroom effect which does not look good on you. The big and puffy growth in almost all the directions will be hard to control or comb. So you need the perfect hairdresser regularly who will be adept to keep the growth under control. At the same time, you should be able to style up your hair in all possible ways.
You must have a positive consultation with your hair stylist. During the discussion, you must share your challenges with hair and explain what you dislike about the look of your hair. Ultimately, let the expert know what do you expect from the stylist. Don't forget to mention which shampoo you use on a regular basis. The stylist will ask you about the time that you spend and will be able to contribute for caressing your hair. If you have colored your hair in the past or have undergone some chemical treatment, mention the details of the process too.
It's essential for you to be comfortable with your hair stylist. When you visit the salon in huntsvillealabama to consult the stylist, don't think that the person will do whatever he or she likes to. The priority will be what you want. Their job is to execute the styling which will match up to your expectation from the consultation. You should ask all the questions that you have in your mind before starting any procedure. It will help to clear your doubts. At the end of the session, you should return as a happy client with the hairdo just as you wanted to have.


Come explore The Arboretum Library and our great story trees! Enjoy plant & nature stories and a take-home craft. It is an indoor/ outdoor program that will go rain or shine, so please dress appropriately. This is a free program for members and free with admission for non-members. This program is recommended for ages 3-6.
February – Are There Sillies in a Garden?
May – Whooo's Here at Night?
This program is recommended for ages 3-6.