text:
\section{Introduction}

The availability of atomic Bose-Einstein condensates (BECs) has
made it possible to create, via photo- or magneto-association,
molecular condensates. In photoassociation, a pair of free atoms
is brought into a bound molecular state by absorption of a photon
through a dipole transition. In magneto-association (or Feshbach
resonance \cite{Feshbach,timmermans99}), atom pairs in an open channel are
converted into bound molecules in a closed channel through
hyperfine spin interaction resonantly enhanced by making the
energies of the two channels close to each other via the Zeeman
effect. Photoassociation creates molecules in excited electronic
level, while magneto-association creates molecules in high
vibrational quantum state. In either case, the resulting molecules
are energetically unstable and suffer from large inelastic loss
rate. Macroscopic coherence between atoms and molecules has been
both studied theoretically \cite%
{timmermans99,drummond98,javanainen99,yurovsky99,heinzen00,kokkelmans02,javanainen02} (also see a recent review article by Duine and Stoof \cite{duine04}), and
demonstrated experimentally \cite{donley02}, nevertheless, long-lived stable molecular
condensates have not been produced by direct association of atomic
condensate \cite{fbecnote}.

\begin{figure}[h]
\centering \includegraphics[width=2.in]{fig1.eps} \caption{The
energy diagrams of three-level atom-molecule system involving
free-bound-bound transitions. Conversion of atoms in $\left\vert
a\right\rangle $ to quasi-bound molecules in $\left\vert
m\right\rangle $ is accomplished by Feshbach resonance, while the
coupling between $|m \rangle$ and the ground molecular state
$|g\rangle$ is provided by laser light.} \label{fig1}
\end{figure}

One possibility to overcome the difficulty is to employ the
two-photon or Raman photoassociation model, as in the proposal for
generating ground molecular condensates \cite{thermal,Mackie00,hope01,Drummond02}
from atomic condensates through the stimulated Raman adiabatic
passage (STIRAP) \cite{Hioe83,stirap}. In spite of the nonlinear
nature of the atom-molecule coupling in the photoassociation
process, it was shown by Mackie \emph{et al}. \cite{Mackie00} that
in the collisionless limit, this model can support a coherent
superposition of free atomic and ground molecular condensates.
This superposition, being immune to the inelastic processes
associated with the excited molecular state, is called the
nonlinear coherent population trapping (CPT) state, in direct
analogy to the CPT state in the linear $\Lambda $ three-level
atomic system \cite{Alzetta76,Gray78}. However, efficient
operation of this scheme typically demands very high laser
intensities in order to overcome the weak coupling caused by the
extremely small Franck-Condon overlap integral for the free
atomic-bound molecular transition \cite{Drummond02}.

To overcome the small Franck-Condon factor, a Feshbach-assisted
STIRAP model is proposed \cite{kokk,mackie}. Here, conversion of
atoms into quasi-bound molecules is accomplished by the much more
efficient Feshbach process. To bring the quasi-bound molecules
into a more deeply bound vibrational state, a two-photon Raman
laser field is applied. In the limit where both optical fields are
tuned far-off resonance from other electronically excited levels,
one can reduce the Raman coupling between the two molecular levels
into an effective one-photon coupling. This leads to a three-level
$\Lambda $ atom-molecule system \cite{Harris02} as shown in
Fig.~\ref{fig1}. In this scheme, the possible source of
decoherence is the decay of the quasi-bound molecular state while
the atomic state and the deeply bound molecular state are both
stable.

In the absence of particle collisions, we expect, from the work of
Mackie \emph{et al}. \cite{Mackie00}, that this system is capable
of supporting a CPT state, owing to the mathematical equivalence
to its photoassociation counterpart. The situation when the
collisions are present is more complicated. It was generally
believed that the nonlinear phase shifts arising from particle
collisions would defeat the two-photon resonance condition
\cite{Mackie00,Drummond02}, a prerequisite for the existence of
CPT states and hence the key for the successful implementation of
STIRAP. In a recent work \cite{ling04}, we have shown that by
appropriately chirping the optical frequency and the Feshbach
detuning, the collision-induced nonlinear phase shifts can be
dynamically compensated, hence generalizing the concept of the CPT
state from the collisionless limit \cite{Mackie00} to situations
where collisions cannot be ignored. In addition, we have also
shown that collisions may lead to dynamical instabilities in the
nonlinear CPT state, in contrast to the CPT states in the linear
$\Lambda $-system which are always stable. Avoiding dynamical
unstable regimes is another key for successful implementation of
the STIRAP process in nonlinear systems.

The purpose of the current paper is to provide a systematic study
of the excitation spectrum of the generalized nonlinear CPT state,
focusing on the characterization of its stability phase diagram.
To the best of our knowledge, a detailed stability analysis
specific to the CPT state of the nonlinear $\Lambda $ system has
never been presented before. Our paper is organized as follows: In
Sec.~II, we present the Hamiltonian and the nonlinear CPT solution
to the coupled Gross-Pitaevskii equations. In Sec.~III, collective
excitations of the CPT state are calculated in the spirit of the
Bogoliubov treatment \cite{bogoliubov47,fetter71}. Sec.~IV focuses
on the stability analysis in the long-wavelength limit based on
the excitation spectrum obtained in Sec.~III. In Sec. V, we study
the dynamics of a CPT state and show that dynamical instabilities
may lead to self-pulsing of a condensate mode or growth of
non-condensate modes. Finally, a summary is provided in Sec. VI.

\section{Hamiltonian, Stationary Equations, and CPT State}

We choose to formulate our theoretical description for the
Feshbach-based $\Lambda $-system as illustrated in
Fig.~\ref{fig1}. The exactly same description can be applied to
the two-photon photoassociation-based $\Lambda $-system, due to
the formal equivalence between the two models \cite{Drummond02}.
In Fig.~\ref{fig1}, we use $|a\rangle $, $|m\rangle $, and
$|g\rangle $ to denote the free atomic, quasi-bound molecular and
ground molecular states, respectively. Levels $|a\rangle $ and
$|m\rangle $ are coupled by a magnetic field through the Feshbach
resonance with a coupling strength $\alpha $ and a detuning
$\epsilon $ ($\epsilon $ is experimentally tunable via the
magnetic field), while levels $|m\rangle $ and $|g\rangle $ are
coupled by a (effective) laser field with a Rabi frequency $\Omega
$ and a detuning $\Delta$. In second quantization, the total
Hamiltonian, including collisions, takes the form
\begin{widetext}
\begin{eqnarray}
\hat{H} &=&\hbar \int d\mathbf{r} \left\{
\sum_{i}\hat{\psi}_{i}^{\dag }\left( \mathbf{r}\right) \left(
-\frac{\hbar }{2m_{i}}\nabla ^{2}\right) \hat{\psi}_{i}\left(
\mathbf{r}\right) +\frac{1}{2}\sum_{i,j}\lambda _{ij}
\hat{\psi}_{i}^{\dag }\left( \mathbf{r}\right)
\hat{\psi}_{j}^{\dag }\left( \mathbf{r}\right)
\hat{\psi}_{j}\left( \mathbf{r}\right) \hat{\psi}
_{i}\left( \mathbf{r}\right) \right. \nonumber \\
&&+\left.\epsilon \, \hat{\psi}_{m}^{\dag }\left(
\mathbf{r}\right) \hat{\psi} _{m}\left( \mathbf{r}\right)
+\frac{\alpha }{2}\left[ \hat{\psi}_{m}^{\dag }\left(
\mathbf{r}\right) \hat{\psi}_{a}\left( \mathbf{r}\right)
\hat{\psi} _{a}\left( \mathbf{r}\right) +h.c.\right] +\left(
\Delta +\epsilon \right) \hat{\psi}_{g}^{\dag }\left(
\mathbf{r}\right) \hat{\psi}_{g}\left( \mathbf{r}\right)
-\frac{\Omega }{2} \left[ \hat{\psi}_{m}^{\dag }\left(
\mathbf{r}\right) \hat{\psi}_{g}\left( \mathbf{r}\right) +h.c.
\right] \right\}, \label{grandH}
\end{eqnarray}
\end{widetext}
where $\hat{\psi}_{i}^{\dag }\left( \mathbf{r}\right) $ [$\hat{
\psi}_{i}\left( \mathbf{r}\right) $] ($i=a$, $m$ and $g$) is the
creation
(annihilation) operator of the bosonic field for species $i$ at location $%
\mathbf{r}$,$\ $and the terms proportional to $\lambda _{ij}$ represent
two-body collisions with $\lambda _{ii}$ $=4\pi \hbar a_{i}/m_{i}$ and $%
\lambda _{ij}=\lambda _{ji}=2\pi \hbar a_{ij}/\mu _{ij}$ for $i\neq j\,$\
characterizing the intra- and inter-state interaction strengths,
respectively, ($a_{i}$ and $a_{ij}$ are $s$-wave scattering lengths, $%
m_{i} $ is the mass of species $i$ with $m_{m}=m_{g}=2m_{a}$, and
$\mu _{ij}$ is the reduced mass between states $i$ and $j$). We
take, without loss of generality, both $\alpha $ and $\Omega $ to
be real as their phase factors can be absorbed by a trivial global
gauge transformation of the field operators. Here, we consider a
uniform system and hence have dropped the external trapping
potentials.

Let us begin our formulation from the Bogoliubov approximation,
which amounts to decomposing $\hat{\psi}_{i}\left(
\mathbf{r}\right) $ as
\begin{equation}
\hat{\psi}_{i}\left( \mathbf{r}\right) \approx \psi _{i}\left( \mathbf{r}%
\right) +\delta \hat{\psi}_{i}\left( \mathbf{r}\right) ,  \label{bogoliubov}
\end{equation}%
where $\psi _{i}\left( \mathbf{r}\right) =\left\langle
\hat{\psi}_{i}\left( \mathbf{r}\right) \right\rangle $ is the
condensate wave function and $\delta \hat{\psi}_{i}\left(
\mathbf{r}\right) $ is a small fluctuation field operator that
obeys the usual bosonic commutation relation
\begin{equation}
\left[ \delta \hat{\psi}_{i}\left( \mathbf{r}\right) ,\delta \hat{\psi}%
_{j}^{\dag }\left( \mathbf{r}^{\prime }\right) \right] =\delta _{ij}\delta
\left( \mathbf{r-r}^{\prime }\right) .  \label{space
communtator relation}
\end{equation}%
As a first step in the Bogoliubov approach, we put
Eq.~(\ref{bogoliubov}) into the grand canonical Hamiltonian
(\ref{grandH})
\begin{equation}
\mathcal{\hat{H}}=\hat{H}-\hbar \mu \int d\mathbf{r}\left[ \hat{\psi}%
_{a}^{\dag }\hat{\psi}_{a}+2\hat{\psi}_{m}^{\dag }\hat{\psi}_{m}+2\hat{\psi}%
_{g}^{\dag }\hat{\psi}_{g}\right] ,  \label{grand grand H}
\end{equation}%
where the chemical potential $\hbar \mu $ is introduced to
conserve the total particle number, and expand Eq.~(\ref{grand
grand H}) up to the second order in fluctuation operators as
\begin{equation}
\mathcal{\hat{H}}\approx \mathcal{\hat{H}}^{\left( 0\right) }+\mathcal{\hat{H%
}}^{\left( 1\right) }+\mathcal{\hat{H}}^{\left( 2\right) },  \label{H 012}
\end{equation}%
where the subscript represents the order in $\delta \hat{\psi}_{i}\left(
\mathbf{r}\right) $.

The zeroth order term, $\mathcal{\hat{H}}^{\left( 0\right) }$, is a constant
depending on the wave functions $\psi_i$ and is irrelevant to our interest.
The first-order term, $\mathcal{\hat{H}}^{\left( 1\right) }$, is required to
vanish in the Bogoliubov formalism. This requirement leads to a set of
coupled stationary equations
\begin{subequations}
\label{steady state equation}
\begin{eqnarray}
\mu \psi _{a}&=& \left( -\frac{\hbar \nabla ^{2}}{2m_{a}} +\sum_{i}\lambda
_{ai}\left\vert \psi _{i}\right\vert ^{2}\right) \psi _{a}+\alpha \psi
_{m}\psi _{a}^{\ast } ,  \label{u equation} \\
2\mu \psi _{m} &=& \left( -\frac{\hbar \nabla ^{2}}{4m_{a}} +\sum_{i}\lambda
_{mi}\left\vert \psi _{i}\right\vert ^{2}\right) \psi _{m}+\epsilon \psi
_{m}+\frac{\alpha }{2}\psi _{a}^{2}  \notag \\
&&\;\;\;\;\;\;-\frac{\Omega }{2}\psi _{g} ,  \label{amplitude} \\
2\mu \psi _{g} &=& \left( -\frac{\hbar \nabla ^{2}}{4m_{a}} +\sum_{i}\lambda
_{gi}\left\vert \psi _{i}\right\vert ^{2}\right) \psi _{g}+\left( \Delta
+\epsilon \right) \psi _{g}  \notag \\
&&\;\;\;\;\;\;-\frac{\Omega }{2}\psi _{m} ,  \label{delta equation}
\end{eqnarray}
which represent the generalized time-independent Gross-Pitaevskii
equations (GPEs). Finally, the second-order term,
$\mathcal{\hat{H}}^{\left( 2\right) } $, which has a more
complicated structure, will be studied separately in the next
section where we calculate the spectra of the collective
excitations.

We now turn our attention to the homogeneous (zero-momentum) solution to the
GPEs. To proceed, we separate each wave function into an amplitude and a
phase according to
\end{subequations}
\begin{equation}
\psi _{j}=\left\vert \psi _{j}\right\vert e^{i\theta _{j}},\;\;j=a,m,g.
\label{steady state ansatz}
\end{equation}%
To seek the CPT solution, we take $\psi _{m}=0$. After
substituting Eq.~(\ref{steady state ansatz}) into Eq.~(\ref{steady
state equation}), we find that under the conditions
\begin{eqnarray}
\theta _{g} &=&2\theta _{a},  \notag \\
\alpha \left\vert \psi _{a}\right\vert ^{2} &=&\Omega \left\vert \psi
_{g}\right\vert ,  \label{amplitude relation}
\end{eqnarray}%
Eq.~(\ref{steady state equation}) possesses a solution consistent
with the requirement that $\psi _{m}=0$. The explicit form of this
solution can be readily found by combining Eq.~(\ref{amplitude
relation} ) with the particle number conservation
\begin{equation}
\left\vert \psi _{a}\right\vert ^{2}+2\left\vert \psi _{g}\right\vert ^{2}=n,
\label{particle number conservation}
\end{equation}%
where $n$ is the total particle density, to yield the particle
density distribution as
\begin{subequations}
\label{CPT distribution}
\begin{eqnarray}
\left\vert \psi _{m}^{0}\right\vert ^{2} &=&0\,, \\
\left\vert \psi _{a}^{0}\right\vert ^{2} &=&\frac{2n}{1+\sqrt{1+8\left(
\frac{\alpha \sqrt{n}}{\Omega }\right) ^{2}}}\,, \\
\left\vert \psi _{g}^{0}\right\vert ^{2} &=&\frac{n}{2}\frac{\sqrt{1+8\left(
\frac{\alpha \sqrt{n}}{\Omega }\right) ^{2}}-1}{1+\sqrt{1+8\left( \frac{%
\alpha \sqrt{n}}{\Omega }\right) ^{2}}}\,.
\end{eqnarray}%
A check for consistency leads to the chemical potential
\end{subequations}
\begin{equation}
\mu =\lambda _{a}\left\vert \psi _{a}^{0}\right\vert ^{2}+\lambda
_{ag}\left\vert \psi _{g}^{0}\right\vert ^{2},  \label{chemical potential}
\end{equation}%
where $\lambda _{i}\equiv \lambda _{ii}$, and the laser detuning
\begin{equation}
\Delta =-\epsilon +\left( 2\lambda _{ag}-\lambda _{g}\right) \left\vert \psi
_{g}^{0}\right\vert ^{2}+\left( 2\lambda _{a}-\lambda _{ag}\right)
\left\vert \psi _{a}^{0}\right\vert ^{2},  \label{Delta arbitrary}
\end{equation}%
which is necessary in order for Eqs.~(\ref{CPT distribution}) to be a
stationary solution of Eqs.~(\ref{steady state equation}).

The vanishing of the population in the quasi-bound molecular state
indicates a destructive interference in the excitation to this
state. A system prepared in the state represented by
Eqs.~(\ref{CPT distribution}) is not subject to the particle loss
suffered by the quasi-bound molecular state. This situation is
reminiscent of the CPT state in a linear $\Lambda $-type atomic
system where the atoms are immune to the spontaneous emission
\cite{Alzetta76,Gray78}. For this reason, we regard the steady
state represented by Eqs.~(\ref{CPT distribution}) as the
nonlinear matter-wave analog of the CPT state in a linear $\Lambda
$-system.

Interestingly, just as in the linear case, the CPT solution
described by Eqs.~(\ref{CPT distribution}) depends only explicitly
on the two coupling strengths, $\alpha $ and $\Omega $, not on the
collisional parameters. The latter, however, play important roles
as evidenced by Eq.~(\ref{Delta arbitrary}). Note that in the
absence of collisions, Eq.~(\ref{Delta arbitrary}) degenerates
into $\Delta +\epsilon =0$, which is just the ordinary
\textquotedblleft two-photon\textquotedblright\ resonance
condition. Thus, Eq.~(\ref{Delta arbitrary}) can be regarded as
the generalization of the two-photon resonance in which the
nonlinear phase shifts due to particle collisions have been
compensated. This CPT solution is therefore a nonlinear
generalization of the one found in the collisionless limit
\cite{Mackie00}.

\section{Collective Excitations of the CPT State}

In this section, we want to calculate the excitation spectra of
the CPT state, especially, the spectra in the long-wavelength
limit, since they determine many important properties, including
the stability, of the CPT state at low temperature.

\subsection{Bogoliubov Equations}

To determine the equations for the collective excitations, we
first take advantage of our system being homogeneous and move to
the momentum space through the expansion
\begin{equation}
\delta \hat{\psi}_{i}\left( \mathbf{r}\right) =\frac{1}{\left( 2\pi \hbar
\right) ^{3/2}}\int d\mathbf{p}\,e^{i\mathbf{p\cdot r/\hbar }}\,\hat{c}%
_{i}\left( \mathbf{p}\right)\,,
\end{equation}%
where $\hat{c}_{i}\left( \mathbf{p}\right) $ [$\hat{c}_{i}^{\dag }\left(
\mathbf{p}\right) $] is the annihilation (creation) operator for a particle
of species $i$ with momentum $\mathbf{p}$ and obey the standard bosonic
commutation relation
\begin{equation}
\left[ \hat{c}_{i}\left( \mathbf{p}\right) ,\hat{c}_{j}^{\dag }\left(
\mathbf{p}^{\prime }\right) \right] =\delta _{ij}\delta \left( \mathbf{p-p}%
^{\prime }\right) ,  \label{c bosonic commutation}
\end{equation}%
in accordance with Eq.~(\ref{space communtator relation}). The third term $%
\mathcal{\hat{H}}^{\left( 2\right) }$ in Eq.~(\ref{H 012}) for our CPT state
can now be put into a compact form
\begin{eqnarray}
\mathcal{\hat{H}}^{\left( 2\right) } &=&\frac{\hbar }{2}\int d\mathbf{p}%
^{\prime }\sum_{nm}\left\{ 2\hat{c}_{n}^{\dag }\left( \mathbf{p}^{\prime
}\right) \left( A\right) _{nm}\hat{c}_{m}\left( \mathbf{p}^{\prime }\right)
\right.  \notag \\
&&+\left. \left[ \hat{c}_{n}\left( \mathbf{p}^{\prime }\right) B_{nm}\hat{c}%
_{m}\left( -\mathbf{p}^{\prime }\right) +h.c.\right] \right\} ,  \label{H^2}
\end{eqnarray}%
where $A$ and $B$ are both real and symmetric matrices, given by
\begin{subequations}
\begin{eqnarray}
A &=&\left(
\begin{array}{ccc}
\epsilon _{p}+\lambda _{a}\left\vert \psi _{a}^{0}\right\vert ^{2} & \alpha
\left\vert \psi _{a}^{0}\right\vert & \lambda _{ag}\left\vert \psi
_{a}^{0}\right\vert \left\vert \psi _{g}^{0}\right\vert \\
\alpha \left\vert \psi _{a}^{0}\right\vert & \frac{\epsilon _{p}}{2}%
+\epsilon ^{\prime } & -\frac{\Omega }{2} \\
\lambda _{ag}\left\vert \psi _{a}^{0}\right\vert \left\vert \psi
_{g}^{0}\right\vert & -\frac{\Omega }{2} & \frac{\epsilon _{p}}{2}+\lambda
_{g}\left\vert \psi _{g}^{0}\right\vert ^{2}%
\end{array}%
\right) , \\
B &=&\left(
\begin{array}{ccc}
\lambda _{a}\left\vert \psi _{a}^{0}\right\vert ^{2} & 0 & \lambda
_{ag}\left\vert \psi _{a}^{0}\right\vert \left\vert \psi _{g}^{0}\right\vert
\\
0 & 0 & 0 \\
\lambda _{ag}\left\vert \psi _{a}^{0}\right\vert \left\vert \psi
_{g}^{0}\right\vert & 0 & \lambda _{g}\left\vert \psi _{g}^{0}\right\vert
^{2}%
\end{array}%
\right) ,
\end{eqnarray}%
\end{subequations}
where we have defined
\begin{subequations}
\begin{eqnarray}
\epsilon _{p} &\equiv &p^{2}/(2m_{a}\hbar )\,,   \\
\epsilon ^{\prime } &=&\epsilon -\Lambda _{a}\left\vert \psi
_{a}^{0}\right\vert ^{2}-\Lambda _{g}\left\vert \psi _{g}^{0}\right\vert
^{2}\,,  \label{shifted epsilon} \\
\Lambda _{a} &=&2\lambda _{a}-\lambda _{am}\,,  \\
\Lambda _{g} &=&2\lambda _{ag}-\lambda _{mg}\,.
\end{eqnarray}
\end{subequations}

The process of finding the quasiparticle spectrum begins by changing Eq.~(%
\ref{H^2}) into a diagonalized form
\begin{equation}
\mathcal{\hat{H}}^{\left( 2\right) }=\hbar \int d\mathbf{p}\sum_{i}\omega
_{i}\left( p\right) \hat{b}_{i}^{\dag }\left( \mathbf{p}\right) \hat{b}%
_{i}\left( \mathbf{p}\right) + c{\rm -number},
\end{equation}%
with the generalized Bogoliubov transformation
\begin{equation}
\hat{b}_{i}\left( \mathbf{p}\right) =\sum_{j}u_{ij}\left( p\right) \hat{c}%
_{j}\left( \mathbf{p}\right) +\sum_{j}v_{ij}\left( p\right) \hat{c}%
_{j}^{\dag }\left( -\mathbf{p}\right) ,  \label{bogoliubov transformation}
\end{equation}%
where $\omega _{i}\left( p\right) $ is the quasiparticle frequency, $%
u_{ij}\left( p\right) $ and $v_{ij}\left( p\right) $ are the transformation
coefficients, and $\hat{b}_{i}\left( \mathbf{p}\right) $ and $\hat{b}%
_{i}^{\dag }\left( \mathbf{p}\right) $ are the respective annihilation and
creation operators for a quasiparticle of frequency $\omega _{i}\left(
p\right) $ and momentum $\mathbf{p}$. To obtain the equations for $\omega
_{i}\left( p\right) $, $u_{ij}\left( p\right) $ and $v_{ij}\left( p\right) $%
, we first note that
\begin{equation}
\left[ \hat{b}_{i}\left( \mathbf{p}\right) ,\mathcal{\hat{H}}^{\left(
2\right) }\right] =\hbar \omega _{i}\left( p\right) \hat{b}_{i}\left( \mathbf{p}%
\right) .  \label{operator equation}
\end{equation}%
Inserting Eq.~(\ref{bogoliubov transformation}) into Eq.~(\ref{operator
equation}) and with the help of Eqs.~(\ref{c bosonic commutation}) and (\ref%
{H^2}) along with the symmetric properties of $A$ and $B$, we arrive at
\begin{widetext}
\begin{equation}
\sum_{m}\sum_{n}\left\{ \left[ A_{mn}u_{in}-B_{mn}v_{in}\right] \hat{c}%
_{m}\left( \mathbf{p}\right) +\left[ B_{mn}u_{in}-A_{mn}v_{in}\right] \hat{c}%
_{m}^{\dag }\left( -\mathbf{p}\right) \right\} =\omega _{i}\sum_{m}\left[
u_{im}\hat{c}_{m}\left( \mathbf{p}\right) +v_{im}\hat{c}_{m}^{\dag }\left( -%
\mathbf{p}\right) \right] .
\end{equation}%
\end{widetext}Equating the coefficients of $\hat{c}_{m}\left( \mathbf{p}%
\right) $ and $\hat{c}_{m}^{\dag }\left( -\mathbf{p}\right) $ on both sides
of the above equation, we arrive at a 6$\times 6$ matrix equation
\begin{equation}
\left(
\begin{array}{cc}
A & -B \\
B & -A%
\end{array}%
\right) \left(
\begin{array}{c}
u_{i} \\
v_{i}%
\end{array}%
\right) =\omega _{i}\left( p\right) \left(
\begin{array}{c}
u_{i} \\
v_{i}%
\end{array}%
\right) ,  \label{collective excitation}
\end{equation}%
where
\begin{equation*}
u_{i}=\left( u_{ia},u_{im},u_{ig}\right) ^{T},\;\;\;v_{i}=\left(
v_{ia},v_{im},v_{ig}\right) ^{T}.
\end{equation*}

A simple manipulation can show that $\omega_i^2(p)$ are the
eigenvalues of the $3 \times 3$ matrix $(A+B)(A-B)$, hence
$\omega_i^2(p)$ satisfy a cubic equation with the form
\begin{equation}
\left( \omega _{i}^{2}\right) ^{3}-a_{1}\left( \omega
_{i}^{2}\right) ^{2}+a_{2}\omega _{i}^{2}-a_{3}=0\,,
\label{eigenvalue equation}
\end{equation}%
where the coefficients are given by
\begin{subequations}
\begin{eqnarray}
a_{1} &=&a_{10}+a_{11}\epsilon _{p}+a_{12}\epsilon _{p}^{2}\,, \\
a_{2} &=&a_{20}+a_{21}\epsilon _{p}+a_{22}\epsilon _{p}^{2}+a_{23}\epsilon
_{p}^{3}+a_{24}\epsilon _{p}^{4}\,, \\
a_{3} &=&a_{31}\epsilon _{p}+a_{32}\epsilon _{p}^{2}+a_{33}\epsilon
_{p}^{3}+a_{34}\epsilon _{p}^{4}+  \notag \\
&&a_{35}\epsilon _{p}^{5}+a_{36}\epsilon _{p}^{6}\,.
\end{eqnarray}
\end{subequations}
The coefficients $a_{ij}$ depend on the coupling and the collisional
constants and are in general quite complicated. We only list three of them
below
\begin{widetext}
\begin{subequations}
\begin{eqnarray}
a_{10} &=&\epsilon ^{\prime 2}+\frac{1}{2}\left( \Omega ^{2}+4\alpha
^{2}\left\vert \psi _{a}^{0}\right\vert ^{2}\right)\, ,  \label{a10} \\
a_{20} &=&\frac{1}{16}\left( \Omega ^{2}+4\alpha ^{2}\left\vert \psi
_{a}^{0}\right\vert ^{2}\right) ^{2}-0.5\epsilon ^{\prime }\alpha
^{2}\left\vert \psi _{a}^{0}\right\vert ^{4}\left( \lambda _{g}-4\lambda
_{ag}+4\lambda _{a}\right)\, ,  \label{a20} \\
a_{31} &=&\frac{1}{8}\left\vert \psi _{a}^{0}\right\vert ^{2}\left( \Omega
^{2}+2\alpha ^{2}\left\vert \psi _{a}^{0}\right\vert ^{2}\right) \times %
\left[ \lambda _{a}\Omega ^{2}+4\alpha ^{2}\left( \lambda
_{g}\left\vert \psi _{g}^{0}\right\vert ^{2}+\lambda
_{ag}\left\vert \psi _{a}^{0}\right\vert ^{2}\right) -8\epsilon
^{\prime }\left( \lambda _{a}\lambda _{g}-\lambda _{ag}^{2}\right)
\left\vert \psi _{g}^{0}\right\vert ^{2}\right] \,, \label{a31}
\end{eqnarray}
\end{subequations}
\end{widetext}as they will be explicitly referenced in later discussions.

\subsection{Excitation Spectra}

Equation~(\ref{eigenvalue equation}) has a cubic form with respect to $%
\omega_i ^{2}$ and hence can be solved analytically. However the
analytical expressions are in general too complicated to provide
much physical insights. Since we are most interested in the
long-wavelength excitations, we proceed to take a
perturbative approach by expanding $\omega _{i}(p)$ in the limit of small $p$%
, or equivalently, small $\sqrt{\epsilon _{p}}$. The perturbative solution
thus takes the form:
\begin{equation}
\omega _{i}\left( p\right) \approx \omega _{i}^{\left( 0\right)
}+d_{i}^{\left( 1\right) }\,\sqrt{\epsilon _{p}}+d_{i}^{\left( 2\right)
}\,\epsilon _{p}+\cdots ,  \label{perturbative ansatz}
\end{equation}%
In order to bring out the effects of collisions more clearly, let
us first consider the collicionless limit.

\subsubsection{Excitation Spectra in the Collisionless Limit}

In the absence of collisions, the matrix $B$ becomes null. The excitation
frequencies $\omega _{i}(p)$ are eigenvalues of the Hermitian matrix $A$ and
therefore must be real. This means that there is no dynamical instability in
spite of the nonlinear nature of the coupling between the atomic and the
quasi-bound molecular state.

To the zeroth order, we find that $\omega _{i}^{\left( 0\right) }$ has three
solutions \cite{comment}
\begin{equation}
\omega _{0}^{\left( 0\right) }=0,\;\;\;\omega _{\pm }^{\left(
0\right) }= \sqrt{\left( a_{10}\pm
\sqrt{a_{10}^{2}-4a_{20}}\right) /2}\,, \label{omega (0)}
\end{equation}%
where the subscripts $\left( 0,-,+\right) $ are used to denote the
three branches of the collective modes. Clearly the 0-branch is
gapless, while the $ (\pm )$-branches are gapped. The reality of
$\omega _{\pm }^{\left( 0\right) }$ is guaranteed because here
both $a_{10}$ and $a_{20}$ are positive with $ a_{10}^{2}\geq
4a_{20}$ [see Eqs.~(\ref{a10}) and (\ref{a20}), in the absence of
collisions.]

For all three branches, we have found that $d_{i}^{(1)}=0$. Hence the next
leading order in $\omega _{i}(p)$ is quadratic in momentum with
\begin{subequations}
\begin{eqnarray}
d_{0}^{(2)} &=&\frac{\Omega ^{2}+2\alpha ^{2}\left| \psi _{a}^{0}\right| ^{2}%
}{\Omega ^{2}+4\alpha ^{2}\left| \psi _{a}^{0}\right| ^{2}}\,, \\
d_{\pm }^{\left( 2\right) } &=&\pm \frac{\left[ \frac{\Omega ^{2}}{8}+\alpha
^{2}\left| \psi _{a}^{0}\right| ^{2}+\frac{(\omega _{\pm }^{(0)})^{2}}{2}%
\right] }{\left[ (\omega _{\pm }^{(0)})^{2}+\frac{\Omega
^{2}}{4}+\alpha ^{2}\left| \psi _{a}^{0}\right| ^{2}\right] }\,,
\end{eqnarray}
\end{subequations}

\begin{figure}[h]
\centering \includegraphics[width=2.5in]{fig2.eps} \caption{The
quasiparticle dispersion spectra for $n=1$, $\Omega =$ $1$, and
$\protect\epsilon =-2$. Solid lines represent the collisionless
case while dotted lines are for a system with $\protect
\lambda _{a}=0.625,$ $\protect\lambda _{m}=\protect\lambda _{g}=\protect%
\lambda _{am}=\protect\lambda _{ag}=\protect\lambda _{mg}=0.3\protect\lambda %
_{a}$. Units are defined in Sec.~\protect\ref{classification
section}.} \label{fig2}
\end{figure}
Solid lines in Fig.~\ref{fig2} represent the three branches in the
collisionless limit. They are obtained by directly solving the cubic
equation (\ref{eigenvalue equation}), and are in good agreement with the
perturbative results (not shown).

\subsubsection{Excitation Spectra with Collisions}

\label{spectra} The spectra including the collisional terms can be
similarly obtained. The zeroth order solutions, $\omega
_{i}^{(0)}$, have exactly the same forms as in the collisionless
limit given by Eq.~(\ref{omega (0)}). Therefore the 0-branch
continues to be gapless. Due to the collision-modified
coefficients $a_{ij}$, the two zeroth order gapped modes $\omega
_{\pm }^{(0)}$ are now, however, no longer guaranteed to be real.
Complex mode frequencies lead to the dynamical instability, which
we will discuss in detail in the next section.

Another important effect of the collisions is the modification of
the gapless 0-branch. While $d_{\pm }^{(1)}$ still remains to be
zero (hence the two gapped branches are still quadratic in $p$ to
the leading order), $ d_{0}^{(1)}$ now takes a finite value given
by
\begin{equation}
d_{0}^{\left( 1\right) }=\sqrt{\frac{a_{31}}{a_{20}}},  \label{d(1)0}
\end{equation}%
Hence the gapless branch, in the long-wavelength limit, is linear in $p$ and
represents the phonon mode with the speed of sound determined by Eq.~(\ref%
{d(1)0}). The dotted lines in Fig.~\ref{fig2} represent the
spectra with collisions in the stable regime. Once again, they are
calculated by solving the cubic Eq.~(\ref{eigenvalue equation}),
but are found in good agreement with the perturbative results (not
shown).

According to Eq.~(\ref{d(1)0}), whenever $a_{31}$ and $a_{20}$ are
opposite in sign, $\omega _{0}$ becomes imaginary for finite $p$
in the long wavelength limit, indicating an unstable 0-branch. We
now turn to a detailed discussion of the collision-induced
dynamical instability. Note that from Eq.~(\ref{d(1)0}), it may
seem that $ \omega _{0}$ is singular for any finite momenta when
$a_{20}=0$. However, when $a_{20}=0,\omega _{0}^{\left( 0\right)
}$ and $\omega _{-}^{\left( 0\right) }$ are degenerate according
to Eq.~(\ref{omega (0)}). Thus, this singularity is only an
artefact, indicating the failure of the nondegenerate perturbative
calculation. Indeed, exact solutions of Eq.~(\ref{eigenvalue
equation}) do not exhibit such a singularity. Despite of this
complication, the perturbative results can be used to identify the
unstable regimes quite accurately.

\section{Classification of Dynamical Instability}

\label{classification section}

Studies in Sec.~\ref{spectra} suggest that complex excitation frequencies
and hence dynamical instabilities can occur under the following situations:

\begin{enumerate}
\item When $a_{31}$ and $a_{20}$ possess opposite signs, 0-branch becomes
unstable [see Eq.~(\ref{d(1)0})];

\item When $a_{20}<0$, $\omega _{-}$ becomes complex [see Eq.~(\ref{omega
(0)})];

\item When $a_{10}^{2}<4a_{20}$, both $(\pm )$-branches become unstable [see
Eq.~(\ref{omega (0)})].
\end{enumerate}

Before moving foward, let us clarify our unit system. We consider
systems with particle densities in the order of $n_{0}=5\times
10^{20}$ m$ ^{-3}$, and $\alpha $ fixed to $4.22\times 10^{-6}$
m$^{3/2}$s$^{-1}$, which corresponds to the atom-molecule coupling
strength for the $^{23}$Na Feshbach resonance at a magnetic field
strength of $85.3$ mT \cite{seaman03}. We then adopt a unit system
in which $n_{0}$ is the unit for density, $ \alpha
/\sqrt{n_{0}}=1.88724\times 10^{-16}$m$^{3}$s$^{-1}$ the unit for
collisional parameters ($ \lambda _{i}$, $\lambda _{ij}$), $\alpha
\sqrt{n_{0}}=9.436\times 10^{4}$ s$ ^{-1}$ the unit for
frequencies $\left( \omega ,\Omega ,\epsilon \right) $, and
$\sqrt{ \left( 2m\hbar \alpha \sqrt{n_{0}}\right) }=8.72\times
10^{-28}$ kg m/s the unit for momentum $p$. Furthermore, for
simplicity, we only consider the situation where all the
scattering lengths are positive.

Our goal is to identify the unstable regimes in the parameter space spanned
by $\epsilon $ and $\Omega $, while the laser detuning $\Delta $ is always
fixed by the resonance condition represented by Eq.~(\ref{Delta arbitrary}).
For cases 1 and 2, we can identify two threshold values for the Feshbach
detuning $\epsilon _{0}^{th}$ and $\epsilon _{1}^{th}$, given by
\begin{eqnarray}
\epsilon _{0}^{th} &=&\Lambda _{a}\left\vert \psi _{a}^{0}\right\vert
^{2}+\Lambda _{g}\left\vert \psi _{g}^{0}\right\vert ^{2}+\frac{2\left(
\frac{\Omega ^{2}}{4|\psi _{a}^{0}|^{2}}+1\right) ^{2}}{4\lambda
_{a}+\lambda _{g}-4\lambda _{ag}},  \label{e 0} \\
\epsilon _{1}^{th} &=&\Lambda _{a}\left\vert \psi _{a}^{0}\right\vert
^{2}+\Lambda _{g}\left\vert \psi _{g}^{0}\right\vert ^{2}  \notag \\
&&\;+\frac{\lambda _{a}\Omega ^{2}+4\left( \lambda _{g}\left\vert \psi
_{g}^{0}\right\vert ^{2}+\lambda _{ag}\left\vert \psi _{a}^{0}\right\vert
^{2}\right) }{8\left\vert \psi _{g}^{0}\right\vert ^{2}\left( \lambda
_{a}\lambda _{g}-\lambda _{ag}^{2}\right) },  \label{e 1}
\end{eqnarray}%
which are obtained from the conditions $a_{20}=0$ and $ a_{31}=0$,
respectively. Case 3 will be discussed at the end of this section.

\begin{figure}[h]
\centering \includegraphics[width=2.5in]{fig3.eps}
\caption{Division of the positive quarter of $\left( \protect\lambda _{g},%
\protect\lambda _{a}\right) $-space into three regions by a straight line: $%
\protect\lambda _{g}=-4\protect\lambda _{a}+4\protect\lambda _{ag}$ and a
hyperbola: $\protect\lambda _{a}\protect\lambda _{g}=\protect\lambda %
_{ag}^{2}$.}
\label{fig3}
\end{figure}

It is not difficult to see that the signs of the denominators of
the last terms of (\ref{e 0}) and (\ref{e 1}), $4\lambda
_{a}+\lambda _{g}-4\lambda _{ag}$ and $ \lambda _{a}\lambda
_{g}-\lambda _{ag}^{2}$, respectively, will play crucial roles in
locating the instability parameter regime. For this reason, we
first decompose the $\left( \lambda _{g}\text{,}\lambda
_{a}\right) $-space into three regions divided by a straight line,
$\lambda _{g}=-4\lambda _{a}+4\lambda _{ag}$, and a hyperbola,
$\lambda _{a}\lambda _{g}=\lambda _{ag}^{2}$, as shown in
Fig.~\ref{fig3}. The line is tangent to the hyperbola at the point
$\left( \lambda _{g}=2\lambda _{ag}\text{,}\lambda _{a}=0.5\lambda
_{ag}\right) $. The three regions are defined as follows: Both
$4\lambda _{a}+\lambda _{g}-4\lambda _{ag}$ and $\lambda
_{a}\lambda _{g}-\lambda _{ag}^{2}$ are negative in Region I and
both are positive in Region II, while in Region III, $4\lambda
_{a}+\lambda _{g}-4\lambda _{ag}<0$ and $\lambda _{a}\lambda
_{g}-\lambda _{ag}^{2}>0$.

\begin{figure}[h]
\centering \includegraphics[width=2.5in]{fig4.eps}
\caption{Classification of the instabilities in the limit of low momentum.
The shaded areas of the diagrams in column one and two correspond to the
instabilities due to the imaginary sound speed and the complex gaps,
respectively. Each set of diagrams in row 1, 2 and 3 are the instability
maps for systems operating in region I, II, and III as specified in Fig. 3,
respectively.}
\label{fig4}
\end{figure}

Next, we choose to construct, for each region defined above, two
instability maps in the $\left( \Omega ,\epsilon \right) $-space;
one for the instability due to the imaginary sound speed (case 1),
and the other for the instability due to the complex gap (case 2).
We represent them by the shaded areas of the diagrams in the first
and second columns of Fig.~\ref{fig4}, respectively. Each set of
diagrams in row 1, 2 and 3 are built upon the collisional
parameters corresponding to Regions I, II, and III of $\left(
\lambda _{g}\text{,} \lambda _{a}\right) $-space, respectively. We
construct these maps based on two principles. First, it can be
shown from Eqs.~(\ref{e 0}) and (\ref{e 1}) that $\epsilon
_{1}^{th}-\epsilon _{0}^{th}\propto 1/\left( 4\lambda _{a}+\lambda
_{g}-4\lambda _{ag}\right) \left( \lambda _{a}\lambda _{g}-\lambda
_{ag}^{2}\right) $ with a positive definite proportionality. \ As
a result, we have $\epsilon _{1}^{th}>\epsilon _{0}^{th}$ in
Regions I and II, and $\epsilon _{1}^{th}<\epsilon _{0}^{th}$\ in
Region III. Second, according to Eq.~(\ref{e0 large Omega})
[Eq.~(\ref{e1 large Omega} )], $\epsilon _{0}^{th}$ [$\epsilon
_{1}^{th}$] is bent downward if $ 4\lambda _{a}+\lambda
_{g}-4\lambda _{ag}$ \thinspace $<0$ [$\lambda _{a}\lambda
_{g}-\lambda _{ag}^{2}$\thinspace $<0$] and vice versa. Note that
the instability of case 2 is solely determined by $\epsilon
_{0}^{th}$; the presence of $\epsilon _{1}^{th}$ in the second
column of Fig.~\ref{fig4} is to make the comparison of the two
maps easier. For example, one can easily identify that the two
instability maps in region II overlap in the band between
$\epsilon _{0}^{th}$ and $\epsilon _{1}^{th}$.

Let us now consider a specific atom-molecule system composed of $^{23}$Na.
The atomic $s$-wave scattering length for sodium is around $a_{a}=3.4$ nm
\cite{abeelen99} which yields $\lambda _{a}=0.625$. No precise knowledge
about the scattering lengths involving molecules exists. Assuming they are
on the same order of magnitude as $a_{a}$, we take $\lambda _{m}=\lambda
_{g}=\lambda _{am}=\lambda _{mg}=\lambda _{ag}=0.3\lambda _{a}$. This set of
parameters puts the system in Region II of Fig.~\ref{fig4}. According to
Eqs.~(\ref{e0 small Omega}) and (\ref{e1 small Omega}), $\epsilon _{0}^{th}$
and $\epsilon _{1}^{th}$ approach $0.5\Lambda _{g}+2\alpha ^{2}/\left(
4\lambda _{a}+\lambda _{g}-4\lambda _{ag}\right) =1.126$ and $0.5\Lambda
_{g}+0.5\alpha ^{2}\lambda _{g}/\left( \lambda _{a}\lambda _{g}-\lambda
_{ag}^{2}\right) =1.237,$ respectively, in the limit of small $\Omega $.

\begin{figure}[h]
\centering \includegraphics[width=2.5in]{fig5.eps}
\caption{Im$\left[ \protect\omega \left( p\right) \right] $ as a
function of $\Omega $ for a fixed momentum $p=0.005$. The curves
from left to right correspond to $\protect\epsilon =1.3$, $
\protect\epsilon =1.5$, $\protect\epsilon =2,$ and
$\protect\epsilon =3$, respectively.
Other parameters are $n=1$, $\protect\lambda _{a}=0.625,$ and $\protect%
\lambda _{m}=\protect\lambda _{g}=\protect\lambda _{am}=\protect\lambda %
_{mg}=\protect\lambda _{ag}=0.3\protect\lambda _{a}$. Units are defined in
Sec.~\protect\ref{classification section}.}
\label{fig5}
\end{figure}

A search using Eq.~(\ref{eigenvalue equation}) above these thresholds and at
small momenta indeed leads to two branches whose frequencies have finite
imaginary parts for certain values of $\Omega $, as illustrated in Fig.~\ref%
{fig5}(a) and (b), respectively. Each curve in Fig.~\ref{fig5}(a) is
restricted between two thresholds, both of which increase as $\epsilon $,
evidently resembling the instability originating from the imaginary sound
speed (case 1). Thus, in the limit of small $p$, we expect it to approach $%
d_{0}^{\left( 1\right) }p$. In fact, one can trace the sharp asymmetry of
each curve to the fact that according to Eq.~(\ref{d(1)0}), $d_{0}^{\left(
1\right) }$ vanishes at the left threshold where $a_{31}$ is close to zero,
while becomes singular at the right threshold where $a_{20}$ is close to
zero. In contrast, each curve in Fig.~\ref{fig5}(b) has only one threshold,
which matches quite closely to the right threshold of the corresponding
curve in Fig.~\ref{fig5}(a). This along with the fact that the value of the
mode frequency is rather insensitive with respect to $p$ (as long as $p$ is
not too large) indicates that it represents the instability from the complex
gapped branch of case 2. Therefore these exact calculations are in good
agreement with the insights gained from the instability map shown in Fig.~%
\ref{fig4}.

\begin{figure}[h]
\centering \includegraphics[width=2.5in]{fig6.eps} \caption{(Color
online) A three-dimensional view of Im$\left[ \protect\omega
\left( p\right) \right] $ as a function of $\Omega $ and
$\protect\epsilon $ for two branches of roots to
Eq.(\protect\ref{eigenvalue equation}). Parameters are same as in
Fig.~\protect\ref{fig5}. Units are defined in Sec.  \protect
\ref{classification section}.} \label{fig6}
\end{figure}

In order to have a more detailed look at various instabilities, we
make a three-dimensional plot in Fig.~\ref{fig6} for the imaginary
parts of the two (non-vanishing) branches of excitation
frequencies. A glance at this 3D view further confirms the
instability map of Fig.~4: one can easily associate the large bump
in Fig.~\ref{fig6}(a) to the instability of case 2, and the curved
band in Fig.~\ref{fig6}(b) to the instability of case 1. The
shapes of these unstable regions completely match with those shown
in Fig.~\ref{fig4}.

However, Fig.~\ref{fig6}(b) also reveals a new feature consisting
of a narrow strip along the $\Omega $ dimension. We can trace this
new feature to the source of instability associated with case 3
which has so far been ignored. An analysis with the help of
Eqs.~(\ref{a10}) and (\ref{a20}) indicates that the condition
$a_{20}>a_{10}^{2}/4$ amounts to
\begin{eqnarray}
&&\epsilon ^{\prime }\left[ \epsilon ^{\prime 3}+\left( \Omega
^{2}+4\alpha ^{2}\left\vert \psi _{a}^{0}\right\vert ^{2}\right)
\epsilon ^{\prime } \right. \nonumber \\ &&\left.+2\alpha
^{2}\left\vert \psi _{a}^{0}\right\vert ^{4}\left( \lambda
_{g}-4\lambda _{ag}+4\lambda _{a}\right) \right] <0, \label{new
instability}
\end{eqnarray}%
from which we find that to a good approximation, the unstable
$\epsilon $ is bounded between
\begin{subequations}
\begin{eqnarray}
\epsilon_1 &=& \Lambda _{a}\left\vert \psi _{a}^{0}\right\vert
^{2}+\Lambda _{g}\left\vert \psi _{g}^{0}\right\vert ^{2}\,, \\
\epsilon_2 & \approx &\epsilon_1-2\alpha ^{2}\left\vert \psi
_{a}^{0}\right\vert ^{4}\,\frac{ \lambda _{g}-4\lambda
_{ag}+4\lambda _{a}}{ \Omega ^{2}+4\alpha ^{2}\left\vert \psi
_{a}^{0}\right\vert ^{2}}\,,
\end{eqnarray}
\end{subequations}
It can be shown that $\epsilon _1$ and $\epsilon _2$
are not very sensitive to $\Omega $, and the they merge together
in the limit of either small or large $\Omega $, which explains
the shape of this new instability region.

\section{Dynamical Signatures of Stable and Unstable CPT states}

In order to determine the fate of an unstable state, we must
examine its dynamical response to fluctuations, which can be
simulated using the time-dependent version of the GPEs, which are
obtained by replacing the left hand sides of Eqs.~(\ref{steady
state equation}) by $i \dot{\psi}_i$.

We take a CPT state associated with a Rabi frequency $\Omega _{0}$
and Feshbach detuning $\epsilon $ to be the initial state. To
simulate small fluctuations, we modify the Rabi frequency as
\begin{equation}
\Omega =\Omega _{0}+\left[ \delta \Omega _{0}+2\delta \Omega _{1}\sin \left(
px/\hbar \right) \right] ,  \label{Omega standing wave}
\end{equation}%
where $\delta \Omega _{0}$ and $\delta \Omega _{1}$ are small
perturbation amplitudes. Note that a position-dependent
perturbation represented by the $ \delta \Omega_1$ term can cause
spatial deformation of the uniform CPT state, making it possible
to trigger the instabilities due to perturbations of finite
momentum.

\begin{figure}[h]
\centering \includegraphics[width=2.5in]{fig7.eps} \caption{The
time evolution of the particle number densities in the condensate
mode probed by a zero-momentum perturbation with $\protect\delta
\Omega _{0}=0.001$ for (a) $\Omega _{0}=1.1$ and (b) $\Omega
=0.9$, while the Feshbach detuning is fixed at $\protect\epsilon
=3$. Other parameters are the same as in Fig.~\protect\ref{fig5}.
Units are defined in Sec.~\protect\ref{classification section}.}
\label{fig7}
\end{figure}
To test the instability of case 2, we take the same collisional parameters
as in Fig.~\ref{fig5}, but fix the Feshbach detuning to $\epsilon =3$. We
apply a zero-momentum perturbation with an amplitude $\delta \Omega
_{0}=0.001$ and $\delta \Omega _{1}=0$ to the CPT state with (a) $\Omega
_{0}=1.1$ and (b) $\Omega _{0}=0.9$, respectively. By inspecting the $%
\epsilon =3$ curve of Fig.~\ref{fig5}(b), which survives even when
$p=0$, we anticipate that the dynamics of the system will change
from a stable to an unstable one as $\Omega _{0}$ varies from
$1.1$ to $0.9$. The results, shown in Fig.~\ref{fig7}(a) and (b),
clearly support our analysis. The stable CPT state does not
respond significantly to the small perturbation; while
Fig.~\ref{fig7}(b) shows a large-amplitude self-pulsing behavior,
characteristic of an unstable system. In addition, there is a
noticeable population in the unstable molecular state. Such
self-pulsing oscillations are also familiar features in the study
of unstable nonlinear optical systems \cite{pierre}.

To demonstrate the instability of case 1, we need to apply a
finite-momentum perturbation since for case 1, the imaginary part
of the excitation frequency $\omega _{0}$ vanishes at $p=0$. The
finite-momentum perturbation will couple together different
momentum modes. To simulate this process, we adopt the Floquet
technique and expand the wave functions as
\begin{equation}
\psi _{i}\left( \mathbf{r},t\right) =\sum_{l=-\infty }^{+\infty
}\phi _{i}\left( l,t\right) e^{ilpx/\hbar }\,,
\end{equation}%
where $\left\vert \phi _{i}\left( l,t\right) \right\vert ^{2}$ represent the
particle density of the $i$th species with momentum $lp$. \ Here, $l$ is an
integer and $p=2\pi \hbar /L$, with $L$ being the length of the condensate
along $x$ dimension. Then the orthonormality condition for the momentum
modes reads
\begin{equation}
\frac{1}{L}\,\int e^{i\left( l-l^{\prime }\right) px/\hbar
}dx=\delta _{l,l^{\prime }}\,.
\end{equation}
This enables us to transform the dynamical GPEs for $\psi _{i}\left( \mathbf{%
r},t\right) $ into the coupled modal equations for $\phi _{i}\left(
l,t\right) $
\begin{widetext}
\begin{subequations}
\begin{eqnarray}
i\frac{d\phi _{a}\left( n\right) }{dt} &=&-\epsilon_p n^{2}\phi
_{a}\left( n\right) +\sum_{l,l^{\prime }}\sum_{s}\lambda _{as}\phi
_{s}\left( l\right) \phi _{s}^{\ast }\left( l^{\prime }\right)
\phi _{a}\left( n+l^{\prime }-l\right) +\alpha \sum_{l}\phi
_{m}\left( l\right) \phi _{a}^{\ast }\left(
l-n\right) \,, \\
i\frac{d\phi _{m}\left( n\right) }{dt}
&=&-\frac{\epsilon_p}{2}n^{2}\phi _{m}\left( n\right)
+\sum_{l,l^{\prime }}\sum_{s}\lambda _{ms}\phi _{s}\left( l\right)
\phi _{s}^{\ast }\left( l^{\prime }\right) \phi
_{m}\left( n+l^{\prime }-l\right) +\epsilon \phi _{m}\left( n\right) \nonumber \\
&&+\frac{\alpha }{2}\sum_{l}\phi _{a}\left( l\right) \phi
_{a}\left( n-l\right) -\frac{\Omega _{0}+\delta \Omega
_{0}}{2}\phi _{g}\left( n\right) -\frac{\delta \Omega
_{1}}{2i}\left[ \phi _{g}\left( n-1\right) -\phi
_{g}\left( n+1\right) \right]\, , \\
i\frac{d\phi _{g}\left( n\right) }{dt}
&=&-\frac{\epsilon_p}{2}n^{2}\phi _{g}\left( n\right)
+\sum_{l,l^{\prime }}\sum_{s}\lambda _{gs}\phi _{s}\left( l\right)
\phi _{s}^{\ast }\left( l^{\prime }\right) \phi _{g}\left(
n+l^{\prime }-l\right) +\left( \Delta +\epsilon \right) \phi
_{g}\left( n\right) \nonumber \\
&&-\frac{\Omega _{0}+\delta \Omega _{0}}{2}\phi _{m}\left( n\right) -\frac{%
\delta \Omega _{1}}{2i}\left[ \phi _{m}\left( n-1\right) -\phi
_{m}\left( n+1\right) \right]\, .
\end{eqnarray}%
\end{subequations}
\end{widetext}
The initial condition is then equivalent to
\begin{equation}
\phi _{a}\left( 0,0\right) =\left\vert \psi _{a}^{0}\right\vert
,\phi _{m}\left( 0,0\right) =0,\phi _{g}\left( 0,0\right)
=\left\vert \psi _{g}^{0}\right\vert \,,
\end{equation}%
and $\phi _{i}\left( l\neq 0,0\right) =0$, where $\left\vert \psi
_{a}^{0}\right\vert $ and $\left\vert \psi _{g}^{0}\right\vert $
are determined from Eqs.~(\ref{CPT distribution}) with $\Omega $
being replaced by $\Omega _{0}$. We now switch to a set of
collisional parameters with $\lambda _{a}=0.625$, $\lambda
_{m}=\lambda _{g}=\lambda _{mg}=0.5\lambda _{a}$, $\lambda
_{am}=\lambda _{a}$, and $\lambda _{ag}=1.4\lambda _{a}$, which
puts the system in Region I, where the two instabilities regions
of case 1 and 2 do not overlap in the same parameter space (see
the two diagrams in the first row of Fig.~\ref{fig4}). We take
$\epsilon =-4$ and $ \Omega _{0}=1$, which puts the system in the
unstable region of case 1. This allows us to demonstrate the
instability due solely to the finite-momentum perturbation.
Indeed, simulations show that the system is stable against
zero-momentum perturbations but is unstable when finite-momentum
perturbations are applied. A dynamical result of a finite-momentum
perturbation is displayed in Fig.~\ref{fig8} which shows that the
instability leads to an irreversible population transfer from the
condensate mode ($p=0$) to non-condensate modes ($p\neq 0$). The
initial growth of the non-condensate modes is approximately
exponential but quickly becomes rather chaotic, which is typical
for an unstable multi-mode system \cite{ling01}.
\begin{figure}[h]
\centering \includegraphics[width=2.8in]{fig8.eps} \caption{The
time evolution of $\left\vert \protect\phi _{g}\left( l,t\right)
\right\vert ^{2}$ ($l=0,1,$ and $2$)\ after a finite-momentum
perturbation with $\protect\delta \Omega _{1}=0.001$ at $p=0.01$
$\left( \protect\delta \Omega _{0}=0\right) $ is applied to a
system initially prepared in a CPT\ state of $\protect\epsilon
=-4$ and $\Omega _{0}=1$. We include modes ranging from
$l=-10$ to 10 in the simulation. Other parameters are $\protect%
\lambda _{a}=0.625$, $\protect\lambda _{m}=\protect\lambda
_{g}=\protect \lambda _{mg}=0.5\protect\lambda _{a}$,
$\protect\lambda _{am}=\protect \lambda _{a}$, $\protect\lambda
_{ag}=1.4\protect\lambda _{a},$ and $n=1$. Units are defined in
Sec.~\protect\ref{classification section}.} \label{fig8}
\end{figure}

\section{Conclusion}

To summarize, using a generalized Bogoliubov approach, we have
calculated the collective excitation spectra of the CPT states
involving atomic and stable molecular condensates. We have found,
through our analysis of the excitation spectra, that collisions
involving atoms and stable molecules strongly affect the stability
properties of the CPT state. We have developed a set of analytical
criteria for classification of various instabilities in the
long-wavelength limit. We have shown that these criteria can be
used to accurately identify the unstable parameter regimes. In
this paper, we studied a homogeneous system. It will be desirable
in the future to extend this work to the inhomogeneous case where
trapping potentials are present, as in typical experimental
situations. Nevertheless, our work here should provide important
analytical insights into the much more complicated inhomogeneous
problem, for which the only viable treatment is perhaps through
numerical simulations.

Just as the usual CPT states in linear three-level $\Lambda
$-systems are behind many important applications in quantum and
atom optics \cite{arimondo96}, we anticipate that the nonlinear
CPT states will also play important roles in applications where
the coherence between atom and molecule condensates becomes
critical. Evidently, the success of these applications depends on
our ability to avoid the unstable regimes of the CPT state. For
example, high efficiency coherent population transfer between the
atomic and molecular condensate using STIRAP can only be achieved
if these unstable regimes are not encountered during the transfer
process \cite{ling04}. With this in mind, we believe that our work
will have important implications in the coupled atom-molecule
condensates \cite{heinzen00}, a system of significant current
interest, with ramifications in other nonlinear systems.

\bigskip

\section{Acknowledgments}

HYL acknowledges the support from the US National Science
Foundation under Grant No. PHY-0307359. We thank Dr. Su Yi for
many discussions and help with some of the figures.

\bigskip
\begin{appendix}%

\section{Asymptotes}

A qualitative understanding of the instability map shown in Fig.~4 depends
on the asymptotic behavior of the threshold Feshbach detunings $%
\epsilon_0^{th}$ and $\epsilon_1^{th}$ which we show below. In the limit of
large $\Omega /\alpha $
\begin{eqnarray}
\epsilon _{0}^{th} &\approx &\Lambda _{a}+\frac{\alpha ^{2}}{8\left(
4\lambda _{a}+\lambda _{g}-4\lambda _{ag}\right) }\left( \frac{\Omega }{%
\alpha }\right) ^{4},  \label{e0 large Omega} \\
\epsilon _{1}^{th} &\approx &\Lambda _{a}+\frac{\lambda _{a}\alpha ^{2}}{%
8\left( \lambda _{a}\lambda _{g}-\lambda _{ag}^{2}\right) }\left( \frac{%
\Omega }{\alpha }\right) ^{4},  \label{e1 large Omega}
\end{eqnarray}
and in the limit of small $\Omega /\alpha $
\begin{widetext}
\begin{eqnarray}
\epsilon _{0}^{th} &\approx &0.5\Lambda _{g}+\frac{2\alpha ^{2}}{4\lambda
_{a}+\lambda _{g}-4\lambda _{ag}}+\left( \frac{2\Lambda _{a}-\Lambda_{g}}{2\sqrt{2}}+\frac{%
\sqrt{2}\alpha ^{2}}{4\lambda _{a}+\lambda _{g}-4\lambda _{ag}}\right) \frac{\Omega }{\alpha },
\label{e0 small Omega} \\
\epsilon _{1}^{th} &\approx &0.5\Lambda _{g}+\frac{0.5\alpha ^{2}\lambda _{g}}{%
\lambda _{a}\lambda _{g}-\lambda _{ag}^{2}}+\frac{\lambda _{ag}^{2}\left(
2\lambda _{ag}-\Lambda _{g}\right) +\lambda
_{a}\lambda _{g}\left(\Lambda _{g}-2\Lambda
_{a}\right) -2\alpha ^{2}\lambda _{ag}}{2\sqrt{2}\left( \lambda
_{ag}^{2}-\lambda _{a}\lambda _{g}\right) }\frac{\Omega }{\alpha },
\label{e1 small Omega}
\end{eqnarray}
\end{widetext}

\end{appendix}%

\setcounter{secnumdepth}{-1}%

redpajama_set_name: RedPajamaArXiv


text:
Katie Yoder
Former Associate Culture Editor
Survey: 80% of Dads Say Media Portrayals of Fatherhood Wrong
Well this is a pleasant surprise: the media incorrectly depict fathers, according to today's dads – and, for once, the media agree.
Just in time for Father's Day, soap brand Dove hired Edelman Berland to survey 1,000 dads for Dove Men+Care. The dads challenged the media's representation of fathers – with 80 percent agreeing the media fail to portray fathers roles accurately. When asked what attributes describe fathers in the media, dads most frequently responded with "disconnected, bumbling and incompetent." Videos Below.
In regards to the media, Dove explained, "Three quarters of dads say they are responsible for their child's emotional well-being, while only 20% of dads see this role reflected in media." Other findings included: "Dads today are caring for their children's emotional well-being (74%) and taking responsibility for their daily needs (51%)."
The company concluded, "It's time to acknowledge the caring moments of fatherhood that often go overlooked."
In response, Dove produced a "Calls for Dad" ad "to overturn antiquated representations of dads and encourage recognition of them as caregivers." A press release highlighted how the short film, now viral with over 2 million views, "features real moments that children of all ages share with their fathers, most of which are regrettably absent from media depictions of dads today."
"Dove Men+Care has a history of celebrating real men and we know nothing is more important to men today than caring for their children," commented Unilever Skincare's (Dove manufacturer) Vice President of Marketing Rob Candelino.
Amazingly, the media aren't ignoring the criticism – in fact, they're reporting on it. The ad, deemed "very moving" by NBC's Willie Geist, aired during "Today" on June 12. Candelino later spoke with Geist and criticized, "Frankly, they're [dads] seeing their role expand, but advertising, TV and the movies aren't depicting that back to that."
He continued: "There's a gap today between what is actually happening at kitchen tables and in backyards across this country and how men are in fact are seeing themselves reflected back in the media. That's what inspired us to create this film, to be honest."
Continuing the theme, NBC co-host Savannah Guthrie later addressed anchor Al Roker saying, "Al, you've been saying this for as long as I've known you that you always have this image in the media of these bumbling dads and that's not like you or any of the dads we know." Roker responded, "No, we're not morons."
Other media including The Huffington Post, TIME, AdWeek, The New Yorker, The Washington Post, Celebuzz, local Fox sites, and Daily Mail either noted the survey or praised the new ad.
The survey also appeared in NBC's "Today" (albeit as a paid advertisement), which summarized the findings as, "In other words, dads want to be portrayed as the capable caregivers they are—and many media portrayals are still getting it wrong."
To share an appreciation for dads, Dove Men+Care asks dads and families "to submit #RealDadMoments @DoveMenCare on Facebook, Twitter and Instagram, which will be heralded on www.Dovemencare.com."

redpajama_set_name: RedPajamaCommonCrawl


text:
package us.nineworlds.serenity.events;

import us.nineworlds.serenity.common.media.model.IMediaContainer;

public class MovieSecondaryCategoryEvent extends SerenityEvent {

  private String key;
  private String category;

  public MovieSecondaryCategoryEvent(IMediaContainer mediaContainer, String key, String category) {
    super(mediaContainer);

    this.key = key;
    this.category = category;
  }

  public String getKey() {
    return key;
  }

  public String getCategory() {
    return category;
  }
}

redpajama_set_name: RedPajamaGithub


text:
Reconsiderations
Cory Panshin on July 27, 2013
Sometimes I feel as though this blog is crawling along one tiny step at a time, but then a dam suddenly breaks and sets off a cascade of reconsiderations. The previous entry triggered one of those avalanches, and I'm still sorting out its implications.
When I initially developed my theory of the visions, I assumed that the basic unit of the cycles was the individual vision and that what kept things moving along was the gradual loss of transcendence as each vision matured. This was adequate as a working hypothesis, but it never answered certain fundamental questions: Precisely what happens when a new vision is born? How and why did the cycle of replacement get started? And why does the same pattern recur in such similar form from one turn of the cycle to the next?
Now I'm realizing that my problems arose from trying to take the visions in isolation and that they are better seen as forming an ecosystem in which each one reflects all the others. When any vision falters, it leaves an empty environmental niche, and the system is thrown out of balance until it can respond by repairing the damage and filling the gap.
This self-maintaining quality might be compared to the ability of DNA to repair itself — an ability that marks the dividing line between non-living matter and living beings. In a similar way, the self-repairing ability of the system of visions might be seen as a crucial accomplishment that pushed us across the threshold from almost-but-not-quite-human to fully human.
At the present time, the ecosystem of the visions is extremely complex, with emerging visions, mature visions, and the remnants of discarded visions all jostling for position. But it started off some 250,000 years ago with a simple triad of highly transcendent visions, one of each of the three types, all working together to present a complete image of reality.
Since last winter, I've been dwelling on what happened when that first triad was disrupted by the arrival of an ice age which forced the oldest vision to assume worldly authority. That's because I've been operating on the assumption that the first three visions were somehow spontaneous and organic and that the mechanisms of the cycle came into play only when successors to those visions had to be invented.
Not so. The human enterprise has demanded profound creativity in response to higher knowledge at every step of the way, and it has never been simple or automatic. The invention of the first triad of visions was, if anything, more of a work of genius than all the replacements that have simply followed in its footsteps.
It's conceivable that the birth of the first vision of the natural world, when our species itself was newly made, was characterized by a kind of childlike innocence. At that time, there were none of the divisions that would later typify human life. People were at one with Nature. The waking world flowed imperceptibly into the world of dream. Humans were few and it was easy for them to maintain their coordination through music, dance, and the marvelous new instrument of language. The mental maps we created of the world around us had ample room for everyone and everything, and they were shared in nearly identical form by the entire community
But as that community expanded, its original unity was lost. People had settled in a variety of locations and were too widely dispersed to come together on a regular basis, so their knowledge and beliefs began to diverge. Their mental maps were no longer identical, and different groups began to perceive one another not as family but as strangers.
The kinship vision was designed to resolve that crisis by reasserting the fact of human unity. It told us that we were all still cousins, great-great-grandchildren of a single First Ancestor, even if we could no longer spell out the precise relationships.
However, the kinship vision was incapable of restoring our original perception of the oneness and harmony of the natural world. Instead, it encouraged people to regard existence as the product of a series of logical oppositions — life and death, water and fire, human and animal — patterned on the same male-female duality that was the organizing principle of human kinship.
This solution represented a great intellectual achievement, but it can't ever have been completely satisfactory. In effect, it said, "The world is broken. Get over it." And most people were prepared to do just that — but not the proto-shamans.
The weirdos and dreamers, the half-crazy ones who heard voices and saw visions, knew from their own experience that the world was larger and more varied than any catalog of dualities could encompass. They were also familiar with the ability of higher knowledge to reconcile opposites and integrate apparent contradictions. There was no way they could settle for a worldview that was wedded to rigid and unresolvable oppositions.
The special knowledge of these proto-shamans alienated them from ordinary society and forced them to follow their own distinctive path. In response, they began to seek one another out and engage in active experimentation. They discovered various shamanistic techniques, almost certainly including the use of psychedelic plants, and as they learned how to induce visionary experiences at will, they developed a unique view of existence as a endless stream of hallucinatory transformations.
At first, this only increased their alienation. They may even have come to doubt the reality of the physical world, as often happens just before the birth of a new inner experience vision. But eventually a period of crisis came along, perhaps as the result of a temporary downturn in the climate, and it allowed society in general to question its assumptions and open up to the whispers of higher knowledge.
The shamans were more than ready to take advantage of this window of opportunity, and they used it to arrive at two remarkable conclusions. One was the concept of a spirit realm that existed alongside the realm of matter but had its own laws and was far more fluctuating and phantasmagorical. The other was the idea that even the realm of matter was not as stable as it appeared but might alter in unforeseen ways or produce prodigies when under the influence of spirit.
Neither of these conclusions was innately obvious. In all probability, neither is true. But they were probably the most powerful and influential ideas that have ever been put forth in the long span of human history, and their invention was a work of almost inconceivable brilliance.
In the previous entry, I suggested that whenever a new triad of visions is assembled, the birth of its youngest member is the final step. But it seems more likely that the appearance of the newest vision is actually the first step — a blast of pure and undiluted transcendence that sets off all the changes which follow.
The initial step in the transition of 250,000 years ago would have been the birth of the spirit vision, based on the premise that shamans were in communication with a realm of being outside the world we know. That was followed by the reformulation of the ancient vision of the natural world to incorporate a tendency to alter abruptly in response to the workings of spirit.
In earlier entries, I've referred to that vision as the "transformation vision," and I've related it to the ancient pastime of string figures and the mysteries of female magic. However, it didn't start out that way. It took on its transformative nature only after it was touched by the power of the spirit vision and began to see the world as a place of flux.
In fact, that may be the trajectory of every vision. They all begin with a belief that they are expressing eternal truths, grudgingly accommodate themselves to worldly imperfection, and finally come back round with a more sophisticated understanding of transcendence as a force that flows into ordinary reality from beyond.
The transformation vision reached that third stage under the influence of the newborn spirit vision, and together they formed a solid basis for perceiving the world as arising from the interaction of matter and spirit. However, there was still one piece that remained to be fitted into the picture in order to encompass the totality of human experience, and that was the kinship vision.
Like holism in the 1920s or horizontalism at the peak of political correctness in the 1970s, kinship was very much caught up in doing its own thing. But although it could not be tightly integrated into the triad, it could be more loosely attached through symbolic equivalences. For example, the spirit beings consulted by the shamans were often considered to be spirits of the ancestors.
But the most powerful means of integration was stories of the Dreamtime. These reaffirmed the new principle of transformation by being set in a time when the world lacked all the familiar dualities. Perhaps death had not yet come into being. Or there were only women and no men. Or there were no sun and moon in the sky, or no dividing line between animal-people and people-people.
The central figures of these stories was a shamanistic trickster figure whose eccentric pranks gradually shaped the world as we now know it. But this same figure was simultaneously identified as the respected First Ancestor of the kinship vision. And such is the reconciling power of higher knowledge that it was possible to believe both things at once and thereby see the world as a whole.
Like any such compromise, this wasn't a perfect solution, and in time it would come unglued. But it did the job well enough, and it formed the template for all the equally imperfect solutions that would come after.
Read the Previous Entry: Triads of Transcendence
Read the Next Entry: Solve et Coagula
Categories: Dance of the Visions, Deep Prehistory


text:
HomeLaw CollegeLaw College in Bihar Soghra Law College, Bihar Shariff, Nalanda
Soghra Law College, Bihar Shariff, Nalanda
COLLEGE NAME: Soghra Law College, Bihar Shariff, Nalanda
On the pressing demands of the muslim intelligentia and the General mass of the locality the members of the general Committee of Soghra Waqf Estate , Bihar-sharif felt the necessity and desirability of etablishing a Muslim Minority college for imparting education to under graduate and post graduate level in Arts, Science, Commerce, Law, Technology, Medical Science, Education, Library Science and Journalism, under the direct management , Control and administration of Soghra Waqf Estate, Biharsharif.
Consequently, the Soghra Waqf Estate General Committee in its meeting held on 5th August, 1973 passed a resolution to establish a muslim minority College in the name of the trustee Bibi Soghra under article,29 and 30 of the Constitution of India and Constituted a Governing Body to Start the College and donated eight acres of land to the College at Biharsharif and deposited the inspection-fee of the College as well as the security money of rupees Seventy Thousand and thirty thousand respectively for the affiliation of the College.
Prof. Shafi Ahmad, the Vice-president of Soghra Wakf Estate was made the Convenor of the College to organise the institution on the premises of Soghra High School. Shri S. Owais Alam, Headmaster, Soghra School was made Prof.-in-charge of the College. The Constitution drafting Committee was given the task to draft a Constitution in the light of article 29 & 30 of the Constitution of India under which religious and linguistic minorities have been given the right to establish manage Control and administer their institutions.
Fee Structure Arts & Commerce(Intermediate)
15/- Per Year
180/- Per Year
Development Fees
Fee Structure Arts & Commerce(Under Graduate)
Science Dev. Fees
A Candidate seeking admission from a recognized school and after completing Graduation in any discipline. Minimum 45% marks in graduation for general and 40% marks for reserved categories.
Maharaja College, Arrah
Min. Vol: 1 undefined
Shri Karpoori Thakur Vidhi Mahavidyalaya, Buxer
T.N.B. Law College, Bhagalpur
Rohtas Mahavidyalaya, Sasaram
A. N. S. College, Barh


*25 PACK* - SAVE- 8 SMD #89 OR #906 FLEXIBLE FLASHER *25 PACK* - SAVE - TWIN 2835 SMD PURPLE PASSION 6.3V AC/DC SUPREME BRIGHTNESS *100 PACK* - SAVE-6 LED 6.3V AC/DC BRILLIANT CRYSTAL FAN SHAPE LED - Only $1.19 Each! *25 PACK* - SAVE- 1 WIRED 6.3V AC/DC ULTIMATE OPTIX SUPER FLUX NO FLICKER, REDUCED STROBING! !


text:
Tags: underway
Home underway
Work is underway on quantum computer cleanroom in Abu Dhabi
31-Mar-2021 Hi Tech Manufacturing Abu Dhabi's Technology Innovation Institute is building the UAE's first quantum computer Work is underway to build a quantum computer in Abu Dhabi, in the United Arab Emirates, ushering an important milestone for the region in this breakthrough era in computing. Technology Innovation Institute (TII), the dedicated 'applied research' pillar of Abu Dhabi's newly established Advanced Technology Research Council (ATRC), has announced that its Quantum Research Centre (QRC) team led by its Chief Researcher Professor José Ignacio Latorre, will construct the quantum computer in the UAE capital, in collaboration with Barcelona-based Qilimanjaro Quantum Tech researchers. "We are at the cusp of a new era with the advent of quantum computing," H.E. Faisal Al Bannai, Secretary General of ATRC, said. "We are...


text:
Summer palace in Winter
Some believe that the development of the picturesque style in Europe was directly inspired by tales from travelers returning from China.
One example of a garden that may have inspired Europeans in this way is the Garden of Clear Ripples created in 1750 by Emperor Qianlong.
The hill and pond structure that we see in this garden, we also see in the scholar gardens of Suzhou and in Japanese gardens. Dominant features in the Garden of Clear Ripples include Kunming Lake, which makes up 75% of the garden, Longevity Hill, and three islands connected by bridges.
Traditional buildings enhance the garden and provide places to pause, relax. The garden is laid out to provide views and vignettes from key vantage points along the path. The overriding theme is harmony with nature; some elements mirror features of the natural scenery found beside the Yangtze River.
A sign along the main garden path explains that this section was inspired by a painting from around 1300 and reminds us that this garden was destroyed in 1860 and rebuilt in 1992. (click on image to read the text)
On the winter day of my visit, the garden was quite busy. I was told by my guide that during the spring and summer this garden is packed. A pile of dragon boats stored in a cove wait for warmer weather to carry passengers into the lake.
The story of this garden follows the modern history of China's relations with the West which resulted in its destruction, and re-creation several times.
In the 1840's, silk, porcelain, and tea exports from China flooded into Europe, but few Western goods made it into China. In an attempt to offset the trade imbalance, the British East India Company began importing opium from India into China via middlemen. Chinese efforts to stop this led to the Opium Wars. During the 2nd Opium War in the 1850's, Anglo-French forces destroyed the garden.
Abandoned for a generation, the garden was rebuilt in the 1890's and renamed the Summer Palace for the Empress Dowager Cixi. When the Empress allied herself with the Boxers, an anti-foreign and anti-Christian group, attacks on missionaries and diplomats resulted in another Western military mission in China. During the invasion in 1900, the garden was destroyed once again. Today the garden holds pride of place for the Chinese and is protected as a Cultural Relic.
Learn more about visiting this garden here.
Freer Gallery of Art and Arthur M. Sackler Gallery. NY Times. October 27, 2013. Empress Dowager Cixi in the gardens of the Summer Palace. Click on photo to check out June Chang's new biography about Empress Cixi.
In Chinese Tags garden history, historic gardens, Summer Palace, Beijing, Empress Cixi, garden stories


text:
Home Android Twitter Outlines What World Leaders Are Restricted To Tweet.
AndroidiPhoneMobileTechnology
Thursday, 17 October 2019, 06:21 EDT
Modified date: Thursday, 17 October 2019, 06:21 EDT
Twitter Outlines What World Leaders Are Restricted To Tweet.
In history, Twitter has been remarkably resistant to take steps against world leaders who skirt throughout the edges of its laws and rules. Now the company has attempted to clarify its reasoning on policing the tweets of elected Politicians and officeholders, and what behaviors would actually pass over the currently technical line.
In the support, Twitter announced it on the side of transmitting the content up if there is a clear public concern in doing so even if tweets by world leaders unless violating content policies. But as published in June, such posts would be set "behind a notice providing circumstances about the rule infringement," and steps would be taken to conquer the tweet's algorithmic limit.
This includes "clear and straightforward" warnings of violence on individuals, supporting terrorism, bestowing private info, posting "private" media without permission, behavior connected to child exploitation or encouraging self-harm.
Don't expect gruff action in other circumstances, though. Twitter repeated that it would drop violating tweets online (with a warning) if there's a "clear public concern" to do so. The aim is to form a space for public discussion and information, the company stated, and it wants to assure that people can participate in that conversation "freely and securely."
It's also not concerned about the intended message of a delivered tweet. Twitter wants to concentrate on the original language of a tweet, not "all possible explanations."
So there you possess it, world leaders on Twitter are assessed against the corresponding set of rules as everyone else on Twitter. The only distinction it seems is that Twitter clearly considers there are situations where even if a world leader violates Twitter's limits, it is in the public interest to devise a tweet public so that leaders can be kept responsible. Only in very severe situations would Twitter take a step against a world leader, as it would be any other account practicing the service.
Hey, I am Abdul From National Capital. I am having a keen interest in technology and the latest innovations and I Love Writing blogs. so for this, I keep myself updated and I am so grateful for sharing my blogs to you via The Digital Wise.
Taika Waititi Is Being Eyed And All Set For Releasing His...
RANJITA PAUL - Modified date: Friday, 17 January 2020, 06:18 EST 0
FIFA: After So Much Speculations In Reality, Why VAR Is Still...
Hulk: Will The Character Ever Come Back? Here Are The Possible...
15-Year-Old Accused Of Opening Fire At A High-School Basketball Game Arrested
20-Year-Old Transgender Brutally Beaten And Left Covered With Blood In Middle...
High School Musical: The Series! Disney+ Release Date, Cast And Other...
Selena Gomez Reveals How It Feels To Date In Hollywood! But...
Legal Secretary Sued A Firm For age Discrimination After They Offered...
The Last Kingdom' Season 4 On Netflix: Release Date And All...
Bus Captures Fire In Fatal Accident, 20 Dead


text:
James A. Rawley, Papers
Title: James A. Rawley, Papers
Creator: Rawley, James A., 1916-2005
Preferred Citation: James A. Rawley, Papers (RG 12-14-16). Archives & Special Collections, University of Nebraska–Lincoln Libraries.
Born in 1916, James A. Rawley was a native of Terre Haute, Indiana. He earned his undergraduate and masters degrees in history from the University of Michigan and his doctorate from Columbia. Rawley taught at Columbia University, Hunter College in New York, and at Sweet Briar College in Virginia prior to starting his career at the University of Nebraska - Lincoln in 1964. He chaired the UNL History Department in 1966-1967 and from 1972 until 1982. Rawley served as Acting Dean of the University Libraries in 1984-1985. He retired from teaching in 1987 as the Carl Adolph Happold Professor of History (Emeritus) at the University of Nebraska - Lincoln. Rawley died on November 29, 2005.
Rawley authored several books and received numerous honors. He was named a Fellow of the Royal Historical Society, the Society of American Historians, and the National Endowment for the Humanities. He served for 18 years as a board member and for two years as president of the Board of Directors for the Nebraska State Historical Society. In addition, Rawley served two years on the Board of Directors for the Nebraska Council for the Humanities Foundation.
The Rawley Papers consist of six boxes of materials covering the years 1938-2001. Included in the collection are correspondence with members of the University of Nebraska History Department and the Nebraska State Historical Society, manuscripts of some of Rawley's published works, articles and reprints, and biographical materials about Rawley. Additional materials have been added to Rawley's papers and an inventory is available from the repository.
Rawley, James A., 1916-2005
United States -- Race relations
University of Nebraska-Lincoln -- History Department
Series Description: Series 1: Correspondence, Box 1, Folder 1-8
This series contains correspondence to and from Rawley. Most of the correspondence relates to the activities of the UNL History Department. Other correspondence relates to his work on the Nebraska State Historical Society Executive Board.
Series 2: Manuscripts, Boxes 1-5
Included in this series are manuscripts, galley proofs, and galleys of several of Rawley's books, including Turning Points of the Civil War (1966), Race and Politics: "Bleeding Kansas" and the Coming of the Civil War (1969), The Transatlantic Slave Trade: a History (1981), and Secession: the Disruption of the American Republic, 1844-1861 (1990).
Series 3: Articles and Reprints, Box 5
This series consists of one folder of articles and reprints authored by Rawley.
Series 4: Biographical Materials, Box 6
Included in series 4 is one folder containing curriculum vitae and a photograph of Rawley as well as news releases and news clippings.
Container List: Series 1: CorrespondenceBox 1. Folder 1. Correspondence, 1965-1974 Box 1. Folder 2. Correspondence, 1975-1976 Box 1. Folder 3. Correspondence, 1977-1979 Box 1. Folder 4. Correspondence, 1980 Box 1. Folder 5. Correspondence, 1981 Box 1. Folder 6. Correspondence, 1982-1986 Box 1. Folder 7. Correspondence, Nebraska State Historical Society, 1985-1986 Box 1. Folder 8. Correspondence, undated Series 2: ManuscriptsBox 1. Folder 9. Turning Points of the Civil War, pgs. 1-150 Box 2. Folder 1. Turning Points of the Civil War, pgs. 151-index Box 2. Folder 2. Race and Politics, pgs. 1-150 Box 2. Folder 3. Race and Politics, pgs. 151-300 Box 2. Folder 4. Race and Politics, pgs. 301-409 Box 2. Folder 4. Race and Politics, final draft, pgs. 1-150 Box 2. Folder 4. Race and Politics, final draft, pgs. 151-300 Box 3. Folder 1. Race and Politics, final draft, pgs. 301-393 Box 3. Folder 2. Transatlantic Slave Trade, pgs. 1-150 Box 3. Folder 3. Transatlantic Slave Trade, pgs. 151-300 Box 3. Folder 4. Transatlantic Slave Trade, pgs. 301-450 Box 3. Folder 5. Transatlantic Slave Trade, pgs. 451-600 Box 3. Folder 6. Transatlantic Slave Trade, pgs. 601-751 Box 4. Folder 1. Transatlantic Slave Trade, proofs, pgs. 1-150 Box 4. Folder 2. Transatlantic Slave Trade, proofs, pgs.151-254 Box 4. Folder 3. Transatlantic Slave Trade, galley proofs, part 1 Box 4. Folder 4. Transatlantic Slave Trade, galley proofs, part 2 Box 4. Folder 5. Transatlantic Slave Trade, galley proofs, part 3 Box 4. Folder 5. Transatlantic Slave Trade, page proofs, part 1 Box 4. Folder 6. Transatlantic Slave Trade, page proofs, part 2 Box 5. Folder 1. Transatlantic Slave Trade, page proofs, part 3 Box 5. Folder 2. Secession, pgs. 1-150 Box 5. Folder 3. Secession, pgs. 151-263 Box 5. Folder 4. Secession, proofs Box 5. Folder 5. Secession, page proofs, part 1 Box 5. Folder 6. Secession, page proofs, part 2 Series 3: Articles and ReprintsBox 5. Folder 7. Articles and reprints, 1938-2001, undated Series 4: Biographical MaterialsBox 6. Folder 1. Biographical materials
Related Material and Resources: Rawley, James A. Abraham Lincoln and a Nation Worth Fighting For. Wheeling, Ill.: Harlan Davidson, c1996. ______________, editor. The American Civil War, an English View. Charlottesville, University Press of Virginia [1964]. ______________. Edwin D. Morgan, 1811-1883; Merchant in Politics. New York, Columbia University Press, 1955. ______________, editor. Lincoln and Civil War Politics. New York, Holt, Rinehart and Winston [1969]. ______________. The Politics of Union : Northern Politics During the Civil War. Lincoln: University of Nebraska Press, 1980, c1974. ______________. Race and Politics : "Bleeding Kansas" and the Coming of the Civil War. Lincoln: University of Nebraska Press, 1979, c1969. ______________. Secession : the Disruption of the American Republic, 1844-1861. Malabar, Fla. : R.E. Krieger Pub. Co., 1990. ______________. The Transatlantic Slave Trade : a History. New York: Norton, c1981. ______________. Turning Points of the Civil War. Lincoln: University of Nebraska Press, 1989, c1966.


text:
The Case of JPMorgan and Jefferson County, Alabama
Posted by Craig in bailout, economy, Financial Crisis, financial reform, financial regulation, Goldman Sachs, Politics, Wall Street
Alabama, Jamie Dimon, Jefferson County, JPMorgan Chase, Looting Main Street, Matt Taibbi, Rolling Stone
In a March 26 letter to shareholders Jamie Dimon, CEO of JPMorgan Chase, wrote:
"The crisis of the past couple of years has had far-reaching consequences, among them the declining public image of banks and bankers…[W]hen we vilify whole industries…we are denigrating ourselves and much of what made this country successful…We also should refrain from indiscriminate blame of any whole group of people…While JPMorgan Chase certainly made its share of mistakes in this tumultuous time, our firm always has remained focused on the fundamentals of banking and the part we can play to support our clients and communities."
One example of JPMorgan's "support" for their "clients and communities" and a reason for the "declining public image of banks and bankers" can be found in another in a long line of excellent pieces by Matt Taibbi at Rolling Stone, entitled "Looting Main Street: How the nation's biggest banks are ripping off American cities with the same predatory deals that brought down Greece"
The article is lengthy, but a must-read, in my opinion. It's the story of bribery, corruption, and fraud in Jefferson County, Alabama. Briefly (or maybe not so briefly), it goes like this.
In the early 90's the EPA sued the county in order to bring its antiquated sewer system into compliance with the Clean Water Act. In 1996 county commissioners decided to build the "Taj Mahal of sewage treatment plants" with cost estimates of $250 million. Taibbi:
"But in a wondrous demonstration of the possibilities of small-town graft and contract-padding, the price tag quickly swelled to more than $3 billion. County commissioners were literally pocketing wads of cash from builders and engineers and other contractors eager to get in on the project, while the county was forced to borrow obscene sums to pay for the rapidly spiraling costs."
Originally the plan was to pay for the project by increasing sewer rates. But as costs continued to escalate county commissioners knew that sooner or later customers would revolt over the ever-increasing rates, so they started looking for "creative financing." That's music to the banksters ears and, true to form, they came riding to the rescue with their gobbledegook of variable rate refinancing and "swaps."
Here's where local JPMorgan rep Charles LeCroy meets crooked politician, with local "wheeler-dealer" Bill Blount as the middle man:
"LeCroy paid Blount millions of dollars, and Blount turned around and used the money to buy lavish gifts for his close friend Larry Langford, who at the time had just been elected president of the county commission…Langford then signed off on one after another of the deadly swap deals being pushed by LeCroy. Every time the county refinanced its sewer debt, JP Morgan made millions of dollars in fees.
Even more lucrative, each of the swap contracts contained clauses that mandated all sorts of penalties and payments in the event that something went wrong with the deal. In the mortgage business, this process is known as churning: You keep coming back over and over to refinance, and they keep "churning" you for more and more fees."
But unbeknownst to LeCroy, Blount had a another suitor, Goldman Sachs. So:
"JP Morgan cut a separate deal with Goldman, paying the bank $3 million to [go away], with Blount taking a $300,000 cut of the side deal."
The payoff for JPMorgan?:
"The deals wound up being the largest swap agreements in JP Morgan's history. Making matters worse, the payoffs didn't even wind up costing the bank a dime. As the SEC explained in a statement on the scam, JP Morgan "passed on the cost of the unlawful payments by charging the county higher interest rates on the swap transactions."
In other words, not only did the bank bribe local politicians to take the [lousy] deal, they got local taxpayers to pay for the bribes. And because Jefferson County had no idea what kind of deal it was getting on the swaps, JP Morgan could basically charge whatever it wanted. According to an analysis of the swap deals commissioned by the county in 2007, taxpayers had been overcharged at least $93 million on the transactions."
As happens sooner or later with all Wall Street scams, the whole thing collapsed in early 2008. And as also happens with Wall Street scams, the banksters got the gold mine and the taxpayers of Jefferson County got the shaft.
But don't think this is an isolated incident. Taibbi concludes:
"The destruction of Jefferson County reveals the basic battle plan of these modern barbarians, the way that banks like JP Morgan and Goldman Sachs have systematically set out to pillage towns and cities from Pittsburgh to Athens. These guys aren't number-crunching whizzes making smart investments; what they do is find suckers in some municipal-finance department, corner them in complex lose-lose deals and flay them alive. In a complete subversion of free-market principles, they take no risk, score deals based on political influence rather than competition, keep consumers in the dark — and walk away with big money."
Any questions about that "declining public image" of banks and bankers, Mr. Dimon?


text:
#pragma once

#include "math_types.h"

namespace altertum
{
namespace aabb
{

/**
 * @file aabb.h
 * Extended implementation of AABB API
 * @note Does not use C++ constructors to maintain aggregate struct status
 */

/** returns true if the 2 boxes intersect, false otherwise*/
inline bool intersect(const AABB & bb1, const AABB & bb2)
{
    return(bb1.max.x > bb2.min.x &&
           bb1.min.x < bb2.max.x &&
           bb1.max.y > bb2.min.y &&
           bb1.min.y < bb2.max.y &&
           bb1.max.z > bb2.min.z &&
           bb1.min.z < bb2.max.z);
}

/** Initializes an AABB type with a minimum and maximum point.*/
AABB aabb(const Vector3 & min, const Vector3 & max)
{
	AABB box;
	box.min = min;
	box.max = max;
	return box;
}

/** Initializes an AABB type from another AABB.*/
AABB aabb(const AABB & box)
{
	AABB newBox;
	newBox.min = box.min;
	newBox.max = box.max;
	return newBox;
}

/** Returns the center point of a given box.*/
Vector3 center(const AABB & box)
{
	Vector3 centerPoint;
	centerPoint.x = (box.max.x + box.min.x) / 2.0f;
	centerPoint.y = (box.max.y + box.min.y) / 2.0f;
	centerPoint.z = (box.max.z + box.min.z) / 2.0f;
	return centerPoint;
}

/** Returns the size of the X value of a given box.*/
float xsize(const AABB & box)
{
	return (box.max.x - box.min.x);
}

/** Returns the size of the Y value of a given box.*/
float ysize(const AABB & box)
{
	return (box.max.y - box.min.y);
}

/** Returns the size of the Z value of a given box.*/
float zsize(const AABB & box)
{
	return (box.max.z - box.min.z);
}

/** Returns the volume of a given box.*/
float volume(const AABB & box)
{
	return (xsize(box) * xsize(box) * zsize(box));
}

/** Returns true if a point is contained within a box*/
bool contains(const AABB & box, const Vector3 & point)
{
	return ((point.x <= box.max.x) && (point.x >= box.min.x)
		&& (point.y <= box.max.y) && (point.y >= box.min.y)
		&& (point.z <= box.max.z) && (point.z >= box.min.z));
}
}; // namespace aabb
}; // namespace altertum


I don't know much about hair. My hair is essentially virgin hair. Aside from the occasional highlights that graced my locks in the early 2000s, it has never been tainted by harsh chemicals or colors. I also haven't spent a whole lot of time trying to learn about my hair. I know it's brown and that I've had it for almost my whole life, but my knowledge is pretty limited besides that. I do know one thing though. I hate cutting it. I can't stand it. It's an exhausting experience for me. I can't stand spending money on things like that. I like spending my money on tangible items that I can enjoy and admire in my grasp. The only thing tangible in getting a hair cut is the hair that took me weeks to grow, laying on the floor, only to get swept up like garbage. It's just terrible.
I understand; however, that getting a hair cut is a non-negotiable necessary evil. You see, when my hair grows, it turns into a beast. It doesn't grow down my back, or in any particular direction at all. It just starts growing all over the place. It feels like I have Medusa's snake head, mixed with a bits of string spun out of lead. One day I asked my barber, "Is there any way to grow this out, but make it do what I want to do?" The response was yes, but I had to train it. I had to have more regular hair cuts make sure that it grows in certain areas and not in others. More hair cuts?? No, thank you. I will let it grow till I'm miserable and then cut it. Kind of like where I'm at now.
But don't let those growing pains discourage you. There will be hurdles and bumps along the way, and that is totally normal. While it may require some changes in your normal workflow (that no one is used to) or a little extra time, always remember that these stages are temporary and can ultimately set you up for great success.
...and NOT the end of the world. I was telling an employee a few weeks ago that every person that has moved up in the company has been very overwhelmed at some point and worked through that. Anyone can be stretched to their breaking point, but you can persevere. Sometimes it may feel like just saying, " I gotta get this cut," but push through and you will reap the benefits.
MediCopy grows because of its employees (which I would confidently say applies to any growing company). Suggestions and ideas can make awkward and stressful stages, and make them more tolerable - possibly easier to get through.
Cutting corners is never a good practice. Making sure that distractions are kept at a minimum and allocating more time for a task at hand will make sure that not only you will get your work done, but also get it done correctly.
Our job here at MediCopy is to provide a good foundation for the business to grow. If we don't follow best practices or provide the best service for our patients, clients and customers, we will not grow. It's really that simple. Make sure you keep this in mind every time you speak with a client or customer or go about your work day.
In conclusion, we want MediCopy to grow to an epic Crystal Gayle scale and not look like Howie Mandel. Just Google those two people and you'll understand. With the proper care, the sky is the limit. Cheers!


text:
Harvard Professor, Author Visits Webster
Remember to RSVP for the talk on Feb. 10
Celebrated author and Harvard professor Sven Beckert is speaking at Webster University Saturday, Feb. 10, 9:30-10:30 a.m. in H. Sam Priest Center 104.
Beckert is the author of Empire of Cotton: A Global History, which won the Bancroft Prize in history and was a finalist for the Pulitzer Prize in 2014. The book is a brilliant retelling of the origins and spread of capitalism, focused on one essential commodity. It demonstrates how imperialism, slavery, and war were central to the creation of a global economy.
In his Webster talk, Sven will reflect on the significance of his book, its challenge to conventional wisdom, and the response of his critics.
The talk is sponsored by the History, Politics, and International Relations Department and the Webster Faculty Speaker Series.
It is free and open to the public, but please rsvp to wrosenbl@webster.edu.
In the News: Kwon on Beyond Ferguson board; Alumni chief diversity officers
Professor Jung Bum Kwon was mentioned in the St. Louis Post-Dispatch for being appointed to the board of Before Ferguson Beyond Ferguson (BFBF). ......


EIWR has piloted two sustainable dry latrines (VIP) at Chefe Jebesa and Bushello primary schools near Hawassa town serving more than 1400 girl students. EIWR used UNICEF, Ministry of Health and Ministry of Water Irrigation and Energy 2010 design guideline as starting point for the latrines but then added modification based on feedback from the community. The latrines will be complete in early July, 2014, but before school lets out EIWR will host an inaugural ceremony with school families where families will be shown the new facility.
Adopted manual rural latrine manual with special consideration to integrate sanitary rooms with the existing latrine mode.


Help Peyton Acquire An Accessible Vehicle!
https://www.hopestrengthens.org/wordpress/media/Peyton-e1500765399452.jpg 420 560 jakeferreira https://www.hopestrengthens.org/wordpress/media/logo-wht2.png jakeferreira2017-07-23 00:01:022017-07-23 00:01:02Help Peyton Acquire An Accessible Vehicle!
https://www.hopestrengthens.org/wordpress/media/unnamed.jpg 640 562 jakeferreira https://www.hopestrengthens.org/wordpress/media/logo-wht2.png jakeferreira2016-12-17 17:02:332016-12-17 17:02:33Back in the saddle!


Today I have a fireworks card to share. I kept the colour scheme minimal for a classic look.
I began by inking a piece of card with distress oxides in black soot, faded jeans and hickory smoke. I put this aside to dry whilst I worked on the sentiment.
Using the MFT Cheers word die, I die cut it three times – twice from black card and one from blue glitter card. I had adhered stick-it adhesive behind each before die cutting, to make it easier to stick them together.
Back to the inked panel, I stamped the MFT Fireworks images and stars, heat embossing them with silver and white embossing powders.
I added the panel to a card base, then the die cut stacked sentiment in the centre. I flicked the card with water for some distress details. To finish, I added some embellishments and rounded two of the corners for interest.
Love the ink blending for your sky, very cool card.


We live in a cruel world. Our profession of software development is very young compared to other fields such as banking, hotels, or carpenters. I truly believe we have taken a couple of wrong turns in our short history. In this blog entry I try to shed some light on it by some seemingly unrelated stories.


Do you believe the words of Jesus, or do you believe those that claim to be 'of Christ' but do not speak the same words as Jesus? Does the Holy Spirit give individual revelation, or does She reveal the meanings of the words that Jesus spoke while He was here on earth? These are the questions that are asked and answered on The Bible On Trial. Join us as we challenge the faith that has become Christianity on the ONLY show on radio that tests religion against the words and teachings of Jesus, that He gave His own disicples while He was here on earth.