text:
\section{Introduction}
Many quantum information protocols involve nonclassical states to achieve their quantum advantages.  For instance, quantum high precision measurements achieve sensitivities beyond the standard quantum limit by utilising nonclassical states.  The standard quantum limit is given by a coherent state, which satisfies the minimum uncertainty relation where quantum fluctuations are equally shared by any two quadrature amplitudes.  One way to break this limit is to squeeze a coherent state~\cite{Yuen}.  A squeezed state exhibits quantum fluctuations below the standard quantum limit in one quadrature at the sacrifice of larger quantum fluctuations in the other, which is directly applicable to achieve high precision measurements. To apply squeezed states to high precision measurements, it is important that squeezing can be achieved relatively easily.  Fortunately, squeezing can be achieved via quadratic Hamiltonian, and hence it does not require higher-order optical nonlinearity such as Kerr effect~\cite{Slusher,Wu}.  Both squeezing and the quantum advantages by squeezing in the high precision measurements have been demonstrated in experiments~\cite{Aasi, Taylor}.

The idea of squeezing has been extended to spin systems~\cite{Kitagawa,Wineland1,Wineland2}.  An ensemble of spins can be considered as a collective spin when it satisfies the symmetry under particle permutations.  An ensemble of spin-$1/2$ systems can be treated as an su($2$) system in a high dimension dependent on the number of spins in the ensemble.  The coherent states of the su($2$) system can be defined as the orbit of the SU($2$) group action on a reference state~\cite{Arecchi}.  Usually, we take the lowest weight state as the reference state, analogous to the vacuum state in the optical coherent states.  Squeezing can then be introduced on the coherent state of the collective su($2$) system.  The spin squeezed states show quantum fluctuations below the standard quantum limit in one degree of freedom, similar to the optical squeezed states.
In the spin-$1$ case~\cite{Mustecaplioglu,Yukawa,Colangelo}, the Hilbert space of a spin-$1$ particle can be spanned by three orthnormal states, and we can consider eight independent observables on the Hilbert space.  They correspond to eight generators of the su($3$) algebra, and hence the collective system inherits the su($3$) structure.  The dimensionality of the su($3$) collective system can be determined by the number of the spin-$1$ particles.
Similarly, if the ensemble is of spin-$J$ systems, the collective spin can be treated as an su($2J+1$) system, where its dimension is determined by the number of the ensemble.  This extension is relevant to current experiments of squeezing on spin ensembles; for instance, squeezing in a spin-$7/2$ atomic gas~\cite{Auccaise} and in spin-1 Bose-Einstein condensates have been observed~\cite{Hald,Sewell,Hamley}.  In view of current and near future experimental developments, it is important to characterize the rich structure of squeezing in the collective su($2J+1$) systems and to systematically classify them based on unitary equivalent classes.

Among the collective su($2J+1$) systems, the su($2$) collective system is simple enough so that squeezing can be understood in comparison with optical squeezing. The representation space based on the SU($2$) coherent states is a sphere, i.e. the Bloch sphere.  Squeezing can be tracked on this two-dimensional space.  Though it is compact,  as the dimensionality of the Bloch sphere is the same as that of the phase space based on the optical coherent states, there are similarities between the SU($2$) squeezed states and the optical squeezed states.  When we extend the former to an ensemble of spin-$J$ systems ($J>1/2$), the structure of squeezing is no longer so simple.  As the collective su($2J+1$) system has $(2J+1)^2-1$ independent observables, there are a number of possible realizations of squeezed states.  In the case of $J=1$, there are the eight independent observables, which can be represented by three spin-vector components and five quadrupolar-tensor components, and squeezing can be implemented in terms of the su($2$) subalgebra among these eight observables.  Then, squeezing can be classified into two classes with the different squeezing limits.

In this paper, we generalize the classification to collective su($2J+1$) systems, where the squeezing can be characterized by
$(2J+1)^2-1$ linearly independent observables.  Following the classification in Ref.~\cite{Yukawa}, we classify squeezing based on the unitary equivalence classes, whose definition is given in Sec. 2.2.  We also derive the structure factor of the su($2$) generators to characterise each class and obtain the squeezing limits via the one-axis twisting interaction.

This paper is organized as follows. In Sec. II, we generalize the classification to the collective su($2J+1$) systems to show that the squeezing can be classified into the unitary equivalence classes of ($2J+1$)-dimensional representations of the su($2$) subalgebras.  In Sec. III, we derive quantum fluctuations for squeezed states of collective su($2J+1$) systems with one-axis twisting and show their squeezing limits.  In Sec. IV we apply our classification to a collective su($4$) system to illustrate the unitary equivalence classes of the squeezing and their squeezing limits, and summarize the main results in Sec. V.
Throughout this paper, a scalar, a vector, and a matrix are respectively represented by a normal letter, a bold letter, and a normal letter with a tilde, as in $A$, $\bi{A}$, and $\tilde{A}$. The operator is denoted by a letter with a caret as in $\hat{A}$.

\section{Classification of squeezing in collective su(2J+1) systems}
\subsection{Observables of the su(2J+1) systems}
Let us identify the linearly independent observables whose quantum fluctuations can be controlled via squeezing.
Suppose there is a collective su($2J+1$) system consisting of $N$ spin-$J$ particles.
The particles can be fermions as well as bosons when the spatial degrees of freedom of each fermion are frozen
and the spin degrees of freedom are separable from the spatial degrees of freedom as in ultracold fermions trapped in an optical lattice~\cite{Martin} or
magnetic impurities in a crystal~\cite{Tyryshkin}.
We consider a squeezed spin state (SSS) which is generated from a coherent spin state (CSS) via a nonlinear interaction such as the one-axis twisting or the two-axis
counter twisting~\cite{Kitagawa}.

In a CSS, all particles are in the same single-spin state~\cite{Kitagawa}.
A single-spin state can be expanded in terms of the rank-$d$ multipoles ($d\in \mathbb{N}$) and it can be described by the spherical harmonics of degree $d$.
In the case of a spin-$1/2$ particle, the three components of the dipole, i.e., the spin vector, are linearly independent and generate the su($2$) algebra.
In the case of a spin-$J$ particle, the $2d+1$ components of the rank-$d$ multipoles ($1 \leq d \leq 2J$) are linearly independent of each other, while the multipoles
of the rank higher than $2J$ can be expressed in terms of the lower-rank multipoles and the identity.
Thus, the spin and multipoles up to the rank of $2J$, which are comprised of $(2J+1)^2-1=4J(J+1)$ observables in total, completely characterize a single spin-$J$ state; hence they
can be chosen as the generators of the su($2J+1$) algebra.
We define the second-quantized forms of the single spin and multipolar observables as
\begin{equation}
	{\hat{\lambda}}_{{\bi n}_j;J,k} = \sum_{m,n=1}^{2J+1} ({\tilde{\lambda}}_{J,k})_{mn} {\hat{c}}^{\dagger}_{{\bi n}_j;J,m} {\hat{c}}_{{\bi n}_j;J,n},
	\label{eq:single-gen}
\end{equation}
where $({\tilde{\lambda}}_{J,k})_{mn}$ represents the $mn$-entry of the $k$-th spin or multipolar matrix ${\tilde{\lambda}}_{J,k}$ of a single spin-$J$
particle, and ${\hat{c}}_{{\bi n}_j;J,m}$ ($ {\hat{c}}_{{\bi n}_j;J,m}^{\dagger}$) denotes the spin-$J$ bosonic or fermionic
annihilation (creation) operator of the spatial mode ${\bi n}_j$ and the magnetic sublevel $m_z=J+1-m$.
Here, we define ${\hat{\lambda}}_{{\bi n}_j;J,k}$ in Eq.~(\ref{eq:single-gen}) so that the first three observables are given by the Cartesian components of the
spin vector, the next five are given by the five independent components of the quadrupolar tensor, the next seven are the seven independent components of
the octupolar tensor~\cite{Shiina}, and so on.
We also note that the matrices ${\tilde{\lambda}}_{J,k}$ are normalized so that their trace norms satisfy
\begin{equation}
	||{\tilde{\lambda}}_{J,k}||_{\mathrm{trace}}^2 = \sum_{m_z=-J}^J m_z^2 = \frac{1}{3} J(J+1)(2J+1). \label{eq:norm}
\end{equation}

A CSS can be completely described by the collective observables of the single spin and multipolar observables given in Eq.~(\ref{eq:single-gen}).
Squeezing can redistribute quantum fluctuations in these collective observables.
The second-quantized forms of the collective observables ${\hat{\Lambda}}_{J,k}$ can be expressed as
\begin{equation}
	{\hat{\Lambda}}_{J,k} = \sum_{j=1}^N {\hat{\lambda}}_{{\bi n}_j;J,k}. \label{eq:coll-gen}
\end{equation}
The observables ${\hat{\Lambda}}_{J,k}$ in Eq.~(\ref{eq:coll-gen}) satisfy the same commutation relations as
${\tilde{\lambda}}_{J,k}$ in Eq.~(\ref{eq:single-gen}).
This implies that they also generate the su($2J+1$) algebra and the matrices $\{ {\tilde{\lambda}}_{J,k} \}$ can be regarded as the irreducible representation
of $\{ {\hat{\Lambda}}_{J,k} \}$ in the basis of $\{ | J, m_z \rangle \}$, which represents the basis of the single-spin magnetic sublevels
with respect to the quantization axis along the $z$ axis.
Thus, a collective observable ${\hat{O}}_J$ of the collective su($2J+1$) system can be expressed by a $(2J+1)$-dimensional matrix representation ${\tilde{O}}_J$
in the representation space of $V(\{ |J, m_z \rangle \})$ as follows:
\begin{equation}
	{\tilde{O}}_J = \sum_{k=1}^{4J(J+1)} v_{J,k} {\tilde{\lambda}}_{J,k}, \label{eq:coll-obs}
\end{equation}
where the real coefficients $v_{J,k}$ satisfy $\sum_{k=1}^{4J(J+1)} v_{J,k}^2 = 1$.

\subsection{Classification based on unitary equivalence classes}
We consider squeezing among three observables $\{ {\hat{O}}_{J,k} \}$ ($k=1,2,3$) of the collective su($2J+1$) system, which form an
su($2$) subalgebra of the su($2J+1$) algebra and satisfy the commutation relations given by
\begin{equation}
	[{\hat{O}}_{J,3}, {\hat{O}}_{J,\pm}] = \pm f {\hat{O}}_{J,3}, \label{eq:sub-su2}
\end{equation}
where ${\hat{O}}_{J,\pm}\equiv {\hat{O}}_{J,1} \pm i {\hat{O}}_{J,2}$ and $f>0$ represents the magnitude of the structure constant.
Note that $f$ in Eq.~(\ref{eq:sub-su2}) is not always $f=1$, since $\pm f$ are equivalent to the structure factors of the su($2J+1$) algebra.

The squeezing among an su($2$) subalgebra $\{ {\hat{O}}_{J,k} \}$ can be classified based on the unitary equivalence class.
The unitary equivalence class of the squeezing among $\{ {\hat{O}}_{J,k} \}$ can be determined by the $(2J+1)$-dimensional matrix representation of
$\{ {\hat{O}}_{J,k} \}$ in the space of $V(\{ |J,m_z \rangle \})$ spanned by the basis $\{ |J,m_z \rangle \}$.
The unitary equivalence class is defined as follows: Suppose $\{ {\tilde{X}}_k \}$ and $\{ {\tilde{X}}^{\prime}_k \}$ are the $n$-dimensional matrix
representations of the semi-simple Lie algebra.
Then, the representations $\{ {\tilde{X}}_k \}$ and $\{ {\tilde{X}}^{\prime}_k \}$ belong to the same unitary equivalence class, if there exists an SU($n$) transformation
matrix $\tilde{U}$ such that $\tilde{U} {\tilde{X}}_k {\tilde{U}}^{\dagger} = {\tilde{X}}^{\prime}_k$ for $\forall k$.

In our case, $\{ {\tilde{O}}_{J,k} \}$ is the $(2J+1)$-dimensional matrix representation of the generators of the su($2$) algebra, which is semi-simple; hence
$\{ {\tilde{O}}_{J,k} \}$ should be completely reducible.
The matrix representation $\{ {\tilde{O}}_{J,k} \}$ and its representation space $V(\{ |J,m_z \rangle \})$ can be decomposed into the direct sum of the
lower dimensional irreducible representations of the su($2$) generators and their representation spaces, respectively.
Suppose the dimension of the $l$-th irreducible representation is $2J_l+1$.
Then, there exists an orthonormal basis set $\{ |J_l, m_l {\rangle}_l \}$
($m_l = -J_l,\cdots ,J_l$) such that the $l$-th irreducible representation of the su($2$) algebra is given by the spin matrices
$\{ {\tilde{\lambda}}_{J_l,1}, {\tilde{\lambda}}_{J_l,2}, {\tilde{\lambda}}_{J_l,3} \}$ for a spin-$J_l$ particle (c.f. Eq.~(\ref{eq:single-gen})).
The state $|J_l, m_l {\rangle}_l$ can be expressed as a linear combination of $|J, m_z \rangle$ ($m_z = -J, \cdots, J$), and $\{ |J_l, m_l{\rangle}_l \}$
and $\{ |J_{l^{\prime}}, m_{l^{\prime}} {\rangle}_{l^{\prime}} \}$ ($l \neq l^{\prime}$) are orthogonal to each other.
Then, the completely reducible representation of $\{ {\tilde{O}}_{J,k} \}$ can be expressed as
\begin{equation}
	{\tilde{O}}_{J,k} = f \bigoplus_{l=1}^{r} {\tilde{\lambda}}_{J_l,k}, \
	V(\{ |J,m_z \rangle \}) = \bigoplus_{l=1}^{r} V(\{ |J_l,m_l {\rangle}_l \}). \label{eq:direct-sum}
\end{equation}
In Eq.~(\ref{eq:direct-sum}), $r$ expresses the number of the irreducible representations and the ``subspins'' $J_l$ satisfy $\sum_{l=1}^{r} (2J_l+1) = 2J+1$.
The structure constant $f$ of $\{ {\tilde{O}}_{J,k} \}$ defined in Eq.~(\ref{eq:sub-su2}) is given by
\begin{equation}
	f = \sqrt{\frac{J(J+1)(2J+1)}{\sum_{l=1}^r J_l (J_l+1) (2J_l+1)}}, \label{eq:f}
\end{equation}
which can be derived from the irreducibility of $\{ {\tilde{\lambda}}_{J_l,k} \}$ and the normalization condition in Eq.~(\ref{eq:norm}).
Here, we note that $\{ {\tilde{\lambda}}_{J_l,k} \}$ and $V(\{ |J_l,m_l {\rangle}_l \})$ are arranged so that $J_l$ satisfies
\begin{equation}
	0 \leq J_r \leq \cdots \leq J_2 \leq J_1 \leq J,  \label{eq:order}
\end{equation}
and we define $\{ {\tilde{\lambda}}_{J_l=0,k} \} = \{ 0, 0, 0 \}$.

If two sets of the generators of the su($2$) subalgebras, $\{ {\hat{O}}_{J,k} \}$ and $\{ {\hat{O}}^{\prime}_{J,k} \}$, belong to the same unitary
equivalence class, $\{ {\hat{O}}^{\prime}_{J,k} \}$ and the representation space $V(\{ |J, m_z \rangle \})$ can be decomposed into
\begin{equation}
	{\tilde{O}}^{\prime}_{J,k} = f^{\prime} \bigoplus_{l=1}^{r^{\prime}} {\tilde{\lambda}}_{J_l^{\prime},k}, \
	V(\{ |J, m_z \rangle \}) = \bigoplus_{l=1}^{r^{\prime}} V(\{ |J_l^{\prime},  m_l^{\prime} {\rangle}_l \}),
\end{equation}
where $f^{\prime} = [J(J+1)(2J+1)/\sum_{l=1}^{r^{\prime}}J_l^{\prime} (J_l^{\prime}+1)(2J_l^{\prime}+1)]^{1/2}$, $m_l^{\prime} =-J_l^{\prime},\cdots ,J_l^{\prime}$,
and
\begin{equation}
	r=r^{\prime} \ \land \ \forall l, J_l = J_l^{\prime}.
	\label{eq:eqn-class}
\end{equation}
Equation~(\ref{eq:eqn-class}) implies that the structure constants $f$ and $f^{\prime}$ are equal.
If two sets of the generators of the su($2$) subalgebras, $\{ {\tilde{O}}_{J,k} \}$ and $\{ {\tilde{O}}^{\prime}_{J,k} \}$, do not belong to the same unitary
equivalence class, Eq.~(\ref{eq:eqn-class}) does not hold, since a unitary matrix transforms the basis but it cannot change $r$ and $J_l$.
The unitary equivalence classes of the su($2J+1$) algebra can be systematically found via the Dynkin diagram of the su($2J+1$) algebra as explained in
\ref{subsec:Dynkin}.

\subsection{\label{subsec:Dynkin}Dynkin diagram and unitary equivalence class}
\begin{figure}[t]
\begin{center}
\includegraphics[width=15cm,clip]{Dynkinsun.eps}
\caption{(Color Online) (a) Dynikin diagram of the su($2J+1$) algebra. The simple root ${\balpha}_k$ expresses the transition from $m_z=J-k$ to $m_z=J-k+1$.
(b) Correspondences between the connected and disconnected simple roots and the lower dimensional irreducible representations of the su($2$) generators.
The filled circles and the gray open circles represent the simple roots that are chosen and  not chosen, respectively.
(i) If the chosen simple roots from ${\balpha}_1$ to ${\balpha}_l$ are connected, then they are substituted by the $(l+1)$-dimensional irreducible representation in the
representation space of $V(\{ |J, m_z \rangle \})$ ($m_z = J, \cdots , J-l$).
(ii) If a magnetic sublevel $J-l^{\prime}$ is isolated from the connected simple roots, then it is substituted by the one-dimensional element, i.e., $0$.}
\label{fig:dynkin-sun}
\end{center}
\end{figure}
The decomposition of the generators $\{ {\tilde{O}}_{J,k} \}$ of the su($2$) subalgebra in Eq.~(\ref{eq:direct-sum}) can be derived from the Dynkin
diagram of the su($2J+1$) algebra.
In the Dynkin diagram of the su($2J+1$) algebra, the $2J$ simple roots are connected as shown in Fig.~\ref{fig:dynkin-sun} (a).
Here, the $k$-th vertex represents the $k$-th simple root ${\balpha}_k$ that corresponds to the raising matrix ${\tilde{A}}_{J,k}$ from the $k$-th sublevel to
the $(k+1)$-th sublevel with respect to the quantization axis determined by the Cartan subalgebra.
For the generators of the Cartan subalgebra, we choose the $z$ component of the spin vector ${\tilde{\lambda}}_{J,3}$ and the other $2J-1$ diagonal matrices.
Then the quantization axis is given by the $z$ axis, which implies that ${\tilde{A}}_{J,k}$ raises the sublevel from $m_z = J-k$ to $m_z=J-k+1$ as follows:
\begin{equation}
	({\tilde{A}}_{J,k})_{mn} \equiv \sqrt{\frac{1}{3}J(J+1)(2J+1)} \ {\delta}_{J-k+1,m} {\delta}_{J-k,n}. \label{eq:simple-root-m}
\end{equation}
The matrix products of ${\tilde{A}}_{J,k}$ and their linear combinations reproduce the spin and multipolar observables ${\tilde{\lambda}}_{J,k}$.

We can construct a complete irreducible representation by choosing $1\leq n\leq 2J$ vertices from the $2J$ vertices and substituting $l$-connected roots
of ${\balpha}_{k}$, ${\balpha}_{k+1}$, $\cdots$, ${\balpha}_{k+l-1}$ ($l=1,\cdots ,2J$) by the $(l+1)$-dimensional irreducible representation $\{ {\tilde{\lambda}}_{J_l,k} \}$
($k=1,2,3$) of the su($2$) generators in the representation space of $V(\{ | J,m_z \rangle \})$ ($m_z = J-k+1, J-k, \cdots , J-k-l+1$) as shown
in Fig.~\ref{fig:dynkin-sun} (b)-(i).
If the magnetic sublevel of $m_z$ is not involved by the connected simple roots, then it is substituted by the one-dimensional element of $0$.
This procedure is equivalent to the decomposition of $V(\{ | J,m_z \rangle \})$ into the subspaces in Eq.~(\ref{eq:direct-sum})
Arranging the irreducible representations so that their dimensions satisfy Eq.~(\ref{eq:order}), we can obtain the decomposition in Eq.~(\ref{eq:direct-sum}).
Since the Dynkin diagram does not depend on the choice of the basis, any $(2J+1)$-dimensional matrix representation can be obtained by rotating
one of the representations derived from the Dynkin diagram via an SU($2J+1$) unitary matrix.

\section{Properties of squeezing determined by unitary equivalence classes}
\subsection{Squeezing parameters}
The properties of the squeezing reflect the structure of the unitary equivalence class, i.e., the subspins and the initial coherent state.
To confirm this, let us consider squeezing among an su($2$) subalgebra $\{ {\hat{O}}_{J,k} \}$, which can be decomposed into Eq.~(\ref{eq:direct-sum})
with the subspins $\{ J_l \}$.
A CSS~\cite{Perelomov,Gilmore,Nemoto,Mathur} can be expressed in terms of two parameters
$\theta \in [0,\pi]$ and $\phi \in [0,2\pi )$ as
\begin{eqnarray}
	&|\theta ,\phi {\rangle}_{\mathrm{tot}}
	\equiv {\left [ \bigoplus_{l=1}^r {\zeta}_l |\theta ,\phi {\rangle}_l \right ]}^{\otimes N}
	\nonumber \\
	&= \sum_{n_1=0}^N \sum_{n_2=0}^{N-n_1} \cdots \sum_{n_{r-1}=0}^{N-n_1-\cdots - n_{r-2}} \sqrt{_NC_{n_1} \ _{N-n_1}C_{n_2} \
	\cdots \ _{N-n_1-\cdots - n_{r-2}}C_{n_{r-1}}} \nonumber \\
	&\times {\zeta}_1^{n_1} {\zeta}_2^{n_2} \cdots {\zeta}_{r-1}^{n_{r-1}} {\zeta}_r^{N-n_1-\cdots - n_{r-1}} \nonumber \\
	&\times \Bigl [  |\theta ,\phi {\rangle}_1^{\otimes n_1} \oplus |\theta ,\phi {\rangle}_2^{\otimes n_2}
	\oplus \cdots \oplus |\theta ,\phi {\rangle}_{r-1}^{\otimes n_{r-1}} \oplus |\theta ,\phi {\rangle}_r^{\otimes N - n_1 - \cdots - n_{r-1}} \Bigr ],
	 \label{eq:CSS-J}
\end{eqnarray}
where $\sum_{l=1}^r |{\zeta}_l|^2 = 1$ and the single particle states $|\theta , \phi {\rangle}_l$ in Eq.~(\ref{eq:CSS-J}) for $J_l \neq 0$ and $J_l = 0$ are
defined in terms of the basis $\{ |J_l, m_l {\rangle}_l \}$ as
\begin{equation}
	\forall J_l \neq 0, \ |\theta , \phi {\rangle}_l \equiv \exp {\left [-\frac{\theta}{2} ( e^{-i\phi } {\tilde{\lambda}}_{J_l,+} - e^{i\phi } {\tilde{\lambda}}_{J_l,-}) \right ]}
	|J_l, J_l {\rangle}_l, \label{eq:one-J}
\end{equation}
with ${\tilde{\lambda}}_{J_l,\pm} \equiv {\tilde{\lambda}}_{J_l,1} \pm i {\tilde{\lambda}}_{J_l,2}$,
and $|\theta ,\phi {\rangle}_l \equiv |J_l = 0, m_l = 0 {\rangle}_l$ ($J_l = 0$), respectively.
The CSS $|\theta ,\phi {\rangle}_{\mathrm{tot}}$ in Eq.~(\ref{eq:CSS-J}) satisfies the minimum uncertainty relation
\begin{equation}
	\forall \nu \in [0,2\pi ), \ \langle ( \Delta O_{J,\nu} )^2 \rangle \langle ( \Delta O_{J,\nu +\frac{\pi}{2}} )^2 \rangle
	= \frac{f^2}{4} \langle {\hat{O}}_{J, \perp} {\rangle}^2, \label{eq:MUR}
\end{equation}
where $\langle \hat{X} \rangle$ represents the expectation value of an observable $\hat{X}$, the quantum fluctuation in $\hat{X}$ is defined
as $\langle (\Delta X )^2 \rangle = \langle {\hat{X}}^2 \rangle - \langle \hat{X} {\rangle}^2$, and
${\hat{O}}_{J,\nu}$ and ${\hat{O}}_{J, \perp}$ are given by
\begin{eqnarray}
	& {\hat{O}}_{J,\perp} \equiv {\hat{O}}_{J,1} \cos {\phi} \sin {\theta} + {\hat{O}}_{J,2} \sin {\phi} \sin {\theta} + {\hat{O}}_{J,3} \cos {\theta},
	\label{eq:O3} \\
	& {\hat{O}}_{J,\nu} \equiv {\hat{O}}_{J,1} ( \cos {\phi} \cos {\theta} \cos {\nu} - \sin {\phi} \sin {\nu} ) \nonumber \\
	&+ {\hat{O}}_{J,2} ( \sin {\phi} \cos {\theta} \cos {\nu} + \cos {\phi} \sin {\nu} )
	- {\hat{O}}_{J,3} \sin {\theta} \cos {\nu},  \label{eq:O1}
\end{eqnarray}
respectively.
The expectation values in Eq.~(\ref{eq:MUR}) can be obtained via the Schwinger-boson approach described in \ref{a:0} as
\begin{eqnarray}
	& \langle {\hat{O}}_{J, \perp} \rangle = fN \sum_{l,J_l\neq 0} J_l |{\zeta}_l|^2, \label{eq:exp-CSS} \\
	& \forall \nu \in [0, 2\pi ), \ \langle ( \Delta O_{J,\nu} )^2 \rangle = \frac{f^2N}{2} \sum_{l,J_l\neq 0} J_l |{\zeta}_l|^2.  \label{eq:fluct-CSS}
\end{eqnarray}
Equations~(\ref{eq:MUR}), (\ref{eq:exp-CSS}) and (\ref{eq:fluct-CSS}) imply that the squeezing among $\{ {\hat{O}}_{J,k} \}$ can suppress
$\langle ( \Delta O_{J,\nu} )^2 \rangle$ below the
coherent-spin-state value of $\frac{f}{2} |\langle {\hat{O}}_{J, \perp} \rangle |$ at the expense of $\langle ( \Delta O_{J,\nu+\frac{\pi}{2}} )^2 \rangle$
enhanced above $\frac{f}{2} |\langle {\hat{O}}_{J, \perp} \rangle |$; hence, the squeezing can be characterized by the squeezing parameter $\xi$ defined as
\begin{equation}
	{\xi}^2 = \left ( 2N \sum_{l,J_l\neq 0} J_l |{\zeta}_l|^2 \right ) \times \frac{\min_{\nu}
	{\langle (\Delta O_{J, \nu} )^2 \rangle} }{\langle {\hat{O}}_{J, \perp} {\rangle}^2},
	\label{eq:sqparam}
\end{equation}
where $\min_{\nu} {\langle (\Delta O_{J, \nu} )^2 \rangle}$ is the quantum fluctuations in Eq.~(\ref{eq:O1}) perpendicular to the $O_{J, \perp}$ plane and minimized with
respect to the angle $\nu$ in Eq.~(\ref{eq:O1}).
Equation~(\ref{eq:sqparam}) is equal to $1$ for the CSS in Eq.~(\ref{eq:CSS-J}) and it implies that a state giving ${\xi}^2 < 1$ is squeezed.
We note that Eq.~(\ref{eq:sqparam}) is equivalent to the Wineland's squeezing parameter~\cite{Wineland1} when the coefficients $\{ |{\zeta}_l |^2 \}$ of the
initial CSS in Eq.~(\ref{eq:CSS-J}) are given by $|{\zeta}_l|^2 = {\delta}_{l,l_0}$ with $l_0$ such that $J_{l_0} \neq 0$.
The squeezing parameter $\xi$ in Eq.~(\ref{eq:sqparam}) is characterized by the subspins and the initial CSS, both of which reflect
the structure of the unitary equivalence class of the spin and multipolar observables $\{ {\hat{O}}_{J,k} \}$ generating the su($2$) subalgebra.

\subsection{Squeezed and anti-squeezed quantum fluctuations for one-axis twisting interactions}
Let us calculate the squeezing parameter $\xi$ in Eq.~(\ref{eq:sqparam}) for an SSS generated via the one-axis twisting interaction~\cite{Kitagawa}.
We consider the one-axis twisting interaction
\begin{equation}
	{\hat{H}}_{\mathrm{OAT}} = \hbar \chi {\hat{O}}_{J,3}^2 \label{eq:OAT}
\end{equation}
with the interaction energy $\chi$, which distribute the quantum fluctuations in the $O_{J,2}$-$O_{J,3}$ plane.
A CSS of the $N$ spin-$J$ particles is given by
$| \theta = \frac{\pi}{2}, \phi = 0 {\rangle}_{\mathrm{tot}}$ in Eq.~(\ref{eq:CSS-J}).
Defining the rescaled evolution time $\mu \equiv 2 \chi f^2 t$, we can express the one-axis-twisted SSS
$| {\Psi}_{\mathrm{OAT}} (J,N;\mu ) {\rangle}_{\mathrm{tot}}$ at $\mu $ as
\begin{equation}
	| {\Psi}_{\mathrm{OAT}} (J,N;\mu ) {\rangle}_{\mathrm{tot}} = \exp {\left [- \frac{i}{2f^2} {\hat{O}}_{J,3}^2 \mu\right ]} \
	|\theta = \frac{\pi}{2}, \phi = 0 {\rangle}_{\mathrm{tot}}. \label{eq:SSS-J}
\end{equation}
In this case, the observable ${\hat{O}}_{J,\perp}$ is given by ${\hat{O}}_{J,1}$ and its expectation value at time $\mu$ can be obtained in a manner similar to
Eqs.~(\ref{eq:exp-CSS}) and (\ref{eq:fluct-CSS}) as
\begin{eqnarray}
	\langle  {\hat{O}}_{J, 1}  \rangle (\mu ) &\equiv \langle  {\Psi}_{\mathrm{OAT}} (J,N;\mu ) | {\hat{O}}_{J,1} | {\Psi}_{\mathrm{OAT}} (J,N;\mu ) {\rangle}_{\mathrm{tot}} \nonumber \\
	&= f N \sum_{l:J_l\neq 0} J_l |{\zeta}_l|^2 {\cos}^{2J_l-1} \frac{\mu}{2} \ {\left [1 - |{\zeta}_l|^2 \left ( 1 - {\cos}^{2J_l} \frac{\mu}{2} \right ) \right ] }^{N-1},
	\label{eq:perp}
\end{eqnarray}
as detailed in \ref{a:0}.
The quantum fluctuations in the plane perpendicular to ${\hat{O}}_{J,\perp}$ can be simplified as a function of $\nu$ as
\begin{equation}
	\langle (\Delta O_{J, \nu} )^2 \rangle (\mu ) = \frac{f^2N}{2}
	\sum_{l:J_l\neq 0} J_l|{\zeta}_l|^2 \left [ 1 +  A_l (1+ \cos {2\nu} ) - B_l \sin {2\nu} \right ].
	\label{eq:fluct-3}
\end{equation}
Here, $A_l$ and $B_l$ are defined as
\begin{eqnarray}
	A_l & \equiv \frac{J_l }{2} (N-1) |{\zeta}_l|^2 \left \{ 1 - {\cos}^{2(2J_l-1)} \mu \ [1-|{\zeta}_l|^2 (1-{\cos}^{2J_l} \mu) ]^{N-2}
	\right \} \nonumber \\
	&+ \frac{1}{2} \left ( J_l - \frac{1}{2} \right ) \{ 1 - {\cos}^{2(J_l-1)} \mu \ [1-|{\zeta}_l|^2 (1-{\cos}^{2J_l} \mu) ]^{N-1} \}, \\
	B_l & \equiv 2 \Biggl \{ J_l (N-1) |{\zeta}_l|^2 {\cos}^{2J_l} \frac{\mu}{2}
	+ \left ( J_l - \frac{1}{2} \right ) \left [1 - |{\zeta}_l|^2 \left (1 - {\cos}^{2J_l} \frac{\mu}{2} \right ) \right ] \Biggr \} \nonumber \\
	& \times \sin {\frac{\mu}{2}} \ {\cos}^{2(J_l-1)} \frac{\mu}{2} \ {\left [1 - |{\zeta}_l|^2 \left (1 - {\cos}^{2J_l} \frac{\mu}{2} \right ) \right ]}^{N-2}. \label{eq:fluct-4}
\end{eqnarray}
Equation~(\ref{eq:fluct-3}) is periodic with respect to $\nu$, so there exist the minimum and the maximum, i.e., the squeezed and anti-squeezed quantum
fluctuations, respectively.
The squeezing parameter ${\xi}^2 (\mu =0) = 1$ for the initial CSS in Eq.~(\ref{eq:CSS-J}) and the spins are said to be squeezed when ${\xi}^2 (\mu )<1$.

The squeezing limit in Eq.~(\ref{eq:sqparam}) can be analytically obtained in the limit of $\mu \ll 1$ and $N \gg 1$, when the subspins $\{ J_l \}$ in Eq.~(\ref{eq:direct-sum})
and the coefficients $\{ |{\zeta}_l|^2 \}$ of the initial coherent state in Eq.~(\ref{eq:CSS-J}) satisfy $|{\zeta}_l|^2 = {\delta}_{l,l_0}$ ($J_{l_0} \neq 0$).
The quantum fluctuations in the $O_{J,2}$-$O_{J,3}$ plane in Eq.~(\ref{eq:fluct-3}) can be simplified as
\begin{eqnarray}
	\langle (\Delta O_{J_{l_0}, \nu} )^2 \rangle (\mu )
	&= \frac{f^2J_{l_0}N}{2} \Biggl \{ 1 + \frac{1}{2} \left (J_{l_0}N - \frac{1}{2} \right ) \nonumber \\
	& \times \Biggl [ ( 1 - {\cos}^{2(J_{l_0}N-1)} \mu ) (1 + \cos {2\nu} ) \nonumber \\
	&- 4 \sin {\frac{\mu}{2}} \ {\cos}^{2(J_{l_0}N-1)} \frac{\mu}{2} \ \sin {2\nu} \Biggr ] \Biggr \}, \label{eq:fluc-r1}
\end{eqnarray}
and the expectation value perpendicular to the $O_{J,2}$-$O_{J,3}$ plane in Eq.~(\ref{eq:perp}) is
\begin{equation}
	\langle {\hat{O}}_{J, 1}  \rangle (\mu ) = fJ_{l_0}N {\cos}^{2JN-1} \frac{\mu}{2}. \label{eq:prep-r1}
\end{equation}
Here, we assume that $\mu$ and $N$ satisfy $\alpha \equiv \frac{1}{2} J_{l_0}N \mu \gg 1$ and $\beta \equiv \frac{1}{4} J_{l_0}N {\mu}^2 \ll 1$.
Then, substituting Eqs.~(\ref{eq:fluc-r1}) and (\ref{eq:prep-r1}) into Eq.~(\ref{eq:sqparam}), we obtain the squeezing parameter for $r=1$ up to the
second order in $\beta$ as:
\begin{equation}
	{\xi}^2 (\mu ) \simeq \frac{1}{4{\alpha}^2} + \frac{2}{3} {\beta}^2 + \frac{\beta}{2{\alpha}^2} + \mathcal{O} (\max {\{ \frac{{\beta}^2}{\alpha}, {\beta}^3 \} }),
	\label{eq:sqpram-r1}
\end{equation}
where $\nu \simeq - \frac{1}{2} \arctan {\frac{1}{\alpha}} + \frac{\pi}{2}$.
The minimum of Eq.~(\ref{eq:sqpram-r1}), i.e., the squeezing limit is achieved at $\mu = {\mu}_{\mathrm{min}} = (12)^{1/6} (J_{l_0}N)^{-2/3}$ are given by
\begin{equation}
	{\xi}_{\mathrm{min}}^2 \equiv {\xi}^2 ({\mu}_{\mathrm{min}}) \simeq \frac{1}{2} {\left ( \frac{3}{2J_{l_0}N} \right )}^{2/3} + \frac{1}{2J_{l_0}N} \propto {(J_{l_0}N)^{-2/3}},
	\label{eq:minsqparam-r1}
\end{equation}
which implies that the squeezing limit monotonically decreases with increasing $J_{l_0}$.

\section{Application to collective su($4$) systems}
\subsection{Complete set of collective spin and multipolar observables}
To examine the squeezing parameter in Eq.~(\ref{eq:sqparam}) for $r>1$, especially the $\{ |{\zeta}_l|^2 \}$-dependence of the squeezing limit, let us consider a collective
su($4$) system consisting of $N$ spin-3/2 particles as an example.
In this case, the observables that can completely characterize collective spin states are the spin vector, the quadrupolar tensor, and the
octupolar tensor.
The Cartesian components of the spin vector ${\hat{\lambda}}_{{\bi n}_j;J=\frac{3}{2},k}$ ($k = 1,2,3$) can be given by
\begin{equation}
	{\hat{\lambda}}_{{\bi n}_j;\frac{3}{2},k} = \sum_{j=1}^N \sum_{m,n=1}^4 ({\tilde{\lambda}}_{\frac{3}{2},k})_{mn} {\hat{c}}^{\dagger}_{{\bi n}_j;\frac{3}{2},m}
	{\hat{c}}_{{\bi n}_j;\frac{3}{2},n}, \label{eq:single-j}
\end{equation}
where ${\tilde{\lambda}}_{\frac{3}{2},k}$ represent the spin-$3/2$ matrices ${\tilde{J}}_{\mu}$ ($\mu = x,y,z$) given by Eq.~(\ref{eq:single-mj}).
The matrix representations of the five independent components of the quadrupolar tensor and the seven independent components of the
octupolar tensor~\cite{Shiina} can be respectively expressed in terms of ${\tilde{J}}_{\mu }$ as
\begin{eqnarray}
	({\tilde{Q}}_{\mu \nu})_{mn}= \frac{\sqrt{15}}{6}  ({\tilde{J}}_{\mu} {\tilde{J}}_{\nu} + {\tilde{J}}_{\nu} {\tilde{J}}_{\mu} )_{mn}, \label{eq:single-q} \\
	({\tilde{D}}_{xy})_{mn} = \frac{\sqrt{15}}{6} ({\tilde{J}}_{x}^2 - {\tilde{J}}_{y}^2 )_{mn}, \label{eq:single-d} \\
	(\tilde{Y})_{mn} = \frac{\sqrt{5}}{6} (-{\tilde{J}}_{x}^2 - {\tilde{J}}_{y}^2 + 2{\tilde{J}}_{z}^2 )_{mn}, \label{eq:single-y}
\end{eqnarray}
where $(\mu ,\nu) = (x,y), (y,z), (z,x)$ in Eq.~(\ref{eq:single-q}), and
\begin{eqnarray}
	({\tilde{T}}^{\alpha}_{\mu})_{mn} = \frac{1}{3} (2 {\tilde{J}}_{\mu}^3 - \overline{{\tilde{J}}_{\mu} {\tilde{J}}_{\nu}^2}
	- \overline{{\tilde{J}}_{\eta}^2 {\tilde{J}}_{\mu}})_{mn}, \label{eq:single-ta} \\
	({\tilde{T}}^{\beta}_{\mu})_{mn} = \frac{\sqrt{15}}{9} (\overline{{\tilde{J}}_{\mu} {\tilde{J}}_{\nu}^2}
	- \overline{{\tilde{J}}_{\eta}^2 {\tilde{J}}_{\mu}})_{mn} \\
	({\tilde{T}}_{xyz})_{mn} = \frac{\sqrt{15}}{9} (\overline{{\tilde{J}}_{x}{\tilde{J}}_{y}{\tilde{J}}_{z}})_{mn}, \label{eq:single-txyz}
\end{eqnarray}
where $(\mu ,\nu ,\eta) = (x,y,z)$, $(y,z,x)$, and $(z,x,y)$ and the overbars above the matrix products are defined as
$\overline{\tilde{A} {\tilde{B}}^2} = \tilde{A} {\tilde{B}}^2 + \tilde{B} \tilde{A} \tilde{B} + {\tilde{B}}^2\tilde{A}$ and
$\overline{\tilde{A} \tilde{B} \tilde{C}} = \tilde{A} \tilde{B} \tilde{C} + \tilde{B} \tilde{C} \tilde{A} + \tilde{C} \tilde{A} \tilde{B} + \tilde{B} \tilde{A} \tilde{C}
+ \tilde{C} \tilde{B} \tilde{A} + \tilde{A} \tilde{C} \tilde{B}$ with respect to the matrices $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$.
Here we note that the matrix representations of the spin and multipolar observables in Eqs.~(\ref{eq:single-j})-(\ref{eq:single-txyz}) are normalized
so that they satisfy the condition in Eq.~(\ref{eq:norm}).
These fifteen spin and multipolar observables in Eqs.~(\ref{eq:single-j})-(\ref{eq:single-txyz}) together form the su($4$) Lie algebra.
Then, the irreducible representations of the collective spin observables describing the symmetric spin state can respectively be
given by the matrix representations of their single-spin counter parts in Eqs.~(\ref{eq:single-j})-(\ref{eq:single-txyz}), whose explicit
expressions are given in Eqs.~(\ref{eq:single-mj})-(\ref{eq:single-mt}).
We define the matrices $\{ {\tilde{\lambda}}_{\frac{3}{2},k} \} \equiv \{ {\tilde{J}}_{\mu}, {\tilde{Q}}_{\mu \nu}, {\tilde{D}}_{xy}, \tilde{Y}, {\tilde{T}}^{\alpha}_{\mu},
{\tilde{T}}^{\beta}_{\mu}, {\tilde{T}}_{xyz} \}$ ($k=1,\cdots ,15$) in the order of Eqs.~(\ref{eq:single-j})-(\ref{eq:single-txyz}).
Then, the matrix representation of any observable can be expressed in terms of $\{ {\tilde{\lambda}}_{\frac{3}{2},k} \}$ ($k=1,\cdots ,15$) as in Eq.~(\ref{eq:coll-obs}).

\subsection{Four types of squeezing}
There exist four unitary equivalence classes of the su($2$) subalgebras in the su($4$) algebra, which can be found as explained in
Sec.~\ref{subsec:Dynkin}.
First, let us construct the Dynkin diagram and consider the relation between the simple roots and the spin and multipolar observables in
Eqs.~(\ref{eq:single-j})-(\ref{eq:single-txyz}).
In collective su($4$) systems, the Dynkin diagram has three simple roots ${\balpha}_1$, ${\balpha}_4$, ${\balpha}_6$ as shown in Fig.~\ref{fig:su4-roots} (b).
Choosing the diagonal matrices ${\tilde{J}}_z = {\tilde{\lambda}}_{\frac{3}{2},3}$, $\tilde{Y} = {\tilde{\lambda}}_{\frac{3}{2},8}$, and
${\tilde{T}}^{\alpha}_z = {\tilde{\lambda}}_{\frac{3}{2},11}$ as the generators of the Cartan subalgebra, we can express the matrices
${\tilde{A}}_{\frac{3}{2},1}$, ${\tilde{A}}_{\frac{3}{2},4}$, and ${\tilde{A}}_{\frac{3}{2},6}$ corresponding to the simple roots as
\begin{eqnarray}
	{\tilde{A}}_{\frac{3}{2},1}
	= \frac{\sqrt{15}}{10} {\tilde{J}}_+ + \frac{1}{2} {\tilde{Q}}_+ - \frac{\sqrt{15}}{20} {\tilde{T}}^{\alpha}_+ - \frac{1}{4} {\tilde{T}}^{\beta}_-,  \label{eq:a1} \\
	{\tilde{A}}_{\frac{3}{2},4}
	= \frac{1}{\sqrt{5}} {\tilde{J}}_+ + \frac{3}{4\sqrt{5}} {\tilde{T}}^{\alpha}_+ + \frac{\sqrt{3}}{4} {\tilde{T}}^{\beta}_-, \label{eq:a4} \\
	{\tilde{A}}_{\frac{3}{2},6}
	= \frac{\sqrt{15}}{10} {\tilde{J}}_+ - \frac{1}{2} {\tilde{Q}}_- - \frac{\sqrt{15}}{20} {\tilde{T}}^{\alpha}_+ - \frac{1}{4} {\tilde{T}}^{\beta}_-, \label{eq:a6}
\end{eqnarray}
where we define ${\tilde{J}}_{\pm} \equiv {\tilde{J}}_x \pm i{\tilde{J}}_y$,
${\tilde{Q}}_{\pm} \equiv {\tilde{Q}}_{zx} \pm i{\tilde{Q}}_{yz}$,
${\tilde{T}}^{\alpha}_{\pm} = {\tilde{T}}^{\alpha}_x \pm i{\tilde{T}}^{\alpha}_y$, and
${\tilde{T}}^{\beta}_{\pm} = {\tilde{T}}^{\beta}_x \pm i{\tilde{T}}^{\beta}_y$, respectively.
The derivation of Eqs.~(\ref{eq:a1})-(\ref{eq:a6}) are detailed in \ref{a:2}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=15cm,clip]{Dynkinsu4.eps}
\caption{(Color Online) (a) The root diagram of the su($4$) algebra, (b) the Dynkin diagram of the the su($4$) algebra, and (c) the four types of the
unitary equivalence classes of the matrix representations of the su($2$) subalgebras. In (c), the chosen simple roots and the omitted simple roots are indicated
by the filled black circles and the open grey circles, respectively. }
\label{fig:su4-roots}
\end{center}
\end{figure}

Then, the four unitary equivalence classes of the su($2$) subalgebras can be found, that is, the types (i)-(iv) as illustrated in Figs.~\ref{fig:su4-roots} (c).
The su($2$) subalgebra $\{ {\tilde{O}}_{\frac{3}{2},k} \}$ ($k=1,2,3$) of these four classes satisfy
$[{\tilde{O}}_{\frac{3}{2},\pm}, {\tilde{O}}_{\frac{3}{2},3}]=\pm f{\tilde{O}}_{\frac{3}{2},\pm}$, where
${\tilde{O}}_{\frac{3}{2},\pm} = {\tilde{O}}_{\frac{3}{2},1} \pm i{\tilde{O}}_{\frac{3}{2},2}$.
Suppose the matrices $\{ {\tilde{O}}_{\frac{3}{2},k} \}$ have the block-diagonalized forms as in Eq.~(\ref{eq:direct-sum}); then
the ladder operator ${\tilde{O}}_{\frac{3}{2},+}$ and the observable ${\tilde{O}}_{\frac{3}{2},3}$ should be expressed in terms of
the linear combinations of ${\tilde{A}}_{\frac{3}{2},k}$ ($k=1, 4, 6$) and ${\tilde{\lambda}}_{\frac{3}{2},k}$ ($k=3,8,11$), respectively, as
\begin{equation}
	{\tilde{O}}_{\frac{3}{2},+} = \sum_{k=1,4,6}^6 c_k {\tilde{A}}_k, \label{eq:rise}
\end{equation}
and
\begin{equation}
	{\tilde{O}}_{\frac{3}{2},3} = d_3 {\tilde{\lambda}}_{\frac{3}{2},3} + d_8 {\tilde{\lambda}}_{\frac{3}{2},8} + d_{11} {\tilde{\lambda}}_{\frac{3}{2},11}, \label{eq:z}
\end{equation}
where $c_k$ and $d_k$ are the solutions of $[{\tilde{O}}_{\frac{3}{2},\pm}, {\tilde{O}}_{\frac{3}{2},3}]=\pm f{\tilde{O}}_{\frac{3}{2},\pm}$.
The solutions, the number of subspaces $r$, the subspins $\{J_l \}$ in Eq.~(\ref{eq:direct-sum}), and the structure factor $f$ are respectively given by
\begin{eqnarray}
	&\mathrm{(i)} \ & {\tilde{O}}_{\frac{3}{2},+}
	= \sqrt{\frac{3}{10}} {\tilde{A}}_{\frac{3}{2},1} + \sqrt{\frac{2}{5}} {\tilde{A}}_{\frac{3}{2},4} + \sqrt{\frac{3}{10}} {\tilde{A}}_{\frac{3}{2},6}, \
	{\tilde{O}}_{\frac{3}{2},3} = {\tilde{\lambda}}_{\frac{3}{2},3}, \nonumber \\  & \ &r = 1, \ \{ J_1 = \frac{3}{2} \}, \ f = 1 \label{eq:type1} \\
	&\mathrm{(ii)} \ & {\tilde{O}}_{\frac{3}{2},+} = \frac{1}{\sqrt{2}} ({\tilde{A}}_{\frac{3}{2},1} \pm {\tilde{A}}_{\frac{3}{2},4}), \\ \nonumber & \ &
	{\tilde{O}}_{\frac{3}{2},3} = \frac{2}{\sqrt{10}} {\tilde{\lambda}}_{\frac{3}{2},3} + \frac{1}{\sqrt{2}} {\tilde{\lambda}}_{\frac{3}{2},8}
	- \frac{1}{\sqrt{10}} {\tilde{\lambda}}_{\frac{3}{2},11}, \nonumber \\ & \ &r = 2, \ \{ J_1 = 1, J_2 = 0 \}, \ f = \sqrt{\frac{5}{2}},
	\label{eq:type2} \\
	&\mathrm{(iii)} \ & {\tilde{O}}_{\frac{3}{2},+} = \frac{1}{\sqrt{2}} ({\tilde{A}}_{\frac{3}{2},1} \pm {\tilde{A}}_{\frac{3}{2},6}), \
	{\tilde{O}}_{\frac{3}{2},3} = \frac{1}{\sqrt{5}} {\tilde{\lambda}}_{\frac{3}{2},3} + \frac{2}{\sqrt{5}} {\tilde{\lambda}}_{\frac{3}{2},11}, \nonumber \\ & \ &r = 2, \
	\{ J_1 = J_2 = \frac{1}{2} \}, \ f = \sqrt{5}, \label{eq:type3} \\
	&\mathrm{(iv)} \ & {\tilde{O}}_{\frac{3}{2},+} = {\tilde{A}}_{\frac{3}{2},1}, \
	{\tilde{O}}_{\frac{3}{2},3} = \frac{1}{\sqrt{10}} {\tilde{\lambda}}_{\frac{3}{2},3} + \frac{1}{\sqrt{2}} {\tilde{\lambda}}_{\frac{3}{2},8}
	+ \sqrt{\frac{2}{5}} \ {\tilde{\lambda}}_{\frac{3}{2},11}, \nonumber \\ & \ &r = 3, \ \{ J_1 = \frac{1}{2}, J_2=J_3 = 0 \}, \ f =  \sqrt{10}. \label{eq:type4}
\end{eqnarray}
The type (i) squeezing in Eq.~(\ref{eq:type1}) is equivalent to the spin squeezing among $\{ {\hat{J}}_x, {\hat{J}}_y, {\hat{J}}_z \}$ and the type (iii) squeezing in
Eq.~(\ref{eq:type4}) is equivalent to the quadrupole-octupole squeezing among $\{{\hat{T}}_z^{\beta}, {\hat{T}}_{xyz}, \hat{Y} \}$ and the quadrupole squeezing
among $\{{\hat{Q}}_{zx}, {\hat{Q}}_{yz}, \hat{Y} \}$.

\subsection{Squeezing limits for four types of squeezing}
In the case of the type (i) in Eq.~(\ref{eq:type1}), $r=1$ and the squeezing limit for the one-axis twisting is given by Eq.~(\ref{eq:minsqparam-r1}) as
\begin{equation}
	{\xi}_{\mathrm{min}}^2 \simeq \frac{1}{2} {\left ( \frac{1}{N} \right )}^{2/3} + \frac{1}{3N}, \label{eq:limit-type1}
\end{equation}
which is achieved at the evolution time of ${\mu}_{\mathrm{min}}= \frac{2}{\sqrt{3}} \times N^{-2/3}$ corresponding to $t_{\mathrm{min}} = \frac{1}{\sqrt{3} \chi} \times N^{-2/3}$.

\begin{figure}[t]
\hspace {0.2cm} (a) \hspace{6.8cm} (b)
\begin{center}
\includegraphics[width=7.5cm,clip]{angle_sqs.eps}
\includegraphics[width=7.5cm,clip]{angle_times.eps}
\caption{(Color Online) (a) The $|{\zeta}_1|^2$-dependence on the squeezing limit ${\xi}^2_{\mathrm{min}}$ and (b) the corresponding evolution time
${\mu}_{\mathrm{min}}$ for $N=10^5$. The horizontal dotted lines at ${\xi}_{\mathrm{min}}^2 = 0.00031$ and $0.00049$ in (a) and at ${\mu}_{\mathrm{min}} = 0.0007$ and
$0.0011$ in (b) respectively indicate the squeezing limits for the type (ii) in Eq.~(\ref{eq:limit-type2}) and the type (iii) in Eq.~(\ref{eq:limit-type4}) and the corresponding evolution
times when $|{\zeta}_1|^2=1$. }
\label{fig:sq(ii)and(iv)}
\end{center}
\end{figure}
In the case of the types (ii)-(iv) in Eq.~(\ref{eq:type2})-(\ref{eq:type4}), the squeezing limits depend on the initial coherent state in Eq.~(\ref{eq:CSS-J}) in general; however,
the squeezing limits for the types (ii) in Eq.~(\ref{eq:type2}) and (iv) in Eq.~(\ref{eq:type4}) can
be calculated in the same manner as the type (i), when $|{\zeta}_l|^2 = {\delta}_{l1}$ in the initial state in Eq.~(\ref{eq:CSS-J}).
They are given by
\begin{eqnarray}
	&\mathrm{(ii)}  \ & {\xi}_{\mathrm{min}}^2 \simeq \frac{1}{2} {\left ( \frac{3}{2N} \right )}^{2/3} + \frac{1}{2N}, \label{eq:limit-type2} \\
	&\mathrm{(iv)} \ & {\xi}_{\mathrm{min}}^2 \simeq \frac{1}{2} {\left ( \frac{3}{N} \right )}^{2/3} + \frac{1}{N}, \label{eq:limit-type4}
\end{eqnarray}
respectively.
The minimum squeezing limits in Eqs.~(\ref{eq:limit-type2}) and (\ref{eq:limit-type4}) are achieved at ${\mu}_{\mathrm{min}} = 12^{1/6} \times N^{-2/3}$
($t_{\mathrm{min}} = \frac{12^{1/6}}{5\chi} \times N^{-2/3}$) and ${\mu}_{\mathrm{min}} = 2 \times 3^{1/6} \times N^{-2/3}$ ($t_{\mathrm{min}} = \frac{3^{1/6}}{10\chi} \times N^{-2/3}$),
respectively.
If $|{\zeta}_l|^2 \neq 0$ for $\exists l>0$, the squeezing limits for types (ii) and (iv) cannot be obtained by the expression in Eq.~(\ref{eq:minsqparam-r1}).
We numerically calculate the $|{\zeta}_1|^2$-dependences of the squeezing limits and their corresponding evolution times and illustrate them in Figs.~\ref{fig:sq(ii)and(iv)}
(a) and (b).
In Figs.~\ref{fig:sq(ii)and(iv)}, we plot the squeezing limit ${\xi}_{\mathrm{min}}^2$ and the
evolution time ${\mu}_{\mathrm{min}}$ with respect to $1-|{\zeta}_1|^2$.
The squeezing limits for the types (ii) and (iv) monotonically decrease with increasing $|{\zeta}_1|^2$.
For $|{\zeta}_1|^2 \simeq 1$, the squeezing limits are almost equal to Eqs.~(\ref{eq:limit-type2}) and (\ref{eq:limit-type4}), respectively; however,
for $|{\zeta}_1|^2 < 0.2$, the minimum squeezing limits sharply increase due to the decreases
in the number of the Schwinger bosons which are nonlinearly interacting via the one-axis twisting interactions in Eq.~(\ref{eq:OAT}).

In the case of the type (iii) in Eq.~(\ref{eq:type3}), $r=2$ and $J_1 = J_2 = 1/2$, the $|{\zeta}_1|^2$-dependence of the minimum squeezing limit is periodic because of the symmetry
with respect to the two subspaces.
To see this, let us derive the expression for the squeezing limit for the type (iii):
\begin{equation}
	{\xi}^2 (\mu ) = \frac{1 + \frac{1}{4}  (N-1) \sum_{l=1}^2 {\Delta}_l(\mu )}{
	 \sum_{l=1}^2 {|{\zeta}_l|}^2 {(1 - 2 {|{\zeta}_l|}^2 {\sin}^2 \frac{\mu}{4})}^{N-1}} , \label{eq:sq-iii}
\end{equation}
where ${\Delta}_l$'s ($l=1,2$) are defined as
\begin{eqnarray}
	& {\Delta}_l(\mu ) = \left [ 1 - {\left (1 - 2 {|{\zeta}_l|}^2 {\sin}^2 \frac{\mu}{2} \right )}^{N-2} \right ] \nonumber \\
	& \times \left \{ 1 - \sqrt{1 + {\left [ \frac{4 {|{\zeta}_l|}^2 \sin {\frac{\mu}{2}} \ {(1 - 2 {|{\zeta}_l|}^2 {\sin}^2 \frac{\mu}{4})}^{N-2}}{
	1 - {(1 - 2 {|{\zeta}_l|}^2 {\sin}^2 \frac{\mu}{2})}^{N-2} } \right ]}^2} \right \}. \label{eq:f-iii}
\end{eqnarray}
When the initial state is given by $|{\zeta}_l|^2 = {\delta}_{l1}$, the squeezing limit is given by Eq.~(\ref{eq:limit-type4}) at
${\mu}_{\mathrm{min}} =  2 \times 3^{1/6} \times N^{-2/3}$, which are same as those for the type (iv) with the initial state of $|{\zeta}_l|^2 = {\delta}_{l1}$,
while the evolution time $t_{\mathrm{min}} = \frac{3^{1/6}}{5\chi} \times N^{-2/3}$ is two times larger than that for the type (iv) with the initial state of $|{\zeta}_l|^2 = {\delta}_{l1}$.
When the initial state is the equal superposition of the two subspaces, i.e., ${|{\zeta}_l|}^2 = \frac{1}{2}$, the squeezing limit can be obtained by
assuming $\alpha \gg 1$ and $\beta \ll 1$ to be
\begin{equation}
	{\xi}_{\mathrm{min}}^2 \simeq \frac{1}{2} {\left ( \frac{6}{N} \right )}^{2/3} + \frac{3}{N} \simeq \frac{1}{2} {\left ( \frac{6}{N} \right )}^{2/3}, \label{eq:limit-type3}
\end{equation}
at the evolution time of ${\mu}_{\mathrm{min}} = 2 \times 3^{1/6} \times (N/2)^{-2/3}$ ($t_{\mathrm{min}} = \frac{48^{1/6}}{5\chi} \times N^{-2/3}$).
Equation~(\ref{eq:limit-type3}) is $6^{2/3} \simeq 3.3$ times larger than the type (i) in Eq.~(\ref{eq:limit-type1}), $4^{2/3} \simeq 2.5$ times larger than the type (ii) in
Eq.~(\ref{eq:limit-type2})
with the initial state of $|{\zeta}_l|^2 = {\delta}_{l1}$, and $2^{2/3} \simeq 1.6$ times larger than the type (iv) in Eq.~(\ref{eq:limit-type4})
with the initial state of $|{\zeta}_l|^2 = {\delta}_{l1}$ and the type (iii) with the initial
state of $|{\zeta}_l|^2 = {\delta}_{l1}$.
The $|{\zeta}_1|^2$-dependence of the squeezing limit and the corresponding evolution time for $N=10^5$ are illustrated in Fig.~\ref{fig:sq(iii)} (a).
The squeezing limit reaches the maximum at $|{\zeta}_1|^2 \simeq 1-\frac{\pi}{4}$ and $\frac{\pi}{4}$.
The dependence of the squeezing limit on the number of spins for $|{\zeta}_1|^2 \simeq 1-\frac{\pi}{4}$ is shown in Fig.~\ref{fig:sq(iii)} (b), which can
be well fitted to
\begin{equation}
	{\xi}^2_{\mathrm{min}} \simeq 0.11 \pm 0.00 + \frac{0.57 \pm 0.00}{N^{0.50 \pm 0.00}} + \frac{3.8 \pm 0.0}{N} \label{eq:approx-xi}
\end{equation}
by the least squared method.
Equation~(\ref{eq:approx-xi}) implies that the scaling of the squeezing limit with respect to $N$ is $0$ for ${|{\zeta}_1|}^2 = \frac{\pi}{4}$ and
$1 - \frac{\pi}{4}$, although the squeezing limit is still below the standard quantum limit of ${\zeta}^2 = 1$.
The evolution time corresponding to the squeezing limit for ${|{\zeta}_1|}^2 = 1 - \frac{\pi}{4}$ can be well fitted to
\begin{equation}
	{\mu}_{\mathrm{min}} \simeq (3.9 \pm 0.0) \times N^{-0.73 \pm 0.00}, \label{eq:approx-t}
\end{equation}
with respect to the number of spins $N$ by the least squared method.
\begin{figure}[t]
\hspace {0.2cm} (a) \hspace{6.8cm} (b)
\begin{center}
\includegraphics[width=7.5cm,clip]{angle_sq_type3.eps}
\includegraphics[width=7.5cm,clip]{max_sq_type3.eps}
\caption{(Color Online) (a) The $|{\zeta}_1|^2$-dependence of the squeezing limit ${\xi}^2_{\mathrm{min}}$ and the corresponding evolution time
${\mu}_{\mathrm{min}}$ for $N=10^5$. The maxima of ${\xi}^2_{\mathrm{min}}$
are achieved at ${|{\zeta}_1|}^2 = 1 - \frac{\pi}{4}$ and $\frac{\pi}{4}$. (b) The $N$-dependence of the
squeezing limit ${\xi}^2_{\mathrm{min}}$ and the corresponding evolution time ${\mu}_{\mathrm{min}}$ for ${|{\zeta}_1|}^2 = 1 - \frac{\pi}{4}$.
The fitting function for ${\xi}^2_{\mathrm{min}}$ and ${\mu}_{\mathrm{min}}$ are given by Eq.~(\ref{eq:approx-xi}) and Eq.~(\ref{eq:approx-t}), respectively. }
\label{fig:sq(iii)}
\end{center}
\end{figure}

\section{Conclusion}
In this paper, we consider the collective su($2J+1$) systems and classify the squeezing among the spin and multipolar observables generating the su($2$) subalgebra of the
su($2J+1$) algebra, based on the unitary equivalence class of the su($2J+1$)-dimensional representations of  the observables.
The matrix representations of the observables and their representation spaces can be decomposed into the direct sums of the lower dimensional irreducible representations of the
su($2$) generators in Eq.~(\ref{eq:direct-sum}).
This implies that if two sets of observables belong to the same unitary equivalence class, they can be decomposed into the same matrix representation in Eq.~(\ref{eq:direct-sum})
whose bases can be transformed to each other via an SU($2J+1$) transformation; hence they are characterized by the same subspins $\{J_l \}$ in Eq.~(\ref{eq:direct-sum}) giving the
structure factor $f$ in Eq.~(\ref{eq:f}).
The unitary equivalence class of the su($2$) subalgebra in the su($2J+1$) algebra can be found by choosing vertices in the Dynkin diagram of the su($2J+1$) algebra as shown
in Fig.~\ref{fig:dynkin-sun}.

The squeezing limits are determined by the dimensionality of the unitary equivalence class of the observables and the initial CSS involved by the squeezing.
Taking the one-axis-twisted SSS for example, we calculate the squeezing limit ${\xi}^2_{\mathrm{min}}$, which is given by the function in Eq.~(\ref{eq:sqparam})
in terms of the subspins $\{ J_l \}$ in the irreducible representations in Eq.~(\ref{eq:direct-sum}) and the coefficients $\{ |{\zeta}_l|^2 \}$ of the initial CSS in
Eq.~(\ref{eq:CSS-J}).
When $|{\zeta}_l|^2 = {\delta}_{l1}$ in Eq.~(\ref{eq:CSS-J}), the squeezing limit ${\xi}^2_{\mathrm{min}}$ in Eq.~(\ref{eq:minsqparam-r1}) for the one-axis twisted
SSS achieved to be proportional to $(J_{l_0}N)^{-2/3}$ at the evolution time of $\mu \equiv 2 \chi f^2 t \propto (J_{l_0}N)^{-2/3}$ in the limit of
$J_{l_0}N \chi f^2 t \gg 1$ and $J_{l_0}N (\chi f^2 t)^2 \ll 1$, which implies that the squeezing among the observables, of which matrix representations are irreducible, gives the
minimum squeezing limit of the collective su($2J+1$) consisting of $N$ spin-$J$ particles.
In the case of $|{\zeta}_{l_0}|^2 < 1$ and $^{\exists} |{\zeta}_{l\neq {l_0}}|^2 \neq 0$, the analytical expressions of the squeezing limits in Eq.~(\ref{eq:sqparam}) cannot be easily
obtained due to the interference between the representation spaces in Eq.~(\ref{eq:direct-sum}).

Finally, we apply our classification to the squeezing in the collective su($4$) systems and obtain the squeezing limits analytically or numerically.
The squeezing can be classified into one of four unitary equivalence classes as shown in Fig.~\ref{fig:su4-roots}.
Their squeezing limits depends on the coefficients $\{ |{\zeta}_l|^2 \}$ in the initial coherent states in Eq.~(\ref{eq:CSS-J}) as well as the subspins $\{J_l \}$, whose behaviors were
numerically calculated as shown in Figs.\ref{fig:sq(ii)and(iv)} (a) and \ref{fig:sq(iii)} (a).
Since the subspins and the initial coherent sate reflect the structure of the unitary equivalence class of the spin and multipolar observables; hence the unitary equivalence class
of the observables can be considered as one of the systematical ways to classify and quantify the squeezing.

\ack
E. Y. thanks Prof. Mark Everitt, Prof. Todd Tilma, Dr. Shane Dooley, Mr. Itsik Cohen, and Ms. Marvellous Onuma-Kalu for fruitful discussions.
This work is supported by MEXT Grant-in-Aid for Scientific Research(S) No. 25220601.

redpajama_set_name: RedPajamaArXiv


We're the premier FAEMA Espresso Equipment Dealer in Southern California. We sell and service just about any type of commercial (and various home) espresso machine. And we can help you choose from today's top espresso machine brands. New Frontier Coffee has over 20 years experience. Your trusted source.
If you're looking for an entry-level unit that's sturdy and reliable, then check out the Two-Group Automatic Dieci A2 Espresso Machine (DIECIA2) by Casadio. Dieci is a versatile machine which is easy to use, adjust, and maintain.


text:
Brand Camp
Crisis Toolbox
GMOs in space? Yes, it's time to step out of the food system
Most of what you hear about biotechnology centers around that big company down in St. Louis. (Come on, you know the one.) Genetically modified organisms (GMOs) in the form of crops that eventually end up as food are a hot topic.
It's a shame that the discussion (more often heated debate) rarely gets to other aspects of biotechnology. I was recently smacked into a different line of thought when a scruffy-faced Ph.D. with a shock of blue hair and a ripped T-shirt walked into room at the convention center in Austin, Texas.
Josiah Zayner is a scientist working for NASA with a bent for, appropriately enough, thinking out there. He was conducting a session at the South by Southwest Interactive Festival (SXSW) on the possibilities of biotech. It was refreshing. It was also somewhat embarrassing to be reminded how infrequently most of us who work in the food system think beyond our little window to the present or the ugly specter of not having enough food in 2050. (Yes, yes, I believe biotech will help us feed the world.)
Zayner left the current debate in the wake of his rocket to Mars and made us think about some things I had never considered. For example, astronauts on a trip to the red planet would produce some 1,500 pounds of plastic waste before their journey had barely begun. What if you could bioengineer bacteria to eat that potential debris and turn it into food, fuel or some other useful product? Scientists like Zayner are working on it.
What if you could bioengineer the bacteria living under astronauts' arms to smell good instead of bad?
What if you could create micro organisms that would turn human waste into (one has to get beyond "yuck" to get to Mars, apparently) food or even rocket fuel?
What if you could add vitamin A to rice to keep children in Africa from losing their vision? Wait, we already have that one. Tragically, turning a blind eye to science and the potential benefits of GMOs has largely prevented us from doing nearly as much as we could on this issue.
Yes, there is much to be decided regarding biotech. We are wise to be thoughtful. We would be foolish to do anything but encourage smart guys like Josiah Zayner as they pursue the real future for this amazing and powerful technology.
Charleston Orwig2022-01-12T15:57:27-06:00April 21st, 2015|Brand Strategy and Development|
Sustainability, Security Drives AgTech Investment Explosion
Trends around sustainable agriculture and biodiversity and the need for a more secure food system in light of COVID-19 are spurring ongoing investments into ag technology from diverse investors.
Brand Strategy and Development
Campaigns and Communiction
Crisis and Risk
Alternative Proteins Research: Consumers committed to fighting climate change one bite at a time.
Alternative Proteins: What are they made of? It's the billion-dollar question.
Alternative Proteins: Gen Z puts the planet first, taste second.
What's nxt: Labor Shortages. Where does the food system go from here?
The time to share your DEI journey is ASAP.
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redpajama_set_name: RedPajamaCommonCrawl


text:
David.Stevens@hklaw.com
Atlanta 404.817.8546
Financial Services | Public Finance
David Stevens is a senior counsel in Holland & Knight's Public Finance Practice Group where he advises local governments, conduit borrowers, credit and liquidity facility providers, and trustee banks in financial transactions involving state and local governmental entities, including in particular the issuance of taxable and tax-exempt bonds. Mr. Stevens has extensive experience with airport financings, as well as general fund, pollution control, school district and pooled financings in a variety of transaction structures, including traditional fixed rate bonds; auction rate bonds; variable rate demand bonds; current, advance and forward refunding bonds; synthetic fixed rate bonds; commercial paper programs; tax and revenue anticipation notes; letters of credit; interest rate swaps; and forward purchase and sale agreements. He has also worked as bond counsel, disclosure counsel or underwriter's counsel on more than $25 billion of municipal bond transactions and has participated in a number of complex public-private partnership transactions.
Prior to joining Holland & Knight, Mr. Stevens was a deputy city attorney with the San Francisco City Attorney's Office for 10 years where he advised the San Francisco International Airport (SFO) on tax-exempt bond and finance matters, professional service contracts, airline terminal leases, government audits and revenue retention matters with the Federal Aviation Administration (FAA). While at the City Attorney's Office, Mr. Stevens drafted the 10-year master lease and use agreement between SFO and more than 40 airlines, and headed the airport's legal team in developing a luxury hotel at SFO. Prior to joining the City Attorney's Office, Mr. Stevens was a public finance attorney for more than 10 years with two large law firms in California and Utah where he represented governmental agencies as bond and disclosure counsel, and investment banks as underwriter's counsel in a wide variety of taxable and tax-exempt public finance transactions.
In the community, Mr. Stevens has been involved for many years with fundraising and leadership with the Boy Scouts of America and was recently awarded the 2015 Unit Scouter of the Year Award.
University of California, Berkeley School of Law, J.D.
Brigham Young University, B.A., Near Eastern Studies, magna cum laude
National Association of Bond Lawyers
Moot Court Commendation for Excellence in Advocacy, 1994


It has no English names ,I can't translate this tool's name .I don't know whether it was published in the BBS.
This tool is an AV company employees works,But the authors in 2009 have no longer update. It's in China and xuetr same famous, But xuetr more stable.


(1) Any person desiring to bid for the performance of any construction contract in excess of $250,000 which the department proposes to let must first be certified by the department as qualified pursuant to this section and rules of the department. The rules of the department shall address the qualification of persons to bid on construction contracts in excess of $250,000 and shall include requirements with respect to the equipment, past record, experience, financial resources, and organizational personnel of the applicant necessary to perform the specific class of work for which the person seeks certification. The department may limit the dollar amount of any contract upon which a person is qualified to bid or the aggregate total dollar volume of contracts such person is allowed to have under contract at any one time. Each applicant seeking qualification to bid on construction contracts in excess of $250,000 shall furnish the department a statement under oath, on such forms as the department may prescribe, setting forth detailed information as required on the application. Each application for certification shall be accompanied by the latest annual financial statement of the applicant completed within the last 12 months. If the application or the annual financial statement shows the financial condition of the applicant more than 4 months prior to the date on which the application is received by the department, then an interim financial statement must be submitted and be accompanied by an updated application. The interim financial statement must cover the period from the end date of the annual statement and must show the financial condition of the applicant no more than 4 months prior to the date the interim financial statement is received by the department. However, upon request by the applicant, an application and accompanying annual or interim financial statement received by the department within 15 days after either 4-month period under this subsection shall be considered timely. Each required annual or interim financial statement must be audited and accompanied by the opinion of a certified public accountant. An applicant desiring to bid exclusively for the performance of construction contracts with proposed budget estimates of less than $1 million may submit reviewed annual or reviewed interim financial statements prepared by a certified public accountant. The information required by this subsection is confidential and exempt from the provisions of s. 119.07(1). The department shall act upon the application for qualification within 30 days after the department determines that the application is complete. The department may waive the requirements of this subsection for projects having a contract price of $500,000 or less if the department determines that the project is of a noncritical nature and the waiver will not endanger public health, safety, or property.
(2) Certification shall be necessary in order to bid on a road, bridge, or public transportation construction contract of more than $250,000. However, the successful bidder on any construction contract must furnish a contract bond prior to the award of the contract. The department may waive the requirement for all or a portion of a contract bond for contracts of $150,000 or less under s. 337.18(1).
(3) Upon the receipt of an application for certification, the department shall examine it, verify its statements when necessary, and determine whether the applicant is competent, is responsible, and possesses the necessary financial resources to perform the desired work.
(4) If the applicant is found to possess the prescribed qualifications, the department shall issue to him or her a certificate of qualification that, unless thereafter revoked by the department for good cause, will be valid for a period of 18 months after the date of the applicant's financial statement or such shorter period as the department prescribes. Submission of an application shall not affect expiration of the certificate of qualification. If the department finds that an application is incomplete or contains inadequate information or information that cannot be verified, the department may request in writing that the applicant provide the necessary information to complete the application or provide the source from which any information in the application may be verified. If the applicant fails to comply with the initial written request within a reasonable period of time as specified therein, the department shall request the information a second time. If the applicant fails to comply with the second request within a reasonable period of time as specified therein, the application shall be denied.
(5) The certificate of qualification shall contain a statement fixing the actual amount of work, in terms of estimated cost, which the applicant will be permitted to have on contract with the department and not completed at any one time, and the certificate may contain a statement by the department limiting such bidder to the submission of bids upon a certain class of work. Subject to such restrictions, the certificate of qualification shall authorize the holder to bid on all work on which bids are taken by the department during the period of time specified in the certificate.
(6) Any applicant for a certificate of qualification who is aggrieved by the action of the department may, within 10 days after receiving notification of such action, request a hearing pursuant to chapter 120.
(7) A "contractor" as defined in s. 337.165(1)(d) or his or her "affiliate" as defined in s. 337.165(1)(a) qualified with the department under this section may not also qualify under s. 287.055 or s. 337.105 to provide testing services, construction, engineering, and inspection services to the department. This limitation does not apply to any design-build prequalification under s. 337.11(7) and does not apply when the department otherwise determines by written order entered at least 30 days before advertisement that the limitation is not in the best interests of the public with respect to a particular contract for testing services, construction, engineering, and inspection services. This subsection does not authorize a contractor to provide testing services, or provide construction, engineering, and inspection services, to the department in connection with a construction contract under which the contractor is performing any work.
(8) This section does not apply to maintenance contracts.
(9)(a) Notwithstanding any other law to the contrary, for contracts in excess of $250,000, an authority created pursuant to chapter 348 or chapter 349 may require that persons interested in performing work under contract first be certified or qualified to do the work. Any contractor may be considered ineligible to bid by the governmental entity or authority if the contractor is behind an approved progress schedule for the governmental entity or authority by 10 percent or more at the time of advertisement of the work. Any contractor prequalified and considered eligible by the department to bid to perform the type of work described under the contract shall be presumed to be qualified to perform the work so described. The governmental entity or authority may provide an appeal process to overcome that presumption with de novo review based on the record below to the circuit court.
(b) With respect to contractors not prequalified with the department, the authority shall publish prequalification criteria and procedures prior to advertisement or notice of solicitation. Such publications shall include notice of a public hearing for comment on such criteria and procedures prior to adoption. The procedures shall provide for an appeal process within the authority for objections to the prequalification process with de novo review based on the record below to the circuit court within 30 days.
(c) An authority may establish criteria and procedures under which contractor selection may occur on a basis other than the lowest responsible bidder. Prior to adoption, the authority shall publish for comment the proposed criteria and procedures. Review of the adopted criteria and procedures shall be to the circuit court, within 30 days after adoption, with de novo review based on the record below.
History.—s. 93, ch. 29965, 1955; s. 14, ch. 57-318; s. 1, ch. 61-501; s. 1, ch. 69-314; ss. 23, 35, ch. 69-106; s. 1, ch. 76-85; s. 1, ch. 78-316; s. 10, ch. 80-374; s. 1, ch. 83-18; s. 146, ch. 84-309; s. 33, ch. 86-243; s. 6, ch. 87-100; s. 2, ch. 89-21; s. 495, ch. 95-148; s. 37, ch. 95-257; s. 37, ch. 96-323; s. 161, ch. 96-406; s. 43, ch. 97-96; s. 80, ch. 2002-20; s. 43, ch. 2003-286; s. 4, ch. 2007-66; s. 33, ch. 2007-196; s. 21, ch. 2010-225; s. 33, ch. 2012-174; s. 10, ch. 2014-223.


text:
Kicking off with the Ravens Marketing Team
Kicking off AIGA Baltimore's Design Week 2014 at ADG Creative, the Ravens in-house marketing team sent in three of their best to talk about how Baltimore's own NFL team thinks creatively about design and marketing. Bryan McDonough, Heather Blocher, and David Lang discussed how they target Baltimore football's wide audience across print, digital, social media, and broadcasting in a fascinating presentation, all while showing off their flashy Ravens World Champion rings.
Marketing the Ravens
Emphasize consistency. That's the team's key secret for seamless collaboration. The Baltimore Ravens' marketing team only joins forces with outside agencies occasionally. Almost all efforts take place in-house, even broadcasting. As a graphic designer for the Ravens, presenter Bryan McDonough described the group as being a "mini agency".
The marketing team has to "churn and burn"—they're ready to react. While most NFL creative teams ask "how does this help us sell tickets?", the Ravens are always sold out, allowing more creative freedom. Each campaign piece still must work across all platforms, including print, digital, broadcast, and environmental. The organization also reaches a wide and inclusive audience of women, men, families, young fans, and seasoned fans.
If that wasn't enough of a challenge, the NFL has strict geographical marketing regions when it comes to things like print collateral and tv ads. The Ravens are uniquely sandwiched between two neighboring teams: the Washington Redskins and the Philadelphia Eagles. Luckily, there are no borders on the digital front. Even with the NFL limiting Ravens from print and tv in areas geographically associated with other teams, the web provides ways to market beyond these borders without regional terms.
How an idea becomes a campaign
Each season's new campaign begins after the playoffs. The success of the players on the field decides when to shift focus to the year ahead. At this point, the design team's first deadline is printing the next season's tickets by May. During the summer months, the team prioritizes working with Ravens sponsors.
Ravens' 2014 season tickets
From the very beginning of each campaign, the marketing department's creative process includes senior management, creative, marketing, and board representatives. Several concepts are narrowed down to the best two or three ideas. The public relations, digital, and print teams collaborate to select which concepts should move forward. Polished into solid presentations, the concepts must be clear to the point where the executives—who don't necessarily have creative backgrounds—understand the direction and ideas.
In charge of overseeing these efforts, event presenter Heather Blocher serves as Senior Manager of Advertising & Branding. Heather started as an intern and officially joined the marketing group as her first job after college, a prime example of how team members are given opportunities for growth.
The one surefire way to win a campaign? Have Coach John Harbaugh buy in. If creative and marketing have his vote, then the players and fans approve. John can be tough, however. Even with free lunch in the cafeteria, Harbaugh recently put everyone on a diet. No more "Pizza Friday." It's just salad. At least everyone gets cake after a win, though. On the other hand, losses do affect morale. The next day everyone is bummed right along with the fans—and a lot goes into game day behind the scenes.
How to "Play Like a Raven"
Give it your all, 110%, on and off of the field. That's what it means to "Play Like A Raven"—the team's tagline and annual marketing campaign since 2009. Praised by coaches, players and fans, the phrase remains one of their most successful campaigns.
Focusing on the human element and illustrating the off-the-field "Play Like A Raven" concept, new photo shoots show players training and behind-the-scenes. Quotes from Coach John Harbaugh integrate into the final design to reflect this more personal tone. Landscape posters using this imagery portrayed the players as heroes and role models. The posters were envisioned as inspirational collectables distributed at stadium practice events leading up to the season opening.
Their 2014 marketing both hints back to the original "Play Like A Raven" campaign and pushes the idea to the next level. Fan shots became a major part of the previous campaign's success. The logical next step uses these even more expansively. Updated photography showcases the vast Baltimore cityscape. A new style guide allows visuals to easily translate from print to web, mobile, and broadcast as the team continues to improve the level of consistency.
Taking the campaign online
Hashtags for the Baltimore Ravens were once inconsistent with mixed messages and sources. Fans generated some with others in the marketing department. For example, #relentless is not specifically related to the Ravens alone. Rather than join these existing and wider conversations, their own threads were started specifically for the Ravens' team. David Lang, as the Ravens' Senior Digital Media Manager, discussed how he oversees the website and digital presence, including social media and broadcasting.
David Lang, discussed how he oversees the Ravens' website and digital presence.
Tackling the confusion, the 2014 campaign includes just one hashtag: #playlikearaven. Since the Ravens are a newer team, creating an abbreviation like the Redskins hashtag—#HTTR for Hail To The Redskins—was deemed less meaningful and effective. Instead, #playlikearaven was trademarked. Beginning in 2014, posters and promotional materials added #playlikearaven, and this encouraged players and fans to share virally on social media.
The marketing team found creative ways to involve the players and the rest of the organization as the official hashtag gained popularity. The most positive and encouraging Facebook posts, tweets and photos were printed after sifting through thousands of social media posts. Displayed at the Ravens' cafeteria entrance, these images help connect the staff and players with their fans. This interactive element that goes beyond an online "like" or "retweet" reinforces the idea that the Ravens marketing is not just a one-way conversation.
Outside of social media, President of the Baltimore Ravens Dick Cass is a driver for old school methods. Even though holiday cards are still printed, the marketing team did take to the streets late one night. With permission, the team stenciled graphics across the city using an environmentally safe paint from Germany to engage with their fans. But most often, the Baltimore Ravens leave street-level tactics to the fans themselves. They're the ones who do guerrilla marketing best.
Photos by ADG Creative
Shannon Crabill is a New Media Specialist at T. Rowe Price. Outside of the Internet, you can find her dancing, riding her motorcycle and binge-watching home improvement shows on HGTV. Tweet her at @shannon_crabill.
Brian E. Young is an art director and magazine designer by day and artist by night. When not painting, he's helping unlock imaginations via his blog and The Uncanny Creativity Podcast. Ask him anything: sketchee.com.
By Shannon Crabill & Brian E. Young
AIGA Baltimore Baltimore Branding Campaign Event Recap Speakers


Fiat Chrysler Automobiles NV said Thursday it will lay off 1,500 workers at its Windsor Assembly Plant in Ontario.
The automaker is cutting its third shift at the facility that produces Chrysler Pacifica minivans to "better align production with global demand," Jodi Tinson, FCA spokeswoman, said in a statement. The action would take effect Sept. 30.
Labor union Unifor Local 444 President Dave Cassidy said the automaker informed the union of the layoffs hours earlier at 2 p.m. Thursday. He demanded a meeting immediately with FCA CEO Mike Manley.
"People's lives — 1,500 direct families depend on us," an emotional Cassidy said during a news conference in Windsor. "We're going to do everything possible to make sure we maintain three shifts. Everyone knows our product in Windsor is No. 1 and if you want to build it right, you want to build it in Windsor."
FCA said it will offer retirement packages to eligible employees and make efforts to place indefinitely laid-off hourly employees in open full-time positions as they become available based on seniority.
The automaker is required by its contact with the union to give a six-month notice of permanent reductions. The plant went from two shifts to three shifts in 1993, Cassidy said.
Cassidy said the decision is based on softening demand of the Pacifica. U.S. sales of the minivan dropped 32 percent year-over-year in February.
Thursday's news follows reports on Wednesday that FCA would idle its plants in Windsor and Brampton, Ontario, for the first two weeks of April to align production with demand. It would be the third time in 2019 that the automaker stopped production in Windsor.
Cassidy said the shift cut is not akin to General Motor Co.'s decision to idle its assembly plant in Oshawa, Ontario, when it ends production of the Chevrolet Impala and Cadillac XTS later this year.
Unifor last month suspended its media campaign against GM after talks between the two parties appeared to be progressing toward a future for the Oshawa plant.
Cassidy also criticized Windsor Mayor Drew Dilkens, who in December emphasized the importance of diversifying the city's economy should FCA decide to leave.
"(Explicit) Drew, seriously," Cassidy said. "I have reached out to Drew Dilkens so many times. I have tried to bridge the gap there. ... This is devastating for the city of Windsor, and I hope Drew Dilkens might reach out to me."
The mayor's office did not immediately return request for comment.


Lechuga's is one of Denver's last remaining north-side red sauce joints. While many other Italian restaurants and bars — Pagliaci's, Carbone's, Longo's Subway Tavern — have succumbed to old age and an influx of development, Lechuga's was saved, at least in name and menu, when the restaurant was purchased after longtime owners Chuck Lechuga and Rachael Vigil decided to retire last year.
And the new owners weren't content with just giving the place a good cleaning and then peddling the famous Little Devil sausage rolls and spaghetti to neighbors. They're operating a Lechuga's kiosk on the 16th Street Mall that will serve those sausage canoli, along with a few other snacks and beverages.


text:
Elizabeth Butler1
F, #192221, b. 6 September 1879
Elizabeth Butler was born on 6 September 1879.2 She is the daughter of Lt.-Gen. Rt. Hon. Sir William Butler and Elizabeth Southerden Thompson.1,3 She married Lt.-Col. Randolph Albert Fitzhardinge Kingscote, son of Major Fitzhardinge Kingscote and Agnes Grant Stuart, on 24 July 1903.1
From 24 July 1903, her married name became Kingscote.1 She lived in 1952 at Warren House, Woodgreen, Fordingbridge, Hampshire, EnglandG.1 She lived in 1976 at Warren House, Woodgreen, Fordingbridge, Hampshire, EnglandG.2
Children of Elizabeth Butler and Lt.-Col. Randolph Albert Fitzhardinge Kingscote
Patrick Bloomfield Fitzhardinge Kingscote1 b. 3 Jun 1904, d. 7 Oct 1990
Nigel Adrian Fitzhardinge Kingscote+1 b. 15 Mar 1906, d. 14 Dec 1983
Edgar Noel Fitzhardinge Kingscote1 b. 24 Jan 1908
Flavia Sybil Adeva Kingscote1 b. 21 May 1911, d. 29 Nov 1946
[S40] L. G. Pine, editor, Burke's Genealogical and Heraldic History of the Landed Gentry, 17th edition, (London, England: Burke's Peerage Ltd, 1952), page 1442. Hereinafter cited as Burke's Landed Gentry, 17th ed.
Sir Robert Peter Burton Burton-Chadwick, 2nd Bt.1
M, #192222, b. 22 June 1911, d. 28 August 1983
Last Edited=4 Jul 2015
Sir Robert Peter Burton Burton-Chadwick, 2nd Bt. was born on 22 June 1911.1 He was the son of Sir Robert Burton-Chadwick, 1st Bt. and Catherine Barbara Williams.1 He married Rosalind Mary Stott, daughter of Harry John Stott, in 1937.1 He and Rosalind Mary Stott were divorced in 1949.1 He married, secondly, Beryl Joan Brailsford, daughter of Stanley Frederick James Brailsford, in 1950.1 He died on 28 August 1983 at age 72.1,2
He was educated at St. George's School, Harpenden, Hertfordshire, EnglandG.1 He fought in the Second World War between 1942 and 1945, with NZ Military Force in North Africa and Italy.1 He succeeded as the 2nd Baronet Chadwick, of Bidston, co. Palatine, Cheshire [U.K., 1935] on 21 May 1951.1
Children of Sir Robert Peter Burton Burton-Chadwick, 2nd Bt. and Rosalind Mary Stott
Diana Burton-Chadwick+1 b. 1938
Sally Virginia Burton-Chadwick+1 b. 1940
Children of Sir Robert Peter Burton Burton-Chadwick, 2nd Bt. and Beryl Joan Brailsford
Wendy Lorraine Burton-Chadwick1 b. 12 Dec 1951
Sir Joshua Kenneth Burton-Chadwick, 3rd Bt. b. 1 Feb 1954
Harry John Stott1
Child of Harry John Stott
Rosalind Mary Stott+1
Rosalind Mary Stott1
Rosalind Mary Stott is the daughter of Harry John Stott.1 She married Sir Robert Peter Burton Burton-Chadwick, 2nd Bt., son of Sir Robert Burton-Chadwick, 1st Bt. and Catherine Barbara Williams, in 1937.1 She and Sir Robert Peter Burton Burton-Chadwick, 2nd Bt. were divorced in 1949.1
From 1937, her married name became Burton-Chadwick.
Children of Rosalind Mary Stott and Sir Robert Peter Burton Burton-Chadwick, 2nd Bt.
Diana Burton-Chadwick1
Last Edited=8 Feb 2011
Diana Burton-Chadwick was born in 1938.1 She is the daughter of Sir Robert Peter Burton Burton-Chadwick, 2nd Bt. and Rosalind Mary Stott.1 She married Andrew William Neilson Gemmill in 1965.1
She lived in 2003 at 18 Lyndhurst Street, Richmond, Melbourne, Victoria, AustraliaG.1
Children of Diana Burton-Chadwick and Andrew William Neilson Gemmill
Tristan John Gemmill1 b. 1967
Giles William Gemmill1 b. 1968
Lucinda Harriet Gemmill1 b. 1972
Andrew William Neilson Gemmill1
Andrew William Neilson Gemmill married Diana Burton-Chadwick, daughter of Sir Robert Peter Burton Burton-Chadwick, 2nd Bt. and Rosalind Mary Stott, in 1965.1
Children of Andrew William Neilson Gemmill and Diana Burton-Chadwick
Tristan John Gemmill1
Tristan John Gemmill was born in 1967.1 He is the son of Andrew William Neilson Gemmill and Diana Burton-Chadwick.1 He married Emily Miranda Ball, daughter of Vincent Ball, on 21 September 2001.1
Vincent Ball1
Vincent Ball lived at Chittaway Point, New South Wales, AustraliaG.1
Child of Vincent Ball
Emily Miranda Ball1
Emily Miranda Ball is the daughter of Vincent Ball.1 She married Tristan John Gemmill, son of Andrew William Neilson Gemmill and Diana Burton-Chadwick, on 21 September 2001.1
From 21 September 2001, her married name became Gemmill.1
Giles William Gemmill1
Giles William Gemmill was born in 1968.1 He is the son of Andrew William Neilson Gemmill and Diana Burton-Chadwick.1


"The ironic thing," said Schmitz of Public Allies, "is that you have donors who took major risks in their own fortunes, but are very risk averse when giving to charity. People rightly want their dollars to have the maximum impact, but don't apply the same logic model that they give to their private sector investment." As a result, he said, many social change groups innovate less often and less wildly.
via The Power of Failure – NYTimes.com.


Very soon you'll be hearing those words in Chico!
Your antiques and collectibles evaluated for free by Witherell's Auction, frequent appraisers on PBS's Roadshow. 2 items per person. Limited to the first 100 people. Wednesday, June 27th, 9am-2pm. monca, The Museum of Northern California Art, 900 Esplanade. Co-hosted by KIXE.


Prime Minister Narendra Modi arrived here from Portugal on Sunday, on the second leg of his three-nation tour, that will also take him to the Netherlands.
"Touchdown! Wait is over, Celebrations begin! PM @narendramodi on US soil," the Indian Embassy in the US tweeted.
"Look forward to welcoming India's PM Modi to @WhiteHouse on Monday. Important strategic issues to discuss with a true friend!" US President Donald Trump tweeted on Saturday.
Talks between Modi and Trump in what would be their first face-to-face meeting would focus on ongoing cooperation, including counter-terrorism and defence partnership in the Indo-Pacific region, besides trade and law enforcement, White House Press Secretary Sean Spicer has said.
Spicer said that talks between the two leaders on Monday would be "robust".
"... The President and the Prime Minister will discuss ongoing cooperation, including counterterrorism, defence partnership in the Indo-Pacific region, global cooperation, burden-sharing, trade, law enforcement, and energy. I think it's going to be a very robust discussion," he said at his regular briefing on Friday.
Modi left New Delhi on Saturday on a three-nation tour that took him first to Portugal. He reached Portugal on Saturday ahead of his visit to the US and will visit the Netherlands on June 27 on the way back home.
Indian Foreign Secretary S. Jaishankar flew into Washington ahead of Modi's visit to meet senior officials in the Trump administration and to lay the groundwork for the Modi-Trump meeting.
On Friday, Jaishankar met US Secretary of State Rex Tillerson and both sides focused on areas of convergence.
Indian envoy to the US Navtej Sarna has termed Modi's visit a "great opportunity for the two leaders to know each other". He said Modi will meet the top 20 CEOs of the US during the visit.


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Note: Not quite a Director yet, Brand Manager role is available as well.


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