text:
\section{Introduction}

The cooling of nano mechanical mirrors by the process of optical
up-conversion has become a standard technique \cite{ar0,bk0,bk01}. However
the same technique does not quite work for cooling of Brilliouin acoustic
modes \cite{ar5} in a bulk material medium. In this case, one has to satisfy
constraints arising from the phase matching conditions and at the same time
avoid the generation of Stokes radiation. In the simplest case it is known
that the Stokes process dominates \cite{bk2,ar3}. The latter becomes
critical because a thermal distribution of phonons would give rise to the
generation of both Stokes (down conversion) and anti-Stokes (up conversion)
fields. The dominant Stokes process would lead to heating. Bahl et. al. \cit
{ar5} solved this problem by using a sophisticated resonator so that the
incident pump and generated anti-Stokes fields were resonant with two
different resonator modes, but the generated Stokes field was nonresonant.
In addition, they found a suitable Brilliouin mode in the spherical
resonator geometry, so that the relevant phase matching condition was
satisfied. More generally phonons are known to have many different types of
dispersion relations \cite{ar1,ar4} which in turn determine the phase
matching conditions and therefore one has to examine the cooling of phonon
modes under more general conditions. It is also very interesting to study
the cooling of two or even several different phonon modes simultaneously,
via anti-Stokes upconversion. This is very relevant in the context of
applications in quantum information science when phonons are used as
carriers of quantum information. In the light of this, in this paper we
develop a more general framework for possible cooling of general phonon
modes.

As mentioned above, our main aim here is to present a general theory of
laser cooling of low frequency phonon modes in matter, via the anti-Stokes
up- conversion process of the inelastic light scattering. Although, our
formulation will begin with an overview of the known case of laser cooling
of a single longitudinal acoustic phonon mode via the anti-Stokes Brillouin
scattering process, it will go on to develop a general theory of cooling of
low frequency phonon modes of arbitrary polarization, whether these are
acoustic or optical phonons, and whether these are modes in an extended bulk
matter or in a confined geometry. In fact, the formulation has been kept
general enough to be applicable not only to phonons but to any type of low
frequency collective excitation modes in matter which can give rise to
inelastic light scattering (Raman scattering) via the modulation of the
linear optical dielectric function. The cooling via the three -wave
anti-Stokes up -conversion is possible in principle for any such mode ,
provided that it has a sufficiently long life time compared to the resonant
cavity anti-Stokes optical mode, and the corresponding Raman tensor coupling
of the mode is not too weak. We will also consider the possibility of
cooling more than one phonon mode simultaneously via the three-wave
up-conversion process, which has not been considered earlier even for the
case of longitudinal acoustic phonons \cite{ar5,ar2}.

The organization of the paper is as follows-In Sec II we introduce the basic
features of the phonon photon interaction, relevant to anti-Stokes
generation. We derive the basic equations for the coupled phonon and
electromagnetic fields. In Sec III we derive the equations for phonon and
photon fields in terms of the relevant resonator modes. The nature of the
phonon dispersion or its frequency spectrum, and its relevance for
phonon-photon interaction are discussed in Sec IV. In Sec V we describe
quantum Langevin equations and show how the single mode cooling emerges \cit
{ar2}. In Sec VI we discuss collective effects in simultaneous cooling of
two modes. The numerical results for the simultaneous cooling of two modes
are given in Sec VII, with a discussion of these results.

\section{Basic Equations for Interaction of Electromagnetic Fields with
Vibrational Fields}

In this section, we present, for completeness, a first principle derivation
of the basic equations for the Stokes and anti-Stokes scattering from
vibrational modes. This basic description would enable us to formulate the
problem of more general three wave interactions in a resonator. For
simplicity, let us first consider the case of longitudinal acoustic waves in
matter, which give rise to the Brillouin scattering. The long wavelength
longitudinal acoustic waves can be described either in terms of density
fluctuations $\Delta \rho (\mathbf{r},t)$ or the displacement field $\mathbf
u}(\mathbf{r},t)$, which are related by
\begin{equation}
\frac{\Delta \rho (\mathbf{r},t)}{\rho _{0}}=-\nabla \cdot \mathbf{u}
\mathbf{r},t),  \label{e1}
\end{equation
where $\Delta \rho $, $\rho _{0}$ are, say, mass densities, and $\rho _{0}$
is the mean mass density of the medium. In terms of the displacement field
\mathbf{u}(\mathbf{r},t)$, the unperturbed Hamiltonian of the acoustic field
is given by
\begin{eqnarray}
H_{S} &=&\int d^{3}\mathbf{r}\mathcal{H}_{S},  \label{e2} \\
\mathcal{H}_{S} &=&\frac{1}{2}\rho _{0}\dot{u}^{2}+\frac{1}{2}\rho
_{0}v_{S}^{2}(\nabla \cdot \mathbf{u})^{2},  \label{e3}
\end{eqnarray
where $v_{S}$ is the sound velocity. Also, for longitudinal sound wave,
\rho _{0}v_{S}^{2}=B$, which is the inverse of the compressibility $c_{S}$.
The equation (\ref{e2}) leads to the wave equation for sound waves
\begin{equation}
\frac{\partial ^{2}\mathbf{u}}{\partial t^{2}}-v_{S}^{2}\nabla ^{2}\mathbf{u
=\frac{\mathbf{f_{\mathrm{ext}}}}{\rho _{0}},  \label{e4}
\end{equation
where $\mathbf{f_{\mathrm{ext}}}$ is external force if any. Because of the
sound wave, the electromagnetic polarization of the medium is modulated by
the density wave. The change $\Delta \mathbf{P}$ in polarization can be
written in terms of the dielectric function as
\begin{eqnarray}
\Delta \mathbf{P} &=&\Delta \left( \frac{\epsilon -1}{4\pi }\right) \mathbf{
}  \notag \\
&=&\frac{1}{4\pi }\left( \frac{\partial \epsilon }{\partial \rho }\right)
_{0}\Delta \rho \mathbf{E}=\frac{\rho _{0}}{4\pi }\left( \frac{\partial
\epsilon }{\partial \rho }\right) _{0}\frac{\Delta \rho }{\rho _{0}}\mathbf{
}  \notag \\
&=&-\frac{\gamma _{e}}{4\pi }(\nabla \cdot \mathbf{u})\mathbf{E,}  \label{e5}
\end{eqnarray
where
\begin{equation}
\gamma _{e}\equiv \rho _{0}\left( \frac{\partial \epsilon }{\partial \rho
\right) _{0}  \label{e6}
\end{equation
is the so called electrostrictive constant. For simplicity we consider an
isotropic medium. The interaction density (interaction between the sound
wave and optical fields) is thus given by \cite{bk1}:
\begin{equation}
\mathcal{H}_{\mathrm{int}}=-\frac{1}{2}\Delta \mathbf{P}\cdot \mathbf{E}
\frac{\gamma _{e}}{8\pi }(\nabla \cdot \mathbf{u})E^{2},  \label{e7}
\end{equation
so that the interaction Hamiltonian $H_{\mathrm{int}}=\int \mathcal{H}_
\mathrm{int}}(\mathbf{r})d^{3}\mathbf{r}$. In view of the volume
integration, Eq. (\ref{e7}) also be written as
\begin{equation}
\mathcal{H}_{\mathrm{int}}=-\frac{\gamma _{e}}{8\pi }(\mathbf{u}\cdot \nabla
)E^{2}.  \label{e8}
\end{equation}

From Eq.(\ref{e8}) we can obtain the force $f_{\alpha }$ acting on the
vibrational mode which would be $-\partial \mathcal{H}/\partial u_{\alpha }
, i.e.
\begin{equation}
\mathbf{f}=\frac{\gamma _{e}}{8\pi }\nabla E^{2}.  \label{e9}
\end{equation
The quantity $-\frac{\gamma _{e}}{8\pi }E^{2}$ can be identified as the
pressure due to electromagnetic waves. Using (\ref{e9}) the wave equation
for $\mathbf{u}$ become
\begin{equation}
\frac{\partial ^{2}\mathbf{u}}{\partial t^{2}}-v_{S}^{2}\nabla ^{2}\mathbf{u
-\Gamma ^{\prime }\nabla ^{2}\frac{\partial \mathbf{u}}{\partial t}=\frac
\gamma _{e}}{8\pi \rho _{0}}\nabla E^{2},  \label{e10}
\end{equation
where we have also introduced the damping parameter $\Gamma ^{\prime }$ for
sound waves.

The electromagnetic field $\mathbf{E}$ satisfies wave equation
\begin{equation}
\nabla ^{2}\mathbf{E}-\frac{\epsilon }{c^{2}}\frac{\partial ^{2}\mathbf{E}}
\partial t^{2}}=\frac{4\pi }{c^{2}}\frac{\partial ^{2}(\Delta \mathbf{P})}
\partial t^{2}},  \label{e11}
\end{equation
where $\epsilon $ is the linear dielectric function of the medium. In
writing the above equation, it has to be noted that we have ignored the
linear dispersion of the dielectric function (which leads to the
corresponding group velocity instead of the phase velocity) and the linear
absorption of optical fields. Later we will however add the linear
absorption term as a phenomenological parameter.

Using Eq. (\ref{e5}), Eq. (\ref{e11}) reduces to
\begin{equation}
\nabla ^{2}\mathbf{E}-\frac{\epsilon }{c^{2}}\frac{\partial ^{2}\mathbf{E}}
\partial t^{2}}=-\frac{\gamma _{e}}{c^{2}}\frac{\partial ^{2}}{\partial t^{2
}\left[ \left( \nabla \cdot \mathbf{u}\right) \mathbf{E}\right] ,
\label{e12}
\end{equation}

The analysis given above can be generalized to any type of vibrational mode
in an arbitrary medium, whether it is an acoustic or optical phonon mode, a
longitudinal or transverse mode in a bulk material, or whether it is a mode
in a confined geometry. We can expand $\epsilon $ as a Taylor series in the
relevant phonon displacement field $\mathbf{Q}(\mathbf{r},t)$ whence
\begin{equation}
(\Delta \mathbf{P})_{i}=\sum_{jk}\frac{1}{4\pi }\frac{\partial \epsilon _{ij
}{\partial Q_{k}}E_{j}Q_{k}=\sum_{jk}R_{ijk}E_{j}Q_{k}.  \label{e13}
\end{equation
where $R_{ijk}$ is the generalized Raman tensor for the mode $\mathbf{Q}$.
If more than one phonon mode is involved, then Eq. (\ref{e13}) is to be
summed over contributions from all the phonon modes. The interaction
Hamiltonian density now becomes
\begin{equation}
\mathcal{H}_{\mathrm{int}}=-\frac{1}{2}\sum_{ijk}R_{ijk}E_{j}E_{i}Q_{k}.
\label{e14a}
\end{equation
It should be emphasized here that we have introduced a new variable $\mathbf
Q}$ for the phonon displacement field for a general phonon mode in matter,
instead of the variable $\mathbf{u}$ used for the low frequency longitudinal
acoustic displacement field, on purpose. Both have the same dimensions of
length. But, whereas $\mathbf{u}$ is related to the density fluctuation via
Eq. (\ref{e1}), and $\nabla \cdot \mathbf{u}=0$ for transverse modes,
\mathbf{Q}$ represents the displacement field for any phonon mode, whether
it is transverse or longitudinal, or whether it is an acoustic mode or an
optical mode. In fact, in the plane-wave representation for the case of bulk
matter, for a mode with wave vector $\mathbf{q}$ and a given general
dispersion relation $\Omega \left( \mathbf{q}\right) $, the displacement
field $\mathbf{Q}$ satisfies the equation, $\frac{\partial ^{2}\mathbf{Q}}
\partial t^{2}}+\Omega ^{2}\left( \mathbf{q}\right) \mathbf{Q}=0$, with its
velocity $\mathbf{v}=\partial \Omega /\partial \mathbf{q}$. For the low
frequency longitudinal acoustic mode, $\mathbf{Q}$ will be the same as
\mathbf{u}$, with $\Omega \left( \mathbf{q}\right) =v_{S}q$. Also, it should
be noted that the dimensions of the generalized Raman tensor $R_{ijk}$
differ from the dimensions of $\gamma _{e}$ of Eq. (\ref{e6}) by a factor of
($1/$length).

In terms of unit polarization vectors, $\hat{e}^{\left( 1\right) }$, $\hat{e
^{\left( 2\right) }$ and $\hat{e}^{\left( Q\right) }$ for the three
interacting modes, namely, the two optical modes and one phonon mode,
respectively, relevant to the process being considered, one can also define
a scalar effective Raman coupling constant for a bulk material, separately
for the Stokes and the anti-Stokes interaction. For example, if $2$ labels
the anti-Stokes optical mode, for the anti-Stokes process one has the
effective bulk coupling constan
\begin{equation}
R^{\left( Q\right) }=\sum_{ijk}R_{ijk}\hat{e}_{i}^{\left( 2\right) \ast
\hat{e}_{j}^{\left( 1\right) }\hat{e}_{k}^{\left( Q\right) }.  \label{e14b}
\end{equation
For the special case of the long wavelength bulk longitudinal acoustic mode
of wave vector $\mathbf{q}$, frequency $\omega _{m}$ and velocity $v_{S}$,
described by Eqs. (\ref{e10}) and (\ref{e11}), one ha
\begin{equation}
4\pi R_{ijk}=\text{i}\gamma _{e}\delta _{ij}q_{k}
\begin{array}{cc}
&
\end{array
4\pi R^{\left( Q\right) }=\text{i}\gamma _{e}q=\text{i}\gamma _{e}\frac
\omega _{m}}{v_{S}},  \label{e14c}
\end{equation
which follows on comparing Eqs. (\ref{e14a}) and (\ref{e14b}) with Eq. (\re
{e8}).

\section{Interaction between Electromagnetic Fields and phonons in a
Resonator}

The coupled equations (\ref{e10}) and (\ref{e12}) are general enough to
describe a variety of situations. Since we are interested here in cooling
issues, we consider the case involving modes in resonators. We also
specialize to the case of the anti-Stokes scattering. Let the
electromagnetic field $\mathbf{E}$ consist of waves $\mathbf{E}_{1}$ and
\mathbf{E}_{2}$ at the frequency $\omega _{1}$ (called pump), and $\omega
_{2}$ (anti-Stokes field). The frequencies $\omega _{1}$ and $\omega _{2}$
would be in the neighborhood of the resonator frequencies $\omega _{\mathrm
c1}}$ and $\omega _{\mathrm{c2}}$. Each resonator frequency would have
certain line width. It is assumed that $\omega _{i}-\omega _{\mathrm{c}i}$
is within the line width of the mode. We can then write the field as
\begin{equation}
\mathbf{E}_{i}(\mathbf{r},t)=\vec{\phi}_{i}\mathcal{E}_{i}(t)\mathrm{e}^{
\mathrm{i}\omega _{i}t}+c.c
\begin{array}{cc}
. &
\end{array}
\label{e15}
\end{equation
In our formulation, we will use the familiar slowly varying amplitude
approximation (SVAA) for finding the temporal variation of the field
amplitudes $\mathcal{E}_{i}(t)$. Here $\vec{\phi}_{i}$ (with $\nabla \cdot
\vec{\phi}_{i}=0$) is the spatial mode function for the electromagnetic mode
in the resonator at the frequency $\omega _{\mathrm{c}i}$ such that $\nabla
^{2}\vec{\phi}_{i}+\frac{\omega _{\mathrm{c}i}^{2}}{c^{2}}\epsilon _{i}\vec
\phi}_{i}=0$. The amplitude $\mathcal{E}_{i}(t)$ is a slowly varying
function of $t$. We use
\begin{eqnarray*}
\nabla ^{2}\mathbf{E}_{i}-\frac{\epsilon _{i}}{c^{2}}\frac{\partial ^{2
\mathbf{E}_{i}}{\partial t^{2}} &\cong &(\nabla ^{2}\vec{\phi}_{i}+\frac
\epsilon _{i}}{c^{2}}\omega _{i}^{2}\vec{\phi}_{i})\mathcal{E}_{i}\mathrm{e
^{-\mathrm{i}\omega _{i}t}+ \\
&+&2\mathrm{i}\frac{\omega _{i}\epsilon _{i}}{c^{2}}\frac{\partial \mathcal{
}_{i}}{\partial t}\mathrm{e}^{-\mathrm{i}\omega _{i}t}\vec{\phi}_{i}+c.c.
\end{eqnarray*
which on using $\omega _{i}^{2}-\omega _{\mathrm{c}i}^{2}\approx 2\omega _
\mathrm{c}i}(\omega _{i}-\omega _{\mathrm{c}i})=-2\omega _{\mathrm{c
i}\Delta _{i}$ reduces to
\begin{eqnarray}
&&\nabla ^{2}\mathbf{E}_{i}-\frac{\epsilon _{i}}{c^{2}}\frac{\partial ^{2
\mathbf{E}_{i}}{\partial t^{2}}  \notag \\
&\approx &2\vec{\phi}_{i}\frac{\epsilon _{i}}{c^{2}}\omega _{\mathrm{c}i
\left[ \mathrm{i}\frac{\partial \mathcal{E}_{i}}{\partial t}-\Delta _{i
\mathcal{E}_{i}\right] \mathrm{e}^{-\mathrm{i}\omega _{i}t}+c.c.,
\label{e16}
\end{eqnarray
$\Delta _{i}=\omega _{\mathrm{c}i}-\omega _{i}$. Note that we have kept only
the terms which are up to the first order in the time derivative or in the
frequency difference $\Delta _{i}$. We next write the phonon mode as
\begin{equation}
\mathbf{u}(\mathbf{r},t)=\vec{\psi}(\mathbf{r})u(t)\mathrm{e}^{-\mathrm{i
\omega _{m}t}+c.c.,  \label{e17}
\end{equation
where, the mode function $\vec{\psi}(\mathbf{r})$ has the dimensions of the
displacement field itself and $u(t)$ is the dimensionless amplitude of the
mode. Note that phonons are interacting with a thermal bath, and depending
on the strength of this interaction, the frequency spectrum of the
amplitudes $u(t)$ for the given mode of frequency $\omega _{m}$ will
represent the line width around the frequency $\omega _{m}$. When such
phonons interact with the external optical fields via the three-wave
interaction, this spectrum would be modified further. For longitudinal
phonons curl $\vec{\psi}(\mathbf{r})=0$. Using the slowly varying amplitude
approximation for the phonon mode amplitude also, the procedure that led to
Eq. (\ref{e16}) now leads t
\begin{equation}
-\nabla ^{2}\mathbf{u}+\frac{\partial ^{2}\mathbf{u}}{v_{S}^{2}\partial t^{2
}\approx 2\frac{\vec{\psi}}{v_{S}^{2}}\omega _{m}\left[ -\mathrm{i}\frac
\partial u}{\partial t}\right] \mathrm{e}^{-\mathrm{i}\omega _{m}t}+c.c
\begin{array}{cc}
. &
\end{array}
\label{e18}
\end{equation
Next we need to work out the interaction terms. Since we are considering the
anti-Stokes process, $\omega _{1}+\omega _{m}\approx \omega _{2}$. This
relation has to be satisfied within the line widths of all three modes
\omega _{\mathrm{c1}}$, $\omega _{\mathrm{c2}}$ and $\omega _{m}$ and hence
all subsequent statements are to be understood within linewidths. Clearly a
term like $\mathcal{E}_{1}^{\ast }\mathcal{E}_{2}$ on the right hand side of
Eq.(\ref{e10}) will lead to the mode $\mathbf{u}$. Then the righthand side
of Eq. (\ref{e10}) is to be approximated by
\begin{equation}
\nabla E^{2}=\mathcal{E}_{1}^{\ast }\mathcal{E}_{2}\nabla \left( \vec{\phi
_{1}^{\ast }\cdot \vec{\phi}_{2}\right) \mathrm{e}^{\mathrm{i}\left( \omega
_{1}-\omega _{2}\right) t}+c.c
\begin{array}{cc}
. &
\end{array}
\label{e19}
\end{equation
Similarly for deriving equation, say, for $\mathcal{E}_{2}$ from Eq. (\re
{e12}), the right hand side is to be approximated b
\begin{eqnarray}
&&-\frac{\gamma _{e}}{c^{2}}\frac{\partial ^{2}}{\partial t^{2}}\left[
\left( \nabla \cdot \mathbf{u}\right) \mathbf{E}\right]   \notag \\
&\cong &\frac{\gamma _{e}}{c^{2}}\omega _{2}^{2}\left( \mathcal{E
_{1}u\left( \nabla \cdot \vec{\psi}\right) \vec{\phi}_{1}\right) \mathrm{e
^{-\mathrm{i}\left( \omega _{1}+\omega _{m}\right) t}+c.c
\begin{array}{cc}
. &
\end{array}
\label{e20}
\end{eqnarray
On combining Eqs. (\ref{e16})-(\ref{e20}) and carrying out all the
simplifications, we get final equations for $\mathcal{E}_{1}$, $\mathcal{E
_{2}$ and $u$
\begin{eqnarray}
\frac{\partial \mathcal{E}_{2}}{\partial t} &=&-\kappa _{2}\mathcal{E}_{2}
\text{i}\Delta _{2}\mathcal{E}_{2}-\text{i}\frac{\chi _{2}}{\epsilon _{2}
\omega _{\mathrm{c}2}u\mathcal{E}_{1}\mathrm{e}^{\mathrm{i}\delta t},
\label{e21} \\
\frac{\partial \mathcal{E}_{1}}{\partial t} &=&-\kappa _{1}\mathcal{E}_{1}
\text{i}\Delta _{1}\mathcal{E}_{1}-\text{i}\frac{\chi _{2}^{\ast }}{\epsilon
_{1}}\omega _{\mathrm{c}1}u^{\ast }\mathcal{E}_{2}\mathrm{e}^{-\mathrm{i
\delta t},  \label{e22} \\
\frac{\partial u}{\partial t} &=&-\Gamma u-\text{i}\frac{\chi _{2}^{\ast
\mathcal{E}_{1}^{\ast }\mathcal{E}_{2}\mathrm{e}^{-\mathrm{i}\delta t}}{4\pi
\rho _{0}\omega _{m}\int \vec{\psi}^{\ast }\left( \mathbf{r}\right) \cdot
\vec{\psi}\left( \mathbf{r}\right) d^{3}\mathbf{r}},  \label{e23} \\
\delta  &=&\omega _{2}-\omega _{1}-\omega _{m},  \label{e24} \\
\chi _{2} &=&\frac{\gamma _{e}}{2}\int \left( \vec{\phi}_{2}^{\ast }\left(
\mathbf{r}\right) \cdot \vec{\phi}_{1}\left( \mathbf{r}\right) \right)
\left( \nabla \cdot \vec{\psi}\left( \mathbf{r}\right) \right) d^{3}\mathbf
r.}  \label{e25}
\end{eqnarray
Here $\chi _{2}$ is the effective three mode coupling which depends on the
overlap of all the modes. The integral in Eq. (\ref{e25}) is the traditional
phase matching integral if plane waves are used for all the three modes in a
bulk material. In writing Eqs. (\ref{e21})-(\ref{e25}), we have added
linewidth terms for all the modes. Note that $\gamma _{e}\left( \nabla \cdot
\mathbf{u}\right) $ is dimensionless and $\chi _{2}$ is dimensionless, the
term $\mathcal{E}^{2}$ has dimensions of energy and $\rho
_{0}v_{S}^{2}\left( \nabla \cdot \vec{\psi}\right) ^{2}u^{2}$ has dimensions
of energy density, keeping in view that $u$ and $\nabla \cdot \vec{\psi}$
are dimensionless. The term $\rho _{0}$ in Eq.(\ref{e23}) can be eliminated
by using the kinetic energy term $\frac{1}{2}\int \rho _{0}^{2}\left\vert
\mathbf{\dot{u}}\right\vert ^{2}d^{3}\mathbf{r}=\frac{1}{2}\rho _{0}\int
\left\vert \mathbf{u}\right\vert ^{2}\omega _{m}^{2}d^{3}\mathbf{r}=\rho
_{0}\left\vert u\right\vert ^{2}\omega _{m}^{2}\int \vec{\psi}^{\ast }\left(
\mathbf{r}\right) \cdot \vec{\psi}\left( \mathbf{r}\right) d^{3}\mathbf{r=
\frac{\hbar \omega _{m}}{2}\left\vert u\right\vert ^{2}$ and henc
\begin{equation}
\rho _{0}\omega _{m}^{2}\int \vec{\psi}^{\ast }\left( \mathbf{r}\right)
\cdot \vec{\psi}\left( \mathbf{r}\right) d^{3}\mathbf{r}=\frac{\hbar \omega
_{m}}{2}.  \label{e26}
\end{equation
We can also introduce a dimensionless quantity $\left( 2\pi \frac{\hbar
\omega }{\epsilon }\right) \left\vert a\right\vert ^{2}=\left\vert \mathcal{
}\right\vert ^{2}$. In terms of the dimensionless quantities $a_{1}$, $a_{2}$
and $u$, the final equations ar
\begin{eqnarray}
&&\frac{\partial a_{2}}{\partial t}=-\kappa _{2}a_{2}-\text{i}\Delta
_{2}a_{2}-\text{i}\beta ua_{1}\mathrm{e}^{\mathrm{i}\delta t},  \label{e27}
\\
&&\frac{\partial a_{1}}{\partial t}=-\kappa _{1}a_{1}-\text{i}\Delta
_{1}a_{1}-\text{i}\beta ^{\ast }u^{\ast }a_{2}\mathrm{e}^{-\mathrm{i}\delta
t},  \label{e28} \\
&&\frac{\partial u}{\partial t}=-\Gamma u-\text{i}\beta ^{\ast }a_{1}^{\ast
}a_{2}\mathrm{e}^{-\mathrm{i}\delta t},  \label{e29}
\end{eqnarray
where the coupling constant $\beta $ is now given b
\begin{equation}
\beta =\frac{\gamma _{e}}{2}\sqrt{\frac{\omega _{\mathrm{c}2}\omega _
\mathrm{c}1}}{\epsilon _{2}\epsilon _{1}}}\int \left[ \vec{\phi}_{2}^{\ast
}\left( \mathbf{r}\right) \cdot \vec{\phi}_{1}\left( \mathbf{r}\right)
\right] \left( \nabla \cdot \vec{\psi}\left( \mathbf{r}\right) \right) d^{3
\mathbf{r}.  \label{e30a}
\end{equation
For a more general phonon mode described by the interaction density in Eq.
\ref{e14a}), one can follow a similar procedure as detailed above for the
longitudinal acoustic mode. Using the same symbols as in the equation (\re
{e17}), one can describe the general vibrational mode by its displacement
fiel
\begin{equation}
\mathbf{Q}(\mathbf{r},t)=\vec{\psi}\left( \mathbf{r}\right) u\left( t\right)
e^{-\text{i}\omega _{m}t}+c.c,  \label{e30b}
\end{equation
where $u\left( t\right) $ is again the slowly varying dimensionless
amplitude of the phonon mode and $\vec{\psi}\left( \mathbf{r}\right) $ is
the corresponding mode function. Instead of introducing too many additional
symbols, note that we are using the same symbols for mode functions and
their amplitudes even for the general phonon mode. It requires a few words
of caution. It should always be remembered that now $\nabla \times \vec{\psi}
$ is not necessarily zero because it may not be representing a longitudinal
phonon mode.

In such a case, one can show that we obtain the same set of final working
equations (\ref{e27}) - (\ref{e29}), except that instead of the expression
\ref{e30a}), the anti-Stokes coupling constant is now defined b
\begin{equation}
\beta ^{\left( Q\right) }=2\pi \sqrt{\frac{\omega _{\mathrm{c}2}\omega _
\mathrm{c}1}}{\epsilon _{2}\epsilon _{1}}}\sum_{ijk}R_{ijk}\int \phi
_{2i}^{\ast }\left( \mathbf{r}\right) \phi _{1j}\left( \mathbf{r}\right)
\psi _{k}\left( \mathbf{r}\right) d^{3}\mathbf{r}.  \label{e30c}
\end{equation}

It has to be noted that the resonator is being driven by a pump field of
frequency $\omega _{1}$. Thus Eq.(\ref{e27}) is to be modified by adding the
effect of the external field. We change $\kappa _{1}a_{1}$ term
\begin{equation}
\kappa _{1}a_{1}\rightarrow \kappa _{1}\left( a_{1}-\mathcal{\tilde{E}
_{1}\right) .  \label{e31}
\end{equation
Here $\mathcal{\tilde{E}}_{1}$ is dimensionless and is determined by the
resonator boundary conditions. The phonon mode is driven by thermal
fluctuations, thus Eq. (\ref{e29}) is to be modified by converting it into a
Langevin equation (see Eq. (\ref{ev5})). The Eq. (\ref{e28}) is also
converted to Langevin equation by adding terms corresponding to the input
vacuum field.

It is quite interesting to note that the basic equations (\ref{e27}) - (\re
{e29}) are quite generic in nature as these are applicable to a variety of
situations. The details of a specific system enter through the coupling
constant $\beta $ and the damping parameter $\Gamma ^{^{\prime }s}$ and
\kappa ^{^{\prime }s}$. The coupling constant Eq. (\ref{e30a}) depends on
the mode under consideration. We also note that in deriving equations (\re
{e27}) - (\ref{e29}) we have ignored the Stokes processes which are expected
to be unimportant as we assume that there is no mode resonant with the
Stokes frequency.

We also note that the coherent terms (i.e. terms without damping) in (\re
{e27}) - (\ref{e29}) can be obtained from the following effective Hamiltonia
\begin{equation}
\mathcal{H}_{\text{eff}}=\hbar \Delta _{1}a_{1}^{\dagger }a_{1}+\hbar \Delta
_{2}a_{2}^{\dagger }a_{2}+\hbar \left( \beta ^{\ast }b^{\dagger
}a_{1}^{\dagger }a_{2}\mathrm{e}^{-\mathrm{i}\delta t}+H.c.\right) ,
\label{e36}
\end{equation
where we have used the second quantized notation $u\rightarrow b$, $u^{\ast
}\rightarrow b^{\dagger }$, $a_{i}^{\ast }\rightarrow a_{i}^{\dagger }$. The
operators $a_{i}^{^{\prime }s}$, $a_{i}^{^{\dagger \prime }s}$, $b$ and
b^{\dagger }$ satisfy Bosonic commutation relations. Under the assumption
that the pump $a_{1}$ remains undepleted and is at resonance with the cavity
mode ($\Delta _{1}\simeq 0$), $a_{1}\rightarrow \mathcal{E}_{1}$, Eq. (\re
{e36}) reduces t
\begin{equation}
\mathcal{H}_{\text{eff}}=\hbar \Delta _{2}a_{2}^{\dagger }a_{2}+\hbar \left(
\beta ^{\ast }b^{\dagger }a_{2}\mathcal{E}_{1}^{\ast }\mathrm{e}^{-\mathrm{i
\delta t}+H.c.\right) .  \label{e37}
\end{equation
It should be borne in mind that the time dependence like $e^{-\text{i}\omega
_{2}t}$ and $e^{-\text{i}\omega _{m}t}$ were separated out from $a_{2}$ and
b$ (Eq. (\ref{e15}) and Eq. (\ref{e17})). If we work in a picture where such
time dependences are kept, then instead of Eq. (\ref{e37}), we will hav
\begin{equation}
\mathcal{H}_{\text{eff}}=\hbar \omega _{\mathrm{c}2}a_{2}^{\dagger
}a_{2}+\hbar \omega _{m}b^{\dagger }b+\hbar \left( \beta ^{\ast }b^{\dagger
}a_{2}\mathcal{E}_{1}^{\ast }\mathrm{e}^{\mathrm{i}\omega _{1}t}+H.c.\right)
.  \label{e38}
\end{equation
It is important to keep track of the fast time dependences. Having derived
Eq. (\ref{e38}) from first principles, we will use it's form to write the
corresponding Hamiltonian for the case of two phonon modes.

\section{PHONON DISPERSION RELATIONS , CAVITY MODES AND DAMPING PARAMETERS}

Let us first consider the general problem of cooling phonon modes in a bulk
material, which are characterized by the dispersion relation $\Omega \left(
\mathbf{q}\right) $, where $\mathbf{q}$ is the wave vector in a bulk
material. Here the mode functions are plane waves. The phonon field is
proportional to $\exp \left\{ \text{i}\mathbf{q}\cdot \mathbf{r}-\text{i
\Omega \left( \mathbf{q}\right) t\right\} $. We could have transverse
phonons as in the case of Raman active optical phonons in solids, or
longitudinal acoustic phonons as, say, in the case of Brillouin scattering.
As already discussed earlier, the physical mechanism for cooling is the
upconversion (anti-Stokes process) of light of frequency $\omega _{1}$ and
wave vector $\mathbf{k}_{1}$. The upconversion is subject to the energy
momentum conservatio
\begin{equation*}
\Omega \left( \mathbf{q}\right) +\omega _{1}\left( \mathbf{k}_{1}\right)
\rightarrow \omega _{2}\left( \mathbf{k}_{2}\right)
\begin{array}{cc}
, &
\end{array
\mathbf{q}+\mathbf{k}_{1}=\mathbf{k}_{2}
\end{equation*
In the upconversion process a thermal phonon of frequency $\Omega \left(
\mathbf{q}\right) $ is quickly removed. Further in order for the
upconversion to be effective, we have to make sure that the process of
downconversion (Stokes process) is avoide
\begin{equation*}
\omega _{1}\left( \mathbf{k}_{1}\right) -\Omega \left( \mathbf{q}\right)
=\omega _{s}\left( \mathbf{k}_{s}\right)
\begin{array}{cc}
, &
\end{array
\mathbf{k}_{1}-\mathbf{q}=\mathbf{k}_{s}
\end{equation*
This is where the use of resonators and cavities becomes important. The form
of the dispersion relation or the frequency spectrum of the phonon modes is
also important. Several interesting forms of dispersion relations in simple
geometries can be noted:

\begin{enumerate}
\item Bulk longitudinal sound waves - Brillouin mode: $\Omega \left(
q\right) =v_{s}q$, where $v_{s}$ is the velocity of sound waves;

\item Bulk Raman active optical modes: \ $\Omega ^{2}\left( q\right) =\Omega
_{0}^{2}-\alpha q^{2}$;

\item Modes in confined geometries: it is well known that if the sample is
confined to a length $d$ in any given direction, phonon modes propagating in
that direction are discrete with spacing proportional to $\pi /d$. If the
sample is confined in two of its dimensions, as in the case of phonons in an
optical fiber, for a given discrete phonon mode in the transverse direction,
the phonon frequency $\Omega \left( q\right) $ in the propagating direction
is given by $\Omega ^{2}\left( q\right) \approx \Omega _{0}^{2}+\alpha q^{2}
, where $\Omega _{0}$ for different discrete modes in the transverse
direction is determined by the corresponding zero of the Bessel functions
and thus it is inversely proportional to the radius of the fiber. In cases
when $\Omega _{0}\neq 0$, the $q$ dependence of the frequency is not very
significant and then the condition for momentum conservation in the
direction of propagation is of a minor consequence. In a spherical
resonator, which is confined in all the three directions, modes (both radial
and azimuthal) are discrete.
\end{enumerate}

One can arrange the resonator structure such that the upconversion process
is resonant with the cavity mode whereas the downconversion process is
non-resonant. Further one can choose the resonant cavity mode at $\omega
_{2} $ to have large damping $\kappa _{2}$ so that the generated anti-Stokes
field is quickly removed from the cavity. We assume that the width $\Gamma
_{m}$ of the phonon mode is much smaller than the width of the cavity mode
at the generated anti-Stokes frequency. We also assume that the pump laser
at $\omega _{1}$ is monochromatic and that the line width of the
corresponding resonator mode at $\omega _{1}$ is much smaller than $\kappa
_{2}$. The situation is shown in the Fig.\ref{Fig1} The anti-Stokes phonon
could be exactly at the line center of the resonator mode at $\omega _{2}$.
It depends on the energy conservation condition.
\begin{figure}[t]
\includegraphics[width=0.49\textwidth]{Fig1.eps}
\caption{Schematic representation of the anti-Stokes generation and the
width of varies lines.}
\label{Fig1}
\end{figure}
The Figure \ref{Fig1} also implies that more than one phonon mode (see for
example, the dotted curve) can be cooled simultaneously as long as these lie
well within the width $2\kappa _{2}$ of the generated anti-Stokes field; in
other words if $\Omega _{1}$ and $\Omega _{2}$ are the center frequencies of
two nearby sharp phonon modes, one should have $|\Omega _{1}-\Omega
_{2}|<2\kappa _{2}$. We consider this in detail in the next section.

\section{Theoretical Description of Bimodal Cooling}

Let us then first consider the case of two phonons modes in a resonator, one
with frequency $\Omega _{1}$ and the other with frequency $\Omega _{2}$. Let
the pump laser be at the frequency $\omega _{1}$. Then as required by the
Fig.\ref{Fig1}, we assume that $\omega _{1}+$ $\Omega _{i}$ $\left(
i=1,2\right) $ lie within the cavity line width $\kappa _{2}$ where the
cavity is centered at the frequency $\omega _{\mathrm{c}}$. Under the
assumption that the pump field remains un-depleted, the Hamiltonian for a
system of two phonon modes $b_{1}$, $b_{2}$ and the cavity mode $a_{2}$ can
be written a
\begin{eqnarray}
H &=&\hbar \omega _{\mathrm{c}}a_{2}^{\dagger }a_{2}+  \label{ev1} \\
&&+\sum_{i=1,2}\left\{ \hbar \Omega _{i}b_{i}^{\dagger }b_{i}+\left( \beta
_{i}^{\ast }b_{i}^{\dagger }a_{2}\mathcal{E}_{1}^{\ast }\mathrm{e}^{-\mathrm
i}\omega _{1}t}+H.c.\right) \right\} ,  \notag
\end{eqnarray
where we have replaced cavity mode $a_{1}$ by $\mathcal{E}_{1}$. The mode
operators $a_{2}$, $b_{1}$ and $b_{2}$ satisfy Bosonic commutation
relations. We assume that the cavity field is built up from the interaction
of the pump field with thermal phonons. Let $\Omega _{0}$ be $\frac{\Omega
_{1}+\Omega _{2}}{2}$ and $\Omega =\frac{\Omega _{1}-\Omega _{2}}{2}$, then
we work in a picture obtained by making unitary transformation with
\begin{equation}
\bar{H}=\hbar \left( \omega _{1}+\Omega _{0}\right) a_{2}^{\dagger
}a_{2}+\hbar \Omega _{0}\left( b_{1}^{\dagger }b_{1}+b_{2}^{\dagger
}b_{2}\right)  \label{ev2}
\end{equation
to obtain an effective Hamiltonia
\begin{eqnarray}
H_{\text{eff}} &=&\hbar \Omega \left( b_{1}^{\dagger }b_{1}-b_{2}^{\dagger
}b_{2}\right) +\hbar \delta a_{2}^{\dagger }a_{2}  \notag \\
&&+\sum_{i=1,2}\left( \beta _{i}^{\ast }b_{i}^{\dagger }a_{2}\mathcal{E
_{1}^{\ast }\mathrm{e}^{\mathrm{i}\delta t}+H.c.\right)  \notag \\
&=&\hbar \Omega \left( b_{1}^{\dagger }b_{1}-b_{2}^{\dagger }b_{2}\right)
+\hbar \delta a_{2}^{\dagger }a_{2}  \notag \\
&&+\sum_{i=1,2}\left( G_{i}^{\ast }b_{i}^{\dagger }a_{2}\mathrm{e}^{\mathrm{
}\delta t}+H.c.\right) ,  \label{ev3} \\
\delta &=&\omega _{\mathrm{c}}-\left( \omega _{1}+\Omega _{0}\right)
\begin{array}{cc}
, &
\end{array
G_{i}=\beta _{i}\mathcal{E}_{1}.  \label{ev4}
\end{eqnarray
This is our working Hamiltonian which we use to obtain Heisenberg equations
for $a_{2}$, $b_{i}$. We now need to introduce the dissipative terms which
account for the leakage of phonons for the resonator and dissipation of the
phonon modes and their then temperature. It is best then to write the
quantum Langevin equations for the Heisenberg operators $a_{2}$ and $b_{i}$
\begin{eqnarray}
&&\dot{a}_{2}=-\text{i}\delta a_{2}-\kappa _{2}a_{2}-\text{i}\left(
G_{1}b_{1}+G_{2}b_{2}\right) +f_{a_{2}}\left( t\right) ,  \notag \\
&&\dot{b}_{1}=-\text{i}\Omega b_{1}-\Gamma _{1}b_{1}-\text{i}G_{1}^{\ast
}a_{2}+f_{b_{1}}\left( t\right) ,  \notag \\
&&\dot{b}_{2}=+\text{i}\Omega b_{2}-\Gamma _{2}b_{2}-\text{i}G_{2}^{\ast
}a_{2}+f_{b_{2}}\left( t\right) .  \label{ev5}
\end{eqnarray
Here $\Gamma _{i}$ denotes the half width of the phonon mode $i$, $2\kappa
_{2}$ denotes the line width of the cavity mode.

Let us note here that the above set of equations are very generic, which can
describe the coupling of any two-mode system to a single optical field. In
this sense, the resulting physics here in our case should be similar to the
studies for other systems \cite{ar6,ar7,ar8,ar9} involving these equations,
including systems of oscillating membranes and trapped atoms.

In what follows, we would work in the limit $\Gamma _{i}\ll \kappa _{2}$;
\Omega _{i}\gg \kappa _{2}$. The force term \ $f_{j}\left( t\right) $ are
the quantum Langevin forces. These are Gaussian forces with zero mean and
with quantum correlations given b
\begin{eqnarray}
\left\langle f_{a_{2}}^{\dagger }\left( t\right) f_{a_{2}}\left( t^{\prime
}\right) \right\rangle  &=&
\begin{array}{cc}
, &
\end{array
\left\langle f_{a_{2}}\left( t\right) f_{a_{2}}^{\dagger }\left( t^{\prime
}\right) \right\rangle =2\kappa _{2}\delta \left( t-t^{\prime }\right) ,
\notag \\
\left\langle f_{b_{i}}^{\dagger }\left( t\right) f_{b_{i}}\left( t^{\prime
}\right) \right\rangle  &=&\frac{2\Gamma _{i}}{\pi }\int_{-\infty }^{\infty
\bar{n}_{i}\left( \omega \right) \mathrm{e}^{-\mathrm{i}\omega \left(
t-t^{\prime }\right) }d\omega ,  \notag \\
\left\langle f_{b_{i}}\left( t\right) f_{b_{i}}^{\dagger }\left( t^{\prime
}\right) \right\rangle  &=&\frac{2\Gamma _{i}}{\pi }\int_{-\infty }^{\infty
}\left( \bar{n}_{i}\left( \omega \right) +1\right) \mathrm{e}^{-\mathrm{i
\omega \left( t-t^{\prime }\right) }d\omega ,  \notag \\
\left\langle f_{b_{1}}\left( t\right) f_{b_{2}}^{\dagger }\left( t^{\prime
}\right) \right\rangle  &=&0.  \label{ev6}
\end{eqnarray
A detailed derivation of the correlation functions in Eq. (\ref{ev6}) can be
found in books on quantum optics \cite{bk3}.

We first analyze the well known case \cite{ar2}\ of the cooling of a single
mode in the limit of large $\kappa _{2}$. We can then make the adiabatic
approximation and writ
\begin{equation}
a_{2}\approx \left( \kappa _{2}+\text{i}\delta \right) ^{-1}\left( -\text{i
G_{1}\right) b_{1},  \label{ev7}
\end{equation
and then
\begin{equation*}
\frac{\partial b_{1}}{\partial t}+\text{i}\Omega b_{1}+\left( \Gamma _{1}
\frac{\left\vert G_{1}\right\vert ^{2}}{\kappa _{2}+\text{i}\delta }\right)
b_{1}=f_{b_{1}},
\end{equation*
i.e
\begin{eqnarray}
&&\frac{\partial b_{1}}{\partial t}+\text{i}\Omega _{1\text{eff
}b_{1}+\Gamma _{1\text{eff}}b_{1}=f_{b_{1}},  \label{ev8} \\
&&\Gamma _{1\text{eff}}=\left( \Gamma _{1}+\frac{\left\vert G_{1}\right\vert
^{2}\kappa _{2}}{\kappa _{2}^{2}+\delta ^{2}}\right) ,  \notag \\
&&\Omega _{1\text{eff}}=\Omega _{1}-\frac{\delta \left\vert G_{1}\right\vert
^{2}}{\kappa _{2}^{2}+\delta ^{2}}.  \label{ev9}
\end{eqnarray
Thus the life time of the phonon mode goes down as $\Gamma _{1\text{eff}}$
increases. The increase of $\Gamma _{1\text{eff}}$ is subject to the
assumption $\Gamma _{1\text{eff}}<\kappa _{2}$. The increase of $\Gamma $
leads to the cooling of the phonon mode, as the strength of the fluctuation
f_{b_{1}}$ is still determined by Equations in (\ref{ev6}), i.e. by the
damping parameter $\Gamma _{1}$. Note further that the fluctuation $f_{a_{2}}
$ associated with the field mode does not contribute to the normally ordered
moments as $\left\langle f_{a_{2}}^{\dagger }\left( t\right) f_{a_{2}}\left(
t^{\prime }\right) \right\rangle =0$, which is the case as long as $\hbar
\omega /k_{B}T\gg 1$.

\section{Mode-Mode Coupling and the Collective Behavior of the phonon Modes}

We will now show that the adiabatic elimination of the cavity field results
in mode-mode coupling. The resulting Langevin equations for phonon modes are
given b
\begin{equation*}
\frac{\partial b_{1}}{\partial t}+\left( \Gamma _{1}+\text{i}\Omega \right)
b_{1}=-\frac{\left\vert G_{1}\right\vert ^{2}b_{1}+G_{1}^{\ast }G_{2}b_{2}}
\kappa _{2}+\text{i}\delta }+f_{b_{1}}\left( t\right) ,
\end{equation*
\begin{equation}
\frac{\partial b_{2}}{\partial t}+\left( \Gamma _{2}-\text{i}\Omega \right)
b_{2}=-\frac{\left\vert G_{2}\right\vert ^{2}b_{2}+G_{2}^{\ast }G_{1}b_{1}}
\kappa _{2}+\text{i}\delta }+f_{b_{2}}\left( t\right) .  \label{ev10}
\end{equation
The mode-mode coupling can be thought of as the proces
\begin{eqnarray*}
&&\text{phonon }1+\text{pump field}\rightarrow \text{anti-Stokes field} \\
&\rightarrow &\text{phonon }2+\text{pump field}
\end{eqnarray*
It should be borne in mind that in the picture we work, all frequencies are
matched due to Eq.(\ref{ev2}). The actual generation of $b_{2}$ depends on
the frequency mismatch factors $\Omega $ and $\delta $ as the solution of
equations in (\ref{ev10}) will obviously show.

Let us now consider the possibility of the collective behavior of the phonon
modes. Let there be two identical phonon modes. This would be the case for
two identical nano-mirrors placed at different nodes of an optical cavity
(Similar to the situation of ref. \cite{ar6}); We can then set $\Omega =0$.
We drop $\Gamma _{i}$ assuming that the coupling field strength is such that
$\Gamma _{\text{eff}}\sim \frac{\left\vert G_{1}\right\vert ^{2}\kappa _{2}}
\kappa _{2}^{2}+\delta ^{2}}$. Then equations in (\ref{ev10}) reduce t
\begin{eqnarray}
\frac{\partial b_{1}}{\partial t} &\approx &-\frac{\left\vert
G_{1}\right\vert ^{2}b_{1}+G_{1}^{\ast }G_{2}b_{2}}{\kappa _{2}+\text{i
\delta }+f_{b_{1}}\left( t\right) ,  \notag \\
\frac{\partial b_{2}}{\partial t} &\approx &-\frac{\left\vert
G_{2}\right\vert ^{2}b_{2}+G_{2}^{\ast }G_{1}b_{1}}{\kappa _{2}+\text{i
\delta }+f_{b_{2}}\left( t\right) .  \label{ev12}
\end{eqnarray
The two noise sources would also be identical. Further under the simplifying
assumption $G_{1}=G_{2}=$rea
\begin{eqnarray}
\frac{\partial b_{+}}{\partial t} &=&-2\frac{G_{1}^{2}b_{+}}{\kappa _{2}
\text{i}\delta }+f_{+}\left( t\right) ,  \notag \\
\frac{\partial b_{-}}{\partial t} &=&0+f_{-}\left( t\right) =0,  \label{ev13}
\\
b_{\pm } &=&\frac{b_{1}\pm b_{2}}{\sqrt{2}
\begin{array}{cc}
, &
\end{array
f_{\pm }=\frac{f_{b_{1}}\pm f_{b_{2}}}{\sqrt{2}}.  \notag
\end{eqnarray
Under the conditions $\Gamma _{1}=\Gamma _{2}$, $\Omega =0$ used in deriving
Eq. (\ref{ev13}), $f_{-}$ does not lead to nonvanishing contribution. Thus
the collective mode $b_{-}$ is sub-radiant whereas the mode $b_{+}$ is
super-radiant. The collective mode $b_{+}$ has a decay rate which is twice
than what it would be in the absence of any collective effects. Under these
very special conditions there is enhanced cooling of only the collective
mode $b_{+}$ where $b_{-}$ mode is unaffected. The collective effects are
known in other contexts, for example in the cooling of two trapped ions \cit
{ar7} and in electro mechanical oscillators \cite{ar6}. The collective modes
in an ensemble of mechanical resonators trapped atoms in a Fabry-Perot
resonator have been studied \cite{ar8,ar9}.

\section{Spectrum of the Generated Anti-Stokes Field and Cooling of the
Phonon Modes}

\begin{figure}[t]
\includegraphics[width=0.49\textwidth]{Fig2p.eps}
\caption{The spectrum $\tilde{S}_{b_{1}}\left( \protect\omega \right) =\frac
\Gamma _{1}S_{b_{1}}\left( \protect\omega \right) }{2n_{1}}$ as a function
of $\protect\omega /\protect\kappa _{2}$ for $\frac{\Gamma _{1}}{\protec
\kappa _{2}}=\frac{\Gamma _{1}}{\protect\kappa _{2}}=0.01$; $\frac{\Omega }
\protect\kappa _{2}}=0.1$ and $\frac{G_{1}}{\protect\kappa _{2}}=0.3$,
G_{2}=0$ (green curve), $\frac{G_{1}}{\protect\kappa _{2}}=0.3$, $\frac{G_{2
}{\protect\kappa _{2}}=0.5$ (blue curve). The black curve represents
spectrum when both $G_{1}=G_{2}=0$. The parameter $R$ (Eq. \protect\ref{ev19
) gives the extent of cooling.}
\label{Fig2}
\end{figure}
The spectrum of phonon fluctuations can be obtained by solving the quantum
Langevin equations (\ref{ev5}). Defining Fourier transformations vi
\begin{equation}
b\left( t\right) =\frac{1}{2\pi }\int_{-\infty }^{\infty }b\left( \omega
\right) \mathrm{e}^{-\mathrm{i}\omega t}d\omega ,  \label{ev14}
\end{equation
so tha
\begin{equation}
\left\langle b^{\dagger }\left( t\right) b\left( t\right) \right\rangle
\frac{1}{2\pi }\int_{-\infty }^{\infty }S_{b}\left( \omega \right) d\omega ,
\label{ev15}
\end{equation
\begin{equation}
2\pi S_{b}\left( \omega \right) \delta \left( \omega ^{\prime }+\omega
\right) =\left\langle b^{\dagger }\left( -\omega ^{\prime }\right) b\left(
\omega \right) \right\rangle .  \label{ev16}
\end{equation}

The quantity $S_{b}\left( \omega \right) $ gives the spectrum of phonon
fluctuations. A reduction in the value of $\left\langle b^{\dagger
}b\right\rangle $ when $G^{\prime s}\neq 0$, would represent cooling. Using
the properties of the Langevin forces we find the final results for the
spectra of the two phonon modes
\begin{subequations}
\label{ev17}
\begin{align}
S_{b_{1}}\left( \omega \right) =& \frac{2\Gamma _{1}n_{1}\left( \omega
\right) \left\vert \left( \mathrm{i}\delta -\mathrm{i}\omega +\kappa
_{2}\right) +\frac{\left\vert G_{2}\right\vert ^{2}}{-\mathrm{i}\Omega
\mathrm{i}\omega +\Gamma _{2}}\right\vert ^{2}}{\left\vert d\left( \omega
\right) \right\vert ^{2}\left[ \left( \Omega -\omega \right) ^{2}+\Gamma
_{1}^{2}\right] }  \notag \\
& +\frac{2\Gamma _{2}n_{2}\left( \omega \right) \left\vert \frac{G_{1}G_{2}}
-\mathrm{i}\Omega -\mathrm{i}\omega +\Gamma _{2}}\right\vert ^{2}}
\left\vert d\left( \omega \right) \right\vert ^{2}\left[ \left( \Omega
-\omega \right) ^{2}+\Gamma _{1}^{2}\right] }  \notag \\
& \underrightarrow{G_{1},G_{2}=0}\text{ }\frac{2\Gamma _{1}n_{1}\left(
\omega \right) }{\left[ \left( \Omega -\omega \right) ^{2}+\Gamma _{1}^{2
\right] },  \label{ev17a}
\end{align
\begin{align}
S_{b_{2}}\left( \omega \right) =& \frac{2\Gamma _{2}n_{2}\left( \omega
\right) \left\vert \left( \mathrm{i}\delta -\mathrm{i}\omega +\kappa
_{2}\right) +\frac{\left\vert G_{1}\right\vert ^{2}}{-\mathrm{i}\Omega
\mathrm{i}\omega +\Gamma _{1}}\right\vert ^{2}}{\left\vert d\left( \omega
\right) \right\vert ^{2}\left[ \left( \Omega +\omega \right) ^{2}+\Gamma
_{2}^{2}\right] }  \notag \\
& +\frac{2\Gamma _{1}n_{1}\left( \omega \right) \left\vert \frac{G_{1}G_{2}}
\mathrm{i}\Omega -\mathrm{i}\omega +\Gamma _{1}}\right\vert ^{2}}{\left\vert
d\left( \omega \right) \right\vert ^{2}\left[ \left( \Omega +\omega \right)
^{2}+\Gamma _{2}^{2}\right] }  \notag \\
& \underrightarrow{G_{1},G_{2}=0}\text{ }\frac{2\Gamma _{2}n_{2}\left(
\omega \right) }{\left[ \left( \Omega +\omega \right) ^{2}+\Gamma _{1}^{2
\right] },  \label{ev17b}
\end{align
\begin{figure}[bt]
\includegraphics[width=0.49\textwidth]{Fig3p.eps}
\caption{Same as in Fig.\protect\ref{Fig2}, except for the spectrum $\tilde{
}_{b_{2}}\left( \protect\omega \right) =\frac{\Gamma _{2}S_{b_{2}}\left(
\protect\omega \right) }{2n_{2}}$ of the $b$ mode. The pink graph is for
G_{1}=0$, $\frac{G_{2}}{\protect\kappa _{2}}=0.5$.}
\label{Fig3}
\end{figure}
where
\end{subequations}
\begin{equation}
d\left( \omega \right) =\left( \mathrm{i}\delta -\mathrm{i}\omega +\kappa
_{2}\right) +\frac{\left\vert G_{1}\right\vert ^{2}}{\mathrm{i}\Omega
\mathrm{i}\omega +\Gamma _{1}}+\frac{\left\vert G_{2}\right\vert ^{2}}{
\mathrm{i}\Omega -\mathrm{i}\omega +\Gamma _{2}}.  \label{ev18}
\end{equation}

The spectrum of fluctuations, given by equations in (\ref{ev17}), should be
compared with the unperturbed spectrum ($G_{1}=G_{2}=0$). For typical
parameters we show the results in Figs. \ref{Fig2} and \ref{Fig3}. The
parameters $\kappa _{2}$, $\Gamma ^{^{\prime }s}$ and $G^{^{\prime }s}$ that
we choose, are similar to the case of a single phonon mode \cite{ar5}.
Although the coupling of the three modes makes $d\left( \omega \right) $
cubic function of $\omega $, the spectrum in the vicinity of $\omega =\Omega
$ ($\omega =-\Omega $) exhibits a single resonance for $\delta =0$, $\Gamma
\ll \kappa $. The Figs. \ref{Fig2} and \ref{Fig3} show the cooling of both
modes. The mode-mode coupling leads to heating, however overall both modes
get cooled. The ratio $R$ gives the ratio of $\left\langle b^{\dagger
}\left( t\right) b\left( t\right) \right\rangle $ in presence of the
coupling to it's value in the absence of coupling i.e. $R$ for example in
Fig. \ref{Fig2} i
\begin{equation}
R=\frac{\left\langle b_{1}^{\dagger }\left( t\right) b_{1}\left( t\right)
\right\rangle }{\bar{n}_{1}}.  \label{ev19}
\end{equation
It should be emphasized that the cooling parameter $R$ introduced here
should not be confused with the effective bulk Raman coupling constant
R^{\left( Q\right) }$ introduced in Eq. (\ref{e14b}). For Fig.\ref{Fig2} for
the values of $G_{1}$ and $G_{2}$ used, $R=0.110$ if $G_{2}=0$; $R=0.288$ if
$G_{2}\neq 0$. The parameters $G_{1}$ and $G_{2}$ [Eq. (\ref{ev4})] depend
on the phase matching parameter $\beta $ and the power of the Stokes field.
Clearly if we increase the power of the Stokes laser i.e. if we increase
G_{,}$ then $R$ goes down. We note that the parameter $R$ also gives the
ratio of the temperature of the cooled phonon mode to its initial
temperature. This is because the factor $\left( e^{\hbar \omega /k_{\text{B
}T}-1\right) ^{-1}\approx \frac{k_{\text{B}}T}{\hbar \omega }$ if $\frac{k_
\text{B}}T}{\hbar \omega }\ll 1$. For phonon frequency, say in the range,
100$MHz \cite{ar5}, the parameter $\frac{k_{\text{B}}T}{\hbar \omega }\ll 1$
even at a temperature like $1$K.

\begin{figure}[t]
\vspace{.5cm} \includegraphics[width=0.49\textwidth]{Fig4p.eps}
\caption{Spectrum of the generated anti-Stokes for $\frac{G_{1}}{\protec
\kappa _{2}}=0.3$, $G_{2}=0$ (green); $G_{1}=0$, $\frac{G_{2}}{\protec
\kappa _{2}}=0.5$ (pink); $\frac{G_{1}}{\protect\kappa _{2}}=0.3$, $\frac
G_{2}}{\protect\kappa _{2}}=0.5$ (blue).}
\label{Fig4}
\end{figure}
The spectrum of the generated anti-Stokes field is given by
\begin{equation}
S_{a}\left( \omega \right) =\frac{2\Gamma _{1}n_{1}\left( \omega \right)
\left\vert \frac{G_{1}}{\mathrm{i}\Omega -\mathrm{i}\omega +\Gamma _{1}
\right\vert ^{2}+2\Gamma _{2}n_{2}\left( \omega \right) \left\vert \frac
G_{2}}{-\mathrm{i}\Omega -\mathrm{i}\omega +\Gamma _{2}}\right\vert ^{2}}
\left\vert d\left( \omega \right) \right\vert ^{2}},  \label{ev20}
\end{equation
which is shown in Fig.\ref{Fig4}. The figure also shows the effect of mode
mixing. The mode mixing arises from the modification of the denominator in
Eq. (\ref{ev20}). The numerator in Eq. (\ref{ev20}) represents the
conversion of phonons into anti-Stokes in the lowest order of the coupling
G_{1}$ and $G_{2}$. Note that the spectrum of the generated antiStokes
radiation has signatures of the phonon spectrum. The spectrum of the
generated radiation can be studied by examining the output field at
frequency $\omega _{2}$. Note that the anti-Stokes photons leak out of the
cavity and the phonon modes are constantly interacting with the thermal
bath, the conservation of photon and phonon numbers in the cavity does not
hold.

If we use the actual numerical parameters of the experiment on the single-
mode cooling of the acoustic phonon modes done by Bahl \textit{et al.} \cit
{ar5}, we can also examine the possibility of bimodal phonon cooling in such
a system. In their experiment, $\kappa _{2}$ was of the order of $5$MHz, and
they had observed cooling of the surface acoustic phonon mode of frequency
95$MHz, with their intrinsic line width of the order of only $8$kHz at the
ambient temperature of $294$K. Thus, if one can find another similar phonon
mode in the system with its frequency within a few MHz away from the $95$MHz
mode, say in the range of $90$ to $100$MHz, which also has an appreciable
coupling $\beta $, defined by Eq. (\ref{e30a}), we should be able to observe
simultaneous cooling of two such modes in the same experimental system.
Otherwise, one has to search for another appropriate phonon system to
observe this effect. Note that in the experimental system discussed above,
if there are more than two phonon modes near the $95$MHz mode within
2\kappa _{2}$ range, each one with appreciable coupling, all those modes are
expected to be cooled simultaneously.

In conclusion, we have presented a general theory of cooling of phonon modes
via their interactions with suitable optical fields. The theory is
formulated in terms of the spatial mode functions for the relevant
interacting modes, in any given geometry, allowing it to be applicable to a
variety of systems with diverse phonon dispersion relations and frequency
spectra. In particular, we show the cooling of two phonon modes at the same
time if the line width of each of the phonon modes is very small compared to
the decay rate $2\kappa _{2}$ of the resonant anti-Stokes cavity mode and
their frequency difference is smaller than the line width $2\kappa _{2}$. In
any experimental set-up similar to that used by Bahl \textit{et al.} \cit
{ar5}, if one can find two nearby sharp phonon modes satisfying the above
conditions , one should be able to observe the bimodal cooling described in
this paper. We also find in this paper the possibility of collectively
enhanced cooling of phonon modes under suitable conditions. However, further
studies are needed to identify such a system with required phonon mode
characteristics. A generalization of the present work to four wave
parametric interactions (for example Hamiltonians which are quadratic in
b_{i}^{^{\prime }s}$) would also be interesting.

One of us (SSJ) acknowledges the hospitality of the Oklahoma State
University, while this work was done.

redpajama_set_name: RedPajamaArXiv


Cookies are short fragments of text (letters and/or numbers) that enable the Web server to memorize about the client (the browser) information to be reused in the course of the same visit to the site (session cookies) or thereafter, even days later (persistent cookies). The cookies are memorized, on the basis of the user's preferences, by the individual browser on the specific device used (computer, tablet, smartphone). Similar technologies, such as web beacons, transparent GIFs and all forms of local storage introduced with HTML5, are usable to collect information on the user's behaviour and on the use of the services. Below in this document we will make reference to cookies and all the similar technologies using simply the term "cookie".
Visitors to a website may receive cookies both from the visited site ("proprietary") and from sites managed by other organizations ("third parties"). A notable example is represented by the presence of the "social plugins" for Facebook, Twitter, YouTube, Google+ and LinkedIn. These are parts of the visited page generated directly by the aforesaid sites and integrated into the page of the host site. The most common use of social plugins is aimed at sharing content on social networks. The presence of these plugins involves the transmission of cookies from and to all the sites managed by third parties. Management of information collected by "third parties" is outlined in by the related information, to which the user is invited to refer. To ensure greater transparency and convenience, we report below the web addresses of the various information notices and of the ways to manage cookies.
(configuration): access your account. Privacy section.
The site Avio.com also includes some components transmitted by Google Analytics, a web traffic analysis serviced provided by Google, Inc. ("Google"). This also involves third party cookies collected and managed anonymously to monitor and improve the performance of the host site (performance cookies). Google Analytics uses the cookies for the anonymous collection and analysis of the information on the behaviour of Avio.com users (including the user's IP address). This information is collected by Google Analytics, which processes it for the purpose of preparing reports for the operators of Avio.com regarding the activities on the websites themselves. This site does not use (and does not permit third parties to use) the Google analysis tool to monitor or collect personal identification information. Google does not associate the IP address with any other data possessed by Google, or seek to connect an IP address with the identity of a user. Google may also communicate this information to third parties where this is required by law or where such third parties handle the aforesaid information on Google's behalf. For further information, see the link indicated hereafter: https://www.google.it/policies/privacy/partners/. The user may disable selectively the action of Google Analytics by installing on his browser the opt-out component provided by Google. To disable the action of Google Analytics, click on the following link: https://tools.google.com/dlpage/gaoptout.
Some cookies (session cookies) remain active only until the browser is closed or the logout command is given. Other cookies "survive" the closing of the browser and are available also on subsequent visits by the user. These cookies are called "persistent" or "stored" cookies and their duration is fixed by the server at the moment of their creation. In some cases a time limit is set, in other cases the duration is unlimited. Avio.com does not make use of persistent cookies. However, visitors browsing the pages of the website Avio.com may interact with sites managed by third parties that can create or modify permanent and profiling cookies.
The user may decide whether to accept cookies or not by using the settings of his browser. Note: total or partial disablement of technical cookies may compromise the use of the functionalities of the site reserved to registered users. On the contrary, the usability of the public content is possible also disabling cookies completely. Disabling "third-party" cookies does not impact on browsability in any way. The setting may be defined in a specific way for the various web sites and applications. In addition, the best browsers make it possible to define different settings for "proprietary" and "third party" cookies.


text:
Cynthia Pooler
Cynthia has combined her interest in the ever evolving new technologies and her passion for politics to create an Internet Radio Talk show that will create a new paradigm in which to inform New Yorkers on what is going on at the State Capital.
Website URL: http://www.focusonalbany.com
Lynne Jackson talks about her Journey for Justice walk
http://www.blogtalkradio.com/cynpooler/2014/05/01/lynne-jackson-talks-about-her-journey-for-justice-walk
Lynne Jackson talks about her Journey for Justice walk in upstate New York to increase awareness about Yassin Aref
Published in Radio Interviews
William Reinhardt talks about the proposed casino in Albany
http://www.blogtalkradio.com/cynpooler/2014/04/30/william-reinhardt-talks-about-the-proposed-casino-in-albany
Bethlehem Town Board member William Reinhardt talks about the proposed casino in Albany
Randy Koniowka on the cannonization of John XXIII and John Paul ll
http://www.blogtalkradio.com/cynpooler/2014/04/29/randy-koniowka-on-the-cannonization-of-john-xxiii-and-john-paul-ll
Local historian Randy Koniowka shares his thoughts on the cannonization of Pope John XXlll and John Paul ll
Zack Fink shares his thoughts on the Moreland Commission
http://www.blogtalkradio.com/cynpooler/2014/04/28/zack-fink-shares-his-thoughts-on-the-moreland-commission
Zack Fink shares his thoughts on the Moreland Commission and what's going on in the State Senate
Roger Grona talks about Branding
http://www.blogtalkradio.com/pooler/2014/04/25/roger-grona-talks-about-branding
Focus on Albany is teaming up with Roger Grona, President of Firebird Businrss Consulting. Our program is called The Successful Entrepreneur and will help business people and entrepreneurs become more successful
Judd Krasher talks about the forum at Griffin on the proposed casino
http://www.blogtalkradio.com/cynpooler/2014/04/17/judd-krasher-talks-about-the-forum-at-griffin-on-the-proposed-casino
Judd Krasher talks about the forum at Griffin Memorial Elementray School on the proposed casino presented by developer David Flaum
Thoughts on Bishop Edward B. Scharfenberger
http://www.blogtalkradio.com/cynpooler/2014/04/15/thoughts-on-bishop-edward-b-scharfenberger-1
Local historian Randy Koniowka shares his thoughts on the new Bishop of the Albany Diocese Edward B. Scharfenberger
What happened to the Moreland Commission?
http://www.blogtalkradio.com/cynpooler/2014/04/14/what-happened-to-the-moreland-commission
John Sullivan offers his views on the disbanding of the Moreland Commission
Judd Krasher talks about the proposed casino in Albany
http://www.blogtalkradio.com/cynpooler/2014/04/13/judd-krasher-talks-about-the-proposed-casino-in-albany
Albany Common Council member Judd Krasher voices his concerns on the proposed casino in Albany
An update about the Port of Albany's Oil Plan
http://www.blogtalkradio.com/cynpooler/2014/04/07/an-update-about-the-port-of-albanys-oil-plan
Albany resident and former instructor of environmental science, Todd Hunsinger updates Focus on Albany listeners on what has been happening in the ongoing developments to heat and ship crude oil at the Port of Albany
© 2013 Focus on Albany. Sitemap
Political Website Design by OpenPotion

redpajama_set_name: RedPajamaCommonCrawl


July, 1896, to June, 1897.
George Stoneman, 39 WARWICK LANE, E.C.
"The Lord's Day," and "The Love Feast," both by Rev. William Elliot, Ashley Terrace, Plymouth; "Strange Fire Among the Primitive Methodists," by John Robertson, City Temple, Glasgow.
Was Peter Ever at Rome?
Shakespeare and the Bible, published by C. E. Bagster and Sons; Music and Worship, by T. F. Agnew, Spottiswoode and Co.; Sunday School Lessons, and rightly Dividing the Word of Truth, both by Dr. Bullinger. Fifth Series, Eyre and Spottiswoode; The Protestant Observer, monthly journal, established in 1888 by John F. Shaw & Co.
"My Master", "The Secret of a Happy Day", and "The Precious Blood"; words by Frances Ridley Havergal, music by R. Alleyne-Harris, Weekes & Co.
The Great Distinction Between the "House of Judah" (the Jews) and the "House of Israel" in the "Latter Days," by Dr. Aldersmith, Nisbet's; The Oldest Christian Church, by Henry Holme, Marshall Brothers; 1834 and 1893 (Read at the Congress of Missions in Chicago, 1893), by the late Miss E. J. Whateley, describes the Society for Promoting Female Education in the East; The Field Lane Refuges and Ragged Schools, 54th Annual Report of the Institution at Vine Street, London; The Times of the Gentiles, by Rev. Theodore Howard, Simpkin, Marshall & Co.; Holy Scripture: The Sole Authoritative Expositor of Divine Truth, by the late Dean Goode, 4th edition, edited by Rev. Dr. Bullinger, Jas. Nisbet & Co.
E. O. A. N. D.
Question 142. Does Paul's teaching in Rom. 6:3-4 refer to baptism by immersion, if so, should it be administered in the name of the Lord Jesus only (Acts 19:5)?


Spectacular, broad spectrum, immune-specific support, especially mouth, GI and urinary tracts. Breakthrough new process delivering pure, stabilized allicin for the first time in history.
Multi-Enzyme digestive support for protein and carbs. Bio-Film buster.


text:
Eremochrysa (Chrysopiella) sabulosa is een insect uit de familie van de gaasvliegen (Chrysopidae), die tot de orde netvleugeligen (Neuroptera) behoort.

Eremochrysa (Chrysopiella) sabulosa is voor het eerst wetenschappelijk beschreven door Banks in 1897.

Gaasvliegen

redpajama_set_name: RedPajamaWikipedia


text:
Fill 1Fill 1MUSEO D'ARTE CONTEMP
Royal Residence
The Manica Lunga was built for Duke Carlo Emanuele I in the first half of the 17th century by the Castellamonte architects, Amedeo and his son Carlo. This long, narrow building, at that time connected to the Castle, was used to house the Duke's collection of pictures.
Like all the buildings in the complex, the Manica Lunga was damaged by the French troops of General Catinat in 1693. In the architect Filippo Juvarra's new plans for the complex, made in the early eighteenth century, it was to be destroyed and replaced by a symmetrical building with a great entrance hall and a ballroom on the first floor. However, due to excessive construction costs, the plan was not carried out, and the Manica Lunga was used as service space. In 1883, when the Savoy family sold the Castle to the city, it was used as a barracks. After World War II, it became a residence for evacuees. These utilisations led to modifications of the structure, with the building of new spaces for hygienic services, as one can see in photographs that document the degradation and dereliction of the building.
The restoration of the Manica Lunga started in 1986 and finished in 2000 with the opening of the exhibition space on the third floor. Thus the building's original vocation was reclaimed. Andrea Bruno, the architect at the center of the restoration, placed staircases and an elevator on the exterior of the building, realized in transparent materials that allow the 147 meter-long, 6 meter-wide structure to remain visible. The metal roof, a completely new addition, is characterized by repetitive ribs, forming a central cap for the entire length of the building. Natural light has been greatly increased thanks to the large windows and the elimination of the attics and the structures realized in more recent times and not pertaining to the Castellamonte period.
Accessibility and Services
Cerruti Collection
The Renovation by Andrea Bruno
Library & Video Library
Museum of advertising
Castello di Rivoli
Piazza Mafalda di Savoia
10098 Rivoli - Torino
info@castellodirivoli.org
Utilizziamo i cookie per essere sicuri che tu possa avere la migliore esperienza sul nostro sito.Ok


text:
Evidence-based analysis and commentary on European politics
Scotland could end up as a kind of post-sovereign polity in a looser union with England
British politicians are still clinging to antiquated notions of sovereignty, argues Michael Keating. While British governments have in recent decades come to accept the idea of Scottish self-determination, they have not been able to see their way beyond traditional ideas of sovereignty, so that Scotland must be subordinate to Westminster or completely independent. The moderate solution to the problem is rejected while the extreme one is reluctantly accepted. It is not necessary to become an independent state in order to forge a distinct welfare settlement but a new division of competences would be required.
There are several ironies in the current constitutional debate in Scotland. One is that both sides are talking the language of union. The label 'unionist', previously a highly charged term associated with the Conservative and Unionist Party and loaded with Irish implications, has been embraced by all UK parties, including Labour and (with the occasional qualification) Liberal Democrats. On the nationalist side, Alex Salmond has argued that there are six unions, of which an independent Scotland would keep five (currency, monarchy, society, Europe and defence). Another irony is that, while the Edinburgh agreement and the referendum question excludes 'third way' options such as 'devolution max', both sides are seeking to position themselves in that very middle ground in the knowledge that occupying it is the key to victory.
Post-Sovereignty
This is not mere political posturing but reflects a profound truth about modern constitutionalism, that the old sovereign nation-state formula no longer works. We are in an era of 'post-sovereignty'. This does not mean that sovereignty has disappeared but that in its traditional monist form it is no longer the only principle of political order.
Scottish Parliament, Credit: ajnabee (CC-BY-SA-3.0)
More profoundly, sovereignty as a principle is in question. At the supranational level, the European Union has become a legal order in its own right and, within its domain, its laws trump national ones. The only redress to this, recognized in the Lisbon Treaty, is secession – but that option is also open to Scotland within the United Kingdom. Indeed the principle of state sovereignty is also in question domestically. Arguments that the Scottish Parliament is the creation of Westminster, which could abolish it tomorrow, carry little political credibility. Moreover, the issue of sovereignty has never been resolved in Scotland, where Blackstone and Dicey's ideas have never commanded consensus. The existence of the Scottish Parliament focuses this alternative idea of where sovereignty might lie. Hence there are, and always have been, two interpretations of the union: as the incorporation of Scotland into a sovereign and unitary state; and as a bargain that can be revisited from time to time by either party. Similar arguments exist in Spain and in Canada, although their legal basis is perhaps less developed.
In such states, it is futile to seek a single foundational principle to underlie constitutional order and define sovereignty. The endless Canada/Quebec search for a settlement and the current constitutional deadlock between the majorities in Catalonia and Spain show this. The only option is a constitutional pluralism that does not seek to dig down to the foundations all the time but recognizes differences in constitutional traditions and aspirations. The Northern Ireland settlement is a rare example of two states accepting this idea. In recent decades, British governments have come to accept the idea of Scottish self-determination, in a way they did not a hundred years ago in the case of Ireland. John Major, Margaret Thatcher and now David Cameron have reinterpreted the union in this way. On the other hand, they have not been able to see their way beyond traditional ideas of sovereignty, so that Scotland must be subordinate to Westminster or completely independent. So the moderate solution to the problem is rejected while the extreme one is reluctantly accepted. Canada and Spain have similarly missed out on opportunities in the Meech Lake Accord and the revised Catalan Statute of Autonomy because they could not accept even the hint of divided sovereignty.
Opinion polls in Scotland have regularly shown a large section of the population (usually a plurality) in favour of extended devolution within the United Kingdom. Since this is almost everyone else's second option, it would seem to be the basis for achieving the widest degree of consensus (based on necessary compromise if not shared premises). It is true that there is some confusion over exactly what people do want and how much variation in policy they would accept across the United Kingdom, but there seems little appetite for a separate military effort or formal status within the United Nations. On the other hand, there is support for more control over taxation and welfare, the big items of domestic policy not already devolved. This is not surprising. Both economic and welfare policy are rescaling upwards and downwards in Europe so that the nation-state is not the sole level for either or for the social compromises that they embody. This is as true of a potentially independent Scotland as it is for the United Kingdom.
Implications for Scotland
Scots do not differ radically from citizens elsewhere in the United Kingdom in their broad preferences for welfare. Public policy, however, does not emerge directly from citizen preferences but from the aggregation and compromise among interests and preferences within political institutions. This has produced a rather different balance within Scotland, notably on matters of universalism and public provision. In the longer run such differences necessarily imply distinct fiscal choices, and indicate the need for mechanisms by which Scotland might recapture the benefits of its own social investments, for example in university education. It also points to changes in the balance of welfare instruments so as to fit local needs and preferences. Measures like the 'bedroom tax/spare room subsidy', designed for conditions in the south of England, may have little relevance in Scotland (or in the north of England for that matter).
As a matter of functional efficiency, the present division between passive welfare (reserved) and active welfare (largely devolved) creates mismatches and disincentives, which are beginning to be recognized. The location of housing benefit at the same level as responsibility for social housing is one instance. Another matter is the interface among unemployment support and disability support with training, economic development, urban regeneration and social work services. There is never an undisputed optimal level for the integration of public services and the idea of 'joined-up government' is often a naïve illusion, but we can probably do a lot better than at present.
It is fruitless to design a reformed system of devolution predicated on the present mode of welfare state. One thing that we do know is that the welfare state is going to undergo some major changes, if only for pressing financial reasons. This provides an opportunity to rethink how Scottish welfare might fit into an emerging multilevel model and how resources might most effectively be deployed.
It is not necessary to become an independent state in order to forge a distinct welfare settlement but a new division of competences would be required. The nearest comparator would be the province of Quebec which, after repeated efforts to arrive at a constitutional settlement, has engaged in the building of a distinct polity within the Canadian federation and use its extensive taxation and welfare responsibilities to forge a distinct welfare settlement, not radically different from that elsewhere in Canada but sufficiently distinct to have made a real difference in outcomes. In the numerous meetings I attend in Scotland on the constitutional issue, it is these social and economic issues that feature most prominently.
It is my belief that Scotland will end up as a kind of post-sovereign polity in a looser union with England and forging its own social compromises within the constraints of markets and wider unions. Our politicians, however, will take us there while denying strenuously that this is what they are doing, and still clinging to antiquated notions of sovereignty.
This article was originally published on the PSA's Insight blog
Please read our comments policy before commenting.
Note: This article gives the views of the author, and not the position of EUROPP – European Politics and Policy, nor of the London School of Economics.
Shortened URL for this post: http://bit.ly/1g6gOOk
Michael Keating – University of Aberdeen / University of Edinburgh
Professor of Politics, Universities of Aberdeen and Edinburgh and Director of ESRC Scottish Centre on Constitutional Change.
March 8th, 2014|Democracy, identity and culture, Michael Keating|3 Comments
martincatt March 9, 2014 at 10:05 am - Reply
Scotland is a wonderful country.i like this country.
Mick March 30, 2014 at 7:17 pm - Reply
I advocate for a post-sovereignty settlement and, despite having no emotional attachment to the UK state, I would have accepted (and voted for) an imaginative compromise, but that has never been on offer from the UK.
A continuing all-island framework along the lines of devo-max but with a thoroughly reformed centre that the constituent countries could directly influence, perhaps through a regionally elected senate to replace the House of Lords alongside a Council of Ministers would have been a start. We could jettison the 18th C. baggage.
More imaginatively, Scots would have accepted continuing single UN representation but independent membership of the supra-national EU – in negotiation with other member states. As it is, such options, or even conversations, were utterly rejected in London from 2007 onwards.
So, I'll vote Yes, enthusiastically convinced that the SNP vision of indy-lite is the best realpolitik option.
Henry Stevens March 30, 2014 at 7:45 pm - Reply
The great problem in this kind of argument, for me, is that it doesn't acknowledge what we're giving up under 'indy-lite'. The SNP's plans mean we have greater autonomy, but in several key areas we'll still be effectively tied to decisions made in UK institutions, without any representation within those institutions.
Westminster making key decisions for local Scottish communities, without those communities having any representation in the decision-making process, is a major problem from a democratic standpoint. It's a problem many times larger than the supposed democratic deficit we suffer from now (that simply amounts to saying we don't like Tories and Tories win half the time).
Under the status quo we might not like Conservative governments, but Scottish constituencies have full representation in decision-making. If a particular policy might disadvantage a local constituency, then there is a direct representative from that constituency to argue the case in Westminster – and if necessary mobilise support among other areas in the rest of the UK.
After independence, we have no such representation. The only avenue for influencing the decision is through the Scottish government via external negotiations – a strategy that is both incredibly difficult to achieve given the disparity in size between Scotland/rUK, and is also premised on the idea that all Scottish local constituencies agree on every issue (when clearly they don't). If a constituency happens to be in the minority on a particular issue in Scotland then there is no representation for that constituency at the UK level, given the Scottish government will negotiate from the majority view among Scottish constituencies.
That's why there's a very real case for saying devo max, full independence, or nothing. Devo max means an incremental increase in our autonomy while still retaining all of the representation we have in UK decision-making. Independence-lite means gaining some autonomy, while losing all of our representation in UK decision-making. As far as I'm concerned, that's the worst compromise of all.
Supported by the LSE's European Institute
In association with European Union Politics
Can Poland's opposition win this year's election?
Why EU states are converging on restrictive migration policies, despite their different political traditions
Will Law and Justice win this year's elections in Poland?
How the European Parliament gained oversight powers over EU crisis legislation
The European elections generated real momentum for renewable energy – it's time for MEPs to rise to the occasion
So why cannot RoI harmionise rules with NI and the EU check...
czarnajama says:
I think that Aleks misses the media transformation ongoing in Poland....
Greek says:
"Syriza as a party of liars, traitors, and incompetent political imposters" Is this...
Follow EUROPP on Twitter
European Institute Events Calendar
Click here for this month's upcoming European Institute events
Archive Select Month July 2019 (16) June 2019 (25) May 2019 (37) April 2019 (29) March 2019 (29) February 2019 (28) January 2019 (26) December 2018 (27) November 2018 (29) October 2018 (32) September 2018 (31) August 2018 (29) July 2018 (31) June 2018 (33) May 2018 (32) April 2018 (33) March 2018 (33) February 2018 (32) January 2018 (34) December 2017 (34) November 2017 (35) October 2017 (41) September 2017 (43) August 2017 (32) July 2017 (36) June 2017 (47) May 2017 (48) April 2017 (44) March 2017 (42) February 2017 (44) January 2017 (42) December 2016 (46) November 2016 (44) October 2016 (43) September 2016 (38) August 2016 (35) July 2016 (41) June 2016 (46) May 2016 (38) April 2016 (37) March 2016 (44) February 2016 (44) January 2016 (35) December 2015 (38) November 2015 (48) October 2015 (47) September 2015 (48) August 2015 (42) July 2015 (46) June 2015 (45) May 2015 (45) April 2015 (41) March 2015 (50) February 2015 (40) January 2015 (40) December 2014 (34) November 2014 (48) October 2014 (52) September 2014 (51) August 2014 (54) July 2014 (58) June 2014 (56) May 2014 (58) April 2014 (53) March 2014 (59) February 2014 (53) January 2014 (56) December 2013 (51) November 2013 (56) October 2013 (58) September 2013 (58) August 2013 (58) July 2013 (59) June 2013 (59) May 2013 (58) April 2013 (57) March 2013 (63) February 2013 (56) January 2013 (58) December 2012 (52) November 2012 (58) October 2012 (66) September 2012 (62) August 2012 (57) July 2012 (57) June 2012 (56) May 2012 (64) April 2012 (55) March 2012 (58) February 2012 (6) January 2012 (14)
Visit our sister blog: LSE USAPP blog
Latest LSE Podcasts
The Levelling: what's next after globalisation [Audio] June 27, 2019
Britain and Europe: how did we get here, and where do we go next? [Audio] June 21, 2019
Global Health and Inequality [Audio] June 18, 2019
Finding My Voice: my journey to the West Wing and the path forward [Audio] June 18, 2019
The Problem of Modernity: reinterpreting decolonisation and the modern? [Audio] June 6, 2019
Visit our sister blog: LSE Impact of Social Sciences
Forthcoming events at the LSE
Populism: causes and responses September 19, 2019
What Citizens of the World Can Learn from Nationalism September 16, 2019
The Sacred Year 1919: women and the professions September 8, 2019
Narrative Economics September 6, 2019
On the Ordination of Women July 18, 2019
Visit our sister blog: LSE British Politics and Policy
Latest articles from BPP
Two troubled premierships: comparing the administrations of John Major and Theresa May July 17, 2019
Politicising and degendering women's part-time work: evidence from Norway July 16, 2019
'Speaking truth to power' in Select Committees: what is the experience like for external evidence-givers? July 15, 2019
The forward march of party members: has the shift in power to the grassroots gone too far? July 14, 2019
Book Review | Heroes or Villains? The Blair Government Reconsidered July 13, 2019
Visit our sister blog: LSE Review of Books
Latest articles from LSE Research Online
The over-securitization of global health: changing the terms of debate July 18, 2019
Edge infrastructures July 18, 2019
The politics of expertise and ignorance in the field of migration management July 18, 2019
Preventable harm: getting the measure right July 18, 2019


text:
InQuicker
AdventHealth Dade City (Change Location)
Directions to AdventHealth Dade City
13100 Fort King Road, Dade City, FL 33525
Call AdventHealth Dade City at 352-521-1100
Bayfront Health Dade City Joins Adventist Health System
Jason Newmyer will serve as administrator of the 120-bed facility.
(April 2, 2018 - Zephyrhills, FL) Adventist Health System, a faith-based non-profit health care organization and the parent company of AdventHealth, today announced that it has finalized an agreement with subsidiaries of Community Health Systems, Inc. that officially brings Bayfront Health Dade City into its system. The agreement includes the 120-bed Bayfront Health Dade City hospital as well as its physician clinic operations and outpatient services.
Our organization is growing, and we are excited about the opportunity to deliver Christ-centered, wholistic care to those in Dade City and the surrounding areas, said Terry Shaw, President/CEO for Adventist Health System. We strive to make the communities we serve better and healthier, and we look forward to the impact our mission of Extending the Healing Ministry of Christ will have in this community.
Effective April 1, 2018, Bayfront Health Dade City was renamed AdventHealth Dade City, and joined Adventist Health Systems West Florida Division, which also includes AdventHealth Carrollwood, AdventHealth Connerton, AdventHealth Lake Placid, AdventHealth Sebring, AdventHealth North Pinellas, AdventHealth Tampa, AdventHealth Wauchula, AdventHealth Wesley Chapel and AdventHealth Zephyrhills.
In addition, the health system has named Jason Newmyer to the position of administrator for the newly named AdventHealth Dade City and Vice President of nearby AdventHealth Zephyrhills. Newmyer will oversee strategic development and operations at the hospital.
Newmyer previously served as Assistant Vice President of specialty service lines for AdventHealth Tampa. He was instrumental in the development of the hospitals Digestive Health Institute, Orthopedic Institute and its Brain and Spine Institute. He also played a key role in forming the recent partnership between AdventHealth and Walgreens, which added 15 retail health care clinic locations in the greater Tampa Bay region. Prior to that, he held several progressive leadership roles with the organization.
It is my honor to serve this hospital as we advance whole-person health care in Dade City, said Newmyer. I look forward to working with our hospital leaders, physicians, team members and community to provide exceptional care.
Bayfront Health Dade City, now AdventHealth Dade City, is an acute care hospital, designated as a primary stroke care center, with 150 physicians representing 27 medical specialties, including bariatrics, cardiology, emergency services, gynecology, neurology, surgery and urology.
With the addition of Bayfront Health Dade City, Adventist Health System now has 27 hospital campuses in the state of Florida.
About Adventist Health System
With a sacred mission of Extending the Healing Ministry of Christ, Adventist Health System (AHS) is a connected system of care for every stage of life and health. More than 80,000 skilled and compassionate caregivers in physician practices, hospitals, outpatient clinics, skilled nursing facilities, home health agencies and hospice centers provide individualized, wholistic care. A Christian mission, shared vision, common values, focus on whole-person health and commitment to making communities healthier unify the systems now 46 hospital campuses and hundreds of care sites in diverse markets throughout nine states.
Our Team and Culture
AdventHealth Pasco County Hospitals Earn All A's in Patient Safety
The health care system is the only one in the county to receive a perfect grade card from the prestigious Leapfrog Group.
AdventHealth Prepares for Hurricane Ian
Hurricane staffing and emergency activations are underway at all locations in Greater Tampa Bay, the Heartland Region and Greater Marion County.
AdventHealth Dade City is nationally recognized for its commitment to providing high-quality stroke care
The American Heart Association presents Get With The Guidelines®-Stroke Gold Plus with Target: Type 2 Diabetes Honor Roll award for proven dedication to ensuring all stroke patients have access to...
Directions to AdventHealth Dade City 13100 Fort King Road
AdventHealth Dade City at 352-521-1100352-521-1100


What keeps us from Serenity? Desire!
Within each piece of art is rolled a history; an undercurrent of unknown mystery, influences of defined desires, and outcomes that give new meaning to the intent. Los Que No Son Gentos may display most of all a misleading mystery that is incapable of certainty, yet honest, and defenseless.
Like many Chubby Wolf albums, there are certainties. Danielle was careful about ensuring titles were appropriately placed, recording locations and instruments were listed, and photography was specifically chosen (in most cases). Yet, always there is mystery closeby. All these materials were simply placed on the bookshelf after completion, stuffed in a random book, or shoved in a drawer full of bank statements. There is no explanation of why the album was made, what the inspiration was, nor were there even efforts to send it out to be released. It was created, completed, and on to the next thing.
Dragon's Eye Recordings is pleased to present Los Que No Son Gentos as is, just as Danielle left it.
5 Against 4, Nicola Catalano, Headphone Commute, & Gary Strasberg (Melody In Mayhem).
Disons le clairement : la musique de l'Américaine est littéralement habitée par une conscience perçante de la proximité de la mort et de la puissance insoupçonnée de l'imagination. Avoir su retranscrire de telles émotions par la musique, et avec une telle acuité, relève bien du miracle que l'on appelle art.
…Los Que No Son Gentos is unadulterated by meaning and allowed to just be.
…Baquet-Long's music is all about changes, not continuity, about surprise not sustain, about liberation through ambiguity rather than comfort in numbers. Nothing is set in stone in her world, everything could literally go into an entirely different direction at any moment.
La magie du disque tient avant tout au savant mélange de ces composantes, donnant l'impression de créer des chœurs, et à une simplicité apparente, héritée du minimalisme des compositions.
The seeming sculptural nature of the 14 pieces would capture the ear of those with a sense of contemporary sound art, the construction of sound within immersive settings and the possibilities of sound for conveying psychological depth.
…the collection manifests a more timbrally austere sound palette than usual for this already tonally-retentive practitioner; less immediately alluring, kind of blue in tenor, if you will, with a smouldering slightly airless quality.
…its soft, Celer-like swells are as crystalline and cathedralesque as anything she released during her lifetime, and Los que No Son Gentos is a solid collection that sits comfortably alongside her other releases.
Disturbing soundscapes come out creating haunting cinematic figures which navigate between dark corridors and colourful landscapes.
C'est la bande-son d'un voyage au désert en quatorze stations de variations climatiques en filigrane.
As sparse as these textures appear, their unadorned composure freely blossoms. Embracing minimalism and attentive arrangement, these 14 pieces paint a stark yet rich canvas encircling the observer.


Shana is a Aesthetic Nurse Injector. She received her certification through Aesthetic Medical Educators Training. She has had extensive training through master injector courses. She specializes in neuromodulators (Botox and Xeomin) and wrinkle fillers such as Juvederm, Voluma, Vobella, Radiesse, Belotero, and Bellafill.
She was named most loved injector of 2017 from RealSelf.
Shana has been a nurse since 1997. She worked in Peds/NICU for 5 years, travel nursing for 3 years, and was a nurse manager at a plasma center for 10 years.
Renee has been an Aesthetic Injection Specialist for over 15 years. She is educated in facial anatomy which is essential for providing the best and safest procedures. She has advanced training in injecting neuromodulators (Botox and Xeomin) and dermal fillers (Radiesse, Belotero Balance, Juvederm Ultra XC, Juvederm Plus XC, Voluma XC, and Bellafill). She is also certified in hand rejuvenation with the dermal filler Radiesse, the only filler FDA approved for hands.
She is passionate about restoring and enhancing natural beauty. She provides a personal and caring experience for all clients.
Elizabeth trained at the Aveda Institute in Minneapolis, MN. She has worked at Day Spas and Medical Spas since becoming licensed. She has been certified in advanced modalities like Coolsculpting, Lasers and Micro-needling since 2013. She loves having so many tools available to assist clients to reach their skincare goals. Education is her passion so if you want to know the WHY and HOW behind products and services, Elizabeth is the person to ask!
Kelsey graduated from the Aveda Institute, Minneapolis with a certification in massage therapy and license in Esthiology. She graduated in 2011 and shortly thereafter, joined the Radiance Medspa team. She has furthered her education with advanced product training in Obagi, SkinCeuticals, Neocutis, Eminence and Jane Iredale mineral makeup. Kelsey has broadened her education in medical procedures such as microdermabrasion, chemical peels, dermal blading and skin tightening.
Kelsey enjoys making others look and feel better about themselves on a daily basis. Whether its a hot stone massage or microdermabrasion, she enjoys getting results and addressing her clients needs.
Michelle gained extensive experience through years of working at a premier day spa, as well as owning a Suite Salon in Oklahoma. She has since pursued advanced training in all medical spa services, staying educated in the latest technology and techniques in the field is very important to her. Michelle is passionate about overall skin health and client education. She enjoys working with her clients to help them achieve their personal skincare goals.
Brigette received her Massage Therapist and Health Educator Certification from the National Holistic Institute in Studio City, California in 2013. She extensively trained in both Western and Eastern modalities, including Swedish, Deep Tissue, Sports, Pregnancy, Shiatsu,Thai massage and Energy Work. After working in some of LA's premier luxury spas she wanted to broaden her education in the field of Aesthetics. Brigette graduated from Saint Paul College in 2016, as a Licensed Aesthetician. She has since furthered her education in Medical Aesthetics, and Nails. Brigette is currently finishing her CIDESCO, which is a prestigious International Certification in massage, skin care and nails. Her passion is overall wellness and she firmly believes that health and beauty truly begin from within.
Sydney graduated Saint Paul College and is cross-trained as an Aesthetician, Massage Therapist, and Nail Technician. With extensive education, she received an Esthetics Diploma, Esthetics Advanced Practice Associate of Applied Science Degree, and CIDESCO Readiness Certificate. CIDESCO is the world's largest International Beauty Association. She takes pleasure in advanced medical procedures, and studied anatomy and physiology, nutrition, and kinesiology. Sydney also has over 8 years of experience in customer service and retail. Sydney fell in love with the healing aspect of skin care, caring for results in others, and holistic self-care.
Rosie received her education from Fedrico Career College in Fresno, California and is licensed in California and Minnesota. Rosie gained extensive experience through years of working at an Aveda Spa in Hanford, California, as well as owning a Suite Salon in Lemoore, California. Rosie is passionate about overall skin health and customer service. Rosie enjoys working as a spa coordinator, ensuring each guest has a positive experience throughout their visits to Radiance.
Although Dawn is new to the spa industry, she brings many years of dedicated customer service to Radiance Medspa. She is passionate about the products and services that Radiance offers. She has furthered her education with advanced product training. She strives to make each client feel warmly welcomed and relaxed after each visit.
Cori is very excited to meet new clients and provide them with outstanding nail services.
Cassie is a licensed esthetician and nail technician, she has recently relocated from Bend, Oregon and is excited to be in Minnesota. Cassie specializes in skin care, focusing on each clients individual needs. She graduated from Esthetics school in 2006, In her past, she has worked with dermatologists and podiatrists. Her specialty is, "Providing skin care and services to all, respecting their requirements, and having them leave feeling satisfied." Cassie's favorite service to perform is the Dermapeel, where clients receive a facial, peel, and dermablading.


text:
EXCLUSIVE: Watch Dierks Bentley Fly on the 'Say You Do' Set
By Sophie Schillaci 10:19 AM PDT, September 22, 2014
Dierks Bentley is going back to basics with his latest single.
The music video for "Say You Do," off his CMA nominated album Riser, hit the web on Monday and only ETonline has a behind the scenes look. Watch the country crooner learn to ride a vintage motorcycle and fly a plane in the video above.
PICS: Country Stars Hit the CMT Awards Red Carpet
"I think it's pretty evident there is a hardcore yearning going on in this song, and personally, it's something I've experienced in the worst ways in pas relationships," said Bentley. "The song is about wanting something so badly even if it's not good for you, just wanting that love back, which is something I think everyone can relate to at some point in their life."
"Say You Do" is Bentley's third single off Riser, following the hits "I Hold On" and "Drunk On a Plane."
"Today is all about tryin' to capture the loneliness, the yearning, the conflict of the song," said director Wes Edwards. What do you think -- did they nail the video? See the finished product below.
Follow Sophie on Twitter: @SophieSchillaci
The CMT Awards' Red Carpet Styles


text:
Jan Láho (22. dubna 1738, Radvaň – 21. listopadu 1790) byl slovenský luterský duchovní působící v Čechách.

Život
Od roku 1767 působil jako učitel na šlechtické škole v Ožďanech, pak jako farář v Brzotíně. Po vydání tolerančního patentu přišel 28. května 1782 jako první luterský pastor do Čech. Téhož roku vydal v Praze katechismus Vejtah některých hlavných otázek a odpovědí z učení křesťanského pro obecný lid evangelický v Čechách. Působil ve sboru v Krucemburku (Křížové). V letech 1784-1785 zastával úřad prvního superintendenta české evangelické církve a. v. V roce 1784 na audienci v Jihlavě žádal císaře Josefa II., aby zrušil placení štólových poplatků katolickým kněžím a podporoval superintendenty ze státních prostředků. V září 1785 ukončil svou službu v Čechách a odešel zpět do Uher.

Bibliografie
 Ottův slovník naučný
 Slovenský biografický slovník
 Hlas Střediska české reformace, 9/2004.

Související články
 Evangelická církev v Rakousku (1781-1918)

Slovenští protestantští duchovní
Superintendenti
Narození v roce 1738
Narození 22. dubna
Narození v Banské Bystrici
Úmrtí v roce 1790
Úmrtí 21. listopadu
Muži