text:
\section{Introduction}
Recently much attention has been devoted to atomic impurities
embedded in ultra-cold atomic media
because of experimental accessibility of such systems in controlled ways
and of observations of various kinds of quasi-particle properties of the impurities
: bosonic \cite{Catani1,Scelle1,Hohmann1,Compagno1,Jrgensen1,Hu1,Rentrop1}
and fermionic \cite{Froehlich1,Scazza1,Cetina1} ones.
For instance, the coupling between impurities and medium
can be tuned using the Fano-Feshbach resonances between atomic hyperfine states,
and the spatial dimensionality or periodicity of the system can also be designed
using the effects of external electromagnetic fields \cite{PethickSmith1,Pitaevskii1}.
The quasi-particle energy,
width and spectral weight of the impurity can be measured
in radiofrequency spectroscopy \cite{Jrgensen1,Hu1,Froehlich1},
and the fine energy splitting of a trapped impurity be measured in the Ramsey spectroscopy
with oscillating fields \cite{Rentrop1}.
Also, the dynamical aspects of the polaron formation can be observed
experimentally \cite{Cetina1}.

Theoretical studies of such systems have been actively performed
prior to the experiments and revealed
that properties of impurity are diverse depending on
the impurity-medium interaction and medium properties:
When the medium is a Bose-Einstein condensate (BEC),
the impurity interacting through the Bogoliubov phonon of the medium is called
a Bose polaron \cite{Cucchietti1,Sacha1,Tempere1,Casteels1,Rath1,Shashi1,Li1,
Levinsen2,Dehkharghani5,Ardila2,Christensen1,Vlietinck1,
Grusdt1,Grusdt3,Shchadilova1,Shchadilova2,Grusdt6,Ashida1,Ardila6,Nielsen3,Mistakidis8,Mistakidis9}
in analogy with that in electron-phonon systems
\cite{polaronreview1,Landau1,FPZ1},
where the atomic impurity is a quasiparticle dressed with a virtual cloud of excited phonons.
In the case of a degenerate Fermi-gas medium,
the impurity is called Fermi polaron
\cite{Chevy1,Massignan1,Schirotzek1,Ku1,Schmidt1,Kohstall1,Koschorreck1,Schmidt6,
Vlietinck2,Trefzger1,Trefzger2,Massignan5,Lan1,Lan2,Nur1,Yi1,Parish5,
Levinsen4,Kamikado1,Kain1,Schmidt5,Mistakidis10},
which is dressed with particle-hole excitations around the Fermi surface.
In these studies
the low energy $s$-wave contact interaction has been frequently used for the impurity-medium interaction.
Other kinds of atomic polarons are also studied
with unconventional impurity-medium interactions, e.g.,
$p$-wave interactions \cite{Levinsen5,Deng1},
dipolar-dipolar interactions \cite{Kain3,Ardila4}.
In the case of the impurity-medium coupling tuned around the unitarity limit,
the impurity and medium atoms can form few-body bound states in the medium  \cite{Levinsen2,Shchadilova2,Levinsen4},
and consequently the quasi-particle residue almost vanishes.
The above mentioned studies entirely assume zero temperature,
but recently thermal evolutions of polarons have been investigated,
where the medium temperature varies from cold degenerate to hot Boltzmann regimes
for Fermi polarons \cite{Tajima1,Yan8,Tajima2},
and, for Bose polarons, the temperature varies over the BEC critical temperature \cite{Levinsen8,Guenther8}.

In many studies of the polaron that have been done so far,
the system is assumed to be spatially uniform,
while the real experiments of the ultra-cold gas are usually done on the systems trapped
in the harmonic potentials.
In the present study,
we consider a Bose polaron in spherically symmetric trap in three dimensions,
where the angular momentum of the polaron gives the conserved quantum numbers
instead of spacial momenta in the uniform system.
In particular,
we calculate the ground-state energy of a trapped Bose polaron
of fixed total angular momentum,
and make clear the distributions of the angular-momentum
and other quantum numbers of the polaron between the impurity and excited bosons in medium.
For this purpose,
we develop a variational method
with the fixed expectation value of the angular momentum operators.

In Sec.~II,
we set up our system, and derive a Fr{\"o}hlich type effective Hamiltonian.
In Sec.~III,
we introduce a cranking gauge transformation,
by which all bosons in medium are cranked to move in the co-rotating frame of impurity.
In Sec.~IV,
we develop a variational method to obtain the energy functional for the cranked Hamiltonian,
and present variational solutions and distribution functions of the excited bosons.
In Sec.~V, numerical results and discussion for them are shown.
Sec.~VI is for the summary and outlook.

\section{Fr{\"o}hlich type effective Hamiltonian}
We consider the system of a single atomic impurity interacting with a dilute Bose gas,
where the impurity and the gas are trapped
in the spherically-symmetric harmonic potentials with the same centers.
The impurity and bosons are all spinless,
so that the total orbital angular momentum of the system is conserved.
We also suppose that bosons are non-interacting,
while the impurity-boson interaction is tuned finite by the Feshbach resonance method.
Thus,
when the interaction is turned off,
all medium bosons occupy the lowest energy state of the trap potential to form a $T=0$ BEC.
This system is described by the following effective Hamiltonian:
\beq
\mathcal{H}(\br)
&=&
H_{ho}(\br)
+\int_{\br'} \, \phi^\dagger(\br')
\left[
-\frac{\hbar^2 {\nabla'}^2}{2 m_b} +\frac{m_b \omega_b^2}{2}{r'}^2
\right]
\phi(\br')
\nn
&&+g \int_{\br'} \, \phi^\dagger(\br')
\delta^{(3)}(\br-\br')
\phi(\br')
\nn
&=&
H_{ho}(\br)
+\sum_{s} E_{s}^{b}b_s^\dagger b_s
+g \sum_{s,s'} \phi_s^{b*}(\br) \phi_{s'}^b(\br)
b_{s}^\dagger b_{s'}
\label{FH1}
\eeq
where the freedoms of the impurity and medium boson are represented
in the first and the second quantized form.
We have used the abbreviated notation for the spacial integral:
$\int_{\br}\equiv \int {\rm d}\br^3$.
The first term $H_{ho}(\br)$ is the Hamiltonian of the impurity
trapped in the harmonic-oscillator potential:
\beq
H_{ho}(\br)
&=&
-\frac{\hbar^2}{2m_I} \frac{1}{r^2} \partial_r\left( r^2 \partial_r \right)
+\frac{\hbar^2{\hat{L}}^2}{2m_I\, r^2}
+\frac{m_I \omega_I^2}{2}r^2
\eeq
where $(r, \theta, \varphi)$ is the spherical coordinate of the impurity,
and $m_I$ and $\omega_I$ is the impurity mass and the angular frequency of the trap.
The squared orbital angular-momentum operator ${L}^2$
is represented by,
\begin{equation*}
{\hat{L}}^2=-\left[ \frac{1}{\sin\theta} \partial_\theta \left( \sin\theta \partial_\theta \right)
+\frac{1}{\sin^2\theta} \partial_\varphi^2 \right],
\end{equation*}
The second term in (\ref{FH1}) represents the Hamiltonian of the medium boson;
the $m_b$ and $\omega_b$ are the mass and the trap angular frequency of the medium boson,
and the coupling constant $g$ of the impurity-medium contact interaction is given
by the s-wave scattering length $a$ as $g=2\pi a/m_r$ in low-temperature approximation.
The second line of (\ref{FH1}) is obtained
with the substitution of the field operator expansion:
$\phi(\br')=\sum_s \phi_s^b(\br') b_s$
where the $\phi_s^b({\bf r})$ are the eigenfunctions
of the harmonic oscillator potential for the eigenvalues $E^b_s$,
and the $b_s$ and $b_s^\dagger$ are the corresponding bosonic annihilation and creation operators.
The label $s$ representing the medium-boson states is
the abbreviated notation for $s=( n, l, m )$:
the principal, the azimuthal, and the magnetic quantum numbers, respectively.

The explicit form of the Harmonic-oscillator eigenfunctions $\phi^\alpha_s({\bf r})$
for the medium boson ($\alpha=b$) and the impurity ($\alpha=I$) are denoted as,
\begin{equation}
     \phi_s^{\alpha}({\bf r}) =R^{\alpha}_{n l}(r) Y_{lm}(\theta, \varphi),
\label{eq:koyukansu}
\end{equation}
where the angular part $Y_{lm}$ is the spherical harmonic $Y$-function,
and the radial part $R^{\alpha}_{nl}(r)$ are,
\begin{align}
R^{\alpha}_{nl}(r) &=
{\mathcal N}_{n,l}
\left( m_\alpha \omega_\alpha\right)^{\frac{3}{4}}
\, \left( \sqrt{m_\alpha \omega_\alpha} r\right)^{l}\,
e^{-\frac{m_{\alpha} \omega_{\alpha}}{2}r^2}
L^{\left( \frac{1+2l}{2}\right)}_n\left( m_\alpha \omega_\alpha r^2 \right),
\label{eq:koyukansu2yabu}\\
{\mathcal N}_{nl}&=\sqrt{\frac{2^{n+l+2} n!}{\sqrt{\pi} \left( 2n+2l+1 \right) !!} }.
\label{eq:normalization}
\end{align}
The Laguerre function $L_n^{(k)}(x)$ that we use in this paper is defined by
\begin{equation*}
     L_n^{(k)}(x) =\frac{e^x x^{-k}}{n!} \frac{{\rm d}^n}{{\rm d}x^n}\left( e^{-x} x^{n+k} \right).

\end{equation*}
The energy-eigenvalue corresponding to the state (\ref{eq:koyukansu}) is
\begin{equation}
E_{nl}^{\alpha}=\frac{\omega_{\alpha}\hbar}{2}( 3+2l+4n ).
\label{eq:energy}
\end{equation}
It should be noticed that we use the unit system of $\hbar=1$ throughout this paper.

\subsection{Bogoliubov approximation and Fr{\"o}hlich type Hamiltonian}
In the case of the small number excitation of medium bosons around the impurity
in comparison with the total condensed boson number $N_0$
\cite{Bruderer1,Nakano2},
we can use the Bogoliubov approximation $b_0\simeq \sqrt{N_0}$,
where $s=0$ corresponds to the lowest energy level ($n=l=0$).
With keeping terms in the interaction part
up to the linear order of the excited boson,
we obtain
\begin{eqnarray}
\mathcal{H}(\br) &\simeq&
H_{ho}(\br)
+E^b_0 N_0
+\sum_{s \neq 0} E_s^b b_s^\dagger b_s
+g N_0 |\phi_0^b(\br)|^2
\nn
&& \ +g\sqrt{N_0} \sum_{s\neq0} \left[ \Phi_s(\br)  b_{s}
+\Phi_s^*(\br) b_{s}^\dagger\right],
\label{FH1y}
\end{eqnarray}
where
\begin{equation*}
     \Phi_{s=\left\{ n,l,m\right\}}(\br)\equiv
     \sqrt{\frac{1}{4\pi}} R_{00}^{b}(r) R_{nl}^b(r) Y_{lm}( \theta, \varphi).
\end{equation*}
The Hamiltonian (\ref{FH1y}) can be transformed
into the same form of the
Fr{\"o}hlich Hamiltonian of the electron-phonon system,
and the electron polaron
was originally studied in \cite{FPZ1} for the polar crystals.
We will use the Hamiltonian (\ref{FH1y}) in the present paper.

\section{Cranking of boson states}
In the present study
we aim to find the lowest energy state of the Hamiltonian (\ref{FH1y})
for given expectation values of the total angular momentum operators.
These states correspond to the yrast states
appeared in the description of rotational collective excitations
of an axially deformed nucleus in nuclear physics,
where the rotation axis is not parallel to that of the axially symmetry,
and the gauge transformation (cranking)
$e^{i\omega_k t \hat{J}_k}$ is introduced conveniently
to shift the state from the normal space-fixed frame
to the co-rotating frame with the nucleons
in which the nucleus wave function is stationary \cite{Rowe1,Inglis1,Thouless1}.
The same method can also be utilized in the present case
to describe the excitations of bosons around the impurity;
we rotate the boson cloud around the impurity collectively
by the gauge transformation ($S$ transformation)
with the solid angle variables $(\theta, \varphi)$ of the impurity:
\beq
S(\varphi,\theta)
&=&
e^{-i\varphi \hat{M}_z} e^{-i\theta \hat{M}_y}
\eeq
where the boson angular-momentum operator is defined by
\beq
\hat{M}_i=\sum_{n,l,m,m'}  b_{nlm}^\dagger \left( \hat{\mathcal{L}}_i\right)_{m,m'}^{(l)} b_{nlm'}
\eeq
where $\left( \hat{\mathcal{L}}_i \right)_{m,m'}^{(l)}$ is the matrix element
of
a general orbital angular momentum operator $\hat{\mathcal{L}}_i$
by the eigen-states of rank $l$ :
\begin{align*}
     & \left( \hat{\mathcal{L}}_i \right)_{m,m'}^{(l)}
   \equiv  \langle l, m| \hat{\mathcal{L}}_i |l, m'\rangle, \\
     &\quad l =0,1,2, \cdots, \quad
            m,m'=-l,-1+1,\cdots, l-1, l.
\end{align*}
The general form of this transformation has been successfully introduced
by Schmidt and Lemeshko to investigate the angular momentum distribution
in the system of a linear rotor impurity
embedded in bosonic environment in free space \cite{Lemeshko3,Lemeshko4,Lemeshko7},
and the simpler version, $S(\varphi,0)$, has been utilized
in the system of Bose polaron in axially symmetric trap potentials
for the study of the angular-momentum drag effect \cite{NYI1}.

\subsection{Cranked angular momentum operators}
The $S$-transformation practically serves as
linear transformations for the boson annihilation operators
and the boson angular-momentum operators:
\beq
S^{-1} b_{nlm} S
&=&
\sum_{m'=-l, \cdots, l}D^l_{m,m'}\left( \varphi, \theta, 0 \right)  b_{nlm'},
\label{eq:S}
\\
S^{-1} \hat{M}_{i} S
&=&
\sum_{j=0,\pm1} {D^1_{i,j}}^*\left( \varphi, \theta, 0 \right) \hat{M}_{j},
\eeq
where we have used the spherical basis: $\hat{M}_0=\hat{M}_z$ and
$\hat{M}_{\pm1}=\mp \frac{1}{\sqrt{2}} \left(\hat{M}_x\pm i\hat{M}_y\right)$,
for vector indices,
and Wigner's $D$ function
with Euler angles $(\alpha,\beta,\gamma)$
for the spacial rotation \cite{Rose1}:
\beq
D^l_{m,m'}\left( \alpha,\beta,\gamma \right) =
\langle l, m| e^{-i\alpha\hat{\mathcal{L}}_z} e^{-i\beta\hat{\mathcal{L}}_y}
e^{-i\gamma\hat{\mathcal{L}}_z} |l, m'\rangle.
\eeq
The $S$ transformation acts as a shift operator for the impurity angular-momentum operators:
\begin{eqnarray}
S^{-1}\hat{L}_0S
&=&
\hat{L}_0+S^{-1}\left( \hat{L}_0 S\right)
=
\hat{L}_0-S^{-1}\hat{M}_z S,
\\
S^{-1}\hat{L}_{\pm1}S
&=&
\hat{L}_{\pm1}+S^{-1}\left( \hat{L}_{\pm1}S\right)
\nn
&=&
\hat{L}_{\pm1}
+\frac{1}{\sqrt{2}}  e^{\pm i \varphi}
S^{-1} \left[
i\,  e^{-i\varphi \hat{M}_z} \hat{M}_y e^{i\varphi \hat{M}_z}
\mp \cot\theta  \hat{M}_z
\right]
S,
\end{eqnarray}
where we have used the spherical-basis representation:
\beq
\hat{L}_0&=&\hat{L}_z=-i\partial_\varphi,
\\
\hat{L}_{\pm1}&=&
\mp \frac{1}{\sqrt{2}} \left( \hat{L}_x\pm i\hat{L}_y\right)
=\frac{1}{\sqrt{2}} e^{\pm i \varphi} \left(
 -\partial_\theta \mp i\, \cot{\theta} \partial_\varphi \right).
\eeq

In the present system,
the total angular momentum operator of the system is given by
\begin{equation}
\hat{J}_i=\hat{L}_i+\hat{M}_i,
\end{equation}
and the $z$-th component $\hat{J}_z$ and
the squared amplitude $\hat{J}^2$ are conserved:
$\left[ \hat{J}_z, {\mathcal H}(\br) \right]
=\left[ \hat{J}^2, {\mathcal H}(\br) \right]
=\left[ \hat{J}_z,\hat{J}^2\right]=0$.
Using the transformation formulas of the angular momentum operators
$\hat{M}_i$ and $\hat{L}_i$,
the $S$-transformed operators of $\hat{J}_z$ and $\hat{J}^2$
becomes,
\beq
S^{-1}\hat{J}_zS
&=&
S^{-1} (\hat{L}_z+\hat{M}_z) S=\hat{L}_z,
\label{traJz}
\\
S^{-1}\hat{J}^2S
&=&
S^{-1} \left( \hat{L}^2 +\hat{M}^2+2\hat{M}\cdot \hat{L}\right) S
\nn
&=&
\hat{L}^2+\hat{O}_L +\hat{M}^2
\nn
&&+2 \sum_{i,j=0,\pm1} \hat{M}_j^\dagger {D^1_{i,j}}(\varphi, \theta, 0)
\left[
S^{-1}(\hat{L}_i S)+ \hat{L}_i
\right],
\label{traJsq}
\eeq
where we used the scalar product
$\hat{M}\cdot \hat{L}=\sum_{i=0,\pm1} \hat{M}_i^\dagger \hat{L}_i$ in the spherical basis representation,
and the shift operator $\hat{O}_L$ of $\hat{L}^2$ is defined by
\beq
\hat{O}_L:=S^{-1}\hat{L}^2S-\hat{L}^2=S^{-1}\left( \hat{L}^2 S\right)+ 2S^{-1}(\hat{L} S)\cdot \hat{L}.
\eeq

We see from the results (\ref{traJz}) and (\ref{traJsq})
that the $z$-component of the total angular momentum is taken over solely
by the impurity after the $S$ transformation,
while the total angular momentum of the system looks complicated.
\footnote{In the case of the linear rotor impurity \cite{Lemeshko3,Lemeshko4}
the total angular momentum operator is transformed to be that of the impurity,
which is by virtue of the intrinsic angular momentum of the rotor itself.}

\subsection{Cranked Hamiltonian}
In similar calculation,
the $S$-transformation of the Hamiltonian (\ref{FH1y}) is obtained,
\begin{eqnarray}
S^{-1} {\mathcal H}(\br) S
&=&
H_{ho}(\br) +\frac{\hat{O}_L}{2m_I r^2}
+ \sum'_{n,l,m}E_{nl}^b
b^\dagger_{nlm} b_{nlm}
\nn
&&
+ E^b_{00} N_0 +g N_0 |\phi_0^b(\br)|^2
\nn
&&
+g\frac{\sqrt{N_0}}{4\pi}R_{00}^{b}(r) \sum'_{n,l}
\sqrt{2l+1}R_{nl}^b(r) \left( b_{nl0}+ b^\dagger_{nl0} \right),
\label{traH}
\end{eqnarray}
where the symbols $\sum'_{n,l,m}$ and $\sum'_{n,l}$ represent the summations
except $n=l=m=0$ and $n=l=0$ respectively.
In the derivation,
we have used the formula
$Y_{lm}^*(\theta,\varphi)=\sqrt{\frac{2l+1}{4\pi}} D_{m,0}^l(\varphi, \theta,0)$
and the orthogonality of the $D$ functions \cite{Rose1}.
The second term including the shift operator $\hat{O}_L$ corresponds
to the rotation energy of the impurity,
which comes from the rotation energy of excited bosons originally before the cranking.
The last term is that of the boson-impurity coupling;
it should be noticed that it includes the coupling with the excited bosons with $m=0$
in the $S$-transformed Hamiltonian\cite{Lemeshko3,Lemeshko4}.

\section{Variational method}
Let's develop the variational method
to obtain the lowest energy states
under the condition that the azimuthal and $z$ (magnetic) components of the total angular momentum are given by the expectation values $(J, J_z)$.
The Hamiltonian (\ref{traH}) shows
that the impurity-boson interaction term includes only the excited bosons with $m=0$
after the $S$-transformation,
so that, as a variational state of excited bosons around the impurity,
we employ the coherent state for the excited bosons with the quantum numbers $s=( n,l,0 )$:
\cite{LLP1,Schweber1},
\beq
\left| b \right\rangle
&=& \exp
\sum'_{n,l} \left( f_{nl} b_{nl0}^\dagger-f_{nl}^*b_{nl0} \right)
\left| 0 \right\rangle,
\label{coh1}
\eeq
where the variational parameters $f_{nl}$ and $f_{nl}^*$ are
eigenvalues of annihilation and creation operators:
$b_{nl0} \left| b \right\rangle =f_{nl}\left| b \right\rangle$,
$\left\langle b \right| b_{nl0}^\dagger=\left\langle b \right| f_{nl}^*$.
The state $\left| b \right\rangle$ is a normalized one: $\langle b|b\rangle=1$.
It would be a good approximation for the heavy impurity trapped in the deep potential;
in the case of heavy mass or high trap-frequency limits of the impurity,
the above coherent state becomes the exact solution
because the impurity becomes localized at the center of trap.
\footnote{A marginal case where $m_I \rightarrow \infty$ as $m_I\omega_I^2$ is kept finite is also soluble.}

Now we use the abbreviated notation for the expectation value of operator
by the coherent state $\left| b \right\rangle$ as
$\langle \cdots \rangle_b \equiv \langle b| \cdots |b\rangle$.
Then that of the transformed Hamiltonian (\ref{traH}) and
the $S$-transformed total angular momentum operators become,
\beq
\langle S^{-1} {\mathcal H}(\br) S \rangle_b
&=&
H_{ho}^f(\br)
+ \sum_{n,l}' \left[ \frac{l(l+1)}{2m_I r^2} + E_{nl}^b \right] |f_{nl}|^2
\nn
&&
+ E^b_{00} N_0 +g N_0 |\phi_0^b(\br)|^2
\nn
&&
+g\frac{\sqrt{N_0}}{4\pi}R_{00}^{b}(r) \sum_{n,l}'
\sqrt{2l+1} R_{n,l}^b(r) \left(  f_{nl} + f^*_{nl} \right),
\label{hb1}
\\
\langle S^{-1}\hat{J}^2S \rangle_b
&=& \hat{L}^2,
\label{L2b1}
\\
\langle S^{-1}\hat{J}_z S \rangle_b &=& \hat{L}_z,
\label{Lzb1}
\eeq
where we have used the expectation value
$\langle \hat{O}_L \rangle_b =\sum_{n,l}' l(l+1)|f_{nl}|^2$\footnote{
For derivations of the expectation values,
see Appendix~A.}.

The expectation value (\ref{hb1}),
where the bosonic degrees of freedom have been eliminated,
provide the effective Hamiltonian of the impurity,
and eqs.~(\ref{L2b1}) and (\ref{Lzb1}) are the corresponding effective total angular-momentum operators
represented with the impurity coordinate.
It is very interesting that the latter are the same with the impurity angular momentum;
it gives an essential advantage in the present variational method
with the condition of the fixed total angular momentum.

\subsection{Variational states of impurity}
Since the total angular momentum operators
(\ref{L2b1}) and (\ref{Lzb1}) are given
by those of the impurity,
the variational state of the impurity can be assumed
as the eigenfunctions (\ref{eq:koyukansu}) of the impurity
with fixed azimuthal and magnetic quantum numbers $(J, J_z)$:
\beq
\Psi_{JJ_z}(\br) = \sum_{n} F_{nJJ_z} \, \phi^I_{nJJ_z}(\br)
\eeq
where $J_z=-J, -J+1, \cdots, J-1, J$
and the coefficients $F_{nJJ_z}$ serve as Ritz-type variational parameters.
Note that we do not consider mixing of different angular momenta,
because of rotational symmetry.
Since the states
with large principal quantum numbers less contribute in the ground state
in the weak coupling regime,
we truncate the variational state up to $n=1$ in the present calculation:
\beq
\Psi_{JJ_z}(\br) = \sum_{n=0,1} F_{nJ} \,\phi^I_{nJJ_z}(\br).
\label{ivs1}
\eeq
Note that the subscript $J_z$ has been omitted in the variational parameters
since the rotational symmetry of the system gives the degeneracy for the direction in real space
and the variational parameters do not depend on $J_z$.
In solving the variational equations,
we impose the normalization condition for the parameters: $|F_{0J}|^2+|F_{1J}|^2=1$.

\subsection{Variational energy functional and solutions}
Now taking the expectation value of the Hamiltonian (\ref{hb1})
with respect to the impurity's variational state (\ref{ivs1}),
we obtain the variational energy functional for the state with the total angular momentum $(J ,J_z)$:
\beq
E\left[ F_{nJ}; f_{nl}\right]
&=& \langle  {\mathcal H}(\br) \rangle_{JJ_z}
\nn
&=&
E^I_{0J}|F_{0J}|^2+E^I_{1J}|F_{1J}|^2
+ E^b_{00} N_0 \left( |F_{0J}|^2+|F_{1J}|^2 \right)
\nn
&&
+ \sum_{n,l}' \left[ E_{nl}^b \left( |F_{0J}|^2+|F_{1J}|^2 \right) +\frac{l(l+1)}{2m_I}G[F_{0J};F_{1J}]\right]
|f_{nl}|^2
\nn
&&
+g\frac{N_0}{4\pi} H[F_{0J};F_{1J}]_{00}
\nn
&&
+g\frac{\sqrt{N_0}}{4\pi} \sum_{n,l}'
\sqrt{2l+1}
\left[
H[F_{0J};F_{1J}]_{nl} f_{nl}
+H[F_{0J};F_{1J}]_{nl}^* f^*_{nl}
\right],
\label{eq:vE}
\eeq
where we have defined the functionals:
\beq
G[F_{0J};F_{1J}] &=& \int_\br \frac{1}{r^2}
\left|\sum_{n=0,1} F_{nJ} \phi^I_{nJJ_z}(\br)\right|^2,
\label{functio1}
\\
H[F_{0J};F_{1J}]_{nl} &=& \int_\br {R_{00}^{b}(r)} R_{nl}^b(r)
\left|\sum_{n=0,1} F_{nJ} \phi^I_{nJJ_z}(\br)\right|^2.
\label{functio2}
\eeq
The variational equation $\delta E/\delta f_{nl}^*=0$
gives the formal solution:
\beq
\bar{f}_{nl}[F_{0J};F_{1J}] =-g\frac{\sqrt{N_0}}{4\pi}
\frac{\sqrt{2l+1} H[F_{0J};F_{1J}]_{nl}^*}
{E_{nl}^b \left( |F_{0J}|^2+|F_{1J}|^2 \right) +\frac{l(l+1)}{2m_I}G[F_{0J};F_{1J}]},
\label{fnl1}
\eeq
and, plugging it back to the variational energy (\ref{eq:vE}),
we obtain
\begin{align*}
E[F_{nJ};\bar{f}_{nl}]
&=E^I_{0J}|F_{0J}|^2+E^I_{1J}|F_{1J}|^2 \nn
&+E^b_{00} N_0 \left( |F_{0J}|^2+|F_{1J}|^2 \right) +g \frac{N_0}{4\pi} H[F_{0J};F_{1J}]_{00}
\nonumber\\
&-g^2\frac{N_0}{(4\pi)^2} \sum_{n,l}'
(2l+1)
\frac{|H[F_{0J};F_{1J}]_{nl}|^2}{E_{nl}^b \left( |F_{0J}|^2+|F_{1J}|^2 \right) +\frac{l(l+1)}{2m_I}G[F_{0J};F_{1J}]}.
\end{align*}
Since the coefficients appearing in the variational energy are all real,
the solutions of $F_{0J}$ and $F_{1J}$ are also found to be real.
Using the normalization condition $F_{0J}=\sqrt{1-F_{1,J}^2}$
and the analytical expression\footnote{see Appendix~B} of $G[F_{0J};F_{1J}]$,
we finally obtain
\begin{align}
E[F_{1J}] &=E^I_{0J}+E^b_{00} N_0 +\left( E^I_{1J}-E^I_{0J}\right) F_{1J}^2
+E_{\rm bg}[F_{1J}]
+E_{\rm int}[F_{1J}],
\label{EF1}
\end{align}
where the background interaction energy is,
\beq
E_{\rm bg}[F_{1J}]
\equiv g \frac{N_0}{4\pi}
\left[ H_{00J}^0+ 2F_{1J} \sqrt{1-F_{1J}^2} H_{00J}^c+F_{1J}^2
\left(  H_{00J}^1- H_{00J}^0\right) \right],
\eeq
which comes from the interaction between impurity and background condensed
bosons corresponding to the term $gN_0|\phi^b_0(\bx)|^2$  in  (\ref{hb1}).
The interaction energy term $E_{\rm int}[F_{1J}]$ is represented as
\beq
E_{\rm int}[F_{1J}]
\equiv-\frac{g^2N_0}{(4\pi)^2} \sum_{n,l}'
(2l+1)
\frac{\left[ H_{nlJ}^0+ 2F_{1J}
\sqrt{1-F_{1J}^2}H_{nlJ}^c+F_{1J}^2\left( H_{nlJ}^1- H_{nlJ}^0\right) \right]^2}
{E_{nl}^b +\frac{l(l+1)}{2J+1} \omega_I
\left[ 1+ 2F_{1J} \sqrt{1-F_{1J}^2}/\sqrt{J+\frac{3}{2}}\, \right]},
\label{int1}
\eeq
which is traced back to the parts including $f_{nl}$ in the second and the last terms in (\ref{hb1}),
and corresponds to the interaction between the impurity and the excited bosons.
The explicit forms of the coefficients $H_{nlJ}^{0}, H_{nlJ}^{c}, H_{nlJ}^{1}$
in $E_{\rm bg}$ and $E_{\rm int}$ are shown also in Appendix~B.

In experiments,
the energy shift,
i.e., the energy with bare impurity and background BEC contributions being subtracted,
is measurable using the radio-frequency spectroscopy \cite{Jrgensen1,Hu1};
in the present theory,
it is given by the formula:
\beq
\Delta E[F_{1J}] \equiv \left( E^I_{1J}-E^I_{0J}\right) F_{1J}^2+E_{\rm bg}[F_{1J}] +E_{\rm int}[F_{1J}].
\label{shift1}
\eeq

\subsection{Comparison with the second order perturbation theory}
In general, the solutions of variational method in the present method
includes non-perturbative effects,
but it is heuristic and interesting to see its perturbative nature
before going into numerical results.
Expanding the variational solution (\ref{EF1}) with the coupling constant $g$,
we obtain
$F_{1J}=-g\frac{N_0}{4\pi} \frac{H_{00J}^c}{ E^I_{1J}-E^I_{0J}}$,
to the leading order of $g$;
then the ground state energy becomes
\begin{eqnarray}
E
&=&
E^I_{0J}+E^b_{00} N_0 +g\frac{N_0}{4\pi}  H_{00J}^0
-g^2\frac{N_0^2}{(4\pi)^2} \frac{\left(  H_{00J}^c\right)^2}{ E^I_{1J}-E^I_{0J}}
\nn
&&
\quad -g^2\frac{N_0}{(4\pi)^2} \sum_{n,l}'
\frac{(2l+1) \left( H_{nlJ}^0 \right)^2}
{E_{nl}^b +\frac{l(l+1)}{2J+1}\omega_I}.
\label{pE1}
\end{eqnarray}
to the order of $g^2$.
The result should be compared with that of the second-order perturbation theory;
for the ground state energy of $J=0$\footnote{
For derivation, see Appendix~C. }:
\beq
\langle {\mathcal H}  \rangle
&\simeq&
E_{00}^I+E_{00}^bN_0+g\frac{N_0}{4\pi}  H_{000}^0
-g^2 \frac{N_0^2}{(4\pi)^2}
\frac{\left(  H_{000}^c \right)^2}{E_{10}^I-E_{00}^I}
\nn
&&
-g^2 \frac{N_0}{\left( 4\pi\right)^2}
\sum_{n\neq 0}
\left(
\frac{\left( H_{n00}^c\right)^2}{E_{10}^I-E_{00}^I+E_{n0}^b}
+ \frac{\left( H_{n00}^0\right)^2}{E_{n0}^b}
\right).
\label{E2nd}
\eeq
Comparing (\ref{pE1}) with (\ref{E2nd}),
we find that differences appear in the $g^2N_0$ term,
which is attributed to the Fr{\"o}hlich-type boson-impurity interaction.
In the denominator of  (\ref{pE1}),
the energy of impurity's intermediate states in (\ref{E2nd}) is replaced
by an averaged rotation energy $\frac{l(l+1)}{2J+1}\omega_I$.
It can be explained from the cranking transformation and the angular momentum conservation:
after cranking transformation, all bosons stop to be in rotating states,
and the impurity rotates instead in order to satisfy the angular momentum conservation;
consequently its effect appears as the rotation energy.
In the perturbation theory for the ground state,
the impurity and bosons intermediate virtual states are taken in the order
from those of low-energy regardless of the angular momentum conservation.

\subsection{Distribution functions of excited bosons and quasiparticle residue}
Since the quasi-particle properties of the Bose polaron are characterized
by the virtual boson excitations around the impurity,
the number of excited bosons around the poralon
is an important quantity.
The excited-boson number $N_{nlm}^{JJ_z}$ of bosons with quantum numbers $( n,l,m )$
for the polaron with the total angular momentum $(J, J_z)$
is given by
\beq
N_{nlm}^{JJ_z} &=&
\langle b_{nlm}^\dagger b_{nlm} \rangle_{JJ_z}
\nn
&=&
\int {\rm d}\theta \sin\theta {\rm d}\varphi \, |Y_{JJ_z}\left( \theta, \varphi \right)|^2
\left|D^l_{m,0}\left( \varphi, \theta, 0 \right) \right|^2
\langle b |  b_{nl0}^\dagger b_{nl0} | b \rangle
\nn
&=&
\left|f_{nl}\right|^2
\sum_{L=|J-l|}^{J+l}
\braket{l 0 ; L 0| J 0}^2 \, \braket{J J_z; l m | L J_z+m}^2.
\label{Nb1}
\eeq
It should be noted that
the dependence on $J_z$ and $m$ in $N_{nlm}^{JJ_z}$ comes
through the Clebsch-Gordan coefficients\footnote{
For the definition of the Clebsch-Gordan coefficients, see \cite{Rose1,Edmonds1}.}
$\braket{l 0 ; L 0| J 0}$ and $\braket{J J_z; l m | L J_z+m}$
which are originated
in the averaged overlap of the coupled angular-momentum states from $J$ and $l$.
The dependence through the Clebsch-Gordan coefficients is not dynamical but kinematical;
it can be understood from
independence of the polaron energy-functional on $J_z$ or $m$.

From (\ref{Nb1}),
we obtain the total excited-boson number by summing up quantum numbers:
\beq
N_b ^{J}&=&\sum'_{n,l}\sum_{m=-l, \cdots, l} N_{nlm}^{JJ_z}
=\sum'_{n,l}\left|f_{nl}\right|^2.
\label{nbtot1}
\eeq
It is clear that the total number does not depend on $J_z$
but it has the implicit $J$-dependence through the variational parameter $F_{1J}$.

The real-space density distribution of the excited bosons is given by
the expectation value:
\beq
\left\langle \phi^{b\, \dagger}(\bx) \phi^b(\bx) \right\rangle_{JJ_z}
&=&
\sum'_{n,l}\sum_{m=-l, \cdots, l}
N_{nlm}^{JJ_z}\, |\phi^{b}_{nlm}(\bx)|^2.
\label{nbx1}
\eeq

For the angular momentum,
the boson contribution is found to vanish,
\beq
\langle \hat{M}_i\rangle_{JJ_z}=0,
\eeq
which implies that the impurity alone bears the contribution for $J_z$;
it shows that no drag effects exist for the angular momentum
unlike the axial symmetric case \cite{NYI1}.
There are two reasons for this property.
First, thanks to the complete rotational symmetry
the energy functional becomes spherically symmetric
and does not depend on $J_z$.
Second, no angular-momentum exchange can happen between impurity and bosons
through the impurity-boson interaction
because a density-density type interaction is employed in this work.
In order for $\langle \hat{M}_i \rangle$ to be finite,
an asymmetry with respect to $m$
is required
in the distribution function of excited bosons,
but there is no sources of the asymmetry in the present case
because of the rotational symmetry.
In the case of axial symmetric trap potentials,
this specific axis provides an asymmetry for the energy functional and the distribution function \cite{NYI1}.
We will come back to this point when we present the numerical results in the next section.

The quasi-particle residue is defined as
\beq
Z_{J}
&=&
\left| \int_\br \phi_{0JJ_z}^{I*}(\br)\langle 0|S|b\rangle \Psi_{JJ_z}(\br) \right|^2
=|F_{0J}|^2 e^{-\sum_{n,l}'|f_{nl}|^2}.
\label{res1}
\eeq
It also quantifies the modification of the impurity
due to the interaction effects,
which is given by the overlap between the bare and interacting impurity states
with the angular momentum $(J, J_z)$.
Eq.~(\ref{res1}) shows that
the residue is factorized
into the ground state component of the impurity wave function $|F_{0J}|^2$
and a weight factor $e^{-\sum_{n,l}'|f_{nl}|^2}$ of the excited bosons,
while in the spatially uniform case it depends only on the latter.

\section{Numerical results and discussion}
In numerical calculation
we take K${}^{40}$ as the impurity immersed in medium bosons of Rb${}^{87}$;
the trap frequencies of the impurity and the medium bosons
and the condensed-boson number that we take are
\beq
&& \omega_I=200\, {\rm Hz}, \quad \omega_b=100\,  {\rm Hz}, \quad  N_0 = 10^4,
\label{ref1}
\eeq
throughout numerical calculations.
We treat the boson-impurity scattering length as a variable parameter,
but neglect the boson-boson scattering length in the present calculation.
In actual experiments of Bose polarons \cite{Jrgensen1,Hu1},
the trap potentials for impurity and medium-bosons are both axially symmetric,
and the boson-boson scattering length is usually set to be a small positive number
to stabilize the boson sector.
In the present theoretical study of the idealized system,
the zero-point energy in the trap supports and stabilizes the system,
and the present system of the negligible boson-boson scattering length
can be potentially realized in real experiments.

The average density of condensed bosons in the trap system is defined as
\beq
\bar{n} &=&N_0 \int_\br |\phi_0^b(\br)|^4
=N_0\left( \frac{m_b\omega_b}{2\pi}\right)^{\frac{3}{2}}.
\eeq
We also introduce the scale factors for momentum and energy:
\beq
k_{\rm ref}=\left( 6\pi^2\bar{n}\right)^{1/3},
\quad
E_{\rm ref}=\frac{k_{\rm ref}^2}{2m_b},
\eeq
as in the case of the uniform systems \cite{Hu1,Rath1,Li1}.

\subsection{The ground state energies for the states of small angular momentum}
In Fig.~\ref{fig1},
we show the scattering-length dependence of the ground state properties of polaron:
the energy shifts (\ref{shift1}),
the calculated values of impurity's variational parameter $F_{1J}$ in (\ref{ivs1}),
the total number of excited bosons (\ref{nbtot1}),
and quasi-particle residue (\ref{res1}),
for the small numbers of the total angular momenta ($J=0,1,2$).
\footnote{
In these calculations we have taken the approximation
to cut the summation in the interaction energy (\ref{int1}) up to $(n,l)=(30,10)$.
We have checked the approximation numerically
by raising the maximum values of $n$ and $l$ by $100\, \%$;
then the numerical results change within a few $\%$,
and the sum of $l$ shows a rapid convergence.
Also, since $H_{nlJ}^{0,c,1} \rightarrow 0$ as $n\rightarrow \infty$ or $l\rightarrow \infty$,
the series of $n,l$ summation in (\ref{int1}) drop faster than the order of $1/n$ ($1/l$)
for large number of $n$ ($l$),
which implies the series is a convergent one.
}
\begin{figure}[H]
  \begin{center}
    \begin{tabular}{cc}
 \resizebox{80mm}{!}{\includegraphics{energyshift1.eps}} &
 \resizebox{80mm}{!}{\includegraphics{F1.eps}} \\
 \resizebox{80mm}{!}{\includegraphics{Ntot1.eps}} &
 \resizebox{80mm}{!}{\includegraphics{res1.eps}} \\
    \end{tabular}
 \caption{The energy shift (a),
 the variational parameter $F_{1J}$ (b),
 the total excited-boson number (c), and
 the quasi-particle residue (d) for $J=0,1,2$,
 as functions of the inverse of scattering length
 with the parameters in (\ref{ref1}).
The inset in the panel (b) is for smaller scattering-length region.
The definitions of these quantities are given in
  (\ref{shift1}),  (\ref{ivs1}),  (\ref{nbtot1}),
and  (\ref{res1}), respectively.}
    \label{fig1}
  \end{center}
\end{figure}
The energy shift obtained here should be comparable
with the experimental result \cite{Hu1}
only for the case of small scattering lengths,
roughly of $1/ak_{\rm ref} <-2$;
it is because
the Bogoliubov approximation (\ref{FH1y}) employed here works
only if the number of excited bosons is less than or equal to the number of impurities,
(it is the unity in the present calculation),
and also the two-level approximation
in the impurity wave function (\ref{ivs1}) is valid
for the smaller values of variational solutions  ($F_{1J}\ll F_{0J}$),
and loses the validity when $F_{1J} \ge 1/\sqrt{2}\sim 0.7$.
\footnote{
Note that the variational solution of $F_{1J}$ is determined
mainly from the first two terms in (\ref{shift1}),
and takes smaller values
in the cases of  the heavier impurity masses or
of the larger trap frequencies than the present ones.
}
Also, the behavior of the residue implies that quasi-particle picture of the polaron
works for about $1/ak_{\rm ref} <-2$ as well.
In the case of the strong coupling regime and around the unitary limit,
i.e., $|1/ak_{\rm ref}|<1$,
we need to include effects of the two-to-two scattering processes
between impurity and excited boson
which were discarded in the Bogoliubov approximation;
they are responsible for the effective in-medium shift
in the unitary limit \cite{Shchadilova2}
and the in-medium few-body bound states \cite{Levinsen2,Shchadilova2}.

\subsection{Distributions of excited bosons}
In Fig.~\ref{fig2} we show the solutions of variational parameter $f_{nl}$
for $J=0,1,2$,
where we have set the impurity-boson scattering length by $a=-5.77 {\rm nm}$
corresponding at $1/ak_{\rm ref}= -9.95$.
The parameter $f_{nl}$ can be interpreted as the probability amplitude
of excited bosons with the quantum numbers $(n,l)$.
\begin{figure}[H]
  \begin{center}
    \begin{tabular}{c}
 \resizebox{90mm}{!}{\includegraphics{f_J0.eps}} \\
 \resizebox{90mm}{!}{\includegraphics{f_J1.eps}} \\
 \resizebox{90mm}{!}{\includegraphics{f_J2.eps}}
    \end{tabular}
 \caption{The variational parameter of excited bosons (\ref{fnl1}) for $J=0,1,2$
and their angular-momentum $l$-dependences.
A set of parameters is given in (\ref{ref1}), and $1/ak_{\rm ref}= -9.95$. }
    \label{fig2}
  \end{center}
\end{figure}
These figures implies that,
for each principal quantum number $n$,
the peak positions of $f_{nl}$ for the quantum number $l$ move to the right
as the total angular momentum $J$ is increased.
This is due to the attractive density-density-type interaction
between impurity and bosons,
which cause the large overlap between their wave functions
to lower the interaction energy.
It can be shown more directly
in the real space distributions (Fig.~\ref{fig4}-\ref{fig6}).

In Fig.~\ref{fig3},
we also show the quantum-number distributions of the excited bosons
$N_{nlm}^{J J_z}$ given by (\ref{Nb1}) for the states of $J=1,2$
as functions of the quantum number $m$ for $l=1,2,3$ and $n=0$,
with the same parameter set as in Fig.~\ref{fig2}.
\begin{figure}[H]
  \begin{center}
    \begin{tabular}{cc}
 \resizebox{90mm}{!}{\includegraphics{N_l1a.eps}} &
\resizebox{90mm}{!}{\includegraphics{N_l1d.eps}} \\
 \resizebox{90mm}{!}{\includegraphics{N_l2b.eps}} &
 \resizebox{90mm}{!}{\includegraphics{N_l2e.eps}} \\
 \resizebox{90mm}{!}{\includegraphics{N_l3c.eps}} &
 \resizebox{90mm}{!}{\includegraphics{N_l3f.eps}}
    \end{tabular}
 \caption{
The angular-momentum $(l,m)$-dependences of the excited-boson numbers
in (\ref{Nb1})  for $n=0$ and $l=1,2,3$.
Panels (a), (b), (c) in left columns
and (d), (e), (f) in  right columns
are for $J=1$ ($J_z=0,\pm1$) and
$J=2$ ($J_z=0,\pm1,\pm2$), respectively.
The parameter set is the same as in Fig.~2. }
    \label{fig3}
  \end{center}
\end{figure}
As expected from the angular-momentum conservation and no drag effect,
i.e., $\langle \hat{M}_z \rangle_{JJ_z}=0$,
in the present calculation,
all plots in the figures show that
the distributions for the quantum number $m$ are symmetric about $m=0$.
In order to understand the result,
let's suppose an impurity prepared in the state with a specific value of $J_z(=L_z)$,
which gives a specific direction in the space.
If the interaction could be turned off between the impurity and surrounding bosons,
the energy of the system should be still degenerate to the value of $J_z$.
However, the presence of the real interaction causes
the same number of virtual boson excitations
with the quantum number $-|m|$ and $|m|$
in order to gain the interaction energy by a maximal overlap with the impurity
(as shown in Fig.~4-6),
which leads to the vanishing $\langle \hat{M}_z \rangle_{JJ_z}$.

For a different value of the principal quantum number $n\neq 0$,
we have confirmed that
the excited-boson number distributions have the completely same shape as that of $n=0$
since distribution shapes are determined by the Clebsch-Gordan coefficients for
a given set of ($J,J_z,l,m$),
which are independent of $n$,
but their intensities decrease with increasing $n$.
The special case is for $J=J_z=0$,
where the factor $N_{nlm}^{00}$
determined from the Clebsch-Gordan coefficients has no $m$ dependence;
their numerical values for $l=1,2,3$ are
$N_{01m}^{00}=9.76\times10^{-4}$,
$N_{02m}^{00}=0.38\times10^{-4}$, and
$N_{03m}^{00}=0.024\times10^{-4}$, respectively.

Finally we show in Figs.~\ref{fig4}-\ref{fig6} the real space distributions (\ref{nbx1})
of excited bosons together
with impurity's probability density obtained from (\ref{ivs1})
for the same parameter set as that in Fig.~\ref{fig2}.
To generate the distributions of excited bosons shown in Fig.~\ref{fig4}-\ref{fig6},
we have evaluated (\ref{nbx1}) in the approximation
of taking quantum numbers up to $(n,l)=(5,5)$
in the summation,
since $f_{nl}$ of higher quantum numbers does not contribute so much
as the variational parameters (Fig.~\ref{fig2}).
\begin{figure}[H]
  \begin{center}
    \begin{tabular}{cc}
 \resizebox{70mm}{!}{\includegraphics{1-1.eps}} &
 \resizebox{70mm}{!}{\includegraphics{1-2.eps}}
    \end{tabular}
 \caption{ The $J=J_z =0$ contour plots of the impurity's probability density
 $|\Psi_{JJ_Z}(\br)|^2$ (left panel)
 and the real-space excited-boson distributions (right panel) defined
 in (\ref{nbx1}),
in the cross-section plane of $y=0$,
where the ordinate is the $z$ axis (the direction of the magnetic quantum number),
 and the abscissa is the $x$ axis.
Note that these plots have the rotational symmetry around the $z$ axis.
The units of axes are $\left( \omega_I m_I\right)^{-1/2}$ (left)
and $\left( \omega_b m_b\right)^{-1/2}$ (right),
and the heights of the contour lines (not shown explicitly) are normalized
by $\left( \omega_I m_I\right)^{3/2}$ (left) and $\left( \omega_b m_b\right)^{3/2}$ (right).
The parameter set is the same as in Fig.~2.
}
    \label{fig4}
  \end{center}
\end{figure}
\begin{figure}[H]
  \begin{center}
    \begin{tabular}{cc}
 \resizebox{70mm}{!}{\includegraphics{2-1.eps}} &
 \resizebox{70mm}{!}{\includegraphics{2-3.eps}} \\
 \resizebox{70mm}{!}{\includegraphics{2-2.eps}} &
 \resizebox{70mm}{!}{\includegraphics{2-4.eps}}
    \end{tabular}
 \caption{The $J=1$ contour plots of the impurity's probability density (left panel)
 and the real-space excited-boson distributions (right panel):
 the top and bottom panels are for $|J_z|=0, 1$.
For other explanations, see the caption in Fig.~\ref{fig4}.  }
    \label{fig5}
  \end{center}
\end{figure}
\begin{figure}[H]
  \begin{center}
    \begin{tabular}{cc}
 \resizebox{70mm}{!}{\includegraphics{3-1.eps}} &
 \resizebox{70mm}{!}{\includegraphics{3-4.eps}} \\
 \resizebox{70mm}{!}{\includegraphics{3-2.eps}} &
 \resizebox{70mm}{!}{\includegraphics{3-5.eps}} \\
 \resizebox{70mm}{!}{\includegraphics{3-3.eps}} &
 \resizebox{70mm}{!}{\includegraphics{3-6.eps}}
    \end{tabular}
 \caption{The $J=2$ contour plots of the impurity's probability density (left panel)
 and the real-space excited-boson distributions (right panel):
 the top, middle, and bottom panels are for $|J_z|=0, 1, 2$.
For other explanations, see the caption in Fig.~\ref{fig4}.}
    \label{fig6}
  \end{center}
\end{figure}
Comparing left and right figures for $J=0,1,2$,
we can observe that the attractive impurity-boson interaction
has the effect that causes overlaps in their distributions,
as discussed just above on the quantum-number distributions.
The impurity's probability is proportional to $|Y_{JJ_z}(\theta,\varphi)|^2$,
thus the figures clearly exhibit the $s$, $p$, and $d$ orbital shapes for $J=0,1,2$, respectively.
On the other hand,
boson's distributions are blurred
because they always include $l=0$ isotropic contributions
as shown in (\ref{nbx1}) with variationally-determined weight factor $|f_{nl}|^2$.

\section{Summary and outlook}
In this paper
we have investigated the ground-state properties of the impurity
interacting with medium-bosons in spherically symmetric trap potentials,
when the total angular momentum $(J, J_z)$ are given.
To this end we have developed a conditional variational method,
and obtained the ground-state energies, quasi-particle residue of polaron,
and the quantum-number and real spaces distributions of excited bosons
for the cases of total angular momenta $J=0,1,2$.
From theoretical consideration,
we have found that
the expectation value
$\langle \hat{J}^2\rangle_{JJ_z}=J(J+1)$ is shared
by the impurity and the excited bosons as
\beq
\langle \hat{L}^2\rangle_{JJ_z} &=& J(J+1)+\sum'_{n,l} l(l+1)|f_{nl}|^2, \nn
\langle \hat{M}^2\rangle_{JJ_z} &=& \sum'_{n,l} l(l+1)|f_{nl}|^2,
\nn
\langle 2 \hat{L}\cdot\hat{M}\rangle_{JJ_z} &=& -2\sum'_{n,l} l(l+1)|f_{nl}|^2,
\eeq
while that of the $z$-th component
$\langle \hat{J}_z\rangle_{JJ_z}=J_z$ comes from the impurity only:
\beq
\langle \hat{L}_z \rangle_{JJ_z} = J_z, \quad \langle \hat{M}_z \rangle_{JJ_z}=0,
\eeq
which implies no drag effect for the polaron in the spherically-symmetric trap potentials.
We have also made numerical calculations based on the variational method,
and,
as shown in Figs.~\ref{fig2}-\ref{fig6},
found that the excited bosons are distributed
so as to make a large overlap with impurity's probability density
in real and quantum-number spaces
because of the attractive impurity-boson interaction.

In the present study
the excited bosons do not move collectively by themselves \cite{Dalfovo1,Japha1}
since no boson-boson interaction is assumed,
so they are in purely quantum regime.
In most of recent experimental researches,
the Bose polarons are realized in the system of the repulsive boson-boson interactions
where the medium-bosons form a superfluid BEC.
In order to analyze such cases,
we need Bogoliubov-de-Gennes-type approaches \cite{Bogol1,Gennes1,Lewenstein4,Nakamura1,Nakamura2,Lampo1}
beyond the Bogoliubov approximation.
Such extensions of the present approach for the trapped polaron
including the boson-boson interactions
should give more detailed polaron's structures
such as a local depletion of BEC around impurity
as well as the excitation spectra of the bosonic sector.

Finally,
we comment a bit on the possibility of experimental observation of
the finite angular momentum states of the trapped Bose polaron discussed in this paper.
To our knowledge,
all experiments have been done with
axial-symmetric traps for both impurity and medium atoms,
and no angular momentum is given to the atoms in total.
To give some finite angular momentum to the system
in axial symmetric trap potentials,
we expect that the experimental methods of creating
a vortex state of the BEC
can be utilized \cite{Vor1,Vor2}:
rotating a very dilute impurity-atom gas
before switching on the interaction with medium bosons,
and then the whole system, as Bose polaron,
finally acquires some finite angular momentum.
Furthermore,
if the axial symmetric trap is deformed adiabatically to the spherical one,
there remains the state with a finite angular momentum,
the quantization axis of which should be the same with that of the original axial symmetry.

For the observation of the angular-momentum distribution of the impurity,
the photon absorption spectra for excitations for the states with different angular momenta
can be utilized,
or the indirect observation of the phase of the impurity's wave function,
which has been done for the vortex state of the BEC \cite{Vor1},
is also an interesting possibility.
At the moment,
we have no fixed idea how to give a definite amount of angular momentum,
but we think that a significant change in boson's distribution
can be observed with the methods as discussed here.
Also, in the observation of the excited bosons,
the photon absorption spectra mentioned above may work out for bosons as well.
In addition, we think that {\it in-situ} experiments may also work to get images of excited bosons \cite{Yan8,insitu7,insitu8,insitu9,insitu10,insitu11},
although it would be a challenge since the total excited-boson number per impurity is quite small.

Acknowledgments

We are grateful to Kei Iida, Junichi Takahashi, and Ryosuke Imai for useful discussions on excitation spectra of interacting bosons in inhomogeneous systems, and to Kota Yanase for comments on
angular momentum structure of excited nuclei.
E.~N. and  H.~Y. are supported by Grants-in-Aid for Scientific Research
through Grants No.~17K05445 and No.~18K03501, respectively,  provided by JSPS.

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Attorney, Banner & Witcoff, Ltd.; Adjunct Professor of Law
Robert F. Altherr, Jr.
B.S.Ch.E, University of Cincinnati; J.D., Capital University; LL.M., Wayne State University; LL.M., George Washington
Professor Altherr is a partner at Banner & Witcoff, Ltd. where he has specialized in intellectual property litigation involving diverse technologies and successfully represented both plaintiffs and defendants as lead trial and appellate counsel. While in private practice, he was appointed Special Attorney to the U.S. Attorney General and served as trial counsel defending the U.S. Government against claims of patent infringement. Prior to joining Banner & Witcoff in 1992, he litigated intellectual property suits as a trial attorney for the U.S. Department of Justice Commercial Litigation Branch, and as a civilian and military attorney for the Department of the Army where he also served as a federal prosecutor and contracts attorney. He is a drafter of the American Bar Association Amicus Curiae brief to the U.S. Supreme Court in eBay v. MercExchange addressing the standard for granting permanent injunctions against patent infringers. Professor Altherr is among the five percent of the attorneys in the Washington DC area selected to the 2011, 2012, 2013 and 2014 Washington DC Super Lawyers lists, for his work in intellectual property litigation.
Back To Faculty Listing
Patent Trial Practice

redpajama_set_name: RedPajamaCommonCrawl


text:
Cruising destination Ao Nang to Krabi Boat Lagoon
November 24, 2015 / Lifestyle & Travel, News
Ao Nang, Krabi 008deg 01.55min N 098deg 49.10min E
This anchorage is suitable in the north east season only and care should be taken on approach into the bay to avoid the numerous bamboo fishing stakes in the water which are fixed to the sea bed. Anchor on the south east side of the bay in front of the main town making sure you will have at least 2 metres under the keel at low tide. The main town has lots of restaurants, basic provisions and a busy night life in the many bars. You can catch a local taxi into Krabi town where there is an excellent morning fresh market and a large Tesco supermarket. Beware that the beach dries out a long way at low tide so check the tide tables before going ashore and be sure to anchor your dinghy securely in case it is carried away on a rising tide. The next small bay along to the south east is only accessible by sea and is dominated by a new 5 star resort.
Rai Le Beach, Krabi 008deg 00.65min N 098deg 49.85min E
This spectacular anchorage is suitable only in the northeast season. Anchor well offshore on a sandy bottom making sure you will have at least 2 metres under the keel at low tide. The beach is separated into two parts by a rocky outcrop. The north west, left hand beach is fairly rocky and caters to backpackers and budget travellers with several bungalow and cheap restaurant developments. The south east, right hand beach has more upmarket resorts like the Rai Le Bay resort and spa. Very few of the restaurants on the beach serve alcohol, as they are owned by Muslim families but you will find a few bars and you can bring alcohol into most restaurants if you ask first. There is an ATM machine behind the Rai Le Bay resort and small supermarket with basic provisions along the small street next to the coffee shop. The only access to these beaches is by longtail from Ao Nang beach as there is no road. It is possible to hire a longtail to Ao Nang or even around the headland to the east and up the river to Krabi town to buy provisions in the excellent fresh market and supermarkets. The trip takes about half an hour and costs around 1200 baht for the whole longtail for a return trip. To the south east of this bay is the beautiful Tham Phra Nang beach. Pass well to the south west of Ko Nang and anchor a good distance of the pretty reef that runs along the entire beachfront and out to Ko Nang. Around the headland to the East is Ao Nam Mao or drunken water bay. As the name suggests this bay can be bumpy in any conditions and is not as pretty as the other two beaches although you can find shelter here in the south west season. The dive boat moorings are private and also extremely long so give them plenty of swinging room. Be careful of the large muddy sand bank that fills the bay and anchor with no less than 4 metres under the keel at low tide. You will need at least 1.5 metres on the tide tables to cross this mud bank in the dinghy.
Ko Dam Hok North 007deg 58.60min N 098deg 48.85min E
Anchor to the east of the small island in 10 metres. There is a restaurant and bungalow development ashore which is often frequented by a large troop of tame monkeys. There is generally good shelter in the south west season but if there is a ground swell you can anchor a little further to the east of the beach but beware of the reef which runs all the way to Ko Dam Khwan. Not suitable overnight in the north east season. Do not attempt to pass between Ko Dam Hok and Ko Dam Khwan in the yacht as the islands are joined by a reef. This reef extends out for some distance with several isolated coral heads so avoid hugging the coast when sailing down the east side of the islands. Take great care to not drop your anchor in coral when anchoring around these islands.
Ko Dam Khwan North 007deg 57.65min N 098deg 48.40min E
Anchor in no less than 12 metres to the west of the small island. A shallow reef extends from Ko Dam Khwan to Ko Dam Hok and is great for snorkelling. Do not anchor here in the south west season but it is sheltered in the north east.
Ko Dam Khwan East 007deg 57.40min N 098deg 48.80min E
Anchor to the east of the island in 10 to 12 metres on sand but beware of large coral heads and make a snorkelling inspection once settled. This is a good spot in the south west season but is not suitable overnight in the north east season. The strange rock tower at the southern end of the island is known as "Chicken head". Just around the corner to the west from Chicken head there is a small sandy bay with room for just a few yachts. There are no facilities at all but this makes a wonderfully quiet overnight stop in the north east season.
Krabi Boat lagoon marina entrance waypoint 007deg 57.50min N 098deg 54.00min E
Like its sister marina on Phuket this marina has been built up a shallow river and so it is very protected but only accessible for a few hours either side of high tide. From the entrance waypoint use the following waypoints 007 57.90 N 098 55.20 E, 007 58.40 N 098 55.70 E, 007 59.10 N 098 55.90 E, 007 59.70 N 098 56.50 E, 008 00.10 N 098 57.30 E, 008 00.80 N 098 57.40 E. The marina entrance is then visible on your starboard side. Call on channel 67 to arrange a berth. This is a great place to refill your water and diesel and you can take a 30 minute taxi to the Tesco Lotus supermarket near Krabi town to reprovision.
Home » News » Cruising destination Ao Nang to Krabi Boat Lagoon


So in a previous post I mentioned my final decision to no longer compete in the sport of bikini or figure bodybuilding.
The following is my "burial" message or eulogy to the past four years of my involvement in the sport and everything that came with it, good and bad.
Now I'm not forsaking fitness or a healthy lifestyle as bodybuilding and competition is honestly neither true fitness or a truly healthy lifestyle. It's actually discrete yo-yo dieting and psychological self-destruction at it's best, because the weight loss is temporary and at unhealthy lows that the body cannot sustain long term.
But as a believer and follower of Christ, I've realized how one can also develop an unbalanced obsession in the pursuit of rising to the top of the sport. The attitudes that develop, the relationships you forsake, and the ungodly attention that it attracts, especially as a woman (it strips your modesty), and certain men prey on that concept.
So in this "Beautiful Eulogy" or Ecclesiastes 12 (quoted at the end of this post) I'm declaring my stance on my current perspective of the priorities, events and activities of my life yesterday and today, tearing down, and forsaking the worthless things and ideas that I once clinged so strongly to, in the pursuit of so called "fame" as one of my former bodybuilding coaches put it; as he'd always say, whenever I felt like giving up on competing "that's the price you pay for fame." A worthless statement in the eyes of God, when your life becomes centered around the pursuit of self satisfying glory and fame in the eyes of people.
So, needless to say that "fleeting fame" is worthlessness, in comparison to the Crown of Life God has promised to those who love Him.
So what are we clinging to? Notoriety, fame, trophies, opinions of people, validation, etc.? Today I'm tearing down those idols and throwing them in the fire! They are worthless in comparison to the Crown of Life and all that God has promised me in this life and the next.
1-2 Honor and enjoy your Creator while you're still young, Before the years take their toll and your vigor wanes, Before your vision dims and the world blurs. And the winter years keep you close to the fire.
3-5 In old age, your body no longer serves you so well. Muscles slacken, grip weakens, joints stiffen. The shades are pulled down on the world. You can't come and go at will. Things grind to a halt. The hum of the household fades away. You are wakened now by bird-song. Hikes to the mountains are a thing of the past. Even a stroll down the road has its terrors. Your hair turns apple-blossom white, Adorning a fragile and impotent matchstick body. Yes, you're well on your way to eternal rest, While your friends make plans for your funeral.
6-7 Life, lovely while it lasts, is soon over. Life as we know it, precious and beautiful, ends. The body is put back in the same ground it came from. The spirit returns to God, who first breathed it.
8 It's all smoke, nothing but smoke. The Teacher says that everything's smoke.
9-10 Besides being wise himself, the Teacher also taught others knowledge. He weighed, examined, and arranged many proverbs. The Teacher did his best to find the right words and write the plain truth.
11 The words of the wise prod us to live well. They're like nails hammered home, holding life together. They are given by God, the one Shepherd.
Fear God. Do what he tells you.
14 And that's it. Eventually God will bring everything that we do out into the open and judge it according to its hidden intent, whether it's good or evil.
The context of Ecc.12 describes my Beautiful Eulogy to yesterday and my Beautiful Beginning for the years to come.
The winter years keep you close to the fire.
When your body no longer serves you so well?
Certain things grind to a halt.
People and places fade away.
You are awakened now by God's-song.
Fleeting ambitions are a thing of the past.
For even a stroll down the road will have its terrors.
And Life, lovely while it lasts, is soon over.
Is put back in the same ground it came from.
And the Spirit returns to God, who first breathed it.


text:
Encyclopedia VA
Topics Books & Literature Civics Cultures & Communities History Science
Programs Encyclopedia VA Fellowships Virginia African American Cultural Resources Task Force
Encyclopedia VA • 77 results found
The Cotton Gin of the Midwest
Many are familiar with the legacy of the cotton gin, the first machine to separate cotton seeds from cotton fiber, patented by Ely Whitney in 1794. The cotton gin made the widespread cultivation of short-staple cotton profitable just as the removal of Indians from vast swaths of the southeast opened up new farming territory for […]
Encyclopedia VA | Cultures & Communities
Virginia Tribal Communities Tell Their Own Stories
In Encyclopedia Virginia
The question of how Indigenous stories are told—and by whom—is part of a long-overdue reckoning with the mainstream historical narrative in VA…
Stories | 20th Century, Native American, Precolonial
The Missed Opportunity of the Emancipation Monument
Our new entry on Archer Alexander, the formerly enslaved man who served as the model for the Emancipation Monument dedicated on the eleventh anniversary of President Abraham Lincoln's assassination, touches on the controversy the monument generated, both when it was dedicated in 1876 and more recently when a replica of the monument was removed from […]
Liberty is Sweet: A Conversation with Woody Holton
Encyclopedia Virginia is thrilled to welcome Woody Holton for a virtual conversation about his much-anticipated new book Liberty is Sweet: The Hidden History of the American Revolution on Friday, October 29, at 12 PM. In this sweeping reassessment of the American Revolution, Holton shows how the founders were influenced by overlooked Americans—women, Native Americans, African Americans, and religious […]
Encyclopedia VA | Books & Literature, History
Join Encyclopedia Virginia editor Patricia Miller in conversation with Woody Holton about his much-anticipated new book Liberty is Sweet: The Hidden History of the American Revolution. In this sweeping reassessment of the American …
October 29, 2021 | 12PM – 1PM | Zoom
Talks & Discussions | Black/African American, Native American, Revolutionary War
Emancipation and Freedom Monument
On September 22, the Virginia Dr. Martin Luther King, Jr. Memorial Commission unveiled the latest addition to Virginia's commemorative landscape: the Emancipation and Freedom Monument on Brown's Island in Richmond. The monument, which features two bronze statues representing a man and a woman and an infant newly freed from slavery, is dedicated to the contributions of […]
The Economics of Slavery
Three new entries in EV shed light on some surprising aspects of the economics of slavery. Doug Sanford illuminates a little-understood aspect of slavery: the Hiring Out of the Enslaved. As he notes, "While less well-known than other facets of institutional slavery, hiring out of the enslaved was a common and long-standing arrangement throughout the […]
Encyclopedia VA | History
Virginia Estelle Randolph
Virginia Humanities grantee Elvatrice Belsches' research into the life of this educator, community leader, and social activist is changing the historical landscape.
Stories | 20th Century, Black/African American, Education
How Washington Outflanked Smallpox
With full FDA approval of the Pfizer COVID-19 vaccine, the Pentagon has announced that all active-duty troops will be required to receive the vaccine as soon as possible—a mandate to protect the health and readiness of the military that stretches back to the very first commander-in-chief, George Washington. His decision to inoculate the Continental Army against smallpox […]
Stories | Health, Revolutionary War
The Woman Who Refused to "Obey"
Sarah Harrison Blair offers a tantalizing glimpse of unlikely female agency in colonial Virginia. She was the daughter of wealthy Surry County tobacco planter, trader, and land speculator Benjamin Harrison II. By the time of Sarah's birth in 1670, the Harrisons were already a political dynasty in the making. Her father served in the House of […]
"Like Reaching for the Moon"
Before Greta Thunberg, before Emma González, before Malala Yousafzai, there was Barbara Johns. Johns kickstarted America's student-led movement for civil rights in education in 1951, when she launched a walkout of her fellow students at the all-Black Robert Russa Moton High School in Farmville. She was sixteen at the time. She planned the walkout to […]
Stories | Black/African American, Civil Rights, Education
The Ledger and the Chain with Joshua Rothman
On June 29, our Encyclopedia Virginia presented this conversation with Joshua Rothman, author of The Ledger and the Chain: How the Domestic Slave Trade Shaped America, and Patti Miller, editor …
Stories | Black/African American


text:
Brainin, Puricelli Earn Honorable Mentions
Springfield, Illinois – The Springfield Jr. Blues, proud member of the North American Hockey League (NAHL), are proud to announce that forwards Max Brainin (Buffalo Grove, IL) and Brandon Puricelli (St. Louis, MO) were named Honorable Mentions for the Midwest Division Star of the Week.
Brainin, who is in his second season here in the capital city, had two goals on the weekend and ended with a plus-1 rating. Brainin now has eight points on the season, four goals and four assists, which is good enough for third on the team. It was Brainin's second two-goal performance in as many weekends as he scored twice against Topeka as well.
Puricelli is in his third campaign in Springfield and had two goals and an assist on the weekend. Puricelli currently leads the Jr. Blues with six goals and his 10 points are second most on the team. He has gotten on the scoresheet in all but two games this season, the season opener against the Maryland Black Bears and against Topeka on 10/6.
Springfield has now had at least one player be mentioned for Star of the Week three out of four possible weeks. The Schmidling brothers (St. Joseph, MO) in week one, none in week two, Nick Techel (Wausau, WI) and Brainin in week three, and now Brainin and Puricelli.
Tagged: Brandon Puricelli, Max Brainin, springfield jr blues
Jr. Blues Fail to Break Magicians Spell
Springfield Jr. Blues Alumni Report Pt. 2
Brainin Commits to St. Norbert College
Puricelli, Jr. Blues Earn Divisional Awards
Jr. Blues Split in Minnesota, Will Host Two Games in Springfield
Jr. Blues, Jets Split Series to Finish Regular Season


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That's right, you can forget cutting up your expensive Clone Strike VRs; all it takes is a rubber band.
Something at my skill level - AWESOME!
I've never been a fan of dual-unique pieces, but this is one that gets my seal of approval (not that it's good for anything)!
Somewhere MacGyver is weeping tears of joy.
..or you could start playing dark side.


The following thoughts are adapted (primarily the preface) from a sermon that I preached from Grace's pulpit. Sometimes a sermon is crafted to deliberately provoke conversation; always such a sermon runs the risk of not adequately engaging the hearers and subsequently becoming one-sided. This may have been one of those sermons.
Whenever a minister officiates at a wedding, she acts as an agent of the state (having been granted this authority from the local municipality via the marriage license). Likewise the church, by hosting a wedding, complies with the legal system of marriage—whether through building rental, by extension via its minister, or in church members' attendance at the wedding, the church acts a witness to the legal binding of a bride and groom. As I see it, this involvement is essentially a conflict of the church-state boundary.
My argument to the church, and my challenge as a minister, is that the participation of the church and minister in this legal institution goes largely unexamined and unqualified. If marriage is as valued and sacred as many Christians claim, then the church should reclaim marriage as a religious covenant and extract marriage from its currently legal entanglement. If, on the other hand, marriage is to remain as a legal institution (with nearly 1,000 laws contingent upon husband-wife designations), then the church should excuse herself from her present church-state compromise and more appropriately speak to any concerns of the "values" of marriage (I would say legal injustices) as an outside voice.
So: here is the hard question for the church to consider amidst all the loveliness and sentimentality of hosting weddings, a question that bubbles to the surface every time I read a marriage license: What's so sacred about marriage?
Before you leap to an answer in your mind, let me elaborate on the question. What is it about getting married that prompts many engaged couples, who are otherwise nominal Christians, to go looking for a church? When couples plan weddings, what's the appeal of having a minister versus having a judge? And on the flip side, what's the interest of a church in deciding who can and who cannot marry within its walls (members, non-members)? What's the church's purpose in hosting a wedding? And—here's what I'm really getting at—what is it about marriage that is so sacred and so valuable to the Christian Church that she has weighed in on the legal (civic/secular) debate over gay marriage in recent years?
Marriage has not always been (and, I suggest, isn't now) an institution of the church. In the span of history, the coming together of two individuals to create a new family unit has typically been secular: either a private family matter or the business of government. For centuries, marriage was governed by local customs. In the Middle Ages, the Church actually had trouble becoming a voice of authority over marriage because marriage traditions were so localized and secular. The Roman Catholic Church finally established marriage as a sacrament in the 13th century, only to have major leaders of the Reformation —including Martin Luther—argue that marriage was a civil contract and not the business of the church (George Chauncey, Why Marriage, 79-80).
Simply said, the debate over who sets the rules for marriage is not new. Whenever gay marriage is debated today, there is an argument in the undercurrent of that debate: an argument of church versus government, sacred versus civil. Churches and church people are lobbying the government for the authority to say who can and who can't marry. And, in Pennsylvania, the government is now making rules for the churches over which ministers can and cannot officiate at weddings.
So let's take the debate in society and lay the questions on the table: What part of marriage—for us as a church, for you as a person of faith–is really important? What aspect of marriage (if any) is worth defending, in your opinion? What's the crux of marriage that we should really value and encourage?
Let's look at an Old Testament reading. It's a familiar story: Moses the shepherd and former prince of Egypt is having an average day watching Jethro's flock of sheep. Suddenly, he spies a burning bush: a green, growing bush that is red and yellow with dancing flames of fire…and yet it is not dying or crumpling into a pile of ashes! Moses says to himself, "I must turn aside and look at this great sight" (Exodus 3:3). Lo and behold, the voice of the LORD speaks to him from the bush, "I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob. I have observed the misery of my people who are in Egypt, and I have heard their cry" (Exodus 3:6-7).
I have already turned aside and bent my ear to hear my people's cries.
is wrapped up in loving and rescuing my people.
in order to care for them as well.
because I have turned aside for all people.
If there is a biblical mandate for marriage, if there is a God-given example for committed relationships, it is not Adam & Eve or Abraham & Sarah & Hagar, it is not Jacob & Rachel & Leah or Hosea & Gomer. If there is a biblical mandate for marriage, it is God's own actions. It is "I AM WHO I AM" turning aside from the business of up-in-the-sky holiness to hear the Hebrew people crying in anguish in Egypt. It is God foolishly loving—over and over and over again—stubborn and selfish people such as us.
There are many examples in the world around us of people turning aside from what they are doing to focus on and care for others: parents turning aside from their work to bandage scraped knees or to rock away their children's pain; friends turning aside from their own schedules to spend time with one another when a tragedy occurs; volunteers turning aside from lucrative jobs to assist total strangers through non-profit organizations. Turning aside for one another—loving our neighbors—is beautiful and certainly holy.
that kind of loving in imitation of God loving is a miracle.
In the sacred/secular battle over marriage, if we—this church or any church—really want to weigh in on the government's rules about marriage, then we should check our bible stories. The example of marriage that is given to us, the story of covenant that is told repeatedly from cover to cover, is not the story of one man and one woman…but the story of God's relationship with humanity. And there's no gender in that relationship! God speaks to Abraham and to Mary both; God appears in fire to the Israelites and to the early Christian Church alike; God turns aside to hear the cries of both Hagar and Job.
If there is something about marriage that is sacred, something about partnership that is holy, it is the faithful repetition of turning aside to see and to hear one's partner, in imitation of God turning aside to care for those most in need, in imitation of Moses turning aside to bask in the glow of God.
None of that is written on the marriage certificate that I sign after officiating a wedding. None of that is written into our state or federal laws. The government does not define what the quality of a married relationship should be or how partners should treat one another….only who can enter a marriage and who cannot.
So I return to the question: What's so sacred about marriage? What's so important about keeping marriage between a man and a woman that many Christian organizations and congregations are spending millions to lobby for legal restrictions? How does gay marriage threaten straight marriage? As an Open & Affirming congregation, can we risk being silent on the issue of marriage—on the meaning of marriage—or do we have something different to offer in this raging debate?
Adam & Eve don't particularly inspire me as the exemplary married couple in the bible; I want to know what their relationship was really like when the honeymoon ended and they were forced to leave the Garden of Eden. Abraham, that great father of monotheism, just wanted a son, and he would take in or throw out whichever woman (Hagar or Sarah) who was most helpful to him or most in his way. I could go on with biblical examples of "married" couples!
and whose constant presence surprises us still.
We are amazed and thankful for the signs of your love among us….
for the tenderness of parents caring for children.
to behold the fire of your presence.
Do not stop calling us, we pray, even if we are slow to listen and slow to look.
We are grateful for people and events that reflect your love for life.
make us bold to break through the limitations and restrictions that have been placed on the gay & lesbian community.
if not for your grace and your example in first loving us.
Blessed be your name, O God, for you do not leave us alone.
Pour out your Spirit on us, full of strength and courage for loving one another.


text:
U.S. President Joe Biden, center, signs H.R. 3684, the Infrastructure Investment and Jobs Act, during a ceremony on the South Lawn of the White House in Washington, D.C., U.S., on Monday, Nov. 15, 2021. Biden has said the $550 billion in fresh infrastructure spending he's signing into law will help ease inflation that's raising consumer alarm and weighing on his approval rating. Photographer: Stefani Reynolds/Bloomberg via Getty Images
GOP Rep. Gary Palmer Takes Credit For Infrastructure Bill He Voted Against
by Brandon GageNovember 20, 2021, 11:25 pmNovember 20, 2021, 11:25 pmNovember 20, 2021, 11:25 pmDaily Digest, News, News Feed >
Earlier this year, as Congress debated President Joe Biden's landmark infrastructure legislation, Chairman of the House Budget Committee John Yarmuth (D-Kentucky) predicted that Republicans would vote against the final bill and then take credit for the benefits it provides to their constituents.
GOP lawmakers did precisely that after former President Barack Obama signed the American Recovery and Reinvestment Act and the Patient Protection and Affordable Care Act (a.k.a. Obamacare) into law in 2009 and after Biden inked his signature onto the American Rescue Plan earlier this year – all of which were necessary to stabilize the collapsing American economy.
"What we are all concerned about on our side is that the Republicans are all going to vote against this, and then they're going to show up at every ribbon cutting, and at every project funded out of this bill, and they're going to pump up their chests and take credit for all of these great benefits that are coming to their citizens," Yarmuth said.
Sure enough, after the Infrastructure Investment and Jobs Act was signed into law by Biden on Monday, one House Republican proved Yarmuth prophetic.
Cue House Freedom Caucus member Rep. Gary Palmer (R-Alabama) – who voted against the bipartisan proposal even though it contained a provision he personally authored and derided it as containing "hundreds of billions of dollars on a Green New Deal wish list and programs under the guise of human infrastructure that simply expand government control of our lives" – issued a statement not long after the president signed it into law in which Palmer boasted about how it will help residents of Alabama.
"The newly passed infrastructure bill, signed into law by President Biden, includes legislation introduced by Congressman Gary Palmer (AL-06) and Congressman David Trone (MD-06) that will provide critical funding to help complete the Northern Beltline. The provision in the infrastructure bill was inserted from the Finish the Appalachian Development Highway System (ADHS) Act and would provide $1.25 billion for construction of incomplete sections of the ADHS. More specifically, Alabama will receive $369 million over the next five years for construction of the Northern Beltline, Alabama's only ADHS project that has yet to be completed," Palmer's congressional website states.
"Funding the Northern Beltline has consistently been one of my top priorities. Birmingham is currently one of the largest metropolitan areas in the country without a complete beltline around it. Completing the Northern Beltline will benefit the entire region and enhance economic development and employment opportunities," Palmer said in a statement of his own.
"The Appalachian Regional Commission has noted the completion will have an annual economic impact exceeding $2 billion in 10 years and has the potential to create 14,000 jobs," Palmer concluded. "This is the opportunity we have been working for as a region and a state. Now is the time for us to take advantage of it and complete the work by finishing the Northern Beltline and building a better future for the Birmingham metro area and central Alabama."
Again, Palmer voted against it.
American Rescue Plan Barack Obama Bipartisan Infrastructure Bill david trone gary palmer Green New Deal House Freedom Caucus Joe Biden john yarmuth Obamacare
Article by Brandon Gage
Treasury Secretary Warns Congress U.S. Will Soon Hit Debt Ceiling


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After restart it will finish installing the driver. Hi Coderiu, Got the file extracted OK. Borrowedwifi Win7 2 posts. Advertising seems to be blocked by your browser. I will try again at the week-end. So the text should start with xMZ and the rest of stuff. We will need to own the file and give ourselves the full control to edit it.
A lot of older laptops inetl this video chipset. The only difference from the standard VGA controller is the brightness control. If your account does not have admin rights you might be prompted to enter username and password. My way of disabling the adapter is to edit the driver file for it "c: Similar help and support threads.
Intel also makes motherboard chipsets, network interface controllers and integrated circuits, flash memory, graphics chips, embedded vm and other devices related to communications and computing. After restart it will finish installing the driver. Great post as this describes the problem I am having.
Find More Posts by coderiu. F Intek will be a confirmation message so type 'y'. I read a lot of tutorials on the net and nothing worked for me.
Remove the first character you put there which was x. Intel is one of the world's largest and highest valued semiconductor chip makers, based on revenue. The point is to make the file vgapnp. How do I get the driver only?
I was able to point the card at the Win2k folder 8255 I got a message saying that I already had the latest drivers I did un-install the drivers first The PC is a Panasonic CF Toughbook and before I upgraded all of the hardware passed the Win 7 check. If it does not prompt click anywhere and wait for message.
The steps above are the key. I was so pleased with the results, that I immediately stopped and created this account on this forum for the express purpose of thanking you for coming up with such a wonderfully tricky and effective fix.
Your file will be usuable again: During this inte, Intel became the dominant supplier of microprocessors for PCs, and was known for aggressive and anti-competitive tactics in defense of its market position, particularly against Advanced Micro Devices AMDas well as a struggle with Microsoft for control over the direction of the PC industry.
Try again and let me know. Intel Corporation Embedded Graphics Controller. Find More Posts by audrinabo. Originally Posted by Borrowedwifi A thousand and one thanks to coderiu!
Drivers video drivers intel Are there any drivers for this chipset to have at least the screensavers work in Windows 7? Do not worry as this is still installing the driver. Hope there would 882855 a better driver for this. Also, check the display driver you just installed [Make sure you downloaded 2 in my post because the problem should be fixed in it] to see if it was installed correctly or not.
And I tried "painfully" to recover the backup, but I always get the blue screen!!! It is the inhel of the x86 series of microprocessors, the processors found in most personal computers.
Click 'Start' and in 'Search' type 'cmd'. If you are paranoid as 88255 and want original permissions restored, I will show you how to restore the permissions to the previous step later on in this tutorial. Point to the driver directory that you downloaded and extracted win I also tried DriverMax.


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Michael Glover, Scott Machado Lead Iona Past Rider, 91-71
College Hoops Player of the Day for Friday, January 20, 2012
Until tournament time comes around in March, Friday nights are reserved for smaller schools and that's where we found the Iona Gaels, playing in one of just five NCAA games on the slate.
The Gaels, seeking a return to the NCAA tournament, a place they haven't been since 2006, when they lost to LSU in first round, 80-64, know they'll likely have to capture the MAAC tournament to receive the automatic bid.
On Friday, they took a step in the right direction, with a 91-71 win over Rider, the team they beat in last season's conference semi-final, only to lose the final to St. Peter's, 62-57, landing them a berth in the Collegeinsider.com tourney.
Leading the charge for Iona was their top scorer and rebounder, senior forward, Michael Glover, who had another superior effort, hitting 8 of 9 shots from the field for a game high 21 points, to go with nine boards and three assists.
Scott Machado, the senior guard who leads the nation in assists (10.6 per game), chipped in with 11 points, nine boards and 15 assists.
The combination of these two leaders with assorted role players has the Gaels atop the MAAC, with a 7-1 record and a 15-4 mark overall, putting them in position to earn a spot at the Big Dance come March.
One part of the Gaels' strategy has been to play a ton of road games. In December, the Gaels spent the entire month on the road, playing eight straight contests in opponents' facilities. They've already played 10 away games and will embark upon a three-game road trip to Sienna, Fairfield and St. Peter's, last year's NCAA entrant, beginning Monday, January 23.
Labels: Iona Gaels, Michael Glover, Scott Machado


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"I've missed out on a good chunk of money, but I'm happy and that's all that matters." - RTÉ Radio 1 Highlights
"I've missed out on a good chunk of money, but I'm happy and that's all that matters."
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Love Island's Greg O'Shea"I've missed out on a good chunk of money, but I'm happy and that's all that matters."
As heard on The Ray D'Arcy Show January 13, 2020
Last year's Love Island winner Greg O'Shea told Ray D'Arcy this afternoon that he doesn't know when the next Irish bombshell will be landing on the Island, but he did say he thinks they are on the hunt for Irish candidates, judging by the amount of people who have got in touch with him for advice on how to clinch the gig:
"A lot of people have got on to me: 'Any tips for the interviews?' and stuff like that, so. They're definitely fishing around Ireland for someone else."
Greg says he tells anyone who asks there's no point trying too hard, they either want you for you, or they don't:
"It sounds clichéd, but I was literally like, there's nothing you can do, you just have to be yourself. They're picking you because of who you are. So don't try to be something you're not."
He was prepared to spill at least one trade secret from behind the scenes at the Love Island shoot, but it's probably not going to come as a surprise to anyone to hear that "casual" conversations on set are largely staged by the directors of the show:
"Do you know when you see them in certain groups and they're lying on the bed having a chat, that's like, they're told to go have those chats. Like you're not told what to say, but like you're told, 'go over there and ask her about her past boyfriends' or what you feel or what you look for in a person. That's not just naturally happening."
In this year's competition, Greg has taken a shine to both Conor and Callum and thinks they are doing well so far, but he says it's much too early days to call a potential winner. Ray suggested to Greg that he didn't exploit his own win the way he could have:
"I definitely missed out on the whole celebrity lifestyle. I could have gone to London and tried to make things work with Amber and lived in that kind of world but it didn't feel right for me."
Greg says he's put so much work into his Ireland Rugby Sevens career and he was keen to get back training if they are to have any hope of making the Olympics. He told Ray today they've a few more hurdles to get through before battling it out with France and Samoa for the final Olympic spot.
You can hear Greg's full interview with Ray D'Arcy about his views on Love Island and his ongoing Rugby Sevens career here.
Ruth Kennedy
"I was taught that you have to stand up even when people say sit down. You have to speak, even when people say be quiet".
Greg O'SheaLove IslandThe Ray D'Arcy Show


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İki dakika – singiel tureckich piosenkarzy İzel Çeliköz, Reyhan Karacy i Cana Uğurluera wydany w 1991 roku. Utwór został napisany przez Aysel Gürel i Şevketa Uğurluera.

W 1991 roku utwór reprezentował Turcję w 36. Konkursie Piosenki Eurowizji dzięki wygraniu w marcu finału krajowych eliminacji po zdobyciu największego poparcia jurorów. 9 maja numer został zaprezentowany przez trio w finale widowiska organizowanego w Rzymie i zajął ostatecznie dwunaste miejsce z 44 punktami na koncie.

Oprócz tureckojęzycznej wersji singla, trio nagrało utwór także w języku angielskim.

Lista utworów
CD single
 "İki dakika" (English Version)
 "İki dakika" (Turkish Version)

Przypisy

Linki zewnętrzne
 Okładka
Single w Konkursie Piosenki Eurowizji
Single wydane w roku 1991

redpajama_set_name: RedPajamaWikipedia


Unless you move frequently, you are likely to experience a major roofing project at some point in your life. A re-roofing, a major restoration, or a complete installation from scratch might become necessary at some point and you will know that it's time for some major work when you are constantly needing repairs, when your roof has severely deteriorated visually, or when your roof has simply surpassed its expected lifespan.
Whenever you are investing in a major roofing project, it's obviously important that things get done correctly. However, it's also important that you aren't paying for more than you need. If you suspect that it's time to replace your roof, don't hesitate to call your local contractors.
Unless you are an expert, it can be difficult to determine exactly what your roof needs. When you get a contractor out to your property, they will be able to evaluate the condition of your roof and determine whether or not you can get by with a repair or if something more comprehensive will be necessary to keep your home safe. They will go over all of your options and restore your roof in the most cost-effective way possible.
Your contractors restore both the appearance and performance of your roofing system so that you can continue without a major replacement for many more years.
When your roof is a little bit further along in its lifetime, you may need a complete re-roofing, which involves the total replacement of all of your roofing tiles.
Your roofers will know when a restoration stops being cost-effective and when a re-roofing is the most appropriate way to go and they will have a variety of materials and colours for you to choose from for your new roof. They will provide a comprehensive service that covers everything from the initial removal of the old materials to the installation of the new materials and they will explain the advantages to each of the materials they supply.
Many roofing contractors can also advise on roofing insulation and help you keep your roof more insulated and reduce your power bills.
There are more options in roofing materials now than there were before so if you haven't replaced your roof in decades, you will be able to upgrade to something more reliable and energy-efficient. Roofs are also available in a variety of colours so you can find the perfect colour to match your home.
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Thermal coating problems land 43 US F-35 stealth fighters
By Boyko Nikolov	 On Jul 16, 2021
WASHINGTON, BM – The fifth-generation stealth fighter and the American flagship for technological progress in military aircraft construction, the F-35, continues to be accompanied by problems, it became clear after hearing Air Force Lt. Gen. Eric Fick before the House Subcommittee on Tactical Air and Land Forces of Congress. According to the latest data released to the public, 15% of the US F-35 fleet cannot fly.
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The reason, as it turns out, is a problem with the thermal coating of the turbine rotor blades, which, contrary to expectations and the expected service life, wear out too quickly. A congressional hearing revealed that 43 American F-35 stealth fighters could not fly for this very reason. This is 15% of the entire US F-35 fleet, which by May 8, 2021, numbered 283 F-35A. Three other fighters from NATO member states (not mentioned which ed.) also have this problem.
This problem is not an isolated case, but part of a rather large list that accompanies the development of the F-35 stealth fighter. In 2019, we published a report from the Pentagon, which lists approximately 870 problems related to the operation of the fighter, 13 of which are in the red zone – urgent to resolve.
These include problems with information transfer, spare parts, incorrect computer data in cold weather, bubbles and cracks on the fighter's hull, frequent display of "green light" in the pilot's helmet, problems with sensors and night vision sights, etc.
Adam Smith said strong words in early March this year. According to him, Washington and the American taxpayer continue to spend on something that has "such low capabilities." Adam Smith is not an accidental man – he is the chairman of the US Armed Forces Committee. Some military experts have begun to compare the F-35 to the Ferrari – you only drive it on Sundays, but not every Sunday.
One of the first public reports of problems with the F-35 arrived in early 2019 from the Australian Air Force, which reported a problem with intergranular corrosion. Australia then hired the auditing and consulting firm KPMG to assess the problem that arose when using fighters in low humidity areas. But back in 2017, Australia was the first country to report various cracks in the body of the machine.
The situation with the problems associated with the development and operation of the F-35 in the US military has led to consideration of whether the Pentagon should continue the order of the F-35 from Lockheed Martin. On February 17, Charles Brown, commander of the US Air Force, told reporters again that Generation 4 ++ fighters could replace the planned production and commissioning of the F-35. There was a lot of talk about F-16 or F-15EX orders.
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Philippines President, Benigno Aquino III, has signed Proclamation No. 655, s. 2013, which lists the annual special and regular non-working public holidays in the Philippines for the calendar year 2014.
The aforementioned announcement of the special and regular non-working public holidays in the Philippines for the calendar year 2014, comes later than last year (01-Sep-2012), although historically, the list of non-working public holidays in the Philippines for the coming year is often released much later (01-Dec-2011, 20-Dec-2010, and 07-Jan-2009).
Proclamation 655 is a de facto reversal of most, if not all, of the "holidays economics" provisions of Republic Act 9492, "An Act Rationalizing the Celebration of National Holidays".
The new annual public holiday for the Chinese New Year, which had been promised 2 years ago (12-Jan-2011 and 03-Jan-2011) is included in the list of special and regular non-working public holidays in the Philippines for the calendar year 2014.