%Numero de paginas en dvi = 9
\documentclass{rmf-d}
\usepackage{nopageno,rmfbib,multicol,times,epsf,amsmath,amssymb,cite}
\usepackage[latin1]{inputenc}
\usepackage[]{caption2}
\usepackage{graphics}
%
%\numero{1}
%
\def\rmfcornisa{RESEARCH \hfill\rmf\ {\bf 64} (2018) 42--50
\hfill JANUARY-FEBRUARY 2018}
\newcommand{\ssc}{\scriptscriptstyle}
%
\newcommand{\nt}{\notag \\  & \times}
\newcommand{\na}{\notag \\  & }
\newcommand{\no}{\notag \\}
%
\def\rmfcintilla{{\it Rev.\ Mex.\ Fis.\/} {\bf 64} (2018) 42--50}
\clearpage \rmfcaptionstyle \pagestyle{myheadings}
\setcounter{page}{42}
\markboth{Shalini Lumb, Sonia Lumb and Vinod Prasad}
{Two-photon transitions in confined hydrogenic atoms}
\begin{document}
\title{Two-photon transitions in confined hydrogenic atoms
\vspace{-6pt}}
\author{Shalini Lumb$^a$, Sonia Lumb$^{b,*}$  and Vinod Prasad$^c$}
\address{$^a$Department of Physics, Maitreyi College, University of Delhi, Delhi-110021, India.
\\
$^b$Department of Physics and Electronics, Rajdhani College, University of Delhi, Delhi-110015, India.
\\
$^*$e-mail: sonia\_lumb@hotmail.com
\\
$^c$Department of Physics, Swami Shraddhanand College, University of Delhi, Delhi-110036, India.}
\maketitle
\recibido{5 September 2017}{27 October 2017
\vspace{-12pt}}
\begin{abstract}
Two-photon transitions from ground state to excited and ionized states are studied. The energy levels and radial matrix elements of an
impenetrable spherically confined hydrogenic atom embedded in plasma environment are
evaluated using accurate Bernstein-polynomial (B-polynomial) method. Transition
probability amplitudes, transparency frequencies and resonance enhancement
frequencies for various transitions, namely, $1s-2s$, $1s-3s$ and $1s-3d$  are evaluated for various values of confining Debye potential parameter. The effect of spherical confinement is studied and explained.
\end{abstract}
\keys{ Two-photon transitions; confinement; Debye plasma; B-polynomials. \vspace{-4pt}}
\pacs{32.30.-r;\ 32.70.-n;\ 32.80.-t \vspace{-4pt}}
\begin{multicols}{2}


\section{Introduction}
\label{intro}

Atomic and molecular systems confined in various plasma environments have
occupied an important place in theoretical and experimental research fields
\cite{Ichimaru,DeSi,ZhHo,YoJu,Sen,KuPrMo,Khordad,HaIkOnZa}. It is well-known that in interpretation of
various data associated with astrophysics, hot plasma etc., basic understanding
of atomic excitation and ionization is required since these processes taking
place in plasma environment provide important information about plasma. The
processes which have recently attracted much attention are laser-assisted
collisions in plasma~\cite{HoJu,BhSi,PaLiHo,MuSi} and response of confined atoms
to short and intense laser pulses \cite{LuLuPrPRA, LuLuPrIJP}. The plasma
environment considered here is Debye plasma where Debye screening length,\linebreak $\lambda_D = \sqrt{k_B T/(4\pi e^2 n)}$, being a function of the plasma temperature $T$ and its density $n$, plays an important role. Various sets of plasma conditions involved in real plasma environments can be simulated for one value of $\lambda_D$ as one can evaluate the plasma temperature for a particular number density and vice-versa by fixing $\lambda_D$ \cite{LuLuPrPLA}.

The screening of interaction potential between the nucleus and the electrons
moving in atomic orbitals plays an important role in a variety of processes. In
case of atoms and molecules confined under various conditions, the potential is
modelled by a so-called Debye-H\"uckel potential \cite{KaHo}. The other form of
screened potential called the exponential cosine screened coulomb potential, has
long been used to describe an ionized impurity inside a semiconductor
heterostructure~\cite{PaHoPRA78,PaHoPRA79,LuLuPrPRA,LuLuPrPLA}. These screened
potentials are prototype for many physical processes such as atoms confined in
Debye or Debye-cosine plasmas, where atomic properties change drastically
compared to free atoms, depending on the screening parameter, in particular \cite{LuLuPrPRA,LuLuPrPLA,BaBa,LuLuPrPScr,LuLuPrEPJD}. The
spectrum of the atom becomes quite interesting as reflected in its response to
external fields. If, in addition to the screening potentials, there happens to
be a spherical confinement, then the atom shows a drastic change in the
spectrum. For example, Lumb \textit{et al}. \cite{LuLuPrIJP,LuLuPrEPJD} have shown that
there are very few bound states if confining radius $r_c$ is very small ($r_c \ll
10$~a.u.). So, spherical confinement in addition to screened potential forms a
new confining potential where atomic and molecular systems are yet to be
explored in detail. Impurities present in quantum dots are an example of a confined system where the spherical boundary represents the cage radius.
In such practical physical situations the existing potential is modified due to interaction with impurities. As reported earlier \cite{PaHoPRA79}, there have been few studies on the scattering processes taking place in confined environment. Two-photon spectroscopy has been an important tool to
study the excitation of atoms and molecules (\cite{QuBaCa} and references
therein). Two photon and three photon transitions in confined atoms under
various kinds of confinements started receiving much attention since starting of
$2000$ \cite{Lukin,PaHoPOP15,PaHoPRA78,PaHoPOP16,
LaLiHo,HoLaLi,PaHoPOP17,Soylu} as tools to
study such transitions became available.

In the present study, we focus on the two-photon transitions in atomic hydrogen
embedded in spherically confined Debye plasma. The model employed in the present case includes the effect of spatial confinement as well as screened Coulomb potential. The hydrogen atom is assumed to be embedded in plasma environment consisting of a finite charge distribution. The boundary condition signifies the finite extent of the charge cloud.The Debye confinement considered here is more likely to occur in many practical situations. In general, plasma environment in space is represented by this potential as is known from available literature. Also, the artificially created short-lived laboratory plasmas always have a finite volume. Such environments simulate the plasma present in interstellar space. Using the present model of confined plasma, simple Debye plasma model can be studied by enlarging the radius of confining sphere to infinity. Understanding of hydrogen atom under such confinement is important as it helps in examining more complex
systems. Here, we consider electron of confined hydrogen atom being excited to
higher states through absorption of two photons. We study excitation from $1s$
to higher states. The effect of confinement on two-photon processes has been
dealt with in detail. The spectrum of confined atom is evaluated using
B-polynomial method \cite{Bhatti,BhPe,LuLuPrJMP,LuLuPrQM}. 

The paper is organized according to the following scheme. The relevant  theoretical details are provided in Sec.~2. The
results of the present work are discussed in Sec.~3. Finally the important
findings are summarized in Sec. 4.


\section{Theory}
\label{theory}
 

The model considered here comprises a hydrogen atom embedded in a Debye plasma
environment. The atom is assumed to be at the center of an impenetrable
spherical cavity of radius $r_0$. This geometrically symmetric arrangement is a
special case of the more general possibility in which the atom may be present at
any position within the cavity. For simplicity, we have chosen the special case.
Our aim here is to study the effect of spherical confinement and surrounding
plasma on the two-photon transition probability amplitudes $D_2$, transparency
frequency $\omega_t$ and resonance enhancement frequencies $\omega_r$.
This in turn requires a knowledge of the energy spectrum and the
dipole matrix elements of the system. The spectrum, oscillator strengths and
other physical quantities of confined systems are known to be highly dependent
on the chosen confinement parameters and hence need an accurate evaluation. The
evaluation of energy spectra and oscillator strengths of confined hydrogen in
Debye plasma environment has been carried out and the results for various
confining radii and Debye lengths characterizing different plasma conditions
have been reported in our earlier works \cite{LuLuPrEPJD,LuLuPrPLA}. The steps
necessary for arriving at these results are summarized below for ready
reference. We have used atomic units throughout our study.

The radial Schr\"odinger equation for the electron of the confined hydrogen atom
is given by
\begin {align}
\bigg[& -\frac{1}{2}\frac{d^2}{d r^2}+\frac{l(l+1)}{2
r^2}
\na
-\frac{1}{r}e^{-r/\lambda_D}+V_c(r)\bigg]U_{nl}(r) = E_{nl} U_{nl}(r)
\label{radsch}
\end{align}
where $-e^{-r/\lambda_D}/r$ is the Debye-H\"uckel potential, $1/\lambda_D$
being the Debye screening parameter \cite{Debye} and $V_c(r)$ is the
confinement potential defined as
\begin {align}
V_c(r) = \left\{
\begin{array}{l l l}
0 \,\, ,& r < r_0\\ \nonumber
\infty \, \, ,& r \ge r_0.
 \end{array}
\right.
\label{Vc}
\end{align}
The radial wave function $R_{n,l}(r) = U_{n,l}(r)/r$. $U_{n,l}(r)$ is expanded
in B-polynomial basis as
\begin {align}
U_{nl}(r) = \sum_{i=0}^n c_iB_{i,n}(r),
\end{align}
where $c_i$s are coefficients of expansion and $B_{i,n}(r)$ are B-polynomials
of degree $n$. The confinement potential being infinite at the boundary, \textit{i.e}.,
$r=r_0$,  forces the wave functions to vanish there. Under these restrictions
the radial Schr\"odinger equation can be recast as a symmetric generalized
eigenvalue equation in matrix form, given by 
\begin {align}
(A + F + G)C = EDC,
\label{afg}
\end{align}
where matrix elements $a_{i,j}, f_{i,j}, g_{i,j}$ and $d_{i,j}$ are defined as
\begin {align}
a_{i,j} &= \frac{1}{2}(B_{i,n}^\prime,B_{j,n}^\prime),\quad f_{i,j} = \frac{l(l+1)}{2}\left(\frac{B_{i,n}}{r^2},B_{j,n}\right), 
\no
g_{i,j} & {=} {-}\left(\frac{B_{i,n}}{r}e^{{-}r/\lambda_D},B_{j,n}\right),\quad
d_{i,j} {=} (B_{i,n},B_{j,n}).
\end{align}
The eigenvalues $E$ provide the energy spectrum and eigenvectors $C$
are used to calculate the corresponding radial wave functions using Eq.
(3). We have used Fortran EISPACK library to solve Eq. (4).

The two-photon transition probability amplitude, $D_2$, of a hydrogen
atom from the initial state $1s$ to a final state $js$ can be evaluated by
using \cite{PaHoPOP15, PaHoPRA78}
\begin {align}
D_2 & = \frac{1}{2}\sum_n \bigg[\frac{1}{-E_{1s} + E_n - \omega_0} 
\na
+
\frac{1}{-E_{js} + E_n + \omega_0}\bigg] \chi_1^n \chi_j^n
\label{d21sjs}
\end{align}
where $n$ represents the intermediate states including continuum, $E_{1s}$ and
$E_{js}$ are the energy eigenvalues of the $1s$ and $js$ states respectively
and $\chi_1^n$ and $\chi_j^n$ are the dipole matrix elements defined as follows
\begin {align}
\chi_k^n = \int\limits_0^\infty{r^3 R_n R_k dr},\hskip 1cm (k = 1,j)
\end{align}
where $R_l$ (with $l=n, k$) represents the radial wave function. The
corresponding formula for calculating the transition probability amplitude for
$1s$ to $jd$ state is given by
\begin {align}
D_2 &= \frac{1}{\sqrt{5}}\sum_n \bigg[\frac{1}{-E_{1s} + E_n - \omega_0} 
\na
+
\frac{1}{-E_{jd} + E_n + \omega_0}\bigg] \chi_1^n \chi_j^n
\label{d21sjd}
\end{align}
The values of $\omega_0$ for which $D_2$ approaches infinity, and  also lie
inside the interval $\Delta E_{if}/2$ and $\Delta E_{if}$ are called the
resonance enhancement frequencies, where $\Delta E_{if}$ is the difference
between final and initial ($1s$) state energies\cite{PaHoPRA78}. Also, the
frequencies for which the transition amplitude vanishes are called as the
two-photon transparency frequencies\cite{PaHoPRA78}.




\end{multicols}


\begin{center}
\includegraphics*[140mm,109.79mm]{fig01.jpg}
\end {center}

\lilahrge{{\sc Figure 1.} Variation of two-photon $1s-2s$ transition amplitude with frequency of
incoming photons, $\omega_0$, for various values of screening length
$\lambda_D$.}
% ********************

\vspace{5pt}



\begin{center}
\includegraphics*[140mm,107.42mm]{fig02.jpg}
\end {center}

\lilahrge{{\sc Figure 2.} Same as Fig. 1 for $1s-3s$ transition.}
% ********************







\begin{multicols}{2}




\section{Results and Discussions}
\label{resdis}

The two-photon transition probability amplitudes of a hydrogen atom placed at
the center of an impenetrable confining sphere of radius $r_0$ and embedded in
a weak plasma characterized by Debye-H\"uckel potential are explored. Since
Debye screening length $\lambda_D$ is dependent on temperature and density of
plasma, its different values represent different conditions of the system. We
have explored the dependence of two-photon transitions on the extent of Debye
screening as well as confinement radius. The probability amplitudes from $1s$
to $js$ ($j = 2,3,4$) states have been calculated using Eq. (\ref{d21sjs}) and
to $jd$ ($j = 3,4)$ states using Eq. (\ref{d21sjd}) for a range of incident
photon frequencies. It may be mentioned that the Hamiltonian being discretized
leads to discreet continuum states, hence the quantum nature of the system is
retained even for small confinements. Therefore, irrespective of the states
being free or bound, the nomenclature of states is assumed to be same. The range
of frequencies selected for studying the variation of two-photon transition
probability amplitudes, transparency frequencies and resonance enhancement
frequencies for various values of confinement parameters $r_0$ and $\lambda_D$
is $\Delta E_{if}/2$ to $\Delta E_{if}$ as mentioned in Sec. \ref{theory} 

The two-photon transition amplitudes as calculated by us show very close
agreement with the previously available results in literature. For example, for
a free hydrogen atom, the contributions to the $1s-2s$ transition amplitudes
due to first few intermediate states taking $\omega_0 = 0.375$ Ryd., given by
Bassani \textit{et al.} \cite{BaFoQu} match well with our results calculated for $r_0 =
50$ a.u. and $\lambda_D = \infty$. It may be noted that the spherical
confinement radius $r_0 = 50$ a.u. is very large as compared to the size of
hydrogen atom and hence corresponds to a nearly free atom. The total
contribution calculated by them is $-11.7805$ and the value obtained in the
present work is $-11.7803$. The energy levels, two-photon transition
amplitudes, absorption coefficients, two-photon transparency frequencies and
resonance enhancement frequencies as calculated by Paul and Ho \cite{PaHoPOP15}
for a Debye plasma screened hydrogen atom are in consonance with our results for
various values of $\lambda_D$. For example, the value of two-photon absorption
coefficient for $1s-2s$ transition based on our calculations is $144.8617$ for
$\lambda_D = 10$ a.u. and $138.8400$ for $\lambda_D = 200$ a.u. taking $\omega_0
= 0.37$ Ryd. This data matches exactly with the results of Paul and Ho
\cite{PaHoPOP15}. This provides a 
check on our calculations. The aim of the
present work is to analyze the effect of spherical confinement on such
properties of the system.

Figures 1-3 show variation of the two-photon transition
probability amplitudes for four different values of $\lambda_D$,
viz., $10$, $20$, $30$ and $200$ a.u. and two values of $r_0$, viz., $10$ a.u.
and $50$ a.u. It is found that the two-photon transition probability amplitudes
$D_2$ depend on both Debye and spherical confinement. Figure 1 which shows $D_2$ elements for $1s-2s$ transitions clearly depicts
the effect of change in Debye as well as spherical confinement. In Figs.
1(a) and (b), it is seen that if $\lambda_D$ is varied over
a wide range from $10$ to $200$ a.u., the resonance enhancement
condition is achieved for small frequencies for smaller $\lambda_D$.
Such variation is not much prominent for small change in $\lambda_D$
as in Figs. 1(c) and (d). It is also evident from Fig. 
1 that the nature of variation of $D_2$ changes with $r_0$. The 
resonance enhancement frequency shifts towards lower $\omega_0$ for 
weaker spherical confinement. The variation with $r_0$ and $\lambda_D$ as
observed in Fig. 1 is also present in the data plotted in Figs. 2 and 3.
These features can be explained on the basis of the change in energy spectrum
and radial matrix elements with Debye as well as spherical confinement. The
detailed structure of energy spectrum for such a confined system has
been described in our earlier works \cite{LuLuPrEPJD,LuLuPrPLA}. The
decrease in number of bound states and increasing separation of the energy
levels for smaller $r_0$ values are responsible for the observed behaviour.
Figure 4 shows similar variation of $D_2$ for $r_0 = 15, 20$ and $40$ a.u. 
and $\lambda_D = \infty$ for $1s-2s$, $1s-3s$, $1s-3d$ and $1s-4d$ transitions. 
It clearly depicts the effect of spherical confinement in absence of Debye 
screening. 

The explicit values of transparency and resonance enhancement frequencies of two photon transition probability amplitudes are presented in Tables I-III. The frequencies corresponding to the transitions to $4s$ and $4d$ have also been included in the tables. The values for transitions to $2s$, $3s$ and $3d$ are also implicit in 
the graphical representation of two photon transition probability amplitudes 
shown in Figs.~\mbox{1-4.}
 The transparency frequency \cite{PaHoPRA78, PaHoPOP15} 
as observed for transitions to $3s$, $4s$ and $4d$ states have been reported in 
Table I. The plasma screening has been found to considerably affect the transparency
frequencies. With decrease in $\lambda_D$ for a fixed $r_0$, the transparency
frequency, $\omega_t$, also decreases as observed by Paul and Ho
\cite{PaHoPOP15}. Similar effect has been observed for variation with
confinement radius. That is, decrease in $r_0$ or increase in confinement leads
to decrease in $\omega_t$ as shown in Table I. For the case of tight
confinement, $r_0 = 10$ a.u., only a single transparency condition is obtained
for $1s$ to $4s$ transition for all $\lambda_D$ as compared to two transparency
frequencies 
for rest of the $r_0$ values. This is due to the fact that transparency 
frequency is limited only to the range $\Delta E_{if}/2$ and $\Delta E_{if}$.

The resonance enhancement frequencies obtained for 
transitions to $2s$, $3s$,$4s$, $3d$ and $4d$ levels are given in Tables 
II and III. These are found to be same for particular confinement 
radius and Debye length irrespective of the final state. The values for $1s-3s$ 
transitions as calculated in the present work match very well with those quoted by Paul
and Ho \cite{PaHoPOP15}. More frequencies correspond to resonance enhancement
condition for transitions from $1s$ to $3s$ or $4s$ states on increasing
the spherical confinement for some $\lambda_D$ as can be seen from the data in
Table III for $r_0 = 30$ and $50$ a.u. An opposite trend is observed
in the transitions from $1s$ to $3d$ or $4d$ states. The trend shows that confinement effect on probability of two-photon transition to a state depends on the shape of its orbital. 




\end {multicols}

\begin{center}
\includegraphics*[140mm,107.8mm]{fig03.jpg}
\end {center}

\lilahrge{{\sc Figure 3.} Same as Fig. 1 for $1s-3d$ transition.}
% ********************
\vspace{5pt}

\begin{center}
\includegraphics*[140mm,108.86mm]{fig04.jpg}
\end {center}

\lilahrge{{\sc Figure 4.} Variation of two-photon transition amplitudes for $1s-2s$, $1s-3s$, 
$1s-3d$ and $1s-4d$ transitions with frequency of incoming photons, $\omega_0$, for screening length $\lambda_D = \infty$.}
% ********************


\begin {multicols}{2}






 \end {multicols}

\longtabletopline\vspace{2pt}\lilahrge{\sc Table I.\ {\rm Two-photon transparency frequencies for various Debye
screening lengths, $\lambda_D$, and confinement radii, $r_0$. The
data is in atomic units. The results for $r_0 = 50$ a.u. have been compared
with those of free hydrogen.}}
\begin{center}
\small{\renewcommand{\arraystretch}{1.2}
\renewcommand{\tabcolsep}{1.93pc}
\begin{tabular}{ccccccc}
\hline
 $r_0$ & $\lambda_D$ & $1s\rightarrow 3s$ & $1s\rightarrow
4s$ & $1s\rightarrow 4d$\\
\hline %inserts single line
%\hline %inserts single line
10	&	10	&	0.6220155	&	1.0418125	&
0.7633365	\\
	&	15	&	0.6374085	&	1.0439695	&
0.7783205	\\
	&	20	&	0.6429075	&	1.0448025	&
0.7839785	\\
	&	30	&	0.6469285	&	1.0454365	&
0.7882275	\\
	&	200	&	0.6501865	&	1.0459845	&
0.7917615	\\
\hline %inserts single line
15	&	10	&	0.6625705	&	0.6671255	&
0.7666505	\\
	&		&		&	0.8144075	&	
\\
	&	15	&	0.6761735	&	0.6771675	&
0.7875435	\\
	&		&		&	0.8422595	&	
\\
	&	20	&	0.6811455	&	0.6808125	&
0.7954565	\\
	&		&		&	0.8532235	&	
\\
	&	30	&	0.6848055	&	0.6834675	&
0.8014125	\\
	&		&		&	0.8616995	&	
\\
	&	200	&	0.6877845	&	0.6855815	&
0.8063935	\\
	&		&		&	0.8690165	&	
\\
\hline %inserts single line
20	&	10	&	0.6690395	&	0.6668105	&
0.7769765	\\
	&		&		&	0.8026365	&	
\\
	&	15	&	0.6819135	&	0.6787535	&
0.8011405	\\
	&		&		&	0.8345025	&	
\\
	&	20	&	0.6865375	&	0.6831165	&
0.8100715	\\
	&		&		&	0.8471905	&	
\\
	&	30	&	0.6899045	&	0.6863195	&
0.8166685	\\
	&		&		&	0.8570665	&	
\\
	&	200	&	0.6926055	&	0.6889105	&
0.8220685	\\
	&		&		&	0.8656685	&	
\\
\hline %inserts single line
30	&	10	&	0.6714915	&	0.6684855	&
0.7913775	\\
	&		&		&	0.8003005	&	
\\
	&	15	&	0.6835095	&	0.6806875	&
0.8172785	\\
	&		&		&	0.8361105	&	
\\
	&	20	&	0.6878315	&	0.6850525	&
0.8256905	\\
	&		&		&	0.8501085	&	
\\
	&	30	&	0.6909785	&	0.6882145	&
0.8313625	\\
	&		&		&	0.8608235	&	
\\
	&	200	&	0.6934965	&	0.6907285	&
0.8355395	\\
	&		&		&	0.8699895	&	
\\
\hline %inserts single line
50	&	10	&	0.6718395	&	0.6704785	&
0.8040815	\\
	&		&	0.672$^{\mathrm{a}}$ &	0.8023525	&	
\\
	&	15	&	0.6836195	&	0.6819315	&
0.8263675	\\
	&		&	0.6835$^{\mathrm{a}}$ &	0.8385935
&	
\\
	&	20	&	0.6878945	&	0.6859735	&
0.8318885	\\
	&		&	0.688$^{\mathrm{a}}$ &	0.8521725
&	
\\
	&	30	&	0.6910215	&	0.6888935	&
0.8354605	\\
	&		&		&	0.8624345	&	
\\
	&	200	&	0.6935135	&	0.6911895	&
0.8380465	\\
	&		&	0.6935$^{*,\mathrm{a}}$ &	0.8711365
&	
\\
\hline %inserts single line
%\hline %inserts single line
	&	$^\mathrm{a}$	&	Reference \cite{PaHoPOP15}.	&	
	&		\\
	&	$^*$	&	 Value is for $\lambda_D = \infty$ &	&	
&	&		\\
\hline
\end {tabular}}
\end {center}
\vspace{10pt}

\begin {multicols}{2}







\end {multicols}

\longtabletopline\vspace{2pt}\lilahrge{\sc Table II.\ {\rm Two-photon resonance enhancement frequencies for various Debye
screening lengths, $\lambda_D$, and confinement radii, $r_0 = 10, 15$ and $20$
a.u. The data is in atomic units.}}
\begin{center}
\small{\renewcommand{\arraystretch}{1.25}
\renewcommand{\tabcolsep}{1.85pc}
\begin{tabular}{ccccccc}
\hline
$r_0$ & $\lambda_D$ & $1s \rightarrow 2s$ & $1s\rightarrow 3s$ & $1s\rightarrow
4s$ & $1s\rightarrow 3d$ & $1s\rightarrow 4d$\\
\hline %inserts single line
%\hline %inserts single line
10	&	10	&	0.738714	&	0.738714	&
1.041836	&	0.738714	&	0.738714	\\
	&		&		&	1.067930	&	1.067930
&		&	1.067930	\\
	&		&		&		&	1.602878
&		&		\\
	&	15	&	0.751158	&	0.751158	&
1.044687	&	0.751158	&	0.751158	\\
	&		&		&	1.083600	&	1.083600
&		&	1.083600	\\
	&		&		&		&	1.617914
&		&		\\
	&	20	&	0.755824	&	0.755824	&
1.045969	&	0.755824	&	0.755824	\\
	&		&		&	1.089686	&	1.089686
&		&	1.089686	\\
	&		&		&		&	1.623756
&		&		\\
	&	30	&	0.759316	&	0.759316	&
1.047029	&	0.759316	&	0.759316	\\
	&		&		&	1.094338	&	1.094338
&		&	1.094338	\\
	&		&		&		&	1.628224
&		&		\\
	&	200	&	0.762209	&	0.762209	&
1.048002	&	0.762209	&	0.762209	\\
	&		&		&	1.098282	&	1.098282
&		&	1.098282	\\
	&		&		&		&	1.632017
&		&		\\
\hline %inserts single line
15	&	10	&		&	0.722295	&	0.722295
&	0.722295	&	0.722295	\\
	&		&		&	0.881457	&	0.881457
&		&	0.881457	\\
	&		&		&		&	1.107375
&		&		\\
	&	15	&		&	0.737215	&	0.737215
&	0.737215	&	0.737215	\\
	&		&		&	0.905802	&	0.905802
&		&	0.905802	\\
	&		&		&		&	1.130544
&		&		\\
	&	20	&		&	0.742775	&	0.742775
&	0.742775	&	0.742775	\\
	&		&		&	0.915543	&	0.915543
&		&	0.915543	\\
	&		&		&		&	1.139876
&		&		\\
	&	30	&		&	0.746927	&	0.746927
&	0.746927	&	0.746927	\\
	&		&		&	0.923137	&	0.923137
&		&	0.923137	\\
	&		&		&		&	1.147195
&		&		\\
	&	200	&	0.750373	&	0.750373	&
0.750373	&	0.750373	&	0.750373	\\
	&		&		&	0.929743	&	0.929743
&		&	0.929743	\\
	&		&		&		&	1.153608
&		&		\\
\hline %inserts single line
20	&	10	&		&	0.721122	&	0.721122
&	0.721122	&	0.721122	\\
	&		&		&	0.834974	&	0.834974
&		&	0.834974	\\
	&		&		&		&	0.954652
&		&		\\
	&	15	&		&	0.736483	&	0.736483
&	0.736483	&	0.736483	\\
	&		&		&	0.865737	&	0.865737
&		&	0.865737	\\
	&		&		&		&	0.984498
&		&		\\
	&	20	&		&	0.742172	&	0.742172
&	0.742172	&	0.742172	\\
	&		&		&	0.878180	&	0.878180
&		&	0.878180	\\
	&		&		&		&	0.996924
&		&		\\
	&	30	&		&	0.746412	&	0.746412
&	0.746412	&	0.746412	\\
	&		&		&	0.887954	&	0.887954
&		&	0.887954	\\
	&		&		&		&	1.006907
&		&		\\
	&	200	&		&	0.749925	&	0.749925
&	0.749925	&	0.749925	\\
	&		&		&	0.896554	&	0.896554
&		&	0.896554	\\
	&		&		&		&	1.015934
&		&		\\
\hline
\end {tabular}}
\end {center}
\vspace{10pt}


\longtabletopline\vspace{2pt}\lilahrge{\sc Table III.\ {\rm Two-photon resonance enhancement frequencies for various Debye
screening lengths, $\lambda_D$, and confinement radii, $r_0 = 30, 50$ a.u. The
data is in atomic units. The results for $r_0 = 50$ a.u. have been compared
with those of free hydrogen.}}
\begin{center}
\small{\renewcommand{\arraystretch}{1.25}
\renewcommand{\tabcolsep}{1.8pc}
\begin{tabular}{ccccccc}
\hline
$r_0$ & $\lambda_D$ & $1s\rightarrow 3s$ & $1s\rightarrow
4s$ & $1s\rightarrow 3d$ & $1s\rightarrow 4d$\\
\hline %inserts single line
%\hline %inserts single line
30	&	10	&	0.721048	&	0.721048
&	0.721048	&	0.721048	\\
	&		&			&	0.814992
&	0.814992	&	0.814992	\\
	&		&			&	0.862225
&		&		\\
	&	15	&		0.736454	&	0.736454
&	0.736454	&	0.736454	\\
	&		&			&	0.852222
&	0.852222	&	0.852222	\\
	&		&			&	0.901888
&		&		\\
	&	20	&		0.742153	&	0.742153
&	0.742153	&	0.742153	\\
	&		&			&	0.867094
&	0.867094	&	0.867094	\\
	&		&			&	0.919349
&		&		\\
	&	30	&		0.746397	&	0.746397
&	0.746397	&	0.746397	\\
	&		&			&	0.878668
&	0.878668	&	0.878668	\\
	&		&			&	0.933938
&		&		\\
	&	200	&		0.749914	&	0.749914
&	0.749914	&	0.749914	\\
	&		&		0.888793	&	0.888793
&		&	0.888793	\\
	&		&			&	0.947785
&		&		\\
\hline %inserts single line
50	&	10	&		0.721044	&	0.721044
&	0.721044	&	0.721044	\\
	&		&		0.721$^{\mathrm{a}}$ & 	
0.811171
&	0.811171	&	0.811171	\\
	&		&			&		&	
&	0.826633	\\
	&	15	&		0.736459	&	0.736459
&	0.736459	&	0.736459	\\
	&		&		0.736$^{\mathrm{a}}$ & 	
0.851124
&	0.851124	&	0.851124	\\
	&		&			&		&	
&	0.876753	\\
	&	20	&		0.742160	&	0.742160
&	0.742160	&	0.742160	\\
	&		&		0.742$^{\mathrm{a}}$ & 	
0.866526
&	0.866526	&	0.866526	\\
	&		&			&		&	
&	0.899727	\\
	&	30	&		0.746409	&	0.746409
&	0.746409	&	0.746409	\\
	&		&			&	0.878360
&	0.878360	&	0.878360	\\
	&		&			&		&	
&	0.918976	\\
	&	200	&		0.749914	&	0.749914
&	0.749914	&	0.749914	\\
	&		&		0.75$^{*,\mathrm{a}}$ & 
0.888621
&	0.888621	&	0.888621	\\
	&		&			&	0.937057
&		&	0.937057	\\
										
		\\
\hline %inserts single line
%\hline %inserts single line
	&	$^\mathrm{a}$	&	Reference \cite{PaHoPOP15}.	&	
	&		\\
	&	$^*$	&	 Value is for $\lambda_D = \infty$ &	&	
&	&		\\
\hline
\end {tabular}}
\end {center}
\vspace{5pt}

\begin {multicols}{2}








\section{Conclusions}
\label{concl}


The effect of spherical confinement on two-photon transition probability amplitudes, transparency frequencies and resonance enhancement frequencies for a Debye plasma embedded hydrogen atom has been explored. The spectrum of the atom has been calculated using B-polynomials. It is understood that experimental data for the confining potential undertaken in the present study is not available. However, the obtained results for loose spherical confinement ($r_0 = 50$ a.u.) have been compared with the theoretical results reported earlier in literature for some values of Debye parameter and no spherical boundary in Tables I and III. A close agreement has been achieved. It is anticipated that with the advancement of technology the present model of confining potential may become a reality in future and the data presented in the paper would be useful for such experimental studies. The spherical confinement is found to play a role analogous to Debye confinement. In particular, the transparency and resonance enhancement frequencies decrease with increase in confinement.



\end{multicols}
\medline
\begin{multicols}{2}
\begin{thebibliography}{99}
%\bibitem{c1}

%1
\bibitem{Ichimaru} S. Ichimaru, {\it Rev. Mod. Phys.} {\bf 54} (1982) 1017.

%2
\bibitem{DeSi} N.C. Deb and N.C. Sil, {\it J. Phys. Murci\'elago Mol. Phys.} {\bf 17} (1984) 3587.

%3
\bibitem{ZhHo} L.B. Zhao and Y.K. Ho, {\it Phys. Plasmas} {\bf 11} (2004) 1695.

%4
\bibitem{YoJu} J.S. Yoon and Y.-D. Jung, {\it Phys. Plasmas} {\bf 3} (1996) 3291.

%5
\bibitem{Sen} {\it Electronic Structure of Quantum Confined Atoms and Molecules} (ed) K D Sen (Switzerland: Springer) (2014)

%6
\bibitem{KuPrMo} R. Kundliya, V. Prasad and M. Mohan, {\it J. Phys. B: At. Mol. Opt. Phys.} {\bf 33} (2000) 5263.

%7
\bibitem{Khordad} R. Khordad, {\it Indian J. Phys.} (2017) doi: 10.1007/s12648-017-0980-8

%8
\bibitem{HaIkOnZa} H. Hassanabadi, A.N. Ikot, C.P. Onyenegecha and S. Zarrinkamar, {\it Indian J. Phys.} (2017). doi: 10.1007/s12648-017-1009-z

%9
\bibitem{HoJu} W. Hong and Y.-D. Jung, {\it Phys. Plasmas} {\bf 3} (1996) 2457.

%10
\bibitem{BhSi} A.K Bhatia and C. Sinha, {\it Phys. Rev. A} {\bf 86} (2012) 053421.

%11
\bibitem{PaLiHo} M.K. Pandey, Y.-C. Lin and Y.K. Ho, {\it Phys. Plasmas} {\bf 22} (2015) 052104.

%12
\bibitem{MuSi} S. Mukhopadhyay and C. Sinha, {\it Phys. Rev. A} {\bf 88} (2013) 033414.

%13
\bibitem{LuLuPrPRA} S. Lumb, S. Lumb and V. Prasad, {\it Phys. Rev. A} {\bf 90} (2014) 032505.

%14
\bibitem{LuLuPrIJP} S. Lumb, S. Lumb and V. Prasad, {\it Indian J. Phys.} {\bf 89} (2015) 13.

%15
\bibitem{LuLuPrPLA} S. Lumb, S. Lumb and V. Prasad, {\it Phys. Lett. A} {\bf 379} (2015) 1263.

%16
\bibitem{KaHo} S. Kar and Y.K. Ho, {\it Phys. Rev. A} {\bf 75} (2007) 062509.

%17
\bibitem{PaHoPRA78} S. Paul and Y.K. Ho, {\it Phys. Rev. A} {\bf 78} (2008) 042711.

%18
\bibitem{PaHoPRA79} S. Paul and Y.K. Ho, {\it Phys. Rev. A} {\bf 79} (2009) 032714. 

%19
\bibitem{BaBa} M. Bassi and K.L. Baluja, {\it Indian J. Phys.} {\bf 86} (2012) 961.

%20
\bibitem{LuLuPrPScr} S. Lumb, S. Lumb, D. Munjal and V. Prasad, {\it Phys. Scr.} {\bf 90} (2015) 095603.

%21
\bibitem{LuLuPrEPJD} S. Lumb, S. Lumb and V. Nautiyal, {\it Eur. Phys. J. D} {\bf 69} (2015) 176.

%22
\bibitem{QuBaCa} A. Quattropani, F. Bassani and S. Carillo, {\it Phys. Rev. A} {\bf 25} (1982) 3079.

%23
\bibitem{LaLiHo} H.F. Lai, Y.C. Lin and Y.K. Ho, {\it Journal of Physics: Conference Series} {\bf 163} (2009) 012096.

%24
\bibitem{PaHoPOP15} S. Paul and Y.K. Ho, {\it Phys. Plasmas} {\bf 15} (2008) 073301.

%25
\bibitem{Lukin} M.D. Lukin, {\it Rev. Mod. Phys.} {\bf 75} (2003) 457.

%26
\bibitem{PaHoPOP16} S. Paul and Y.K. Ho, {\it Phys. Plasmas} {\bf 16} (2009) 063302.

%27
\bibitem{HoLaLi} Y.K. Ho, H.F. Lai and Y.C. Lin, {\it Journal of Physics: Conference Series} {\bf 194} (2009) 022023.

%28
\bibitem{PaHoPOP17} S. Paul and Y.K. Ho, {\it Phys. Plasmas} {\bf 17} (2010) 082704.

%29
\bibitem{Soylu} A. Soylu, {\it Phys. Plasmas} {\bf 19}  (2012) 072701.

%30
\bibitem{Bhatti} M.I. Bhatti, {\it Adv. Studies Theor. Phys.} {\bf 3} (2009) 451.

%31
\bibitem{BhPe} M.I. Bhatti and W.F. Perger, {\it J. Phys. B: At., Mol. Opt. Phys.} {\bf 39} (2006) 553.

%32
\bibitem{LuLuPrJMP} S. Lumb, S. Lumb and V. Prasad, {\it J. Mod. Phys.} {\bf 4} (2013) 1139.

%33
\bibitem{LuLuPrQM} S. Lumb, S. Lumb and V. Prasad, {\it Quantum Matter} {\bf 2} (2013) 314.

%34
\bibitem{Debye} P. Debye and E. H\"uckel, {\it Z. Phys.} {\bf 24} (1923) 185.

%35
\bibitem{BaFoQu} F. Bassani, J.J. Forney and A. Quattropani, {\it Phys. Rev. Lett.} {\bf 39} (1977) 1070.



\end{thebibliography}
\end{multicols}
\end{document}
%
%Notas:
%
%\tabletopline\vspace{2pt}\lilahf{\sc Table I.\ {\rm Table caption}}
%\begin{center}
%\small{\renewcommand{\arraystretch}{1.3}
%\renewcommand{\tabcolsep}{1.35pc}
%\begin{tabular}{cccc}
%\hline
%\hline
%\end{tabular}}
%\end{center}
%
