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\begin{document}
\begin{titlepage}
\begin{center}
{\Large\bf Some effects of non-classical light on an atom}
\footnote{This paper is dedicated to Professor Jan Pe\v{r}ina on the
  occasion of his 60th birthday}\\[36pt] 
A.~Messikh and R.~Tana\'s\\[12pt]
{\em Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz
  University,\\
Umultowska 85, 61-614 Pozna\'n, Poland}
\begin{abstract}
We consider a three-level atom in a V configuration damped to a
broadband squeezed vacuum and driven on one of its allowed transitions
(pump transition) by a coherent laser field. The resonance
fluorescence spectrum and the probe absorption spectrum on the other
allowed transition (probe transition) is studied for two cases:
(i)~carrier frequency of squeezed vacuum is tuned to the probe
transition, and (ii)~carrier frequency of the squeezed vacuum is tuned 
to the pump transition. 
\end{abstract}
\end{center}
\end{titlepage}

%----------------------------------------------------------------------- 
\section{Introduction}
Non-classical light exhibits a number of features that cannot be
observed with classical light. The non-classical light can be generated
in nonlinear optical processes. Properties of such non-classical light
have been studied for many years by Professor Jan Pe\v{r}ina and his
collaborators in Olomouc. The subject and results of many research
works in this field have been covered in details in an excellent book
of Professor Pe\v{r}ina on ``{\em Quantum Statistics of Linear and
Nonlinear Optical Phenomena}'', the second edition of which appeared in
1991~\cite{per91}. The literature related to non-classical light
contains already enormous number of papers and is still
growing. References to a substantial part of the literature can be
found in Professor's Pe\v{r}ina book. Beside the problems associated
with the properties of 
non-classical light itself that are a subject of intensive research, the
old problems that involved `classical' fields (ordinary vacuum, for
example, is considered in this respect as classical) are reconsidered
with the non-classical light replacing the classical one (squeezed
vacuum, for example, instead of ordinary vacuum), which leads to new
and sometimes unexpected behavior of the well known systems.
In this paper we consider a three-level atomic system in the V
configuration which is coherently pumped on one of the allowed
transitions (pump transition) and is damped to the squeezed vacuum
instead of the ordinary vacuum. Moreover, we assume that the two allowed atomic
transitions are coupled by a non-zero coherence transfer rate. We
study resonance fluorescence and probe absorption spectra on undriven
transition (probe transition).
 
The Mollow treatment \cite{Mollow72} of a driven system of two level
atoms with an off resonant field predicted that the absorption
spectrum contains one absorption and one emission peak at the Rabi
sidebands and a small dispersion profile at the central frequency. 
However, for a resonant driving field the central component of the 
spectrum disappears and the absorption spectrum exhibit 
dispersion-like profiles located at the Rabi sidebands.  When
atoms are damped into a squeezed  vacuum, they interact with a
modified electromagnetic vacuum and thus,  
their radiative properties change. It is well known
\cite{gar86,clw87,par90,fd93,ss92,zt88} that there are essential
differences between the atomic decay in an ordinary vacuum and the
corresponding decay in a squeezed vacuum. For example, the atomic
dipole decay rate depends on its phase difference with respect to the
squeezed vacuum, the resonance fluorescence and probe absorption
spectra also depend on the relative phase between the coherent
driving field and the squeezed vacuum. The line-widths of the spectra
can be drastically reduced for a particular choice of this phase
difference. Recently, it has been shown 
\cite{SS93}, that the fluorescence spectrum can be asymmetric even if 
the atoms are driven by a resonant field. This feature 
arises from the fact that the relative phase between the squeezed vacuum and 
the driving field, if chosen between $0$ and $\pi$, makes unequal
populations of the dressed states.  Moreover a small change in the
Rabi frequency, phase, and detuning will result in the spectrum loosing
its dispersive profiles.   
Ficek et al. \cite{FiSS95} has studied the asymmetric 
probe-absorption spectrum and amplification without population 
inversion of a two level atom damped by a broadband squeezed vacuum. 
They have shown that a resonant driving field which is out of phase  
with the squeezed vacuum can be asymmetric. This feature is again due to 
the fact that unequal populations of the dressed states of the 
driven system have been produced. They have also shown that the 
amplification of the probe field at the central frequency is not due 
to any population inversion in both bare  and dressed 
states, but it is rather related to the coherent population
oscillations, and it cannot appear in the absence of the squeezed vacuum.

Since the squeezed vacuum is characterized by correlated pairs of
photons, it is quite natural to expect that two-photon transitions in
multilevel atoms should be significantly affected by the presence of
the squeezed vacuum. Such a situation occurs, for instance, when the
the carrier frequency of the squeezed vacuum is chosen so as to its
double is tuned to resonance between the lower and upper states of the
three-level cascade system. The two-photon correlations present in the
squeezed vacuum have in this case dramatic influence on the
steady-state populations of the atom~\cite{fd91,fd91a,bkk91,fd93}. The
results for three-level systems driven by two independent laser beams
show~\cite{jll91,jp92,ss92} that the relative heights and widths of
the peaks in the resonance fluorescence spectra can be sub-natural or
supernatural depending on the relative phase between the driving
fields and the squeezed vacuum, similarly as in two-level systems. 
Recently, Ferguson~et~al.~\cite{ffd95} have examined the effect of a
single broad-band squeezed vacuum on the stationary populations and
coherences in a three-level atom in lambda configuration that is
driven by two independent laser fields. They also included the
coherence transfer rates that couple the two allowed atomic
transitions. The squeezed vacuum together with the nonzero atomic
coherence transfer rate lead to the nonzero steady-state atomic
coherences~\cite{mt96} in a three-level atom under certain conditions.

In this paper, we will study the effect of the squeezed  vacuum on the
fluorescence and absorption spectra for a three level V-type system
driven by a coherent laser field on one of the allowed transitions,
which we call the pump transition, and damped  
to a squeezed vacuum which is tuned to either the probe transition or to
the pump transition. The spectra are calculated for the atomic
transition that is not pumped by the coherent field, which we call the
probe transition. We also take into account the atomic coherence
transfer rate. We  discuss some new features of the spectra that are
due to the squeezed vacuum. 

%-------------------------------------------------------------------------
\section{The model and equations of motion}
We consider an atom with three levels in a V configuration shown
schematically in Fig.\ref{fig1}.
The system has the lower state $|1\rangle$, and the two upper states
$|2\rangle$ and $|3\rangle$. 
The atom is pumped by a coherent field with the Rabi frequency
$\Omega$ (our $\Omega$ is in fact one half of the resonant Rabi frequency)
on the transition $|1\rangle\leftrightarrow |3\rangle$ and probed by a
weak field with the strength $\Omega_{p}$ on the 
transition $|1\rangle \leftrightarrow |2\rangle$.
Moreover, the atom is damped to a  broadband  
squeezed vacuum with the frequency $\omega_{s}$.  The master equation
for such a system takes the form   
%------------------------------------------------------ 
\begin{eqnarray}
\label{mastereq}
\frac{d\rho }{dt}&=&-{i\over \hbar}[H,\rho ]+{\cal L} \rho\, ,
\end{eqnarray}
%------------------------------------------------------
where H is the Hamiltonian of the system, which in the
rotating wave approximation is given by
%------------------------------------------------------
\begin{eqnarray}
H&=&\hbar\omega_{21}|2\rangle\langle2|+ \hbar\omega_{31}|3\rangle\langle3|+
\hbar\Omega {\rm e}^{i\,\omega_{1}\,t}\mid 1\rangle\langle3 \mid
  +\hbar\Omega_{p} {\rm e}^{i\,\omega_{2}\,t}\mid 1\rangle\langle2 \mid 
+{\rm H.c.}.
\label{hamiltonian}
\end{eqnarray}
%-------------------------------------------------------
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfysize=8cm\epsfbox{vsys.eps}
    \caption{Schematic diagram of the system}
    \label{fig1}
  \end{center}
\end{figure}
%------------------------------------------------------------------
In equation (\ref{hamiltonian}) $\omega_{i1}$ is the energy of the level
$i$ (level $1$ is assumed to have zero energy), $\omega_{1}$ is
the frequency of the pump field acting on the atomic transition $1$,
i.e., the transition $\mid 1\rangle \leftrightarrow \mid 3\rangle$,
$\omega_{2}$ is the frequency of the probe field acting on the atomic
transition $2$, i.e., the transition  $\mid 1\rangle\leftrightarrow\mid
2\rangle$. The second term
${\cal L}\rho$ which describes various damping terms, is the effect of
the reservoir on the atom and is given by~\cite{ffd95}
%--------------------------------------------------------
\begin{eqnarray}
 \label{damp}
{\cal L}\rho&=&
-\frac{1}{2}\,(\,N+1\,)\sum_{i,j}{\Gamma_{ij}}(\rho \sigma^{+}_{i}
\sigma^{-}_{j}+{\sigma}^{+}_{i} {\sigma}^{-}_{j}\rho-2 {\sigma}^{-}_{j} \rho {\sigma}^{+}_{i})\nonumber\\
&&-\frac{1}{2}\,N\, \sum_{i,j}{\Gamma_{ij}}(\rho {\sigma}^{-}_{i}
{\sigma}^{+}_{j}+{\sigma}^{-}_{i} {\sigma}^{+}_{j}\rho-2 {\sigma}^{+}_{j} \rho {\sigma}^{-}_{i})\nonumber\\
&&-\frac{1}{2} \,M\,\sum_{i,j}\,{\Gamma_{ij}}(\rho {\sigma}^{+}_{i}
{\sigma}^{+}_{j}+{\sigma}^{+}_{i} {\sigma}^{+}_{j}\rho-2
{\sigma}^{+}_{i} \rho {\sigma}^{+}_{j}) {\rm e}^{-i
2\,{\omega}_{s} t}\\ &&-\frac{1}{2}\,M^{*}\,
\sum_{i,j}\,{\Gamma_{ij}}(\rho {\sigma}^{-}_{i}
{\sigma}^{-}_{j}+{\sigma}^{-}_{i} {\sigma}^{-}_{j}\rho-2
{\sigma}^{-}_{i} \rho {\sigma}^{-}_{j}) {\rm e}^{i 2\,{\omega}_{s}
  t}\, .\nonumber{}
\end{eqnarray}
%---------------------------------------------------------
The parameters $\Gamma_{ij}$ are the relaxation rates; for $i=j$ they describe
 spontaneous damping rates $\Gamma_{11}=\gamma_{1}$,
 $\Gamma_{22}=\gamma_{2}$ from the level 
$|3\rangle \leftrightarrow |1\rangle$ and $|2\rangle \leftrightarrow
|1\rangle$, respectively,  
for $i\neq j$ we have the cross relaxation rate
$\Gamma_{12}=\Gamma_{21}=\gamma_{c}$, that  
couples the two atomic transitions and is given by~\cite{ffd95}
\begin{eqnarray}
  \label{e5}
  \gamma_{c}={{\mbox{\boldmath$\mu$}}_{13}\cdot{\mbox{\boldmath$\mu$}}_{12} 
\over {12\pi\epsilon_{0}\hbar c^{3}}}(\omega_{31}^{3}+\omega_{21}^{3}).  
\end{eqnarray}
The operators ${\sigma}_{i}^{-}$, ${\sigma}_{i}^{+}$ ($i=1,2$) are
the atomic lowering  and raising operators for the transition $i$,
i.e., ${\sigma}_{1}^{-}=\mid 1\rangle\langle 3\mid$,
${\sigma}_{2}^{-}=\mid 1\rangle\langle 2\mid$, etc. 
The parameter $N$ is the mean photon number  
of the reservoir, which in our case is the broad-band squeezed vacuum, 
and $M=|M|\exp{i\phi}$ is a complex number describing the photon
correlations present in the squeezed vacuum  with the carrier
frequency $\omega_{s}$ and the phase $\phi$. The
parameter $M$ must satisfy the inequality $|M|\leq\sqrt{N\,(\,N+1\,)}$.

The master equation (\ref{mastereq}), under the assumption that
$\Omega_{p}=0$, leads to the following set of equations for the atomic
density matrix elements (in a frame rotated with respect to
$\omega_{1}$ and $\omega_{2}$)  
%-------------------------------------------------------
\begin{eqnarray}
\label{mastereq1}
\frac {{ \partial}}{{ \partial}{t}}\,{{
    \rho}_{11}}&=&-(\,N\,(\gamma_{1}+\gamma_{2}) +
(\,N+1\,)\,\gamma_{1}\,)\,{{ \rho}_{11}}
+ {i}\,{ \Omega}\,{{ \rho}_{13}} + (\,{ N}\,+1\,)\,(\,\gamma_{1}+\gamma_{2}
\,)\,{{ \rho}_{22}}\, , \nonumber\\
& & +(\,{N}\,+1\,)\,{ \gamma_{c}}\,{\rm
 e}^{(\,{i}\,\Delta_{12}\,{t}\,)} {{ \rho}_{23}} - {i}\,{
\Omega}\,{{ \rho}_{31}} + (\,{ N}+1\,)\,{ \gamma_{c}}\,
{\rm e}^{(\,-i\,\Delta_{12}\,t)}\,{{ \rho}_{32}}
+ (\,{N}+1\,)\,{ \gamma_{1}}\, ,\nonumber\\
\frac {{ \partial}}{{ \partial}{t}}\,{{ \rho}_{12}}&=&-(
\,{\gamma_{12}} +{i}\,{ \delta_{2}}\,)\,{{ \rho}_{12}} -
\,{\displaystyle \frac {1}{2}}\,{ (N+1)}\,{ \gamma_{c}}\,
{\rm e}^{(\,\Delta_{12}\,{t} \,)} \,{{ \rho}_{13}}
+ {\it M^{*}}\,{ \gamma_{2}}\,
{\rm e}^{(\,2\,{i}\,{\Delta_{s1}}\,{t}\, )}\,{{ \rho}_{21}}\, ,\nonumber\\
& & +{\it M^{*}}
\,{ \gamma_{c}}\,{\rm e}^{(\,2\,{i}\,{\Delta }\,{t}) }\,{{ \rho}_{31}}
- {i}\,{ \Omega}\,{{ \rho}_{32}}\, ,\nonumber\\
\frac {{ \partial}}{{ \partial}{t}}\,{{ \rho}_{13}}&=&{2\,i
}\,{ \Omega}\,{{ \rho}_{11}} - \,
{\displaystyle \frac {1}{2}}\,{( N+1)}\,{ \gamma_{c}}\,{\rm e}^{(
\, - {i}\,{\Delta_{12}}\,{t}\,)} \,{{ \rho}_{12}}
- (\, { \gamma_{13}} + {i}\,{ \delta_{1}}\,)\,{{ \rho}_{
13}}\, ,\\
& &+ i\,\Omega\,\rho_{22}+ {\it M^{*}}\,{ \gamma_{c}}\,{\rm e}^{(\,2\,{i}\,{
\Delta}\,{t}\,)}\,{{ \rho}_{21}}
+{\it M^{*}}\,{ \gamma_{1}}\,{\rm
e}^{(\,2\,{i}\,{\Delta_{s2}}\,{t}\,)}\, {{ \rho}_{31}} - {i}\,{
\Omega}\, ,\nonumber\\
\frac {{ \partial}}{{ \partial}{t}}\,{{\rho}_{22}}&=
&{N}\,{ \gamma_{2}}\,{{ \rho}_{11}} - {(\,
 N+1)}\,{ \gamma_{2}} \,{{ \rho}_{22}} -{\displaystyle \frac
{1}{2}}\, {(\,N+1\,)}\,{ \gamma_{c}}\,{\rm
e}^{(\,{i}\,{\Delta_{12}}\,{t}\,)}\,{{ \rho}_{23}}\, ,\nonumber\\ 
& & \mbox{}-{\displaystyle \frac {1}{2}}\, {(\,N+1\,)}\,{ \gamma_{c}}\,{\rm
e}^{(\, - {i}\,{ \Delta_{12}}\,{t}\,)} ) \,{{ \rho}_{32}}\, ,\nonumber\\
\frac {{ \partial}}{{ \partial}{t}}\,{{\rho}_{23}}&=&
{\frac{1}{2}\,(\,3\,N+1\,)}\,{ \gamma_{c}}\,{\rm
 e}^{-i\,\Delta_{12}\,t}\,{{ \rho}_{11}} + {i}\,{ \Omega}\,{{
\rho}_{21}} -{(\,N+1\,)}\,{
\gamma_{c}}\,{\rm e}^{-i\,\Delta_{12}\,{t}} \,{{ \rho}_{22}}\, ,\nonumber\\ & &
 \mbox{} - (\, { \gamma_{23}} - {i}\,(\,{ \delta_{2}} - { \delta_{1}}))
\,{{ \rho}_{23}} -{\displaystyle \frac {
1}{2}}\,{(\,N+1\,)}\,{\gamma_{c}}\, {\rm e}^{-i\,\Delta_{12}\,{t}}\, .\nonumber{}
\end{eqnarray}
%----------------------------------------------------------------
where the parameters $\gamma_{ij}$ are given by:
\begin{eqnarray}
\label{gammas}
\gamma_{12}&=& {(\,N+1\,)}\,{ \gamma_{2}} + {\displaystyle \frac {1}{2}}
\,{ N}\,{\gamma_{1}}\, ,\nonumber{}\\
\gamma_{13}&=&{(\,N+1\,)}\,{\gamma_{1}} + {\displaystyle \frac {1}{2}}
\,{ N}\,{ \gamma_{2}}\, ,\\
\gamma_{23}&=& {\displaystyle \frac {1}{2}}\,{(\,N+1\,)}\,
(\,\gamma_{2}+ \gamma_{1}\,)\, ,\nonumber{}
\end{eqnarray}
%-----------------------------------------------------------------
 $\delta_{1}=\omega_{1}-\omega_{31}$,
$\delta_{2}=\omega_{2}-\omega_{21}$,  
 $\Delta_{12}=\omega_{1}-\omega_{2}$,
$\Delta_{si}=\omega_s-\omega_i $,
$\Delta=2\,\omega_s-\omega_{1}-\omega_{2}$, $\rho_{ji}$ is the
complex conjugate of $\rho_{ij}$,
and the trace condition $\rho_{11}+\rho_{22}+\rho_{33}=1$ have been
used in (\ref{mastereq1}). Since
we have assumed $\Omega_{p}=0$, we have actually assumed that the
atomic coherences $\rho_{12}$ and $\rho_{21}$ have been rotated with
respect to the atomic frequency $\omega_{21}$ instead of the frequency
$\omega_{2}$, so $\delta_{2}=0$, and we will simplify the notation
using $\delta_{1}=\delta$ later on.

%---------------------------------------------------------------------
\section{Resonance fluorescence and probe absorption spectrum}
%---------------------------------------------------------------------

We study here the radiative properties of the atom associated with the
atomic transition $\mid 1\rangle\leftrightarrow\mid 2\rangle$, which is
not pumped by the coherent light. The only mechanism that can populate
the state $\mid 2\rangle$ is due to the nonzero value of the mean
number of photons $N$ in the squeezed vacuum. However, the squeezed
vacuum introduces some extra couplings to the equations
(\ref{mastereq1}) through the parameter $M$, which modify the
radiative properties of the atom.
We calculate the resonance fluorescence  as well as absorption
spectrum of light on the probe transition showing the appearance of an
additional structure that is introduced by the squeezed vacuum. 

The fluorescence spectrum is defined by the Fourier transform of the
correlation function 
\begin{eqnarray}
{\cal F}(t,t+\tau)=\langle{E^{(-)}(r,t)\,E^{(+)}(r,t+\tau)}\rangle 
\label{F1}
\end{eqnarray}
of the electric field emitted by the atom,
where $E^{(\pm)}$ are the positive and negative frequency parts of
the total field operator. The field correlation function (\ref{F1}) is
proportional to the atomic correlation function
\begin{eqnarray}
{\cal F}(t,t+\tau) \sim \langle\sigma^{+}(t) \sigma^{-}(t+\tau)\rangle\, .
\label{F2}
\end{eqnarray}
According to the quantum regression theorem, this two-time correlation
function can be calculated from the equations of motion for the
expectation values of the atomic lowering operator 
$\langle\sigma^{-}(t+\tau)\rangle$. Once the atomic correlation
function (\ref{F2}) is known, the steady-state resonance fluorescence
spectrum can be calculated as the Fourier transform of this function
from the formula 
%---------------------------------------------------------
\begin{eqnarray}
F(\omega)={\rm Re}
\int_{0}^{\infty}\langle\sigma^{+}(t)\sigma^{-}(t,t+\tau)\rangle {\rm
  e}^{i \omega\tau}{\rm d}\tau\, .
\label{eqfl}
\end{eqnarray}
%--------------------------------------------------------
Similarly, the absorption spectrum can be calculated according to the
formula~\cite{Mollow72} 
\begin{eqnarray}
S(\omega)={\rm Re}
\int_{0}^{\infty}{\langle[\sigma^{+}(t+\tau),\sigma^{-}(t)]\rangle}
{\rm e}^{i \omega \tau}{\rm d}\tau .
\label{eqab}
\end{eqnarray}
%-----------------------------------------------------------
The correlation functions (\ref{eqfl}) and (\ref{eqab}) are calculated
using the quantum regression theorem in the absence of the probe field
using  equations (\ref{mastereq1}). 
Generally, equations (\ref{mastereq1}) cannot be solved directly
because the matrix of the coefficients  is time dependent. To get rid
of this time dependence we  
choose the carrier frequency $\omega_s$ of the squeezed vacuum to be
resonant, or nearly resonant, with one of the allowed transitions, and we
discard in equations (\ref{mastereq1}) all rapidly oscillating terms
(secular approximation).

%---------------------------------------------------------------------
\subsection{Squeezed vacuum frequency tuned to the probe transition}
%---------------------------------------------------------------------

We assume  that the atom is non-degenerate, so the fast oscillating
terms that oscillate at the frequency $\Delta_{12}$ can be neglected,
and if the squeezed vacuum is tuned to the probe
transition ($\Delta_{s1}=0$), the terms that oscillate at the
frequencies $\Delta_{s2}$ and $\Delta$ are also fast oscillating,  and they
can thus be neglected.  
As a result the  equations (\ref{mastereq1}) can be written in the form
%-----------------------------------------------------------
\begin{eqnarray}
\label{eq1}
\frac {{ \partial}}{{ \partial}{t}}\,{{
    \rho}_{11}}&=&-(\,N\,(\,\gamma_{1}+\gamma_{2}\,)+ 
(\,N+1\,)\,\gamma_{1}\,)\,{{ \rho}_{11}} + {i}\,{ \Omega}\,{{
\rho}_{13}} + (\,{ N}\,+1\,)\,(\,\gamma_{2}+\gamma_{1} \,)\,{{
\rho}_{22}}\, ,\nonumber\\ & & - {i}\,{\Omega}\,{{ \rho}_{31}} + (\,{N}+1\,)\,{
\gamma_{1}}\, ,\nonumber\\ \frac {{ \partial}}{{ \partial}{t}}\,{{
\rho}_{12}}&=&-\,{\gamma_{12}}\,{{ \rho}_{12}}
+ {\it M^{*}}\,{ \gamma_{2}}\,{{ \rho}_{21}}-{i}\,{ \Omega}\,{{
\rho}_{32}}\, ,\nonumber\\ \frac {{ \partial}}{{ \partial}{t}}\,{{
\rho}_{13}}&=&{2\,i }\,{ \Omega}\,{{ \rho}_{11}}- (\, {\gamma_{13}}
+ {i}\,{ \delta}\,)\,{{ \rho}_{ 13}}+ i\,\Omega\,\rho_{22}-
{i}\,{\Omega}\, ,\\ 
\frac {{ \partial}}{{
\partial}{t}}\,{{\rho}_{22}}&= &{N}\,{ \gamma_{2}}\,{{ \rho}_{11}} -
{(\, N+1)}\,{ \gamma_{2}} \,{{ \rho}_{22}}\, ,\nonumber\\ \frac {{ \partial}}{{
\partial}{t}}\,{{\rho}_{23}}&=& {i}\,{ \Omega}\,{{\rho}_{21}}- (\,
{\gamma_{23}} + {i}\,{ \delta}\,)
 \,{{\rho}_{23}}\, .\nonumber{}
 \end{eqnarray}
%-----------------------------------------------------------------

The steady-state solution to equations (\ref{eq1}) can be found by
putting their left hand side equal to zero and  solving the resulting
algebraic set of equations. In this way we get formulas for the
steady-state values  of the populations $\rho_{22}$, $\rho_{11}$  
and the coherence $\rho_{31}$. The steady-state value for the
coherence $\rho_{12}$ is equal to zero, as one could expect because
this transition is not pumped.
The results are found to be 
%----------------------------------------------------
\begin{eqnarray}
{\rho_{11}}&=&\frac {(\,N + 1\,)((\,N+1\,)\,{
    \gamma_{1}}\,{\gamma_{13}}+ 2\,{\Omega}^{2})} 
{(\,N+1\,)(\,3\,N+1\,)\,\gamma_{1}\,{\gamma_{13}}+2\,(\,3\,N+2\,)\,
{\Omega}^{2}}\, ,\nonumber{}\\
{\rho_{22}}&=&\frac{N((\,N+1\,)\,{\gamma_{1}}\,{\gamma_{13}}+2\,{\Omega}^{2})} 
{(\,N+1\,)\,(\,3\,N+1\,)\,\gamma_{1}\,\gamma_{13}+2\,(\,3\,N+2\,)\,
{\Omega}^{2}}\label{ss1}\, ,\\
\rho_{31}&=&\frac { - {i}\,(\,N +
  1\,)\,{\gamma_{1}}\,{\Omega}}{(\,N+1\,)(\,3\,N+1\,)\,\gamma_{1}\, 
\gamma_{13}+2\,(\,3\,N+2\,)\,{\Omega}^{2}}\, .  
\nonumber{}
\end{eqnarray}
It is clear from the solutions (\ref{ss1}) that they depend on the
mean number of photons in the squeezed vacuum $N$, but they do not
depend on the parameter $M$ describing correlations in the squeezed
field. For $\Omega=0$, $\rho_{31}=0$, and the atomic populations are
the same as in a thermal field with the same mean number of
photons. This means that the steady-state expectation values of the
atomic operators are not affected by the presence of photon
correlations in the squeezed vacuum. The results do not depend also on
the coherence transfer rate $\gamma_{c}$, given by (\ref{e5}). The
situation can, however, be different when the fluorescence and/or
absorption spectrum is calculated. The spectra are defined by the
two-time correlation functions (\ref{eqfl}) and (\ref{eqab}) that can
be sensitive to the presence of correlations in the squeezed light. We
will show that this is really the case.

We calculate the  fluorescence spectrum and the
absorption spectrum for the probe transition, using  formulas
(\ref{eqfl}) and (\ref{eqab}), for the case when the squeezed vacuum frequency
is resonant with the atomic transition $|1\rangle \leftrightarrow
|2\rangle$. This means that we calculate the spectra for the atomic
transition that is not pumped by the laser field. The mechanism that
populates state $\mid 2\rangle$ is due to the squeezed vacuum field
with the mean number of photons $N$, as it is clear from (\ref{ss1}).   
The spectra are given by
%--------------------------------------------------------------------
\begin{eqnarray}
F(\omega)&=&{\rm Re}
\left.
\left\{
\frac{\displaystyle(z+\gamma_{23}-i \delta)\,\rho_{22}}
{A_{-}(z,\Omega)-
\displaystyle{\frac{\gamma_{2}^{2}((z+\gamma_{23})^2+\delta^2)}
{A_{+}(z,\Omega)}|M|^2}}
\right|_{z=i\,(\omega -\omega_{21})}
\right\}\, ,
\label{fl2}\\
S(\omega)&=&{\rm Re}
\left.
\left\{
\frac{\displaystyle
(z+\gamma_{23}-i\delta)\,(\,\rho_{11}-\rho_{22}\,)
-i\,\Omega\,\rho_{31}}
{A_{-}(z,\Omega)-
{\displaystyle\frac{\gamma_{2}^{2}((z+\gamma_{23})^2+\delta^2)}
{A_{+}(z,\Omega)}|M|^2}}
\right|_{z=i\,(\omega -\omega_{21})}
 \right\}\, ,
\label{ab2}
\end{eqnarray}
%---------------------------------------------------------------------
where
%---------------------------------------------------------------------
\begin{eqnarray}
A_{\pm}(z,\Omega)&=&z^2 +(\gamma_{12}+\gamma_{23})\,z
+\gamma_{12}\gamma_{23} +\Omega^2\pm {i} \delta (z +\gamma_{23})\, .
\end{eqnarray}
%---------------------------------------------------
%-----------------------------------------------------------------
\begin{figure}[ht]
\centerline{\epsfysize=8cm\epsfbox{azum.eps}}
\caption{{The fluorescence spectrum on the probe transition when the 
squeezed vacuum frequency is tuned to the probe transition for 
different values of $\Omega=1.7,\, 7.7,\, 13.7$. The other parameters 
are: $N=1,M=\protect\sqrt{2}$, $\gamma_{1}=\gamma_{2}=1$, $\Omega=10$, and 
$\delta=0$. Solid line represents the spectrum for the squeezed vacuum 
and dotted line for thermal field.}}  
\label{flv21pro} 
\end{figure} 
%--------------------------------------------------------------------
It is clear from (\ref{fl2}) and (\ref{ab2}) that for $M=0$, the
denominator is a polynomial of the second 
order in $z$. This means that for the ordinary vacuum or thermal light
the spectrum can show only two Lorentzian peaks. The situation is
different for $M\ne 0$ in which case there can generally be four
different Lorentzian contributions to the fluorescence spectrum.
 
In the case when the pump field is resonant with the atomic transition
$|1\rangle-|3\rangle$ ($\delta=0$)  the fluorescence  spectrum can be
rewritten in the simple form
%--------------------------------------------------
\begin{eqnarray}
\label{fleq0}
F(\omega) &=&{\rho_{22} \over 2}\,{\rm Re}
\left\{
\left[\frac{z+{
\gamma_{23}}}
{(z+\gamma_{12}+M
  \gamma_{2})(z+\gamma_{23})+\Omega^2}\right.\right.\nonumber{}\\ 
&&+\left.\left.\left.\frac{z+{\gamma_{23}}}
{(z+\gamma_{12}-M\gamma_{2})(z+\gamma_{23})+\Omega^2}\right]\right|_{z=i\,(\omega-\omega_{21})}\right\}.
\end{eqnarray}
It is clear from the expression (\ref{fleq0}), which is a sum of two
terms each of which has the second order polynomial in the denominator
and thus represents two lines, that generally the
spectrum is composed of four lines which are symmetrically disposed
with respect to the frequency $\omega_{21}$.
To confirm this statement we decompose the spectrum (\ref{fleq0}) into
the four Lorentzians explicitly. Beside the four Lorentzians the
spectrum contains also some dispersion-like terms, which, however, are
small in the strong field limit. The strong field limit means in this
case that the following condition is satisfied
\begin{eqnarray}
  \label{fl3}
  4\,(\gamma_{12}\,\gamma_{23}+\Omega^{2}) > (\gamma_{12}+\gamma_{23}+|M|\gamma_{2})^{2}.
\end{eqnarray}
In the strong field limit the spectrum (\ref{fleq0}) can be rewritten
in the form 
\begin{eqnarray}
  \label{lorentz}
F(\omega)&=&{\rho_{22} \over 4}\left\{\left[{\gamma_{+}
    \over{(\omega-\omega_{21}-\Omega_{+})^{2}}+\gamma_{+}^{2}}+{\gamma_{+}
    \over{(\omega-\omega_{21}+\Omega_{+})^{2}}+\gamma_{+}^{2}}\right]\right.
\nonumber{}\\ 
&&+\left[{\gamma_{-}
    \over{(\omega-\omega_{21}-\Omega_{-})^{2}}+\gamma_{-}^{2}}+{\gamma_{-}
    \over{(\omega-\omega_{21}+\Omega_{-})^{2}}+\gamma_{-}^{2}} \right]
\nonumber{}\\
&&+{1 \over
  \Omega_{+}}\left[{{(\gamma_{+}-\gamma_{23})(\omega-\omega_{21}
 -\Omega_{+})}\over
(\omega-\omega_{21}-\Omega_{+})^{2}+\gamma_{+}^{2}}    
-{{(\gamma_{+}-\gamma_{23})(\omega-\omega_{21} 
 +\Omega_{+})}\over
(\omega-\omega_{21}+\Omega_{+})^{2}+\gamma_{+}^{2}}\right]\\ 
&&+{1 \over
  \Omega_{-}}\left.\left[{{(\gamma_{-}-\gamma_{23})(\omega-\omega_{21}
 -\Omega_{-})}\over
(\omega-\omega_{21}-\Omega_{-})^{2}+\gamma_{-}^{2}}    
-{{(\gamma_{-}-\gamma_{23})(\omega-\omega_{21} 
 +\Omega_{-})}\over
(\omega-\omega_{21}+\Omega_{-})^{2}+\gamma_{-}^{2}}\right]\right\}\, ,
\nonumber{}
\end{eqnarray}
where
\begin{eqnarray}
  \label{lor1}
  \gamma_{\pm}&=&{1\over 2}(\gamma_{12}+\gamma_{23}\pm
  |M|\gamma_{2})\, ,\\
\Omega_{\pm}&=&{1\over
  2}\sqrt{4\,(\gamma_{12}\,\gamma_{23}+\Omega^{2})-\gamma_{\pm}^{2}}\, .
\label{lor2} 
\end{eqnarray}
%-----------------------------------------------------------------
\begin{figure}[ht]
\subfigure[]{\label{absp1}\epsfysize=7cm\epsfbox{absp1.eps}}
\subfigure[]{\label{flsp1}\epsfysize=7cm\epsfbox{flusp1.eps}}
\centering
\caption{(a)--The absorption spectrum $S(\omega-\omega_{21})$ on the
  transition  $1{\leftrightarrow}2$ for the squeezed vacuum frequency
  tuned to the probe transition 
(b)--Fluorescence spectrum $F(\omega-\omega_{21})$ for the transition
1$\leftrightarrow$2.   
 The parameters are:
$\gamma_{1}=0.01$, $\gamma_{2}=1$,
$\Omega_{2}=14$, $\delta_2=-20$, $N=0.8$, $M=\protect\sqrt{N(N+1)}$.}
\label{fp101}
\end{figure}
%--------------------------------------------------------------------
It is clear from equation (\ref{lorentz}) that for the squeezed vacuum
($|M|\ne 0$) the resonance fluorescence spectrum on the probe
transition consists of four Lorentzian lines located symmetrically with
respect to the transition frequency $\omega_{21}$ at the frequencies
$\Omega_{\pm}$ and having the widths $\gamma_{\pm}$. Beside the four
Lorentzians the spectrum contains some dispersion-like terms, which
are of the order of $\Omega_{\pm}^{-1}$ and are negligible if
$\Omega_{\pm}\gg \gamma_{\pm}$. The presence of the four lines in the
fluorescence spectrum is the result of the squeezed vacuum. This
effect is associated with the different damping rates for the two
quadrature components of the atomic dipole moment in the squeezed
vacuum. In the
case of thermal field, $|M|=0$, with the mean number of photons $N$
 the four lines merge into two lines. Since the difference
$\Omega_{+}-\Omega_{-}$ in the strong field limit is small and the
width $\gamma_{+}$ is large, the peaks associated with the frequencies
$\Omega_{+}$ and $\Omega_{-}$ cannot be separated in practice. The two
peaks merge into one peak, but the resulting peak is narrower than the
peak obtained for the thermal field with the same mean number of photons. 
Similar decomposition can be performed for the absorption spectrum.

Examples of the fluorescence spectrum are shown in Fig.\ref{flv21pro}.
It is seen that in the presence of the squeezed vacuum the
fluorescence spectrum is narrower than the corresponding spectrum for
thermal light with the same number of photons. For strong driving
fields the spectrum splits into two lines. Each of the two lines is
composed of two components with different widths, but effectively they
appear as a single line because the splitting due to the different
widths resulting in the squeezed vacuum are too small to be resolved
within the line-width. To see the presence of the extra structure in
the spectrum the driving field must be considerably detuned from the
resonance. In the case of large detuning the lines should be separated
and one can expect the appearance of additional structure in the
spectrum. That it is really the case we have shown in Fig.\ref{fp101},
where we have plotted the fluorescence as well as absorption spectra
for large detuning to make the extra structure visible. As it is seen
there are really small dips that appeared in the spectra. The
additional structure is, however, 
hardly visible, and the most essential result of the squeezed vacuum
is the narrowing of the spectrum that is clearly seen in Fig.\ref{flv21pro}.

%------------------------------------------------------------------------
\subsection{Squeezed vacuum frequency tuned to the pump transition}
%-------------------------------------------------------------------------
In the case when the squeezed vacuum frequency is close to resonance with the
pump transition  $|1\rangle{\leftrightarrow}|3\rangle$, we
substitute $\Omega_{p}=0$ and $\Delta_{s2}=0$ in  equation
(\ref{mastereq1}), and we neglect the fast oscillating terms that 
oscillate at the frequencies  $\Delta_{12}$ and  $\Delta$.
The master equation (\ref{mastereq1}) gives us then
\begin{eqnarray}
\label{eq2}
\frac {{ \partial}}{{ \partial}{t}}\,{{ \rho}_{11}}&=&-(\,N\,(\,\gamma_{1}+\gamma_{2}\,)+(\,N+1\,)\,\gamma_{1}\,)\,{{ \rho}_{11}} + {i}\,{ \Omega}\,{{
\rho}_{13}} + (\,N\,+1\,)\,(\,\gamma_{2}+\gamma_{1} \,)\,{{
\rho}_{22}}\, ,\nonumber\\ & & - {i}\,{\Omega}\,{{ \rho}_{31}} + (\,{N}+1\,)\,{
\gamma_{1}}\, ,\nonumber\\ 
\frac {{ \partial}}{{ \partial}{t}}\,{{
\rho}_{12}}&=&-\,{\gamma_{12}}\,{{ \rho}_{12}}
-{i}\,{ \Omega}\,{{\rho}_{32}}\, ,\nonumber\\
\frac {{ \partial}}{{ \partial}{t}}\,{{
\rho}_{13}}&=&{2\,i }\,{ \Omega}\,{{ \rho}_{11}}- (\, {\gamma_{13}}
+ {i}\,{ \delta}\,)\,{{ \rho}_{ 13}}+
i\,\Omega\,\rho_{22}+M^{*}\,\gamma_{1}\,\rho_{31}-{i}\,{\Omega}\, ,\\
\frac {{ \partial}}{{\partial}{t}}\,{{\rho}_{22}}&= &{N}\,{
  \gamma_{2}}\,{{ \rho}_{11}} - {(\, N+1)}\,{ \gamma_{2}} \,{{
    \rho}_{22}}\, ,\nonumber\\ 
\frac {{ \partial}}{{\partial}{t}}\,{{\rho}_{23}}&=& {i}\,{
  \Omega}\,{{\rho}_{21}}-(\,{\gamma_{23}} +
{i}\,\delta)\,{{\rho}_{23}}\, .\nonumber{}
 \end{eqnarray}
The steady-state solutions to (\ref{eq2}) can be found analytically in
general form, but the formulas are rather involved, so we only adduce
them here for the special case when  $\delta=0$.
\begin{eqnarray}
\label{ss2}
\rho_{11}&=&
\frac
{(\,N+1\,)\,\left[\,2\,\gamma_{13}^{(-)}(\phi)\,\Omega^2+(\,N+\,1)\,
\gamma_{23}^{(-)}(0)\,\gamma_{23}^{(+)}(0)\,\gamma_{1}\,\right]}
{2\,(\,3\,N\,+2\,)\,\gamma_{13}^{(-)}(\phi)\,\Omega^2+
(\,3\,N+1\,)\,(\,N+1\,)\gamma_{13}^{(-)}(0)\,\gamma_{13}^{(+)}(0)\,\gamma_{1}
}\, ,\nonumber{}\\
\rho_{22}&=&\frac
{N\,\left[\,2\,\gamma_{13}^{(-)}(\phi)\,\Omega^2+(\,N+\,1)\,
\gamma_{23}^{(-)}(0)\,\gamma_{23}^{(+)}(0)\,\gamma_{1}\,\right]}
{2\,(\,3\,N\,+2\,)\,\gamma_{13}^{(-)}(\phi)\,\Omega^2+
(\,3\,N+1\,)\,(\,N+1\,)\gamma_{13}^{(-)}(0)\,\gamma_{13}^{(+)}(0)\,\gamma_{1}
}\, ,\\
\rho_{31}&=&\frac
{-i\,(\,N+1\,)(\,|M| \gamma_{1}\,e^{i\,\phi}\,-\gamma_{13}\,)\,\gamma_{1}\,\Omega}
{2\,(\,3\,N\,+2\,)\,\gamma_{13}^{(-)}(\phi)\,\Omega^2+
(\,3\,N+1\,)\,(\,N+1\,)\gamma_{13}^{(-)}(0)\,\gamma_{13}^{(+)}(0)\,\gamma_{1}
}\, ,\nonumber{}
\end{eqnarray}
where
\begin{eqnarray}
  \label{gam}
 \gamma_{mn}^{(\pm)}&=&\gamma_{mn}\pm |M|\,\gamma_{1}\,\cos \phi\, .
\end{eqnarray}
It is seen from (\ref{ss2}) that when the squeezed vacuum acts on the
same transition as the coherent field the atomic density matrix
elements depend on the phase $\phi$ of the squeezed light. This is the
effect known from the two-level system damped to the squeezed vacuum
\cite{clw87,par93}, which leads to different damping rates for the two
quadrature components of the atomic dipole associated with the
transition $\mid 1\rangle\leftrightarrow \mid 3\rangle$.
 
%-----------------------------------------------------------------
\begin{figure}[ht] 
\centerline{\epsfysize=8cm\epsfbox{flv21.ps}} 
\caption{{ The fluorescence spectrum on the probe transition
when the squeezed vacuum frequency is tuned to the pump transition. There is
no difference in the scale of the figure between the thermal field and
the squeezed vacuum. The parameters are: $\gamma_{1}=1$, $\gamma_{2}=0.001$,
$\delta=0$, $\Omega=10$, $\phi=\pi /10$, $N=5$, and $M=\protect\sqrt{30}$.}}
 \label{flv21pu}
\end{figure}
%--------------------------------------------------------------------
\begin{figure}[ht]
\centerline{\epsfysize=8cm\epsfbox{gain.eps}}
\caption{The absorption spectrum on the probe
transition when the squeezed vacuum frequency is tuned to the pump
 transition for $M=\protect\sqrt{30}$ (solid line), and for $M=0$ 
 (dashed line).  The other parameters are the same
as in Fig.\ref{flv21pu}.} 
\label{abv21pu} 
\end{figure} 
The fluorescence and absorption spectra for this case are given by
\begin{eqnarray}
F(\omega)&=&{\rm
    Re}\,\left\{\left.\frac{(z+\gamma_{23}-i\,\delta)\rho_{22}} 
{(z+\gamma_{12})(z+\gamma_{23}-i\,\delta))+\Omega^2}\right|_{z=i(\omega
-\omega_{21})}\right\}\, ,
\label{fl22}\\ 
S(\omega)&=&{\rm Re}\,\left\{\left.\frac
{(z+\gamma_{23}-i\,\delta)(\rho_{11}-\rho_{22})-i \Omega
  \rho_{31}}
{(z+\gamma_{12})(z+\gamma_{23}-i\,\delta)+\Omega^2}\right|_{z=i(\omega
-\omega_{21})}\right\}\, .
\label{ab22}
\end{eqnarray}
From equations (\ref{fl22}) and (\ref{ab22}) it is clear that when the
squeezed vacuum acts on the pump transition, both the fluorescence and
absorption spectra on the probe transition do not exhibit, contrary to
the previous case, any additional structure. Their dependence on the
squeezed vacuum parameter $M$ is only through the atomic populations
and coherence given by (\ref{ss2}). 
For high Rabi frequencies the population $\rho_{22}$ of the upper lasing level
is independent of $M$ as well as damping rates, and it is given by
%-------------------------------------------------------------------
\begin{eqnarray}
\rho_{22}=\frac{N}{3\,N+2}\, .
\end{eqnarray}
%----------------------------------------------------------------------
The same is also true for low Rabi frequencies, for which the
population $\rho_{22}$ is equal to
%--------------------------------------------------------------------
\begin{eqnarray}
\rho_{22}=\frac{N}{2\,(\,3\,N+1\,)(\,3\,N+2\,)}\, .
\end{eqnarray}
%--------------------------------------------------------------------
The examples of the fluorescence and absorption spectra for the pump
transition are shown in Fig.\ref{flv21pu} and Fig.\ref{abv21pu}. The
fluorescence spectrum shown in Fig.\ref{flv21pu} exhibits two peaks
the presence of which is caused by the splitting of the state
$|1\rangle$ in the strong field. For the choice of atomic parameters
in Fig.\ref{flv21pu} there is no visible difference between the
fluorescence in the squeezed vacuum and a thermal field with the same
number of photons. The absorption spectrum illustrated in
Fig.\ref{abv21pu} also reveals the two-peak structure but with
absorption on one side of the resonance and gain (negative absorption)
on the other side. This is the gain without population inversion in the bare
atomic states. Since the atomic populations (\ref{ss2}) depend on the
phase of the squeezed vacuum, the amplification of the probe light
depends crucially on this phase. The structure of the spectra is
better understood in  the dressed-state picture discussed in the next Section.
%-----------------------------------------------------------------

\section{Dressed state picture}
In the strong field limit the dressed-atom picture \cite{CR77,B-C77}
provides useful description of the interaction of atom with strong
fields. In this 
approach, eigenstates of the coupled atom+driving field system, 
referred to as the dressed states, serve as the basis for the 
whole system.  For our model the semi-classical dressed states are the 
eigenvectors $|2\rangle$ and $|\pm\rangle$ of the Hamiltonian 
\begin{eqnarray} 
H=\hbar\,\left[ \begin{array}{ccc} 0&0&\Omega\\ 0&\omega_{21}&0\\ 
 \Omega&0&-\delta
 \end{array}
\right]\, .
\end{eqnarray}
\begin{figure}[h]
\centerline{
\epsfysize=8cm\epsfbox{dress.eps}}
\caption{The semi-classical dressed states of the atom pumped by the
  strong field with the Rabi frequency $\Omega_{R}$ on the
  $1\leftrightarrow 3$ transition and the possible transitions leading
  to fluorescence on the $1 \leftrightarrow 2$ transition}
\label{qr}
\end{figure}
The dressed states are given by
\begin{eqnarray}
|\pm\rangle=\frac{1}{\sqrt{2}}\,\sqrt{1\pm {\delta \over
    \Omega_{R}}}\,|1\rangle\,\pm{1 \over \sqrt{2}}\,\sqrt{1\mp {\delta
    \over \Omega_{R}}}\,|3\rangle
\end{eqnarray}
with eigenvalues equal, respectively, to
\begin{eqnarray}
E_2&=&\hbar \omega_{21}\nonumber{}\, ,\\
E_{\pm}&=&-\frac{1}{2}\,
\hbar\delta\,\pm \,{1\over 2}\hbar \Omega_{R}\, ,
\end{eqnarray}
and the Rabi frequency $\Omega_{R}$ equal to
\begin{eqnarray}
\Omega_{R}&=&\sqrt{\delta^{2}+4 \Omega^{2}}\, .
\end{eqnarray}

In the case when the squeezed vacuum is tuned to the probe transition
there are two peaks in the fluorescence spectrum that can be easily
explain as the transitions from the atomic level $|2\rangle$ to the two
dressed states $|\pm\rangle$. For thermal field, when $|M|=0$, and for
$\delta=0$, these are exactly two peaks as it is seen from the
analytical formula (\ref{lorentz}). In the squeezed vacuum each of the peaks
is composed of two separate peaks with different widths. It is,
however, difficult to separate the two components of each peak because
of the small distance between the components and the large width of
 one of them. A very weak additional structure in the fluorescence as
well as absorption spectrum can be seen from Fig.\ref{fp101} for large
values of the detuning $\delta$. 
%--------------------------------------------------------------------
\begin{figure}[ht]
\centerline{\epsfysize=8cm\epsfbox{gph.eps}}
\caption{{The absorption spectrum at the frequency $\omega=\omega_{21}-\Omega$ 
(solid line) and at $\omega=\omega_{21}+\Omega$ (dashed line) versus
the squeezed vacuum phase, when the squeezed vacuum is tuned to the
pump transition. The parameters are: $\gamma_{1}=1,\gamma_2=0.001,
\Omega=10,\delta=0,N=5,M=\protect\sqrt{30}$.}} 
\label{abr} 
\end{figure}
%---------------------------------------------------------------------
\begin{figure}[ht]
\centerline{\epsfysize=8cm\epsfbox{dph1.eps}}
\caption{{The dressed-state population inversion
    $\rho_{--}-\rho_{22}$ (solid line) and $\rho_{++}-\rho_{22}$ 
(dashed line) versus the squeezed vacuum 
phase, when the squeezed vacuum is tuned to the pump transition. The 
parameters are are the same as Fig.\ref{abr}.}} 
\label{din} 
\end{figure}
%---------------------------------------------------------------------
In the case when the squeezed vacuum frequency is tuned to the pump transition
the fluorescence spectrum on the probe transition  shows only two
peaks that reflect the structure of dressed states and the form of the
spectrum is not affected by the squeezed vacuum parameter $M$. The
squeezed vacuum modifies in this case only the steady-state solutions
for the atomic populations and coherences, as it is seen from formulas
(\ref{ss2}). 

The absorption spectrum shown in Fig.\ref{abv21pu} reflects the same
structure of the dressed states exhibiting two absorption peaks in the
thermal field. In the squeezed vacuum, however, we can observe
narrowing of the absorption peak on one side, and the negative
absorption, i.e., the gain on the other side of the resonance.
Amplification without population inversion can occur on this side.
The amplification of the probe field
depends strongly on the phase of the squeezed vacuum and the mechanism
behind this effect is shown in Fig.\ref{din}. There is population
inversion between in the  dressed states leading to the gain, and
there is no population inversion on the absorption side.
At low $N$ the absorption spectrum shows absorption of the probe
field. By increasing $N$ we can get amplification or absorption
depending on the phase of the squeezed vacuum. 

We can see from our formulas for the fluorescence and absorption
spectra, for the two 
cases considered in this paper, that for the case when the squeezed
vacuum is tuned to the probe transition, the spectrum itself depends on
 $|M|$, but the steady-state populations and coherences do not depend on the 
squeezed vacuum parameter $M$. Quite differently, in the case when the
squeezed  vacuum frequency is tuned to the pump transition, the
spectrum on the probe transition depends on the squeezed vacuum
parameter $M$ through the steady-state populations and coherence only.
%---------------------------------------------------------------------
\section{Conclusions}
In this paper we have studied the radiative properties of a
three-level atom in $V$ configuration that is pumped by a coherent
laser field on one of the allowed transitions and probed on the other
transition. The main idea of the paper was to study resonance
fluorescence and absorption spectra on the probe transition taking
into account the fact that the other transition is coherently pumped
and assuming that the atom is damped to the squeezed vacuum instead of
the ordinary vacuum or thermal field. Moreover, we have included the
non-zero coherence transfer rate that couples the two atomic
transitions. We have considered two different situations in which the
carrier frequency of the squeezed vacuum was tuned either to the probe
transition or to the pump transition. When the squeezed vacuum
frequency is tuned to a particular atomic transition we have in fact
the situation similar to the two-level atom damped to the squeezed
vacuum, but the presence of the other atomic transition changes,
sometimes in an essential way,  the radiative properties of the atom with
respect to those observed in the two-level atom. We have found
analytical formulas for the fluorescence and absorption spectra and
illustrated them graphically for certain choices of the atomic and
field parameters. Our results show some similarities and differences
with respect to the two-level atom spectra. We have found narrowing of
the spectra in the squeezed vacuum, which is known from the two-level
case, but we have also found a weak additional structure in the
spectra. When the frequency of the squeezed vacuum is tuned to the
pump transition and the pump field is strong, the absorption spectrum
on the probe transition has two peaks at Rabi sidebands: one of them
shows absorption and the other amplification of the probe field. In
this case we get absorption without inversion in the bare atomic
states, but as our analysis has shown, the mechanism for this gain is
related to the population inversion in the dressed states. We have
also shown that for the two cases of the squeezed vacuum considered in
this paper the atomic coherence transfer rate has no influence on the
atomic spectra. In the strong-field limit the dressed states appeared
to be a good tool for explaining the features of the spectra.

\section*{Acknowledgments}
This work was supported by the Polish Committee for Scientific
Research (KBN) under grant No. 2 P03B 128 8
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\end{document}









