\section{Introduction}

One of the most remarkable effects predicted by quantum field theory
is the Lamb shift, the shift of the energy levels of an atom which is
caused by the coupling to the quantum vacuum. It is known that this
level shift can be modified by external influences like a cavity
\cite{Meschede92}, for example. Its presence alters the mode structure of
the vacuum and leads to a Lamb shift which is different from its
free-space value.

In this paper, we study the effect of acceleration on radiative
energy shifts. It may not seem obvious at first sight why acceleration
should lead to a modification of the Lamb shift. To see this, one has to
combine results from different subfields of physics. First we note that
for a uniformly accelerated observer, the Minkowski vacuum appears as a thermal
heat bath of so-called ``Rindler particles''. This is usually interpreted as
a consequence of the
non-equivalence of the particle concept in inertial and accelerated frames
\cite{Fulling73,Unruh76,Birrell82}. The second ingredient we need is
the fact that the presence of photons leads to additional energy shifts
for atomic systems. This effect is called AC Stark shift or light shift
\cite{Cohen-Tannoudji92} and is connected with the virtual absorption and
emission of real photons. In particular, a thermal photon field causes
such an effect \cite{Barton72,Knight72,Farley81}. Consequently, taking these
results together, we can gain a heuristic insight why the Lamb shift of an
atom is modified if the atom is uniformly accelerating. A corresponding
effect is to be expected for other types of acceleration.

The first aim of this paper is the
calculation of radiative energy shifts. It can be carried out in an
elegant manner using the formalism of Dalibard, Dupont-Roc, and
Cohen-Tannoudji (DDC) \cite{Dalibard82,Dalibard84}. This approach has
the advantage that it allows also to separate the
contributions of vacuum fluctuations and radiation reaction to the
energy shifts. The independent treatment of these two effects
has a tradition in Heisenberg picture quantum electrodynamics
\cite{Ackerhalt73,Senitzky73,Milonni73,Ackerhalt74,Milonni75}
and beyond \cite{Barut90}.
In a previous paper \cite{Audretsch94}, we have studied the influences
of vacuum fluctuations and radiation reaction on the spontaneous transitions
of a uniformly accelerated atom moving through the Minkowski vacuum
It leads to a modified value of the Einstein A coefficient for spontaneous
emission. In addition we have shown how the lack of a balance between
vacuum fluctuations and radiation reaction causes a spontaneous excitation
from the ground state. This gives an interpretation of the physics
underlying the Unruh effect. For the radiative shift of atomic levels, the
separate discussion of vacuum fluctuations and radiation reaction may
also be interesting from a conceptual point of view, since in heuristic
pictures the Lamb shift has been often associated with the notion of
vacuum fluctuations alone \cite{Welton48}. This discussion is the second
aim of this paper.

The fully realistic calculation of the influence of acceleration on the Lamb
shift would require to deal with a multilevel atom coupled to the
electromagnetic field. This will not be done in the present paper.
To keep the discussion as clear and transparent as possible, we will
restrict to the simplest nontrivial example, a two-level atom in
interaction with a massless scalar quantum field. However, as we will see,
most of the essential features of the full problem are also present in
the simple model. The structure of the results can be seen clearly.
Furthermore, we will discuss only the nonrelativistic contribution
to the energy shift, i. e. we neglect the effects which are due to
the quantum nature of the electron field. However, it is well known
that the nonrelativistic part gives the dominant contribution to the
Lamb shift. Since we are only interested in the structure of the
results, we will concentrate on this part of the problem.

Our treatment applies the formalism of DDC to the case of a two-level
atom coupled to a scalar quantum field and generalizes it to an
arbitrary stationary trajectory of the atom. We will proceed as
follows: We consider the time evolution of an arbitrary atomic observable
$G$ as given by the Heisenberg equations of motion. Since we are only
interested in atom variables, we trace out the field degrees of freedom
in the part of the Heisenberg equations that is due to the coupling with
the field. It is then possible to identify in the resulting expressions
unambiguously the part that acts as an effective Hamiltonian with respect
to the time evolution of $G$. The expectation value of this operator
in an atom state $|b \rangle$ gives the radiative energy shift of this
level.

We must emphasize that the modification of radiative level shifts by
acceleration which is considered here must carefully be distinguished
from the direct influence of acceleration or curvature on the energy levels
of an atom \cite{Parker80,Audretsch93}. These corrections are obtained
in the simplest case by the inclusion of a term $amx$ into the Dirac
equation of the atom. They are assumed to be already included in the
otherwise unperturbed energies $\pm {1\/ 2} \omega_0$ of the atom.
In contrast to this, the effects considered in
this paper are true radiative corrections caused by the interaction of
the atom with the quantum field.

The organization of the paper is the following: In Sec. 2, we define
our model and set up the Heisenberg equations of motion, which are
solved formally. We apply the formalism of DDC to our model in Sec. 3
and generalize it to an atom moving on an arbitrary stationary trajectory
through the Minkowski vacuum.
In Sec. 4, we calculate the level shift for a two-level atom at rest.
We find that for the symmetric operator ordering adopted here, the only
contribution to relative energy shifts comes from vacuum fluctuations,
while the effect of radiation reaction is the same for both levels.
The result obtained for the scalar theory are then compared to the
standard results for the electromagnetic field and a similar structure
is found. Sec. 5 deals with a uniformly accelerated atom. Because the
analysis becomes more involved in this case, we use some methods from
quantum field theory in accelerated frames, which are discussed in the
Appendix. As a result, we find that the contribution of vacuum fluctuations
to the level shift is altered by the appearance of a thermal term with
the Unruh temperature $T= \hbar a/(2\pi c k)$. The contribution of
radiation reaction
is the same as for an atom at rest and does not contribute to relative
shifts. Finally, we point out the similarity of the results to those
obtained for the Lamb shift in a thermal heat bath
\cite{Barton72,Knight72,Farley81}.

\section{Two-level atom interacting with a massless scalar quantum field}

To investigate how the radiative energy shifts of atoms are modified by
acceleration, we choose the simplest nontrivial example: a two-level
atom in interaction with a real massless scalar field. We consider an
atom on a timelike Killing trajectory $x(\tau) =( t(\tau), {\vec x}(\tau))$, which
is parametrized by the proper time $\tau$. It will be called {\it stationary
trajectory} \cite{Letaw81}. The important consequence of the stationarity
is that the level spacing $\omega_0$ of the two states $|- \rangle$ and
$|+ \rangle$ of the atom does not depend on $\tau$. The zero of energy
is chosen so that that the energies of the two stationary states are
$-{1\/2}\omega_0$ and $+{1\/2}\omega_0$. As mentioned in the Introduction,
a possible constant modification of the energy level spacing which is
directly caused by the acceleration is assumed to be already included.
The free Hamiltonian of the two-level atom which generates the
atom's time evolution with respect to the proper time $\tau$ is given by
\begin{equation} H_A (\tau) =\omega_0 R_3 (\tau) \label{eq1} \end{equation}
where we have written $R_3 = {1\/2} |+ \rangle \langle + | - {1\/2}
|- \rangle \langle - |$, following Dicke \cite{Dicke54}.

The free Hamiltonian of the quantum field is
\begin{equation} {\tilde H}_F (t)  = \int d^3 k \, \omega_{\vec k}\, a^\dagger_{\vec k} a_{\vec k} .
\label{eq2}\end{equation}
where $a^\dagger_{\vec k}$, $a_{\vec k}$ are creation and annihilation operators
`photons' with momentum ${\vec k}$. The Hamiltonian (\ref{eq2}) governs the
time evolution of the quantum field with respect to the time variable $t$
of the inertial frame. However, to derive the Heisenberg equations of
motion of the coupled system, we have to choose a common time variable.
It turns out that it is most reasonable to refer generally to the atom's
proper time $\tau$. The free Hamiltonian of the field with respect to
$\tau$ can be obtained by a simple change of the time variable in the
Heisenberg equations:
\begin{equation} H_F (\tau) = \int d^3 k\, \omega_{\vec k} \,a^\dagger_{\vec k} a_{\vec k}
	{dt\/d \tau}. \label{eq3}\end{equation}

We couple the atom and the quantum field by the interaction Hamiltonian
\begin{equation} H_I = \mu R_2 (\tau) \phi ( x(\tau)) . \label{eq5}\end{equation}
which is a scalar model of the electric dipole interaction. Here we
have introduced $R_2 = {1\/2} i ( R_- - R_+)$, where $R_+ = |+ \rangle
\langle - |$ and $R_- = |- \rangle \langle +|$ are raising and lowering
operators for the atom. $\mu$ is a small coupling constant. The field
operator in (\ref{eq5}) is evaluated along the world line $x(\tau)$ of
the atom.

In the solutions of the Heisenberg equations of the atom and field
variables, two physically different
contributions can be distinguished: (1) the {\it free part}, which is
the part of the solution that goes back to the free Hamiltonians (\ref{eq1})
and (\ref{eq3}) and which is present even in the absence of the interaction,
(2) the {\it source part} which is caused by the coupling between atom
and field and represents their mutual influence:
\begin{eqnarray} R_\pm (\tau) &=& R^f_\pm (\tau) + R^s_\pm (\tau),  \\
	R_3(\tau) &=& R^f_3 (\tau) + R_3^s (\tau),  \\
	\phi(x(\tau)) &=& \phi^f (x(\tau)) + \phi^s (x(\tau)).\label{eq5a}\end{eqnarray}
Their explicit form has been given in \cite{Audretsch94}.

\section{Radiative energy shifts: The contributions of vacuum fluctuations
and radiation reaction}

To determine the Lamb shift of an accelerated atom, we use
the physically appealing formalism of Dalibard, Dupont-Roc,
and Cohen-Tannoudji (DDC) \cite{Dalibard82,Dalibard84}. We will generalize
it for calculating level shifts to an atom on an arbitrary
stationary world line $x(\tau)$. One attractive feature of the
formalism is that it allows a separate
discussion of the two physical mechanisms which both contribute to
radiative energy shifts: the contributions of {\it vacuum fluctuations} and
of {\it radiation reaction}. Let us first discuss how these two parts
can be separated (for a more detailed discussion see \cite{Audretsch94}):
Because of the coupling (\ref{eq5}), the field operator
$\phi$ will appear in the Heisenberg equation of motion of an
arbitrary atomic variable $G$. According to (\ref{eq5a}), $\phi$
can be split into its free and source part. The following physical
mechanisms can be connected with these two contributions:
(1) the part of the Heisenberg equation which contains $\phi^f$
represents the change
in G due to {\it vacuum fluctuations}. It is caused by the fluctuations
of the field which are present even in the vacuum. (2) The term
containing $\phi^s$ represents the influence of the atom on the field.
This part of the field in turn reacts back on the atom.
This mechanism is called {\it radiation reaction}. Explicitly, the
contributions of vacuum fluctuations and radiation reaction to
$dG/d \tau$ are:
\begin{equation} \left( {d G(\tau) \/ d \tau} \right)_{vf} = {1\/2}i \mu \Bigl( \phi^f
	(x(\tau)) [R_2 (\tau), G (\tau)] + [R_2 (\tau), G (\tau)]
	\phi^f (x(\tau)) \Bigr), \label{eq15}\end{equation}
\begin{equation} \left( {d G(\tau) \/ d \tau} \right)_{rr} = {1\/2}i \mu \Bigl( \phi^s
	(x(\tau)) [R_2 (\tau), G (\tau)] + [R_2 (\tau), G (\tau)]
	\phi^s (x(\tau)) \Bigr), \label{eq16}\end{equation}
where we followed DDC \cite{Dalibard82} in choosing a symmetric ordering
between atom and field operators. Because we are interested only in
observables of the atom, we take the average of (\ref{eq15})
and (\ref{eq16}) in the vacuum state of the quantum field. In a
perturbative treatment, we take into account only terms up to
second order in $\mu$. Proceeding in a similar manner as in Refs.
\cite{Dalibard84,Audretsch94}, it is then possible to identify in
Eqs. (\ref{eq15}) and (\ref{eq16}) an {\it effective Hamiltonian}
for atomic observables that governs the time evolution with respect to
$\tau$ in addition to the free hamiltonian (\ref{eq1}). The averaged
equations (\ref{eq15}) and (\ref{eq16}) can be written
\begin{equation} \langle 0| \left( {d G (\tau) \/ d\tau} \right)_{vf,rr} |0\rangle =
	i [H^{eff}_{vf,rr} (\tau), G(\tau)] + \hbox{non-Hamiltonian terms}.
	\label{eq26}\end{equation}
where in order $\mu^2$
\begin{equation} H_{vf}^{eff} (\tau) = {1\/ 2} i \mu^2 \int_{\tau_0}^\tau d \tau'
  C^F(x(\tau),x(\tau')) [R_2^f(\tau'),R_2^f(\tau)] \label{eq33}\end{equation}
\begin{equation} H_{rr}^{eff} (\tau) = -{1\/ 2} i \mu^2 \int_{\tau_0}^\tau d \tau'
  \chi^F(x(\tau),x(\tau')) \{ R_2^f(\tau'),R_2^f(\tau)\} \label{eq34}\end{equation}
These effective Hamiltonians depend on the field variables only through
the simple statistical functions
\begin{equation} C^{F}(x(\tau),x(\tau')) = {1\/2} \langle 0| \{ \phi^f (x(\tau)), \phi^f
	(x(\tau')) \} | 0 \rangle, \label{eq24} \end{equation}
\begin{equation} \chi^F(x(\tau),x(\tau')) = {1\/2} \langle 0| [ \phi^f (x(\tau)), \phi^f
	(x(\tau'))] | 0 \rangle \label{eq25}\end{equation}
which are called the {\it symmetric correlation function} and the {\it linear
susceptibility of the field}.
The non-Hamiltonian terms in (\ref{eq26}) describe the effects of relaxation.

The expectation values of the effective Hamiltonians (\ref{eq33}) and
(\ref{eq34})
in an atomic state $|b \rangle$ represent the energy shift of the atomic
level $|b\rangle$ caused by the coupling to the quantum field. The total
shift contains the contribution of vacuum fluctuations as well as the
contribution of radiation reaction. Taking the expectation values of
(\ref{eq33}) and (\ref{eq34}), we obtain the radiative energy shifts of the
 level
$|b\rangle$ due to vacuum fluctuations
\begin{equation} (\delta E_b)_{vf} = - i \mu^2 \int_{\tau_0}^\tau d \tau' \,
	C^F(x(\tau),x(\tau')) \chi^A_b(\tau,\tau'), \label{eq35} \end{equation}
and due to radiation reaction
\begin{equation} (\delta E_b)_{rr} = - i \mu^2
	\int_{\tau_0}^\tau d \tau' \, \chi^F(x(\tau),x(\tau'))
	C^A_b(\tau,\tau'), \label{eq36}\end{equation}
where we have defined the {\it symmetric correlation function}
\begin{equation} C^{A}_b(\tau,\tau') = {1\/2} \langle b| \{ R_2^f (\tau), R_2^f (\tau')\}
	| b \rangle \label{eq38}\end{equation}
and the {\it linear susceptibility of the atom}
\begin{equation} \chi^A_b(\tau,\tau') = {1\/2} \langle b| [ R_2^f (\tau), R_2^f (\tau')]
	| b \rangle \label{eq37}\end{equation}
As a result, we note that
Eqs. (\ref{eq35}) and (\ref{eq36}) for the radiative energy shifts
for atoms that move on an arbitrary stationary trajectory differs from
the results for atoms at rest \cite{Dalibard82,Dalibard84} only in that
the statistical functions of the field (\ref{eq24}) and (\ref{eq25}) are
evaluated along the world line of the atom which may now be an accelerated one.
The statistical functions of the atom do not depend on the trajectory
$x(\tau)$.

Below we will need the explicit forms of the different statistical functions.
Those refering to the atom can be written
\begin{eqnarray} C^{A}_b(\tau,\tau') & =& {1\/2} \sum_d |\langle b| R_2^f (0) | d
	\rangle |^2 \left( e^{i \omega_{bd}(\tau - \tau')} + e^{-i \omega_{bd}
	(\tau - \tau')} \right), \label{eq39}\\
\chi^A(\tau,\tau')_b & =& {1\/2} \sum_d |\langle b| R_2^f (0) | d \rangle |^2
	\left(e^{i \omega_{bd}(\tau - \tau')} - e^{-i \omega_{bd}(\tau -
\tau')}
	\right), \label{eq40}\end{eqnarray}
where $\omega_{bd}= \omega_b-\omega_d$ and the sum extends over a complete
set of atomic states.

\section{Radiative energy shift for an atom at rest}

Let us first reproduce the standard result for the Lamb shift for
an atom at rest in the scalar theory. It can be compared afterwards
with the corresponding
result for a uniformly accelerated atom. The statistical functions of
the field for the trajectory
\begin{equation} t(\tau) =\tau, \qquad {\vec x} (\tau)=0 \label{eq41}\end{equation}
can be easily evaluated. We obtain
\begin{eqnarray}
C^F(x(\tau),x(\tau')) &=& {1\/ 8\pi^2}\int d \omega \, \omega
	\left( e^{-i \omega
    (\tau-\tau')} + e^{i \omega (\tau-\tau')}\right) \label{eq42} \\
\chi^F(x(\tau),x(\tau')) &=& {1\/ 8\pi^2}\int d \omega\, \omega
	\left( e^{-i \omega
  	(\tau-\tau')} - e^{i \omega (\tau-\tau')}\right). \label{eq43}
\end{eqnarray}

The contribution of vacuum fluctuations to the radiative shift of level
 $|b\rangle$ can be calculated from (\ref{eq35})
\begin{equation} (\delta E_b)_{vf} = {\mu^2\/ 8\pi^2} \sum_d |\langle b|R_2^f(0)|
     d\rangle|^2 \int_0^\infty d\omega \,\omega \left({{\cal P}\/ \omega +
     \omega_{bd}} - {{\cal P}\/ \omega -\omega_{bd}} \right),
     \label{eq47}\end{equation}
where $\cal P$ denotes the principle value.
The integral is logarithmically divergent, as expected for a nonrelativistic
calculation of radiative shifts \cite{Bethe47}. As is well known, the
introduction of a cutoff
is therefore necessary. The summation over $d$ displays the role of virtual
transitions to other levels.

The {\it relative} energy shifts of the two levels due to vacuum fluctuations
can be calculated by evaluating the $d$ summation for $|b\rangle =
|\pm\rangle$. The resulting expression is
\begin{eqnarray} \Delta_{vf} &=& (\delta E_+)_{vf} - (\delta E_-)_{vf} \\
     &=& {\mu^2\/ 16\pi^2} \int_0^\infty d\omega \,\omega \left({{\cal P}\/
 \omega
     +\omega_{bd}} - {{\cal P}\/ \omega -\omega_{bd}} \right).
     \label{eq48}\end{eqnarray}

For the contribution of radiation reaction we obtain
\begin{equation} (\delta E_b)_{rr} = -{\mu^2\/ 8\pi^2} \sum_d |\langle b|R_2^f(0)|
     d\rangle|^2 \int_0^\infty d\omega \,\omega \left({{\cal P}\/ \omega +
     \omega_{bd}} + {{\cal P}\/ \omega -\omega_{bd}} \right).
     \label{eq49}\end{equation}
This expression diverges linearly. However, as can be seen by explicitly
evaluating the sum, radiation reaction does not give a contribution to
the relative shift of the two levels:
\begin{equation} \Delta_{rr} = (\delta E_+)_{rr} - (\delta E_-)_{rr} = 0.
     \label{eq50}\end{equation}
Thus the {\it Lamb shift} as the the relative radiative energy shift of the
two-level atom at rest is caused entirely by vacuum fluctuations:
\begin{eqnarray} \Delta_0  &\equiv& \Delta_{vf} + \Delta_{rr} \nonumber\\
     &=& {\mu^2\/ 16\pi^2} \int_o^\infty d\omega \,\omega \left({{\cal P}\/
	 \omega +\omega_{bd}} - {{\cal P}\/ \omega -\omega_{bd}} \right).
     \label{eq51}\end{eqnarray}
This expression agrees structurally with the standard result for the
Lamb shift of a two-level atom
\cite{Ackerhalt73,Senitzky73,Milonni73,Ackerhalt74,Milonni75}. The
modifications are caused by the differences between the electromagnetic
and the scalar theory.

We mention that Welton's picture of radiative shifts \cite{Welton48}, who
tried to interpret the Lamb shift only in terms of vacuum fluctuations
conforms with the fact that the energy shift (\ref{eq51}) is caused
solely by vacuum fluctuations. This feature is a peculiarity of the simple
model of a two-level atom, however. For a real multilevel atom,
the contribution of radiation reaction will be different for different
levels, and a mass renormalization will become necessary.

\section{Radiative energy shifts for a uniformly accelerated atom}

Let us now consider the case of a uniformly accelerated two-level atom.
It moves on the trajectory
\begin{equation} t(\tau)={1\/ a}\sinh a \tau, \qquad z(\tau)={1\/ a}\cosh a \tau,
     \label{eq52}\end{equation}
$$ x(\tau)=y(\tau)=0, $$
where $a$ is the proper acceleration. The calculation of the statistical
functions of the field turns out to be much more complicated for an
accelerated atom than for an atom at rest. It is most convenient to
employ methods from the quantum field theory in accelerated frames.
In order to keep the discussion transparent, we have put the
calculation into the Appendix and simply quote the results here:
\begin{eqnarray} C^F (x(\tau),x(\tau')) &=& {1\/ 8\pi^2} \int_0^\infty d\omega'
   \,\omega' \coth\left({\pi\omega'\/a}\right)\left( e^{-i \omega'
   (\tau-\tau')} + e^{i \omega'(\tau-\tau')}\right) \label{eq53}\\
\chi^F (x(\tau),x(\tau')) &=& {1\/ 8\pi^2} \int_0^\infty d\omega'
   \,\omega' \left( e^{-i \omega' (\tau-\tau')} -
   e^{i \omega'(\tau-\tau')}\right) \label{eq54}\end{eqnarray}
We see that the expression (\ref{eq54}) for the linear susceptibility of the
field is formally identical for an accelerated atom and an atom at rest.
However, with regard to the cutoff prescription to be given below, it is
important to note that $\omega'$ in (\ref{eq53}) and
(\ref{eq54}) denotes the energy in the accelerated frame, i. e. as measured
by an observer on the trajectory (\ref{eq52}).

The remaining calculation is straightforward. By substituting the
statistical functions (\ref{eq39}), (\ref{eq40}), (\ref{eq53}), and
(\ref{eq54})
into the general formulas (\ref{eq35}) and (\ref{eq36}) for the level
shifts, we find
\begin{eqnarray} (\delta E_b)_{vf} &=& {\mu^2\/8\pi^2} \sum_d|\langle b|R^f_2 (0)|d
    \rangle|^2 \nonumber\\
&&\qquad\times \int_0^\infty d\omega'\,\omega' \coth\left({\pi\omega'\/a}
    \right) \hbox{Im}\left\{ \int_0^\infty du \left( e^{i(\omega'
    +\omega_{bd})u} - e^{i(\omega'-\omega_{bd})u}\right)\right\},
    \label{eq55}\\
(\delta E_b)_{rr} &=& -{\mu^2\/8\pi^2} \sum_d|\langle b|R^f_2 (0)|d
    \rangle|^2 \nonumber\\
&&\qquad\times\int_0^\infty d\omega'\,\omega'
    \hbox{Im}\left\{ \int_0^\infty du \left( e^{i(\omega'
    +\omega_{bd})u} + e^{i(\omega'-\omega_{bd})u}\right)\right\}.
    \label{eq56}\end{eqnarray}

The result for the radiative energy shift for a uniformly accelerated atom
can now be obtained by evaluating the integrals. The contribution of vacuum
fluctuations is
\begin{equation} (\delta E_b)_{vf} =  {\mu^2\/8\pi^2} \sum_d|\langle b|R^f_2 (0)|d
    \rangle|^2 \int_0^\infty d\omega'\,\omega'\left( {1} + {2\/
    e^{2\pi\omega'/a}-1}\right)\left({{\cal P}\/\omega'+\omega_{bd}}
    - {{\cal P}\/\omega'-\omega_{bd}}\right).  \label{eq57}\end{equation}
Comparing this formula to Eq. (\ref{eq47}) for an atom at rest, we note
that the acceleration-caused correction is additive and contains a
 characteristic thermal term
with the Unruh temperature $T = \hbar a/(2\pi kc)$.
On the other hand, the contribution of radiation reaction,
\begin{equation} (\delta E_b)_{rr} = -{\mu^2\/8\pi^2} \sum_d|\langle b|R^f_2 (0)|d
    \rangle|^2 \int_0^\infty d\omega'\,\omega' \left({{\cal
	P}\/\omega'+\omega_{bd}}
    + {{\cal P}\/\omega'-\omega_{bd}}\right)  \label{eq58}\end{equation}
is  exactly the same as for an atom at rest. The uniform acceleration
has no effect on the shift caused by radiation reaction and we find
again no relative energy shift: $\Delta_{rr}=0$.

Thus the {\it Lamb shift} $\Delta$ can be obtained from the contribution of
vacuum fluctuations (\ref{eq57}) by evaluating the summation over $d$ for
each of the two levels separately:
\begin{eqnarray} \Delta &=& \Delta_{vf}= (\delta E_+)_{vf} - (\delta E_-)_{vf}\nonumber\\
	&=& {\mu^2\/ 16\pi^2}\int_0^\infty d \omega'\, \omega' \left[ {1} +
{2\/
    e^{2\pi\omega'/a}-1}\right]\left({{\cal P}\/\omega'+\omega_0}
    - {{\cal P}\/\omega'-\omega_0}\right). \label{eq59}\end{eqnarray}
{}From the two terms in square brackets, we can distinguish two contributions
in (\ref{eq59}). The first one has
the same functional form as $\Delta_0$ of (\ref{eq51}) for an atom at rest.
Because it is logarithmically divergent, the introduction
of a cutoff is necessary. The only sensible way to do this is to
impose the cutoff frequency in the rest frame of the atom. Since
$\omega'$ in (\ref{eq59}) is the frequency in the accelerated system,
this can be done quite naturally in the present formalism.
The second contribution in (\ref{eq59}) represents the modification of the
Lamb shift caused by the acceleration and contains the thermal term.
Inspection of the integral shows that this correction is finite. Equation
(\ref{eq59}) can thus be written
\begin{equation} \Delta = \Delta_0 + {\omega_0\mu^2 \/ 16\pi^2} \,G \left({a\/ 2\pi
	\omega_0} \right), \label{eq60}\end{equation}
where $\Delta_0$ is the Lamb shift (\ref{eq51}) for $a=0$ and $G(u)$ is
defined by
\begin{equation} G(u) = u \int_0^\infty dx {x\/ e^x -1} \left({{\cal P}\/ x+u^{-1}} -
	{{\cal P}\/ x-u^{-1}} \right). \label{eq61}\end{equation}
The evaluation of the integral must be done numerically. The result
is shown in Fig. 1. In the limit of small $u=a/(2\pi \omega_0)$,
(\ref{eq60}) can be approximated by
\begin{equation} \Delta = \Delta_0 + {a^2\mu^2 \/ 192 \pi^2}{1 \/ \omega_0}.
	\label{eq62}\end{equation}
The dependence on $a$ can be read off from Fig. 1. To estimate the order of
magnitude of the acceleration needed for an appreciable effect, we demand
$(\Delta -\Delta_0)/\omega_0 = \mu^2/(16 \pi)$ and therefore $G(u)=1$.
This amounts to $a \approx c \omega_0$, which gives $a \approx 10^{24}
\hbox{m/s}^2$ for an 1 eV transition.

\begin{figure}[t]
\epsfysize=8cm
\hspace{1.5cm}
\epsffile{shift.eps}
\caption{
Plot of the function $G(u)$ defined in (\protect\ref{eq61}).
The inset shows the
behaviour at small values of $u$ (cf. (\protect\ref{eq62})).}
\end{figure}

Finally, we note the structural similarity of the equations (\ref{eq59})
and (\ref{eq60}) to the corresponding expressions obtained for the
Lamb shift in a thermal heat bath \cite{Barton72,Knight72,Farley81}.
They have in common the appearance of the factor in square brackets
under the integral in (\ref{eq59}) with $T= a/(2\pi)$. A
heuristic connection between the two physical situations has already been
given in the Introduction.
The differences are due to the properties of the scalar theory and the
fact that we considered a two-level atom instead of a real atom.

\section{Conclusion}

In this paper, we have discussed two main points. First, we studied the
influence of acceleration on the radiative energy shift of atoms. The
formalism can be applied to an atom on an arbitrary stationary trajectory.
We considered here the special case of uniformly accelerated motion.
The resulting expression for the Lamb shift of a uniformly accelerated
two-level atom is given in Eq. (\ref{eq59}) and (\ref{eq60}).
Comparison with the corresponding formula (\ref{eq51}) for an atom at rest
shows that the correction consists in the appearance of the factor in
square brackets, which contains the thermal term. This modification
is structurally the same as for the Lamb shift in a thermal heat bath.

The second main goal of the paper has been the identification of the two
physical mechanisms responsible for the radiative shifts. Using the
formalism of DDC, we were able to discuss the contributions of vacuum
fluctuations and radiation reaction separately. The effect of radiation
reaction is the same for a uniformly accelerated atom as for an atom at
rest. It does not affect relative energy shifts. It is interesting to note
that also in the case of other radiative phenomena like the Unruh effect and
spontaneous emission, radiation reaction is not influenced by acceleration
\cite{Audretsch94}. The contribution of vacuum fluctuation, however, is
modified by the thermal terms and gives the total effect.


\section{Introduction}
One of the most prominent concerns of wireless communication systems is to ensure the security of the users against eavesdropping attacks. In last years, wireless physical layer security has gained a lot of attention from the research community. Compared to the traditional cryptographic techniques, physical layer security is performed with less complexity and is more convenient for the emerging ad-hoc and decentralized networks and the next wave of innovative systems known as the Internet of Things. The information theoretic security was firstly introduced by Shannon in \cite{shannon01}. Many years later, Wyner proposed a new model for the wiretap channel \cite{wyner01}. In his seminal work, Wyner presented the degraded wiretap channel, where a source exploits the structure of the medium channel to transmit a message reliably to the intended receiver while leaking asymptotically no information to the eavesdropper. Ulterior works generalized Wyner's work to the case of non-degraded channels \cite{csiszar01}, Gaussian channels \cite{leung11}, and fading channels \cite{gopala01,liang01}.

Recent research interest has been given to analyzing the secrecy capacity of multi-users systems. For the broadcast multi-users scenario, the secrecy capacity of parallel and fading channels, assuming perfect main CSI at the transmitter, was considered in~\cite{ashish01} and\cite{ulukus1}. The case of imperfect main channel estimation was elaborated in~\cite{hyadi2}. For the multiple access scenario, the authors in~\cite{tekin01} and \cite{ulukus3} investigated the secrecy capacity of degraded wiretap channels. The problem of analyzing the secrecy capacity of multiple antenna systems has also been of great interest. The secrecy capacity for the multiple-input single-output (MISO) wiretap Gaussian channel was proven in \cite{ashish04} and \cite{li2}. Another work \cite{ashish02} characterized the secrecy capacity for the MISO case, with a multiple-antenna eavesdropper, when the main and the eavesdropping channels are known to all terminals. The secrecy capacity of the multiple-input multiple-output (MIMO) transmission with a multiple-antenna eavesdropper was considered in \cite{oggier11} and \cite{ashish11} when the channel matrices are fixed and known to all terminals. The secrecy capacity region of the Gaussian MIMO broadcast wiretap channel was derived in \cite{ulukus2}.

Taking full advantage of the ability of the physical layer to secure wireless transmissions, requires a complete knowledge of the channel state information (CSI) at the transmitter (CSIT); which is difficult to have in practical scenarios. One way to overcome this challenge is by using feedback \cite{heath01,love11}. This is a natural setting as CSI feedback is incorporated in most if not all communication standards. For the case of a single user transmission, an upper and a lower bounds on the secrecy capacity were presented, in \cite{rezki11}, for block-fading channels with finite rate feedback. For the MIMO case, the work in \cite{lin11} and \cite{liu01} evaluated the impact of quantized channel information on the achievable secrecy rate, for multiple-antenna wiretap channels, using artificial noise.

In this paper, we investigate the ergodic secrecy capacity of a broadcast wiretap channel where a source transmits to multiple legitimate receivers in the presence of an eavesdropper. In particular, we analyze the case of block-fading channels with limited CSI feedback. More specifically, we consider that the transmitter is unaware of the channel gains to the legitimate receivers and to the eavesdropper and is only provided a finite CSI feedback. This feedback is sent by the legitimate receivers through error-free links with limited capacity. Both the common message transmission, where the same message is broadcasted to all the legitimate receivers, and the independent messages transmission, where the source broadcasts multiple independent messages, are considered. Assuming an average power constraint at the transmitter, we provide an upper and a lower bounds on the ergodic secrecy capacity for the common message case, and an upper and a lower bounds on the secrecy sum-rate for the independent messages case. For the particular case of infinite feedback, we prove that our bounds coincide.

The paper is organized as follows. Section~\ref{model} describes the system model. The main results along with the corresponding proofs are introduced in section~\ref{BCM} for the common message transmission and in section~\ref{BIM} for the independent messages case. Finally, selected simulation results are presented in section~\ref{NR}, while section~\ref{conclusion} concludes the paper.

\textit{Notations:}
Throughout the paper, we use the following notational conventions. The expectation operation is denoted by $\mathbb{E}[.]$, the modulus of a scalar $x$ is expressed as $|x|$, and we define $\{\nu\}^+{=}\max (0,\nu)$. The entropy of a discrete random variable $X$ is denoted by $H(X)$, and the mutual information between random variables $X$ and $Y$ is denoted by $I(X;Y)$. A sequence of length $n$ is denoted by $X^n$, $X(k)$ represents the $k$-th element of $X$, and $X(l,k)$ the $k$-th element of $X$ in the $l$-th fading block.
\section{System Model}\label{model}
We consider a broadcast wiretap channel where a transmitter~$\text{T}$ communicates with $K$ legitimate receivers $(\text{R}_1,\cdots,\text{R}_K)$ in the presence of an eavesdropper $\text{E}$ as depicted in Fig.~\ref{fig:model}. Each terminal is equipped with a single antenna for transmission and reception. The respective received signals at each legitimate receiver $\text{R}_k; k\in\{1,\cdots,K\},$ and the eavesdropper, at fading block $l$, $l{=}1,{\cdots ,}L$, are given by
\begin{equation}\label{Sys_Out}
\begin{aligned}
&Y_k(l,j)=h_k(l)X(l,j)+v_k(l,j)\\
&Y_\text{e}(l,j)\hspace{0.06cm}=h_\text{e}(l)X(l,j)\hspace{0.05cm}+w_\text{e}(l,j),
\end{aligned}
\end{equation}
where $j{=}1,{\cdots ,}\kappa$, with $\kappa$ representing the length of each fading block, $X(l,j)$ is the $j$-th transmitted symbol in the $l$-th fading block, $h_k(l)\!\in\!\mathbb{C}$, $h_\text{e}(l)\!\in\!\mathbb{C}$ are the complex Gaussian channel gains corresponding to each legitimate channel and the eavesdropper's channel, respectively, and $v_k(l,j)\!\in\!\mathbb{C}$, $w_\text{e}(l,j)\!\in\!\mathbb{C}$ represent zero-mean, unit-variance circularly symmetric white Gaussian noises at $\text{R}_k$ and~$\text{E}$, respectively. We consider a block-fading channel where the channel gains remain constant within a fading block, i.e., $h_k(\kappa l)=h_k(\kappa l-1)=\cdots =h_k(\kappa l-\kappa +1)=h_k(l)$ and $h_\text{e}(\kappa l)=h_\text{e}(\kappa l-1)=\cdots =h_\text{e}(\kappa l-\kappa +1)=h_\text{e}(l)$. We assume that the channel encoding and decoding frames span a large number of fading blocks, i.e., $L$ is large, and that the blocks change independently from a fading block to another.
An average transmit power constraint is imposed at the transmitter such that
\begin{equation}\label{pavg}
\frac{1}{n}\sum_{t=1}^n\mathbb{E}\left[|X(t)|^2\right]\leq P_{\text{avg}},
\end{equation}
with $n{=}\kappa L$, and where the expectation is over the input distribution.

The channel gains $h_k$ and $h_\text{e}$ are independent, ergodic and stationary with bounded and continuous probability density functions (PDFs). In the rest of this paper, we denote $|h_k|^2$ and $|h_\text{e}|^2$ by $\gamma_k$ and $\gamma_\text{e}$, respectively. We assume perfect CSI at the receiving nodes. That is, each legitimate receiver is instantaneously aware of its channel gain $h_k(l)$, and the eavesdropper knows $h_\text{e}(l)$, with $l{=}1,\cdots ,L$. The statistics of the main and the eavesdropping channels are available to all nodes. Further, we assume that the transmitter is not aware of the instantaneous channel realizations of neither channel. However, each legitimate receiver provides the transmitter with $b$-bits CSI feedback through an error-free orthogonal channel with limited capacity. This feedback is transmitted at the beginning of each fading block and is also tracked by the other legitimate receivers. The eavesdropper knows all channels and also track the feedback links so that they are not sources of secrecy.

The adopted feedback strategy consists on partitioning the main channel gain support into $Q$ intervals $[\tau_1, \tau_2), \cdots , [\tau_q, \tau_{q+1}), \cdots , [\tau_Q, \infty)$, where $Q{=}2^b$. That is, during each fading block, each legitimate receiver $\text{R}_k$ determines in which interval, $[\tau_q, \tau_{q+1})$ with $q{=}1,\cdots ,Q$, its  channel gain $\gamma_k$ lies and feedbacks the associated index $q$ to the transmitter. At the transmitter side, each feedbacked index $q$ corresponds to a power transmission strategy $P_q$ satisfying the average power constraint. We assume that all nodes are aware of the main channel gain partition intervals $[\tau_1, \tau_2), \cdots , [\tau_q, \tau_{q+1}), \cdots , [\tau_Q, \infty)$, and of the corresponding power transmission strategies $\{P_1, \cdots , P_Q\}$.

\begin{figure}[t]
\psfrag{t}[l][l][1.9]{$\text{T}$}
\psfrag{e}[l][l][1.9]{$\text{E}$}
\psfrag{r1}[l][l][1.9]{$\text{R}_1$}
\psfrag{r2}[l][l][1.9]{$\text{R}_k$}
\psfrag{r3}[l][l][1.9]{$\text{R}_K$}
\psfrag{h1}[l][l][1.5]{$h_1$}
\psfrag{h2}[l][l][1.5]{$h_k$}
\psfrag{h3}[l][l][1.5]{$h_K$}
\psfrag{g}[l][l][1.5]{\hspace{-0.1cm}$h_\text{e}$}
\psfrag{feedback}[l][l][1.5]{\hspace{0.3cm}$b$-bits Feedback Link}
\psfrag{l}[l][l][2]{$\left. \rule{0pt}{2.2cm} \right\}$}
\psfrag{transmitter}[l][l][1.7]{Transmitter}
\psfrag{eavesdropper}[l][l][1.7]{Eavesdropper}
\psfrag{receivers}[l][l][1.7]{$\hspace{-0.1cm}K$ Legitimate Receivers}
\begin{center
\scalebox{0.4}{\includegraphics{system_v1.eps}}
\end{center
\caption{Broadcast wiretap channel with limited CSI feedback.}\vspace{-0.2cm}
\label{fig:model}
\end{figure}

The transmitter wishes to send some secret information to the legitimate receivers. In the case of common message transmission, a $(2^{n\mathcal{R}_\text{s}}, n)$ code consists of the following elements:
\begin{itemize}
\item A message set $\mathcal{W}=\left\lbrace 1,2,{\cdots},2^{n\mathcal{R}_\text{s}}\right\rbrace$ with the messages $W\in\mathcal{W}$ independent and uniformly distributed over $\mathcal{W}$;
\item A stochastic encoder $f: \mathcal{W}\rightarrow\mathcal{X}^n$ that maps each message $w$ to a codeword $x^n\in\mathcal{X}^n$;
\item A decoder at each legitimate receiver $g_k: \mathcal{Y}_k^n\rightarrow\mathcal{W}$ that maps a received sequence $y_k^n\in\mathcal{Y}_k^n$ to a message $\hat{w}_k\in\mathcal{W}$.
\end{itemize}

A rate $\mathcal{R}_\text{s}$ is an \textit{achievable secrecy rate} if there exists a sequence of $(2^{n\mathcal{R}_\text{s}}, n)$ code such that both the average error probability at each legitimate receiver
\begin{equation}
P_{\text{e}_k}=\frac{1}{2^{n\mathcal{R}_\text{s}}}\sum_{w=1}^{2^{n\mathcal{R}_\text{s}}}\text{Pr}\left[W\neq\hat{W}_k\big|W=w\right],
\end{equation}
and the leakage rate at the eavesdropper
\begin{equation}
\frac{1}{n} I(W;Y_e^n,h_e^L,h_1^L,\cdots,h_K^L,q_1^L,\cdots,q_K^L),
\end{equation}
go to zero as $n$ goes to infinity. The \textit{secrecy capacity} $\mathcal{C}_\text{s}$ is defined as the maximum achievable secrecy rate, i.e., $\displaystyle{\mathcal{C}_\text{s}\triangleq\sup\mathcal{R}_\text{s},}$ where the supremum is over all achievable secrecy rates. In the case of independent messages transmission, the achievable secrecy rate tuple $\left(\mathcal{R}_1,\mathcal{R}_2,\cdots,\mathcal{R}_K\right)$ can be analogously defined.
\section{Broadcasting a Common Message}\label{BCM}
In this section, we examine the case of common message transmission when a unique confidential information is broadcasted to all the legitimate receivers in the presence of an eavesdropper. Taking into account the adopted system model, we present an upper and a lower bounds on the ergodic secrecy capacity.
\subsection{Main Results}\label{MR1}
\textit{Theorem 1:} The common message secrecy capacity, $\mathcal{C}_{\text{s}}$, of a block fading broadcast channel, with an error free $b$-bits CSI feedback sent by each legitimate receiver, at the beginning of each fading block, is bounded by
\begin{equation}\label{th1}
\mathcal{C}_{\text{s}}^-\leq \mathcal{C}_{\text{s}}\leq \mathcal{C}_{\text{s}}^+,
\end{equation}
such as\vspace{-0.3cm}
\begin{subequations}
\begin{align}\label{th1_a}
\mathcal{C}_{\text{s}}^-{=}\hspace{-0.2cm}\min_{1\leq k\leq K}\max_{\{\tau_q;P_q\}_{q=1}^Q}\sum_{q=1}^Q\Theta_{\tau_q}^{\gamma_k}\underset{\gamma_\text{e}}{\mathbb{E}}\!\left[\left\lbrace\log\!\left(\frac{1{+}\tau_qP_q}{1{+}\gamma_\text{e}P_q}\right)\right\rbrace^{\hspace{-0.1cm}+}\right],
\end{align}
and\vspace{-0.3cm}
\begin{align}\label{th1_b}
\mathcal{C}_{\text{s}}^+&{=}\hspace{-0.1cm}\min_{1\leq k\leq K}\max_{\{\tau_q;P_q\}_{q=1}^Q}\sum_{q=0}^Q\Theta_{\tau_q}^{\gamma_k}\nonumber\\
&\qquad\times\underset{\gamma_\text{e},\gamma_k}{\mathbb{E}}\!\left[\left\lbrace\!\log\!\left(\frac{1{+}\gamma_kP_q}{1{+}\gamma_\text{e}P_q}\right)\!\right\rbrace^+\!\Bigg|\tau_q\!\leq\!\gamma_k\!<\!\tau_{q+1}\right],
\end{align}
where $Q{=}2^b$, $\{\tau_q~\!|~\!0{=}\tau_0{<}\tau_1{<}\cdots{<}\tau_Q\}_{q=1}^Q$ represent the reconstruction points describing the support of $\gamma_k$ with $\tau_{Q+1}{=}\infty$ for convenience, $\{P_q\}_{q=1}^Q$ are the power transmission strategies satisfying the average power constraint, and $\displaystyle{\Theta_{\tau_q}^{\gamma_k}{=}\text{Pr}\left[\tau_q{\leq}\gamma_k{<}\tau_{q+1}\right]}$ for all $q\in\{1,{\cdots},Q\}$.
\end{subequations}\vspace{0.2cm}

\textit{Proof:} A detailed proof of Theorem 1 is provided in the following subsection.\vspace{0.2cm}

\noindent It is worth mentioning that the main difference between our bounds is that the transmission scheme, for the achievable secrecy rate, uses the feedback information to adapt both the rate and the power in such a way that the transmission rate is fixed during each fading block. Also, Theorem~1 states that even with 1-bit feedback, sent by each legitimate receiver at the beginning of each fading block, a positive secrecy rate can still be achieved.

\textit{Corollary 1:} The common message secrecy capacity of a block fading broadcast channel with perfect main CSIT, and the average power constraint in \eqref{pavg}, is given by
\begin{align}\label{cr1}
\mathcal{C}_{\text{s}}=\min_{1\leq k\leq K}\max_{P(\gamma_k)}\underset{\gamma_k,\gamma_\text{e}}{\mathbb{E}}\!\left[\left\lbrace\log\!\left(\frac{1{+}\gamma_kP(\gamma_k)}{1{+}\gamma_\text{e}P(\gamma_k)}\right)\right\rbrace^+\right],
\end{align}
with $\mathbb{E} [P(\gamma_k)]\leq P_\text{avg}$.

\vspace{0.3cm}
\textit{Proof:} Corollary 1 results directly from the expressions of the achievable rate in \eqref{th1_a} and the upper bound in \eqref{th1_b}, by letting $\displaystyle{\Theta_{\tau_q}^{\gamma_k}=\frac{1}{Q}}$ and taking into consideration that as $Q\rightarrow\infty$, the set of reconstruction points, $\{\tau_1,{\cdots},\tau_Q\}$, becomes infinite and each legitimate receiver $\text{R}_k$ is basically forwarding $\gamma_k$ to the transmitter. $\hfill \square$

To the best of our knowledge, this result has not been reported in earlier works. For the special case of single user transmission, the secrecy capacity in corollary 1 coincides with the result in Theorem~2 from reference \cite{gopala01}.
\subsection{Ergodic Capacity Analysis}\label{ECA1}
In this subsection, we establish the obtained results for the ergodic secrecy capacity presented in the previous subsection.
\subsubsection{Proof of Achievability in Theorem 1}\label{PLB_CM}
Since the transmission is controlled by the feedbacked information, we consider that, during each fading block, if the main channel gain of the receiver with the weakest channel gain falls within the interval $[\tau_q,\tau_{q+1})$, $q\in\{1,{\cdots},Q\}$, the transmitter conveys the codewords at rate $\mathcal{R}_q=\log\left(1{+}\tau_qP_q\right).$ Rate $\mathcal{R}_q$ changes only periodically and is held constant over the duration interval of a fading block. This setup guarantees that when $\gamma_\text{e}{>}\tau_q$, the mutual information between the transmitter and the eavesdropper is upper bounded by $\mathcal{R}_q$. Otherwise, this mutual information will be $\log\left(1{+}\gamma_\text{e}P_q\right)$.

Besides, we adopt a probabilistic transmission model where the communication is constrained by the quality of the legitimate channels. Given the reconstruction points, $\tau_1{<}\tau_2{<}\cdots{<}\tau_Q{<}\tau_{Q+1}{=}\infty$, describing the support of each channel gain $\gamma_k; k\in\{1,\cdots ,K\}$, and since the channel gains of the $K$ receivers are independent, there are $M{=}Q^K$ different states for the received feedback information. Each of these states, $\mathcal{J}_m; m\in\{1,\cdots ,M\},$ represents one subchannel. The transmission scheme consists on transmitting an independent codeword, on each of the $M$ subchannels, with a fixed rate. We define the following rates
$$\displaystyle{\mathcal{R}_\text{s}^-=\sum_{m=1}^M\text{Pr}\left[\mathcal{J}_m\right]\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_m^\text{min}P_m}{1{+}\gamma_\text{e}P_m}\right)\right\rbrace^+\right],}$$
and $\displaystyle{\mathcal{R}_{\text{e},m}=\underset{\gamma_\text{e}}{\mathbb{E}}\left[\log\left(1{+}\gamma_\text{e}P_m\right)\right],}$
where $\tau_m^\text{min}$ is the quantized channel gain corresponding to the weakest receiver in state $\mathcal{J}_m$ and $P_m$ is the associated power policy satisfying the average power constraint.

\textit{Codebook Generation:}
We construct $M$ independent codebooks $\mathcal{C}_1$, $\cdots$, $\mathcal{C}_M$, one for each subchannel, constructed similarly to the standard wiretap codes. Each codebook $\mathcal{C}_m$ is a $(n,2^{n\mathcal{R}_\text{s}^-})$ code with $2^{n(\mathcal{R}_\text{s}^-{+}\mathcal{R}_{\text{e},m})}$ codewords randomly partitioned into~$2^{n\mathcal{R}_\text{s}^-}$ bins.

\textit{Encoding and Decoding:}
Given a particular common message $w{\in}\{1,2,{\cdots},2^{n\mathcal{R}_\text{s}^-}\}$, to be transmitted, the encoder selects $M$ codewords, one for each subchannel. More specifically, if the message to be sent is $w$, then for each subchannel $m$, the encoder randomly selects one of the codewords $U_m^n$ from the $w$th bin in $\mathcal{C}_m$. During each fading block, of length $\kappa$, the transmitter experiences one of the events $\mathcal{J}_m$. Depending on the encountered channel state, the transmitter broadcasts $\kappa\mathcal{R}_q$ information bits of $U_m^{n}$ using a Gaussian codebook. By the weak law of large numbers, when the total number of fading blocks $L$ is large, the entire binary sequences are transmitted with high probability. To decode, each legitimate receiver considers the observations corresponding to all $M$ subchannels. And since $\log\left(1{+}\tau_m^\text{min}P_q\right){<}\log\left(1{+}\gamma_kP_q\right)$ is valid for all fading blocks, the receivers can recover all codewords, with high probability, and hence recover message~$w$.
The expression of $\mathcal{R}_\text{s}^-$ can then be reformulated as
\begin{align}
&\mathcal{R}_\text{s}^-=\sum_{m=1}^M\text{Pr}\left[\mathcal{J}_m\right]\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_m^\text{min}P_m}{1{+}\gamma_\text{e}P_m}\right)\right\rbrace^+\right]\\
&=\hspace{-0.2cm}\min_{1\leq k\leq K}\sum_{m=1}^M\text{Pr}\left[\mathcal{J}_m\right]\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_{k,m}P_m}{1{+}\gamma_\text{e}P_m}\right)\right\rbrace^+\right]\label{stp1}\\
&=\hspace{-0.2cm}\min_{1\leq k\leq K}\sum_{m=1}^M\sum_{q=1}^Q\text{Pr}\left[\mathcal{J}_m,\tau_{k,m}{=}\tau_q\right]\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_qP_q}{1{+}\gamma_\text{e}P_q}\!\right)\!\right\rbrace^{\hspace{-0.1cm}+}\right]\label{stp2}\\
&=\hspace{-0.2cm}\min_{1\leq k\leq K}\sum_{q=1}^Q\text{Pr}\left[\tau_q{\leq}\gamma_k{<}\tau_{q+1}\right]\underset{\gamma_\text{e}}{\mathbb{E}}\!\left[\!\left\lbrace\!\log\!\left(\!\frac{1{+}\tau_qP_q}{1{+}\gamma_\text{e}P_q}\!\right)\!\right\rbrace^{\hspace{-0.1cm}+}\right],\label{stp3}
\end{align}
where \eqref{stp1} results since the logarithm function is monotonic and the sum and the expectation are taking over positive terms, \eqref{stp2} is obtained by noting that $\tau_m^\text{min}\in\{\tau_1,{\cdots},\tau_Q\}$ and applying the total probability theorem, and \eqref{stp3} comes from the fact that $\sum_{m=1}^M\text{Pr}\left[\mathcal{J}_m,\tau_{k,m}{=}\tau_q\right]{=}\text{Pr}\left[\tau_q{\leq}\gamma_k{<}\tau_{q+1}\right]$.

Since each user gets to know the feedback information of the other legitimate receivers, our proof is also valid when the reconstruction points $\{\tau_q\}_{q=1}^Q$ and the transmission strategies $\{P_q\}_{q=1}^Q$, associated with each legitimate receiver, are different. That is, we can choose these quantization parameters to satisfy~\eqref{th1_a}.

\textit{Secrecy Analysis:}
We need to prove that the equivocation rate satisfies $\mathcal{R}_\text{e}\geq\mathcal{R}_\text{s}^--\epsilon$. Let $\Gamma^L{=}\left\{\gamma_1^L,\gamma_2^L,{\cdots},\gamma_K^L\right\}$
and $F^L{=}\left\{F_1^L,F_2^L,{\cdots},F_K^L\right\}$, with $F_k(l)\in\{\tau_1,{\cdots},\tau_Q\}$ being the feedback information sent by receiver $k$ in the $l$-th fading block. We have\vspace{-0.1cm}
\begin{align}
&n\mathcal{R}_\text{e}=H(W|Y_\text{e}^n,\gamma_\text{e}^L,\Gamma^L,F^L)\\
&\geq I(W;X^n|Y_\text{e}^n,\gamma_\text{e}^L,\Gamma^L,F^L)\\
&=H(X^n|Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\Gamma^L\!,\!F^L){-}H(X^n|Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\Gamma^L\!,\!F^L\!,\!W).\label{lb1_step0}
\end{align}  \vspace{-0.1cm}
On one hand, we can write
\allowdisplaybreaks
\begin{align}
&H(X^n|Y_\text{e}^n,\gamma_\text{e}^L,\Gamma^L,F^L)\nonumber\\
&=\sum_{l=1}^L\hspace{-0.1cm}\resizebox{.38\textwidth}{!}{$\displaystyle{H(X^\kappa(l)|Y_\text{e}^\kappa(l)\!,\!\gamma_\text{e}(l)\!,\!\gamma_1(l)\!,\!{\cdots}\!,\!\gamma_K(l)\!,\!F_1(l)\!,\!{\cdots}\!,\!F_K(l))}$}\label{lb1_step1}\\
&\geq\!\sum_{l\in\mathcal{D}_L}\hspace{-0.15cm}\resizebox{.38\textwidth}{!}{$\displaystyle{H(X^\kappa(l)|Y_\text{e}^\kappa(l)\!,\!\gamma_\text{e}(l)\!,\!\gamma_1(l)\!,\!{\cdots}\!,\!\gamma_K(l)\!,\!F_1(l)\!,\!{\cdots}\!,\!F_K(l))}$}\label{lb1_step2}\\
&\geq \!\sum_{l\in\mathcal{D}_L}\hspace{-0.15cm}\kappa\!\resizebox{.37\textwidth}{!}{$\displaystyle{\left(\min_{1\leq k\leq K}\sum_{q=1}^Q\Theta_{\tau_q}^{\gamma_k(l)}\left(\mathcal{R}_q{-}\log\left(1{+}\gamma_\text{e}(l)P_q\right)\right){-}\epsilon'\!\right)}$}\\
&=\!\sum_{l=1}^L\hspace{-0.1cm}\kappa\!\left(\!\min_{1\leq k\leq K}\sum_{q=1}^Q\hspace{-0.1cm}\Theta_{\tau_q}^{\gamma_k(l)}\!\left\lbrace \!\mathcal{R}_q\!-\!\log\!\left(1{+}\gamma_\text{e}(l)P_q\right)\!\right\rbrace^{+}\hspace{-0.1cm}\!-\!\epsilon'\!\right)\\
&=n\min_{1\leq k\leq K}\sum_{q=1}^Q\Theta_{\tau_q}^{\gamma_k}~\!\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace \mathcal{R}_q{-}\log\left(1{+}\gamma_\text{e}P_q\right)\right\rbrace^+\right]-n\epsilon'\label{lb1_step3}\\
&=n\mathcal{R}_\text{s}^- -n\epsilon',\label{lb1_step3p1}
\end{align}
where \eqref{lb1_step1} results from the memoryless property of the channel and the independence of the $X^\kappa(l)$'s, \eqref{lb1_step2} is obtained by removing all the terms corresponding to the fading blocks $l\not\in\mathcal{D}_L$, with $\mathcal{D}_L=\cup_{k\in\{1,{\cdots},K\}}\left\lbrace l\in\{1,{\cdots},L\}:F_k(l)>h_\text{e}(l)\right\rbrace$, $\displaystyle{\Theta_{\tau_q}^{\gamma_k(l)}{=}\text{Pr}\left[\tau_q{\leq}\gamma_k(l){<}\tau_{q+1}\right]}$, and \eqref{lb1_step3} follows from the ergodicity of the channel as $L\rightarrow\infty$.

On the other hand, using list decoding argument at the eavesdropper side and applying Fano's inequality~\cite{gopala01}, $\frac{1}{n}H(X^n|Y_\text{e}^n,\gamma_\text{e}^L,\Gamma^L,F^L,W)$ vanishes as $n\rightarrow\infty$ and we can write
\begin{equation}\label{lb1_step4}
H(X^n|Y_\text{e}^n,\gamma_\text{e}^L,\Gamma^L,F^L,W)\leq n\epsilon''.
\end{equation}
Substituting \eqref{lb1_step3p1} and \eqref{lb1_step4} in \eqref{lb1_step0}, we get $\mathcal{R}_\text{e}\geq \mathcal{R}_\text{s}^- -\epsilon$, with $\epsilon=\epsilon'+\epsilon''$, and $\epsilon'$ and $\epsilon''$ are selected to be arbitrarily small.
This concludes the proof.$\hfill \square$

\subsubsection{Proof of the Upper Bound in Theorem 1}\label{PUB_CM}
To establish the upper bound on the common message secrecy capacity in~(\ref{th1_b}), we start by supposing that the transmitter sends message $w$ to only one legitimate receiver $\text{R}_k$. Using the result in \cite{rezki11}, for single user transmission with limited CSI feedback, the secrecy capacity of our system can be upper bounded as
\begin{equation}\label{UB_CM_stp1}
\mathcal{C}_{\text{s}}\!\leq\!\hspace{-0.2cm}\max_{\{\tau_q;P_q\}_{q=1}^Q}\sum_{q=0}^Q\Theta_{\tau_q}^{\gamma_k}\underset{\gamma_\text{e},\gamma_k}{\mathbb{E}}\!\left[\!\left\lbrace\!\log\!\left(\!\frac{1{+}\gamma_kP_q}{1{+}\gamma_\text{e}P_q}\!\right)\!\right\rbrace^{\hspace{-0.1cm}+}\!\Bigg|\tau_q\!\leq\!\gamma_k\!<\!\tau_{q+1}\!\right],
\end{equation}
with $\displaystyle{\Theta_{\tau_q}^{\gamma_k}{=}\text{Pr}\left[\tau_q{\leq}\gamma_k{<}\tau_{q+1}\right]}$.

\noindent Since the choice of the receiver to transmit to is arbitrary, we tighten this upper bound by choosing the legitimate receiver $\text{R}_k$ that minimizes this quantity, yielding
\begin{align}\label{UB_CM_stp2}
\mathcal{C}_{\text{s}}^+&=\min_{1\leq k\leq K}\max_{\{\tau_q;P_q\}_{q=1}^Q}\sum_{q=0}^Q\Theta_{\tau_q}^{\gamma_k}\\
&\qquad\quad\times\underset{\gamma_\text{e},\gamma_k}{\mathbb{E}}\!\left[\!\left\lbrace\!\log\!\left(\!\frac{1{+}\gamma_kP_q}{1{+}\gamma_\text{e}P_q}\!\right)\!\right\rbrace^{\hspace{-0.1cm}+}\!\Bigg|\tau_q\!\leq\!\gamma_k\!<\!\tau_{q+1}\!\right].\qquad \square\nonumber
\end{align}

\section{Broadcasting Independent Messages}\label{BIM}
In this section, we consider the independent messages case when multiple confidential messages are broadcasted to the legitimate receivers in the presence of an eavesdropper. Taking into account the adopted system model, we present an upper and a lower bounds on the ergodic secrecy sum-capacity.
\subsection{Main Results}\label{MR2}
\textit{Theorem 2:} The secrecy sum-capacity, $\widetilde{\mathcal{C}}_{\text{s}}$, of a block fading broadcast channel, with an error free $b$-bits CSI feedback sent by each legitimate receiver, at the beginning of each fading block, is bounded by
\begin{equation}\label{th2}
\widetilde{\mathcal{C}}_{\text{s}}^-\leq \widetilde{\mathcal{C}}_{\text{s}}\leq \widetilde{\mathcal{C}}_{\text{s}}^+,
\end{equation}
such as\vspace{-0.3cm}
\begin{subequations}
\begin{align}\label{th2_a}
\widetilde{\mathcal{C}}_{\text{s}}^-{=}\max_{\{\tau_q;P_q\}_{q=1}^Q}\sum_{q=1}^Q\Theta_{\tau_q}^{\gamma_\text{max}}\hspace{0.2cm}\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_qP_q}{1{+}\gamma_\text{e}P_q}\right)\right\rbrace^+\right],
\end{align}
and\vspace{-0.3cm}
\begin{align}\label{th2_b}
&\widetilde{\mathcal{C}}_{\text{s}}^+{=}\max_{\{\tau_q;P_q\}_{q=1}^Q}\sum_{q=0}^Q\Theta_{\tau_q}^{\gamma_\text{max}}\nonumber\\
&\times\underset{\gamma_\text{e},\gamma_\text{max}}{\mathbb{E}}\!\left[\left\lbrace\!\log\!\left(\frac{1{+}\gamma_\text{max}P_q}{1{+}\gamma_\text{e}P_q}\right)\!\right\rbrace^+\Bigg|\tau_q\!\leq\!\gamma_\text{max}\!<\!\tau_{q+1}\right],
\end{align}
where $\displaystyle{\gamma_\text{max}{=}\max_{1\leq k\leq K}\gamma_k}$, $Q{=}2^b$, $\{\tau_q~\!|~\!0{=}\tau_0{<}\tau_1{<}\cdots{<}\tau_Q\}_{q=1}^Q$ represent the reconstruction points describing the support of $\gamma_\text{max}$ with $\tau_{Q+1}{=}\infty$ for convenience, $\{P_q\}_{q=1}^Q$ are the power transmission strategies satisfying the average power constraint, and $\Theta_{\tau_q}^{\gamma_\text{max}}{=}\text{Pr}\left[\tau_q{\leq}\gamma_\text{max}{<}\tau_{q+1}\right]$ for all $q\in\{1,{\cdots},Q\}$.
\end{subequations}

\textit{Proof:} A detailed proof of Theorem 2 is provided in the following subsection.
Theorem~2 states that even with 1-bit feedback, sent by the strongest legitimate receiver at the beginning of each fading block, a positive secrecy rate can still be achieved.\vspace{0.2cm}

\textit{Remarks:}
\begin{itemize}
\item The presented results, for both common message and independent messages transmissions, are also valid in the case when multiple non-colluding eavesdroppers conduct the attack. In the case when the eavesdroppers collude, the results can be extended by replacing the term $\gamma_\text{e}$ with the squared norm of the vector of channel gains of the colluding eavesdroppers.
\item In the analyzed system, we assumed unit variance Gaussian noises at all receiving nodes. The results can be easily extended to a general setup where the noise variances are different.
\end{itemize}

\textit{Corollary 2:} The secrecy sum-capacity of a block fading broadcast channel with perfect main CSIT, and the average power constraint in \eqref{pavg}, is given by
\begin{align}\label{cr2}
\widetilde{\mathcal{C}}_{\text{s}}=\max_{P(\gamma_\text{max})}\underset{\gamma_\text{max},\gamma_\text{e}}{\mathbb{E}}\!\left[\left\lbrace\log\!\left(\frac{1{+}\gamma_\text{max}P(\gamma_\text{max})}{1{+}\gamma_\text{e}P(\gamma_\text{max})}\right)\right\rbrace^+\right],
\end{align}
with $\gamma_\text{max}{=}\max_{1\leq k\leq K}\gamma_k$, and $\mathbb{E}[P(\gamma_\text{max})]\leq P_\text{avg}$.

\vspace{0.3cm}
\textit{Proof:} Corollary 2 results directly by following a similar reasoning as for the proof of Corollary 1. $\hfill \square$

To the best of our knowledge, this result has not been reported in earlier works. For the special case of single user transmission, the secrecy sum-capacity in corollary 2 coincides with the result in Theorem~2 from reference \cite{gopala01}.

\subsection{Ergodic Capacity Analysis}\label{ECA2}
In this subsection, we establish the obtained results for the ergodic secrecy sum-capacity
\subsubsection{Proof of Achievability in Theorem 2}\label{PLB_IM}
The lower bound on the secrecy sum-capacity, presented in \eqref{th2_a}, is achieved using a time division multiplexing scheme that selects periodically one receiver to transmit to. More specifically, we consider that, during each fading block, the source only transmits to the legitimate receiver with the highest $\tau_q$, and if there are more than one, we choose one of them randomly. Since we are transmitting to only one legitimate receiver at a time, the achieving coding scheme consists on using independent standard single user Gaussian wiretap codebooks.

During each fading block, the transmitter receives~$K$ feedback information about the CSI of the legitimate receivers. Since the channel gains of the $K$ receivers are independent, there are $M{=}Q^K$ different states for the received feedback information, as discussed in the proof of achievability of Theorem 1. Each of these states, $\mathcal{J}_m; m\in\{1,\cdots ,M\},$ represents one subchannel. The transmission scheme consists on sending an independent message, intended for the receiver with the highest $\tau_q$, on each of the $M$ subchannels, with a fixed rate. Let $\tau_m^\text{max}$ be the maximum received feedback information on channel $m$. The overall achievable secrecy sum-rate can be written as
\begin{align}
&\widetilde{\mathcal{R}}_\text{s}^-=\sum_{m=1}^M\text{Pr}[\mathcal{J}_m]~\!\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_m^\text{max}P(\tau_m^\text{max})}{1{+}\gamma_\text{e}P(\tau_m^\text{max})}\right)\right\rbrace^+\right]\label{Rs_IM_stp1}\\
&=\sum_{q=1}^Q\text{Pr}[\tau_q{\leq}\gamma
_\text{max}{<}\tau_{q+1}]~\!\underset{\gamma_\text{e}}{\mathbb{E}}\left[\left\lbrace\log\left(\frac{1{+}\tau_qP_q}{1{+}\gamma_\text{e}P_q}\right)\right\rbrace^+\right],\label{Rs_IM_stp2}
\end{align}
where \eqref{Rs_IM_stp1} is obtained by using a Gaussian codebook with power $P(\tau_m^\text{max})$, satisfying the average power constraint, on each subchannel $m$ \cite{gopala01}, and \eqref{Rs_IM_stp2} follows by using the fact that $\tau_m^\text{max}\in\{\tau_1,\cdots ,\tau_Q\}$ and rewriting the summation over these indices. Also, we note that the probability of adapting the transmission with $\tau_q$ corresponds to the probability of having $\tau_q{\leq}\gamma
_\text{max}{<}\tau_{q+1}$, with $\gamma_\text{max}=\max_{1\leq k\leq K}\gamma_k$.
Maximizing over the main channel gain reconstruction points $\tau_q$ and the associated power transmission strategies $P_q$, for each $q\in\{1,\cdots ,Q\}$, concludes the proof.$\hfill \square$
\subsubsection{Proof of the Upper Bound in Theorem 2}\label{PUB_IM}
To prove that $\widetilde{\mathcal{C}}_{\text{s}}^+$, presented in \eqref{th2_b}, is an upper bound on the secrecy sum-capacity, we consider a new genie-aided channel whose capacity upper bounds the capacity of the $K$-receivers channel with limited CSI feedback. The new channel has only one receiver that observes the output of the strongest main channel. The output signal of the genie-aided receiver is given by $Y_\text{max}(t)=h_\text{max}(i)X(t)+v(t)$, at time instant $t$, with $h_\text{max}$ being the channel gain of the best legitimate channel, i.e., $|h_\text{max}|^2{=}\gamma_\text{max}$ and $\gamma_\text{max}{=}\max_{1\leq k\leq K}\gamma_k$.
Let $\tau_q; q\in\{1,{\cdots},Q\}$ be the feedback information sent by the new receiver to the transmitter about its channel gain, i.e., $\tau_q$ is feedbacked when $\tau_q{\leq}\gamma_\text{max}{<}\tau_{q+1}$. First, we need to prove that the secrecy capacity of this new channel upper bounds the secrecy sum-capacity of the $K$-receivers channel with limited CSI. To this end, it is sufficient to show that if a secrecy rate point $(\mathcal{R}_1,\mathcal{R}_2,{\cdots},\mathcal{R}_K)$ is achievable on the $K$-receivers channel with limited CSI feedback, then, a secrecy sum-rate $\sum_{k=1}^K\mathcal{R}_k$ is achievable on the new channel.

Let $(W_1,\!W_2,\!{\cdots},\!W_K)$ be the independent transmitted messages corresponding to the rates $(\mathcal{R}_1,\!\mathcal{R}_2,\!{\cdots},\!\mathcal{R}_K)$, and $(\hat{W}_1,\hat{W}_2,{\cdots},\hat{W}_K)$ the decoded messages. Thus, for any $\epsilon{>}0$ and $n$ large enough, there exists a code of length $n$ such that $\text{Pr}[\hat{W}_k\!\neq\!W_k]\!\leq\!\epsilon$ at each of the $K$ receivers, and
\begin{equation}
\frac{1}{n}H\!(W_k|W_1\!,\!\cdots\!,W_{k\!-\!1}\!,\!W_{k\!+\!1}\!,\!\cdots\!,\!W_K\!,\!Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\!F^L)\!\geq\!\mathcal{R}_k\!-\!\epsilon ,\label{sc23}
\end{equation}
with $F^L{=}\{F_1^L,F_2^L,{\cdots},F_K^L\}$, and $F_k(l)~\!{\in}~\!\{\tau_1,{\cdots},\tau_Q\}$ is the feedback information sent by receiver $k$ in the $l$-th fading block. Now, we consider the transmission of message $W{=}(W_1\!,\!W_2\!,\!\cdots\!,\!W_K)$ to the genie-aided receiver~using the same encoding scheme as for the $K$-receivers case. Adopting a decoding scheme similar to the one used at each~of the $K$ legitimate receivers, it is clear that the genie-aided~receiver can decode message $W$ with a negligible probability of error, i.e., $\text{Pr}(\hat{W}\!\neq\!W)\!\leq\!\epsilon$. For the secrecy condition, we have
\begin{align}
&\frac{1}{n}H\!(W|Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L,\!F_\text{max}^L)\nonumber\\
&=\frac{1}{n}H\!(W_1\!,\!W_2\!,\!\cdots\!,\!W_K|Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L,\!F_\text{max}^L)\\
&\geq\hspace{-0.1cm}\sum_{k=1}^K\!\frac{1}{n}H\!(W_k|W_1,\!\cdots\!,W_{k\!-\!1}\!,\!W_{k\!+\!1}\!,\!\cdots\!,\!W_K\!,\!Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L,\!F_\text{max}^L)\\
&\geq\hspace{-0.1cm}\sum_{k=1}^K\!\frac{1}{n}H\!(W_k|W_1,\!\cdots\!,W_{k\!-\!1}\!,\!W_{k\!+\!1}\!,\!\cdots\!,\!W_K\!,\!Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L,\!F^L)\label{UB_IM_v0}
\end{align}\vspace{-0.4cm}
\begin{equation}
\geq\hspace{-0.1cm}\sum_{k=1}^K\mathcal{R}_k{-}K\epsilon ,\label{UB_IM_v01}\hspace{5.5cm}
\end{equation}
where $F_\text{max}^L{=}\{F_\text{max}(1),{\cdots},F_\text{max}(L)\}$ and $F_\text{max}(l)$ is the feedback information sent by the genie-aided receiver in the $l$-th fading block, \eqref{UB_IM_v0} follows from the fact that $F_\text{max}{\in}\{F_1,{\cdots},F_K\}$ and that conditioning reduces the entropy, and where \eqref{UB_IM_v01} follows from the secrecy constraint \eqref{sc23}.

Now, we need to prove that $\widetilde{\mathcal{C}}_\text{s}^+$ upper bounds the secrecy capacity of the genie-aided channel. Let $\widetilde{\mathcal{R}}_\text{e}$ be the equivocation rate of the new channel. We have
\allowdisplaybreaks
\begin{align}
&n\widetilde{\mathcal{R}}_\text{e}=H(W|Y_\text{e}^n,\gamma_\text{e}^L,\gamma_\text{max}^L,F_\text{max}^L)\label{Re_step1}\\
&=I(W\!;\!Y_\text{max}^n|Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L\!,\!F_\text{max}^L){+}H(W|Y_\text{max}^n\!,\!Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L\!,\!F_\text{max}^L)\\
&\leq I(W;Y_\text{max}^n|Y_\text{e}^n,\gamma_\text{e}^L,\gamma_\text{max}^L,F_\text{max}^L){+} n\epsilon \label{Re_step2}\\
&=\sum_{l=1}^L\sum_{k=1}^\kappa\hspace{-0.1cm}I(W\!;\!Y_\text{max}(l,k)|Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L,F_\text{max}^L,Y_\text{max}^{\kappa(l\!-\!1)\!+\!(k\!-\!1)}){+}n\epsilon \\
&=\sum_{l=1}^L\sum_{k=1}^\kappa H(Y_\text{max}(l,k)|Y_\text{e}^n,\gamma_\text{e}^L,\gamma_\text{max}^L,F_\text{max}^L,Y_\text{max}^{\kappa(l\!-\!1)\!+\!(k\!-\!1)})\nonumber\\
&-H(Y_\text{max}(l,k)|W\!,\!Y_\text{e}^n\!,\!\gamma_\text{e}^L\!,\!\gamma_\text{max}^L\!,\!F_\text{max}^L\!,\!Y_\text{max}^{\kappa(l\!-\!1)\!+\!(k\!-\!1)}){+} n\epsilon \\
&\leq \sum_{l=1}^L\sum_{k=1}^\kappa H(Y_\text{max}(l,k)|Y_\text{e}(l,k),\gamma_\text{e}(l),\gamma_\text{max}(l),F_\text{max}^l)\\
&-H(Y_\text{max}(l,k)|W\!,\!X(l,k)\!,\!Y_\text{e}^n,\gamma_\text{e}^L\!,\!\gamma_\text{max}^L\!,\!F_\text{max}^L\!,\!Y_\text{max}^{\kappa(l\!-\!1)\!+\!(k\!-\!1)}){+} n\epsilon\nonumber\\
&=\sum_{l=1}^L\sum_{k=1}^\kappa H(Y_\text{max}(l,k)|Y_\text{e}(l,k),\gamma_\text{e}(l),\gamma_\text{max}(l),F_\text{max}^l)\\
&-H(Y_\text{max}(l,k)|X(l,k),Y_\text{e}(l,k),\gamma_\text{e}(l),\gamma_\text{max}(l),F_\text{max}^l){+} n\epsilon\nonumber\\
&= \sum_{l=1}^L\sum_{k=1}^\kappa\hspace{-0.1cm}I(X(l\!,\!k);\!Y_\text{max}(l\!,\!k)|Y_\text{e}(l,k)\!,\!\gamma_\text{e}(l)\!,\!\gamma_\text{max}(l)\!,\!F_\text{max}^l)\!+\!n\epsilon
\end{align}\vspace{-0.8cm}
\begin{align}
&\leq \sum_{l=1}^L\sum_{k=1}^\kappa \left\lbrace I(X(l,k);Y_\text{max}(l,k)|\gamma_\text{max}(l),F_\text{max}^l)\right.\hspace{2cm}\nonumber\\
&\left.\hspace{1.6cm}-I(X(l,k);Y_\text{e}(l,k)|\gamma_\text{e}(l),F_\text{max}^l)\right\rbrace^+ {+} n\epsilon \label{Re_step3}\\
&= \sum_{l=1}^L\kappa \left\lbrace I(X(l);Y_\text{max}(l)|\gamma_\text{max}(l),F_\text{max}^l)\right.\nonumber\\
&\left.\hspace{1.6cm}-I(X(l);Y_\text{e}(l)|\gamma_\text{e}(l),F_\text{max}^l)\right\rbrace^+ {+} n\epsilon ,\label{Re_step3p1}
\end{align}
where inequality \eqref{Re_step2} follows from the fact that $H(W|Y_\text{max}^n,Y_\text{e}^n,\gamma_\text{e}^L,\gamma_\text{max}^L,F_\text{max}^L){\leq}H(W|Y_\text{max}^n,\gamma_\text{max}^L,F_\text{max}^L),$ and Fano's inequality $H(W|Y_\text{max}^n,\gamma_\text{max}^L,F_\text{max}^L){\leq}n\epsilon ,$ and \eqref{Re_step3} holds true by selecting the appropriate value for the noise correlation to form the Markov chain $X(l){\rightarrow}Y_\text{max}(l){\rightarrow}Y_\text{e}(l)$ if $\gamma_\text{max}(l){>}\gamma_\text{e}(l)$ or $X(l){\rightarrow}Y_\text{e}(l){\rightarrow}Y_\text{max}(l)$ if $\gamma_\text{max}(l){\leq}\gamma_\text{e}(l)$, as explained in \cite{liang01}.

The right-hand side of \eqref{Re_step3p1} is maximized by a Gaussian input. That is, taking $X(l){\sim}\mathcal{CN}\left(0,\omega_l^{1/2}(F_\text{max}^l)\right)$, with the power policy $\omega_l(F_\text{max}^l)$ satisfying the average power constraint, we can write
\begin{align}
&n\widetilde{\mathcal{R}}_\text{e}\leq\kappa\sum_{l=1}^L\mathbb{E}\!\left[\!\left\lbrace\log\left(\frac{1{+}\gamma_\text{max}(l)\omega_l(F_\text{max}^l)}{1{+}\gamma_\text{e}(l)\omega_l(F_\text{max}^l)}\right)\right\rbrace^+\right]{+}n\epsilon \\
&{=}\kappa\sum_{l=1}^L\mathbb{E}\!\left[\!\mathbb{E}\!\resizebox{.35\textwidth}{!}{$\displaystyle{\left[\!\left\lbrace\!\log\!\left(\!\frac{1{+}\gamma_\text{max}(l)\omega_l(F_\text{max}^l)}{1{+}\gamma_\text{e}(l)\omega_l(F_\text{max}^l)}\!\right)\!\right\rbrace^{\hspace{-0.15cm}+}\!\bigg|F_\text{max}(l),\gamma_\text{max}(l),\gamma_\text{e}(l)\right]}$}\right]\hspace{-0.15cm}{+}n\epsilon \\
&{\leq}\kappa\sum_{l=1}^L\mathbb{E}\!\left[\resizebox{.36\textwidth}{!}{$\displaystyle{\!\left\lbrace\log\!\left(\!\frac{1{+}\gamma_\text{max}(l)\mathbb{E}\!\left[\omega_l\scriptstyle{(F_\text{max}^l)|F_\text{max}(l),\gamma_\text{max}(l),\gamma_\text{e}(l)}\right]}{1{+}\gamma_\text{e}(l)\mathbb{E}\!\left[\omega_l\scriptstyle{(F_\text{max}^l)|F_\text{max}(l),\gamma_\text{max}(l),\gamma_\text{e}(l)}\right]}\right)\!\right\rbrace^{\hspace{-0.15cm}+}}$}\right]\hspace{-0.15cm}{+}n\epsilon \label{Re_step4} \\
&=\kappa\sum_{l=1}^L\mathbb{E}\!\left[\left\lbrace\log\left(\frac{1{+}\gamma_\text{max}(l)\Omega_l(F_\text{max}(l))}{1{+}\gamma_\text{e}(l)\Omega_l(F_\text{max}(l))}\right)\!\right\rbrace^{\hspace{-0.15cm}+}\right]\hspace{-0.1cm}{+}n\epsilon \label{Re_step5} \\
&=\kappa\sum_{l=1}^L\mathbb{E}\left[\left\lbrace\log\left(\frac{1{+}\gamma_\text{max}\Omega_l(F_\text{max})}{1{+}\gamma_\text{e}\Omega_l(F_\text{max})}\right)\right\rbrace^+\right]\!+\!n\epsilon ,\label{Re_step6}
\end{align}
where \eqref{Re_step4} is obtained using Jensen's inequality, $\Omega_l(F_\text{max}(l))$ in \eqref{Re_step5} is defined as $\displaystyle{\Omega_l(F_\text{max}(l)){=}\mathbb{E}\left[\omega_l(F_\text{max}^l)|F_\text{max}(l),\gamma_\text{max}(l),\gamma_\text{e}(l)\right],}$ and where \eqref{Re_step6} follows from the ergodicity and the stationarity of the channel gains, i.e., the expectation in \eqref{Re_step5} does not depend on the block fading index. Thus, we have
\begin{align}
&\widetilde{\mathcal{R}}_\text{e}\leq\frac{1}{L}\sum_{l=1}^L\mathbb{E}\!\left[\left\lbrace\log\left(\frac{1{+}\gamma_\text{max}\Omega_l(F_\text{max})}{1{+}\gamma_\text{e}\Omega_l(F_\text{max})}\right)\right\rbrace^+\right]\!+\!\epsilon\\
&\leq\mathbb{E}\left[\left\lbrace\log\left(\frac{1{+}\gamma_\text{max}\Omega(F_\text{max})}{1{+}\gamma_\text{e}\Omega(F_\text{max})}\right)\right\rbrace^+\right]+\epsilon ,\label{Re_step7}
\end{align}
where \eqref{Re_step7} comes from applying Jensen's inequality once again, with $\displaystyle{\Omega(F_\text{max}){=}\frac{1}{L}\sum_{l=1}^L\Omega_l(F_\text{max}).}$
Maximizing over the main channel gain reconstruction points $\tau_q$ and the associated power transmission strategies $P_q$, for each $q\in\{1,\cdots ,Q\}$, concludes the proof.$\hfill \square$
\section{Numerical Results}\label{NR}
In this section, we provide selected simulation results for the case of Rayleigh fading channels with $\mathbb{E}\left[\gamma_\text{e}\right]{=}\mathbb{E}\left[\gamma_k\right]{=}1$; $k~\!{\in}~\!\{1,{\cdots},K\}$. Figure~\ref{fig:fig1} illustrates the common message achievable secrecy rate $\mathcal{C}_\text{s}^-$, in nats per channel use (npcu), with $K{=}3$ and various $b$-bits feedback, $b{=}6,4,2,1.$ The secrecy capacity $\mathcal{C}_\text{s}$, from Corollary 1, is also presented as a benchmark. It represents the secrecy capacity with full main CSI at the transmitter. We can see that, as the capacity of the feedback link grows, i.e., the number of bits $b$ increases, the achievable rate grows toward the secrecy capacity $\mathcal{C}_\text{s}$.

The independent messages transmission case is illustrated in figure~\ref{fig:fig2}. Two scenarios are considered; the transmission of three independent messages to three legitimate receivers, $K{=}3$, and the transmission of six independent messages with $K{=}6$. Both the achievable secrecy sum-rate $\widetilde{\mathcal{C}}_\text{s}^-$, with 4-bits CSI feedback, and the secrecy capacity $\widetilde{\mathcal{C}}_\text{s}$, with perfect main CSI, are depicted. The curves are presented in npcu. From this figure, we can see that the secrecy throughput of the system, when broadcasting multiple messages, increases with the number of legitimate receivers $K$.

\begin{figure}[t]
\psfrag{}[l][l][1]{}
\psfrag{rate}[l][l][1.3]{\hspace{-1cm}Secrecy Rate (npcu)}
\psfrag{snr}[l][l][1.3]{SNR (dB)}
\psfrag{Secrecy Capacity with Perfect Main CSI--------}[l][l][1.2]{Secrecy Capacity $\mathcal{C}_\text{s}$ with Perfect Main CSI}
\psfrag{Secrecy Rate with 6-bits CSI Feedback}[l][l][1.2]{Secrecy Rate $\mathcal{C}_\text{s}^-$ with 6-bits CSI Feedback}
\psfrag{Secrecy Rate with 4-bits CSI Feedback}[l][l][1.2]{Secrecy Rate $\mathcal{C}_\text{s}^-$ with 4-bits CSI Feedback}
\psfrag{Secrecy Rate with 2-bits CSI Feedback}[l][l][1.2]{Secrecy Rate $\mathcal{C}_\text{s}^-$ with 2-bits CSI Feedback}
\psfrag{Secrecy Rate with 1-bit CSI Feedback}[l][l][1.2]{Secrecy Rate $\mathcal{C}_\text{s}^-$ with 1-bit CSI Feedback}
\vspace{-0.7cm}
\begin{center}%
\scalebox{0.55}{\includegraphics{CM_Pavg_Q.eps}}
\end{center}
\vspace{-0.7cm}
\caption{Common message secrecy rate, for Rayleigh fading channels, with $K{=}3$ and various $b$-bits feedback, $b{=}6,4,2,1.$}
\label{fig:fig1}
\end{figure}

\begin{figure}[ht]
\psfrag{}[l][l][1]{}
\psfrag{rate}[l][l][1.3]{\hspace{-1.3cm}Secrecy Sum-Rate (npcu)}
\psfrag{snr}[l][l][1.3]{SNR (dB)}
\psfrag{Secrecy Capacity with Perfect Main CSI and k=3--------}[l][l][1.15]{Secrecy Capacity $\widetilde{\mathcal{C}}_\text{s}$ with Perfect Main CSI and $K{=}3$}
\psfrag{Secrecy Capacity with Perfect Main CSI and k=6}[l][l][1.15]{Secrecy Capacity $\widetilde{\mathcal{C}}_\text{s}$ with Perfect Main CSI and $K{=}6$}
\psfrag{Secrecy Rate with 4-bits CSI Feedback and k=3}[l][l][1.15]{Secrecy Rate $\widetilde{\mathcal{C}}_\text{s}^-$ with 4-bits CSI Feedback and $K{=}3$}
\psfrag{Secrecy Rate with 4-bits CSI Feedback and k=6}[l][l][1.15]{Secrecy Rate $\widetilde{\mathcal{C}}_\text{s}^-$ with 4-bits CSI Feedback and $K{=}6$}
\vspace{-0.55cm}
\begin{center}%
\scalebox{0.55}{\includegraphics{IM_Pavg_K.eps}}
\end{center}
\vspace{-0.7cm}
\caption{Independent messages secrecy sum-rate, for Rayleigh fading channels, with $b{=}4$ and two different values of the total number of legitimate receivers, $K{=}3$ and $K{=}6$.}\vspace{-0.4cm}
\label{fig:fig2}
\end{figure}

\section{Conclusion}\label{conclusion}
In this work, we analyzed the ergodic secrecy capacity of a broadcast block-fading wiretap channel with limited CSI feedback. Assuming full CSI on the receivers' side and an average power constraint at the transmitter, we presented lower and upper bounds on the ergodic secrecy capacity and the sum-capacity when the feedback link is limited to $b$ bits per fading block. The feedback bits are provided to the transmitter by each legitimate receiver, at the beginning of each coherence block, through error-free public links with limited capacity. The obtained results show that the secrecy rate when broadcasting a common message is limited by the legitimate receiver having, on average, the worst main channel link, i.e., the legitimate receiver with the lowest average SNR. For the independent messages case, we proved that the achievable secrecy sum-rate is contingent on the legitimate receiver with the best instantaneous channel link. Furthermore, we showed that the presented bounds coincide, asymptotically, as the capacity of the feedback links become large, i.e. $b\rightarrow\infty$; hence, fully characterizing the secrecy capacity and the sum-capacity in this case. As an extension of this work, it would be of interest to examine the behavior of the secrecy capacity bounds in the low and the high SNR regimes. It would also be interesting to look at the scaling laws of the system as the number of legitimate receivers increases.

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