%The entropy stored in inflationary spectra Inflationary %primordial spectra and partially decohered distributions
David Campo
cpPeyresqv23.tex


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\vskip 1truecm

\title{
%The entropy stored in inflationary spectra
Inflationary
%primordial
spectra and partially decohered distributions}
%, decoherence,
%entropy,and two-mode coherent states}
\vskip 1truecm
\author{David Campo}
\email[]{dcampo@math.uwaterloo.ca}
 \affiliation{Department of Applied Mathematics, University of Waterloo,
 University Avenue, Waterloo, Ontario, N2L 3G1 Canada}
%\email[]{campo@phys.univ-tours.fr}
%\affiliation{Laboratoire de Math\'{e}matiques et Physique
%Th\'{e}orique, CNRS UMR 6083,
%Universit\'{e} de Tours, 37200 Tours, France}
\author{Renaud Parentani}
%\email[]{parenta@celfi.phys.univ-tours.fr}
\email[]{Renaud.Parentani@th.u-psud.fr}
%\affiliation{Laboratoire de Math\'{e}matiques et Physique
%Th\'{e}orique, CNRS UMR 6083,
%Universit\'{e} de Tours, 37200 Tours, France}
\affiliation{Laboratoire de Physique Th\'{e}orique, CNRS UMR 8627,
B\^atiment 210,
Universit\'{e} Paris XI, 91405 Orsay Cedex, France}
%\date{D-28 Novembre, 2003}
\maketitle


\vskip 1truecm
\centerline{{\bf Abstract }}
\vskip 0.5truecm
%We re-examine the question of the entropy stored in the
%decoherence of the
%distribution of primordial density fluctuations.
 It is generally expected that decoherence processes will erase the
 quantum properties
 %which originally characterized
 of the inflationary
 %distribution of
 primordial spectra.
 %fluctuations.
 However, given the weakness of gravitational interactions,
 one might end up with a distribution which is only
 partially decohered.
 % and which has kept some quantum properties.
% We analyze these distributions and
% find that their properties
% fall into two classes: they are either extremely robust or
% fragile to a small change of the distribution. Nevertheless, b
 Below a certain critical change, we show that the inflationary
 distribution retains quantum %(i.e. fragile)
 properties.
% which can lead to violation of Bell's inequalities.
 %prevent to consider it as stochastic distributions.
% There exist four such properties
We identify four of these: a squeezed spread
 in some direction of phase space, non-vanishing off-diagonal matrix
 elements, and two properties used in quantum optics called
 non-$P$-representability and non-separability.
 The last two are necessary conditions to violate Bell's inequalities.
 The critical value above which all these properties are lost
 is associated to the `grain' of coherent states.
 The corresponding value of the entropy
 is equal to half the maximal (thermal) value.
 Moreover it coincides with the
 entropy of the effective distribution obtained by neglecting the
 decaying modes. By considering backreaction effects, we also
 provide an upper bound for this entropy at the onset of the
 radiation dominated era.



 %Their analysis reveals first that their classical (quantum)
 %statistical properties are robust (fragile) with respect to perturbations.
 %Second, statistical superpositions of two-mode coherent states
 %play a crucial role: they retain the classical properties of
 %the original distribution
 %%, that is the power spectrum and the temporal
 %%coherence of the modes at horizon re-entry. Second, they
 %and they determine the level of decoherence from which all quantum
 %properties are erased.
 %Third, their entropy is very sensitive to the residual level of
 %quantum coherence. Moreover, minimally decohered distributions
 %have a high entropy, half the maximal (thermal) value, and it coincides with the
 %entropy of the effective distribution obtained by neglecting the
 %decaying modes. By considering backreaction effects, we also
 %provide an upper bound for this entropy at the onset of the
 %adiabatic era.



%We analyze these distributions and we identify their classical and
%their residual quantum properties.
%% and their classical ones,
%%it is appropriate to use
%Our analysis rests on two-mode coherent states.
%%On one hand, these states easily express
%%%R13 c'est lourdingue both
%%the semi-classical character of highly amplified modes
%%%R05areas well as
%%%and their temporal coherence upon horizon re-entry.
%%%incorporate the%Dhomogeneity and
%%% isotropy of the distribution as well as the temporal coherence and
%%%the semi-classical character of highly amplified modes.
%%%We shall establish that
%%On the other hand, t
%These semi-classical states play a critical role: they determine
%the decoherence level from which all quantum properties are
%erased. The corresponding value of the entropy is high: it is half
%the maximal (thermal) value. Moreover it coincides with the
%entropy of the effective distribution obtained by neglecting the
%decaying modes. By considering backreaction effects, we also
%provide an upper bound for this entropy at the onset of the
%adiabatic era.




\vskip 1truecm

\section{Introduction}

Inflation tells us that the primordial density fluctuations arise
from the amplification of vacuum fluctuations
 \cite{Mukhanov81,Staro79,H,G,B}. As a result of this amplification, the
initial vacuum state becomes a product of highly squeezed two-mode
states \cite{Grishchuk90}.
 In spite of their
 intricate character,
%R26complexity of the writing of these states in the occupation number basis,
expectation values
%R26 , such as the two-point function,
are rather simple.
For instance, when evaluated on the Last Scattering Surface,
besides the small deviations from a scale invariant spectrum
\cite{LiddleLythPhysRep}, the non-trivial information delivered by
the two-point function concerns the temporal coherence of the
modes \cite{Albrecht00}. This last property follows from the fact
that its Fourier transform, i.e. the power spectrum, displays
oscillations and zeros. Since we are dealing with an ensemble, the
presence of these zeros tells us that {\it all} realizations of
the ensemble with a given wavenumber have the same temporal phase.
 In classical terms, this coherence can be easily
 enforced by neglecting the decaying mode. The residual random properties of
the distribution then only concern the amplitude of the
 %D non-decaying (ne rafinons pas, restons simples)
 growing modes.
In the high occupation number limit, these amplitudes can be
treated as stochastic
%R26 Gaussian j'ai supprime car la gauss ne decoulep as de high occ n. limit
variables.


 In brief, when
considering the physics which took place near
 the recombination,
%late time behaviour of the two-point function
it is convenient and sufficient to neglect the decaying mode.
%,thereby ignoring the quantum properties of the original distribution.
However this simplified description has several
drawbacks. Indeed, the settings are too restrictive to describe
distributions which result from decoherence processes (or more
generally any non-linear process) taking place in the early
universe. In particular, generic modifications of two-point
function from the standard result
%induced non-Gaussianities
%will destroy the factorizability of the state
%into uncoupled sectors of different wave-vectors.
%introducing non-Gaussianities,
%and altering the quantum coherence characterizing squeezed states.
% Second, and more phenomenologically, the simplified
%settings are also unable
%degenerate in that they cannot be used
cannot be parameterized in these simplified terms.
% the generic
%phenomenological description of
% deviations from the standard results,
This is true even for modifications
%which results from some unknown process but
which preserve
the isotropy and the Gaussianity of the distribution.

The aim of this paper is to analyze quantum distributions characterized by
an arbitrary level of coherence. We identify the parameter
which governs this level and determine how the properties of the
distribution depend on it.
%In particular, we shall see that
At fixed occupation number, two-mode squeezed states
%inflationary distribution
give the most coherent distribution, and thermal states the least one.
Starting with squeezed states in the high occupation number limit,
when slightly modifying this parameter,
%R40 we shall see that
the properties are either extremely robust,
and therefore accessible in the classical regime, or
extremely fragile.
%Nevertheless, below a certain critical value,
%distributions retain these fragile properties.
% when slightly modifying this parameter.
%R23 perturbing the distribution. In
%fact the higher the occupation number, the neater is the
%separation.
One finds that the fragile properties are quantum mechanical in
character. They concern the squeezed spread in some direction in
phase space (also called the sub-fluctuant direction), the quantum
coherence of macroscopically different states in the perpendicular
direction, as well as two refined properties called the
non-separability \cite{Werner89} and the non-$P$-representability
\cite{Glauber69} of the distribution. We show that both the last
properties are necessary to have expectation values which can
violate Bell's inequalities. Below a critical value of the
decoherence parameter, the distributions retain quantum
properties, and can therefore violate these inequalities.

Two-mode coherent states provide an appropriate basis to perform
this analysis. The reasons are the following. First, as well
known, a coherent state provides the quantum counterpart of a
particular classical realization of an ensemble. This
correspondence is well defined
%these states naturally incorporate
%the temporal coherence and
%in the semi-classical character of
for the highly excited (amplified) modes we are dealing with,
because the spread in position and momentum is negligible with
respect to the ``displacement'' in phase space which encodes the
occupation number. Remember that the observed temperature
anisotropies of relative amplitude $10^{-5}$ fix the mean
occupation number to be of the order of $10^{100}$.

 Second, homogeneity and isotropy restrict the non-vanishing matrix
elements of the distribution to {\it two-mode} elements with
opposite wave vectors ${\bf k}$. Hence, superpositions of
 {\it two-mode} coherent states
 can characterize Gaussian, homogeneous, and
 isotropic distribution with arbitrary level of coherence.
 Two important relationships should be noticed. First, the
 level of coherence is fixed by the expectation value
 of two destruction operators with opposite
 wave vectors.
 (This expectation value
 determines the number of quanta in {\it entangled pairs}.)
 %By that we mean that the distributions relevant for describing
 %primordial fluctuations are not diagonal in
 %${\bf k}$-space, but rather show correlations between
 %opposite modes.
% This follows from
%the requirement of isotropy and homogeneity which
%%drastically
%restricts the non-vanishing matrix elements
%of the distribution: only {\it two-mode} elements with opposite wave vectors
%could be different from zero.
Second, the residual coherence
of the distribution is related to the residual value of the decaying mode.
Therefore the `quality' of temporal coherence of the modes at horizon
re-entry can be (in principle) traced back to the quantum distribution.

 Third, two-mode coherent states allow to make contact
 with the general remark of \cite{Zurekbook,Zurek93}
according to which squeezed states rapidly decay into a
statistical mixture of coherent states when the oscillators are
weekly coupled to an environment.
%R19We expect that, w
When no longer neglected, small non-linearities \cite{Maldacena03}
amongst modes in inflationary models will
also induce some decoherence.
%R24produce a (je prefere etre vague)
%R23 also similar decay.
(Let us make
clear that the apparent violation of unitarity follows,
as usual, from the neglect of the
%R19 non-gaussianities
correlations among modes
 when evaluating the entropy, see footnote p. 188 in \cite{gottfried}. If
one keeps track of these non-linearities, the non-linear evolution
would give no entropy increase.) When applying the considerations
of \cite{Zurekbook,Zurek93} to two-mode squeezed states, the
distribution becomes diagonal in the basis of two-mode coherent
states, but not of one-mode coherent states. As we shall see, this
guarantees that the temporal coherence of modes is preserved while
the quantum properties are progressively erased.


The distribution which is diagonal in the two-mode coherent states
basis plays a critical role
because it separates distributions which have
kept their quantum properties from those which have lost them. The
corresponding value of the entropy
%For each mode, the lower bound of the entropy is very large since it
is $1/2$ the maximal (thermal) value, given the occupation
number.

To establish the {\it critical} character of this decoherence
scheme we explore the space of (partially) decohered Gaussian
distributions. When dealing with macroscopic occupation numbers,
the properties of these distributions fall into two separate
classes: they are either extremely robust or extremely sensitive
to the level of coherence. (In more dynamical terms, this would
translate into a insensitivity/sensitivity to a weak coupling to
an environment.) The entropy is found to be a sensitive variable
whereas the power spectrum is a robust one:
 the relative modifications of the latter are
$1/n$ whereas the changes of the former are in $\ln n$.
(Throughout the paper we shall evaluate these entropies exactly by
exploiting the fact that all Gaussian distributions can be
expressed in terms of thermal distributions of new oscillators,
see Appendix C.) Similarly, the residual squeezing and the quantum
coherence are both sensitive whereas the temporal coherence of the
modes is the second robust property of the distribution. In fact,
we shall see that {\it all} quantum properties are lost for a
critical value of the residual coherence of the distribution. This
critical value is that given by the {\it grain} of coherent
states.

We then exploit the sensitivity of the entropy to establish a
correspondence between the above distribution and the effective
one obtained by neglecting the decaying mode and used in numerical
codes such as CMBFAST \cite{cmbfast}. Since their entropy
coincide, and since the entropy faithfully traces the residual
coherence, this agreement tells us the quantum distributions
 which
correspond to the effective one are very `close' to the diagonal
distribution in the two-mode coherent state basis.

Finally, we also provide an upper bound for the entropy by
considering backreaction effects at the end of the inflationary
period. This upper bound is given by $3/4$ of the maximal entropy.

Having established these properties, two questions should be
confronted. The first concerns the efficiency of decoherence
processes which occurred in nature: {\it Were they powerful enough
to reach the critical decoherence level associated with coherent
states ?} This question is currently under investigation.
Preliminary results indicate that the critical value is indeed
reached thereby implying that the resulting entropy is larger than
%R40(or equal to)
the above bound. The second question is more
important. It concerns the impact of the decoherence level on
structure formation (i.e. non-linear evolution).
%R40
This is a difficult question but we can already acknowledge
the fact that the critical
decoherence level associated with the loss of quantum coherence is
so small that it could hardly have any impact on this evolution.
Therefore the question of the existence of residual quantum
properties seems to be of pure academic delight.

 We conclude this introduction by mentioning
 that closely related questions have been already analyzed
 in several papers, see
 \cite{MukhaBrandyProkopecg,Albrecht94,Giovanninig,Matacz94,PolStar96,KiefPolStar98a,KiefPolStar00,cpP1}.
 %D50 %R50
% We comment on those works in section V.C.
 What we add\footnote{The present work is based
 on our former paper \cite{cpP1}.
%\it Inflationary spectra, decoherence, and two-mode coherent states:
% astro-ph/$0404021$.
 For completeness and to ease the reading, we
 have included
 some of its material and shall no longer refer to it.}
 is a further clarification of the matters,
the critical role played by coherent states in separating density
matrices, the upper bound of the entropy, and the algebraic
relationships between the entropy and the other quantum properties.
%R34 Before addressing the above mentioned questions, we
%believed it was necessary to first provide the {\it appropriate
%tools} to describe partially decohered quantum distributions and
%to identify their properties.
%R34 seems obvious now.



\section{review of the standard derivation of primordial spectra}

\subsection{Quantum distribution of two-mode states}

In this subsection we recall how the amplification of modes
of a massless field propagating in a FRW universe translates
in quantum settings in the fact that
%propagation of a quantum field in a FRW universe initially in its
the initial ground state
%homogeneous, non-stationary background
evolves into a product of
%to a quantum distribution of
highly squeezed {two-mode states}.
%i.e. entangled, squeezed states.
%The properties of the spectrum of primordial
%fluctuations follows. Indeed,
Before proceeding, we remind the reader that
it has been shown that the evolution of linearized cosmological perturbations
(metric and density perturbations) reduces to the
propagation of
%real,
massless
%, minimally coupled
scalar fields in a FRW spacetime \cite{MBF92}. In this article, we
shall only consider the massless scalar test field since the
transposition of the results to physical fields represents no
difficulty. Indeed, when preserving the linearity of the
evolution, the only modification concerns the late time dependence
of the modes \cite{cmbslow}.



Let us work in a flat FRW universe. The line element is:
 \ba \label{metric}
  ds^2 &=& a(\eta)^2 \left[ -d\eta^2 +  \delta_{ij}dx^{i}dx^{j} \right] \, .
 \ea
For definiteness and simplicity, we consider a
cosmological evolution which starts with an inflationary de Sitter
phase and ends
with a %matter dominated period after having experienced a
radiation dominated period. When using the conformal time $\eta$
to parameterize the evolution, the scale factor is respectively
given by
 \sba \label{aeta}
  a(\eta) &=& - \frac{1}{H(\eta-2 \eta_r)} \, , \qquad
  {\mbox{for\, \, }} -\infty < \eta < \eta_r  \, , \\
  a(\eta) &=& \frac{1}{H\eta_r^2} \, \eta  \, , \qquad
  {\mbox{for\, \, }}  \eta > \eta_r \, ,
  % \\
  % a(\eta) &=&  \frac{1}{4H\eta_r^2}
  % \frac{(\eta-4\eta_r+\eta_{eq})^2}{\eta_{eq}-2\eta_r} \, , \qquad
  %{\mbox{for\, \, }} \eta_{eq} < \eta \, ,
 \sea
where $\eta_r> 0 $ %and $\eta_{eq}$
designates  % respectively
the end of inflation.
%The shift in the parenthesis in the second %and third lines are
%line is necessary to parameterize the two periods by a single conformal
%time $\eta$.
The transition
%pasting of these periods
is such that
the scale factor and the Hubble parameter are
continuous functions.
%and the time of equilibrium between radiation and matter.
This approximation based on an instantaneous transition is
perfectly justified for modes relevant for CMB physics. Indeed,
their wave vector $k$ obeys
%$k/Ha(\eta_r) =
$k \eta_r  \sim 10^{-25}\sim e^{-60}$ when inflation lasts for
$60$ e-folds. Hence, the phase shift they would accumulate in a
more realistic smoothed out transition would be completely
negligible.


Let $\xi(\eta,{\bf x})$ be a massless test scalar field
propagating in this background metric. It is convenient to work
with the rescaled field $\phi = a \xi $ and to decompose it into
Fourier modes
 \ba
   \phi(\eta, \bold x) &=& \int\!\!d^3k \,
   \frac{e^{i \bold k \bold x}}{(2\pi)^{3/2}}
   \phi_{\bf k}(\eta) \, .
 \ea
The time dependent mode $\phi_{\bold k}$ obeys
 \ba \label{PropPhi_k}
  \partial_{\eta}^2 \phi_{\bf k}  +
  \left(k^2 - \frac{\partial_\eta^2 a}{a} \right) \phi_{\bf k} = 0 \, ,
 \ea
where $k=\abs{\bold k}$ is the norm of the conformal wave vector.
%In Appendix A we recall how this equation gets modified when considering
%density perturbations. The rescaled $\bold k$-modes of gravitational waves
%obey \Eqref{PropPhi_k}\cite{??}.
%and where the time-dependent frequency is given
%b sound velocity $c_s^2=\delta p / \delta\rho$ \cite{MBF92}, e.g. $1/3$
%and $0$ in purely radiation and purely matter eras respectively. }
%\ba \label{omega_k}
% \omega_{k}^2(\eta) = {\bold k}^2 - \frac{a''}{a} \, .
%\ea

In our background solution, ${k}^2 - {\partial_\eta^2 a}/{a}$ is
negative during the de Sitter period when the wavelength is larger
that the Hubble radius. This leads to a large amplification of
$\phi_k$. In quantum terms this mode amplification translates into
spontaneous pair production characterized by correspondingly large
occupation numbers.

To obtain the final distribution of particles, one should
introduce two sets of positive frequency solutions of Eq.
\Eqref{PropPhi_k}. The {\it in} modes are defined at asymptotic
early time, and the {\it out} ones at late time. Both have unit
positive Wronskian in conformity with the usual particle
interpretation \cite{B&D}. One gets
 \sba \label{modes}
  \phi_k^{in}(\eta) &=& \frac{1}{\sqrt{2k}}
  \left(1-\frac{i}{k(\eta - 2 \eta_r)}\right)
  e^{-ik(\eta - 2 \eta_r)} \, , \quad
  %\label{dsmodes}
  \mbox{for\, \,} \eta < \eta_r   \, , \\
  \phi_k^{out}(\eta) &=&
  \frac{1}{\sqrt{2k}} e^{-ik\eta} \, ,\quad
  \mbox{for\, \, }  \eta > \eta_r  \, .
  %\label{outmods}
  %  \\
  % \phi_k^{mat}(\eta) &=& \frac{1}
  %%e^{i\phi_{mat}}}
  %{\sqrt{2k}}
  %   \left(1-\frac{i}{k(\eta - \eta_m)}\right)e^{-ik\eta} \, , \quad
  % \mbox{for\, \,} \eta_{eq} < \eta \, ,
 \sea
In spite of the time dependence of the background, these two
positive frequency modes are unambiguously defined (up to an
arbitrary constant phase which drops in all expectation values and
which has here been chosen so as to simplify the forthcoming
expressions). In the de Sitter epoch, there is no ambiguity for
relevant modes if inflation lasts more than $70$ e-folds, see
\cite{NPC} for the evaluation of the small corrections one obtains
when imposing positive frequency at some finite early time.
Moreover, there is no ambiguity for the initial state of relevant
modes: at the onset of inflation they must be in their ground
state \cite{MBF92,cmbParenta}.

In the radiation dominated era there is no ambiguity either
because the conformal frequency is constant since $\partial^2_\eta
a = 0$. In this era, the decaying and the growing modes
correspond, respectively, to the real and the imaginary part of
the out modes Eq. (\ref{modes}b),
 as can be seen by considering the limit $k \eta \ll 1$.
 % to define the the
 %D22 Je ne suis pas convaincu: avec cette definition on ne
 %comprends pas pourquoi l'un est decay et l'autre grow.
 %R23 what do you mean ??? one ne doit PAS TOUT expliquer, es tu d'ac que
 %le sine EST le growing mode ? si oui, je pense qui'il n'y a pas de proble
 %
 %D24
 %Et bien non, je ne suis pas d'accords.
 %C'est ce que je me tues a t'expliquer depuis une semaine.
 %La solution qui croit pendant l'inflation est Im(phi^{in}), et son prolongement
 % pour t>t_reh est Im(alpha \phi^{out} + beta \phi^{out}*),
 % qui differe de sin par une petite phase.
 %
 %%%R24 quelle est LA solution qui croit pendant l'ere de rad ????
 %D25 je ne comprends pas cette question : les solutions oscillent.
 %%oui mais la convention est de prendre le comportmt k eta <<<1 comme ref
 %%R25 cad deep inside car dans cette limite k drops out au leading order
 %De plus, le mode physique est divise par etat, donc son envoloppe diminue...
 %%pas pour k eta <<<1 ou il est constant
 %%R24 peux tu l'ecrire en terme de z ??
 %%n'y a til pas de mixing lors du reaheating ??
 %D25 Do you agree that the effect of reh sur la phase
 %est O(k \Delta \eta_reh) ? Is it smaller than (k\eta_r)^3 ???
 %
 %%R24 ce qui me gene dans ta reponse c'est quelle depend de eta_r
 %D25 ce qui me gene dans la tienne c'est qu'il me semble que ton
 %mode croissant n'est pas continu au reh. Have I done a mistake ?
 %If no, how can it be ?
 %Pourquoi 'la solution des equation du mouvement' n'est il pas LE
 %SEUL ET UNIQUE critere ?
 % le sine est sol dans l'ere de rad. POINT 1.
  %% il est le mode croiss. pour keta <<1
 %R24 hors a durant la rad. est prop to eta,
 %D25 peux tu me reexpliques pourquoi la phase du growing doit
 %etre propto a stp.
 %Is it light cone propag from reh (CP2).
 %In that case, why can't I choose the phase at reh as I am pleased ?
 %Is it your comment about NPC ? Please explain
 %
 %D24 Certes, un cosmologue ne verra pas la difference entre les 2 def,
 %mais adopter la tienne nous force a expliquer qu'il y a 2 powers differents
 %(et tu viens de dire qu'on ne doit pas tout expliquer !!???!!) : le power du cos et
 %le power de l'eigenmode subfluct. Ce qui est se fatiguer pour rien a mon avis.
 %
 %%R24 la je ne suis pas du tout d'ac. reprend NPC,
 %% en modi la def de l'etat on modifie la power du dec mode
 %%es tu d'accord ????
 %D25 je suis d'accord avec ce statement.
 % Mais je ne comprends pas le lien avec mon precedent comment
 % en quoi choisir une CL des modes out plutot qu'une autre modifie le vide in ????
 %
 %%%R24 le deca mode ne peut etre defini que par a(eta) durant l'ere de rad
 %D25 je suis perdu (voir mes questions/comments plus haut)
%D22 Growing is la solution qui croit, je ne comprend qu'il
 %puisse y avoir une autre definition, meme si tout ce qui est ecrit dans la
 %parenthese ci dessous est juste.
 (When considering only the leading order terms (in $1/k \eta_r$)
 of expectation values,
 several definitions of the growing modes can be used
  and lead to the same results. However
  in order to obtain unambiguous answers to the questions
  we shall ask (which concern next to leading order terms),
  % given that we are interested
  %to relate the degree of coherence of a distribution
  %to the value of certain quantities,
  %such as the residual power of the decaying mode,
  we must
  %be able to compute the corrections, one must
  adopt a precise definition.
  %R24
  In the sequel, we define the growing mode
  by the imaginary part of $\phi_k^{out}$.
  Such a definition is necessary
  to relate, for instance,
   the power of the decaying mode to that of
   the so-called subfluctuant mode.)
  % in other
  %. Given that we are interested
  %to relate the degree of coherence to the value of observables, such as
  %the residual power of the decaying mode,
  %we must adopt the second option to
  %get unambiguous answers.
  %This was not always done in the literature.)
%R21 on laisse cette phrase ??
 %D22 I think we should

%Since Eq. \Eqref{PropPhi_k} is homogeneous and  linear,
The {\it in} and {\it out} modes are related by a Bogoliubov transformation
\ba \label{inoutmodes}
 \phi_k^{in}(\eta) = \alpha_k \phi_k^{out}(\eta) +
             \beta_k^* \phi_k^{out\, *}(\eta) \, ,
\ea
where the Bogoliubov coefficients are given by the Wronskians
\ba \label{abcoef}
  \alpha_{k} = \left( \phi_{k}^{out} , \phi_{k}^{in} \right)
  \mbox{, \ \ }
  \beta_{k}^{*} = -  \left( \phi_{k}^{out\, *} , \phi_{k}^{in} \right) \, .
\ea
These overlaps should be evaluated at transition time $\eta_r$
since modes satisfy different equations in each era.
One gets
 \sba \label{bogolincosmo}
   \alpha_k &=& - \frac{ e^{2i k \eta_r}}{2k^2\eta_r^2}
   \left( 1 - 2 i{k\eta_r}- 2k^2\eta_r^2 \right)
   %\left( 1 + \frac{i}{k\eta_r}-\frac{1}{2k^2\eta_r^2} \right)
   = \frac{-1}{2k^2\eta_r^2} \, ( 1 + O(k\eta_r)^3) \, , \\
   \beta_k^{*} &=&
   %\frac{e^{-2i k \eta_r}}
   \frac{1}{2k^2\eta_r^2} \, .
 \sea
Thus, for relevant modes, $k\eta_r \sim 10^{-25}$, the in modes
are enormously amplified. Concomitantly, they are dominated by the
sine during the radiation dominated era because the phase of
$-\beta_k^{*} / \alpha_k$ is much smaller than one:
 \ba \label{neglectdecay}
  \phi_k^{in}(\eta)
   &=& \frac{i}{k^{2}\eta_r^2} \left[ \frac{\sin k\eta}{\sqrt{2 k}} \,
       + \, O(k\eta_r)^3 \,
       \frac{\cos k\eta}{\sqrt{2k}}\right]
       \, , \qquad  \eta \geq \eta_r \, .
 \ea
To leading order, i.e. neglecting the cosine in Eq.
\Eqref{neglectdecay}, the physical modes $\phi^{in}_k/a$
correspond to the growing modes. They possess a well-defined
temporal behavior, e.g. they are constant until they start
oscillating as they re-enter the Hubble radius when $k \eta \simeq
1$.



Lets now see how these considerations translate in second
quantized settings. Each mode operator is decomposed twice
 \ba \label{hatphik}
  \hat \phi_{\bold k}(\eta) =
  \hat a_{\bold k}^j  \phi_{k}^j(\eta) +
  \hat a_{-\bold k}^{j\, \dagger} \phi_{k}^{j \, *}(\eta) \, ,
 \ea
where $j$ stands for both the in and out basis.
%Eq. \Eqref{inoutmodes}. A family of 'in' and 'out'
%annihilation and creation operators is
%defined for each set of modes respectively :
%\ba
% \hat a_{\bold k}^{in} = \left(\phi_{k}^{in}\, , \hat \phi \right) \, , \qquad
%\hat a_{-\bold k}^{in \, \dagger} = - \left(\phi_{k}^{in\, *}\, , \hat \phi
%\right) \,  ,
%\ea
%and similar expressions hold for the out operators. In and out
The operators so defined are related by the
transformation
 \ba \label{inoutoperator}
   \hat a_{\bold k}^{in} = \alpha_k^{*} \, \hat a_{\bold k}^{out} -
              \beta_k \, \hat a_{-\bold k}^{out \, \dagger} \, .
 \ea
%Because of the homogeneity of the background,
This transformation
%is $2\times 2$ block-diagonal as it
couples ${\bf k}$ to $-{\bf k}$ only.
Hence, when starting from the in vacuum (the state annihilated by the
$\hat a_{\bold k}^{in}$ operators),
every out particle
of momentum ${\bf k}$ will be accompanied by a partner
of momentum $-{\bf k}$. Moreover, pairs characterized
by different momenta are incoherent
%in the in vacuum
(in the sense that in expectation values any
product of annihilation and creation operators of different momenta
will factorize).

These two properties are explicit when writing the in vacuum in
terms of out states (i.e. states with a definite out particle
content). From Eq. \Eqref{inoutoperator}, one gets (see
 \cite{MukhaBrandyProkopecg,PhysRep95})
 \ba \label{inoutvac}
  \ket{0,\, in}
  &=&
      \widetilde{\prod\limits_{\bold k}} \otimes
      \kett{0,\, \bold k, in}
  \nonumber\\
  &=&
      \widetilde{\prod_{\bold k}}  \otimes
      \left(  \inv{\vert \alpha_{k} \vert} \exp \left( z_{k}\,
      \hat  a_{\bold k}^{out \, \dagger}  \,
      \hat a_{-\bold k}^{out \, \dagger} \right)
      \ket{0,\, \bold k, out}\otimes \ket{0,\, -\bold k, out} \right) \, .
 \ea
The tilde tensorial product takes into account only half the
modes. It must be introduced because the squeezing operator acts
both on the $\bf k$ and the $-\bf k$ sectors. The definition of
this product requires the introduction of an arbitrary wave vector
to divide the modes into two sets. The sign of $k_z$ can be used.
Notice that a rigorous definition of $\widetilde{\prod_{\bold k}}$
requires to consider a discrete set of modes normalized with
Kroneckers (that is, to normalize the modes in a finite conformal
3-volume). To be explicit, the two-mode state $\kett{0,\, \bold k,
in} $ is given by
 \ba \label{in2mode}
   \kett{0,\, \bold k, in} =
   \ket{0,\, \bold k, in} \otimes \ket{0,\, -\bold k, in} \, ,
 \ea
where $\ket{0,\, \bold k, in}$ is the ground state of
 the $\bold k$-th mode at the onset of inflation. The complex parameter $z_k$
appearing in the squeezing operator in Eq. \Eqref{inoutvac} is
given by the ratio of the Bogoliubov coefficients
 \ba \label{z}
    z_k &=& \frac{\beta_k}{\alpha_k^*} = - e^{-2i k \eta_r}
    \frac{1}{\left( 1 + i2{k\eta_r}- 2k^2\eta_r^2 \right)}
    \nonumber \\
    &=& -  1 +  O(k\eta_r)^3  \, .
 \ea
 The high occupation number limit corresponds to
 $\abs{z_k} \to 1^-$.

It has to be emphasized that none of the out states in Eq.
\Eqref{inoutvac} carries 3-momentum. Hence, the distribution is
homogeneous in a strong sense: at late time the 3-momentum
operator is still annihilated by the state of Eq.
\Eqref{inoutvac}. (This property is not satisfied by incoherent
distributions such as thermal baths. In those cases, the
3-momentum fluctuates and vanishes only in the mean.) The present
distribution is also isotropic since the Bogoliubov coefficients
are functions of the norm $k$ only. Finally, it is a Gaussian
distribution, as can be seen from Eq. \Eqref{inoutvac}.
%This follows from the fact that
%the hamiltonian is quadratic \cite{Grishchuk90,Albrecht94,PolStar96}.
%since the state Eq. \Eqref{inoutvac} can be shown to
%results from the unitaty evolution
%of the in-vacuum \cite{albrecht,PolStar}.

 To appreciate the peculiar properties of the distribution of Eq.
 \Eqref{inoutvac},
 and as a preparation for the analysis of the problem of
 decoherence and entropy,
 it is interesting to consider the most general homogeneous,
 isotropic, and Gaussian distributions of out quanta.
 A detailed description of these states is given in Appendix D.
 Their properties are completely specified by three real functions of
 the norm $k$ (one real and one complex) through the following
 expectation values
 \sba \label{momenta}
   && \langle \hat a_{\bold k}^{out} \rangle =  0 \, , \\
   &&\langle \hat a_{\bold k}^{out\, \dagger} \,
   \hat  a_{\bold k'}^{out} \rangle
   = n_k   \, \delta^3(\bold k - \bold k' ) \, ,   \\
   &&\langle \hat a_{\bold k}^{out}  \,
   \hat a_{\bold k'}^{out} \rangle
   = c_k \, \delta^3(\bold k + \bold k' )\, .
 \sea
In the second line, $n_k$ is the mean occupation number. In the
third one, the complex number $c_k$ characterizes the quantum
 coherence of the distribution.
%It is to be noticed that the functions $n_k$ and $c_k$ can be used to characterize
%the most general homogeneous, isotropic, and Gaussian distribution.
The degree of two-mode coherence is given by $|c_k|/(n_k+1/2)$,
see Appendix C. It is bounded by 1. For a thermal (incoherent)
distribution, one has $c_k \equiv 0$: no 2-mode coherence.

 In the  case of pair production from vacuum, one has
 \ba \label{cterm}
   n_k = \abs{\beta_k}^2 = \frac{\abs{z_k}^2}{1- \abs{z_k}^2}\, ,
   \qquad
   c_k = \alpha_k \beta_k =
   \frac{z_k}{1- \abs{z_k}^2} \, .
 \ea
 Therefore, when considering relevant modes in inflation, using Eq.
 \Eqref{bogolincosmo}
 %\Eqref{z},
 one has
  \sba \label{cterm2}
    n_k &=& \frac{1}{4 (k \eta_r)^4 } \simeq 10^{100}
    \, , \\
    c_k &=& - \left[ n_k + \frac{1}{2}+O(\frac{1}{n}) \right]
    e^{i2\theta_k} \, , \qquad
    \theta_k =  \frac{10}{3}(k\eta_r)^3  + O({n_k^{-5/4}}) \, .
 \sea
 That is, the distribution which results from inflation is highly
 populated \footnote{We recall how $n_k \simeq 10^{100}$ is obtained.
 For the test field $\hat \xi = \hat \phi/a$, using Eq.
 \Eqref{neglectdecay}, the power
 spectrum at the onset of the radiative era is, in physical units,
 \ba \label{defprimpower}
   {\cal P}_{\xi}(k; \eta_r) = \frac{k^3}{2\pi^2} \,
   \bra{0 in} \hat \xi_{\bf k}(\eta_r)
   \hat \xi_{\bf k}^{\dagger}(\eta_r) \ket{0 in} =
   \frac{k^3}{2\pi^2} n_k \times \frac{\hbar}{2k}
   \frac{4(k \eta_r)^2}{a_r^2} = \frac{k^3}{2\pi^2} \, \frac{\hbar}{2k}
    \, \frac{H^2}{k^2} \nonumber
    \, ,
 \ea
 where $a_r^{-1}= H \eta_r$.
% $\hbar/2k \times (k \eta_r)^2 / a_r^2$
% is the contribution of each excitation of the mode
% $k$, see Eq.
%s. \Eqref{Ginx} and \Eqref{Gxi}.
 The power spectra of
 the gravitational potential $\Psi$ and the gravitons $h_{ij}^{TT}$,
 are obtained from
 the above. For slow-roll inflation \cite{MBF92},
 a mode by mode integration yields
 \ba \label{primpower}
   {\cal P}_{\Psi} = \frac{9}{25 \epsilon} \frac{4\pi \hbar G}{c^3}
   \left( \frac{H_k}{2\pi} \right)^2
   \left( 1 + O(\epsilon,\, \frac{d \ln \epsilon}{dN}) \right)
   \, , \qquad
   {\cal P}_{h_{ij}^{TT}} = 16 \epsilon {\cal P}_{\Psi} \, .
   \nonumber
 \ea
 Here $H_k$ designates the value of the Hubble parameter
 at horizon exit:
 $a(\eta_k) H(\eta_k) = k$.
 The data from COBE normalize the power spectrum
 %i.e. fix the product $H L_{Pl}$,
 to be $ \delta T_k / T \vert_{rms} \simeq \Psi_{k}/3
  \vert_{rms} \sim 10^{-5}$.
 The energy scale of inflation is thus
  $ (L_{Pl} H )^2 \simeq 10^{-10} \epsilon $,
 with $\epsilon = 10^{-2} - 10^{-1}$.
 The mean occupation number $n_k$ must therefore obey
 \ba
    n_{k} \simeq  \epsilon (\lambda_r H)^4 \, , \nonumber
 \ea
 where $\lambda_r = a_r/k$ is the physical wavelength of
 a mode $k$ at reheating. Assuming that reheating happened at
 redshift $\sim 10^{28}$,
 i.e. with a temperature $\simeq 10^{14} \rm{GeV}$,
 modes corresponding to large
 structures today, e.g. $100 {\rm Mpc} \sim 10^{21} {\rm m}$, have
 a mean occupation number $\simeq 10^{96}$.}
 and, more importantly, maximally coherent.
 These two properties go hand in hand for two-mode squeezed
 states.
 %verrry good
 When computing the two-point function,
 the mean occupation number determines the
 {\it primordial}
 power spectrum, while
 the two-mode coherence of
the distribution manifests itself in the {\it temporal coherence}
of the modes, as we now explain.



\subsection{Two-point function and the neglect of the decaying mode}

 %D27
 After the reheating,
 when expressed in terms of {\it out} modes, the (Weyl ordered) two-point
 function associated with the general distribution
 specified by Eqs. \Eqref{momenta} is
 \ba \label{Ginx}
    G^{in} (\eta,{\bf x}, \eta', {\bf x'}) &=&
     \int_0^{\infty} \frac{d k}{k} \frac{k^3}{2\pi^2}
    %\int_0^{\infty} \frac{dk k^2}{\pi^2}
    \frac{\sin(k\vert {\bf x} - {\bf x}'\vert)}{k\vert {\bf x} - {\bf x}'\vert}
    \left[ \left( n_k + \frac{1}{2} \right) \, \phi_k(\eta) \phi_k^*(\eta')
    +  c_k \, \phi_k(\eta) \phi_k(\eta') + c.c.
    \right]  \, . \nonumber \\
 \ea
We have dropped the superfix 'out' on the modes for lisibility. We
have chosen to work with the anti-commutator in order to obtain a
symmetrical function in $\eta,\, \eta'$ and to analyze the
classical limit, namely the high occupation number limit
 $n_k \gg 1$.

In order to discard the contribution of the decaying modes, two
conditions must be met.
 First, the sum in brackets should factorize.
 For a general distribution, it does not.
 However it factorizes
 for coherent states, see Eq. \Eqref{CohWaveFunct} in Appendix A,
 for distributions resulting from pair production from vacuum
 in the high occupation number limit,
 and, more generally, whenever the following inequality is satisfied
 \ba \label{neqqq}
   \frac{n_k+1/2 - \abs{c_k} }{n_k + 1/2} \ll 1 \, .
 \ea
 Let us write
 \ba \label{epsilontheta}
  c_k = - e^{i2\theta_k} (n_k+ \frac{1}{2})\, (1-\epsilon_k)
  \, ,
 \ea
 where $\epsilon_k$ and $\theta_k$ are real.
 Eq. \Eqref{neqqq} is equivalent to  $\abs{\epsilon_k} \ll 1$. Then
 one can write the %R21 term inside the
 bracket of \Eqref{Ginx} as
% %R21 je n'aime pas ton texte car tu mets l'accent sur la facto
% %et pas sur growng decaying. Le mode growing est deja def a l'eq 9
% % pourquoi ne pas ecrire le braket in terms of sin and cos
% %and call the sin THE grow mode ?
% \footnote{Notice that these linear combinations of the modes correspond to
% the diagonalization of the covariance matrix of Appendices C and D.4.}
 \ba \label{brackets}
   \left( n_k + \frac{1}{2} \right)\left[
   \left( e^{i\theta_k} \phi_k(\eta) - c.c. \right) \,
   \left( e^{-i\theta_k} \phi^{*}_k(\eta') -  c.c. \right) \, + \,
   \epsilon_k  \left( e^{i2\theta_k} \phi_k(\eta) \phi_k(\eta') +
   c.c. \right)
   \right] \, .
   \label{powern}
 \ea
One thus verifies that Eq. \Eqref{neqqq} is sufficient to
factorize the two-point function.


The second condition for discarding the decaying mode
 arises from the fact that the mode in parenthesis,
 \ba \label{nonalwaysgrowingmode}
    \phi_k(\eta) = i2Im\left( e^{i\theta_k} \phi^{out}_k(\eta) \right) \, ,
 \ea
will not, in general, be the growing mode. Indeed
it is only for $\theta_k \ll 1$ that one
can approximate it by the sine function.
%D24 as tu compris que nous ne sommes pas d'accords sur ce point ?
%%R24 agian comme le growing mode ne peut dep QUE de a
%D25 Peux tu me reexpliquer ce point stp
%%ce mode ce ne PEUT etre le growing mode pour theta arbitraire
%%quele que soit la def adoptee....
%%%following Peebles large scale structure
%%%prend un a(eta) de ref, sol de Friedmann eq.
% perturbe la a' = a + delta a
%%delta a obeit une eq du second ordre. l'une est le grow l'autre le dec,
%%dans ces eqs k , n'apparait pas, par construct
%mais on retrouve ces eqs pour les modes des fluct dans le lim k eta <<1

In brief, to discard the cosines, both $\epsilon_k \ll 1$ and
$\theta_k \ll 1$ must be satisfied. When considering pair creation
from vacuum \Eqref{bogolincosmo}, we get
 $\epsilon_k \propto 1/n_k$ and
 $\theta_k \propto 1/n_k^{3/4}$. Hence it is perfectly legitimate
 to discard the cosines.
% When we shall consider the corrections associated to
% partially decohered distributions, only $\epsilon_k $ will vary.
%corresponds to discard the cosines.
In that case, the two-point
function of the physical field $\xi = \phi /a$ reduces to
 \ba \label{Gxi}
   \bra{0 in} \{ \, \hat \xi(\eta, {\bf x}),\,
   \hat \xi(\eta', {\bf x}') \,\}\ket{0 in}
   &=&
   \int_0^{\infty}\!\! \frac{d k}{k} \,
   \frac{\sin(k\vert {\bf x} - {\bf x}'\vert)}{
          k\vert {\bf x} - {\bf x}'\vert} \,
          {\cal P}^{0}_{\xi}(k) \,
          \frac{\sin (k\eta )}{k \eta}
          \frac{\sin (k \eta')}{k\eta'}
   \, . %\nonumber \\
 \ea
% decaying mode respectively
% \footnote{The terminology comes from the fact that after horizon
% exit during the inflationnary era, the in mode (\ref{modes}a)
% splits into a growing and a decaying solution, as seen by writing
% as the sum of a sine and a cosine.
% The terms in \Eqref{Gxi} are the continuation of these solutions in
% the adiabatic era.
% %R21 est ce bien cela la def standard ?
% %je prefereais utiliser l'ere rad dom pour def growing/dec
% }.
% When the condition \Eqref{neqqq} applies, the second term in
% \Eqref{brackets} is subdominant since $\abs{\epsilon_k} \ll 1$.
%  In particular, for squeezed vacuum states \Eqref{inoutvac}, its value is
% $\epsilon_k  = O(1/n^2)$, see \Eqref{cterm2}.
% Hence the
% integrand of the two-point function \Eqref{Ginx} factorizes into
% the product of two-growing modes
% %R21 je suis en desaccord total si psi est grand ce n'est PAS le gr mode
% \ba \label{growingmode}
%   \phi_k^{(g)}(\eta) = i2Im\left( e^{i\psi_k} \phi^{out}_k(\eta) \right) \, .
% \ea
% When considering our cosmological model \Eqref{aeta},
% the two-point function of the physical field $\xi = \phi /a$ is,
% after neglecting the decaying mode,
% \ba \label{Gxi}
%   \bra{0 in} \{ \, \hat \xi(\eta, {\bf x}),\,
%   \hat \xi(\eta', {\bf x}') \,\}\ket{0 in}
%   &=&
%   \int_0^{\infty}\!\! \frac{d k}{k} \,
%   \frac{\sin(k\vert {\bf x} - {\bf x}'\vert)}{
%          k\vert {\bf x} - {\bf x}'\vert} \,
%          {\cal P}^{0}_{\xi}(k) \,
%          \frac{\sin (k\eta )}{k \eta}
%          \frac{\sin (k \eta')}{k\eta'} \, , \nonumber \\
% \ea
% where we have neglected the phase shift introduced by $c_k$ in
% \Eqref{growingmode} since
% $\psi_k = k\eta_r \sim 10^{-25}$ according to
% Eq. (\ref{cterm2}b).
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %R24 le mystere is gone

  The remaining statistical properties of the distribution are the
 power spectrum and the temporal coherence of the growing mode.
 First, the primordial power spectrum
 \ba \label{primpowerxi}
    {\cal P}^{0}_{\xi}(k) = \left( \frac{H}{2\pi} \right)^2 \, ,
 \ea
 is proportional to
 the mean occupation number $n_k$,
 %R28 as can be seen in \Eqref{Ginx} in the limit $\eta=\eta' \rightarrow
 % \eta_r$ (see also
 see footnote 2.
 Here, it is scale
 invariant because the inflationary background has been approximated by
 de Sitter space.
 Second, the time dependent function which appears in \Eqref{Gxi} is
 $(\sin k\eta)/(k\eta)$,
 where $\eta$ is proportional to the
 scale factor given by Eq. (\ref{aeta}b).
 This is how the temporal coherence of modes obtains
 from the two-mode coherence of the distribution. Consequently,
 the power spectrum ${\cal P}_{\xi}(k; \eta)$ at time
 $\eta$, defined by the Fourier transform of the two-point
 function,
 is the product of the primordial power spectrum and the
 square of the mode function evaluated at that time:
 \ba \label{power}
   {\cal P}_{\xi}(k; \eta) &=& {\cal P}^{0}_{\xi}(k) \,
   \left( \frac{\sin k \eta}{k\eta} \right)^2 \, .
 \ea
The power
 spectrum at a given time, as a function of k, oscillates and has
 zeros. This behaviour
 is a necessary condition for
 the existence of acoustic oscillations and anti-correlations in
 the temperature anisotropy TT and cross-correlation TE power
 spectra respectively \cite{WMAP1}.


 Hence, once the cosine is neglected, the quantum distribution
 can be
 effectively replaced by a stochastic Gaussian distribution of
 classical fluctuations
 \ba \label{sineeq}
   \xi_{\bf k}(\eta) = \xi^0_{\bf k} \, \frac{\sin k\eta}{k\eta}  \, ,
 \ea
 with locked temporal argument, and random amplitudes with
 variances given by
 \ba \label{Aeff}
   \av{\xi^0_{\bf k} \, \xi_{\bf k'}^{0\, *}}_{e\!f\!f} =
   \av{\xi^0_{\bf k}\, \xi^0_{-\bf k'}}_{e\!f\!f} =
   {\cal P}_{\xi}^{0}(k) \, \delta^{3}({\bf k}-{\bf k}') \, .
 \ea
 Being Gaussian, the effective probability distribution is simply
 \ba \label{Peff}
  {\cal P}_{e\!f\!f} = \widetilde{\prod_{\bold k}} \,
  \frac{1}{{\cal P}_{\xi}^{0}(k)}
  \exp\left(-\frac{\abs{\xi^0_{\bf k}}^2}{{\cal P}_{\xi}^{0}(k)} \right) \, .
 \ea
 To avoid double counting, one must again
 use the tilde product which takes into account half the modes
 only, as was done in the quantum distribution of Eq. \Eqref{inoutvac}.
 This counting becomes crucial when computing the entropy, see Section IV.C.


 We emphasize that the reality of the field $\phi(\eta,{\bf x})$
 has nothing to do with the temporal coherence of the %R28 growing
 modes of Eq. \Eqref{nonalwaysgrowingmode}.
 Nevertheless, once having neglected the decaying mode, it is true that
 the first equality
 in Eq. \Eqref{Aeff}
 is imposed by the reality of $\phi(\eta,{\bf x})$.
 However, this should not be confused with the relation between
 the expectation values of the creation and destruction operators
 in the squeezed vacuum state prior to have discarded the
 decaying mode,
 \ba \label{newn}
   \langle \hat a_{\bf k}^{out\, \dagger}
   \, \hat a_{\bf k'}^{out} \rangle = -
   \langle \hat a_{\bf k}^{out}  \, \hat a_{-\bf k'}^{out} \rangle
   (1 + O(k \eta_r)^3)\, .
 \ea
 The latter follows from Eqs. \Eqref{momenta}-\Eqref{cterm2}, and
 is the expression of the {\it two-mode coherence} of the
 {\it in} vacuum (or more generally of strongly correlated
 two-mode distributions, see Section V).
The (complex) Eq. \Eqref{newn} guarantees
 that both conditions $\epsilon_k \ll 1, \, \theta_k \ll 1$ are
 satisfied, thereby allowing
 to factorize the 2-point function
 and to discard the contribution of the
 %D27 cosines.
 %R23 (i.e. the
 decaying modes.
 %je prefere decaying mode qui est plus general


\subsection{Additional remarks}


First we remind the reader why a random distribution of both the
sine and the cosine does not give rise to temporal coherence
 \cite{Grishchuk90}. In fact such a distribution  corresponds to an
incoherent (thermal) distribution.


Consider the incoherent distribution:
 \ba \label{incoh}
   \langle \hat a_{\bf k}^{\dagger} \hat a_{{\bf k}'}
   \rangle_{i\!n\!c} &=& n_k \, \delta^{3}({\bf k}-{\bf k}') \, ,
   \nonumber \\
   \langle \hat a_{\bf k} \hat a_{{\bf k}'} \rangle_{i\!n\!c} &=&
   c_k \, \delta^{3}({\bf k}+{\bf k}') = 0 \, .
 \ea
The last equation implies that temporal coherence of the modes is
lost, as can be seen from the temporal behaviour of the bracket in
the integrand of Eq. \Eqref{Ginx} : when $\eta = \eta'$  the
bracket no longer exhibits oscillations in $k$.
 (This absence can also be understood by
 considering
 field amplitudes as classical stochastic variables
 rather than quantum ones.
 Writing the mode in terms of its norm and its phase
  \cite{Grishchuk90,Allen00}
 \ba
   \phi_{\bf k} &=& \Phi_{\bf k} \, \sin(k\eta + \theta_{\bf k})
   \, ,
 \ea
 one can treat $\Phi_{\bf k}$ and $\theta_{\bf k}$ as stochastic
 variables. One verifies that the distribution for the phase
 $\theta_{\bf k}$ is uniform over the interval
 $\left[0,\, 2\pi \right]$.
 Hence, taking the ensemble average to compute the power spectrum,
 no temporal coherence could obtain.)


 It is of value to estimate what has been neglected when
 discarding the cosine in Eq. \Eqref{sineeq}.
  To this end, we
 write the field modes
 directly in terms of the growing and decaying modes
 \ba \label{gdmodes}
   \hat \phi_{\bf k} &=& \hat a_{\bf k}^{out} \phi_k^{out} +
   \hat a_{-\bf k}^{\dagger \, out} \phi_k^{out \, *} \, \nonumber \\
   &=& \hat g_{\bf k} \frac{\sin(k\eta)}{\sqrt{k}}
   +\hat d_{\bf k} \frac{\cos(k\eta)}{\sqrt{k}}\,  .
 \ea
 The leading non-vanishing order in $n_k$ of their fluctuations
 and cross correlation are
 \sba \label{ABvariances}
   \langle \hat g_{\bf k} \hat g_{{\bf k}'}^{\dagger} \rangle_{in} &=&
   \left( n_k + \frac{1}{2} - Re(c_k) \right) \, \delta^{3}({\bf k} - {\bf k}')
   = 2 n_k  \,
    \left( 1 + O(n_k^{-1})  \right) \, \delta^{3}({\bf k} - {\bf k}')
   \, , \\
   \langle \hat d_{\bf k} \hat d_{{\bf k}'}^{\dagger} \rangle_{in} &=&
   \left( n_k + \frac{1}{2} + Re(c_k) \right) \, \delta^{3}({\bf k} - {\bf k}')
   =  O(n_k^{-1/2})
    \, \delta^{3}({\bf k} - {\bf k}')
   \, , \\
   \langle \{ \hat d_{\bf k}, \hat g_{{\bf k}'}^\dagger \} \rangle_{in} &=&
   Im(c_k) \, \delta^{3}({\bf k} - {\bf k}')
   = O(n_k^{1/4})
   \, \delta^{3}({\bf k} - {\bf k}')
   \, .
 \sea
 Notice that the cross-correlation,
 $\langle \{ \hat d_{\bf k}, \hat g_{{\bf k}'}^\dagger \} \rangle_{in}$,
 the Weyl ordered product (i.e. the anticommutator
 divided by 2),
 is non-zero. This means
 that the growing and the decaying mode
 are not eigenmodes of the distribution (see Appendix C).

 %D22 C'est bien d'avoir preciser que la litterature est flou quant a la
 %phase du growing mode. Ceci dit je continue a penser qu'avec ce choix
 %on marche sur la tete.

 When divided by the variance of $\hat g_{\bf k}$, the variance
 of %R24 $\hat d_{\bf k}$ and
 the cross-correlation is of order
 $n_k^{-3/4} \sim (k\eta_r)^3 \sim 10^{-75}$. %R25
 The power of the decaying mode is even smaller.
 One can therefore
 safely use the effective distribution Eq. \Eqref{Peff} in replacement of the
 original quantum distribution Eqs. \Eqref{inoutvac} or \Eqref{ABvariances}
 when calculating the power spectrum. However this is {\it not} the
 case for the entropy.


 It is illustrative to compare the in-vacuum
 correlations, Eqs. \Eqref{ABvariances},
 to those
 of the incoherent distribution \Eqref{incoh}.
 In that case
 $\hat d_{\bf k}$ and $\hat g_{\bf k}$ are uncorrelated Gaussian
 operators with equal variance:
 \ba \label{incoherent}
    \langle  {\hat g_{\bf k}\hat g_{{\bf k}'}^\dagger}\rangle_{i\!n\!c}&=&
    \langle {\hat  d_{\bf k}\hat d_{{\bf k}'}^\dagger}\rangle_{i\!n\!c}
   =
   \left( n_k + \frac{1}{2} \right) \, \delta^{3}({\bf k}-{\bf k}')
   \, , \nonumber \\
   \langle
   \{\hat d_{\bf k} , \hat g_{{\bf k}'} \} \rangle_{i\!n\!c} &=& 0
   \, .
 \ea
These differences with Eqs. \Eqref{ABvariances}
are particularly clear.









\subsection{Drawbacks of the simplified description}



The simplified description in terms of a statistical ensemble of
sine standing waves, see Eq. \Eqref{Peff}, has several
shortcomings. It is of value to describe them with some attention

First, there is the discrepancy of the value of the entropy
mentioned after Eq. \Eqref{ABvariances}. Second, it should be
pointed out that, in the early universe, non-linear processes,
however weak they may be, will modify the density matrix obtained
by making use of free fields. To be able to parameterize the
modifications of the power spectrum, one must return to
 two-mode distributions \footnote{As mentioned in the Introduction,
 we shall neglect
 non-Gaussianities when computing the entropy.
 %D27
%R28  We will also neglect the non-homogeneities.
 % je en vois pas bien comment le justifier
 In other words, we replace the actual (non-Gaussian) distribution
 by the Gaussian one which possesses the same
 moments \Eqref{momenta}
 and therefore the same power spectrum.
 The justifications for this substitution are the following.
 First, it leads to a drastic simplification of the analysis, and
 {\it all} Gaussian (isotropic) distributions can be considered.
 Second, when focussing on a given wave vector, one anyway traces
 over the
 non-Gaussianities relating this scale to other wave vectors.
 Third, since non-Gaussianities are expected to be
 weak \cite{Maldacena03,Matarrese}, it is a well defined
 (mean field/Hartree) and useful
 approximation to replace the actual distribution
 by the Gaussian one associated to it.}.
 Indeed, the classical ensemble of sine waves
 cannot describe these modifications. It should be considered only
 as an effective description of the density matrix when the
 decaying
 mode is strictly negligible.

The appropriate basis to describe density matrices is provided by
coherent states. The reasons are the following. First, they
constitute the quantum counterparts of classical configurations in
phase space. Therefore they provide an adequate basis for studying
the semi-classical limit. In particular, when non-diagonal terms
are still present in this basis, it is the signal that some
quantum coherence has been kept,
 see Section V and Appendix D.3.
 Secondly,
 they are the preferred basis in which squeezed states
decohere when weak interactions are taken into account,
 see Section IV.A and \cite{Zurek93} for further details.

 %R25 je l'enleve car on le dit trop de fois
 % and this is NOT a drawback...
 %This second remark provides the dynamical
 %justification of using these states for the study
 %of decoherence.

%(Entre nous le point cle dans Zurek est que les temps longs efface
%toute direction provilegiee in phase space. Il me semble donc que
%l'on doit essayer de demontrer si nos modes $k' \gg k$ eux aussi
%ont une tendance a randomiser en la phase) Qu'en penses tu ??
%R23je suis parfaitement d'acc. c'est un challenge et c'est peut
% etre pas le cas en cosmo

\section{Two-mode coherent states}



 When using coherent states in inflationary cosmology, one must pay attention to
 the entanglement between $\bf k$ and $-{\bf k}$ modes.
 As we shall see this leads to the notion of
 'two-mode coherent states'.
 On the contrary, a naive use of coherent states assigning
 amplitudes to each mode separately
 will erase the entanglement and therefore suppress the
 temporal coherence of the modes.
 For completeness this is shown in the next subsection.




 \subsection{Naive use of coherent states and the loss of
 temporal phase}


 If assuming that each mode decoheres separately, the matrix
 density of the ${\bf k}, {\bf -k}$ pair of modes factorizes
 and the modes are independent:
 \ba \label{naive}
   \hat \rho^{Naive}
   = \left( \int\!\!\frac{d^2 v}{\pi} \, G(v)
   \ket{v, \, {\bf k}} \bra{v, \, {\bf k}} \right)
   \otimes
   \left( \int\!\!\frac{d^2 w}{\pi} \,
    G(w) \ket{w,\, - {\bf k}} \bra{w,\, - {\bf k}} \right) \, ,
 \ea
 where $G(v) = n_k^{-1} \exp\left( - \abs{v}^2/n_k \right)$
 is the normalized Gaussian distribution giving the
 appropriate power-spectrum,
 %D27 couper coller du paragraphe suivant
 and where $\ket{v, \, {\bf k}}$ and
 $\ket{w,\, - {\bf k}}$ are coherent states
 constructed with out operators, see Appendix A for their definition.
 The expectation values in this decohered density matrix are:
 \ba \label{RLcoherence}
   {\rm Tr}(\hat \rho^{Naive}  \hat a_{\bf k}^{\dagger} \hat a_{\bf k} ) &=&
   n_k   \, ,  \nonumber\\
   {\rm Tr}(\hat \rho^{Naive}  \hat a_{\bf k} \hat a_{-{\bf k}} ) &=& 0\,  .
 \ea


 One thus verifies that Eq. \Eqref{naive}
 corresponds to the incoherent distribution of Eq. \Eqref{incoh}
 written in the basis of one-mode coherent states.
 The two-mode correlations are completely
 erased and the temporal coherence of the modes is lost.
 In fact Eq. \Eqref{naive} is maximally decohered,
 since it is a tensorial product of two thermal states
 (because $G(v)$ is Gaussian and isotropic in the complex $v$ space).



 \subsection{Representation of the in vacuum with two-mode coherent states}

 To understand the usefulness
 of two-mode coherent states it is appropriate to first mention
the following properties \cite{CP04}. Consider a mode ${\bf k}$ in
a coherent state $\ket{v,\, {\bf k}}$. Then compute the one-mode
reduced state obtained by projecting it on the two-mode initial
vacuum of Eq. \Eqref{in2mode}. You will get:
 \ba \label{remar}
   \langle v,\, {\bf k} \kett{0 in,\, {\bf k}}  =
   {\cal A}_k(v) \, \ket{z_k v^*, \, -{\bf k} } \, .
 \ea
It is remarkable that the state of the $-{\bf k}$ mode which is
entangled to $\ket{v,\, {\bf k}}$ is also a coherent state. Its
amplitude is given by the complex conjugate of $v$ times $z_k$
characterizing the pair creation process. These facts follow from
the EPR correlations in the initial vacuum displayed in Eq.
\Eqref{inoutvac}. The prefactor ${\cal{A}}_k(v)$ is
 \ba \label{A(v)}
   {\cal{A}}_k(v) &=& \inv{\abs{\alpha_k}}
   \exp{\left(-\frac{\abs{v}^2}{2\abs{\alpha_k}^2} \right)}  \, .
 \ea
It is the probability amplitude to find the mode ${\bf k}$ in
the coherent state $\ket{v, \, {\bf k}}$ given that we start with vacuum at the
onset of inflation.

Using the representation of the identity with coherent states, Eq.
\Eqref{coh1}, Eq. \Eqref{remar} permits to decompose the two-mode
{\it in} vacuum as a {\it single} sum of {\it two-mode coherent
states}. It should be stressed that two independent integrations
over one-mode coherent states are necessary to describe a generic
two-mode state. The fact that only one integration is sufficient
is therefore a direct consequence of the two-mode coherence of the
initial state $\kett{0 in,\, {\bf k}}$. Explicitly we have
 \ba \label{insuperpcoh}
   \kett{0 in,\, {\bf k}}  =
   \int\!\!\frac{d^2v}{\pi} \,
   {\cal{A}}_k(v) \, \,  \ket{v,\, {\bf k}}  \otimes \ket{z_k v^*, \, -{\bf k}}
   \, .
 \ea
 This result is exact
 \footnote{Notice however that the above
 decomposition is not unique since the coherent states are not
 orthogonal, compare with \cite{Kim00}.
 Eq. \Eqref{insuperpcoh} has the advantage to be directly related to
 the detection
 of a quasi-classical configuration in the ${\bf k}$ sector.}
 and applies even for low occupation numbers.
 It is the consequence of the coherence of
 the in vacuum and
 holds for every homogeneous pair creation process.


 Another important consequence of Eq. \Eqref{remar} is that the
 probability to find {\it simultaneously} the ${\bf k}$-mode with
 coherent amplitude $v$ and its partner with amplitude $w$ is
 \ba \label{vwdistrib}
   {\cal P}_{2, \, k}(v,w) =
   \abs{\bra{v,\, {\bf k}}\bra{w,\, -{\bf k}} 0 in, \, {\bf k} \rangle_2}^2 =
   \abs{{\cal{A}}_k(v)}^2 \times e^{-\abs{w-z_k v^{*}}^2} \, .
 \ea
 The second factor follows from the overlap between the reduced
 state in the r.h.s. Eq. \Eqref{remar} and the bra
 $\bra{w,\, -{\bf k}}$
 (we also used the coherent state overlap:
 $\abs{\langle u \vert v \rangle}^2 = \exp({-\abs{u-v}^2}) $).

Equation \Eqref{vwdistrib} implies that once the amplitude of the
${\bf k}$-mode is fixed, the conditional probability to find its
partner in a coherent state $\ket{w}$ is centered around
 \ba \label{meanw}
  \bar w(v)=z_k v^{*} \, .
 \ea
Moreover, in the high occupation number limit we are dealing with,
the spread ($=1$) around this mean value is negligible when
compared to the spread in $v$ which is given by $\abs{\alpha_k}^2
= n_k + 1$. Therefore, when computing expectation values to
leading order in $n_k$, the conditional probability acts as
 $\delta^{(2)}(w- \bar w(v))$.
Hence both the real and the imaginary parts of $w$ are fixed. This
is how the EPR correlations in the in-vacuum translate in the
coherent states basis: double integrations over one-mode coherent
states can be replaced by single integrations over two-mode
states.



 Furthermore, the strict correspondence between the coherent
 state amplitudes in the sectors ${\bf k}$ and $-{\bf k}$
 leads, in the high-squeezing limit,
 to the temporal coherence of modes since the only random variable
 is their common amplitude
 \cite{CP04}.
 As we shall see below, this result is
 crucial as it allows to erase the quantum
 features of the distribution (such as the existence of a
 sub-fluctuant direction in phase space)
 while preserving the temporal coherence
 of modes upon horizon re-entry (which can be classically interpreted).

 To complete this analysis, and in preparation for studying decoherence,
 it is also interesting to explicitly write the non-diagonal matrix
 elements of the in vacuum density matrix. One has
 \ba \label{vwrhov'w'}
   \bra{v}\bra{w} \hat \rho_{in} \ket{v'} \ket{w'} &=&
   {\cal A}_{2, \, k}(v,w) \, {\cal A}_{2, \, k}(v',w')^{*} \, ,
 \ea
 where the two-mode amplitude is
 \ba
   {\cal A}_{2, \, k}(v,w) &=& {\cal A}_k(v) \,
   e^{-\frac{1}{2}\abs{w-z_k v^*}^2 }\,
   e^{i\, {\rm Im}(w^*z_k v^*)} \, .
 \ea
Since the initial vacuum is a pure state and since
$\abs{\alpha_k}^2 = n_k + 1 \gg 1$, the above matrix elements do not vanish
when $w = z_k v^*, \, w' = z_k {v'}^*$ even for $\abs{v-v'}^2 \simeq n_k$,
i.e. they do not vanish
{\it even for macroscopically different coherent states}.
(Remember that $n_k \simeq 10^{100}$.)


Because of this macroscopic quantum coherence, $\hat \rho_{in}$
does not describe a classical ensemble (a mixture) of coherent
states. Fortunately, as we shall see below, this coherence is
unstable to small modifications of the distribution. To describe
this decoherence we shall proceed in two steps. First we shall
consider a specific example of decohered density matrix and argue
why it plays a critical role, namely, it separates distributions
which have kept some quantum coherence from those which are
mixtures (i.e. distributions for which the off-diagonal matrix
elements vanish for $\abs{v-v'}^2 \geq 1$.) Second, we shall
describe partially decohered density matrices in general and more
technical terms. In particular we shall sort out the robust
properties from its unstable ones. In one word, when using
$|c_k|/(n_k + 1/2)$ as a parameter for characterizing the two-mode
coherence of the distribution, we shall see that the power
spectrum and the temporal coherence are robust whereas the
entropy, the above quantum coherence, and the residual squeezing
are extremely sensitive to this parameter. Even though the value
of this parameter will be introduced from the outset, similar
results obtain when performing a dynamical analysis.




 \section{Minimal decoherence scheme of primordial spectra}

\subsection{Zurek {\it et al.} analysis and minimal decoherence scheme}


In general, it is a difficult question to determine into what
mixture an initial density matrix will evolve when
 taking into account some interactions with other modes
 and tracing over these extra degrees of freedom so as
 to obtain an effective (reduced) density matrix.
There exist  however several cases where clear conclusions can be
drawn. First, when one can neglect the free Hamiltonian, the
preferred states (that is the basis into which the reduced density
matrix will become diagonal) are the eigenstates of the
interaction Hamiltonian \cite{gottfried,Zurek81,Zurek82}. This
approach has been applied in \cite{KiefPolStar98a}, to primordial
density fluctuations when the (physical) modes are almost constant
because their wave length is much larger than the Hubble radius.
The conclusion is that the preferred basis is provided by
amplitude (position) eigenstates. However this conclusion leaves
some ambiguity and might lead to some difficulties. First,
position eigenstates are not normalized. Second, and more
importantly, the spread in momentum is infinite for these states.
Therefore, the velocity field would not be well defined when the
modes re-enter the horizon. Moreover, as pointed out in
 \cite{KiefPolStar00}, some additional decoherence could be
obtained as they re-enter the horizon. In this case, the momentum
should be treated in the same footing as the position. To cure
these problems,
 some finite spread in position should be introduced.
One then needs a physical criterion to choose this spread.



To introduce a critical spread,
 to identify the various regimes of decoherence,
 and to determine their range, one should use
coherent states, as we now
argue. (We reserve for a forthcoming publication a proper
justification of the physical relevance of these states in a
cosmological context.)
 First, as shown in \cite{Zurek93}, when considering
harmonic oscillators weakly interacting with an environment,
coherent states provide the basis in which the density matrix
decoheres. In particular,
 when the initial state is
 a squeezed state, there is a phase of rapid growth of the entropy
 which sends the system into a mixture of coherent states.
 This is then followed by a period of slower increase accompanying
 dissipation or thermalization.
 %(The same applies
 Second, in inflationary cosmology, fluctuation modes
 are weakly interacting harmonic oscillators \cite{Maldacena03}.
 Indeed, given that the relative density
 fluctuations have small amplitude ($\sim 10^{-5}$), the hypothesis
 of weak interactions is perfectly legitimate.
 Therefore, the modes with ${\bf k'} \neq {\bf k} $ will effectively
 act as an environment for a given two-mode state.
A novelty with respect to the analysis of \cite{Zurek93} is that
in cosmology, one deals with two-mode squeezed states. In this
case one ends up with statistical mixtures of two-mode coherent
states, but not with mixtures of one-mode coherent states. The
procedure to define these distributions
%R27, here after called minimally decohered distributions,
is explained in Appendix B.
However, because coherent states are
%D27 not orthogonal
overcomplete, there is an inherent ambiguity (albeit small in the large $n$
limit) in determining this distribution,
{\it c.f.} footnote 4 for a related %R40
ambiguity. The procedure
we shall adopt in this Section is to use Eq. \Eqref{vwdistrib} to
define it.

%R23 on l'a deja dit
%This diagonal distribution plays a special role:
% it gives the lowest
% value of the entropy stored in the fluctuation spectra
% when no quantum properties remain.
% More precisely, consider the set of final
% (Gaussian) distributions which
% result from the initial distribution of Eq. \Eqref{vwrhov'w'}
%% and which have loss their squeezing
% no longer possess fluctuations
% whose spread, in some direction in phase space,
% is smaller than that of the vacuum.
% In the space of these decohered distributions,
% the  diagonal distribution possesses the smallest entropy.
% This follows from the fact that
% coherent states have minimal, isotropic, and constant spreads
% (given quantum uncertainties and when considering free evolution,
% see Appendix A).


 So unless the interactions taking place in the early cosmology
 are sufficiently weak (or sufficiently anisotropic
 in phase space
 as in the case they act only for a fraction of a period)
 so as to keep some quantum coherence,
 we can infer that the
 effective entropy (see footnote 3)
 of the final distribution should be higher than (or equal
 to) that stored in
 the diagonal distribution of two-mode coherent states,
 here after called minimally decohered.



 In the next two subsections, we shall explicitly write down this
 distribution, compute the entropy it carries, and, very
 importantly, show that the temporal coherence of the modes is kept
 whereas the macroscopic quantum coherence of the
 original distribution, see Eq. \Eqref{vwrhov'w'}, is lost, thereby
 allowing a classical interpretation of the residual statistical
 properties.


\subsection{The minimally decohered distribution}



The minimally decohered distribution which corresponds to Eq.
\Eqref{vwrhov'w'} is
 \ba \label{rhodec}
   \hat \rho_{min} = \int\!\!\frac{d^2v}{\pi} \frac{d^2w}{\pi}\,
   {\cal{P}}_{2, \, k}(v,w) \, \,
   \ket{v,\, {\bf k}}\bra{v,\, {\bf k}} \otimes
   \ket{w, \, -{\bf k} }\bra{w, \, -{\bf k}} \, .
 \ea
The probability distribution  ${\cal{P}}_{2, \, k}(v,w)$
is given in Eq. \Eqref{vwdistrib}.
%In Appendix B we show that Eq. \Eqref{rhodec} is indeed the
%resulting normalized distribution. The technical point which
%requires clarification is the extension of the above mentioned
%minimal scheme to two-mode squeezed states.

By having replaced the original distribution
$\hat \rho_{in}= \ket{0, in}\bra{0, in}$ by the decohered one
$\hat \rho_{min}$, some entropy has been introduced.
%One verifies indeed that ${\rm Tr}(\hat \rho_{min}^2) < 1$.
%One is drawn to the conclusion that in order to take into account both the
%inevitable decoherence of the initial quantum distribution into a mixture of
%coherent states and the necessity to preserve the temporal coherence of the
%modes, one must write a two-mode distribution of coherent states.
%In fact, this is both a necessary and a natural refinement of the naive
%distribution considered above, because of the entanglement between
%${\bf k}$ and $-{\bf k}$ modes in the initial distribution.
%The first important point is that this decohered distribution has
%retained all the information about the temporal coherence of the
%modes.
Indeed, by direct computation, one gets
 \ba \label{RLcoherence}
   {\rm Tr}(\hat \rho_{min} \hat a_{\bf k}^{\dagger} \hat a_{\bf k} ) &=&
   \abs{\alpha_k}^2 = n_k + 1
   \, ,
   \nonumber\\
   {\rm Tr}(\hat \rho_{min} \hat a_{\bf k} \hat a_{-{\bf k}} ) &=&
   z_k \abs{\alpha_k}^2 = c_k \,  .
 \ea
The first line shows the occupation number increased by one unit.
Thus the relative change is only $1/n_k$. In the second line one
finds that that the
 coherence term
 is equal {\it in phase and amplitude} to that of the
original distribution, see Eqs. (\ref{momenta}, \ref{cterm}).
These two results imply, first, that the (relative) degree of
coherence has been reduced and therefore some entropy has indeed
been gained. Secondly, in the high occupation number limit we are
dealing with, the two-point function of Eq. \Eqref{Ginx} is not
affected by this loss of coherence since, for relevant modes,
 the relative change is of the order of $1/n_k \sim 10^{-100}$.
 Let us emphasize that the smallness of this change
 follows from the fact that the phase of
 ${\rm Tr}(\hat \rho_{min} \hat a_{\bf k} \hat a_{-{\bf k}} )$
 stays equal to that of the r.h.s of Eq. (\ref{momenta}c).
 %D22
 Hence, the fact that Eq. \Eqref{neqqq} still applies leads
 to the preservation of the temporal coherence of modes.
 %and the fact that the phase of $c_k$ is unchanged
 %leads to the preservation of the phase of the growing mode.






 %D19 le texte n'est pas clair, tu meles factorization de G
 %et phase des modes.
 %que veux tu dire ?
 %R20 il me semble que c'est toi qui parle de la factoris. pas le texte
 %
 %D20 oui, et c'est toi qui a rajoute le changement de G. Il y a donc 2 statements
 %qui ne sont pas clairement differences dans ce texte ecrit a deux mains :
 %
 %1) la factorisation de G (ET la coherence temporelle, ie le decaying mode est negligeable).
 %Une CNS est 19, parce que la phase de c, quelle qu'elle soit,
 %est absorbee dans le sinus : $\sin^2(k\eta - \psi_k)$. Si tu me dis que c'est faux, alors
 %je ne comprends pas ce que tu appelles le 'subfluctuant mode in phase space' dans V.A.2.
 %
 %2) le non-changement de G (ET de la phase des modes) dans le passage c_sq -> c_decoh,
 %qui requiere en plus de 19 que la phase de c ne change pas.
 %Apparemment tu n'as pas lu mon commentaire dans V.B.
 %Sommes nous d'accord que le decaying mode est la variable subfluctuante. Si oui, alors
 %1) la phase de c sert a definir le mode croissant
 %2) le power du decaying mode est la plus petite des 2 valeurs propres de la
 %matrice de covariance, i.e. n+1/2 - \abs{c} = 1/n pour un etat squeeze, =1 pour delta=1.
 %3) il faut corriger le texte en I.B, I.C
 %Do we understand each other now ?
%%R21 oui mais on n'est pas d'acc avec la def de grwoing mode.
%% donc NON la phase de c ne sert PAS a definir le mode croissant
%% NON le power du decaying mode n'est PAS
%% il faut corriger le texte en I.B, I.C let's see later
 %encore une fois, je m'occupe de la non-modif de la 2pt ftion,
 %de l'absnec de phase shift et de la coh des modes
 %remarque que 19 est une condit nec mais pas suffis
 %il faut aussi que la phase de c n'ai pas change
 %en as tu pris conscience ?
 %
 %D20 oui, et je suis parfaitement d'accord avec ce statement.
 %je ne me suis jamais occupe de la non modif de G car
 %1) on a jamais touche a la phase de c
 %2) bien que c_eff n'a a priori pas la meme phase que c_sq, elle est
 %parfaitement definie (i.e. pas random) donc on peut definir un mode croissant et
 %un mode decroissant. (cf aussi mon 2eme email)
%%R21 on est manitenenat aware de nos deux defs differentes
%% je ne peux accepter la tienne car si psi = pi/2 we are up-side down

 In fact, the main modification associated with the replacement
 of
 $\hat \rho_{in}= \ket{0, in}\bra{0, in}$ by
 $\hat \rho_{min}$ concerns the elimination of the
 non-vanishing off-diagonal matrix elements.
 Indeed, in the high
 squeezing limit, we have, see also Appendix D.$2$,
 \ba
   &&\bra{v}\bra{w} \hat \rho_{min} \ket{v'}\ket{w'}
   %D22
   \simeq \frac{1}{3 \abs{n_k}} \times \nonumber \\
   &&\exp-\frac{1}{2} \left[ \abs{v}^2 + \abs{v'}^2 + \abs{w}^2 +
   \abs{w'}^2 - \frac{4}{3}\left(v^* v' + w^* w' -
   \frac{z_k}{2}v^* w^* - \frac{z_k^*}{2}v w \right) \right]
   \, .
 \ea
 (We have set $\abs{z_k} = 1$ at the end
 of calculations to simplify the expression.)
 The above result implies that along the
 big axis of the ellipse, see Fig. 1,
 i.e. for $w=z_kv^*$, the term in the exponential is $ (4/3) \abs{v-v'}^2$.
 %D22 je ne trouve pas d'erreur
 Therefore,
 two coherent states separated by $\abs{v' - v} > 1$ will no
 longer interfere quantum mechanically.

 Moreover, each one of the coherent states in Eq. \Eqref{rhodec} is
 stable under the evolution of the quadratic Hamiltonian, i.e., the
 expectation values of
 $\hat \phi_{\bf k}, \, \partial_{\eta} \hat \phi_{\bf k}$ in these states
 evolve according to the classical equations of motion, see
 Eqs. \Eqref{cltrajincohstate}.
 Hence, the
 diagonal distribution \Eqref{rhodec} can safely
 be interpreted in classical statistical mechanics, i.e. as a
 (stable) statistical ensemble of (non-interacting) classical states.


Finally, when computing expectation values in leading order in
$n_k$, this distribution can be further simplified by replacing
the Gaussian factor of unit spread of Eq. \Eqref{vwdistrib} by a
double delta function
 as discussed
 after Eq. \Eqref{meanw}.
 Using this equation,
 one gets the simplified distribution
 \ba \label{rhodec2}
   \hat \rho_{simpl} = \int\!\!\frac{d^2v}{\pi} \,
   \abs{{\cal A}_k(v)}^2  \,\,
   \ket{v,\, {\bf k}} \bra{v,\, {\bf k}}
   \otimes
   \ket{\bar w(v), \, -{\bf k}} \bra{\bar w(v), \, -{\bf k}}
   \, .
 \ea
It is now written as a single sum of two-mode coherent states,
thereby making explicit the fact that we are dealing with a
statistical superposition of two-mode states (and not a double
superposition of one-mode coherent states).
%R34 trop de renvois, en voila un de moins. As we shall see in Appendix D5,
% the distributions Eqs. (\ref{rhodec}, \ref{rhodec2})
%  should be conceived
% as quantum counterparts of the effective distribution of sine functions
%of Eq. \Eqref{Peff}.

 \begin{figure}[ht] \label{decoherence}
 \epsfxsize=15.0truecm
 \epsfysize=12.0truecm
 \centerline{{\epsfbox{decoherence.eps}}}
 \caption{{\it Decoherence  of a one-mode squeezed state in the coherent
 state basis.}
 Given the 2-dimensional character of this sheet of paper, it is
 simpler to represent the phase space of a one-mode squeezed state
  than that of a two-mode squeezed state. To this end, one should
 first decompose the two-mode squeezed states we are dealing with into two
 copies of one-mode squeezed states, as exposed in Appendices B and D.4.
 The large red ellipse represents the
 $1-\sigma$ contour of a one-mode squeezed state in $(q_1,\, p_1)$ space,
 with $\omega = 1$.
 The squeezed state is characterized by the squeezing parameters
 $r_k$ et $\phi_k$, where $n_k = \sh^2 r_k$ and $\phi_k = \arg(c_k)/2$.
 The small green circles represent the $1-\sigma$
 contour of coherent states, which have no privileged direction
 in phase-space. When replacing $\hat \rho_{in}$ by $\hat \rho_{min}$,
 two modifications obtain. First, the squeezed spread in the direction of the
 small axis of the ellipse is replaced by that of the vacuum. Second,
 and concomitantly, the macroscopic quantum
 coherence along the big axis, see Eq. \Eqref{vwrhov'w'},
 is removed and replaced by that of coherent states.}
 \end{figure}



\subsection{Minimal entropy and the neglect of the decaying mode}

Our aim is to compute the entropy stored in Eq. \Eqref{rhodec}.
The entropy of any Gaussian two-mode distribution,
%$ \hat \rho_{2,\, k}$
$\hat \rho_{{\bf k},\, -{\bf k}}$,  can be exactly calculated
\cite{Zeh85,Serafini04} by using the fact that its
density matrix is unitarily %R34 you see you wrote UNITARY
%D35 yes, because it acts on \rho, and it's the exponential of a symplectic
%operator which acts on C, the log of rho
%R35 oui je suis d'ac. mais dans l'app C tu n'utilisais pas le mot unitaire
%D36 so, what is your question ?
equivalent to the tensorial product of
two thermal density matrices of auxilliary oscillators, see Appendix C.
%R34 We shall name $a$ and $b$ these two real oscillators. One has
 These two real oscillators can be taken to be the real
 and the imaginary parts of $\phi_{\bf k}$ we used in Appendix B.
 One can indeed write
 \ba \label{map}
%D35
   \hat \rho_{{\bf k},\, -{\bf k}} = {\cal M}^{\dagger} \, \hat \rho_{th,\, 1} \otimes
   \hat \rho_{th,\, 2} \, {\cal M} \, ,
 \ea
 where $\cal M$ is a unitary operator acting on the two-mode Hilbert space.
 The expression of the (von Neumann) entropy immediately follows:
 \ba
   S\left[ \hat \rho_{{\bf k},\, -{\bf k}} \right] &=& S\left[ \hat \rho_{th,\, 1} \right] +
   S\left[ \hat \rho_{th,\, 2} \right] \, ,
 \ea
 where the entropy of a thermal bath with mean occupation $\bar n$ is
 \ba  \label{Stherma}
   S\left[ \hat \rho_{th} \right] = (\bar n +1 ) \ln(\bar n +1 ) -
   \bar n \ln(\bar n) \, .
 \ea
When considering %only
distributions preserving homogeneity and
isotropy, the occupation numbers of the thermal matrices are equal
and given by
 \ba \label{tilde n}
   \bar n_{k} + \frac{1}{2} =
   \left( (n_{k} + \frac{1}{2})^2 - \abs{c_{k}}^2 \right)^{1/2} \, ,
 \ea
where $n_k$ and $c_k$ are defined in Eq. \Eqref{momenta}. It
should be noticed that the phase of $c_k$ (which is essential for
the temporal coherence of modes) drops out from this expression.
Hence the quantum purity is not univocally related to the temporal
coherence.


Let us apply Eqs. (\ref{Stherma}, \ref{tilde n}) to several cases.
First, for the two-mode in vacuum of Eq. \Eqref{inoutvac}, the
occupation number and the coherence term are related by Eq.
\Eqref{cterm}, one has
 $\bar n_k = 0$, as expected. Hence the entropy vanishes.

For the decohered matrix Eq. \Eqref{rhodec}, using Eq.
(\ref{RLcoherence}), the occupation number of the two thermal
baths are
 \ba \label{barn}
   \bar n_{k} =
   \frac{1}{2} \left( -1+\sqrt{8(n_k +1 ) +1}\right) \sim
   \sqrt{2  n_k} \, ,
 \ea
where the last term is the leading order when $n_k \gg 1$. The
two-mode entropy of this mixture is then
 \ba \label{Smin}
   S_{{\bf k},\, -{\bf k}}\left[ \hat \rho_{min} \right] &=&
   2 \, S\left[ \hat \rho_{th}
   \right] = 2 \ln \bar n_k + O(1) \, ,
   %2 \left[(\bar n_k +1 ) \ln(\bar n_k +1 ) - \bar n_k \ln(\bar n_k) \right]
   \nonumber \\
   &=& \ln n_k + O(1) = 2\, r_k +  O(1) \, ,
   \nonumber \\
   &\simeq& 100 \ln(10)  \, .
 \ea
  In the second line, we have expressed the occupation number in
term of the squeezing parameter $r_k$ : $n_k = \sh^2 r_k$. Hence,
a two-mode squeezed vacuum state which decoheres in the two-mode
coherent basis goes along with an entropy of $S_{{\bf k},\, -{\bf
k}} = 2 \, r_k$ {\it per two-mode}. This value is large, but not
maximal. Indeed, when $c_k = 0$ one would have found the maximal
value of the entropy. It is twice the above value, i.e. \ba
S_{{\bf k},\, -{\bf k}}^{max}= S_{{\bf k},\, -{\bf k}}^{inc} = 4
\, r_k\, , \ea or $S^{inc}_{\bf k} = 2 \, r_k$ per mode ${\bf k}$.


It is interesting to notice that the entropy associated with the
effective distribution Eq. \Eqref{Peff} of sine functions equals
that of Eq. \Eqref{rhodec}, up to an arbitrary constant which
arises from the usual ambiguity of attributing an entropy to a
classical distribution. (This ambiguity can be lifted when
introducing $\hbar$ to normalize the phase space integral.) Using
this trick, the entropy associated with Eq. \Eqref{Peff} is
$S_{e\!f\!f}= 2 \, r_k$ for each {\it independent} mode (the
entropy is maximal because the state is Gaussian).
 However, 'for each {independent} mode' here means for each two-mode
 since the mode $-{\bf k }$ is no longer independent
 once the cosines have been neglected, see Eq.
 \Eqref{Aeff}.
 From this equality of entropies
 to leading order in $n_k$,
 we conclude that the quantum density
 matrix which corresponds to Eq. \Eqref{Peff}
 can be taken to be that given by Eq. \Eqref{rhodec}.
 A priori one might have thought that many quantum distributions
 can be associated with
 %the classical distribution
 Eq. \Eqref{Peff}.
 This is not the case when imposing that Gaussianity is preserved
 and that the entropies coincide. Indeed,
 in the high occupation number limit, the entropy is an
 extremely sensitive function of the residual coherence.
 As we shall see in
 Appendix D7,
 this drastically restricts the space of quantum density matrices
 in correspondence with Eq. \Eqref{Peff}.

 Having obtained a lower bound for the entropy from
 quantum considerations, we now provide an upper bound for the
 entropy which could have resulted from processes in the
 inflationary phase.


 \subsection{Maximal entropy and backreaction effects}


The bound follows from the fact that increasing the decoherence
implies increasing the power of the decaying mode, see the power
of the cosines in Eqs. \Eqref{ABvariances} and \Eqref{RRR2}. One
thus obtains an upper bound on the decoherence level when
requiring that the power of the decaying mode be smaller than that
of the growing mode at the onset of the
%D40 adiabatic era
 radiation dominated era. This requirement follows from the
fact that the r.m.s. value of the primordial fluctuations (of the
Bardeen potential) cannot be much higher than the contribution of
the growing mode from in vacuum because otherwise this would
invalidate the whole framework of linear metric perturbations.
Notice that this type of bound also occurs in the context of the
trans-Planckian question \cite{MB,NP,NPC}, or whenever one is
exploring the upper value of the deviations from the standard
predictions of inflation \cite{cmbParenta}.


 %D50
 %Using Eq.\Eqref{Ginx} evaluated at $\eta_r$ and the
 %parametrization of Eq. \Eqref{paramd}, one obtains
 Using Eqs. \Eqref{gdmodes}, \Eqref{RRR2} and \Eqref{RRR3}, and
 expanding the sine and cosine as
 $\cos(k\eta_r) = 1 + O(n^{-3/2})$ and
 $\sin(k\eta_r) = O(n^{-1/4})$
 where we used $n_k \propto (k\eta_r)^{-4} \gg 1$, the condition
 on the power of the decaying mode
 stated above reads $\delta_k \leq O({n_k}^{1/2})$, or
 equivalently
 \ba
   \bar n_k
   \leq  n_k^{3/4} \, , \qquad
   %D31 S^{upper}_{{\bf k},\, -{\bf k}}  \leq
   S_{{\bf k},\, -{\bf k}}  \leq
    3 r_k + O(1) \, .
 \ea
 If no further decoherence is added in the radiation dominated
era, this is the maximum amount of entropy stored in the
primordial spectrum. Notice that when evaluated at recombination,
the two-point function is still unaffected by this modification of
the coherence because at that time the decaying mode has still
further decreased. Indeed the relative modifications are then of
the order of $n_k^{-1/2} \sim 10^{-50}$.

 From this analysis we (re-)learn that the dynamical processes
 occurring in the inflationary era which are compatible with a
 linearized treatment of perturbations will, in general,
 leave no significant
 imprint on the spectra at recombination.
 %R26 citer uzan-bernardeau non-Gauss inflationite ??
 %D27 Maldacena n'est il pas suffisant ?
 %R28 uzan-bernardeau is MULT fields inflation, pas malda
%R26 an IMPORTANT QUESTION ?
%% considerons la gauss la plus generale avec n fixe, d'ou param par
% epsilon et theta. si on s'impose que le power ne depasse pas 1,
%no large back reaction effects,
%what is the excluded domain in the theta, epsilon plane ??
%by considering the infla era, and (then) the rad. era.
%si tu comprend la question posons la a jens.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




 \section{Homogeneous Gaussian distributions with high occupation numbers}

 The purpose of this section is to identify the quantum and classical
 statistical properties of  homogeneous Gaussian distributions
 when the two-mode coherence is not perfect.
 In particular, we shall determine the
 two interesting ranges of decoherence level.
 First there is a narrow range wherein modifications
 of the initial density matrix are so small than the distribution keeps
 quantum properties. Second there is a much wider range
 wherein the distribution, on the one hand,
%D27 looses
 has lost its quantum properties so as
 to be interpretable in classical terms,
 but, on the other hand,
 has preserved the temporal coherence of modes.
 We shall only present the results.
 The interested reader will find the technical details
 in Appendix D.




 The most general statistically homogeneous and isotropic Gaussian
 distribution is characterized by its second moments given in Eqs.
 \Eqref{momenta}.
 They are characterized by three real quantities.
 However when focussing on the coherence of the distribution,
 only one parameter
 matters, namely that governing the norm of
 the coherence term.
 In this paper, we shall not discuss the possible modifications
 of the spectrum which are due to a change of the phase of $c_k$.
 Firstly because this change induces no entropy change.
 Secondly because these modifications have been discussed in the context
  of the trans-Planckian question \cite{NP,NPC,MB}. In that case
  indeed  the
  phase shifts result from a change of the Bogoliubov coefficients
  which preserve the purity of the state. Therefore those changes
  are {orthogonal} to the ones considered
  in the present paper.

  A convenient parametrization of the coherence level is provided by
 $\delta_k$ which is defined by
 \ba \label{paramd}
   \abs{c_k}^2 = (n_k + 1)(n_k  - \delta_k) \, .
 \ea
 $\delta_k$ ranges between $0$ and $n_k$.
 They are three characteristic values:
 $\delta_k =0$ obviously corresponds to the pure squeezed state: the
 original {\it in}-vacuum,
 $\delta_k = 1$ corresponds to the critical value above which
 density matrices loose their quantum properties,
 %R34 comme il y a renvoi on peut se paser de la footnote \footnote{Notice
 %that the decoherence scheme based on \Eqref{rhodec}
 %gives $\delta_k = 2$. However, as shown in Appendix D.7,
 %the matrices associated with $\delta_k = 1$ and $\delta_k = 2$
 %belong to the same class.}
 % In fact, using Eq. \Eqref{tilde n}, one gets
 %$ \bar n_k = (1/4 + (n_k + 1)\delta_k)^{1/2}$. Hence all density
 %matrices with $\delta_k = O(1)$ give, to leading order,
 % the same entropy, namely $S= \ln(n_k) + O(1)$.}
 see Appendices D3-6,
 and $\delta_k = n_k$
 corresponds to the thermal case with maximal entropy.

  %R23This orthogonality
  %explains why the same criterium of small backreaction effects
  %leads to different results, compare the above relative change of
  %the two-point functions to those obtained in \cite{NP}.




% Q vois tu que pas grand chose change si on jouait aussi avec la phase ??
% ca relance notre draft en suspend non ?
%R14 plaese note pour le futur l'orthog des diff modif of the standard
%results is un BON critere to classify them

 \subsection{Quantum and classical properties}



 The analysis of Appendix D
 reveals that the statistical properties
 of the distribution
 fall in two categories,
 according to whether they are
 {\it robust} or {\it fragile} when the distribution is slightly perturbed.
 By definition the ``classical'' properties of the distribution, i.e.
 the physical properties observable in the classical regime,
 are the robust ones.


 \subsubsection{Classical properties}

 By robust, we designate the statistical
 properties of
 the original distribution $\hat \rho_{in}$ which
 are preserved in a wide class of partially decohered distributions.
 More precisely, to leading order in $n_k$,
 these properties are unaffected by a
 modification of the coherence term $\Delta |c_k|/|c_k| \ll 1$.
 Thus one has:
 \ba \label{robust}
   {\rm  classical} \qquad \Longleftrightarrow \qquad
   {\rm robust} \qquad \Longleftrightarrow \qquad {\rm unaffected} \,  \,
   {\rm when } \, \, \, \delta_k \ll n_k
   \, .
 \ea
Since the phase of $c_k$ is unchanged, $\delta_k \ll n_k$
guarantees that the power spectrum and the temporal
 coherence of the modes are robust properties of the state.
 As we shall see below, (together with the Gaussianity
which is here exactly preserved) they are the only ones.

 For high occupation numbers, the condition  $\delta_k \ll n_k$
 is a very mild constraint. Indeed, robust
 properties will be lost only when
 the state is close to a thermal state (and thus completely
 incoherent). When starting from a squeezed state, this regime
 can only be reached dynamically in the presence of strong non-linearities.
 Given the amplitudes of primordial fluctuations $\simeq 10^{-5}$,
 this possibility seems excluded in inflation.


 It is interesting to notice that the condition $\delta_k \ll n_k$
 can be interpreted in two complementary ways.
 First, with the mean values. In this case, $\delta_k \ll n_k$
 gives $(\abs{c_k} - n_k) /n_k \ll 1$. This
  coincides with Eq. \Eqref{neqqq}
 which is the
 condition for the two-point function to factorize.
 Alternately, one can consider the distribution itself (see Appendix D.2).
 One gets
 \ba \label{Qtexte}
   \bra{v,\,  {\bf k}}\bra{w,\, - {\bf k}} \hat \rho
   \ket{w,\,  -{\bf k}} \ket{v,\,  {\bf k}}
   \propto \exp\left[ - \frac{\abs{v}^2}{n_k+1} -
   \frac{\abs{w - \bar w}^2}{1+ \delta_k}  \right]
   \, ,
 \ea
%R19 where $\Delta_k = (n_k+1)^2 - \abs{c_k}^2=(n_k + 1) ( 1+ \delta_k ) $, and
 where $\bar w = c_k\, v^*/(n+1)$ designates
 the most probable value of $w$ given $v$.
 It generalizes Eq. \Eqref{meanw}.
 When $\delta_k \ll n_k$
 the spread of $w$ around $\bar w $ is negligible
 with respect to the spread in $v$, the power $(=n_k+1)$.
 Thus the condition
 $\delta_k \ll n_k$
 means that the modes ${\bf k}$ and
 $-{\bf k}$ are still tightly correlated in phase and amplitude,
 as they were in the original distribution, see discussion
 bellow  Eq. \Eqref{vwdistrib}.
 These distributions thus are (to leading order in $n$)
 statistical mixtures of  two-mode coherent states
 $\ket{v} \ket{\bar w}$ as in Eq. \Eqref{rhodec2}.

 \subsubsection{Quantum properties}

 As discussed in Appendix D, the ``quantum features'' of the distribution are
 the following:
 \begin{itemize}
 \item
 The correlations between macroscopically separated
 semi-classical states, see
  discussion after Eq. \Eqref{vwrhov'w'} and Appendix D.3,


 \item
  the existence of a sub-fluctuant
  mode in phase-space, see
 Appendix D.4,

 \item the non-$P$-representability
 of the state, see Appendix D.5,

 \item
 the non-separability of the state
 and the violation of Bell inequalities, see Appendix D.6, and

 \item
 an entropy smaller than that of the minimal scheme, see Eq. \Eqref{Smin}
 and Appendix D.7.
 \end{itemize}



The important fact is that all these properties are erased when
$\delta_k \geq 1$, {\it independently} of $n_k$ when $n_k \gg 1$.
One can thus say
 \ba
  {\rm quantum}  \qquad \Longleftrightarrow \qquad {\rm fragile }
  %\,\,\, quantum \,\,\, statistics}
   \qquad \Longleftrightarrow \qquad
   {\rm erased \,\,\, iff}
   \quad \delta_k \geq 1 \, .
 \ea


 Let us emphasize the important lessons.
 The first is that these quantum properties are
 still present when $\delta_k < 1$. Therefore $\delta_k = 1$ is indeed
 the critical value
 from which quantum properties are lost.
 %D20 see paragraph bellow
 %In this sense, the distribution of Eq. \Eqref{rhodec} {\it is} the minimally
 %decohered distribution, and two-mode coherent states do provide
 %a minimal coarse graining.
%R21 le probleme avec ce choix est le suivant
%qules sont les etats qui donnent delta = 1
%on PART des etats coh.
%mais ensuite on (tu) les a
%laisse tomber pour un AUTRE critere (le quel ???)
%dont on ne connait pas les etats associes
%%quelle est la logique ??
%
%D21 c'est juste. Paper I is inductif : c'est un guess. App D de paper II
%est un resultat mathematique qui repond a la question :
%pour quelle valeur de delta les prop quantiques disparaissent ?
%Et la reponse est unique, delta = 1.
%
%J'ai donc en effet change de point de vue par rapport a l'annee derniere.
%Du nouveau point de vue, la question des etats me semble irrelevante. Neanmoins,
%on a un element de reponse : Equation D18, la representation P.
%L'inconvenient est qu'elle ne correps pas comme P_2 du rho_{min} a un proba de
%mesurer le 2-mode dans la paire (v,w), ce qui est la facon commune de concevoir la decoh.
%Mais ca n'est qu'une approximation : l'esprit de Zurek, Eisert... c'est
%justement que la reponse a la question 'dans quelle base rho est elle diag'
%est tres non-triviale car n'admet pas une reponse univoque (base robuste ne signifie pas
%base indestructible). Il y a toujours une certaine indetermination.
%
%Enfin, ne perdons pas de vue que nos distrib sont
%des Gaussiennes effectives.
%Both n_eff et c_eff seront a priori diff de n_sq et c_sq.
%
%En conclusion, les deux criteres ('rho diag avec P_2 assoc a une mesure'
%ou 'pas de prop quantiques') sont des criteres phenomenologiques, et on a
%donc aucune bonne raison de choisir un critere plutot qu'un autre.
%
%Maintenant que j'y vois plus clair, je propose donc qu'on annonce clairement la couleur
%1) c'est phenomenologique
%2) donc on a plusieurs choix pour 'minimal', tous d'ailleurs equiv
%du point de vue de l'entropie (delta = O(1))
%3) on choisit tel critere, mais on pourrait aussi bien choisir l'autre.
%
%What do you think ?
%R22 entierement d'acc


 Second,
 the density matrix which corresponds to $\delta_k = 1$
 is that given in Eq. \Eqref{rhodec2}. From Eq. \Eqref{lim},
 one clearly sees that it is the least decohered separable
 distribution.



 Third,
 for $n_k \gg 1$, the quantum properties are very
 sensitive to a slight perturbation of the system since
 $\delta_k = O(1)$
 corresponds to
 a relative change of order $1/n_k$
 of the power spectrum  Eq. \Eqref{powern}.
 Therefore,
 %in the large occupation number limit,
 there is a wide range
 of $\delta_k$ (from $1$ to $n_k^{\gamma}$ with $\gamma < 1$) where the robust
 properties are still unaffected (to leading order in $n_k$)
 while the quantum properties are completely erased.
 %D24 %R24 right this is no longer needed
% \footnote{The distribution \Eqref{rhodec} belongs to this class
% of distributions, since in that case $\delta_k = 2$. It is in a
% one of the minimally decohered distributions in the sense of
% footnote 7.
% }.


 Fourth, for $\delta_k \leq 1$, the product
 of the residual squeezed spread
 times the residual width of the correlations between
 semi-classical states in the
 orthogonal direction
 stays approximately constant whereas it increases linearly with $\delta_k$
 for $\delta_k \gg 1$, see Eq. \Eqref{interprod} and
 Fig $3$, see also \cite{Morikawa}. This means that
 these two spreads are equivalent expressions of
 the residual quantum properties of the state.

 Fifth, the entropy (which is a measure of the degree of purity of the state)
 is also very sensitive
 to the level of coherence, see Fig. $5$. Therefore
 it can be used
 to parameterize the level of coherence
  and to
  order different distributions, see Appendix D7.

%R11 : entre nous 1. n'est ce pas la `preuve` que la decoh se fait
%graduellemnt mais en utilisant des MINIMAL states (i.e. des moins
%squeezes) until les coherent states sont atteints ?? (si c'est
%vrai se serait genial) \\
%
%D12 : Je ne sais pas mais ca merite d'etre creuse. \\
%
%2. que dit la condition 14 si on utilise des moins squeezes states
%comme je la'i suggere avant que tu ne partes au canada ?? l'exclu
%t'elle ????????????????????? \\
%
%D12 : Il faut que j'y travaille. Ma premiere reponse est que la
%condition pour avoir D14 est $\abs{c} \leq n$, et c'est bien sur
%independant de la facon dont on ecrit l'etat. Mais attention, ca
%n'est vrai que pour les etats Gassiens homog. Si on permet
%d'autres expect. values, alors la condition sur les moments en est
%une autre. Par consequent, 1) soit ca veut dire que le pavement
%avec des etats sq est equivalent a celui avec les etats coherents,
%2) soit ils ne le sont pas, i.e. les etats sq ne decrivent pas des
%distrib homog.
%
%Comments ? \\
%
%R12 : ce que tu ecris me rend perplexe.
% Je ne comprends pas pq des
%etats moins sq, mais bien alignes, et superposes STATISTMT peuvent
%briser quoi que se soit. d'ou peut etre 1) est correct. \\
%
%D13 : Je suis d'accord, par contre je ne comprends toujours pas
%pourquoi tu ne considere que le cas aligne. J'ai pris conscience
%que la decoh due aux non-lin produit certainement des $\langle
%a_{\bf}^2 \rangle \neq 0$. Si c'est vrai, alors l'image est un
%pavement avec des etats de moins en moins squeeze qui ne sont pas
%alignes avec $\rho_{in}$. Do you agree ? \\
%
%
%Je proposerais d'ecrire explicitement l'equiv de D15 avec des
%etats
%sq et de calculer n et c, pour un given P(v,w) gaussien. \\
%
%D13: C'est ce que j'avais fais avant mon depart, et j'avais trouve
%un $<< v^2 >> \neq 0$. Une autre facon de voir est celle la :
%prend D13, et sandwiches avec des operateurs de squeezing. On a
% \ba
%   {\rm Tr}( a_{\bf k} S \rho S^{\dagger}) =
%   {\rm Tr}( S^{\dagger} a_{\bf k} S \rho ) =
%   \langle (\alpha_{\xi} a_{\bf k} +
%   \beta_{\xi} a_{\bf k}^{\dagger} )^2 \rangle \neq 0
% \ea
% unless some fine tuning des phases des Bogol (qui doit sans
% doute etre le cas aligne). \\
%
%R13. Je suis d'accord que les squeezes alignes ne representent pas
%le cas generiques; Mais bien une deco MINIMALE d'ou en accord avce
%notre philo.
%Et c'est dans CE cas LA, que je pose ma question. ceci dit ces
%questions sont beyond the present paper.//es tu d'ac. ?\\
%
% D14 : J'aimerais traiter completement les aspects cinematiques
% pour ne pas les melanger avec la dynamique qui sera deja assez
% complique. Ca serait instructif de determiner de 1) ou 2) lequel
% est vrai. On aurait ainsi une tres belle image de la decoherence.
%
%D15: je n'en suis pas encore certain, mais il semble a present que
% il n'y a Pas de distrib NON-separable qui s'ecrive comme un
% ens stat d'etats squeezes deplaces.
% Je reflechis donc a savoir si l'image de la decoh avec des etats de
% moins en moins squeeze est correcte, et si non, a t'on un autre
% concept semblable.
%
%R14: j'ai une question (pour le futur)
%%quest ce qui fixe le signe de  ${\rm Tr}( a_{\bf k}^2 S \rho S^{\dagger}$
%D15: c'est un nombre complex non ?
%R14: il y a la une brisure d'une invariance
%%a mon avis il n'y a pas de reponse universelle
%%cf nos papier sur les transPl.
%%a part ca on est d'acc et ce papier ci me semble complet avec ce que j'ai
%rajoute today
%D15 je vais essayer de faire des fig 3d pour visualiser le spread.


 \subsection{Discussion}

 First, the above results show that
 the following statements are
 {\it equivalent} in the high occupation number limit,
 for decoherence levels obeying $\delta_k \ll n_k$ :
 \begin{itemize}

 \item
 The distribution of the modes ${\bf k}$ and $-{\bf k}$
 is strongly correlated in phase and amplitude,
 where strong means that, given a value %D27 $v_{\bf k}$
 $v$ of the amplitude of the mode ${\bf k}$, the conditional probability
 to find %D27 $w_{-{\bf k}}$
 ${-{\bf k}}$ with amplitude $w$ is centered around a value
 %D27 $\bar w  (v_{\bf k})$
 $\bar w  (v)$ with a
 spread $=1 + \delta_k \ll n_k$, see Eq. \Eqref{Qtexte}.

  \item
  All realizations of
 the modes contributing to the two-point function
 have the same temporal
 phase. This phase is fixed by $\arg(\bar w)$ of Eq.
 \Eqref{Qtexte} and is not affected by the level of decoherence.
 Indeed the modifications of $\arg(\bar w)$ preserve the
 quantum purity  of the distribution and are orthogonal
 to those considered here.
 The absence of phase shift
 guarantees  that
 the decaying mode stays subdominant for $\delta_k \ll n_k$, in the sense that
 its power
 $(= \delta_k/2+ O(1/n))$ stays $\ll n_k$, see Eq. \Eqref{RRR2}.
%R23 je pense qu'il faudrait DETAILLER ce resultat dans un des App
 %D24 Si on adopte ta def de growing, je pense en effets que c'est necessaire.
 %Ce que je ne suis pas pres a faire vu que je ne comprends pas cette definition.
%R24 look the job est deja fait dans 32. les modes out ne dep QUE de a(eta)
%R24  je pense que  $O(n^{1/4})$. est INCORRECT
%D25 Now I agree that your def is acceptable. But I don't understand any more
%why the phase should be $a(\eta)$.

%
%R19 please check \\
%
%D20 (regide avant que je ne modifie II.B)
%from 9, je trouve $n^{1/2}$, mais ce resultat me laisse tres
%perplexe, car le power devrait se lire sur $Q(v,w)$ (c'est ce que
%j'ai fait la 1ere fois, et si ce n'est pas juste je ne comprends
%plus rien). look at Eq 18 : on negige $G_{out}$, donc pour etre
%coherent il faut negliger tous les termes $O(1)$, i.e. ecrire
%$c=n*e^{i2\psi_k}$, ce qui donne Eq. 20 sans qu'on l'ai dit. Par
%consequent, si on calcul le power a partir de D9, il faut aussi
%soustraire un terme, ie utilise $\alpha - 1$ au lieu de $\alpha$,
%ou qqch dans ce genre. Evidemment on ne comprends pas trop quoi.
% Donc, si on veut le power du decaying mode, je pretends qu'on ne l'a pas en lisant
%Eq 9 (au passage, je n'ai jamais compris ce ($k\eta_r)^3$, qu'on
%ne retrouve d'ailleurs qu'avec les operateurs S et C. Si on
%choisit les phases des modes telles que $Im(c) = 0$, alors le
%power du cos est $1/n$. Je pense donc qu'on a pas bien absorbe la
%phase de c dans les modes quand on a ecrit Eq. 30). La seule
%methode ou on est sur de ne pas faire de connerie est de regarder
%$G_{in}$ COMPLETE (plus rigoureusement le Weyl ordered, mais a
%temps egaux ca ne change rien), ou de maniere equiv la Wigner
%$W(v,w)$. On a alors, en ecrivant $c=\abs{c}e^{-i2\psi_k}$,
% \ba
%   G_{in} = \int \, ... (n+1/2) \left[
%   1 - \zeta Re\left(e^{-i2\psi_k} e^{i2k\eta} \right) \right]
% \ea
%where
% \ba
%   \zeta = \frac{\abs{c}}{n+1/2} = 1-\frac{\delta_r}{2n} -
%   \frac{1+\delta_r^2}{8n^2} + O(n^{-3}) \, .
% \ea
% Par consequent le power du decaying mode est $n\times 1/n^2 = 1/n$
% si on est dans l'etat squeeze ($\delta = 0$), et $\delta$ sinon.
% En conclusion, je persiste et signe par un \delta \\Morikawa
%
% Do you agree ?

 \item
  The two-point function factorizes in a product of two growing modes.

\end{itemize}

 Second, both the power spectrum and the temporal coherence
 are fairly robust, in the sense
 that they are lost only for $\delta_k \simeq n_k$,
 i.e. through a significant regeneration of the decaying mode.
  They are the statistical properties of the state
 which can be interpreted classically.
 The fact that the criterion of robustness singles out
 certain statistical properties
 should be conceived as a {\it definition} of
 the classical (statistical)
 information contained in the quantum density matrix,
 in agreement with Gottfried's analysis \cite{gottfried}.


% D13 Est ce qu'on ne tempererais pas en commencant
% avec un ''we believe that'' \\
%
% R16 n'est pas explained in Gottfried ??\\
%
% D17 : il parle de notre $\sigma_v$.
%
% Il dit que pour toute observable realiste portant sur
% tout le syst (objet$+$appareil), apres la mesure,
% $Tr(rho A) = Tr(rho' A)$, ou $rho'$ ne contient que les elements diagonaux de rho.
% Donc les prop stat sont interpretables classiquement (des proba et non des amp de
% proba).\\
%
% 1) ce dont parle Gottfried, c'est notre $\sigma_v$ :
% de maniere effective (ie corrections ($1/n$)), quand on mesure des observables locales,
% $Tr(\rho_{in} A) = Tr (\rho_{min} A)$ \\
%  2) le sens de mon statement est donc : (robust $<->$ cl) $<->$ (use rho' instead of rho
% because you won't see the difference) \\
%
% La difference avec Gottfried est que notre base n'est pas
% orthogonale.
% Q: Y a t'il une base orthogonale ou l'info sur la phase
% est preservee ? \\
%
% on dit plusieurs choses : 1)les prop qui sont intrepretables classiquement persistent
% pour $\delta > 1$. C'est grace au fait que la phase de c n'est pas effacee, ce qui n'est a mon
% avis possible que dans une base overcomplete.
% ce n'est pas de la decoherence naive. \\
%  2) il n'y a plus que des corr classiques pour $\delta > 1$ (c'est la separabilite)
% ca non plus c'est pas trivial \\
%  3) mon statement etait que cette distinction des prop de l'etat, basee sur le
%  module de c, sert de critere pour distinguer les prop stat
%  q/cl\\



% To be more precise,
% we emphasize that {\it both} $n_k \gg 1$ and the vanishing of
% the quantum properties of the state must be met so that
% one can interpret the statistical properties in classical terms.
 %D24
 %This is not a very clever comment. Supprimerait on la parenthese ?
 %R24 je supprimerais les 2 paragr
 %car on l'a deja dit n fois avant
 % et car c'est vague ``must be met''
 %D25 right
 %R24 par contre if it is NOT met, so what ????
% est une question plus importante
% (Remember that in a laboratory, the observation of a quantum behavior
% by a system with a large number of particles requires to isolate
% it,
 %R23(lasers, BEC, superconductors and superfluids),
% and the larger the occupation number is, the more difficult it becomes to
% isolate the system.)




In the case where $\delta_k < 1$, the distribution cannot be
interpreted in classical terms. Nevertheless, when probing (or
observing) the momenta of this distribution with a relative
uncertainty larger than $1/n_k$, the above robustness prevents the
existence of signals telling us that some quantum properties are
still present. Indeed, quantum properties of Gaussian
distributions can {\it only} be put into evidence
in two cases. Either one must be able
to measure the momenta of distribution $\Eqref{momenta}$
with a precision higher than $1/n_k$. This amounts to be able
to measure the amplitude of the sub-fluctuant mode
and hence that of the decaying mode, see Eq. \Eqref{RRR2}.
Or one must have a set of
%unusual, i.e. unlike those
%``occuring in nature'' see \cite{gottfried} p. 186, %R40
non-quadratic operators. A first exemple of such operator
giving rise to violations of Bell's inequalities
%based on
%related to %R40 the parity of the particle number
can be found in \cite{Wodkievicz}, see also
Appendix D6; a second is discussed in \cite{CP4}.
The consistency of these two alternatives
is understood when noticing that the measures involved
in the second case amount to distinguish configurations
with a precision higher than $1/n_k$.


%D27 have a look, c'est le plus relevant que je connaisse
%R26 as tu une ref pour cela ??
%cad pour des operat a la Bell pour des etats gauss

These considerations imply that the non-linear processes giving
rise to structure formation will also be insensitive to the
existence of a residual degree of quantum coherence. (In this we
disagree with the claim \cite{HuCalzetta} according to which only
the decohered part of the power spectrum will participate to
structure formation).
%R26,
%and that of \cite{LiddleLyth} stating that a quantum
%distribution will manifest itself by ``Schroedinger cat'' type of behaviour.
 %D25 who said that ? are you sure that's what poeple say
 %about primordial fluctuations ??
%%R25 je l'ai lu dans Liddle Lyth I shall have a look
%%R26 je l'ai retrouve page 64. c'est suff vague pour ne pas etre attaquable
The question of the residual quantum coherence (i.e. the value of
$\delta_k$ if $\delta_k < 1$) seems therefore of purely academic
interest.



 % t should be also stressed that nothing
% We have seen that the power spectrum
% A similar situation happens in cosmology.
% Pure squeezed states are obtained only
% when using %R23  result of the
% an idealized quadratic Hamiltonian which results from
% the neglect of non-linearities.
% %evolution emanating from the
% % {\it linear approximation} (the two-mode sectors decouple).
% Because of the weakness of gravitational interactions, the
% non-linearities
% %properly taken into account
% will not significantly alter the power spectrum
% the linearized treatment.
% However, the non-linearities will effectively
% engender a non-vanishing $\delta_k$. The open question is then:
  % it seems unlikely that they w Will produce of
 %will $\delta_k$ be smaller than $1$ ?
%will they destroy the quantum properties
%knowing that in the high occupation number limit,
%the quantum features are extremely sensitive.

%R23 We stress again that the important fact is that when
 %considering the large occupation number limit,
%i.e. the usual semi-classical limit $n_k+1 \simeq n_k$,
%the separation of the two classes of features becomes very neat.
%What is some times called the 'quantum to classical transition' is
%therefore a consequence of the fact that the properties fall
%into two classes.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





 \section{Conclusion}

 In this paper we
 analyzed  partially decohered distributions
 susceptible to describe the primordial fluctuations
 in inflationary scenarios.

 First we have
 identified the parameter governing the level of decoherence
 and computed the associated entropy, the residual value of the
 squeezing, and the residual coherence encoded in
 off-diagonal matrix elements.

 Second, we showed
 that the
 correlations of modes in the
 initial distribution and its decohered versions,
 which is at the origin of the temporal coherence of the
 mode at horizon re-entry,
 is properly taken into account by considering superpositions
 of 'two-mode coherent states'.

 Third, we discussed how, in the large occupation number limit, the
 fragility of quantum properties of the distribution to
 small modifications
 lead to a critical decoherence scheme above which all quantum
 properties are erased. The associated entropy is very large
 (and equal to $1/2$ the thermal entropy) even though
 the relative change of the
 power spectrum is only $1/n_k$.
 For $\delta_k > 1$, the remaining statistical properties can
 be interpreted classically.

 The open question concerns the calculation
 of the residual degree of coherence (the value of $\delta_k$).
 This is a difficult  question which
 calls for further work.
 During the inflationnary period,
 the dynamics is indeed not trivial because of the conjunction of
 mode amplification and non-linear effects.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%r%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vskip .5 truecm

 {\bf Acknowledgments:}
 We are grateful to Serge Massar for having
 explained to us the meaning of decoherence by drawing Fig. 1 many
 years ago in Brussels. We are also grateful to Dani Arteaga,
 Claus Kiefer, Jihad Mourad and Alexei Starobinsky for useful remarks.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{appendix}


\section{Coherent states}

This appendix aims to present the properties which we shall use in
the body of the manuscript. For more details, we refer to
\cite{Glauber63a,Glauber63b,Zhang90}.

Coherent states (of a real oscillator) can be defined as
eigenstates of the annihilation operator:
 \ba \label{DefCoh1}
   \hat a \ket{v} = v \ket{v} \, ,
 \ea
where $v$ is a complex number. In Fock basis it is written as
 \ba \label{DefCoh2}
   \ket{v} = e^{-\frac{\vert v \vert^2}{2}}  \sum_{n=0}^{\infty}
   \frac{v^n}{\sqrt{n!}} \ket{n} \, ,
 \ea
where the exponential prefactor guarantees that the state is normalized
to unity $\langle v \vert v \rangle=1$.
They are also obtained by the action of a displacement operator on the vacuum :
 \ba \label{displacement}
   \ket{v} = \hat D(v) \ket{0} = e^{v^* \hat a - v \hat a^{\dagger}}
   \ket{0} \, .
 \ea

The first interesting property of coherent states is that they correspond
to states with a well defined complex amplitude $v$.
Indeed, by definition \Eqref{DefCoh1},
the expectation values of the annihilation and creation operators are
 \ba
   \bra{v} \hat a \ket{v} = v \, , \quad
   \bra{v} \hat a^{\dagger} \ket{v} = v^{*}
   \, .
 \ea
Thus the mean occupation number is
 \ba \label{occupnumber}
   \bra{v} \hat a^{\dagger} \hat a \ket{v} = \abs{v}^2 \, .
 \ea
It is to be also stressed that the variances vanish:
 \ba \label{nullvariance}
   \Delta \hat a^2 = \bra{v} \hat a^2 \ket{v} - \bra{v} \hat a \ket{v}^2 = 0 \, ,
   \quad \Delta \hat a^{\dagger \, 2} = \bra{v} \hat a^{\dagger \, 2} \ket{v} -
   \bra{v} \hat a^{\dagger} \ket{v}^2 =  0 \, .
 \ea

 From these properties one sees that the expectation values of the
 position and momentum operators (in the Heisenberg picture)
 \ba
   \hat q(t)=\sqrt{\frac{\hbar}{2\omega}}\left(\hat a e^{-i \omega t} +
        \hat a^{\dagger} e^{i \omega t}\right) \, , \qquad
   \hat p(t)=-i\sqrt\frac{\hbar\omega}{2}(\hat a e^{-i \omega t} -
                \hat a^{\dagger}e^{i \omega t})
   \, , \nonumber
 \ea
 are
 \ba \label{cltrajincohstate}
   \bar q(t) &=& \bra{v} \hat q(t) \ket{v} =
   \sqrt{\frac{\hbar}{2\omega}}( v e^{-i \omega t} + v^* e^{i \omega t})
            = \sqrt{\frac{2\hbar}{\omega}} \abs{v} \cos(\omega t - \phi_v)
   \, , \nonumber \\
   \bar p(t) &=& \bra{v} \hat p(t) \ket{v} =
   -i\sqrt{\frac{\hbar\omega}{2}} ( v e^{-i \omega t} - v^* e^{i \omega t})
            = -\sqrt{2\hbar\omega} \abs{v} \sin(\omega t - \phi_v)
        = \partial_t \bar q(t) \, .
 \ea
We have used the polar decomposition $v=\abs{v} e^{i \phi_v}$.
These expectation values have a well defined amplitude and phase
and follow a classical trajectory of the oscillator. This is due
to the ``stability'' of coherent states under the evolution of the
free Hamiltonian $2 H_0= p^2 + \omega^2 q^2$: if the state is
$\ket{v}$ at time $t_0$, one immediately gets from \Eqref{DefCoh2}
that at a later time $t$, the state is given by $\ket{v(t)} =
\ket{ve^{-i\omega (t-t_0)}}$. Notice that the variances of the
position and the momentum are
 \ba
   \Delta \hat q^2 = \frac{\hbar}{2\omega} \, , \qquad
   \Delta \hat p^2  = \frac{\hbar\omega}{2} \, .
 \ea
 They minimize the Heisenberg uncertainty relations and are
time-independent. Hence, in the phase space $(q,p)$, a coherent
state can be considered as a unit quantum cell $2\pi \hbar$ in
physical units (see also \Eqref{measure} for the measure of
integration over phase space) centered on the classical position
and momentum of the harmonic oscillator $(\bar q(t),\bar p(t))$.
In the large occupation number limit $\abs{v}\gg 1$, coherent
states can therefore be interpreted as classical states since
$\Delta \hat q / \sqrt{\bar q^2+ \bar p^2/\omega^2} = \Delta \hat
p / \sqrt{\omega^2\bar q^2+ \bar p^2} = 1/2\abs{v}$. This is a
special application of the fact that coherent states can in
general be used to define the classical limit of a quantum theory,
see \cite{Zhang90} and references therein.

One advantage of coherent  states \cite{Glauber63a} is that the
calculations of Green functions resembles closely to those of the
corresponding classical theory (i.e. treating the fields not as
operators but as c-numbers) provided either one uses normal
ordering, or one considers only the dominant contribution when
$\abs{v}\gg 1$. We compute the Wightman function in the coherent
state $\ket{v}$
 \ba
   \widetilde G_v(t,t') &=& \bra{v} \hat q(t) \hat q(t') \ket{v} \nonumber \\
           &=& \langle :\hat q(t)\hat q(t') : \rangle_v +
       \frac{\hbar}{2\omega} e^{i\omega(t-t')} \, ,
 \ea
where we have isolated the contribution of the vacuum.
The normal ordered correlator is order $\abs{v}^2$:
 \ba \label{cohgreen}
   \langle :\hat q(t)\hat q(t') : \rangle_v &=&
   \frac{\hbar}{\omega} \,
   Re\left[\langle \hat a^2 \rangle_v e^{-i\omega(t+t')} +
    \langle  \hat a^{\dagger} \hat a \rangle_v e^{i\omega(t-t')} \right]
    \nonumber  \\
   &=& \frac{2\hbar}{\omega} \vert v \vert^2 \cos(\omega t - \phi_v)
     \cos(\omega t' -\phi_v)   = \bar q(t) \,  \bar q(t') \, .
 \ea
We see that the perfect coherence of the state, namely
$\abs{\langle \hat a  \hat a  \rangle_v }
= \langle \hat a^\dagger  \hat a  \rangle_v  $
is necessary to
combine the contributions of the diagonal and the interfering term so as to
bring the time-dependent classical position $ \bar q(t)$ in
Eq. \Eqref{cohgreen}.


The wave-function of a coherent state in the coordinate
representation is given by
 \ba \label{CohWaveFunct}
    \psi_{v}(q) = \left(\frac{\omega}{\pi\hbar}\right)^{1/4}
    \exp\left(-\frac{\omega}{2\hbar}(q-\bar q)^2 -i \frac{\bar p q}{\hbar} \right)
    \, ,
 \ea
 where $v=(\omega \bar q + i \bar p)/\sqrt{2\omega\hbar}$. This
follows from the definition $\bra{q} \hat a \ket{v} = v \langle q
\vert v \rangle$. From this equation one notes that two coherent
states are not orthogonal. The overlap between two coherent states
is
 \ba \label{overlap}
   \langle v \vert w \rangle = \exp\left(v^{*} w - \inv{2}\abs{v}^2 -
   \inv{2}\abs{w}^2 \right)  \, .
 \ea
 Nevertheless they form an (over)complete basis of the Hilbert
space in that the identity operator in the coherent state
representation $\{ \ket{v} \}$ reads
 \ba \label{coh1}
   {\bf 1} = \int\!\!\frac{d^2v}{\pi} \, \ket{v} \bra{v} \, .
 \ea
The measure is
 \ba \label{measure}
   \frac{d^2v}{\pi} = \frac{d({\mathrm Re} v) d({\mathrm Im} v)}{\pi} =
   \frac{d\bar q d\bar p}{2\pi \hbar} \, .
 \ea
The representation of identity can be established by calculating
the matrix elements of both sides of
the equality in the coordinate representation $\{ \ket{q} \}$,
with the help of \Eqref{CohWaveFunct}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Application of Zurek \& al. results to the cosmological case}

In this appendix we show that Eq. \Eqref{rhodec} is %indeed
the minimally decohered distribution in the sense of \cite{Zurek93}.
To this end, we decompose the complex mode $\hat \phi_{\bf k}$
into two real oscillators $\hat \phi_{1}$ and $\hat \phi_{2}$
given by its real and imaginary parts. Since the two-mode
Hamiltonian is quadratic and Hermitian, it splits into the sum of two identical
one-mode oscillator Hamiltonians for $\hat \phi_{1}$ and $\hat
\phi_{2}$ separately. Notice that the annihilation operators of
these two real oscillators,
\ba
\hat a_1 = \left(\hat a_{\bf k} +
\hat a_{-\bf k}\right)/\sqrt{2} , \, \quad  \hat a_2 = -i\, \left(\hat
a_{\bf k} - \hat a_{-\bf k}\right)/\sqrt{2}, \,
\label{artop}
\ea
 mix ${\bf k}$ and
$-{\bf k}$ annihilation operators. Hence they can easily enforce
correlations between these two modes. However, for our decoherence
procedure to be valid, as noticed in \cite{KiefPolStar98a},
it is important that the interactions
% do not break the
%R34  factorizability condition \Eqref{neqqq}, nor
%D36
 do not mix $\hat \phi_{1}$ and $\hat \phi_{2}$,
 %D40
 %or equivalently
 so that they
 do not break the homogeneity.
 %R37
 %translation invariance.
% preserve the
% phase coherence between $\bf k$ and $\bf -k$ modes, or
% equivalently do not mix $\hat \phi_{1}$ and $\hat \phi_{2}$.
%D36je ne suis pas sur de comprendre ce que tu veux dire par 'phase coherence'
%%  significantly change/scatter the phase of $c$.
%%%R34 on est un peu coince car on dit
 % primo $\hat \phi_{1}$ and $\hat \phi_{2}$ ne doivent pas etre
 % mixes
 % deuzio $\hat \phi_{1}$ and $\hat \phi_{2}$ ne sont pas le bon subset
 % alors que k -k le sont. Est ce compatible ?????
 %D36 c'est compatible car les NL sont faiblent. C'est un peu comme le systeme
 % des Kaons :
 %les egsts de l'Ham QCD sont K^0 et \bar K^0, mais les egsts de l'inter faible (decay)
 %sont sont diff (K^0_1 et K^0_2) car ils sont egstates de CP.
 %
 %D35 oui et non, cette analyse est trop cinematique :
 %on traite la decoh en supposant que la distrib
 %reste homogene, je suis encore sceptique a ce sujet. Si tel est le cas
 %(et tu m'as soutenu que oui l'autre jours),
 %on ne peut pas dire a la
 %fois que k est privilegier par les NL, et que 1,et 2 ne sont pas melanger.
 % apres reflection: non in the mean, yes si on etudie des outcomes post selectionnes
 %% est tu d'ac ??????????????????
 %D35 je ne sais pas pour la postselec
%R35  %on traite la decoh en supposant que la distrib
 %reste homogene, je suis encore sceptique a ce sujet. ecris tu
 %je continue a penser que si on part de couplage de type mentionne
 %dans la non-sep., alors en tracant sur les autres modes, il n'y pas de terme
 %en a_k^2 car la fct a deux point dependraitr du lieu
 %%en fait impose a G d'etre transl inv. et deduis en les contraintes sur les
 %moments.
 %%%ne me dis pas que G trace sur les non-lin se metttrait a dep de x,
%R35
%pour ce qui est de la post select relis CP04.

A two-mode squeezed state $\kett{0 in, {\bf k}}$ can always be
written as the tensorial product of the two one-mode squeezed
states \cite{schum}. In our case, the one-mode squeezed states are
those of the oscillators $1$ and $2$ because the Hamiltonian
separates. Thus we have
 \ba \label{factor12}
   \kett{0 in, {\bf k}} = \ket{0 in, 1} \otimes \ket{0 in, 2} \, .
 \ea
 The one-mode squeezed states are governed by the same parameter $z/2$ :
 \ba
   \ket{0 in, 1} = \frac{1}{\sqrt{\abs{\alpha}}} \sum\limits_{n=0}^{\infty}
   \left( \frac{z}{2} \right)^n \frac{\sqrt{2n !}}{n!} \ket{2 n,\, 1} \, .
 \ea
 The same expression holds for the ket $\ket{0 in, 2}$.


 The overlap of this one-mode squeezed state with a one-mode coherent state is
 \ba
   \langle v ,\, 1 \vert 0 in, 1  \rangle =
   %\int\!\!\frac{d^2 v_1}{\pi} \,
   \frac{1}{\sqrt{\abs{\alpha}}} \exp\left(- \abs{v_1}^2/2+ zv_1^{*\, 2}/2 \right) \, .
 \ea

According to \cite{Zurek93}, when taking small interactions into
account, the density matrix of a one-mode squeezed state will
preferably decohere into the mixture
 \ba
   \hat \rho_{red,\, 1} =  \int\!\!\frac{d^2 v_1}{\pi} \,
   {\cal P}_1(v_1)  \,
   \ket{v_1} \bra{v_1} \, .
 \ea
 where the statistical weight is given by the probability to find a coherent
 state starting from the initial state: %R24 vacuum :
 \ba \label{proba1}
    {\cal P}_1(v_1) = \abs{\langle v ,\, 1 \vert 0 in, 1  \rangle }^2 &=&
    \frac{1}{\abs{\alpha}} \exp\left(- \abs{v_1}^2 + Re(zv_1^{*\, 2}) \right)
   \, .
 \ea
 It should be noticed that this choice of the probability is not unique.
 %R28 ,because of operator ordering ambiguities.
 The procedure adopted here
 consists in taking the so-called $Q$-representation of the original density
 matrix
 %D26
 (see Appendix D.1 for its definition)
 and to treat it as a $P$-representation to define the
  the decohered matrix density. A slightly more coherent
  distribution is given at the end
  of Appendix D5. Since the ambiguity results from
  %R27 ordering procedure
  the fact that coherent states are
  %D27 not orthogonal,
  %ou  overcomplete que preferes tu ??
   overcomplete,
%R28    the
no differences show up to leading order in $n_k$.
   Hence, the various distributions belong to the same class:
   %R28 . This explains why, to leading order in the occupation number,
   they have the same entropy, see Appendix D7.



%%R24 is this correct or is it our invention-guess ??
%D25 Zurek dit 'the density matrix of a one-mode squeezed state will
%preferably decohere into the mixture', et nous on a rajoute
%'where the statistical weight is given by the probability to find a coherent
% state starting from the initial state', parce que c'est comme ca qu'on apprend la decoh
% dans Gottfried. Mais comme je te le disais la semaine derniere,
% 1) Gottfried c'est des bases ortho,
% alors que la decoh pas naive (Zurek, Eisert,POVM), c'est des bases overcomplete.
% So is the generalisation obvious ? I don't think so.
% 2) L'app D nous apprend que changer le power de $delta = O(1)$ ne modifira pas
% les quantites classiques. So its a guess, certes, mais a good one.
% Does any of this bother you ??
%%R25 on est sur la meme longueur d'onde
%%il nous reste a nous expliquer clairement

To complete the proof we need to show that product of two one-mode
coherent states $1$ and $2$ is also the product of a coherent
state for the ${\bf k}$ and $-{\bf k}$ modes. This is the case:
 \ba
   \ket{v_1}\otimes\ket{v_2} &=&
   \hat D_1(v_1) \hat D_2(v_2) \ket{0 in, 1} \otimes \ket{0 in, 2}
   \, , \nonumber \\
   &=& \hat D_{\bf k}(v) \hat D_{-{\bf k}}(w)
   \ket{0 in, {\bf k}} \otimes \ket{0 in, -{\bf k}}
   \, , \nonumber \\
   &=&  \ket{v, {\bf k}} \otimes \ket{w, -{\bf k}} \, ,
 \ea
 %D40
 where the displacement operator $\hat D$ is defined at
 Eq. \Eqref{displacement}, and
 where the amplitudes are related by
 \ba
   v=\frac{v_1+iv_2}{\sqrt{2}} \, , \qquad w=\frac{v_1-iv_2}{\sqrt{2}} \, .
 \ea
Finally, the product of the probabilities \Eqref{proba1} gives the probability
Eq. \Eqref{vwdistrib}. Performing the change of variables of integration
from $(v_1,\, v_2)$ to $(v,\, w)$ completes the proof.

%R20 si on introduit deux pahses shits diff. pour 1 et 2
%D20 Do you mean ecrire
%$\phi_{\bf k} = (e^{i \theta} \phi_1 + i e^{-i\theta} \phi_2) / \sqrt{2}$ ??
%Si oui, alors le terme de squeezing $\phi_k \pi_{-k} + hc$ contient
%des termes $\pi_1\phi_2$ pour \theta \neq 0
%peut on repasser a v et w ???

%As a final comment, we wish to draw the attention to the following
%point. When using the real oscillators $\phi_1$ and $\phi_2$
%the highly squeezed character show up in a certain direction
%whose orientation depends of arbitrary conventions. For instance
%a shift in the spatial origin introduces a relative phase
%shift of the operators $\hat a_{\bf k} $ and $\hat a_{-\bf k}$
%which enter in $\hat a_1$. ???????????
%Still requires clarification........




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{The covariance matrix}

%R30 a comment
%%l'origine du fait que ta matrice de cov
%n'est PAS la meme que Zoller-PRL est due au fait que
% le commut phi et de pi est canonical pour pi_-k PAS pi_k
% d'ou pour eux les deux premieres lignes-colones def les var canon conjug,
% pas chez toi. DO you agree ??
%D31 oui
%%%en fait etant donne l'app 6 sur la non-sepa
%%ton choix ici est en contradiction
% car par les matrices `locales-campo' tu DIAGONALISES C
%%je pense que nous devons permutter la col/ligne de pi_-k  et pi_k
%ceci dit C3,4 sont corrects
%%autre chose C5-7 n'a rien avoir avec C3-4 DO you agree ??
%D31 le seul lien est la valeur du determinant


%R31 il y a confusion des concepts, symboles , dou des resultats


 Let us consider a given pair of modes $({\bf k}, \, - {\bf k})$
 in a Gaussian state \Eqref{momenta}. We prove the
 relations (\ref{map}-\ref{tilde n}). This is best done with the
 covariance matrix of the state. We also define the sub- and
 super-fluctuant modes.%, which will also be useful in Appendix D.

 \subsection{Definition of the covariance matrix}

%R33
To conform ourselves to the usual definition of the
covariance matrix \cite{Simon00},
 we introduce canonically conjugated and
 {\it hermitian} 'position' and 'momentum'
 variables for each mode, i.e.
 $\hat a_{{\bf k}} = (\hat %R31 phi
   q_{{\bf k}} +
 i \hat p_{{\bf k}})/\sqrt{2 }$,
 %D31 with $\hat p_{{\bf k}} = \partial_{\eta} \hat q_{{\bf k}}$,
 %%R33 la SEULE raison que j'ai mis le k c'etait a coause de cela
 %%maintenant que tu l'enveles, plus besoin de k
 %%on pouvait la maintenir car on travaille avec des outmodes, mais soit.
 and similarly
 $\hat a_{-{\bf k}} = ( \hat q_{-{\bf k}} +
 i\hat p_{-{\bf k}})/\sqrt{2 }$.
 % with
 %$\hat \phi_{-{\bf k}} = \hat \phi_{{\bf k}}^{\dagger}$.
 All the %R33statistical
 information of the Gaussian state
 is encoded in the covariance matrix.
 In terms of the %column
 $4$ vector
 \ba
   {\bf \zeta} =
   \left(
     \begin{array}{c}
     %\hat \phi_{{\bf k}} \\
     %\hat \pi_{{\bf k}} \\
     %\hat \phi_{-{\bf k}} \\
     %\hat \pi_{-{\bf k}}
     \hat q_{{\bf k}} \\
     \hat p_{{\bf k}} \\
     \hat q_{-{\bf k}} \\
     \hat p_{-{\bf k}}
     \end{array}
   \right) \, ,
   %\nonumber
 \label{zeta}
 \ea
 the covariance matrix has
 %D31, by definition \cite{Glauber69},
 %\footnote{Given the ambiguity of ordering, this definition is a
 %convention. It is usually chosen because of its similitude with the
 %covariance matrix in probability theory (
 %characteristic functions  \cite{Glauber69}).
 %},
%R30 PLUS important ne faut il pas que la matrice simplect soit tjrs la meme,
%see les PRL ??
%D32 que veux tu dire ???
%R33 see eq. 3 PRL simon
%D34 je ne comprends pas ta question : je peux choisir les 4 operateurs
%que je veux, du moment qu'il satisfont les rel de commut can
%(soit q_k,... ou \phi_k,... ou encore a_k,... ) et ecrire
%des equations (cov matrix, Heis uncert, critere de separabilite) avec les
%expect values de ces 4 operateurs.
%je pense que la seule raison pour qu'ils utilisent (q_k,...) dans les PRL est le fait
%qu'ils veulent utiliser les prop du groupe simplectique.
 the matrix elements
 %D31
 $C_{ij}= %\frac{1}{2}
 {\rm Tr}\left(
 \hat \rho \left\{ \, \hat {\bf \zeta}_i , \,
 \hat {\bf \zeta}_j^{T} \right\} \right)$,
 where $\left\{ \, , \, \right\}$ is the
 Weyl ordered product, i.e.
 the anticommutator divided by 2.
 For the states with variances \Eqref{momenta}, one has
 \ba \label{covmatrix}
   C =
   \left(
     \begin{array}{c c c c}
      n + \frac{1}{2} & 0 &  c_r & c_i  \\
      0 & n + \frac{1}{2} & c_i & -  c_r  \\
      c_r & c_i & n + \frac{1}{2} & 0 \\
      c_i & - c_r  & 0 & n + \frac{1}{2}
     \end{array}
   \right) \, ,
 \ea
 %D31
 where $c_r = Re(c) = - \vert c \vert \cos 2\theta$,
 and $c_i = Im(c) = - \vert c \vert \sin 2\theta$,
 see Eq. \Eqref{epsilontheta}.
 %R31
 This covariance matrix is not bloc diagonal
 %D31. It expresses the fact that
 %we are dealing with two-modes squeezed states.
 since the modes ${\bf k}$ and $- {\bf k}$ are correlated.
 %D32 But each bloc on the diagonal is a diagonal matrix because of
 %homogeneity.
 As we shall see in Appendix D6, this will lead to
 the notion of non-separability.


 \subsection{The sub- and super-fluctuant modes}


 %R33
 It is appropriate to return to complex modes to express the
 %eigenvalues
 eigenmodes and the eigenvalues of the covariance matrix \Eqref{covmatrix}.
 They
  %are obtained from  Let us define the rotated (complex) modes:
  are given by the rotated  modes:
 \ba \label{rotatedvar}
   \Phi_{\bf k} = \frac{1}{\sqrt{2 }} (\hat a_{\bf k}e^{-i\theta} +
   \hat a_{-\bf k}^{\dagger}e^{+i\theta}) \, , \qquad
   \Pi_{\bf k} = -i \frac{1}{\sqrt{2}} (\hat a_{\bf k}e^{-i\theta} -
   \hat a_{-\bf k}^{\dagger}e^{+i\theta})
   \, .
 \ea
 %The eigenvalues correspond to the
 Their expectation values are %of these rotated variables:
 \ba \label{eigenvalues}
  \langle \{ \Phi_{\bf k} , \, \Phi^\dagger_{\bf k} \} \rangle =
   %\frac{1}{k}\left[
   \left( n+\frac{1}{2} \right) + \abs{c} \, , \quad
   \langle \{ \Pi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle =
   %k\left[
   \left( n+\frac{1}{2} \right) - \abs{c} \, ,  \quad
   \langle \{ \Phi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle = 0
  \, . \nonumber \\
 \ea
% each twice degenerate.
%R35
The first two r.h.s. expressions % expectation values
give the eigenvalues of \Eqref{covmatrix}.
They are twice degenerate because of homogeneity.
In the case of pair creation from the vacuum,
$n=\sh^2(r)$ and $\abs{c} = \ch r \, \sh r$,
they reduce respectively to $e^{2r}/2$ and $e^{-2r}/2$.
 %%R35
 They are therefore
 called the super-fluctuant and sub-fluctuant modes respectively.
 It is worth noticing
 that their product gives $1/4$ as in the vacuum. This is
 the expression of the purity of the state. %R36

 \subsection{Proof of Eq. \Eqref{map}}

%R36
An efficient way to prove
Eq. \Eqref{map} and
Eq. \Eqref{tilde n} consists in using
the %R35 fictitious
the real and the imaginary parts of $\phi_{\bf k}$,
%parthermitian modes $\phi_1$ and $\phi_2$ of
see Eq. \Eqref{artop}.
 %They
 %R34 that mix ${\bf k}$ and $-{\bf k}$ operators.
 The passage from $({\bf k},\, -{\bf k}) \mapsto (1,\, 2)$
 is performed by making
 %first a local squeezing
 %$S_{loc} = S_{\bf k} \otimes  S_{-{\bf k}}$,
 %where $S_{\pm{\bf k}}$ acts on the $\pm{\bf k}$ sector alone,
 %then
 a {\it global} rotation $R_{g   }$ of angle $\pi/4$.
 %both
 %of determinant 1.
 %R33
 By global we mean
 %a tensor product of $2\times 2$ transformations
 %i.e.
 a $4 \times 4$ transformation  mixing
 ${\bf k}$ and $-{\bf k}$ sectors. Explicitly, one has
 %rotation of angle $\pi/4$
 \ba \label{U}
   \left(
     \begin{array}{c}
    \sqrt{k} \, \hat \phi_{1}  \\
    \hat \pi_{1}/\sqrt{k} \\
    \hat \pi_{2}/\sqrt{k} \\
    - \sqrt{k} \,\hat \phi_{2}
     \end{array}
   \right) =
   %\left(
   %  \begin{array}{c}
   % \hat \phi_{1}  \\
   % \hat \pi_{1} \\
   % \hat \phi_{2} \\
   % \hat \pi_{2}
   %  \end{array}
   %\right) = S_{loc} U \zeta
 %R34 R_{ g  }     \, \zeta
    %\left(
    % \begin{array}{c c c c}
    %  1/\sqrt{k} & 0 & 0 & 0 \\
    %  0 & \sqrt{k} & 0 & 0 \\
    %  0 & 0 & 1/\sqrt{k} & 0 \\
    %  0 & 0 & 0 & \sqrt{k}
    % \end{array}
   %\right)
   \frac{1}{\sqrt{2}}
   \left(
     \begin{array}{c c c c}
      1 & 0 & 1 & 0 \\
      0 & 1 & 0 & 1 \\
      -1 & 0 & 1 & 0 \\
      0 & -1 & 0 & 1
     \end{array}
   \right)
   \left(
     \begin{array}{c}
     \hat q_{{\bf k}} \\
     \hat p_{{\bf k}} \\
     \hat q_{-{\bf k}} \\
     \hat p_{-{\bf k}}
     \end{array}
   \right) \, .
 \ea
 The covariance matrix becomes bloc diagonal:
 %D34in such a way that the covariance
 %matrix becomes bloc diagonal:
 \ba \label{covmatrixfictious}
   C \mapsto C_{(1,2)} = R_{g} C R_{g}^T =
   \left(
     \begin{array}{c c c c}
      n_{k} + \frac{1}{2} + c_r & c_i & 0 & 0 \\
      c_i & n_{k} + \frac{1}{2} - c_r & 0 & 0 \\
      0 & 0 & n_{k} + \frac{1}{2} - c_r & c_i \\
      0 & 0 & c_i & n_{k} + \frac{1}{2} + c_r
     \end{array}
   \right) \, .
 \ea
 %D31
 %where
 %Therefore %R33
 %D34 Any Gaussian state is a tensor product
 The state is a tensor product
 $\hat \rho = \hat \rho_1 \otimes \hat \rho_2$
 %Correct ?
 %D34 No, homog <=> bloc diag + identical blocs
 %And when the state is homogeneous,
 and the  matrices $\hat \rho_1$
 and $\hat \rho_2$ coincide. These properties
 %generalize , and
 follow from the homogeneity of the state.
%R35 This factorized form %factorisability of the state
 %must not however be confused with the notion of
 %separability, see Appendix D.6.
 %trop de renvoi.

% Each bloc of Eq. \Eqref{covmatrixfictious} is made diagonal
% by the application of
% $2 \times 2 $ (local)
% %squeezing $S \otimes  S$ and
% rotations $R(\theta) \otimes  R(\theta)$  where
% \ba \label{SR}
%   % S =
%   % \left(
%   %  \begin{array}{c c}
%   %  \sqrt{k} & 0 \\
%   %  0 &  1/\sqrt{k}
%   % \end{array}
%   % \right)
%   % \, , \qquad
%    R(\varphi) &=&
%    \left(
%     \begin{array}{c c}
%     \cos \varphi & \sin \varphi \\
%     -\sin \varphi & \cos \varphi
%     \end{array}
%    \right)  \, , \qquad
%    \varphi = \frac{1}{2} \arg(c) = \frac{\pi}{2} + \theta
%    \, .
% \ea
% The eigenvalues are given at \Eqref{eigenvalues}.
% This provides us with a picture of the noise in the state, see
% Fig 1.
% %D31 We have represented the 1-$\sigma$ contour
% %of the probability ${\cal P}_1(v_1) $ of Eq. \Eqref{proba1}.
% It is represented by an ellipse whose principal axis are defined by
% the eigenvectors of the covariance matrix \Eqref{covmatrix}:
% their orientation in the phase-space plane
% $(\phi_{1}, \pi_{1})$  is
% %D31defined by the argument of the coherence term,
% $\arg(c)/2 = \varphi$, and their lengths are
% %D31
% the value of the eigenvalues \Eqref{eigenvalues}.
% %equal to $e^{\pm 2r}/2$ respectively.


 %The entropy of the state \Eqref{momenta} can be easily
 %calculated from \Eqref{covmatrixfictious}.
 The transformation Eq. \Eqref{map} amounts to bring $C$ under the
 form
 %D32
 \ba
   C = M T  M^T \, , \qquad
   T =  \left( \bar n + \frac{1}{2}\right) {\bf 1} \, ,
 \ea
 %R36
 where the matrix $T$ is the covariance matrix of
 the product of the two thermal
 density matrices $\hat \rho_{th,\, 1} \otimes \hat \rho_{th,\, 2}$
 in Eq. \Eqref{map}. The matrix $M$ is the product of three %four
 special matrices %R35 that preserve the determinant:
 %D36 c'est le point important de la preuve
 \ba
   M = S_{loc}(\kappa) \, R_{loc}(\varphi) \, %S_{loc}(r) \,
   R_{g   } \, .
 \ea
 %D32
 First, the global rotation $R_{g   }$ of \Eqref{U}.
 Second, the %R36 application of a
 product of
 $2 \times 2 $, i.e. {\it local},
 %squeezing $S_{loc}(r) = S(r) \otimes  S(r)$ followed by
 %a local
 rotations, $R_{loc}(\varphi) = R(\varphi) \otimes  R(\varphi)$,  where
 \ba \label{SR}
    %S &=&
    %\left(
    % \begin{array}{c c}
    % e^{r/2} & 0 \\
    % 0 &  e^{-r/2}
    %\end{array}
    %\right)
    %\, , \qquad r = \ln k \, , \nonumber \\
    R(\varphi) &=&
    \left(
     \begin{array}{c c}
     \cos \varphi & \sin \varphi \\
     -\sin \varphi & \cos \varphi
     \end{array}
    \right)  \, , \qquad
    \varphi = \frac{1}{2} \arg(c) = \frac{\pi}{2} + \theta
    \, ,
 \ea
 brings $C_{(1,\,2)}$ into a diagonal form:
 ${\rm diag}(\lambda_+,\, \lambda_-,\, \lambda_-,\, \lambda_+)$
 with $\lambda_\pm = n+1/2 \pm \abs{c}$,
 see \Eqref{eigenvalues}.
 %One can verify that the
 %eigenvectors are
 Third two local squeezing $S(\kappa) \otimes S(-\kappa)$
 %similar to
 %\Eqref{SR}, with
 \ba \label{S}
    S(\kappa) =
    \left(
     \begin{array}{c c}
     e^{\kappa/2} & 0 \\
     0 &  e^{-\kappa/2}
    \end{array}
    \right)  \, , \qquad
    \th \kappa = - \abs{c}/(n + 1/2)
    \, ,
 \ea
 rescale the eigenvalues to a common value $\bar n$.
 %The eigenvalue $\bar n$ of the thermal matrices is easily
 %obtained
 The latter is fixed by the conservation of the determinant:
 \ba
   %{\rm det} \, C_{(1,2)} =
   \left(\bar n + \frac{1}{2}\right)^4 =
   {\rm det} \, C = \left( (n + \frac{1}{2})^2 - \abs{c}^2 \right)^2
   \, .
 \ea












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{General description of homogeneous Gaussian states}

 We review the statistical properties of two-mode (or
 bi-party) Gaussian states. They have received much attention
 because of their importance in the contexts of Quantum Optics and
 Quantum Information, see for instance \cite{Wodkiewiczlecture} for
 a comprehensive review.

 Since in cosmology one deals with statistically homogeneous
 and isotropic distributions,
 one needs to consider only a
 sub-class of Gaussian states. These
 symmetries considerably simplify
 the discussion since the
 density matrices are characterized by
 only three real parameters
 given in Eqs. \Eqref{momenta}.
 %D50 maybe it's not necessary: %R50 I agree
% This is easily seen from the requirement that the Green functions
% evaluated in homogeneous states depend on the difference of the
% space arguments only.
 It should be noticed that the squeezed
 states \Eqref{inoutvac}
 (including the case of zero squeezing, i.e. the vacuum)
 are the only pure states. They are characterized by only two
 independent parameters.
 This is because
 squeezed vacuum states minimize
 the Heisenberg uncertainty relations, see Eqs. \Eqref{positivity}
 and \Eqref{Heis}.
 The thermal state is
 the most decohered distribution and corresponds to $c=0$.
 As in the body of the text,
 we shall parameterize the level of decoherence by $\delta$
 defined by
 \ba \label{deltafine}
   \abs{c}^2 = (n+1)(n - \delta) \, ,
 \ea
and leave the phase of $c$ unchanged.
 %R40
 Nevertheless,
 for completeness, we consider in Appendix D.8 the effects of a
 smearing of the {phase} of $c$.



% \begin{figure}[ht] \label{etatsgaussienhomog}
% \epsfxsize=8.0truecm
% \epsfysize=10.0truecm
% \centerline{{\epsfbox{etatsgaussienhomog.eps}}}
% \caption{{\it Parameter space representations of the homogeneous
% two-mode Gaussien states.}
% Each point ($n,\zeta$) in the plane is a Gaussien state, where $n$ is
% the occupation number and $\zeta={\abs c}/(n+1)$ is a measure of
% the degree of
% coherence of the distribution. Thermal states form the abscisse
% $\zeta=0$, and vacuum squeezed states are the upper curve. The latter
% also delimit the ensemble of positive-definite density matrices,
% i.e. such that $n(n+1) - \abs{c}^2 \geq 0$.
% Both ensembles are of null measure. The vacuum
% is at the origin. States above (below) the lower curve are
% entangled (separable). The ratio of the
% area between the two curves by the total area goes to zero as
% $n \rightarrow \infty$.
% }
% \end{figure}




 \subsection{The $Q$-representation of a density matrix}


 To proceed, it is convenient to work with a representation of the
 density matrix $\hat \rho$ in phase-space. We choose the so-called
 $Q$-representation $Q_{\rho}$.
 We recall that it is well defined
 for every hermitian, positive definite operator (density matrix).
 %(contrary to the
 %diagonal $P$-representation).
 %, and because of it is intuitive and simple
 %(contrary to the Wigner function which needs some getting used to).
 For a two-mode state, $Q_{\rho}$ is the expectation value of
 $\hat \rho$ in a pair of coherent states
 \ba \label{defQrepr}
   Q_{\rho}(v,w) = \bra{v, \, {\bf k}}\bra{w, \, -{\bf k}} \hat \rho_{\bf k,\, -{\bf k}}
   \ket{w, \, -{\bf k}}\ket{v, \, {\bf k}} \, .
 \ea
 It is a function over phase space with the convention
 $v=(k \phi_{\bf k} + i \pi_{\bf k})/ \sqrt{2k}$, and
 $w=(k \phi_{-{\bf k}} + i \pi_{-{\bf k}})/ \sqrt{2k}$, see
 Appendix A.
 It is remarkable that although \Eqref{defQrepr} is an expectation value, the
 knowledge of
 the function $Q_{\rho}(v,w)$ is completely equivalent to that of
 $\hat \rho$. In particular the information about the off-diagonal matrix
 elements $\bra{v}\bra{w} \hat \rho \ket{w'}\ket{v'},\, v\neq v',\,
 w \neq w'$ is encoded in \Eqref{defQrepr} \cite{Glauber69,Nussenzveig}.
 This is due to the
 redundant (overcomplete) character of the basis of coherent
 states
 \footnote{Let us work with one-mode states for simplicity.
 Consider the expansion in the Fock representation of
 a (bounded) operator $\hat O = \sum\limits_{n,m=0}^{\infty}
   \ket{n} O_{nm} \bra{m}$. Using
 \Eqref{DefCoh2}, one sees that the function
 \ba
   {\cal O}(v^*, v') =
   \exp{\left[\frac{1}{2}\left(\abs{v}^2 + \abs{v'}^2 \right)
   \right]} \bra{v} \hat O \ket{v'} =
   \sum\limits_{n,m=0}^{\infty}
   O_{nm} \frac{(v^*)^n (v')^m}{\sqrt{n!m!}}
   \, , \nonumber
 \ea
 is an entire analytic function of both variables $(v^*, v')$.
 A theorem for functions of several complex variables states that if
 ${\cal O}(v^*, v')$ vanishes on the line $v' = v^*$, then it
 vanishes identically \cite{Nussenzveig}.
 Consider then two density matrices
 $\hat \rho_1$ and $\hat \rho_2$ with identical diagonal
 matrix elements in the basis of coherent states. Setting,
 $\hat O = \hat \rho_1 - \hat \rho_2$, and using the previous
 theorem, one concludes that $\hat \rho_1 = \hat \rho_2$.
 }, see Appendix A.


 In addition, for Gaussian states, the knowledge of the expectation values
 \Eqref{momenta} is all one needs to write the $Q$-representation
 of the density matrix. The general method of characteristic
 functions is given in every textbook of
 Quantum Optics, see for instance
 \cite{Glauber69,Nussenzveig,Wodkiewiczlecture}.
 The case of homogeneous distributions can be worked out by hand,
 and one finds
 \ba \label{Qrepr}
   Q_{\rho}(v,w) = \frac{1}{\Delta} \exp\left[-\frac{1}{\Delta}
   \left( (n+1)(\abs{v}^2 + \abs{w}^2 ) - 2 Re(c^*vw) \right) \right]
   \, ,
 \ea
 where
 \ba
   \Delta = (n+1)^2 - \abs{c}^2 = (n+1)(1 + \delta )\, .
 \ea
 One verifies that \Eqref{Qrepr} gives back  \Eqref{vwdistrib}
 when considering two-mode squeezed vacuum states, i.e. $\Delta = n+1$.

 A necessary and sufficient condition for \Eqref{Qrepr}
 to be the expectation value of a
 density matrix (a positive hermitian operator) is
 \ba \label{positivity}
   n(n+1) - \abs{c}^2 \geq 0 \, .
 \ea
 This is easily seen from the fact that $\hat \rho$ is unitarily
 equivalent to a tensor product of thermal states, see Eq.
 \Eqref{map} and Appendix C. From \Eqref{tilde n},
 the positivity condition $\bar n \geq 0$
 gives \Eqref{positivity}. It should be stressed that \Eqref{positivity}
 {\it is} the Heisenberg's uncertainty relations
 \footnote{We recall that the general form of the uncertainty
 relations is, for all hermitian operators $\hat A$ and $\hat B$,
 $\Delta \hat A^2 \Delta \hat B^2 \geq
 \abs{\langle \left[\hat A , \, \hat B \right]\rangle}^2/4$.
 To get \Eqref{Heis} use
 $\hat A = \hat \phi_{\bf k} \hat \phi_{\bf k}^{\dagger}$ and
 $\hat B = \hat \pi_{\bf k} \hat \pi_{\bf k}^{\dagger}$.
 },
 \ba \label{Heis}
   \langle \hat a_{{\bf k}} \hat a_{{\bf k}}^{\dagger} \rangle
   \langle \hat a_{-{\bf k}}^{\dagger} \hat a_{-{\bf k}} \rangle \geq
   \langle \hat a_{{\bf k}} \hat a_{-{\bf k}} \rangle
   \langle \hat a_{{\bf k}}^{\dagger} \hat a_{-{\bf k}}^{\dagger} \rangle
   \,  .
 \ea
 Notice that the lower bound is reached for the squeezed states
 \Eqref{inoutvac}.


 Incidentally, the positivity condition \Eqref{positivity}
 furnishes a clear way of distinguishing classical phase-space
 distributions (i.e. Probability Distributions Functions, the states
 being points in phase-phase),
 and phase-space {\it representations} of quantum states. Were
 $Q_{\rho}(v,w)$ be a PDF, the positivity criterion would have
 been $\Delta
 %(n+1)^2 - \abs{c}^2
 \geq 0$
 %, see Eq. \Eqref{},
 which is less restrictive
 %D21
% \footnote{The inquisitive reader might wonder about the validity
% of this statement given that the $Q$-representation is one
% phase-space representation amongst many. This is also true for
% every well-defined representation of the density matrix. More
% precisely, Glauber introduced the technique of $s$-ordering of
% the density matrix \cite{Glauber69}, where $s$ is a continuous parameter in
% $\left[ -1, \, +1 \right]$. In particular, $s=-1,\, 0,\,+1$
% corresponding to anti-normal, Weyl, and normal ordering. Then,
% one can associate to each ordering a given phase-space
% representation that serves to calculate the expectation values
% of operators with opposite ordering ($-s$), see footnote $15$.
%It can be defined as the convolution of $Q$-representation with a Gaussian
% \ba
%   W_s(v,w) &=& \frac{2}{1+s} \,
%   \int\!\!\frac{d^2u}{\pi}\frac{d^2u}{\pi}\, Q(v,w)
%   \exp\left[ - \frac{2}{1+s}\left( \abs{v}^2 + \abs{w}^2 \right)
%   \right]  \nonumber \\
%   &=& \frac{1}{\Delta_s} \exp\left\{ -\frac{1}{\Delta_s}
%   \left[\left( n+\frac{1-s}{2}\right)(\abs{v}^2 + \abs{w}^2 )
%   - 2 Re(c^*vw) \right] \right\}
%   \, . \nonumber
% \ea
% The values $s=-1,\, 0,\,+1$ correspond to
% the $Q$, Wigner, and $P$-representation respectively.
% A given function $W_s$ is well defined if, and only if,
% $\Delta_s = (n+(1-s)/2)^2 - \abs{c}^2 \geq 0$, or equivalently
% $\abs{c}^2 \leq n(n+1) + f(s)$, where
% $f(s) = [s^2 - 4s(n+1/2) + 1]/4 \geq 0$.
% The function $f$ is non-negative if, and only if,
% $-1 \leq s \leq s_{-}(\abs{c}) = 2 \left[(n+1/2) - \abs{c} \right]$.
% For squeezed vacuum states, the upper bound becomes arbitrarily
% small as the squeezing increases since
% $s_{-}(\abs{c_{sq}}) = e^{-2r}$.
% In conclusion, for all the well-defined phase-space distributions
% $W_{s < s_{-}}$, the function $f(s) > 0$, and therefore the condition
% $\Delta_s \geq 0$ is therefore less restrictive than the
% Heisenberg inequalities \Eqref{positivity}.
% }
 than \Eqref{positivity}.

 The off-diagonal matrix elements are \cite{Glauber69,Wodkiewiczlecture}
 \ba \label{offdiag}
   \bra{v}\bra{w} \hat \rho \ket{w'}\ket{v'} =
   &&\frac{1}{\Delta} \exp\left[-\frac{1}{2} \left(\abs{v}^2 + {\abs v'}^{2}
   + \abs{w}^2 + {\abs w'}^{2}\right) \right] \nonumber \\
   &&\times
   \exp\left[\left(1-\frac{n+1}{\Delta}\right)
   \left( v^* v' + w^* w'\right) + \frac{1}{\Delta} \left(
   c^* v'w' + c v^{*} w^{*} \right) \right] \, ,
 \ea
 and the density matrix itself
 is
 \ba \label{rho}
   \hat \rho &=& \frac{1}{\Delta} \exp\left[ \frac{c}{\Delta}
   \hat a_{\bf k}^{\dagger} \hat a_{-{\bf k}}^{\dagger} \right]
   \, : \exp\left[ - \frac{n+1}{\Delta}
   \left( \hat a_{\bf k}^{\dagger} \hat a_{\bf k} +
   \hat a_{-{\bf k}}^{\dagger} \hat a_{-{\bf k}} \right) \right]
   : \,  \exp\left[ \frac{c^*}{\Delta}
   \hat a_{\bf k} \hat a_{-{\bf k}} \right]
   \, , \nonumber
 \ea
 where ``$: \, : $`` stands for normal ordering.
This completes the proof that the knowledge of $c$ and $n$
allows to reconstruct the {\it quantum} distribution.


 \subsection{Factorisability of the two-point function, the
 classical statistical properties of the state, and two-mode
 coherent states}

 The function $Q_{\rho}$ can be re-written as the product
 %D21
% \footnote{Using the representation of the identity with coherent states
% \Eqref{coh1}, one immediately checks that these definitions
% follow from the postulates of Quantum Mechanics :
% \ba
%   Q_v(v) = {\rm Tr}\left[ \hat \rho \,
%   \ket{v,\,{\bf k}}\bra{v,\,{\bf k}} \otimes {\bf 1}_{-{\bf k}}
%   \right]
%   \, , \qquad
%   Q_{w|v}(w) = \frac{{\rm Tr} \left[ \hat \rho \,
%   \ket{v,\,{\bf k}}\bra{v,\,{\bf k}} \otimes
%   \ket{w,\,-{\bf k}}\bra{w,\,-{\bf k}} \right]}{
%   \int\!\!\frac{d^2 w}{\pi} \, {\rm Tr}
%   \left[ \hat \rho \,\ket{v,\,{\bf k}}\bra{v,\,{\bf k}}
%   \otimes \ket{w,\,-{\bf k}}\bra{w,\,-{\bf k}}\right]} \, . \nonumber
% \ea
% }
 of a
 marginal probability $Q_v(v) = \int\!\!\frac{d^2 w}{\pi} \, Q(v,w)$ and
 a conditional probability $Q_{w|v}(w)$
 \ba \label{Q}
   Q_{\rho}(v,w) &=& Q_{v}(v) \times Q_{w|v}(w) \, , \nonumber \\
   &=& \frac{1}{n+1} \exp\left[ - \frac{\abs{v}^2}{(n+1)} \right]
   \times   \frac{1}{1+ \delta} \exp\left[ -
   \frac{\abs{w - \bar w(v)}^2}{1+\delta}  \right]
   \, .
 \ea
 Both $Q_v$ and $Q_{w|v}$ are normalized.
 The quantity $\bar w(v) = cv^*/(n+1)$ is the
 conditional mean value of $w$ given $v$.
 One recovers Eq. \Eqref{meanw}
 for a squeezed vacuum state.
 The conditional width around this conditional mean value is
 \footnote{The means and variances can either be read from \Eqref{Q}, or calculated using
 $v$ and $w$ as random variable whose PDF is \Eqref{Q}. One immediately
 gets
 \ba
   &&\av{v} =
   \int\!\!\frac{d^2 v}{\pi}\frac{d^2 w}{\pi} \, v Q_{\rho}(v,w) =
   0 \, , \qquad
   \av{\abs{v}^2} =
   \int\!\!\frac{d^2 v}{\pi}\frac{d^2 w}{\pi} \, \abs{v}^2
   Q_{\rho}(v,w) =  n+1
   \, . \nonumber
 \ea
 Similar equations hold for the averages $\av{w} = 0$ and
 $\av{\abs{w}^2}=n+1$ since \Eqref{Qrepr} is symmetric in $v$ and $w$.
 The {\it conditional} means and variance of $w$ are calculated
 with the conditional PDF,
 \ba
   \langle w \rangle = \int\!\!\frac{d^2 w}{\pi} \, w
   Q_{w|v}(w) = \bar w(v) \, , \qquad
   \Delta_v w^2 =  \langle \abs{w}^2 \rangle - \abs{\bar w}^2 =
   \int\!\!\frac{d^2 w}{\pi} \,
   \left(\abs{w}^2 - \abs{\bar w}^2 \right) Q_{w|v}(w)  = 1+\delta   \, . \nonumber
 \ea
 Notice that this spread is independent of the value of $v$. This
 guarantees the temporal coherence in the mean
 $\Delta w^2 = \av{\abs{w}^2} - \av{\abs{\bar w}^2} = 1 + \delta$,
 see discussion below \Eqref{strongpeak}.
 For subtleties concerning operator ordering  using
 phase-space representations of the density matrix, we refer
 to \cite{Glauber69,Nussenzveig}.
 }
% \footnote{An interesting subtlety appears when
% treating $v$ and $w$ as classical stochastic variables and
% $Q_{\rho}$ as a PDF.
% Notice that the second moments $\av{\abs{v}^2}$ and
% $\av{\abs{w}^2}$
% do not give the expectation value of the
% occupation number operators
% $\hat n_k = \hat a_{\pm{\bf k}}^{\dagger}
% \hat a_{\pm{\bf k}}$, but rather their antinormally ordered versions.
% This discrepancy is indeed a resurgence of the ambiguity of ordering
% in Quantum Mechanics.
% Put differently, $\hat n_k$ does not commute with the projector
% $\ket{v}\bra{v} \otimes \ket{w}\bra{w}$, see \cite{CP04}
% for further details.
% Hence, one sees that the $Q$-representation, which is naturally
% associated with the normally ordered expression of $\hat \rho$
% as seen on \Eqref{rho},
% is meant to calculate expectation values of
% anti-normally ordered operators \cite{Glauber69}.
% Of course, since this discrepancy rests on commutators, it becomes
% irrelevant for high occupation numbers.
% }
% \footnote{These values can either be read from \Eqref{Q}, or calculated using
% $w$ as a random variable whose PDF is $Q_{\rho}$. One immediately
% gets
% %D19 j'ai fait une grossiere erreur, puisque $\bar w$ contient un $v$ !
% \ba
%   &&\av{w} =
%   \int\!\!\frac{d^2 v}{\pi}\frac{d^2 w}{\pi} \, w Q_{\rho}(v,w) =
%   0 \, , \qquad
%   \av{\abs{w}^2} =
%   \int\!\!\frac{d^2 v}{\pi}\frac{d^2 w}{\pi} \, \abs{w}^2
%   Q_{\rho}(v,w) =
%   %D19
%   n+1 \, , \qquad \nonumber\\
%   &&\av{\abs{w}^2} - \av{\abs{\bar w}^2} = {1+ \delta}
%   %R19 \frac{\Delta}{n+1}
%   \, . \nonumber
% \ea
% Similar equations hold for the averages $\av{v} = 0$ and
% $\av{\abs{v}^2}=n+1$.
% %D19 evidemment puisque k et -k sont symetriques, see D2.
% An interesting subtlety appears when
% treating $v$ and $w$ as classical stochastic variables and
% $Q_{\rho}$ as a PDF.
% Notice that the second moments $\av{\abs{v}^2}$ and
% $\av{\abs{w}^2}$
% do not give the expectation value of the
% occupation number operators
% $\hat n_k = \hat a_{\pm{\bf k}}^{\dagger}
% \hat a_{\pm{\bf k}}$, but rather their antinormally ordered versions.
% This discrepancy is indeed a resurgence of the ambiguity of ordering
% in Quantum Mechanics.
% Put differently, $\hat n_k$ does not commute with the projector
% $\ket{v}\bra{v} \otimes \ket{w}\bra{w}$, see \cite{CP04}
% for further details.
% Hence, one sees that the $Q$-representation, which is naturally
% associated with the normally ordered expression of $\hat \rho$
% as seen on \Eqref{rho},
% is meant to calculate expectation values of
% anti-normally ordered operators \cite{Glauber69}.
% Of course, since this discrepancy rests on commutators, it becomes
% irrelevant for high occupation numbers.}
 \ba
   \Delta_v w^2 = \langle \abs{w}^2 \rangle - \abs{\bar w(v)}^2
   = 1+\delta \, .
 \ea
 In the case of two-mode squeezed states ($\delta = 0$),
 this is exactly $1$. (Thus the second term of the exponent is the overlap
 of coherent states
 $\abs{\langle w \vert \bar w \rangle}^2 = e^{-\abs{w-\bar w}^2}$,
 and one recovers again the pure state
 Eqs. (\ref{insuperpcoh}, \ref{vwdistrib})).
 For the
 %D19 elle n'est pas minimally puisque delta = 2 > 1
 %minimally
 %R20 QUI decide si c'est 1 ou 2 qui est critique ???
 %D20 personne, c'est la physique
 %% c'est a nous de chosir le critere et d'exposer nos raisons pour ce choix
 % il me semble mainetenat que 1 est devenu `tabou'
 %D20 ABSOLUMENT PAS !!!!!!
 %R19 so why do you want to stick to 1 ?????
 %D20 Minimal a toujours ete pour moi delta=1 parce que
 %les prop quantique disparaissent. Il n'y a AUCUNE RAISON de prefere delta = 2 qui est un
 %cas particuler. Delta =2 est plus decohere que delta=1.
 %La valeur critique est 1 parce que c'est la reponse a une question physique.
 %Au contraire, c'est rho_{min}
 %qui est obsolete. Son seul interet est que c'est un resultat intuitif, que tout le monde peut comprendre
 %sans se farcir les sophistications de l'app D. C'est, et ca doit rester, un cas particulier,
 %une illustration d'un resultat beaucoup plus general et rigoureux.
 decohered  distribution \Eqref{rhodec}
 with parameter $\delta = 2$,
 one has $\Delta w^2 = 3$. This value should be compared with the
 corresponding width of the centered distribution in $v$,
 that is $\Delta v^2 = \av{\abs{v}^2} = n+1$.
 The ratio of the spreads is
 \ba
   \frac{\Delta w^2}{\Delta v^2} =
   \frac{1+\delta}{n+1}
   %%%R19   \frac{n\delta}{n+1}
   \, .
   %\, , \qquad  \abs{c}^2 = n(n+1 - \delta)
   %(n+1)(1 - \delta)
 \ea
 Therefore we obtain
 \ba \label{strongpeak}
   \frac{\Delta_v w^2}{\Delta v^2} \ll 1 \qquad
   \Longleftrightarrow \qquad \delta \ll n
   \, .
 \ea
 First, when it is verified, the conditional probability
 to find $w$ given $v$
 is sharply
 peaked around the conditional value $\bar w$. Therefore the modes
 ${\bf k}$ and $-{\bf k}$ are
 strongly correlated in phase and amplitude
 as discussed below \Eqref{vwdistrib}.
 Consequently, the power spectrum and the temporal coherence, see
 Eq. \Eqref{power}, are preserved whenever decoherence processes
 induce $\delta \ll n$.
 Put differently, whenever \Eqref{strongpeak} applies,
 these
 statistical properties of the distribution are as well
 described by statistical mixtures of entangled
 two-mode coherent states
 $\ket{v} \ket{\bar w}$, as in Eq. \Eqref{rhodec2}.

 Second, \Eqref{strongpeak}
 {\it is} the condition of factorizability of the two-point
 function Eq. \Eqref{neqqq}.
 What we shall learn is that this is a rather mild
 constraint which includes a wide range of density matrices.
 Indeed as we shall see below,
 this condition is satisfied
 by density matrices
 which have completely lost their quantum properties.





 \subsection{Macroscopic correlations}


 As discussed after Eq. \Eqref{vwrhov'w'},
 the pure squeezed vacuum state cannot be
 interpreted as a classical distribution because
 macroscopically separated
 ($n \geq \abs{v-v'}^2 \gg 1$)  semi-classical
 configurations are correlated.
 We now show that this property is very sensitive to the value of $\delta$.

 The width of these correlations is
 governed by
 the coefficient of the cross terms $v^*v',\, w^*w'$
 in Eq. \Eqref{offdiag}, i.e.
 $1 - (n+1)/\Delta  = %R19 [n(n+1) - \abs{c}^2]/\Delta \geq 0$,
 \delta/(\delta + 1)$.
 It belongs to the interval
 $\left[ 0 , \, n/(n+1) \right]$.
 The lower and upper bounds are reached for
 squeezed vacuum states and thermal states respectively.
 To better understand its physical meaning, it is
 instructive to pose and examine these two limiting cases. The
 former is presented in Section III.B.
 For a one-mode
 thermal state, one has
 \ba
   \abs{\bra{v} \hat \rho \ket{v'}} = \frac{1}{n+1} \exp\left[
   -\frac{1}{4}\left( 1+\frac{n}{n+1}\right)\abs{v-v'}^2 -
   \frac{1}{4(n+1)} \abs{v+v'}^2  \right]\, . \nonumber
 \ea
 The first term in the exponent is bounded by
 %D40 one
 $1/2$, which means
 that the two different semi-classical configurations are
 uncorrelated. The second term is the power spectrum.

%R19
%NB ce que je voulais dire dans mon dernier mail
%c'est que l'on melange des OH complexes et des reels
%d'ou il y a des facteurs de 2 qui different.
%ATTENZIONE
%D19 non, puisqu'on divise par \sqrt{2}
%R20 je ne pretend pas que les eqs sont fausses
% mais qu'il y a un arbitraire dans le choix des spreads = 1 car nous avons
%2 OH reels.
%D20 Il me semble que non puisque les incertitudes de Heisenberg sont
%1/4 comme pour 1 OH reel. De plus, le spread pour un thermal state est $\sigma_v=1$.

%R22 je re-essaye
%en mettant w=bar w, en faisant varier v, on traine DEUX OH, d'ac ?
%dou pour moi il n'et pas evid que en traitant separemnt 1 et 2
%on obtienne le meme critere.
%je te pose donc une question
%prend le vide. calcule pour cet etat D12 pour un OH
%que vaut sigma^2 ?
%maintenant D12 dans le texte est un produit, dou sigma^2 esyt DIVISE par 2
%d'ou qu'obtient on ?????????????????????????


 We are now ready to interpret the results for \Eqref{offdiag}.
 To simplify the discussion,
 we calculate the width of the off-diagonal matrix elements
 \Eqref{offdiag} when
 $w=\bar w = cv^*/(n+1)$ and $w'=\bar w'=cv'^{*}/(n+1)$,
 i.e. for the most probable value of the conditional
 amplitudes of the $-{\bf k}$ mode.
 Then one has
 \ba \label{D12e}
   &&\abs{\bra{v}\bra{\bar w} \hat \rho \ket{v'}\ket{\bar w'}} =
   \frac{1}{\Delta} \exp\left[ -\frac{\abs{v-v'}^2}{2\sigma_{v}^2(\delta)} -
   \frac{\abs{v+v'}^2}{4(n+1)}  \right] \, , \nonumber \\
   && \frac{1}{\sigma_{v}^2(\delta)} = \frac{1}{2(n+1)} +
   \frac{\delta}{1+\delta}\left[ 2 -\frac{1+\delta}{n+1} \right]
  \, .
 \ea
 The width $\sigma_{v}$ on the line $v-v'$ is written as
 a function of $\delta$.
 Its inverse is plotted on Fig. $2$
 for a value of $n=100$. For large occupation numbers, $\sigma^2_{v}(\delta)$
 drops from its maximum value $2(n+1)$ (for the two-mode squeezed state)
 to $1$ over a range of
 $\delta = 1 + O(1/n)$.
 In conclusion,
 \ba
   {\rm{uncorrelated \,\,\, semiclassical \,\,\, states}}
   \quad \Longleftrightarrow \quad \sigma_{v}
   \leq 1   \quad \Longleftrightarrow \quad
   \delta \geq 1  \,  .
 \ea


 \begin{figure}[h] \label{macrospread}
 \vbox{ \hbox to\linewidth{\hss
    \resizebox{7.5cm}{6.5cm}{\includegraphics{macrospreadzeta.eps}}
 \hspace{5mm}
        \resizebox{7.5cm}{6.5cm}{\includegraphics{macrospreadNL.eps}}}
\hss}
 \caption{{\it The inverse width $\sigma_{v}^{-2}$ of the off-diagonal
 matrix elements as a function of the degree of coherence of
 the distribution}
 for $n=10^1$ (lower, blue),
 $n=10^2$ (middle, red), and $n=10^3$ (upper, black).
 Left panel, as a function of
  $\zeta = \delta/n$,
 one clearly sees the extreme sensitivity of $\sigma_{v}$
 with respect to $\delta$ for small values.
 On the right panel, the same spread as
 function of $x=\arctan(\ln(\delta))$.
 This further shows that, %R28
 for {\it all} values of $n \gg 1$, the critical regime is
 centered around $\delta = 1$,
  i.e. the `grain' of
  coherent states.}
\end{figure}

% \begin{figure}[ht] \label{macrospread}
% \epsfxsize=10.0truecm
% \epsfysize=10.0truecm
% \centerline{{\epsfbox{macrospread.eps}}}
% \caption{{\it The width $\sigma_{v}$ of the off-diagonal matrix elements
% as a function of
% the degree of coherence $r$ of the distribution for $n=100$.}
% }
% \end{figure}



 \subsection{Sub-fluctuant mode and the power of the decaying mode}




 From the second equation in \Eqref{eigenvalues},
 we see that the sub-fluctuant mode behaves as
 \ba  \label{RRR}
   \langle \{ \Pi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle
   =  \left( \frac{\delta}{2} + \frac{(1+\delta)^2}{8n} + O(n^{-2})
   \right)
   \, .
 \ea
 Therefore one looses the squeezed spread,
 i.e. $\langle \{ \Pi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle \geq 1/2$,
  when $\delta \geq 1$. In conclusion, one has
 \ba
   {\rm{no \,\,\, subfluctuant \,\,\, variable}} \quad
   \Longleftrightarrow \quad \delta \geq 1  \, .
 \ea

 It is interesting to notice that the
 %D40 residual
 value
 of the quantum coherence $\sigma_v$ and that of the squeezed spread
 are correlated.
 Indeed, their product obeys
 \ba \label{interprod}
   \sigma_v^2 \,
    \langle \{ \Pi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle
   = \frac{1+\delta}{4}\left(1 + O(\frac{1}{n}, \frac{\delta}{n})\right) \, ,
 \ea
 for  $0 \leq \delta \ll n$. (This result is non-trivial for $\delta \leq 1/n$.)
 In Fig. 3, we have represented the plot of this product for the whole range of
 $\delta$.
 It           is growing with $\delta$, the
 lower bound $1/4$ being reached for pure (squeezed) states.
 More importantly it           stays
 almost constant
 for $\delta \leq 1$ before growing linearly with $\delta$ for $\delta \gg 1$.

 The first regime corresponds to the loss of the quantum coherence
 whereas the second corresponds to the linear growth of the power
 of the subfluctuant mode
 (since $ \sigma_v^2 $ stays $\simeq  1$).
At this point it is important to notice that this second regime
also corresponds to a linear growth of the power
 of the decaying mode. Indeed, the powers of
 the subfluctuant mode and the decaying mode are related by
 \ba  \label{RRR2}
   \langle \{ d_{\bf k} , \, d^\dagger_{{\bf k}} \} \rangle
   &=&\langle \{ \Pi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle
  + \vert c \vert (1 - \cos 2 \theta)
   \nonumber \\
   &=&  \langle \{ \Pi_{\bf k} , \, \Pi^\dagger_{\bf k} \} \rangle
  + O(n^{-1/2})
 = \frac{\delta}{2} + O(n^{-1/2})
   \, ,
 \ea
 where $\theta$
 %R29 and $\epsilon$ are
 is defined in Eq.
 \Eqref{epsilontheta}.
 In the second line, we have used the inflationary value of
 $\theta$ ($\propto n^{-3/4}$)
 and the fact that $\theta$ is unchanged when increasing $\delta$.

 %R25 cette fct est tres interesss il y a 3 regimes, d < 1/n, 1/n < d < 1, d >1
%R26 au point ou on en est autnat achever le travail (cf Prevert)
Similarly, the cross-correlation is
 \ba  \label{RRR3}
   \langle \{ d_{\bf k} , \, g^\dagger_{{\bf k}} \} \rangle
    = - %R30 2
    \sin(2\theta) \vert c \vert \,
    \propto n^{1/4} \left( 1 + \frac{1-\delta}{2n} + O(n^{-2}) \right)
   \, .
 \ea
 One sees that the cross term is unaffected as long as
 $\delta \ll n$.


 \begin{figure}[ht]
 \epsfxsize=10.0truecm
 \epsfysize=10.0truecm
 \centerline{{\epsfbox{prod.eps}}}
 \caption{{\it The product of the width $\sigma_v^2$ with the
 power  of the subfluctuant mode},
 for $n=10^1$ (upper, blue),
 $n=10^2$ (middle, red), and $n=10^3$ (lower, black).
 For the horizontal axis, we used the non-linear scale
 $x=\arctan(\ln(\delta))$ of Fig 2. For
 the vertical axis, we used the non-linear scale
 $y=\arctan(\sigma_v^2 \langle \{ \Pi_{\bf k} , \, \Pi_{-\bf k} \}
 \rangle)$. %R40 , which is approximately linear for $\delta \leq 1$.
 One clearly sees that %R28
 for all $n \gg 1$, the product of Eq. \Eqref{interprod}
 stays between $1/4$ and $1/2$ as long
 as $\delta \leq 1$.
 %R40 Moreover, the shape of
 One also verifies that it is asymptotically $n$-independent
 for all values of $\delta \ll n$.
 % sees that this curves
 %becomes universal in the large $n$ limit.
 Finally we recall that the power of the decaying mode
 coincides, to leading order, with
 $\langle \{ \Pi_{\bf k} , \, \Pi_{-\bf k} \}
 \rangle $, see Eq. \Eqref{RRR2}.}
 \end{figure}





 \subsection{$P$-representability of partially
 decohered density matrices}

  We now discuss the
 notion of $P$-representability of a density matrix \cite{Glauber63b}.
 It is an intrinsic property of states which enlightens the
 distinct behaviours of quantum and classical distributions.

 A one-mode density matrix
 is said to be $P$-representable if it can be written as a
 mixture of coherent states
 \ba \label{defPrepr}
   \hat \rho_{P-\rm{repr}} = \int\!\!\frac{d^2v}{\pi}
   \, P(v) \, \ket{v,\, {\bf k}}\bra{v,\, {\bf k}} \, ,
 \ea
 where the weight function $P(v)$ is a
 tempered distribution \cite{Cahill65,Wodkiewiczlecture}.
 %Gaussian function \cite{Cahill65,Wodkiewiczlecture}.
 %tempered distribution .
 Therefore, when considering $P$-representable {\it Gaussian density matrices},
 the weight function $P(v)$ is simply a Gaussian function.
 The generalization of this definition
 to two-mode (Gaussian) distributions is straightforward:
 \ba \label{twomodeP}
   \hat \rho_{P-\rm{repr}} = \int\!\!\frac{d^2v}{\pi}
   \frac{d^2w}{\pi}
   \, P(v,w) \, \ket{v,\, {\bf k}}\bra{v,\, {\bf k}}
   \otimes
   \ket{w,\, -{\bf k}}\bra{w,\, -{\bf k}}
   \, .
 \ea
 The $P$-representation of the distribution
 characterized by Eqs. \Eqref{momenta} can easily be calculated.
 From \Eqref{twomodeP}, one calculates the expectation values
 ${\rm Tr}(\hat n_k \hat \rho_{P-\rm{repr}}) = n$ and
 ${\rm Tr}(\hat a_{\bf k} \hat a_{-{\bf k}} \hat \rho_{P-\rm{repr}}) =
 c$, and one gets
 \ba \label{Puvp}
   P(v,w) = \frac{1}{\Delta'} \exp\left[-\frac{\abs{v}^2}{n}
   - \frac{n}{\Delta'} \abs{w - (cv^*/n)}^2 \right] \, ,
 \ea
 where $\Delta' = n^2 - \abs{c}^2$ must be positive.
 The function $P$ is well defined if,
 and only if, $n \geq \abs{c}$,
 i.e. $\delta \geq n/(n+1)$.
 Hence, in the high occupation number limit,
 \ba \label{Prepr}
   {\rm{P-representability}}
   \quad \Longleftrightarrow
   \quad \delta \geq 1 \, .
 \ea
 This condition
 coincides with that for the
 non-existence of a sub-fluctuant variable,
 see Eq. \Eqref{RRR}, as well as that for the vanishing of
 the off-diagonal matrix elements, see Eq. \Eqref{D12e}.
 It also coincides with the condition obtained from the criterion
 of separability we shall discuss in the next subsection,
 see Eq. \Eqref{separability}.
 %D22 et c'est la ou ton delta est tout pourrit parce que celui
 %qui essaie de faire le calcul ne trouvera jamain un Dirac !!!!
 %R30 One thus sees
 This proves that {\it all} quantum properties are lost %R40 for
 if, and only if, $\delta \geq 1$.

 It is therefore of interest to further analyze this case
 and to determine how it is related  to two-mode coherent states.
 Consider the limiting case $\Delta' \to 0^+$,
 i.e. $\delta \to n/(n+1) $ from above.
 In this limit $P(v,w)$ of Eq. \Eqref{Puvp} becomes
 \ba
 \label{lim}
   P_{lim}(v,w) =
   \frac{1}{n} \exp\left[-\frac{\abs{v}^2}{n} \right] \,
   \delta^{(2)}\left(w - \frac{c}{n}v^* \right) \, .
 \ea
The presence of the two Dirac functions constraining
the real and imaginary parts of $w$
clearly expresses the critical character of this distribution.

One can now notice that this distribution corresponds to
the simplified distribution of Eq. \Eqref{rhodec2}.
%D50 On l'a dit 5 lignes au dessus, je le supprimerais
%R50 je ne vois rien 5 lignes au dessus d'ou je propose de maintenir
 %In other words
 %\Eqref{rhodec2} is {\it the} quantum distribution above which all
 %quantum properties are lost.
 %D50 je ne sais plus ce qui m'est passer par la tete,
 %mais la logique ne me semble pas correcte
 %
 %Since $\abs{c} = n$, the
 %Dirac functions force the decaying mode (the cosine)
 %to vanish. C'est faux !
 % Hence, the distribution \Eqref{lim}
 %is the quantum conterpart of the classical effective
 %classical distribution Eqs. (\ref{sineeq}-\ref{Peff})
 %obtained by setting the decaying mode to zero.
 %This explains the equality of the entropies.
 %, see Appendix D.7.
 More precisely, the equality of the entropies, see Appendix D.7,
 suggest that \Eqref{lim} is the quantum counterpart of the
 classical effective classical distribution
 Eqs. (\ref{sineeq}-\ref{Peff})
 obtained by setting the decaying mode to zero.
 The (badly named) {\it minimally decohered} distribution
 \Eqref{rhodec} is in fact slightly more decohered as can be
 understood from the replacement of the Dirac's
 $\delta^{(2)}\left(w - \frac{c}{n}v^* \right)$ by a Gaussian with
 a width equal to 1. This subtle difference is better appreciated
 when one realizes that the simplified distribution of Eq.
 \Eqref{rhodec2} and that of \Eqref{rhodec},  which are both
 diagonal in two-mode coherent states, belong to the {\it same
 class} of decohered matrices when considering the entropy.
 %, see Appendix D7.




%R30 ce qui precede est inambigu, correct et comporensible et donne delta = 1.
%cela n'interdit rien en ce qui concerne la sep. mais
%c'est AUTRE chose



 \subsection{Separability of two-mode homogeneous distributions
 and violations of Bell's inequalities}


 The non-separability
 is another important criterion for distinguishing
distributions which cannot be viewed as classical stochastic distributions.
Its physical relevance comes from the fact that the non-separability
is a necessary property to have expectation values violating
Bell's inequalities\cite{Werner89}.
At the end of this subsection, we have included
a brief discussion concerning the relationship
between decoherence and the associated loss of
the violation of Bell's inequalities.
For more details we refer to \cite{CP4}.

 A word of caution is in order.  Unlike the
 $P$-representability
 %which is independent of the choice of the $4$ canonical variables in each two-mode sector,
 %the criterion of
 the non-separability rests on a division of the
 $4$ canonical variables into two subsets of $2$ canonical variables.
 %D37 In the sequel
 To define the two subsets, we use creation and destruction operators
 carrying a momentum ${\bf k}$ or ${\bf -k}$. The criterion should
 therefore be called the  $({\bf k}, {\bf -k})$-separability.
 The justification of the choice of the two subsets
 is presented after having explained the concept of separability.
%D37 In order to explicitize the choice of the two subsets,
% one should call it %D37 this concept
% the  $({\bf k}, {\bf -k})$-separability.
%D37 n'est il pas trop tot pour ca
% We shall see below that it %D37 the  $({\bf k},{\bf -k})$-separability
% defines an equivalent class of density matrices
% %D37
% through {\it local} transformations only, i.e.
% %because  only {\it local} transformations, i.e.
% transformations which do not mix ${\bf k}$ and ${\bf -k}$
% operators.
% %will be considered.
% %D37
% Hence, the $({\bf k}, {\bf -k})$-separability cannot be confused
% with the $(\phi_1 , \phi_2)$-separability (which is a
% characterization of the homogeneity of the state), since
% the transformation from the operators with a definite
% momentum ${\bf k}$ to $\phi_1$ and $\phi_2$ is global, see
% \Eqref{U}.
% %and as such it brings the distribution outside the equivalent class.see Appendix C.


 Following Werner \cite{Werner89},
 a distribution of two-modes
 $\bf k$ and $-{\bf k}$ is said to be
 separable if, and only if,
 it can be written as a convex sum of
 products of distributions, i.e.
 \ba \label{Defseparability}
   \hat \rho_{sep} = \sum_{l} p_l \, \hat \rho_{\bf k}^{(l)} \otimes
   \hat \rho_{-{\bf k}}^{(l)} \, , \qquad p_l \geq 0 \, , \qquad
   \sum_l p_l = 1 \, .
 \ea
 Notice that the definition \Eqref{Defseparability}
 is not the statement that a separable
 state is
 a product of
 %D31
 its reduced density matrices
 %it cannot be anything else, so autant le dire
 (in which case one would obtain uncorrelated sub-systems:
 $p_l = 0$ for all $l$ but one).

 The key point is that
  all states of the form
 \Eqref{Defseparability} can be obtained by
 the following classical protocol: when a random generator
 produces the number $l$ with probability $p_l$,
 two distant observers performing separate
 measurements on the subsystems
 $\bf k$ and $-{\bf k}$ respectively, prepare them
 into the states $\hat \rho_{\pm {\bf k}}^{(l)}$.
 Separable states are said classically correlated because, by construction,
 their statistical properties can be interpreted
 classically. In particular they cannot violate Bell's inequalities.
 States which are not separable are called ``entangled''
 (such as a two-mode squeezed vacuum states, or a singlet
 of $1/2$ spins) and can only be produced by letting the two
 parts of the system interact.

%R20 je ne comprend pas le lien avce la suite
%D20 la phrase du dessous est l'illustration de la phrase du dessus, par contraposee.
%Mais ca devient redondant avec ta nouvelle phrase.
%R20 je ne comprend pas non plus le lien entre v,w et 1,2
% car pour les OH 1 et 2, la il y a FCATORISATION de leur matrix density.
%D20 Tu fais bien d'en parler c'est un peu subtile. J'avais aussi eu un doute.
%Ma reponse est que
%1 et 2 sont des oscillateurs fictifs, qui couplent chacun k et -k,
%comme on la ecrit dans App B. Le passage (k,-k) -> (1,2) N'EST PAS un changement de
%base : c'est le point important a mon avis car un entangled state
%l'est dans n'importe qu'elle base, c'est une propriete intrinseque.
%Le fait qu'on factorise rho en l'ecrivant sous la forme 1 \otimes 2 ne signifie
%donc en rien que la matrice est separable.
%Suis je clair ?
%R21 absulment jet kje l'avais parfmt compris
%mais est ce qcla emerge du texte est la question ?
%D21 non, ca ne transparait pas dans le texte. ceci etant, 1) le concepte de
%matrice densite reduite n'est pas ambigue, donc le paragraphe ci-dessus non plus.
%2) (k,-k) est le systeme etudie : on se pose la question de
%l'entanglement de k et -k, pas de 1 et 2.
%
%R21 acr on parle de k -k entagn alors que c'est a l'`interieur`  de 1 et 2
%que cela se passe

% The modes ${\bf k}$ and $-{\bf k}$ are statistically
% correlated (i.e. expectation values such as
% $\langle \hat a_{\bf k} \hat a_{-{\bf k}} \rangle$ do not factorize).
%%is it relevant ??
%%D20 of course,
%R21 mais pas pour sepaarble !!! ne melangeons pas tout
%%otherwise the state is thermal. C'est juste un rappel
% This is important since the reduced density matrix of \Eqref{Q} is
% \ba
%   Q_{{\bf k}}^{red}(v) = \int\!\!\frac{d^2 w}{\pi} \, Q_{\rho}(v,w)
%   = \frac{1}{n+1} \exp\left(-\frac{\abs{v}^2}{n+1} \right)
%   \, , \nonumber
% \ea
% whatever the value of $\abs{c}$ may be.
% One checks that this is the
% $Q$-representation of a
% %R20 chaotic field
% thermal state with mean occupation number $n$, i.e. this is \Eqref{naive}
% Second, \Eqref{Defseparability} is
% different from the condition of
% factorizability of the two-point function. The latter requires only
% Eq. \Eqref{neqqq},
% which is satisfied for
% %in particular by
% two-mode squeezed states,
% which are not separable, as explained below.
%R22 je propose de ne pas soulever des points `negatifs' ceci n';est pas cela...

 Although it may be very difficult to decipher the separability of
 a general two-mode state,
 there is a criterion for Gaussian
 states. We briefly outline the proof of this theorem,
 the complete proof can be found in \cite{Simon00}.
 First, one notices on \Eqref{Defseparability} that
 the
 separability is a property
 that is not affected by
 %D31 so-called
 %car que l'on parte de ma matrice de cov ou de celle des PRL,
 %les modes k et -k ne sont jamais melanges, so its local in k
 {\it local} unitary transformations
 $U_{loc} = U_{{\bf k}} \otimes \tilde U_{-{\bf k}}$.
 %r30 j'ai mis un tilde car rien n'impose que ce soit la meme transf, right ?
 %D31 oui, mais a mon avis ce n'est pas ambigue car les indices sont diff
 These local
 transformations are of two types, %R30 local
 rotations and squeezing %R30  \Eqref{R}
 which do not mix $\bf k$ and $\bf -k$.
 %, and and local squeezing \Eqref{S}).
 Therefore,
% i.e.
 if $\hat \rho$ is a separable state, then all states
 $\hat \rho\, '= U_{loc} \, \hat \rho \, U_{loc}$
  are also separable.
 Second, one notices that if $\hat \rho$ is a $P$-representable state
 \Eqref{twomodeP}, it is separable. The non trivial part of the
 proof consists then to show that {\it all} Gaussian separable
 states can be constructed by applying local unitary transformations
 on a $P$-representable state.
 In brief, the criterion for separability for two-mode
 Gaussian states is
 {\it a Gaussian state $\hat \rho$ is separable if, and only if
 there is a local unitary
 transformation $U_{loc}$ which brings it to a $P$-representable
 state}.

 These local transformations are very constrained for  homogeneous
 states because any local squeezing would break the homogeneity of the state
 (one would have either $\langle \hat a_{\bf k}^2 \rangle \neq 0$
 or $\langle \hat a_{\bf  -k}^2 \rangle \neq 0$,
 or both).
 Hence, $U_{loc}$ can only be a product of local rotations.
 These preserve
 the non-negativity of the Gaussian function $P(v,w)$.
 As a result, separability and
 $P$-representability are equivalent properties of
 homogeneous Gaussian states.
 %D38
 Given \Eqref{Prepr}, one concludes %In conclusion,
  \ba \label{separability}
   ({\bf k}, -{\bf k})-{\rm{{separability
   \,\,\, of \,\,\, homogeneous\,\,\, states}}}
   \quad \Longleftrightarrow
   \quad \delta \geq 1 \, .
 \ea

 Notice that the $({\bf k}, {\bf -k})$-separability and
 the $(\phi_1 , \phi_2)$-separability
 are two independent properties of the state.
 The former %R40
 implies the above inequality on the decoherence level.
 %$2^{nd}$ momenta of the
 %distribution which essentially prevent the non-existence
 %of sub-fluctuant modes.
 As shown in Appendix C, the latter is simply a consequence of
 homogeneity of the distribution.
 The independence of these two concepts can be understood by
 noticing that the separability
 defines an equivalent class of density matrices
 through {\it local} transformations only.
 In fact,
 the transformation from the operators with a definite
 momentum ${\bf k}$ to ($\phi_1$, $\phi_2$) is a global, $4$ by $4$,
 transformation, see \Eqref{U}.

 We should now explain the selection of the two subsets.
 %because  only {\it local} transformations, i.e.
 %D37 We have chosen
 The choice of $(\hat a_{\bf k}, \, \hat a^\dagger_{\bf k})$ and
 $(\hat a_{\bf -k}, \, \hat a^\dagger_{\bf -k})$, and not
 $(\hat a_{1}, \, \hat a^\dagger_{1})$ and $(\hat a_{2}, \, \hat
 a^\dagger_{2})$, is dictated by the
 nature of the interactions amongst modes. When keeping non-linear terms,
 the Lagrangian density is a sum of products of
 fields evaluated at the same point.
 %D38
% (a bilinear coupling between $\hat \phi(\eta,{\bf x})$
% and a real stochastic field $\Psi (\eta,{\bf x})$
% does not affect the separability since it does not change
% the second moments of the distribution).
 %D37 it produces a coherent state, do you agree ?
 This leads to terms of the form
 $\phi_{{\bf k}} \phi_{{\bf p}} \phi_{-{\bf k}-{\bf p}}$
 which are governed by the Fourier
 transform of the field $\phi(\eta,{\bf x})$. These non-linear terms
 give rise to momentum exchanges
 which  cannot be described by $\hat a_1$ or $\hat a_2$ separately.
 (It should be noticed that these hermitian operators obey unusual commutation
 relations with the momentum operator $\hat {\bf P}$:
 $[\hat {\bf P},\, \hat a_{1,\,2}] = \mp i {\bf k} \,  \hat a_{2,\,1} $).
 %D37
 %On the other hand, since homogeneity is preserved by non-linear interactions,
 %all the distributions we consider are
 %$(\phi_1,\, \phi_2)$-factorizable, see Appendix C.


%D38
 Notice that contrary to non-linear interactions, bilinear
 couplings do not permit to %R39 distinguish between the two-subsets.
 dismiss $\phi_1, \, \phi_2$.
%We must consider non-linearities because bilinear couplings
%% do not affect the separability since they not change
%% the second moments of the distribution.
 %
 %%R38 ce que tu ecris me semble faux car al decoh peut etre introd par Psi
 %qui change le c term
 %D38 right, j'ai eu une absence
 %
 %do not break the $(\phi_1,\, \phi_2)$-factorizability of the state.
 %D38 ca reste faux car ca veut dire implicitement que les non-linearites la brise
 %
 %D37 Probleme:
 %1) bilin coupling is irrelevant for separability because it is
 %a displacement,
 %%%%%FAUX tu oublie l'int fctionlle  sur Psi
 %whereas separability
 %concerns uniquely the second momenta (ie the existence of squeezed directions)
 %2) cette phrase contredit deux statements qu'on a ecrits plus haut
 %i) (1,2) facto <=> homog et
 %ii) non-lin do not break homog.
 %La logique a l'endroit est:
 %1) theorem : separability is the following ineq satisfied by the 2nd moments
 %2) Q: pq (k,-k) et pas (1,2) ? R: la forme des inter non-lin qui echangent les moments
 %(ou bien plus subtile, la postselec)
 %3) Q: pq pas bilin coupling ? R: parce que
 %3.1 )ca n'affecte pas la separabilite de l'etat
 %3.2) tous les etats que l'on considere sont (1,2) facto, que le couplage soit lin ou pas
 %Do you agree ?
 %
 Indeed, consider a  bilinear
 interaction with a random field by adding a coupling term
 $\int\!\!d^3x \, \hat \phi(\eta,{\bf x}) \, \Psi (\eta,{\bf x})$
 where $\Psi (\eta,{\bf x})$ is a real stochastic field.
 %D38 je n'aime pas cette phrase, 'prefered' n'est pas le bon mot car dans le cas des
 %couplages bilin seuls, il n'y a pas moyen de determiner si (k,-k) est plus ou moins physique
 %que (1,2).
 %%R39 je ne suis pas d'acc : en 1,2 cela factorize, en k,-k cela se separe, au
% mieux.
 %D. Il me semble que c'est ce qu'on veut dire. Do you agree ??
 %R: NON
 Then the preferred basis
 are still the real and imaginary parts of $\phi(\eta,{\bf x})$,
 see \cite{KiefPolStar98a}.
 To verify this it suffices to
 decompose the interaction term in Fourier transform, use the
 reality condition of the stochastic field $\Psi (\eta,{\bf x})$
%%tu penses que c'est necess  ?
%in order
to get
$\int\!\!\widetilde{d^3k} \, \left[\hat \phi_1(\eta, {\bf k}) \, \Psi_1(\eta, {\bf k})
 + \hat \phi_2(\eta, {\bf k}) \, \Psi_2(\eta, {\bf k})\right]$,
%R39  One should
and then use the homogeneous and isotropic character
of the moments of $\Psi (\eta,{\bf x})$.
% and one
In this we recover the fact that
 homogeneity implies that all Gaussian density matrices
 are $(\phi_1,\, \phi_2)$-factorizable.


 It is nevertheless possible to consider observables
 which cannot be written in terms of $\hat a_1$ or $\hat a_2$ separately
 without introducing explicitely non-linear interactions. %R40
 To this end one should exploit the cosmic variance,
 namely the fact we do not observe the mean value, but rather
 some realization of the ensemble.
 (We refer  to Chap. V of
 \cite{CP04} for the procedure to extract some realization
 from the distribution, to Chap. VI for
 the calculation of observables associated with a given realization,
 and to the references of this work for the relationship
 between the role played by the interaction Hamiltonian in non-linear
 theories and the role of
 the projector introduced to extract some configurations
 from the mean.)
 In particular, upon specifying that a localized classical field
 configuration possessing a definite {\it velocity} (e.g. a
 localized wave packet made with running waves)
 has been realized/detected, the conditional
 expectation value of $\hat \phi(\eta, {\bf x})$ describes a pair
 of wave packets moving away from each other
 and which are spatially well separated.
 These wave packets localized both in ${\bf x}$ and ${\bf k}$ space
 are built with $\hat a_{\bf k}$ and $\hat a_{-{\bf k}}$ separately. Hence
 they cannot be written in terms of $\hat a_1$ or $\hat a_2$ separately.

% The equivalence between this criterion \cite{Simon00}
% and the one by Peres \cite{Peres} is easily verified
% for homogeneous Gaussian states. Peres noticed that
% if $\hat \rho_{sep}$ is a separable state as defined in
% \Eqref{Defseparability}, then its partial transpose
% $\hat \rho_{sep}^{PT} = \sum_{l} p_l \hat \rho_{\bf k}^{(l)} \otimes
% \hat \rho_{-{\bf k}}^{(l)\, T}$, where
% $\bra{q} \hat \rho_{-{\bf k}}^{(l)\, T} \ket{q'} =
% \bra{q'} \hat \rho_{-{\bf k}}^{(l)} \ket{q}$,
% is a viable density matrix (of a separable state). This necessary
% condition is also sufficient for Gaussian states.
% First, one checks that in a partial transpose,
% $\hat a_{-{\bf k}}^{\dagger} \mapsto \hat a_{-{\bf k}}$

\smallskip

{ \bf{Loss of violations of Bell inequalities}}

\noindent
In \cite{Wodkievicz}, the authors point out that
%phase space representations
the projectors on a given field configuration
%furnish operators violating
can violate
Bell's inequalities when evaluated in two-mode squeezed states.
In \cite{CP4},
we extend their analysis by replacing the pure states
by the partially decohered distributions of Eq. \Eqref{Qrepr}.
We plot in Fig. $4$ the combination $(n+1)\cal C$ of Eq. ($11$)
of \cite{CP4}
as a function of the square field amplitude
% ($={\cal J}e^{2r}$ in their notations)
for $n=100$ and for three different values of $\delta$.
%, namely $0, 0.05, 0.1$.
The points over the horizontal line
$(n+1){\cal C}=2$ give combinations of operators violating Bell's inequalities.
We observe that for $\delta > 0.1$, there is no more violations.
However,
%as noted in \cite{Wodkievicz},
the combination $\cal C$ does not provide the
operator which maximizes the violation.
Hence, there might still be operators which violate Bell's inequalities for
$0.1 \leq \delta < 1$.

\begin{figure}[h]
    \resizebox{7.5cm}{6.5cm}{\includegraphics{bellviolation.eps}}
 \caption{{\it The loss of violation as decoherence increases.}
 The function is normalized to its maximal classical value $=2$.
 The variable $x$ gives the square field amplitude.
 % of the configurations
 %$\phi_{\pm {\bf k}}$ measured along the big axis of the ellipse
 %which enters in the combination of Eq. ($7$) in \cite{Wodkievicz}.
 The occupation number is $n=100$, and the three values of $\delta$
 are $0$ (red, upper), the pure state, $0.05$ (blue, middle), and
 $0.1$ (brown, lower), the border regime. It is also worth to point out that
 the function $(n+1){\cal C}(x)$ is independent of $n$ in the large $n$ limit
 as what is found in Figs. 3, 5, and 6.}
 \end{figure}


  \subsection{Entropy of partially decohered distributions}



 \begin{figure}[h] \label{macrospread}
 \vbox{ \hbox to\linewidth{\hss
    \resizebox{7.5cm}{6.5cm}{\includegraphics{entropy.eps}}
 \hspace{5mm}
        \resizebox{7.5cm}{6.5cm}{\includegraphics{entropylogscale.eps}}}
\hss}
 \caption{{\it Entropy as a function degree of coherence
 for three values of $n$.} The function is normalized to it maximal value.
 The occupation number is $n=10^2$ (lower, blue),
 $n=10^3$ (middle, red), and $n=10^5$ (upper, black).
 On the left panel, as a function of the ratio
 $\zeta = \abs{c}/(n+1/2)$, %R40
 one clearly see the sensitivity for %R40 values of $\delta \leq 1$.
 $\zeta \to 1^{-}$. On the right panel, as a
 function of $x=\arctan(\ln(\delta))$, one verifies
 that the critical value $S_{max}/2$ %R40
 corresponds to
 $\delta =1$ for all $n$. %R40 separates to regimes of growth of the entropy.
 }
 \end{figure}




 To characterize the entropy of the general distribution
 \Eqref{Qrepr},
 we write $\delta = A n^\gamma,\, A> 0, \,  1 \geq \gamma \geq 0$.
 As we shall see, in the high occupation number limit,
 $\gamma$ is tightly constrained.

 The thermal occupation number defined
 in Eq. \Eqref{tilde n} and the entropy respectively scale as
 \ba
   \bar n = \sqrt{A} n^{(1+\gamma)/2} + O(n^{-(1+\gamma)/2}) \, ,
 \ea
 and
 \ba \label{paramd2}
   S^{(\gamma, A)}_{{\bf k},\, -{\bf k}} =
   (1+\gamma) 2 r + \left( 2 + \ln A \right)
   + O(n^{-(1+\gamma)/2}) \, .
 \ea
 We have used the squeezing parameter $n = \sh^2(r)$.
 From this analysis we see that the uncertainty in the definition of the
 quantum distributions which gives rise to the entropy of the effective
 distribution of sine functions of \Eqref{Peff}
 is very limited.
 (The entropy of that distribution is $S_{{\bf k},\, -{\bf k}}= 2 r$,
 see Eq. \Eqref{Smin} and the discussion in the following paragraphs.)
 Indeed, the linear dependence of the entropy in $r$
 implies that the distributions with entropy $= 2 r$
 all have
 \ba
  \gamma \ll \frac{1}{\ln n} \, .
 \ea


 When refining the analysis by taking into account the term in $\ln A$,
 and requiring that $\vert S_{{\bf k},\, -{\bf k}} - 2(r+1) \vert \leq B$,
 one gets
 $\vert \ln \delta  \vert \simeq \delta -1 \leq B$,
 independently of $n$.
 In the large $n$ limit, this
 fixes the value of the coherence term $c$
 with a precision $B/n$,
 thereby tightly restricting the quantum density matrices which
 can be put in correspondence with  Eq. \Eqref{Peff}.


Nevertheless, the criterium  $S_{{\bf k},\, -{\bf k}} = 2 r + O(1)$ %R30
does not discriminate between the
 distribution of Eq. \Eqref{rhodec} ($\delta = 2$)
 and that of  Eq. \Eqref{rhodec2} ($\delta = 1$).
 Hence the distribution \Eqref{rhodec} should be conceived
 as {\it minimal} only as %R26
 being a member of the {\it equivalent class
 of minimally decohered matrices}, i.e.
 those which have lost their quantum properties ($\delta > 1$)
 and which possess an entropy equal to $2 r$.



 \begin{figure}[h] \label{macrospread}
 \vbox{ \hbox to\linewidth{\hss
    \resizebox{7.5cm}{6.5cm}{\includegraphics{SvsSigmaPP3n.eps}}
 \hspace{5mm}
        \resizebox{7.5cm}{6.5cm}{\includegraphics{SvsSigmaPPzoom3n.eps}}}
\hss}
 \caption{{\it Parametric curves}
 We plot the product
 $\sigma^2_v \langle \{ \Pi_{\bf k} ,\, \Pi_{-{\bf k}} \} \rangle $
 versus the entropy. The latter (horizontal axis)
 has been normalized to its maximum value $S_{max}$.
 The occupation number is $n=10^2$ (upper, blue),
 $n=10^3$ (middle, red), and $n=10^5$ (lower, black).
 On the left panel, $\delta$ goes from $0$ to $n$ and
 the product
 $\sigma^2_v \langle \{ \Pi_{\bf k} ,\, \Pi_{-{\bf k}} \} \rangle $
 has been normalized to its maximum
 value $n$.
 On the right panel, we show a zoom on the region $\delta \leq 1$,
 and the product is not normalized.
 One sees the two regimes.
 First as long as the entropy has not reached
 half its maximum value, the product stays nearly constant.
 On the contrary, for $S > S_{max}/2$, the product explodes and reaches
 $n$. One can also admire
 the fact that the curves meet near $(1/2, 1/2)$ irrespectively
 of the value of $n$. %R28
 Moreover in the limit $n \to \infty$ the curve becomes an Heavyside
 function, thereby cleary separating the two regimes: the first one
 where quantum properties are progressively lost, and the second one
 where the power of the decaying mode linearly grows with $\delta$.}
 \end{figure}


 \subsection{Partial randomization of the phase}

 To complete the analysis we now
 assume that the phase of the squeezed vacuum
 state is randomly scattered.
  The density matrix
 $\hat \rho_{in}$ is then replaced by the statistical mixture
 \ba \label{rhophase}
   \hat \rho(r,\varphi_0) = \int_{0}^{2\pi}\!\!d\varphi \,
   f(\varphi;\, \varphi_0,\, \sigma) \hat \rho_{in}(r,\,\varphi) \, ,
 \ea
 where each pure state
 $\hat \rho_{in}(r,\,\varphi)$ is parameterized by the squeezing
 parameters $r$ and $\varphi$, and $f(\varphi;\, \varphi_0,\,
 \sigma)$ is the PDF of $\varphi$ centered at $\varphi_0$
 with a spread $\sigma$.
 We  assume that the mean phase shift vanishes. Therefore
 the mean value of $\varphi$ is the phase of the original distribution
 \Eqref{inoutvac}, namely $\varphi_0 = \pi + 2k\eta_r$.

 Since all the states in the ensemble \Eqref{rhophase} have the
 same value of $r$, the expectation value of the occupation number
 is still
 $\langle \hat a_{\pm {\bf k}}^{\dagger} \hat a_{\pm {\bf k}} \rangle =
 \sh^2 r = n$.
%R40
However the effective coherence term is reduced since it is affected
by %the result of an
 averaging over the phase :
 \ba
   \av{c}_{\varphi} =
   {\rm Tr}\left( \hat a_{{\bf k}} \hat a_{-{\bf k}}
   \hat \rho(r,\varphi_0) \right) =
   \sqrt{n(n+1)} \int_{0}^{2\pi}\!\!d\varphi \,
   f(\varphi) e^{i\varphi} \, .
 \ea
 Let us assume that the function $f(\varphi)$ is sharply peaked.
 Then we can extend the bounds of integration to infinity. Let us
 choose for instance a Gaussian PDF,
 $f(\varphi) = (2\pi\sigma^2)^{-1/2} \exp\left(
 -(\varphi - \varphi_0)^2/2\sigma^2\right)$. One gets
 \ba
   \av{c}_{\varphi} = \sqrt{n(n+1)} e^{i\varphi_0} e^{-\sigma^2/2}
   \, .
 \ea
 Hence, we have that $\abs{\av{c}_{\varphi}} =
 \abs{c_{sq}(r)} e^{-\sigma^2/2} <  \abs{c_{sq}(r)}$, i.e.
 the norm of the effective coherence term is smaller,
 as expected.
 A value $\delta_{cr} = 1$ corresponds to
 an extremely narrow distribution:  $\sigma_{cr}^2 = 1/(n+1)
 = 4(k\eta_r)^4 \sim 10^{-100}$.
 %R40
 It should be noticed that these changes affecting the
 coherence are much smaller than those obtained
 when considering the trans-Planckian problem, see \cite{NP,NPC,MB},
 which concerned modifications of the power $n(k)$.



% \begin{figure}[h] \label{macrospread}
% \vbox{ \hbox to\linewidth{\hss
%    \resizebox{7.5cm}{6.5cm}{\includegraphics{3dsq.eps}}
% \hspace{5mm}
%        \resizebox{7.5cm}{6.5cm}{\includegraphics{3dmin.eps}}}
%\hss}
% \caption{{\it Wigner function of two distributions}, for
% $n=1000$. Left panel, $\abs{c} = \sqrt{n(n+1)}$, right panel,
% $\abs{c} = n$.
% }
%\end{figure}


\end{appendix}


\begin{thebibliography}{9}

\bibitem{Mukhanov81} V. Mukhanov, C. Chibisov,
JETP Lett. ${\bf 33}$, No. 10, 532 (1981), astro-ph/0303077.

\bibitem{Staro79} A.A. Starobinsky,
JETP Lett. ${\bf 30}$, 682 (1979),
 {\it and} Phys.Lett. B ${\bf 117}$, 175 (1982).

 \bibitem{H} S. Hawking, Phys.Lett. B ${\bf 115}$, 295 (1982).

%    Hawking: The Development Of Irregularities In A Single Bubble Inflationary Universe
%\bibitem{Phys.Lett. B117 (1982) 175
%    Starobinsky: Dynamics Of Phase Transition In The New Inflationary Universe Scenario And Generation Of Perturbations

\bibitem{G} A. Guth and S.Y. Pi,
Phys.Rev.Lett. ${\bf  49}$, 1110 (1982).
%    Guth: Fluctuations In The New Inflationary Universe
\bibitem{B} J.M. Bardeen,  P.J. Steinhardt, M.S. Turner,
Phys.Rev. D${\bf 28}$, 679 (1983).
%Bardeen: Spontaneous Creation Of Almost Scale - Free Density Perturbations In An Inflationary Universe


\bibitem{Grishchuk90} L.P. Grishchuk and Yu.V. Sidorov,
Phys. Rev. D$\bold{42}$, 3413 (1990).

\bibitem{LiddleLythPhysRep}  A.R. Liddle, D.H. Lyth,
Phys.Rept. {\bf 231}, 1 (1993).


\bibitem{Albrecht00} A. Albrecht,
Presented at NATO Advanced Study Institute: Structure Formation in
the Universe, Cambridge, England, 26 Jul - 6 Aug 1999,
astro-ph/0007247.



 \bibitem{Werner89} R.F. Werner,
 Phys.Rev.A $\bold{40}$, 4277 (1989).


 \bibitem{Glauber69} K.E. Cahill and R.J. Glauber,
 Phys. Rev.{\bf 177}, 1882 (1969).

\bibitem{Zurekbook} W.H. Zurek, in
{\it The Physics of Time Asymmetry}, edited by J.J. Halliwell, J.
Perez-Mercader, and W.H. Zurek, (Cambridge University Press,
Cambridge, UK, 1993): Theor. Phys.

\bibitem{Zurek93} W.H. Zurek, S. Habib, J. P. Paz,
Phys.Rev.Lett. $\bold{70}$, 1187 (1993).

\bibitem{Maldacena03} J. Maldacena,
 JHEP $\bf 0305$, 013 (2003).


\bibitem{gottfried} K. Gottfried, Chap. IV in
{\it{Quantum Mechanics, Vol. I}}, Benjamin, New York, 1966.



 \bibitem{cmbfast}  U. Seljak, M. Zaldarriaga,
 Astrophys.J. {\bf 469}, 437 (1996).

 \bibitem{MukhaBrandyProkopecg}
 R. Brandenberger, V. Mukhanov, and T. Prokopec,
 Phys.Rev.Lett. $\bold{69}$, 3606 (1992);
 {\it and} Phys.Rev. D$\bold{48}$, 2443(1993).

 \bibitem{Albrecht94} T. Prokopec,
 Class.\ Quant.\ Grav.\  {\bf 10} 229 (1993);
 {\it and} A. Albrecht, P. Ferreira, M. Joyce, and T. Prokopec,
 Phys.Rev. D $\bold{50}$, 4807 (1994).

\bibitem{Matacz94} A.L. Matacz,
Phys.Rev. D$\bold{49}$, 788 (1994).

\bibitem{Giovanninig} M.Gasperini and M.Giovannini,
Phys. Lett. B$\bold{301}$, 334 (1993); {\it and} Class.\ Quant.\
Grav.\ $\bold{10}$, L133 (1993); {\it and} {\it 'String Theory in
Curved space Times'}, (World Scientific, Singapore, 1998), p.249.


\bibitem{PolStar96}  D. Polarski and A.A. Starobinsky,
Class. Quant. Grav. $\bold{13}$, 377 (1996).

\bibitem{KiefPolStar98a} C. Kiefer, D. Polarski, and A.A. Starobinsky,
Int.J.Mod.Phys. D$\bf 7$, 455 (1998);
 {\it and} C. Kiefer and  D. Polarski,
 Annalen Phys. $\bf 7$, 137 (1998).

\bibitem{KiefPolStar00} C. Kiefer, D. Polarski, and A.A. Starobinsky,
Phys.Rev. D$\bold{62}$, 043518 (2000).

 \bibitem{cpP1} D. Campo and R. Parentani,
 {\it Inflationary spectra, decoherence, and two-mode coherent
 states}, astro-ph/$0404021$.

%\bibitem{Penrose} R. Penrose in {\it General Relativity, An Einstein
%Centenary Survey}, Ed. by S.W. Hawking and W. Israel, Cambridge
%university Press (1979).

\bibitem{MBF92} V. Mukhanov, H. Feldman, and R. Brandenberger,
Phys.Rep. $\bold{215}$, 203 (1992).

\bibitem{cmbslow} V. Mukhanov, astro-ph/0303072.


\bibitem{B&D} N.D. Birrel and P.C.W. Davis, in
{\it{Quantum Fields in Curved Space}}
(Cambridge University Press, Cambridge, UK, 1984).



  \bibitem{NPC} J.C. Niemeyer, R. Parentani, and D. Campo,
 Phys.Rev. D$\bf{66}$, 0083510 (2002).


 \bibitem{cmbParenta} R. Parentani,
 C.R. Physique $\bf 4$, 935 (2003).


\bibitem{PhysRep95} R. Brout, S. Massar, R. Parentani, and Ph. Spindel,
Phys.Rept. $\bold{260}$, 329 (1995).


 \bibitem{WMAP1} H.V. Peiris \& {\it al}.,
Astrophys.J.Suppl.{\bf 148} 213 (2003).


\bibitem{Allen00} B. Allen, E. E. Flanagan, and M. A. Papa,
Phys.Rev. D$\bold{61}$, 024024 (2000).



 \bibitem{Matarrese} N. Bartolo, E. Komatsu, S. Matarrese,
 and A. Riotto,
 Phys.Rept. $\bf 402$, 103 (2004).


\bibitem{CP04} D. Campo and R. Parentani,
 Phys.Rev. D$\bf{70}$, 105020 (2004).


\bibitem{Kim00} H. Jeong, J. Lee, and M. S. Kim,
 Phys.Rev. A$\bold{61}$, 052101 (2000).


\bibitem{Zurek81} W.H. Zurek,
Phys.Rev. D$\bold{24}$, 1516 (1981).

\bibitem{Zurek82} W.H. Zurek,
Phys.Rev. D$\bold{26}$, 1862 (1982).


\bibitem{Zeh85} E. Joss and H. Zeh,
Z. Phys. B$\bold{59}$,223 (1985).

\bibitem{Serafini04}  A. Serafini, F. Illuminati, and S. De Siena,
J.Phys. B$\bold{37}$,L21 (2004).

 \bibitem{MB} J. Martin and R. Brandenberger,
 Phys. Rev. D$\bf{63}$, 123501 (2001),
 {\it ibid} $\bf{68}$, 063513 (2003).


 \bibitem{NP}  J.C. Niemeyer and R. Parentani,
 Phys.Rev. D$\bf{64}$, 101301(R) (2001).


 \bibitem{Morikawa} M. Morikawa,
 Phys.Rev. D$\bold{42}$, 2929 (1990).


 \bibitem{Wodkievicz} K. Banaszek and K. W\'odkiewicz,
 %Phys.Rev. A$\bf 58$, 4345 (1998).
 Acta Phys.Slov. ${\bf 49}$, 491 (1999).

 \bibitem{CP4} D. Campo and R. Parentani,
 {\it Inflationary spectra and violations of Bell's inequalities},
 astro-ph/0505376;
 {\it Quantum correlations in inflationary spectra and
 violation of Bell inequalities},
 Proceedings of the conference "100 years in relativity", Sao Paulo,
 August 22-24, to appear in a special issue of the
 Brazilian Journal of Physics (BJP), astro-ph/0510445,


 \bibitem{HuCalzetta} E. Calzetta and B.L. Hu,
 Phys.Rev. D$\bf 52$, 6770 (1995).


\bibitem{Glauber63a} R.J. Glauber, Phys.Rev. $\bold{130}$, 2529 (1963).

\bibitem{Glauber63b} R.J. Glauber, Phys.Rev. $\bold{131}$, 2766 (1963).

\bibitem{Zhang90} W.M. Zhang, D.H. Feng, and R. Gilmore,
Rev.Mod.Phys. $\bold{62}$, 867 (1990).

 \bibitem{schum} B. Schumaker and C. Caves,
 Phys. Rev. $\bold{A31}$, 3068; 3093 (1985).

 \bibitem{Simon00} R. Simon,
 Phys.Rev.Lett. ${\bf 84}$, 2726 (2000);
 L-M. Duan, G. Giedke, J.I. Cirac, and P. Zoller,
 Phys.Rev.Lett. ${\bf 84}$, 2722 (2000);
 B-G. Englert, and K. W\'odkiewicz,
 Phys.Rev.A ${\bf 65}$, 054303 (2002).

 \bibitem{Wodkiewiczlecture} B-G. Englert, and K. W\'odkiewicz,
 International Journal of Quantum Information {\bf 11}, No. 2, 153
 (2003).


  \bibitem{Nussenzveig} H.M. Nussenzveig,
 {\it Introduction to Quantum Optics}, Gordon and Breach Science
 Publishers, New York, N.Y. (1973).

 \bibitem{Cahill65} K.E. Cahill,
 Phys.Rev. $\bf{138}$, 1566 (1965).




\end{thebibliography}

\end{document}

astro-ph0505379.extracted_bib
\begin{thebibliography}{9}

\bibitem{Mukhanov81} V. Mukhanov, C. Chibisov,
JETP Lett. ${\bf 33}$, No. 10, 532 (1981), astro-ph/0303077.

\bibitem{Staro79} A.A. Starobinsky,
JETP Lett. ${\bf 30}$, 682 (1979),
 {\it and} Phys.Lett. B ${\bf 117}$, 175 (1982).

 \bibitem{H} S. Hawking, Phys.Lett. B ${\bf 115}$, 295 (1982).

%    Hawking: The Development Of Irregularities In A Single Bubble Inflationary Universe
%\bibitem{Phys.Lett. B117 (1982) 175
%    Starobinsky: Dynamics Of Phase Transition In The New Inflationary Universe Scenario And Generation Of Perturbations

\bibitem{G} A. Guth and S.Y. Pi,
Phys.Rev.Lett. ${\bf  49}$, 1110 (1982).
%    Guth: Fluctuations In The New Inflationary Universe
\bibitem{B} J.M. Bardeen,  P.J. Steinhardt, M.S. Turner,
Phys.Rev. D${\bf 28}$, 679 (1983).
%Bardeen: Spontaneous Creation Of Almost Scale - Free Density Perturbations In An Inflationary Universe


\bibitem{Grishchuk90} L.P. Grishchuk and Yu.V. Sidorov,
Phys. Rev. D$\bold{42}$, 3413 (1990).

\bibitem{LiddleLythPhysRep}  A.R. Liddle, D.H. Lyth,
Phys.Rept. {\bf 231}, 1 (1993).


\bibitem{Albrecht00} A. Albrecht,
Presented at NATO Advanced Study Institute: Structure Formation in
the Universe, Cambridge, England, 26 Jul - 6 Aug 1999,
astro-ph/0007247.



 \bibitem{Werner89} R.F. Werner,
 Phys.Rev.A $\bold{40}$, 4277 (1989).


 \bibitem{Glauber69} K.E. Cahill and R.J. Glauber,
 Phys. Rev.{\bf 177}, 1882 (1969).

\bibitem{Zurekbook} W.H. Zurek, in
{\it The Physics of Time Asymmetry}, edited by J.J. Halliwell, J.
Perez-Mercader, and W.H. Zurek, (Cambridge University Press,
Cambridge, UK, 1993): Theor. Phys.

\bibitem{Zurek93} W.H. Zurek, S. Habib, J. P. Paz,
Phys.Rev.Lett. $\bold{70}$, 1187 (1993).

\bibitem{Maldacena03} J. Maldacena,
 JHEP $\bf 0305$, 013 (2003).


\bibitem{gottfried} K. Gottfried, Chap. IV in
{\it{Quantum Mechanics, Vol. I}}, Benjamin, New York, 1966.



 \bibitem{cmbfast}  U. Seljak, M. Zaldarriaga,
 Astrophys.J. {\bf 469}, 437 (1996).

 \bibitem{MukhaBrandyProkopecg}
 R. Brandenberger, V. Mukhanov, and T. Prokopec,
 Phys.Rev.Lett. $\bold{69}$, 3606 (1992);
 {\it and} Phys.Rev. D$\bold{48}$, 2443(1993).

 \bibitem{Albrecht94} T. Prokopec,
 Class.\ Quant.\ Grav.\  {\bf 10} 229 (1993);
 {\it and} A. Albrecht, P. Ferreira, M. Joyce, and T. Prokopec,
 Phys.Rev. D $\bold{50}$, 4807 (1994).

\bibitem{Matacz94} A.L. Matacz,
Phys.Rev. D$\bold{49}$, 788 (1994).

\bibitem{Giovanninig} M.Gasperini and M.Giovannini,
Phys. Lett. B$\bold{301}$, 334 (1993); {\it and} Class.\ Quant.\
Grav.\ $\bold{10}$, L133 (1993); {\it and} {\it 'String Theory in
Curved space Times'}, (World Scientific, Singapore, 1998), p.249.


\bibitem{PolStar96}  D. Polarski and A.A. Starobinsky,
Class. Quant. Grav. $\bold{13}$, 377 (1996).

\bibitem{KiefPolStar98a} C. Kiefer, D. Polarski, and A.A. Starobinsky,
Int.J.Mod.Phys. D$\bf 7$, 455 (1998);
 {\it and} C. Kiefer and  D. Polarski,
 Annalen Phys. $\bf 7$, 137 (1998).

\bibitem{KiefPolStar00} C. Kiefer, D. Polarski, and A.A. Starobinsky,
Phys.Rev. D$\bold{62}$, 043518 (2000).

 \bibitem{cpP1} D. Campo and R. Parentani,
 {\it Inflationary spectra, decoherence, and two-mode coherent
 states}, astro-ph/$0404021$.

%\bibitem{Penrose} R. Penrose in {\it General Relativity, An Einstein
%Centenary Survey}, Ed. by S.W. Hawking and W. Israel, Cambridge
%university Press (1979).

\bibitem{MBF92} V. Mukhanov, H. Feldman, and R. Brandenberger,
Phys.Rep. $\bold{215}$, 203 (1992).

\bibitem{cmbslow} V. Mukhanov, astro-ph/0303072.


\bibitem{B&D} N.D. Birrel and P.C.W. Davis, in
{\it{Quantum Fields in Curved Space}}
(Cambridge University Press, Cambridge, UK, 1984).



  \bibitem{NPC} J.C. Niemeyer, R. Parentani, and D. Campo,
 Phys.Rev. D$\bf{66}$, 0083510 (2002).


 \bibitem{cmbParenta} R. Parentani,
 C.R. Physique $\bf 4$, 935 (2003).


\bibitem{PhysRep95} R. Brout, S. Massar, R. Parentani, and Ph. Spindel,
Phys.Rept. $\bold{260}$, 329 (1995).


 \bibitem{WMAP1} H.V. Peiris \& {\it al}.,
Astrophys.J.Suppl.{\bf 148} 213 (2003).


\bibitem{Allen00} B. Allen, E. E. Flanagan, and M. A. Papa,
Phys.Rev. D$\bold{61}$, 024024 (2000).



 \bibitem{Matarrese} N. Bartolo, E. Komatsu, S. Matarrese,
 and A. Riotto,
 Phys.Rept. $\bf 402$, 103 (2004).


\bibitem{CP04} D. Campo and R. Parentani,
 Phys.Rev. D$\bf{70}$, 105020 (2004).


\bibitem{Kim00} H. Jeong, J. Lee, and M. S. Kim,
 Phys.Rev. A$\bold{61}$, 052101 (2000).


\bibitem{Zurek81} W.H. Zurek,
Phys.Rev. D$\bold{24}$, 1516 (1981).

\bibitem{Zurek82} W.H. Zurek,
Phys.Rev. D$\bold{26}$, 1862 (1982).


\bibitem{Zeh85} E. Joss and H. Zeh,
Z. Phys. B$\bold{59}$,223 (1985).

\bibitem{Serafini04}  A. Serafini, F. Illuminati, and S. De Siena,
J.Phys. B$\bold{37}$,L21 (2004).

 \bibitem{MB} J. Martin and R. Brandenberger,
 Phys. Rev. D$\bf{63}$, 123501 (2001),
 {\it ibid} $\bf{68}$, 063513 (2003).


 \bibitem{NP}  J.C. Niemeyer and R. Parentani,
 Phys.Rev. D$\bf{64}$, 101301(R) (2001).


 \bibitem{Morikawa} M. Morikawa,
 Phys.Rev. D$\bold{42}$, 2929 (1990).


 \bibitem{Wodkievicz} K. Banaszek and K. W\'odkiewicz,
 %Phys.Rev. A$\bf 58$, 4345 (1998).
 Acta Phys.Slov. ${\bf 49}$, 491 (1999).

 \bibitem{CP4} D. Campo and R. Parentani,
 {\it Inflationary spectra and violations of Bell's inequalities},
 astro-ph/0505376;
 {\it Quantum correlations in inflationary spectra and
 violation of Bell inequalities},
 Proceedings of the conference "100 years in relativity", Sao Paulo,
 August 22-24, to appear in a special issue of the
 Brazilian Journal of Physics (BJP), astro-ph/0510445,


 \bibitem{HuCalzetta} E. Calzetta and B.L. Hu,
 Phys.Rev. D$\bf 52$, 6770 (1995).


\bibitem{Glauber63a} R.J. Glauber, Phys.Rev. $\bold{130}$, 2529 (1963).

\bibitem{Glauber63b} R.J. Glauber, Phys.Rev. $\bold{131}$, 2766 (1963).

\bibitem{Zhang90} W.M. Zhang, D.H. Feng, and R. Gilmore,
Rev.Mod.Phys. $\bold{62}$, 867 (1990).

 \bibitem{schum} B. Schumaker and C. Caves,
 Phys. Rev. $\bold{A31}$, 3068; 3093 (1985).

 \bibitem{Simon00} R. Simon,
 Phys.Rev.Lett. ${\bf 84}$, 2726 (2000);
 L-M. Duan, G. Giedke, J.I. Cirac, and P. Zoller,
 Phys.Rev.Lett. ${\bf 84}$, 2722 (2000);
 B-G. Englert, and K. W\'odkiewicz,
 Phys.Rev.A ${\bf 65}$, 054303 (2002).

 \bibitem{Wodkiewiczlecture} B-G. Englert, and K. W\'odkiewicz,
 International Journal of Quantum Information {\bf 11}, No. 2, 153
 (2003).


  \bibitem{Nussenzveig} H.M. Nussenzveig,
 {\it Introduction to Quantum Optics}, Gordon and Breach Science
 Publishers, New York, N.Y. (1973).

 \bibitem{Cahill65} K.E. Cahill,
 Phys.Rev. $\bf{138}$, 1566 (1965).




\end{thebibliography}