%  29-9-1998
\magnification\magstep1
\input mssymb.tex

\centerline{\bf Teleportation of general quantum states, squeezing and }

\centerline{{\bf  the decoherence problem in quantum
computers}\footnote{$^{(*)}$}{Invited talk to the: Fourth International Conference on
Quantum Communication, Measurement, and computing, Norris University Center,
Northwestern University E\-van\-ston, Illinois, USA August 22-27, 1998.}}
\bigskip\bigskip

\centerline{{\bf Luigi Accardi}\footnote{$^{(1)}$}{Graduate School of
Polymathematics, Nagoya University, Chi\-ku\-sa--ku, Na\-go\-ya, 464--01, Japan,
and Centro V. Volterra, Universit\`a degli Studi di Roma ``Tor
Vergata'' -- 00133 Rome, Italy, email:
accardi@volterra.mat.uniroma2.it, accardi@math.nagoya-u.ac.jp}}
\bigskip
\bigskip

\centerline{\bf Abstract}
\bigskip

In sections (1), (2) a channel theoretical formulation of the teleportation
problem for quantum states, is given and a general solution, valid in
arbitrary finite dimensions is proposed. The problem of uniqueness of the
hidden key is discussed and a classification is obtained. The results in
these sections survey recent joint work with Masanori Ohya.

It is shown that the stochastic limit of quantum theory suggests the 
existence of two types of squeezing states of the quantum EM field and 
correspondingly of two types of squeezing white noises. 
These states are stationary but usually non equilibrium states of the 
field and the stochastic limit evidentiates some qualitatively new 
phenomenon that arises in connection with these states.
As a particular case we shall illustrate how one could take advantage of 
these new physical phenomena to prevent decoherence in quantum computers.
The results in the second part of the paper survey recent joint work with 
S. Kozyrev, Y.G. Lu and I. Volovich.
\bigskip

\beginsection{1.) Teleportation of general quantum states }

\bigskip
Quantum teleportation has been by introduced by Benett and Brassard [BeBr84]
and discussed by several authors (cf. [Eke91], [BBCJPW93], [BBPSSW96]
and references therein) in the framework of the singlet state.
Our starting point shall be the following channel theoretical generalization
of the Benett and Brassard scheme proposed by [IOS98]:
\item{ STEP 0}:
Alice has a unknown quantum state $\rho ^{(1)}$ on a
Hilbert space ${\cal H}_{1}$, and she wants to teleport it to Bob.
\item{ STEP 1}:
Prepare a state 
$$\sigma _{23}\in {\cal S}\left( {\cal H}_{2}\otimes {\cal H}_{3}\right) $$ 
Hilbert spaces attached to Alice and to Bob, respectively.
\item{STEP 2}:
After choosing a projection valued measure
$$\left\{F_{k}^{(12)}\right\} $$
 on the Hilbert space ${\cal H}_{1}\otimes {\cal H}_{2}$,
 Alice performs the incomplete measurement corresponding to the $k$-th
projection $F_{k}^{(12)}.$
After Alice's measurement, Bob's state becomes $\rho ^{(3)}.$
\item{ STEP 3}:
Bob is informed which projection (the number $k$) was used by Alice.
This information is transmitted from Alice to Bob without disturbance.
\item{ STEP 4}:
Bob reconstructs the unknown Alice's state $\rho ^{(1)}$ from $\rho^{(3)}$
using a key agreed in advance.
\medskip 
The above procedure can be realized by a channel 
$$\Lambda ^{*[k]}:{\cal S}\left( {\cal H}_{1}\right) \longrightarrow 
{\cal S}\left( {\cal H}_{1}\right) $$ 
constructed by the following four maps (channels):

(1) a trivial (i.e. product) lifting in the sense of [AcOh92]
$$\gamma ^{*}:{\cal S}\left( {\cal H}_{1}\right) \longrightarrow 
{\cal S}\left( {\cal H}_{1}\otimes {\cal H}_{2}\otimes {\cal H}\right) 
\eqno(1) $$
$$\gamma ^{*}\left( \rho^{(1)}\right) =
\rho^{(1)}\otimes \sigma^{(23)}\quad
\forall\, \rho^{(1)}\in {\cal S}\left( {\cal H}_{1}\right) $$
expressing a free coupling of an initial state $\rho^{(1)}$ with the entangled 
state $\sigma ^{(23)}$.

(2) The second step is described by a channel of the form
$$\pi _{k}^{*}:{\cal S}\left( {\cal H}_{1}\otimes {\cal H}_{2}
\otimes {\cal H}_{3}\right) \longrightarrow {\cal S}\left( {\cal H}_{1}
\otimes {\cal H}_{2}\otimes {\cal H}_{3}\right) \eqno(2) $$ 
where, $\forall \rho ^{(123)}\in {\cal B}\left( {\cal H}_{1}
\otimes {\cal H}_{2}\otimes {\cal H}_{3}\right)$, 
$$\pi _{k}^{*}\left( \rho ^{(123)}\right) :=
{ \left( F_{k}^{(12)}\otimes I^{(3)}\right) \rho ^{(123)}
\left( F_{k}^{(12)}\otimes I^{(3)}\right) \over 
{\rm tr}_{123}\left( F_{k}^{(12)}\otimes I^{(3)}\right) 
\rho^{(123)}\left( F_{k}^{(12)}\otimes I^{(3)}\right) }  $$ 

corresponding to an incomplete first kind measurement 
describing the state change due to the nonclassical effect determined by 
Alice's measurement:

(3) the third step is defined by the channel 
$a^{*}:{\cal S}\left( {\cal H}_{1}\otimes {\cal H}_{2}\otimes {\cal H}_{3}
\right) \longrightarrow {\cal S}\left( {\cal H}_{3}\right) $
defined by
$$\rho^{(3)}=a^{*}\left( \rho^{(123)}\right) =
{\rm tr}_{12}\rho^{(123)}\quad \forall \rho^{(123)}\in 
{\cal S}\left( {\cal H}_{1}\otimes \cal{H}_{2}\otimes {\cal H}_{3}\right) $$
Here ${\rm tr}_{12}$ is the partial trace on the Hilbert space
${\cal H}_{1}\otimes {\cal H}_{2}$ defined by
$$<\Phi _{1},{\rm tr}_{12}\,Q\Phi _{2}>\,\,\equiv 
\sum\limits_{n}<\Psi_{n}\otimes \Phi _{1},\,Q\Psi _{n}\otimes \Phi _{2}
>,\hbox{ }Q\in B({\cal H}_{1}\otimes {\cal H}_{2}\otimes {\cal H}_{3}) $$
for any CONS $\,\{\Psi _{n}\}$ of ${\cal H}_{1}\otimes {\cal H}_{2}$
and any $\Phi _{1},\Phi _{2}\in {\cal H}_{3}$.
This channel $a^{*}$ is a reduction from the state $\rho ^{(123)}$ to Bob's
state $\rho ^{(3)}$ after Alice's measurement.

(4) In the teleportation scheme of Benett and Brassard , there exists
a unitary channel
$$u_{k}^{*}:{\cal S}\left( {\cal H}_{3}\right) \longrightarrow {\cal S}
\left( {\cal H}_{1}\right) $$
$$u_{k}^{*}\left( \rho ^{(3)}\right) =
U_{k}\rho ^{(3)}U_{k}^{*}\quad \forall\rho ^{(3)}\in
{\cal S}\left( {\cal H}_{3}\right)$$
where $u_{k}^{*}$ is Bob's unitary operation (key) corresponding to Alice's
measurement $F_{k}^{(12)}$.

Thus the whole teleportation process is written by the channel 
$$\Lambda^{*[k]}:{\cal S}\left( {\cal H}_{1}\right) \longrightarrow 
{\cal S}\left({\cal H}_{1}\right) $$
$$\Lambda ^{*[k]}=u_{k}^{*}\circ \Lambda _{A\to B}^{*[k]}=
u_{k}^{*}\circ a^{*}\circ \pi _{k}^{*}\circ \gamma ^{*}$$
The above subscript ``$[k]$'' means that the channels $\Lambda ^{*[k]}$ and 
$\Lambda _{A\to B}^{*[k]}$ depend on the choice of Alice's measurement 
$F_{k}^{(12)}$. More precisely,
$$\Lambda ^{*[k]}\rho ^{(1)} \equiv U_{k}\left[ 
{\rm tr}_{12}\left[ {\left( F_{k}^{(12)}\otimes 
I^{(3)}\right) \left( \rho ^{(1)}\otimes \sigma^{(23)}\right) 
\left( F_{k}^{(12)}\otimes I^{(3)}\right) \over {\rm tr}_{123}
\left( F_{k}^{(12)}\otimes I^{(3)}\right) \left( \rho ^{(1)}\otimes
\sigma ^{(23)}\right) \left( F_{k}^{(12)}\otimes I^{(3)}\right) }\right]
\right] U_{k}^{*}  \eqno(3)$$
$$\forall \rho ^{(1)}\in {\cal S}\left( {\cal H}%
_{1}\right)$$
Note that the channel $\Lambda ^{*[k]}$ is generally non linear.
\bigskip

\beginsection{2.) Problem I.}

\bigskip
Let
$$N=\dim {\cal H}_{1}=\dim {\cal H}_{2}=\dim {\cal H}_{3}<+\infty $$
we want to find unit vectors $\psi\in{\cal H}_2\otimes{\cal H}_3$
and $\xi\in{\cal H}_1\otimes{\cal H}_2$ such that, denoting
$$F=|\xi\rangle\langle\xi|\otimes1_3\in
\hbox{ Proj }({\cal H}_1\otimes{\cal H}_2\otimes{\cal H}_3)\eqno(1)$$
there exists a unitary transformation $U:{\cal H}_3\to{\cal H}_1$ such
that for any rank one density matrix $\rho\in\hbox{ Dens }({\cal H}_1)$
one has
$$U\cdot{\overline{Tr}_{1,2}(F(\rho\otimes|\psi\rangle\langle\psi|)F)\over
Tr(F(\rho\otimes|\psi\rangle\langle\psi|)F)}U^*\,=\rho\eqno(2)$$
A possible solution is the following.
\bigskip
By our assumption there exists a unit vector $\eta\in{\cal H}_1$ such that
$$\rho=|\eta\rangle\langle\eta|\eqno(3)$$
Let $(\varepsilon_\alpha)$ be an arbitrary orthonormal basis of ${\cal H}_3$
and let $(\lambda_{\gamma\alpha})$ be any $N\times N$ complex unitary matrix,
so that also
$$\left(\sum_\alpha\lambda_{\gamma\alpha}\varepsilon_\alpha\right)_\gamma=:
(\varphi_\gamma)\eqno(4)$$
is an o.n. basis of ${\cal H}_3$.
Let $(\varepsilon'_h)$ be an arbitrary o.n. basis of ${\cal H}_2$ and define
$$\psi:=\sum\lambda_{h\alpha}|\varepsilon'_h\rangle\otimes|\varepsilon_\alpha\rangle
\in{\cal H}_2\otimes{\cal H}_3\eqno(5)$$
then
$$\|\psi\|^2=\sum_{h,\alpha}|\lambda_{h\alpha}|^2=\sum_h 1=\dim
{\cal H}_2=\dim{\cal H}_3\eqno(6)$$
It is at this point that we use in a crucial way the finite dimensionality of
${\cal H}_2$ and ${\cal H}_3$.
Let $(\varepsilon''_n)$ be an arbitrary o.n. basis of ${\cal H}_1$ and choose
$$\xi={1\over  N^{1/2}}\,\sum_\mu\varepsilon''_\mu\otimes\varepsilon'_\mu\ $$
Thus, for $F$ as in (1), one  has, with the convention that we sum over
repeated indices:
$$F(\rho\otimes|\psi\rangle\langle\psi|)F=\lambda_{h\alpha}\overline\lambda_{k\beta}
F(|\eta\rangle\langle\eta|\otimes|\varepsilon'_h\rangle\langle\varepsilon'_k|)F\otimes|
\varepsilon_\alpha\rangle\langle\varepsilon_\beta|=$$
$$={1\over  N}\,\lambda_{h\alpha}\overline\lambda_{k\beta}
|\varepsilon''_\mu\rangle\langle\varepsilon''_\mu , \eta\rangle
\langle\eta,\varepsilon''_{\mu'}\rangle\langle\varepsilon''_{\nu'}|\otimes
|\varepsilon'_\mu\rangle\langle\varepsilon'_\nu,\varepsilon'_h\rangle\langle
\varepsilon'_k,\varepsilon'_{\mu'}\rangle\langle\varepsilon'_{\nu'}
|\otimes|\varepsilon_\alpha\rangle\langle\varepsilon_\beta|=$$
$$={1\over  N}\,\lambda_{h\alpha}\overline\lambda_{k\beta}\delta_{\nu h}
\delta_{k\mu'}\langle\varepsilon''_\nu,\eta\rangle\langle\eta,
\varepsilon''_{\mu'}\rangle
|\varepsilon''_\mu\rangle\langle\varepsilon''_{\nu'}|\otimes|\varepsilon'_\mu
\rangle\langle\varepsilon'_{\nu'}|\otimes|\varepsilon_\alpha\rangle
\langle\varepsilon_\beta|$$
$$={1\over N}\,
\lambda_{h\alpha}\overline\lambda_{k\beta}\langle\varepsilon''_h,
\eta\rangle\langle\eta,\varepsilon''_k\rangle|\varepsilon''_\mu\rangle
\langle\varepsilon''_{\nu'}|\otimes|\varepsilon'_\mu\rangle
\langle\varepsilon'_{\nu'}|\otimes|\varepsilon_\alpha\rangle\langle
\varepsilon_\beta|$$
Taking trace over ${\cal H}_1\otimes{\cal H}_2$ this gives
$${1\over N}\,\lambda_{h\alpha}\overline\lambda_{k\beta}\delta_{\mu,\nu'}
\langle\varepsilon''_h,\eta\rangle \langle\eta,\varepsilon''_k
\rangle|\varepsilon_\alpha\rangle\langle\varepsilon_\beta|=
\langle\varepsilon''_h,\eta\rangle \langle\eta,\varepsilon''_k\rangle
|\varphi_h\rangle\langle\varphi_k|\eqno(7)$$
Notice that by taking the trace over ${\cal H}_3$ of the above
expression one finds
$$\langle\eta,\varepsilon''_k\rangle
\langle\varepsilon''_h,\eta\rangle =\langle\eta,\eta\rangle =1$$

Now, if $U:{\cal H}_3\to{\cal H}_1$ is the unitary operator such that
$$U\varphi_h=\varepsilon''_h$$
then we have
$$UF(\rho\otimes|\psi\rangle\langle\psi|)FU^*=
U \langle\varepsilon''_h,\eta\rangle \langle\eta,\varepsilon''_k\rangle
|\varphi_h\rangle\langle\varphi_k|U^*=
|\varepsilon''_h\rangle\langle\varepsilon''_h,\eta\rangle
\langle\eta,\varepsilon''_k\rangle\langle\varepsilon''_k|=
|\eta\rangle\langle \eta|$$
This solves Problem I.\bigskip

\noindent{\bf The stochastic limit of quantum theory} 

\smallskip

In the following I shall quickly report on joint work done with several 
authors in the past ten years, in particular: Irina Arefeva, Alberto 
Frigerio, Lu Yun Gang, Igor Volovich, Sergei Kozyrev,...

Both in classical and in quantum theory, two main techniques are used to
deal with non integrable systems:
\item{i)} scattering theory, dealing with long times $(t\to\infty)$
\item{ii)} perturbation theory, dealing with small parameter $(\lambda\to0)$

By putting together these two techniques one obtains the {\it stochastic 
limit of QT} which deals with:
\item{iii)} long times \& small parameter, more precisely:
$$t\to\infty\ ;\quad\lambda\to0\ ;\quad\lambda^2 t=\tau\ =\hbox{ const }
\Leftrightarrow t=\hbox{ cost}/\lambda^2$$
The physical intuition behind the stochastic limit is:
\item{--} the effects of the interaction are ``small'' $(\lambda\to0)$
\item{--} but they have a lot of time to cumulate $(t\to\infty)$ and to
produce a nontrivial effect.

As a bonus of the theory one finds that the dominating contribution to the 
interaction {\it resum to give a new unitary evolution\/}. Although the 
mathematics may become quite involved, the basic prescriptions to apply 
the stochastic limit of quantum theory are very simple and they can be 
condensated into the following {\bf stochastic golden rule }
(a generalization of the Fermi golden rule):
\item{1.)} Start from Schr\"odinger's equation in interaction
representation
$$\partial_tU^{(\lambda)}_t=-i\lambda H_I(t)U^{(\lambda)}_t\ ;
\quad U^{(\lambda)}_0=1$$
$$H_{\hbox{tot}}=H_0+H_I\ ;\quad H_I(t)=e^{itH_0}H_Ie^{-itH_0}$$
\item{2.)} By a simple change of variables one verifies that 
to rescale time $t\to t/\lambda^2$ is equivalent to rescale energy as follows:
$$\partial_tU^{(\lambda)}_{t/\lambda^2}=-i{1\over\lambda}\,H_{I
\lambda^2}(t)U^{(\lambda)}_{t/\lambda^2}\eqno(1)$$
\item{3.)} One can prove that, in an appropriate meaning of the 
convergence, $U^{(\lambda)}_{t/\lambda^2}\to U_t$ unitary
\item{4.)} Moreover the limit satisfies the following white noise equation
$$\partial_tU_t=-iH_tU_t\eqno(2)$$ 
\item{5.)} The normally ordered form of the white noise equation (4) is
a quantum stochastic differential equation.

Summing up: the stochastic limit of QT explains the microscopic
(Hamiltonian) origins of many phenomenological (stochastic) equations
studied in the past years. It also provides, apparently for the first 
time, the microscopic structure of the
so called {\it quantum noises\/}.\medskip

\noindent The Physical picture of the stochastic limit of quantum
theory is the following:

One starts from a ``Large'' quantum system where ``Large'' is meant in 
the sense that the
time correlations for the free dynamics are integrable, i.e.
$\int dt|\langle
A(0)B(t)\rangle|<+\infty$ (purely discrete spectrum excluded)

Inside such a system we distinguish two classes of motions: the slow
ones and the fast ones. If we look at the fast degrees of freedom with 
the clock of
the slow degrees of freedom then what we perceive is {\it pure noise\/}
i.e. maximum chaotic behaviour which is described mathematically by an
independent increment process. 
The simplest example of such a process is white noise!\bigskip

\noindent{\bf Typical example}\smallskip

Consider a discrete system $S$ (e.g. an oscillator or a 2--level system) 
interacting with a continuous one $R$ (e.g. a field or an infinite gas) by 
means of an Hamiltonian $H_I$. In the stochastic limit the system 
remains unaltered while the field converges to a quantum white noise.

Summing up: just like statistical mechanics gives a microscopic 
interpretation of the thermodynamical quantities
(``Those kind of motions we call heat''), the stochastic limit of
quantum theory gives a microscopic interpretation of the quantum noises
(``Those kind of fields we call noise'').
\bigskip

\noindent{\bf Dissipation and decoherence}\smallskip

A typical explicit form for the white noise equation (2) is
$$\partial_tU_t=-(iDb^+_t + D^+b_t)U_t\eqno(3)$$ 
where $b_t$, $b^+_t$ is a Boson white noise satysfying 
$[b_t,b^+_{t'}]=\gamma\delta(t-t')$. The normally ordered form of the 
white noise equation (3) can be proved to be
$$\partial_tU_t=-iDb^+_tU_t-iD^+U_tb_t-YU_t\eqno(4)$$
where 
$$Y=\cases{
\hbox{Ito correction term (mathematical interpretation)}\cr
\hbox{operator valued transport coefficient (physical interpretation)}\cr}$$


The physical meaning of $Y$ is best illustrated by the simplest example:
the vacuum transition amplitude associated to the evolution defined by
equation (4). Let
$\langle\cdot\rangle$ denote vacuum expectation. Then from (4) we
deduce:
$$\partial_t\langle U_t\rangle=-\langle YU_t\rangle=-Y\langle
U_t\rangle$$
$$\langle U_t\rangle=e^{-tY}=e^{-t\{\hbox{Re }(Y)
+i\hbox{ Im }(Y)\}}$$
Here $\hbox{Re }(Y)$ can be proved to be a positive operator, so it
describes damping, while $\hbox{Im }(Y)$, is a self--adjoint operator
describing energy shift. Since
$$|\langle U_t\rangle|^2=\hbox{Prob }\{\hbox{vacuum at $t|$ vacuum at }
0\}$$
is the conditional probability of remaining in the vacuum at time $t$ if
one was in the vacuum at time zero, we conclude that:
\item{--} The real part of the operator transport coefficient drives
decoherence.
\item{--} The imaginary part descrives: oscillatory behaviour\bigskip

\noindent{\bf How to control decoherence}\smallskip

Basic idea:

Since $U_t$ collects the dominating contributions of the Hamiltonian
dynamics, if we are able to control a quantum system so that $\hbox{Re
}(Y)=0$, then
$$\langle U_t\rangle=e^{-it\hbox{ Im }(Y)}$$
so we have a purely oscillatory behaviour i.e.: {\it no decoherence\/}!
\item{--} The stochastic limit scheme is quite general but: the specific form
of $Y$ depends on:
\item{i)} the state of the field (Fock, temperature, squeezing,...)
\item{ii)} $H_I$
\item{iii)} The free field dynamics
\item{iv)} The free system dynamics

However it does not depend on the state of the system!

The simplest example is described by the following ingredients:
\item{--} Boson field $a_k$, $a^+_k$ interacting with a system
\item{--} Fock state of this field
\item{--} interaction
$$H_I(t)=\int_{\Bbb R^3}dk D^+\otimes\overline g_ke^{it(\omega_k-
\omega_0)}a_k+h.c.$$
\item{--} $D^+$ arbitrary system operator (e.g. $D^+=a^+$, $\sigma_+$,
$\sigma_x,\dots$)
\item{--} $\overline g_k$ form factor $\equiv$ control parameter
\item{--} $(\omega_k)$ Field energy
\item{--} $(\omega_0)$ resonant frequency (of the system)
\item{--} here we are in Dipole + RWA

Under these assumptions one can prove that
$$Y=\gamma_-D^+D$$
where $\gamma_-$ is given by:
$$\gamma_-=\int^0_{-\infty}dt\int_{\Bbb R^3}e^{-it(\omega_k-\omega_0)}
|g_k|^2dk$$
Using the identity
$$\int^0_{-\infty}e^{-it\omega}dt=\pi\delta(\omega)-iP.P.{1\over\omega}$$
we get
$$\gamma_-=\int_{\Bbb R^3}dk|g_k|^2\delta(\omega_k-\omega_0)-iP.P.
\int{|g_k|^2\over\omega_k-\omega_0}\,dk$$
which is a generalization of the Kramers--Kr\"onig dispersion relations.
Summing up:
$$\hbox{Re }Y=D^+D\int_{\Bbb R^3}dk|g_k|^2\delta(\omega_k-\omega_0)=
D^+D\hbox{ Re }\gamma_-$$
$\omega_k=\omega_0$ resonant surface
and from this condition we see that
$\hbox{Re }Y=0\Leftrightarrow\hbox{ supp }g_k\cap\{\omega_k=
\omega\}=\emptyset$ (off--resonance condition)
where supp $g_k$ denotes the support of the form factor. This in
principle can be controlled by controlling the frequency $b$ and of the
original Boson field.

But the general theory of stochastic limit proves that:
$$\hbox{Re }\gamma_-=\hbox{ covariance of the noise}$$
Therefore
$$\hbox{ Re }\gamma_-=0\Leftrightarrow\hbox{ no noise}$$
Now, recalling that, in the stochastic limit
$$\hbox{noise }=\lim_{\lambda\to0}\hbox{ field}$$
we conclude that: no noise $\Leftrightarrow\lim\limits_{\lambda\to0}$ field
$=0$. But physical effects of the field remain under the form of an
(operator) energy shif in the system so that the limit white noise
equation is in fact deterministic and looks like.
$$\partial_tU_t=i(\hbox{Im }\gamma_-)D^+DU_t$$
In the paper [AcKoVo97] this was called a: {\it Cheshire cat effect\/}.
\bigskip

\beginsection{3.) Squeezing and the decoherence problem in quantum computers}

\bigskip
Let $a_k$, $a^+_k$ be a quantum field
and let $\langle \cdot \rangle$ be a mean zero
gauge invariant gaussian state on the $a_k,a^+_{k'}$--algebra
with
covariance 
$$\pmatrix{
\langle a^+_ka_{k'}\rangle&0\cr
0&\langle a_ka^+_{k'}\rangle\cr}=
\pmatrix{
n(k)&0\cr
0&m(k)\cr}\delta(k-k')
\eqno(1)$$
Then defining
$$b_k:=\overline c_ka_k+s_ka^+_{-k}  \qquad ;\qquad
b^+_k:=c_ka^+_k+\overline s_ka_{-k}  \eqno(2)$$
the state $\langle \cdot \rangle$ also gives a mean zero
gauge invariant gaussian state on the $b_k,b^+_{k'}$--algebra with
covariance matrix
$$\pmatrix{
\langle b^+_kb_{k'} \rangle   &   \langle b_kb_{k'}\rangle \cr
\langle b^+_kb^+_{k'}\rangle  &   \langle b_kb^+_{k'}\rangle \cr} = \eqno(3)$$
$$= \pmatrix{
(|c_k|^2 n(k) + |s_k|^2  m(k) )  \delta(k-k') &
(c_k\overline s_{-k} n(k) +\overline s_kc_{-k} m(k) )\delta(k+k') \cr
(\overline c_ks_{-k} n(k) +s_k\overline c_{-k} m(k) )\delta(k+k') \rangle  &
(|c_k|^2 m(k) + |s_k|^2  n(k) )  \delta(k-k') \cr}$$

Notice the $\delta(k+k')$ in the off--diagonal elements. This means that
in this state one can have simultaneous creation (resp. annihilation) of
pairs, but momentum has to be conserved. The positivity of the state
is guaranteed by construction.

We suppose that the state $\langle \cdot \rangle$, a fixed quantum state (Fock,
equilibrium, $\ldots$), and that it is mean zero gaussian and
invariant under the second quantized evolution associated to a given
one--particle dynamics defined by a function of momentum $\omega (k)$ (for
example $\omega (k)=|k|$)
$$a_k\mapsto a(t,k)=e^{-it\omega_k}a_k\eqno(4)$$
In this case the process
$$b(t,k):= \overline c_ke^{-it\omega_k}a_k+s_ke^{it\omega_{-k}}a^+_{-k}
  \qquad ;\qquad
b^+(t,k):=c_ke^{it\omega_{k}}a^+_k+\overline s_ke^{-it\omega_{-k}}a_{-k}
\eqno(5)$$
is mean zero, gaussian and time translation invariant, because it is a
linear combination of processes with these properties. The time
dependent field $b(t,k), b^+(t,k)$ is called a {\it squeezed stationary
field}. In particular
$$\pmatrix{
\langle b^+(t,k)b(t',k') \rangle   &   \langle b(t,k)b(t',k')\rangle \cr
\langle b^+(t,k)b^+(t',k')\rangle  &   \langle b(t,k)b^+(t',k')\rangle
\cr}\!=\!\pmatrix{
\langle b^+(0,k)b(t'-t,k') \rangle   &   \langle b(0,k)b(t'-t,k')\rangle \cr
\langle b^+(0,k)b^+(t'-t,k')\rangle  &   \langle b(0,k)b^+(t'-t,k')\rangle \cr} $$
Explicitly the diagonal part of the covariance is equal to
$$ \pmatrix{
\langle b^+(0,k)b(t,k') \rangle   &   0 \cr
0  &   \langle b(0,k)b^+(t,k')\rangle \cr} =\delta(k-k')\cdot \eqno(6)$$
$$\cdot \pmatrix{
|c_k|^2 e^{-it\omega_{-k}}n(k)
+ |s_k|^2e^{it\omega_{k}}m(k)  &  0\cr
0 &
|c_k|^2 e^{it\omega_{-k}}m(k)
+ |s_k|^2e^{-it\omega_{k}}n(k)\cr} $$
while the off diagonal part of the covariance is equal to
$$ \pmatrix{
0   &   \langle b(0,k)b(t,k')\rangle \cr
\langle b^+(0,k)b^+(t,k')\rangle  &   0 \cr}=\delta(k+k')\cdot \eqno(7)$$
$$\cdot \pmatrix{
0  &  c_k\overline s_{-k}e^{-it\omega_{k}}n(k)
+ c_{-k}\overline s_ke^{it\omega_{-k}}m(k)\cr
\overline c_ks_{-k}e^{it\omega_{k}}n(k)
+ \overline c_{-k}s_ke^{-it\omega_{-k}}m(k)& 0\cr} $$
\bigskip
\noindent{\bf Remark}. Instead of the $b$--fields (2), one could also
define the fields
$$d_k:=\overline c_ka_k+s_ka^+_{k}  \qquad ;\qquad
d^+_k:=c_ka^+_k+\overline s_ka_{k}  \eqno(8)$$
leading to the covariance
$$\pmatrix{
\langle d^+_kd_{k'} \rangle   &   \langle d_kd_{k'}\rangle \cr
\langle d^+_kd^+_{k'}\rangle  &   \langle d_kd^+_{k'}\rangle \cr} =
\pmatrix{
(|c_k|^2 n(k) + |s_k|^2  m(k) )   &
(c_k\overline s_{k} n(k) +\overline s_kc_{k} m(k) ) \cr
(\overline c_ks_{k} n(k) +s_k\overline c_{k} m(k) ) \rangle  &
(|c_k|^2 m(k) + |s_k|^2  n(k) )  \cr} \delta(k-k')
\eqno(9)$$
and also in this case, by the same argument as above, one would
obtain a mean zero, gaussian time translation invariant, field.
However it is not clear if these states can be physically realized since
for them simultaneous pair creation (resp. annihilation) would take
place without conservation of momentum.

Now suppose that the field interacts with a system $S$ whose
free Hamiltonian $H_S$ has a discrete spectrum
$$H_S=\sum_n\varepsilon_nP_n\eqno(10)$$
where $\varepsilon_n$ are the energy levels and $P_n$ the spectral
projections. Let the interaction Hamiltonian (x.1) evolved with the
free evolution (interaction representation) be:
$$H_I(t):= \int dk [D^+\otimes \bar g_ke^{-it \omega (k) }a_k +
D\otimes g_k e^{it\omega (k) }a^+_k] =: D^+a_t +Da^+_t\eqno(1)$$
where in (4), and often in the following, we omit the symbol $\otimes$
and where we have introduced the shorthand notations:
$$a_t:= \int dk \bar g_ke^{-it \omega (k) }a_k =A(S_tg) \eqno(12)$$
$$ S_t:= S^1_te^{it \omega_o} = e^{-it(\omega_1(k) - \omega_o)}=:
e^{-it\omega(k)} \eqno(13)$$
$$\omega (k) := \omega_1(k) - \omega_o \eqno(14)$$
The stochastic limit studies the rescaled Schr\"odinger equation
$$ \partial _t U^{(\lambda)}_{t/\lambda^2}\,=-iH^{(\lambda)}_I(t)
U^{(\lambda)}_{t/\lambda^2}\eqno(15)$$
where the {\it rescaled interaction Hamiltonian} is defined by:
$$H^{(\lambda)}_I(t):=
{1\over\lambda}\,H_I(t/\lambda^2)=D^+\left({1\over\lambda}\,a_{t/
\lambda^2}\right)+D\left({1\over\lambda}\,a^+_{t/\lambda^2}\right)
=: D^+a^{(\lambda)}_t  + Da^{(\lambda)+ }_t \eqno(16)$$
and the {\it rescaled fields} are
$$a^{(\lambda)}_t={1\over\lambda}\,a_{t/\lambda^2}={1\over\lambda}\,
\int_{\bf R^d}dk\overline g_ke^{-i(t/\lambda^2)\omega(k)}
a_k\eqno(17)$$
The rescaled fields (17)
converge in the sense of correlators to a quantum white noise $b_t$, $b^+_t$
whose covariance is determined by the reference state of the $a$--field.
In symbols
$$a^{(\lambda)}_t=
{1\over\lambda}\,a_{t/\lambda^2}\to b_t\quad;\qquad a^{(\lambda)+}_t=
{1\over\lambda}\,a^+_{t/\lambda^2}\to b^+_t\eqno(18)$$
$$\lim_{\lambda\to0}{1\over\lambda}\,H_I\left({t\over\lambda^2}\right)
=H_t=D^+b_t+Db^+_t\eqno(19)$$
and the white noise Hamiltonian equation associated to the right hand
side of (19) is
$$ \partial _t U_{t}\,=-iH_I(t) U_{t} =-i(D^+b_t+Db^+_t)U_{t}\eqno(20)$$
Equation (20) can be given a meaning and
admits a unique unitary solution $U_t$. Moreover
$\lim_{\lambda\to0}U^{(\lambda)}_{t/\lambda^2}=U_t$
where convergence is meant in the sense of the matrix elements with
respect to the collective vectors. Now suppose that the
{\it generalized rotating wave approximation} condition is satisfied,
i.e. there exists
a fixed strictly positive real number:
$$\omega _o >0 \eqno(21)$$
such that for any $t\in {\bf R}$ one has
$$e^{itH_S}D  e^{-itH_S}=e^{-i\omega _o t}D \eqno(22)$$

Then, if the reference state of the $a$--field
is the Fock vacuum, then the normally ordered form of the
white noise equation (20) is the following equation:
$$\partial_tU_t\,=-i(D^+U_tb_t+Db^+_tU_t) -YU_t \eqno(23)$$
where
$$ Y = \gamma_-D^+D $$
is the operator transport coefficient and where $\gamma_-$ is given by
the following integral
$$\gamma_- := \int^0_{- \infty} \langle g,S_s g\rangle  ds
=\int^0_{-\infty}dt \int_{{\bf R}^d}e^{-it(\omega_1(k) -\omega_o)} \mid g(k)\mid^2
=\int^0_{- \infty} ds e^{-it\omega_o} G_{-+}(s) \eqno(24)$$
The normally ordered form of the above white noise equation is
equivalent to the quantum stochastic differential equation obtained by
multiplying both sides of (23) by $dt$ and by the identification
$$b^\pm_t = dB^\pm_t$$
and $dB^\pm_t$ is the usual quantum Brownian motion.

Using the normally ordered equation (23) for $U_t$ one can easily
calculate the matrix elements $\langle \psi, U_t \phi \rangle $  for
a large class of states $\psi , \phi $ in the limit space (number or
coherent vectors, $\ldots$). As an illustration we
compute some vacuum--to--vacuum amplitude of the evolution operator
$U_t$ (we mean the vacuum for the operators $b_t$, $b^+_t$).
The equation for this amplitude is
$${d\over dt}\langle U_t\rangle\,=-\gamma_-D^+D
\langle U_t\rangle\eqno(25)$$
The explicit form of the solution is
$$\langle \psi,e^{-t\gamma_-D^+D}\psi\rangle=\langle\psi\otimes\Phi,U_t
\psi\otimes\Phi\rangle\eqno(26)$$

Since both the operator $D^+D$ and the real part of $\gamma_-$ are positive we
see that the generic behavior one should expect from the amplitude (26)
is of an {\bf exponential decay driven by the real part of $\gamma_-$
and modulated by its imaginary part}. The operator $\gamma_-D^+D$
could be interpreted as an {\it operator inverse life--time of the vacuum
in the limit space} in the sense that, on each spectral subspace of
$D^+D$, the inverse lifetime is uniquely defined. So the generic
behaviour is
$$\langle  \alpha, U_t \alpha \rangle  \sim
e^{i t \Delta \varepsilon_\alpha/\hbar} \eqno(27)$$
where $\Delta \varepsilon_\alpha$ is a complex number with a {\bf positive
imaginary part}:
$$\Delta \varepsilon_\alpha = - { 1 \over 2i}\, \Gamma_\alpha +
\Omega_\alpha\ ;\quad\Gamma_\alpha >0\ ;\quad
\Omega_\alpha\in {\bf R}\eqno(28)$$
Replace now the vacuum by an exponential vector $\psi(f)$, $D$ and $D^+$
by $a$, $a^+$ (1--mode creation and annihilation) and denote
$$\beta:=\xi\otimes\psi(f)\ ;\quad\alpha=\xi\otimes\psi(g)$$
where $\xi=|z>$ is alos a coherent vector for the $a$, $a^+$ field.

Then equation (25) becomes
$${d\over dt}\,\langle\beta,U_t\alpha\rangle=(-i\overline zg(t)-iz
\overline f(t)-\gamma_-|z|^2)\langle\beta,U_t\alpha\rangle$$
So, if we could choose $f,g$ and $z$ so that
$$\hbox{Im }(\overline zg(t)+z\overline f(t))=(\hbox{Re
}\gamma_-)|z|^2\eqno(29)$$
we would obtain a purely oscillatory behaviour of the amplitude
$\langle\beta,U_t\alpha\rangle$. Such a choice
cannot be done for any time $t$, because it is physically impossible to
realize {\it coherent vectors with constant test functions}
(plane waves), however we can obtain reasonably good approximations by
taking $f$ and $g$
constant in an arbitrarily large
interval of time, say $[0,T]$ and in this interval the transition
amplitude $\langle\beta,U_t\alpha\rangle$ shall exhibit a purely
oscillatory behaviour.

Another example of purely oscillatory behaviour can be obtained
from the stochastic limit for an Hamiltonian of a two--state system
coupled to an environment and not satisfying
the generalized rotating wave approximation,
Models of this kind arise in several contexts in particular in
connection with
{\it macroscopic quantum coherence phenomena}, related to the SQUID
model [AcKoVo96]. Also here one can prove the emergence of two interesting
regimes for the system:
the {\it purely oscillating} and the {\it purely damping} one.
By experimentally controlling the spectral function of the field one
should be able to bring the system into the purely oscillatory non
decaying regime and this suggests a way to solve the decoherence problem.
 \bigskip

\bigskip
\centerline{\bf Bibliography}

\bigskip
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Compound channels, transition expectations and liftings,
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Dynamics of dissipative two--state systems in the stochastic approximation.
Phys. Rev. A 56 N. 3 (1996).
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Teleportation of general quantum states. in preparation (1998).
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Quantum Theory and its Stochastic Limit.
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\end
