Content-Type: multipart/mixed; boundary="-------------0311040932646" This is a multi-part message in MIME format. ---------------0311040932646 Content-Type: text/plain; name="03-480.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-480.comments" 46 pages ---------------0311040932646 Content-Type: text/plain; name="03-480.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-480.keywords" quantum probability, continuous measurement, stochastic differential equations ---------------0311040932646 Content-Type: application/x-tex; name="attal_pautrat_1.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="attal_pautrat_1.tex" \magnification=1200 \vsize=187mm \hsize=125mm \hoffset=4mm \voffset=10mm %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX \abovedisplayskip=4.5pt plus 1pt minus 3pt \abovedisplayshortskip=0pt plus 1pt \belowdisplayskip=4.5pt plus 1pt minus 3pt \belowdisplayshortskip=2.5pt plus 1pt minus 1.5pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \def\megaskip{\vskip12mm plus 4mm minus 4mm} %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX % Les Hauts de Pages %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX \newif\ifpagetitre \pagetitretrue \newtoks\hautpagetitre \hautpagetitre={\hfil} \newtoks\baspagetitre \baspagetitre={\hfil\tenrm\folio\hfil} \newtoks\auteurcourant \auteurcourant={\hfil} \newtoks\titrecourant \titrecourant={\hfil} \newtoks\chapcourant \chapcourant={\hfil} \newtoks\hautpagegauche \newtoks\hautpagedroite \hautpagegauche={\vbox{\it\noindent\the\chapcourant\hfill\the\auteurcourant\hfill { }\smallskip\smallskip\vskip 2mm\line{}}} \hautpagedroite={\vbox{\hfill\it\the\titrecourant\hfill{ } \smallskip\smallskip\vskip 2mm\line{}}} \newtoks\baspagegauche \newtoks\baspagedroite \baspagegauche={\hfil\tenrm\folio\hfil} \baspagedroite={\hfil\tenrm\folio\hfil} \headline={\ifpagetitre\the\hautpagetitre \else\ifodd\pageno\the\hautpagedroite \else\the\hautpagegauche\fi\fi} \def\nopagenumbers{\def\folio{\hfil}} \footline={\ifpagetitre\the\baspagetitre \global\pagetitrefalse \else\ifodd\pageno\the\baspagedroite \else\the\baspagegauche\fi\fi} %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX \auteurcourant={} \titrecourant={} %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX % MACROS PERSONNELLES %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX \font\itsept=cmitt10 scaled 700 \font\itonze=cmitt10 scaled 1100 \font\timequatorze=cmr10 scaled 1440 \font\timeonze=cmr10 scaled 1100 \font\timedouze=cmr12 \font\timeneuf=cmr9 \font\timehuit=cmr8 \font\timesept=cmr7 \font\timecinq=cmr10 scaled 500 \font\bfquarante=cmbx10 scaled 4000 \font\bftrente=cmbx10 scaled 3000 \font\bfvingt=cmbx10 scaled 2000 \font\bfdixsept=cmbx10 scaled 1700 \font\bfquinze=cmbx10 scaled 1500 \font\bfquatorze=cmbx10 scaled\magstep2 \font\bftreize=cmbx10 scaled 1300 \font\bfdouze=cmbx12 scaled\magstep1 \font\bfonze=cmbx10 scaled 1100 \font\bfneuf=cmbx9 scaled 900 \font\bfhuit=cmbx8 scaled 800 \font\bfsept=cmbx7 scaled 700 \font\goth=eufm10 \def\ZZ{{\mathord{Z\!\!\! 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\int}}}} \def\qq{\qquad} \def\Ker{\mathop{{\rm Ker}}} \def\Ran{\mathop{{\rm Ran}}} \def\Dom{\mathop{{\rm Dom}}} \def\cd{\cdot} \def\und#1{{\underline{#1}}} \def\aa{a.a.\thsp} \def\sm{\smallskip} \def\bg{\bigskip} \def\frac#1#2{{{#1}\over{#2}}} \def\seq#1{{(#1_n)}_{n\in\NN}} \def\vect#1{\vec{#1}} \def\rA{{\cal A}}\def\rB{{\cal B}}\def\rC{{\cal C}}\def\rD{{\cal D}} \def\rE{{\cal E}}\def\rF{{\cal F}}\def\rG{{\cal G}}\def\rH{{\cal H}} \def\rI{{\cal I}}\def\rJ{{\cal J}}\def\rK{{\cal K}}\def\rL{{\cal L}} \def\rM{{\cal M}}\def\rN{{\cal N}}\def\rO{{\cal O}}\def\rP{{\cal P}} \def\rQ{{\cal Q}}\def\rR{{\cal R}}\def\rS{{\cal S}}\def\rT{{\cal T}} \def\rU{{\cal U}}\def\rQ{{\cal Q}}\def\rV{{\cal V}}\def\rW{{\cal W}} \def\rX{{\cal X}}\def\rY{{\cal Y}}\def\rZ{{\cal Z}} \def\rl{\ell} \def\ut{\tilde \phi} \def\vt{\tilde \psi} \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon} \def\f{\phi} \def\s{\sigma} \def\t{\tau} \def\r{\rho} \def\w{\omega} \def\p{\psi} \def\x{\chi} \def\m{\mu} \def\n{\nu} \def\nn{\eta} \def\o{\omega} \def\G{\Gamma} \def\D{\Delta} \def\L{\Lambda} \def\S{\Sigma} \def\O{\Omega} \def\P{\Psi} \def\Y{\Psi} \def\N{\nabla} \def\T{\tau} \def\F{\Phi} \def\NNE{{\NN^\ast}} %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX \def\derp#1#2{\frac{\partial #1}{\partial #2}} \auteurcourant={St\'ephane ATTAL and Yan PAUTRAT} \titrecourant={From repeated to continuous quantum interactions} \titredeux{From repeated to continuous}{quantum interactions}{St\'ephane ATTAL and Yan PAUTRAT} \bigskip \bigskip \sspa{}{Abstract}{\sevenrm We consider the general physical situation of a quantum system $\rH_0$ interacting with a chain of exterior systems $\bigotimes_\NNE \rH$, one after the other, during a small interval of time $h$ and following some Hamiltonian $H$ on $\rH_0\otimes\rH$. We discuss the passage to the limit to continuous interactions ($h\rightarrow 0$) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems $\bigotimes_{\Rp}\rH$. Surprisingly, the passage to the limit shows the necessity for 3 different time scales in $H$. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. We show that the three time scales correspond to the normal regime, the weak coupling limit and the low density limit. Our approach allows to consider these two physical limits altogether for the first time. Their combination produces an effective Hamiltonian on the small system, which had never been described before. We apply these results to give an Hamiltonian description of the von Neumann measurement. We also consider the approximation of continuous time quantum master equations by discrete time ones. In particular we show how any Lindblad generator is obtained as the limit of completely positive maps.} \def\ch#1{\widehat{#1}} \def\abs#1{\left| #1 \right| } \def\absca#1{\left| #1 \right| ^2} \def\norme#1{\left\| #1 \right\|} \def\normeca#1{{\left\| #1 \right\|}^2} \def\normei#1{\left\| #1 \right\| _{\infty}} \def\normeica#1{{\left\| #1 \right\| _{\infty}}^2} \def\pr#1{#1 ^{\prime}} \def\somu#1{\sum_{\scriptstyle{#1}}} \def\conj#1{\overline{#1}} \def\rac#1{\left( #1 \right)^{\frac{1}{2}}} \def\intk#1{\int _{t_#1} ^{t_{#1+1}} \!\!} \vfill\eject \spa{}{Contents} \bigskip \noindent I. {\bf Introduction} \smallskip \noindent II. {\bf Discrete dynamics on the atom chain} II.1 Repeated quantum interactions II.2 Structure of the atom chain II.3 Unitary dilations of completely positive maps \smallskip \noindent III. {\bf From the atom chain to the atom field} III.1 Structure of the atom field III.2 Quantum noises III.3 Embedding and approximation by the atom chain III.4 Quantum Langevin equations \smallskip \noindent IV. {\bf Convergence theorems} IV.1 Convergence to quantum Langevin equations IV.2 Typical Hamiltonian: weak coupling and low density IV.3 Hamiltonian description of von Neumann measurements IV.4 One example IV.5 From completely positive maps to Lindbladians \bigskip \bigskip \spa{I.}{Introduction} Quantum Langevin equations as a model for quantum open systems have been considered for at least 40 years (for example [FKM], [FLO], [AFL]). They have been given many different meanings in terms of several definitions of quantum noises or quantum Brownian motions (for example [G-Z], [H-P], [GSI]). One of the most developed and useful mathematical languages developed for that purpose is the quantum stochastic calculus of Hudson and Parthasarathy and their quantum stochastic differential equations ([H-P]). The quantum Langevin equations they allow to consider have been used very often to modelize typical situations of quantum open systems: continual quantum measurement ([Ba1], [B-B]), quantum optics ([F-R], [FRS] [Ba2]), electronic transport [BRSW], thermalization ([M-R], [L-M]), etc. The justification for such quantum Langevin equation is often given in terms of some particular approximations of the true Hamiltonian interaction dynamic: rotating wave approximation, Markov approximation, large band approximation (cf [G-Z] chapter 11). They are also often justified as natural dilations of quantum master equations on the small system. That is, for any (good) semigroup of completely positive maps on the small system (with Lindblad generator $\rL$), one can dilate the small system with an appropriate Fock space, and obtain an explicit quantum stochastic differential equation on the whole space. The unique solution of this equation is a unitary evolution (in interaction picture) such that the trace on the small system of the induced evolution yields the original semigroup. This corresponds, at the quantum level, to the well-known way of realizing a concrete Markov process from a given semigroup (or generator) by adding a noise space to the (classical) system space and solving an adequate stochastic differential equation. Some quantum stochastic differential equations have also been obtained in the so-called {\it stochastic limit} from explicit Hamiltonian dynamics ([A-L], [AGL], [ALV]). This shows some similarities with the results described here, but the limits considered in those articles are in the sense of the convergence of processes living in a different space than the one of the Hamiltonian dynamic. \bigskip In this article we consider the effective Hamiltonian dynamic describing the repeated interactions, during short time intervals of length $h$, of a small system $\rH_0$ with a chain of exterior systems $\otimes_\NNE \rH$. We embed all these chains as particular subspaces, attached to the parameter $h$, of a continuous field $$\bigotimes_{\Rp} \rH$$ in such a way that the subspaces associated to the chain increase and fill the field when $h$ tends to 0. By developing an appropriate language of the chain $\otimes_\NNE\rH$ and of the field $\otimes_{\Rp}\rH$ and by describing the discrete time Hamiltonan evolution generated by the repeated interactions, we are able to pass to the limit when $h\rightarrow 0$ and prove that the limit evolution equation is solution of a quantum stochastic differential equation. This limit is obtained in the weak topology of operators and in a single space: the continuous field $\bigotimes_{\Rp}\rH$. \bigskip Of course, such a limit cannot be obtained without assumptions on the elementary interaction Hamiltonian $H$. This is similar to the central limit theorem: a random walk gives a trivial limit when its time step $h$ goes to zero and it is only when suitably renormalized (by a factor $\sqrt h$) that it yields a Gaussian. Other normalizations give either trivial limits or no limit at all. In our Hamiltonian context the situation is going to be the same. For a non-trivial limit of these repeated interactions to exist, we will need the Hamiltonian $H$ to satisfy some renormalization properties. The surprise here is that the necessary renormalization factor is not global, it is different following some parts of the Hamiltonian operator. We identify 3 different time scales in $H$: one of order 1, one of order $\sqrt h$, one of order $h$. We describe a class of Hamiltonian which seems to be typical for the above conditions to be satisfied. These typical Hamiltonians are clearly a combination of free evolution, weak coupling limit typical hamiltonians and low density limit typical Hamiltonians. This physically explains the three different time scales. But the originality of our approach allows to consider both limits together; to our knowledge this constitutes a novelty in the literature. As a consequence, the combination of the two limits shows an effective Hamiltonian for the small system which is very surprising: it contains a new term $$ V^\ast D^{-2}(\sin D-D)V $$ which comes from the presence of both the weak coupling and the low density limit in the Hamiltonian. It seems that such a term had never been described before. \bigskip This article is structured as follows. In section II we present the exact mathematical model of repeated quantum interactions and end up with the associated evolution equation (subsection II.1). We then introduce a mathematical setup for the study of the space $\bigotimes_\NNE\rH$ which will help much for passing to the continuous field. In particular this includes a particular choice of an orthonormal basis of the phase space and a particular choice of a basis for the operators on that phase space (subsection II.2). Finally we show how the typical evolution equations obtained in II.1 are the general model for the unitary dilation of any given discrete semigroup of completely positive maps (subsection II.3). Section III is devoted to presenting the whole formalism of the continuous atom field. In subsection III.1 we present the space which is candidate for representing the continuous field limit of the atom chain. It is actually a particular Fock space on which we develop an unusual structure which clearly shows the required properties. In subsection III.2 we present the natural quantum noises on the continuous field and the associated quantum stochastic integrals, the quantum Ito formula and the quantum stochastic differential equations. In subsection III.3 we concretely realize the atom chain of section II as a strict subspace of the atom field. Not only do we realize it as a subspace, but also realize the action of its basic operators inside the atom field. All these atom chain subspaces are related to a partition of $\Rp$. When the diameter of the partition goes to 0, we show that the corresponding subspace completely fills the continuous field and the basic operators of the chain converge to the quantum noises of the field (with convenient normalizations). Finally, considering the projection of the continuous atom field onto an atom chain subspace, we state a formula for the projection of a general quantum stochastic integral. In section IV all the pieces of the puzzle fit together. By computing the projection on the atom chain of a quantum stochastic differential equation we show that the typical evolution equation of repeated interactions converges in the field space to the solution of a quantum Langevin equation, assuming the fact that the associated Hamiltonian satisfies some particular renormalization property corresponding to three different time scales. It is to that result and to some of its extensions that subsection IV.1 is devoted. In subsection IV.2 we describe a family of Hamiltonians which seems to be typical of the conditions obtained above. We show that this family of Hamiltonians describes altogether free evolution, weak coupling limit and low density limit terms. Computing the associated quantum Langevin equation at the limit, we obtain an effective Hamiltonian on $\rH_0$ which contains a new term. This new term appears only when weak coupling and low density limits are in presence together in the Hamiltonian. In subsection IV.3, we apply these results to describe the von Neumann measurement apparatus in the Hamiltonian framework of repeated quantum interactions. In subsection IV.4 we explicitly compute a simple example. In subsection IV.5 we show that our approximation theorem puts into evidence a natural way that completely positive maps have to converge to Lindblad generators. \spa{II.}{Discrete dynamics on the atom chain} \sspa{II.1}{Repeated quantum interactions} We here give a precise description of our physical model: repeated quantum interactions. \bigskip We consider a small quantum system $\rH_0$ and another quantum system $\rH$ which represents a piece of environment, a measuring apparatus, incoming photons$\ld$ We consider the space $\rH_0\otimes \rH$ in order to couple the two systems, an Hamiltonian $H$ on $\rH_0\otimes\rH$ which describes the interaction and the associated unitary evolution during the interval $[0,h]$ of time: $$ \LL=e^{-ihH}. $$ This single interaction is therefore described in the Schr\"odinger picture by $$ \r\mapsto \LL\,\r\,\LL^\ast $$ and in the Heisenberg picture by $$ X \mapsto \LL^\ast X \LL.$$ \bigskip Now, after this first interaction, we repeat it but this time coupling the same $\rH_0$ with a new copy of $\rH$. This means that that new copy was kept isolated until then; similarly the previously considered copy of $\rH$ will remain isolated for the rest of the experience. One can think of many physical examples where this situations arises: in repeated quantum measurement where a family of identical measurement devices are presented one after the other before the system (or a single device is refreshened after every use), in quantum optics where a sequence of independent atoms arrives one after the other to interact with a field in some cavity for a short time. More generally it can be seen as a good model if it is assumed that perturbations in $\rH$ due to the interaction are dissipated after every time $h$. The sequence of interactions can be described in the following way: the state space for the whole system is $$\rH_0\otimes\bigotimes_\NNE\rH $$ Index for a few lines only the copies of $\rH$ as $\rH _1$, $\rH _2$, $\ld$ Define then a unitary operator $\LL _n$ as the canonical ampliation to $\rH_0\otimes \rH _1 \otimes \rH _2 \otimes \ldots$ of the operator which acts as $\LL$ on $\rH _0 \otimes \rH_n$; that is, $\LL _n$ acts as the identity on copies of $\rH$ other than $\rH _n$. The effect of the n-th interaction in the Schr\"odinger picture writes then $$ \r \mapsto \LL _n \, \r\, \LL _n ^*,$$ for every density matrix $\r$, so that the effect of the $n$ first interactions is $$ \r \mapsto u_n\, \r\, u_n^*$$ where $(u_n)_{n\in\NN}$ is a sequence in $\rB(\rH _0 \otimes \bigotimes _{\NNE}\rH)$ which satisfies the equations $$ \cases{ u_{n+1} = \LL_{n+1} \, u_n \cr \hp{{}_{1}}u_0 \hp{{}_{+}} = \ I.} \eqno{(1)} $$ It is evolution equations such as (1) that we are going to study in this article. \sspa{II.2}{Structure of the atom chain} We here describe some useful mathematical structure on the space $\otimes_\NNE\rH$ which will constitute the main ingredient of our approach. \bigskip Let us fix a particular Hilbertian basis $(X^i)_{i\in\L \cup \{0\}}$ for the Hilbert space $\rH$, where we assume (for notational purposes) that $0 \not\in \L$. This particular choice of notations is motivated by physical interpretations: indeed, we see the $X^i$, $i \in \L$, as representing for example the different possible excited states of an atom. The vector $X^0$ represents the ``ground state" or ``vacuum state" of the atom and will usually be denoted $\Omega$. Let $\TF$ be the tensor product $\bigotimes _{\NNE} \rH$ with respect to the stabilizing sequence $\Omega$. In other words, this means simply that an orthonormal basis of $\TF$ is given by the family $$ \{ X_A; \ A \in \rP_{\NNE,\L}\}$$ where \def\seqe#1{{{(#1_n)}_{n\in\NNE}}} -- the set $\rP _{\NN, \L}$ is the set of finite subsets $$ \{ (n_1, i _1), \ldots, (n_k, i_k)\}$$ of $\NNE \times \L$ such that the $n_i$'s are mutually different. Another way to describe the set $\rP_{\NNE,\L}$ is to identify it to the set of sequences $\seqe A$ with values in $\L \cup \{0\}$ which take a value different from 0 only finitely often. -- $X_A$ denotes the vector $$ \O\otimes\ld\otimes\O\otimes X^{i_1}\otimes \O\otimes\ld\otimes\O\otimes X^{i_2}\otimes\ld $$ where $X^{i_1}$ appears in $n_1$-th copy of $\rH$... \smallskip The physical signification of this basis is easy to understand: we have a chain of atoms, indexed by $\NNE$. The space $\TF$ is the state space of this chain, the vector $X_A$ with $A=\{ (n_1, i_1), \ldots, (n_k, i _k)\}$ representing the state in which exactly $k$ atoms are excited: atom $n_1$ in the state $i _1$, etc, all other atoms being in the ground state. \bigskip This particular choice of a basis gives $\TF$ a particular structure. If we denote by $\TF_{n]}$ the space generated by the $X_A$ such that $A\subset\{1,\ld,n\}\times\L$ and by $\TF_{[m}$ the one generated by the $X_A$ such that $A\subset\{m,m+1,\ld\}\times\L$, we get an obvious natural isomorphism between $\TF$ and $\TF_{n-1]}\otimes \TF_{[n}$ given by $$ [f\otimes g](A)=f\left(A\cap\{1,\ld,n-1\}\times\L\right)\, g\left(A\cap\{n,\ld\}\times\L\right). $$ Put $\{a^i_j;i,j \in \L\cup\{0\} \}$ to be the natural basis of $\rB(\rH)$, that is, $$ a^i_j(X^k)=\d_{ik}X^j. $$ We denote by $a^i_j(n)$ the natural ampliation of the operator $a^i_j$ to $\TF$ which acts on the copy number $n$ as $a^i_j$ and the identity elsewhere. That is, in terms of the basis $X_A$, $$ a^i_j(n)X_A=\indic_{(n,i)\in A}X_{A\setminus (n,i)\cup (n,j)} $$ if neither $i$ nor $j$ is zero, and $$ \eqalign{ a^i_0 (n) X_A &= \indic_{(n,i) \in A} X_{A \setminus (n,i)}, \cr a^0_j(n)X_A &=\indic_{(n,0) \in A} X_{A\cup (n,j)},\cr a^0_0 (n) X_A &= \indic_{(n,0) \in A} X_{A},\cr} $$ where $(n,0)\in A$ actually means ``for any $i$ in $\L$, $(n,i)\not\in\L$''. \bigskip \sspa{II.3}{Unitary dilation of completely positive maps} The evolution equations $$ u_n=\LL_n\ld \LL_1 $$ obtained in the physical setup of repeated quantum interactions are actually of mathematical interest on their own for they provide a canonical way of dilating discrete semigroups of completely positive maps into unitary automorphisms. \bigskip The mathematical setup is the same. Let $\LL$ be any operator on $\rH_0\otimes\rH$. Let $\TF=\otimes_{\NN^\ast}\rH$ and $(\LL_n)_{n\in\NNE}$ be defined as in the above section. We then consider the associated evolution equations $$ u_n=\LL_n\ld \LL_1\eqno{(1)} $$ with $u_0=I$. \bigskip The following result is obvious. \prp{1.}{\it The solution $\seq u$ of (1) is made of unitary (resp. isometric, contractive) operators if and only if $\LL$ is unitary (resp. isometric, contractive).} \qed \bigskip Note that if $\LL$ is unitary, then the mappings $$ j_n(H)=u_n^\ast H u_n $$ are automorphisms of $\rB(\rH_0\otimes\rH)$. \bigskip Let $\EE_0$ be the partial trace on $\rH_0$ defined by $$ \ps{\f}{\EE_0(H)\,\psi}=\ps{\f\otimes\O}{H\,\psi\otimes\O} $$ for all $\f,\psi\in\rH_0$ and every operator $H$ on $\rH_0\otimes \TF$. Unitary dilations of completely positive semigroups are obtained in the following theorem. Recall that, by Kraus' theorem, any completely positive operator $\ell$ on $\rB(\rH _0)$ is of the form $$ \ell(X) = \sum _{i\in \NN} A_i^\ast X A_i$$ where the summation ranges over $(\L \cup \{0\})^2$, the $A_i$ are bounded operators and the sum is strongly convergent. Conversely, any such operator is completely positive. \smallskip \noindent{\bf Remark}: Of course the Kraus form of an operator is {\it a priori} indifferent to the specificity of the value $i=0$. The special role played by one of the indices will appear later on. \th{2.}{\it Let $\LL$ be any unitary operator on $\rH_0\otimes\rH$. Consider the coefficients $(\LL^i_j)_{i,j\in\L\cup\{0\}}$, which are operators on $\rH_0$, of the matrix representation of $\LL$ in the basis $\O,X^i$, $i\in\L$ of $\rH$. Then, for any $X\in\rB(\rH_0)$ we have $$ \EE_0[j_n(X\otimes I)]=\ell^n(X) $$ where $\ell$ is the completely positive map on $\rB(\rH_0)$ given by $$ \ell(X)=\sum_{i\in\L\cup\{0\}} (\LL^0_i)^\ast X \LL^0_i. $$ Conversely, consider any completely positive map $$ \ell(X)=\sum_{i\in\L\cup\{0\}} A_i^\ast XA_i $$ on $\rB(\rH_0)$ such that $\ell(I)=I$. Then there exists a unitary operator $\LL$ on $\rH_0\otimes\rH$ such that the associated unitary family of automorphisms $$ j_n(H)=u_n^\ast Hu_n $$ satisfies $$ \EE_0[j_n(X\otimes I)]=\ell^n(X), $$ for all $n\in\NN$. } \prf Consider $\LL=(\LL^i_j))_{i,j\in\L\cup\{0\}}$ such as in the above statements. Consider the unitary family $$ u_n=\LL_n\ld \LL_1. $$ Note that $$ u_{n+1}=\LL_{n+1}u_n. $$ Put $j_n(H)=u_n^\ast H u_n$ for every operator $H$ on $\rH_0\otimes\rH$. Then, for any operator $X$ on $\rH_0$ we have $$ j_{n+1}(X\otimes I)= u_n^\ast \LL_{n+1}^\ast(X\otimes I)\LL_{n+1}u_n. $$ When considered as a matrix of operators on $\rH_0$, in the basis $\O,X^i$, $i\in\L$ of $\rH$, the matrix associated to $X\otimes I$ is of diagonal form. We get $$ \di{ \LL_{n+1}^\ast(X\otimes I)\LL_{n+1}=\hf\cr \hf=\left(\matrix{(\LL^0_0)^\ast&(\LL^0_1)^\ast&\ld\cr (\LL^1_0)^\ast\ecarte&(\LL^1_1)^\ast&\ld\cr \vdots&\vdots&\ddots\cr}\right) \left(\matrix{X&0&\ld\cr 0&\ecarte X&\ld\cr \vdots&\vdots&\ddots\cr }\right)\left(\matrix{\LL^0_0&\LL^1_0&\ld\cr \LL^0_1&\ecarte\LL^1_1&\ld\cr \vdots&\vdots&\ddots}\right)\cr } $$ \def\bLL{{\bf\LL}} which is the matrix $\bLL_{n+1}(X)={(B^i_j(X))}_{i,j\in\L\cup\{0\}}$ with $$ B^i_j(X)=\sum_{k\in\L\cup\{0\}} (\LL^j_k)^\ast X\LL^i_k. $$ Note that the operator $\bLL_{n+1}(X)$ acts non trivialy only on the tensor product of $\rH_0$ with the $(n+1)$-th copy of $\rH$. When represented as an operator on $$ \rH_0\otimes \TF_{n+1]}=\left(\rH_0\otimes \TF_{n]}\right)\otimes\rH $$ as a matrix with coefficients in $\rB(\rH_0\otimes \TF_{n]})$ it writes exactly in the same way as above, just replacing $B^i_j(X)$ (which belongs to $\rB(\rH_0)$) by $$B^i_j(X)\otimes I_{\vert\TF_{n]}}.$$ Also note that, as can be proved by an easy induction, the operator $u_n$ acts on $\rH_0\otimes \TF_{n]}$ only. As an operator on $\rH_0\otimes \TF_{n+1]}$ it is represented by a diagonal matrix. Thus $j_{n+1}(X)= u_n^\ast\bLL_{n+1}(X)u_n$ can be written on $\rH_0\otimes \TF_{(n+1)]}=\rH_0\otimes \TF_{n]}\otimes\rH$ as a matrix of operators on $\rH_0\otimes \TF_{n]}$ by $$ \left(j_{n+1}(X\otimes I)\right)^i_j=j_n(B^i_j(X)\otimes I). $$ Note that $B^0_0(X)=\sum_{i\in\L\cup\{0\}}(\LL^0_i)^\ast X \LL^0_i$ which is the mapping $\ell(X)$ of the statement. Put $T_n(X)=\EE_0[j_n(X\otimes I)]$. We have, for all $\f,\Y\in\rH_0$ $$ \eq{ \ps{\f}{T_{n+1}(X)\Y}&=\ps{\f\otimes\O}{j_{n+1}(X\otimes I)\,\Y\otimes\O}\cr &=\ps{\f\otimes\O}{\left(j_{n}(B^i_j(X)\otimes I)\right)_{i,j}\Y\otimes\O}\cr &=\ps{\f\otimes\O_{\,\TF_{n]}}\otimes\O_{\,\rH}}{\left(j_{n}(B^i_j(X)\otimes I)\right)_{i,j}\Y\otimes\O_{\,\TF_{n]}}\otimes\O_{\,\rH}}\cr &=\ps{\f\otimes\O_{\,\TF_{n]}}}{j_{n}(B^0_0(X)\otimes I)\Y\otimes\O_{\,\TF_{n]}}}\cr &=\ps{\f}{T_n(\ell(X))\Y}.\cr } $$ This proves that $T_{n+1}(X)=T_n(\ell(X))$ and the first part of the theorem is proved. \smallskip Conversely, consider a decomposition of a completely positive map $\ell$ of the form $$ \rL(X)=\sum_{i\in\L\cup\{0\}} A_i^\ast XA_i $$ for a familly $(A_i)_{i\in\L\cup\{0\}}$ of bounded operators on $\rH_0$ such that $$\sum_{i\in\L\cup\{0\}} A_i^\ast A_i=I.$$ We claim that there exists a unitary operator $\LL$ on $\rH_0\otimes \rH$ of the form $$ \LL=\left(\matrix{A_0&\ld&\ld\cr A_1&\ld&\ld\cr\vdots&\vdots&\ddots\cr }\right). $$ Indeed, the condition $\sum_{i\in\L\cup\{0\}}A_i^\ast A_i=I$ guarantees that the first columns of $\LL$ are made of orthonormal vectors of $\rH_0\otimes \rH$. We can thus complete the matrix by completing it into an orthonormal basis of $\rH_0\otimes \rH$. This makes out a unitary matrix $\LL$ the coefficients of which we denote by ${(A^i_j)}_{i,j\in\L\cup\{0\}}$. Note that $A^0_i=A_{i+1}$. We now conclude easily by the first part of the theorem. \qed \spa{III}{From the atom chain to the atom field} \sspa{III.1}{Structure of the atom field} We now describe the structure of the continuous version of $\TF$. The structure we are going to present here is rather original and not much expanded in the literature. It is very different from the usual presentation of quantum stochastic calculus ([H-P]), but it actually constitutes a very natural language for our purpose: approximation of the atom field by atom chains. This approach is taken from [At1]. We first start with a heuristical discussion. By a continuous version of the atom chain $\TF$ we mean a Hilbert space with a structure which makes it the space $$ \F=\bigotimes_{\RR^+}\rH. $$ We have to give a meaning to the above notation. This could be achieved by invoquing the framework of continous tensor products of Hilbert spaces (see [Gui]), but we prefer to give a self-contained presentation which fits better with our approximation procedure. Let us make out an idea of what it should look like by mimicking, in a continuous time version, what we have described in $\TF$. The countable orthonormal basis $X_A, A\in\rP_{\NNE, \L}$ is replaced by a continuous orthonormal basis $d\x_\s,\, \s\in\rP_{\RR, \L}$, where $\rP_{\RR, \L}$ is the set of finite subsets of $\RR^+\times \L$. With the same idea as for $\TF$, this means that each copy of $\rH$ is equipped with an orthonormal basis $\O,d\x^i_t$, $i\in\L$ (where $t$ is the parameter attached to the copy we are looking at). The orthonormal basis above is just the one obtained by specifying a finite number of sites $t_1,\ld,t_n$ which are going to be excited, the other ones being supposed to be in the fundamental state $\O$, and by specifying their level of excitation. The representation of an element $f$ of $\TF$: $$ \eqalign{ f&=\sum_{A\in\rP_{\NNE, \L}} \! f(A)\, X_A\cr \normca f&=\sum_{A\in\rP_{\NNE,\L}} \! \ab{f(A)}^2\cr } $$ is replaced by an integral version of it in $\F$: $$ \eq{ f&=\int_{\rP_{\RR,\L}} \! f(\s)\, d\x_\s\cr \normca f&=\int_{\rP_{\RR,\L}} \! \ab f^2\, d\s.\cr } $$ This last integral has to be explained: the measure $d\s$ is a ``Lebesgue measure'' on $\rP_{\RR,\L}$, as will be explained later. From now on, the notation $\rP$ will denote, depending on the context, spaces of the type $\rP_{\NNE,\L}$ or $\rP_{\RR,\L}$. A good basis of operators acting on $\F$ can be obtained by mimicking the operators $a^i_j(n)$ of $\TF$. We will here have a set of infinitesimal operators $da^i_j(t)$, $i,j \in\L\cup\{0\}$, acting on the ``t-th" copy of $\rH$ by: $$ \eq{ da^0_0(t)\, d\x_\s&=d\x_{\s}\, dt\, \indic_{t\not\in \s}\cr da^0_i(t)\, d\x_\s&=d\x_{\s\cup \{(t,i)\}}\, \indic_{t\not\in \s}\cr da^i_0(t)\, d\x_\s&=d\x_{\s\setminus \{(t,i)\}}\, dt\, \indic_{(t,i)\in \s}\cr da^i_j(t)\, d\x_\s&=d\x_{\s\setminus\{(t,i)\}\cup\{(t,j)\}}\, \indic_{(t,i)\in \s}\cr } $$ for all $i,j \in \L$. \bigskip We shall now describe a rigourous setup for the above heuristic discussion. \def\pcc{\rP} \def\rb{\RR} \def\cb{\CC} \def\pcn{\rP_n} \def\fc{\rF} \def\vi{\emptyset} We recall the structure of the bosonic Fock space $\F$ and its basic structure (cf [At1] for more details and [At3] for a complete study of the theory and its connections with classical stochastic processes). Let $\rH$ be, as before, a Hilbert space with an orthonormal basis $X^i$, $i \in \L \cup \{ 0\}$ and let $\rH '$ be the closed subspace generated by vectors $X^i$, $i\in\L$ (or simply said, the orthogonal of $X^0$). Let $\F=\G_s(L^2(\Rp, \rH '))$ be the symmetric (bosonic) Fock space over the space $L^2(\Rp, \rH ')$. We shall here give a very efficient presentation of that space, the so-called {\it Guichardet interpretation} of the Fock space. Let $\pcc$ ($=\pcc_{\rb, \L}$) be the set of finite subsets $\{(s_1,i_1),\ld,(s_n,i_n)\}$ of $\rb^+\times \L$ such that the $s_i$ are two by two different. Then $\pcc = \cup_n \pcn$ where $\pcn$ is the subset of $\pcc$ made of $n$-elements subsets of $\ \rb^+\times \L$. By ordering the $\Rp$-part of the elements of $\s\in\pcn$, the set $\pcn$ can be identified to the increasing simplex $\Sigma _n = \{0\vert \leq\cr &\leq\ab{\ps{ a \otimes \e(\f)}{(U_{t_k} - U_s) b \otimes \e(\psi) }}\cr & \ \ +\ab{\ps{ a \otimes \e(\f)}{(\es U_{t_k}\es - U_{t_k}\es) b \otimes \e(\psi)}}\cr &\ \ + \ab {\ps{a \otimes \e(\f)}{( U_{t_k}\es - U_{t_k})b \otimes \e(\psi) }}\cr &\leq\sum_{i,j}\int_{t_k}^s\ab{\bar\f_i(u)}\ab{\psi_j(u)}\ab{\ps{a\otimes\e(\f)}{L^i_j U_ub\otimes\e(\psi)}}\, du \cr &\ \ +\norme{(I-\es)a\otimes\e(\f)}\norme{U_{t_k}\es b\otimes\e(\psi)}\cr &\ \ + \norme{ U_{t_k}^\ast a\otimes\e(\f)}\norme{(I-\es) b \otimes \e(\psi)}\cr } $$ and we conclude as in the previous lemma. \qed \bigskip \noindent We can now prove Theorem 13. \smallskip \noindent{\bf Proof of Theorem 13} Let $\o^i_j(h)$ be such that $$ \LL^i_j(h)-\d_{ij}I=h^{\e_{ij}}(L^i_j+\o^i_j(h)) $$ for all $i,j$ in $\La \cup \{0\}$. In particular we have that, $$ \sum _{i,j \in \La \cup \{0\}} \normeca{\o^i_j(h)}$$ converges to 0 when $h$ tends to 0. Consider the solution $\seqe u$ of $$ u_{n+1}=\LL_{n+1}u_n $$ with the notations of section II.3. Note that if $A$ denotes the matrix $\LL-(\d_{ij}I)_{i,j}$ we then have $$ u_{n+1}-u_n=A_{n+1}u_n. $$ Let $F$ be the matrix $(h^{\e_{ij}}L^i_j+\wh\d_{ij}hL^0_0)_{i,j}$ where $$ \wh\d_{ij}=\cases{1 & if $i=j$ and $(i,j)\not=(0,0)$,\cr 0&if $i\not=j$ or $(i,j)=(0,0)$\cr} $$ and consider the solution $\seq v$ of the equation $$ v_{n+1}-v_n=F_{n+1}v_n. $$ Note that $$ A_{n+1}=\sum_{i,j}A^i_ja^i_j(n+1) $$ and similarly for $F_{n+1}$. Also note that $a^i_j(n+1)$ commutes with $u_n$ ({\it resp.} $v_n$), for they do not act on the same part of the space $\TF$. Thus we get another useful way to write the above equations in terms of the basis $a^i_j(n)$: $$ u_{n+1}-u_n=\sum_{i,j}A^i_ju_n\, a^i_j(n+1). $$ and $$ v_{n+1}-v_n=\sum_{i,j}\left(h^{\e_{ij}}L^i_j+\wh\d_{ij}hL^0_0\right)v_n\, a^i_j(n+1). $$ From the above lemma it is enough to prove the convergence to zero of $u_n - w_n$. We actually start with $w_n - v_n$. From the fact that $$ U_{t_{k+1}} - U_{t_k} = \sum_{i,j}\intk k L^i_j U_s\, da^i_j(s) $$ and thanks to the formulas for projections of Fock space integrals onto the toy Fock space in Theorem 11, one obtains the following expression for $w_{k+1} - w_k$ (be careful that the $da^0_0(t)$ integrals gives rise to $a^i_i(k)$ terms for {\it all} $i$, for $I=\sum_i a^i_i$): $$ \eq{ w_{k+1}& - w_k= \sum_{i,j\not=(0,0)} h^{\e_{ij}}L^i_j \left( \frac{1}{h} \, \es \intk k P_{t_k} U_t \,dt\right)\, a^i_j(k+1) \es \cr & \qq + \sum_{i}\ h L^0_0 \left(\frac1h \es \intk k U_t \, dt \right) a^i_i (k+1)\es \cr & \qq+ \sum_{i\in\La}\sum_{j\in\La}\es \left( \frac{1}{h} \intk k \left(L^i_0 P_{t_k} U_t(a^j_0(t)-a^j_0(t_k))\right) \,dt\right.\cr & \qq\left.+ \frac{1}{h} \intk k\left(L^0_j P_{t_k} U_t (a^0_i(t)-a^0_i(t_k))\right) \,dt \right) a^i_j(k+1) \es.\cr} $$ As a consequence $$ \eq{ w_n - v_n &= \somu {k