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\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.}
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\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