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arXiv:math-ph/0612055v1 17 Dec 2006
THE LANGEVIN EQUATION
FOR A QUANTUM HEAT BATH
St´ ephane ATTAL
1& Alain JOYE
2
1Institut C. Jordan
Universit´ e C. Bernard, Lyon 1
21, av Claude Bernard
69622 Villeurbanne Cedex
France
2Institut Fourier
Universit´ e de Grenoble 1
100, rue des Maths, BP 74
38402 St Martin d’Heres
France
Abstract
We compute the quantum Langevin equation (or quantum stochastic differential equation) repre-
senting the action of a quantum heat bath at thermal equilibrium on a simple quantum system. These
equations are obtained by taking the continuous limit of the Hamiltonian description for repeated
quantum interactions with a sequence of photons at a given density matrix state. In particular we spe-
cialise these equations to the case of thermal equilibrium states. In the process, new quantum noises
are appearing: thermal quantum noises. We discuss the mathematical properties of these thermal
quantum noises. We compute the Lindblad generator associated with the action of the heat bath on
the small system. We exhibit the typical Lindblad generator that provides thermalization of a given
quantum system.
I. Introduction
The aim of Quantum Open System theory (in mathematics as well as in
physics) is to study the interaction of simple quantum systems interacting with
very large ones (with infinite degrees of freedom). In general the properties that
one is seeking are to exhibit the dissipation of the small system in favor of the
large one, to identify when this interaction gives rise to a return to equilibrium or
a thermalization of the small system.
There are in general two ways of studying those system, which usually repre-
sent distinct groups of researchers (in mathematics as well as in physics).
The first approach is Hamiltonian. The complete quantum system formed by
the small system and the reservoir is studied through a Hamiltonian describing
the free evolution of each component and the interaction part. The associated
unitary group gives rise to a group of *-endomorphisms of a certain von Neumann
algebra of observables. Together with a state for the whole system, this constitutes
a quantum dynamical system. The aim is then to study the ergodic properties of
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that quantum dynamical system. This can be performed via the spectral study of
a particular generator of the dynamical system: the standard Liouvillian. This is
the only generator of the quantum dynamical system which stabilizes the self-dual
cone of the associated Tomita-Takesaki modular theory. It has the property to
encode in its spectrum the ergodic behavior of the quantum dynamical system.
Very satisfactory recent results in that direction were obtained by Jaksic and Pillet
([JP1], [JP2] and [JP3]) who rigorously proved the return to equilibrium for Pauli-
Fierz systems, using these techniques.
The second approach is Markovian. In this approach one gives up the idea of
modelizing the reservoir and concentrates on the effective dynamics of the small
system. This evolution is supposed to be described by a semigroup of completely
positive maps. These semigroups are well-known and, under some conditions,
admit a generator which is of Lindblad form:
L(X) = i[H,X] +1
2
i
The first order part of L represents the usual quantum dynamic part, while the
second order part of L carries the dissipation. This form has to be compared with
the general form, in classical Markov process theory, of a Feller diffusion generator:
a first order differential part which carries the classical dynamics and a second
order differential part which represents the diffusion. For classical diffusion, such
a semigroup can be realized as resulting of a stochastic differential equation. That
is, a perturbation of an ordinary differential equation by classical noise terms such
as a Brownian motion usually. In our quantum context, one can add to the small
system an adequate Fock space which carries quantum noises and show that the
effective dynamics we have started with is resulting of a unitary evolution on the
coupled system, driven by a quantum Langevin equation. That is, a perturbation
of a Schr¨ odinger-type equation by quantum noise terms.
?
(2L∗
iXLi− L∗
iLiX − XL∗
iLi).
Whatever the approach is, the study of the action of quantum thermal baths is
of major importance and has many applications. In the Hamiltonian approach, the
model for such a bath is very well-known since Araki-Woods’ work ([A-W]). But
in the Markovian context, it was not so clear what the correct quantum Langevin
equation should be to account for the action of a thermal bath. Some equations
have been proposed, in particular by Lindsay and Maassen ([L-M]). But no true
physical justification of them has ever been given. Besides, it is not so clear what
a “correct” equation should mean?
A recent work of Attal and Pautrat ([AP1]) is a good candidate to answer
that problem. Indeed, consider the setup of a quantum system (such as an atom)
having repeated interactions, for a short duration τ, with elements of a sequence
of identical quantum systems (such as a sequence of photons). The Hamiltonian
evolution of such a dynamics can be easily described. It is shown in [AP1] that in
the continuous limit (τ → 0), this Hamiltonian evolution spontaneously converges
to a quantum Langevin equation. The coefficient of the equation being easily
computable in terms of the original Hamiltonian. This work has two interesting
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consequences:
– It justifies the Langevin-type equations for they are obtained without any
probabilistic assumption, directly from a Hamiltonian evolution;
– It is an effective theorem in the sense that, starting with a naive model for
a quantum field (a sequence of photons interacting one after the other with the
small system), one obtains explicit quantum Langevin equations which meet all
the usual models of the litterature.
It seems thus natural to apply this approach in order to derive the correct
quantum Langevin equations for a quantum heat bath. This is the aim of this
article.
We consider a simple quantum system in interaction with a toy model for
a heat bath. The toy model consists in a chain of independent photons, each of
which in the thermal Gibbs state at inverse temperature β, which are interacting
one after the other with the small system. Passing to the continuous interaction
limit, one should obtain the correct Langevin equation.
One difficulty here is that in [AP1], the state of each photon needed to be
a pure state (this choice is crucial in their construction). This is clearly not the
case for a Gibbs state. We solve this problem by taking the G.N.S. (or cyclic)
representation associated to that state. If the state space of one (simplified) photon
was taken to be n-dimensional, then taking the G.N.S. representation brings us
into a n2-dimensional space. This may seem far too big and give the impression
we will need too many quantum noises in our model. But we show that, in all
cases, only 2n chanels of noise resist to the passage to the limit and that they can
be naturally coupled two by two to give rise to n “thermal quantum noises”. The
Langevin equation then remains driven by n noises (which was to be expected!)
and the noises are shown to be actually Araki-Woods representations of the usual
quantum noises. Furthermore, the Langevin equation we obtain is very similar to
the model given in [L-M].
Altogether this confirms we have identified the correct Langevin equation
modelizing the action of a quantum heat bath.
An important point to notice is that our construction does not actually use
the fact that the state is a Gibbs-like state, it is valid for any density matrix.
This article is organized as follows. In section II we present the toy model for
the bath and the Hamiltonian description of the repeated interaction procedure.
In section III we present the Fock space, its quantum noises, its approximation by
the toy model and the main result of [AP1]. In section IV we detail the G.N.S.
representation of the bath and compute the unitary operator, associated with the
total Hamiltonian, in that representation. In section V, applying the continuous
limit procedure we derive the limit quantum langevin equation. In the process, we
identify particular quantum noises that are naturally appearing and baptize them
“thermal quantum noises”, in the case of a heat bath. The properties of those
thermal quantum noises are studied in section VI; in particular we justify their
name. In section VII, tracing out the noise, we compute the Lindblad generator of
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the induced semigroup on the small system. In section VII, being given any finite
dimensional quantum system with its Hamiltonian, we show how to construct a
Lindblad generator, representing some interaction with a heat bath, such that the
quantum system thermalizes.
II. The toy model
We describe here the physical model of repeated interactions with the bath
toy model.
The quantum system (we shall often call “small system”) to be put in contact
with the bath is represented by a separable Hilbert space HS, as state space, and
a self-adjoint operator HS, as Hamiltonian.
The toy model for the heat bath is the chain
?
of copies of CN+1, where N ≥ 1 is a fixed integer. Each copy of CN+1represents
the (simplified) state space of a photon. By this countable tensor product we
mean the following. We consider a fixed orthonormal basis {e0,e1,...,eN} of
CN+1, corresponding to the eigenstates of the photon (e0being the ground state);
the countable tensor product is taken with respect to the ground state e0. Together
with this structure we consider the associated basic matrices ai
acting on CN+1by
ai
jek= δikej
and their natural ampliations to ⊗k∈I N∗CN+1given by
ai
j(k) =
I on the other copies.
The Hamiltonian of one photon is the operator
k∈I N∗
CN+1
j, i,j = 1,...,N,
?
ai
j
on the k-th copy of CN+1
HR=
N
?
i=0
γia0
iai
0,
where the γi’s are real numbers. Here notice two points.
We have assumed the Hamiltonian HR to be diagonal in the chosen basis.
This is of course not actually a true restriction, for one can always choose such
a basis.Note that HRdescribe the total energy of a single photon, not the whole
field of photon. For this we differ from the model studied in [AJ1].
The second point is that γ0is the ground state eigenvalue, it should then be
smaller than the other γi. One usually assumes that it is equal to 0, but this is not
actually necessary in our case, we thus do not specify its value. The only hypothesis
we shall make here is that γ0< γi, for all i = 1,...N. This hypothesis means
that the ground eigenspace is simple, it is not actually a necessary assumption, it
only simplifies our discussion. At the end of section V we discuss what changes if
we leave out this hypothesis.
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Finally, notice that the other eigenvalues γineed not be simple in our discus-
sion.
When the system and a photon are interacting, we consider the state space
HS⊗ CN+1together with the interaction hamiltonian
N
?
where the Vi’s are bounded operators on HS. This is a usual dipole-type interaction
Hamiltonian. The total Hamiltonian for the small system and one photon is thus
HI=
i=1
?Vi⊗ a0
i+ V∗
i⊗ ai
0
?,
H = HS⊗ I + I ⊗ HR+
N
?
i=1
?Vi⊗ a0
i+ Vi⊗ ai
0
?.
Finally, the state of each photon is fixed to be given by a density matrix ρ which
is a function of HR. We have in mind the usual thermal Gibbs state at inverse
temperature β :
ρβ=1
Ze−βHR,
where Z = tr(e−βHR), but our construction applies to more general states ρ.
Note that ρβis also diagonal in our orthonormal basis. Its diagonal elements
are denoted by {β0,β1,...,βn}.
We shall now describe the repeated interactions of the system HS with the
chain of photons. The system HS is first in contact with the first photon only
and they interact together according to the above Hamiltonian H. This lasts for
a time length τ. The system HSthen stops interacting with the first photon and
starts interacting with the second photon only. This second interaction is directed
by the same Hamiltonian H on the corresponding spaces and it lasts for the same
duration τ, and so on... This is mathematically described as follows.
On the space HS⊗ CN+1, consider the unitary operator representing the
coupled evolution during the time interval [0,τ]:
U = e−iτH.
This single interaction is therefore described in the Schr¨ odinger picture by
ρ ?→ U ρU∗
and in the Heisenberg picture by
X ?→ U∗XU.
After this first interaction, we repeat it but coupling the same HSwith a new copy
of CN+1. This means that this new copy was kept isolated until then; similarly
the previously considered copy of CN+1will remain isolated for the rest of the
experience.
The sequence of interactions can be described in the following way: the state
space for the whole system is
?
HS⊗
I N∗
CN+1.
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Consider the unitary operator Uk which acts as U on the tensor product of HS
and the k-th copy of CN+1, and which acts as the identity on all the other copies
of CN+1.
The effect of the k-th interaction in the Schr¨ odinger picture is
ρ ?→ UkρU∗
for every density matrix ρ on HS⊗∗
interactions is
ρ ?→ VkρV∗
where Vk= UkUk−1...U1.
k,
I NCn+1. In particular the effect of the k first
k
Such a Hamiltonian description of the repeated interaction procedure has no
chance to give any non-trivial limit in the continuous limit (τ → 0) without asking
a certain renormalization of the interaction. This renormalization can be thought
of as making the Hamiltonian depend on τ, or can be also seen as renormalizing
the field operators a0
0of the photons. As is shown is [AP1] (see the detailed
discussion in section III), for our repeated interaction model to give rise to a
Langevin equation in the limit, we need the interaction part of the Hamiltonian
to be affected by a weight 1/√τ. Hence, from now on, the total Hamiltonians we
shall consider on HS⊗ CN+1are
1
√τ
j,ai
H = HS⊗ I + I ⊗ HR+
N
?
i=1
?Vi⊗ a0
i+ V∗
i⊗ ai
0
?. (1)
In [AJ1], one can find a discussion about this time renormalization and its inter-
pretation in terms of weak coupling limit for repeated quantum interactions.
III. The continuous limit setup
We present here all the elements of the continuous limit result: the structure
of the corresponding Fock space, the quantum noises, the approximation of the
Fock space by the photon chain and [AP1]’s main theorem.
III.1 The continuous tensor product structure
First, as a guide to intuition, let us make more explicit the structure of the
photon chain. We let TΦ denote the tensor product ⊗I N∗CN+1with respect to
the stabilizing sequence e0. This simply means that an orthonormal basis of TΦ
is given by the family
{eσ;σ ∈ PI N∗,N}
where
– the set PI N,N is the set of finite subsets
{(n1,i1),...,(nk,ik)}
of I N∗× {1,...,N} such that the ni’s are mutually different;
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– eσdenotes the vector
Ω ⊗ ... ⊗ Ω ⊗ ei1⊗ Ω ⊗ ... ⊗ Ω ⊗ ei2⊗ ...
where ei1appears in n1-th copy of H, where ei2appears in n2-th copy of H... Here
Ω plays the same role as e0in the toy model.
This is for a vector basis on TΦ. From the point of view of operators, we
denote by ai
copy number k as ai
jand the identity elsewhere. That is, in terms of the basis eσ,
ai
j(k)eσ= 1 l(k,i)∈σe(σ\(k,i))∪(k,j)
if neither i nor j is zero, and
ai
0(k)eσ= 1 l(k,i)∈σeσ\(k,i),
a0
j(k)eσ= 1 l(k,0)∈σeσ∪(k,j),
a0
0(k)eσ= 1 l(k,0)∈σeσ,
where (k,0) ∈ σ actually means “for any i in {1,...,N}, (k,i) ?∈ σ”.
We now describe the structure of the continuous version of the chain of pho-
tons. 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 quan-
tum 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 heuristic discussion.
By a continuous version of the atom chain TΦ we mean a Hilbert space with
a structure which makes it the space
?
We have to give a meaning to the above notation. This could be achieved by in-
voquing 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 ap-
proximation procedure.
Let us make out an idea of what it should look like by mimicking, in a con-
tinuous time version, what we have described in TΦ.
The countable orthonormal basis eσ,σ ∈ PI N∗,N is replaced by a continuous
orthonormal basis dχσ, σ ∈ PI R+,N, where PI R+,Nis the set of finite subsets of
I R+× {1,...,N}. With the same idea as for TΦ, this means that each copy of
CN+1is equipped with an orthonormal basis {Ω,dχ1
parameter attached to the copy we are looking at).
Recall the representation of an element f of TΦ:
?
||f||2=
σ∈PI N∗,N
j(k) the natural ampliation of the operator ai
jto TΦ which acts on the
Φ =
I R+
CN+1.
t,...,dχN
t} (where t is the
f =
σ∈PI N∗,N
?
f(σ)eσ,
|f(σ)|2,
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it is replaced by an integral version of it in Φ:
f =
?
?
PI R+,N
f(σ)dχσ,
||f||2=
PI R+,N
|f|2dσ.
This last integral needs to be explained: the measure dσ is a “Lebesgue measure”
on PI R+,N, as will be explained later.
From now on, the notation P will denote, depending on the context, the set
PI N∗,N or PI R+,N.
A good basis of operators acting on Φ can be obtained by mimicking the
operators ai
{0,1,...,N}, acting on the “t-th” copy of CN+1by:
da0
0(t)dχσ= dχσdt1 lt?∈σ
da0
i(t)dχσ= dχσ∪{(t,i)}1 lt?∈σ
dai
0(t)dχσ= dχσ\{(t,i)}dt1 l(t,i)∈σ
dai
j(t)dχσ= dχ(σ\{(t,i)})∪{(t,j)}1 l(t,i)∈σ
for all i,j ∈ {1,...,N}. We shall now describe a rigourous setup for the above
heuristic discussion.
j(k) of TΦ. We have here a set of infinitesimal operators dai
j(t), i,j ∈
We recall the structure of the bosonic Fock space Φ and its basic structure
(cf [At1] for more details and [At2] for a complete study of the theory and its
connections with classical stochastic processes).
Let Φ = Γs(L2(I R+,CN)) be the symmetric (or bosonic) Fock space over the
space L2(I R+,CN). We shall give here a very efficient presentation of that space,
the so-called Guichardet interpretation of the Fock space.
Let P (= PI R+,N) be the set of finite subsets {(s1,i1),...,(sn,in)} of I R+×
{1,...,N} such that the siare two by two different. Then P = ∪kPkwhere Pk
is the subset of P made of k-elements subsets of I R+× {1,...,N}. By ordering
the I R+-part of the elements of σ ∈ Pk, the set Pk can be identified with the
increasing simplex Σk = {0 < t1< ··· < tk} × {1,...,N} of I Rk× {1,...,N}.
Thus Pkinherits a measured space structure from the Lebesgue measure on I Rk
times the counting measure on {1,...,N}. This also gives a measure structure on
P if we specify that on P0= {∅} we put the measure δ∅. Elements of P are often
denoted by σ, the measure on P is denoted by dσ. The σ-field obtained this way
on P is denoted by F.
We identify any element σ ∈ P with a family {σi, i ∈ {1,...,N}} of (two by
two disjoint) subsets of I R+where
σi= {s ∈ I R+;(s,i) ∈ σ}.
The Fock space Φ is the space L2(P,F,dσ). An element f of Φ is thus a
measurable function f : P → C such that
||f||2=
P
?
|f(σ)|2dσ < ∞.
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Finally, we put Ω to be the vacuum vector of Φ, that is, Ω(σ) = δ∅(σ).
One can define, in the same way, P[a,b]and Φ[a,b]by replacing I R+with
[a,b] ⊂ I R+. There is a natural isomorphism between Φ[0,t]⊗Φ[t,+∞[and Φ given
by h⊗g ?→ f where f(σ) = h(σ ∩[0,t])g(σ∩[t,+∞[). This is, with our notations,
the usual exponential property of Fock spaces. Note that in the sequel we identify
Φ[a,b]with a subspace of Φ, the subspace
{f ∈ Φ;f(σ) = 0 unless σ ⊂ [a,b]}.
We now define a particular family of curves in Φ, which is going to be of great
importance here. Define χi
t∈Φ by
χi
t(σ) =
0
Then notice that for all t ∈ I R+we have that χi
have much more than that:
χi
s∈ Φ[s,t]for all s ≤ t.
This last property can be checked immediately from the definitions, and it is
going to be of great importance in our construction. Also notice that χi
are orthogonal elements of Φ as soon as i ?= j. One can show that, apart from
trivialities, the curves (χi
t)t≥0are the only ones to share these properties.
These properties allow to define the so-called Ito integral on Φ. Indeed, let
g = {(gi
and {1,...,N}, such that
i) t ?→ ?gi
ii) gi
t∈Φ[0,t]for all t,
iii)?N
N
?
to be the limit in Φ of
N
?
where S = {tj, j∈I N} is a partition of I R+which is understood to be refining
and to have its diameter tending to 0, and (? gi
Note that by assumption we always have that ? gi
Also note that, as an example, one can take
?
?1 l[0,t](s) if σ = {(s,i)}
otherwise.
tbelongs to Φ[0,t]. We actually
t− χi
tand χj
s
t)t≥0, i ∈ {1,...,N}} be families of elements of Φ indexed by both I R+
t? is measurable, for all i,
?∞
i=1
0?gi
t?2dt < ∞,
then one says that g is Ito integrable and we define its Ito integral
i=1
?∞
0
gi
tdχi
t
i=1
?
j∈I N
? gi
tj⊗
?
χi
tj+1− χi
tj
?
(2)
·)iis an Ito integrable family in Φ,
tis a step process, and which converges to (gi
such that for each i, t ?→ ? gi
χi
tjbelongs to Φ[tj,tj+1], hence the tensor product symbol in (2).
·)i in
L2(I R+× P).
tjbelongs to Φ[0,tj]and χi
tj+1−
? gi
t=
tj∈S
1
tj+1− tj
?tj+1
tj
Ptjgi
sds1 l[tj,tj+1[(t)
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if t ∈ [tj,tj+1], where Ptdenotes the orthogonal projection onto Φ[0,t].
One then obtains the following properties ([At1], Proposition 1.4), where ∨σ
means max{s ∈ I R+;(s,k) ∈ σ for some k} and where σ− denotes the set σ\(∨σ,i)
if (∨σ,i) ∈ σ.
Theorem 1.–The Ito integral I(g) =?
I(g)(σ) =
gi
∨σ(σ−)
It satisfies the Ito isometry formula:
???
In particular, consider a family f = (fi)N
L2(I R+× {1,...,N}), then the family (fi(t)Ω), t ∈ I R+, i = 1,...,N, is clearly
Ito integrable. Computing its Ito integral we find that
?∞
is the element of the first particle space of the Fock space Φ associated with the
function f, that is,
?fi(s)
Let f∈L2(Pn), one can easily define the iterated Ito integral on Φ:
In(f) =
Pn
by iterating the definition of the Ito integral:
?
We obtain this way an element of Φ which is actually the representant of f in the
n-particle subspace of Φ, that is
?fi1,...,in(t1,...,tn)
Finally, for any f ∈ P we put
?
to denote the series of iterated Ito integrals
?∞
i
?∞
0gi
tdχi
t, of an Ito integrable family
g = (gi
·)N
i=1, is the element of Φ given by
?0 if σ = ∅
otherwise.
||I(g)||2=
N
?
i=1
?∞
0
gi
tdχi
t
???
2
=
N
?
i=1
?∞
0
????gi
t
????2dt . (3)
i=1which belongs to L2(P1) =
I(f) =
N
?
i=1
0
fi(t)Ωdχi
t
I(f)(σ) =
if σ = {(s,i)}
otherwise.0
?
f(σ)dχσ
In(f) =
i1,...,in∈{1,...,N}
?∞
0
?tn
0
...
?t2
0
fi1,...,in(t1,...,tn)Ω dχi1
t1... dχin
tn.
[In(f)](σ) =
if σ = {(t1,i1) ∪ ... ∪ (tn,in)}
otherwise.0
P
f(σ)dχσ
f(∅)Ω +
∞
?
n=1
N
?
i1,...,in=1
0
?tn
0
...
?t2
0
fi1,...,in(t1,...,tn)Ωdχi1
t1... dχin
tn.
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We then have the following representation ([At1], Theorem 1.7).
Theorem 2. [Fock space chaotic representation property]–Any element f of Φ
admits a Fock space chaotic representation
?
satisfying the isometry formula
?
This representation is unique.
f =
P
f(σ) dχσ
(4)
?f?2=
P
|f(σ)|2dσ. (5)
The above theorem is the exact expression of the heuristics we wanted in order
to describe the space
Φ =
?
I R+
H.
Indeed, we have, for each t ∈ I R+, a family of elementary orthonormal elements
{Ω,dχ1
basis of Φ (formula (4)) and, even more, form an orthonormal continuous basis
(formula (5)).
t,...,dχN
t} (a basis of H) whose (tensor) products dχσform a continuous
III.2 The quantum noises
The space Φ we have constructed is the natural space for defining quantum
noises. These quantum noises are the natural, continuous-time, extensions of the
basis operators ai
j(n) we met in the atom chain TΦ.
As indicated in the heuristic discussion above, we shall deal with a family of
infinitesimal operators dai
j(t) on Φ which act on the continuous basis dχσin the
same way as their discrete-time counterparts ai
version of the above heuristic infinitesimal formulas easily gives an exact formula
for the action of the operators ai
j(t) on Φ:
?
s≤t
?t
[ai
j(t)f](σ) =
s∈σi
s≤t
[a0
0(t)f](σ) = tf(σ)
for i,j ?= 0.
All these operators, except a0
0(t), are unbounded, but note that a good com-
mon domain to all of them is
?
11
j(n) act on the eσ. The integrated
[a0
i(t)f](σ) =
s∈σi
f(σ \ (s,i)),
[ai
0(t)f](σ) =
0
f(σ ∪ (s,i)) ds,
?
f ((σ \ (s,i)) ∪ (s,j))
D =f∈Φ ;
?
P
|σ| |f(σ)|2dσ < ∞
?
.
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This family of operators is characteristic and universal in a sense which is close
to the one of the curves χi
t. Indeed, one can easily check that in the decomposition
of Φ ≃ Φ[0,s]⊗ Φ[s,t]⊗ Φ[t,+∞[, the operators ai
I ⊗ (ai
This property is fundamental for the definition of the quantum stochastic integrals
and, in the same way as for (χi
·), these operator families are the only ones to share
that property (cf [Coq]).
This property allows to consider Riemann sums:
?
where S = {0 = t0< t1< ... < tk< ...} is a partition of I R+, where (Ht)t≥0is a
family of operators on Φ such that
– each Ht is an operator of the form Ht⊗ I in the tensor product space
Φ = Φ[0,t]⊗ Φ[t,+∞[(we say that Htis a t-adapted operator and that (Ht)t≥0is
an adapted process of operators),
– (Ht)t≥0is a step process, that is, it is constant on intervals:
Ht=Htk1 l[tk,tk+1](t).
j(t) − ai
j(s) are all of the form
j(t) − ai
j(s))|Φ[s,t]⊗ I.
k
Htk
?ai
j(tk+1) − ai
j(tk)?
(6)
?
k
In particular, the operator product Htk
product of operators
?ai
j(tk+1) − ai
j(tk)?.
j(tk)?
is actually a tensor
Htk⊗?ai
j(tk+1) − ai
Thus this product is commutative and does not impose any new domain constraint
on the operators apart from the ones attached to the operators Htand ai
ai
j(tk) themselves. The resulting operator associated to the Riemann sum (6) is
denoted by
?∞
One can compute the action of T on a “good” vector f of its domain and obtain
explicit formulas which are not worth developping here (cf [At1] for more details).
For general operator processes (Ht)t≥0(still adapted but not a step process any-
more) and for a general f, these explicit formulas can be extended and they are
kept as a definition for the domain and for the action of the operator
?∞
The maximal domain and the explicit action of the above operator can be de-
scribed but also are not worth developing here (cf [A-L]). The main point with
these quantum stochastic integrals is that, when composed, they satisfy a Ito-type
integration by part formula. This formula can be summarized as follows, without
taking care at all of domain constraints. Let
?∞
j(tk+1)−
T =
0
Hsdai
j(s).
T =
0
Hsdai
j(s).
T =
0
Hsdai
j(s), S =
?∞
0
Ksdak
l(s).
12
Page 13
For every t ∈ I R+put
Tt=
?∞
?∞
?δil
0
Hs1 l[0,t](s)dai
j(s)
and the same for St. We then have
?∞
where
TS =
0
HsSsdai
j(s) +
0
TsKsdak
l(s) +
?∞
0
HsKs?δildak
j(s), (7)
?δil=
if (i,l) ?= (0,0)
if (i,l) = (0,0).0
The last term appearing in this Ito-type formula is often summarized by saying
that the quantum noises satisfy the formal formula:
dai
j(s)dak
l(s) =?δildak
j(s).
III.3 Embedding and approximation by the Toy Fock space
We now describe the way the chain and its basic operators can be realized as a
subspace of the Fock space and a projection of the quantum noises. The subspace
associated with the atom chain is attached to the choice of some partition of I R+
in such a way that the expected properties are satisfied:
– the associated subspaces increase when the partition refines and they con-
stitute an approximation of Φ when the diameter of the partition goes to 0,
– the associated basic operators are restrictions of the others when the parti-
tion increases and they constitute an approximation of the quantum noises when
the diameter of the partition goes to 0.
Let
S = {0 = t0 < t1 < ··· < tn < ···} be a partition of
δ(S) = supi|ti+1− ti| be the diameter of S. For S fixed, define Φn= Φ[tn−1,tn],
n∈I N∗. We clearly have that Φ is naturally isomorphic to the countable tensor
product ⊗n∈I N∗Φn(which is again understood to be defined with respect to the
stabilizing sequence (Ω)n∈I N).
For all n∈I N∗, define for i,j ∈ {1,...,N}
χi
tn−1
√tn− tn−1
ai
√tn− tn−1
ai
I R+and
ei(n) =
tn− χi
∈ Φn,
0(n) =ai
j(n) = P1]◦?ai
a0
0(tn) − ai
0(tn−1)
◦ P1],
j(tn−1)?◦ P1],
j(tn) − ai
i(tn) − a0
√tn− tn−1
i(n) = P1]◦a0
a0
0(n) = P0],
i(tn−1)
,
where for i = 0,1 and Pi]is the orthogonal projection onto L2(Pi). The above
definitions are understood to be valid on Φn only, the corresponding operator
acting as the identity operator I on the others Φm’s.
13
Page 14
For every σ ∈ P = PI N∗,N, define eσfrom the ei(n)’s in the same way as for
TΦ:
eσ= Ω ⊗ ... ⊗ Ω ⊗ ei1(n1) ⊗ Ω ⊗ ... ⊗ Ω ⊗ ei2(n2) ⊗ ...
in ⊗n∈I N∗Hn. Define TΦ(S) to be the space of f∈Φ which are of the form
f =
?
σ∈P|f(σ)|2< ∞ is automatically satisfied).
σ∈P
f(σ)eσ
(note that the condition ?f?2=?
The space TΦ(S) is a closed subspace of Φ. We denote by PS the operator of
orthogonal projection from Φ onto TΦ(S).
The main point is that the above operators ai
way as the the basic operators of TΦ (cf [AP1], Proposition 8).
The space TΦ(S) can be clearly and naturally identified to the spin chain TΦ.
j(n) act on TΦ(S) in the same
Proposition 3.–We have, for all i,j = 1,...,N
?ai
?ai
?a0
?a0
0(n)ej(n) = δijΩ
ai
0Ω = 0
j(n)ek(n) = δikej(n)
ai
jΩ = 0
i(n)ej(n) = 0
a0
i(n)Ω = ei(n)
0(n)ek(n) = 0
a0
0Ω = Ω.
Thus the action of the operators ai
action of the corresponding operators on the spin chain of section II; the operators
ai
j(n) act on TΦ(S) exactly in the same way as the corresponding operators do
on TΦ. We have completely embedded the toy Fock space structure into the Fock
space.
jon the ei(n) is exactly the same as the
We are now going to see that the Fock space Φ and its basic operators ai
i,j ∈ {0,1,...,n} can be approached by the toy Fock spaces TΦ(S) and their
basic operators ai
j(n). We are given a sequence (Sn)n∈I N of partitions which are
getting finer and finer and whose diameter δ(Sn) tends to 0 when n tends to +∞.
Let TΦ(n) = TΦ(Sn) and Pn= PSn, for all n∈I N. We then have the following
convergence result (see [AP1], Theorem 10), where the reader needs to recall the
domain D introduced in section III.2.
j(t),
Theorem 4.–
i) The orthogonal projectors Pn converge strongly to the identity operator I
on Φ. That is, any f ∈ Φ can be approached in Φ by a sequence (fn)n∈I Nsuch
that fn∈ TΦ(n) for all n ∈ I N.
14
Page 15
ii) If Sn= {0 = tn
1,...,n the operators
0< tn
1< ··· < tn
?
?
?
and
k< ···}, then for all t∈I R+, all i,j =
k;tn
k≤t
ai
j(k),
k;tn
k≤t
?
?
(tn
tn
k− tn
k−1ai
0(k),
k;tn
?
k≤t
tn
k− tn
k−1a0
i(k)
k;tn
k≤t
0(t), a0
k− tn
k−1)a0
0(k)
converge strongly on D to ai
We have fulfilled our duties: not only the space TΦ(S) recreates TΦ and its
basic operators as a subspace of Φ and a projection of its quantum noises, but,
when δ(S) tends to 0, this realisation constitutes an approximation of the space
Φ and of its quantum noises.
j(t), ai
i(t) and a0
0(t) respectively.
III.4 Quantum Langevin equations
In this article what we call quantum Langevin equation is actually a restricted
version of what is usually understood in the physical literature (cf [G-Z]); by this we
mean that we study here the so-called quantum stochastic differential equations as
defined by Hudson and Parthasarathy and heavily studied by further authors ([H-
P], [Fag]). This type of quantum noise perturbation of the Schr¨ odinger equation
is exactly the type of equation which we will get as the continuous limit of our
Hamiltonian description of repeated quantum interactions.
Quantum stochastic differential equations are operator-valued equations on
HS⊗ Φ of the form
dUt=
i,j=0
with initial condition U0 = I. The above equation has to be understood as an
integral equation
?t
i,j=0
the operators Li
jbeing bounded operators on HS alone which are ampliated to
HS⊗ Φ.
The main motivation and application of that kind of equation is that it gives
an account of the interaction of the small system HSwith the bath Φ in terms of
quantum noise perturbation of a Schr¨ odinger-like equation. Indeed, the first term
of the equation
dUt= L0
N
?
Li
jUtdai
j(t),
Ut= I +
0
N
?
Li
jUtdai
j(t),
0Utdt + ...
15
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