# The Langevin equation for a quantum heat bath

**ABSTRACT** We compute the quantum Langevin equation (or more exactly, the quantum stochastic differential equation) representing 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 quantum systems at a given density matrix state. In particular we specialise 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.

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**ABSTRACT:**We systematically compare the Hamiltonian and Markovian approaches of quantum open system theory, in the case of the spin-boson model. We first give a complete proof of the weak coupling limit and we compute the Lindblad generator of this model. We study properties of the associated quantum master equation such as decoherence, detailed quantum balance and return to equilibrium at inverse temperature 0Séminaire de Probabilités. 01/2009; XLII:397-431. - [Show abstract] [Hide abstract]

**ABSTRACT:**A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This unifies and extends previous work on repeated-interactions models, including that of Attal and Pautrat (Ann Henri Poincaré 7:59-104 2006) and Belton (J Lond Math Soc 81:412-434, 2010; Commun Math Phys 300:317-329, 2010). When the random-walk generator acts by ampliation and either multiplication or conjugation by a unitary operator, it is shown that the quantum stochastic cocycle which arises in the limit is driven by a unitary process.Communications in Mathematical Physics 01/2014; 328(2). · 1.90 Impact Factor - SourceAvailable from: Michal Gnacik
##### Thesis: Quantum Random Walks

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**ABSTRACT:**In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree representations and the corresponding quasifree stochastic calculus are also described. In particular, we study in detail gauge-invariant and squeezed quasifree states. We describe repeated quantum interactions between a `small' quantum system and an environment consisting of an infinite chain of particles. We study different cases of interaction, in particular those which occur in weak coupling limits and low density limits. Under different choices of scaling of the interaction part we show that random walks, which are generated by the associated unitary evolutions of a repeated interaction system, strongly converge to unitary quantum stochastic cocycles. We provide necessary and sufficient conditions for such convergence. Furthermore, under repeated quantum interactions, we consider the situation of an infinite chain of identical particles where each particle is in an arbitrary faithful normal state. This includes the case of thermal Gibbs states. We show that the corresponding random walks converge strongly to unitary cocycles for which the driving noises depend on the state of the incoming particles. We also use conditional expectations to obtain a simple condition, at the level of generators, which suffices for the convergence of the associated random walks. Limit cocycles, for which noises depend on the state of the incoming particles, are also obtained by investigating what we refer to as `compressed' random walks. Lastly, we show that the cocycles obtained via the procedure of repeated quantum interactions are quasifree, thus the driving noises form a representation of the relevant CCR algebra. Both gauge-invariant and squeezed representations are shown to occur.05/2014, Degree: PhD, Supervisor: Alexander C. R. Belton, J. Martin Lindsay

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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) =

Ion 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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Page 5

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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Page 6

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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Page 7

– 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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Page 8

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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Page 9

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)

9

Page 10

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.

10

Page 11

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 < ∞

?

.

Page 12

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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- Available from Stéphane Attal · Sep 10, 2014
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