TWO-PHOTON ABSORPTION AND EMISSION PROCESS

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Infinite Dimensional Analysis, Quantum Probabilityand Related TopicsVol. 8, No. 4 (2005) 573–591c© World Scientific Publishing Company

TWO-PHOTON ABSORPTION AND EMISSION PROCESS

FRANCO FAGNOLA

Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano,

Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

[email protected]

ROBERTO QUEZADA

Departamento de Matematicas, UAM-Iztapalapa,

Av. San Rafael Atlixco 186, Col Vicentina,

09340 Mexico D.F., Mexico

[email protected]

Received 25 March 2005Communicated by L. Accardi

We analyze the two-photon absorption and emission process and characterize the sta-tionary states at zero and positive temperature. We show that entangled stationarystates exist only at zero temperature and, at positive temperature, there exists infinitelymany commuting invariant states satisfying the detailed balance condition.

Keywords: Quantum dynamical semigroup; detailed balance; KMS condition.

AMS Subject Classification: 46L55, 82C10, 60J27

1. Introduction

The two-photon absorption and emission process is one of the most basic radiation-

matter interaction mechanisms. The first steps in the development of the physical

theory go back to the work of M. Goppert-Mayer17 in 1931 but the phenomenon of

two-photon absorption was not observed until 1961 (see Ref. 16), after the advent

of the laser, in fact two-photon absorption is one of the first phenomena demon-

strated with the aid of laser radiation. Since then, this phenomenon has been studied

intensively. Gilles and Knight,15 introduced a model based on a quantum Markov

semigroup and found nonclassical stationary states. Here we consider a general-

ization of this model. In Sec. 8 we outline the deduction of our model for the

two-photon absorption or absorption and emission processes, from the stochastic

limit of the evolution of the system (one-mode EM field) weakly coupled with a

boson reservoir (see Ref. 3 for the general theory of the stochastic limit).

573

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574 F. Fagnola & R. Quezada

The main features of our model are the following: nonclassical (entangled) sta-

tionary states exist only at zero temperature; there exist infinitely many commuting

invariant states at positive temperature and all of them satisfy the detailed balance

condition; it is possible to determine explicitly the attraction domain of any invari-

ant state.

The paper is organized as follows. In Sec. 2 we define the formal Lindblad gen-

erator of the model and prove the conservativity of the corresponding minimal

quantum dynamical semigroup. In Sec. 4 we show that there are two natural in-

variant subspaces for our evolution, namely the space of states supported by even

or odd number states, and discuss the ergodicity of the restricted even and odd

evolutions. Later in Sec. 5 we discuss the detailed balance condition with respect

to a family of invariant states and show, in Sec. 6, that these are the only invariant

states at positive temperature. In Sec. 7 we discuss the approach to equilibrium,

i.e., the convergence of any initial state to some invariant state and determine the

attraction domain of every invariant state.

2. The Model

In this section we introduce the Lindbladian for the two-photon creation and anni-

hilation process.

Let h be the Hilbert space h = `2(N) and let a, a+ and N be the annihilation

and creation operators. Denote by (ek)k≥0 the canonical orthonormal basis of h.

Let G be the operator defined on the domain Dom(N 2) of the square of the number

operator by

G = −λ2

2a2a+2 − µ2

2a+2a2 − iωa+2a2

with λ ≥ 0, µ > 0, ω ∈ R and let L1, L2 be the operators defined on Dom(N) by

L1 = µa2 , L2 = λa+2 .

Clearly G generates a strongly continuous semigroup of contractions (Pt)t≥0 with

Pt = e−t(λ2(N+1)(N+2)+(µ2+2iω)N(N−1))/2 .

For every x ∈ B(h) the Lindblad formal generator is a sesquilinear form defined by

L−(x)[u, v] = 〈Gu, xv〉 +

2∑

`=1

〈L`u, xL`v〉 + 〈u, xGv〉 , (2.1)

for u, v ∈ Dom(G) = Dom(N2). One can easily check that conditions for construct-

ing the minimal quantum dynamical semigroup (QDS) associated with the above

G, L1, L2 ((H-min) in Ref. 7) hold and this semigroup T = (Tt)t≥0 satisfies the so

called Lindblad equation

〈v, Tt(x)u〉 = 〈v, P ∗t xPtu〉 +

2∑

`=1

∫ t

0

〈L`Pt−sv, Ts(x)L`Pt−su〉ds , (2.2)

for all u, v ∈ Dom(G).

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Two-Photon Absorption and Emission Process 575

We have that L−(N + 1) ≤ 4µ2(N + 1) if λ ≤ µ, therefore L− satisfies a well-

known criterion for conservativity, see Ref. 7. Moreover, for λ > µ the formal

generator satisfies a simple criterion for nonconservativity, see Ref. 14, Example 2.

Then the minimal QDS is Markov (or conservative) if and only if λ ≤ µ. It follows

from conservativity that the minimal QDS is the unique solution of Eq. (2.2).

Moreover, an operator x ∈ B(h) belongs to the domain of the generator L if and

only if the sesquilinear form L−(x) is bounded (see Ref. 8 Lemma 1.1, or Ref. 7

Proposition 3.33). It follows that the action of L on the linear manifold M =

span{|ej〉〈ek| : j, k ≥ 0} of finite range operators is given by

L(x) = iω∑

j,k

(j[j − 1] − k[k − 1])xjk |ej〉〈ek|

+∑

j,k

|ej〉〈ek|(

µ2k1

2 [k − 1]1

2 j1

2 [j − 1]1

2 xj−2k−2 −µ2

2(k[k − 1] + j[j − 1])xjk

+ λ2(k + 1)1

2 (k + 2)1

2 (j + 1)1

2 (j + 2)1

2 xj+2k+2 − λ2

2((j + 1)(j + 2)

+ (k + 1)(k + 2))xjk

)

, (2.3)

where [k − 1] = max{k − 1, 0}, [j − 1] = max{j − 1, 0}.

3. Invariant States

When ν = λ/µ < 1 we can easily find two invariant states. Indeed, a straightforward

computation yields the states ρe, ρo defined by

ρe = (1 − ν2)∑

k≥0

ν2k|e2k〉〈e2k| , ρo = (1 − ν2)∑

k≥0

ν2k|e2k+1〉〈e2k+1| .

Proposition 3.1. The states ρe and ρo are invariant.

Proof. Let L∗ be the generator of the predual semigroup T∗ = (T∗t)t≥0, acting on

the Banach space L1(h) of trace class operators on h. Consider the approximations

ρe,n = (1 − ν2)∑n

k=0 ν2k |e2k〉〈e2k|, of ρe by finite range operators.

The operators ρe,n belong to the domain of L∗ and we have L∗(ρe,n) = λ2(2n+

1)(2n+2)ν2n(|e2n+1〉〈e2n+1|− |e2n〉〈e2n|). Thus, for all m < n, we have ‖L∗(ρe,n −ρe,m)‖1 = 2λ2((2n+1)(2n+2)ν2n +(2m+1)(2m+2)ν2m) which converges to zero

as n, m → ∞ since ν < 1. Similar computations and conclusion hold for ρo,n. This

proves that ρe, ρo ∈ Dom(L∗) and L∗(ρe) = 0 = L∗(ρo).

In Sec. 6 we shall prove that, when λ > 0, all the invariant states are convex

combination of ρo and ρe. When λ = 0, we can easily find all the invariant states.

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576 F. Fagnola & R. Quezada

Proposition 3.2. If λ = 0 all invariant states have the form αρe + (1 − α)ρo +

z|e0〉〈e1| + z|e1〉〈e0|, with α ∈ [0, 1] and |z|2 ≤ α(1 − α).

Proof. Let σ be an invariant state. Then L∗(σ) = 0, i.e.

∑

j,k

(µ2((k + 2)(k + 1)(j + 2)(j + 1))1

2 σj+2k+2

− (µ2(k[k − 1] + j[j − 1]))/2 + iω(k[k − 1] − j[j − 1])))σjk) = 0 .

Since x is arbitrary, the diagonal terms of the sum yield σj+2j+2 = j(j−1)(j+1)(j+2)σjj .

Thus σjj = 0 for all j ≥ 2. Moreover, since the operator σ is positive, we have the

inequality |σjk |2 ≤ σjjσkk . We find that σjk = 0 for any j, k ≥ 2. The invariant

states are then of the claimed form. Condition |z|2 ≤ α(1 − α) assures positivity.

4. The Even and Odd Subalgebras

The above discussion makes clear that “even” and “odd” states play a special role.

In this section we shall study the behavior of the restrictions of the semigroup

T to the “even” and “odd” algebra determined by the support projections of the

invariant states ρe and ρo.

Recall that the support projection p of an invariant state is T -subharmonic,

i.e. Tt(p) ≥ p for all t ≥ 0 (see Fagnola and Rebolledo9 Theorem 2.16 and also

Ref. 10). Subharmonic projections are characterized by the following:

Theorem 4.1. A projection p is T -subharmonic if and only if its range R(p) is

an invariant subspace for the operators Pt (t ≥ 0) and L`pu = pL`u for all u ∈Dom(G) ∩ R(p) and all ` ≥ 1.

As an application we have the following:

Theorem 4.2. Subharmonic projections for the two-photon absorption emission

QMS T are

(i) If λ > 0 : 0, 1l and

pe =∑

k≥0

|e2k〉〈e2k| , po =∑

k≥0

|e2k+1〉〈e2k+1| .

(ii) If λ = 0 : 0, 1l, pe, po,

n∑

k≥0

|e2k〉〈e2k| ,m∑

k≥0

|e2k+1〉〈e2k+1| ,

n, m ∈ N, and sums of orthogonal projections of the above class.

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Two-Photon Absorption and Emission Process 577

In both cases the projections pe and po are T -invariant and the hereditary sub-

algebras Ae = peApe, the even subalgebra, and Ao = poApo, the odd subalgebra, are

T -invariant.

Proof. Any invariant subspace of a normal compact operator is generated by eigen-

vectors (e.g. by Theorem 4, p. 272 in Ref. 19). The operators (Pt)t≥0 are clearly

compact and normal, and all the ek’s are eigenvectors. Therefore any invariant

subspace IK is generated by a collection {ek : k ∈ K} where K ⊂ N.

Assume λ > 0. If K contains only even (resp. odd) numbers, then IK is invariant

under L1 = µa2 and L2 = λa+2 if and only if K contains all the even (resp. odd)

numbers. On the other hand, if K contains at least an odd and an even number, it

is clear that IK is invariant under L1 and L2 if and only if it coincides with h.

If λ = 0 and K contains only even (resp. odd) numbers, then the action of

a2pK (pK denotes the projection onto IK) moves the indices two places backward,

while pKa2pK moves the indices two places backward and kills the smallest index.

Therefore IK is invariant under L1 if and only if K is finite of the form K =

{0, 2, . . . , 2n} (resp. {1, 3, . . . , 2m + 1}) with n ≥ 1, or K coincides with N, 2N or

2N + 1.

This proves (i) and (ii). Moreover, a simple and straightforward computation

yields L−(po) = L−(pe) = 0. Therefore, by Lemma 1.1, p. 563 of Ref. 8, both po and

pe belong to the domain of L and we have Tt(po) = po, Tt(pe) = pe.

Finally, for all self-adjoint x ∈ B(h), we have

−‖x‖pe = Tt(−‖x‖pe) ≤ Tt(pexpe) ≤ Tt(‖x‖pe) = ‖x‖pe .

It follows that Tt(pexpe) belongs to Ae for all t ≥ 0. By decomposing an arbitrary

x ∈ B(h) into its self-adjoint and anti-self-adjoint parts we find that Tt(Ae) ⊆ Ae

for all t ≥ 0. The invariance of Ao is proved in the same way.

Denote by T e and T o the restrictions of T to Ae and Ao, respectively.

In the remaining part of this section we establish the asymptotic behavior of T e

and T o. To this end first recall the following results (see Frigerio and Verri11,13):

Theorem 4.3. Let S be a QMS on a von Neumann algebra A with a faithful

normal invariant state ω and let F(S), N (S) be the von Neumann subalgebras of

A

F(S) = {x ∈ A|St(x) = x, ∀ t ≥ 0} ,

N (S) = {x ∈ A|St(x∗x) = St(x

∗)St(x),St(xx∗) = St(x)St(x∗), ∀ t ≥ 0} .

Then:

(i) F(S) is contained in N (S),

(ii) if F(S) = N (S), then limt→∞ S∗t(σ) exists for all normal state σ on A,

(iii) if F(S) = C 1l, then ω is the unique S-invariant state,

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578 F. Fagnola & R. Quezada

(iv) if N (S) = F(S) = C 1l, then limt→∞ S∗t(σ) = ω for all normal state σ

on A.

The following result by Fagnola and Rebolledo,8,9 allows us to determine easily

F(T e), F(T o), N (T e) and N (T o) and apply Theorem 4.3.

Theorem 4.4. Suppose that both minimal QDS T associated with the operators

G, L` and T associated with the operators G∗, L` are Markov. Moreover, suppose

that there exists D ⊂ h dense which is a common core for G and G∗ such that the

sequence (nG∗(n − G)−1)u)n≥1 converges for all u ∈ D. Then N (T ) = {Lk, L∗k :

k ≥ 1}′ and F(T ) = {H, Lk, L∗k : k ≥ 1}′.

Here the {X1, X2, . . .}′ denotes the generalized commutator of the (possibly

unbounded) operators X1, X2, . . .. This is the subalgebra of B(h) of all the operators

y such that yXk ⊆ Xky (i.e. Dom(Xk) ⊆ Dom(Xky) and yXku = Xkyu for all

u ∈ Dom(Xk)) for all k ≥ 1.

We can now prove the main result of this section

Theorem 4.5. Let T e (resp. T o) be the restriction of the QMS T to the subalgebra

Ae (resp. Ao). Then N (T e) = N (T e) = C 1l (resp. N (T o) = N (T o) = C 1l). It

follows that, for any even state σe on Ae (resp. odd state σo on Ao), we have

limt→∞

T e∗t(σe) = ρe , lim

t→∞T o∗t(σo) = ρo .

Proof. Let us prove that N (T e) = C 1l. The proof of N (T o) = C 1l is identical. If

x ∈ N (T e) is a self-adjoint operator we have that xmn = 0 for all odd m, n ≥ 0.

Moreover, if a2x ⊂ xa2 and a+2x ⊂ xa+2, after simple computations we obtain

((2m)!)1

2 〈e2m, xe2n〉 = 〈e0, a2mxen〉 = 〈e0, xa2men〉 = 0

for all n < m. Being self-adjoint, this proves that x is diagonal. Now for any m,

n ≥ 0 after direct computations we obtain

(2m + 2)1

2 (2m + 1)1

2 x2m+22n = 〈e2m, a2xe2n〉

= 〈e2m, xa2e2n〉 = (2n − 1)1

2 (2n)1

2 x2m2n−2 ,

hence with n = m + 1 we obtain that x2n2n = x2n−22n−2 for all n ≥ 1. This proves

that x is a multiple of the identity operator. For a general element x ∈ N (T e) we

can use its decomposition as a linear combination of self-adjoint operators both in

N (T e). Then we have that F(T e) = N (T e) = C 1l. The conclusion follows then

from Theorem 4.3(iv).

5. Detailed Balance

A QMS T with a faithful normal invariant state ρ satisfies the quantum detailed

balance condition introduced by Frigerio, Gorini, Kossakowski and Verri12 if there

exists another QMS T such that

tr(ρyTt(x)) = tr(ρTt(y)x)

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Two-Photon Absorption and Emission Process 579

for all x, y ∈ B(h). The QMS T , in our case, is the QMS associated with G∗, L1,

L2 therefore it has the same generator as T up to the sign of ω. In particular, for

all x ∈ M, the generators satisfy

L(x) − L(x) = 2iω[a+2a2, x] . (5.1)

In this section we shall prove that the two-photon absorption and emission QMS

with λ > 0 satisfies the quantum detailed balance with respect to any invariant state

ρα = αρe + (1 − α)ρo , α ∈ ]0, 1[ . (5.2)

Indeed, we will show that, for all θ ∈ [0, 1], x, y ∈ B(h), we have

tr(ρ1−θα yρθ

αTt(x)) = tr(ρ1−θα Tt(y)ρθ

αx) . (5.3)

The identity (5.1) follows from a simple algebraic computation. However, since

we do not know whether M is an essential domain for both L and L, or even if

these operators have a common essential domain we cannot deduce (5.3) directly

from (5.1). In order to circumvent this difficulty we associate with our QMS certain

semigroups on the Hilbert space L2(h) of Hilbert–Schmidt operators on h endowed

with the scalar product 〈y, x〉 = tr(y∗x). For each ρα with α ∈ ]0, 1[ define the

embedding of B(h) into L2(h),

ι : B(h) → L2(h) , ι(x) = ρθ

2

αxρ1−θ

2

α .

The map ι is an injective contraction with a dense range and it is a completely

positive map for θ = 1/2. We now define T αt (ι(x)) = ι(Tt(x)) for every t ≥ 0 and

x ∈ B(h). The operators T αt can be extended to the whole L2(h) and they define

a unique strongly continuous contraction semigroup T α = (T αt )t≥0 on L2(h) (see

Carbone,5 Theorem 2.0.3). Moreover, if Lα is the infinitesimal generator of T α,

then ι(D(L)) is contained in the domain of Lα and

Lα(

ρθ

2

αxρ1−θ

2

α

)

= ρθ

2

αL(x)ρ1−θ

2

α

for every x in the domain D(L) of L. Notice that T α∗t (ρ

1

2

α) = ρ1

2

α for t ≥ 0, indeed

tr(ρ1

2

αT αt (ι(x))) = tr(ραTt(x)) = tr(ραx) = tr(ρ

1

2

α ι(x)) ,

for all x ∈ B(h). The generator Lα is characterized as follows.

Proposition 5.1. The linear manifold ι(M) is contained in the domain of Lα and

is a core for Lα.

Proof. The linear manifold M is contained in the domain of L, thus the weak∗

limit for t → 0+ of t−1(Tt(x) − x) exists. Exploiting this fact it is easy to check

that, for each x ∈ M, the weak limit of t−1(T αt (ι(x)) − ι(x)) for t → 0+ exists. It

follows that ι(x) belongs to the domain of Lα.

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580 F. Fagnola & R. Quezada

In order to show that ι(M) is a core for Lα we construct a sequence Lαn (n ≥ 1)

of bounded approximations such that, calling Mn the submanifold of M generated

by the |ej〉〈ek| with 0 ≤ j, k ≤ 2n, we have

Lαn(ι(Mn)) ⊆ ι(Mn+1) , ‖(Lα − Lα

n)|ι(Mn)‖ ≤ 8λµn .

Then, since Lα is dissipative and closed as the generator of a contraction semigroup,

it follows from Theorem 3.1.34, p. 193 of Ref. 4 that ι(M) = ∪n≥1ι(Mn) is a core

for Lα.

For each n ≥ 1 we denote by Nn the bounded approximation of the number

operator N ∧ (2n) of the number operator defined by truncation, i.e. Nnek = kek

for k ≤ 2n and Nnek = 2nek for k > 2n. Let an, a+n be the bounded approximations

of creation and annihilation operators

an = S∗N1/2n , a+

n = N1/2n S .

Let Ln be the approximated Lindblad generator defined as in (2.1) replacing a and

a+ by an and a+n . The operator Ln is bounded and, moreover, L(Mn) is contained

in Mn+1 for all n ≥ 1. A straightforward computation shows that, for all x ∈ Mn,

we have

Lα(ι(x)) − Lαn(ι(x)) = µ2ρ

θ

2

α (a+2xa2 − a+2n xa2

n)ρ1−θ

2

α .

Since x =∑

0≤j,k≤2n xjk |ej〉〈ek|, the operator a+2xa2 − a+2n xa2

n can be written as

2n∑

j=2n−1

2n−2∑

k=0

xjk((k + 1)(k + 2))1/2(((j + 1)(j + 2))1/2 − 2n)|ej+2〉〈ek+2|

+

2n∑

k=2n−1

2n−2∑

j=0

xjk((j + 1)(j + 2))1/2(((k + 1)(k + 2))1/2 − 2n)|ej+2〉〈ek+2|

+

2n∑

j=2n−1

2n∑

k=2n−2

xjk(((j + 1)(j + 2)(k + 1)(k + 2))1/2 − 4n2)|ej+2〉〈ek+2| .

Then the elementary inequalities,∣

∣((k + 1)(k + 2))1/2(((j + 1)(j + 2))1/2 − 2n)∣

∣ ≤ 2(3n + 1) ,∣

∣((j + 1)(j + 2)(k + 1)(k + 2))1/2 − 4n2∣

∣ ≤ 2(3n + 1)

for 2n − 1 ≤ j, k ≤ 2n lead to the estimate∥

∥

∥ρθ

2

α (a∗2xa2 − a∗2n xa2

n)ρ1−θ

2

α

∥

∥

∥

2

2≤ 4ν2(3n + 1)2‖ι(x)‖2

2 .

Now, since 2(3n+1) ≤ 8n for n ≥ 1, we can apply Theorem 3.1.34, p. 193 of Ref. 4

to conclude.

Starting from T , we can define in the same way T α and Lα by T αt (ρ

1−θ

2

α xρθ

2

α ) =

ρ1−θ

2

α Tt(x)ρθ

2

α . The linear manifold ι(M) is also a core for Lα (Lα and Lα differ only

by the sign of ω).

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Two-Photon Absorption and Emission Process 581

The following result establishes the duality between T α and T α.

Theorem 5.1. The semigroups T α, T α are the dual on L2(h), i.e. for all t ≥ 0

and all x, y ∈ L2(h) we have

tr(T αt (y)x) = tr(yT α

t (x)) . (5.4)

In particular (5.3) holds for all x, y ∈ B(h).

Proof. Since ρα is a function of the number operator, it commutes with the oper-

ators G and G∗. Moreover, an algebraic computation using the canonical commu-

tation relations shows that

ρθ/2α L∗

1 = ν2θ−1L2ρθ/2α , L1ρ

(1−θ)/2α = ν1−2θρ

(1−θ)/2α L∗

2 ,

ρθ/2α L∗

2 = ν1−2θL1ρθ/2α , L1ρ

(1−θ)/2α = ν2θ−1ρ

(1−θ)/2α L∗

1 .(5.5)

Therefore, for all x, y ∈ M, we have the identities

tr(ρ(1−θ)/2α yρθ/2

α (G∗x + xG)) = tr(ρ(1−θ)/2α (Gy + yG∗)ρθ/2

α x) ,

tr(ρ(1−θ)/2α yρθ/2

α (L∗1xL1 + L∗

2xL2)) = tr(ρ(1−θ)/2α (L∗

1yL1 + L∗2yL2)ρ

θ/2α x)

showing that tr(ρ(1−θ)α yρθ

αL(x)) = tr(ρ(1−θ)α L(y)ρθ

αx) and, for all r > 0,

tr((r − Lα)(ι(y))ι(x)) = tr(ι(y)(r − Lα)(ι(x))) .

It follows then that, for all x ∈ Dom(Lα) and y ∈ Dom(Lα), we have

tr((r − Lα)(y)x) = tr(y(r − Lα)(x)) .

Taking the resolvents, we find the identity

tr(y(r − Lα)−1(x)) = tr((r − Lα)−1(y)x)

for all x, y ∈ L2(h). Therefore, for all t > 0 and n ≥ 1 we obtain

tr(y(nt−1 − Lα)−n(x)) = tr((nt−1 − Lα)−n(y)x) .

The duality formula (5.4) follows from the Trotter–Kato formula letting n tend to

infinity. Replacing the operators x, y by ρθ/2α xρ

(1−θ)/2α , ρ

(1−θ)/2α yρ

θ/2α with x, y ∈ M,

in this formula we find

tr(ρ(1−θ)α T α

t (y)ρθαx) = tr(ρ(1−θ)

α yρθαT α

t (x)) .

Now (5.3) follows from the weak* density of M in B(h).

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582 F. Fagnola & R. Quezada

6. Characterization of the Invariant States for λ > 0

In this section we show that ρe and ρo determine all invariant states of T . Indeed,

we prove the following

Theorem 6.1. If λ > 0, then all invariant states of T are convex linear combina-

tions of ρe and ρo.

Let σ a fixed normal T -invariant state and let (σjk)j,k≥0 be its matrix elements

in the canonical orthonormal basis. As a first step we show that the diagonal part

of σ has the desired form.

Lemma 6.1. Let σ be a normal T -invariant state. Then we have

peσpe = tr(σpe)ρe , poσpo = tr(σpo)ρo . (6.1)

As a consequence, the diagonal part σd =∑

j σjj |ej〉〈ej | of σ is given by

σd = tr(σpe)ρe + tr(σpo)ρo .

Moreover, we have the inequalities |σjk | ≤ c(σ, ν)ν(j+k)/2 for all j, k, with c(σ, ν)

constant depending only on σ, ν.

Proof. We know from Theorem 4.5 that

w∗ − limt→∞

T et (pexpe) = tr(ρex)pe , w∗ − lim

t→∞T o

t (poxpo) = tr(ρox)po

for any x ∈ B(h) since Ae (resp. Ao) is the dual of states supported in pe (resp. po).

By the invariance of σ we have then

tr(σpexpe) = tr(σTt(pexpe)) = limt→∞

tr(σT et (pexpe)) = tr(ρex)tr(σpe) .

In the same way we find tr(σpoxpo) = tr(ρox)tr(σpo). Therefore, since x is arbitrary,

we obtain (6.1).

The positivity of σ implies |σjk |2 ≤ σjjσkk for all j, k. Therefore, since the

diagonal part of σ is a convex combination of ρe and ρo, the last claim follows.

We are now in a position to prove Theorem 6.1.

Proof of Theorem 6.1. Let σ be a T -invariant state. Then L∗(σ) = 0. Writing

σ =∑

j,k≥0 σjk |ej〉〈ek| we find

L∗(σ) =∑

j,k≥0

|ej〉〈ek|{

iω(k[k − 1] − j[j − 1])σjk − µ2

2(k[k − 1] + j[j − 1])σjk

+ µ2(k + 1)1

2 (k + 2)1

2 (j + 1)1

2 (j + 2)1

2 σj+2k+2

+ λ2k1

2 [k − 1]1

2 j1

2 [j − 1]1

2 σj−2k−2

− λ2

2((j + 1)(j + 2) + (k + 1)(k + 2))σjk)

}

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

Two-Photon Absorption and Emission Process 583

with the understanding that [k − 1] = max{(k − 1), 0}, [j − 1] = max{(j − 1), 0}.We shall prove that σjk = 0 for all j 6= k. Clearly, since σ is hermitian, it suffices

to check that σjk = 0 for all j > k. Letting j = k + n, with n ≥ 1, the identity

L∗(σ) = 0 yields

0 = −iωn(2k + n − 1)σk+n,k − µ2(k[k − 1] + (k + n)(k + n − 1))σk+n,k

−λ2((k + 1)(k + 2) + (k + n + 1)(k + n + 2))σk+nk

+ 2µ2(k + 1)1

2 (k + 2)1

2 (k + n + 1)1

2 (k + n + 2)1

2 σk+n+2k+2

+ 2λ2k1

2 [k − 1]1

2 (k + n)1

2 (k + n − 1)1

2 σk+n−2k−2

for all k, n ≥ 0. Fix n and put yk = ν−k/2σk+nk . Notice that the sequence (yk)k≥0

is square summable because, by Lemma 6.1, |yk| ≤ c(σ, ν)ν(k+n)/2. Multiplying by

yk, summing on k and taking the real part, the above equation reads as

0 = −µ2∑

k≥0

k[k − 1]|yk|2 − µ2∑

k≥0

(k + n)(k + n − 1)|yk|2

+ 2λµ Re∑

k≥0

(k + 1)1

2 (k + 2)1

2 (k + n + 1)1

2 (k + n + 2)1

2 ykyk+2

−λ2∑

k≥0

(k + 1)(k + 2)|yk|2 − λ2∑

k≥0

(k + n + 1)(k + n + 2)|yk|2

+ 2λµ Re∑

k≥0

k1

2 [k − 1]1

2 (k + n)1

2 (k + n − 1)1

2 ykyk−2

= −µ2n(n − 1)|y0|2 − µ2n(n + 1)|y1|2

−µ2∑

k≥2

k[k − 1]|yk|2 − µ2∑

k≥0

(k + n + 1)(k + n + 2)|yk+2|2

+ 2λµ Re∑

k≥0

(k + 1)1

2 (k + 2)1

2 (k + n + 1)1

2 (k + n + 2)1

2 ykyk+2

−λ2∑

k≥0

(k + 1)(k + 2)|yk|2 − λ2∑

k≥2

(k + n − 1)(k + n)|yk−2|2

+ 2λµ Re∑

k≥2

k1

2 [k − 1]1

2 (k + n)1

2 (k + n − 1)1

2 ykyk−2 .

Reconstructing a square from the three sums for k ≥ 0 and another from the three

sums on k ≥ 2 we find

0 = −µ2n(n − 1)|y0|2 − µ2n(n + 1)|y1|2

−∑

k≥0

∣

∣

∣λ(k + 1)

1

2 (k + 2)1

2 yk − µ(k + n + 1)1

2 (k + n + 2)1

2 yk+2

∣

∣

∣

2

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

584 F. Fagnola & R. Quezada

−∑

k≥2

∣

∣

∣λ(k + n − 1)1

2 (k + n)1

2 yk−2 − µk1

2 [k − 1]1

2 yk

∣

∣

∣

2

= −µ2n(n − 1)|y0|2 − µ2n(n + 1)|y1|2

−∑

k≥0

∣

∣

∣λ(k + 1)1

2 (k + 2)1

2 yk − µ(k + n + 1)1

2 (k + n + 2)1

2 yk+2

∣

∣

∣

2

−∑

k≥0

∣

∣

∣λ(k + n + 1)1

2 (k + n + 2)1

2 yk − µ(k + 2)1

2 (k + 1)1

2 yk+2

∣

∣

∣

2

.

Now, for n > 1, this implies y0 = y1 = 0 and then, by induction yk = 0 for all

k ≥ 0. If n = 1 then y1 = 0 and yk = 0 for all odd k’s by induction. Moreover, if

the two sums of squares are 0, then

yk+2 = ν(k + 1)1

2 (k + 3)−1

2 yk , yk+2 = ν(k + 1)−1

2 (k + 3)1

2 yk

for all k ≥ 0. This can happen only if yk = 0. Therefore σ is diagonal.

7. Approach to Equilibrium

In this section we study convergence of states T∗t(σ) towards an invariant state and

determine the domains of attraction of the invariant states. As a first step, with

the notation of Sec. 4, we prove the following.

Lemma 7.1. Suppose λ > 0. Then F(T ) = N (T ).

Proof. The QMS T has a faithful normal invariant state. Therefore the inclusion

F(T ) ⊆ N (T ) always holds. It suffices then to prove the opposite.

By Theorem 4.4 we have N (T ) = {a2, a+2}′ and F(T ) = {a2, a+2, a+2a2}′. A

x ∈ N (T ), satisfies then a2x ⊆ xa2 and a+2x ⊆ xa+2. It follows that

(a+2a2)x = a+2(a2x) ⊆ a+2(xa2) = (a+2x)a2 ⊆ (xa+2)a2 = x(a+2a2) .

Therefore x belongs also to F(T ).

Proposition 7.1. Suppose λ > 0. Then, for any normal state σ, we have

limt→∞

T∗t(σ) = tr(σpe)ρe + tr(σpo)ρo .

Proof. With the same notation as in the proof of Theorem 6.1 write σ = ασe +

(1 − α)σo + σr, where σe = peσpe and σo = poσpo are the even and odd diagonal

parts of σ, α = tr(peσ) ∈ [0, 1] and σr is the off-diagonal part of σ.

By Lemma 7.1 and Theorem 4.3(ii) the family of states T∗t(σ) converges to an

invariant state σ∞, which is diagonal, σ∞ = γρe + (1 − γ)ρo say, by Theorem 6.1.

Therefore by Theorem 4.5 and the Tt-invariance of pe we have

γ = tr(pe(γρe +(1−γ)ρo)) = limt→∞

tr(peT∗t(ασe +(1−α)σo +σr)) = αtr(peσe) = α .

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

Two-Photon Absorption and Emission Process 585

Applying again Theorem 4.5 we find then

αρe+(1−α)ρe = σ∞ = limt→∞

T∗t(ασe+(1−α)σo+σr) = αρe+(1−α)ρo+ limt→∞

T∗t(σr) .

This shows that limt→∞ T∗t(σr) = 0 and completes the proof.

Proposition 7.2. Suppose λ = 0. Then for any normal state σ we have

limt→∞〈ej , T∗t(σ)ek〉 = 0 for all j, k with max{j, k} ≥ 2 and

limt→∞

〈e0, T∗t(σ)e0〉 = tr(σpe) , limt→∞

〈e1, T∗t(σ)e1〉 = tr(σpo) ,

limt→∞

|〈e0, T∗t(σ)e1〉|2 ≤ tr(σpe)tr(σpo) .

Proof. If pjk(t) = 〈ej , T∗t(σ)ek〉, we have that p′jk(t) = tr(L(|ej〉〈ek|)T∗t(σ)). Com-

puting L(|ej〉〈ek|) we obtain the system of differential equations

p′jk(t) = µ2((k + 2)(k + 1)(j + 2)(j + 1))1

2 pj+2k+2(t)

−(

µ2

2(k[k − 1] + j[j − 1]) − iω(k[k − 1] − j[j − 1])

)

pjk(t) .

Now, putting dm(t) = pj+mk+m(t), the above system takes the form

d′m(t) = µ2((k + m + 2)(k + m + 1)(j + m + 2)(j + m + 1))1

2 dm+2(t)

− ((2−1µ2((k + m)(k + m − 1) + (j + m)(j + m − 1))

− iω((k + m)(k + m − 1) − (j + m)(j + m − 1)))dm(t) .

For j, k ≥ 2 fixed and 0 ≤ m ≤ n denote rm = 2−1((k + m)(k + m − 1) +

(j + m)(j + m− 1))− iωµ−2((k + m)(k + m− 1)− (j + m)(j + m− 1)), and sm =

((k+m+2)(k+m+1)(j+m+2)(j+m+1))1

2 . We have then an equation of the form

d′(t) = µ2(M+N)d(t), with d(t) = (d1(t), . . . , dn(t)), M = diag(−r0,−r1,−r2, . . .),

N =

0 s0 0 0 0 · · ·0 0 s1 0 0 · · ·0 0 0 s2 0 · · ·· · · · · · · · · · · · · · · · · ·

.

For the diagonal part the explicit solution is given by the exponential matrix

eµ2Mt = diag(e−r0µ2t, e−r1µ2t, e−r2µ2t, . . .)

satisfying the estimate ‖eµ2Mt‖1 ≤ e−Re(r0)µ2t, where ‖ · ‖1 denotes the norm of an

operator acting on `1(C).

When σ is a state with compact support, i.e. such that 〈ej , σek〉 = 0 for all j, k

larger than a fixed integer n, using the perturbation expansion

eµ2(M+N)tσ =

n∑

l=0

∫ t

0

ds1 · · ·∫ sl−1

0

dsleµ2M(t−s1)Neµ2M(s1−s2)N · · · eµ2Mslσ

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

586 F. Fagnola & R. Quezada

we obtain the estimate ‖e(M+N)t‖1 ≤ e−Re(r0)µ2t∑n

l=0(l!)−1(‖N‖1t)

l. Since

Re(r0) > 0, it follows that 〈ej , T∗t(σ)ek〉 → 0 as t → ∞, for any j, k ≥ 2. For

an arbitrary σ the conclusion follows by approximation with a compact support σ.

Now let us consider the matrix elements 〈ej , T∗t(σ)ek〉 with 0 ≤ j, k ≤ 1. By

Theorem 4.2 the subalgebras Ae and Ao are T -invariant. Therefore for a compactly

supported σe (since λ = 0 the support of T∗t(σe) is contained in that of σe for all

t ≥ 0) we have then

limt→∞

〈e0, T e∗t(σe)e0〉 = 1 − lim

t→∞

∑

k≥2

〈ek, T e∗t(σe)ek〉 = 1 .

This fact also holds for an arbitrary σe by approximation in trace norm

with a compact support σe. Moreover, in the same way we can prove that

limt→∞〈e1, T o∗t(σo)e1〉 = tr(σo) = 1.

Given an arbitrary σ we can write the decomposition σ = ασe + (1−α)σo + σr

with α = tr(σpe). It follows that

limt→∞

〈e0, T∗t(σ)e0〉 = α limt→∞

〈e0, T e∗t(σe)e0〉 = α ,

and limt→∞〈e1, T∗t(σ)e1〉 = (1 − α) limt→∞〈e1, T o∗t(σo)e1〉 = 1 − α. For the off-

diagonal terms we find ddt〈e0, T∗t(σ)e1〉 = 2

√3〈e2, T∗t(σ)e3〉, then, for all t, s > 0

|〈e0, (T∗t − T∗s)(σ)e1〉| ≤ 2√

3

∫ t

s

|〈e2, T∗τe3〉|adτ ≤ 2√

3(e−ksp(s) − e−ktp(t)) ,

where p is a polynomial and k a positive constant. Therefore there exists the limit

z = limt→∞〈e0, T∗t(σ)e1〉 and |z|2 ≤ α(1 − α), by positivity of limt→∞ T∗t(σ).

Proposition 7.3. Suppose λ = 0. The attraction domain of the invariant state

ρα,z = α|e0〉〈e0| + (1 − α)|e1〉〈e1| + z|e0〉〈e1| + z|e1〉〈e0| ,with |z|2 ≤ α(1 − α) is given by

D(ρα,z) =

σ = (σjk)|α =∑

m≥0

σ2m2m , z =∑

m≥0

c2m

√2m + 1σ2m2m+1

,

where c2m = 2−2m (2m)!m! Πm

j=1(j − iωµ2 )−1.

Proof. With the notation in the proof of Proposition 7.2 let θm(t) =√m + 1pmm+1(t), then θm satisfies the following differential equation

θ′m(t) = µ2(m + 2)(m + 1)θm+2(t) − (µ2m2 − 2iωm)θm(t) .

Since λ = 0, the linear manifold M is T∗t-invariant. Thus we can define for

σ ∈ M the function f(σ, t) =∑

m≥0 c2mθ2m(t), where c2m are coefficients to be

determined. We have that

f ′(σ, t) =∑

m≥0

(µ2c2m(2m + 2)(2m + 1) − c2m+2(µ2(2m + 2)2 − 4iω(m + 1)))θ2m ,

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

Two-Photon Absorption and Emission Process 587

therefore f ′(σ, t) = 0 for every σ ∈ M if and only if c2m

c2m−2

= 2−1(2m − 1)(m −iωµ2 )−1, for m ≥ 0. Taking c0 = 1 we obtain after some computations that c2m =

2−2m (2m)!m! Πm

j=1(j − iωµ2 )−1. With this choice of the coefficients c2m, we have that

f(σ, t) is a constant function of t for every σ ∈ M.

Now using Wallis’s product formula one can show that the sequence√

2m + 1c2m

is bounded by a positive constant c. Then for a positive σ ∈ M, the inequality

|pjk(t)|2 ≤ pjj(t)pkk(t) yields

f(σ, t) ≤ c∑

m≥0

|p2m,2m+1(t)| ≤ c∑

m≥0

p1

2

2m(t)p1

2

2m+1(t)

≤ c

∑

m≥0

p2m(t)

1

2

∑

m≥0

p2m+1(t)

1

2

≤ c tr(σ) .

Since every positive σ ∈ L1(h) can be approximated by an increasing sequence of

positive elements (σn) ⊂ M, f(σ, t) can be extended continuously to the whole

L1(h). We obtain that f(σ, t) is also a constant function of t and consequently

z = limt→∞

θ0(t) = limt→∞

fσ(t) = f(σ, 0) =∑

m≥0

c2m

√2m + 1〈e2m, σe2m+1〉 .

8. Deduction from the Stochastic Limit

In this section we outline the deduction of the two-photon absorption quantum

Markov semigroup from the stochastic limit of the evolution of a system (one-mode

EM field) coupled with a boson, zero temperature reservoir.

The state space of the system, a one-mode electromagnetic field, is the complex

separable Hilbert space h = `2(N). The free evolution of the system is given by

a strongly continuous unitary group (e−itHS )t∈R, where HS = E(N) is a positive

function E (E : Z → [0, +∞[) of the number operator N = a+a. The state space

of the reservoir (zero temperature boson gas) is the boson Fock space F over a

complex separable Hilbert space k1 (the one-particle space of the reservoir). The

free evolution of one particle in the reservoir is given by a strongly continuous one

parameter group (S0t )t∈R of unitary operators on k1 enjoying the following property:

there exists a dense subspace k of k1 such that

∫

R

|〈g, S0t f〉|dt < ∞

for all f, g ∈ k. The free evolution of the reservoir is given by the unitary group

obtained by second quantization (Γ(S0t ))t∈R of the unitaries S0

t on k1. This is a

strongly continuous unitary group and its generator HR, the self-adjoint operator

on F such that e−itHR = Γ(S0t ) for all t ∈ R, is the Hamiltonian of the reservoir.

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

588 F. Fagnola & R. Quezada

The evolution of the whole system is given by the unitary group generated by

the total Hamiltonian

Hλ = HS ⊗ 1F + 1S ⊗ HR + λV ,

where λ is a real positive parameter and V is an interaction operator such that Hλ

is self-adjoint for all λ > 0.

Suppose that the interaction operator (of dipole type) has the form

Vg = i(ad ⊗ A∗(g) − a+d ⊗ A(g)) ,

where d ∈ N∗, A(g), A∗(g) are creation and annihilation operators on F with g ∈ k1.

A straightforward computation using the commutation relations

eitE(N)ad = adeitE(N−d)

shows that generalized rotating wave approximation

eitHSade−itHS = e−iω0tad ,

where ω0 > 0 (see Ref. 3, Definition 4.10.1 on p. 125) holds, with ω0 = E(n− d)−E(n) for all n ∈ N if and only if E is linear. This is the case, for example, when

HS = a+a.

Suppose that the generalized rotating wave approximation holds. The sesquilin-

ear form on k

(f |g) :=

∫

R

〈g, S0t f〉dt

is positive (see Ref. 3). Therefore it defines a pre-scalar product on k. We denote

by K the Hilbert space obtained by quotient and completion; the scalar product

will be denoted by (·|·). Defining

U(λ)t = eitH0e−itHλ

a straightforward computation shows that the family of unitaries (U(λ)t )t≥0 on h⊗F

satisfies the differential equation

d

dtU

(λ)t = −iλVg(t)U

(λ)t , U

(λ)0 = 1l ,

where Vg(t) = i(D ⊗ A∗(Stg) − D∗ ⊗ A(Stg)) and St = eitω0S(0)t .

Let W (f) (f ∈ k1) denote the unitary Weyl operators on F acting on exponen-

tial vectors as

W (f1)e(f2) = e−‖f1‖2/2−〈f1,f2〉e(f1 + f2) .

The basic idea of Accardi, Frigerio and Lu1 was to study the result of small

interactions (λ → 0) on a large time scale (time goes to infinity). This was realized

by scaling time by λ2 and space by λ and letting λ tend to 0. As a result of the

limiting procedure, the state space of the whole system also changes.

The following result allows us to find the structure of the space of the limit

evolution.

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

Two-Photon Absorption and Emission Process 589

Proposition 8.1. For all n, n′ ∈ N and all f1, . . . , fn, f ′1, . . . , f

′n′ ∈ K,

s1, t1, . . . , sn, tn, s′1, t′1, . . . , s

′n′ , t′n′ ∈ R with sk ≤ tk, s′k ≤ t′k for all k denote

W (f1, . . . , fn) = W

(

λ

∫ λ−2t1

λ−2s1

Sr1f1dr1

)

· · ·W(

λ

∫ λ−2tn

λ−2sn

Srnfndrn

)

.

We have then

limλ→0

〈W (f1, . . . , fn)0, W (f ′1, . . . , f

′n)0〉

= 〈W (f1 ⊗ 1[s1,t1]) · · ·W (fn ⊗ 1[sn,tn])0 ,

W (f ′1 ⊗ 1[s′

1,t′

1]) · · ·W (f ′

n′ ⊗ 1[s′

n′,t′

n′])e(0)〉 ,

where W (f1 ⊗ 1[s1,t1]), . . . , W (f ′n′ ⊗ 1[sn′ ,tn′ ]) are Weyl operators in the boson Fock

space over L2(R+;K).

It follows that the state space of the limit evolution is the tensor product of the

initial space h with the boson Fock space Γ(L2(R+;K)). The limit of unitaries is

given in the following theorem.

Theorem 8.1. For all v, u ∈ h, n, n′ ∈ N and all f1, . . . , fn, f ′1, . . . , f

′n′ ∈ K, s1,

t1, . . . , sn, tn, s′1, t′1, . . . , s′n′ , t′n′ ∈ R with sk ≤ tk, s′k ≤ t′k we have

limλ→0

〈vW (f1, . . . , fn)e(0), U(λ)

λ−2tuW (f ′1, . . . , f

′n)e(0)〉

= 〈W (f1 ⊗ 1[s1,t1]) · · ·W (fn ⊗ 1[sn,tn])e(0) ,

UtuW (f ′1 ⊗ 1[s′

1,t′

1]) · · ·W (f ′

n′ ⊗ 1[s′

n′,t′

n′])e(0)〉 ,

where U is the unique unitary process satisfying the quantum stochastic differential

equation on h ⊗ Γ(L2(R+;K))

dUt = (addA∗t (g) − a+ddAt(g) − (g|g)−a+daddt)Ut , U0 =1l

with (g|g)− =∫ 0

−∞〈g, Stg〉dt. Moreover, for all x ∈ B(h), we have

limλ→0

〈vW (f1, . . . , fn)e(0), U(λ)∗λ−2t(x⊗ 1lF)U

(λ)λ−2tuW (f ′

1, . . . , f′n)e(0)〉

= 〈W (f1 ⊗ 1[s1,t1]) · · ·W (fn ⊗ 1[sn,tn])e(0) ,

U∗t (x⊗ 1lF)UtuW (f ′

1 ⊗ 1[s′

1,t′

1]) · · ·W (f ′

n′ ⊗ 1[s′

n′,t′

n′])e(0)〉 .

As a consequence, applying results on quantum stochastic differential equations

and quantum flows in Ref. 7, we have

Theorem 8.2. Suppose that HS = a+a. Then the quantum dynamical semigroup

of the quantum flow

jt(x) = U∗t (x⊗ 1lF )Ut

is the minimal quantum dynamical semigroup associated with

L−(x) = −(g|g)−a+dadx + 2 Re(g|g)−a+dxad − (g|g)−xa+dad .

November 9, 2005 11:49 WSPC/102-IDAQPRT 00211

590 F. Fagnola & R. Quezada

Several of the above assumptions (rotating wave approximation, boundedness

of D, . . .) can be removed or weakened. Moreover, other reservoirs and other types

of interaction can be studied. In particular, for d = 2, the stochastic limit of the

above system coupled with a boson reservoir at positive temperature leads to the

model introduced in Sec. 2. We refer to Ref. 3 for the proof.

Acknowledgments

The authors want to thank L. Accardi and J. C. Garcıa for several stimulat-

ing discussions. The financial support from the project Mexico-Italia “Dinamica

Estocastica”, the EU RTN Network “Quantum Probability with Applications to

Physics, Information Theory and Biology” and the Grant 37491-E of CONACYT-

Mexico, are gratefully acknowledged.

References

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2. L. Accardi and S. Kozyrev, Lectures on quantum interacting particle systems, Quan-

tum Interacting Particle Systems, eds. L. Accardi and F. Fagnola (World Scientific,2002), p. 1–195.

3. L. Accardi, Y. G. Lu and I. Volovich, Quantum Theory and Its Stochastic Limit

(Springer-Verlag, 2002).4. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechan-

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