Lectures on the Qualitative Analysis of Quantum Markov Semigroups

LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS

FRANCO FAGNOLA AND ROLANDO REBOLLEDO

Contents Lecture 1. Quantum Markov Semigroups and Master Equations 2 1.1. Preliminaries 2 1.2. Examples of Master Equations 3 1.3. The minimal quantum dynamical semigroup 4 Lecture 2. The existence of Stationary States 5 2.1. Tightness 6 2.2. A general result 6 2.3. Conditions on the generator 7 2.4. Applications 8 Lecture 3. Faithful stationary states and irreducibility 10 3.1. The support of an invariant state 10 3.2. Subharmonic projections 12 3.3. Applications 13 Lecture 4. The convergence towards the equilibrium 16 4.1. A result due to Frigerio and Verri 16 4.2. Quantum Markovian Cocycles 16 4.3. Main results 17 4.4. Applications 20 References 20

This research has been partially supported by the “Cátedra Presidencial en Análisis Cualitativo de Sistemas Dinámicos Cuánticos”, FONDECYT grant 1990439 and MURST Research Program “Probabilità Quantistica e Analisi Infinito Dimensionale” 2001-2002. 1 2 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

Lecture 1. Quantum Markov Semigroups and Master Equations The notion of a Quantum Markov Semigroup (QMS), from the mathematical point of view, is a non- commutative generalisation of that of classical Markov semigroup. In addition, it is physically motivated by contemporary research on open quantum systems. In this framework, states are represented by density matrices whose evolution is given by a Master Equation leading to a QMS. Several applications and examples will be discussed below. Related material has been presented in [2, 10, 30, 31] too. These lectures are aimed at discussing simple and effective methods to analyse qualitatively the behaviour of a given Quantum Markov Semigroup. In particular we focus our attention on the existence of normal stationary states, the characterisation of their support and eventually, the return to equilibrium. All of our applications come from Quantum Optics. Thus, the mathematical framework is different from that chosen by other authors of this volume who are mostly interested in Quantum Statistical Mechanics. 1.1. Preliminaries. We start by introducing the class of Quantum Markov Semigroup which will be studied in these lectures. Let us denote h a complex separable Hilbert space, endowed with a scalar product h·, ·i antilinear in the first variable, linear in the second. We denote by B(h) the von Neumann algebra of all linear bounded operators on h and that of all trace-class operators, by I1(h). For any A ∈ I1(h), we write tr (A) its trace. Definition 1.1. Let A be a von Neumann algebra on h.A Quantum Markov Semigroup is a w∗- continuous semigroup T = (Tt)t≥0 of normal completely positive linear maps on A such that T0(·) is the identity map of A and Tt(1) = 1, for all t ≥ 0, where 1 denotes the identity operator in A.

We recall that a linear map Φ defined on A is completely positive if for any finite collection ak, bk ∈ A, the element X ∗ ∗ bkΦ(akak)bk, k is positive. This notion is equivalent to positivity whenever A is abelian. Moreover, note that A is the dual of a Banach space (see for instance [4]), thus, the w∗-topology has an obvious meaning. Finally, if B denotes another von Neumann algebra, a positive linear map Φ : A → B is normal if for any increasing net (aα)α of positive elements in A, Φ(l.u.b. aα) = l.u.b. Φ(aα). In general these semigroups are termed Quantum Dynamical Semigroups too, however, we will reserve that name for the most general category of semigroups including both Markov and sub-Markov semigroups for which the weaker condition Tt(1) ≤ 1 holds. Definition 1.2. The infinitesimal generator of T is the operator L(·) with domain D(L) which is the ∗ −1 vector space of all elements a ∈ A such that the w –limit of t (Tt(a) − a) exists and, for a ∈ D(L), L(a) is the above limit. To go further, we remind the following well known concepts. Definition 1.3. A positive normal linear map ω : A → C such that ω(1) = 1 is called a normal state. Moreover, ω is faithful if ω(a) > 0 for all non-zero positive element a ∈ A. A state ω is invariant or stationary for a QMS T if ω(Tt(a)) = ω(a), for any a ∈ A, t ≥ 0. The w∗ or σ–weak topology of B(h) is the weaker topology for which all maps x 7→ tr (ρx) are continuous, where ρ ∈ I1(h) and tr (·) denotes the trace. The predual algebra of a von Neumann algebra A, which is the space of all σ-weakly continuous linear functionals on A, is denoted A∗, in particular, B(h)∗ = I1(h). Any quantum Markov semigroup T on A induces a predual semigroup T∗ on A∗ defined by

T∗t(ω)(a) = ω (Tt(a)) , (1.1) for all ω ∈ A∗, a ∈ A, t ≥ 0. + The space of normal states is S = {ω ∈ A∗ : ω(1) = 1}, where 1 denotes the identity in A. Normal states on B(h) are in one-to-one correspondence with density matrices, that is, positive trace- class operators with unit trace. The so-called Master Equations are nothing but evolution equations on density matrices (thus, associated with the predual semigroup) written in the form dρ t = L (ρ ), dt ∗ t where ρt = T∗ t(ρ) and ρ is a density matrix, L∗(·) being the generator of the predual semigroup. Given the generator of a QMS through a Master Equation, we are interested in answering the following questions: (i) Does there exist a stationary state of the QMS?, (ii) How its support looks like? LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 3

(iii) Is the semigroup returning to equilibrium? In a number of applications A is simply B(h) the von Neumann algebra of all bounded linear operators in h. The generator of a QMS is then represented in the form 1 X L(X) = i[H,X] + (L∗L X − 2L∗XL + XL∗L ) , (1.2) 2 k k k k k k k under suitable hypotheses on H,Lk (see [25, 29] for H,Lk bounded and [11, 6] for H,Lk unbounded). In all these cases we will provide simple conditions on H,Lk allowing to answer the above questions (i), (ii), (iii). To be more concrete, we list three physical models that will be studied throughout these lectures. 1.2. Examples of Master Equations. 1.2.1. Absorption and stimulated emission. This model has been introduced by Gisin and Percival [23]. Here h is l2(N) and the generator is formally given through (1.2) with ∗ ∗ L1 = νa a, L2 = µa, H = ξ(a + a), where µ, ν > 0, ξ ∈ R and a and a∗ are the usual annihilation and creation operators on h. 1.2.2. Squeezed oscillator. The squeezed oscillator has been obtained in the weak coupling limit of an oscillator coupled with a squeezed bath in [16]. In this case h is the same Hilbert space as before and

ν + η L(x) = − (a∗ax − 2a∗xa + xa∗a) 2 ν − (aa∗x − 2axa∗ + xaa∗) 2 ζ − (a∗2x − 2a∗xa∗ + xa∗2) (1.3) 2 ζ¯ − (a2x − 2axa + xa2) + 2iδ[a∗a, x], 2 where ν, η > 0, δ ∈ R and ζ ∈ C such that |ζ|2 ≤ ν(ν + η). It can be shown (see for instance [14] Section 4.3) that the above formal generator can be written in the form (1.2), with ∗ ∗ √ H = 2δa a, L1 = za + wa ,L2 = ν + η − µ a, (1.4) where 0 ≤ µ ≤ ν + η and z, w ∈ C satisfy |z|2 = µ, |w|2 = ν, zw¯ = ζ. 1.2.3. Multimode Dicke laser. This is a more complex model proposed by Alli and Sewell in [3] for a Dicke laser or maser. The system consists of N identical two-level atoms coupled with a radiation ﬁeld corresponding to n modes. Therefore, one can choose the Hilbert space h which consists of the tensor product of N copies of C2 and n copies of l2(N). To simplify notations we simply identify any operator acting on a factor of the above tensor product with its canonical extension to h. Let σ1, σ2, σ3 be the Pauli matrices and deﬁne the spin raising and lowering operators σ± = (σ1 ± iσ2)/2. The atoms are located at the sites r = 1,...,N of a one dimensional lattice, so that we denote by σ,r ( = 1, 2, 3, +, −) the spin component of the atom at the site r. The free evolution of the atoms is described by a generator Lmat which is bounded and given in Lindblad form as

3N 1 X L (x) = i[H, x] − (V ∗V x − 2V ∗xV + xV ∗V ), (1.5) mat 2 j j j j j j j=1 PN where H = (ε/2) r=1 σ3,r, and the Vj’s are the operators:

Vj = c±σ±, r, if j = 3r ± 1

Vj = c3σ3,r, if j = 3r, with ε, c+, c− > 0, c3 ≥ 0. ∗ Moreover, we denote by aj , aj, the creation and annihilation operators corresponding to the j-th mode of the radiation, (j = 1, . . . , n). These operators satisfy the canonical commutation relations: ∗ [aj, ak] = δjk1, [aj, ak] = 0. The free evolution of the radiation is given by the formal generator n X ∗ ∗ ∗ ∗ Lrad(x) = (κ`(−a` a`x + 2a` xa` − xa` a`) + iω`[a` a`, x]) , (1.6) `=1 4 FRANCO FAGNOLA AND ROLANDO REBOLLEDO where κ` > 0 are the damping and ω` ∈ R are the frequencies corresponding to the `-th mode of the radiation. The coupling between the matter and the radiation corresponds to a Hamiltonian interaction of the form:

N n i X X H = λ (σ a∗e−2πik`r − σ a e2πik`r), (1.7) int N 1/2 ` −,r ` +,r ` r=1 `=1 where k` is the wave number of the `-th mode and the λ’s are real valued, N independent coupling constants. With the above notations, the formal generator of the whole dynamics is given by

L(x) = Lmat(x) + Lrad(x) + i[Hint, x] (1.8)

To identify L` in our notations, we use in force the convention on the abridged version of tensor products with the identity. That is, here we ﬁnd √ L` = κ` a`, (` = 1, . . . , n) (1.9)

All the remaining L`’s are bounded operators. Among them a finite number (indeed at most 3N) coincides with some of the Vj’s appearing in (1.5) and the other vanish. 1.3. The minimal quantum dynamical semigroup. We finish this lecture by discussing a construction of the semigroup general enough to deal with most of applications like those mentioned before. P ∗ When H and the Lk’s are bounded and the series k LkLk is strongly convergent, then the generator given by (1.2) is also bounded and generates a norm continuous semigroup. Unfortunately, this is not sufficient for our purposes. Here we recall a procedure which allows to construct a semigroup, named the Minimal Quantum Dynamical Semigroup, associated with possibly unbounded H and Lk’s. A function t : D(t) × D(t) → C, where D(t) is a subspace of h, is a sesquilinear form over a Hilbert space h if t(v, u) is antilinear in v and linear in u. The set of all sesquilinear forms over h is denoted F (h). The form is said to be densely defined if its domain D(t) is dense, symmetric if t(v, u) = t(u, v), for all u, v ∈ D(t), and positive if t(u, u) ≥ 0, for all u ∈ D(t). We consider the von Neumann algebra A = B(h) and we rephrase, for easier reference the crucial result which allows to construct a quantum dynamical semigroup starting from a generator given as a sesquilinear form. For further details on this matter we refer to [14], section 3.3, see also [7]. Let G and L`,(` ≥ 1) be operators in h which satisfy the following hypothesis: • (H-min) G is the infinitesimal generator of a strongly continuous contraction semigroup in h, D(G) is contained in D(L`), for all ` ≥ 1, and, for all u, v ∈ D(G), we have ∞ X hGv, ui + hL`v, Lùi + hv, Gui = 0. `=1 Under the above assumption (H-min), for each x ∈ B(h) let −L(x) ∈ F (h) be the sesquilinear form with domain D(G) × D(G) defined by ∞ X −L(x)(v, u) = hGv, xui + hL`v, xLùi + hv, xGui. (1.10) `=1 It is well-known (see e.g. [11] Sect.3, [14] Sect. 3.3) that, given a domain D ⊆ D(G), which is a core for G, it is possible to built up a quantum dynamical semigroup, called the minimal QDS, satisfying the equation: Z t hv, Tt(x)ui = hv, xui + −L(Ts(x))(v, u)ds, (1.11) 0 for u, v ∈ D. This equation, however, in spite of the hypothesis (H-min) and the fact that D is a core for G, does not necessarily determine a unique semigroup. The minimal QDS is characterized by the following property: ∗ (min) for any w -continuous family (Tt)t≥0 of positive maps on B(h) satisfying (1.11) we have Tt (x) ≤ Tt(x) for all positive x ∈ B(h) and all t ≥ 0 (see e.g. [14] Th. 3.21). (min) (min) Let T∗ denote the predual semigroup on I1(h) with infinitesimal generator L∗ . It is worth (min) noticing here that T∗ is a weakly continuous semigroup on the Banach space I1(h), hence it is strongly (min) continuous. The linear span V of elements of I1(h) of the form |uihv| is contained in the domain of L∗ . Thus we can write the equation (1.11) as follows Z t (min) tr (|uihv|Tt(x)) = tr (|uihv|x) + tr L∗ (|uihv|)Ts(x) ds. 0 LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 5

(min) This equation reveals that the solution to (1.11) is unique whenever the linear manifold L∗ (V) is big enough. Indeed, the following characterization holds. Proposition 1.4. Under the assumption (H-min) the following conditions are equivalent: (min) (i) the minimal QDS is Markov (i.e. Tt (1) = 1), (min) ∗ (ii) (Tt )t≥0 is the unique w -continuous family of positive contractive maps on B(h) satisfying (1.11) for all positive x ∈ B(h) and all t ≥ 0, (min) (iii) the domain V is a core for L∗ . We refer to [11] Th. 3.2 or [14] Prop. 3.31 (resp. [14] Th. 3.21) for the proof of the equivalence of (i) and (iii) (resp. (i) and (ii)). Assume that the minimal QDS is Markov and call (Pt)t≥0 the semigroup of contractions generated by G, then the equation (1.11) may be written in an equivalent form as X Z t hv, Tt(x)ui = hPtv, Ptui + hL`Pt−sv, Tt(x)L`Pt−suids, (1.12) `≥1 0

(n) Moreover, the solution to (1.12) is obtained as the supremum of an approximating sequence (T )n∈N deﬁned recursively as follows on positive elements x ∈ B(h):

(0) ∗ Tt (x) = Pt xPt (1.13) (n+1) hu, Tt (x)ui = hPtu, xPtui (1.14) Z t X (n) + hL`Pt−su, Tt (x)L`Pt−suids, (u ∈ D(G)). `≥1 0 For any positive element x ∈ B(h) the above sequence is increasing with n. This allows to deﬁne (n) Tt(x) = supn Tt (x). T is the Minimal Quantum Markov semigroup, (see for instance [14]). The above discussion has shown the importance of getting a minimal quantum dynamical semigroup which preserves the identity, that is, a quantum Markov semigroup. Recently A.M. Chebotarev and the ﬁrst author have obtained easier criteria to verify the Markov property. For instance, the following result ([6] Th. 4.4 p.394) which will be good enough to be applied in our framework. Proposition 1.5. Under the hypothesis (H-min) suppose that there exists a self-adjoint operator C in h with the following properties: (a) the domain of G is contained in the domain of C1/2 and is a core for C1/2, 2 1/2 (b) the linear manifold L`(D(G )) is contained in the domain of C , (c) there exists a self-adjoint operator Φ, with D(G) ⊆ D(Φ1/2) and D(C) ⊆ D(Φ), such that, for all u ∈ D(G), we have

X 2 1/2 2 −2

where b is a positive constant depending only on G, L`, C. Then the minimal QDS is Markov. As shown in [14] the domain of G2 can be replaced by a linear manifold D which is dense in h, is a core 1/2 for C , is invariant under the operators Pt of the contraction semigroup generated by G, and enjoys the properties: 1/2 1/2 R(λ; G)(D) ⊆ D(C ),L` (R(λ; G)(D)) ⊆ D(C ) where R(λ; G)(λ > 0) are the resolvent operators. Moreover the inequality (1.15) must be satisﬁed for all u ∈ R(λ; G)(D).

Lecture 2. The existence of Stationary States The Ergodic Theory for positive semigroups on von Neumann algebras started longtime ago. This theory received many contributions, among them, those of Kovacs and Sz¨ucs[27], Størmer [39], Guichardet [24], Dang Ngoc [9], Frigerio [21], K¨ummerer and Nagel [28] (see [35] and the references therein). Here below we will be concerned with Markov semigroups on the von Neumann algebra B(h). Furthermore, the search for stationary states involves the σ(I1(h), B(h))-convergence of sequences of states of the form 6 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

1 Z tn T∗ s(σ)ds; n ≥ 0 , (2.1) tn 0 where σ ∈ S. From now on, weak convergence will mean always σ(I1(h), B(h))-convergence. Moreover, ∗ since I1(h) is the dual space of ﬁnite rank operators, the w -convergence on I1(h) has an obvious meaning. We have, indeed, the following well-known result. ∗ Proposition 2.1. For any σ ∈ S the w -limit of sequences of the form (2.1), with tn → ∞, is a T∗ t-invariant element of I1(h), for all t ≥ 0. Moreover the weak limit in addition an element of S. Proof. Call ρ one of these limit points, i.e.

Z tn ∗ 1 ρ = w − lim T∗ s(σ)ds. n tn 0 Then

Z tn ∗ 1 T∗ t(ρ) = w − lim T∗ s+t(σ)ds n tn 0 Z tn+r ∗ 1 = w − lim T∗ s(σ)ds n tn r Z tn Z tn+r Z r ∗ 1 = w − lim T∗ s(σ)ds + T∗ s(σ)ds − T∗ s(σ)ds n tn 0 tn 0 Z tn ∗ 1 = w − lim T∗ s(σ)ds. n tn 0 This shows that ρ is invariant. Clearly, it is positive, however it might not be a state since the property tr (ρ) = 1 could fail. If the convergence is weak, then

1 Z tn tr (ρ) = lim tr T∗ s(σ)ds 1 = 1. n tn 0 Therefore in this case ρ ∈ S.

2.1. Tightness. The above discussion stresses the fact that one can always ﬁnd w∗-convergent sequences of the form (2.1) by the Banach-Alaglou theorem, since I1(h) is the dual of the space of operators of ﬁnite rank. However, to have a state as a limit we need to preserve the unit trace property. This leads to introducing the concept of tightness.

Definition 2.2. A sequence of states (ωn)n is tight if for any > 0 there exists a finite rank projection p ∈ A and n0 ∈ N such that ωn(p) ≥ 1 − , for all n ≥ n0. Theorem 2.3. Any tight sequence of states on B(h) admits a weakly convergent subsequence. The reader is referred to [34] Theorem 2 p.27 (see also [30] Appendix 1.4) for the proof. A detailed exposition of this kind of results is contained in [8], where the link with tightness and narrow convergence in classical probability theory is done. The next two subsections are aimed at finding a criterion for the existence of stationary states for a quantum Markov semigroup whose generator is unbounded and known as a sesquilinear form.

2.2. A general result. We begin by introducing a notation for truncations of self-adjoint operators. Given a self-adjoint operator Y , bounded from below, we denote by Y ∧ r the truncated operator ⊥ Y ∧ r = YEr + rEr (2.2) where Er denotes the spectral projection of Y associated with the interval ] − ∞, r]. The following is our ﬁrst result on the existence of normal stationary states. Theorem 2.4. Let T be a quantum Markov semigroup. Suppose that there exist two self-adjoint operators X and Y with X positive and Y bounded from below and with ﬁnite dimensional spectral projections associated with bounded intervals such that Z t hu, Ts(Y ∧ r)uids ≤ hu, Xui (2.3) 0 for all t, r ≥ 0 and all u ∈ D(X). Then the QMS T has a normal stationary state. LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 7

Proof. Let −b (with b > 0) be a lower bound for Y . Note that, for each r ≥ 0 we have ⊥ Y ∧ r ≥ −bEr + rEr = −(b + r)Er + r1 so that (2.3) yields Z t 2 −(b + r) hu, Ts(Er)uids + rtkuk ≤ hu, Xui 0 for all u ∈ D(X). Normalize u and denote by |uihu| the pure state with unit vector u. Dividing by t(b + r), for all t, r > 0 we have then 1 Z t r hu, Xui tr (T∗s(|uihu|)Er) ds ≥ − . t 0 b + r t(b + r) It follows that the family of states 1 Z t T∗s(|uihu|) ds, t > 0 t 0 is tight. The conclusion follows then from Theorem 2.3.

It is worth noticing that we wrote the inequality (2.3) truncated (integral on [0, t], and Y ∧ r) to cope with two difficulties: the divergence of the integral and the unboundedness of Y . Defining appropriately the supremum of a family of self-adjoint operators and then the potential U for positive self-adjoint operators, the formula (2.3) can be written as U(Y ) ≤ X. This also throws light on the classical potential-theoretic meaning of our condition which is currently under investigation. In the applications, however, the inequality (2.3) is hard to check, since often the QMS is not explicitly given. We shall look for conditions involving the infinitesimal generator. 2.3. Conditions on the generator. Here we use the notations and hypotheses of the previous lecture yielding to the construction of the minimal quantum dynamical semigroup associated to a given form- generator. Definition 2.5. Given two selfadjoint operators X,Y , with X positive and Y bounded from below, we write −L(X) ≤ −Y on D, whenever ∞ X 1/2 1/2 hGu, Xui + hX Lù, X Lùi + hXu, Gui ≤ −hu, Y ui, (2.4) `=1 for all u in a linear manifold D dense in h, contained in the domains of G, X and Y , which is a core for 1/2 X and G, such that L`(D) ⊆ D(X ), (` ≥ 1). Theorem 2.6. Assume that the hypothesis (H-min) of the previous lecture holds and that the minimal QDS associated with G, (L`)`≥1 is Markov. Suppose that there exist two self-adjoint operators X and Y , with X positive and Y bounded from below and with finite dimensional spectral projections associated with bounded intervals, such that (i) −L(X) ≤ −Y on D; (ii) G is relatively bounded with respect to X; −1 1/2 (iii) L`(n + X) (D) ⊆ D(X ),(n, ` ≥ 1).

Then the minimal quantum dynamical semigroup associated with G, (L`)`≥1 has a stationary state. It is worth noticing that the above sufficient conditions always hold for a finite dimensional space h. Indeed, by the hypothesis (H-min), it suffices to take X = 1, Y = 0 and D = h. Here below we sketch the main idea of the proof, the reader is referred to [19] for the details. Proof. [Sketch] We outline the formal algebraic computation (which can be made rigorous through an appropriate approximation procedure). Differentiate d Z t X − Tt(X) − Ts(Y ∧ r)ds dt 0 = −Tt (−L(X) + Y ∧ r) ≥ 0. Therefore, Z t X − Ts(Y ∧ r)ds ≥ Tt(X) ≥ 0 0 for all t ≥ 0. 8 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

Thus the hypothesis (2.3) is fulﬁlled.

2.4. Applications.

2.4.1. Absorption and stimulated emission. This example corresponds to a family of models introduced by Gisin and Percival in [23]. The framework is given by the Hilbert space h = l2(N) where, as usual, we call (en)n≥0 the canonical orthonormal basis. The operators deﬁning the form-generator −L(·) are ∗ ∗ L1 = νa a, L2 = µa, H = ξ(a + a), where µ, ν > 0 and ξ ∈ R. Thus, 1 G = −iξ(a∗ + a) − ν2(a∗a)2 + µ2a∗a , 2 Let us check the existence of an invariant state by means of Theorem 2.6. Take, for instance X = (N + 1)2, where N = a∗a is the number operator. A straightforward computation yields −L(X) = iξ ((a − a∗)(N + 1) + (N + 1)(a − a∗)) + µ2N(1 − 2N). (2.5) We ﬁrst study the term iξ ((a − a∗)(N + 1) + (N + 1)(a − a∗)), which corresponds to i[H,X]. Call D the linear manifold generated by (en)n≥0. For any u ∈ D, we have |hu, i[H,X]i| = |ξ| h(a − a†)u, (N + 1)ui

1 † ≤ (a − a )u k(N + 1)uk 2 ≤ N 1/2u k(N + 1)uk 2 2 ≤ hu, Nui + hu, (N + 1)2i. 2 2 Thus,

2 2 −L(X) ≤ −2µ2N 2 + µ2N + (N + 1)2 + N. 2 2 Finally, if we call 2 2 2 Y = 2µ2 − N 2 − µ2 + 2 + N − 1, 2 2 2 we obtain

−L(X) ≤ −Y. In a similar way one can verify as well the inequalities which show that the minimal quantum dynamical semigroup is Markov. To summarize, Corollary 2.7. The quantum semigroup which corresponds to the model of absorption and stimulated emission here before is Markov and has a stationary state.

2.4.2. Squeezed oscillator. We start the study of this model proving that the generator associated with (1.3) defines a Markov semigroup. Indeed, the associated quadratic form can be written in the form (1.10) with G having the same domain as the number operator and given by 1 1 G = − L∗L − L∗L − 2iδa∗a. (2.6) 2 1 1 2 2 2 This operator clearly generates a strongly continuous contraction semigroup by [4], Thm.3.1.8, p.179. Indeed, each k-particle vector ek is an analytic vector for G since it satisfies an inequality of the form n n kG ekk ≤ Ck n!, where Ck is a constant depending on k, ν, η, µ, δ. Thus (H-min) follows easily. Now we proceed to verify that the minimal semigroup is Markov and then our form generator determines a unique QMS. We obtain this by an application of Proposition 1.5 with C = Φ = a∗a + 1. Within this extremely simple framework all domain assumptions are easily checked. The basic inequality (1.15) follows essentially from the algebraic computation L(a∗a + 1) = −ηa∗a + ν1. (2.7) The same computation allows to find X, Y and D satisfying Definition 2.5. Namely, D coincides with the domain of the number operator a∗a, X = a∗a + 1, Y = ηa∗a − ν1. The remaining hypotheses of Theorem 2.6 can be easily checked as well, thus, there exists a normal stationary state for the QMS. LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 9

2.4.3. Multimode Dicke laser. To identify L` and G in our notations, we use in force the convention on the abridged version of tensor products with the identity. That is, here we ﬁnd √ L` = κ` a`, (` = 1, . . . , n) (2.8)

All the remaining L`’s are bounded operators. Among them a finite number (indeed at most 3N) coincides with some of the Vj’s appearing in (1.5) and the other vanish. So that the operator G becomes formally: 1 X X G = − L∗L − i ω a∗a − iH − iH , (2.9) 2 ` ` ` ` ` int ` ` where the sum contains only a finite number of non zero terms. To make the above expression rigorous some preliminary work is needed. Call (fm)m≥0 the canonical orthonormal basis on the space l2(N). In the radiation space, which consists of the tensor product of n copies of l2(N), we denote f = f (1) ⊗ ... ⊗ f (n), α α1 αn (`) 2 where α = (α1, . . . , αn) and fα` is an element of the canonical basis of the ` copy of l (N). Thus, n (fα)α∈N is the canonical orthonormal basis of the radiation space. With these notations we have ( √ √ ∗ α`fα−1` if α` > 0 a` fα = α` + 1fα+1` , a`fα = , (2.10) 0 if α` = 0 where 1` is the vector with a 1 at the `th coordinate and zero elsewhere. 2N Thus, the operator G is well defined over vectors of the form ufα where u ∈ C and the symbol of tensor product is dropped. It is well known (see [26], Thm. 2.7 p.499) that a perturbation of a negative selfadjoint operator, relatively bounded with relative bound less than 1, is the infinitesimal generator of a contraction semigroup. Pn ∗ Therefore, we choose X formally given by X = `=1 a` a`. That is, Xufα = |α|ufα, where, |α| = α1 + ... + αn. k k Since X ufα = |α| ufα it follows that the linear span of vectors of the form ufα is a dense subset of the analytic vectors for X. Therefore, by a theorem of Nelson (see e.g. [37]), X is essentially self-adjoint on the referred domain. From now on we identify X with its closure which is selfadjoint. We show now that Hint is relatively bounded with respect to X. Let ξ be a finite linear√ combination√ √ of elements of the√ form ufα. By Schwarz’ inequality, and elementary inequalities like t + s ≤ t + s ≤ p2(t + s), 2 ts ≤ t + −1s, we obtain N n 1 X X kH ξk ≤ |λ |(ka∗ξk + ka ξk) int N 1/2 ` ` ` r=1 `=1 N n 1 X X 1/2 ≤ |λ | 4hξ, a∗a ξi + 2kξk2 N 1/2 ` ` ` r=1 `=1 " N n N n # 1 X X √ X X ≤ 2(|λ |2hξ, a∗a ξi)1/2 + 2 |λ |kξk N 1/2 ` ` ` ` r=1 `=1 r=1 `=1 " N n N n # 1 X X √ X X ≤ (ka∗a ξk + −1|λ |2kξk) + 2 |λ |kξk N 1/2 ` ` ` ` r=1 `=1 r=1 `=1 Pn √ Pn Finally, by the elementary inequality `=1 ks`k ≤ nk `=1 s`k, it follows

N n N n n1/2 X X 1 X X √ kH ξk ≤ k a∗a ξk + ( 2 + −1|λ |)|λ |kξk int N 1/2 ` ` N 1/2 ` ` r=1 `=1 r=1 `=1 N n 1 X X √ ≤ N 1/2n1/2kXξk + ( 2 + −1|λ |)|λ |kξk, N 1/2 ` ` r=1 `=1 −1/2 thus, choosing < (Nn) , the above inequality yields the required relative boundedness of Hint with respect to X. 1 As a result, the operator G appears as a dissipative perturbation of − 2 X, relatively bounded with respect to X, with bound strictly less than 1. Therefore, G is the generator of a contraction semigroup. Moreover, the domain of G coincides with that of X and hypothesis (H-min) easily checked. To apply our main result, we fix the domain D as the space of vectors ξ which are finite linear combinations of the form ufα. Notice that this is an invariant for X, G, and all the L`’s. To identify an 10 FRANCO FAGNOLA AND ROLANDO REBOLLEDO appropriate operator Y to have −L(X) ≤ −Y , we first perform the computation of −L(X). For the sake of clarity, we avoid handling forms in the computations below. However, the reader may easily notice that all the expressions are well defined since the domain D is invariant under the action of the operators X, G and L`. 2 Firstly, it holds Lmat(X) = 0, since the Vj’s act on the tensor product of N copies of C and leave the domain D invariant. Secondly, a straightforward computation using the canonical commutation relations, yields n X ∗ Lrad(X) = −2 κà` a`. `=1 Another easy computation leads us to

N n −1/2 X X ∗ −2πik`r 2πik`r i[Hint,X] = −iN λ`(σ−,ra` e + σ+,ra`e ). r=1 `=1 Summing up,

n N n X i X X −L(X) = −2 κ a∗a − λ (σ a∗e−2πik`r + σ a e2πik`r). ` ` ` N 1/2 ` −,r ` +,r ` `=1 r=1 `=1

To identify Y it suﬃces to control the term i[Hint,X]. For each ξ ∈ D, it follows

−1/2 So that choosing 0 < < 2N min κ` the required operator Y may be taken as 1/2 Y = (2 min κ` − N )(X + c), where c > 0 is a suitable constant. The spectrum of X coincides with N. For each m ∈ N, the corresponding eigenspace is generated by the fα with |α| = m. Therefore, it follows that all spectral projections of X and Y associated with bounded intervals are ﬁnite dimensional. Similar arguments allow to check the hypotheses of Proposition 1.5 (with C = X) to show that the minimal QDS associated to the operators G and L`, (1 ≤ ` ≤ n), is Markov (this was also proved in [3] by another method). To summarize, our main theorem implies the following Corollary 2.8. There exists an invariant state for the multimode Dicke model.

Lecture 3. Faithful stationary states and irreducibility In our previous lecture we discussed a useful criterion to prove the existence of a stationary state for a given QMS. Now we concentrate on the following question. If a semigroup admits a stationary state, how its supports looks like? In particular is it a faithful state?, is it a pure state? Indeed, the existence of a faithful stationary state is a crucial hypothesis in most of the ergodic studies developed in this ﬁeld (see for instance [21], [22], [17]). In this lecture we study this problem taking inspiration from classical probability, where a notion of irreducible Markov semigroups has been fruitfully introduced.

3.1. The support of an invariant state.

Deﬁnition 3.1. Given a Quantum Markov Semigroup (Tt)t≥0 on B(h), a positive operator a ∈ B(h) is subharmonic (resp. superharmonic, resp. harmonic) for the semigroup if Tt(a) ≥ a, (resp. Tt(a) ≤ a, resp. Tt(a) = a), for all t ≥ 0. LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 11

Subharmonic events play a fundamental role in the Potential Theory of classical Markov semigroups. They are related to stationarity, recurrence, supermartingale properties. In our framework, we will start by showing a relation between invariant states and subharmonics projections. We will frequently use the following elementary fact, which we state as a lemma whose trivial proof is omitted. Lemma 3.2. Let p be a projection on h, and x ∈ B(h) a positive element. If pxp = 0, then p⊥xp = pxp⊥ = 0. As a ﬁrst result we have Theorem 3.3. The support projection of a stationary state for a quantum Markov semigroup is subharmonic.

Proof. Let p be the support projection of a given stationary state ρ of (Tt)t≥0. That is, p is the projection on the closure of the rank of ρ, thus ρp = pρ = ρ, and T∗t(ρ) = ρ, (for all t ≥ 0). Let be given an arbitrary t ≥ 0. We ﬁrst notice that pTt(p)p ≤ p, since p ≤ 1. Therefore,

tr (ρ(p − pTt(p)p)) = tr (ρ(p − Tt(p))) = 0, and, since ρ is faithful on the subalgebra pB(h)p, it follows

pTt(p)p = p. On the other hand, ⊥ pTt(p )p = pTt(1)p − pTt(p)p = p − p = 0. ⊥ Moreover, since Tt(p ) is positive, the previous lemma yields ⊥ ⊥ ⊥ ⊥ pTt(p )p = 0 = p Tt(p )p. To summarize, ⊥ ⊥ Tt(p) = p + p Tt(p)p , and the projection p is subharmonic.

Proposition 3.4. For any Quantum Markov semigroup T the following propositions are equivalent: a) p is subharmonic for T . b) The subalgebra p⊥B(h)p⊥ is invariant for T . ⊥ c) For any normal state ρ such that pρp = ρ, it holds tr ρTt(p ) = 0, for all t ≥ 0.

Proof. Assume that condition a) holds. By hypothesis, we have that pTt(p)p = p, thus

⊥ pTt(p )p = pTt(1)p − pTt(p)p = 0. ⊥ ⊥ Therefore, for any positive x ∈ p B(h)p it follows pTt(x)p = 0 since ⊥ 0 ≤ pTt(x)p ≤ kxk pTt(p )p = 0. ⊥ ⊥ From Lemma 3.2, pTt(x)p = p Tt(x)p = 0, since Tt(x) is positive. ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ Thus Tt(x) = p Tt(x)p ∈ p B(h)p . The same conclusion holds for any arbitrary x ∈ p B(h)p since all those elements may be decomposed as a linear combination of four positive elements of p⊥B(h)p⊥. ⊥ ⊥ ⊥ ⊥ Now we prove that b) implies c). By hypothesis, for any x ∈ p B(h)p , it holds Tt(x) = p yp for ⊥ some y ∈ B(h). Clearly, y = Tt(x) since hv, Tt(x)ui = hv, yui for all vectors u and v for which u = p u, v = p⊥v. ⊥ ⊥ Therefore, Tt(x) = p Tt(x)p . In particular, ⊥ ⊥ ⊥ ⊥ Tt(p ) = p Tt(p )p . From this it follows that, given a state ρ such that pρ = ρp = ρ, we have p⊥ρ = ρp⊥ = 0 which yields ⊥ tr ρTt(p ) = 0. We ﬁnally prove that c) implies a). From c) we obtain that ⊥ tr ρpTt(p )p = 0,

⊥ thus pTt(p )p = 0. As a result, by Lemma 3.2, ⊥ ⊥ ⊥ ⊥ ⊥ Tt(p ) = p Tt(p )p ≤ p , ⊥ which gives Tt(p) = Tt(1 − p ) ≥ p.

Deﬁnition 3.5. We say that a quantum Markov semigroup is irreducible if there is no non-trivial subharmonic projection. 12 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

3.2. Subharmonic projections. Now we study the support of the stationary state of a minimal QMS, obtained from an unbounded generator given as a sesquilinear form (1.10). The hypothesis (H-min) is in force throughout this section, moreover, we suppose that the minimal semigroup is Markov. In what follows, we use the same notation p for both, a closed subspace and the projection determined by this subspace.

Lemma 3.6. Let (Pt)t≥0≥ 0 be the semigroup generated by G. A closed subspace Rk(p) is invariant for the operators Pt ( t ≥ 0) if and only if D(G) ∩ Rk(p) is dense in Rk(p), (λ − G)(D(G) ∩ Rk(p)) = Rk(p) for any λ > 0 and Gu = pGu for any u ∈ D(G) ∩ Rk(p).

Proof. Suppose ﬁrst that Rk(p) is invariant for the operators Pt ( t ≥ 0). Then Rk(p) is also invariant for the resolvent operators R(λ; G)(λ > 0). Therefore, for each u ∈ Rk(p), the vector uλ = λR(λ; G)u belongs to D(G) ∩ Rk(p) and, by the well-known properties of the Yosida approximations, uλ converges to u as λ goes to inﬁnity. Thus D(G) ∩ Rk(p) is dense in Rk(p). Moreover, for any u ∈ D(G) ∩ Rk(p) we have −1 −1 −1 pt (Ptu − u) = t (Ptpu − pu) = t (Ptu − u) . Therefore, letting t → 0, we obtain Gu = pGu. Finally, since R(λ; G)(Rk(p)) ⊆ D(G) ∩ Rk(p), Rk(p) = (λ − G)R(λ; G)Rk(p) ⊆ (λ − G)(D(G) ∩ Rk(p)) = p(λ − G)(D(G) ∩ Rk(p)) ⊆ Rk(p). It follows that (λ − G)(D(G) ∩ Rk(p)) = Rk(p). Conversely, since R(λ; G)Rk(p) = R(λ; G)(λ − G)(D(G) ∩ Rk(p)) = D(G) ∩ Rk(p), the closed subspace Rk(p) is invariant for R(λ; G) for any λ > 0. Therefore, since −1 −1 n Pt = strong − lim nt R(nt ; G) , (3.1) n→∞ it follows that Rk(p) is Pt-invariant.

Theorem 3.7. Under (H-min) suppose in addition that the minimal semigroup is Markov. Let (Pt)t≥0 denote the strongly continuous contraction semigroup on h generated by G. A projection p is subharmonic for T if and only if p is Pt-invariant for all t ≥ 0 and

L`u = pL`u, (3.2) for all u ∈ D(G) ∩ Rk(p), ` ≥ 1.

Proof. We start assuming that p is subharmonic, thus Tt(p) ≥ p for all t ≥ 0. From equation (1.12) we obtain ⊥ ⊥ ∗ ⊥ p ≥ Tt(p ) ≥ Pt p Pt. Therefore, for all u ∈ Rk(p), ∗ ⊥ ⊥ 2 hu, Pt p Ptui = p Ptu = 0, ⊥ that is p Ptp = 0. Thus Ptp = pPtp, for all t ≥ 0. Then Lemma 3.6 implies that Gu = pGu for all u ∈ D(G) ∩ Rk(p). In addition, the equation satisfied by the minimal semigroup yields   Z t ⊥ X ⊥ ⊥ hGu, Ts(p )ui + hLù, Ts(p )Lùi + hu, Ts(p )Gui ds ≤ 0, 0 `≥1 for all t ≥ 0 and all u ∈ D(G). As a result, computing the derivative in 0 of the above equation, we obtain: ⊥ X ⊥ ⊥ hGu, p ui + hLù, p Lùi + hu, p Gui ≤ 0. `≥1 Now, if u ∈ D(G) ∩ Rk(p) the above inequality gives

X ⊥ 2 p L`pu ≤ 0, `≥1 ⊥ that is p L`pu = 0 or, equivalently, pL`pu = pL`u, for all ` ≥ 1 and u ∈ D(G) ∩ Rk(p). LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 13

Conversely, we assume condition (3.2). We will prove that p is subharmonic by an induction argument (n) which relays on the sequence (T )n∈N used in the construction of T . Firstly, p is subharmonic for T (0) since

(0) ⊥ ∗ ⊥ ⊥ ∗ ⊥ ⊥ ⊥ Tt (p ) = Pt p Pt = p Pt p Ptp ≤ p . Secondly, assume that p is subharmonic for T (n), we prove that it is subharmonic for T (n+1) too. Indeed, for all u ∈ D(G) ∩ Rk(p), the deﬁnition of T (n+1) and the induction hypothesis yield Z t (n+1) ⊥ ∗ ⊥ X ⊥ hu, Tt (p )ui ≤ hu, Pt p Ptui + hL`Pt−su, p L`Pt−suids = 0, `≥1 0 for any t ≥ 0. (n+1) ⊥ ⊥ (n+1) ⊥ (n+1) ⊥ ⊥ It follows that pTt (p )p = 0 and Lemma 3.2 implies p Tt (p )p = pTt (p )p = 0. Therefore, (n+1) ⊥ ⊥ Tt (p ) ≤ p , for all t ≥ 0 and p is subharmonic for T (n). Hence, p is subharmonic for the minimal semigroup T and the proof is complete.

3.3. Applications.

3.3.1. Absorption and stimulated emission, continued. To continue the analysis of this model, we want to characterise all common invariant subspaces for Pt, which satisfy additionally (3.2). We will repeatedly use the following result based on a simple modiﬁcation of Theorem 3.2, p.81 of [32]. A detailed proof has been spelled out in [20]. Theorem 3.8. Let G be the generator of the strongly continuous contraction semigroup P and let B an operator on h such that: (i) D(B) ⊇ D(G), (ii) kBuk ≤ α kGuk + β kuk, for u ∈ D(G), with 0 ≤ α < 1 and β ≥ 0, (iii) G + rB is dissipative for each r ∈ [0, 1]. Then, G + rB generates a strongly continuous contraction semigroup P (r). Moreover, if the range Rk(p) of a projection p is an invariant subspace for the operators Pt and (r) B(D(G) ∩ Rk(p)) ⊆ Rk(p), then Rk(p) is also an invariant subspace for the operators Pt , for each r ∈ [0, 1]. A standard computation yields

2 2 ∗ ∗ L1u = 2G + µ (a a) + 2iξ(a + a) u 2 −2 ∗ ≤ 2 kGuk + µ ν kL1uk + 4 |ξ| ka uk µ2 2 |ξ| ≤ 2 kGuk + + kL uk + 4 |ξ| kuk ν2 ν2 1 2 µ + 2 |ξ| 2 1/2 1/2 ≤ 2 kGuk + L u kuk + 4 |ξ| kuk ν2 1 " 2 2 # 2 µ + 2 |ξ| ≤ 2 kGuk + L u + + 4 |ξ| kuk , 1 ν2 for any u ∈ D(N 2). Therefore, 2 2 2 −4 −2 (1 − ) L1u ≤ 2 kGuk + (µ + 2 |ξ|) ν + 4 |ξ| kuk , for any u ∈ D(N 2), 0 < < 1. Thus, for all 0 ≤ λ < 1, there exists 0 ≤ α < 1 and β > 0 such that

k(λ/2)L1uk ≤ α kGuk + β kuk . 2 (λ) Clearly, Gλ = G + (λ/2)L1 is dissipative and generates a semigroup P , by Theorem 3.8. Moreover, (λ) the same theorem shows that all our invariant subspaces are also invariant under the operators Pt . 2 2 Now we let λ → 1. Since D(N ) is a core for G1 and Gλu → G1u as λ → 1 for any u ∈ D(N ) it (λ) (1) follows from [32], Th. 4.5, p.88, that Pt u → Pt u, uniformly in t for t in bounded intervals. As a (1) result, all our invariant subspaces are Pt -invariant as well. 14 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

Repeating the above argument, it turns out that the sought invariant subspaces are invariant under ∗ the semigroup generated by G1 + (µ/2)L1 too. This generator is a restriction of −iξ(a + a). It follows from Lemma 3.6 that (a∗ + a)(D(N) ∩ Rk(p)) ⊆ Rk(p). ∗ At this point, we can apply again Theorem 3.8, adding iξ(a + a) to G1, so that the desired invariant subspaces are also invariant under the operators e−tN , for all t. These operators are compact and self- adjoint, therefore their invariant subspaces are generated by eigenvectors, i.e. they are the subspaces IK , spanned by {ek : k ∈ K} where K ⊆ N. However, notice that if K is non-empty and k0 = min K, for IK to be invariant also for L2, it must include all of the ek for k ≤ k0, since L2 is a multiple of the annihilation operator. In the above argument ∗ ∗ we obtained also that (a + a)(IK ) ⊆ IK . This, together with a(IK ) ⊆ IK , yields a (IK ) ⊆ IK , for all K ⊆ N, if ξ 6= 0. As a result, since e0 ∈ IK , if ξ 6= 0, the only invariant subspaces are {0} and h. If ξ = 0, it is easy to show that the full collection of invariant subspaces is

{0} , h, I{0,...,k}, for all k ∈ N. We summarize the above results in the following corollary. Corollary 3.9. If ξ 6= 0, the quantum Markov semigroup which corresponds to the model of absorption and stimulated emission here before is Markov and has at least a stationary state. Moreover, all stationary states are faithful.

3.3.2. Squeezed oscillator. Now we turn back to the example of the squeezed oscillator, where the existence of at least a stationary has already been established. Here we want to go a step further by proving the irreducibility of the minimal Markov semigroup. We apply Theorem 3.7 with G given by (2.6) and L1, L2 given by (1.4). Suppose, for simplicity, that |ζ|2 < ν(ν + η). ∗ ∗ Let p denote a subharmonic projection. Clearly, both L1 and L2 can be written as a linear combination of L1 and L2. Indeed, for instance ∗ −1 2 −1 −1/2 L1 =zw ¯ L1 + (w − |z| w )(ν + η − µ) L2. ∗ ∗ Therefore both L1 and L2 satisfy (3.2) on D(G) ∩ Rk(p). ∗ Recall that the domain of G coincides with that of the number operator (a a and that of the L`, ∗ ∗ L` , with the square root of the number operator. A standard approximation argument shows that L1 ∗ ∗ 1/2 −1 ∗ and L2 satisfy (3.2) on D((a a) ∩ Rk(p). It follows that, letting B = 2 (L1L1) all the assumptions of Theorem 3.8 hold as soon as we check the inequality (ii) therein. This will be done in the following lemma. Lemma 3.10. For all u ∈ D(G): −1 ∗ 2 (L1L1)u ≤ kGuk + β kuk , (3.3) where β is a suitable constant which depends only on the parameters of the model. Proof. The algebraic computations below makes sense on the domain of the square of the number operator. However to reduce the clutter of the notations we will not write the vectors on which the operators act. Clearly, 1 i G∗G = |L∗L + L∗L |2 − [H,L∗L + L∗L ] + H2. (3.4) 4 1 1 2 2 2 1 1 2 2 A tedious but simple computation using the CCR yields

∗ ∗ 2 ∗ 2 ∗ 2 ∗ ∗ ∗ ∗ |L1L1 + L2L2| = (L1L1) + (L2L2) + L1L1L2L2 + L2L2L1L1 ∗ 2 ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ = (L1L1) + (L2L2) + 2L2(L1L1)L2 + [L1L1,L2]L2 + L2[L2,L1L1] ∗ 2 ∗ 2 ∗ ∗ ∗ = (L1L1) + (L2L2) + 2L2(L1L1)L2 + 2(ν + η − µ)(L1L1 − ν1). Furthermore,

∗ ∗ ∗2 ¯ 2 [H,L1L1 + L2L2] = 4δ(ζa − ζa ). As a result, (3.4) becomes

∗ ∗ 2 ∗ 2 ∗ ∗ 4G G = (L1L1) + (L2L2) + 2L2(L1L1)L2 (3.5) ∗ ∗2 ¯ 2 2 + 2(ν + η − µ)(L1L1 − ν1) − 8iδ(ζa − ζa ) + 4H . LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 15

Now, the linear combination ζa∗2 − ζa¯ 2 can be bounded from below by a∗a + aa∗. Indeed, write ζ in polar representation as ζ = |ζ| exp(2iϕ). On the other hand,

iϕ ∗ −iϕ 2 0 ≤ e a ± ie a = (e−iϕa ∓ ieiϕa∗)(eiϕa∗ ± ie−iϕa) = ∓i(e2iϕa∗2 − e−2iϕa2) + a∗a + aa∗. Therefore, irrespectively of the sign of δ, we obtain iδ(ζa∗2 − ζa¯ 2) ≤ |δ| |ζ| (a∗a + aa∗) = |δ| |ζ| (2(a∗a + 1). Moreover, by Schwartz inequality, the right-hand side of the above equation can be bounded above by the operator 1 2 |δ|2 (a∗a)2 + (|δ| |ζ| + |ζ|2)1. 2 Thus from (3.5), we ﬁnally obtain 1 1 1 G∗G ≥ (L∗L )2 + (L∗L )2 + L∗(L∗L )L 4 1 1 4 2 2 2 2 1 1 2 1 + (ν + η − µ)(L∗L − ν1) − 4(|ζ|2 + 2 |δ| |ζ|)1 2 1 1 1 ν ≥ (L∗L )2 − (ν + η − µ) + 4(|ζ|2 + 2 |δ| |ζ|) 1. 4 1 1 2 1/2 Let β = 2−1ν(ν + η − µ) + 4(|ζ|2 + 2 |δ| |ζ|) . The above inequality reads as

2 1 ∗ 2 2 2 L L1u ≤ kGuk + β kuk . 2 1 √ √ √ To end the proof of the lemma it suﬃces to apply the elementary inequality x + y ≤ x + y, for x, y ≥ 0.

Following now the arguments used in the previous application, one obtains that the QMS of the squeezed oscillator is irreducible.

3.3.3. Multimode Dicke laser. To prove that there is a normal faithful stationary state we need to study all the projections p satisfying the conditions of Theorem 3.7. This is not a trivial problem at all because the annihilation operators admit plenty of non trivial invariant subspaces. Indeed, the linear span of any ﬁnite family of exponential vectors provides such an invariant subspace. Notice that by Theorem 3.7 every invariant projection p satisﬁes, for each r,

σ+,rp = pσ+,rp, σ−,rp = pσ−,rp. Taking the adjoints yields

pσ+,r = pσ+,rp, pσ−,r = pσ−,rp. It follows that p commutes with all the raising and lowering operators. Therefore it is of the form 1 ⊗ p0, where p0 is a projection on l2(N). For simplicity we identify p with p0 in what follows. Now, 1 ⊗ p must be invariant under the operators Pt of the semigroup generated by G. We apply P ∗ Theorem 3.8 as we did in the previous example to remove iH + (1/2) `>n L` L` from G, to end up with n 1 X G = − (κ + 2iω )a∗a − iH . 1 2 ` ` ` ` int `=1 (1) Theorem 3.8 shows that 1 ⊗ p is invariant under the operators Pt of the semigroup generated by G1 too. Now we state another easy modiﬁcation of a well-known result on multiplicative perturbations of a semigroup generator. Theorem 3.11. Let G be the generator of the strongly continuous contraction semigroup P and let Rk(p) be an invariant subspace under the operators Pt. Suppose that B is a bounded operator on h such that: (i) the domain of G is invariant under both B and B∗, (ii) the closure of B∗GB generates a strongly continuous contraction semigroup P B, (iii) Rk(p) is an invariant subspace for both B and B∗. B Then Rk(p) is an invariant subspace for the operators Pt . 16 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

∗ Applying Theorem 3.11 with B = σ+,1 . . . σ+,N , since B HintB = 0, the sought projections are also B ∗ ∗ Pn ∗ invariant under the operators Pt of the semigroup generated by 2B G1B = −B B `=1(κ` + 2iω)a` a`. Pn ∗ Therefore, being of the form 1⊗p, it follows that p is an invariant projection for exp (−t `=1(κ` + 2iω)a` a`), for all t. These are normal operators but, surprisingly, normality alone does not guarantee that the eigenspaces are generated by eigenvectors (see [33]). However, the above operators have the additional property of being compact. Thus, Theorem 4 p.272 in [40] applies, and shows that all the invariant subspaces are generated by eigenvectors. √ At this point, since the eigenvectors are explicitly known and L` = κ` a`, with κ` > 0, we can repeat the argument of the previous example to conclude that p must be trivial.

Lecture 4. The convergence towards the equilibrium Within this lecture we assume again the hypotheses leading to the construction of a minimal quantum Markov semigroup introduced in Lecture 1.3. Moreover, we also assume that the quantum Markov semigroup has a faithful normal stationary state ρ. Our aim here is to derive conditions under which T∗t(σ) converges in the weak topology towards ρ as t → ∞, for any initial state σ. We start giving a useful criterion to characterize the domain of the generator, assuming two hypothesis: (H-min) of section 1.3 and that the minimal QDS is Markov. We recall that, as in section 1.3, D is a core for G. Moreover, V the linear manifold generated by the rank-one operators |uihv|, where u, v ∈ D, is a core for L∗. Lemma 4.1. Under the above hypotheses the domain of L is given by all the elements X ∈ B(h) for which the application (v, u) 7→ −L(X)(v, u) is norm–continuous in the product Hilbert space.

Proof. We remark that X ∈ D(L) if and only if the linear form ρ 7→ tr (L∗(ρ)X), deﬁned on D(L∗), is ∗ continuous for the norm k · k1 of I1(h), since L = (L∗) . So that the essential of the proof consists in establishing the equivalence of the above property with the continuity of (v, u) 7→ −L(X)(v, u) as stated. Moreover, the reader will agree that the latter is a necessary condition for X being an element of D(L), so that it remains to prove the suﬃciency. Indeed, if L(X) is bounded then it is represented by Y = L(X) ∈ B(h). Then for any ρ ∈ V, the computation of tr (L∗(ρ)X) yields

|tr (L∗(ρ)X)| = |tr (ρY )| ≤ kρk1 kY k . The proof is then completed by a standard argument based on the core property of V.

4.1. A result due to Frigerio and Verri. Suppose that the semigroup T has a faithful stationary state ρ. Then there exists a conditional expectation X 7→ T∞(X) in the sense of Umegaki, defined over the von Neumann algebra of invariant elements under the action of T . We recall an early result of Frigerio and Verri ([22], Theorem 3.3, p.281) proved in this framework. ∗ ∗ Definition 4.2. We denote N (T ) the set of elements X ∈ B(h) for which Tt(X X) = Tt(X )Tt(X) and ∗ ∗ Tt(XX ) = Tt(X)Tt(X ), for all t ≥ 0. Theorem 4.3 (Frigerio–Verri). If the semigroup T has a faithful stationary state ρ and the set of fixed points of T coincides with N (T ), then ∗ w − lim Tt(X) = T∞(X), (4.1) t→∞ for all X ∈ B(h). The basic idea of the proof consists in associating with T a strongly continuous contraction semigroup on the Hilbert space of the GNS representation based on the state ρ. To look for a more friendly criterion based on the generator of the semigroup we need to introduce first some auxiliary notions and results on Quantum Markovian Cocycles. 4.2. Quantum Markovian Cocycles. Within this section we will built up a concrete dilation of the 2 QMS, as follows. Our basic Hilbert space is the tensor product H = h ⊗ Γ(L (R+; K)) where the first 2 factor is the initial space and the second, the Fock space associated to L (R+; K), where K is a complex separable Hilbert space with an orthonormal basis denoted by (zk; k ≥ 1). 2 2 The exponential vector on Γ(L (R+; K)), associated to a function f ∈ L (R+; K), is denoted e(f). Furthermore, we introduce a canonical projection E defined by Eu ⊗ e(f) = u ⊗ e(0). For simplicity we write the vector u ⊗ e(f) in the form ue(f), and we identify any operator on h with its canonical extension to the whole space H. A Quantum Markovian Cocycle (QMC) can be associated to our quantum dynamical semigroup. This notion was first introduced by Accardi and was later used in the Quantum Stochastic Calculus developed by Hudson and Parthasarathy to dilate Quantum Dynamical Semigroups with bounded generators. LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 17

In the case of unbounded generators, there is a need for some additional hypotheses (see [13]) which are listed below: • (H-D) There exists a domain D which is a core for both G and G∗; • (H-R) For all u ∈ D, the image R(n; G)u by the resolvent of G, belongs to D(G∗) and the ∗ sequence (nG R(n; G)u)n≥1 strongly converges. Under the hypotheses (H-D) and (H-R) it follows that the minimal quantum dynamical semigroup is connected with a unique contractive cocycle (see e.g. [13]) V = (Vt; t ≥ 0) through the relationship ∗ Tt(X) = EVtXVt E (4.2) and the equation: ∞ ! X ∗ † ∗ dVt = Vt [LkdAk(t) − LkdAk(t)] + G dt , (4.3) k=1 † in D, where (Ak; k ≥ 1), (Ak; k ≥ 1) denote annihilation and creation operators (see e.g. [31]) defined by Z t Ak(t)e(f) = ( fk(s)ds)e(f), (4.4) 0 d A† (t) = e(f + z 1 )| , (4.5) k d k [0,t] =0 2 for all f ∈ L (R+, K), for which fk denotes its k–th component, (t ≥ 0, k ≥ 1). ∗ The dual cocycle Ve = (Vet; t ≥ 0), is given by Vet = RtVt Rt , where Rt denotes the unitary time reversal operator on the Fock space, (t ≥ 0), (see eg. [30], ch.VI, 4.9). In addition, the equation satisfied ∗ by Ve is obtained by replacing Lk by −Lk and G by G in equation (4.3). This dual cocycle is associated to another semigroup Te as we did before: ∗ Tet(X) = EVetXVet E, (t ≥ 0). (4.6) The property of preservation of the identity by the semigroup and that of being an isometry for the cocycle, are related as it follows from a known result ([13], Th.5.3) quoted below for further easy reference. Proposition 4.4. Under the hypotheses (H-min), (H-D), (H-R), the cocycle V (respectively the dual cocycle Ve) is an isometry if and only if the semigroup Te (resp. T ) preserves the identity. From now on we further assume, • (H-Markov2) Both semigroups, T and Te, preserve the identity. Definition 4.5. We say that a quantum dynamical semigroup is natural if it satisfies hypotheses (H- min), (H-D), (H-R), (H-Markov2).

4.3. Main results. We begin by establishing a property of the space N (T ). Proposition 4.6. Given a natural quantum Markov semigroup, the space N (T ) is a von Neumann ∗ 0 algebra contained in the generalized commutator algebra of L = (Lk,Lk; k ≥ 1), denoted by L . Proof. Let X be any element in N (T ). Then for all ﬁxed t ≥ 0, ∗ ∗ Tt(X X) = Tt(X )Tt(X). (4.7) ∗ We introduce the quantum ﬂow jt associated to the cocycle V through the relationship jt(X) = VtXVt . ∗ ∗ This is an homomorphism and we obtain jt(X X) = jt(X )jt(X), which yields to ∗ ∗ Ejt(X X)E = Ejt(X )jt(X)E. (4.8) The left hand side of both (4.8) and (4.7) are equivalent. Moreover, the right–hand side of (4.7) can ∗ be written Ejt(X )Ejt(X)E, and it follows ∗ ∗ Ejt(X )Ejt(X) = Ejt(X )jt(X)E. (4.9) Let E⊥ denote the projection orthogonal to E, since E + E⊥ is equal to the identity of H, it yields ∗ ∗ ⊥ Ejt(X )Ejt(X) = Ejt(X )(E + E )jt(X)E, so that ∗ ⊥ Ejt(X )E jt(X)E = 0, (4.10) ⊥ that is, the operator E jt(X)E is null. ∗ ⊥ ∗ Replacing X by X in (4.7), (4.8), (4.9), (4.10), one obtains similarly that E jt(X )E is null. 2 Hence, for any f in L (R+; K), u, v ∈ h, and all t ≥ 0:

0 = hvRtf, jt(X)ue(0)i = hVetvf, XVetue(0)i, 18 FRANCO FAGNOLA AND ROLANDO REBOLLEDO and the equation for Ve yields Z t 0 = [hVesGvf, XVesue(0)i + hVesvf, XVesGue(0)i 0 d X + hVesLkvf, XVesLkue(0)i]ds k=1 d Z t X ∗ ¯ + [hVesve(0),XVesLkue(0)i − hVesLkve(0),XVesue(0)i]fk(s)ds, k=1 0 where we have adopted the notation fk for the eventually zero components of f according to the basis (zk; k ≥ 1). In particular, if a continuous f is chosen and the derivative of both members of the above equation is performed:

0 = hVetGvf, XVetue(0)i + hVetvf, XVetGue(0)i d X + hVetLkvf, XVetLkue(0)i k=1 d X ∗ ¯ + hVetve(0),XVetLkue(0)i − hVetLkve(0),XVetue(0)i]fk(t). k=1

Now, for all k ≥ 1 ﬁxed, we choose a function f such that fk(0) 6= 0, and make t → 0. Then the previous equation becomes ∗ hv, XLkui − hLkv, Xui = 0 (4.11) From (4.11) we ﬁrst deduce that Xu ∈ D(Lk) for all k ≥ 1, u ∈ D, thus, for all u ∈ D(G). In addition,

hv, XLkui = hv, LkXui, (4.12) from which we obtain for all k ≥ 1, XLk ⊆ LkX, since Lk is the closure of its restriction to the domain D(G). ∗ ∗ Similarly, X Lk ⊆ LkX is proven and it follows (see e.g. [41], Th.3, p.195): ∗ ∗ ∗ ∗ ∗ ∗ LkX = (X Lk) ⊇ (LkX ) ⊇ XLk for all k ≥ 1 and the proof is over.

It is an interesting problem to determine under which conditions the equality of N (T ) and L0 holds. To answer this question, we ﬁrst notice that from the equality L(1) = 0 it follows that G can be decomposed as a sum ∞ 1 X G = − L∗L − iH, 2 k k k=1 where the series weakly converges in D and H is a symmetric operator deﬁned on that domain. Theorem 4.7. A natural quantum Markov semigroup converges in the sense that ∗ w − lim Tt(X) = T∞(X), (4.13) t→∞ for all X ∈ B(h), whenever the generalized commutator L0 is reduced to the trivial algebra C1. Proof. This result follows straightforward from the Theorem of Frigerio and Verri, since N (T ) is contained in L0.

Proposition 4.8. Under the above hypotheses, if in addition the closure of H is self–adjoint, then the set A(T ) of ﬁxed elements for the semigroup is given by ∗ 0 A(T ) = {Lk,Lk,H; k ≥ 1} (4.14) Proof. Since A(T ) ⊂ N (T ), and N (T ) is contained in L0, therefore A(T ) ⊂ L0. So that, for all X ∈ A(T ) and any u, v ∈ D it holds: 0 = −L(X)(v, u) ∞ ∞ 1 X 1 X = − hL∗L v, Xui + hL v, XL ui 2 k k 2 k k k=1 k=1 ∞ ∞ 1 X 1 X + hL v, XL ui − hv, XL∗L ui 2 k k 2 k k k=1 k=1 − hiHv, Xui − hv, iXHui. LECTURES ON THE QUALITATIVE ANALYSIS OF QUANTUM MARKOV SEMIGROUPS 19

We now study the right hand side of the above equation. Since XLk ⊆ LkX, the ﬁrst two terms cancel and the computation

∗ ∗ ∗ ∗ hLkv, XLkui = hX Lkv, Lkui = hLkX v, Lkui = hX v, LkLkui, shows that the third and fourth terms cancel as well. From the above we deduce XH ⊆ HX. Therefore, X belongs to

∗ 0 {Lk,Lk,H; k ≥ 1} . ∗ 0 Reciprocally, if X ∈ {Lk,Lk,H; k ≥ 1} , the equation for −L(X)(v, u) gives 0 and from Lemma 4.1 we obtain that X is a ﬁxed point of T .

Proposition 4.9. For any natural quantum Markov semigroup which satisfies either (a) H is bounded; or (b) H is selfadjoint and eitH (D) ⊂ D(G), it holds, ∗ 0 N (T ) = {Lk,Lk; k ≥ 1} . (4.15) Proof. Consider first the case (a) of a bounded operator H. Then X ∈ D(L), and −L(X)(v, u) = hv, i(H∗X − XH)ui, (4.16) so that, iHt −iHt Tt(X) = e Xe . (4.17) Since L0 is a ∗–algebra, it contains both X∗ and X∗X. Then (4.17) holds as well for those elements, and ∗ iHt ∗ −iHt ∗ Tt(X )Tt(X) = e X Xe = Tt(X X), for all t ≥ 0. Therefore, X ∈ N (T ). Now, in case we have hypothesis (b), H being self-adjoint and exp(itH)(D) ⊆ D(G), take u, v ∈ D. The algebra N (T ) is invariant under the action of the semigroup by its own definition. Then, d hv, e−isH T (X)eisH ui = hiHeisH v, T (X)eisH ui ds t−s t−s isH isH + he v, Tt−s(X)(iH)e i isH isH − −L(Tt−s(X))(e v, e u) = 0. It is worth noticing that the hypothesis (b) on the core D implies that eitH u ∈ D(H). From the above equations it holds that −isH isH hv, e Tt−s(X)e ui is constant in s. Therefore X ∈ N (T ).

The corollary which follows is easily derived from the propositions and theorem before. Corollary 4.10. For any natural quantum Markov semigroup, the convergence towards the equilibrium ∗ 0 ∗ 0 holds if {Lk,Lk,H; k ≥ 1} = {Lk,Lk; k ≥ 1} and either (a) H is bounded; or (b) H is selfadjoint and eitH (D) ⊆ D(G). Remark. The suﬃcient condition obtained for proving the convergence towards the equilibrium is necessary, at least for a wide class of operators H, as we state in the following theorem. Theorem 4.11. Under the above notations, given a natural quantum Markov semigroup for which H is a self–adjoint operator with pure point spectrum and either (a) H is bounded; or (b) H is selfadjoint and eitH (D) ⊆ D(G).

Then Tt(·) converges towards the equilibrium if and only if ∗ 0 ∗ 0 {Lk,Lk,H; k ≥ 1} = {Lk,Lk; k ≥ 1} . (4.18) 20 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

Proof. From the corollary before, (4.18) is a suﬃcient condition for the convergence towards the equilibrium. We will prove below that it is a necessary condition as well. Indeed, the hypotheses assumed imply that itH −itH Tt(X) = e Xe , for all X ∈ N (T ). For any two diﬀerent eigenvalues λ and µ of H, choose corresponding eigenvector v, u ∈ h. Then, he−itH v, Xe−itH ui = eit(µ−λ)hv, Xui, converges when t → ∞. Therefore, hv, Xui = 0 and X commute with H. Consequently,

∗ 0 ∗ 0 {Lk,Lk,H; k ≥ 1} = {Lk,Lk; k ≥ 1} .

4.4. Applications. Now we complete the study of the models introduced in these lectures. It is worth noticing that in all these examples the convergence towards the equilibrium holds under the hypotheses on the parameters introduced in Lecture 3. Indeed, the reader can easily verify that in all the aforementioned models ∗ 0 ∗ 0 {H,Lk,Lk; k = 1, 2} = {Lk,Lk; k = 1, 2} = C1, thus, Theorem 4.7 applies straightforward. Moreover, by another result contained in [22], the faithful stationary state obtained in each of the applications before is unique, since the ﬁxed-point algebra is trivial.

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Dipartimento di Matematica, Universita` degli Studi di Genova, Via Dodecaneso 35, I-16146, Genova, Italia

Facultad de Matematicas,´ Pontificia Universidad Catolica´ de Chile, Casilla 306, Santiago 22, Chile