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Lectures on the Qualitative Analysis of Quantum Markov Semigroups

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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 “C´atedra Presidencial en An´alisis Cualitativo de Sistemas Din´amicos Cu´anticos”, FONDECYT grant 1990439 and MURST Research Program “Probabilit`a 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.
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