A View on Stochastic Differential Equations Derived from Quantum Optics

A VIEW ON STOCHASTIC DIFFERENTIAL EQUATIONS DERIVED FROM QUANTUM OPTICS

FRANCO FAGNOLA AND ROLANDO REBOLLEDO

Quantum Optics has had an impressive development during the last two decades. Laser–based devices are invading our every–day life: printers, pointers, a new surgical technology replacing the old scalpel, transport of information, among many other applications which give the idea of a well established theory. Even though, various theoretical problems remain open. In particular Quantum Optics continues to inspire mathematicians in their own research. This survey is aimed at giving a taste of some mathematical problems raised by Quantum Optics within the ﬁeld of Stochastic Analysis. Espe- cially, we want to show the interplay between classical and non commutative stochastic diﬀerential equations associated to the so called master equations of Quantum Optics.

Contents 1. Introduction 2 1.1. Notations 3 1.2. Deriving the master equation 4 2. The associated Quantum Stochastic Diﬀerential Equation 7 2.1. Quantum noises in continuous time 8 2.2. The master equation for the quantum cocycle 9 2.3. The Quantum Markov Flow 10 3. Unravelling: classical stochastic processes related to quantum ﬂows 12 3.1. The algebra generated by the number operator 12 3.2. Position, momentum and their algebras 13 3.3. The evolution on AN , Aq, Ap. 15 4. Stationary state and the convergence towards the equilibrium 16 References 18

Research partially supported by FONDECYT grant 1960917 and the “CátedraPresi- dencial en Análisis Estocástico y F´ısica Matemática”. 1 2 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

1. Introduction A major breakthrough of Probability Theory during the twentieth cen- tury is the advent of Stochastic Analysis with all its branches, which is being currently used in modelling phenomena in different sciences. Par- ticularly, in Physics, it is well known the probabilistic nature of Quantum Mechanics, since its very beginning. Moreover, it is dramatically striking that both, Probability, and Quantum Mechanics were given ground for the first time contemporaneously. Indeed, Kolmogorov in the thirties, provided the first rigourous axiomatic setting for Probability Theory. In parallel, von Neumann, inspired by the research of Planck, Schrödinger, Heisenberg, Bohr and all the founders of Quantum Mechanics, looked for another theory of probability which could cope with the experimental evidence stated in Heisenberg’s uncertainty principle. Quantum Optics is perhaps one of the fields where the probabilistic nature of quantum physics is most self evident. Indeed, mathematical models for quantum optics are often labelled as examples of the so called open systems in Quantum Physics. In those models, it is considered that the main phenomenon is affected by the interaction of the “free” system with the thermal bath in which the system is embedded. Naturally, the existence of a thermal bath is immediately associated to stochastic noises. However, physicists use to describe the evolution of an open system by mimicking very laboriously the customary setting of a closed system evolution. Indeed, that approach is strongly based in determining the generator, or Hamiltonian of a unitary group of transformations on the state space. Unfortunately, such a Hamiltonian should consider the description of position and momentum for an infinite number of particles, which is far from being rigorously tractable. Several different procedures have been developed by physicists to reduce the complexity of the above Hamiltonian, deriving the so called master or Langevin equations (e.g. coarse-graining, weak coupling limit, spectral tech- niques). Perhaps one of the most interesting approaches is that of weak coupling limit and its extension to the concept of stochastic limit which has been deeply studied by Accardi, Frigerio and Lu in several articles (see eg. [7]). The current article being not primarily addressed to physicists but to classical probabilists, will choose the less sophisticated physical approach which is, in our opinion, that of the coarse graining, to derive the basic master equation. We are most interested in the interplay between classical and quantum stochastic differential equations associated to the aforementioned master equation. There are four basic operations related to particles which we are faced to describe in a suitable mathematical language: creation, annihilation, the gauge or number of particles and the time shift. In a classical probabilistic approach, these operations may be seen first by means of point or counting processes, for instance. Indeed, creation may STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 3 be thought of a positive jump on the total number of particles at a given time, annihilation, on the contrary, is a negative jump of the same process. Someone could perhaps argue that from the point of view of a (classical) mechanical description, what counts is a couple of data, the position and velocity of each individual particle. In that case, the use of diffusion processes should be more appropriate to write the evolution of a single particle. From there on, one has to consider the interaction between particles as they perform their own evolution, measure-valued processes arising then in a very natural way to characterize the motion of the system as a whole. However, interactions in quantum theory may be of more fundamental a nature, since they can deeply change the observed objects. This fact naturally leads to deal with non-commuting operations, that is, to work with linear operators defined on a given Hilbert space. Furthermore, the seminal work of Wiener on chaos decomposition, provided a deep understanding of the important role played by some classes of Hilbert spaces which contain, intrinsically, a model for Brownian motion or other remarkable stochastic “noises”. Therefore, to built up stochastic models for quantum physics one needs an additional amount of functional analysis and algebra. After all, from a very heuristic point of view, Itôcalculus could be algebraically interpreted through three relationships on “differentials”: dtdWt = 0 = dWtdt, dtdt = 0, dWtdWt = dt, where W denotes a Brownian motion. As well as Canonical Commutation Relations (CCR) in quantum theory have a precise algebraic statement in terms of the commutator of the position and the momentum operators (that is the commutator of position and momentum is equal to i~, where ~ is the Planck constant). So that, intuitively, a mathematical structure supporting at least a non commutative version of Itôcalculus as well as the CCR is needed. To start with, we consider an elementary Hilbert space and will introduce therein four important operators which will support our first approach to modelling stochastic noises.

2 1.1. Notations. We ﬁx h = l (Z) as the initial space, with its canonical basis (en). B(h) denotes the space of bounded linear operators acting on h.

Deﬁnition 1.1. We deﬁne the following fundamental operators on h: (1) The annihilation operator (√ n en−1 , if n > 0 (1) aen = 0 , if n ≤ 0,

for all n ∈ Z. The domain of a is D(a) = {x = (xn)n ∈ h : P 2 n |n||xn| < ∞} 4 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

(2) The creation operator (√ † n + 1 en+1 , if n ≥ 0, (2) a en = 0 , if n < 0,

† † for all n ∈ Z. The domain of a , is also D(a ) = D(a).

(3) The number operator ( nen , if n > 0, (3) Nen = 0 , if n ≤ 0,

for all n ∈ Z. The domain of N is D(N) = {x = (xn)n ∈ h : P 2 2 n |n| |xn| < ∞}.

(4) The right–shift operator

(4) Sen = en+1, (n ∈ Z).

The above operators are related by the following relationship: (5) a† = N 1/2S = S(N + 1)1/2,

(6) a = (N + 1)1/2S∗ = S∗N 1/2.

Furthermore, given any function ϕ : N → C, we deﬁne a function of the number operator as follows: ( ϕ(n)en , if n > 0 (7) ϕ(N) en = ϕ(0)en , if n ≤ 0 Here below, we use the above operators to deal with the so called “coarse graining method” for deriving the master equation of quantum optics. It is worth noticing that for a classical probabilist, an eﬀort is demanded to change his mind about elementary notions like events (which will now be assimilated to projections on a given Hilbert space), random variables (which become observables or self-adjoint operators on a Hilbert space), probability measures (replaced by states, which means positive trace-class operators with unit trace).

1.2. Deriving the master equation. The statement of master equations of quantum optics follows some general principles which we make explicit below for better understanding: • The device is based upon the emission of photons due to the interaction of an electromagnetic ﬁeld and a collection of atoms. The electromagnetic ﬁeld acts within a cavity, STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 5

• The atoms are injected in the cavity at different times. They interact with the electromagnetic field, but they do not interact with each other, • The energy level of each atom is affected by the interaction with the field. This is a the basic framework for a microscopic approach to laser and maser description(see [36]). We are now interested in deriving (at least heuristically) an equation describing the evolution of the field state. What we mean by the state at time t is a positive trace-class operator ρ(t) with unit trace, defined over the Hilbert space h introduced before. This type of equation is named after Langevin. The first known research on this type of equations in quantum theory is due to Senitzky, and further used by Haken [27] and Lax [33] to study quantum optics. The Master Equation for the Laser was obtained by Scully and Lamb by the so called coarse–grained approximation [50], [49], which we recall briefly below. Let (ti; i ∈ N) denote the sequence of atom injection times inside the laser cavity. So that ∆ti = ti − ti−1 is the length of the time interval between two injections. During such a time interval, the effective interaction between the injected atom and the field begins at instant τi−1 and ends at τi. Thus, ∆τi < ∆ti, and the temporal evolution of ρ(t) within [ti−1, ti] is given by the expression: (8) ρ(t ) = exp(Lf,r∆t )T ga ρ(t ), (i ≥ 1), i i ∆τi i−1 where Lf,r is an operator which describes the interaction between the field and the reservoir with nT thermal photons and loss rate γc, given by (see [27]) γ Lf,rρ = − c (n + 1)(a†aρ + ρa†a − 2aρa†)(9) 2 T γ − c n (aa†ρ + ρaa† − 2a†ρa), 2 T ga and (Tt ; t ≥ 0) is the family of gain operators defined by Jaynes–Cummings in [31] which we will replace by an approximation. Indeed, if we assume the interaction length period ∆τi = ∆τ to be constant and that atoms are injected at their upper energy level, then it has been shown in [40] that the gain operator can be expressed as: √ √ T gaρ = cos(g(∆τ)∆τ aa†)ρ cos(g(∆τ)∆τ aa†)(10) ∆τ √ √ sin(g(∆τ)∆τ aa†) sin(g(∆τ)∆τ aa†) + a† √ ρ √ a, aa† aa† where, g(∆τ) is the coupling coefficient between field and atoms, which depends on the length ∆τ of the effective interaction time. To handle the equation we will need to introduce some additional as- ga sumptions. Indeed, since γc∆ti << 1 and that ρ(ti−1) − T∆ τρ(ti−1) is 6 FRANCO FAGNOLA AND ROLANDO REBOLLEDO small enough when ∆ti is close to 0, since it represents the variation of the gain for one atom, during the short interval of time [ti−1, ti], we obtain f,r ga (11) ρ(ti) = ρ(ti−1) + L ∆tiρ(ti−1) + [T∆τ − 1]ρ(ti−1). by a series expansion up to the order one in (8). We rewrite (11) as ga ∆ρ(ti) f,r [T∆τ − 1]ρ(ti−1) ∆τ (12) = L ρ(ti−1) + . ∆ti ∆τ ∆ti

Now let ∆ti decrease to 0 together with ∆τ in such a way that ∆τ → R2 < 1 and g(∆τ)∆τ → φ, (φ is the phase constant), ∆ti we obtain a diﬀerential equation of the form

d µ2 ρ(t) = (−a†aρ(t) + 2aρ(t)a† − ρ(t)a†a)(13) dt 2 λ2 + (−aa†ρ(t) + 2a†ρ(t)a − ρ(t)aa†) 2 2 + R (ϕ1(N)ρ(t)ϕ1(N) − ρ(t)) 2 ∗ + R Sϕ2(N)ρ(t)ϕ2(N)S , √ p where λ = γcnT , µ = γc(nT + 1), where ϕ1, ϕ2 are given by √ (14) ϕ (N) = cos(φ N + 1), 1 √ (15) ϕ2(N) = sin(φ N + 1).

Equivalently, we have a semigroup of operators (Tt) which has a generator L(X) defined over a domain D(L) to be made precise later, given by µ2 L(X) = (−NX + 2N 1/2SXS∗N 1/2 − XN) 2 λ2 + (−(N + 1)X + 2(N + 1)1/2S∗XS(N + 1)1/2 − X(N + 1))(16) 2 2 2 ∗ + R (ϕ1(N)Xϕ1(N) − X) + R ϕ2(N)S XSϕ2(N). We call (13) the Master Equation of Quantum Optics, and L given by (16), the Generator of the Quantum Optics Semigroup. It should be remarked that in the physical literature, the derivation of the master equation include a further approximation procedure consisting in a Taylor expansion of trigonometric functions which appear in the gain operator (see [50], [48]). The semigroup (Tt)t≥0 is a prototype of a so called Quantum Dynamical Semigroup. Let us quote here the precise definition of this class of semigroups. In general they may be defined on von Neumann subalgebras of B(h). STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 7

Deﬁnition 1.2. A Quantum Dynamical Semigroup (QDS) of a von Neu- mann algebra M ⊂ B(h) is a weakly*–continuous one–parameter semigroup (Tt)t≥0 of completely positive linear normal maps of M into itself which preserve the identity. In addition, it is assumed that T0 coincides with the identity map I. The property of being completely positive for a map T : M → M means that for any ﬁnite collection Xi,Yi,(i = 1, . . . , n) of elements of M, the Pn ∗ ∗ operator i,j=1 Yi T (Xi Xj)Yj is positive. A QDS is a straightforward generalisation of a Markov Semigroup. In- deed, given a measurable space (E, E), and a probability space (Ω, F, P), a Markov semigroup (Pt)t≥0 corresponding to a Markov process (Xt)t≥0 with states in E is given by:

Pt(f)(x) = E(f(Xt)/X0 = x), for any f ∈ M = L∞(E, E), t ≥ 0, x ∈ E. So that, the semigroup may be naturally deﬁned over a von Neumann algebra, and the property of being completely positive is nothing but positivity. The general structure of a QDS has been derived after the application of a well known result of Stinespring (see [37], [42] for a brief account of the theory). The general form of the generator of a uniformly continuous QDS, or Liouvillian, has been obtained by Lindblad in [34], (indeed, for this rea- son some authors, call “Lindbladian” the above generator). To summarize, what physicists call a “master equation”, is nothing but an expression giving the generator L (the Lindbladian or Liouvillian) of the Quantum Dynamical Semigroup describing the evolution of the physical phenomenon. To end with this brief survey of remarkable properties of QDS, a fundamental con- tribution of Accardi to this theory was the notion of a Quantum Markovian cocycle connected with the representation of the semigroup. We will return to this general representation later, since we will need to introduce some additional notations and concepts.

2. The associated Quantum Stochastic Differential Equation The search for a stochastic process associated to a given semigroup is connected, like in the classical Markov theory, with the construction of a bigger canonical space, where the process could be obtained in some cases as a solution of a stochastic diﬀerential equation. This is our task now. We need to built up a mathematical structure which could give an account of the evolution in terms of quantum ﬂows or processes. Before to proceed, we want to say a word about the choice of canonical spaces for quantum processes. The chaos decomposition property of the Wiener space is well known to probabilists. Indeed, that property establish a remarkable isomorphism between the L2-space built up on the Wiener probability space and the direct sum of symmetric tensor products of a Hilbert space, which is the 8 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

L2-closure of the vector space generated by the Wiener process. Therefore, that direct sum of symmetric tensor products of a Hilbert space contains intrinsically the proper Wiener process. Now, given a complex separable J Hilbert space K, denote by K n the n-th symmetric tensor power of that space. The Boson-Fock space associated to K is deﬁned by M J Γ(K) = K n. n≥0 As a result, the chaos representation property of the Wiener space could be restated by saying that this space is an example of a Fock space. But, is more important to notice that, as well as in the Wiener case, a Fock space contains stochastic noises intrinsically.

2.1. Quantum noises in continuous time. We begin by introducing our fundamental quantum noises. We let H be the Hilbert space defined as the tensor product 2 4 H = h ⊗ Γ(L (R+, C )), 2 2 4 where h = l (Z) is the initial space we have taken before and Γ(L (R+, C )) 4 is the usual Boson–Fock space of C –valued functions. 4 We will work now with operators defined on H. This leads to consider C – valued functions. In order to make explicit the action over each component 4 4 we take the canonical basis (uk)k=1 of C . As usual, |ulihum| denotes the projection |ulihum|z = hum, ziul for any z ∈ C. Moreover, given any t > 0, the multiplication operator by the characteristic function 1]0,t[, denoted by the 2 same symbol, defines a projection on the space L (R+, C). As a result, since 2 4 2 4 L (R+, C ) is isomorphic to L (R+, C) ⊗ C , the operator 1]0,t[ ⊗ |ulihum| is 2 4 2 4 a projection on L (R+, C ). The usual notation e(f) ∈ Γ(L (R+, C )) will be adopted for the exponential vector corresponding to the test function 2 4 f ∈ L (R+, C ), that is X 1 J e(f) = √ f n. n≥0 n! In the sequel we will write elements of the form u ⊗ e(f), (u ∈ h, f ∈ 2 4 L (R+, C )) by ue(f). On the other hand, given an operator X on h, we extend it to H as X ⊗ I, where I is the identity operator on the Fock space. In addition, we introduce the canonical projection E : H → h defined as Eue(f) = ue(0) (where h is identified with the space {ue(0) : u ∈ h}). The above projection o H induces a projection E on the algebras as follows:

(17) E(Z) = EZE ∈ B(h), for any Z ∈ B(H). STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 9

2 4 For all t ≥ 0 consider the operators on Γ(L (R+; C )) deﬁned by l Λm(t)e(f) = Λ(1]0,t[ ⊗ |ulihum|)e(f), d (18) = −i e(exp(iε1]0,t[ ⊗ |ulihum|)f) . dε ε=0 Λ is known as the Number operator.

0 † Λm(t)e(f) = A (1]0,t[ ⊗ hum|)e(f) d (19) = e(f + ε1]0,t[um) , dε ε=0 A† is called the Creation operator.

l Λ0(t)e(f) = A(1]0,t[ ⊗ |uli)

(20) = hul1]0,t[, fie(f). A is named the Annihilation operator.

0 (21) Λ0(t)e(f) = te(f), which is simply tI. For these operators the following Itˆo’stable holds: k l ˆ l (22) dΛm(t)dΛj(t) = δk,jdΛm(t), where, ( ˆ 1 if k = j > 0, (23) δk,m = 0 otherwise 2.2. The master equation for the quantum cocycle.

Definition 2.1. A Quantum cocycle V = (V (t))t≥0 connected to the semigroup T is a family of contractions on H, such that the semigroup is repre- sented in the form: ∗ ∗ Tt(X) = E(V (t)XV (t) ) = EV (t)(X ⊗ I)V (t) E, for all t ≥ 0 and X ∈ B(h). The master equation derived in the previous section yields to an equation for the cocycle which represents the evolution of physical phenomenon. We first derive formally the equation from the master equation of quantum optics, to continue then studying the problem of existence and uniqueness of their solutions. The coefficients of our equation are taken as the closure of the following operators defined over the domain D which is the linear span of the basis (em)m of h. 1 1/2 2 1/2 3 4 L0 = −µN ,L0 = −λ(N + 1) ,L0 = −Rϕ1(N),L0 = −Rϕ2(N), 10 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

0 1/2 0 1/2 ∗ 0 0 ∗ L1 = µN S, L2 = λ(N + 1) S ,L3 = Rϕ1(N),L4 = Rϕ2(N)S ,

1 2 ∗ 4 ∗ L1 = S − I,L2 = S − I,L4 = S − I, 1 L0 = − (µ2N + λ2(N + 1) + R2),L1 = 0 otherwise. 0 2 m With the above notations we can restate the master equation in the form: 4 X l m (24) dV (t) = V (t)LmdΛl (t), l,m=0 (25) V (0) = I. This is our fundamental Quantum Stochastic Diﬀerential Equation. It should be remarked that the following two conditions are necessary for V to be unitary: 4 l m ∗ X l m ∗ (26) (Lm + (Ll ) + Lk(Lk ) )u = 0 (u ∈ D), k=1

4 l m ∗ X k ∗ k (27) (Lm + (Ll ) + (Ll ) Lm)u = 0, (u ∈ D). k=1 l 4 Since D is a common invariant domain for the operators (Lm)l,m=0 we can apply the main theorem in [15] to prove the existence of a solution to (24). Theorem 2.1 (see [18]). There exists a unique unitary cocycle V which is a solution to the Quantum Stochastic Diﬀerential Equation (24) associated to the Master Equation in Quantum Optics. Furthermore, V is strongly 2 4 continuous on vectors of the form ue(f) with u ∈ D, f ∈ L (R+; C ) The solution of this master equation gives the evolution of all the observables of the quantum system and it is worth noticing that some of them are related to well-known classical stochastic processes. We will particularly concentrate on some of those classical processes. However, a word on the concept of quantum process becomes necessary before.

2.3. The Quantum Markov Flow. In classical theory, a process is un- derstood as a family (Xt)t≥0 of random variables defined over a given probability space (Ω, F, P), taking values on a state space (E, E). To produce an algebraic version of this concept, one introduces two algebras of complex valued bounded measurable functions: A = L∞(Ω, F) and M = L∞(E, E). Secondly, for any t ≥ 0, the map jt : M → A defined as jt(f) = f(Xt), becomes a ∗–homomorphism of algebras, where the ∗ operation means sim- ∗ ply the complex conjugation. The family of –homomorphisms j = (jt)t≥0, called the stochastic flow, contains all the information on the stochastic STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 11 process (Xt)t≥0. It is worth noticing that, a classical stochastic differential equation induces, through the application of Itô’s formula, a stochastic differential equation on jt(f), for f on a suitable domain included in M. The extension of the above concept is provided by a Quantum Markov Flow or Quantum Markov Process. In our case, we define the quantum Markov flow as given by the expression:

∗ (28) jt(X) = V (t)XV (t) , for all t ≥ 0 and X ∈ B(h). The quantum ﬂow has the important property of sending any element on the initial algebra to a bigger algebra depending on time. Indeed, ﬁxing a t ≥ 0 and X ∈ B(h), the image jt(X) is an element of the algebra B(Ht), where Ht is the space obtained as

2 4 Ht = h ⊗ Γ(L ([0, t]; C )).

Here again the role played by the Fock space is crucial, since it allows to extend the notion of filtration in classical probability, because the total space 2 4 Γ(L ([0, ∞[; C )) may be, (isomorphically), expressed as a tensor product 2 4 2 4 of two pieces: Γ(L ([0, t]; C )) and Γ(L (]t, ∞[; C )) for any t ≥ 0. Notice that the above property in the classical case translates in jt(f) ∈ ∞ L (Ω, Ft), where Ft is an element of a filtration to which the process X is adapted. Moreover, the quantum flow is connected to the quantum dynamical semigroup T through the relation

Tt(X) = E(jt(X)), for all X ∈ B(h) and t ≥ 0. We ﬁnish this discussion by stating a corollary of Theorem 2.1.

Corollary 2.1. For all X ∈ B(h) for which X(D(N)) ⊂ D(N) the quantum Markov process (jt(X))t≥0 satisﬁes the stochastic equation

4 X l l (29) djt(X) = jt(θm(X))dΛm(t), j0(X) = X, l,m=0

l where the structure maps θm are given by

4 l l m ∗ X l m ∗ (30) θm(X) = LmX + X(Ll ) + LkX(Lk ) , (0 ≤ l, m ≤ 4). k=1

Proof. It suﬃces to apply Ito’s formula to compute the diﬀerential of jt(X) 12 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

3. Unravelling: classical stochastic processes related to quantum flows A quantum Markov flow, when restricted to invariant abelian W ∗-subalgebras, gives raise to different classical stochastic processes. We call this property the unravelling of the quantum process. ∗ Indeed, assume that A is a W -commutative algebra such that Tt(A) ⊂ A, for all t ≥ 0, which implies L(A) ⊂ A. We first notice that A is isomorphic to some space L∞(E, E) and the elements X ∈ A may be identified with a ∞ multiplication operator Mf by a function of L (E, E). Then a semigroup ∞ structure is induced on L (E, E) by defining Ttf like MTtf = Tt(Mf ), for all t ≥ 0, or, by introducing a generator L through the relation MLf = L(Mf ), (f ∈ L∞(E, E)). This leads to a classical Markov semigroup and therefore, to an associated classical Markov process (Ω, F, (Ft)t≥0, (Px)x∈E, (Xt)t≥0) which verifies Ttf(x) = Ex(f(Xt)), for all t ≥ 0, x ∈ E. We will illustrate this fact with the quantum optics flow. To this end, we summarize, for easy reference, our previous results. The free quantum Liouvillian is the unbounded operator L0 describing the interaction between the field and the reservoir given by µ2 L (X) = (−NX + 2N 1/2SXS∗N 1/2 − XN) 0 2 λ2 + (−(N + 1)X + 2(N + 1)1/2S∗XS(N + 1)1/2 − X(N + 1)) 2 on the appropriate domain. In this section we describe some invariance properties of L0. The same invariance properties of the corresponding free quantum dynamical semigroup can be easily proved rigorously using an in- tegral equation satisfied by this semigroup (see, for instance, [17]). 3.1. The algebra generated by the number operator. Let us consider the abelian W ∗-subalgebra of B(h) generated by the operator N, that is ∞ AN = { ϕ(N) | ϕ ∈ l (N; C) } .

Proposition 3.1. The restriction of the free generator L0 to the abelian subalgebra AN coincides with the generator of a classical birth and death process. Proof. A straightforward computation using the well-known commutation relations yields

2 2 L0 (ϕ(N)) = µ N (ϕ(N − 1) − ϕ(N)) + λ (N + 1) (ϕ(N + 1) − ϕ(N)) , for all appropriate functions ϕ. Therefore, the restriction of L0 to AN agrees with the inﬁnitesimal gener- 2 ator of a classical birth-and-death process with birth rates λ (k + 1) k≥0 STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 13

2 and death rates µ k k≥0.

3.2. Position, momentum and their algebras. The momentum (or electric field) operator is defined by i (31) D(p) = D(a) = D(a†), p = √ a† − a . 2 The position (or magnetic field) operator is given by 1 (32) D(q) = D(a) = D(a†), q = √ a† + a . 2 The above operators satisfy the canonical commutation relation [q, p] = qp − pq = iI, (since we take the Planck constant ~ = 1 as a convention). In addition to these two operators we consider the family of Weyl operators. The unitary Weyl operator corresponding to z ∈ C is defined by (33) W (z) = exp za† − za¯ , The unitary Weyl operators satisfy the canonical commutation relations (34) W (z)W (u) = W (z + u) exp (i=(¯uz)) , which imply the identities (35) W (z)∗ = W (−z), (36) W (z)W (u)W (z)∗ = exp(2i=(¯uz))W (u). All the above computations have been done in the Heisenberg representation of the CCR over C. However it is well known that it is equivalent to the Schrödinger representation. More precisely, letting (Hn)n≥0 be the orthonor- 2 −1/4 2 mal sequence of the Hermite polynomials in L (R; π exp(−x /2)dx) the unitary operator 2 2 −1/4 2 U : l (N) → L (R; π exp(−x /2)dx), Uen = Hn allows to “pass from one to the other representation”. We refer to Meyer’s book [37] for more details on this subject. Here we shall use the following “conversion table”

Heisenberg Schr¨odinger d p −i dx q x 2 N 1 − d + x2 − 1 2 dx2 √ W (iu), u ∈ R exp(i 2ux) 14 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

∗ Let Aq, Ap be the following abelian W -subalgebras of B(h) ∞ Aq = { ϕ(q) | ϕ ∈ L (R; C) } , ∞ Ap = { ϕ(p) | ϕ ∈ L (R; C) } , ∞ where L (R; C) denotes the vector space of all complex-valued bounded measurable functions on R. Now we will compute, formally, the action of L0 on the abelian algebra Aq, Ap.

Proposition 3.2. The restriction of L0 to the abelian algebra Aq coincides with the inﬁnitesimal generator of an Ornstein-Uhlenbeck process; in particular, this process reduces to a Brownian motion with variance λ2 if λ = µ. Proof. We will use the commutation relations 1 h i 1 (37) [a, ϕ(q)] = √ ϕ0(q), a†, ϕ(q) = −√ ϕ0(q), 2 2 i h i i (38) [a, ϕ(p)] = √ ϕ0(p), a†, ϕ(p) = √ ϕ0(p). 2 2 that can be easily checked working in the Schr¨odinger representation. 2 Let ϕ ∈ Cb (R; C), we have then, identifying ϕ(q) with the corresponding 2 −1/4 2 multiplication operator on L (R; π exp(−x /2)dx), µ2 L (ϕ(q)) = −a†aϕ(q) + 2a†ϕ(q)a − ϕ(q)a†a 0 2 λ2 + −aa†ϕ(q) + 2aϕ(q)a† − ϕ(q)aa† 2 µ2 h i = −a† [a, ϕ(q)] + a†, ϕ(q) a 2 λ2 h i + −a a†, ϕ(q) + [a, ϕ(q)] a† 2 µ2 λ2 = √ −a†ϕ0(q) − ϕ0(q)a + √ aϕ0(q) + ϕ0(q)a† 2 2 2 2 µ2 = √ −(a + a†)ϕ0(q) + a, ϕ0(q) 2 2 λ2 h i + √ (a + a†)ϕ0(q) − a†, ϕ0(q) 2 2 µ2 λ2 = −2qϕ0(q) + ϕ00(q) + 2qϕ0(q) + ϕ00(q) 4 4 λ2 + µ2 λ2 − µ2 = ϕ00(q) + qϕ0(q). 4 2 Thus L0 transforms the multiplication operator by ϕ into the multiplication operator by the function STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 15

λ2 + µ2 λ2 − µ2 q → ϕ00(q) + qϕ0(q), 4 2 i.e. the restriction of L0 to the abelian algebra Aq agrees with the inﬁnites- imal generator of an Ornstein-Uhlenbeck process (a Brownian motion with variance λ2 if λ = µ).

2 A similar computation shows that, for all ϕ ∈ Cb (R; C), we have λ2 + µ2 λ2 − µ2 L (ϕ(p)) = ϕ00(p) + pϕ0(p). 0 4 2

Proposition 3.3. The restriction of L0 to the abelian algebra Ap coincides with the inﬁnitesimal generator of an Ornstein-Uhlenbeck process (a Brownian motion with variance λ2 if λ = µ).

3.3. The evolution on AN , Aq, Ap. The quantum evolution is obtained by adding to the free Liouvillian L0 the bounded perturbation Lga (called the “gain operator”) given by 2 ∗ Lga(X) = R (ϕ1(N)Xϕ1(N) + ϕ2(N)S XSϕ2(N) − X) .

The bounded operator Lga on B(h) is the infinitesimal generator of a norm continuous, identity preserving, semigroup on B(h). This semigroup is completely positive because the infinitesimal generator can be written in the Lindblad form as 1 1 L (X) = R2 − X(ϕ (N))2 + ϕ (N)Xϕ (N) − (ϕ (N))2X ga 2 1 1 1 2 1 1 1 − X(ϕ (N))2 + ϕ (N)S∗XSϕ (N) − (ϕ (N))2X . 2 2 2 2 2 2 ∗ In [18] it has been shown that the restriction of L to the W -algebra AN generated by the number operator agrees with the infinitesimal generator of a birth and death process. In fact it is easy to see that the restriction to AN of the operator Lga agrees with√ the infinitesimal generator of a pure birth 2 process with birth rates (sin (φ k + 1))k≥0. Therefore, the restriction of the whole evolution to the algebra generated by the number operator correspond to a birth and death process with birth rates √ 2 2 2 (39) λk = λ (k + 1) + R sin (φ k + 1) if k ≥ 0, and death rates 2 (40) µk = µ k if k ≥ 0.

In this subsection we shall show that the abelian algebras Aq, Ap generated by the position (or magnetic ﬁeld) and momentum (or electric ﬁeld) operator are not invariant under the action of Lga. Consider, for example, the algebra Aq. Using the formulae on the Weyl operators it is easy to see that, for all u, v ∈ R − {0}, we have

W (iv)Lga(W (iu))W (−iv) 6= Lga(W (iu)). 16 FRANCO FAGNOLA AND ROLANDO REBOLLEDO

This shows that the operators Lga(W (iu)) and W (iv) do not commute. Therefore the operator Lga(W (iu)) does not belong to the algebra Aq which is the maximal abelian W ∗-subalgebra of B(h) generated by the operators W (iv) with v ∈ R. The same computation, with Weyl operators of the form W (−u) with u ∈ R, shows that also the algebra Ap is not invariant for Lga.

4. Stationary state and the convergence towards the equilibrium In [18] it has been shown that, when λ < µ, T has a stationary state given by ∞ X ρ∞ = πn|enihen| n=0 where (πn)n≥0 is the sequence defined by n √ Y λ2k + R2 sin2(φ k) π = c, π = c (n ≥ 1). 0 n µ2k k=1 where c is a suitable normalization constant. We will reproduce below the proof of this statement. To obtain a precise characterisation of invariant states corresponding to a quantum evolution is not in general an easy matter. We want to stress here the importance of classical probability models in obtaining the stationary state of the quantum optics semigroup. Indeed, since the semigroup leave invariant the algebra AN of the number operator, we first looked for an invariant probability measure corresponding to the restriction of the semigroup to that algebra. Once we obtained such an invariant measure, we looked for conditions to extend it to a stationary state of the whole evolution. The semigroup restricted to AN corresponds to a birth and death process. For this kind of processes, invariant measures have been completely studied, (see e.g. [32]), and as an application we obtain the following proposition. Proposition 4.1. If λ < µ a stationary distribution for the classical birth- and-death process with birth rates (39) and death rates (40) is given by n √ Y λ2k + R2 sin2(φ k) (41) π = c, π = c (n ≥ 1). 0 n µ2k k=1 where c is a suitable normalization constant. If λ ≥ µ the classical birth-and-death process with birth rates (39) and death rates (40) has not a nontrivial stationary distribution. Proof. In the case λ 6= µ, it suffices to apply the ratio test for positive numerical series. In the case λ = µ it suffices to remark that each factor in the product (41) is greater or equal to 1. STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 17

Definition 4.1. Let j be the quantum flow (3.9), let T be the associated semigroup on B(h) defined by (3.12) and let T∗ be the predual semigroup on the Banach space of trace class operators on h, that is tr T∗t(ρ)X = tr ρTt(X) for any trace class operator ρ and any X ∈ B(h). A positive operator ρ on h with unit trace is a stationary distribution for j if T∗t(ρ) = ρ for all t ≥ 0.

We have the following result

Theorem 4.1. Suppose λ < µ. Let (πn)n≥0 be the sequence deﬁned by (41). Then the following operator on h:

∞ X (42) ρ = πn|enihen|, n=0 is a stationary distribution for j. Moreover, the semigroup converges towards the equilibrium, which means that

∗ (43) w − lim Tt(X) = T∞(X), t→∞ for all X ∈ B(h), where T∞ is a norm one projection (a conditional expec- tation in the sense of Umegaki), or, equivalently,

∗ (44) w − lim T∗t(ρ0) = ρ, t→∞ for any positive trace class operator with unit trace ρ0.

We quote below a lemma proved in [18], which is an important tool in proving the above proposition.

Lemma 4.1. Let L∗ be the inﬁnitesimal generator of T∗. The orthogonal projection |uihu| belongs to the domain of L∗ for all u ∈ D.

With this property in hands, we go through the proof of Proposition 4.1, which was published in full detail in [18]. Proof. [Theorem 4.1]. Let L∗ be the inﬁnitesimal generator of T∗. Thus the operator L∗ is closed. For all integer n ≥ 1 the operator

(ρn)n≥0, (L∗(ρn))n≥0 converge for the trace norm and the ﬁrst one converges to ρ. Thus ρ belongs to the domain of L∗ and

L∗(ρ) = lim L∗(ρn) = 0. n→∞ Therefore we have d T (ρ) = T (L (ρ)) = 0. dt ∗t ∗t ∗ Finally, the last part of the theorem is an easy application of the main result in [23].

Acknowledgement. This survey was the subject of a conference lectured by R. Rebolledo at the Fifth Mexican Symposium on Probability Theory, held in Guanajuato in March 1998. Rebolledo want to express his deep gratitude to the organizers and other participants of the meeting for their warm hospitality and very stimulating discussions.

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Universita` di Genova, Dip. di Matematica, via Dodecaneso 35, 16146 GEN- OVA, ITALY E-mail address: [email protected]

Facultad de Matematicas,´ Pontificia Universidad Catolica´ de Chile, Casilla 306, Santiago 22 E-mail address: [email protected]