aa r X i v : . [ m a t h - ph ] J un December 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Repeated Interaction Quantum Systems: Deterministic andRandom
Alain Joye
Institut FourierUniversit´e de GrenobleBP 7438402 Saint Martin d’H`eres, France
This paper gives an overview of recent results concerning the long time dy-namics of repeated interaction quantum systems in a deterministic and ran-dom framework. We describe the non equilibrium steady states (NESS) suchsystems display and we present, as a macroscopic consequence, a second lawof thermodynamics these NESS give rise to. We also explain in some detailsthe analysis of products of certain random matrices underlying this dynamicalproblem.
Keywords : Non equilibrium quantum statistical mechanics, Repeated interac-tion quantum systems, Products of random matrices
1. Introduction and Model
A repeated interaction quantum system consists of a reference quantumsubsystem S which interacts successively with the elements E m of a chain C = E + E + · · · of independent quantum systems. At each moment in time, S interacts precisely with one E m ( m increases as time does), while the otherelements in the chain evolve freely according to their intrinsic dynamics.The complete evolution is described by the intrinsic dynamics of S and ofall the E m , plus an interaction between S and E m , for each m . The latter ischaracterized by an interaction time τ m >
0, and an interaction operator V m (acting on S and E m ); during the time interval [ τ + · · · + τ m − , τ + · · · + τ m ), S is coupled to E m only via V m . Systems with this structure are importantfrom a physical point of view, since they arise naturally as models for funda-mental experiments on the interaction of matter with quantized radiation.As an example, the “One atom maser” provides an experimental setup inwhich the system S represents a mode of the electromagnetic field, whereasthe elements E k describe atoms injected in the cavity, one by one, which ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 interact with the field during their flight in the cavity. After they leave thecavity, the atoms encode some properties of the field which can be mea-sured on these atoms , For repeated interaction systems considered as ideal , i.e. such that all atoms are identical with identical interactions andtimes of flight through the cavity, corresponding mathematical analyses areprovided in , To take into account the unavoidable fluctuations in theexperiment setup used to study these repeated interaction systems, mod-elizations incorporating randomness have been proposed and studied in and. With a different perspective, repeated quantum interaction modelsalso appear naturally in the mathematical study of modelization of openquantum systems by means of quantum noises, see and references therein.Any (continuous) master equation governing the dynamics of states on asystem S can be viewed as the projection of a unitary evolution driving thesystem S and a field of quantum noises in interaction. It is shown in howto recover such continuous models as some delicate limit of a discretizationgiven by a repeated quantum interaction model. Let us finally mention for results of a similar flavour in a somewhat different framework.Our goal is to present the results of the papers , and on (random)repeated interaction quantum systems, which focus on the long time be-haviour of these systems.Let us describe the mathematical framework used to describe thesequantum dynamical systems. According to the fundamental principles ofquantum mechanics, states of the systems S and E m are given by normal-ized vectors (or density matrices) on Hilbert spaces H S and H E m , respec-tively, . We assume that dim H S < ∞ , while dim H E m may be infinite.Observables A S and A E m of the systems S and E m are bounded opera-tors forming von Neumann algebras M S ⊂ B ( H S ) and M E m ⊂ B ( H E m ).They evolve according to the Heisenberg dynamics R ∋ t α t S ( A S ) and R ∋ t α t E m ( A E m ), where α t S and α t E m are ∗ -automorphism groups of M S and M E m , respectively, see e.g. We now introduce distinguished referencestates , given by vectors ψ S ∈ H S and ψ E m ∈ H E m . Typical choices for ψ S , ψ E m are equilibrium (KMS) states for the dynamics α t S , α t E m , at inversetemperatures β S , β E m . The Hilbert space of states of the total system isthe tensor product H = H S ⊗ H C , a A normalized vector ψ defines a “pure” state A
7→ h ψ, Aψ i = Tr( ̺ ψ A ), where ̺ ψ = | ψ ih ψ | . A general “mixed” state is given by a density matrix ̺ = P n ≥ p n ̺ ψ n , wherethe probabilities p n ≥ ψ n are normalized vectors. ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 where H C = N m ≥ H E m , and where the infinite product is taken withrespect to ψ C = N m ≥ ψ E m . The non-interacting dynamics is the prod-uct of the individual dynamics, defined on the algebra M S N m ≥ M E m by α t S N m ≥ α t E m . It will prove useful to consider the dynamics in the Schr¨odinger picture , i.e. as acting on vectors in H . To do this, we firstimplement the dynamics via unitaries, satisfying α t ( A ) = e i tL A e − i tL , t ∈ R , and L ψ = 0 , (1)for any A ∈ M , where S or E m . The self-adjointoperators L S and L E m , called Liouville operators or “positive temperatureHamiltonians”, act on H S and H E m , respectively. The existence and unique-ness of L satisfying (1) is well known, under general assumptions on thereference states ψ . We require these states to be cyclic and separating .In particular, (1) holds if the reference states are equilibrium states. Let τ m > V m ∈ M S ⊗ M E m be the interaction time and interaction oper-ator associated to S and E m . We define the (discrete) repeated interactionSchr¨odinger dynamics of a state vector φ ∈ H , for m ≥
0, by U ( m ) φ = e − i e L m · · · e − i e L e − i e L φ, (2)where e L k = τ k L k + τ k X n = k L E n (3)describes the dynamics of the system during the time interval [ τ + · · · + τ k − , τ + · · · + τ k ), which corresponds to the time-step k of our discreteprocess. Hence L k is L k = L S + L E k + V k , (4)acting on H S ⊗ H E k . We understand that the operator L E n in (3) actsnontrivially only on the n -th factor of the Hilbert space H C of the chain.As a general rule, we will ignore tensor products with the identity operatorin the notation.A state ̺ ( · ) = Tr( ρ · ) given by density matrix ρ on H is called a normalstate . Our goal is to understand the large-time asymptotics ( m → ∞ ) ofexpectations ̺ ( U ( m ) ∗ OU ( m )) = ̺ ( α m ( O )) , (5)for normal states ̺ and certain classes of observables O that we specifybelow. We denote the (random) repeated interaction dynamics by α m ( O ) = U ( m ) ∗ OU ( m ) . (6) ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Van Hove Limit Type Results
A first step towards understanding the dynamics of repeated interactionquantum systems reduced to the reference system S was performed inthe work. This paper considers
Ideal Repeated Quantum Interaction Sys-tems which are characterized by identical elements E k ≡ E in the chain C , constant interaction times τ k ≡ τ and identical interaction operators V k ≡ V ∈ M S ⊗ M E between S and the elements E of the chain. Inthis setup, a Van Hove type analysis of the system is presented, in sev-eral regimes, to describe the dynamics of observables on S in terms of aMarkovian evolution equation of Lindblad type. Informally, the simplestresult of reads as follows. Assume the interaction operator V is replacedby λV , where λ > m , the number of inter-actions during the time T = mτ , scale like m ≃ t/λ , where 0 ≤ t < ∞ and τ are fixed. Assume all elements of the chain are in a same thermal stateat temperature β . Then, the weak coupling limit λ → O acting on S obtained by tracing out the chain degrees offreedom from the evolution (6) satisfies, after removing a trivial free evolu-tion, a continuous Lindblad type evolution equation in t . The temperaturedependent generator is explicitely obtained from the interaction operator V and the free dynamics. The asymptotic regimes of the parameters ( λ, τ )characterized by τ → τ λ ≤ giving rise to dif-ferent Lindblad generators which all commute with the free Hamiltonian on S . The critical situation, where τ → τ λ = 1 yields a quite generalLindblad generator, without any specific symmetry. In particular, it showsthat any master equation driven by Lindblad operator, under reasonableassumptions, can be viewed as a Van Hove type limit of a certain explicitrepeated interaction quantum system.By contrast, the long time limit results obtained in , and that wepresent here are obtained without rescaling any coupling constant or pa-rameter, as is usually the case with master equation techniques. It is pos-sible to do without these approximations, making use of the structure ofrepeated interaction systems only, as we now show.
2. Reduction to Products of Matrices
We first link the study of the dynamics to that of a product of reduced dy-namics operators. In order to make the argument clearer, we only considerthe expectation of an observable A S ∈ M S , and we take the initial state of ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 the entire system to be given by the vector ψ = ψ S ⊗ ψ C , (7)where the ψ S and ψ C are the reference states introduced above. We’ll com-ment on the general case below. The expectation of A S at the time-step m is h ψ , α m ( A S ) ψ i = D ψ , P e i e L · · · e i e L m A S e − i e L m · · · e − i e L P ψ E , (8)where we introduced P = 1l H S O m ≥ P ψ E m , (9)the orthogonal projection onto H S ⊗ C ψ C . A first important ingredient in ouranalysis is the use of C -Liouvilleans introduced in , which are operators K k defined by the propertiese i e L k A e − i e L k = e i K k A e − i K k , (10) K k ψ S ⊗ ψ C = 0 , (11)where A in (10) is any observable of the total system. The identity (10)means that the operators K k implement the same dynamics as the e L k whereas relation (11) selects a unique generator of the dynamics amongall operators which satisfy (10). The existence of operators K k satisfying(10) and (11) is rooted to the Tomita-Takesaki theory of von Neumannalgebras, c.f. and references therein. It turns out that the K k are non-normal operators on H , while the e L k are self-adjoint. Combining (10) with(8) we can write h ψ , α m ( A S ) ψ i = (cid:10) ψ , P e i K · · · e i K m P A S ψ (cid:11) . (12)A second important ingredient of our approach is to realize that the inde-pendence of the sub-systems E m implies the relation P e i K · · · e i K m P = P e i K P · · · P e i K m P. (13)Identifying P e i K k P with an operator M k on H S , we thus obtain from (12)and (13), h ψ , α m ( A S ) ψ i = h ψ S , M · · · M m A S ψ S i . (14)It follows from (11) that M k ψ S = ψ S , for all k , and, because the operators M k = P e i K k P implement a unitary dynamics, we show (Lemma 4.1) thatthe M k are always contractions for some suitable norm ||| · ||| on C d . Thismotivates the following ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Definition:
Given a vector ψ S ∈ C d and a norm on ||| · ||| on C d , we call Reduced Dynamics Operator any matrix which is a contraction for ||| · ||| and leaves ψ S invariant. Remark:
In case all couplings between S and E k are absent, V k ≡ M k = e iτ k L S is unitary and admits 1 as a degenerate eigenvalue.We will come back on the properties of reduced dynamics operators(RDO’s, for short) below. Let us emphasize here that the reduction processto product of RDO’s is free from any approximation, so that the set ofmatrices { M k = P e i K k P } k ∈ N encodes the complete dynamics. In particu-lar, we show, using the cyclicity and separability of the reference vectors ψ S , ψ E k , that the evolution of any normal state, not only h ψ , · ψ i , can beunderstood completely in terms of the product of these RDO’s.We are now in a position to state our main results concerning the asymp-totic dynamics of normal states ̺ acting on certain observables O . Theseresult involve a spectral hypothesis which we introduce in the next Definition:
Let M ( E ) denote the set of reduced dynamics operators whosespectrum σ ( M ) satisfies σ ( M ) ∩ { z ∈ C | | z | = 1 } = { } and 1 is simpleeigenvalue.We shall denote by P ,M the spectral projector of a matrix M corre-sponding to the eigenvalue 1. As usual, if the eigenvalue 1 is simple, withcorresponding normalized eigenvector ψ S , we shall write P ,M = | ψ S ih ψ | for some ψ s.t. h ψ | ψ S i = 1.
3. Results3.1.
Ideal Repeated Interaction Quantum System
We consider first the case of
Ideal Repeated Interaction Quantum Systems ,characterized by E k = E , L E k = L E , V k = V, τ k = τ for all k ≥ ,M k = M, ∀ k ≥ . (15) Theorem 3.1.
Let α n be the repeated interaction dynamics determined byone RDO M . Suppose that M ∈ M ( E ) so that P ,M = | ψ S ih ψ | . Then, forany < γ < inf z ∈ σ ( M ) \{ } (1 − | z | ) , any normal state ̺ , and any A S ∈ M S , ̺ ( α n ( A S )) = h ψ, A S ψ S i + O ( e − γn ) . (16) ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Remarks:
1. The asymptotic state h ψ | · ψ S i and the exponential decay rate γ are both determined by the spectral properties of the RDO M .2. On concrete examples, the verification of the spectral assumption on M can be done by rigorous perturbation theory, see. It is reminiscent of aFermi Golden Rule type condition on the efficiency of the coupling V , seethe remark following the definiton of RDO’s.3. Other properties of ideal repeated interaction quantum systems arediscussed in , e.g. continuous time evolution and correlations.For deterministic systems which are not ideal, the quantity ̺ ( α n ( A S ))keeps fluctuating as n increases, which, in general, forbids convergence, seeProposition 5.3. That’s why we resort to ergodic limits in a random setup,as we now explain. Random Repeated Interaction Quantum System
To allow a description of the effects of fluctuations on the dynamics ofrepeated interaction quantum systems, we consider the following setup.Let ω M ( ω ) be a random matrix valued variable on C d defined on aprobability space (Ω , F , p). We say that M ( ω ) is a random reduced dynamicsoperator (RRDO) if(i) There exists a norm ||| · ||| on C d such that, for all ω , M ( ω ) is acontraction on C d for the norm ||| · ||| .(ii) There exists a vector ψ S , constant in ω , such that M ( ω ) ψ S = ψ S ,for all ω .To an RRDO ω M ( ω ) on Ω is naturally associated a iid randomreduced dynamics process (RRDP) ω M ( ω ) · · · M ( ω n ) , ω ∈ Ω N ∗ , (17)where we define in a standard fashion a probability measure d P on Ω N ∗ byd P = Π j ≥ dp j , where dp j ≡ dp , ∀ j ∈ N ∗ . We shall write the expectation of any random variable f as E [ f ].Let us denote by α nω , ω ∈ Ω N ∗ , the process obtained from (6), (14),where the M j = M ( ω j ) in (14) are iid random matrices. We call α nω therandom repeated interaction dynamics determined by the RRDO M ( ω ) = P e i K ( ω ) P . It is the independence of the successive elements E k of the chain C which motivates the assumption that the process (17) be iid. ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Theorem 3.2.
Let α nω be the random repeated interaction dynamics de-termined by an RRDO M ( ω ) . Suppose that p( M ( ω ) ∈ M ( E ) ) > . Thenthere exists a set Ω ⊂ Ω N ∗ , s.t. P (Ω) = 1 , and s.t. for any ω ∈ Ω , anynormal state ̺ and any A S ∈ M S , lim N →∞ N N X n =1 ̺ ( α nω ( A S )) = h θ, A S ψ S i , (18) where θ = P ∗ , E [ M ] ψ S .Remarks:
1. Our setup allows us to treat systems having various sourcesof randomness. For example, random interactions or times of interactions,as well as random characteristics of the systems E m and S such as randomtemperatures and dimensions of the E m and of S .2. The asymptotic state h θ, · ψ S i is again determined by the spectraldata of a matrix, the expectation E [ M ] of the RRDO M ( ω ). Actually, ourhypotheses imply that E [ M ] belongs to M ( E ) , see below.3. The explicit computation of the asymptotic state, in this Theoremand in the previous one, is in general difficult. Nevertheless, they can bereached by rigorous perturbation theory, see the examples in , and. Instantaneous Observables
There are important physical observables that describe exchange processesbetween S and the chain C and, which, therefore, are not represented byoperators that act just on S . To take into account such phenomena, weconsider the set of observables defined as follows. Definition:
The instantaneous observables of S + C are of the form O = A S ⊗ rj = − l B ( j ) m , (19)where A S ∈ M S and B ( j ) m ∈ M E m + j .Instantaneous observables can be viewed as a train of l + r + 1 observables,roughly centered at E m , which travel along the chain C with time.Following the same steps as in Section 2, we arrive at the followingexpression for the evolution of the state ψ acting on an instantaneousobservable O at time m : h ψ , α m ( O ) ψ i = h ψ , P M · · · M m − l − N m ( O ) P ψ i . (20) ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Here again, P is the orthogonal projection onto H S , along ψ C . The operator N m ( O ) acts on H S and has the expression (Proposition 2.4 in ) N m ( O ) ψ = (21) P e i τ m − l e L m − l · · · e i τ m e L m ( A S ⊗ rj = − l B ( j ) m )e − i τ m e L m · · · e − i τ m − l e L m − l ψ . We want to analyze the asymptotics m → ∞ of (20), allowing for ran-domness in the system. We make the following assumptions on the randominstantaneous observable:(R1) The operators M k are RRDO’s, and we write the corresponding iidrandom matrices M k = M ( ω k ), k = 1 , , · · · , .(R2) The random operator N m ( O ) is independent of the M k with 1 ≤ k ≤ m − l −
1, and it has the form N ( ω m − l , . . . , ω m + r ), where N : Ω r + l +1 → B ( C d ) is an operator valued random variable.The operator M k describes the effect of the random k -th interaction on S , as before. The random variable N in (R2) does not depend on the timestep m , which is a condition on the observables. It means that the natureof the quantities measured at time m are the same. For instance, the B ( j ) m in (19) can represent the energy of E m + j , or the part of the interactionenergy V m + j belonging to E m + j , etc. Both assumptions are verified in awide variety of physical systems: we may take random interaction times τ k = τ ( ω k ), random coupling operators V k = V ( ω k ), random energy levelsof the E k encoded in L E k = L E ( ω k ), random temperatures β E k = β E ( ω k ) ofthe initial states of E k , and so on.The expectation value in any normal state of such instantaneous ob-servables reaches an asymptotic value in the ergodic limit given in the next Theorem 3.3.
Suppose that p( M ( ω ) ∈ M ( E ) ) = 0 . There exists a set e Ω ⊂ Ω N ∗ of probability one s.t. for any ω ∈ e Ω , for any instantaneousobservable O , (19), and for any normal initial state ̺ , we have lim µ →∞ µ µ X m =1 ̺ (cid:0) α mω ( O ) (cid:1) = h θ, E [ N ] ψ S i , E [ N ] ∈ M S . (22) Remarks
1. The asymptotic state in which one computes the expectation(w.r.t the randomness) of N is the same as in Theorem 3.2, with θ = P ∗ , E [ M ] ψ S .2. In case the system is deterministic and ideal, the same result holds,dropping the expectation on the randomness and taking θ = ψ , as in The-orem 3.1, see. ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Energy and Entropy Fluxes
Let us consider some macroscopic properties of the asymptotic state. Thesystems we consider may contain randomness, but we drop the variable ω from the notation.Since we deal with open systems, we cannot speak about its total en-ergy; however, variations in total energy are often well defined. Using anargument of one gets a formal expression for the total energy which isconstant during all time-intervals [ τ m − , τ m ), and which undergoes a jump j ( m ) := α m ( V m +1 − V m ) (23)at time step m . Hence, the variation of the total energy between the instants0 and m is then ∆ E ( m ) = P mk =1 j ( k ). The relative entropy of ̺ with respectto ̺ , two normal states on M , is denoted by Ent( ̺ | ̺ ). Our definitionof relative entropy differs from that given in by a sign, so that in ourcase, Ent( ̺ | ̺ ) ≥
0. For a thermodynamic interpretation of entropy andits relation to energy, we assume for the next result that ψ S is a ( β S , α t S )–KMS state on M S , and that the ψ E m are ( β E m , α t E m )–KMS state on M E m ,where β S is the inverse temperature of S , and β E m are random inversetemperatures of the E m . Let ̺ be the state on M determined by the vector ψ = ψ S ⊗ ψ C = ψ S N m ψ E m . The change of relative entropy is denoted∆ S ( m ) := Ent( ̺ ◦ α m | ̺ ) − Ent( ̺ | ̺ ). This quantity can be expressed interms of the Liouvillean and interaction operators by means of a formulaproven in. One checks that both the energy variation and the entropy variationscan be expressed as instantaneous observables, to which we can apply theresults of the previous Section. Defining the asymptotic energy and entropyproductions by the limits, if they exist,lim m →∞ ̺ (cid:18) ∆ E ( mm (cid:19) =: d E + and lim m →∞ ∆ S ( m ) m =: d S + , (24)we obtain Theorem 3.4 ( nd law of thermodynamics). Let ̺ be a normal stateon M . Then d E + = (cid:10) θ, E (cid:2) P ( L S + V − e i τL ( L S + V )e − i τL ) P (cid:3) ψ S (cid:11) a.s. d S + = (cid:10) θ, E (cid:2) β E P ( L S + V − e i τL ( L S + V )e − i τL ) P (cid:3) ψ S (cid:11) a.s. The energy- and entropy productions d E + and d S + are independent of theinitial state ̺ . If β E is deterministic, i.e., ω -independent, then the systemsatisfies the second law of thermodynamics: d S + = β E d E + . ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Remark:
There are explicit examples in which the entropy production canbe obtained via rigorous perturbation theory and is proven to be strictlypositive, a sure sign that the asymptotic state is a NESS, see .As motivated by (14), the theorems presented in this Sections all relyon the analysis of products of large numbers of (random) RDO’s. The restof this note is devoted to a presentation of some of the key features suchproducts have.
4. Basic Properties of RDO’s
Let us start with a result proven in as Proposition 2.1. Lemma 4.1.
Under our general assumptions, the set of matrices { M j } j ∈ N ∗ defined in (14) satisfy M j ψ S = ψ S , for all j ∈ N ∗ . Moreover, to any φ ∈ H S there corresponds a unique A ∈ M S such that φ = Aψ S . ||| φ ||| := k A k B ( H S ) defines a norm on H S , and as operators on H S endowed with this norm,the M j are contractions for any j ∈ N ∗ . Again, the fact that ψ S is invariant under M j is a consequence of (11) andtheir being contractions comes from the unitarity of the quantum evolutiontogether with the finite dimension of H S .As a consequence of the equivalence of the norms k · k and ||| · ||| , we get Corollary 4.1.
We have ∈ σ ( M j ) ⊂ { z | | z | = 1 } and sup {k M j n M j n − · · · M j k , n ∈ N ∗ , j k ∈ N ∗ } = C < ∞ Actually, if a set of operators satisfies the bound of the Corollary, itis always possible to construct a norm on C d relative to which they arecontractions, as proven in the next Lemma 4.2.
Let R = { M j ∈ M d ( C ) } j ∈ J , where J is any set of indicesand C ( R ) ≥ such that k M j M j · · · M j n k ≤ C ( R ) , ∀{ j i } i =1 , ··· ,n ∈ J n , ∀ n ∈ N . (25) Then, there exists a norm ||| · ||| on C d , which depends on R , relative towhich the elements of R are contractions. Proof:
Let us define T ⊂ M d ( C ) by T = ∪ n ∈ N ∪ ( j ,j , ··· j n ) ∈ J n M j M j · · · M j n . (26) ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Obviously R ⊂ T , but the identity matrix I does not necessarily belong to T . Moreover, the estimate (25) still holds if the M j i ’s belong to T insteadof R . For any ϕ ∈ C d we set ||| ϕ ||| = sup M ∈ T ∪ I k M ϕ k ≥ k ϕ k , (27)which defines a bona fide norm. Then, for any vector ϕ and any element N of T we compute ||| N ϕ ||| = sup M ∈ T ∪ I k M N ϕ k ≤ sup M ∈ T ∪ I k M ϕ k = ||| ϕ ||| , (28)from which the result follows. Remark.
If there exists a vector ψ S invariant under all elements of R , it isinvariant under all elements of T and satisfies k ψ S k = ||| ψ S ||| = 1.
5. Deterministic Results
In this section, we derive some algebraic formulae and some uniform boundsfor later purposes. Since there is no probabilistic issue involved here, we shalltherefore simply denote M j = M ( ω j ). We are concerned with the productΨ n := M · · · M n . (29) Decomposition of the M j With P ,M j the spectral projection of M j for the eigenvalue 1 we define ψ j := P ∗ ,j ψ S , P j := | ψ S ih ψ j | . (30)Note that h ψ j | ψ S i = 1 so that P j is a projection and, moreover, M ∗ j ψ j = ψ j .We introduce the following decomposition of M j M j := P j + Q j M j Q j , with Q j = 1l − P j . (31)We denote the part of M j in Q j C d , by M Q j := Q j M j Q j . It easily followsfrom these definitions that P j P k = P k , Q j Q k = Q j , (32) Q j P k = 0 , P k Q j = P k − P j = Q j − Q k . (33) Remark.
If 1 is a simple eigenvalue, P ,M j = P j and (31) is a (partial)spectral decomposition of M j . ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Proposition 5.1.
For any n , Ψ n = | ψ S ih θ n | + M Q · · · M Q n , (34) where θ n = ψ n + M ∗ Q n ψ n − + · · · + M ∗ Q n · · · M ∗ Q ψ (35)= M ∗ n · · · M ∗ ψ (36) and where h ψ S , θ n i = 1 . Proof.
Inserting the decomposition (31) into (29), and using (32), (33),we have Ψ n = n X j =1 P j M Q j +1 · · · M Q n + M Q · · · M Q n . Since P j = | ψ S ih ψ j | , this proves (34) and (35). From (33), we obtain forany j, k , M Q j M Q k = M Q j M k = Q j M j M k . (37)Hence, Ψ n = P M · · · M n + Q M · · · M n = | ψ S ih M ∗ n · · · M ∗ ψ | + M Q · · · M Q n , which proves (36). (cid:3) Uniform Bounds
The operators M j , and hence the product Ψ n , are contractions on C d forthe norm |||·||| . In order to study their asymptotic behaviour, we need someuniform bounds on the P j , Q j , . . . Recall that k ψ S k = 1. Proposition 5.2.
Let C be as in Corollary 4.1. Then, the followingbounds hold(1) For any n ∈ N ∗ , k Ψ n k ≤ C .(2) For any j ∈ N ∗ , k P j k = k ψ j k ≤ C and k Q j k ≤ C .(3) sup {k M Q jn M Q jn − · · · M Q j k , n ∈ N ∗ , j k ∈ N ∗ } ≤ C (1 + C ) .(4) For any n ∈ N ∗ , k θ n k ≤ C . Proof.
It is based on Von Neumann’s ergodic Theorem, which states that P ,M j = lim N →∞ N N − X k =0 M kj . ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 The first two estimate easily follow, whereas the third makes use of (37) toget M Q jn M Q jn − · · · M Q j = Q j n M j n M j n − · · · M j , so that k M Q jn M Q jn − · · · M Q j k ≤ k Q j n k C ≤ C (1 + C ) . Finally, (36) and the above estimates yield k θ n k ≤ C k ψ k ≤ C . (cid:3) Asymptotic Behaviour
We now turn to the study of the asymptotic behaviour of Ψ n , starting withthe simpler case of Ideal Repeated Interaction Quantum Systems .That means we assume M k = M, ∀ k ≥ . (38)If 1 is a simple eigenvalue of M , then P ,M = | ψ S ih ψ | , for some ψ s.t. h ψ | ψ S i = 1, and Ψ n = M n = | ψ S ih ψ | + M nQ (39)Further, if all other eigenvalues of M belong to the open unit disk, M nQ converges exponentially fast to zero as n → ∞ .Consequently, denoting by spr( N ) the spectral radius of N ∈ M d ( C ), Lemma 5.1.
If the
RDO M belongs to M ( E ) , Ψ n = | ψ S ih ψ | + O ( e − γn ) , (40) for all < γ < − spr( M Q ) . Two things are used above, the decay of M nQ and the fact that θ n = ψ isconstant, see (34). The following result shows that in general, if one knows a priori that the products of M Q j ’s in (34) goes to zero, Ψ n converges ifand only if P n = | ψ S ih ψ n | , does. Proposition 5.3.
Suppose that lim n →∞ sup {k M Q jn · · · M Q j k , j k ∈ N ∗ } = 0 . Then θ n converges if and only if ψ n does. If they exist, thesetwo limits coincide, and thus lim n →∞ Ψ n = | ψ S ih ψ ∞ | , where ψ ∞ = lim n →∞ ψ n . Moreover, | ψ S ih ψ ∞ | is a projection. In general, we cannot expect pointwise convergence of the θ n , but wecan consider an ergodic average of θ n instead. This is natural in terms ofdynamical systems, a fluctuating system does not converge. ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 The previous convergence results relies on the decay of the productof operators M Q j . Conditions ensuring this are rather strong. However,Theorem 6.1 below shows that in the random setting , a similar exponentialdecay holds under rather weaker assumptions.
6. Random Framework6.1.
Product of Random Matrices
We now turn to the random setup in the framework of Section 3.2. For M ( ω ) an RRDO, with probability space (Ω , F , p), we consider the RRDPon Ω N ∗ given by Ψ n ( ω ) := M ( ω ) · · · M ( ω n ) , ω ∈ Ω N ∗ . We show that Ψ n has a decomposition into an exponentially decayingpart and a fluctuating part. Let P ( ω ) denote the spectral projection of M ( ω ) corresponding to the eigenvalue one (dim P ( ω ) ≥ P ∗ ( ω )be its adjoint operator. Define ψ ( ω ) := P ( ω ) ∗ ψ S , (41)and set P ( ω ) = | ψ S ih ψ ( ω ) | , Q ( ω ) = 1l − P ( ω ) . (42)The vector ψ ( ω ) is normalized as h ψ S , ψ ( ω ) i = 1. We decompose M ( ω ) as M ( ω ) = P ( ω ) + Q ( ω ) M ( ω ) Q ( ω ) =: P ( ω ) + M Q ( ω ) . (43)Taking into account this decomposition, we obtain (c.f. Proposition 5.1)Ψ n ( ω ) := M ( ω ) · · · M ( ω n ) = | ψ S ih θ n ( ω ) | + M Q ( ω ) · · · M Q ( ω n ) , (44)where θ n ( ω ) is the Markov process θ n ( ω ) = M ∗ ( ω n ) · · · M ∗ ( ω ) ψ ( ω ) (45)= ψ ( ω n ) + M ∗ Q ( ω n ) ψ ( ω n − ) + · · · + M ∗ Q ( ω n ) · · · M ∗ Q ( ω ) ψ ( ω ) ,M ∗ ( ω j ) being the adjoint operator of M ( ω j ). We analyze the two parts inthe r.h.s. of (44) separately. Theorem 6.1 (Decaying process).
Let M ( ω ) be a random reduced dy-namics operator. Suppose that p( M ( ω ) ∈ M ( E ) ) > . Then there exist a set Ω ⊂ Ω N ∗ and constants C, α > s.t. P (Ω ) = 1 and s.t. for any ω ∈ Ω ,there exists a random variable n ( ω ) s.t. for any n ≥ n ( ω ) , k M Q ( ω ) · · · M Q ( ω n ) k ≤ C e − αn , (46) and E [ e αn ] < ∞ . Moreover, E [ M ] ∈ M ( E ) . ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Remarks.
1. The sole condition of M having an arbitrarily small, non-vanishing probability to be in M ( E ) suffices to guarantee the exponentialdecay of the product in (46) and that E [ M ] belongs to M ( E ) .2. Actually, E [ M ] ∈ M ( E ) is a consequence of spr( E [ M Q ]) <
1, whichcomes as a by product of the proof of Theorem 6.1. From the identities E [ M ] = | ψ S ih E [ ψ ] | + E [ M Q ] , h E [ ψ ] | ψ S i = 1 , E [ M Q ] ψ S = 0 , (47)which do not correspond to a (partial) spectral decomposition of E [ M ],and this estimate, we get E [ M ] n = | ψ S ih E [ ψ ] + E [ M Q ] ∗ E [ ψ ] + · · · E [ M Q ] ∗ n − E [ ψ ] | + E [ M Q ] ∗ n → n →∞ | ψ S ih ( I − E [ M Q ] ∗ ) − E [ ψ ] | ≡ P , E [ M ] . (48)3. Our choice (41) makes ψ ( ω ) an eigenvector of M ∗ ( ω ). Other choices of(measurable) ψ ( ω ) which are bounded in ω lead to different decompositionsof M ( ω ), and can be useful as well. In particular, if M ( ω ) is a bistochasticmatrix, ψ ( ω ) can be chosen as an M ∗ ( ω )-invariant vector which is indepen-dent of ω . A Law of Large Numbers
We now turn to the asymptotics of the Markov process (46).
Theorem 6.2 (Fluctuating process).
Let M ( ω ) be a random reduceddynamics operator s.t. that p( M ( ω ) ∈ M ( E ) ) > . There exists a set Ω ⊂ Ω N ∗ s.t. P (Ω ) = 1 and, for all ω ∈ Ω , lim N →∞ N N X n =1 θ n ( ω ) = θ, (49) where θ = lim n →∞ E [ θ n ] = P ∗ , E [ M ] E [ ψ ] = P ∗ , E [ M ] ψ S . (50) Remarks:
1. The ergodic average limit of θ n ( ω ) does not depend on theparticular choice of ψ ( ω ). This follows from the last equality in (50).2. The second equality in (50) stems from E [ θ n ] = n − X k =0 ( E [ M Q ]) k E [ ψ ] , (51)by independence, and which converges to P ∗ , E [ M ] E [ ψ ] by (48). The thirdequality follows from (47). ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10
3. Comments on the proof of these Theorems are provided below.Combining Theorems 6.1 and 6.2 we immediately get the following result.
Theorem 6.3 (Ergodic theorem for RRDP).
Let M ( ω ) be a randomreduced dynamics operator. Suppose p( M ( ω ) ∈ M ( E ) ) > . Then thereexists a set Ω ⊂ Ω N ∗ s.t. P (Ω ) = 1 and, for all ω ∈ Ω , lim N →∞ N N X n =1 M ( ω ) · · · M ( ω n ) = | ψ S ih θ | = P , E [ M ] . (52) Remarks
1. If one can choose ψ ( ω ) ≡ ψ to be independent of ω , then wehave by (36) that θ n ( ω ) = ψ , for all n, ω . Thus, from (44)-(46), we getthe stronger result lim n →∞ M ( ω ) · · · M ( ω n ) = | ψ S ih ψ | , a.s., exponentiallyfast.2. This result can be viewed as a strong law of large numbers for thematrix valued process Ψ n ( ω ) = M ( ω ) · · · M ( ω n ). Comments:
The existence of (ergodic) limits of products of random op-erators is known for a long time and under very general conditions, seee.g. , . However, the explicit value of the limit depends on the detailedproperties of the set of random matrices considered. The point of our anal-ysis is thus the explicit determination of the limit (52) which is crucial forthe applications to the dynamics of random repeated interaction quantumsystems.The more difficult part of this task is to prove Theorem 6.1. The ideaconsists in identifying matrices in the product Ψ n ( ω ) which are equal (orclose) to a fixed matrix M that belongs to M ( E ) . Consecutive productsof M give an exponential decay, whereas products of other matrices areuniformly bounded. Then one shows that the density of long strings of con-secutive M ’s in a typical sample is finite. Once this is done, a self-containedproof of Theorem 6.3, is not very hard to get .On the other hand, given Theorem 6.1 and the existence result of ,we can deduce Theorem 6.3 as follows. Let us state the result of Beck andSchwarz in our setup. Let T denote the usual shift operator on Ω N ∗ definedby ( T ω ) j = ω j +1 , j = 1 , , · · · . Theorem 6.4 (Beck and Schwartz ). Let M ( ω ) be a random reduceddynamics operator on Ω . Then there exists a matrix valued random variable L ( ω ) on Ω N ∗ , s.t. E [ k L k ] < ∞ , which satisfies almost surely L ( ω ) = M ( ω ) L ( T ω ) , (53) ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 where T is the shift operator, and lim N →∞ N N X n =1 M ( ω ) · · · M ( ω n ) = L ( ω ) . (54)Further assuming the hypotheses of Theorem 6.1, and making use of thedecomposition (44), we get that L can be written as L ( ω ) = | Ψ S ih θ ( ω ) | , (55)for some random vector θ ( ω ). Now, due to (53) and the fact that ψ S isinvariant, θ ( ω ) satisfies θ ( ω ) = θ ( T ω ) a.s. (56)The shift being ergodic, we deduce that θ is constant a.s., so that θ ( ω ) = E [ θ ] a.s. (57)which, in turn, thanks to Proposition 5.2 and Lebesgue dominated conver-gence Theorem, allows to get from (46) E [ θ ] = lim n →∞ E [ θ n ] = P ∗ , E [ M ] Ψ S . (58) Limit in Law and Lyapunov Exponents
We present here results for products “in reverse order” of the form Φ n ( ω ) := M ( ω n ) · · · M ( ω ), which have the same law as Ψ n ( ω ). They also yield infor-mation about the Lyapunov exponent of the process. The following resultsare standard, see e.g. . The limitsΛ Φ ( ω ) = lim n →∞ (Φ n ( ω ) ∗ Φ n ( ω )) / n and Λ Ψ ( ω ) = lim n →∞ (Ψ n ( ω ) ∗ Ψ n ( ω )) / n exist almost surely, the top Lyapunov exponent γ ( ω ) of Λ Φ ( ω ) coincideswith that of Λ Ψ ( ω ), it is constant a.s., and so is its multiplicity. It is ingeneral difficult to prove that the multiplicity of γ ( ω ) is 1. Theorem 6.5.
Suppose p( M ( ω ) ∈ M ( E ) ) > . Then there exist α > , arandom vector η ∞ ( ω ) = lim n →∞ ψ ( ω ) + M ∗ Q ( ω ) ψ ( ω ) + · · · + M ∗ Q ( ω ) · · · M ∗ Q ( ω n − ) ψ ( ω n )(59) and Ω ⊂ Ω N ∗ with P (Ω ) = 1 such that for any ω ∈ Ω and n ∈ N ∗ (cid:13)(cid:13)(cid:13) Φ n ( ω ) − | ψ S ih η ∞ ( ω ) | (cid:13)(cid:13)(cid:13) ≤ C ω e − αn , for some C ω . (60) As a consequence, for any ω ∈ Ω , γ ( ω ) is of multiplicity one. ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 Comments:
While the Theorems above on the convergence of asymptoticstates give us the comfortable feeling provided by almost sure results, it isan important aspect of the theory to understand the fluctuations aroundthe asymptotic state the system reaches almost surely. In our iid setup, thelaw of the product Ψ n ( ω ) of RRDO’s coincides with the law of Φ n ( ω ) whichconverges exponentially fast to | ψ S ih η ∞ ( ω ) | . Therefore, the fluctuations areencoded in the law of the random vector η ∞ ( ω ). It turns out it is quitedifficult, in general, to get informations about this law. There are partialresults only about certain aspects of the law of such random vectors incase they are obtained by means of matrices belonging to some subgroupsof Gl d ( R ) satisfying certain irreducibility conditions, see e.g. However,these results do not apply to our RRDO’s.
Generalization
A generalization of the analysis performed for observables acting on S onlydescribed above allows to establish the following corresponding results wheninstantaneous observables are considered.The asymptotics of the dynamics (20), in the random case, is encodedin the product M ( ω ) · · · M ( ω m − l − ) N ( ω m − l , . . . , ω m + r ) , where N : Ω r + l +1 → M d ( C ) is given in assumption (R2). Theorem 6.6 (Ergodic limit of infinite operator product).
Assume M ( ω ) is a RRDO and (R2) is satisfied. Suppose that p( M ( ω ) ∈M ( E ) ) = 0 . Then E [ M ] ∈ M ( E ) . Moreover, there exists a set Ω ⊂ Ω N ∗ ofprobability one s.t. for any ω = ( ω n ) n ∈ N ∈ Ω , lim ν →∞ ν ν X n =1 M ( ω ) · · · M ( ω n ) N ( ω n +1 , . . . , ω n + l + r +1 ) = | ψ S ih θ | E [ N ] , where θ = P ∗ , E [ M ] ψ S . As in the previous Section, a density argument based on the cyclicity andseparability of the reference vector ψ allows to obtain from Theorem 6.6the asymptotic state for all normal initial states ̺ on M given as Theorem3.3 Acknowledgements.
I wish to thank I. Beltita, G. Nenciu and R. Purice who organized the10th edition of QMath for their kind invitation and L. Bruneau and M.Merkli for a very enjoyable collaboration. ecember 11, 2018 21:48 WSPC - Proceedings Trim Size: 9in x 6in qmath10 References
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