Repeated Interaction Quantum Systems: van Hove Limits and Asymptotic States
Abstract.
We establish the existence of two weak coupling regime effective dynamics for an open quantum system of repeated interactions (vanishing strength and individual interaction duration, respectively). This generalizes known results [3] in that the von Neumann algebras describing the system and the chain element may not be of finite type. Then (but now assuming that the small system is of finite type), we prove that both effective dynamics capture the longterm behavior of the system: existence of a unique asymptotic state for them implies the same property for the respective exact dynamics—provided that the perturbation parameter is sufficiently small. The zeroth order term in a power series expansion in the perturbation parameter of such an asymptotic state is given by the asymptotic state of the effective dynamics. We conclude by working out the case in which the small system and the chain element are spins.
Key words and phrases:
repeated interaction quantum systems, van Hove limit, asymptotic state, perturbation theory.2000 Mathematics Subject Classification:
47A55, 47N55, 82C10, 81V80.1. Introduction
Recall that an open quantum system consists of a socalled small system immersed in a reservoir , and that one is usually interested (perhaps by necessity) only in the observables of . In the repeated interaction model one assumes that the reservoir is an infinite chain of identical subsystems , called chain elements, which interact with sequentially, one at a time, in the order given by their labels . Here we will suppose that:

The time that spends interacting with each —which could depend on or even be random—is actually constant, equal to .

The way in which interacts with each is also independent of .

All chain elements are initially in the same state.
Repeated interaction systems (RISs) have been used in connection with several domains, including quantum optics [15] (in particular, regarding quantum state preparation [16]) and quantum noises [2, 4, 5]. From an open systems point of view, they are interesting because of their mixture of simplicity—they have, by construction, a markovian nature—and thermodynamical nontriviality. Since not much is known about statistical physics far from equilibrium, that makes them a promising source of examples and inspiration; nevertheless, their rigorous study is just in its beginnings. In this article, we focus on their perturbative analysis: we address the question of existence of van Hove effective dynamics and its use in studying the eventual asymptotic states, as we explain in what follows.
To place things in context, let us recall some known results about open systems with timeindependent hamiltonian. In general, the evolution restricted to the small system satisfies a complicated integrodifferential equation, and one is interested in finding asymptotic regimes in which the resulting effective dynamics is simpler. One possibility is to assume that the coupling between the small system and its environment is small, in which case one must rescale time so as to see the effects of the interaction: the dynamics is, then, composed of a fast part coming from the free evolution, and a slow part coming from the interaction. As it turns out, those dynamics decouple in the limit: the slow part, called van Hove limit, becomes markovian; the fast one becomes noise, which is the reason why the weak coupling regime is also called stochastic limit [1]. The mathematical study of the van Hove limit was begun by Davies [10] in 1974. The fact that the slow dynamics exists (at least in some cases) can be seen as one justification for the use of master equations when studying open systems. The procedure which gives the generator of the effective dynamics can be understood as a dynamical Fermi golden rule; see [12] for an exposition of the subject. An interesting, somewhat unexpected result is the following: if the original system has an asymptotic state, then it is well approximated by the asymptotic state of its van Hove limit. Additional information on the subject can be found in [14].
The study of weak coupling regimes in the case of RISs was begun by Attal and Joye [3]. As we will see later, there are at least two such regimes in this context: calling the strength of the interacion, one has the cases , and as . In [3], the existence of the slow dynamics is established for both regimes, under the hypothesis that both the small system and the chain element are finitedimensional. They also study a third regime ( while is kept constant) which is not perturbative anymore; it has the interesting feature that one can always adjust the model in such a way that the effective dynamics is generated by any prescribed Lindbladian.
Our objective in this article is twofold:

To generalise the results in [3] to the infinitedimensional case.

To study the extent to which the previously described relation between asymptotic states of a given system and its van Hove limit holds for RISs.
The precise meaning of asymptotic state in this context is provided by Bruneau, Joye and Merkli [9] who have proved, assuming that the small system is finitedimensional and under an ergodicity hypothesis, that any given initial state of the small system converges, when , towards a unique timeperiodic state. It is to be noted that this is not a state of thermal equilibrium, to start with because it is not constant, but above all because it has a nonvanishing entropy production; this justifies the claim above about the thermodynamical nontriviality of RISs.
2. Mathematical setup
Let and be two von Neumann algebras, meant to describe the small system and one individual chain element. Let and be the weaklycontinuous groups of automorphisms which correspond to their free evolutions. We will suppose that and are mutually commuting subalgebras of a larger von Neumann algebra which is generated by them.^{1}^{1}1This amounts to identifying , and letting . This permits to extend ; we denote the derivations which generate these extended groups by and , respectively, and we denote simply by . We write and for the set of normal states of and , respectively.
Given a selfadjoint element , consider the perturbed dynamics generated by the derivation . It is explicitely given by the convergent series
(1) 
where the are given by the weaklyconvergent integrals
(2) 
We are interested in the repeated interaction evolution restricted to the small system, under the assumption that all chain elements are initially in the KMS state . Therefore, we consider
(3) 
where , , and is the conditional expectation given by
(4) 
Remark 2.1.
The existence of follows from the fact that, under the isomorphisms , and , it can be written as the composition
Equation (3) defines a weakly continuous family of completely positive maps. Observe that the semigroup property fails, since gives the correct time evolution only if we start at times which are integer multiples of . Note, however, that one can define in the obvious way a twoparameter family satisfying and , for all . This is also related to the fact that our intuitively correct formula for time evolution can be obtained by exponentiation of a timedependent hamiltonian—which would be somewhat more rigorous. In fact, one could consider the von Neumann algebra which describes simultaneously the small system and the entire chain, and define there a hamiltonian which, depending on the instant of time, makes the small system interact with the adequate chain element. One would obtain a piecewise constant generator whose exponentiation, after composition with the right conditional expectation projecting onto the small system, coincides with . We will omit the simple but lengthy and notationally involved proof of this fact, because it does not give any insight on the problems which concern us in this work. For more details, see [3, 9].
To simplify the study of the weak coupling regime, we will impose a condition on the perturbation which ensures that there are no first order effects:

There exists a projection , invariant under , such that
Remark 2.2.
First order effects (as can be seen from the Dyson series) do not reflect an influence from the environment: they come from the part of the perturbation which can be interpreted as modifying the free dynamics of the small system.
One can think of as the projection onto the first eigenspace of , which could be interpreted as an absolute vacuum state. In this case, Hypothesis (H1) is loosely saying that the small system and the chain element interact only through creation and annihilation processes at the chain level. To see this in more detail, we refer the reader to [3], where an interaction which precisely falls within this description is considered. But Hypothesis (H1) can perfectly apply in other, different situations, where the interpretation just given is not adequate. We should warn, however, against one potentially tempting interpretation: by GNS construction we can always assume that , with belonging to a Hilbert space on which acts. The projection cannot take the role of because it does not belong to .
Proposition 2.1.
The linear operator
(5) 
is completely positive, normal and . Moreover, given , the map is analytic and, if the hypothesis (H1) holds, it is also even.
Proof.
The convergence of the Dyson series shows that is analytic; it follows that is analytic too, since
is linear and bounded (observe that , being a conditional expectation, has norm 1). Complete positivity and normality are a consequence of the fact that and have these properties. Since , by general properties of completely positive maps we also have that .
Let us check the parity. Under the hypothesis (H1), the invariance of under the free evolution implies—thanks to the KMS condition—that
Hence,
Using this, all we have to do is prove that, for all odd and ,
where is defined in (2). But this follows again from the invariance of and the relations
which are a consequence of the fact that and commute. ∎
3. Van Hove limit
Schematically, we are concerned with the study of an operator of the form
(6) 
where is a projection, the generator of a group of isometries, a perturbation and . Note that the parameter that determines the perturbative nature of a given regime is ; thus, we can immediately identify three different perturbative regimes:

is kept constant, in which case must go to zero.

. Now, can go to zero, remain bounded or even diverge—provided .

and .
In this article we treat the first two cases. The third one, which is a priori out of the reach of our method, seems to oscilate with (the example of Section 5 gives some evidence of this).
To identify the adequate time scale of an effective dynamics in each of these regimes, note that the approximation (6) is likely to become useless when —that is, when . Therefore, the appropriate time scale should be , irrespective of the perturbative regime which is being considered.
3.1. A preliminary result
Here we state a simple generalization of a theorem by Davies [11], which is an abstract weak coupling dynamics existence result.
Theorem 3.1.
Let be a Banach space, the generator of a strongly continuous group of isometries and a normcontinuous map. Suppose that
exists and denote it by . Then, defining , we have that
for any and .
Proof.
Davies proved this result when is actually constant; we will get the general case as a consequence, by showing that
and using the triangle inequality. By Duhamel’s formula,
Now, apply the Dyson expansion and use Remark A.1 to get the estimate
which by continuity is bounded uniformly in . Similar considerations apply to
from which the claim follows. ∎
Remark 3.1.
The strong limit
is the socalled spectral averaging of with respect to the spectrum of . There are at least two known conditions which ensure its existence [12], namely:

admits a total set of eigenvectors, and

is compact and is a Hilbert space.
In the first case, is equal to
where the ’s are the spectral projections of and the sum converges strongly. Observe that it is, in a sense, the part of which commutes with —and this interpretation holds whenever the strong limit exists.
3.2. The regime
To use Theorem 3.1 in the repeated interaction case we start by restricting our attention to the discrete semigroup consisting of integer powers of ; otherwise said, we regard only times which are integer multiples of . The only problem then is to “interpolate” the semigroup to continuous time.
Theorem 3.2.
Suppose that Hypothesis (H1) holds, as well as

The spectrum of is not dense in the circle .
Let be a curve with which encircles the spectrum of , choose a branch of logarithm analytic in the interior of , and define
Assume, finally, that

exists.
Then, the normcontinuous contraction semigroup
satisfies
for all . Here, denotes the integer part of its argument.
Proof.
Recall that , whence and there exists an such that the curve encircles the spectrum of , for all . Define by
which gives an analytic function since the dependence of in is quadratic. Assuming that
exists, Theorem 3.1 would provide the conclusion with . Therefore, we have to prove that exists and is equal to . To do that, recall that
Hence,
The integration order can be reversed, since the integrand and the domain of integration are both bounded; the same argument justifies the exchange of strong limit and complex integral. Note that Hypothesis (H3a) ensures the existence of the limit. ∎
Remark 3.2.
The spectral projections of (which is always bounded) do not necessarily coincide with those of , so that and do not necessarily commute. An extreme case of this would be a harmonic oscilator with energy spectrum . Then, if we take with , we get .
Remark 3.3.
In [3], Attal and Joye prove Theorem 3.2 when the Hilbert spaces and upon which and act, respectively, are finite dimensional. Their method consists in solving explicitely the equation
(7) 
where and are the unknowns. Our method, although conceptually simpler, is essentially the same. Note that the use of a logarithm makes things easier but does not provide an optimal result, since in infinite dimension it might be possible that equation (7) admits a solution, even if the spectrum of is dense in the unit circle.
Theorem 3.2 actually allows one to understand the behavior of for and arbitrary ; in other words, the restriction to times which are integer multiples of is immaterial.
Corollary 3.3.
Under the same hypothesis of Theorem 3.2, the contraction semigroup satisfies also
3.3. The regime ,
This regime is, analitically, somewhat more delicate, because one has to control the dependence in of the error as . That prevents us from just using functional calculus as in the previous subsection. In [3], Attal and Joye use a refined, but finite dimensional, version of Theorem 3.1 to deal with this; however, their proof cannot be easily extended to the infinite dimensional case. We take a different approach, which consists essentially in regrouping the error terms so that one can apply Theorem 3.1 directly.
Lemma 3.4.
Given constants , we say that and are admissible if
Suppose that there exists some such that, for all admissible and ,
where is such that is norm continuous. Then, again for all admissible and ,
where is arbitrary.
Proof.
Thanks to the Dyson series, with in the notation of Appendix A,
where, using the function defined in (8),
Moreover, by continuity of ,
and we conclude that, for all admissible and ,
where the constant depends only on , , and . Now, a standard telescope expansion shows that
But we also have, this time using the Dyson series with , that
whence
We conclude by observing that, writing with ,
which is of for all admissible and . ∎
Theorem 3.5.
Suppose that Hypothesis (H1) holds, as well as

exists,

is norm continuous.
Let be two sequences such that . Then, the semigroup satisfies
for all fixed and .
Proof.
Observe, first, that by continuity of one has
since linear operator composition is normcontinuous. Therefore, using Dyson’s expansion and the evenness of in , one finds that
where we have used the fact that and is, actually, when . To apply Lemma 3.4 we have to check that
is continuous, which is direct by hypothesis.
To conclude we would like to use Theorem 3.1, but the group
is only weaklycontinuous; we have to show that it admits a predual, which then by definition would be strongly continuous. But we know that admits a predual (the generator of the strongly continuous group ), and therefore it suffices to see that leaves the subspace of ultraweakly continuous forms invariant. Now, for that it is enough that be ultraweakultraweak continuous, and, since is positive and normal, all we have to do is prove that the operations of left and right multiplication by elements of are ultraweakultraweak continuous—which is an elementary property of the ultraweak topology, concluding the proof. ∎
Remark 3.4.
When the small system and the chain element are finitedimensional, the hypothesis on the continuity of always holds; hence, this theorem is a generalization of the one in [3].
4. Asymptotic state
In this section we will suppose that the von Neumann algebra is of finite type —that is, isomorphic to . Recall that in this case all semigroups are automatically normcontinuous.
The expression “asymptotic state” in the context of quantum dynamics presupposes that the system is being studied in the Schrödinger picture; if we actually have a completely positive semigroup , the evolution of states is given by
Now, the convergence for every state implies the weak convergence of towards a limit which defines a linear function . Note that must be a multiple of the identity, because otherwise would depend on ; therefore,
Conversely, the convergence of to a rankone projection (whose range must be since ) implies the existence of a unique asymptotic state.
In the case of repeated interaction systems, one must take into account the fact that the asymptotic state, if it exists, is, in general, periodic [9]; an obvious necessary condition for its existence is, then, that . In the next subsection we study this situation from an abstract viewpoint.
4.1. On the analytic perturbation theory of matrices
In this subsection we will suppose that is an analytic function such that and . The classical reference for this material is [13]. We start with a lemma which lies at the heart of the section.
Lemma 4.1.
For each , let be the spectral projection of . Suppose that when . Then,

. Let be its spectral projection.

, so is a projection, too.

exists and is a subprojection of .
Proof.
Let be an analytic extension of to a complex neighbourhood of zero . We want to prove, in the terminology of [13], that 0 is not a branch point of . Since exceptional points are isolated, in any case we can suppose that there exist analytic functions , which are all spectral projections of , such that
Now, one of these spectral projections, say , must correspond to the eigenvalue and must therefore be an analytic extension of . Suppose, by contradiction, that is a branch point of of order . We know (see [13, Theorem 1.9]) that, in this case, admits a Laurent expansion in powers of which necessarily contains negative powers. However, by continuity of the norm, we have
This means that, if we approach through the real positive axis, . This is a contradiction, and we conclude that is not a branch point of . In particular, can be further extended to an analytic continuation of defined on a complex neighbourhood of 0 and exists.
Making use of , each yields an analytic choice of eigenvectors of with eigenvalue 1. Now, the first order term in in the equation is
which, premultiplied by , gives . In particular,
Let be its spectral projection. This means that for all , and therefore that .
Next, we show that . This follows from applying the same reasoning above to the (real) analytic function . In fact: it satisfies the hypothesis of the lemma; the spectral projection of is ; and we have that
Therefore, we conclude that .
Finally, since is obtained by spectral calculus from and