Quantum Measurements of Scattered Particles

Abstract

We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables. We link the asymptotic properties of this process to spectral characteristics of the dynamics. We show that the process has decaying time correlations and that a zero-one law holds. We deduce that if the incoming probes are not sharply localized with respect to the spectrum of the measurement operator, then the process does not converge. Nevertheless, the scattering modifies the measurement outcome frequencies, which are shown to be the average of the measurement projection operator, evolved for one interaction period, in an asymptotic state. We illustrate the results on a truncated Jaynes–Cummings model.

1. Introduction and Main Results

We consider a scattering experiment in which a beam of probes is directed at a scatterer. The probes are sent to interact sequentially, one by one. Before the scattering process, they are identical and independent. The interaction of each probe with the system is governed by a fixed interaction time τ > 0 and a fixed interaction operator V. After interacting with the scatterer, a quantum measurement is performed on each “outcoming” probe. The result of the measurement of the n-th probe is a random variable, denoted X_n. The stochastic process {X_n}_n≥1 is the measurement history. Due to entanglement of the probes with the scatterer, the X_n are not independent random variables. We analyse the asymptotic properties of this process.

A concrete physical set-up is given by atoms (being the probes) shot through a cavity containing an electromagnetic field, the modes that interact with the atoms forming the scatterer. We assume that the incoming probe states are stationary with respect to their isolated dynamics.

We study systems with only finitely many degrees of freedom involved in the scattering process. This means that the Hilbert spaces of pure states, both of the system and each probe, are finite-dimensional. The measurement of a probe is a von Neumann, or projective, measurement associated with a self-adjoint probe measurement observable M. The eigenvalues m of M are the possible measurement outcomes. Due to finite dimensionality, the random variables X_n have a finite range.

The present work can be viewed as the continuation of recently developed techniques for the mathematical analysis of repeated interaction quantum systems [1–4]. In these references, asymptotic properties of the scatterer have been investigated, without considering the fate of the outcoming probes and without quantum measurements. While the set-up of our present work is similar to the one in the given references, our focus here is on the measurement outcomes process. We show that, generically, this process does not converge. We describe the fluctuations on the measurement history, provoked by the scattering process, by analysing the measurement frequencies. A more detailed comparison to related works is given at the end of this section.

As explained in the references above, in absence of quantum measurements on probes and under a generic ergodicity assumption, one shows that the scatterer approaches a so-called repeated interaction asymptotic state after many interactions. We keep this assumption in the present work.

(A) Assume that if no measurement is performed $(M=1)$, then, under the repeated interaction with the probes, the scatterer approaches a final state. The convergence is exponentially quick in time.

The precise mathematical formulation of this assumption is given in Section 2.4; see before Equation (27). It is a condition on the spectrum of a reduced dynamics operator and necessitates the introduction of some technicalities, which we want to avoid in this introduction. Condition (A) is generically satisfied and is not hard to be verified explicitly, and one even calculates the rate of convergence for concrete models (see the above references).

We now explain our main results. Denote by σ(X_r,…, X_s) the sigma-algebra generated by the random variables X_r,…, X_s, 1 ≤ r ≤ s ≤ ∞. We denote by P the probability measure associated with the process {X_n}_n≥1.

Theorem 1 (Decay of correlations). Suppose that Condition (A) holds. There are constants c, γ′ > 0, such that for 1 ≤ k ≤ l < m ≤ n < ∞, A ∈ σ(X_k,… X_l) and B ∈ σ(X_m,…, X_n), we have:

$$|P(A\cap B)-P(A)P(B)|\le cP(A){\mathrm{e}}^{-{\mathrm{\gamma }}^{\prime }(m-l)}$$

(1)

We give a proof of the theorem in Section 4.1. Intuitively, the system starts relaxation to its asymptotic state during the time m — l between two consecutive measurements and, hence, erases correlations between the two measurements. The rate γ′ in Equation (1) is linked to the convergence rate in Assumption (A); see Section 2.4.

The tail sigma-algebra is defined by $\mathcal{T}={\cap }_{n\ge }{}_1\sigma \left(X_n,X_{n+}{}_1,\dots \right)$. Decaying correlations imply the following zero-one law.

Corollary 2 (Zero-one law). Assume that Condition (A) holds. Any tail event A ∈ $\mathcal{T}$ satisfies P(A) = 0 or P(A) = 1.

In textbooks, the Kolmogorov zero-one law is usually presented for independent random variables [5]. However, an adaptation of the proof yields the result for random variables with decaying correlations; see [6] (and also Section 2.4). The tail sigma-algebra captures convergence properties. For instance, given any outcome m ∈ spec(M), the set {lim_nX_n = m} is a tail event; hence, according to Corollary 2, it has probability zero or one.

We now explain why fluctuations in the process persist generically, for all times. Let ω_in be the state of the incoming probes; denote by E_S the spectral projection of the measurement operator M associated with S ⊂ spec(M), and denote E_m = E_{m} for m ∈ spec(M). In absence of interaction (V = 0 or τ = 0), the X_j are independent random variables. We show in Proposition 12 that the dependence generated by the interaction with the scatterer is small for small interactions, uniformly in time. Therefore, since P(X_n = m) = ω_in(E_m) + O(∥V∥), we have P(X_n+1 = m, X_n = m) = P(X_n+1 = m)P(X_n = m) + O(∥V∥), and consequently,

$$P(X_{n+1}=X_n)={\displaystyle \sum _{\mathrm{m}\in \mathrm{spec}(M)}{\omega }_{\mathrm{in}}^2(E_m)+O}\left(\Vert V\Vert \right)$$

The numbers ω_in(E_m) are probabilities. Thus, ${\displaystyle {\sum }_m{\omega }_{\mathrm{in}}^2(E_m)=1}$ if and only if for a single m₀, we have ${\omega }_{\mathrm{in}}(E_{m_0})=1$, while for all other m, ω_in(E_m) = 0. This means that P(X_n+1 = X_n) < 1 for small V, whenever there are several m with ω_in(E_m) > 0. Together with the zero-one law, this implies that P(X_n converges) = 0 whenever the incoming state is not localized in a single subspace of M (and V is small enough). If m is a simple eigenvalue of M with associated eigenvector ψ_m, then ω_in(E_m) = 1 is equivalent to ω_in(·) = 〈ψ_m,·ψ_m〉. Statistical fluctuations in the incoming probes (mixture of states localized w.r.t. measurement values) thus get transferred to outcoming probes, even in the limit of large times. The following is a more general statement of this fact.

Theorem 3. Assume Condition (A) holds. There is a constant C s.t, for any S C spec(M) with ω_in(ES) ≠ 1, if ∥V∥ ≤ C(1 − ω_in(ES)), then:

$$P(X_n\in S\mathrm{eventually})=0$$

The result on non-convergence of X_n explained before Theorem 3 is a special case of Theorem 3, when S = {m}, m ∈ spec(M). We mention that our analysis also gives a condition under which P(X_n ∈ S eventually) = 1; see Lemma 11.

The process X_n carries information about the scattering process, encoded in the relative occurrence of a particular measurement outcome. We define the frequency of m ∈ spec(M) by:

$$f_m=\underset{n\to \infty }{\lim }\frac{1}{n}\{\mathrm{number}\mathrm{of}k\in \{1,\dots ,n\}\mathrm{s}.\mathrm{t}.X_k=m$$

f_m is a priori a random variable, and the existence (and type) of limit has to be clarified.

Theorem 4 (Frequencies). Assume Condition (A) holds, and denote the final scatterer state under the evolution without measurements by ω₊. Let H be the interacting Hamiltonian of a single probe with the scatterer. Then, the frequencies f_m exist (as almost everywhere limits) and are deterministic (not random), given by:

$$f_m={\omega }_{+}\otimes {\omega }_{\mathrm{in}}\left({\mathrm{e}}^{\mathrm{i}\tau H}{E}_m{\mathrm{e}}^{-\mathrm{i}\tau H}\right)$$

Remark: he proof of Theorem 4, given in Section 4.3, can be readily generalized to yield the following result: For any m ≥ 1, S₁,…, S_m ⊂ spec(M),

$$\begin{array}{l}\underset{n\to \infty }{\lim }\frac{1}{n}\left\{\mathrm{number}\mathrm{of}j\le n\mathrm{s}.\mathrm{t}.X_j\in S_1,\dots ,X_{j+m}\in S_m\right\}\\ ={\omega }_{+}\otimes {\omega }_{\mathrm{in}}\cdots \otimes {\omega }_{\mathrm{in}}\left({\mathrm{e}}^{\mathrm{i}\tau H_1}\cdots {\mathrm{e}}^{\mathrm{i}\tau H_1}{E}_{S_1}\cdots E_{S_m}{\mathrm{e}}^{-\mathrm{i}\tau H_1}\cdots {\mathrm{e}}^{-\mathrm{i}\tau H_1}\right)\end{array}$$

Here, H_j is the Hamiltonian describing the free motion of $\mathcal{S}$ and m probes $\mathcal{P}$, plus the interaction of $\mathcal{S}$ with the j-th probe.

The next result describes the process $\left\{{\overline{X}}_n\right\}$ of the empirical average:

$${\overline{X}}_n=\frac{1}{n}(X_1+\cdots +X_n)$$

Theorem 5 (Mean). Assume Condition (A) holds and adopt the notation of Theorem 4. We have a law of large numbers,

$$\underset{n\to \infty }{\lim }{\overline{X}}_n={\mu }_{\infty }:={\omega }_{+}\otimes {\omega }_{\mathrm{in}}\left({\mathrm{e}}^{\mathrm{i}\tau H}M{\mathrm{e}}^{-\mathrm{i}\tau H}\right)$$

The limit is in the almost everywhere sense. Note that µ_∞ = ∑_m mf_m.

Relation to other work: The literature on repeated interaction systems we are aware of can be classified into two categories. In the first one, effective evolution equations are derived by taking continuous interaction limits [7–13]; and in a second category, the dynamics is left discrete, but the time-asymptotics is investigated [1–4,14–17]. To our knowledge, repeated interaction systems have been first proposed in [18,19] and in [7,8] as approximations for system-environment type models, where it is proven that the discrete evolution converges to that of a quantum Langevin equation, in the limit of ever shorter system-probe interaction times. The study of the time-asymptotics was initiated in [1] (and continued in [2–4]). It is shown there that the (reduced) system state converges to a final state, and this state’s thermodynamic properties were analysed rigorously.

The mathematical formalism in [1–4] is based on a spectral approach to the time-asymptotics of open quantum systems, expressed in the language of operator algebraic quantum theory. The present work extends this formalism to the setting of repeated probe quantum measurements. A review of the above-cited works, as well as an announcement of some of the results of the present work, can be found in [20].

In [21], a result similar to Corollary 2 is shown. There, the authors discuss ergodic properties of quantum counting processes, described by unravelling of a Lindblad generator. The analysis is based on the assumption that the dynamics, without taking into account the counting, is ergodic (i.e., converges in the mean to an equilibrium state). On the other hand, our system is mixing due to Condition A (i.e., it converges ‘pointwise’).

In [14,15], an energy conserving model is considered, where the dynamics is assumed to be such that there exists a basis of special stationary states (‘pointer states’). Then, the system converges to one of these states, determined by the measurement outcomes. Many physical systems do not fall into this class. Whenever there is the exchange of energy between probes and the system, there is only one invariant state (under generic conditions). An important example of an energy-exchange system is the ‘one atom maser’, where atoms (probes) interact with modes of the electromagnetic field in a cavity (system) by exciting the field modes, leading to subsequent photon emission [22]. A famous mathematical model describing this situation is the Jaynes–Cummings model, a simplification of which we discuss in Section 3.

2. Quantum Dynamical System Set-Up

2.1. Formalism

The general formalism of quantum dynamical systems is presented in [23,24] (see also [1]). Both $\mathcal{S}$ and $\mathcal{P}$ are described as quantum (W*) dynamical systems in standard form. The states of such a system are given by unit vectors in a Hilbert space $\mathscr{H}$, observables form a von Neumann algebra $\mathfrak{M}\subset ℬ(\mathscr{H})$ and the dynamics given by a group of *automorphisms α^t on $\mathfrak{M},t\in ℝ$. There is a distinguished (reference) vector $\psi \in \mathscr{H}$, which is cyclic and separating for $\mathfrak{M}$ and such that the dynamics is represented as α^t (A) = e^itLAe^−itL, for $A\in \mathfrak{M}$, and where L is a self-adjoint operator on $\mathscr{H}$ satisfying Lψ = 0. This operator is called the standard Liouville operator.

Accordingly, the system $\mathcal{S}$ is determined by a Hilbert space ${\mathscr{H}}_{\mathcal{S}}$, a von Neumann algebra ${\mathfrak{M}}_S$, a cyclic and separating vector ψ_S and a dynamics ${\alpha }_S^t=e^{\mathrm{i}t{L}_S}\cdot e^{-\mathrm{i}t{L}_S}$, with L_SψS = 0. A single probe is described by the same ingredients (with index $\mathcal{S}$ replaced by $\mathcal{P}$). We assume throughout the paper that dim ${\mathscr{H}}_{\mathcal{S}}<\infty$, dim ${\mathscr{H}}_{\mathcal{P}}<\infty$. The Hilbert space of the chain of all probes is the tensor product ${\mathscr{H}}_{\mathcal{C}}={\otimes }_{n\ge }{}_1{\mathscr{H}}_{\mathcal{P}}$, stabilized on the reference vector ${\psi }_{\mathcal{P}}\in {\mathscr{H}}_{\mathcal{P}}$. The von Neumann algebra of observables of C is ${\mathfrak{M}}_c={\otimes }_{\mathrm{n}\ge 1}{\mathfrak{M}}_P$ and its dynamics is ${\alpha }_C^t={\otimes }_{n\ge 1}{\alpha }_P^t$.

The full system is described by the Hilbert space:

$$\mathscr{H}={\mathscr{H}}_{\mathcal{S}}\otimes {\mathscr{H}}_{\mathcal{C}}$$

(2)

on which acts the von Neumann algebra of observables:

$${\mathfrak{M}=\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{C}}$$

(3)

The non-interacting Liouville operator is given by:

$${\tilde{L}}_0=L_{\mathcal{S}}+{\displaystyle \sum _{n\ge 1}{L}_{n,\mathcal{P}}}$$

(4)

where $L_{n,\mathcal{P}}$ is the operator acting trivially on all factors of $\mathscr{H}$, except on the n-th factor of ${\mathscr{H}}_{\mathcal{C}}$, on which it acts as $L_{\mathcal{P}}$. The interaction between $\mathcal{S}$ and the n-th probe lasts for a duration of τ > 0 and is determined by an operator V_n, acting trivially on all factors of $\mathscr{H}$, except ${\mathscr{H}}_{\mathcal{S}}$, and the n-th one in ${\mathscr{H}}_{\mathcal{C}}$, where it acts as a fixed self-adjoint operator:

$$V=V*\in {\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{P}}$$

(5)

Let ${\mathrm{\Psi }}_0\in \mathscr{H}$ be an initial state of the full system. Then, the state of the system at time-step n is given by the vector:

$${\mathrm{\Psi }}_n=U_n\cdots U_2{U}_1{\mathrm{\Psi }}_0$$

(6)

where:

$$U_n={\mathrm{e}}^{-\mathrm{i}\tau ({\tilde{L}}_0+V_n)}$$

(7)

is the unitary generating the one-step time evolution at instant n.

We consider initial states of the form:

$${\mathrm{\Psi }}_0={\mathrm{\Psi }}_{\mathcal{S}}{\otimes }_{n\ge 1}B{\psi }_{\mathcal{P}}$$

(8)

where B is an operator in the commutant von Neumann algebra:

$${M^{\prime }}_{\mathcal{C}}=\{A\in ℬ({\mathscr{H}}_{\mathcal{P}}):AX=XA\forall X\in {\mathfrak{M}}_{\mathcal{C}}\}$$

such that $\Vert B{\psi }_{\mathcal{P}}\Vert =1$ and $L_{\mathcal{P}}B=B{L}_{\mathcal{P}}$ (normalized, invariant state). Since ${\psi }_{\mathcal{P}}$ is cyclic for ${{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$ (⇔ separating for ${\mathfrak{M}}_{\mathcal{P}}$, where ${{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$ the commutant of ${\mathfrak{M}}_{\mathcal{P}}$) and since dim ${\mathscr{H}}_{\mathcal{P}}$ is finite, every $\psi \in {\mathscr{H}}_{\mathcal{P}}$ is exactly represented as $\psi =B{\psi }_{\mathcal{P}}$ for a unique $B\in {{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$.

Remarks: 1. Since both ${\mathscr{H}}_{\mathcal{S}}$ and ${\mathscr{H}}_{\mathcal{P}}$ are finite-dimensional, one can work in a less general framework and analyse the measurement process based on a density matrix description of the system and the probes, as opposed to work in the (GNS) Hilbert space representation. Such an analysis can be carried out even for some systems having infinitely many discrete energy levels; see [25]. Nevertheless, we adhere to the present set-up. It will come in handy (and necessary) when one considers models where the system is large (e.g., some membrane or screen) subjected to an incoming beam of scattering probes.

2. We have presented in this section the chain Hilbert space as an infinite tensor product of single-probe spaces and similarly for the evolution and reference state. All quantities we are examining involve the whole system up to arbitrary, but finite times, involving only finitely many probes. Consequently, we could define our Hilbert space to be time-dependent, having n probe factors only, but with n arbitrarily large. The expressions for the probabilities of the measurement process, or the reduced state of the scatterer, or the exited probes would not change. Whether we take an infinite tensor product or a finite one, with arbitrary factors, does not influence the physical properties we describe. Nevertheless, mathematically, there is a slight difference between these two cases; see e.g., [1–4,20]. As an example, strictly speaking, the vector Ψ₀ given in Equation (8) does not belong to the Hilbert space $\mathscr{H}$ unless $B=1$ (because ${\mathscr{H}}_{\mathcal{C}}$ is “stabilized” along ψ_P). What we mean is that, at arbitrary time n, the vector Ψ₀ has the form ${\mathrm{\Psi }}_{\mathcal{S}}\otimes B{\psi }_{\mathcal{P}}\otimes B{\psi }_{\mathcal{P}}\otimes \cdots \otimes B{\psi }_{\mathcal{P}}$ (n probe factors) and, as such, belongs to the Hilbert space $H_{\mathcal{S}}\otimes H_{\mathcal{P}}\otimes H_{\mathcal{P}}\otimes \cdots \otimes H_{\mathcal{P}}$.

Let J and Δ denote the modular conjugation and the modular operator associated with the cyclic and separating vector ${\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}$ for the von Neumann algebra ${\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{P}}$. By the Tomita-Takesaki theorem [5], we know that ${\mathrm{\Delta }}^{\mathrm{i}t}({\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{P}}){\mathrm{\Delta }}^{-}{}^{\mathrm{i}t}={\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{P}}$ for all t ∈ ℝ (Δ^−it is a group of unitaries) and that $J({\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{P}})J={{\mathfrak{M}}^{\prime }}_{\mathcal{S}}\otimes {{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$. Consequently, $J{\mathrm{\Delta }}^{\mathrm{i}t}V{\mathrm{\Delta }}^{-it}J\in {{\mathfrak{M}}^{\prime }}_{\mathcal{S}}\otimes {{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$, where V is the interaction operator defined in Equation (5). In the finite-dimensional case, as considered here, an easy analyticity argument shows that the last relation stays valid for any t ∈ ℂ In particular,

$$J{\mathrm{\Delta }}^{-1/2}V{\mathrm{\Delta }}^{-1/2}J\in {{\mathfrak{M}}^{\prime }}_{\mathcal{S}}\otimes {{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$$

(9)

It will be convenient to represent the joint dynamics of the system $\mathcal{S}$ and the probe $\mathcal{P}$ interacting at the given moment with $\mathcal{S}$ by the following operator acting on ${\mathscr{H}}_{\mathcal{S}}\otimes {\mathscr{H}}_{\mathcal{P}}$ (see Subsection 2.3),

$$K=L_{\mathcal{S}}+L_{\mathcal{P}}+\mathrm{\lambda }V-\mathrm{\lambda }J{\mathrm{\Delta }}^{1/2}V{\mathrm{\Delta }}^{-}{}^{1/2}J$$

(10)

Here, λ ∈ ℝ is a coupling constant. Due to Property (9), we have:

$${\mathrm{e}}^{\mathrm{i}tK}A{\mathrm{e}}^{-\mathrm{i}tK}={\mathrm{e}}^{\mathrm{i}t[L_{\mathcal{S}}+L_{\mathcal{P}}+\mathrm{\lambda }V]}A{\mathrm{e}}^{-\mathrm{i}t[L_{\mathcal{S}}+L_{\mathcal{P}}+\mathrm{\lambda }V]}$$

(11)

for all A ∈ M_S ⊗ M_P and all t ∈ ℝ, as is not hard to see, for instance, by using the Trotter product formula. The term −λJΔ^1/2V Δ^−1/2 J in Equation (10) is introduced in order to have the property:

$$K{\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}=0$$

(12)

The latter relation follows from $\left(L_{\mathcal{S}}+L_{\mathcal{P}}\right){\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}=0$ and the fact that $J{\mathrm{\Delta }}^{1/2}V{\mathrm{\Delta }}^{-}{}^{1/2}J{\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}=V{\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}$ (which, in turn, is implied by Δ^−1/2 J = JΔ^1/2 and $J{\mathrm{\Delta }}^{1/2}A{\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}=A*{\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}$ for all $A\in {\mathfrak{M}}_{\mathcal{S}}\otimes {\mathfrak{M}}_{\mathcal{P}}$). The operator K has been used in [26] for the study of non-equilibrium open quantum systems and in the setting of repeated interaction open systems in [1–4].

2.2. Multi-Time Measurement Process

In this subsection, we describe the process of multi-time measurement of the outcoming probes. We refer to [27–30] for a detailed introduction to quantum measurement theory, outlining here only a few particularities pertaining to repeated interaction systems. Consider a quantum system described by a density matrix ρ and a self-adjoint “measurement operator” M having eigenvalues spec(M) = {m₁,…,m_µ} with corresponding eigenprojections P₁,…, P_µ. When the system undergoes interaction with the measurement apparatus and we know that the measurement result is one of the eigenvalues from a set S ⊂ spec(M), then the state after measurement is given by:

$${\rho }^{\prime }={\displaystyle \sum _{\{j:m_j\in S\}}{P}_j\rho }{P}_j$$

(13)

This corresponds to a non-selective measurement [29]. In the repeated interaction setting, the following experiment is performed: the entire system evolves according to U₁ (interaction between the first probe and the system), and then, a measurement of the observable M is made on the outcoming probe, yielding a value in S₁ ⊂ spec(M). Then, the system evolves according to U₂ (interaction with second probe), and after this evolution, a measurement of M is made on the outcoming probe and yields a result lying in S₂ ⊂ spec(M); this procedure is repeated n times. Let $\text{Let}{\tilde{\mathcal{U}}}_j={\mathrm{e}}^{-\mathrm{i}\tau {\tilde{H}}_j}$, where ${\tilde{H}}_j=H_{\mathcal{S}}+H_{1,\mathcal{P}}+\cdots +H_{n,\mathcal{P}}+V_j$ and V_j is the interaction operator, acting on the system and the j-th probe only. The expectation of a system observable O_S after n probe measurements have been performed, knowing that measurement j has yielded a result in the set S_j ⊂ spec(M), is given by:

$${\omega }_{S_1,\dots ,S_n}(O_{\mathcal{S}})=\frac{1}{P(S_1,\dots ,S_n)}{\displaystyle \sum _{\{j_n:m_{j_n}\in S_n\}}\cdots {\displaystyle \sum _{\{j_1:m_{j_1}\in S_1\}}\mathrm{Tr}\left(P_{j_n}{\tilde{\mathcal{U}}}_n\cdots P_{j_1}{\tilde{\mathcal{U}}}_1\rho {\tilde{\mathcal{U}}}_1^{*}{P}_{j_1}\cdots {\tilde{\mathcal{U}}}_n^{*}{P}_{j_n}{O}_{\mathcal{S}}\right)}}$$

(14)

Here, P(S₁,…, S_n) is the probability that the measurement outcomes at time j lies in the set S_j, for j = 1,…, n, and is determined by ${\omega }_{S_1,\dots ,S_n}(1)=1$. Using the cyclicity of the trace and the structure of the Hamiltonians ${\tilde{H}}_j,j=1,\dots ,n$, one readily sees that:

$${\omega }_{S_1,\dots ,S_n}(O_{\mathcal{S}})=\frac{1}{P(S_1,\dots ,S_n)}\mathrm{Tr}\left(\rho {({\mathcal{U}}_n^{+})}^{*}{\mathcal{U}}_1^{*}{E}_{S_1}\cdots {\mathcal{U}}_n^{*}{E}_{S_n}{O}_{\mathcal{S}}{E}_{S_n}{\mathcal{U}}_n\cdots E_{S_1}{\mathcal{U}}_1{\mathcal{U}}_n^{+}\right)$$

(15)

where $E_S={\displaystyle {\sum }_{\{j:m_j\in S\}}{P}_j,{\mathcal{U}}_n^{+}={\mathrm{e}}^{-\mathrm{i}\tau {\displaystyle {\sum }_{j=2}^n(j-1)H_{j,\mathcal{P}}}}}$ and ${\mathcal{U}}_j={\mathrm{e}}^{-\mathrm{i}\tau (H_{\mathcal{S}}+H_{j,\mathcal{P}}+V_j)}$. It follows directly from Equation (15) that if we do not know about the outcome of any of the measurements, which corresponds to S_j = spec(M) and $E_{S_j}=1$, for all j = 1,…, n, then the dynamics of the system is the same as if no interaction with the measurement apparatus took place. This phenomenon is particular to the set-up of repeated interactions and is not true for general quantum systems. Indeed, Relation (13) with the sum extended over all j does not yield ρ′ = ρ in general.

Relation (15) implies that:

$$P(S_1,\dots ,S_n)=\mathrm{Tr}\left(\rho {({\mathcal{U}}_n^{+})}^{*}{\tilde{\mathcal{U}}}_1^{*}{E}_{S_1}\cdots {\tilde{\mathcal{U}}}_n^{*}{E}_{S_n}{\tilde{\mathcal{U}}}_n\cdots E_{S_1}{\tilde{\mathcal{U}}}_1{\tilde{\mathcal{U}}}_n^{+}\right)$$

(16)

The stochastic process associated with the measurements is constructed as follows. Let:

$$\mathrm{\Omega }={\sum }^{\mathbb{N}}=\{\omega =({\omega }_1,{\omega }_2,\dots ):{\omega }_j\in \mathrm{spec}(M)\}$$

and let $ℱ$ be the σ-algebra of subsets of Ω generated by all cylinder sets of the form:

$$\{\omega \in \mathrm{\Omega }:{\omega }_1\in S_1,\dots ,{\omega }_n\in S_n,n\in \mathbb{N},S_j\subset \mathrm{spec}(M)\}$$

On $(\Omega ,ℱ)$, we define the random variables X_n : Ω → spec(M) by X_n(ω) = ω_n, for n = 1,2,…. The random variable X_n represents the outcome of the measurement at time-step n. The finite-dimensional distribution of the process {X_n}_n≥1 is given by:

$$P(X_1\in S_1,\dots ,X_n\in S_n)=P(S_1,\dots ,S_n)$$

(17)

for any n ∈ ℕ, any subsets S₁,…,S_n of spec(M) and where the right-hand side is defined in Equation (16). P extends uniquely to a probability measure on $(\Omega ,ℱ)$ by the Kolmogorov extension theorem.

2.3. Representation of Joint Probabilities

In the (GNS) Hilbert space setting, the multi-time measurement process introduced in the previous paragraph is formulated as follows. Let $M\in {\mathfrak{M}}_{\mathcal{P}}$ be a self-adjoint “measurement” operator on ${\mathscr{H}}_{\mathcal{P}}$ with spectrum spec(M) = {m₁,…,m_µ}, where 1 ≤ µ ≤ dim ${\mathscr{H}}_{\mathcal{P}}$ (distinct eigenvalues). Let S be any subset of spec(M), and denote by E_S the spectral projection of M associated to S. Suppose that the entire system is in a state ${\mathrm{\Psi }}_0\in \mathscr{H}$ initially (see also Remark 2 in the previous paragraph). The state of the system after n measurements, viewed as a state of M_S, is given by (see Equation (15)) A ↦ 〈A〉_n = 〈Ψn, AΨ_n〉, where:

$${\mathrm{\Psi }}_n={\Vert {\tilde{\mathrm{\Psi }}}_n\Vert }^{-1}{\tilde{\mathrm{\Psi }}}_n\mathrm{and}{\tilde{\mathrm{\Psi }}}_n=E_{S_n}{\mathrm{e}}^{-\mathrm{i}\tau L_n}\cdots E_{S_1}{\mathrm{e}}^{-i\tau L_1}{U}_n+{\mathrm{\Psi }}_0$$

(18)

Here,

$$L_j=L_{\mathcal{S}}+L_{j,\mathcal{P}}+V_j$$

(19)

acts non-trivially only on the Hilbert space of $\mathcal{S}$ and the j-th probe Hilbert space, and we introduced the unitary:

$$U_n^{+}=\exp \left[-\mathrm{i}\tau {\displaystyle \sum _{j=2}^n(j-1)L_{j,\mathcal{P}}}\right]$$

(20)

Recall that ${\mathrm{\Psi }}_0=(1_{\mathcal{S}}\otimes B\otimes B\cdots ){\mathrm{\Psi }}_{\mathrm{ref}}$, with ${\mathrm{\Psi }}_{\mathrm{ref}}={\psi }_{\mathcal{S}}\otimes {\psi }_{\mathcal{P}}\otimes {\psi }_{\mathcal{P}}\cdots$. The operator $C_j={\mathrm{e}}^{-\mathrm{i}\tau (j-1)L_{j,\mathcal{P}}}B{\mathrm{e}}^{\mathrm{i}\tau (j-1)L_{j,\mathcal{P}}}$ belongs to the commutant ${{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$, since B does and since the dynamics generated by $L_j{}_{,\mathcal{P}}$ leaves ${{\mathfrak{M}}^{\prime }}_{\mathcal{P}}$ invariant. We obtain:

$$\begin{array}{l}\langle {\tilde{\mathrm{\Psi }}}_n,A{\tilde{\mathrm{\Psi }}}_n\rangle \\ =\langle C_1\cdots C_n{\mathrm{\Psi }}_{\mathrm{ref}},{\mathrm{e}}^{\mathrm{i}\tau L_1}{E}_{S_1}\cdots {\mathrm{e}}^{\mathrm{i}\tau L_n}{E}_{S_n}A{E}_{S_n}{\mathrm{e}}^{-\mathrm{i}\tau L_n}\cdots E_{S_1}{\mathrm{e}}^{-\mathrm{i}\tau L_1}{C}_1\cdots C_n{\mathrm{\Psi }}_{\mathrm{ref}}\rangle \\ =\langle {\mathrm{\Psi }}_{\mathrm{ref}},C_1^{*}{C}_1\cdots C_n^{*}{C}_n{\mathrm{e}}^{\mathrm{i}\tau K_1}{E}_{S_1}\cdots {\mathrm{e}}^{\mathrm{i}\tau L_n}{E}_{S_n}A{\mathrm{\Psi }}_{\mathrm{ref}}\rangle \\ =\langle {\mathrm{\Psi }}_{\mathrm{ref}},\left[P_1{C}_1^{*}{C}_1{\mathrm{e}}^{\mathrm{i}\tau K_1}{E}_{S_1}{P}_1\right]\cdots \left[P_n{C}_n^{*}{C}_n{\mathrm{e}}^{\mathrm{i}\tau K_n}{E}_{S_n}{P}_n\right]A{\mathrm{\Psi }}_{\mathrm{ref}}\rangle \end{array}$$

(21)

where ${\mathcal{P}}_j$ is the projection acting trivially on all factors of $\mathscr{H}$, except on the j-th ${\mathscr{H}}_{\mathcal{P}}$, where it acts as the rank-one orthogonal projection onto ${\psi }_{\mathcal{P}}$. We define the operator $T_j=P_j{C}_j^{*}{C}_j{\mathrm{e}}^{\mathrm{i}\tau K_j}{E}_{S_j}{P}_j$, which, under the hypothesis that ${\mathrm{e}}^{\mathrm{i}s{L}_{\mathcal{P}}}B{\mathrm{e}}^{-\mathrm{i}s{L}_{\mathcal{P}}}=B$, becomes $T_j=P_{j}B*B{\mathrm{e}}^{\mathrm{i}\tau K_j}{E}_{S_j}{P}_j$. We identify T_j as an operator on the Hilbert space ${\mathscr{H}}_{\mathcal{S}}$ and, as such, write:

$$T_S=PB*B{\mathrm{e}}^{\mathrm{i}\tau K}{E}_{S}P$$

(22)

where P the orthogonal projection onto ${\psi }_{\mathcal{P}}\otimes {\mathscr{H}}_{\mathcal{C}},S\subset$ spec(M) determines the measurement performed at the given time-step, and where K is given in Equation (10). We write simply T for T_spec(M). Remark that since $E_{S_j}\in {\mathfrak{M}}_{\mathcal{P}}$ and ${\psi }_{\mathcal{P}}$ is separating for ${\mathfrak{M}}_{\mathcal{P}}$, we have $E_{S_j}{P}_j\ne 0$ for all j. With this definition, we arrive at:

$$\langle {\tilde{\mathrm{\Psi }}}_n,A{\tilde{\mathrm{\Psi }}}_n\rangle =\langle {\psi }_{\mathcal{S}},T_{S_1}{T}_{S_2}\cdots T_{S_n}A{\psi }_{\mathcal{S}}\rangle$$

(23)

In particular, measurement probability can be expressed as:

$$P(S_1,\dots ,S_n)=\langle {\psi }_{\mathcal{S}},T_{S_1}\cdots T_{S_n}{\psi }_{\mathcal{S}}\rangle$$

(24)

2.4. Analysis of Joint Probabilities

Lemma 6. The spectrum of TS, Equation (22), lies in the closed unit disk centred at the origin of the complex plane. For S = spec(M), i.e., , we have in addition$T{\psi }_{\mathcal{S}}={\psi }_{\mathcal{S}}$.

We consider the probability P(X_n ∈ S eventually), for S ⊂ spec(M). This quantity can be expressed using the Riesz spectral projections Π and Π_S of the operators T and T_S associated with the eigenvalue of one. They are defined by:

$${\prod }_S=\frac{1}{2\mathrm{\pi }\mathrm{i}}{\displaystyle \oint {(z-T_S)}^{-1}\mathrm{d}z,}\prod ={\prod }_{\mathrm{spec}(M)}$$

(25)

where the integral is over a simple closed contour in the complex plane encircling no spectrum of T_S, except Point1. If one is not an eigenvalue, then Π_S = 0. For the next result, we recall the following definition,

$$\{X_n\in S\mathrm{eventually}\}=\{\omega |\mathrm{there}\mathrm{exists}\mathrm{a}k\mathrm{s}.\mathrm{t}.X_n(\omega )\in S\mathrm{for}\mathrm{all}n\ge k\}$$

Lemma 7.$WehaveP(X_n\in Seventually)=\langle {\psi }_{\mathcal{S}},\mathrm{\Pi }{\mathrm{\Pi }}_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle$.

Proof: The set {X_n ∈ S eventually} is the increasing union of {X_n ∈ S ∀n ≥ k}, so P(X_n ∈ S eventually) = lim_k→∞ P(X_n ∈ S ∀n ≥ k). Next, {X_n ∈ S ∀n ≥ k} is the intersection of the decreasing sequence {X_n ∈ S, n = k,…, k + l}, so:

$$P(X_n\in S\mathrm{eventually})=\underset{k\to \infty }{\lim }\underset{l\to \infty }{\lim }P(X_n\in S,n=k,\dots ,k+l)$$

(26)

We have $P(X_n\in S,n=k,\dots ,k+l)=\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_S^{l+1}{\psi }_{\mathcal{S}}\rangle$. Since for each k fixed, the limit of $\langle {\psi }_{\mathcal{S}}{T}^{k-1}{T}_S^{l+1}{\psi }_{\mathcal{S}}\rangle$ as l → ∞ exists (it is the probability P(X_n ∈ S ∀n ≥ k)), we have:

$$\underset{l\to \infty }{\lim }\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_S^{l+1}{\psi }_{\mathcal{S}}\rangle =\underset{L\to \infty }{\lim }\frac{1}{L}{\displaystyle \sum _{l=1}^L\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_S^{l+1}{\psi }_{\mathcal{S}}\rangle }=\langle {\psi }_{\mathcal{S}},T^{k-1}{\tilde{\prod }}_S{\psi }_{\mathcal{S}}\rangle$$

where ${\tilde{\prod }}_S$ is the ergodic projection of T_S associated with the eigenvalue of one ( ${\tilde{\prod }}_S=0$if one is not an eigenvalue of T_S). Arguing in the same way for the limit k → ∞, we obtain $P(X_n\in S\mathrm{eventually})=\langle {\psi }_{\mathcal{S}},\tilde{\prod }{\tilde{\prod }}_S{\psi }_{\mathcal{S}}\rangle$. Invoking Lemma 21, we replace the ergodic projections with the Riesz projections. ■

The probability is given entirely by information on the spectrum of T and T_S at Point 1 (Riesz projections). The result holds even if T or T_S have spectrum on the unit circle other than possibly at one.

Finer information about the asymptotic dynamics depends on all of the spectrum of T on the unit circle. We make the following ergodicity assumption (compare with Lemma 6).

Condition A: The point z = 1 is the unique eigenvalue of T with |z| = 1, and this eigenvalue is simple (with eigenvector ${\psi }_{\mathcal{S}}$). We define the gap by:

$$\mathrm{\gamma }=1-\sup \{|z|:z\in \mathrm{spec}(T),z\ne 1\}$$

(27)

For λ = 0, the operator T is ${\mathrm{e}}^{\mathrm{i}\tau L_{\mathcal{S}}}$ and has spectrum on the unit circle, with degenerate eigenvalue of one. Assumption A is verified in practice typically by perturbation theory (λ small, non-zero). It is sometimes called a “Fermi golden rule condition”. In this setting, Condition A implies γ > 0 for small non-zero λ. Condition A implies the dynamical behaviour of Assumption (A) stated before Theorem 1.

The random variables X_n are not independent, but their correlations decay.

Theorem 8 (Decay of correlations). Suppose Condition A holds. For any ϵ > 0, there is a constant C_ϵ, such that for 1 ≤ k ≤ l < m ≤ n < ∞, A ∈ σ(X_k,… X_l) and B ∈ σ(X_m,…, X_n), we have:

$$|P(A\cap B)-P(A)P(B)|\le C_{\in }P(A){\mathrm{e}}^{-(m-l)[\ln (\frac{1}{1-\mathrm{\gamma }})-\in ]}$$

(28)

We give a proof of Theorem 8 in Section 4.1. For fixed $\in <\ln (\frac{1}{1-\mathrm{\gamma }})$, define the function $\mathcal{C}:\mathbb{N}\to {ℝ}_{+}$ by:

$$\mathcal{C}(d)=C_{\in }{\mathrm{e}}^{-d[\ln (\frac{1}{1-\mathrm{\gamma }})-\in ]}$$

(29)

Theorem 8 implies that the random variables X_n have decaying correlations in the following sense: for all 1 ≤ k ≤ l < m ≤ n, all A ∈ σ(X_k,… X_l) and all B ∈ σ(X_m,…, X_n), we have:

$$|P(A\cap B)-P(A)P(B)|\le \mathcal{C}(m-l)$$

(30)

where the function $\mathcal{C}$ is independent of A, B, k, l, m, n and satisfies $\mathcal{C}\left(d\right)\to 0$ as d → ∞.

Let X_k, k = 1, 2,… be a sequence of random variables. We denote by σ(X_n,X_n+1,…) the sigma algebra generated by {X_k}_k≥n. The tail sigma algebra of the process {X_n}_n≥1 is defined by $\mathcal{T}={\cap }_{n\ge 1}\sigma (X_n,X_{n+1}\dots )$.

The following result is a generalization of the Kolmogorov zero-one law, valid for a process {X_n}, where the random variables are not independent, but have decaying correlations.

Theorem 9 (Extended Kolmogorov zero-one law). Let X_n be a sequence of random variables with decaying correlations, as in Equation (30). Then, we have P(A) = 0 or P(A) = 1 for any tail event A ∈ $\mathcal{T}$.

A proof of this result can be obtained by extending proofs of the Kolmogorov zero-one law for independent variables; see [6]. Under Condition A, we write the rank-one Riesz projection of T associated with z = 1 as:

$$\prod =|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|$$

(31)

where $T{\psi }_{\mathcal{S}}={\psi }_{\mathcal{S}},T*{\psi }_{\mathcal{S}}^{*}={\psi }_{\mathcal{S}}^{*}$ and $\langle {\psi }_{\mathcal{S}}^{*},{\psi }_{\mathcal{S}}\rangle =1,\Vert {\psi }_{\mathcal{S}}\Vert =1$. (T* is the adjoint of T.)

Corollary 10. Assume that Condition A holds, and let S ⊂ spec(M). Then:

$$P(X_n\in Seventually)=\langle {\psi }_{\mathcal{S}}^{*},{\prod }_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle \in \{0,1\}$$

Remark: Both ${\psi }_{\mathcal{S}}^{*}$ and ${\prod }_{\mathcal{S}}$ depend on λ $({\psi }_{\mathcal{S}}dose\mathrm{not})$. If Condition A holds for λ ∈ I\{0} for some neighbourhood I c ℝ of zero, then for λ sufficiently small, we have $P(X_n\in S\mathrm{eventually})={\lim }_{\mathrm{\lambda }\to 0}\langle {\psi }_{\mathcal{S}}^{*}(\mathrm{\lambda }),{\prod }_{\mathcal{S}}(\mathrm{\lambda }){\psi }_{\mathcal{S}}\rangle$. This follows from the facts that the mapping $\mathrm{\lambda }\mapsto \langle {\psi }_{\mathcal{S}}^{*},{\prod }_S{\psi }_{\mathcal{S}}\rangle$ is continuous in a deleted neighbourhood of λ = 0 and that the image is discrete. We point out that the map $\mathrm{\lambda }\mapsto \langle {\psi }_{\mathcal{S}}^{*},{\prod }_S{\psi }_{\mathcal{S}}\rangle$ is actually holomorphic in a punctured neighbourhood of the origin and stays bounded there (the only possible image points being 0 or 1, even for complex λ, by the identity principle). Hence, zero is a removable singularity of this map. As objects on their own, ${\psi }_{\mathcal{S}}^{*}$ and ${\prod }_{\mathcal{S}}$ are not holomorphic at the origin in general (eigenvalue splitting), but their combination as in the inner product is.

Theorem 3 gives a criterion for P(X_n ∈ S eventually) = 0. The next result is a characterization of when this probability is one.

Lemma 11. Assume that Condition A holds. If$\langle {\psi }_{\mathcal{S}},{\prod }_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle \ne 0$ for some small enough λ, then P(X_n ∈ S eventually) = 1 for sufficiently small values of λ.

Proof of Lemma 11: The inequality P(X_n ∈ S eventually) ≥ P(X_n ∈ S ∀n ≥ 1) is the same as $\langle {\psi }_{\mathcal{S}},\prod {\prod }_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle \ge \langle {\psi }_{\mathcal{S}},{\prod }_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle$ (see also Appendix A). The result now follows from the fact that the left side can only take the values zero or one, independently of λ for λ sufficiently small (guaranteeing that $\langle {\psi }_{\mathcal{S}},\prod {\prod }_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle$ is continuous). ■

Proposition 12. Let A_j ∈ σ(X_j), j ≥ 1. We have:

$$\underset{n\ge 1}{\sup }|P(A_n,\dots ,A_{n+k})-P(A_n)\cdots P(A_{n+k})|\le C_k\Vert V\Vert$$

for any k ≥ 1 and for some constant C_k.

Proof of Proposition 12: It suffices to show that:

$$P(X_n\in S_n,\dots ,X_{n+k}\in S_{n+k})-P(X_n\in S_n)\cdots P(X_{n+k}\in S_{n+k})=O\left(\Vert V\Vert \right)$$

uniformly in n and where S_j ⊆ spec(M). Since $T_S=PB*B{\mathrm{e}}^{\mathrm{i}\tau K}{E}_{S}P={\mathrm{e}}^{\mathrm{i}\tau L_{\mathcal{S}}}{\omega }_{\mathrm{in}}\left(E_S\right)+O\left(\Vert V\Vert \right)$, the joint probability on the left side is:

$$\langle {\psi }_{\mathcal{S}},T^{n-1}{T}_{S_n}\cdots T_{S_{n+k}}{\psi }_{\mathcal{S}}\rangle ={\omega }_{\mathrm{in}}\left(E_{S_n}\right)\cdots {\omega }_{\mathrm{in}}\left(E_{S_{n+k}}\right)+O\left(\Vert V\Vert \right)$$

Similarly, $P\left(X_j\in S_j\right)={\omega }_{\mathrm{in}}\left(E_{S_j}\right)+O\left(\Vert V\Vert \right)$, and so, the result follows. ■

Let ω_n be the state of $\mathcal{S}$ at time step n (obtained by reducing the state of the entire system to $\mathcal{S}$).

Lemma 13 (Evolution of the averaged system state). The system state at time step n, ω_n, is a random variable (determined by the random measurement history). Its expectation, $\mathbb{E}[{\omega }_n]$, equals the state obtained by evolving the initial condition according to the dynamics without measurement.

Proof of Lemma 13: For a given measurement path X₁ = m₁,…, X_n = m_n, the system state is:

$${\omega }_n(A)=\frac{\langle {\psi }_{\mathcal{S}},T_1\cdots T_{n}A{\psi }_{\mathcal{S}}\rangle }{\langle {\psi }_{\mathcal{S}},T_1\cdots T_n{\psi }_{\mathcal{S}}\rangle }$$

where A is any system observable and T_j = T_{mj}. Since $P(X_1=m_1,\dots ,X_n=m_n)=\langle {\psi }_{\mathcal{S}},T_1\cdots T_n{\psi }_{\mathcal{S}}\rangle$, this yields:

$$\begin{array}{l}\mathbb{E}[{\omega }_n(A)]={\displaystyle \sum _{m_1,\dots ,m_n}P(X_1=m_1,\dots ,X_n=m_n)\frac{\langle {\psi }_{\mathcal{S}},T_1\cdots T_{n}A{\psi }_{\mathcal{S}}\rangle }{\langle {\psi }_{\mathcal{S}},T_1\cdots T_n{\psi }_{\mathcal{S}}\rangle }}\\ ={\displaystyle \sum _{m_1,\dots ,m_n}\langle {\psi }_{\mathcal{S}},T_1\cdots T_{n}A{\psi }_{\mathcal{S}}\rangle }\\ =\langle {\psi }_{\mathcal{S}},T^n\cdots A{\psi }_{\mathcal{S}}\rangle \end{array}$$

In the last step, we have used that:

$${\displaystyle \sum _m{T}_{\left\{m\right\}}}={\displaystyle \sum _{m}P{B}^{*}B{\mathrm{e}}^{\mathrm{i}\tau K}{E}_{\left\{m\right\}}P=P{B}^{*}B{\mathrm{e}}^{\mathrm{i}\tau K}P=T}$$

The right-hand side is the single-step dynamics operator of the system without probe measurements.

Lemma 13 shows in particular that the expectation of the system state converges to the repeated interaction state,

$$\underset{n\to \infty }{\lim }\mathbb{E}[{\omega }_n(A)]=\langle {\psi }_{\mathcal{S}}^{*},(A\otimes 1_{\mathcal{S}}){\psi }_{\mathcal{S}}\rangle$$

see also Equation (31). This, of course, does not mean that ω_n itself converges, in general. However, if the measurement process converges, then we do have the following result.

Lemma 14 (Asymptotic state of $\mathcal{S}$). Suppose that the measurement outcomes:

$$X_1\in S_1,\dots ,X_{n-1}\in S_{n-1},X_k\in S,k\ge n$$

(32)

are observed for some n ≥ 1 and that one is a simple eigenvalue of T_S with Riesz projection Π_S = |ψ〉 〈ψ*|. We have for any observable A of$\mathcal{S}$:

$$\underset{n\to \infty }{\lim }{\omega }_n(A)={\omega }_{\infty }(A)=\frac{\langle \psi *,A{\psi }_{\mathcal{S}}\rangle }{\langle \psi *,{\psi }_{\mathcal{S}}\rangle }$$

This is an inverse scattering result: knowing that the scattered particles are measured to lie in S, we can deduce the state of the scattering object $\mathcal{S}$. The final state does not depend on the initial outcomes X_n for n < k, any k. However, we show in the proof of Lemma 14 that if the eigenvalue of one of T_S is not simple, then the system converges to a final state, which depends on the whole measurement path X₁, X₂,…

Proof of Lemma 14: The asymptotic state of the system is:

$${\omega }_{\infty }(A)=\underset{l\to \infty }{\lim }\frac{\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}{T}_S^{l}A{\psi }_{\mathcal{S}}\rangle }{\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}{T}_S^l{\psi }_{\mathcal{S}}\rangle }$$

see Equation (23). Now, $\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}{T}_S^l{\psi }_{\mathcal{S}}\rangle$ converges to the non-zero probability of observing Equation (32). Therefore, by (the proof of) Lemma 21:

$$\underset{l\to \infty }{\lim }\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}{T}_S^l{\psi }_{\mathcal{S}}\rangle =\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}{\tilde{\prod }}_S{\psi }_{\mathcal{S}}\rangle =\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}{\prod }_S{\psi }_{\mathcal{S}}\rangle$$

By simplicity of the eigenvalue of one of T_S,

$${\omega }_{\infty }(A)=\frac{\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}\psi \rangle \langle \psi *,A{\psi }_{\mathcal{S}}\rangle }{\langle {\psi }_{\mathcal{S}}{T}_1\cdots T_{n-1}\psi \rangle \langle \psi *,{\psi }_{\mathcal{S}}\rangle }$$

If one is not a simple eigenvalue of T_S, so that ${\prod }_S={\displaystyle {\sum }_{j=1}^r|{\psi }_j\rangle \langle {\psi }_j^{*}|}$, the final state is:

$${\omega }_{\infty }(A)=\frac{{\displaystyle {\sum }_{j=1}^r\langle {\psi }_{\mathcal{S}},T_1\cdots T_{n-1}{\psi }_j\rangle \langle {\psi }_j^{*},A{\psi }_{\mathcal{S}}\rangle }}{{\displaystyle {\sum }_{j=1}^r\langle {\psi }_{\mathcal{S}},T_1\cdots T_{n-1}{\psi }_j\rangle \langle {\psi }_j^{*},{\psi }_{\mathcal{S}}\rangle }}$$

The final state then depends on the whole measurement history. ■

3. The Truncated Jaynes–Cummings Model

We consider a simple system where both the scatterer and the probes have only two degrees of freedom participating in the scattering process. The pure state space of $\mathcal{S}$ and $\mathcal{P}$ is ${ℂ}^2$, and the Hamiltonians are given by the Pauli σ_z operator,

$$H_{\mathcal{S}}=H_{\mathcal{P}}=\left[\begin{array}{ll}1\hfill & 0\hfill \\ 0\hfill & -1\hfill \end{array}\right]$$

(33)

The interaction between $\mathcal{S}$ and $\mathcal{P}$ is determined by the operator:

$$\mathrm{\lambda }V=\lambda (a_{\mathcal{S}}^{*}\otimes a_{\mathcal{P}}+a_{\mathcal{S}}\otimes a_{\mathcal{P}}^{*})$$

(34)

with coupling constant $\lambda \in ℝ$, and where:

$$a=\left[\begin{array}{ll}0\hfill & 0\hfill \\ 1\hfill & 0\hfill \end{array}\right],a^{*}=\left[\begin{array}{ll}0\hfill & 1\hfill \\ 0\hfill & 0\hfill \end{array}\right]$$

(35)

are the annihilation and creation operators. In the usual Jaynes–Cummings model (used, e.g., in quantum optics), the system $\mathcal{S}$ has infinitely many levels (harmonic oscillator); see, e.g., [30] and also [25]. Our model is a truncation, but it still describes energy exchange between $\mathcal{S}$ and $\mathcal{P}$. In what follows, we can treat all values of λ, not necessarily small ones only. This is so since the model is essentially exactly solvable. The total Hamiltonian $H=H_{\mathcal{S}}+H_{\mathcal{P}}+\mathrm{\lambda }V$ describes the exchange of energy between $\mathcal{S}$ and $\mathcal{P}$, while the total number of excitations, $N=a_{\mathcal{S}}^{*}{a}_{\mathcal{S}}+a_{\mathcal{P}}^{*}{a}_{\mathcal{P}}$, is conserved (commutes with H). This allows for a treatment of the system separately in the invariant sectors N = 0,1,2.

For an arbitrary probe observable $X\in ℬ\left({ℂ}^2\right)$, we write X_ij = 〈φ_i,Xφ_j〉, where φ₁, φ₂, are the orthonormal eigenvectors of $H_{\mathcal{P}}$ (with $H_{\mathcal{P}}{\phi }_1={\phi }_1$). Incoming states are determined by p ∈ [0,1] via:

$${\omega }_{\mathrm{in}}(X)=p{X}_{11}+(1-p)X_{22}$$

(36)

where $X\in ℬ\left({ℂ}^2\right)$ is an arbitrary probe observable.

We will use the notation and definitions of Section 2 in what follows. In particular, the single step operator T_S is defined in Equation (22). For the following explicit formula, we take the reference state ${\mathrm{\Psi }}_{\mathcal{S}}$ to be the trace state.

Theorem 15 (Explicit reduced dynamics operator). Set φ_ij = φ_i ⊗ φ_i, and let X be any operator of$\mathcal{P}$. In the basis {φ₁₁, φ₁₂, φ₂₁, φ₂₂}, we have:

$$\begin{array}{l}PB*B{\mathrm{e}}^{\mathrm{i}\tau K}XP={\omega }_{\mathrm{in}}(X){\mathrm{e}}^{\mathrm{i}\tau L_{\mathcal{S}}}+\\ [\begin{array}{cc}(1-p)X_{22}a& (1-p)X_{21}b\\ -p{X}_{12}{\mathrm{e}}^{2\mathrm{i}\tau }\mathrm{i}\sin (\lambda \tau )& {\mathrm{e}}^{2\mathrm{i}\tau }(\cos (\lambda \tau )-1){\omega }_{\mathrm{in}}(X)\\ p{X}_{21}{\mathrm{e}}^{-2\mathrm{i}\tau }\mathrm{i}\sin (\lambda \tau )& 0\\ -p{X}_{22}a& -p{X}_{21}b\end{array}\\ \begin{array}{cc}-(1-p)X_{12}b& -(1-p)X_{11}a\\ 0& (1-p)X_{12}{\mathrm{e}}^{2\mathrm{i}\tau }\mathrm{i}\sin (\lambda \tau )\\ {\mathrm{e}}^{-2\mathrm{i}\tau }(\cos (\lambda \tau )-1){\omega }_{\mathrm{in}}(X)& -(1-p)X_{21}{\mathrm{e}}^{-2\mathrm{i}\tau }\mathrm{i}\sin (\lambda \tau )\\ p{X}_{12}b& p{X}_{11}a\end{array}]\end{array}$$

(37)

where a = − sin²(λτ), b = − i sin(λτ) cos(λτ).

We point out that the vector [p, 0,0,1 − p]^t is an eigenvector of the adjoint of Equation (37) with eigenvalue ω_in(X).

Proof: The proof is obtained by an explicit calculation. Since $H_{\mathcal{S}}$ and $H_{\mathcal{P}}$ commute with I = V – V′, where V′ = JΔ^1/2V Δ^−1/2 J (see Equation (10)), it suffices to calculate:

$$PB*B{\mathrm{e}}^{i\tau I}XP={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(i\tau \right)}^n}{n!}}{\displaystyle \sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right){\left(-1\right)}^{n-k}PB*B{V}^k{\left(V^{\prime }\right)}^{n-k}XP}$$

Here, it is understood that all operators are considered in the “doubled” (GNS) Hilbert space, e.g.,

$$V=a_{\mathcal{S}}^{*}{1}_{\mathcal{S}}\otimes a_{\mathcal{P}}\otimes 1_{\mathcal{P}}+a_{\mathcal{S}}\otimes 1_{\mathcal{S}}\otimes a_{\mathcal{P}}^{*}{1}_{\mathcal{P}}$$

Powers of V and V′ can be calculated explicitly. For instance, for k ≥ 2 even, we have $V^k={\stackrel{⌢}{n}}^{k/2}\otimes 1\otimes {(1-\stackrel{⌢}{n})}^{k/2}\otimes 1+{(1-\stackrel{⌢}{n})}^{k/2}\otimes 1\otimes {\stackrel{⌢}{n}}^{k/2}\otimes 1$, where $\stackrel{⌢}{n}=a*a$. One obtains similar expressions for k odd and for (V′)^l. Using these expression in the above series, one gets the result of Theorem 15. ■

Resonant and non-resonant system: If λτ is a multiple of π, then Equation (37) reduces to PB*Be^iτKXP = ω_in(X) diag(1, ±1, ±1,1) with plus and minus signs if the multiple is even and odd, respectively. Then, by using the expression P(X₁ ∈ S₁,…, X_n ∈ S_n) given in Equation (24), it is readily seen that the random variables X_j are independent, and P(X_j E S) = ω_in(E_S). When $\mathrm{\lambda }\tau \in \mathrm{\pi }\mathbb{Z}$, we call the system resonant [25]; otherwise, we call it non-resonant. One can understand the resonant regime as follows: consider the dynamics generated on $\mathcal{S}$ and a single probe $\mathcal{P}$ by the Hamiltonian $H=H_{\mathcal{S}}+H_{\mathcal{P}}+\lambda V$. The probability of transition from the initial state ${\phi }_2^{\mathcal{S}}\otimes {\phi }_1^{\mathcal{P}}$, where the $\mathcal{S}$ is in the ground state and $\mathcal{P}$ in the excited state, to the opposite state ${\phi }_1^{\mathcal{S}}\otimes {\phi }_2^{\mathcal{P}}$, at time t, is given by $P_t={|\langle {\phi }_1^{\mathcal{S}}\otimes {\phi }_2^{\mathcal{P}},{\mathrm{e}}^{\mathrm{i}tH}{\phi }_2^{\mathcal{S}}\otimes {\phi }_1^{\mathcal{P}}\rangle |}^2={\sin }^2(\mathrm{\lambda }t)$. For $\mathrm{\lambda }t\in \mathrm{\pi }\mathbb{Z}$, this probability vanishes. If the interaction time τ in the repeated interaction system is a multiple of π/λ, then interaction effects are suppressed. It is not hard to see that in this case, the system does not feel the interaction with the probes in the sense that ${\omega }_n(A)={\omega }_0({\alpha }_n^{\mathcal{S}}(A))$ for all n ≥ 1, where ${\alpha }_n^{\mathcal{S}}(A)$ is the reduced dynamics of $\mathcal{S}$ alone. We focus now on the non-resonant situation.

Asymptotics of the measurement process: We suppose the incoming state of the probes is given by ω_in = |φ₁〉〈φ₁|, i.e., they are in the pure spin-up state. This corresponds to p = 1 in Equation (36). Let M be a measurement operator, S ⊂ spec(M), and let E_S be the projection onto the corresponding spectral subspace. The operator Equation (37) with X = E_S has spectrum:

$$\mathrm{spec}(T_S)={(E_S)}_{11}\{1,{\mathrm{e}}^{2\mathrm{i}\tau }\cos (\mathrm{\lambda }\tau ),{\mathrm{e}}^{-2\mathrm{i}\tau }\cos (\mathrm{\lambda }\tau ),{\cos }^2(\mathrm{\lambda }\tau )\}$$

(38)

Since E_S is a projection, we have 0 ≤ (E_S)₁₁ = 〈φ₁, E_Sφ₁〉 ≤ 1.

• ○ The equality (E_S)₁₁ = 1 holds if and only if E_Sψ₁ = ψ₁, so if and only if either $E_S=1$ or E_S = |φ₁〉〈φ₁|. The case $E_S=1$ is discarded, since it corresponds to not making a measurement. Hence, (E_S)₁₁ = 1 in the presence of a measurement implies E_S = |φ₁〉〈φ₁|. This forces the measurement operator M to be diagonal in the basis {φ₁, φ₂}, i.e., M = diag(m₁, m₂). Conversely, if M is diagonal and E_S = |φ₁〉〈φ₁|, then (ES)₁₁ = 1. It follows that T_S has an eigenvalue of one if and only if M = m₁|φ₁〉〈φ₁|+m₂|φ₁〉〈φ₂| and E_S = |φ₁〉〈φ₂|. In this case, the associated Riesz spectral projection is $\prod =\sqrt{2}|{\psi }_{\mathcal{S}}\rangle \langle {\phi }_{11}|$ (see Equation (25)), and we have P(X_n = m₁ eventually) = 1.
• ○ If the measurement operator M is not diagonal in the basis {φ₁, φ₂}, then (E_S)₁₁ < 1 for any S with |S| = 1 (and again, if |S| = 2, then $E_S=1$, which means we do not make a measurement). Then, one is not an eigenvalue of T_S, according to Equation (38), and so P (X_n ∈ S eventually) = 0.

We conclude that the measurement process converges if and only if the incoming state is pure and localized with respect to the measurement operator (i.e., if and only if it is given by an eigenvector of M, and we measure the corresponding eigenvalue). In the situation where the measurement outcomes converge, we can determine the asymptotic state of the scatterer $\mathcal{S}$ from Lemma 14. It is given by ${\omega }_{\infty }(A)={\omega }_{\mathrm{in}}(A),A\in \mathcal{B}({ℂ}^2)$; thus, the state of the incoming probe is copied onto the scatterer. This copying mechanism has been described before as “homogenization” in [31] (our analysis is more complete than previous ones, as it describes the entire system of scatterer and probes). Note also that the asymptotic mean is given by µ_∞ = ω_in(M). The frequencies are f_m = ω_in(E_m) (Expression (55)). This suggests that the cavity becomes ‘transparent’ for large times (no effect on incoming probes).

Large deviations for the mean: The logarithmic moment generating function [32] is defined by:

$$\mathrm{\Lambda }(\alpha )=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\mathbb{E}[\mathrm{e}}^{n\alpha \overline{X}n}]$$

(39)

for α ∈ ℝ s.t., the limit exists as an extended real number. Using Expression (66) and Theorem 15 (with p = 1), we find that Λ(α) = log ω_in(e^αM), for α ∈ ℝ. The Legendre transformation of Λ(α),

$${\mathrm{\Lambda }}^{\ast }(x)=\underset{\alpha \in ℝ}{\sup }\alpha x-\mathrm{\Lambda }(\alpha )$$

(40)

x ∈ ℝ, is called the rate function. Its usefulness in the present context is due to the Gärtner–Ellis theorem [32], which asserts that for any closed set F ⊂ ℝ and any open set G ⊂ ℝ, we have:

$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }\frac{1}{n}\log P({\overline{X}}_n\in F)\le -\underset{x\in F}{\inf }{\mathrm{\Lambda }}^{*}(x)\\ \underset{n\to \infty }{\lim \inf }\frac{1}{n}\log P({\overline{X}}_n\in G)\ge -\underset{x\in G\cap ℱ}{\inf }{\mathrm{\Lambda }}^{*}(x)\end{array}$$

(41)

Here, $ℱ$ denotes the set of ‘exposed points of Λ^*’ (see [32]).

Proposition 16. Suppose that Var(M) = ω_in(M²) − ω_in(M)², the variance of M in the state ω_in does not vanish. Then, Λ^* is holomorphic at x = ω_in(M), and:

$$\mathrm{\Lambda }*(x)=\frac{{\left(x-{\omega }_{\mathrm{in}}(M)\right)}^2}{2\mathrm{Var}(M)}+O\left({\left(x-{\omega }_{\mathrm{in}}(M)\right)}^4\right)$$

Proof: Note that Λ is twice differentiable, and the second derivative w.r.t. α of the argument of the supremum in Equation (40) is less than or equal to zero. Therefore, for fixed x, the supremum is taken at α ∈ ℝ satisfying:

$$x={\mathrm{\Lambda }}^{\prime }(\alpha )=\frac{{\omega }_{\mathrm{in}}(M{\mathrm{e}}^{\alpha M})}{{\omega }_{\mathrm{in}}({\mathrm{e}}^{\alpha M})}$$

(42)

For α = 0, we have x = ω_in(M). If Λ″(0) = Var(M) ≠ 0, then Equation (42) has an implicit solution α = α(x), locally around x = ω_in(M). Since Λ′ (α) is holomorphic at α = 0, the implicit solution is holomorphic at x = ω_in(M) (see, e.g., [33], p. 163, Equation (12.4)). The Taylor expansion of Equation (42) is:

$$x={\omega }_{\mathrm{in}}(M)+\alpha \mathrm{Var}(M)+c{\alpha }^2+O({\alpha }^3)$$

(43)

where $c=\frac{1}{2}\left\{{\omega }_{\mathrm{in}}(M^3)-3{\omega }_{\mathrm{in}}(M^2){\omega }_{\mathrm{in}}(M)+2{\omega }_{\mathrm{in}}{\left(M\right)}^3\right\}$. We solve Equation (43) implicitly for α = α(x), which is the point where the supremum in Equation (40) is taken. The explicit formula for the supremum given in Proposition 16 follows. ■

Example: Measuring the outgoing spin angle. Since ω_in is the state ‘spin up’, we have ω_in(M) = M₁₁ and Var(M) = |M₁₂|². Imagine an experiment where we measure the angle of the spins as they exit the scattering process. Let θ ∈ [0, π/2) and ϕ ∈ [0, 2π) be the angles measuring the altitude (θ = 0 is spin-up) and azimuth (ϕ = 0 is the plane orthogonal to the axis of the cavity). The measurement operator “spin-in direction (θ, ϕ)” is given by:

$$M=\left[\begin{array}{cc}\cos \theta & {\mathrm{e}}^{-\mathrm{i}\varphi }\sin \theta \\ {\mathrm{e}}^{\mathrm{i}\varphi }\sin \theta & -\cos \theta \end{array}\right]$$

see, e.g., [34], Chapter IV, (A-19). The eigenvectors of M associated with the eigenvalues ±1 of M are:

$$\begin{array}{l}\chi +={\mathrm{e}}^{-\mathrm{i}\varphi /2}\cos (\theta /2){\phi }_1+{\mathrm{e}}^{-\mathrm{i}\varphi /2}\sin (\theta /2){\phi }_2\\ \chi -=-{\mathrm{e}}^{-\mathrm{i}\varphi /2}\sin (\theta /2){\phi }_1+{\mathrm{e}}^{\mathrm{i}\varphi /2}\cos (\theta /2){\phi }_2\end{array}$$

The eigenprojection E₊ measures the spin in the positive direction (θ, ϕ). By using Lemma 7, it is easy to see that:

$$P(X_n\text{is in direction}(\theta ,\varphi )\text{eventually})=\{\begin{array}{cc}1& \text{if}\theta =0\\ 0& \text{if}\theta \ne 0\end{array}$$

This is another manifestation of the asymptotic transparency of the cavity.

We obtain from Theorem 5 (with µ_∞ = cos θ) that for any ϵ > 0,

$$\underset{n\to \infty }{\lim }P(|{\overline{X}}_n-\cos \theta |\ge ϵ)$$

The speed of convergence can be estimated using Equation (41) and Proposition 16. It is easy to see that the logarithmic generating function and the rate function associated with the shifted random variable ${\overline{X}}_n-\cos \theta$ are given by Λ_shift (α) = Λ(α) − α cos θ and ${\mathrm{\Lambda }}_{\mathrm{shift}}^{*}(x)={\mathrm{\Lambda }}^{*}(x+\cos \theta )$, respectively. Next, we note that all points in the vicinity of zero belong to the set ${ℱ}_{\mathrm{shift}}$ shift, the set of exposed points of ${\mathrm{\Lambda }}_{\mathrm{shift}}^{*}$. Indeed, if $x={{\mathrm{\Lambda }}^{\prime }}_{\mathrm{shift}}(\alpha )$ for some α ∈ ℝ, then $x\in {ℱ}_{\mathrm{shift}}$ ([32], Lemma 2.3.9). However, $x=0=\mathrm{\Lambda }{\prime }_{\mathrm{shift}}(0)$, and $\mathrm{\Lambda }{\prime }_{\mathrm{shift}}$ is invertible around zero (as ${{\mathrm{\Lambda }}^{″}}_{\mathrm{shift}}(0)\ne 0$). This shows that ${ℱ}_{\mathrm{shift}}$ contains a neighbourhood of the origin.

Take 0 < ϵ < ϵ′ ≪ 1; set G = (−ϵ′, − ϵ) ∪ (ϵ, ϵ′); and let F be the closure of G. Then, ${\inf }_{x\in F}{\mathrm{\Lambda }}_{\mathrm{shift}}^{*}(x)={\inf }_{x\in G\cap {ℱ}_{\mathrm{shift}}}{\mathrm{\Lambda }}_{\mathrm{shift}}^{*}(x)=\frac{{ϵ}^2}{2\mathrm{Var}(M)}+O\left({\left({ϵ}^{\prime }\right)}^4\right)$ (we use Proposition 16). Combining this with the two bounds Equation (41) (for the shifted random variable), we obtain:

$$P(ϵ\le |{\overline{X}}_n-\cos \theta |\le {ϵ}^{\prime })~\exp \left[-n\left\{\frac{{ϵ}^2}{2{\sin }^2\theta }+O\left({\left({ϵ}^{\prime }\right)}^4\right)\right\}\right],n\to \infty$$

which is a large deviation statement for the average ${\overline{X}}_n$.

4. Proofs

4.1. Proof of Theorems 1 and 8

Theorem 8 is a stronger version of Theorem 1, so it suffices to prove the former.

Let A ∈ σ(X_k, …, X_l) and B ∈ σ(X_m, …, X_n). The range Σ of the X_n is finite, so σ(X_k, …, X_l) consists of the collection of all sets of the form {ω : (X_k(ω), …, X_l(ω)) ∈ H}, where H ⊆ Σ^l−k+1 ([5], Theorem 5.1). Therefore, we have:

$$\begin{array}{l}A={\displaystyle \underset{j=1}{\overset{J}{\cup }}{X}_k^{-1}(\{s_k^{(j)}\})\cap \cdots \cap X_l^{-1}(\{s_l^{(j)}\})}=:{\displaystyle \underset{j=1}{\overset{J}{\cup }}{A}_j}\\ B={\displaystyle \underset{i=1}{\overset{I}{\cup }}{X}_m^{-1}(\{s_m^{(i)}\})\cap \cdots \cap X_n^{-1}(\{s_n^{(i)}\})}=:{\displaystyle \underset{i=1}{\overset{I}{\cup }}{B}_i}\end{array}$$

where s^(j)r, s⁽ⁱ⁾r ∈ Σ, and A_i ∩ A_j = ∅, B_i ∩ B_j = ∅ for i ≠ j. Thus,

$$P(A)={\displaystyle \sum _{j}P(A_j),}P(B)={\displaystyle \sum _{i}P(B_i),P(A\cap B)}={\displaystyle \sum _{i,j}P(A_j\cap B_i)}$$

Setting $T_m^{(n)}:=T_{S=\left\{s_m^{(n)}\right\}}$, we have:

$$P(A\cap B)={\displaystyle \sum _{i,j}\langle {\psi }_S,T^{k-1}{T}_k^{(j)}\cdots T_l^{(j)}{T}^{m-l-1}{T}_m^{(i)}\cdots T_n^{(i)}{\psi }_S\rangle }$$

(44)

We now approximate T ^m−l−1 by its value for large m − l. To do so, let $P_{\mathcal{S}}$ denote the Riesz spectral rank-one projection onto the eigenvalue of one of T, and let ${\overline{P}}_{\mathcal{S}}=1_{\mathcal{S}}-P_{\mathcal{S}}$. We have $P_{\mathcal{S}}=|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|$, where ${\psi }_{\mathcal{S}}^{*}\in {\mathscr{H}}_{\mathcal{S}}$ satisfies $T*{\psi }_{\mathcal{S}}^{*}={\psi }_{\mathcal{S}}^{*}$ and $\langle {\psi }_{\mathcal{S}}^{*},{\psi }_{\mathcal{S}}\rangle =1$. The operator T has the spectral representation ([35], I§5):

$$T=|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|+{\displaystyle \sum _{r=1}^d\{z_r{P}_r+D_r\}}$$

(45)

where P_r is the Riesz projection associated with the eigenvalue z_j and D_r is the associated eigen-nilpotent. We have $D_r^{v_r}=0$ (ν_r is the index of z_r). Consequently,

$$T^k=|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|+{\displaystyle \sum _{r=1}^d{\displaystyle \sum _{q=0}^{v_r-1}\left(\begin{array}{c}k\\ q\end{array}\right)z_r^{k-q}{P}_r{D}_r^q=:|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|+R_k}}$$

(46)

Note that it suffices to consider the non-zero z_r in Equation (46). Using the bound ${\displaystyle {\sum }_{q=0}^{v-1}\left(\begin{array}{c}k\\ q\end{array}\right)}\le v{k}^{v-1}\le k^v$, and the fact that for any ϵ > 0, there is a constant C_ϵ s.t. k^ν ≤ C_ϵe^ϵ for all k ≥ 1, we obtain:

$$||R_k||\le C_{\epsilon }{\mathrm{e}}^k\underset{1\le r\le d}{\max }|z_r{|}^k\le C{\mathrm{e}}^{\epsilon k}{(1-\gamma )}^k$$

(47)

where we invoke Assumption A, |z_r| ≤ 1 − γ. We now replace T ^m−l−1 in Equation (44) using Equation (46),

$$P(A\cap B)=P(A){\displaystyle \sum _i\langle {\psi }_S^{*},T_m^{(i)}\cdots T_n^{(i)}{\psi }_S\rangle }$$

(48)

$$+{\displaystyle \sum _{i,j}\langle {\psi }_S,T^{k-1}{T}_k^{(j)}\cdots T_l^{(j)}{R}_{m-l-1}{T}_m^{(i)}\cdots T_n^{(i)}{\psi }_S\rangle }$$

(49)

The sum in Equation (48) equals:

$$\begin{array}{l}{\displaystyle \sum _i\langle {\psi }_S^{*},T_m^{(i)}\cdots T_n^{(i)}{\psi }_S\rangle }={\displaystyle \sum _i\langle {\psi }_{\mathcal{S}},|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|T_m^{(i)}\cdots T_n^{(i)}{\psi }_S\rangle }\\ ={\displaystyle \sum _i\langle {\psi }_{\mathcal{S}},(T^{m-1}-R_{m-1})T_m^{(i)}\cdots T_n^{(i)}{\psi }_{\mathcal{S}}\rangle }\\ =P(B)-{\displaystyle \sum _i\langle {\psi }_{\mathcal{S}},R_{m-1}{T}_m^{(i)}\cdots T_n^{(i)}{\psi }_{\mathcal{S}}\rangle }\end{array}$$

(50)

Lemma 17. There is a constant C s.t.$\Vert {\displaystyle {\sum }_i{T}_m^{(i)}\cdots T_n^{(i)}}\Vert \le C$, independently of the range of values of i, m, n and of the$s_l^{(i)}$ defining the$T_l^{(i)}$.

We give a proof below. Lemma 17 together with Equations (48)–(50) gives

$$\begin{array}{l}|P(A\cap B)-P(A)P(B)|\le CP(A)\Vert R_{m-1}\Vert \\ +|{\displaystyle \sum _{i,j}\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_k^{(j)}\cdots T_l^{(j)}{R}_{m-l-1}{T}_m^{(i)}\cdots T_n^{(i)}{\psi }_{\mathcal{S}}\rangle }|\end{array}$$

(51)

The remainder R_m−l−1 is given by the sum in Equation (46) (with k = m − l − 1). We expand:

$$P_r{D}_r^q={\displaystyle \sum _{s,s^{\prime }}\omega (s,s^{\prime })|{\chi }_S\rangle \langle {\chi }_{S^{\prime }}|}$$

(52)

in an orthonormal basis {χ_s}, where $\omega (\mathrm{s},{\mathrm{s}}^{\prime })\in ℂ$ are matrix elements (also depending on r, q). The modulus of the sum in Equation (51) is bounded above by:

$$\begin{array}{c}C{\displaystyle \sum _{r=1}^d{\displaystyle \sum _{q=0}^{v_r-1}\left(\begin{array}{c}m-l-1\\ q\end{array}\right)|z_r{|}^{m-l-1-q}}}{\displaystyle \sum _{s,s^{\prime }}|\omega (s,s^{\prime })|}\\ \times {\displaystyle \sum _j|\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_k^{(j)}\cdots T_l^{(j)}{\chi }_{\mathcal{S}}\rangle |}\end{array}$$

(53)

Lemma 18. Let$A_s\in {\mathfrak{M}}_{\mathcal{S}}$ be the unique operator s.t.${\chi }_s=A_s{\psi }_{\mathcal{S}}$. We have:

$${\displaystyle \sum _j|\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_k^{(j)}\cdots T_l^{(j)}{\chi }_{\mathcal{S}}\rangle |}\le ||A_{\mathrm{s}}||P(A)$$

We give a proof of Lemma 18 below. Using the result of the Lemma in Equation (53), we obtain, for any ϵ > 0, the upper bound P (A)C_ϵ e ^ϵ(m−l)(1 − γ)^m−l for the sum in Equation (51). Thus, for all ϵ > 0, there is a C_ϵ s.t.:

$$|P(A\cap B)-P(A)P(B)|\le C_{ϵ}P(A)\{{(1-\gamma )}^m{e}^{ϵm}+{(1-\gamma )}^{m-l}{e}^{ϵ(m-l)}\}$$

Since m ↦ (1 − γ)^me^ϵm is decreasing (for ϵ small enough), we get Equation (28). This concludes the proof of Theorem 8 modulo proofs of Lemmata 17 and 18.

Proof of Lemma 17: Any vector $\psi \in {\mathscr{H}}_{\mathcal{S}}$is of the form $\psi =Ã{\mathrm{\Psi }}_{\mathcal{S}}$, for some $\tilde{A}\in {\mathfrak{M}}_{\mathcal{S}}$. We have:

$$\begin{array}{l}\Vert {\displaystyle \sum _i{T}_m^{(i)}\cdots T_n^{(i)}\tilde{A}{\mathrm{\Psi }}_{\mathcal{S}}}\Vert =\underset{\Vert \tilde{B}{\mathrm{\Psi }}_{\mathcal{S}}\Vert =1}{\sup }|\langle \tilde{B}{\mathrm{\Psi }}_{\mathcal{S}},{\displaystyle \sum _i{T}_m^{(i)}\cdots T_n^{(i)}\tilde{A}{\mathrm{\Psi }}_{\mathcal{S}}}\rangle |\\ =\underset{\Vert \tilde{B}{\mathrm{\Psi }}_{\mathcal{S}}\Vert =1}{\sup }|\langle \tilde{B}{\mathrm{\Psi }}_{\mathcal{S}}\otimes B{\mathrm{\Psi }}_{{\mathcal{P}}_m}\otimes \cdots \otimes B{\mathrm{\Psi }}_{{\mathcal{P}}_m},{\mathrm{e}}^{i\tau L_n}\cdots {\mathrm{e}}^{i\tau L_m}Q\tilde{A}\times \\ \times {\mathrm{e}}^{-i\tau L_m}\cdots {\mathrm{e}}^{-i\tau L_n}{\mathrm{\Psi }}_{\mathcal{S}}\otimes B{\mathrm{\Psi }}_{{\mathcal{P}}_m}\otimes \cdots \otimes B{\mathrm{\Psi }}_{{\mathcal{P}}_n}\rangle |\end{array}$$

where $Q:={\displaystyle {\sum }_i{E}_m(\{s_m^{(i)}\})\cdots E_n(\{s_n^{(i)}\})}$

$$||{\displaystyle \sum _i{T}_m^{(i)}\dots T_n^{(i)}\tilde{A}{\mathrm{\Psi }}_s}||\le ||\tilde{A}||$$

The result now follows from the uniform boundedness principle. This proves Lemma 17.

Proof of Lemma 18: By using the definition $T_k^{(j)}=P_k{B}^{\ast }B{\mathrm{e}}^{\mathrm{i}\tau K}{E}_k^{(j)}{P}_k$, where $E_k^{(j)}=E_k(\{s_k^{(j)}\})$ (see Equation (22)), we see that:

$$\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_k^{(j)}\cdots T_k^{(j)}{A}_s{\psi }_{\mathcal{S}}\rangle =\langle {\mathrm{\Psi }}^{(j)},A_s{\mathrm{\Psi }}^{(j)}\rangle$$

where ${\mathrm{\Psi }}^{(j)}=E_k^{(j)}\cdots E_l^{(j)}{\mathrm{e}}^{\mathrm{i}\tau L_l}\cdots {\mathrm{e}}^{-}{}^{\mathrm{i}\tau L_1}{\psi }_{\mathcal{S}}\otimes B{\psi }_{\mathcal{P}}\cdots \otimes B{\psi }_{\mathcal{P}}$ (l probes). To arrive at the form of Ψ^(j), we replace the action of e^iτK in the operators $T_k^{(j)}$ (see just above) by the action of the Liouville operators ${\mathrm{e}}^{\mathrm{i}\tau L_k}\cdot {\mathrm{e}}^{-}{}^{\mathrm{i}\tau L_k}$; see Equations (11) and (21). Since a positive linear functional on ${\mathfrak{M}}_{\mathcal{S}}$ is bounded, with the norm equal to its value for the observable $1$ ([24] Proposition 2.3.11), we have:

$$|\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_k^{(j)}\cdots T_k^{(j)}{A}_s{\psi }_{\mathcal{S}}\rangle |\le \Vert A_s\Vert \langle {\psi }_{\mathcal{S}}{T}^{k-1}{T}_k^{(j)}\cdots T_k^{(j)}1{\psi }_{\mathcal{S}}\rangle$$

Note that the scalar product on the r.h.s. is non-negative, as it is a probability. It now follows that:

$${\displaystyle \sum _j|\langle {\psi }_{\mathcal{S}},T^{k-1}{T}_k^{(j)}\cdots T_k^{(j)}{\mathrm{\chi }}_s\rangle |}\le \Vert A_s\Vert {\displaystyle \sum _j\langle {\psi }_{\mathcal{S}}{T}^{k-1}{T}_k^{(j)}\cdots T_k^{(j)}{\psi }_{\mathcal{S}}\rangle }$$

the latter sum being P (A). ■

4.2. Proof of Theorem 3

We have $P(X_n\in S\mathrm{eventually})=\langle {\psi }_{\mathcal{S}},\mathrm{\Pi }{\mathrm{\Pi }}_{\mathcal{S}}{\psi }_{\mathcal{S}}\rangle$, where Π and Π_S are the Riesz projections of T and T_S respectively, associated with Point 1; see Lemma 7. We know that T_S = ω_in(E_S) + O(||V||). Since E_S is an orthogonal projection and ω_in is a state, ω_in(E_S) ≠ 1 means ω_in(E_S) < 1. Thus, for small enough V, T_S does not have an eigenvalue at Point 1, and consequently, Π_S = 0. ■

4.3. Proof of Theorem 4

The proof is in two parts. First, we show that the expectation of f_m has the indicated limit. Then, we upgrade this convergence to almost everywhere convergence, using the decay of correlations; Theorem 1.

Step 1. We show the convergence of the expectation of the frequency,

$$F_m:\mathbb{E}[f_m]=\underset{n\to \infty }{\lim }\frac{1}{n}\mathbb{E}[\mathrm{number}\mathrm{of}k\in \{1,\dots ,n\}\mathrm{s}.\mathrm{t}.X_k=m]$$

(54)

Let m ∈ spec(M) be fixed, and define χ by χ(x) = 1 if x = m and χ(x) = 0, otherwise. Then, the expectation in Equation (54) equals:

$$\begin{array}{l}{\displaystyle \sum _{m_1,\dots ,m_n}\left[{\displaystyle \sum _{j=1}^n\mathrm{\chi }(m_j)}\right]}P(X_1=m_1,\dots ,X_n=m_n)\\ ={\displaystyle \sum _{j=1}^n{\displaystyle \sum _{m_k,k\ne j}P(X_1=m_1,\dots ,X_j=m,\dots ,X_n=m_n)}}\\ ={\displaystyle \sum _{j=1}^n\langle {\psi }_{\mathcal{S}}{T}^{j-1}{T}_m{\psi }_{\mathcal{S}}\rangle }\end{array}$$

We have $\frac{1}{n}{\displaystyle {\sum }_{j=0}^{n-1}{T}^j\to \prod }$ as n → ∞ (the ergodic projection that equals the Riesz projection of T associated with one; see Appendix A) and, due to Condition A (see Equation (27)), $\prod =|{\psi }_{\mathcal{S}}\rangle \langle {\psi }_{\mathcal{S}}^{*}|$, where ${\psi }_{\mathcal{S}}^{*}$ is the unique vector in ${\mathscr{H}}_{\mathcal{S}}$ satisfying $T*{\psi }_{\mathcal{S}}^{*}={\psi }_{\mathcal{S}}^{*}$ and $\langle {\psi }_{\mathcal{S}}^{*},{\psi }_{\mathcal{S}}\rangle =1$. Note that the asymptotic state ω₊ of the dynamics without scattering is ${\omega }_{+}(\cdot )=\langle {\psi }_{\mathcal{S}}^{*},\cdot {\psi }_{\mathcal{S}}\rangle$. We have:

$$F_m=\langle {\psi }_{\mathcal{S}},\prod T_m{\psi }_{\mathcal{S}}\rangle =\langle {\psi }_{\mathcal{S}}^{*},T_m{\psi }_{\mathcal{S}}\rangle$$

(55)

Recall that T_m = PB*Be^iτKE_mP, so that:

$$\begin{array}{l}{F}_m=\langle {\psi }_{\mathcal{S}}^{*}\oplus {\psi }_{\mathcal{P}},(B*B{\mathrm{e}}^{i\tau K}{E}_m){\psi }_{\mathcal{S}}\oplus {\psi }_{\mathcal{P}}\rangle \\ =\langle {\psi }_{\mathcal{S}}^{*}\oplus {\psi }_{\mathcal{P}},(B*B{\mathrm{e}}^{i\tau L}{E}_m{e}^{-i\tau L}){\psi }_{\mathcal{S}}\oplus {\psi }_{\mathcal{P}}\rangle \\ ={\omega }_{+}\oplus {\omega }_{\mathrm{in}}({\mathrm{e}}^{i\tau H}{E}_m{e}^{-i\tau H})\end{array}$$

(56)

where $L=L_{\mathcal{S}}+L_{\mathcal{P}}+\lambda V$; see Equation (11). This finishes the first step of the proof, showing that:

$$\underset{n\to \infty }{\lim }\frac{1}{n}\mathbb{E}[\text{number of}k\in \left\{1,\dots ,n\right\}\mathrm{s}.\mathrm{t}.X_k=m]={\omega }_{+}\oplus {\omega }_{\mathrm{in}}({\mathrm{e}}^{i\tau H}{E}_m{e}^{-i\tau H})$$

(57)

Step 2. We upgrade Equation (57) to almost sure convergence, by using a classical fourth moment method. Introduce the random variable:

$$Z_n=\{\text{number of}k\in \left\{1,\dots ,n\right\}\mathrm{s}.\mathrm{t}.X_k=m\}$$

We are going to show below that, for any ϵ > 0,

$${\displaystyle \sum _{n\ge 1}P({|Z_n/n-\mathbb{E}[Z_n]/n|}^2\ge ϵ)<\infty }$$

(58)

Then, by the (first) Borel–Cantelli lemma, $P\left(|Z_n/n-\mathbb{E}\left[Z_n\right]/n{|}^2\ge ϵ\mathrm{i}.\mathrm{o}.\right)=0$, i.e., there is a set Σ of measure one, s.t. for all ω ∈ Σ, there exists a k with:

$${|Z_n/n-\mathbb{E}[Z_n]/n|}^2<\epsilon ,\forall n\ge k$$

(59)

From (57), we know that $\mathbb{E}[Z_n]/n$ converges to ν := ω₊ ⊗ ω_in(e^iτH E_me^−iτH), so Equation (59) implies that Z_n/n converges to ν almost everywhere. It remains to prove the summability (58). By Chebyshev’s inequality,

$$P({|Z_n/n-\mathbb{E}[Z_n]/n|}^2\ge \epsilon )\le \frac{1}{{\epsilon }^2{n}^4}\mathbb{E}[{|Z_n-\mathbb{E}[Z_n]|}^4]$$

(60)

We get an upper bound on the r.h.s. Set:

$$Z_n-\mathbb{E}[Z_n]={\displaystyle \sum _{i=1}^n\{\chi (X_i=m)-P(X_i=m)\}=:{\displaystyle \sum _{i=1}^n{Y}_i}}$$

so that $\mathbb{E}\left[Y_i\right]=0$. Here, χ(X_i = m) = 1 if X_i = m and χ(X_i = m) = 0, otherwise. We have:

$$\mathbb{E}[{|Z_n-\mathbb{E}[Z_n]|}^4]\le {\displaystyle \sum _{i,j,k,l=1}^n\mathbb{E}[|Y_i||Y_j||Y_k||Y_l|]}$$

(61)

The idea is to control the sum by using that if the indices i, j, k, l are far apart from each other, then the expectation is small due to the decay of correlations. Thus, only a few terms in the sum contribute to its value. Let Λ ≥ 0 be a given integer “length” scale. All combinations of values of the four indices i, j, k, l belong to exactly one of the following five cases: (1) all indices lie inside an interval of length Λ; (2) three indices lie within Λ, the fourth does not; (3) two pairs of indices are separated by more than Λ, but within each pair, the indices are apart at most by Λ; (4) one pair lies within Λ; the other two indices are apart from each other and from the close pair by more than Λ; and (5) all four indices are apart from each other by at least Λ. Let n_j, j = 1, …, 5, be the number of terms in the sum that satisfy Cases (1) to (5) above. We have n₁ ≤ 4!nΛ³, n₂ ≤ 4!n²Λ², n₃ ≤ 4!n²Λ², n₄ ≤ 4!n³Λ, n₅ ≤ 4!n⁴. Next, due to the separation of indices and Theorem 1, each term of Case (5) is of the form:

$$\mathbb{E}[|Y_i||Y_j||Y_k||Y_l|]\le \mathbb{E}[|Y_i|]\mathbb{E}[|Y_j|]\mathbb{E}[|Y_k|]\mathbb{E}[|Y_l|]+O(e^{-{\gamma }^{\prime }\mathrm{\Lambda }})=O(e^{-{\gamma }^{\prime }\mathrm{\Lambda }})$$

as $\mathbb{E}\left[|Y_i|\right]=0$ for all i. Similarly, each term of Cases (4), (3) and (2) has the same upper bound. Each term of Case (1) is bounded above by one. We conclude that:

$$\mathbb{E}[{|Z_n-\mathbb{E}[Z_n]|}^4]\le C(n_2+n_3+n_4+n_5)e^{-{\gamma }^{\prime }\mathrm{\Lambda }}+n_1$$

Choose now Λ = n^α, with 0 < α < 2/3. Then, using the above bounds on n_j, we have that $n^{-4}\mathbb{E}\left[{|Z_n-\mathbb{E}\left[Z_n\right]|}^4\right]$ is summable over n ≥ 1, i.e., by Equation (60), the inequality Equation (58) holds. This completes the proof of Theorem 4.

4.4. Proof of Theorem 5

We proceed as in the proof of Theorem 4. We have:

$$\begin{array}{l}\mathbb{E}[{\overline{X}}_n]=\frac{1}{n}{\displaystyle \sum _{m_1,\dots ,m_n}(m_1+\cdots +m_n)P(X_1=m_1,\dots ,X_n=m_n)}\\ =\frac{1}{n}{\displaystyle \sum _{j=1}^n\langle {\psi }_{\mathcal{S}}{T}^{j-1}PB*B{\mathrm{e}}^{i\tau L}MP{\psi }_{\mathcal{P}}\rangle }\end{array}$$

(62)

Using that ${\lim }_{n\to \infty }\frac{1}{n}{\displaystyle {\sum }_{j=0}^{n-1}{T}^j}=\prod$ and proceeding as in Equation (56), we arrive at ${\lim }_{n\to \infty }\mathbb{E}[{\overline{X}}_n-{\mu }_{\infty }]=0$. By proceeding as in Step 2 of the proof of Theorem 4 above, one upgrades this to almost everywhere convergence, ${\overline{X}}_n-{\mu }_{\infty }$ a.e.

The result follows.