Samir Adly
email: samir.adly@unilim.fr

Loïc Bourdin
email: loic.bourdin@unilim.fr

Fabien Caubet
email: fabien.caubet@math.univ-toulouse.fr

On a decomposition formula for the proximal operator of the sum of two convex functions

Keywords: convex analysis, proximal operator, Douglas-Rachford operator, Forward-Backward operator AMS Classification: 46N10, 47N10, 49J40, 49Q12

The main result of the present theoretical paper is an original decomposition formula for the proximal operator of the sum of two proper, lower semicontinuous and convex functions f and g. For this purpose, we introduce a new operator, called f -proximal operator of g and denoted by prox f g , that generalizes the classical notion. Then we prove the decomposition formula prox f +g = prox f • prox f g . After collecting several properties and characterizations of prox f g , we prove that it coincides with the fixed points of a generalized version of the classical Douglas-Rachford operator. This relationship is used for the construction of a weakly convergent algorithm that computes numerically this new operator prox f g , and thus, from the decomposition formula, allows to compute numerically prox f +g . It turns out that this algorithm was already considered and implemented in previous works, showing that prox f g is already present (in a hidden form) and useful for numerical purposes in the existing literature. However, to the best of our knowledge, it has never been explicitly expressed in a closed formula and neither been deeply studied from a theoretical point of view. The present paper contributes to fill this gap in the literature. Finally we give an illustration of the usefulness of the decomposition formula in the context of sensitivity analysis of linear variational inequalities of second kind in a Hilbert space.

Introduction, notations and basics 1.Introduction

The proximal operator (also known as proximity operator ) of a proper, lower semicontinuous, convex and extended-real-valued function was first introduced by J.-J. Moreau in 1962 in [START_REF] Moreau | Fonctions convexes duales et points proximaux dans un espace hilbertien[END_REF][START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF] and can be viewed as an extension of the projection operator on a nonempty closed and convex subset of a Hilbert space. This wonderful tool plays an important role, from both theoretical and numerical points of view, in applied mathematics and engineering sciences. This paper fits within the wide theoretical literature dealing with the proximal operator. For the rest of this introduction, we use standard notations of convex analysis. For the reader who is not acquainted with convex analysis, we refer to Section 1.2 for notations and basics.

Motivations from a sensitivity analysis. The present work was initially motivated by the sensitivity analysis, with respect to a nonnegative parameter t ≥ 0, of a parameterized linear variational inequality of second kind in a Hilbert space H, with a corresponding function h ∈ Γ 0 (H), where Γ 0 (H) is the set of proper, lower semicontinuous and convex functions from H into R∪{+∞}. More precisely, for all t ≥ 0, we consider the problem of finding u(t) ∈ H such that u(t), z -u(t) + h(z) -h(u(t)) ≥ r(t), z -u(t) ,

for all z ∈ H, where r : R + → H is assumed to be given and smooth enough. In that framework, the solution u(t) ∈ H (which depends on the parameter t) can be expressed in terms of the proximal operator of h denoted by prox h . Precisely it holds that u(t) = prox h (r(t)) for all t ≥ 0. As a consequence, the differentiability of u(•) at t = 0 is strongly related to the regularity of prox h . If h is a smooth function, one can easily compute (from the classical inverse mapping theorem for instance) the differential of prox h , and then the sensitivity analysis can be achieved. In that smooth case, note that the variational inequality (1) can actually be reduced to an equality. On the other hand, if h = ι K is the indicator function of a nonempty closed and convex subset K ⊂ H, then prox h = proj K is the classical projection operator on K. In that case, the work of F. Mignot in [10, Theorem 2.1 p.145] (see also the work of A. Haraux in [8, Theorem 2 p.620]) provides an asymptotic expansion of prox h = proj K and permits to obtain a differentiability result on u(•) at t = 0.

In a parallel work (in progress) of the authors on some shape optimization problems with unilateral contact and friction, the considered variational inequality [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF] involves the sum of two functions.

Precisely, h = f + g where f = ι K (K being a nonempty closed and convex set of constraints), and where g ∈ Γ 0 (H) is a smooth function (derived from the regularization of the friction functional in view of a numerical treatment). Despite the regularity of g, note that the variational inequality [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF] cannot be reduced to an equality due to the presence of the constraint set K. In that framework, in order to get an asymptotic expansion of prox h = prox f +g , a first and natural strategy would be to look for a convenient explicit expression of prox f +g in terms of prox f and prox g . Unfortunately, this theoretical question still remains an open challenge in the literature. Let us mention that Y.-L. Yu provides in [START_REF] Yu | On decomposing the proximal map[END_REF] some necessary and/or sufficient conditions on general functions f , g ∈ Γ 0 (H) under which prox f +g = prox f • prox g . Unfortunately, as underlined by the author himself, these conditions are very restrictive and are not satisfied in most of cases (see, e.g., [18, Example 2] for a counterexample).

Before coming to the main topic of this paper, we recall that a wide literature is already concerned with the sensitivity analysis of parameterized (linear and nonlinear) variational inequalities. We refer for instance to [START_REF] Bensoussan | Inéquations quasi-variationnelles dépendant d'un paramètre[END_REF][START_REF] Haraux | How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities[END_REF][START_REF] Qiu | Sensitivity analysis for variational inequalities[END_REF][START_REF] Shapiro | Sensitivity analysis of parameterized variational inequalities[END_REF] and references therein. The results in are considered in very general frameworks. We precise that our original objective was to look for a simple and compact formula for the derivative u (0) in the very particular case described above, that is, in the context of a linear variational inequality and with h = f + g where f is an indicator function and g is a smooth function. For this purpose, we were led to consider the proximal operator of the sum of two functions in Γ 0 (H), to introduce a new operator and finally to prove the results presented in this paper.

Introduction of the f -proximal operator and main result. Let us consider general functions f , g ∈ Γ 0 (H). In order to avoid trivialities, we will assume in the whole paper that dom(f ) ∩ dom(g) = ∅ when dealing with the sum f + g.

Section 2 is devoted to the introduction (see Definition 2.1) of a new operator prox f g : H ⇒ H called f -proximal operator of g and defined by

prox f g := I + ∂g • prox f -1 . (2)
This new operator can be seen as a generalization of prox g in the sense that, if f is constant for instance, then prox f g = prox g . More general sufficient (and necessary) conditions under which prox f g = prox g are provided in Propositions 2.13 and 2.15. We prove in Proposition 2.5 that the domain of prox f g satisfies D(prox f g ) = H if and only if ∂(f + g) = ∂f + ∂g. Note that prox f g is a set-valued operator a priori. We provide in Proposition 2.18 some sufficient conditions under which prox f g is single-valued. Some examples illustrate all the previous results throughout the section (see Examples 2.2, 2.3, 2.4, 2.7 and 2.17).

Finally, if the additivity condition ∂(f + g) = ∂f + ∂g is satisfied, the main result of the present paper (see Theorem 2.8) is the original decomposition formula

prox f +g = prox f • prox f g . (3)
It is well-known in the literature that obtaining a theoretical formula for prox f +g is not an easy task in general, even if prox f and prox g are known. We give a more precise description of the difficulty to obtain an easy computable formula of prox f +g in Appendix A, which claims that there is no closed formula, independent of f and g, allowing to write prox f +g as a linear combination of compositions of linear combinations of I, prox f , prox g , prox -1 f and prox -1 g . In the decomposition formula (3), it should be noted that the difficulty of computing prox f +g is only transferred to the computation of prox f g which is not an easier task. Note that other rewritings, which are not suitable for an easy computation of prox f +g neither, can be considered such as Relationship with the classical Douglas-Rachford operator. Recall that the proximal operator prox f +g is strongly related to the minimization problem argmin f + g, since the set of solutions is exactly the set of fixed points of prox f +g denoted by Fix(prox f +g ). In the sequel, we will assume that the above minimization problem admits at least one solution. The classical Douglas-Rachford operator, introduced in [START_REF] Douglas | On the numerical solution of heat conduction problems in two and three space variables[END_REF] and denoted here by DR f,g (see Section 3 for details), provides an algorithm x n+1 = DR f,g (x n ) that is weakly convergent to some x * ∈ H satisfying prox f (x * ) ∈ argmin f + g.

prox f +g = (prox -1 f + prox -1 g -I) -1 = (prox -1 2f + prox -1 2g ) -1
Even if the Douglas-Rachford algorithm is not a proximal point algorithm in general, in the sense that DR f,g is not equal to prox ϕ for some ϕ ∈ Γ 0 (H) in general, it is a very powerful tool since it allows to solve the above minimization problem, requiring only the knowledge of prox f and prox g . We refer to [2, Section 28.3 p.517] for more details.

Section 3 deals with the relations between the Douglas-Rachford operator DR f,g and the fproximal operator prox f g introduced in this paper. Precisely, we prove in Proposition 3.2 that

prox f g (x) = Fix DR f,g (x, •) ,
for all x ∈ H, where DR f,g (x, •) denotes a x-dependent generalization of the classical Douglas-Rachford operator DR f,g , in the sense that DR f,g (y) = DR f,g (prox f (y), y) for all y ∈ H. We refer to Section 3 for the precise definition of DR f,g (x, •) that only depends on the knowledge of prox f and prox g .

Let us show that the above statements, in particular the decomposition formula (3), allow to recover in a concise way the well-known inclusion

prox f (Fix (DR f,g )) ⊂ argmin f + g = Fix(prox f +g ). (4) Indeed, if x * ∈ Fix(DR f,g ), then x * ∈ Fix(DR f,g (prox f (x * ), •)) = prox f g (prox f (x * )).
From the decomposition formula (3), we conclude that

prox f (x * ) = prox f • prox f g (prox f (x * )) = prox f +g (prox f (x * )).
This proof of only few lines is an illustration of the theoretical interest of the decomposition formula [START_REF] Bensoussan | Inéquations quasi-variationnelles dépendant d'un paramètre[END_REF]. Note that the above inclusion (4) is, as well-known, an equality (see Section 3.3 and Proposition 3.8 for details).

The f -proximal operator prox f g introduced in this paper is also of interest from a numerical point of view. Indeed, if x ∈ D(prox f g ), we prove in Theorem 3.3 that the fixed-point algorithm y k+1 = DR f,g (x, y k ), denoted by (A 1 ), weakly converges to some y * ∈ prox f g (x). Moreover, if the additivity condition ∂(f + g) = ∂f + ∂g is satisfied, we get from the decomposition formula (3) that prox f (y * ) = prox f +g (x). In that situation, we conclude that Algorithm (A 1 ) allows to compute numerically prox f +g (x) with the only knowledge of prox f and prox g . It turns out that Algorithm (A 1 ) was already considered, up to some translations, and implemented in previous works (see, e.g., [START_REF] Combettes | Dualization of signal recovery problems[END_REF]Algorithm 3.5]), showing that the f -proximal operator prox f g is already present (in a hidden form) and useful for numerical purposes in the existing literature. However, to the best of our knowledge, it has never been explicitly expressed in a closed formula such as (2) and neither been deeply studied from a theoretical point of view. The present paper contributes to fill this gap in the literature. Some other applications and forthcoming works. Section 4 can be seen as a conclusion of the paper. Its aim is to provide a glimpse of some other applications of our main result (Theorem 2.8) and to raise open questions for forthcoming works. This section is splitted into two parts.

In Section 4.1 we consider the framework where f , g ∈ Γ 0 (H) with g differentiable on H. In that framework, we prove from the decomposition formula (3) that prox f +g is related to the classical Forward-Backward operator (see [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]Section 10.3 p.191] for details) denoted by FB f,g . Precisely, we prove in Proposition 4.1 that

prox f +g (x) = Fix FB f,g (x, •) ,
for all x ∈ H, where FB f,g (x, •) denotes a x-dependent generalization of the classical Forward-Backward operator FB f,g . We refer to Section 4.1 for the precise definition of FB f,g (x, •) that only depends on the knowledge of prox f and ∇g. From this point, one can develop a similar strategy as in Section 3. Precisely, for all x ∈ H, one can consider the algorithm y k+1 = FB f,g (x, y k ), denoted by (A 2 ), in order to compute numerically prox f +g (x), with the only knowledge of prox f and ∇g. Convergence proof (under some assumptions on f and g) of (A 2 ) should be the topic of a future work.

In Section 4.2 we turn back to our initial motivation, namely the sensitivity analysis of the parameterized variational inequality [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF]. Precisely, under some assumptions (see Proposition 4.3 for details), we derive from the decomposition formula (3) that if

u(t) := prox f +g (r(t)),
for all t ≥ 0, where f := ι K (where K ⊂ H is a nonempty closed convex subset) and where g ∈ Γ 0 (H) and r : R + → H are smooth enough, then

u (0) = prox ϕ f +ψg (r (0)),
where ϕ f := ι C (where C is a nonempty closed convex subset of H related to K) and where ψ g (x) := 1 2 D 2 g(u(0))(x), x for all x ∈ H. It should be noted that the assumptions of Proposition 4.3 are quite restrictive, raising open questions about their relaxations (see Remark 4.5). This also should be the subject of a forthcoming work.

Notations and basics

In this section we introduce some notations available throughout the paper and we recall some basics of convex analysis. We refer to standard books like [START_REF] Bauschke | Convex analysis and monotone operator theory in Hilbert spaces[END_REF][START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF][START_REF] Rockafellar | Convex analysis[END_REF] and references therein.

Let H be a real Hilbert space and let •, • (resp. • ) be the corresponding scalar product (resp. norm). For every subset S of H, we denote respectively by int(S) and cl(S) its interior and its closure. In the sequel we denote by I : H → H the identity operator and by L x : H → H the affine operator defined by L x (y) := x -y, for all x, y ∈ H.

For a set-valued map A : H ⇒ H, the domain of A is given by

D(A) := {x ∈ H | A(x) = ∅}.
We denote by A -1 : H ⇒ H the set-valued map defined by

A -1 (y) := {x ∈ H | y ∈ A(x)}, for all y ∈ H. Note that y ∈ A(x) if and only if x ∈ A -1 (y), for all x, y ∈ H. The range of A is given by R(A) := {y ∈ H | A -1 (y) = ∅} = D(A -1 ).
We denote by Fix(A) the set of all fixed points of A, that is, the set given by

Fix(A) := {x ∈ H | x ∈ A(x)}.
Finally, if A(x) is a singleton for all x ∈ D(A), we say that A is single-valued.

For all extended-real-valued functions g : H → R ∪ {+∞}, the domain of g is given by dom(g

) := {x ∈ H | g(x) < +∞}.
Recall that g is said to be proper if dom(g) = ∅.

Let g : H → R∪{+∞} be a proper extended-real-valued function. We denote by g * : H → R∪{+∞} the conjugate of g defined by g * (y

) := sup z∈H { y, z -g(z)},
for all y ∈ H. Clearly g * is lower semicontinuous and convex.

We denote by Γ 0 (H) the set of all extended-real-valued functions g : H → R ∪ {+∞} that are proper, lower semicontinuous and convex. If g ∈ Γ 0 (H), we recall that g * ∈ Γ 0 (H) and that the Fenchel-Moreau equality g * * = g holds. For all g ∈ Γ 0 (H), we denote by ∂g : H ⇒ H the Fenchel-Moreau subdifferential of g defined by

∂g(x) := {y ∈ H | y, z -x ≤ g(z) -g(x), ∀z ∈ H},
for all x ∈ H. It is easy to check that ∂g is a monotone operator and that, for all x ∈ H, 0 ∈ ∂g(x) if and only if x ∈ argmin g. Moreover, for all x, y ∈ H, it holds that y ∈ ∂g(x) if and only if x ∈ ∂g * (y). Recall that, if g is differentiable on H, then ∂g(x) = {∇g(x)} for all x ∈ H.

Let A : H → H be a single-valued operator defined everywhere on H, and let g ∈ Γ 0 (H). We denote by VI(A, g) the variational inequality which consists of finding y ∈ H such that -A(y) ∈ ∂g(y), or equivalently, A(y), z -y + g(z) -g(y) ≥ 0, for all z ∈ H. Then we denote by Sol VI (A, g) the set of solutions of VI(A, g). Recall that if A is Lipschitzian and strongly monotone, then VI(A, g) admits a unique solution, i.e. Sol VI (A, g) is a singleton.

Let g ∈ Γ 0 (H). The classical proximal operator of g is defined by

prox g := (I + ∂g) -1 .
Recall that prox g is a single-valued operator defined everywhere on H. Moreover, it can be characterized as follows:

prox g (x) = argmin g + 1 2 • -x 2 = Sol VI (-L x , g),
for all x ∈ H. It is also well-known that Fix(prox g ) = argmin g.

The classical Moreau's envelope M g : H → R of g is defined by

M g (x) := min g + 1 2 • -x 2 ,
for all x ∈ H. Recall that M g is convex and differentiable on H with ∇M g = prox g * . Let us also recall the classical Moreau's decompositions

prox g + prox g * = I and M g + M g * = 1 2 • 2 .
Finally, it is well-known that if g = ι K is the indicator function of a nonempty closed and convex subset K of H, that is, ι K (x) = 0 if x ∈ K and ι K (x) = +∞ if not, then prox g = proj K , where proj K denotes the classical projection operator on K.

2 The f -proximal operator

Definition and main result

Let f , g ∈ Γ 0 (H). In this section we introduce (see Definition 2.1) a new operator denoted by prox f g , generalizing the classical proximal operator prox g . Assuming that dom(f ) ∩ dom(g) = ∅, and under the additivity condition ∂(f + g) = ∂f + ∂g, we prove in Theorem 2.8 that prox f +g can be written as the composition of prox f with prox f g .

Definition 2.1 (f -proximal operator). Let f , g ∈ Γ 0 (H). The f -proximal operator of g is the set-valued map prox f g : H ⇒ H defined by

prox f g := (I + ∂g • prox f ) -1 . (5)
Note that prox f g can be seen as a generalization of prox g since prox c g = prox g for all constant c ∈ R.

Example 2.2. Let us assume that H = R. We consider f = ι [-1,1] and g(x) = |x| for all x ∈ R.

In that case we obtain that ∂g • prox f = ∂g and thus prox f g = prox g .

Example 2.2 provides a simple situation where prox f g = prox g while f is not constant. We provide in Propositions 2.13 and 2.15 some general sufficient (and necessary) conditions under which prox f g = prox g .

Example 2.3. Let us assume that H = R. We consider f = ι {0} and g(x) = |x| for all x ∈ R.

In that case we obtain that ∂g 1 for graphical representations of prox g and prox f g in that case. Example 2.4. Let us assume that H = R. We consider f (x) = g(x) = |x| for all x ∈ R. In that case we obtain that ∂g 2 for graphical representations of prox g and prox f g in that case.

• prox f (x) = [-1, 1] for all x ∈ R. As a consequence prox f g (x) = [x -1, x + 1] for all x ∈ R. See Figure
• prox f (x) = -1 for all x < -1, ∂g • prox f (x) = [-1, 1] for all x ∈ [-1, 1] and ∂g • prox f (x) = 1 for all x > 1. As a consequence prox f g (x) = x + 1 for all x ≤ -2, prox f g (x) = [-1, x + 1] for all x ∈ [-2, 0], prox f g (x) = [x -1, 1] for all x ∈ [0, 2] and prox f g (x) = x -1 for all x ≥ 2. See Figure
Examples 2.3 and 2.4 provide simple illustrations where prox f g is not single-valued. In particular it follows that prox f g cannot be written as a proximal operator prox ϕ for some ϕ ∈ Γ 0 (H). We provide in Proposition 2.18 some sufficient conditions under which prox f g is single-valued. Moreover, Examples 2.3 and 2.4 provide simple situations where ∂g • prox f is not a monotone operator. As a consequence, it may be possible that D(prox f g ) H. In the next proposition, a necessary and sufficient condition under which D(prox f g ) = H is derived.

∂(f + g) = ∂f + ∂g, ( C 1 )
is satisfied.

Proof. We first assume that ∂(f + g) = ∂f + ∂g. Let x ∈ H. Defining w = prox f +g (x) ∈ H, we obtain that x ∈ w + ∂(f + g)(w) = w + ∂f (w) + ∂g(w). Thus, there exist w f ∈ ∂f (w) and w g ∈ ∂g(w) such that x = w + w f + w g . We define y = w + w f ∈ w + ∂f (w). In particular we have w = prox f (y). Moreover we obtain x = y + w g ∈ y + ∂g(w) = y + ∂g(prox f (y)). We conclude that y ∈ prox f g (x). Without any additional assumption and directly the definition of the subdifferential, one can easily see that the inclusion ∂f (w) + ∂g(w) ⊂ ∂(f + g)(w) is always satisfied for every w ∈ H. Now let us assume that D(prox f g ) = H. Let w ∈ H and let z ∈ ∂(f + g)(w). We consider x = w + z ∈ w + ∂(f + g)(w). In particular it holds that w = prox f +g (x). Since D(prox f g ) = H, there exists y ∈ prox f g (x) and thus it holds that x ∈ y + ∂g(prox f (y)). Moreover, since y ∈ prox f (y) + ∂f (prox f (y)), we get that x ∈ prox f (y) + ∂f (prox f (y)) + ∂g(prox f (y)) ⊂ prox f (y) + ∂(f +g)(prox f (y)). Thus it holds that prox f (y) = prox f +g (x) = w. Moreover, since x ∈ prox f (y)+ ∂f (prox f (y)) + ∂g(prox f (y)), we obtain that x ∈ w + ∂f (w) + ∂g(w). We have proved that z = x -w ∈ ∂f (w) + ∂g(w). This concludes the proof.

In most of the present paper, we will assume that Condition (C 1 ) is satisfied. It is not our aim here to discuss the weakest qualification condition ensuring that condition. A wide literature already deals with this topic (see, e.g., [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF][START_REF] Ekeland | Analyse convexe et problèmes variationnels[END_REF][START_REF] Rockafellar | Convex analysis[END_REF]). However, we recall in the following remark the classical sufficient condition of Moreau-Rockafellar under which Condition (C 1 ) holds true (see, e.g., [2, Corollary 16.48 p.277]), and we provide a simple example where Condition (C 1 ) does not hold and D(prox f g ) H.

Remark 2.6 (Moreau-Rockafellar theorem). Let f , g ∈ Γ 0 (H) such that dom(f )∩int(dom(g)) = ∅.

Then ∂(f + g) = ∂f + ∂g.

Example 2.7. Let us assume that H = R. We consider f = ι R -and g(x) = ι R + (x) -√ x for all x ∈ R. In that case, one can easily check that dom(f ) ∩ dom

(g) = {0} = ∅, ∂f (0) + ∂g(0) = ∅ R = ∂(f + g)(0) and D(prox f g ) = ∅ H.
We are now in position to state and prove the main result of the present paper.

Theorem 2.8. Let f , g ∈ Γ 0 (H) such that dom(f ) ∩ dom(g) = ∅. If ∂(f + g) = ∂f + ∂g, then the decomposition formula prox f +g = prox f • prox f g ( 6
)
holds true. In other words, for every x ∈ H, we have prox f +g (x) = prox f (z) for all z ∈ prox f g (x).

Proof. Let x ∈ H and let y ∈ prox f g (x) constructed as in the first part of the proof of Proposition 2.5. In particular it holds that prox f (y) = prox f +g (x). Let z ∈ prox f g (x). We know that x -y ∈ ∂g(prox f (y)) and x -z ∈ ∂g(prox f (z)). Since ∂g is a monotone operator, we obtain that

(x -y) -(x -z), prox f (y) -prox f (z) ≥ 0.
From the cocoercivity (see for instance [2, Definition 4.10 p.72]) of the proximal operator, we obtain that 0

≥ y -z, prox f (y) -prox f (z) ≥ prox f (y) -prox f (z) 2 ≥ 0.
We deduce that prox f (z) = prox f (y) = prox f +g (x). The proof is complete.

Remark 2.9. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅ and such that ∂(f + g) = ∂f + ∂g and let x ∈ H. Theorem 2.8 states that, even if prox f g (x) is not a singleton, all elements of prox f g (x) has the same value through the proximal operator prox f , and this value is equal to prox f +g (x).

Remark 2.10. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅. Note that the additivity condition ∂(f + g) = ∂f + ∂g is not only sufficient, but also necessary for the validity of the equality prox f +g = prox f • prox f g . Indeed, from Proposition 2.5, if ∂f + ∂g ∂(f + g), then there exists x ∈ H such that prox f g (x) = ∅ and thus prox f +g (x) = prox f • prox f g (x).

Remark 2.11. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅ and such that ∂(f + g) = ∂f + ∂g. From Theorem 2.8, we deduce that R(prox f +g ) ⊂ R(prox f ) ∩ R(prox g ). If the additivity condition ∂(f + g) = ∂f + ∂g is not satisfied, this remark does not hold true anymore. Indeed, with the framework of Example 2.7, we have R(prox f +g ) = {0} while 0 / ∈ R(prox g ).

Example 2.12. Following the idea of Y.-L. Yu in [18, Example 2], let us consider H = R and f (x) = 1 2 x 2 for all x ∈ R. Since prox γf = 1 1+γ I for all γ ≥ 0 and prox f f = 2 3 I, we retrieve that

1 3 I = prox 2f = prox f +f = prox f • prox f f = 1 3 I = 1 4 I = prox f • prox f ,
which illustrates Theorem 2.8.

Properties

Let f , g ∈ Γ 0 (H). We know that prox f g is a generalization of prox g in the sense that prox f g = prox g if f is constant for instance. In the next proposition, our aim is to provide more general sufficient (and necessary) conditions under which prox f g = prox g . We will base our discussion on the following conditions:

∀x ∈ H, ∂g(x) ⊂ ∂g(prox f (x)), (C 2 ) ∀x ∈ H, ∂g(prox f (x)) ⊂ ∂g(x). (C 3 )
Note that Condition (C 2 ) has been introduced by Y.-L. Yu in [START_REF] Yu | On decomposing the proximal map[END_REF] as a sufficient condition under which prox

f +g = prox f • prox g . Proposition 2.13. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅. (i) If Condition (C 2 ) is satisfied, then prox g (x) ∈ prox f g (x) for all x ∈ H.
(ii) If Conditions (C 1 ) and (C 3 ) are satisfied, then prox f g (x) = prox g (x) for all x ∈ H.

In both cases, Condition (C 1 ) is satisfied and the equality prox f +g = prox f • prox g holds true.

Proof. Let x ∈ H. If Condition (C 2 ) is satisfied, considering y = prox g (x), we get that x ∈ y + ∂g(y) ⊂ y + ∂g(prox f (y)) and thus y ∈ prox f g (x). In particular, it holds that D(prox f g ) = H and thus Condition (C 1 ) is satisfied from Proposition 2.5. Secondly, if Conditions (C 1 ) and (C 3 ) are satisfied, then D(prox f g ) = H from Proposition 2.5. Considering y ∈ prox f g (x), we get that x ∈ y + ∂g(prox f (y)) ⊂ y + ∂g(y) and thus y = prox g (x). The last assertion of Proposition 2.13 directly follows from Theorem 2.8.

In the first item of Proposition 2.13 and if prox f g is set-valued, we are in the situation where prox g is a selection of prox f g . Proposition 2.15 specifies this selection in the case where ∂(f + g) = ∂f + ∂g.

Lemma 2.14. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅. Then prox f g (x) is a nonempty closed and convex subset of H for all x ∈ D(prox f g ).

Proof. The proof of Lemma 2.14 is provided after the proof of Proposition 3.2 (required).

Proposition 2.15. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅ and such that ∂(f

+ g) = ∂f + ∂g and let x ∈ H. If prox g (x) ∈ prox f g (x), then prox g (x) = proj prox f g (x) (prox f +g (x)).
Proof. If prox g (x) ∈ prox f g (x), then x ∈ D(prox f g ) and thus prox f g (x) is a nonempty closed and convex subset of H from Lemma 2.14. Let z ∈ prox f g (x). In particular we have prox f (z) = prox f +g (x) from Theorem 2.8. Using the fact that x -prox g (x) ∈ ∂g(prox g (x)) and x -z ∈ ∂g(prox f (z)) = ∂g(prox f +g (x)) together with the monotonicity of ∂g, we obtain that

prox f +g (x) -prox g (x), z -prox g (x) = prox f +g (x) -prox g (x), (x -prox g (x)) -(x -z) ≤ 0. Since prox g (x) ∈ prox f g (x)
, we conclude the proof from the characterization of proj prox f g (x) .

Remark 2.16. Let f = ι {ω} with ω ∈ H and let g ∈ Γ 0 (H) such that ω ∈ int(dom(g)). Hence the additivity condition ∂(f + g) = ∂f + ∂g is satisfied from Remark 2.6. From Remark 2.11 and since prox f = proj {ω} , we easily deduce that R(prox f +g ) = {ω}. Let x ∈ H such that prox g (x) ∈ prox f g (x). From Proposition 2.15 we get that prox g (x) = proj prox f g (x) (ω).

If moreover ω = 0, we deduce that prox g (x) is the particular selection that corresponds to the element of minimal norm in prox f g (x) (also known as the lazy selection). The following example is in this sense.

Example 2.17. Let us consider the framework of Example 2.3. In that case, Conditions (C 1 ) and (C 2 ) are satisfied. We deduce from Proposition 2.13 that prox g (x) ∈ prox f g (x) for all x ∈ R. From Remark 2.16, we conclude that prox g (x) is exactly the element of minimal norm in prox f g (x) for all x ∈ R. This result is clearly illustrated by the graphs of prox g and prox f g provided in Figure 1.

Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅ and such that ∂(f + g) = ∂f + ∂g. From Theorem 2.8, one can easily see that, if prox f is injective, then prox f g is single-valued. Since the injection of prox f is too restrictive, other sufficient conditions under which prox f g is single-valued are provided from Theorem 2.8 in the next proposition.

Proposition 2.18. Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅ and such that ∂(f + g) = ∂f + ∂g. If either ∂f or ∂g is single-valued, then prox f g is single-valued. Proof. Let x ∈ H and let z 1 , z 2 ∈ prox f g (x)
. From Theorem 2.8, it holds that prox f (z 1 ) = prox f (z 2 ) = prox f +g (x). If the operator ∂f is single-valued, we obtain that z 1 = prox f +g (x) + ∂f (prox f +g (x)) = z 2 . If the operator ∂g is single-valued, we get x -z 1 = ∂g(prox f (z 1 )) = ∂g(prox f (z 2 )) = x -z 2 and thus z 1 = z 2 .

Relations with the Douglas-Rachford operator

Let f , g ∈ Γ 0 (H). The Douglas-Rachford operator DR f,g : H → H associated to f and g is usually defined by DR f,g (y) := y -prox f (y) + prox g (2prox f (y) -y), for all y ∈ H. We refer for instance to [2, Section 28.3 p.517] where details can be found on this classical operator.

One aim of this section is to study the relations between the f -proximal operator prox f g introduced in this paper and the Douglas-Rachford operator DR f,g . For this purpose, we introduce an extension DR f,g : H × H → H of the classical Douglas-Rachford operator defined by DR f,g (x, y) := y -prox f (y) + prox g (x + prox f (y) -y), for all x, y ∈ H. Note that DR f,g (y) = DR f,g (prox f (y), y) for all y ∈ H, and that the definition of DR f,g only depends on the knowledge of prox f and prox g .

Several characterizations of prox f g

Let f , g ∈ Γ 0 (H). In this subsection, our aim is to derive several characterizations of prox f g in terms of solutions of variational inequalities, of minimization problems and of fixed point problems (see Proposition 3.2). [2, Proposition 24.8 p.416]) and from Moreau's decompositions.

Lemma 3.1. Let f , g ∈ Γ 0 (H). It holds that DR f,g (x, •) = prox g * •Lx • prox f * , for all x ∈ H. Proof. Let x ∈ H. Lemma 3.1 directly follows from the equality prox g * •Lx = L x • prox g * • L x (see
Proposition 3.2. Let f , g ∈ Γ 0 (H). It holds that prox f g (x) = Sol VI (prox f , g * • L x ) = argmin (M f * + g * • L x ) = Fix(DR f,g (x, •)),
for all x ∈ H.

Proof. In this proof we will use standard properties of convex analysis recalled in Section 1.2. Let x ∈ H. One can easily prove that ∂(g

* • L x ) = -∂g * • L x . For all y ∈ H, it holds that y ∈ prox f g (x) ⇐⇒ x -y ∈ ∂g(prox f (y)) ⇐⇒ prox f (y) ∈ ∂g * (x -y) ⇐⇒ -prox f (y) ∈ ∂(g * • L x )(y).
Moreover, since dom(M f * ) = H and from Remark 2.6, we have

-prox f (y) ∈ ∂(g * • L x )(y) ⇐⇒ 0 ∈ ∇M f * (y) + ∂(g * • L x )(y) ⇐⇒ 0 ∈ ∂(M f * + g * • L x )(y).
Finally,

-prox f (y) ∈ ∂(g * • L x )(y) ⇐⇒ prox f * (y) ∈ y + ∂(g * • L x )(y) ⇐⇒ y = prox g * •Lx • prox f * (y).
This concludes the proof from Lemma 3.1.

Proof of Lemma 2.14. Let x ∈ D(prox f g ). In particular prox f g (x) is not empty. From Proposition 3.2, we have prox

f g (x) = argmin (M f * + g * • L x ). Since M f * + g * • L x ∈ Γ 0 (H),
one can easily deduce that prox f g (x) is closed and convex.

A weakly convergent algorithm that computes prox f g numerically

Let f , g ∈ Γ 0 (H). In this section, our aim is to derive from Proposition 3.2 an algorithm, that depends only on the knowledge of prox f and prox g , allowing to compute numerically an element of prox f g (x) for all x ∈ D(prox f g ). We refer to Algorithm (A 1 ) in Theorem 3.3. Moreover, if the additivity condition ∂(f + g) = ∂f + ∂g is satisfied, it follows from Theorem 2.8 that Algorithm (A 1 ) is an algorithm allowing to compute numerically prox f +g (x) for all x ∈ H with the only knowledge of prox f and prox g . Theorem 3.3. Let f , g ∈ Γ 0 (H) and let x ∈ D(prox f g ) be fixed. Then, Algorithm (A 1 ) given by y 0 ∈ H,

y k+1 = DR f,g (x, y k ), ( A 1 )
weakly converges to an element y * ∈ prox f g (x). Moreover, if dom(f ) ∩ dom(g) = ∅ and ∂(f + g) = ∂f + ∂g, it holds that prox f (y * ) = prox f +g (x).

Proof. From Lemma 3.1, DR f,g (x, •) coincides with the composition of two firmly non-expansive operators, and thus of two non-expansive and • prox f * ) = ∅. We conclude from [2, Theorem 5.23 p.100] that Algorithm (A 1 ) weakly converges to a fixed point y * of DR f,g (x, •). From Proposition 3.2, it holds that y * ∈ prox f g (x). Finally, if dom(f )∩dom(g) = ∅ and ∂(f + g) = ∂f + ∂g, we conclude that prox f (y * ) = prox f +g (x) from Theorem 2.8. Remark 3.4. As already mentioned in the introduction, it turns out that Algorithm (A 1 ) was already considered, up to some translations, and implemented in previous works (see, e.g., the socalled dual forward-backward splitting in [4, Algorithm 3.5]), showing that the f -proximal operator prox f g is already present (in a hidden form) and useful for numerical purposes in the existing literature. However, to the best of our knowledge, it has never been explicitly expressed in a closed formula such as [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF] and neither been deeply studied from a theoretical point of view. Remark 3.5. Let us discuss with more details the relationship between the present work and the one proposed in [START_REF] Combettes | Dualization of signal recovery problems[END_REF]. Let x ∈ H. In [START_REF] Combettes | Dualization of signal recovery problems[END_REF]Proposition 3.4], the authors prove

that if v ∈ H is a solution to argmin (M f * • L x + g * ) (7)
then prox f (x -v) = prox f +g (x). Combining this result with Proposition 3.2 easily constitutes an alternative proof of the new decomposition formula (6) derived in this paper. Moreover, in [4, Algorithm 3.5], the authors consider the so-called dual forward-backward splitting given by

v k+1 = prox g * (v k + prox f (x -v k )),
which is related to Algorithm (A 1 ) by setting y k = x -v k . From Proposition 3.2, the present work points out that the operator given in [START_REF] Ekeland | Analyse convexe et problèmes variationnels[END_REF] actually coincides, up to a translation, with a generalization of the classical proximal operator, that is exactly the f -proximal operator introduced and studied from a theoretical point of view in this paper. In this section we also prove that Algorithm (A 1 ) actually coincides with a fixed-point algorithm associated to a generalized version of the classical Douglas-Rachford operator.

Relations with the Forward-Backward operator

Let f , g ∈ Γ 0 (H) such that g is differentiable on H. In that situation, note that the additivity condition ∂(f + g) = ∂f + ∂g is satisfied from Remark 2.6, and that Proposition 2.18 implies that prox f g is single-valued. In that framework, the classical Forward-Backward operator FB f,g : H → H associated to f and g is usually defined by FB f,g (y) := prox f (y -∇g(y)), for all y ∈ H. We refer to [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]Section 10.3 p.191] for more details. Let us introduce the extension FB f,g : H × H → H defined by

FB f,g (x, y) := prox f (x -∇g(y)),
for all x, y ∈ H. In particular, it holds that FB f,g (y) = FB f,g (y, y) for all y ∈ H. The following result follows from the decomposition formula (6) in Theorem 2.8.

Proposition 4.1. Let f , g ∈ Γ 0 (H) such that g is differentiable on H. Then prox f +g (x) = Fix(FB f,g (x, •)),
for all x ∈ H.

Proof. Let x ∈ H. Firstly, let z = prox f +g (x) and let y = prox f g (x). In particular, we have x = y + ∇g(prox f (y)). From the decomposition formula (6), we get that z

= prox f (y) = prox f (x - ∇g(prox f (y))) = prox f (x -∇g(z)) = FB f,g (x, z). Conversely, let z ∈ Fix(FB f,g (x, •)), that is, z = prox f (x -∇g(z)
). Considering y = x -∇g(z), we have z = prox f (y) and thus x = y + ∇g(prox f (y)), that is, y = prox f g (x). Finally, from the decomposition formula (6), we get that z = prox f • prox f g (x) = prox f +g (x).

From Proposition 4.1, we retrieve the following classical result (see, e.g., [2, Proposition 26.1]).

Proposition 4.2. Let f , g ∈ Γ 0 (H) such that g is differentiable on H. Then

argmin f + g = Fix(FB f,g ). Proof. Let x ∈ H. It holds that x ∈ argmin f + g ⇐⇒ x = prox f +g (x) ⇐⇒ x ∈ Fix(FB f,g (x, •)) ⇐⇒ x = FB f,g (x, x) = FB f,g (x) ⇐⇒ x ∈ Fix(FB f,g ).
The proof is complete.

Let f , g ∈ Γ 0 (H) such that g is differentiable on H. The classical Forward-Backward algorithm x n+1 = FB f,g (x n ) is a powerful tool since it provides an algorithm, only requiring the knowledge of prox f and ∇g, that weakly converges (under some conditions on g, see [2, Section 28.5 p.522] for details) to a fixed point of FB f,g , and thus to a minimizer of f + g.

From Proposition 4.1, and for all x ∈ H, one can consider the algorithm (potentially weakly convergent) given by y 0 ∈ H, y k+1 = FB f,g (x, y k ), (A 2 ) in order to compute numerically prox f +g (x), with the only knowledge of prox f and ∇g. Convergence proof (under some assumptions on f and g) of Algorithm (A 2 ) should be the topic of a future work.

Application to sensitivity analysis for variational inequalities

As a conclusion of the present paper, we turn back to our initial motivation, namely the sensitivity analysis, with respect to a nonnegative parameter t ≥ 0, of some parameterized linear variational inequalities of second kind in a real Hilbert space H. More precisely, for all t ≥ 0, we consider the variational inequality which consists of finding u(t) ∈ K such that u(t), z -u(t) + g(z) -g(u(t)) ≥ r(t), z -u(t) , for all z ∈ K, where K ⊂ H is a nonempty closed and convex set of constraints, and where g ∈ Γ 0 (H) and r : R + → H are assumed to be given and smooth enough. The above problem admits a unique solution given by u(t) = prox f +g (r(t)), where f = ι K is the indicator function of K.

Our aim is to provide from Theorem 2.8 a simple and compact formula for the derivative u (0) under some assumptions (see Proposition 4.3 for details). Following the idea of F. Mignot in [START_REF] Mignot | Contrôle dans les inéquations variationelles elliptiques[END_REF] (see also [8, (i) r is differentiable at t = 0;

(ii) g is twice differentiable on H;

(iii) O v(0) is dense in C v(0) ;

(iv) u is differentiable at t = 0;

for all f , g ∈ Γ 0 (H), where denotes finite composition of operators.

Then, let us consider the one-dimensional setting H = R with f (x) = g(x) = γ 2 x 2 for all x ∈ R and all γ ≥ 0. In that case prox f +g is the linear function with slope for all γ ≥ 0. We deduce from Equality (8) that

1 1 + 2γ = m i=1 λ i   n j=1 (b ij + c ij ) + a ij (1 + γ) + (d ij + e ij )(1 + γ) 2 1 + γ   ,
for all γ ≥ 0, where denotes now the classical finite product of real numbers. We get that

(1 + γ) n = (1 + 2γ) m i=1 λ i n j=1 (b ij + c ij ) + a ij (1 + γ) + (d ij + e ij )(1 + γ) 2 ,
for all γ ≥ 0. We easily deduce that the above polynomial equality can be extended to all γ ∈ R, and thus it raises a contradiction for γ = -1 2 .

Figure 1 :

 1 Figure 1: Example 2.3, graph of prox g in bold line, and graph of prox f g in gray.

Figure 2 :

 2 Figure 2: Example 2.4, graph of prox g in bold line, and graph of prox f g in gray.

Proposition 4 . 3 .

 43 Theorem 2 p.620]), we first introduce the following setsO v := {w ∈ H | ∃λ > 0, proj K (v) + λw ∈ K} ∩ [v -proj K (v)] ⊥ , C v := cl {w ∈ H | ∃λ > 0, proj K (v) + λw ∈ K} ∩ [v -proj K (v)] ⊥ ,for all v ∈ H, where ⊥ denotes the classical orthogonal of a set. Let v(t) := r(t)-∇g(u(t)) for all t ∈ R. If the following conditions are satisfied:

then the derivative u ( 0 )

 0 is given by u (0) = prox ϕ f +ψg (r (0)),where ϕ f := ι C v(0) and ψ g (x) := 1 2 D 2 g(u(0))(x), x for all x ∈ H.for all µ = (a, b, c, d, e) ∈ R 5 , all f , g ∈ Γ 0 (H) and all Hilbert spaces H. Let us assume by contradiction that for all Hilbert spaces H, there exist m, n ∈ N * , λ = (λ i ) i ∈ R m and (µ ij ) ij ∈ (R 5 ) m×n such that prox f +g =

  1 2 -averaged operators (see [2, Remark 4.34(iii) p.81]). Since x ∈ D(prox f g ), it follows from Proposition 3.2 and Lemma 3.1 that Fix(prox g * •Lx

  ij + c ij ) + a ij (1 + γ) + (d ij + e ij )(1 + γ) 2 1 + γ ,

	1 1+2γ and each P µij f,g is the linear
	function with slope
	(b

Remark 3.6. Let f , g ∈ Γ 0 (H) and let x ∈ D(prox f g ). Algorithm (A 1 ) consists in a fixed-point algorithm from the characterization given in Proposition 3.2 by prox f g (x) = Fix(DR f,g (x, •)). Actually, one can easily see that Algorithm (A 1 ) also coincides with the well-known Forward-Backward algorithm (see [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]Section 10.3 p.191] for details) from the characterization given in Proposition 3.2 by prox f g (x) = argmin (M f * + g * • L x ). Indeed, we recall that M f * is differentiable with ∇M f * = prox f . We also refer to Section 4.1 for a brief discussion about the Forward-Backward algorithm.

Recovering a classical result from the decomposition formula

Let f , g ∈ Γ 0 (H) with dom(f ) ∩ dom(g) = ∅ and such that ∂(f + g) = ∂f + ∂g. Our aim in this section is to recover in a simple way, with the help of the decomposition formula (6), the well-known equality argmin f + g = prox f (Fix(DR f,g )).

This result can be found for example in [2, Proposition 26.1].

Proof. Let y ∈ Fix(DR f,g ). Then y ∈ Fix(prox f g • prox f ) from Lemma 3.7. Thus y ∈ prox f g • prox f (y). From the decomposition formula (6), we get that prox f (y) = prox f +g (prox f (y)) and thus prox f (y) ∈ argmin f + g.

. The proof is complete.

Some other applications and forthcoming works

This section can be seen as a conclusion of the paper. Its aim is to provide a glimpse of some other applications of our main result (Theorem 2.8) and to raise open questions for forthcoming works. This section is splitted into two parts.

Proof. Note that v is differentiable at t = 0 with v (0) = r (0) -D 2 g(u(0))(u (0)).

Note that prox f g is single-valued from Proposition 2.18 and Remark 2.6. From the decomposition formula (6) in Theorem 2.8, one can easily obtain that v(t) = prox f g (r(t)), and thus

, we use the asymptotic expansion of F. Mignot [10, Theorem 2.1 p.145] and we obtain that

We deduce that

Since g is convex and since C v(0) is a nonempty closed convex subset of H, we deduce that ϕ f , ψ g ∈ Γ 0 (H). Moreover ∂(ϕ f + ψ g ) = ∂ϕ f + ∂ψ g from Remark 2.6 and prox

ψg is single-valued from Proposition 2.18. It also should be noted that ∇ψ g = D 2 g(u(0)). As a consequence, we have obtained that

ψg (r (0)). We conclude the proof from the equality u (0) = prox ϕ f (v (0)) and from Theorem 2.8. Remark 4.4. Proposition 4.3 provides an expression of u (0) in terms of the proximal operator of a sum of two proper, lower semicontinuous and convex functions. Hence, it could be numerically computed from Algorithm (A 1 ), requiring the knowledge of proj C v(0) and prox ψg . Alternatively, if the convergence is proved, one can also consider Algorithm (A 2 ) requiring the knowledge of proj C v(0) and ∇ψ g = D 2 g(u(0)).

Remark 4.5. The relaxations in special frameworks of the assumptions of Proposition 4.3 should be the subject of future works. In particular, it would be relevant to provide sufficient conditions ensuring that u is differentiable at t = 0. A promising idea in this sense is to invoke the concepts of twice epi-differentiability and proto-differentiability introduced by R.T. Rockafellar in [START_REF] Rockafellar | Maximal monotone relations and the second derivatives of nonsmooth functions[END_REF][START_REF] Rockafellar | Proto-differentiability of set-valued mappings and its applications in optimization[END_REF].

The application of Proposition 4.3 in the context of some shape optimization problems with unilateral contact and friction is the subject of a forthcoming research paper (work in progress).

A A nonexistence result for a closed formula

The aim of the present appendix is to prove that there is no closed formula, independent of f and g, allowing to write prox f +g as a linear combination of compositions of linear combinations of I, prox f , prox g , prox -1 f and prox -1 g . For this purpose, let us introduce the elementary operator P µ f,g : H ⇒ H defined by P µ f,g := a I + b prox f + c prox g + d prox -1 f + e prox -1 g ,


Emmanuel Lassalle

Caroline Champenois

Brian Stout

Vincent Debierre
email: vincent.debierre@mpi-hd.mpg.de

Thomas Durt

Conditions for anti-Zeno effect observation in free-space atomic radiative decay

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

One of the more peculiar features of quantum mechanics is that the measurement process can modify the evolution of a quantum system. The archetypes of this phenomenon are the quantum Zeno effect (QZE) and the quantum anti-Zeno effect (AZE) [1,2]. The QZE refers to the inhibition of the decay of an unstable quantum system due to frequent measurements [3], and was observed experimentally for the first time with trapped ions [4,5] and more recently in cold neutral atoms [6]. The opposite effect, where the decay is accelerated by frequent measurements, was first called the AZE in Ref. [7], and was discovered theoretically for spontaneous emission in cavities [8,9], and first observed in a tunneling experiment with cold atoms (along with the QZE) [10], and recently with a single superconducting qubit coupled to a waveguide cavity [11]. However, despite predictions that the AZE should be much more ubiquitous than the QZE in radiative decay processes [1], it has never been observed to our knowledge for atomic radiative decay (spontaneous emission) in free space.

Here, we investigate the case of hydrogen-like atoms, for which the exact expression of the coupling between the atom and the free radiative field (cf. [12,13]) allows us to derive an analytical expression for the measurement-modified decay rate. From this, we find that only non-electric-dipole transitions can exhibit the AZE in free space (i.e. non-dipole electric transitions and magnetic transitions of any multipolar order), which drastically limits the experimental possibilities to observe this effect. We start with a brief review of the general formal results about the measurement-modified decay rate in Sec. II, and we then apply, in Sec. III, this general framework to the case of electronic transi-tions in hydrogen-like atoms to derive an analytical expression of the measurement-modified decay rate in free space. Then, we discuss the experimental realizability of the described phenomenon in Sec. IV, and we identify a potential candidate: the electric quadrupole transition between D 5/2 and S 1/2 in Ca + or Sr + . Conclusions are finally given in Sec. V.

II. MONITORED SPONTANEOUS EMISSION: GENERAL ANALYSIS

We consider a two-level atom in free space, consisting of a ground state |g and an excited state |e separated by the Bohr energy ω 0 , and initially prepared in |e . Due to the coupling with the modes of the electromagnetic (EM) reservoir, the atom will naturally decay to the ground state |g , with a survival probability to stay in the excited state |e given by P (t) = exp(-Γt) (Wigner-Weisskopf decay [START_REF] Cohen-Tannoudji | Photons and atoms: introduction to quantum electrodynamics[END_REF]). For the free dynamics (i.e. without measurements), the decay rate Γ is given by the Fermi's golden rule (FGR) [START_REF] Cohen-Tannoudji | Photons and atoms: introduction to quantum electrodynamics[END_REF], and will be denoted by Γ 0 in the following.

In Ref. [1], Kofman and Kurizki showed that frequent measurements on an excited two-level atom, i.e. repeated instantaneous projections onto the state |e , lead to a broadening of its energy level, analogous to collisional broadening. Therefore, the atom probes a larger range of EM modes in the reservoir spectrum, and these new decay channels might modify the dynamics. Specifically, it was shown, within the rotating-wave approximation (RWA), that if frequent measurements are performed at short intervals τ , the dynamics still follows an exponential decay, but with a measurement-modified decay rate given by [

1] Γ = 2π ∞ 0 dω F τ (ω -ω 0 ) R (ω) . (1)
The effects of the RWA on the QZE and AZE have been discussed in Refs. [START_REF] Zheng | [END_REF]16], showing no essential differences between the predictions made with and without the RWA in the case of the reservoir that we shall consider here. Moreover, for a discussion about a non-exponential decay, see Ref. [17]. In Eq. ( 1), the function R(ω) represents the reservoir coupling spectrum and is written

R(ω) = -2 k | e, 0| ĤI |g, 1 k | 2 δ(ω -ω k ) (2)
where |g,

1 k = |g ⊗ |1
F τ (ω -ω 0 ) = τ 2π sinc 2 (ω -ω 0 ) τ 2 (3)
with sinc(x) ≡ sin(x)/x. Note that the spectral profile function can be generalized to the case where no assumption is made beforehand about the state that is being repeatedly prepared [18]. In Fig. 1 (a) (orange line), the function F τ (ω -ω 0 ) is shown, centered on ω 0 and with a width of about 2πν. When ν → 0, F τ (ω -ω 0 ) → δ (ω -ω 0 ) and Eq. (1) gives: Γ → 2πR(ω 0 ), which is the natural decay rate given by the FGR Γ 0 ≡ 2πR(ω 0 ), where only the single photon states of frequency ω 0 contribute to the decay. From Eq. (1), we can see that the measurementmodified decay rate corresponds to the overlap between the functions R(ω) and F τ (ω -ω 0 ), and therefore depending on the profile of R(ω) in the interval around ω 0 , the system may experience an acceleration (Γ > Γ 0 , AZE) or a deceleration (Γ < Γ 0 , QZE) of the decay compared to the measurement-free decay. In the following, we aim at investigating the case of hydrogen-like atoms coupled to the free space EM field, for which the function R(ω) can be calculated analytically. This will allow us to highlight the conditions for an AZE observation in such systems. Before doing so, however, it is worth mentionning that in the perturbative treatment that we use, Eqs. ( 1) and ( 2) are valid to the first order (i.e. only one-photon processes are considered), and do not include higher-order contributions (i.e. two-photon and manyphoton processes). For this approximation to be valid, we need to ensure that, compared to the spontaneous single-photon emission of the |e → |g transition considered, two-photon processes, which involve other atomic levels, are negligible. This can only be checked on a caseby-case basis for specific atoms. In Sec. IV, we consider the specific case of the electric quadrupole transition of Ca + , and we check that the single-photon emission is the dominant decay channel from the relevant excited state (in Sec. IV A).

III. QUANTUM ANTI-ZENO EFFECT IN HYDROGEN-LIKE ATOMS A. Reservoir coupling spectrum for hydrogen-like atoms

For hydrogen-like atoms, it is useful to write the states of the atom in terms of the multipolar modes |g = |n g , l g , m g and |e = |n e , l e , m e where each atomic state is described by three discrete quantum numbers n i , l i and m i which are respectively the principal, angular momentum and magnetic quantum numbers. Similarly, it is useful to write the one-photon states in the energy-angularmomentum basis [12,13]

|1 k = |J, M, λ, ω
, where a photon is characterized by its angular momentum and magnetic quantum numbers J and M , respectively, and also its helicity λ and frequency ω. Based on the exact calculations of the matrix elements in (non-relativistic) hydrogen-like atoms in free space (initiated by Moses [12] and completed by Seke [13]), the reservoir (2) can be obtained analytically and depends on the type of the multipole transition |e → |g considered (see Appendix A for details)

R(ω) = |le+lg| J=|le-lg| N J r=0 D Jr ω η J +2r-1 X ω η J +2r 1 + ω ωX 2 µ (4)
where η J = 1 + 2J for magnetic transitions, and η J = -1 + 2J for electric transitions with J starts at 1 for a dipole transition (l e -l g = 1), at 2 for a quadrupole transition (l e -l g = 2) and so on; µ = 2 (n g + n e -1); D Jr are dimensionless constants involving the Clebsch-Gordan coefficients of the transition under consideration; and ω X is the non-relativistic cutoff frequency that emerges naturally from calculations [19,20] and reads [13]:

ω X = 1 n g + 1 n e c a 0 Z (5)
with a 0 the Bohr radius and Z the atomic number. Finally, the index at which the sum is terminated is N J = 2 (n e + n g ) -4 -J -l e -l g -with = 0 for electric transitions and = 1 for magnetic transitions.

For simplicity, we first consider electric transitions ( = 0) between an excited state of maximal angular momentum (l e = n e -1) and the ground state 1S (n g = 1, l g = 0). In that case, N J = 0 and the two sums disappear in Eq. ( 4) which reduces to

R(ω) = D ω η-1 X ω η 1 + ω ωX 2 µ (6)
where we defined D ≡ D J0 and η ≡ η J . This reservoir coupling spectrum is sketched on Fig. 1 (a). The parameters η, µ and ω X corresponding to the electric transitions 2P -1S (dipole), 3D-1S (quadrupole) or 4F -1S (octupole) are given in Table I. I. Parameters of the reservoir spectrum given by Eq. ( 6) for the electric transitions 2P -1S (dipole), 3D-1S (quadrupole) or 4F -1S (octupole) in the hydrogen atom.

B. Analytical results for ω0 ωX

In this section, we want to derive an analytical expression of the decay rate (1) to see how it scales with the measurement rate ν when the reservoir coupling spectrum is of the form of Eq. ( 6), in the case ω 0 ω X which is always respected for low-Z atoms. Indeed, using the Bohr formula for ω 0 , the ratio between ω 0 and the cutoff frequency ω X can be written from Eq. ( 5) as

ω 0 ω X = 1 2 (Zα) 1 n g - 1 n e , (7)
with α the fine structure constant of electrodynamics of approximate value α 1/137, whence we can see that the assumption ω 0 ω X makes sense for atoms with Z moderately small.

The details of our derivation are given in Appendix B, and we present the main ideas here. In the integral Γ (Eq. ( 1)), we start by expanding (to all orders) the numerator ω η of the reservoir function R(ω) (Eq. ( 6)) around the transition frequency ω 0 . This binomial expansion yields a series of terms of the type (ω -ω 0 ) k with k integers between 0 and η. We can then consider that the total decay rate in Eq. ( 1) results from two contributions. (i) A 'resonant' contribution Γ res coming from the k = 0 and k = 1 terms of the binomial expansion of ω η , for which only the part of F τ (ω -ω 0 ) that probes the reservoir R (ω) in a frequency range of width ∼ ν around ω 0 contributes. This amounts to making the ap-proximation that F τ (ω -ω 0 ) = 1/ (2πν) in the interval -πν < ω -ω 0 < πν and vanishes elsewhere. With the hierarchy ω 0 ω X in mind, the resonant contribution can then be calculated (see Appendix B)

Γ res 2π D ω η-1 X ω η 0 (8)
and is found to be equal to the natural decay rate Γ 0 = 2πR(ω 0 ) 2πDω η 0 /ω η-1 X computed by the FGR. (ii) The 'tail' contribution Γ tail , which only exists if η > 1, comes from all the terms with order 1 < k ≤ η, for which F tail τ (ω -ω 0 ) ∝ 1/(ω -ω 0 ) 2 probes the entire reservoir and has a non-negligible contribution. By approximating the square sine by its mean value 1/2, we can then compute the tail contribution (see Appendix B)

Γ tail D ν B 1 -η 2 + µ, - 1 -η 2 (9)
where B refers to Euler's Beta function and is a simple numerical prefactor (roughly of the order of unity). Finally, the measurement-modified decay rate Γ = Γ res + Γ tail normalized by the natural decay rate Γ 0 yields the result (partially obtained in Ref. [1]):

Γ Γ 0        1 for η = 1, 1 + 1 2π ν ω0 ωX ω0 η-1 B 1-η 2 + µ, -1-η 2 for η > 1. (10)
In Appendix C, we show how this expression can be extended to the general form of R(ω) given by Eq. ( 4): the result is similar in terms of scaling with the different parameters η, ν, ω 0 and ω X ; and the Beta function is simply replaced by a more complex numerical prefactor (see Eq. (C3)).

C. Comparison between numerical and analytical calculations and discussion

Before commenting on the scope of this result, we first compare in Fig. 2 the analytical approximation of Γ given by Eq. ( 10) to the numerical computation Γ num of Eq. (1) (using (3) and ( 6)) for three different reservoir coupling spectra R(ω) corresponding to the electric dipole (η = 1, in green), quadrupole (η = 3, in red) and octupole (η = 5, in blue) transitions whose parameters are given in Table I. We can see a very good agreement for the quadrupole and octupole transitions up to ν 100 ω 0 , and for the dipolar transition up to ν ω 0 . Note that in practice, it may not be feasible to reach such high measurement rates as ν ∼ ω 0 (particularly for optical transitions, cf. Sec. IV), and moreover, for ν ω 0 , the RWA is not valid anymore. Therefore, the analytical results are revealed to be excellent in the regime of interest ν ω 0 with a relative error (Γ num -Γ)/Γ num less than 2% for ν/ω 0 < 10 -2 in the three cases represented on the plot.

Concerning the AZE, we can see that in the case of the electric dipole transition, the AZE trend (Γ > Γ 0 ) appears only for ν ω 0 (green curve) -which is not interesting for experimental observations as just discussed, whereas for the other transitions (red and blue curves), the AZE is obtained already for ν ω 0 and can be very strong. This has been overlooked in the past and constitutes our main result: within the natural hierarchy ω 0 ω X , we predict from our general Eq. ( 10) that electric dipole transitions (η = 1) will not exhibit the AZE, whereas the AZE can be expected for all other types of electronic transitions (η > 1). On the one hand, as electric dipole transitions are arguably the most standard and studied type of electronic transitions in atoms, these predictions make the AZE much less ubiquitous than what had been stated in Ref. [1]. On the other hand, we see from Eq. ( 10) that for all other transitions, the ratio ω X /ω 0 1 may give rise, despite the ratio ν/ω 0 1, to a strong anti-Zeno effect Γ Γ 0 , particularly for high-order multipolar transitions. The goal of the next section is to identify realistic systems suitable for an AZE observation.

IV. EXPERIMENTAL PROPOSAL A. Transition choice

The search for a possible candidate to observe the AZE is framed by experimental constraints. Even if the AZE is expected to be observable on magnetic dipolar transitions and even more effective on electric octupolar transitions, the very long natural lifetime (of the order of one year or more) of the excited states involved in these transitions makes them very inappropriate to lifetime measurement. Therefore, in what follows, we focus on demonstrating the AZE on an electric quadrupolar transition.

The first choice candidate to confirm the predictions I.

derived for hydrogenic atoms is the hydrogen atom itself, by transferring the atomic population to the lowest D-state (the 3D-state would play the role of the excited state |e ), and frequently monitoring the excited state. A major limit lies in the level scheme of hydrogen which allows an atom in the 3D-state to decay to the 2P -states by a strong dipolar transition. The lifetime of the 3D-state is then conditioned by its dipolar coupling to 2P and is not limited by its quadrupolar coupling to 1S. Therefore, no measurable reduction of the lifetime due to the AZE is expected. The same problem arises with Rydberg states, which were originally proposed as promising candidates [1] for AZE observation due to their transitions in the microwave domain that favor the scaling in (1/ω 0 ) η of Eq. ( 10) compared to optical frequencies.

To circumvent this problem of unwanted transitions, it is then essential to identify a metastable D-state, which has no other decay route to the ground state than the quadrupolar transition. This can be found in the alkaliearth ions like Ca + or Sr + , where the lowest D level is lower in energy than any P -level. The order of magnitude of the lifetime of these D-levels ranges from 1 ms to 1 s. The contribution to the D-level spontaneous emission rate of two-photon decay, allowed by second-order perturbation theory based on non-resonant electric-dipole transitions, has been calculated in [21,22] for Ca + and Sr + . The results show that the two-photon decay channel contributes to 0.01% to the lifetime of the lowest Dstates of Ca + and Sr + . As a consequence, the spontaneous emission from the lowest D-level in Ca + and Sr + can be considered to be due only to electric quadrupolar transition and we then focus on these two atomic systems in the following.

B. Measurement scheme and read-out

Concerning the measurements of the frequently monitored excited state, ideal instantaneous projections on |e are not strictly required. Indeed, they amount in effect to dephasing the level |e , that is, make the phase of state |e completely random [1]. Different schemes were proposed to emulate projective measurements in Refs. [1,23,24] and performed in Ref. [11], for which Eq. ( 1) still holds. Here, we propose an alternative protocol in the same spirit of the "dephasing-only measurement" of Ref. [11]. In this scheme, state |e is the metastable state D 5/2 and the dephasing measurement is driven by the transition from D 5/2 to D 3/2 , by two lasers through the strong electric dipolar transitions to the common excited state P 3/2 (see Fig. 3) using a stimulated Raman adiabatic passage (STIRAP) process [25]. If the two-photon Raman condition is fulfilled (identical detuning for the two transitions), the intermediate P 3/2 -state is not populated and the population is trapped in a coherent superposition of the two states D 5/2 and D 3/2 . By changing the laser power on each transition with appropriate time profile and time delay, the atomic population can be transferred between the two metastable D-states, like demonstrated in Ref. [26]. After one transfer and return, state |e thus acquires a phase related to the phase of the two lasers. By applying a random phase jump on one laser between each completed STIRAP transfer, the phase coherence of the excited state |e is washed out, and a "dephasing" measurement of the level |e is performed.

To measure the effective lifetime of the D 5/2 -state, the read-out of the internal state must be based on electronic states which do not interfere with D 5/2 . For that purpose, the electron-shelving scheme first proposed by Dehmelt can be used [27]. It requires two other lasers, coupling to the S 1/2 → P 1/2 and to the D 3/2 → P 1/2 transitions (see Fig. 3). When shining these two lasers simultaneously, the observation of scattered photons at the S 1/2 → P 1/2 transition frequency is the signature of the decay of the atom to the ground state [28]. This read-out scheme is switched on during a short time compared to the lifetime of the D 5/2 -state, at a time when the STIRAP process has brought back the electron to D 5/2 .

C. Calculation for 40 Ca +

We now try to see whether the AZE might be observable in Ca + , which is not strictly speaking hydrogenic, but is alkali-like in a sense that it has a single valence electron, and can be seen as a single electron orbiting around a core with a net charge +2e. Ca + is the lightest of the alkali-earth ions having the appropriate levelscheme required for the proposed experimental protocol (see Fig. 3). Therefore, we assume that it still makes sense to use Eq. ( 10) (derived for hydrogen-like atoms) and we apply it to the electric quadrupole (η = 3) tran-FIG. 3. Dephasing measurement and read-out schemes for AZE observation in 40 Ca + . The transition used for the AZE is the electric quadrupole transition at 729 nm (red solid arrow). The dephasing measurement can be performed using a STIRAP process between the D 5/2 and D 3/2 states via two strong electric dipole transitions D 5/2 → P 3/2 at 854 nm and D 3/2 → P 3/2 at 850 nm, both detuned from resonance (brown dashed arrows) [26]. The read-out consists in observation of laser induced fluorescence if the atom has decayed to the ground state [27] (purple solid arrows).

sition 3D 5/2 → 4S 1/2 to find Γ -Γ 0 Γ 0 = A ν ω 0 ω X ω 0 2 (11)
where the numerical pre-factor A cannot be computed for such an electronic system (see Appendix C for a calculation of A in the simpler case of hydrogen-like atoms).

To be observable, the AZE must induce a lifetime relarger than 1%, the best precision reached in recent 3D 5/2 -lifetime measurements in Ca + [28]. We evaluate ω X using Eq. ( 5) with n g = 4 and n e = 3 and by replacing the atomic number Z by the effective number of charges Z eff = 2. Using the frequency of this transition ω 0 = 2π×411 THz, this gives a ratio (ω X /ω 0 ) 2 6.6•10 6 . If the unknown pre-factor A is assumed to be of the order of unity, one would need ν ∼ 4 MHz to meet the observation requirement.

The transfer between the states D 5/2 and D 3/2 has been demonstrated in 40 Ca + with a STIRAP process [26], where a complete one-way transfer duration of 5 µs was observed for 420 mW/mm 2 on the 850 nm 3D 3/2 → 4P 3/2 transition and 640 mW/mm 2 on the 854 nm 3D 5/2 → 4P 3/2 transition, with both lasers detuned by ∆ = 600 MHz from resonance (see Fig. 3). To reduce the duration of the dephasing measurement to time scale smaller than 1 µs, one can increase the laser intensity by stronger focusing and/or larger power, but we can also consider that a complete STIRAP transfer is not required to achieve a dephasing of the excited state. Furthermore, a close inspection of Tables I and II in Ref. [13] suggests that the pre-factor A could be much larger than unity, making the constraint on a high measurement rate less stringent for AZE observation.

Even if the experimental requirements for AZE observation on quadrupole transition in Ca + are more demanding than today's best achievements, realistic arguments show that they can be met in a dedicated experimental set-up. This experimental challenge would benefit from theoretical insight concerning the still unknown pre-factor scaling the lifetime reduction.

V. CONCLUSION

Based on well-established results for hydrogen-like atoms, we derived an analytical expression of the decay rate modified by frequent measurements which allows us to highlight the main condition for an observable AZE in atomic radiative decay in free space: all transitions except electric-dipole transitions will exhibit an AZE under sufficiently rapid repeated measurements. This analytical formula also indicates how the AZE scales with the measurement rate. We then identified a suitable level scheme in the alkali-earth ions Ca + and Sr + for AZE observation, involving the electric quadrupole transition between D 5/2 and S 1/2 , and using a new "dephasing" measurement protocol based on the STIRAP technique. Other suitable experimental schemes might exist, and we encourage further proposals in this sense.

which can be recast in the expression given in the main text by Eq. ( 4), where we introduced dimensionless coefficients D Jr involving the Clebsch-Gordan coefficients and the other constants and the sums over M and λ. This is true unless D J0 vanishes. This is the case for instance of the electric dipole transitions ( = 0, J = 1) between levels sharing the same principal quantum number (see Table I in Ref. [13]), due to the special properties of these dipolar transitions. That D J0 vanishes can be shown rather easily by using the orthogonality properties of the Gegenbauer polynomials (see Ref. [29] for a derivation of the momentum-space wave functions of hydrogen in terms of these polynomials). However, we do not focus on this special case here. In the generic case (D J0 = 0), the decay rate under frequent observations for a specific J will be

Γ τ J 2π D J0 ω η J 0 ω η J -1 X + ν × N J r=0 D Jr θ η J + 2r - 3 2 × B 1 -η J -2r 2 + µ, - 1 -η J -2r 2 . (C2)
with θ the Heaviside step function. It thus appears that, for given J, all terms in the sum over r in (4) have a contribution to the modified decay rate that is of the same order of magnitude. All that remains to be done is to sum over J. This sum is resolved quite differently for the free decay rate on the one hand, and the modified decay rate on the other. Namely, for the former, we see from (C1) that the hierarchy ω 0 ω X ensures that the contribution from the smallest possible J is dominant. This value is equal to |l e -l g | ≡ J min , and we will write η min ≡ η Jmin . For the latter, however, we are forced to keep the double sum over J and N J : all (sufficiently large) powers of the frequency in the coupling contribute to the modified decay rate on the same level, with numerical prefactors as the sole difference. Namely, writing Γ 0 = le+lg J=|le-lg| Γ 0J and Γ τ = le+lg J=|le-lg| Γ τ J , we have obtained

Γ τ Γ 0 1 + 1 2π ν ω 0 ω X ω 0 ηmin-1 le+lg J=|le-lg| N J r=r0 D Jr D Jmin0 × B 1 -η J -2r 2 + µ, - 1 -η J -2r 2 (C3)
where we have introduced

r 0 ≡ max 3 4 - η J 2 , 0 . (C4)
Despite the more complicated appearance of this expression, we see that the parametric dependence of the ratio of the decay rates is independent of the details of the matrix elements: the important parameter is η min -1. There is a competition between ν/ω 0 1 and ω X /ω 0 1 but, for η min ≥ 3, we can expect that the second factor will dominate, especially for low values of Z [see Eq. ( 7)]. This second factor becomes all the more dominant for higher values of η min , that is, for transitions with high difference between the orbital angular momenta of the initial and final levels. Only for transitions where |l e -l g | = 1 does the second factor (ω X /ω 0 ) 0 = 1 fail to play a role, so that they verify Γ τ Γ 0 . These transitions are often called "electric dipole transitions" (including by us in our Sec. I), although in most cases they are accompanied by emission of photons of angular momentum J > 1 (as well of course as J = 1). Indeed, as we have recalled with Eq. (C1), it is always the photons with the smallest allowed value J min of J that dominate the spontaneous emission in an electronic transition.

FIG. 1 .

 1 FIG. 1. (a) Scheme of the broadened spectral profile Fτ (ω -ω0) (orange line) of an atom with transition frequency ω0 due to repeated measurements at a rate ν = 1/τ with τ the interval between each measurement, and reservoir coupling spectrum R(ω) (blue line) of the form of Eq. (6) with a cutoff frequency ωX ω0. (b) Scheme of the broadened spectral profile Fτ (ω -ω0) (orange line) and its resonant [F res τ (ω -ω0) = 1/(2πν) for -πν < ω -ω0 < πν (green dashed line)] and tail [F tail τ (ω -ω0) = ν/[π(ω -ω0) 2 ] for ω -ω0 > πν (red dashed line)] approximations. The inset shows that the energy broadening of |e , induced by the frequent measurements at rate ν, modifies the decay into the EM reservoir.

FIG. 2 .

 2 FIG. 2. Comparison between numerical (full lines) and analytical (dotted lines) calculations of ln(Γ/Γ0) as a function of the normalized measurement rate ν/ω0 for three different electric transitions: dipole (η = 1, in green), quadrupole (η = 3, in red) octupole (η = 5, in blue). The associated parameters used for the function R(ω) corresponding to these transitions are displayed in TableI.

  k is the outer product between the atomic state |g and the state of the EM field |1 k containing one photon in the mode labelled by k, |e, 0 = |e ⊗ |0 is the outer product between the atomic state |e and the vacuum state of the EM field |0 , and ĤI is the interaction Hamiltonian. The function F τ (ω -ω 0 ), on the other hand, corresponds to the broadened spectral profile of the atom due to the frequent measurements at a rate ν = 1/τ , and takes the form

ACKNOWLEDGMENTS

We thank Siddartha Chattopadhyay and David Wilkowski for helpful discussions. CC acknowledges fruitful discussions with Gonzalo Muga (UPV/EHU). EL would like to thank the Doctoral School "Physique et Sciences de la Matière" (ED 352) for its funding.

Appendix A: Form of the reservoir coupling spectrum for hydrogen-like atoms in free space Using the notations introduced in Sec. III A, the reservoir coupling spectrum (2) is given by

× | n e , l e , m e ; 0| ĤI |n g , l g , m g ; J, M, λ, ω | 2 . (A1)

Here, the density of states is ρ (ω) = 1, on account of the normalisation J, M, λ, ω | J , M , λ , ω = δ JJ δ M M δ λλ δ (ω -ω ) (this can be understood by dimensional considerations). In the non-relativistic approximation, Seke calculated in [13] the exact matrix elements n e , l e , m e ; 0| ĤI |n g , l g , m g ; J, M, λ, ω for hydrogen-like atoms in free space, using the interaction Hamiltonian (in SI units)

with e the elementary electric charge, m e the electron mass, x and p the position and the linear momentum operators of the electron respectively and  the vector potential operator of the quantized EM field. By employing these exact matrix elements (Eqs. (17)(18)(19) in [13]) in Eq. (A1), one gets the following analytical form for the reservoir coupling spectrum

where c the speed of light in vacuum, α is the fine structure constant of electrodynamics, l g , J, m g , M |l g , J, l e , m e are the Clebsch-Gordan coefficients of the transition of interest, and ω X is the non-relativistic cutoff frequency given by Eq. ( 5). The coefficients d Jr are numerical coefficients that have been calculated for certain transitions in [13] (note that the coefficients d Jr here correspond to the coefficients d 00 d r in Eq. ( 18) in Ref. [13]). The index at which the sum is terminated is

with = 0 for electric transitions and = 1 for magnetic transitions. Eq. (A3) can be recast in the form

where N J = 2 (n e + n g ) -4 -J -l e -l g -and d Jr are combinations of the previous d Jr coefficients. Moreover, as a consequence of the conservation of the angular momentum, the values of J and M must verify

which are the exact selection rules. Therefore, the full reservoir takes the form

Appendix B: Derivation of the AZE scaling in the simple case of the reservoir (6)

Here we derive an analytical form of the integral of Eq. ( 1) with the simplified form of the reservoir (6). Keeping in mind the hierarchy ω 0 ω X , we will proceed to derive an approximate analytical expression of the general integral

(B1) in terms of which the measurement-modified decay rate (1) is straightforwardly expressed: Γ τ = DI ηµ (τ ) /ω η-1 X . Using the binomial expansion of ω η , we first rewrite our integral as

The k = 0 and k = 1 terms in the sum may be treated in a specific way. Namely, we make the following approximation of the square cardinal sine function in F τ (ω), that is illustrated in Fig. 1 (b):

1 for ω 0 -πν < ω < ω 0 + πν , 0 otherwise. (B3) This approximation is sufficient for k = 0 and k = 1 only, as the integrand in (B2) decays sufficiently fast when one moves away from ω 0 so that the frequency ranges outside the door function (B3) can be ignored. In addition to this, in the frequency range of interest here (that is, a small range of width ∼ ν centered on ω 0 ), we can consider that ω/ω X ∼ 0 (which is justified by the hierarchy ω 0 ω X ). Using these approximations, we can write the low-k contribution to the integral as

Now we turn to the terms for which k ≥ 2 and that will only exist if η > 1. For these terms, replacing the square cardinal sine by a rectangle function is no longer valid, as the growth of (ω -ω 0 ) k is not overridden by the decrease of (ω -ω 0 ) -2 that comes from the square cardinal sine, and therefore we must consider the whole frequency range. We therefore need to find another way to approximate the integral (B2), and we may simply replace the square sine by its mean value 1/2 here to get

Also note, that we should not have ω 0 /ν excessively large, lest the square cardinal sine converges to the Dirac δ distribution, and replacing the square sine with its average value is no longer valid. We then compute the resulting integral, which, in the limit ω 0 /ω X → 0, acceptable for the transitions that interest us, reads

where B refers to Euler's Beta function. As can be checked from (B2) and (B6), of all the contributions for k ≥ 2, the one for which k = η is easily the largest (this is due, again, to the hierarchy ω 0 ω X ). As such, we can rewrite (B2) as

where the second summand on the r.h.s. of (B7) will only exist for η > 1. Comparison with the natural decay rate

Note that this result had been (partially) obtained in [1], where the authors found that for a reservoir of the form R(ω) ∝ ω η with η > 1: Γ ∝ ν ω η-1 X , in the approximation ω η 0 /ω η-1 X ν ω X (cf. Eq. ( 20) in Ref. [1]).

Appendix C: Derivation of the AZE scaling in the complete case of the reservoir (4)

In this section, we extend the previous result found for a reservoir of the simple form (6) to the general form (4). Let us first sum over r, and then over J. In the generic case, the FGR decay rate will be, for the reservoir coupling spectrum (4), given by