L Kallmeyer

W Maier
email: wmaier@sfs.uni-tuebingen.de

Y Parmentier
email: parmenti@loria.fr

J Dellert
email: jdellert@sfs.uni-tuebingen.de

TuLiPA -Parsing Extensions of TAG with Range Concatenation Grammars

In this paper we present a parsing framework for extensions of Tree Adjoining Grammars (TAG) called TuLiPA (Tübingen Linguistic Parsing Architecture). In particular, besides TAG, the parser can process Tree-Tuple MCTAG with shared nodes (TT-MCTAG), a TAG-extension that has been proposed to deal with scrambling in free word order languages such as German. The central strategy of the parser is such that the incoming TT-MCTAG (or TAG) is transformed into an equivalent Range Concatenation Grammar (RCG) which, in turn, is then used for parsing. The RCG parser is an incremental Earley-style chart parser. In addition to the syntactic anlysis, TuLiPA computes also an underspecified semantic analysis for grammars that are equipped with semantic representations.

Introduction

The starting point of the work presented here is the aim to implement a parser for a German TAG-based grammar that computes syntax and semantics. As a grammar formalism for German we chose a multicomponent extension of TAG called TT-MCTAG (Multicomponent TAG with Tree Tuples) which has been first introduced by Lichte [START_REF] Timm Lichte | An MCTAG with Tuples for Coherent Constructions in German[END_REF]. Instead of implementing a specific TT-MCTAG parser we follow a more general approach by using Range Concatenation Grammars (RCG) as a pivot formalism. The TT-MCTAG (or TAG) is transformed into a strongly equivalent RCG that is then used for parsing. The motivation for the passage via RCG is that the RCG directly represents the set of derivation trees of the original grammar. Consequently, it abstracts away from traversals of elementary trees without substitutions or adjunctions and the output of the RCG parser can be directly interpreted as the TT-MCTAG (or TAG) derivation forest.

We have implemented the conversion into RCG, the RCG parser and the retrieval of the corresponding TT-MCTAG analyses. The parsing architecture comes with graphical input and output interfaces, and an XML export of the result of parsing. It is freely available under the GPL. 1In this paper, we present this parsing architecture focussing on the following aspects: first, we introduce the TT-MCTAG formalism (section 2). Then, we present successively the RCG formalism and the conversion of TT-MCTAG into RCG (section 3). Section 4 explains the parsing algorithm we use for the specific RCGs we obtain from TAG and TT-MCTAG.

TT-MCTAG

Tree Adjoining Grammars (6) are a formalism based on tree rewriting. A Tree Adjoining Grammar (TAG) is a tuple G = (V N ,V T , S, I, A) where V N and V T are disjoint alphabets of non-terminal and terminal symbols, respectively, S ∈ V N is the start symbol, and I and A are finite sets of initial and auxiliary trees, respectively. Trees in I ∪ A are called elementary trees. The internal nodes in the elementary trees are labeled with non-terminal symbols, the leaves with non-terminal or terminal symbols. As a special property, each auxiliary tree β has exactly one of its leaf nodes marked as the foot node, having the same label as the root. Such a node is denoted by * . Leaves with non-terminal labels that are not foot nodes are called substitution nodes.

In a TAG, larger trees can be derived from the elementary trees by subsequent applications of the operations substitution and adjunction. The substitution operation replaces a substitution node η with an initial tree having root node with the same label as η. The adjunction operation replaces an internal node η in a previously derived tree γ with an auxiliary tree β having root node with the same label as η. The subtree of γ rooted at η is then placed below the foot node of β. Only internal nodes can allow for adjunction, adjunction at leaves is not possible.

TAG derivations are represented by derivation trees that record the history of how the elementary trees are put together. A derivation tree is an unordered tree whose nodes are labeled with elements in I ∪ A and whose edges are labeled with Gorn addresses of elementary trees. 2 Each edge in a derivation tree stands for an adjunction or a 1 substitution. In the following, we write a derivation tree D as a directed graph V, E, r where V is the set of nodes, E ⊂ V × V the set of arcs and r ∈ V the root. For every v ∈ V , Lab(v) gives the node label and for every

v 1 , v 2 ∈ E, Lab( v 1 , v 2 )
gives the edge label. TT-MCTAGs ( 9) are multicomponent TAGs (MCTAG) where the elementary tree sets consist of one lexicalized tree γ, the head tree and a set of auxiliary trees β 1 , ..., β n , the argument trees. We write these sets as tuples γ, {β 1 , ..., β n } . During derivation, the argument trees have to attach to their head, either directly or indirectly via node sharing. The latter means that they are linked by a chain of root-adjunctions to a tree adjoining to their head. A derivation tree D = V, E, r in the underlying TAG G T is licensed in G if and only if the following conditions (MC) and (SN-TTL) are both satisfied.

(MC): For all sets Γ from G and for all γ 1 , γ 2 in Γ, we have

|{v | v ∈ V, Lab(v) = γ 1 }| = |{v | v ∈ V, Lab(v) = γ 2 }|.
(SN-TTL): For all argument trees β with head h(β)

and n ≥ 1, let v 1 , . . . , v n ∈ V be pairwise different nodes with Lab(v i ) = h(β), 1 ≤ i ≤ n. Then there are pairwise different nodes u 1 , . . . , u n ∈ V with Lab(u i ) = β, 1 ≤ i ≤ n. Furthermore, for 1 ≤ i ≤ n either v i , u i ∈ E,
or else there are u i,1 , . . . , u i,k with auxiliary tree labels such that

u i = u i,k , v i , u i,1 ∈ E and, for 1 ≤ j ≤ k -1, we have u i, j , u i, j+1 ∈ E with Lab( u i, j , u i, j+1 ) = ε.
TT-MCTAG has been proposed to deal with free word order languages. An example from German is shown in Fig. 1. Here, the NP nom auxiliary tree adjoins directly to verspricht (its head) while the NP acc tree adjoins to the root of a tree that adjoins to the root of a tree that adjoins to reparieren.

TT-MCTAG can be further restricted, such that at each point of the derivation the number of pending β-trees is at most k. This subclass is called k-TT-MCTAG. Both, TT-MCTAG and the restricted k-TT-MCTAG generate only polynomial langugaes (7; 8).

Transforming TT-MCTAG into RCG

The central idea of our parsing strategy is to use RCG (1; 2) as a pivot formalism. RCGs are grammars that rewrite predicates ranging over parts of the input by other predicates. E.g., a clause S(aXb) → S(X) signifies that S is true for a part of the input if this part starts with an a, ends with a b, and if, furthermore, S is also true for the part between a and b.

A RCG is a tuple G = N, T,V, S, P such that a) N is an alphabet of predicates of fixed arities; b) T and V are disjoint alphabets of terminals and of variables; c) S ∈ N is the start predicate (of arity 1) and d) P is a finite set of clauses A 0 (x 01 , . . . , x 0a 0 ) → ε or A 0 (x 01 , . . . , x 0a 0 ) → A 1 (x 11 , . . . , x 1a 1 ) . . . A n (x n1 , . . . , x na n ) with n ≥ 1, A i ∈ N, x i j ∈ (T ∪V ) * and a i being the arity of A i .

When applying a clause with respect to a string w = t 1 . . .t n , the arguments in the clause are instantiated with substrings of w, more precisely with the corresponding ranges. 3 The instantiation of a clause maps all occurrences of a t ∈ T in the clause to an occurrence of a t in w and consecutive elements in a clause argument are mapped to consecutive ranges.

If a clause has an instantiation wrt w, then, in one derivation step, the left-hand side of this instantiation can be replaced with its right-hand side. The language of an RCG G is L(G) = {w | S( 0, |w| ) * ⇒ ε wrt w}. An RCG is simple if in all clauses, all righthand side arguments are only single variables and every variable in the lefthand side occurs exactly once in the righthand side and vice versa. The RCGs we are dealing with are all simple. Simple RCGs are equivalent to linear context-free rewriting systems (LCFRS).

The transformation of a given k-TT-MCTAG into a strongly equivalent simple RCG is an extension of the TAGto-RCG transformation proposed by Boullier [START_REF] Boullier | On TAG Parsing[END_REF]. The idea of the latter is the following: the RCG contains predicates α (X) and β (L, R) for initial and auxiliary trees respectively. X covers the yield of α and all trees added to α, while L and R cover those parts of the yield of β (including all trees added to β) that are respectively to the left and the right of the foot node of β. The clauses in the RCG reduce the argument(s) of these predicates by identifying those parts that come from the elementary tree α/β itself and those parts that come from one of the elementary trees added by substitution or adjunction.

For the transformation from TT-MCTAG into RCG we use the same idea. There are predicates γ... for the elementary trees (not the tuples) that characterize the contribution of γ. We enrich these predicates in a way that allows to keep track of the "still to adjoin" argument trees and constrain thereby further the RCG clauses. The pending arguments are encoded in a list that is part of the predicate name. The yield of a predicate corresponding to a tree γ contains not only γ and its arguments but also arguments of predicates that are higher in the derivation tree and that are adjoined below γ via node sharing. In addition, we use branching predicates ad j and sub that allow computation of the possible adjunctions or substitutions at a given node in a separate clause.

α rep , / 0 (L zu reparieren R) → ad j, α rep , ε, {β acc } (L, R) ad j, α rep , ε, {β acc } (L, R) → β acc , / 0 (L, R) | β v , {β acc } (L, R) β acc , / 0 (L X, R) → ad j, β acc , ε, / 0 (L, R) sub, β acc , 1 (X) sub, β acc , 1 (X) → α es , / 0 (X) α es , / 0 (es) → ε β v , {β acc } (L, verspricht R) → ad j, β v , ε, {β nom , β acc } (L, R)
Figure 2: Some clauses of the RCG correponding to the TT-MCTAG in Fig. 1 As an example see Fig. 2. The first clause states that the yield of the initial α rep consists of the left and right parts of the root-adjoining tree wrapped around zu reparieren. The ad j predicate takes care of the adjunction at the root (address ε). It states that the list of pending arguments contains already β acc , the argument of α rep . According to the second clause, we can adjoin either β acc (while removing it from the list of pending arguments) or some new auxiliary tree β v .

RCG Parsing

The input sentence is parsed using the RCG computed from the input TT-MCTAG via the conversion algorithm introduced in the previous section. Note that the TT-MCTAG to RCG transformation is applied to a subgrammar selected from the input sentence since the cost of the conversion depends heavily on the size of the grammar (all licensed adjunctions have to be computed while taking into account the state of the list of pending arguments). 4The RCGs one obtains from transforming a k-TT-MCTAG are always ordered simple RCGs. A simple RCG is ordered if the order of variables is the same in the lhs and rhs predicates of a clause. In particular, this means that for every clause instantiation, the order of arguments of a predicate is the same as the order of the correponding ranges in the input string.

The algorithm used in our parser is a modification of (3) and is very close to the strategy adopted in [START_REF] Villemonte De | Parsing mildly context-sensitive languages with thread automata[END_REF]. It is an incremental Earley-style chart parser.

The general idea is as follows: We process the arguments of lefthand sides of clauses incrementally, starting from an S-clause. Whenever we reach a variable, we move into the clause of the correponding righthand side predicate. The first time, this is done by predicting the righthand side predicate and starting the traversal of its first argument. When moving into the clause of a righthand side predicate that was already started, we resume it. Whenever we reach the end of an argument, we suspend this clause and move into the parent clause that has called the current one. In addition, we treat the case where we reach the end of the last argument and move into the parent as a special case. Here, we first convert the item into a passive one and then complete the parent item with this passive item. This allows for some additional factorization. We have implemented this strategy as a chart parser.

As a result of the parser, we obtain a RCG derivation forest that can be directly interpreted as the underlying TAG derivation forest since the RCG predicates represent the elementary trees used in the derivation and the substitutions and adjunctions performed.

The parsing architecture introduced here has been extended to support the syntax/semantics interface of Gardent and Kallmeyer [START_REF] Gardent | Semantic Construction in FTAG[END_REF]. The underlying idea of this interface is to associate each tree with flat semantic formulas. The arguments of these formulas are unification variables co-indexed with features labelling the nodes of the syntactic tree. During derivation, trees are combined via adjunction and/or substitution, each triggering the unifications of the feature structures labelling specific nodes. As a result of these unifications, the arguments of the semantic formulas associated with the trees involved in the derivation get unified.

  (1) ... dass es der Mechaniker zu reparieren versucht ... that it the mechanic to repair tries '... that the mechanic tries to repair it'

	Tree tuples for (1):
		VP VP * verspricht	,	VP NP nom VP *	NP nom der Mech.	, {}
		VP zu reparieren	,		VP NP acc VP *	NP acc es	, {}
	derived tree: VP						derivation tree: reparieren ε
	NP acc	VP					verspricht ε
	es	NP nom	VP				NP nom	←	argument of verspricht
							1	ε
	der Mech.	VP	verspricht	Mechaniker NP acc ← argument of reparieren
		zu reparieren				1

es Figure 1: TT-MCTAG analysis of 1

http://sourcesup.cru.fr/tulipa/

In this convention, the root address is ε and the jth child of a node with address p has address p • j.

A range i, j with 0 ≤ i < j ≤ n corresponds to the substring between positions i and j, i.e., to t i+1 ...t j .

We do not have a proof of complexity of the conversion algorithm yet, but we conject that it is exponential in the size of the grammar since the adjunctions to be predicted depend on the adjunctions predicted so far and on the auxiliary trees adjoinable at a given node.


Mazyar Mirrahimi
email: mazyar.mirrahimi@inria.fr

Pierre Rouchon
email: pierre.rouchon@ensmp.fr

REAL-TIME SYNCHRONIZATION FEEDBACKS FOR SINGLE-ATOM FREQUENCY STANDARDS *

Keywords: quantum Monte-Carlo trajectories, extremum seeking, feedback, synchronization, quantum systems AMS subject classifications. 34F05, 93D15, 37N20

Simple feedback loops, inspired from extremum-seeking, are proposed to lock a probefrequency to the transition frequency of a single quantum system following quantum Monte-Carlo trajectories. Two specific quantum systems are addressed, a 2-level one and a 3-level one that appears in coherence population trapping and optical pumping. For both systems, the feedback algorithm is shown to be convergent in the following sense: the probe frequency converges in average towards the system-transition one and its standard deviation can be made arbitrarily small. Closed-loop simulations illustrate robustness versus jump-detection efficiency and modeling errors.

1. Introduction. The SI second is defined to be "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom" [START_REF]13ème Conférence Générale des Poids et Mesures[END_REF]. A primary frequency standard is a device that realizes this definition. Extremum seeking techniques (see, e.g, [START_REF] Ariyur | Real-Time Optimization by Extremum-Seeking Control[END_REF] for a recent exposure) are usually used in high precision spectroscopy to achieve frequency lock with an atomic transition frequency. For micro atomic-clocks [START_REF] Kitching | A microwave frequency reference based on vcsel-driven dark line resonance in cs vapor[END_REF] synchronization is achieved when the output signal of a photo-detector is maximum (or minimum). This characterizes perfect resonance between the probe laser frequency with the atomic one. As sketched on figure 1.1, such synchronization feedback schemes are based on a modulation of the probe frequency, the input u, with a sinusoidal variation a sin(ωt) of small amplitude a and fixed (low) frequency ω, on a high-pass filtering (transfer s s+h ) of the photo-detector signal (the output y), on a multiplier and finally an integrator giving the mean input value. In such synchronization scheme the system corresponds to a population of identical quantum systems with few mutual interactions (the vapor cell) having reached its asymptotic statistical regime described by a density matrix solution of a static Lindbald-Kossakovski master equation. In this paper, we propose to adapt this feedback strategy to a single quantum system. Such a system cannot be described by a static non-linear input/output map but it obeys a stochastic jump dynamics [START_REF] Dalibar | Wave-function approach to dissipative processes in quantum optics[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF]. The output signal is no more continuous since it corresponds to a counter giving the jump times. As shown in [START_REF] Cohen-Tannoudji | Single-atom laser spectroscopy. looking for dark periods in fluorescence light[END_REF], all the spectroscopic information and in particular the value of the atomic transition frequency are contained in the statistics of these jump-time series. Thus it is not surprising that such feedback loops are possible. The contribution of this paper is to propose for the first time (as far as we know) a real-time synchronization feedback scheme that can be implemented on electronic circuits of similar complexity to those used for extremum-seeking loops. In the feedback loop, we avoid thus the use of quantum filters [START_REF] Van Handel | Feedback control of quantum state reduction[END_REF] and records of jump-times sequences required by usual statistical treatments.

We consider here two kinds of quantum systems. The first system is the simplest one we can imagine. It has a stable ground state and an excited unstable one. These two sates are in interaction with a quasi-resonant electromagnetic field characterized by a complex amplitude u + ıv and a frequency Ω close to the transition frequency between the ground and excited states. The measure corresponds then to the photon emitted by the excited state when it relaxes to the ground state by spontaneous emission. The complexe amplitude is then modulated according to ū + ıv cos(ωt) ((ū, v) positive parameters, modulation frequency ω Ω). The synchronization feedback (playing the role of the integrator in figure 1.1) corresponds essentially to the recurrence Ω N +1 = Ω N -δ sin(ωt N ) where N is the jump-index, t N the jump-time, and Ω N the probe frequency between time t N -1 and time t N (δ positive parameter). The second system corresponds to a typical Λ-system appearing in coherent population trapping phenomena and optical pumping [START_REF] Arimondo | Coherent population trapping in laser spectroscopy[END_REF]. Such 3-level configurations are also presented in micro atomic clocks. The synchronization feedback is very similar to the previous one (see subsection 3.2). Both feedback schemes are illustrated by closed-loop quantum Monte-Carlo simulations and rely on two formal results (theorems 2.1 and 3.1) ensuring the convergence of the mean frequency de-tuning to 0 with a standard deviation that can be made arbitrary small.

The paper is organized as follows. Section 2 is devoted to the two-level system: the stochastic jump dynamics are depicted in subsection 2.1; the synchronization feedback is detailed in subsection 2.2; the remaining two subsections dealt with closed-loop simulations illustrating theorem 2.1. Section 3 deals with the Λ-system and admits exactly the same structure as section 2. The two last sections 4 and 5 are devoted to the proofs of the two main results, theorems 2.1 and 3.1.

The authors thank Guilhem Dubois from LKB for interesting discussions and suggestions.

2. The two-level system.

2.1. Monte-Carlo trajectories. This 2-level system is defined on the Hilbert space H = span {|g , |e }: the ground state |g is stable whereas the excited state |e is unstable with life time 1/Γ and relaxes to |g . The system is submitted to a near-resonant laser field whose complex amplitude is assumed to be slowly variable with respect to the transition frequency. Its dynamics are stochastic with quantum Monte-Carlo trajectories [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF] described here below.

In the absence of quantum jump, the density matrix ρ evolves through the dy-namics

d dt ρ = -ı[ H , ρ] - 1 2 L † L, ρ + Tr L † Lρ ρ
where L † L, ρ = L † Lρ + ρL † L stands for the anti-commutator.

The Hamiltonian H = ∆ 2 σ z + uσ x + vσ y is attached to the conservative part of the dynamics: (σ x , σ y , σ z ) are the Pauli matrices; ∆ denotes the laser-atom detuning; u and v are the real coefficients of the complex laser amplitude. The jump operator L = √ Γ |g e| , is associated to the dissipative dynamics with Γ > 0 denoting the decoherence rate.

At each time step dt the system may jump on the ground state |g g| with a probability given by

p jump (ρ → |g g|) = Tr L † Lρ dt = ΓTr (|e e| ρ) dt = Γ e| ρ |e dt.
Each jump is associated to the spontaneous emission of a photon that is detected by the photo-detector: the measurement is just a simple click and we know that just after the click the system is at the ground state, i.e., ρ = |g g|.

In the sequel, we will use this stochastic dynamics in the Γ-scale. This just consists in replacing u by uΓ, v by vΓ, ∆ by ∆Γ, and t by t/Γ in the equations. In this de-coherence time-scale, the density matrix ρ evolves through the dynamics

d d t ρ = -ı ∆ 2 σ z + uσ x + vσ y , ρ - 1 2 {|e e| , ρ} + e| ρ |e ρ (2.1)
and the jump probability between t and t + dt reads

p jump (ρ → |g g|) = e| ρ |e dt. (2.2)
Just after each jump, ρ coincides with |g g|. The atom/laser detuning is ∆ and the laser complex amplitude is u + ıv.

2.2. The synchronization feedback. We consider here the two-level system described, in the decoherence time-scale, by (2.1) (2.2). The quantum jumps lead to the emission of photons that will be detected with a certain efficiency η ∈ (0, 1].

The main goal of this paper is to provide a real-time algorithm so that, using the information obtained through the detected photons, we can synchronize the laser with the atomic transition frequency and therefore make ∆ converge to zero.

Note that, in practice we have a certain knowledge of the transition frequency and therefore, we can always tune our laser so that the detuning |∆| does not get larger than a fixed constant C.

In the aim of providing a synchronization algorithm inspired from extremumseeking, we consider a laser field amplitude of the form

u = ū, v = v cos(ωt)
where the modulation frequency ω is of order 1 but where ū and v are small: ū, v 1. The main strategy for the correction of the detuning is to wait for the matured quantum jumps (clicks of the photo-detector). This means that we choose a certain time constant T 1 and if the distance between two jumps is more than T , we will correct the detuning according to the time when the second jump happens. Note that, one can easily show that these matured quantum jumps, almost surely, happen within a finite horizon. Here is the explicit feedback algorithm:

∆ N +1 = ∆ N -δ sin(ωt) i f|∆ N -δ sin(ωt)| ≤ C, ∆ N +1 = C, otherwise
and go back to the step 2. Here we have chosen the correction gain δ 1. Our claim is that such an algorithm provides an approximate synchronization of laser frequency: given any small , we can adjust the design parameters ū, v and δ small enough such that with the above algorithm, the detuning ∆ N converges in average to an O( 2 )-neighborhood of 0 with a deviation of order O( ) (in the Γ-scale): according to theorem 2.1, it suffices to take ū, v ∼ , δ ∼ 2 and (ω, C) such that 4ω 2 > 1 + 4C 2 to ensure such convergence. Notice that, such algorithm is very simple and can be implemented via a standard electronic circuit.

Numerical simulations.

Let us now show the performance of this algorithm on some simulations. For the simulations of Figure 2.1, we take (Γ = 1 in the decoherence time-scale) C = 1/2, η = 0.9, ū = v = 6.0 10 -2 , ω = 1.0, δ = 9.0 10 -4 . The first plot provides the number of clicks (quantum jumps) while the second one gives the evolution of the detuning ∆ N . As it can be noted, the detuning converge to 0 in average with a standard deviation of order (here ∼ 10 -2 ). In these simulations, we take the parameters T = 0. In theorem 2.1, this "dead-time" T is chosen mostly for technical reasons during the proof of theorem 2.1. It is related to the convergence time for the jump-free dynamics (2.1) starting with |g g| towards an 4 -neighborhood of its asymptotic regime. Since the convergence is exponential, T is linear in -log , this explain the fact that we can choose T around 1, even if is very small. In simulation, we have observed no convergence difference between T > 0 large (around 10) and T = 0.

Formal result.

The proof of the following theorem underlying the above simulations is given in section 4. we have the following theorem Theorem 2.1. Consider the Monte-Carlo trajectories described by (2.1)-(2.2). Assume a perfect jump-detection efficiency η = 1 and take the synchronizationfeedback presented in subsection 2.2 with We can fix then the dead-time T in the algorithm large enough so that:

u, v ∼ 1. (2.3) Assume that δ ∼ 2 , 4C 2 + 1 < 4ω 2 . ( 2
lim sup N →∞ E(∆ 2 N ) ≤ O( 2 ). (2.5)
Corollary 2.2. Under the assumptions of the Theorem 2.1, one has lim sup

N →∞ P (|∆ N | > √ ) ≤ O( ). (2.6)
This corollary results from the Markov inequality:

P (|∆ N | > √ ) = P (∆ 2 N > ) ≤ E(∆ 2 N ) .
Therefore applying (2.5), one deduces (2.6). Remark 1. The assumption (2.4) is not so restrictive. Indeed, for an a priori knowledge of the detuning magnitude, by taking a large enough frequency ω, one can ensure the relevance of this assumption.

3. The Λ-system.

Monte-Carlo trajectories.

Here, we consider a three time-scale system where a laser irradiates a 3-level Λ-system. The system is composed of 2 (fine or hyperfine) ground states |g 1 and |g 2 having energy separation in the radio-frequency or microwave region, and an excited state |e coupled to the lower ones by optical transitions at frequencies ω 1 and ω 2 . The decay times for the optical coherences are assumed to be much shorter than those corresponding to the ground state transitions (here assumed to be metastable).

Applying near-resonant laser fields and under the rotating wave approximations, while assuming the transition frequencies ω 1 and ω 2 much higher than the other frequencies, we can remove one of the time scales. Then, the quantum Markovian master equation of Lindblad type, modeling the evolution of a statistical ensemble of identical systems given by Figure 3.1, reads (see [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF], chapter 4, for a tutorial and exposure on such master equation):

d dt ρ = - ı [ H, ρ] + 1 2 2 j=1 (2Q j ρQ † j -Q † j Q j ρ -ρQ † j Q j ), (3.1)
where

H = ∆ 2 (|g 2 g 2 | -|g 1 g 1 |) + ∆ e + ∆ 2 (|g 1 g 1 | + |g 2 g 2 |) + Ω 1 |g 1 e| + Ω * 1 |e g 1 |) + Ω 2 |g 2 e| + Ω * 2 |e g 2 |)
and decoherence rates Γ 1 and Γ 2 much larger than the Rabi frequencies | Ω 1 |, | Ω 2 |, and the detuning frequencies ∆ and ∆ e , we may apply the singular perturbation theory to remove these fast and stable dynamics. Indeed, the dynamics corresponding to the excited state |e represent the fast dynamics and can be removed in order to obtain a system living on the 2-level subspace span{|g 1 , |g 2 }. The reduced Markovian master equation is still of Lindblad type and reads (see [START_REF] Mirrahimi | Singular perturbations and lindblad-kossakowski differential equations[END_REF] for a detailed proof)

Q j = Γ j
d dt ρ = - ı [H, ρ] + 1 2 2 j=1 (2L j ρL † j -L † j L j ρ -ρL † j L j ), (3.2)
where the reduced slow-Hamiltonian H is given, up-to a global phase change, by

H = ∆ 2 (|g 2 g 2 | -|g 1 g 1 |) = ∆ 2 σ z (3.3)
and

L j = γj |g j b e Ω with γj = 4 | Ω 1 | 2 + | Ω 2 | 2 (Γ 1 + Γ 2 ) 2 Γ j . (3.4)
Here, |b Ω represents the bright state (in the coherent population trapping)

b e Ω = Ω 1 | Ω 1 | 2 + | Ω 2 | 2 |g 1 + Ω 2 | Ω 1 | 2 + | Ω 2 | 2 |g 2 .
From now on, we deal with the 2-level system (3.2) instead of (3.1).

In order to characterize the Monte-Carlo trajectories of the system, we note that in the absence of the quantum jumps the reduced slow system evolves through the dynamics:

d dt ρ = -ı ∆ 2 [σ z , ρ] - 1 2 2 j=1 L † j L j , ρ + 2 j=1
Tr L † j L j ρ ρ, the Lindblad operators L j being given by (3.4). Since

L † j L j = γj b e Ω b e Ω we have, with γ = γ1 + γ2 , 1 γ d dt ρ = -ı ∆ 2γ [σ z , ρ] - 1 2 b e Ω b e Ω , ρ + Tr b e Ω b e Ω ρ ρ. (3.5)
At each time step dt the system may jump towards the state |g j g j | with a probability given by:

P jump (ρ → |g j g j |) = Tr L † j L j ρ dt = γj Tr b e Ω b e Ω dt, j = 1, 2. (3.6)
As it can be seen this probability is proportional to the population of the bright state b e Ω (this is actually the reason to call b e Ω the bright state). 3.2. The synchronization feedback. In this subsection, we consider the 2level system as the slow subsystems of the Λ-system presented in subsection 3.1. The only change we admit is that instead of constant amplitude laser fields Ω 1 and Ω 2 , we consider amplitudes varying with a frequency much lower than the decoherence rate Γ 1 and Γ 2 . Consider two positive constant Rabi-frequencies Ω 1 and Ω 2 (Ω 1 , Ω 2 Γ 1 , Γ 2 ) and take the following modulations

Ω 1 = Ω 1 + ı Ω 2 cos(ωt), Ω 2 = Ω 2 -ı Ω 1 cos(ωt) ( 3 . 7 )
with 1 and ω Γ 1 , Γ 2 . Following subsection 3.1, consider the orthogonal basis

|b = Ω 1 |g 1 + Ω 2 |g 2 Ω 2 1 + Ω 2 2 , |d = Ω 2 |g 1 -Ω 1 |g 2 Ω 2 1 + Ω 2 2 (3.8)
and set

γ j = 4 Ω 2 1 + Ω 2 2 (Γ 1 + Γ 2 ) 2 Γ j , for j = 1, 2 and γ = γ 1 + γ 2 . (3.9)
Here, |b = |b Ω (resp. |d = |d Ω ) denote the bright (resp. dark) state of the unperturbed non-oscillating system. If we replace ∆/γ by ∆, ω/γ by ω and γt by t in the stochastic dynamics (3.5) and jump probability (3.6), we get the quantum jump dynamics in the 1/γ scale, the optical-pumping scale, that reads:

• In the absence of quantum jumps, the systems density matrix ρ evolves through the dynamics • At each time step dt the system may jump on the ground state |g j (j = 1, 2) with a probability given by

d dt ρ = -ı ∆ 2 σ z , ρ - 1
p jump (ρ → |g j g j |) = γ j γ Tr (|b + ı cos(ωt)d b + ı cos(ωt)d| ρ) dt (3.11)
This quantum jump leads to the emission of a photon that will be detected with certain efficiencies: η j ∈ (0, 1] for the jumps to the state |g j . We assume a broad band detection process and thus the only information available with such measure is just the jump time. The type of jump (either to |g 1 or |g 2 ) is not available here. Thus the total jump probability reads

p jump = Tr (|b + ı cos(ωt)d b + ı cos(ωt)d| ρ) dt (3.12)
After each jump, ρ coincides with

|g 1 g 1 | or |g 2 g 2 |.
Similarly to the last subsection, we are interested in synchronizing the lasers with the system's frequencies and therefore make ∆ converge to zero. As for the twolevel case, we have a certain knowledge of the system's frequencies and therefore, we can always tune our lasers so that the detuning |∆| does not get larger than a fixed constant C.

Assume that 1 ω and consider the following synchronization algorithm: 1. Start with a certain detuning ∆ 0 with |∆ 0 | ≤ C and set the switching parameter S = 0 and the counter N = 0. 2. Wait for a first click and meanwhile evolve the switching parameter through and go back to the step 2. Here, we have chosen the correction gain δ 1. Similarly to the last subsection, we claim that, given any small , we can adjust the parameters ω large and δ small enough such that with the above algorithm, the detuning ∆ N converges in average to an O( 2 )-neighborhood of 0 with a deviation of order O( ).

Numerical simulations.

Let us now show the performance of this algorithm on some simulations. In the simulations of Figure 3.2, we apply the above synchronization strategy directly on the main Λ-system (and not on the slow 2-level subsystem).

We take the parameters C = 0.5, Ω 1 = Ω 2 = 1 (i.e. α = π/4), Γ 1 = Γ 2 = 3.0 (i.e. γ 1 = γ 2 = 0.6667), η 1 = 0.9, η 2 = 1.0, = 0.03, γ/ω = 0.05 and δ = 0.015. The simulations of Figure 2.1, then, illustrate 10 random trajectories of the system starting at ∆ 0 = .5 and ρ 0 = |d d| where |d = 1 √ 2 (|g 1 -|g 2 ). The first plot provides the number of clicks (quantum jumps) while the second one gives the evolution of the detuning ∆ N . As it can be noted, the detuning converges to a small neighborhood of zero within at most 1000 clicks. We can fix then the time constant T in the algorithm large enough so that: one can show that, the detuning reaches an O( )-neighborhood of 0 with a deviation of order √ . This is actually the assumption (3.17) that is relevant for the real system and that is considered in the simulations of the subsection 3.2. In fact, through this assumption the slow/fast approximation of [START_REF] Mirrahimi | Singular perturbations and lindblad-kossakowski differential equations[END_REF] is still available and therefore the system (3.10) is a relevant approximation of the real Λ-system.

lim sup N →∞ E(∆ 2 N ) ≤ O( 2 ). ( 3
Remark 3. Similarly to the simple two-level case, we have lim sup

N →∞ |E(∆ N )| ≤ O( 2 ),
and lim sup

N →∞ P (|∆ N | > √ ) ≤ O( ).
4. Proof of theorem 2.1. In order to simplify the notations, we assume

u = κ 1 , v = κ 2 , δ= 2 κ 3 ,
where κ 1 , κ 2 , κ 3 ∼ 1.

We proceed the proof of the Theorem 2.1 in two main steps:

Step 1 We consider the evolution in the absence of the quantum jumps through the system 2.1. We study the asymptotic regime of the dynamics. The constant time T will then be chosen to ensure the non-jumping system to reach an 4 -neighborhood of the limit regime.

Step 2 In the second step, applying the result of the first step, we calculate the conditional expectation of ∆ N +1 having fixed ∆ N . Finally, we sum up all these results in order to find the limit (2.5).

4.1.

Step 1: asymptotic regime of the non-jumping system. We are interested in the dynamics of the system In this aim, we apply the averaging theorem (see e.g. [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF], page 168). The un-perturbed dynamics, given by the first line of (4.1), admits an asymptotically stable hyperbolic equilibrium given by |g g|. Therefore, applying the averaging theorem, for small enough , the perturbed system (4.1) admits an asymptotically stable hyperbolic periodic orbit in an -neighborhood of |g g|. The main objective through the first step of the proof is to characterize this periodic orbit. Before going any further and in order to simplify the computations, we change the language to the Bloch sphere coordinates. Taking X = Tr (σ x ρ) , Y = Tr (σ y ρ) , Z = Tr (σ z ρ) , the system (4.1) reads

d dt ρ = -ı ∆ 2 σ z , ρ - 1
dX dt = -∆Y - 1 2 X + 1 2 (1 + Z)X + 2 κ 2 cos(ωt)Z, (4.2) dY dt = ∆X - 1 2 Y + 1 2 (1 + Z)Y -2 κ 1 Z, (4.3
)

dZ dt = - 1 2 (1 -Z)(1 + Z) + 2 κ 1 Y -2 κ 2 cos(ωt)X. (4.4)
We proceed the characterization of the periodic orbit through a perturbative development similar to the Kapitsa shortcut method (see e.g. [START_REF] Landau | Mechanics. Mir[END_REF], page 147). We are looking for a periodic orbit of the form   

X(t) Y (t) Z(t)    =   0 0 -1   +   X 1 (t) Y 1 (t) 0   + 2   X 2 (t) Y 2 (t) Z 2 (t)   + 3   X 3 (t) Y 3 (t) Z 3 (t)   + O( 4 ),
where for the first order approximation, (X 1 , Y 1 , Z 1 ), Z 1 is taken to be 0 as the vector must be orthogonal to the unit sphere at (0, 0, -1). Similarly to the Kapitsa method, we choose X 1 (t) and Y 1 (t) to be of the form

X 1 = α 1 cos(ωt) + β 1 sin(ωt) + γ 1 and Y 1 = α 2 cos(ωt) + β 2 sin(ωt) + γ 2 . (4.5)
Inserting (4.5) in (4.2), developing and considering just the first order terms while regrouping the sin and cos terms, we find the following system:

8 > > > > > > > > > < > > > > > > > > > : ∆β2 + 1 2 β1 -ωα1 = 0 -∆α2 - 1 2 α1 -ωβ1 = 2κ2 -∆β1 + 1 2 β2 -ωα2 = 0 ∆α1 - 1 2 α2 -ωβ2 = 0 and 8 > < > : ∆γ2 + 1 2 γ1 = 0 -∆γ1 + 1 2 γ2 = 2κ1
This system admits for solution

α1 = -4κ2 4ω 2 + 4∆ 2 + 1 Ξ , α2 = 8κ2∆ 4ω 2 -4∆ 2 -1 Ξ , β1 = -8κ2ω 4ω 2 -4∆ 2 + 1 Ξ , β2 = -32κ2ω∆ 1 Ξ , γ1 = -8κ1∆ 1 1 + 4∆ 2 , γ2 = 4κ1 1 1 + 4∆ 2 , (4.6)
where

Ξ = 16∆ 4 + 8∆ 2 + 1 + 8ω 2 + 16ω 4 -32∆ 2 ω 2 .
Let us go further and consider the second order terms now. Through the requirement of

X 2 + Y 2 + Z 2 = 1, one easily has Z 2 (t) = X 2 1 (t) + Y 2 1 (t) 2 .
Moreover, developing the two first equations of (4.2) up to the second order terms, we have

d dt X 2 = -∆Y 2 - 1 2 X 2 , d dt Y 2 = ∆X 2 - 1 2 Y 2 .
The functions X 2 (t) and Y 2 (t) being periodic the only possibility is

X 2 (t) = Y 2 (t) = 0.
Finally, we develop only the third equation of (4.2) up to the third order terms to obtain

d dt Z 3 = -Z 3 ,
and as Z 3 is periodic the only possibility is Z 3 (t) ≡ 0. Thus we have

Z(t) = -1 + 2 X 2 1 + Y 2 1 2 + O( 4 ).
This yields

Z(t) = -1+ 2 C 1 + 2 C 2 cos(2ωt)+ 2 C 3 sin(2ωt)+ 2 C 4 cos(ωt)+ 2 C 5 sin(ωt)+O( 4 ), (4.7)
where

C 1 = α 2 1 + α 2 2 4 + β 2 1 + β 2 2 4 + γ 2 1 + γ 2 2 2 , C 2 = α 2 1 + α 2 2 4 - β 2 1 + β 2 2 4 , C 3 = (α 1 β 1 + α 2 β 2 ), C 4 = 2(α 1 γ 1 + α 2 γ 2 ), C 5 = 2(β 1 γ 1 + β 2 γ 2 ). (4.8)
This periodic orbit being hyperbolically stable, we have proved the following lemma: Lemma 4.1. Consider the system (4.1) with ρ(0) = |g g|. For any small enough > 0, there exists a time constant T > 0 such that

Tr (σ z ρ(t)) = Z(t) + O( 4 ), for t ≥ T, (4.9)
where Z is given by (4.7).

We are now ready to attack the real quantum system with its jumps. The time constant T in the tuning algorithm of the subsection 2.2 is fixed through the Lemma 4.1.

4.2.

Step 2: conditional evolution of detuning. We are interested in the conditional expectations of ∆ N +1 and ∆ 2 N +1 knowing the value of ∆ N . Due to the synchronization algorithm ∆ N +1 = ∆ N -δ sin(ωt), the value of ∆ N +1 only depends on the phase φ = ωt mod (2π). We update ∆ N +1 only if the time interval with respect to the previous jump is large enough to ensure that the solution of the nojump dynamics (2.1) has reached its asymptotic regime. Thus (1 + Z)/2 = e|ρ|e is given by (4.7) and the jump probability defined by (2.2) depends only on φ = ωt mod (2π). Since the probability of having a phase φ during the update ∆ N to ∆ N +1 is proportional to e|ρ|e , this probability admits a density with respect to the Lebesgue measure on [0, 2π], given by

P ϕ,N = 1 2π + C 2 2πC 1 cos(2ϕ) + C 3 2πC 1 sin(2ϕ) + C 4 2πC 1 cos(ϕ) + C 5 2πC 1 sin(ϕ) + O( 2 ).
(4.10) Here the index N denotes the dependence through (4.8) and (4.6) of the constants {C j } j=1,...,5 on the detuning ∆ N .

Removing the threshold C in the algorithm by allowing the detuning to get large, the value of ∆ N +1 , having fixed ∆ N , is given as follows

∆ N +1 = ∆ N -δ sin(ϕ) ( 4 . 1 1 )
with a probability density P ϕ,N . Thus

E(∆ N +1 | ∆ N ) = ∆ N -δ 2π 0 sin(ϕ)P ϕ,N dϕ = ∆ N -δ C 5 2C 1 + O( 2 ).
Similarly for ∆ 2 N +1 one has

∆ 2 N +1 = ∆ 2 N -2δ sin(ϕ)∆ N + δ 2 sin 2 (ϕ) ( 4 . 1 2 )
with a probability density P ϕ,N . Inserting (4.10) into (4.12) and with δ = κ 3 2 , we have

E(∆ 2 N +1 | ∆ N ) = ∆ 2 N -2δ∆ N 2π 0 sin(ϕ)P ϕ,N dϕ + δ 2 2π 0 sin 2 (ϕ)P ϕ,N dϕ = ∆ 2 N -κ 3 2 C 5 C 1 ∆ N + O( 4 ), (4.13)
where E(∆ 2 N +1 | ∆ N ) denotes the conditional expectation of ∆ 2 N +1 having fixed ∆ N . Applying (4.8) and (4.6), we have

C 5 = 128κ 1 κ 2 ω(4ω 2 -4∆ 2 N -1) Ξ N (1 + 4∆ 2 N ) ∆ N . (4.14) with Ξ N = 16∆ 4 N + 8∆ 2 N + 1 + 8ω 2 + 16ω 4 -32∆ 2 N ω 2
. Now, taking into account the threshold C for the growth of the detuning ∆ N +1 , applying the assumption (2.4) and through some simple computations, we have

4ω 2 -4∆ 2 N -1 Ξ N (1 + 4∆ 2 N ) ≥ 4ω 2 -4C 2 -1 (16ω 4 + 8ω 2 + 8C 2 + 1)(1 + 4C 2 ) > 0, α 2 1 ≤ 16κ 2 2 , α 2 2 ≤ 16κ 2 2 C 2 , β 2 1 ≤ 64κ 2 2 ω 2 , β 2 2 ≤ 64κ 2 2 C 2 , γ 2 1 ≤ 16κ 2 1 , γ 2 2 ≤ 16κ 2 1 .
Note in particular that the last line imply

0 < C 1 ≤ 4κ 2 2 (1 + 5C 2 + 4ω 2 ) + 16κ 2 1 =: .
Therefore, we can change the equation (4.13) to the inequality

E(∆ 2 N +1 | ∆ N ) ≤ ∆ 2 N -2 ς∆ 2 N + O( 4 ) ( 4 . 1 5 )
where

ς = 64κ 1 κ 2 κ 3 ω 4ω 2 -4C 2 -1 (4C 2 + 1) (16ω 4 + 8ω 2 + 1 + 8C 2 ) > 0. (4.16)
Taking now the expectation of the both sides of (4.15), we have

E(∆ 2 N +1 ) ≤ 1 -2 ς E(∆ 2 N ) + O( 4 ), ( 4
E(∆ 2 N ) ≤ 1 -2 ς N ∆ 2 0 + O( 2 ),
where the positive constant ς is given in (4.16). This trivially finishes the proof of the Theorem 2.1 and we have lim sup

N →∞ E(∆ 2 N ) ≤ O( 2 ).
5. Proof of theorem 3.1. We proceed the proof of the Theorem 3.1 in a similar way to that of the Theorem 2.1. We assume

1/ω = 2 κ 1 , δ= 3 κ 2 ,
where κ 1 , κ 2 ∼ 1.

As for theorem 2.1, the proof admits 2 main steps:

Step 1 We consider the evolution in the absence of the quantum jumps through the system (3.10). We study the asymptotic regime of the dynamics. The constant time T will then be chosen to ensure the non-jumping system to reach an 3 -neighborhood of the limit regime.

Step 2 In the second step, applying the result of the first step, we calculate the conditional expectation of ∆ N +1 having fixed ∆ N . Finally, we sum up all these results in order to find the limit (3.16).

5.1.

Step 1: asymptotic regime of the non-jumping system. We are interested in the dynamics of the system (3.10). In this aim, we apply the Kapitsa shortcut method. Note that, Applying the Kapitsa method, the variable ρ may be developed as

ρ = ρ + O( ω ) = ρ + O( 3 ), (5.1)
where ρ represents the unperturbed trajectory. In the next part, we study the dynamics of the unperturbed part ρ.

5.1.1. Unperturbed no-jump dynamics on the Bloch Sphere. The unperturbed part, ρ, satisfies the dynamics:

d dt ρ = -ı ∆ 2 [σ z , ρ] - 1 2 |b b| + 2 2 |d d| , ρ + Tr |b b| + 2 2 |d d| ρ ρ. (5.2)
The second point comes from the negative invariance of S 2 + , positive invariance of S 2 -and the Poincare-Bendixon theory for autonomous systems on the sphere: an hypothetic limit cycle C cannot intersect S 2 + and S 2 -simultaneously and thus must be included in S 2 + or S 2 -; strict surface dilation (resp. contraction) in S 2 + (resp. S 2 -) is incompatible with the existence of C ⊂ S 2 + (resp. C ∈ S 2 -) because of the Gauss theorem; since there is no limit cycle and since there exist only two equilibrium points, M + exponentially unstable in all direction and M -exponentially stable, the attraction domain of M -is the all sphere without the unstable point M + .

Remark 4. It is tempting to conjecture that, for all values of the parameters p and β ensuring two separate equilibria M + and M -defined here above, we have a quasi-global convergence towards M -, the locally exponentially stable equilibrium. This is not true since for β = π/2 and |p| > 1 we have the coexistence of the periodic orbit X 2 + Y 2 = 1 with Z = 0 with the two equilibria

X ± = 0, Y ± = - 1 p , Z ± = ± 1 - 1 p 2
and thus a trajectory starting with Z > 0 remains with Z > 0 for all the time and cannot converge to M -since Z -< 0.

5.1.2. Perturbed no-jump dynamics. Under the assumption of the Theorem 3.1 on C, we know that the detuning ∆ can not get larger than 1/2 and therefore in the above notations p < 1. This trivially implies |p sin β| < 1 and therefore we are in the settings of the second point of the Lemma 5.1. Hence, the system (5.3) admits two distinct equilibria ρ -and ρ + given by (5.6) in the Bloch sphere coordinates. Moreover the trajectories of the system, not starting at ρ + , necessarily converge towards the equilibria ρ -.

Applying this characterization of the dynamics, one easily gets Lemma 5.2. Under the assumption of the Theorem 3.1 for |b and the assumption |∆| < 1 2 , and for small enough , the system (5.2) admits a locally asymptotically stable equilibrium ρ of the form

ρ = ρ -+ 2 ρ 1 + O( 4 ),
where ρ -is given by (5.6) in the Bloch sphere coordinates. Moreover the trajectories starting at |g 1 g 1 | or |g 2 g 2 | converge towards this equilibrium.

For the proof of this lemma, note that, as α = 0, |g 1 g 1 | and |g 2 g 2 | are not the equilibriums of the system (5.3). Thus, taking small enough, they will not be an equilibrium of (5.2) neither, and therefore the trajectories starting at |g 1 g 1 | and |g 2 g 2 | necessarily converge towards the perturbed asymptotically stable equilibrium ρ .

The Lemma 5.2, together with (5.1), implies that the trajectories ρ(t) of the system (3.10) starting at |g 1 g 1 | or |g 2 g 2 | converge to an O( 3 )-neighborhood of ρ -+ 2 ρ 1 .

We may therefore choose the time constant T in the synchronization algorithm of the Subsection 3.2 such that ρ(t) = ρ -+ 2 ρ 1 + O( 3 ), ∀t > T.

(5.7)

5.2.

Step 2: conditional evolution of detuning. Similarly to the last section, we are interested in the conditional expectations of ∆ N +1 and ∆ 2 N +1 knowing the value of ∆ N . Due to the synchronization algorithm ∆ N +1 = ∆ N -δ sin(2α) cos(ωt), the value of ∆ N +1 only depends on the phase φ = ωt mod (2π). We update ∆ N +1 only if the time interval with respect to the previous jump is large enough to ensure that the solution of the no-jump dynamics (3.10) has reached its asymptotic regime (5.7). Thus Tr (|b + ı cos(ωt)d b + ı cos(ωt)d| ρ) is given inserting the limit (5.7). The jump probability defined by (3.12) depends only on φ = ωt mod (2π). Since the probability of having a phase φ during the update ∆ N to ∆ N +1 is proportional to Tr (|b + ı cos(φ)d b + ı cos(φ)d| ρ), this probability admits a density with respect to the Lebesgue measure on [0, 2π], given by where the index N in P φ,N denotes, in particular, the dependence of ρ -and ρ 1 to the detuning ∆ N . Furthermore, the constant Z N ( ) > 0 is a normalization constant given by the integral over [0, 2π] of the term between parentheses. In particular, one easily has 0 < O( 2 ) < Z N ( ) < O(1).

Removing the threshold C in the algorithm by allowing the detuning to get large, the value of ∆ N , having fixed ∆ N , is given as follows ∆ N +1 = ∆ N -δ sin(2α) cos(ϕ) ( 5 . 9 )

with a probability density P ϕ,N . Similarly for ∆ 2 N +1 one has ∆ 2 N +1 = ∆ 2 N -2δ sin(2α) cos(ϕ)∆ N + δ 2 sin 2 (2α) cos 2 (ϕ), (5.10) with a probability density P ϕ,N . Inserting (5.8) into (5.10), we have

E(∆ 2 N +1 | ∆ N ) = ∆ 2 N -π δ Z N ( ) Θ N 2 + O δ 2 Z N ( ) + O δ 3 Z N ( ) , (5.11)
where

Θ N = 4∆ 2 N + 1 -(4∆ 2 N -1) 2 + 16∆ 2 N cos 2 (2α).
Note, in particular, that Θ N > 0 as α = 0. Now, taking into account the threshold C for the growth of the detuning ∆ N +1 , we can easily see that

Θ N = 16∆ 2 N sin 2 (2α) 2 + 16∆

Fig. 1 . 1 .

 11 Fig. 1.1. The basic extremum seeking feedback loop for a non-linear static system y = f (u) (s = d dt is the Laplace variable and (h, k, a, ω) are constant design parameters).

1 .

 1 Start with a certain detuning ∆ 0 with |∆ 0 | ≤ C and set the switching parameter S = 0 and the counter N = 0. 2. Wait for a first click and meanwhile evolve the switching parameter through d dt S = 1. 3. If the click happens while S ≤ T then switch the parameter S to zero and go back to the step 2. 4. If the click happens while S > T then switch the parameter S to zero, change the counter value to N + 1, correct the detuning ∆ N as follows:

Figure 2 .

 2 Figure 2.1 correspond to 10 random trajectories of the system starting with the same initial condition ρ 0 = |g g| and detuning ∆ 0 = C = 1/2.The first plot provides the number of clicks (quantum jumps) while the second one gives the evolution of the detuning ∆ N . As it can be noted, the detuning converge to 0 in average with a standard deviation of order (here ∼ 10 -2 ). In these simulations, we take the parameters T = 0. In theorem 2.1, this "dead-time" T is chosen mostly for technical reasons during the proof of theorem 2.1. It is related to the convergence time for the jump-free dynamics (2.1) starting with |g g| towards an 4 -neighborhood of its asymptotic regime. Since the convergence is exponential, T is linear in -log , this explain the fact that we can choose T around 1, even if is very small. In simulation, we have observed no convergence difference between T > 0 large (around 10) and T = 0.

Fig. 2 . 1 .

 21 Fig. 2.1. The detuning evolution versus the number of quantum jumps for the synchronization algorithm of Subsection (2.2).

  Fig. 3.1. Relevant energy levels, transition and decoherence terms for the Λ-system (3.1).

2

 2 {|b + ι cos(ωt)d b + ι cos(ωt)d| , ρ} + Tr (|b + ι cos(ωt)d b + ι cos(ωt)d| ρ) ρ. (3.10) with |b = cos α |g 1 + sin α |g 2 , |d = -sin α |g 1 + cos α |g 2 (α ∈ 0, π 2 is the argument of Ω 1 + ıΩ 2 ).

d dt S = 1 . 3 .

 13 If the click happens while S ≤ T then switch the parameter S to zero and go back to the step 2. 4. If the click happens while S > T then switch the parameter S to zero, change the counter value to N + 1, correct the detuning ∆ N as follows: ∆ N +1 = ∆ N -δ sin(2α) cos(ωt) i f|∆ N -δ sin(2α) cos(ωt)| ≤ C, ∆ N +1 = C, otherwise

Fig. 3 . 2 .

 32 Fig. 3.2. The detuning evolution versus the number of quantum jumps for the synchronization algorithm of Subsection (2.2).

3. 4 .Moreover, we assume perfect detection efficiency η 1 = η 2

 412 Formal result. The proof of the following theorem is given in section 5. Theorem 3.1. Consider the Monte-Carlo trajectories described by (3.10)-(3.11) where |b = cos α |g 1 + sin α |g 2 with 0

2

 2 {|e e| , ρ} + Tr (|e e| ρ) ρ + [κ 1 σ x + κ 2 cos(ωt)σ y , ρ] . (4.1)

2

 2 |b + ι cos(ωt)d b + ι cos(ωt)d| = |b b| + 2 |d d| + ι cos(ωt)(|b d| -|d b|) + 2 2 cos(2ωt) |d d| .

P φ,N = 1 Z

 1 N ( ) Tr |b b| ρ -+ 2 cos 2 (ϕ)Tr |d d| ρ -+ 2 Tr (|b b| ρ 1 ) -cos(ϕ)Tr σ y ρ -+ O( 3 ) , φ∈ [0, 2π), (5.8)

  Considering the Monte-Carlo trajectories described by (2.1)-(2.2) and applying the synchronization algorithm of Subsection 2.2, we have

	yield to the following lemma:
	Lemma 4.2.
		.17)
	where we have applied the relation E(E(X|Y )) = E(X). Simple computations with
	1 + (1 -2 ς) + . . . + (1 -2 ς) N ≤	1 2 ς

* This work was partially supported by the "Agence Nationale de la Recherche" (ANR), Projet Blanc CQUID number 06-3-13957.

In order to study the asymptotic behavior of (5.2), we begin with the case ≡ 0 and we study first the system

{|b b| , ρ} + Tr (|b b| ρ) ρ.

(5.

3)

The dynamics in the Bloch sphere coordinates, X = Tr (σ x ρ), Y = Tr (σ y ρ), Z = Tr (σ z ρ), are given as follows: we have the following dynamical system

(5.4) living on R 3 . Since the two transformations (X, Y, Z, p, β) → (-X, -Y, -Z, β + π) and (X, Y, Z, p, β) → (X, Y, -Z, π -β) leave the above equations unchanged, we can always consider, for the study of this dynamical system versus the parameter p and β, that the angle β ∈ [0, π 2 ] and p ∈ R. Since X 2 + Y 2 + Z 2 = 1 is invariant, these 3 differential equations define a dynamical system on the two dimensional sphere S 2 , the Bloch sphere.

Consider the element of Euclidian length δs 2 = (δX) 2 + (δY ) 2 + (δZ) 2 and its evolution along the dynamics defined by (5.4) on S 2 . We have δs 2 = 2 (δXδX + δY δY + δZδZ ) with (δX , δY , δZ ) given by the first variation of (5.4): δX = -pδY + (sin βX + cos βZ)δX + X(sin βδX + cos βδZ) δY = pδX + (sin βX + cos βZ)δY + Y (sin βδX + cos βδZ) δZ = (sin βX + cos βZ)δZ + Z(sin βδX + cos βδZ).

Since XδX + Y δY + ZδZ = 0, we obtain the simple relation δs 2 = 2(sin βX + cos βZ)δs 2 .

(5.5)

Thus S 2 splits into two hemispheres: the open hemisphere S 2 + corresponding to sin βX + cos βZ > 0 and where the dynamics is a strict dilation in any direction; the open hemisphere S 2 -corresponding to sin βX + cos βZ < 0 where the dynamics is a strict contraction (see [START_REF] Lohmiler | On metric analysis and observers for nonlinear systems[END_REF]). The boundary between these two hemispheres is given by the intersection of the plane sin βX + cos βZ = 0 with S 2 . We have

Thus, when |p sin β| ≤ 1, S 2 + is negatively invariant and S 2 -positively invariant. Assume first that p = 0 and β ∈]0, π 2 [ and consider the equilibrium on S 2 . Simple computations prove that we have only two equilibria associated to the point

-of coordinates (X + , Y + , Z + ) and (X -, Y -, Z -) given by

(5.6)

When p = 0, the above formula can be extended by continuity to get the two equilibria:

When β = 0, similarly we obtain the two equilibria

When β = π 2 the situation is slightly different: • for |p| < 1 we have two equilibria

• for |p| = 1 we have a unique equilibrium

• for |p| > 1 we have two equilibria

With all the above properties we deduce the following lemma Lemma 5.1. Consider the differential equations (5.4) defining an autonomous dynamical system on the Bloch Sphere S 2 with the parameters p ∈ R and β ∈ [0, π 2 ]. Then 1. for (|p|, β) = (1, π/2), we have two distinct equilibrium points M + and M - defined here above by (5.6). The two Lyapounov exponents at M + (resp. M -) have strictly positive (resp. negative) real parts: M + is locally exponentially unstable (in all direction) and M -is locally exponentially stable. 2. For |p sin β| < 1, all the trajectories (except the unstable equilibrium M + ) converge asymptotically to the equilibrium point M -that is exponentially stable: the attraction region of M -is S 2 /{M + }. Proof of Lemma 5.1. The first point result from (5.5) applied locally around M + and M -and from sin βX + + cos βZ + > 0 whereas sin βX -+ cos βZ -< 0.

where δ = κ 2 3 , we have

(5.13)

Taking now the expectation of the both sides, we have

where we have applied the relation

the system (5.14) is a contracting one and a similar computation to that of the last sub-section yields the following lemma: Lemma 5.3. Considering the Monte-Carlo trajectories described by (3.10)-(3.11) and applying the synchronization algorithm of the Subsection 3.2, we have

where the positive constant ς is given in (5.12). This trivially finishes the proof of the Theorem 3.1 and we have lim sup

Furthermore, note that as the detuning ∆ N gets near 0, the normalization constant Z N ( ) converges to an O( 2 ). This, in particular, leads to a higher convergence rate in the Lemma 5.3.

6. Concluding remark. For 2-level and Λ-systems, we have proposed synchronization feedback loops to lock the probe-frequency to the atomic one. Simulations illustrate the interest and robustness of such simple feedbacks. Theorems 2.1 and 3.1 constitute a first tentative proving the stability and convergence under assumptions that seem to be conservative since they can be relaxed in simulations. In particular assumptions relative to 100% detection efficiently and relative to the minimum timedelay T between two successive jumps are not fulfilled in simulations of figures 2.1 and 3.2. We have observed no difference when they are satisfied. Thus, we conjecture that extension of theorems 2.1 and 3.1 to partial detection and T = 0.