1 Singular perturbations and Lindblad-Kossakowski differential equations Mazyar Mirrahimi and Pierre Rouchon Abstract—Weconsideranensembleofquantumsystemswhose we loose the physical interpretation of the master equation average evolution is described by a density matrix, solution of in terms of Hamiltonian and jump operators, explained in [6, a Lindblad-Kossakowski differential equation. We focus on the chapter 4]. special case where the decoherence is only due to a highly 8 unstable excited state and where the spontaneously emitted 0 The main contribution of this note is to propose a more photons are measured by a photo-detector. We propose a sys- 0 intrinsic elimination of the fast part of the dynamics by tematic method to eliminate the fast and asymptotically stable 2 dynamics associated to the excited state in order to obtain using only matrix manipulations for systems with a structure n another differential equation for the slow part. We show that sketched on figure 1. The main theoretical guide is the a this slow differential equation is still of Lindblad-Kossakowski geometric theory of singularly perturbed differential systems J type, that the decoherence terms and the measured output initiated in [5] and center manifold techniques to approximate 0 depend explicitly on the amplitudes of quasi-resonant applied the invariant slow manifold [7], [3]. Such theoretical guides 1 field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, have been already used in [4] in the context of reduction of ] we also provide a physical interpretation of the result in the kineticscombustionschemes.Theseguidesavoidheretheuse h context of coherence population trapping via dark states and of the coherence vector and provide a slow dynamics that is p decoherence-free subspaces. Numerical simulations illustrate the also a Markovian master equation of Lindbald-Kossakowski - accuracy of the proposed approximation for a 5-level systems. h type with a slow Hamiltonian and slow jump operators. This t Index Terms—Quantum systems, Lindblad-Kossakowski mas- slow master equation describes the dynamics of the density a ter equation, singular perturbations, optical pumping, coherent m matrix of the open quantum system that lives in the Hilbert population trapping, adiabatic approximation. space spanned by the stable states. As far as we know, such [ formulationoftheslowdynamicsisnew,eveninthephysicist 1 I. INTRODUCTION community, and could be of some interest for the control. In v particular, the controls appear explicitly in the decoherence 2 Under the usual assumptions of optical pumping and/or co- terms and the output map. 0 herentpopulationtrapping,theLindblach-Kossakowskimaster 6 equation describing the dynamics of the density operator Thenoteisorganizedasfollows.SectionIIisdevotedtothe 1 . admits multiple time-scales. In this paper, we are studying modeling of systems depicted on figure 1, to the three time- 1 the fast/slow structure resulting from a separation between scales structure and to the elimination (by averaging, i.e., by 0 8 • the life-times of the excited and unstable states assumed the rotating wave approximation usually used by physicists) 0 to be short. of the fastest time-scale attached to the transition frequencies v: • the oscillation periods, assumed to be long, associated to between the stable and unstable states. The resulting model, i the energies of the other stable states and to the Rabi equation (8) with complex value controls Ωk and measured X pulsation generated by the control, coupling the unstable outputy,stilladmitstwotime-scales,anasymptoticallystable r and stable states in a quasi-resonant way. fast part and a slow part. Extraction of the slow part is the a Usually, the elimination of the fast dynamics is performed in object of section III where the approximation Theorem 1 is terms of coherence vector gathering in a single column the proved. For readers not interested by these technical devel- coefficients of the density matrix. In this form, the system opments, we have summarized at the end of this section III is not written in a standard form [7, chapter 9] (also called the main formula for deriving the slow master equation (23) Tikhonovnormalform)withaclearsplittingofthecoherence from the original slow/fast one. In section IV, we compute vector into two sub-vectors: a fast sub-vector and a slow one. the slow approximation when the Hamiltonian H corresponds Some tedious linear algebra and changes of variables are then to (8) and provides physical interpretations in terms of slow needed to put the system into the standard form in order to Hamiltonianandslowjumpoperatordependingdirectlyonthe perform the adiabatic (quasi-static approximation) elimination controlamplitudesΩk.InsectionV,wecompare,numerically of the fast dynamics. Moreover with such coherence vector on a five-level system, the slow/fast master equation with the slow one. M.MirrahimiiswiththeSISYPHEteam,INRIARocquencourt,Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France. e-mail: maz- A preliminary version of these results can be found in [8]. [email protected] The authors thank Guilhem Dubois from LKB for several P. Rouchon is with the Centre Automatique et Syste`mes, Ecole des discussions on the physics underlying coherence population Mines de Paris, 60 Bd Saint-Michel, 75272 Paris cedex 06, France, e-mail: [email protected] trapping. 2 II. THETHREETIME-SCALEMASTEREQUATION theµk beingcouplingandconstantparameters.Moreover,the quantumjumpoperatorsQ correspondingtothespontaneous Such master equations typically describe coherent pop- k emission from the state |e(cid:105) towards |g (cid:105) are given as follows ulation trapping when a laser irradiates an (N + 1)-level k system [2], [1]. The system is composed of N (fine or Q =|g (cid:105)(cid:104)e|. hyperfine) ground states (|g (cid:105))N having energy separations k k k k=1 in the radio-frequency or microwave region, and an excited One easily has the following relations state |e(cid:105) coupled to the lower ones by optical transitions at frequencies(ω )N (seeFigure1).Naturally,thedecaytimes Q Q =0, Q†Q =P =|e(cid:105)(cid:104)e| ∀k (cid:54)=l∈{1,··· ,N}. k k=1 k l k k for the optical coherences are assumed to be much shorter (3) than those corresponding to the ground state transitions (here The transition frequencies, ω =λ−λ (where λ and λ are k k k assumed to be metastable). the eigenvalues of H0 corresponding to the energy levels |e(cid:105) (cid:126) and |g (cid:105), respectively), are supposed to be much larger than k the decoherence rates Γ . The control field u(t) is assumed k to be in the quasi-resonant regime with respect to the natural frequencies of the system: N (cid:88) u(t)= ukeı(ωk−δk)t+u∗ke−ı(ωk−δk)t (4) k=1 wherethecomplexamplitudesu ∈Carevaryingslowlyand k where δ are the small de-tuning frequencies. We have thus k three different time scales: 1) The very fast time-scale associated to the optical fre- quencies ω . k 2) The fast time-scale associated to the life times of the excited state |e(cid:105), Γ . k 3) The slow time-scale associated to the laser amplitude |µ u | and to the other atomic transition frequencies k k Fig. 1. relevant energy levels, transitions and decoherence rates for the ω =ω −ω , k (cid:54)=l. kl k l consideredmodel(1) We are interested here by the slow time-scale of system (1) wherethecontrolu(t)isgivenby(4)withthefollowingtime- The quantum Markovian master equation of Lindblad- scales separation: Kossakowski type, modeling the evolution of a statistical (cid:12) (cid:12) ensembleofidenticalsystemsgivenbyFigure1,reads(see[6, (cid:12)d (cid:12) chapter 4] for a tutorial and up-to-date exposure on such |ωkl|,|µkuk|(cid:28)Γk(cid:48) (cid:28)ωk(cid:48)(cid:48) and (cid:12)(cid:12)dtuk(cid:12)(cid:12)(cid:28)Γk(cid:48)|uk| master equations): with k,l,k(cid:48),k(cid:48)(cid:48) ∈{1,...,N}, k (cid:54)=l. d ı Elimination of the fastest time-scales is standard. It corre- ρ=− [H +uH ,ρ] dt (cid:126) 0 1 sponds to the so-called rotating wave approximation and can 1(cid:88)N be justified by averaging techniques. This is not the object + 2 Γk(2QkρQ†k−Q†kQkρ−ρQ†kQk) (1) of this note and thus we just recall here the resulting master k=1 equation: N (cid:88) (cid:16) (cid:17) y = Γ Tr Q†Q ρ (2) N k k k dρ=−ı[H,ρ]+ 1(cid:88)Γ (2Q ρQ† −Q†Q ρ−ρQ†Q ). k=1 dt (cid:126) 2 k k k k k k k where u ∈ R is the controlled input (laser field) and y ≥ 0 k=1 (5) is the measured output (number of photons per time unit Calculating the secular terms of ue−ıH(cid:126)0tH1eıH(cid:126)0t, the effec- spontaneously emitted from the excited state |e(cid:105)). Here the tive Hamiltonian H is given as follows: Hermitian operators H and H are, respectively, the free 0 1 Hamiltonian and the interaction Hamiltonian with a coherent H (cid:88)N source of photons u(t)∈R: (cid:126) = δk|gk(cid:105)(cid:104)gk|+Ωk|gk(cid:105)(cid:104)e|+Ω∗k|e(cid:105)(cid:104)gk|. (6) k=1 N H0 =λ|e(cid:105)(cid:104)e|+(cid:88)λ |g (cid:105)(cid:104)g | with Ωk = µkuk. Note that the measured output remains (cid:126) k k k unchanged k=1 H1 =(cid:88)N µ (|g (cid:105)(cid:104)e|+|e(cid:105)(cid:104)g |) (cid:32)(cid:88) (cid:33) (cid:32)(cid:88) (cid:33) (cid:126) k k k y = Γk Tr(Pρ)= Γk Tr(|e(cid:105)(cid:104)e|ρ). (7) k=1 k k 3 We are led to the following master equation where 1 only appears in first equation defining dρ . There- (cid:15) dt f (cid:34) N (cid:35) fore ρf is associated to the fast part of the dynamics and ρs ddtρ=−ı k(cid:88)=1δk|gk(cid:105)(cid:104)gk|+Ωk|gk(cid:105)(cid:104)e|+Ω∗k|e(cid:105)(cid:104)gk| , ρ represents (cid:16)thPe sNkl=o1wΓkp(cid:17)art. The fast part is asymptotically stable because− (ρ +Pρ P)definesanegativedefinite +(cid:88)N Γk(2(cid:104)e|ρ|e(cid:105)|g (cid:105)(cid:104)g |−|e(cid:105)(cid:104)e|ρ−ρ|e(cid:105)(cid:104)e|) super-operator o2(cid:15)n the spface offHermitian operators: 2 k k k=1 Tr(−(ρ +Pρ P)ρ )=−((cid:107)ρ (cid:107)2+(cid:107)Pρ P(cid:107)2). (8) f f f f f (cid:32) N (cid:33) Moreover its inverse is given by : (cid:88) y = Γ Tr(|e(cid:105)(cid:104)e|ρ) k 1 X (cid:55)→X− PXP. (14) k=1 2 where the Ω ’s are the slowly varying complex amplitudes k Here we can apply the slow manifold approximation (25) (controlledinputs),theδ ’sarethelaserde-tuningsandwhere k describedintheAppendixA.Computingthefirstorderterms, the two time-scales separation results from: wefindthefollowingapproximationforρ withrespecttoρ : f s (cid:12) (cid:12) (cid:12)d (cid:12) −2ı (cid:15) |δk|,|Ωk|(cid:28)Γk(cid:48) and (cid:12)(cid:12)dtΩk(cid:12)(cid:12)(cid:28)Γk(cid:48)|Ωk| ρf = (cid:126)(cid:16)(cid:80)N Γ (cid:17) (PHρs−ρsHP)+O((cid:15)2). (15) k=1 k for k,k(cid:48) ∈{1,...,N}. Inserting now the equations (11) into the equation (13), we have: III. SLOW/FASTREDUCTION d ı ı ρ =− (1−P)[H,ρ ](1−P)− (1−P)[H,ρ ](1−P) WecanthereforetakeΓk =Γk/(cid:15)where(cid:15)isasmallpositive dt s (cid:126) s (cid:126) f parameter.Thuswehaveamasterequationwiththefollowing + ı (1−P)XN Γ [H,Q ρ Q†](1−P) structure: (cid:126)“PN Γ ” k k f k k=1 k k=1 N ddtρ=−(cid:126)ı[H,ρ]+k(cid:88)=1Γ2(cid:15)k(2QkρQ†k−Q†kQkρ−ρQ†kQk), (9) − (cid:126)“PNı Γ ”XN Γk Qk[H,ρf]Q†k, k=1 k k=1 where Γ ’s and H (given by (6) for example) have the same k where we have used (3) and orders of magnitude but where (cid:15)>0 is a small parameter. Define, with P =|e(cid:105)(cid:104)e|, Qkρs =ρsQ†k =0. (16) ρ =Pρ+ρP −PρP Applying now the first order approximation (15), and after f some simple but tedious computations, we have N 1 (cid:88) ρs =(1−P)ρ(1−P)+ (cid:16)(cid:80)Nk=1Γk(cid:17) k=1Γk QkρQ†k. ddtρs =−(cid:126)ı(1−P)[H,ρs](1−P) (10) 2(cid:15) (cid:16) − (1−P)HPH(1−P)ρ (cid:16) (cid:17) s We have (cid:126)2 (cid:80)Nk=1Γk (cid:17) ρ=ρ +ρ − 1 (cid:88)N Γ Q ρ Q† (11) +ρs(1−P)HPH(1−P) s f (cid:16)(cid:80)N Γ (cid:17) k k f k N k=1 k k=1 + 4(cid:15) (cid:88)Γ Q Hρ HQ† +O((cid:15)2). (17) (cid:16) (cid:17)2 k k s k and therefore ρ (cid:55)→ (ρf,ρs) is a bijective map. This map is a (cid:126)2 (cid:80)Nk=1Γk k=1 sort of “change of variables” decoupling the slow part from the fast part of the dynamics. Note that, in the slow part, ρ , Here, we have in particular applied (16) as well as the fact s we have somehow removed the fast dynamics associated to that QkP =Qk. Continuing the computations, we have the optical state |e(cid:105). Indeed, this change of variable leads to a d ı standard form: ρ =− [H,ρ ]+ dt s (cid:126) s (cid:16) (cid:17) ddtρf =− (cid:80)Nk2=(cid:15)1Γk (ρf +PρfP) 2(cid:15)(cid:88)N Γk(cid:16)2QkρsQ†k−Q†kQkρs−ρsQ†kQk(cid:17) (18) ı k=1 − (P[H,ρ]+[H,ρ]P −P[H,ρ]P), (12) (cid:126) where we have defined d ı(cid:126) ρ =(1−P)[H,ρ](1−P) H =(1−P)H(1−P) (19) dt s + 1 (cid:88)N Γ Q [H,ρ]Q†. (13) and 1 (cid:16)(cid:80)Nk=1Γk(cid:17)k=1 k k k Qk = (cid:126)(cid:16)(cid:80)Nk=1Γk(cid:17)(1−P)QkH(1−P). (20) 4 (cid:95) Note that Denoting by δρ =ρ −ρ , we have s (cid:101)s s † 1 d (cid:18) (cid:95) 2(cid:19) QkQk = (cid:126)2(cid:16)(cid:80)N Γ (cid:17)2(1−P)HPH(1−P), dtTr δρs ≤ k=1 k (cid:88)N (cid:18) (cid:16) (cid:95) (cid:95) (cid:17) (cid:18) (cid:95) 2(cid:19)(cid:19) 8(cid:15) Γ Tr Q δρ Q†δρ −Tr Q†Q δρ which, in particular, allows us passing from (17) to (18). k k s k s k k s The situation is different for the measured output y. We k=1 (cid:16) (cid:95) (cid:17) have: +Tr O((cid:15)2)δρ . s (cid:16)(cid:80)N Γ (cid:17) (cid:16)(cid:80)N Γ (cid:17) This, together with Cauchy-Schwartz inequality, implies k=1 k k=1 k y(t)= (cid:15) −T2rı(Pρ)= (cid:15) Tr(Pρf) Tr(cid:18)δ(cid:95)ρs2(t)(cid:19)≤Tr(cid:18)δ(cid:95)ρs2(0)(cid:19)+(cid:15)L(cid:90) tTr12 (cid:20)δ(cid:95)ρs4(τ)(cid:21)dτ = Tr(P(PHρ −ρ HP))+O((cid:15)). 0 (cid:126) s s (cid:90) t (cid:20) (cid:95) 2 (cid:21) +(cid:15)2C Tr12 δρs (τ) dτ, But Tr(P(PHρs−ρsHP)) = 0. We should therefore con- 0 sider the second order terms otherwise the first order approxi- (cid:95) 2 mationyieldsy =0.UsingtheAppendixA,simplebuttedious where L and C are positive constants. Note that, δρs being computations end up by the following natural approximation: definite positive, we have (cid:20) (cid:95) 4 (cid:21) (cid:20) (cid:95) 2 (cid:21) y(t)=4(cid:15)(cid:16)(cid:88)N Γ (cid:17)Tr(cid:0)Pρ (cid:1)+O((cid:15)2), (21) Tr21 δρs (τ) ≤Tr δρs (τ) . k s (cid:115) k=1 (cid:20) (cid:95) 2 (cid:21) Therefore, noting ξ = Tr δρ (τ) , we have s where we have defined † 1 (cid:90) t (cid:90) t P =QkQk = (cid:16) (cid:17)2(1−P)HPH(1−P). ξ2(t)≤ξ2(0)+(cid:15)L ξ2(τ)dτ +(cid:15)2C ξ(τ)dτ. (cid:126)2 (cid:80)Nk=1Γk 0 0 Denoting ζ =ξ(t)+ C (cid:15), some simple computations lead to 2L In order to derive (21), we only need to apply (26) with the appropriate values of the functions given in (12) and (13) and (cid:90) t C2 C2 ζ2(t)≤2ξ2(0)+2(cid:15)L ζ2(τ)dτ − (cid:15)3 t+ (cid:15)2 the inverse map given in (14). 2L 2L2 0 We can therefore prove the following theorem: C2 (cid:90) t ≤ξ2(0)+ (cid:15)2+2(cid:15)L ζ2(τ)dτ. Theorem 1. ConsiderρthesolutionoftheMarkovianmaster 2L2 0 equation (9) and ρs the solution of the slow master equa- Applying the Gronwall lemma, we have tion (18) with ((cid:112)19) and (20). Assume for the initial states (cid:20) C2 (cid:21) |ρ(0)−ρ (0)|= Tr((ρ(0)−ρ (0))(ρ(0)−ρ (0)))=O((cid:15)). ζ2(t)≤ ξ2(0)+ (cid:15)2 e2(cid:15)Lt. s s s 2L2 Then Noting that, by the Theorem’s assumption, ξ(0) = O((cid:15)), we (cid:112) |ρ(t)−ρs(t)|= Tr((ρ(t)−ρs(t))(ρ(t)−ρs(t)))=O((cid:15)) have ζ(t) = O((cid:15)) on a time scale of t ∼ 1/(cid:15). As ξ(t) = ζ(t)+O((cid:15)), this trivially finishes the proof. on a time scale t∼1/(cid:15). Fromapracticalpointofview,themainresultofthissection Moreover the output y(t) of the system (given by (7)) may is as follows. The correct slow approximation (also called by be written as in (21). physicistsadiabaticapproximation)ofthesystemdescribedby Remark 1. Note that, the approximation of this theorem is d ı ρ=− [H,ρ] stronger than the usual one only ensuring an error of order dt (cid:126) O((cid:15))onafinitetimeT ratherthanonatimescaleoft∼T/(cid:15). N This stronger result is due to the Hamiltonian structure of the +(cid:88)Γk (cid:16)2Q ρQ† −Q†Q ρ−ρQ†Q (cid:17) 2 k k k k k k dominant part of the dynamics. k=1 with Q = |g (cid:105)(cid:104)e| and where the Γ ’s are much larger than Proof:Applying(11)andthesingularperturbationtheory k k k (cid:16) (cid:17) of the appendix A, we have ρ(t)=ρ(cid:101)s(t)+O((cid:15)) where H(cid:126) and where the output reads y = (cid:80)kΓkTr Q†kQkρ , is given by d ı ρ =− [H,ρ ]+ d ı dt(cid:101)s (cid:126) (cid:101)s ρ =− [H ,ρ ] dt s (cid:126) s s N 2(cid:15)(cid:88)Γk(cid:16)2Qkρ(cid:101)sQ†k−Q†kQkρ(cid:101)s−ρ(cid:101)sQ†kQk(cid:17)+O((cid:15)2), + (cid:88)N 4Γk (cid:16)2Q ρ Q† −Q† Q ρ −ρ Q† Q (cid:17) k=1 2 s,k s s,k s,k s,k s s s,k s,k k=1 ρ (0)=ρ(0). (22) (cid:101)s (23) 5 where ρs is the density operator associated to the space V. NUMERICALSIMULATIONS spanned by the |g (cid:105)’s, where the slow Hamiltonian is k Finallyletuschecktherelevanceoftheadiabaticreduction result of the Section IV in a simulation. Here, we consider a H =(1−P)H(1−P) s (4+1)-level system given by the following parameters and the slow jump operators are δ =0.5 δ =1.2 δ =0.7 δ =1.0 1 2 3 4 H Qs,k =Qk(cid:126)Γ(1−P). Ω1 =1.0 Ω2 =1.2 Ω3 =1.1 Ω4 =1.3 Γ =5.0 Γ =4.0 Γ =7.0 Γ =5.0 (24) 1 2 3 4 (cid:80) Here, we have set Γ = Γ and P = |e(cid:105)(cid:104)e|. The slow k k ThesimulationsofFigure2illustratetheoutputsignalderived approximation of the measured output is still given by the from the reduced slow dynamics, y , versus the slow/fast standard formula s dynamics, y. The simulation time is taken to be 2.5T , where n s y =(cid:88)4Γ Tr(cid:16)Q† Q ρ (cid:17). Ts, the time scale of the slow system is given by s k s,k s,k s k=1 T = Γ1+Γ2+Γ3+Γ4 . s Ω2+Ω2+Ω2+Ω2 1 2 3 4 IV. PHYSICALINTERPRETATION The initial conditions are identical ρ(0) = ρ (0) = s In this section, we provide a physical interpretation of P4k=1|gk(cid:105)(cid:104)gk|. We observe, for the slow/fast dynamics, a first 4 the last section’s result for the particular Hamiltonian of the fast transient corresponding to the relaxation time of the fast system (8). We get dynamicsfortbetween0and1/Γ ,i.e.t∈[0,1/4],andthen k (cid:88) twoslowtransientsverysimilarforbothmasterequations:the H = δ |g (cid:105)(cid:104)g | s k k k slow approximation is clearly valid only for time-scale much k larger than 1/Γ . k and Q = (cid:112)(cid:80)l|Ωl|2 |g (cid:105)(cid:104)b | with |b (cid:105)= (cid:80)lΩl|gl(cid:105) . 0.35 s,k (cid:80)lΓl k Ω Ω (cid:112)(cid:80)l|Ωl|2 0.3 Let us set 0.25 ys (cid:80) |Ω |2 γ = l l 4Γ 0.2 k ((cid:80) Γ )2 k l l 0.15 the slow master equation reads 0.1 (cid:34) (cid:35) 0.05 y d (cid:88) ρ =−ı δ |g (cid:105)(cid:104)g | , ρ 0 dt s k k k s 0 1 2 3 4 tim5 e 6 7 8 9 k +(cid:88)γk(2(cid:104)b |ρ|b (cid:105)|g (cid:105)(cid:104)g |−|b (cid:105)(cid:104)b |ρ−ρ|b (cid:105)(cid:104)b |) Fig.2. Theoutputofthereducedslowdynamics(23)(solidline)versusthe 2 Ω Ω k k Ω Ω Ω Ω outputforthe(4+1)-levelsystem(8)(dashedline).Theparametersarelisted k in (24). Even if the time-scale separation is not so large with an (cid:15) around with ys = ((cid:80)kγk)(cid:104)bΩ|ρs|bΩ(cid:105). Thus, whenever all the de- d14y,nsaumchicsa,daiasbsattaicteadpbpyrotxhiemoaretimon1c.apturesquitepreciselytheslowpartofthe tunings δ vanish, the unitary state |b (cid:105) is the bright state k Ω and the vector-space orthogonal to |b (cid:105) is a decoherence free Ω spacesinceonthissub-space,theLindbald-Kossakowskiterms CONCLUSION identically vanish and the output y is null. Notice that the controls Ω appear only in the decoherence terms and have We observed that for an ensemble of independent and k disappeared form the slow Hamiltonian. identical quantum systems, and whenever the decoherence If we restrict ourselves to the case of a 3-level Λ-system dynamicsduetothemeasurementismuchfasterthantheother (N =2), we have H = δ(|g (cid:105)(cid:104)g |−|g (cid:105)(cid:104)g |), dynamics, the adiabatic approximation helps us to find the s 2 1 1 2 2 slow dynamics as well as the measurement result with respect Ω Ω |b (cid:105)= 1 |g (cid:105)+ 2 |g (cid:105) to the slow dynamics. Note that in this new system, the deco- Ω (cid:112) 1 (cid:112) 2 |Ω1|2+|Ω2|2 |Ω1|2+|Ω2|2 herence term can be removed in a first order approximation. We obtain therefore a system of the form is the bright state of the Λ-system, as in the context of the coherent population trapping. As it can be seen easily, d ı ρ =− [H ,ρ ], whenever no de-tuning is admitted (δ =0), the dark state dt s (cid:126) s s Ω∗ Ω∗ where the control appears linearly in the reduced Hamiltonian |d(cid:105)=|b(cid:105)⊥ = (cid:112)|Ω |2+2 |Ω |2 |g1(cid:105)− (cid:112)|Ω |2+1 |Ω |2 |g2(cid:105) Hs. This system corresponds to a bilinear system with the 1 2 1 2 wavefunctionsasstatevariables.Furthermore,wehaveaccess is the only equilibrium state of the slow dynamics. to a measurement y given by the reduced slow evolution. We 6 can henceforth consider a control problem with continuous measurement associated to this system. Notice that, by some simple but tedious computations, one can extend the result of this paper to the more general case of an N+M-level system with N metastable ground states and M highly unstable excited states. REFERENCES [1] E.Arimondo. Coherentpopulationtrappinginlaserspectroscopy. Progr. Optics,35:257,1996. [2] E. Arimondo and G. Orriols. Nonabsorbing atomic coherences by coherenttwo-photontransitionsinathree-levelopticalpumping. Nuovo CimentoLett.,17:333–338,1976. [3] J.Carr. ApplicationofCenterManifoldTheory. Springer,1981. [4] P. Ducheˆne and P. Rouchon. Kinetic scheme reduction via geometric singular perturbation techniques. Chem. Eng. Science, 51:4661–4672, 1996. [5] N.Fenichel. Geometricsingularperturbationtheoryforordinarydiffer- entialequations. J.Diff.Equations,31:53–98,1979. [6] S.HarocheandJ.M.Raimond. ExploringtheQuantum:Atoms,Cavities andPhotons. OxfordUniversityPress,2006. [7] H.K.Khalil. NonlinearSystems. MacMillan,1992. [8] M. Mirrahimi and P. Rouchon. Continuous measurement of a statistic quantumensemble. InIEEEConferenceonDecisionandControl,pages 2465–2470,2006. APPENDIX This appendix has for goal to remind an approximation technique that can be perfectly justified using the geometrical tools of singular perturbation and the center manifold [7], [3], [4]. Consider the slow/fast system (x and y are of arbitrary dimensions, f and g are regular functions) d d 1 x=f(x,y), y =− Ay+g(x,y) dt dt (cid:15) where x and y are respectively the slow and fast states (Tikhonov coordinates), all the eigenvalues of the matrix A have strictly positive real parts, and (cid:15) is small strictly positive parameter. Therefore the invariant attractive manifold admits for equation y =(cid:15)A−1g(x,0)+O((cid:15)2) (25) and the restriction of the dynamics on this slow invariant manifold reads d x=f(x,(cid:15)A−1g(x,0))+O((cid:15)2) dt ∂f =f(x,0)+(cid:15) | A−1g(x,0)+O((cid:15)2). ∂y (x,0) TheTaylorexpansionofg canbeusedtofindthehigherorder terms. For example, the second order term in the expansion of y is given by: y =(cid:15)A−1g(x,0)+ (cid:18) (cid:19) ∂g ∂g (cid:15)2A−1 | A−1g(x,0)−A−1 | f(x,0) +O((cid:15)3), ∂y (x,0) ∂x (x,0) (26) and so on.