Semi-Global Approximate stabilization of an infinite dimensional quantum stochastic system

Semi-Global Appro ximate stabilizatio n of an infinite dimensional quan tum sto c hastic system Ram Somara ju a,b , Mazy ar Mirrahimi a,b , Pierre Rouc ho n c a INRIA R o c quenc ourt, Domaine de V oluc e au, B.P. 105 , 78153 L e Chesnay c e dex, F r anc e, (r am.somar aju, mazyar.mirr ahimi)@inria. fr b R am Somar aju and Mazya r Mirr ahimi acknow le dge supp ort fr om “A genc e Nationale de la R e cher che” (ANR), Pr ojet Jeunes Ch er cheurs EPOQ2 numb er ANR-09-JCJC-0070. c P. R ouchon is with Mines P arisT e ch, Centr e Automatique et Syst´ emes, Math´ ematiques et Syst´ emes, 60 Bd Saint Michel, 75272 Paris c e dex 06, F r anc e, pierr e.r ouchon@mines- p ariste ch.fr d Pierr e R ouchon acknow le dges supp ort fr om ANR (CQUID ) . Abstract In this pap er w e study the semi-global (approximate) state feed bac k stabiliza- tion of a n infinite dimensional quan tum sto c hastic system tow ards a ta rget state. A discrete-time Mark o v c hain on an infinite-dimensional Hilb ert space is used to mo del the dynamics of a quan t um optical ca vity . W e can ch o ose an (un b ounded) strict Ly a puno v function that is minimized at eac h time-step in order to prov e (w eak- ∗ ) con v ergence of probability measures to a final state that is concentrated on the targ et state with (a pre-sp ecified) probabilit y that may b e made a r bitrarily close to 1. The feedbac k parameters and the Ly apunov function are c hosen so that the sto c hastic flo w that describ es the Mark ov pro cess ma y b e show n to b e tigh t (concen tra ted on a compact set with probabilit y arbitrarily close to 1 ). W e then use Prohorov’s theorem and prop erties of the Ly a puno v function to prov e the desired con vergenc e result. Keywor ds: Quan tum control, Ly apunov stabilization, Sto c hastic stabilit y, One-parameter semigroups 1. In tro duction In this pap er w e consider the stabilization o f a discrete-time Marko v pro- cess (Equations ( 3 ) and ( 4 )) that is defined o n the unit sphere on an infinite- dimensional Hilb ert (F o c k) space H at a ta rget state whic h is a sp ecific Pr eprint su bmitte d to Elsevier Septemb er 22, 2018 unit v ector in H corr e sp onding to a photo n-n um b er state. W e consider a Ly apunov function based state- fee dba ck controller tha t driv es our quan tum system to the target state with probabilit y greater than some pre-sp ecified probabilit y 1 − ǫ fo r all 1 > ǫ > 0 1 . The sp ec ific ph ysical system under consideration uses Quantum Non- Demolition (QND) measuremen ts to detect and/or pro duce highly nonclas- sical states of ligh t in trapp ed sup er-conducting ca vities [ 3 , 4 , 5 ] (see [ 6 , Ch. 5] for a description o f suc h quan tum electro-dynamical systems and [ 7 ] for detailed phy sical mo dels with QND measures of ligh t using atoms). In this pap er w e examine the feedbac k stabilization of suc h exp erimen tal setups near a pre-sp ecified target photon n umber state. Suc h photon num b er states, with a precisely defined num b er of photo ns , a re highly non-classical and ha ve p oten tial applications in quan tum infor ma t io n and computation. As the Hilb ert space H is infinite dimensional it is difficult to design feed- bac k con tr o llers to drive the system tow ar ds a target state (b ecause closed and b ounded subsets of H are not compact). In [ 1 , 2 ] a controller was designed b y approx ima t ing the underlying Hilb ert space H with a finite- dimensional G alerkin approx imatio n H N max . Ph ysically t his approximation leads t o a n artificial b ound N max on the ma ximum num b er o f photons that ma y b e inside t he ca vity . In this paper w e wish to design a con troller for the full Hilb ert space H without using the finite dimensional approxima- tion. Simulations (see [ 8 ]) indicate that the controller in Theorem 3.1 b elo w p erforms b etter than the one designed using the finite-dimensional approxi- mation in [ 1 , 2 ]. Con trolling infinite dimensional quan tum syste ms hav e previously b een examined in the deterministic setting of partial differential equations, whic h do not in volv e quan tum measuremen ts. V arious approa c hes hav e been used to o vercome the non-compactness of closed and b ounded sets. O ne approac h consists of prov ing appro ximate con v ergence results whic h sho w conv ergence to a neighborho o d of the target state for example in [ 9 , 1 0 ]. Alternativ ely , one examines w eak conv ergence for example, in [ 11 ]. Other approac hes suc h as using strict Lyapuno v functions o r strong con v ergence under restrictions on p ossible t ra jectories to compact sets hav e also b een used in the contex t 1 The proble m o f output feedback control has b een examined in the finite-dimens io nal context in [ 1 , 2 ] using a quantum adaptation of the Kalman filter . W e do not dis cuss the problem of estimating the state of the system and r efer the rea der to [ 1 ] for further details on des igning a state estimator. 2 of infinite dimensional state-space for example in [ 12 , 13 ]. The situation in our pap er is differen t in the sense that the system under consideration is inherently sto c hastic due to quan tum meas uremen ts. The system w e consider may b e describ ed using a discrete time Marko v pro cess on the set of unit v ectors in the state Hilb ert space H as explained in Sub- section 2.3 . W e use a strict Ly apunov function that restricts the system tra jectories with high probability to compact sets as explained in Section 3 . W e use the prop erties of w eak-conv ergence of measures to sho w approximate con v ergence (i.e. with probabilit y of con vergenc e a ppro ac hing one) o f the discrete time Mark ov pro cess tow ards the ta r g et state. 1.1. Outline The remainder of the pap er is orga nised as follo ws: in the follow ing Sec- tion 2 we intro duce some notation and the syste m mo del of the discrete- time Mark ov pro cess. W e a lso recall some results concerning the (w eak- ∗ )- con v ergence o f probabilit y measures. In Section 3 w e state the main result of our pap er (Theorem 3.1 ) concerning the a pproximate semi-global stabiliz- abilit y of the Mark ov pro cess a t our ta rget state. W e a ls o provide a pro of of the main result using sev eral Lemmas. W e then presen t o ur conclusions in the final Section. 2. Definitions and System Description W e intro duc e some notation that will b e used to describ e the discrete-time Mark ov pro cess that c ha racteriz es the sys t em. 2.1. Notation In this pap er, we use Dirac’s Br a -k et notat ion commonly used in Ph ysics literature 2 . The system Hilb ert space asso ciated with the quan tum ca vity is a F o c k space whic h w e denote by H with inner-pro duct h· |·i H and norm k · k H . W e drop the subscript, for ease of notation, if this causes no confusion. Let the set 3 {| n i : n ∈ Z + 0 } 2 See e.g . [ 14 , Sec. 8.3] for more details o f the quantum Harmo nic oscilla tor mo del and notation used her e. 3 Z , Z + and Z + 0 denote the s et of in teg ers, p ositive integers a nd non-nega tiv e integers, resp ectiv ely . 3 denote the canonical ba sis of the F o c k space H . Ph ysically , the stat e | n i represen ts a cav ity state with precisely n photons. Let a and a † b e the annihilatio n and creation op erators defined o n do- mains D ( a ) ⊂ H a nd D ( a † ) ⊂ H , resp ectiv ely a nd N = a † a b e the num b er op erator with domain D ( N ). These unbounded op erators satisfy the rela- tions a | n i = √ n | n − 1 i , a † | n i = √ n + 1 | n + 1 i , N | n i = n | n i (1) for all n ∈ Z + 0 . F or all x ∈ H and ǫ > 0 denote by B ǫ ( x ) the op en ball in H cen tered at x and of radius ǫ . Also denote b y ¯ B 1 = { x ∈ H : k x k = 1 } the closed set of unit v ectors in H with the top ology inherited from H . Let B ′ ǫ ( x ) denote the set ¯ B 1 ∩ B ǫ ( x ) for all x ∈ ¯ B 1 and ǫ > 0 . Let C = ( C ( ¯ B 1 ) , k · k ∞ ) b e the Banac h space of con tinuous functions on ¯ B 1 with the suprem um norm k · k ∞ . W e denote by B = B ( ¯ B 1 ) the Borel σ -algebra of ¯ B 1 and b y M 1 the set of a ll probabilit y measures on the measure space ( ¯ B 1 , B ). F or all µ ∈ M 1 and B -measurable functions f defined o n ¯ B 1 w e denote b y E µ [ f ] = Z x ∈ ¯ B 1 f ( x ) dµ ( x ) the exp ec tation v alue of the function f with resp ect to measure µ . The supp ort of a probability measure µ is defined to b e the set supp( µ ) = { x ∈ ¯ B 1 : µ ( V ) > 0 fo r all op en neighborho o ds V of x } . 2.2. T op olo gy on M 1 In this pap er w e study the we ak- ∗ con v ergence of proba bility measures. It can b e shown ( s ee e.g. [ 15 ]) that the space M 1 is a subset o f t he unit ba ll in the (con tinuous linear) dual space C ∗ of C through the relation µ ∈ M 1 7→ Λ µ ∈ C ∗ where Λ µ ( f ) = E µ [ f ] for all f ∈ C . When we refer to the conv ergence of a sequence of measures , w e mean con v erges with resp ec t to the w eak- ∗ top ology of C ∗ . Definition 2.1. We say that a se quenc e of pr ob ability me asur e { µ n } ∞ n =1 ⊂ M 1 c onver ges (we ak- ∗ ) to a pr ob ability me as ur e µ ∈ M 1 if fo r al l f ∈ C lim n →∞ E µ n [ f ] = E µ [ f ] 4 and we write µ n ֒ → µ . In the w eak- ∗ top ology on the set of proba bilit y measures, compactness is related to the notion of tig h tness of measures. A set of probability measures S ⊂ M 1 is said to b e tight [ 16 , p. 9] if for all ǫ > 0 there exists a compact set K ǫ ⊂ ¯ B 1 suc h tha t for all µ ∈ S , µ [ ¯ B 1 \ K ǫ ] < ǫ. W e recall b elo w Pro ho r o v’s theorem (see e.g. [ 15 ]). Theorem 2.1 (Prohoro v’s theorem) . Any tight se quenc e of pr ob ability me a- sur e s ha s a (we ak- ∗ ) c onver ging subse quenc e. 2.3. Discr ete-time Markov pr o c ess W e no w describ e the ev olution of our quantum system whic h is gov erned b y a Marko v pro cess o n the state space ¯ B 1 . W e in tro duce b elo w the Marko v pro cess mo del with minimal references to the actual ph ysical system under consideration. W e refer the in terested reader to [ 6 , Ch. 5] and references therein for a description of the ph ysical system a nd the approximations in- v olv ed in deriving the Mar ko v pro cess mo del (also see [ 1 , 2 ]). Define the displacemen t o p erator D α : H → H and measuremen t op era- tors M s : H → H where α ∈ R and s ∈ { g , e } as D α = exp { α ( a † − a ) } , M g = cos ( θ + N φ ) and M e = sin ( θ + N φ ) . Here, θ and φ are exp erime ntally determined real n umbers and the op erators a , a † and N are defined in Equation ( 1 ). R e call that b ecause the o perator i ( a † − a ) is self-adjo in t, w e ma y conclude from Stone’s theorem that the set of op erators { D α : α ∈ R } form a strongly-con tinuous unitary group ( s ee e.g. [ 17 , Sec V111.4]), i.e. lim α → α 0 k D α | ψ i − D α 0 | ψ i k H = 0 , ∀ ψ ∈ H . (2) Denote by | ψ k i ∈ ¯ B 1 the state o f the system at time-step k . Giv en the state | ψ k i at time-step k the state | ψ k +1 i at time-step k + 1 is a ra ndo m v ar iable whose distribution is giv en by the following tw o-step equation   ψ k +1 / 2  = M s | ψ k i k M s | ψ k i k H with probabilit y P s = k M s | ψ k i k 2 H , (3) | ψ k +1 i = D α k   ψ k +1 / 2  . (4) Here s ∈ { e, g } and t he con tro l α k ∈ R . 5 Remark 2.1. The time e v olut i on fr om the step k to k + 1 , c onsists of two typ es of evolutions: a pr oje ctive me asur em ent by the op er ators M s and a c ontr ol p art invo lving o p er ator D α . F or the s a ke of sim plicity, we wil l use the notation of   ψ k +1 / 2  to il lustr ate this interme diate step. The Marko v j ump probabilities may also b e written in t erms of densit y op erators. Give n any | ψ i denote by ρ ψ the densit y op erator | ψ i h ψ | . The n | h ψ | n i | 2 = T r { ρ ψ | n i h n |} and k M g | ψ i k 2 = T r  M 2 g ρ ψ  . Here T r {·} is the trace of a trace-class op erator on H . If w e set ρ k = | ψ k i h ψ k | then w e can write the Marko v jump probabilities ( 3 ), ( 4 ) using the equiv alen t densit y op erator description. ρ k +1 / 2 = M s ( ρ k ) T r { M s ( ρ k ) } with probabilit y P s = T r { M s ( ρ k ) } ρ k = D α k ( ρ k ) . Here, D α ( · ) , D α · D − α and M s ( · ) = M s · M s are sup er-op erators. W e switc h b et w een the tw o equiv alen t descriptions throughout the pap er dep ending on con v enience. Remark 2.2. Equations ( 3 ) and ( 4 ) determine a sto chastic flow in M 1 and we denote by Γ k ( µ 0 ) the pr ob a bility distribution of | ψ k i , given µ 0 , the pr ob a b ility dis tribution of | ψ 0 i . 2.4. Some useful formulas W e recall b elo w some useful results. The Bak er-Campb ell-Hausdorff for- m ula, whic h will b e used to ev aluate the deriv ativ es of our Ly apunov function, states (see e.g. [ 18 , p. 291]) exp( αH ) A exp( − αH ) = ∞ X n =0 α n n ! C n , (5) where A, H and C n are linear op erators on H and α ∈ C . The C n are defined recursiv ely with C 0 = A and C n +1 = [ H , C n ] for n ≥ 0. Let X n b e a Mark ov pro cess o n some state space X. Supp ose that there is a non-negat ive function V on X satisfying E [ V ( X 1 ) | X 0 = x )] − V ( x ) ≤ 0, then Do ob’s inequality states P  sup n ≥ 0 V ( X n ) ≥ γ | X 0 = x  ≤ V ( x ) γ . (6) 6 3. Main R esu lt s W e pro v e the ma in results of our pap er in this Section. W e wish to use the con tro l α k to driv e the system into a pre-sp ecified target state | ¯ n i where ¯ n ∈ Z + 0 . W e use a strict Ly apuno v function V : ¯ B 1 → [0 , ∞ ] defined 4 V ( | ψ i ) = ∞ X n =0 σ n |h ψ | n i| 2 + δ  cos 4 ( φ ¯ n ) + sin 4 ( φ ¯ n ) − k M g | ψ ik 4 − k M e | ψ ik 4  . (7) Here φ n = θ + nφ = cos − 1 ( k M g | n i k ) = sin − 1 ( k M e | n i k ) , n = 0 , 1 , 2 , . . . . δ > 0 is a small po s it ive n um b er and σ n =        1 8 + P ¯ n k =1 1 k − 1 k 2 , if n = 0 P ¯ n k = n +1 1 k − 1 k 2 , if 1 ≤ n < ¯ n 0 , if n = ¯ n P n k = ¯ n +1 1 k + 1 k 2 , if n > ¯ n (8) W e set D ( V ) ⊂ ¯ B 1 to b e the set of all | ψ i ∈ ¯ B 1 where the a b ov e Ly apuno v function is finite. Remark 3.1. We note that c oher ent states | ξ α i = exp ( −| α | 2 / 2) P ∞ n =0 α n n ! | n i , for α ∈ C ar e in D ( V ) . These c oher ent states na tur al ly o c cur in o ptic al c avities and ar e gener al ly the initial c ondition | ψ 0 i is a c oher ent state in exp eriments. W e c ho ose a feedbac k that maximises the exp ectation v alue of the Ly a- puno v function in eve ry time-step k - α k = argmin α ∈ [ − ¯ α, ¯ α ] V  D α   ψ k +1 / 2  (9) for some p ositiv e constant ¯ α . 4 W e choose this Lyapuno v function ass uming ¯ n ≥ 2. The Lyapunov function may easily b e mo dified for the cas e ¯ n = 0 , 1 and a ll the pro ofs in this pap er may be applied to that case as well. 7 Remark 3.2. T h e Lyapunov function and fe e db ack α k ar e chosen to b e this sp e c i fi c form to serve thr e e purp oses - 1. We cho ose the se quenc e σ n → ∞ as n → ∞ . This guar ante es that if we cho ose α k to minim ize the exp e ctation value of the Lyapunov function then the tr aje ctories of the Markov pr o c ess ar e r estricte d to a c omp act set in ¯ B 1 with pr ob ability arbitr arily close to 1. This implies that the ω -limit set of the pr o c ess is non-empty (se e L emma 3.11 ). 2. The term − δ ( k M g | ψ i k 4 + k M g | ψ i k 4 ) is c h osen such that the Lyapunov function is a strict Lyapunov f unction s for the F o ck states. This im- plies that the supp ort of the ω -limit set o n ly c ontains F o ck states (se e L emma 3.12 ). 3. The r elative magnitudes o f the c o efficients σ n have b e en chosen such that V ( | ¯ n i ) is a strict glob al minimum of V . Mor e ov er given any M > ¯ n we c an cho ose δ, ¯ α such that for al l M ≥ m 6 = ¯ n , and for al l | ψ i in some neigh b orho o d of | m i , V ( D α ( | ψ i ) do es not ha v e a lo c al minimum at α = 0 . This implies that i f | ψ k i is in this neighb orho o d of | m i then we c an cho ose an α k ∈ [ − ¯ α, ¯ α ] to de cr e ase the Lyapunov function and move | ψ k i aw ay fr om | m i by s o me finite distanc e with pr ob ab i l i ty that c an b e made arbitr arily close to 1 by an appr o p riate choic e of ¯ α and δ (se e pr o of of L emma 3.13 ). W e mak e the fo llo wing assumption 5 . A1 The eigenv alues of M g and M e are non-degenerate. This is equiv a len t to t he assumption that π /φ is not a rational num b er. This implies that the only eigenv ectors o f M g and M e are the F o c k states {| n i : n ∈ Z + 0 } . The f o llo wing Theorem is our main result. Theorem 3.1. Supp o s e as s ump t ion A1 is true. F or any initial me asur e µ , let Γ k ( µ ) = Γ ¯ α,δ k ( µ ) b e the Markov flow induc e d by Equations ( 3 ) , ( 4 ) and c ontr ol α k given in Equation ( 9 ) with Lyapunov function V in Equation ( 7 ) . Her e, ¯ α de t e rmines the c ontr ol sign al and δ determines the Lyapunov function. Given any ǫ > 0 and C > 0 , ther e exist c onstants δ > 0 and ¯ α such that for al l µ satisfying E µ [ V ] ≤ C , Γ k ( µ ) c o nver ges (w e ak- ∗ ) to a limit set Ω . Mor e over for al l µ ∞ ∈ Ω , | ψ i ∈ supp( µ ∞ ) onl y if | ψ i is one of the F o ck states | n i and µ ∞ ( { ¯ n } ) ≥ 1 − ǫ. 5 See Remark 3.3 b elow, to see how w e may weaken this assumption. 8 3.1. Overview of Pr o of The pro of of the Theorem uses sev eral Lemmas that are prov en in the next t w o subsections. W e outline the cen tral idea of the pro of now . The Ly a punov function is suc h tha t c ho osing α k = 0 ensures that the exp ectatio n v alue of the Ly apunov function is non-increasing. Therefore b y c ho osing α k to b e the α ∈ [ − ¯ α, ¯ α ], that minimizes the expectatio n v alue of the Ly apunov function in eac h step, w e ensure that V ( | ψ k i ) is a sup er-martingale (Lemma 3.10 ). The set {| ψ i ∈ ¯ B 1 : P n σ n | h ψ | n i | 2 ≤ C } is a compact subset of ¯ B 1 (Lemma 3.4 ). W e use this fa ct and the sup er-martingale prop ert y of V ( | ψ k i ) to sho w that with probability approac hing 1, | ψ k i is in a compact set for all k . This implies the tightne ss of the sequenc e Γ n ( µ ). Hence, w e can use Prohoro v’s Theorem 2.1 to prov e the existence of a conv erging subseque nce (Lemma 3.11 ). Supp ose Γ k l ( µ ) is some subsequence that conv erges to a measure µ ∞ . W e sho w that the second term in the Ly apunov function V 2 ( | ψ i ) , δ (cos 4 ( θ ¯ n ) + sin 4 ( θ ¯ n ) − k M g | ψ i k 4 − k M e | ψ i k 4 ) is a strict Ly apunov function for F o c k states. That is E [ V 2 ( | ψ k +1 i ) | | ψ k i ] − V 2 ( | ψ k i ) ≤ 0 with equality if and only if | ψ k i is a F o c k state. This implies that t he supp ort set of a ll µ ∞ only consists of F o c k states (Lemma 3.12 ). Finally w e note that the first part of the Ly apuno v function V 1 ( | ψ i ) = P n σ n | h ψ | n i | 2 satisfies ∇ α V 1 ( | m i ) = 0 for all m . Moreov er ∇ 2 α V 1 ( D α | m i ) < 0 f or m 6 = ¯ n and ∇ 2 α V 1 ( D α | ¯ n i ) > 0. T his implies | m i , m 6 = ¯ n is a lo cal maxim um for the Lyapuno v function. Therefore, w e can find a f e edbac k α to mo ve the system aw a y from F o c k states | m i if m 6 = ¯ n with high probability (Lemma 3.13 ). This therefore implies t hat w e con v erge to our tar g et F o c k state with high probability . W e pro v e in the followin g tw o sections the con ve r g en ce result r igorously . W e first establish some prop erties of the Lyapuno v function. 3.2. Pr op erties of the Lyapunov function Lemma 3.2. F or δ sma l l enough V ( | ψ i ) is non-ne gative on ¯ B 1 . 9 Pr o of. W e ha ve , cos 4 ( φ ¯ n ) − k M g | ψ i k 4 = (cos 2 ( φ ¯ n ) − k M g | ψ i k 2 )(cos 2 ( φ ¯ n ) + k M g | ψ i k 2 ) ≥ cos 2 ( φ ¯ n ) − ∞ X n =0 cos 2 ( φ n ) | h ψ | n i | 2 ! cos 2 ( φ ¯ n ) ≥ − X n =0 n 6 = ¯ n | h ψ | n i | 2 . Using a similar analysis for sin 4 ( φ ¯ n ) − k M e | ψ i k 4 w e get V ( | ψ i ) ≥ X n ( σ n − 2 δ (1 − δ n ¯ n )) | h ψ | n i | 2 . Here, δ n ¯ n is the Kronec ke r-delta. Because σ n ≥ min { σ ¯ n +1 , σ ¯ n − 1 } > 0 for all n 6 = ¯ n , for δ small enough, V ≥ 0. F or t he remainder of this pap er, we assume that δ has b een c hosen small enough to ensure that V is non-negative. Also note that for δ small enough V ( | ψ i ) = 0 if and only if | ψ i = | ¯ n i . There f ore | ¯ n i is a strict global minim um for V . Lemma 3.3. The function V is lower semi-c ontinuous on D ( V ) . Pr o of. Because M g and M e are b ounded o perators, k M g | ψ i k and k M e | ψ i k are con t inuous functions on D ( V ). So w e only need t o prov e the low er semi- con tinuit y of k | ψ i k σ , P ∞ n =0 σ n | h ψ | n i | 2 . Let | ψ 0 i = P ∞ n =0 c n | n i ∈ D ( V ). L et ǫ > 0 b e g iv en. Then the finiteness of V ( | ψ 0 i ) implies that there exists an N suc h t ha t ∞ X n = N σ n | h ψ | n i | 2 < ǫ 2 . W e can choose κ small enough suc h that for all c ′ n satisfying | c ′ n − c n | 2 < κ , N − 1 X n =0 σ n   | c ′ n | 2 − | c n | 2   < ǫ 4 . 10 Therefore, for all | ψ i ∈ B κ ( | ψ 0 i ) ∞ X n =0 σ n | h ψ | n i | 2 ≥ N − 1 X n =0 σ n | h ψ | n i | 2 > N − 1 X n =0 σ n | h ψ 0 | n i | 2 − ǫ 2 > ∞ X n =0 σ n | h ψ 0 | n i | 2 − ǫ. Lemma 3.4. F or al l C ≥ 0 , the set V C = {| ψ i ∈ ¯ B 1 : V ( | ψ i ) ≤ C ] } is c omp act in ¯ B 1 with r es p e ct to the top olo gy inherite d fr om H . Pr o of. Let | ψ 1 i , | ψ 2 i , . . . b e a sequence in V C and let | ψ l i = ∞ X m =0 c l,m | m i . Because, k M g | ψ i k , k M e | ψ i k ≤ 1 and σ m is strictly increasing f or m > ¯ n , w e hav e ∞ X m ′ = m | c l,m ′ | 2 ≤ C − 4 σ m for all l = 1 , 2 , . . . and m > ¯ n . Therefore, using a diagonalization argumen t w e kno w tha t there exists a subsequence   ψ l p  and a set of n um b ers c ∞ ,m , m ∈ Z + 0 suc h tha t P ∞ m ′ = m | c ∞ ,m ′ | 2 ≤ C − 4 σ m if m > ¯ n (10) | c ∞ ,m | 2 ≤ 1 if m ≤ ¯ n (11) c l,m → c ∞ ,m . (12) W e claim tha t | ψ l n i → P ∞ m =0 c ∞ ,m | m i . T o see this supp ose ǫ > 0 is giv en. Then b ecause σ m → ∞ as m → ∞ , there exists an M suc h that ( C − 4) /σ M < ǫ/ 2 . (13) 11 Also b ecause c l p ,m → c ∞ ,m as p → ∞ , there exists a P large enough suc h that for all p ≥ P , | c l p ,m − c ∞ ,m | 2 < ǫ 2 M for m = 1 , 2 , . . . , M − 1 (14) Com bining Equations ( 10 )-( 14 ) w e get for p ≥ P        ψ l p  − ∞ X m =0 c ∞ ,m | m i      < ǫ. The low er semi-con tin uit y of V implies that the set V C is closed. Therefore P ∞ m =0 c ∞ ,m | m i ∈ V C . Hence V C is compact. Let { s n } b e any sequence of p ositiv e n umbers with s n → ∞ a s n → ∞ . Define an inner pro duct h x | y i s = ∞ X n =0 s n h x | n i H h n, y i H . Here the h· |·i s is defined on the linear subspace H s of H on whic h k · k s , p h·|·i s is finite. Giv en a linear op erator A , let R λ ( A ) = ( λ I − A ) − 1 b e its resolve nt op er- ator. W e recall b elo w a Theorem in [ 19 , Th. 12.31]. Theorem 3.5. A close d, densely define d op er ator A in some Hilb ert sp ac e X is the gener a t o r of an analytic semigr oup exp( α A ) (w. r. t. the uniform top olo gy of the set of b ounde d op er ators on X ) if and only if ther e exis t s ω ∈ R such that the half-plane R e { λ } > ω is c ontaine d in the r e solvent set of A and, mor e over, ther e is a c ons tant C such that k R λ ( A ) k ≤ C / | λ − ω | . W e sho w that if w e consider ( a † − a ) to b e an op erator on some domain in H s then exp( α ( a † − a )) is a n analytic semigroup on H s . Lemma 3.6. The op er ator i ( a − a † ) is symmetric on H s . 12 Pr o of. Let | ψ i , | φ i ∈ D ( a − a † ) ⊂ H s and let | ψ i = P ∞ n =0 c n | n i and | φ i = P ∞ n =0 d n | n i . Then , ( a − a † ) | ψ i = ∞ X n =0 c n ( a − a † ) | n i = ∞ X n =0 c n ( √ n | n − 1 i − √ n + 1 | n + 1 i ) = ∞ X n =0 c n +1 √ n + 1 | n i − ∞ X n =1 c n − 1 √ n | n i Therefore,  i ( a − a † ) ψ | φ  s = ∞ X p =0 s p  i ( a − a † ) ψ | p  H h p | φ i H = − i ∞ X p =0 s p ∞ X n =0 c n +1 √ n + 1 δ np − ∞ X n =1 c n − 1 √ nδ np ! d p = − i ∞ X p =0 s p ( c p +1 d p − c p d p +1 ) p p + 1 Similarly ,  ψ | i ( a − a † ) φ  s = i ∞ X p =0 s p ( d p +1 c p − d p c p +1 ) p p + 1 Therefore t he op erator i ( a − a † ) is symmetric. The op erators a † − a is w ell defined on the finite linear span of the F o c k basis | n i and this finite linear span of the F o c k basis is obvious ly dense in the space H s . The fact t ha t a † − a is closable follo ws from the fact that i ( a − a † ) is symmetric (see e.g. [ 20 , p. 270]). The r efo r e , we can assume that a † − a is a closed op erator (if it is not closed then w e can alwa ys redefine a † − a to b e its closed linear extension). Lemma 3.7. The semigr oup D α = exp( α ( a † − a )) is analytic in the set of b ounde d op er ators on H s with r esp e ct to the uniform o p er ator top olo gy of b ounde d op er ators on H s . 13 Pr o of. Because i ( a − a † ) is symmetric, we ha ve for all complex n um b ers λ (see e.g. [ 20 , p. 270]) k i ( a − a † ) − λ I k ≥ | Im { λ }| . Therefore, k R λ ( a † − a ) k ≤ 1 | Re { λ }| . Therefore with ω = 0 and C = 1 and b ecause ( a † − a ) is closed and densely defined, the conditions of Theorem 3.5 are satisfied. Hence the semigroup generated b y ( a † − a ) is analytic. Lemma 3.8. L et m 6 = ¯ n and C > V ( | m i ) b e given. Then ther e exists a c onstant ¯ ǫ = ¯ ǫ ( | m i ) > 0 such that for al l ǫ ∈ (0 , ¯ ǫ ) and | ψ i ∈ B ′ ǫ ( | m i ) satisfying V ( | ψ i ) ≤ C , V ( D α | ψ i ) − V ( | ψ i ) = f 1 ( | ψ i ) α + f 2 ( | ψ i ) α 2 + O ( ¯ α 3 ) + O ( δ ) (15) for | α | < ¯ α = ¯ α ( C ) . Mor e over, f 2 ( | ψ i ) < γ m < 0 for so me c onstant γ m . Pr o of. Because M e and M g are b ounded op erators on H and D α is an analytic semigroup on H , w e ha ve V ( D α | ψ i ) − V ( | ψ i ) = k D α | ψ i k 2 σ − k | ψ i k 2 σ + O ( δ ) . where k · k σ , P ∞ n =0 σ n | h·| n i | 2 . W e kno w f rom Lemma 3.7 that D α is analytic with resp ec t to the uniform- op erator top ology induced b y the semi-norm 6 k · k σ . Hence, for all | ψ i suc h that k | ψ i k σ ≤ V ( ψ ) ≤ C we can write, using the T a ylor series expansion at α = 0, k D α | ψ i k 2 σ − k | ψ i k 2 σ = f 1 ( | ψ i ) α + f 2 ( | ψ i ) α 2 + O ( ¯ α 3 ) for | α | < ¯ α and ¯ α o nly depends on C . 6 Even though the Lemma 3.7 was prov en for sequences s n > 0, we can apply it to the case where one of the s n is zer o. In fact small changes in s n do not change the analycity of D α . W e can prov e the same Lemma by considering the quotient spa c es of the semi-nor m k · k σ . 14 W e now prov e the required b ound on f 2 using the Baker-Campbell-Hausdorff form ula ( 5 ). By noting that k D α | ψ i k 2 σ = ∞ X n =0 σ n T r  exp( α ( a † − a )) | ψ i h ψ | exp( − α ( a † − a )) | n i h n |  and setting H = a † − a and A = | ψ i h ψ | in Equation ( 5 ), w e get f 2 ( | ψ i ) = 1 2 ∞ X n =0 σ n T r  [ a † − a, [ a † − a, | ψ i h ψ | ] | n i h n |  = 1 2 ∞ X n =0 σ n T r  T 2 | ψ i h ψ | + | ψ i h ψ | T 2 − 2 T | ψ i h ψ | T  | n i h n |  . ( 16) Where we hav e set T = a † − a . If w e let | ψ i = P n c n | n i , then 1 2 T r  T 2 | ψ i h ψ | + | ψ i h ψ | T 2  | n i h n |  = 1 2 h n | T 2 | ψ i h ψ | n i + h ψ | T 2 | n i h n, ψ i = Re { p ( n + 1)( n + 2) c n +2 c ∗ n + p n ( n − 1) c n − 2 c ∗ n } − (2 n + 1) | c n | 2 (17) and T r { T | ψ i h ψ | T | n i h n |} = h n | T | ψ i · h ψ | T | n i = p n ( n + 1)Re { c n − 1 c ∗ n +1 } − n | c n − 1 | 2 − ( n + 1) | c n +1 | 2 . (18) Substituting Equations ( 17 ) and ( 18 ) into ( 16 ) and rearranging terms, we get f 2 ( | ψ i ) = ∞ X n =0 | c n | 2  ( n + 1) σ n +1 + nσ n − 1 − (2 n + 1) σ n  + ∞ X n =0 Re { c n − 1 c ∗ n +1 } p n ( n + 1)( σ n − 1 + σ n +1 − 2 σ n ) (19) 15 If 2 ≤ n 6 = ¯ n then substituting for σ n from Equation ( 8 ) w e get ( n + 1) σ n +1 + nσ n − 1 − (2 n + 1) σ n = − 1 n ( n + 1) and f o r n = 0 , 1 w e get ( n + 1) σ n +1 + nσ n − 1 − (2 n + 1) σ n = − 1 4 Substituting this into ( 19 ), w e hav e for 2 ≤ m 6 = ¯ n , f 2 ( | ψ i ) ≤ − | c m | 2 κ m + | c ¯ n | 2 (( ¯ n + 1) σ ¯ n +1 + ¯ nσ ¯ n − 1 ) + ∞ X n =0 Re { c n − 1 c ∗ n +1 } p n ( n + 1)( σ n − 1 + σ n +1 − 2 σ n ) (20) where κ m =  m ( m + 1) if 2 ≤ m 6 = ¯ n, 4 if m = 1 , 2 F or all | ψ i ∈ B ′ ǫ ( | m i ), the first term in Equation ( 20 ) will b e less than − 3 / (4 κ m ) for ǫ small enough. W e show that fo r ǫ small enough, for all | ψ i ∈ B ′ ǫ ( | m i ) suc h that V ( | ψ i ) ≤ C , 1 2 κ m ≥ | c ¯ n | 2 (( ¯ n + 1) σ ¯ n +1 + ¯ nσ ¯ n − 1 ) +      ∞ X n =0 Re { c n − 1 c ∗ n +1 } p n ( n + 1)( σ n − 1 + σ n +1 − 2 σ n )      (21) F or all | ψ i ∈ B ′ ǫ ( | m i ) w e ha ve | c ¯ n | 2 < ǫ 2 and hence w e can c ho ose a small enough ǫ suc h that | c ¯ n | 2 (( ¯ n + 1) σ ¯ n +1 + ¯ n σ ¯ n − 1 ) ≤ 1 8 κ m . (22) W e ha v e | p n ( n + 1)( σ n − 1 + σ n +1 − 2 σ n ) | = p n ( n + 1) n 2 + 3 n + 1 n 2 ( n + 1) 2 , h ( n ) 16 The function h ( n ) defined in the ab o ve Equation is of or der O (1 /n ). Because σ ( n ) is order O (ln ( n )) and b ecause the series P n | c n | 2 σ n < C con v erges, w e kno w tha t the series P n | c n | 2 h ( n ) con ve rg es . Henc e, P ∞ n = M | c n | 2 h ( n ) can b e made arbitrarily small for large M . By Cauc h y-Sc hw a rtz, P n | c n | 2 ≥ P n Re { c n − 1 c n +1 } and hence M can b e c hosen large enough suc h that for all | ψ i satisfying V ( | ψ i ) ≤ C , w e ha ve      ∞ X n = M Re { c n − 1 c ∗ n +1 } p n ( n + 1)( σ n − 1 + σ n +1 − 2 σ n )      ≤ 1 8 κ m . (23) Because P n | c n | 2 ≥ P n Re { c n − 1 c ∗ n +1 } , w e can choose ǫ small enough so tha t for all | ψ i ∈ B ǫ ( | m i ),      M − 1 X n =0 Re { c n − 1 c ∗ n +1 } p n ( n + 1)( σ n − 1 + σ n +1 − 2 σ n )      ≤ 1 4 κ m . (24) Equations ( 22 ), ( 23 ) and ( 24 ) pro v e Equation ( 21 ). Substituting ( 21 ) in to ( 20 ) w e get the required b ound, f 2 ( | ψ i ) ≤ γ m < 0 where γ m = − 1 / (4 κ m ). W e ha v e the follo wing Corollary to the ab o v e Lemma. Corollary 3.9. Given an y m 6 = ¯ n and C > V ( | m i ) , the c onstants ¯ α and δ in Equations ( 9 ) and ( 7 ) , r esp e ctive l y, c an b e ch o sen to b e smal l enough such that ther e exists an ǫ > 0 and c > 0 such that if | ψ k i ∈ W ǫ , B ′ ǫ ( | m i ) ∩ {| ψ i : C ≥ V ( | ψ i ) > V ( | m i ) − ǫ } then V ( | ψ K +1 i ) − V ( | ψ k i ) < − c, with pr ob ability 1. Pr o of. Let ¯ ǫ b e as in Lemma 3.8 and c ho ose ǫ ′ ∈ (0 , ¯ ǫ ). Becaus e M g and M e are b ounded o perators on H and M g | m i = | m i and M e | m i = | m i , ǫ can b e c hosen small enough such that if | ψ k i ∈ B ′ ǫ ( | m i ) then   ψ k +1 / 2  ∈ B ′ ǫ ′ ( | m i ). Moreo v er, w e can sho w using the same argumen t used to prov e Lemma 3.10 b elo w, we kno w that if V ( | ψ k i ) < C then V (   ψ k +1 / 2  ) < C . Therefore, if | ψ k i ∈ W ǫ then   ψ k +1 / 2  ∈ W ǫ ′ with probabilit y 1. 17 No w w e choose ¯ α small enough such that the term of or der O ( ¯ α 3 ) in Equation ( 15 ) is smaller than γ m ¯ α 2 2 . Then from Lemma 3.8 w e kno w that for all | ψ i ∈ W ǫ ′ and for either α = + ¯ α or α = − ¯ α , w e ha v e V ( D α | ψ i ) − V ( | ψ i ) ≤ − γ m ¯ α 2 2 . W e can c ho ose δ small enough suc h that the O ( δ ) term in Lemma 3.8 is smaller than γ m ¯ α 2 4 . Therefore, fo r all | ψ k i ∈ W ǫ the statemen t of the Corollar y is satisfied b y c ho osing c = γ m ¯ α 2 4 . 3.3. Pr o o f of Conver genc e W e wish t o establish that V ( | ψ k i ) is sup er-martingale. W e hav e, E  V ( | ψ k +1 i )   | ψ k i = | ψ i  − V ( | ψ i ) = E  V  D α k    ψ k +1 / 2    | ψ k i = | ψ i  − V ( | ψ i ) = E " argmin α ∈ [ − ¯ α, ¯ α ] V  D α    ψ k +1 / 2      | ψ k i = | ψ i # − V ( | ψ i ) = K 1 ( | ψ i ) + K 2 ( | ψ i ) (25) where w e set K 1 ( | ψ i ) , E " argmin α ∈ [ − ¯ α, ¯ α ] V  D α    ψ k +1 / 2    | ψ k i = | ψ i # − E  V (   D 0 ( ψ k +1 / 2  )   | ψ k i = | ψ i  , (26) K 2 ( | ψ i ) , E  V  D 0    ψ k +1 / 2    | ψ k i = | ψ i  − V ( | ψ i ) = E  V    ψ k +1 / 2    | ψ k i = | ψ i  − V ( | ψ i ) (27) Ob viously K 1 ( | ψ i ) is non-p ositiv e for all | ψ i . In order to calculate K 2 ( | ψ i ) w e set ρ = ρ ψ = | ψ i h ψ | and w e note that M g and M e comm ute and satisfy M 2 g + M 2 e = I , where I is the iden t ity op erator. Therefore, k M e | ψ i k 4 = T r  M 2 e ρ  2 =  T r  ( I − M 2 g ) ρ  2 =  1 − T r  M 2 g ρ  2 = 1 − 2T r  M 2 g ρ  + T r  M 2 g ρ  2 = 1 − 2 k M g | ψ i k 2 + k M g | ψ i k 4 18 But, E  k M g   ψ k +1 / 2  k 2   | ψ k i = ψ  = k M g | ψ i k 2     M g M g | ψ i k M g | ψ i k     2 + k M e | ψ i k 2     M g M e | ψ i k M e | ψ i k     2 = T r  M 4 g ρ  + T r  M 2 g M 2 e ρ  = T r  M 2 g ρ  = k M g | ψ k i k 2 . Similarly , w e get for all | n i , E [ |  n, ψ k +1 / 2  | 2   ψ k = | ψ i ] = | h n, ψ k i | 2 Therefore, K 2 ( | ψ i ) = 2  k M g | ψ i k 4 − E  k M g   ψ k +1 / 2  k 4   | ψ k i = | ψ i  W e ha v e, E  k M g   ψ k +1 / 2  k 4   | ψ k i = | ψ i  = T r  M 2 g ρ  T r ( M 2 g M g ρM g T r  M 2 g ρ  ) 2 + T r  M 2 e ρ  T r  M 2 g M e ρM e T r { M 2 e ρ }  2 = T r  M 4 g ρ  2 T r  M 2 g ρ  + T r  M 2 g M 2 e ρ  2 T r { M 2 e ρ } = T r  M 4 g ρ  2 − T r  M 2 g ρ  T r  M 4 g ρ  2 + T r  M 2 g ρ   T r  M 2 g ρ  − T r  M 4 g ρ  2 T r  M 2 g ρ  T r { M 2 e ρ } = T r  M 4 g ρ  2 + T r  M 2 g ρ  3 − 2T r  M 2 g ρ  2 T r  M 4 g ρ  T r  M 2 g ρ  T r { M 2 e ρ } Therefore, K 2 ( | ψ i ) = 2  k M g | ψ i k 4 − E  k M g | ψ k +1 i k 4   | ψ k i = | ψ i  = 2 T r  M 2 g ρ  2 − T r  M 4 g ρ  2 + T r  M 2 g ρ  3 − 2T r  M 2 g ρ  2 T r  M 4 g ρ  T r  M 2 g ρ  T r { M 2 e ρ } ! = − 2  T r  M 4 g ρ  − T r  M 2 g ρ  2  2 T r  M 2 g ρ  T r { M 2 e ρ } (28) 19 The following Lemma is a consequence of the ab o v e calculation. Lemma 3.10. V ( | ψ k i ) is a sup er-martingale and satisfies E  V ( | ψ k +1 i )   | ψ k i = | ψ i  − V ( | ψ i ) ≤ − 2  T r  M 4 g ρ  −  T r  M 2 g ρ  2  2 T r  M 2 g ρ  T r { M 2 e ρ } . Her e ρ = | ψ i h ψ | . Pr o of. Firstly from the low er semi-con tin uity of V (Lemma 3.3 ) w e kno w that V is measurable function of | ψ i . Therefore E µ [ V ] is w ell defined. The sup er-martingale prop ert y follow s from Equations ( 25 ), ( 28 ) and the fa ct that K 1 ( | ψ i ) is not p ositiv e b ecause 0 ∈ [ − ¯ α , ¯ α ]. Lemma 3.11. If E µ [ V ] < ∞ then the se quenc e Γ n ( µ ) has a (we ak- ∗ ) c on- ver ging subse quenc e. Pr o of. W e pro ve tha t the seq uence of measures Γ n ( µ ) is tigh t . Let ǫ > 0 b e giv en. W e need to sho w that there exits a compact set K ǫ suc h tha t [Γ n ( µ )]( K ǫ ) ≥ 1 − ǫ for a ll n . By Lemma 3.4 , the set V =  | ψ i ∈ ¯ B : V ( | ψ i ) ≤ E µ [ V ] ǫ  is compact. W e prov e that [Γ n ( µ )]( V ) ≥ 1 − ǫ . Because V ( | ψ k i ) is a sup er-martingale, w e ha ve E Γ n ( µ ) [ V ] ≤ E µ [ V ] . Hence, by applying Do ob’s ineq ua lit y , t he probability that V ( | ψ n i ) > E µ [ V ] /ǫ is [Γ n ( µ )]( ¯ B 1 \ V ) < ǫ. Hence, the sequence Γ n ( µ ) is tight a nd by Prohorov ’s Theorem 2.1 has a (w eak- ∗ ) conv erging subsequence. Let Ω denote the limit set o f Γ n ( µ ). i.e. Ω = { µ ∞ ∈ M 1 : Γ n m ( µ ) ֒ → µ, for some subsequenc e Γ n m of Γ n } . (29) Lemma 3.12. Supp ose assumption A 1 i s true. I f | ψ i is not a F o c k state | n i for so me n ∈ Z + 0 then | ψ i / ∈ supp( µ ∞ ) for al l µ ∞ ∈ Ω . 20 Pr o of. Supp ose Γ k p [ µ ] ֒ → µ ∞ . W e know fro m Equation ( 25 ) a nd Lemma 3.10 that K 1 ( | ψ k i ) + K 2 ( | ψ k i ) → 0 as k → ∞ . Because b oth K 1 and K 2 are non-p ositiv e, w e need K 2 ( | ψ k i ) → 0 as k → ∞ . Therefore, for a ll initial distributions µ lim p →∞ E Γ k p ( µ ) [ K 2 ] = 0 . Also note that the f unction K 2 in Equation ( 28 ) is a contin uous function of | ψ i ∈ ¯ B 1 with resp ect to the top ology inherited from H . Therefore, b y the definition of (w eak- ∗ ) con vergenc e of measures 2.1 , w e ha v e E µ ∞ [ K 2 ] = lim p →∞ E Γ k p ( µ ) [ K 2 ] = 0 . (30) But from Equation ( 28 ), w e know that if K 2 ( | ψ i ) = 0 t hen T r  M 4 g ρ  = T r  M 2 g ρ  2 . But the Cauc hy-Sc h w ar t z inequalit y implies T r  M 4 g ρ  = T r  M 4 g ρ  T r { ρ } ≥ T r  M 2 g ρ  2 with equality if and only if M 4 g ρ and ρ are co-linear. That is if and only if ρ is a pro jection o ve r the eigenstate o f M 4 g . Therefore, by assumption A 1, K 2 ( | ψ i ) ≤ 0 with equality if and only if | ψ i is a F o c k state. No w if | ψ i is not a F o c k state then there is an op en neighborho o d W of | ψ i suc h that the neigh b orho o d do es not con tain a F o ck stat e. T herefore, µ ∞ ( W ) = 0 otherwise E µ ∞ [ K ] < 0 con tradicting ( 30 ). Therefore | ψ i / ∈ supp µ ∞ . Lemma 3.13. Given the a ssumptions of The or em 3.1 , for any F o ck state | m i , m 6 = ¯ n and for al l κ, C > 0 , ther e exist c onstants ¯ α and δ > 0 and a neighb orho o d V of | m i such that i f E µ [ V ] ≤ C then µ ∞ [ V ] < κ for a l l µ ∞ ∈ Ω . Pr o of. Supp ose Γ p q ( µ ) is some subsequence of Γ p ( µ ) that con verges to µ ∞ ∈ Ω. Giv en an y ν > 0, the lo w er semi-con tin uity of V implies that the set V ν , B ′ ν ( | m i ) ∩ {| ψ i ∈ ¯ B 1 : V ( | ψ i ) > V ( | m i ) − ν } is an op en neigh b orho o d o f | m i in ¯ B 1 . Because V ν is op en, w e kno w from the (w eak- ∗ ) con v ergence of Γ p q ( µ ) → µ ∞ that lim inf q →∞ [Γ p q ( µ )]( V ν ) ≥ µ ∞ ( V ν ) . 21 Therefore, in order to prov e the Lemma, w e need to sho w t ha t for some ν > 0, lim q →∞ Γ p q ( V ν ) ≤ κ. Because E µ [ V ] ≤ C we kno w from Do ob’s inequalit y 6 that for all p and all ν > 0, [Γ p ( µ )]  | ψ i ∈ V ν ⊂ ¯ B 1 : V ( | ψ i ) > C κ/ 2  < κ 2 . (31) W e sho w that there exists a ν > 0 suc h that lim p →∞ [Γ p ( µ )]( ¯ V ν ) = 0 , where ¯ V ν ,  | ψ i ∈ V ν ⊂ ¯ B 1 : V ( | ψ i ) ≤ C κ/ 2  . (32) W e know from Coro llary 3 .9 that there exists an ǫ > 0 and c > 0 suc h that if | ψ p i ∈ W ǫ , B ′ ǫ ( | m i ) ∩  | ψ i : C κ/ 2 ≥ V ( | ψ i ) > V ( | m i ) − ǫ  then V ( | ψ p +1 i ) − V ( | ψ p i ) < − c, with probability 1. Therefore if | ψ p i ∈ W ǫ , then for some finite P satisfying p < P ≤ l C κ/ 2 − ǫ  /c m , w e hav e | ψ P i ∈ {| ψ i : V ( | ψ i ) ≤ V ( | m i ) − ǫ } with probability 1 . W e choose ν = ǫ/ 2. Then if | ψ p i ∈ ¯ V ν then within a finite nu m b er o f steps less than l C κ/ 2 − ǫ  /c m the system state is in the set {| ψ i : V ( | ψ i ) ≤ V ( | m i ) − ǫ } . But b ecaus e {| ψ i : V ( | ψ i ) ≤ V ( | m i ) − ǫ } ∩ ¯ V ν is empt y , w e know that if | ψ p i ∈ ¯ V ν then the pro ces s is outside ¯ V ν within a finite num b er of steps less than l C κ/ 2 − ǫ  /c m . So lim p →∞ [Γ p ( µ )]( ¯ V ν ) 6 = 0 if and only if the Mark ov pro cess jumps bac k and forth b et w een the sets {| ψ i : V ( | ψ i ) ≤ V ( | m i ) − ǫ } and ¯ V ν infinitely often. But the sup ermartigale prop ert y of V ( | ψ p i ) and D oob’s inequality 6 implies prob  V ( | ψ p i ) < V ( | m i ) − ǫ a nd inf p ′ >p V (   ψ ′ p  ) ≥ V ( | m i ) − ǫ 2  < 1 − V ( | m i ) − ǫ V ( | m i ) − ǫ/ 2 < 1 . 22 As the probabilit y of a single jump is less than 1, the probability of in- finitely many jumps is zero. Therefore lim p →∞ [Γ p ( µ )]( ¯ V ν ) = 0 and from Equations ( 31 ) and ( 32 ) w e get µ ∞ ( V ν ) ≤ lim inf q →∞ [Γ p q ( µ )]( V ν ) ≤ κ. W e no w finally pro v e Theorem 3.1 . Pr o of of The or em 3.1 . Let ǫ > 0 and C > 0 b e given . W e know from Lemma 3.12 that the supp ort set of µ ∞ only consists o f the F o c k states | m i . Because σ n → ∞ , there exists a n M suc h t ha t σ M > C / ( ǫ/ 2) + δ . Because for all m > M , V ( | m i ) > σ m − δ , the sup ermartingale prop ert y of V ( | ψ k i ) and Do ob’s inequalit y ( 6 ) implies fo r a ll k ∈ Z + 0 , Γ k ( µ )( {| m i : m ≥ M } ) < ǫ 2 . If w e set κ = ǫ/ 2 M in Lemma 3.13 , then w e know there exist constants ¯ α > 0 and δ > 0 and neigh b orho ods V ( | m i ) of | m i for 0 ≤ m < M , m 6 = ¯ n , suc h tha t µ ∞ ( {| m i} ) < ǫ 2 M . Therefore, µ ∞ ( | ¯ n i ) = 1 − µ ∞ ( {| m i : m 6 = ¯ n } ) ≥ 1 − ǫ . Remark 3.3. I n L emma 3.12 we show that the only ve ctors in the s upp ort o f µ ∞ ar e those c orr esp onding to eigenve ctor of M s . We then use d assumption A 1 in the p r o of of the L emma to claim that the only eigenve ctors of M s ar e the F o ck states. We c an ho w ever we aken this assumption to the fol lo wing: for so me lar ge M such that σ M > 2 C /ǫ , eigenvalues c orr esp onding to eigen - ve ctors | m i , m < M ar e non-de gener ate. This is b e c ause, we c an show that if some eigenve ctor | ψ i is in the sp an of the set {| M i , | M + 1 i , . . . } then using the same ar gument as that use d for | m i , m > M , we c an show that the pr ob a b ility of | ψ i i s smal l. This is signific ant for c ases w her e M g is a mor e c omplic ate d non-line ar func tion of N , as is the c ase in a pr actic al system. Remark 3.4. In this p ap er we pr ove (we ak- ∗ ) c onver genc e of Γ k ( µ ) to some µ ∞ . The q u antum exp e ctation value of an observable (self - a djoint line ar op er- ators) T : H → H is s o me state | ψ i , define d h T i | ψ i , h ψ | T ψ i is a c ontinuous function of | ψ i if T is b ounde d. The r efor e, if we think of h T i : | ψ i 7→ h T i | ψ i 23 as a r andom variable on the state s p ac e ¯ B 1 , then by the (we ak- ∗ ) c onver genc e of Γ k l ( µ ) to µ ∞ , we hav e lim l →∞ E Γ k l ( µ ) [ h T i ] = E µ ∞ [ h T i ] . In p articular if T = | ¯ n i h ¯ n | then lim l →∞ E Γ k l ( µ ) [ h| ¯ n i h ¯ n |i ] = E µ ∞ [ h| ¯ n i h ¯ n |i ] ≥ 1 − ǫ, wher e ǫ is as in The o r em 3.1 . But h| ¯ n i h ¯ n |i is pr e cis e l y the (q uantum) pr ob- ability that the system is in state | ¯ n i . Ther ef o r e, the pr ob ability that the quantum system is in the state | ¯ n i mayb e made arbitr arily close to 1 . Similar statements mayb e made ab out the standar d deviation of b ounde d observables T : H → H . Sim ulations ha ve b een p erformed with the controller in Theorem 3.1 b y truncating the con troller using a Galerkin approxim ation. The sim ulations indicate that the con troller designed using the infinite dimensional Hilb ert space provides p erformance impro v ements (of a bout 4-5%) in the proba bility of conv ergence to the target state when compared t o the con t r o ller designed using the finite dimensional appro ximation [ 8 ]. Moreov er a s sho wn in The- orem 3.1 the feedback parameters ¯ α and δ maybe chosen to increase t he probabilit y o f con ve r g en ce to t he target state. 4. Conclusion In this pap er we examine the semi-global, appro ximate stabilization of the Mark ov pro cess in Equations ( 3 ) and ( 4 ) at a photo n- n um b er target state. The Mark ov pro cess is defined on the set of all unit v ectors in the infinite dimensional Hilb ert space H . The non-compactness of this set of unit v ectors dictates t he use of a sp ecial Lyapuno v f unc tio n ( 7 ) to sho w the follow ing in Theorem 3.1 - provided t he initial measures µ satisfies certain initial condi- tions, fo r all ǫ > 0 w e can choose f ee dback suc h that with probability greater than 1 − ǫ , the Mark o v pro ces s con v erges to the target F o c k state. 5. A c kno wledgmen ts The autho rs thank M. Brune, I. D otsenk o, S. Haro c he and J.M. Raimond for enligh tening discuss io ns and advices. 24 References [1] M. Mirrahimi, I. Dotsenk o , P . Ro uchon, in: 48th IEEE Conference on Decision and Con t r ol, IEEE, Shangha i, 2010, pp. 1451 – 14 5 6. [2] I. Do tse nko, M. Mirrahimi, M. Brune, S. Haro c he, J.-M. R a imond, P . Rouc hon, Phy sical Review A 80 ( 2 009) 013805–0 13813. [3] S. Del´ eglise, I. Dotsenk o, C. Sa yrin, J. Bern u, M. Brune, J.-M. Raimond, S. Haro c he, Nature 4 55 (2008) 510–51 4. [4] S. Gleyzes, S. Kuhr, C. Guerlin, J. Bern u, S. Del ´ eglise, U. B. Hoff, M. Brune, J.-M. Raimond, S. Har o che , Nature 446 (2007) 297–30 0. [5] C. Guerlin, S. D. J. Bernu , C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.-M. Raimond, S. Haro c he, Na ture 448 (2007) 889 –893. [6] S. Har o che , J. R aimond, Exploring the Quan tum: A toms, Ca vities and Photons, Oxford Univ ersity Press, 2 0 06. [7] M. Brune, S. Haro c he, J.-M. Raimond, L. Davido vic h, N. Zagury , Ph ys- ical Review A 45 (1 9 92) 5 193–5214. [8] R. Somara ju, M. Mirrahimi, P . Ro uc hon, a rXiv:1103.1724v1 (2 011). [9] K. Beauchard, M. Mirrahimi, SIAM J. Contr. Optim. 48 (2009) 1179– 1205. [10] M. Mirra himi, Ann. IHP Nonlinear Analysis 2 (2009) 17 43–1765. [11] K. Beauc hard, V. Nersesy an, ArXiv e-prints (201 0 ). [12] J.-M. Coron, B. dAnd ´ rea Nov el, IEEE T ra ns . Automat. Con trol 43 (1998) 608– 618. [13] J.-M. Coron, B. dAndr ´ ea No vel, G. Bastin, IEEE T ransactions on Au- tomatic Con t r o l 52 (200 7) 2–11. [14] G . T esc hl, Mathematical Methods in Q uan tum Mec hanics: With Ap- plications to Sc hr¨ odinger Op erators, volume 99 of Gr aduate Studies i n Mathematics , American Mathematical So cie t y , 2009. [15] M. Merkle, Zb. rado v a Mat. Inst. 9 (2000) 235 –274. 25 [16] P . Billingsley , Con v ergence of Probability Measures, John Wiley & Sons, Inc., 1999 . [17] M. Reed, B. Simon, Metho ds of mo dern mathematical ph ysics, v olume 1. F unctional Analysis, Academic Press, 1980. [18] M. A. Nielsen, I. L. Ch uang, Quan tum Computation a nd Quantum In- formation, Cambridge Univ ersity Press, 20 0 0. [19] M. Renardy , R. C. Ro gers , An Intro duction to P ar t ia l D iffere ntial Equa- tions, Springer-V erlag, 2 edition, 2004. [20] Kat o, P erturbation theory for linear op erators, Springer, 1966. 26