UMPC mercredi 19 avril 2017

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1 UMPC ercrei 19 avril 017 M Mathéatiques & Applications UE ANEDP, COCV: Analyse et contrôle e systèes quantiques Contrôle es connaissances, urée heures. Sujet onné par M. Mirrahii et P. Rouchon Les ocuents sont autorisés. Les accès aux réseaux internet et obiles sont interits. Exercise 1 Consier the tensor prouct H = H 3 H c where H 3 C 3 aits g, e, f as Hilbert basis an H c L R, C l C aits n n 0 as Hilbert basis Fock basis. Take the following Hailtonian on H ω g, ω e, ω f, ω c, χ real paraeters H = ω g g g + ω e e e + ω f f f I c + ω c I 3 N + Ic + χ g f + f g + e f + f e N + Ic where I 3 an I c are ientity operators on H 3 an H c, N = a a is the photon nuber operator on H c. We consier the Schröinger equation t ψ = ih ψ where ψ H. 1. With a = 1 x + x an ψ ψg, ψ e, ψ f L R, C L R, C L R, C give the partial ifferential forulation of the Schröinger equation.. With ψ = n 0 ψ g,n g n + ψ e,n e n + ψ f,n f n give the infinite set of orinary ifferential equations satisfie by ψ g,n, ψ e,n, ψ f,n n 0. Exercise Consier the 3-level syste of Hilbert space H C 3 with g, e, f as Hilbert basis with the following Hailtonian Ht = ω e e e +ω f f f +ut µ ge g e + e g +µ ef e f + f e +µ fg f g + g f where t ut R is the control input an ω e, ω f, µ ge, µ ef, µ fg are constant real paraeters. Consier the Schröinger equation t ψ = iht ψ with ω f > ω e > 0 an 0 < µ ge, µ ef, µ fg inω e, ω f ω e. 1. Take the passage to the interaction frae ψ φ = e it ω e e e +ω f f f ψ an copute the interaction Hailtonian H int t governing the Schröinger ynaics of φ : t φ = ih intt φ.. Assue that ut = ūe iω f t + ū e iω f t of constant aplitue ū C/{0} with ū 1. Justify that one can approxiate the tie evolution of φ by t φ = ih φ where H is a constant Hailtonian an provie its explicit expression. 1

2 3. We assue now that the state f is unstable an relaxes towars g or e with rates κ g, κ e > 0 uch saller that inω e, ω f ω e. This open quantu quantu is escribe by the Linbal aster equation for the ensity operator ρ in the interaction frae: [ t ρ = i H, ρ ]+κ g L g ρl g 1 L gl g ρ + ρl gl g +κ e L e ρl e 1 L el e ρ + ρl el e with L g = g f an L e = e f. Show that for any initial ensity operator ρ 0 = ρ0, the liit of ρt when t tens to + is the pure state e e Hint: use the Lyapunov function V ρ = 1 e ρ e an LaSalle s invariance principle. Proble We consier a quantu haronic oscillator efine on the Hilbert space { } H c = c n n c n l C n=0 where n correspons to the Fock state with n photons. Driving it at its resonance, the Hailtonian in the interaction frae is given by H c = iū a ūa. where ū C is a coplex aplitue an a is the photon annihilator operator. As illustrate in the course, this Hailtonian generates uring T 0 a unitary evolution U T = D α = e it Hc = e αa α a with α = T ū. Through this proble, we will stuy the situation where this Hailtonian evolution is accopanie by frequent easureents of a certain observable O 1 = 1 1. Inee, we will assue that this ynaics is perfore in steps of length T/ an labele fro k = 0 to k = 1, together with a easureent after each step. In this ai, we consier the easureent operators M g = I 1 1, M e = 1 1. The ynaics of the syste is oele by the Markov chain of state ψ k H c an easureent outcoes y k {g, e} at step k: ψ k+1/ = D α ψ k, M g ψ k+1/ with y k = g, probability ψ k+1/ M gm g ψ k+1/ ; ψ k+1/ M gm g ψ k+1/ ψ k+1 = M e ψ k+1/ with y k = e, probability ψ k+1/ M em e ψ k+1/. ψ k+1/ M em e ψ k+1/ Furtherore, we assue the initial state to be given by ψ 0 = 0. Physically ψ correspons then to the wave function at tie T. 1. Show that the operators M g an M e represent an eligible Kraus ap. Show that this easureent is quantu non-eolition for an observable O if an only if n O 1 = 0 for all n 1.. Provie the state ψ g r of the syste conitione on r easureents giving as result y k = g for all k = 0,, r 1.

3 3. Show that the probability p g r of easuring y k = g for all k = 0,, r 1 is given by p g r = M g D α r Now, we ai at stuying the liits li p g an li ψ. g Show that 1 M g D α 0 0 = O. 5. Deuce that li pg = 1 an li ψg = 0 strongly in H c. Hint: Use the fact that D α/ is a unitary an that M g is a projection, an therefore they o not increase the nor of a state in H c. 6. Provie a siple an physical interpretation of the above liits. 7. Now we consier a ifferent easureent process base on the observable O =. We consier the associate Kraus operators M g = I an M e =. Also, for siplicity sakes, we assue α to be real. a Take c 0, c 1 R such that c 0 + c 1 = 1, an consier the wave functions ψ = c c 1 1 an ψ = c 0 αc 1 / 0 + c 1 + αc 0 / α /. Show that M g D α ψ ψ = O 1 Hint: Calculate Dα/ 1 by noting that 1 = a 0 an using the coutation relations. b Deuce the liits li p g an li ψ g p g an ψ g are the probability to etect y k = g for k = 0,, 1 an the corresponing quantu state at step starting fro ψ 0 = 0. c Provie a siple an physical interpretation of the above liits. 3

4 UMPC Mercrei 19 avril 017 M Mathéatiques & Applications UE ANEDP, COCV: Analyse et contrôle e systèes quantiques Corrigé u Contrôle es connaissances M. Mirrahii et P. Rouchon Exercise 1 1. We have i ψ g t i ψ e t i ψ f t = ω g ψ g + ω c x x ψ g + χ x x ψ f = ω e ψ e + ω c x x ψ e + χ x x ψ f = ω f ψ f + ω c x x ψ f + χ x x ψ g + χ x x ψ e. We have i t ψ g,n = n + 1/ω c + ω g ψ g,n + χn + 1/ψ f,n i t ψ e,n = n + 1/ω c + ω e ψ e,n + χn + 1/ψ f,n i t ψ f,n = n + 1/ω c + ω f ψ f,n + χn + 1/ψ g,n + χn + 1/ψ e,n Exercise 1. We have H int t = utµ ge e iωet g e + e iωet e g + utµ ef e iω f ω et e f + e iω f ω et f e + utµ fg e iω f t f g + e iω f t g f.. Since µ ge, µ ef, µ fg inω e, ω f ω e, we can use the rotating wave approxiation an keep only the non-oscillating ters secular ters in H int t where ut is replace by ūe iω f t + ū e iω f t. This yiels to H = µ fg ū f g + ū g f φ. 3. Since ρt reains non-negative an of trace one, V ρ reains between 0 an 1. Moreover V ρ = 0 eans that ρ = e e. Since [ ] t ρ = iµ fg ū f g + ū g f, ρ + f ρ f κ g g g + κ e e e κg+κe f f ρ + ρ f f 4

5 we have t V ρ = κ e f ρ f 0. Thus V is a ecreasing tie function. Since the set of ensity operators is copact an V 0, we can apply LaSalle s invariance principle: the trajectories converge towars the largest invariant set of ensity operators satisfying tv = 0. When f ρ f = 0 we have ρ f = 0 an f ρ = 0 since ρ is a ensity operator therefore non-negative. Then we have t ρ = iµ fg ū f g ρ ū ρ g f an we get by ifferentiating ρ f = 0 with respect to t: t ρ f = 0, i.e. µ fgū ρ g = 0. This eans that f an g are in the kernel of ρ. This iplies that ρ is necessarily the projector on e since it ust be of trace one an non-negative. This Linbal equation is the siplest ynaical oel escribing optical puping, a siple an powerful iea ue to Alfre Kastler Physics Nobel Prize 1966 for preparing an stabilizing pure states. Proble 1. It is easy to check that M gm g + M em e = I an therefore they represent an eligible Kraus ap. The easureent is non-eolition for an observable O, if the Kraus operators M g an M e coute with O. It is easy to check that this conition is equivalent to O 1 = 0, 1.. The state at the step r is given by ψ g r = M gd α ψg r 1 M g D α ψg r 1. Therefore by inuction, it is easy to see that M ψr g g D α r 0 = M g D α r The probability for the first easureent to give y 0 = g is clearly M g D α 0. The probability to achieve r easureents giving all y k = g is given by But an p g r = Py r 1 = g, y r = g, y 0 = g = Py r 1 = g y r = g, y 0 = gpy r = g, y r 3 = g, y 0 = g. Py r 1 = g y r = g, y 0 = g = M g D α ψg r 1 = The proof is clear by inuction. Py r = g, y r 3 = g, y 0 = g = p g r 1. 5 M g D α M g D α r 0 r 1 0

6 4. We have M g D α 0 = I 1 1 α α = e I 1 1 = e α 0 + e α k= 1 α k k! k k. k=0 1 α k k! k k We note that e α = 1 + O1/ an that 1 α k α k k! k k= k= Noting that the series is convergent, the result is clear. 5. One can write 1 α k k! k. M g D α 0 = 0 + O 1 χ 0, where χ 0 is a noralize state in H c. Therefore M g D α 0 = M g D α 0 + O 1 χ 0 We note that M g D α χ 0 1, as D α = 0 + O 1 χ 0 + O 1 M gd α χ 0. is a unitary therefore conserving the nor an M g is a projection therefore reucing the nor. Thus O 1 M g D α χ 0 can be written as O 1 χ 1 for a noralize state χ 1 in H c. In the sae anner M g D α 0 = 0 + O 1 1 χ k, where χ k s are noralize states in H c. Therefore Furtherore p g = M g D α 0 = 0 + O 1 1 χ k 1 as. k=0 ψ g M g D α 0 = 0 0 p g k=0 0 as. 6. We have illustrate that, whenever we easure frequently the observable O 1 uring the unitary evolution, we freeze the state at tie T T > 0 being arbitrary in 0 an reove the effect of the riving Hailtonian. This is calle the quantu Zeno effect. 7. a We have M g D α ψ = c 0M g α + c 1M g D α 1. In orer to calculate D α 1, we note that D α 1 = D α a 0 = D α a D α D α 0 = a α α. 6

7 As M g = I, we have M g D α ψ = c α 1! 1 + c 1M g a α α 1! 1 + O 1 χ 0, where χ 0 is a noralize state in H c. Therefore M g D α ψ = c α 1 + c 1M g a α α 1! 1 + O 1 χ 0 = c α 1 + c 1M g 1 + α α 0 + O 1 χ 1 = c α 1 + c 1 1 α 0 + O 1 χ 1, where χ 0 is a noralize state in H c. This proves the relation as 1/ 1 + α / = 1 + O1/. b One has ψ = R θ ψ, where M g D α ψ ψ = O1/, 1 α/ R θ = 1+α / 1+α / α/ 1 1+α / 1+α / is a rotation atrix with θ = arctanα/ in the space span{ 0, 1 }. Siilarly to the question 5, we have li p g = 1. Also, we have ψ g = R arctanα/ 0 + O1/ χ = R arctanα/ 0 + O1/ χ, where χ is a noralize state in H c. Now, note that R arctanα/ 0 converges to R α 0 for tening to infinity. c We have shown that the easuring frequently the observable O, we confine the ynaics of the haronic oscillator to the two-iensional subspace spanne by 0 an 1. A unitary isplaceent of the cavity state is therefore replace by a Rabi oscillation for this effective two-level syste. This is calle Quantu Zeno Dynaics. 7