Closed loop observer-based parameter estimation of quantum systems with a single population measurement

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1 Close loop observer-base parameter estimation of quantum systems with a single population measurement Zaki Leghtas To cite this version: Zaki Leghtas. Close loop observer-base parameter estimation of quantum systems with a single population measurement. 11 pages <hal > HAL I: hal Submitte on 1 Nov 008 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 Close loop observer-base parameter estimation of quantum systems with a single population measurement Zaki Leghtas November 18, 008 Abstract An observer-base Hamiltonian ientification algorithm for quantum systems has been propose in []. In this paper we propose another observer enabling the ientification of the ipole moments of a multi-level case, an having access to the population of the groun state only. Keywors: Nonlinear systems, Quantum systems, Parameter ientification, Asymptotic observers, Averaging, Feeback. 1 Introuction This work is the result of a summer internship I i at INRIA Rocquencourt, with Mazyar Mirrahimi 1 an Pierre Rouchon. Our goal was to improve the result given in [] in orer to enable an experimentator to estimate the ipole moments of a quantum system measuring continuously the population on the first state. First, we will explain the problem an expose the moel of the consiere quantum system. In the next section, we explicit the observer algorithm an look at some simulations. Then, using some averaging arguments an neglecting some secon orer terms, we prove the convergence of the observer. We finally come to the interesting conclusion that a feeback is require. The three level system.1 The problem We have a quantum system an we wish to estimate its ipole moment matrix. Ecole es Mines e Paris, 60 B Saint-Michel, 757 Paris ceex 06, FRANCE. zaki.leghtas@ensmp.fr 1 INRIA Rocquencourt, Domaine e Voluceau, Rocquencourt B.P. 105, Le Chesnay Ceex, FRANCE. mazyar.mirrahimi@inria.fr Ecole es Mines e Paris, Centre Automatique et Systèmes, 60 B Saint-Michel, 757 Paris ceex 06, FRANCE. pierre.rouchon@ensmp.fr 1

3 Figure 1: Diagram of the close loop ientification algorithm. 1. We excite the system with a laser. We are free to moulate its amplitue an it s phase.. We only measure continuously the population on it s groun state. In this paper, we propose a metho to extract the esire information (the ipole moments) from this output.. The moel We consier a three level quantum system, whose ynamics is escribe by the Schröinger equation: t Ψ = i (H 0 + A(t)H 1 )Ψ, y = 1 ΨΨ 1, where Ψ is the wavefunction, A(t) is the laser fiel, H 0 is the sum of the kinetic energy operator an the potential energy operator without the laser fiel, H 1 is the ipole moment matrix an y is the measurement output. Our aim is to estimate this matrix: λ µ 1 0 H 0 = 0 λ 0, H 1 = µ 1 0 µ λ 3 0 µ 3 0 The goal is to estimate µ 1 an µ 3. Within the ensity language matrix ρ = Ψ Ψ, an taking = 1, we have: 3 The observer 3.1 The laser fiel ρ = i[h, ρ]. We choose an electromagnetic fiel which is the sum of two lasers:

4 1. Laser 1: Ā 1 cos((λ λ 1 )t) with Ā1 constant an of orer 1. Laser : A 3 cos((λ 3 λ )t) with A 3 = Ā3 cos(ˆθ) an ˆθ = Ā 1 ˆµ 1, Ā3 constant of orer 1 an ˆµ 1 being the estimate value of µ 1 an is explicitly efine below. This amplitue moulation can be as slow as esire by choosing a small Ā 1 as long as Ā1 >> ɛ. We then sum these two lasers, putting a small factor ɛ in front of the secon one. This gives the following laser fiel: A(t) = Ā1cos((λ λ 1 )t) + ɛā3cos(ˆθ)cos((λ 3 λ )t) Which leas us to a ynamics written with the effective hamiltonian to: ρ = iā1µ 1 [σx 1, ρ] iɛā3µ 3 cos(ˆθ)[σx 3, ρ] 3. The propose metho We use observers which make the estimate values converge to the real ones. We suppose that H 1 =Ĥ1, an see what output ŷ it woul give. The error: e=y-ŷ is then use to correct the estimation of H 1. The ifficulty is to fin an aequate way to eal with this error in orer to make Ĥ1 converge to H 1. The other ifficulty is the choice of the laser amplitue A(t) which enables us to extract the right information from the system. The estimation metho is the following: We esign an observer such that estimator of µ 1 converges rapily towars µ 1. We a some small perturbation terms so that, once the estimation of µ 1 is accurate enough, the secon orer terms will make the estimator of µ 3 converge to µ 3. We o this by using the estimate values of µ 1 to moulate the amplitue of Ā the secon laser. The moulation is one with a frequency of 1µ 1. Note that this frequency Ā1µ1 can be chosen as small as neee, through the choice of Ā 1 (which is itself a control egree of freeom). The notion of feeback can lea some ifficulties in the realization of the experiment. However, to avoi suc complications, one can first apply the laser 1, an once the value of µ 1 is known, he can switch on the laser an moulate its amplitue with a signal of frequency A1µ1. This will ensure the convergence of ˆµ 3 to µ 3 Intuitively, one can preict that by measuring the population of state 1, one can ientify the ipole moment between state 1 an any other state k (k = or k = 3), by exciting the system with a resonant laser with the transition frequency between state 1 an k. On the other han, ientifying the ipole moment between any two states k an m (m an k ifferent from 1), with only the population of state 1 seems ifficult to achieve. Hence, we excite the system with a laser resonant with transition 1 an m, an m an k. We make sure the laser coupling 1 an m is much more powerful than the secon one. We can therefore consier states 1 an m as being equivalent to a unifie average state. An therefore, we are back to a situation where we have states, knowing the population of the first one. The ifferent scales of amplitues between the two 3

5 lasers an the notion of average systems[4][3] are key notions in the esign of the propose observer. 3.3 The observer We propose the following observer: ˆµ 1 ˆρ = iā1 [σ 1 ˆµ 3 x, ˆρ] iɛā3 cos(ˆθ)[σx 3, ˆρ] +ɛγ 1 (y ŷ)(σz 1 ˆρ + ˆρσz 1 Tr(σz 1 ˆρ)ˆρ) +ɛ Γ (y ŷ)[(1 + cos(ˆθ))(q 1 ˆρ + ˆρQ 1 Tr(Q 1 ˆρ)ˆρ) +(1 cos(ˆθ))(q ˆρ + ˆρQ Tr(Q ˆρ)ˆρ) (Q 3 ˆρ + ˆρQ 3 Tr(Q 3 ˆρ)ˆρ)] ˆθ = Ā1 ˆµ 1 ˆµ 1 = i Ā 1 ɛ γ 1 Tr(σ 1 z[σ 1 x, ˆρ])(y ŷ) ˆµ 3 = i Ā 3 ɛ 4 (y ŷ)γ [(1 cos(ˆθ))tr(q [Σ 3 x, ˆρ]) Tr(Q 3 [Σ 3 x, ˆρ])] With: Γ 1, Γ, γ 1, γ are constants of orer 1. An ɛ is a small parameter. Explicit expressions of the matrices σx ij an so on are given in the appenix. We have consiere the notations: ˆΩ ij = 1 ˆµ ijāij y = Tr(P 1 ρ) ŷ = Tr(P 1 ˆρ) an: where: Σ 1 x = L σ 1 x L Σ 3 z = L σ 3 z L Σ 3 x = L σ 3 x L Q 1 = L P 1 L Q = L P L Q 3 = L P 3 L 3.4 Simulations 1 iˆθσ L = e x Before entering the etails of the convergence proof of the observer, let us look at some simulations. N.B: Ω e an y e stan respectively for ˆΩ an ŷ The simulation was one with the following parameters: ɛ = 1/10 Γ 1 = γ 1 = Γ = 1.6 γ = 1.8 With an initial state: Ψ 0 = i.6 +.1i i We initialise the observer at: ˆΨ0 = i.6 +.i i 4

6 Figure : Output. Measure output error an parameter estimation Figure 3: Output with a 0% gaussian noise. Measure output error an parameter estimation 5

7 The ipole moment matrix is: H 1 = We start the observer at Ĥ1 = µ 1 converges much faster (about 1 ɛ times faster) than µ 3. We notice that the ouput error is about 10 which is the orer of the noise. 3.5 Ientifiability It has been prove in [1] that if one has access to the population of all states of the quantum system, it is possible to ientify the ipole moment matrix. In this paper, we only measure the population of the groun state. But in fact, the neee information is coe in the average system. Inee, by writing the ynamics of the system in a frame such as the terms of orer 1 vanish, hence by consiering: η = e iω1t ρe iω1t, we obtain the following output: y(t) = 1 (Tr(I 1η) + cos(ω 1 t)tr(σ 1 zη) + sin(ω 1 t)tr(σ 1 y η)) The first orer terms in η are slow compare to cos(ω 1 t) an sin(ω 1 t). Notice that in average: y(t)(1 + cos(ω 1 t)) = Tr(P 1 η) y(t)(1 cos(ω 1 t)) = Tr(P η) 1 y(t) = Tr(P 3 η) Hence, we have transforme the problem from a system where we only measure the population of its first groun state, to a system where, in average, we know the populations of all the states. Consiering that ˆΩ 1 has converge to Ω 1, we obtain the following system: t η = i ɛ Ω 3[σx 3, η] y j (t) = Tr(P j η) is known in average for j = 1,, 3. This, accoring to [1], is ientifiable. 4 Convergence analysis Let us write the observer as follows: ˆρ = iˆω 1 [σx 1, ˆρ] iɛˆω 3 cos(ˆθ)[σx 3, ˆρ] + ɛγ 1 (y ŷ)(σz 1 ˆρ + ˆρσz 1 Tr(σz 1 ˆρ)ˆρ) [ + ɛ Γ (y ŷ) (1 + cos(ˆθ))(q 1 ˆρ + ˆρQ 1 Tr(Q 1 ˆρ)ˆρ) ] + (1 cos(ˆθ))(q ˆρ + ˆρQ Tr(Q ˆρ)ˆρ) (Q 3 ˆρ + ˆρQ 3 Tr(Q 3 ˆρ)ˆρ) ˆθ = ˆΩ 1 ˆΩ 1 = iɛ γ 1 Tr(σz[σ 1 x 1, ˆρ])(y ŷ) ˆΩ 3 = iɛ 4 γ (y ŷ)γ [(1 cos(ˆθ))tr(q [Σ 3 x, ˆρ]) Tr(Q 3 [Σ 3 x, ˆρ])]. 6

8 As we can see in the simulation, an seeing the esign of the observer, the convergence is one in two steps: 1. We consier the ynamics of the system neglecting the terms in iɛˆω 3 cos(ˆθ)[σx 3, ˆρ] in front of iˆω 1 [σx 1, ˆρ] an neglecting the term in ɛ Γ in front of ɛγ 1. We then prove that (a) lim t + ˆΩ1 = Ω 1 (b) lim t + Tr(σ 1 y ˆη) = Tr(σ 1 y η) (c) lim t + Tr(σ 1 z ˆη) = Tr(σ 1 zη). Once all these terms have converge, we consier the higher orer terms of the system an prove that: (a) lim t + ˆΩ3 = Ω 3 (b) lim t + Tr(P 1 ˆζ) = Tr(P1 ζ) (c) lim t + Tr(P ˆζ) = Tr(P ζ) () lim t + Tr(P 3 ˆζ) = Tr(P3 ζ) η an ζ being equal to ρ in some aequately chosen frames. Their expressions will be given below. 4.1 The convergence of the first orer system We now write the system consiering the approximations of step 1, mentione above. Which leas us to the system: ρ = iω 1 [σ 1 x, ρ] ˆρ = iˆω 1 [σ 1 x, ˆρ] + ɛγ 1 (y ŷ)(σ 1 z ˆρ + ˆρσ 1 z Tr(σ 1 z ˆρ)ˆρ) ˆΩ 1 = iɛ γ 1 Tr(σ 1 z[σ 1 x, ˆρ])(y ŷ) We are now tempte to write these equation in a rotating frame, such as the ominant term in Ω 1 vanishes. We will then be able to see which terms have a first or higher orer influence. Let us efine: η = R ρr an ˆη = R ˆρR where R = e iω1σ1 x t Here are some results which will be useful throughout our calculations: R σ 1 zr = cos(ω 1 t)σ 1 z + sin(ω 1 t)σ 1 y (1) LP 1 L = 1 (I 1 + cos(ω 1 t)σ 1 z + sin(ω 1 t)σ 1 y ) () 7

9 Which leas us to: t η = 0 t ˆη = i(ˆω 1 Ω 1)[σx 1, ˆη] h i Tr(I 1(η ˆη)) + cos(ω 1t)Tr(σz(η 1 ˆη)) + sin(ω 1t)Tr(σy 1 (η ˆη)) h i h i + cos(ω 1t) σz 1 ˆη + ˆησ z 1 Tr(σz 1 ˆη)ˆη + sin(ω 1t) σy 1 ˆη + ˆησ y 1 Tr(σy 1 ˆη)ˆη + ɛ Γ1 t ˆΩ 1 = iɛ γ 1 ˆTr(I1(η ˆη)) + cos(ω 1t)Tr(σz(η 1 ˆη)) + sin(ω 1t)Tr(σy 1 (η ˆη)) + (cos(ω 1t)Tr(σz[σ 1 x 1, ˆη]) + sin(ω 1t)Tr(σy 1 [σx 1, ˆη])) After neglecting the highly oscillating terms of average 0 in front of the first orer secular ones, an rearranging the terms, we obtain the following equations: t η = 0 h t ˆη = i(ˆω 1 Ω 1)[σx 1, ˆη] + ɛγ1 Tr(σz(η 1 ˆη))[σz 1 ˆη + ˆησ z 1 Tr(σz 1 ˆη)ˆη] 4 i + Tr(σy 1 (η ˆη))[σy 1 ˆη + ˆησ y 1 Tr(σy 1 ˆη)ˆη] t ˆΩ 1 = iɛ γ 1 1 `Tr(σ z (η ˆη))Tr(σz[σ 1 x 1, ˆη]) + Tr(σy 1 (η ˆη))Tr(σy 1 [σx 1, ˆη]). 4 N.B: Note that ˆΩ 1 Ω 1 of orer ɛ is an essential hypothesis. That is why it is important to choose, for t = 0, ˆΩ 1 close to Ω 1. Let us use the following notations: We have X = Tr(σx 1 η) Y = Tr(σy 1 η) Z = Tr(σzη) 1 ˆX = Tr(σx 1 ˆη) Ŷ = Tr(σy 1 ˆη) Ẑ = Tr(σz 1 ˆη). Ẋ = Ẏ = Ż = 0 ˆX = ɛ Γ 1 4 [(Z Ẑ)(iŶ + Ẑ ˆX) + (Y Ŷ )(iẑ + Ŷ ˆX)] Ŷ = (ˆΩ 1 Ω 1 )Ẑ + ɛγ 1 4 [(Z Ẑ)(i ˆX Ŷ Ẑ) (Y Ŷ )(1 Ŷ )] Ẑ = (ˆΩ 1 Ω 1 )Ŷ + ɛγ 1 4 [(Z Ẑ)(1 Ẑ ) (Y Ŷ )(i ˆX + Ŷ Ẑ)] ˆΩ 1 = iɛ γ 1 ((Z Ẑ)Ŷ (Y Ŷ )Ẑ) 4 We are now going to prove that the propose observer enables us to estimate Ω 1, X, Y an Z (we assume everywhere in this paper that Y an Z are both ifferent from zero). Let s consier the Lyapounov function: V : ˆΩ 1 Ŷ Ẑ R 3 R 4 ɛ γ 1 (ˆΩ 1 Ω 1 ) + (Ẑ Z) + (Ŷ Y ) V = 4 ɛ γ 1 ˆΩ1 (ˆΩ 1 Ω 1 ) + Ẑ(Ẑ Z) + Ŷ (Ŷ Y ) 8

10 Let us etail some calculations which will help us compute the latter expression σ 1 y σ 1 y = σ 1 zσ 1 z = I 1 σ 1 y σ 1 z = iσ 1 x = σ 1 y σ 1 z Notice that in a first approximation, we have Tr(I 1ˆη) = Tr(I 1ˆη ) = 1 Which leas us to: V Γ = [(Ẑ Z) + (Ŷ Y ) ] + ((Ẑ Z)Ẑ + (Ŷ Y )Ŷ ). Accoring to the Cauchy Schwartz inequality, we have: V Γ [(Ẑ Z) + (Ŷ Y ) ] + ((Ẑ Z) + (Ŷ Y ) )(Ẑ + Ŷ ) [(Ẑ Z) + (Ŷ Y ) ][1 Ẑ Ŷ ]. Obviously, [(Ẑ Z) + (Ŷ Y ) ] 0 an since Tr(I 1ˆη ) 1, we have 1 Ẑ Ŷ case of equality: case where 1 Ẑ Ŷ = 0 an K in R such that Ẑ = KZ an Ŷ = KY : In this situation, ˆη = ˆη ± = 1 (I ± ( Y Y + Z σ1 y + Z Y + Z σ1 z) +...) One can see that (ˆη ±, Ω 1 ) are equilibrium points. But by choosing ˆη(0) η < ˆη ± η, ˆη(0) being the initial state of the observer, the Lyapounov function will keep ecreasing, an the observer will never reach this equilibrium point. case where (Ẑ Z) = (Ŷ Y ) = 0: The ynamics of the system at this point become: an: t η = 0 t ˆη = i(ˆω 1 Ω 1 )[σx 1, ˆη] t ˆΩ 1 = 0 t η = 0 t (Ẑ Z) = (ˆΩ 1 Ω 1 )Ŷ t (Ŷ Y ) = (ˆΩ 1 Ω 1 )Ẑ t ˆΩ 1 = 0 9

11 Notice that by assuming Y an Z both non-zeros, we have exclue the case Ŷ = Ẑ = 0 an therefore necessarily ˆΩ 1 = Ω 1. We conclue that the LaSalle invariant set is given by Ẑ = Z, Ŷ = Y an ˆΩ 1 = Ω 1. Hence ˆΩ 1 Ŷ Ẑ converges to Ω 1 Y Z. 4. The ientification of µ 3 The main ifficulty here, compare to the solution propose by [1] is that we only have access to the population on the groun state. We now consier the average system with all the terms, in which we neglect the terms in Z Ẑ, Y Ŷ an Ω 1 ˆΩ 1 : t η = iω 3ɛ[σx 3, η] t ˆη = iˆω 3 ɛ[σx 3, ˆη] +ɛ Γ [Tr(P 1 (η ˆη))(P 1ˆη + ˆηP 1 Tr(P 1ˆη)ˆη) +Tr(P (η ˆη))(P ˆη + ˆηP Tr(P ˆη)ˆη) +Tr(P 3 (η ˆη))(P 3 ˆρ + ˆρP 3 Tr(P 3 ˆρ)ˆρ)] t ˆΩ 3 = iɛ 4 γ (Tr(P (η ˆη))Tr(P 1 [σx 3, ˆη] + Tr(P 3 (η ˆη))Tr(P 3 [σx 3, ˆη]) We are now confronte to the problem solve in [], where the populations on all the states are known. Please refer to the latter paper for an explanation about the convergence of this observer. The scheme of the explanation is: 1. We write the equations in the rotating frame by efining a new variable: ζ = e iω3σ3 x t ηe iω3σ3 x t. We then neglect the highly oscillating terms 3. After efining a Lyapounov function an using the Lasalle invariance principle an the averaging theorem, we conclue that (a) lim t + ˆΩ3 = Ω 3 (b) lim t + Tr(P 1 ˆζ) = Tr(P1 ζ) (c) lim t + Tr(P ˆζ) = Tr(P ζ) () lim t + Tr(P 3 ˆζ) = Tr(P3 ζ) References [1] C. Le Bris, M. Mirrahimi, H. Rabitz, an G. Turinici. Hamiltonian ientification for quantum systems: well-poseness an numerical approaches. ESAIM: Control, Optimization an Calculus of Variations, 13 (): ,

12 [] S. Bonnabel, M. Mirrahimi, P. Rouchon Observer-base Hamiltonian ientification for quantum systems. Automatica, Accepte, ArXiv: mathph/ [3] P. Rouchon. Quantum systems an control. Conference in honor of Claue Lobry, Saint Louis, Senegal, September 007 [4] L. Lanau an E. Lifchitz. Volume 1. Mecanics ; Mir itions. [5] M. Mirrahimi an P. Rouchon. Singular perturbations an Linbla- Kossakowski ifferential equations. submitte, preliminary version: ArXiv: v1. A Gell-Mann matrices Let s recall the expressions of the Gell-Mann Matrices: an We also efine σx kl = k l + l k, σy kl = i k l + i l k σ z1 = σ z = P 1 = 0 0 0, σz 3 =