Gerardo Cardona
email: gerardo.cardona@mines-paristech.fr

Alain Sarlette
email: alain.sarlette@inria.fr

Pierre Rouchon
email: pierre.rouchon@mines-paristech.fr

Exponential stabilization of quantum systems under continuous non-demolition measurements

Keywords: Quantum systems, quantum control, stochastic systems, control Lyapunov function, exponential stabilization, measurement-based feedback, low-order quantum filters, quantum non demolition measurement

We present a novel continuous-time control strategy to exponentially stabilize an eigenstate of a quantum measurement operator. In open-loop, the system converges to a random eigenstate of the measurement operator. The role of the feedback is to prepare a prescribed eigenstate with unit probability. To achieve this we introduce the use of Brownian motion to drive the unitary control actions; the feedback loop just adapts the amplitude of this Brownian noise input as a function of the system state. Essentially, it "shakes" the system away from undesired eigenstates by applying strong noise there, while relying on the open-loop dynamics to progressively reach the target. We prove exponential convergence towards the target eigenstate using standard stochastic Lyapunov methods. The feedback scheme and its stability analysis suggest the use of an approximate filter which only tracks the populations of the eigenstates of the measurement operator. Such reduced filters should play an increasing role towards advanced quantum technologies.

Introduction

The progress of methods for measuring and controlling quantum systems [START_REF] Sayrin | Real-time quantum feedback prepares and stabilizes photon number states[END_REF][START_REF] Campagne-Ibarcq | Using spontaneous emission of a qubit as a resource for feedback control[END_REF] now allows the physics community to implement building blocks of quantum information processors [START_REF] Reinier | Cavity state manipulation using photon-number selective phase gates[END_REF][START_REF] Matthew D Reed | Realization of three-qubit quantum error correction with superconducting circuits[END_REF][START_REF] Ofek | Extending the lifetime of a quantum bit with error correction in superconducting circuits[END_REF] that pose challenging control problems. One of the elementary building blocks is the stabilization of a quantum system onto a target eigenstate of a measurement operator. In a quantum computer [START_REF] Michael | Quantum computation and quantum information[END_REF], such stabilization could be used e.g. for initializing the input states, or for providing auxiliary states that enable particular operations, like entangled states for quantum teleportation [START_REF] Bennett | Purification of noisy entanglement and faithful teleportation via noisy channels[END_REF][START_REF] Yamamoto | Feedback control of quantum entanglement in a two-spin system[END_REF][START_REF] Sarlette | Loss-tolerant parity measurement for distant quantum bits[END_REF] and metrology [START_REF] Cox | Deterministic squeezed states with collective measurements and feedback[END_REF] or magic states for performing T-gates [START_REF] Bravyi | Universal quantum computation with ideal clifford gates and noisy ancillas[END_REF]. Moreover, quantum error correction protects information by encoding it in a larger state space and rejecting deviations from the nominal code-space [START_REF] Gottesman | Stabilizer codes and quantum error correction[END_REF]. In this sense, strategies for stabilizing a state are a stepping stone towards stabilizing a code subspace and thus protecting in-formation towards quantum information processing. An extension of the present setting towards quantum error correction can be found in [START_REF] Ahn | Continuous quantum error correction via quantum feedback control[END_REF][START_REF] Cardona | Continuous-time quantum error correction with noiseassisted quantum feedback[END_REF].

The defining element of our control setting is continuous quantum measurement, where a continuous-time signal provides at each time infinitesimal information, associated to infinitesimal backaction, on the quantum state [START_REF] Barchielli | Measurements continuous in time and a posteriori states in quantum mechanics[END_REF]. In practice this can be understood as measuring the system indirectly, by actually measuring an auxiliary system (usually a traveling electromagnetic field mode) that interacts with it. In general, this setting allows to obtain measurements of non-commuting observables in parallel, although thus only providing partial information and backaction about each channel, and to have measurement channels associated to non-Hermitian operators like the one describing energy loss [START_REF] Campagne-Ibarcq | Observing quantum state diffusion by heterodyne detection of fluorescence[END_REF]. The most standard measurement for feedback control though is the continuous-time counterpart to the projective measurement [START_REF] Haroche | Exploring the Quantum: atoms, cavities and photons[END_REF], namely the case where the measurement signal(s) give infinitesimal information about a (set of commuting) Hermitian measurement operator(s); following stochastic dynamics, the system progressively converges towards one of the eigenstates of this measurement operator, and the asymptotic distribution is in perfect agreement with the projective measurement postulate. All eigenstates of the measurement operator are thus invariant under the dynamics; it is therefore sometimes called quantum non-demolition (QND) measurement, although this term has been used with different meanings. The measurement itself can be considered as a conditional preparation tool [START_REF] Dotsenko | Quantum feedback by discrete quantum nondemolition measurements: Towards on-demand generation of photon-number states[END_REF], where the resulting state is accepted or rejected depending on measurement results, since the system converges to one of the eigenstates at random. An additional feedback control is necessary for preparing and stabilizing a particular target eigenstate unconditionally. We here address how to add this feedback layer in continuous time.

Stabilization of measurement operator eigenstates with continuous-time measurements has been investigated in the literature extensively [START_REF] John K Stockton | Deterministic dicke-state preparation with continuous measurement and control[END_REF][START_REF] Van Handel | Feedback control of quantum state reduction[END_REF][START_REF] Mirrahimi | Stabilizing feedback controls for quantum systems[END_REF][START_REF] Tsumura | Global stabilization at arbitrary eigenstates of n-dimensional quantum spin systems via continuous feedback[END_REF][START_REF] Liang | On exponential stabilization of spin-1/2 systems[END_REF]. Our proposal is meant to improve these results on three aspects. First and foremost, while previous proposals prove convergence in a proper probabilistic sense, their analysis technique does not provide an estimate of the rate of convergence. By using a Lyapunov technique and a novel control approach, we here prove exponential convergence. This is not unexpected since the measurement process in open loop converges exponentially towards the set of eigenstates. Second, the feedback laws in existing work a priori depend on the full state ρ, which must be estimated in real-time. Our feedback law only depends on eigenstate populations which leads to a reduced filter. Third, existing feedback laws rely on conditional pulses, inspired by optimal discrete-time strategies, yet with the danger that such abrupt signals excite spurious dynamics in the fragile quantum system. Our strategy instead uses continuous control signals, consisting of Brownian noise whose gain is adapted as a function of the estimated state. We call this noise-assisted quantum feedback.

In a nutshell, the strategy that we propose is rather simple to summarize. We let the system evolve in open loop until it gets close to one of the measurement operator eigenstates. If it is the good one, we are done. If it is another one, as we get closer to it, we increase noise input on the system, effectively shaking it away from the bad eigenstate such that it has another chance to converge to the good one. This strategy works with generic conditions on the control Hamiltonian and simple control logic. The detail of the convergence proof is somewhat challenging, but transparent to the end user. The use of a noisy input signal can also be understood as a necessity to induce global exponential convergence on a compact set.

The structure of this paper is as follows: Section 2 presents the dynamical model for continuous-time quantum measurements and introduces the control problem. Section 3 provides a brief overview of current feedback designs. In section 4 we introduce the use of Brownian motion to drive the controls and the standard control design that will be followed. Section 5 presents the main convergence theorem, proving exponential convergence towards a target eigenstate with a closed-loop Lyapunov function. In section 6 we present an approximated filter to estimate the eigenstate populations, inspired from the stability analysis of the previous section. Lastly, in section 7 we make numerical simulations on a spin J system, illustrating the robustness of the control approach.

2 Open-loop dynamics and feedback goal Consider a quantum system of finite dimension n. The state space is the set of density matrices S = {ρ ∈ C n×n : ρ = ρ † , ρ positive semidefinite, Tr (ρ) = 1}. Here Tr ( • ) denotes the trace, A † is the complex conjugate transpose of A. The open-loop system for a continuous-time quantum measurement with a single measurement channel is governed by the Itō stochastic differential equations [START_REF] Barchielli | Quantum trajectories and measurements in continuous time: the diffusive case[END_REF]:

dρ t = D L (ρ t )dt + √ ηM L (ρ t )dW t , (1)
dY t = √ η Tr (L + L † )ρ t dt + dW t . (2)
with the super-operators

D L (ρ) = LρL † -1 2 (L † Lρ + ρL † L) M L (ρ) = Lρ + ρL † -Tr ρ(L + L † ) ρ .
Here L ∈ C n×n is the measurement operator, W is a standard Brownian motion, Y ∈ R corresponds to the measurement process and η ∈ [0, 1] to its efficiency. The second equation describes the stochastic measurement output signal, the first one describes the corresponding measurement backaction. This backaction describes how the act of measuring the quantum system, perturbs it; as we gain infinitesimal information at each time step, the backaction on the system is also infinitesimal.

A standard case, which comes down to a continuoustime unraveling of the projective quantum measurement postulate, corresponds to a Hermitian measurement operator L = L † . This case is also called quantum nondemolition (QND) measurement by some authors, because an eigenstate of L will remain untouched by the measurement process. Note that other authors have used QND with different meanings. Consider the spectral decomposition L =

p k (ρ t ) := Tr (ρ t Π k ) ≥ 0 , (3)
with the property

d k=1 p k (ρ t ) = 1.
The following Lemma summarizes the asymptotic behavior of (1) (without mentioning the corresponding measurement signal). It is based on an original exponential Lyapunov function V o providing an estimate of the convergence rate towards the set of stationary states.

Lemma 1

Consider the open-loop system (1)with initial condition ρ 0 ∈ S. Then any realization of (1) remains in S. Moreover (i) For any k, the subspace population p k (ρ t ) is a martingale, i.e. E[p k (ρ t )] = p k (ρ 0 ). (ii) If there exists k ∈ {1, 2, ..., d} such that p k (ρ 0 ) = 1, then ρ 0 is a steady state of (1). (iii) The Lyapunov function

V o (ρ) = 1≤k<k ≤d p k (ρ) p k (ρ), decreases exponentially as ∀t ≥ 0, E[V o (ρ t )] ≤ exp(-rt)V o (ρ 0 ) with rate r = η 2 min k,k (λ k -λ k ) 2 .
In this sense, the open-loop system (1) converges, for all initial states, towards the set of invariant states described in point (ii).

PROOF. The fact that S is positively invariant for (1) is standard [START_REF] Barchielli | Quantum trajectories and measurements in continuous time: the diffusive case[END_REF].

(i) For each k, the subspace populations p k (ρ) follow the Itō SDE:

dp k (ρ t ) = 2 √ η λ k - d k =1 λ k p k (ρ t ) p k (ρ t )dW t .
Taking the expectation yields

d dt E[p k (ρ t )] = 0, so indeed E[p k (ρ t )] = p k (ρ 0 ), ∀t ≥ 0. (ii) Take ρ 0 such that p k (ρ 0 ) = 1. Plugging into (1) we have D L (ρ 0 ) = Lρ 0 L -1 2 L 2 ρ 0 -1 2 ρ 0 L 2 = λ 2 k ρ - 1 2 (2λ 2 k ρ 0 ) = 0 and M L (ρ 0 ) = √ η(Lρ 0 + ρ 0 L - Tr (2L ρ 0 ) ρ 0 ) = √ η(2λ k ρ 0 -Tr (2λ k ρ 0 ) ρ 0 ) = 0.
Thus ρ 0 is a steady state of (1).

(iii) V o is a positive definite function on S and it equals 0 only when p (ρ) = 1 for some . It remains to check that it is a supermartingale with exponential decay. By Itō's formula (A.2), the variable

ξ k := √ p k satisfies dξ k = -1 2 η(λ k -(ξ)) 2 ξ k dt+ √ η(λ k -(ξ))ξ k dW, (4)
with

(ξ) = d k=1 λ k ξ 2 k . Since V o (ρ) = d 1≤k<k ≤d ξ k ξ k , consider the computa- tion with Itō's formula: d(ξ k ξ k ) = (dξ k )ξ k + ξ k (dξ k ) + (dξ k )(dξ k ) = -1 2 η(λ k -(ξ)) 2 ξ k ξ k dt + √ η(λ k -(ξ))ξ k ξ k dW -1 2 η(λ k -(ξ)) 2 ξ k ξ k dt + √ η(λ k -(ξ))ξ k ξ k dW + η(λ k -(ξ))(λ k -(ξ))ξ k ξ k dt = -1 2 η(λ k -λ k ) 2 ξ k ξ k dt+ √ η(λ k +λ k -2 (ξ))ξ k ξ k dW.
The Markov generator A (see (A.3)) following from the expectation of this equation thus yields:

AV o = -η 2 d k =1 k <k (λ k -λ k ) 2 ξ k ξ k .
Since each ξ k (t) remains non-negative for all t, this readily yields

AV o ≤ -η 2 min k ,k =k (λ k -λ k ) 2 V o .
By Theorem 3 in Appendix, V o decays exponentially towards zero, concluding the proof.

The open-loop system corresponding to the measurement ( 1),( 2) thus converges towards an eigenstate of L, satisfying p k (ρ) = 1 for some k, for each realization. However, the particular eigenstate k will be random, with correlated measurement results indicating which state has been chosen; from (i), the probability to converge towards the particular eigenstate k is equal to p k (ρ 0 ).

The control objective is to ensure convergence to a target eigenstate, indexed by ∈ {1, . . . , d}, for all realizations. More precisely, we will design a continuous stochastic real feedback process v, depending on the state ρ, such that lim t→∞ E[p (ρ t )] = 1 with exponential convergence rate, for any initial condition ρ 0 ∈ S. The feedback action is modeled by means of a unitary control operation

U t = e -iHudt during the infinitesimal interval [t, t + dt],
where H = H † is the actuator Hamiltonian and dv = udt is the feedback input signal. Applied on (1), the closedloop system reads then:

ρ t+dt = e -iHudt (ρ t + dρ t )e iHudt (5)
with measurement process Y still given by (2). Regarding exponential convergence, we aim to provide a global Lyapunov function V (ρ) such that in closed-loop, V (ρ t ) is a supermartingale with exponential decay for all t ≥ 0 and all ρ 0 ∈ S. Providing feedback controls that ensure decay of V in this sense has remained so far an open issue.

Existing feedback designs

Measurement-based quantum feedback has been investigated thoroughly in the literature, considering a wide array of applications such as state preparation [START_REF] John K Stockton | Deterministic dicke-state preparation with continuous measurement and control[END_REF][START_REF] Mirrahimi | Stabilizing feedback controls for quantum systems[END_REF], state purification [START_REF] Combes | Rapid state reduction of quantum systems using feedback control[END_REF][START_REF] Howard | Reconsidering rapid qubit purification by feedback[END_REF] or continous-time quantum error correction [START_REF] Ahn | Continuous quantum error correction via quantum feedback control[END_REF][START_REF] Sarovar | Practical scheme for error control using feedback[END_REF]. Nevertheless, to our knowledge, control design has been done mainly on two feedback architectures: static output feedback [START_REF] Howard M Wiseman | Bayesian feedback versus markovian feedback in a two-level atom[END_REF] (so-called Markovian feedback in the physics literature) and Bayesian feedback [START_REF] Andrew C Doherty | Quantum feedback control and classical control theory[END_REF][START_REF] Andrew | Feedback control of quantum systems using continuous state estimation[END_REF][START_REF] Mirrahimi | Stabilizing feedback controls for quantum systems[END_REF] involving a full state estimate. Intermediate approaches have attracted much less attention and promise an interesting research area.

Static output feedback

Static output feedback corresponds to quantum feedback of the form dv = udt = f dt + σdY , where f and σ are constant. From ( 5) the dynamics in closed-loop read (see e.g. [START_REF] Howard M Wiseman | Quantum theory of continuous feedback[END_REF][START_REF] Wang | Feedback-stabilization of an arbitrary pure state of a two-level atom[END_REF]):

dρ = -i (f [H , ρ] + √ ησ[H , Lρ + ρL]) dt + D L (ρ)+σ 2 D H (ρ) dt+ ( √ ηM L (ρ) -iσ[H , ρ]) dW.
The simplicity of the feedback scheme makes it attractive for experimental implementations, since it avoids any overhead associated to dynamical computations in the feedback loop; in particular it needs no quantum state observer. With proper tuning of the constants f, σ, Markovian feedback allows to exponentially stabilize a range of target states, with direct algebraic proofs [START_REF] Howard M Wiseman | Bayesian feedback versus markovian feedback in a two-level atom[END_REF][START_REF] Wang | Feedback-stabilization of an arbitrary pure state of a two-level atom[END_REF][START_REF] Ticozzi | Quantum markovian dynamics: Invariant subsystems, attractors, and control[END_REF][START_REF] Ticozzi | Analysis and synthesis of attractive quantum markovian dynamics[END_REF][START_REF] Ticozzi | Stabilizing quantum states by constructive design of open quantum dynamics[END_REF].

Unfortunately, an invariance argument shows that precisely the eigenstates of a Hermitian measurement operator, which are of particular interest in quantum engineering, are not stabilizable using this static output feedback with f, σ just constants ( [START_REF] Cardona | Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control[END_REF]).

It seems that a more involved controller is needed in order to bias the stochastic evolution towards a prescribed measurement operator eigenstate. One solution is to turn to state feedback, assuming an underlying quantum state observer.

State feedback

The standard state feedback takes the form dv = udt = f (ρ)dt . Thus the control signal is a deterministic scalar function of the state ρ. The closed-loop model for this quantum state feedback just takes the form:

dρ = -if (ρ)[H, ρ]dt + D L (ρ)dt + √ ηM L (ρ)dW.
Stabilization of a target eigenstate under this controlled dynamics has been treated extensively in the literature [START_REF] Ahn | Continuous quantum error correction via quantum feedback control[END_REF][START_REF] Van Handel | Feedback control of quantum state reduction[END_REF][START_REF] Yamamoto | Feedback control of quantum entanglement in a two-spin system[END_REF][START_REF] Mirrahimi | Stabilizing feedback controls for quantum systems[END_REF][START_REF] Tsumura | Global stabilization of n-dimensional quantum spin systems via continuous feedback[END_REF][START_REF] Tsumura | Global stabilization at arbitrary eigenstates of n-dimensional quantum spin systems via continuous feedback[END_REF][START_REF] Liang | On exponential stabilization of spin-1/2 systems[END_REF]. These results have succeeded in proving asymptotic convergence, in proper probabilistic settings. There is however room for improvement on a few aspects.

A first aspect is the convergence speed. Most papers do not provide any convergence rate; as the strongest result so far to our knowledge, [START_REF] Liang | On exponential stabilization of spin-1/2 systems[END_REF] provides an estimate of the Lyapunov exponent for a qubit, valid for the final approach of the target state after an unspecified final initial transient. By Lemma 1, the open-loop system, just under measurements, converges towards the set of its steady states at global exponential speed. The absence of a proven similar property for the selection of one target steady state thus appears as an avoidable gap.

A second aspect is that full state feedback, as is used in the above proposals, may appear unnecessary for this application. Running a quantum state estimate in real-time does pose experimental challenges, given the very short timescales involved (nanoseconds) and the plan to ultimately scale quantum computers to high-dimensional systems. In the present task, the asymptotic behavior is directly visible on the measurement signal. Indeed, the measurement signal corresponding to p k (ρ t ) = 1 for all t ≥ 0 is Y t = Y 0 + 2λ k t + W t , with thus an expectation that directly informs on the eigenstate via the drift λ k t, and a standard deviation in √ t. This suggests that simple filtering should allow to essentially solve the task too. The necessity of keeping a full state observer, including quantum coherences, comes from the actuation strategy. This could be improved.

A third aspect, more related to model uncertainties, is that many existing proposals work with short pulses, inspired from the discrete-time counterpart. In actual implementations, it may be more cautious to use smoother control signals in order to avoid exciting spurious dynamics.

With respect to these two main approaches, we thus aim for an intermediate solution using a reduced estimator and smoother controls, while providing an exponential convergence guarantee. For this we resort to a control signal with a novel structure, called noise-assisted feedback.

Noise-assisted feedback scheme

In our feedback scheme, the control signal dv = udt still depends on (part of) the state like in the Bayesian feedback approach, but instead of just involving a deterministic function we drive dv by an exogenous Brownian noise dB independent of W t :

dv t = u t dt = σ(ρ t )dB t . (6)
The feedback control occurs by making the noise gain σ(ρ) a continuously differentiable function of ρ. Using in [START_REF] Barchielli | Quantum trajectories and measurements in continuous time: the diffusive case[END_REF] the Baker-Campbell-Hausdorff (BCH) formula e xH ρe -xH = j T j x j j! , where T 0 = ρ and T j+1 = [H, T j ] for j ≥ 0, and applying the Itō rules (dv) 2 = σ 2 (ρ)dt, dtdv = dtdB = dW dB = dt 2 = 0, yields the following closed loop dynamics with two independent Wiener processes W and B :

dρ = D L (ρ) + σ(ρ) 2 D H (ρ) dt + √ ηM L (ρ) dW + σ(ρ)i[ρ, H] dB. ( 7
)
The main idea comes down to noise discouraging the system to converge towards an eigenstate different from p (ρ) = 1. Indeed, as the open-loop dynamics stochastically converges to one of the measurement operator eigenstates, but on the average does not move closer to any particular one, it is sufficient to activate noise only when the state is close to a bad equilibrium in order to "shake it away" and induce global convergence to the target. Accordingly, we consider the feedback (6) with the following gain law:

σ(ρ) = σ ϕ max k = p k (ρ) -p min p max -p min (8)
where ϕ ≥ 0 is a smooth saturating function on [0, 1], i.e.

ϕ(] -∞, 0]) = {0} and ϕ([1, +∞[) = {1}, with parameters σ > 0 and 1 > p max > p min > 1 2 . Since k p k = 1, each p k ≥ 0 and p min > 1/2, the argument of the max can only change when max k = p k (ρ) -p min < 0. Therefore, the function ρ → σ(ρ) is smooth despite the use of a max in its definition.

Exponential stabilization via noise-assisted feedback

We now construct a Lyapunov function and prove that our feedback design ensures its exponential convergence.

A particular point for n-level systems, compared to 2level systems like in [START_REF] Cardona | Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control[END_REF][START_REF] Liang | On exponential stabilization of spin-1/2 systems[END_REF], is to take the limited actuation into account.

Inspired by [START_REF] Amini | Design of strict control-lyapunov functions for quantum systems with qnd measurements[END_REF], we consider the d × d real symmetric matrix ∆ with components

∆ k,k = Tr (Π k D H (Π k )) , (9)
combining the spectral decomposition L = k λ k Π k with the actuator Hamiltonian H. Its off-diagonal elements are non negative since ∆ k,k = Tr (Π k HΠ k H) ≥ 0 for k = k . Its diagonal elements are non positive and

∆ k,k = Tr (Π k HΠ k H) -Tr Π k H 2 = Tr (Π k HΠ k H) -Tr (Π k H(Π 1 + . . . + Π d )H) = - k =k Tr (Π k HΠ k H) .
Thus ∆ is a Laplacian matrix.

Theorem 2 Assume that L is nondegenerate, i.e., d = n and each projector Π k is a rank one projector. Consider the closed-loop system (7) with feedback gain σ(ρ) given by (8) for a given projector Π with ∈ {1, . . . , d}. Assume that the graph associated to the Laplacian matrix ∆ defined in (9) is connected. Then there exists p ∈] 1 2 , 1[ such that for any choice of parameter σ > 0 and parameters 1 > p max > p min ≥ p, the closed-loop trajectories converge exponentially to ρ = Π , in the sense that: there exist constants ν > 0 and C > 0 (depending on σ, p max , p min ) for which E 1 -p (ρ t ) ≤ Ce -νt 1 -p (ρ 0 ) for any initial state ρ 0 ∈ S.

If the graph associated to ∆ is not fully connected then there exists a partition of {1, . . . , d} = I ∪ J (I, J = ∅, I ∩ J = ∅) such that H = Π I HΠ I + Π J HΠ J with Π I = k∈I Π k and Π J = k∈J Π k . Then any trajectory ρ t of (5) with any feedback scheme starting from Tr (ρ 0 Π I ) = 0 (resp. Tr (ρ 0 Π J ) = 0), satisfies Tr (ρ t Π I ) = 0 (resp. Tr (ρ t Π J ) = 0) for all t > 0. Thus, closed-loop convergence to p = 1 with ∈ I is impossible when Tr (ρ 0 Π I ) = k∈I p k (ρ 0 ) < 1. In this sense the above connectivity condition on the graph of ∆ cannot be weakened.

PROOF. (-Lyapunov function construction -)

We do not use directly 1 -p (ρ) as a closed-loop Lyapunov function. Instead we construct a closed-loop Lyapunov function V (ρ) equivalent to √ 1 -p (i.e. c * V (ρ) ≤ 1 -p (ρ) ≤ c * V (ρ) with 0 < c * < c * ) such that AV ≤ -rV . More precisely, we use

V α (ρ) = s∈{1,...,d}\{ } k∈{1,...,d}\{ } α s,k p k (ρ) .
The positive parameters α s,k will be given by solving d -1 linear systems, indexed by s:

k ∆ k,k α s,k = -β s,k
with β s,k > 0 for k = and β s, =k = β s,k . Standard arguments used in [START_REF] Amini | Design of strict control-lyapunov functions for quantum systems with qnd measurements[END_REF] guarantee under the connectivity assumption that there exists, for each s, a unique solution (α s,k ) such that α s,k > 0 for k = and α s, = 0.

(see, e.g. [START_REF] Lowell W Beineke | Topics in algebraic graph theory[END_REF]Chapter 4]). When the (d -1) × d matrix β is chosen of maximal rank d -1, the obtained matrix α is also of maximal rank d -1. Since k∈{1,...,d}\{ } p k = 1 -p , one has (-expression of the stochastic Markov generator -) The rest of the proof consists in showing that for any such choice of maximal rank matrix β, the resulting V α becomes an exponential Lyapunov function as soon as p is close enough to 1 and p min > p. This is based on the following simple but slightly tedious computations of AV α :

c * V α (ρ) ≤ 1 -p (ρ) ≤ c * V α (ρ)
AV α (ρ) = σ 2 (ρ) 2 f α (ρ) -η 2 g α (ρ) -σ 2 (ρ) 8 h α (ρ) (10)
with

f α (ρ) = s Σ k α s,k Tr (Π k D H (ρ)) Σ k α s,k p k g α (ρ) = s Σ k α s,k (λ k -Tr (Lρ))p k Σ k α s,k p k 2 Σ k α s,k p k h α (ρ) = s (Σ k α s,k Tr (i[Π k , H]ρ)) 2 (Σ k α s,k p k ) 3/2
.

These expressions are obtained with the following general formula based on Itō rules (a k > 0 constant)

d k a k p k = k a k dp k 2 k a k p k - ( k a k dp k ) 2 8 ( k a k p k ) 3/2 with E[dp k |ρ] = σ 2 (ρ) Tr (Π k D H (ρ)) dt and E[(Σ k a k dp k ) 2 |ρ] = σ 2 (Σ k a k Tr (i[Π k , H]ρ)) 2 dt + 4η (Σ k a k (λ k -Tr (Lρ))p k ) 2 dt. (-closed-loop essential contribution -) For p ∈ [0, 1] let S p ρ ∈ S | ∃j = , p j (ρ) ≥ p .
Take ρ ∈ S p with p = 1. Then there exists j ∈ {1, . . . , d} \ { } such that ρ = Π j , thus k α s,k p k (ρ) = α s,j > 0 and

Σ k α s,k Tr (Π k D H (ρ)) = -β s,j < 0. Consequently f α (ρ) = -s∈{1,...,d}\{ } βs,j √ αs,j < 0 with V α (ρ) = s √ α s,j > 0.
By continuity of f α /V α on S 1/2 , there exist f > 0 and p ∈]1/2, 1[ such that

f α (ρ) ≤ -f V α (ρ) for all ρ ∈ S p .
Taking p min > p, we ensure that the feedback will only be turned on when it contributes a negative term to AV α .

(-open-loop essential contribution -) For all ρ ∈ S \ {Π }, χ(ρ) = g α (ρ)/V α (ρ) is well defined. Since Tr (ρL) = k λ k p k (ρ), the function χ(ρ) depends only on the populations p k . Consider the following parametrization exploiting the degree 0 homogeneity of χ in the populations:

r = 1 -p , x k = p k /(1 -p ) for k = .
For ρ ∈ S \ {Π }, the function χ admits the following smooth expression with the variables r ∈]0, 1] and

x k ∈ [0, 1] satisfying k = x k = 1: χ(r, x) = s = Σ k = α s,k (λ k -(r,x))x k Σ k = α s,k x k 2 Σ k = α s,k x k s = Σ k = α s,k x k with (r, x) = (1 -r)λ + r k = λ k x k .
We study its extension to the compact set with r ∈ [0, 1]. Clearly χ(r, x) ≥ 0. Consider the solutions (r, x) of χ(r, x) = 0. Necessarily, they satisfy

∀s = , k = α s,k (λ k -(r, x))x k = 0. Since the (d -1) × d matrix α s,k is of maximal rank d -1, ∀k = we have (λ k -(r, x))x k = 0. Taking the sum over k = , one gets (1 -r) λ -k = λ k x k = 0 implying two possibilities: • if λ = k = λ k x k ,
then (r, x) = λ and the condition before summing requires (λ k -λ )x k = 0 for k = . Since λ k = λ this implies that x k = 0 ∀k = . But this is not possible since k = x k = 1.

• if r = 1, then (r, x) = k = λ k x k . For k = the conditions become λ k -k = λ k x k x k = 0 before summing. Since λ k = λ k for k = k and x k ∈ [0, 1] with k = x k = 1,
there necessarily exists j = such that x j = 1 and x k = 0 for k / ∈ { , j}.

Consequently, the nonnegative smooth function χ(r, x) vanishes only at d -1 isolated points, where r = 1 and x k = δ kj for some j = . By continuity, for any p ∈]0, 1[, there exists θ p > 0 such that ∀ρ ∈ S \ (S p ∪ {Π }), we have χ(ρ) ≥ θ p . This proves that

g α (ρ) ≥ θ p V α (ρ) ∀ρ ∈ S \ S p .
(-bringing all pieces together -) To conclude, consider AV α given in [START_REF] Campagne-Ibarcq | Observing quantum state diffusion by heterodyne detection of fluorescence[END_REF]. Since h α ≥ 0, we have

∀ρ ∈ S, AV α (ρ) ≤ σ 2 (ρ) 2 f α (ρ) -η 2 g α (ρ).
Consider the feedback gain σ(ρ) with p min > p. Then from the closed-loop essential contribution,

AV α (ρ) ≤ - σ2 2 f V α (ρ) ∀ρ ∈ S pmax
and from the open-loop essential contribution,

AV α (ρ) ≤ -η 2 θ pmax V α (ρ) ∀ρ ∈ S/S pmax .
Thus for all ρ in S, one has AV α (ρ) ≤ -νV α (ρ) where

ν is equal to min σ2 f 2 , ηθp max 2 . A direct application of Theorem 3 recalled in Appendix ensures E(V α (ρ t )] ≤ V α (ρ 0 )e -νt .

Observer and approximate quantum filtering

The feedback is based on the value ρ t of the quantum state, which is not directly measured. One has to reconstruct in real-time this quantum state via a quantum filter, a quantum state-observer. For [START_REF] Bennett | Purification of noisy entanglement and faithful teleportation via noisy channels[END_REF], it reads (see e.g. [START_REF] Bouten | An introduction to quantum filtering[END_REF]):

dρ t = D L (ρ t )dt + √ ηM L (ρ t ) dY t -2 √ η Tr (ρ t L) dt + σ(ρ t ) 2 D H (ρ t )dt -iσ(ρ t )[H, ρ t ]dB t .
A standard result [START_REF] John K Stockton | Deterministic dicke-state preparation with continuous measurement and control[END_REF][START_REF] Van Handel | Feedback control of quantum state reduction[END_REF][START_REF] Mirrahimi | Stabilizing feedback controls for quantum systems[END_REF][START_REF] Amini | Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays[END_REF] ensures that the resulting observer/controller closed-loop system

dρ t = D L (ρ t )dt + √ ηM L (ρ t )dW t + σ( ρ t ) 2 D H (ρ t )dt -iσ( ρ t )[H, ρ t ]dB t . dY t = 2 √ η Tr (ρ t L) dt + dW t d ρ t = D L ( ρ t )dt + √ ηM L ( ρ t ) dY t -2 √ η Tr ( ρ t L) dt + σ( ρ t ) 2 D H ( ρ t )dt -iσ( ρ t )[H, ρ t ]dB t .
converges almost surely towards the stationary state ρ = ρ = Π as soon as Tr (ρ 0 ρ 0 ) > 0 and the convergence conditions of Theorem 2 are satisfied.

Practical implementation of such quantum filter could be a problematic issue: when the dimension n of the system is large, it requires to store and update in real-time the n(n -1)/2 components of the operator ρ. Since the feedback depends only on the populations, i.e., the diagonal of ρ in the eigenbasis of the measurement operator L, the development of a reduced-order (possibly approximate) quantum filter depending only on these populations is suggested.

A first reduction consists in replacing ρ t in the feedback law by t corresponding to the Bayesian estimate of ρ t knowing ρ 0 and Y τ for τ ∈ [0, t]. One thus discards here the knowledge of B t . Then, one can prove that t obeys to the following stochastic differential equation:

d t = D L ( t )dt + √ ηM L ( t ) dY t -2 √ η Tr ( t L) dt + σ( t ) 2 D H ( t )dt. (11)
This filter involves less computations but still the full matrix . Since the feedback law only depends on the p k ( ) = Tr ( Π k ), it would be ideal to have a reduced filter involving only those variables. In cases like [START_REF] Cardona | Continuous-time quantum error correction with noiseassisted quantum feedback[END_REF], the filter (11) can be directly and exactly reduced to an autonomous system on the p k ( ), as we discard the random coherences among eigenstates induced by dB t . However, when a single noise process dB t drives many levels at once, the filter [START_REF] Campagne-Ibarcq | Using spontaneous emission of a qubit as a resource for feedback control[END_REF] has to model correlations among the various random coherences and this precludes an exact reduction. Nevertheless, by neglecting those feedbackinduced correlations (recalling that feedback actuation is often turned off), we can set coherences k| |k , k = k to zero, and propose an approximate filter for populations p k to estimate p k ( ). This amounts to replacing D H ( ) by a population transfer via the Laplacian matrix ∆ defined in [START_REF] Bravyi | Universal quantum computation with ideal clifford gates and noisy ancillas[END_REF]. It yields:

d p k = 2 √ η p k λ k -( p) dY t -2 √ η ( p)dt + σ 2 ( p) d k =1 ∆ k,k p k dt (12)
where

( p) = d k =1 λ k p k .
This approximate filter requires to store and update in real-time only d real numbers. For any measurement trajectory Y t , the components of p remain nonnegative and their sum equal to one. In open-loop (σ ≡ 0), this population filter is exact.

Simulation and robustness issues

Theorem 2 ensures for p min close enough to 1 exponential closed-loop convergence of (5) with noise-assisted feedback [START_REF] Lowell W Beineke | Topics in algebraic graph theory[END_REF], [START_REF] Bouten | An introduction to quantum filtering[END_REF]. This section is devoted to numerical estimation of closed-loop convergence rates and investigation of the related robustness on a specific quantum system already considered in [START_REF] Mirrahimi | Stabilizing feedback controls for quantum systems[END_REF][START_REF] John K Stockton | Deterministic dicke-state preparation with continuous measurement and control[END_REF]: a spin J system of dimension n = 2J + 1 where the measurement operator is

L = 2J m=0 (J -m)|J-m J-m|
and the actuator Hamiltonian is a tridiagonal matrix

H = 2J-1 m=0 √ (m+1)(2J-m) 2i (|J-m J-m-1| -|J-m-1 J-m|)
The Hilbert space is spanned by the 2J + 1 orthonormal vectors |J-m for m = 0, . . . , 2J.

All the simulations below correspond to detection efficiency η = 0.8 and J = 2 (n = 5), for which:

L =          2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -2          , H = 1 2i          0 -2 0 0 0 2 0 - √ 6 0 0 0 √ 6 0 - √ 6
∆ =          -1 1 0 0 0 1 -5 2 3 2 0 0 0 3 2 -3 3 2 0 0 0 3 2 -5 2 1 0 0 0 1 -1         
, inherits the tridiagonal structure of H and thus admits a connected graph, so Theorem 2 predicts exponential convergence with our controller using a sufficiently high value of p min and the filter [START_REF] Campagne-Ibarcq | Using spontaneous emission of a qubit as a resource for feedback control[END_REF].

We first illustrate the exponential convergence rate ν. The proof of Theorem 2 only provides a very loose bound on both ν and the necessary p min , so we here stick to numerical simulations. From the system analysis, the qualitative trend should be that lower values of p min imply more frequent feedback corrections and thus faster convergence. Figure 1 illustrates a simulation set with p min = 0.9, which fits an exponential convergence at rate ν ≈ 0.04. In a second simulation set, see Figure 2, we have pushed this to p min = 0.6 and observed a much faster convergence rate around ν ≈ 0.2, i.e. one half of the open-loop convergence rate. This suggests that such noise-assisted feedback can be tuned to achieve convergence rates similar to the open-loop one.

We next investigate the behavior with the approximate reduced filter on the population vector p. The simulations of Fig. 3 differ from the ones of Fig. 2 just by replacing in the feedback law the ideal populations p by the approximated ones p solutions of [START_REF] Cardona | Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control[END_REF]. One observes still an exponential convergence, with a reasonable decrease of the convergence rate from 0.19 to 0.12, illustrating the practical interest of such low-dimensional filters.

Finally, on Fig. 4 we further investigate robustness of the control strategy by using the reduced filter [START_REF] Cardona | Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control[END_REF] with a feedback delay of 0.5 time units in the closed-loop simulations. Despite a decrease by a factor two of the convergence rate, the fact a feedback latency of 1/4 of the openloop convergence time does not destabilize this feedback scheme suggests promising robustness properties.

Concluding remarks

We have approached the problem of stabilizing a quantum measurement operator eigenstate in continuous- The present work still leaves room for improvement in, at least, the following directions:

• While our proof of exponential convergence can provide an estimate of the convergence rate, we did not try to optimize the Lyapunov function nor the other control parameters in order to maximize the speed of convergence. We have shown numerically that the closed-loop convergence rate apparently can be made similar to the open-loop convergence rate with our approach, suggesting promising results in a precise analysis of convergence rate. • In numerical simulations, the reduced approximate filter [START_REF] Cardona | Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control[END_REF] appears good enough to achieve global exponential stabilization. We conjecture that this can be proven, for this filter and possibly for even simpler filters based on direct output signal filtering or sparse p. • The capability of performing several quantum measurements and of applying different unitary feedback controls on a single quantum system motivates the study of multi-input multi-output (MIMO) quantum feedback schemes. A MIMO version of static output feedback was introduced in [START_REF] Chia | Quantum theory of multiple-input-multiple-output markovian feedback with diffusive measurements[END_REF] and an implementation was made in [START_REF] Campagne-Ibarcq | Using spontaneous emission of a qubit as a resource for feedback control[END_REF]. In addition, we have presented, in the context of quantum error correction [START_REF] Cardona | Continuous-time quantum error correction with noiseassisted quantum feedback[END_REF], a MIMO scheme of this noise-assisted feedback to stabilize a manifold of quantum states. A general MIMO extension of Theorem 2 should be feasible along the same lines as the present work.

From a more general viewpoint, the following elements may be worth recalling towards further control theory applications. The use of a Lyapunov approach has allowed us to simplify and clarify the convergence analysis, transferring the difficulty to algebraic analysis. In designing the Lyapunov function, the fact that it has a negative second-order derivative ensures that the Itô correction from the feedback helps. We note that achieving local or almost-global negativity of the Lyapunov function's stochastic derivative is not difficult, but also not conclusive in a stochastic setting: in principle almost all trajectories could eventually escape from the local region. Thus, deforming the Lyapunov function to obtain truly global validity is a key point in the analysis. To use noise-assisted feedback, a main point appears to be that the target is an equilibrium in open-loop, locally stable in some sense. In fact the approach can be extended to the situation where the target can be locally stabilized by a feedback loop; this enables for instance to stabilize target states which are not measurement operator eigenstates, much like in Markovian feedback [START_REF] Howard M Wiseman | Bayesian feedback versus markovian feedback in a two-level atom[END_REF] but with faster convergence rate thanks to noise-assistance.

A notable property of this stochastic system is its natural property of achieving global convergence on a compact homogeneous manifold (strictly, for measurement efficiency 1), while for deterministic systems this is forbidden by a topological obstruction. Finally, the reduced filter is obtained by neglecting some correlations which are not essential for the system to converge. Such approximation is standard in e.g. the context of robotics [START_REF] Thrun | Probabilistic Robotics[END_REF]; a somewhat original twist here is that the neglected knowledge is a part of our input commands.

d

  k=1 λ k Π k where λ 1 , ...λ d are the distinct (d ≤ n), real eigenvalues of L with corresponding orthogonal projection operators Π 1 , ..., Π d resolving the identity, i.e.

  d k=1 Π k = I. The population of the eigenspace k is denoted by

  ,...,d}\{ } α s,k and α * = max s,k∈{1,...,d}\{ } α s,k .

.

  According to Lemma 1, the open-loop convergence rate is η/2 = 0.4. As a generic control goal we choose to stabilize the state Π = |0 0| associated to the zero eigenvalue of L. For the noise-assisted feedback gain (8), we take σ = √ 5η, p max = p min + 0.05, p min = 0.9 or p min = 0.6 with the saturation function ϕ(s) = min(1, max(0, s)). For each case we simulate a set of 1000 realizations starting from the fully depolarized state ρ 0 = I/5. We estimate the evolution of E[ 1 -Tr (Π ρ t )] ≡ E[ 1 -0|ρ t |0 ] by taking the ensemble average over these 1000 realizations. The Laplacian matrix

Fig. 1 .

 1 Fig. 1. Ideal closed-loop simulations with pmin = 0.9. In gray: selection of 200 individual trajectories t → 1 -Tr (Π ρt); In red: average over 1000 realizations, showing exponential convergence at a rate ν ≈ 0.04.

Fig. 2 .

 2 Fig. 2. Ideal closed-loop simulations with pmin = 0.6. In gray: selection of 200 individual trajectories t → 1 -Tr (Π ρt); In red: average over 1000 realizations, showing exponential convergence at a rate ν ≈ 0.2.

Fig. 3 .

 3 Fig. 3. Closed-loop simulations with pmin = 0.6 with the approximate population filter (12). In gray: selection of 200 individual trajectories t → 1 -Tr (Π ρt); In red: average over 1000 realizations, showing exponential convergence at a rate ν ≈ 0.12.

Fig. 4 .

 4 Fig. 4. Closed-loop simulations with pmin = 0.6 with the approximate population filter (12) and feedback delay of 0.5. In gray: selection of 200 individual trajectories t → 1 -Tr (Π ρt); In red: average over 1000 realizations, showing exponential convergence at a rate ν ≈ 0.06.time by introducing Brownian noise to drive the control. The use of noise provides a simple controller that shakes away spurious steady states in closed loop and thus achieves exponential stabilization of a target eigenstate.

This work has been partially supported by ANR grant JCJC-HAMROQS. The authors want to thank Birgitta Whaley and Leigh Martin for stimulating discussions.

A Tools from stochastic stability

We refer the reader to [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF] for further reference on these fundamental results. We consider concrete instances of Itō stochastic differential equations (SDEs) on R n of the form

where W t is a standard Brownian motion on R k , and µ, θ are regular functions of x with image in R n and R n×k respectively, satisfying the usual conditions for existence and uniqueness of solutions [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF] on S, a compact and positively invariant subset of R n .

Let I := {x ∈ S : µ(x) = θ(x) = 0} be an invariant set of (A.1). Let V (x), a nonnegative real-valued twice continuously differentiable function on S \ I, with V (x) = 0 implying x ∈ I. Itō's formula on V yields [START_REF] Oksendal | Stochastic differential equations: an introduction with applications[END_REF] dV

The Markov generator A of the SDE (A.1) is defined for any function V in its domain by

It is related to the SDE (A.2) by [START_REF] Oksendal | Stochastic differential equations: an introduction with applications[END_REF]Chapter 7]

The stochastic counterpart of Lyapunov's second method provides sufficient conditions for stochastic stability by analyzing the generator AV , e.g.:

Theorem 3 (Khasminskii [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF]) If for some r > 0 the Markov generator satisfies

) is a supermartingale with exponential decay. Thus lim t→∞ E[V (x(t))] = 0; since V (x) = 0 implies x ∈ I, this implies convergence towards I of all the solutions of Eq. (A.1) starting in S.


Anna Baccaglini-Frank
email: anna.baccaglinifrank@unipi.it

Carla Finesilver
email: carla.finesilver@kcl.ac.uk

Samet Okumus
email: samet.okumus@erdogan.edu.tr

Michal Tabach
email: tabachm@post.tau.ac.il

Introduction to the work of TWG24: Representations in mathematics teaching and learning

Introduction

This Thematic Working Group explicitly welcomed papers from a variety of different theoretical approaches and methodological frameworks addressing the role of representations of different types in teaching and learning processes, in particular those involving visualization (considered here as defined by [START_REF] Arcavi | The role of visual representations in the learning of mathematics[END_REF]). From its first appearance at CERME10 when it was made up of 24 participants from 13 countries with 16 accepted papers and 2 posters, at CERME11 TWG24 grew: there were 31 participants (authors, co-authors, and some other participants), from 16 countries with 18 accepted papers and 4 accepted posters. The structure of the working sessions allocated for discussion of each paper or poster was designed to stimulate interaction and collaboration among participants: each paper or poster was allocated to a working session in which a same or similar theme was addressed (to the extent possible); all participants were asked to read all papers ahead of time, concentrating on those they found most appealing; presenting authors were asked to prepare a short presentation, followed by open questions and a brief analysis of how the work presented contributed to the open questions in the TWG's call for proposals. After the short presentation by the presenting author, a general discussion was initiated and conducted for a total of 30 minutes for papers and 15 minutes for posters. The last session was completely devoted to summing up the main issues that had emerged from the group discussions. The four themes are: 1) theories, language and categorizations used to study representations (theories, language and categorization), 2) embodied and enactive perspectives on the use of representations in mathematics education (embodied and enactive perspectives), 3) forms of representation implemented through technological approaches (technological approaches), 4) pedagogical implications in the choice and use of representations in the teaching and learning of mathematics (pedagogical implications).

Theories, language, categorization

The researchers in TWG24 did not share a common theoretical framework for the use of the term representations. We see the diversity as a source of strength of our community, as different theoretical emphasis may illuminate various aspects of teaching and learning mathematics. We do share the need for having an explicit theoretical stance towards what is representation in our research.

The role of theory for the researchers in the group varied. Some articulated a theoretical perspective for designing tasks (e.g., Günster; Johnson et al.; Tabach & Koichu), while others view theory as a tool to guide their data analysis (e.g., Miragliotta; Lisarelli, Antonini & Baccaglini-Frank). Some of us consider as representation what is accepted as such by the mathematical community (e.g., Duijzer et al.), while others see them as a collection of inscriptions produced by students for a given mathematical situation (e.g., what Dahl called 'self-invented representations'). We discussed tensions that might arise, during the learning process, between students' unique-created representations and the need for students to be familiar and work with classical representations.

The group also discussed the issue of internal/external representations: we cannot directly access internal representations, hence it is difficult to assess; across papers and posters there was variation in the extent to which internal versus external representation was theorized. This methodologicaltheoretical issue is important to acknowledge, even if we do not see a direct way to resolve it.

Moreover, discussions highlighted a general move away from dichotomous categories of representation towards continuous spectrums, in both research design and analysis of data.

The advance of technological tools and our ability to design complex and dynamic learning environments seems to be increasing. Our theoretical language, however, is not yet well-enough developed to allow consistent "talking about" dynamism in representations. This might be a challenge to handle in the coming years. For example, do we have a theoretical language allows us to differentiate between representing a mathematical phenomenon with dragging a finger over a touch screen, versus representing the same mathematical phenomenon with ones' whole body?

Embodied and enactive perspectives

In our call for papers we explicitly noted that 'representations' should be interpreted broadly to encompass pictures, gestures, sounds, stories, metaphors and more (as well as more conventional classroom formats). Several presenters included material on the embodied and enactive representations employed by diverse participants (children and adults, from very high to very low mathematical attainment, with and without disabilities) within a wide variety of mathematical topics and contexts (educational, professional and recreational).

Several papers considered the salience of how hands, in particular, are used in embodied mathematical thinking. Miragliotta's work on geometrical predictions analysed both discursive and gestural aspects of older school and university students' geometrical problem-solving activity, highlighting the interplay both between language and motion, and perception and reasoning processes. One of the foci of Finesilver's microgenetic case study of a teenager with severe numeracy difficulties was the significance of the regular pattern of hand movements in 'dealing' out units into equal groups. In keeping with the emerging theme of continua, this motion aspect, along with mixed-mode representation, enabled transition from concrete to graphic representation. In contrast, Wille and Schreiber focused not on informal gesture but the communication of mathematics in sign language, with a comparison of how explanations of geometrical terminology function in a visual signed language versus a spoken mode of communication. The subsequent discussion clarified that while sign language should be considered distinct from gesture, it may be productive to consider a continuum between them.

The intertwining of enaction and language was also considered by Arnoux and Soto-Andrade, with a focus on the significance of metaphor in understanding and representing mathematics. Their paper on moving between concrete and abstract forms also addressed affective concerns, and highlighted the way that metaphors may be particular to communities based on local phenomena and experiences. Meanwhile, O'Brien and de Freitas presented research from one particular community; that of those engaged in textile arts who, while they may not consider themselves to be 'doing mathematics', are creating complex haptic and visual representations of mathematical patterns. Their fibre mathematics includes technical-aesthetic considerations of topological dimension and connectivity as well as space, mechanics and computation. This year's working group took a particular interest in the relationships between different representational forms, and discussion of enactive aspects also arose in the various discussions on graphical representations of time (Lisarelli et al.), motion (Duijzer, Van den Heuvel-Panhuizen, Veldhuis & Doorman), and relationships between quantities (Johnson, McClintock & Gardner). The discussions considered the different characteristics, affordances and limitations of creating moving representations with one's hands, manipulating external physical objects, and elements displayed on a screen. There was particular interest in how changing the subject of operation (objects, the environment, oneself) could affect perception of mathematical relationships.

The embodiment and enaction of mathematics can work in many different ways. However, being so fundamental to human experience, it may particularly help learners to ground concepts and form connections between mathematical ideas that they tend to see as isolated.

Technological approaches

A number of contributions presented studies in which different forms of technology were used to create or interact with representations of mathematical concept. Indeed, through technological artifacts, representations can become both dynamic and interactive. For example, Duijzer, Van den Heuvel-Panhuizen, Veldhuis and Doorman studied students creating distance-time graphs by describing their own movements in front of a motion sensor; Lisarelli, Antonini and Baccaglini-Frank analyzed students' written discourse about an experience in a dynamic interactive digital environment in which functions were represented in one dimension, as dynagraphs; and Johnson McClintock and Gardner explore students' transfer of covariational reasoning intertwined with their creation and interpretation of dynamic graphs on the Cartesian plane. Dynamismseen as interactive, controlled motioncan be very useful for learning about particular mathematical concepts related to functions: covariation, input-output relationship, effect of parameters.

The studies also suggested that dynamical software influence students' practice and, afterwards, their drawings and verbal representations. This seems to be the case both when the whole body in involved and when only some parts of the body are used to interact with the technological artifact. For example, when a student walks in front of motion sensors and software captures and represents whole body movement, new expressions emerge such as "walking the graph" (e.g., Duijzer et al.). However, also after experiencing covariation in one-dimensional dynagraphs students use arrows in their drawing to describe movement, and words such as 'motion', 'then', 'before', 'dragging ' (e.g., Lisarelli et al.). Futhermore, studies suggested that interacting with technological artifacts that produce dynamical interactive representations may also change the way students think of mathematical concepts (Duijzer et al.; Lisarelli et al.; Johnson et al.; Miragliotta; Tabach & Koichu). The group discussed how these could eventually become psychological tools as in the case of "dragging" (e.g. [START_REF] Baccaglini-Frank | From conjecture generation by maintaining dragging to proof[END_REF].

The use of dynamic representations was also discussed within the proposal of a theoretical framework providing the foundation to design tasks aimed at promoting students' functional thinking (Günster). Finally, a different technological approach to representations was provided by O'Brien and de Freitas who used the loom to unpack the relationship between making, mathematics, and technology.

Much work remains to be done regarding how to employ technology's full potential in learning trajectories (including those for other topics than the ones discussed), and on how to foster the development of these experiences into formal mathematics.

Pedagogical implications

Representations play a vital role in mathematics teaching and learning. However, as researchers, we experience a constraint in communicating our findings in 'snapshots'we noticed how many studies begin as small scale endeavors. Thus, researchers need ways to scale up from small scale studies and to bring what we are learning to teachers and their students. During the presentations and discussions, the group attempted to establish connections between representations and the pedagogical implications in the choice and use of such representations. For example, Milinkovic, Mihajlovic and Dejic found that students who managed to solve mathematical problems were able to connect different mathematical representations. However, in mathematics classrooms, teachers generally stimulate algebraic/symbolic representations to solve mathematical problems more than other types. Pressures may exist for teachers to move quickly to help students have swift ways to solve problems and represent processes promoting one of the representations more than others, and these pressures may prematurely curtail students' intuitive approaches when engaging in problem solving. Some discussions focused on how teachers should be able to use incorrect answers as an opportunity for mathematical investigation. While some teachers may prefer to ignore or dismiss an incorrect answer and focus on teaching correct ways to represent mathematical problems, others decide to use an incorrect strategy to engage in a class discussion. Tabach and Koichu used the potential of incorrect strategies for the solution of a problem with a novel method, namely in a "who is right" form in which students are given scenarios and asked to pick their sides (if any) for the correct solution of a problem. Okumus and Dede introduced students misleading representations without letting them know that graphs tweaked information. Alternatively, some may prefer to be more explicit and provide specific representations (e.g., Hough et al.; Böcherer-Linder et al.). Teachers pedagogical decisions will influence how these tasks move students' thinking forward or how they will hinder their learning.


Martin Levenius

Matteo Conforti

Fabio Baronio

Valdas Pasiskevicius

Fredrik Laurell

Costantino De Angelis

Katia Gallo
email: gallo@kth.se

Multistep quadratic cascading in broadband optical parametric generation

come   L'archive ouverte pluridisciplinaire

A wide range of applications, encompassing ultrafast spectroscopy, high-harmonic and attosecond pulse generation, frequency combs and remote sensing, spur current research on broadband optical sources in the infrared [1][2]. Coherent ultra-broadband parametric gain suitable for ultrashort pulse generation and amplification can be afforded by quadratic frequency down-conversion in Quasi Phase-Matched (QPM) materials operated close to their zero group velocity dispersion point. Based on this principle, broadband optical parametric generation (OPG) has been achieved in periodically patterned GaAs [3], KTiOPO4 [4], LiNbO3 [5], and, recently, LiTaO3, with a record gain bandwidth of 180 THz [6].

The spectral flatness of the parametric gain curves is a feature almost as important as the breadth of its spectral coverage. However, multistep χ (2) processes triggered by broadband OPG, such as sum-frequency generation (SFG), often disrupt the flatness of gain profiles in QPM media [3][4][5][6].

In this Letter we investigate theoretically and experimentally the impact of multistep χ (2) processes on broadband OPG in the pulsed regime. With reference to SFG, we show how quadratic cascading [7] can be used to engineer the OPG spectral profiles. Specifically, we demonstrate that the SFG-induced gain dips commonly observed in broadband OPG [3][4][5][6] can be turned into peaks by suitably choosing the pulse duration of the OPG pump. Experiments conducted in periodically poled MgOdoped stoichiometric LiTaO3 (PPMgSLT) [6] provide evidence for theoretical predictions based on a model accounting rigorously for broadband OPG and multistep processes [8].

The multistep quadratic interactions which can affect ultra-broadband parametric generation in a periodically poled medium are schematically illustrated by Fig. 1a. In the χ (2) material, the optical pump at λP excites parametric generation in the infrared region (λOPG) via first order QPM. High-order resonances of the QPM grating can simultaneously enable SFG between the pump and specific components (λS in Fig. 1) within the broadband OPG spectrum. Fig. 1. a) Interaction paths of multistep χ (2) processes triggered by broadband optical parametric generation (OPG) in a QPM medium. SFG = sum frequency generation, DFG = difference frequency generation. (b) Spectral signatures of OPG + SFG in the spectral domain (experimental data ref. [6]).

The SFG process (1/λP + 1/λS  1/λSF) subtracts energy from the OPG band, carving a gain dip at λS, as highlighted on the spectrum of Fig. 1b. This is the situation typically encountered in the experiments [3][4][5][6].

The unwanted effects of parasitic SFG processes on the parametric gain could in principle be alleviated through more sophisticated QPM grating designs, yet this option becomes ever more challenging as the OPG spectra get wider. Alternatively, instead of trying to remove the QPM resonances responsible for upconversion, one could exploit SFG cascading to re-route power into the OPG band via difference frequency generation (DFG, compare Fig. 1a).

In principle, according to the theory of quadratic cascading for non-degenerate frequency upconversion [9], the direction of energy flow in the SFG step is reversible (even at phase-matching) as long as depletion of one of the waves (either λS or λP) is locally achieved in the device. In our configuration (Fig. 1a) the signal (λS) is generated by the OPG pump, therefore the control of SFG cascading ultimately resides only in the latter (i.e. the input at λP).

To assess the impact of SFG cascading on broadband parametric generation, we performed a systematic numerical analysis based on a recently developed mathematical model [8], which can account rigorously for all the spectral components and multiple interactions of Fig. 1. We explored the device response in the pulsed regime under experimentally viable conditions [6], referring to OPG in a 1 cm-long PPMgSLT crystal, doped with 1 mol% MgO, with a QPM grating with a period of 25 µm and a duty cycle of 42.5 %, operated at a temperature Τ = 80 ο C and pumped close its zerodispersion point (λP = 860 nm) by Gaussian pulses at 1kHz, with peak intensities in the range 1-10 GW/cm 2 and pulse durations ranging between 0.5 and 30 ps.

The simulations employed the following quadratic nonlinear envelope equation [8]:

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑖𝑖𝑖𝑖𝑖𝑖 = - 𝑖𝑖𝑖𝑖 (2) 𝜔𝜔 0 2 4𝛽𝛽 0 𝑐𝑐 2 �1 - 𝑖𝑖 𝜔𝜔 0 𝜕𝜕 𝜕𝜕𝜕𝜕 � �𝑖𝑖 2 𝑒𝑒 𝑖𝑖𝑖𝑖 + 2|𝑖𝑖| 2 𝑒𝑒 -𝑖𝑖𝑖𝑖 � (1)
where A(z, t) is the broadband complex electric field envelope at a reference pulsation 𝜔𝜔 0 , 𝜙𝜙(𝑧𝑧, 𝑡𝑡) = 𝑖𝑖𝜔𝜔 0 𝜏𝜏 -𝑖𝑖(𝛽𝛽 0 -𝑘𝑘 1 𝜔𝜔 0 )𝑧𝑧, 𝛽𝛽 0 = 𝑅𝑅𝑒𝑒[𝑘𝑘(𝜔𝜔 0 )], 𝑘𝑘(𝜔𝜔) = (𝜔𝜔/𝑐𝑐)�1 + 𝜒𝜒(𝜔𝜔) being the propagation constant and 𝜒𝜒(𝜔𝜔) the linear electric susceptibility. (2) is the second order nonlinearity, 𝑧𝑧 the propagation coordinate, and 𝜏𝜏 = 𝑡𝑡 -𝑘𝑘 1 𝑧𝑧 the coordinate system, moving with the pump group velocity. The model refers to a plane-wave configuration, with no dependence on the transverse coordinates. Material dispersion is accounted for based on ref. [10]. We simulated OPG from a classical Gaussian pump and a semiclassical pulse shot noise of one photon per mode with random phase on each spectral discretization bin. The OPG response was statistically evaluated over the ensemble of multiple simulations, performed with different random noise seeds. We typically used an average of 30 realizations, which compares well with the experimental conditions described in what follows.

𝑖𝑖 = ∑ 1 𝑚𝑚! ∞ 𝑚𝑚=2 𝑘𝑘 𝑚𝑚 (-𝑖𝑖 𝜕𝜕 𝜕𝜕𝜕𝜕 ) 𝑚𝑚 is the dispersion operator, 𝑘𝑘 𝑚𝑚 = 𝜕𝜕 𝑚𝑚 𝑘𝑘 𝜕𝜕𝜔𝜔 𝑚𝑚 (𝜔𝜔 0 ), 𝜒𝜒
Fig. 2 illustrates key features of the calculated response, by considering OPG in PPMgSLT with a pump at 860 nm. In each picture, the upper plot shows the broadband OPG spectrum (sweeping from 1 to 3.5 µm) at the output, while the contour plot illustrates its evolution in propagation (along z) within the PPMgSLT crystal. Fig. 2a and2b concern two different pump pulse durations, namely 10 and 0.7 ps, and peak intensities, namely 7 and 15 GW/cm 2 , respectively.

Fig. 2a refers to the situation generally encountered in the experiments, where SFG induces localised dips in the gain spectra. In this specific case the dips, located at 1.4, 1.5, 1.95 and 2.2 µm, are the signatures of two distinct SFG processes occurring in the PPMgSLT device, namely:

• SFG between λP and λS1 = 1.4 µm, generating a wave at λSF1 = 533nm via 3 rd -order QPM; • SFG between λP and λS2 = 1.95 µm, generating a wave at λSF2 = 600nm via 2 nd -order QPM. The two SFG processes affect also the OPG gain at the conjugate wavelengths [ λS* = λS⋅λP / (λS -λP)], located at λS1* = 2.2 µm and λS2* = 1.5 µm, respectively. 1), averaged over 30 realizations. Fig. 2b shows simulations for the same device of Fig. 2a, this time pumped with pulses of a much shorter duration (0.7ps). As apparent from the output spectrum, a gain enhancement now occurs at the very same spectral locations (λS1, λS2, λS1*, λS2* ) where SFG was carving dips in the gain curve of Fig. 2a. This happens as a consequence of the onset of SFG cascading, which brings power back into the OPG band by activating the DFG channel of Fig. 1a.

The simulation results point out to a crucial role played by pulse walk-off on the onset of SFG cascading. As the pump pulse duration is decreased, the walk-off among the spectral components involved in the SFG process (λP, λSF and λS) increases. Accordingly, as these components lose their overlap upon propagation, the SF wave begins to back-convert its power to λP and λS, yielding an enhancement of the OPG output. This agrees well with what would be expected for a single three-wave mixing interaction, where periodic up-and down-conversion takes place with a spatial period which depends on field intensities and temporal walk off [11].

The above intuitive explanation for the results of Fig. 2 is also consistent with the evolution of the interacting fields observed in the time domain, illustrated by Fig. 3. As seen in Fig. 3a, with the longer input pulses (10 ps), since the pump and the generated visible and infrared waves maintain their overlap, the temporal distribution of the total pulse exhibits no significant offset with respect to the input pump (t=0). This leads to efficient SFG, hence depletion of the OPG outputs at λS1,2 and λS1,2* (Fig. 2a). On the other hand, with shorter pulses (0.7ps) the visible and infrared components generated in the device get away faster from the pump. This effect can be recognised in the time domain propagation plot of Fig. 3b, where a skewed trace (associated to components walking-off from the pump) clearly emerges for z > 6mm. The progressive reduction of the temporal overlap among the spectral components involved in SFG locally alters the frequency conversion dynamics, starting to feed power back in the infrared region via DFG (Fig. 2b). Fig. 3. Averaged total field evolution in the time domain, corresponding to the configurations of Fig. 2, namely: Gaussian pump pulses of a) 10ps and b) 0.7ps.

To confirm the above predictions, we then set to investigate the response of ultra-broadband OPG in PPMgSLT [6] under varying pumping conditions, seeking experimental evidence for SFG cascading. The samples used for these experiments were 500 µm-thick, 1 cm-long, z-cut substrates poled with a constant period of 25 µm. The pump source was a tunable picosecond Ti:sapphire amplifier system, consisting of a Nd:YVO4-pumped femtosecond source which seeded a regenerative chirped pulse amplifier, delivering micro-joule pulses at 1 kHz with bandwidths of 2 nm, tunable from 820 to 900 nm. We varied the pump pulse durations by adjusting the group delay dispersion introduced by the final compressor stage of the amplifier. We used a 300 mm lens to loosely focus the pump beam to a radius of 100 µm at the centre of the PPMgSLT grating. At the sample ends the radius increased only by 10%, justifying the plane-wave approximation used in the model. The crystal was kept at 80ºC. At its output, after rejecting the residual pump with a dichroic filter, we recorded the signal spectra with a 0.55 m spectrometer (Horiba Jobin Yvon iHR550) sensitive up to 4.8 µm, averaging over 900 pulses.

Within the limits of our experimental setup, we systematically investigated the impact of input pulse durations and powers on the output OPG spectra. Fig. 4 summarises the key results, by comparing two measurements, taken for the same pump peak intensity (6 GW/cm 2 ), but with two different input pulse durations, namely 6.1 and 1.7 ps.

A comparison with the spectral responses expected with long and short pump pulses, respectively (Fig. 2a-b), confirms the good agreement between our experimental observations (Fig. 4) and theory. The spectrum recorded with long pump pulses (6.1 ps, red curve in Fig. 4) bears the characteristic signatures of SFG, exhibiting a marked depletion of the OPG output at four signal wavelengths (1.4, 1.5, 1.95 and 2.2 µm, highlighted by the arrows), which match well the theoretical predictions (Fig. 2a). On the other hand, when pulse durations are reduced (1.7 ps, black curve in Fig. 4), the OPG output is locally enhanced at the same spectral locations where the gain reduction was apparent with long pulses, as predicted by theory (Fig. 2b), due to SFG cascading.

Fig. 2 .

 2 Fig. 2. Simulated OPG response of a 1cm-long PPMgSLT device in the presence of cascaded processes, with a Gaussian pump at λP = 860 nm. Input pulse durations: a) 10ps and b) 0.7ps. Peak intensities: a) 7 GW/cm 2 and b) 15 GW/cm 2 . Output spectra (1D plots) and OPG signal evolution inside the device (contour plots). Numerical solutions of Eq. (1), averaged over 30 realizations.

Fig. 4 .

 4 Fig. 4. OPG spectra measured with a pulsed pump at 860 nm, with Gaussian pulse durations of 6.1 ps (red line) and 1.7 ps (black line), at 6 GW/cm 2 in a 1 cm PPMgSLT crystal. In conclusion, we studied the interplay of multistep quadratic processes with broadband OPG in the pulsed regime. We demonstrated the possibility to counteract the impact of SFG on the parametric gain by triggering SFGcascading to channel energy back into the OPG band. Simulations and experiments in PPMgSLT show how the energy flow in the SFG step can be reversed by reducing the pump pulse duration so to drive walk-off initiated down-conversion of the SF wave, turning the gain dips typically observed during broadband OPG in QPM media into peaks. Simulations predict also the possibility to access an intermediate regime, without dips nor peaks, with significant improvements in the flatness of the OPG spectra, by fine-tuning the input pulse duration in relation to the sample length. Ultimately, these results point out to new possibilities to engineer all-optically the response of broadband parametric sources and amplifiers in QPM materials. The present research is supported at KTH by the Swedish Research Council (VR fellowship 622 2010 526, VR grant 621 2011 4040 and ADOPT Linnée Centre) and in Brescia by MIUR (PRIN 2009P3K72Z) and Fondazione CARIPLO (2010-0595).

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