M Bauer

D Bernard
email: denis.bernard@ens.fr

T Jin

Monitoring continuous

Introduction

Upon monitoring a quantum system, one progressively extracts random information and backacts randomly on the system [START_REF] Jin | Mathematical foundations of quantum mechanics[END_REF]. There are of course various way to monitor a quantum system [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF]. Let us assume that the observation process is such that if the monitoring is strong enough the result of these back-actions is to project the system state close to one of the eigenstates of the monitored observable. Then, if the monitoring process and the system dynamics are compatible -that is, if the flows they induce are commuting -, the net effect of the combined system dynamics plus monitoring is this projection mechanism. However if the system dynamics is not compatible with the monitoring process, the resulting dynamical evolution is more subtle and depends on the relative competition between these two processes.

In the case of finite dimensional systems, the effective dynamics is simple enough. The observable eigenstates are well-defined normalisable states, often called pointer states [START_REF] Zurek | Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?[END_REF]. In absence of system dynamics, strongly monitoring the system results in non-linearly projecting its state onto one of the pointer state according to Born's rules. This is usually referred to as non-demolition measurement [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities and Photons[END_REF][START_REF] Guerlin | Progressive field-state collapse and quantum non-demolition photon counting[END_REF]. Adding a non trivial system dynamics, be it coherent or dissipative, induces quantum jumps between pointer states [START_REF] Bohr | On the constitution of atoms and molecules[END_REF]. These jumps occurs randomly and are governed by a Markov chain whose transition rates depend on the system dynamics and the nature of the monitored observable [START_REF] Adler | Martingale Models for Quantum State Reduction[END_REF]. These jumps processes are dressed by quantum spikes but we shall here neglect these finer effects [START_REF] Bauer | Spikes in quantum trajectories[END_REF]. One noticeable difference between coherent and dissipative system dynamics is that the former is Zeno frozen [START_REF] Itano | Quantum zeno effect[END_REF] under monitoring so that the transition rates of the quantum jump Markov chain asymptotically depend on the strength of the monitoring process while they do not in the latter case.

The picture is a priori more complicated in the case of the monitoring of an observable with continuous spectrum [START_REF] Caves | Quantum-mechanical model for continuous position measurements[END_REF], because then the observable eigenstates are not normalisable -they are generalised eigenstates -, so that the result of the monitoring process cannot be the projection of the system state onto one of those non-normalisable states and the result of the combined system plus monitoring dynamics cannot be a simple Markov chain on those generalised eigenstates.

The dynamical process and the statistical aspects involved in monitoring an observable with continuous spectrum has recently been precisely described in ref. [START_REF] Ballesteros | Non-demolition measurements of observables with general spectra[END_REF], in the absence of internal dynamics. The aim of this paper is to show that, at least for some classes of system dynamics, one can present a precise formulation of the resulting stochastic processes describing the combined system plus observation dynamics in the limit of strong monitoring of an observable with continuous spectrum. Naively, we may expect that those processes are some kind of diffusion processes on the spectrum of the monitored observable. The aim of this paper is to make this statement more precise. To fix the framework, we shall assume that the monitored observable is the position observable for a massive particle on flat space, but the generalisation to other observables is clear. We shall deal with dissipative and hamiltonian dynamics. In both the Lindbladian and the Hamiltonian case we start from an exact treatment of a simple model (related in both cases, though for different reasons, to an harmonic oscillator) and then extend our results to more general situations.

The case of dissipative dynamics is dealt with in Section 3. There we shall prove that, in the limit of strong monitoring of the position observable, the resulting processes are driven by stochastic differential equations on the position space, if the Lindblad operators generating the system dynamics are quadratic in the impulsion operators. In other words, under such hypothesis, quantum trajectories are mapped into stochastic trajectories -a kind of quantum to classical transition induced by monitoring.

The case of hamiltonian dynamics is dealt with in Section 4. There we take a look at the well-documented problem of a quantum massive particle in a smooth potential whose position is monitored continuously [START_REF] Caves | Quantum-mechanical model for continuous position measurements[END_REF][START_REF] Belavkin | tsov, Semiclassical asymptotics of quantum stochastic equations[END_REF]. As previously analysed [START_REF] Belavkin | tsov, Semiclassical asymptotics of quantum stochastic equations[END_REF][START_REF] Kolokol | tsov, Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate[END_REF][START_REF] Bassi | Collapse models: analysis of the free particle dynamics[END_REF][START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF], in such a case the resulting dynamics possesses three different regimes: (a) a collapse regime in which the wave function localizes in space in agreement with non-demolition measurement, (b) a classical regime in which the localized wave function moves in space according to classical dynamics, and (c) a diffusive regime in which the wave function diffuses randomly and significantly. However, as pointed out in ref. [START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF], "it is not an easy task to spell out rigorously these regimes and their properties". The aim of this section is to argue that we may define a double scaling limit which maps quantum trajectories onto solutions of a Langevin equation describing precisely enough the cross-over from the classical regime to the diffusive regime.

How to formulate position monitoring in quantum mechanics and what is the associated notion of quantum trajectories is recalled in the Section 2. This Section is also used to describe fine statistical aspects of position monitoring. In particular we explain how one may adopt two different points of view to describe those statistical properties depending on which information one has access to. Some mathematical results related to this last point -i.e. related to possible changes of filtrations in observing the monitoring process -are given in the Appendix.

2 QND measurement of a continuous spectrum observable.

The general rules of quantum mechanics (including the measurement postulates) enable to show that the evolution of the density matrix under continuous monitoring of the observable X (later we shall take X to be the position observable, hence the name) but in absence of any internal evolution read [START_REF] Caves | Quantum-mechanical model for continuous position measurements[END_REF] (see the lecture notes [START_REF] Bernard | Statistical Aspects of Quantum State Monitoring for (and by) Amateurs[END_REF] for a simple derivation):

dρ t = - γ 2 2 [X, [X, ρ t ]]dt + γ Xρ t + ρ t X -2 X t ρ t dW t , (1)
dS t = 2γ X t dt + dW t ,
with X t := Tr(Xρ t ) and W t a normalized Brownian motion, with dW 2 t = dt, and S t the output signal. The dimensionful parameter γ codes for the rate at which information is extracted. One may alternatively write the above evolution equation eq.( 1) for the density matrix kernel ρ t (x, y) = x|ρ t |y , with |x the generalized eigenstate of X with eigenvalue x, as:

dρ t (x, y) = - γ 2 2 (x -y) 2 ρ t (x, y)dt + γ (x + y -2 X t )ρ t (x, y) dW t , (2)
with X t := dx xρ t (x, x).

One way to derive these equations is to look at a discrete time version where the monitoring is the effect of repeated interactions of probes with the system, with subsequent "yes-no" von Neumann measurement coupled to X on the probes. The process S is the bookkeeping device to record the monitoring and in the discrete-time viewpoint, a "yes" (resp. a "no") for a probe measurement leads to an increase (resp. decrease) of S for the corresponding time step. Probe measurements with more than two possible outcomes lead to introduce vectors S and W .

The purpose of the next subsection is to exhibit two equivalent, but rather different looking, descriptions of the monitoring process. We state the main results, but the details are relegated to the appendix, together with a quick discussion of the discrete time situation which the reader should refer to for motivation.

Diagonal evolution and probabilistic interpretation

The point of view we develop now is the following: during the monitoring process of the observable X, the observer gains information little by little and is able at large times to infer a value, say x, that characterizes the asymptotic state of the system. It turns out that the description of the process is substantially simpler for a "cheater" who uses from the beginning the knowledge of the result of the measurement. Though we do not claim that this is "the" interpretation of the density matrix, it is convenient for the forthcoming discussion to talk about it (in particular about its diagonal with respect to the spectral decomposition of X) using a Bayesian vocabulary, i.e. view it as expressing the best guess for the observer with the information available for him at a certain time.

If X is (as suggested by the name) the position of the particle, the "eigenstates" of X are not normalizable, so that the matrix elements of the density matrix between those "eigenstates", also called pointer states, may not be well-defined: in general the diagonal of the density matrix is not a function of the position variable x but a probability measure dµ t (x). In terms of the density matrix kernel we have dµ t (x) = ρ t (x, x) dx.

We choose to concentrate on the diagonal of the density matrix in the pointer states basis for the forthcoming discussion. Happily, the time evolution of the pair (dµ t , S t ) t≥0 , which is easily deduced from the general equations above, eqs.(1,2), remains autonomous and reads1 :

ddµ t (x) = 2γ x -y dµ t (y) dµ t (x) dW t , (3)
dS t = 2γ y dµ t (y) dt + dW t . (4)
The measure dµ 0 , i.e. the diagonal of the density matrix before the monitoring starts, is assumed to be known to the experimenter (e.g. via preparation). We shall see that dµ t can be interpreted as a best guess at time t for the experimenter knowing only dµ 0 and S u , u ∈ [0, t].

To simplify the notation a little bit, we set A := 2γX and use α to denote the variables in the spectrum of A. The above coupled stochastic differential equations are not totally trivial to analyze. However, the following holds:

Proposition: Let (B t ) t≥0 be a Brownian motion and A be an independent random variable with distribution dµ 0 (α) defined on some common probability space. Let S t be the process S t := B t +At. Let G t := σ{A and S u , u ∈ [0, t]} = σ{A and B u , u ∈ [0, t]} be the filtration describing the knowledge accumulated by knowing A since t = 0 and B u or S u for u ∈ [0, t]. Let H t := σ{S u , u ∈ [0, t]} be the filtration describing the knowledge accumulated by knowing S u for u ∈ [0, t]. Then: -If h is an arbitrary measurable function such that h(A) is integrable then

E[h(A)|H t ] = dµ 0 (β)h(β) e βSt-β 2 t/2 dµ 0 (β) e βSt-β 2 t/2 .
-The process (W t ) t≥0 defined by

W t := S t - t 0 E[A|H u ] du
is an H t -adapted Brownian motion.

-The pair (dµ t , S t ) t≥0 where µ t is defined by

E[h(A)|H t ] =: dµ t (β)h(β),
i.e. dµ t (α) = dµ 0 (α)e αSt-α 2 t/2 dµ 0 (β)e βSt-β 2 t/2 (5)

solves the system

ddµ t (α) = α -β dµ t (β) dµ t (α) dW t , (6)
dS t = β dµ t (β) dt + dW t , (7)
A detailed proof in given in the appendix, together with some motivations from the discrete time situation. The proof we give is a pedestrian one, based on explicit translations in Gaussian integrals. This is also the main intuition behind Girsanov's theorem, which is indeed the right general framework in which the proposition fits.

Let us now explain the meaning of this result. The important thing to realize is that the process S t can be analyzed under a number of filtrations, but its definition and properties are independent of the choice of filtration.

For instance, from the initial decomposition S t = B t + At we infer that St t converges at large times to the random variable A with probability 1, because by the law of large numbers for Brownian motion Bt t converges at large times to 0 with probability 1. By the law of large numbers for Brownian motion again Wt t converges at large times to 0 with probability 1. Hence, the H t -measurable random variable 1 t t 0 E[A|H u ] du which, by eq.( 5), equals

1 t t 0
β dµ u (β) du converges at large times to the random variable A with probability 1. Now E[A|H u ] is, by the very definition of conditional expectations, the best H u -measurable approximation of A. Thus the connection between the definition S t = B t + At and the system of eqs. [START_REF] Adler | Martingale Models for Quantum State Reduction[END_REF][START_REF] Bauer | Spikes in quantum trajectories[END_REF] has an interpretation in the field of statistical estimation: It is equivalent to sample A at time 0 (with dµ 0 of course) and observe the process (S t ) t≥0 (i.e. to use the filtration G t ) or to observe only (S t ) t≥0 (i.e. to use the filtration H t ) and retrieve A asymptotically using Bayesian inference. Another road to this result is to substitute S t = B t + At in the formula for dµ t (α) to find that the numerator e αSt-α 2 t/2 is strongly peaked around α = A at large times.

The striking fact is that the systems [START_REF] Zurek | Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?[END_REF][START_REF] Guerlin | Progressive field-state collapse and quantum non-demolition photon counting[END_REF] and [START_REF] Adler | Martingale Models for Quantum State Reduction[END_REF][START_REF] Bauer | Spikes in quantum trajectories[END_REF] are the same (mutatis mutandi, with the substitution A = 2γX). But the first system results from the application of the rules of quantum mechanics, while the second one has a purely statistical estimates content as explained above. From the point of view of quantum mechanics, the natural situation of an experimenter is that she/he observes only the result of monitoring i.e. the process (S t ) t≥0 and infers/measures a more and more accurate value of X when time gets large. But there is also an interpretation when a cheater measures X at time 0 (with the outcome distributed as dµ 0 of course) and then gives the system to the experimenter. The cheater has G t at its disposal, and in particular knows in advance which value for X the experimenter will infer/measure after an infinite time from the sole result of monitoring.

The explicit formula for dµ t as a function of S t can be quickly recovered by linearization of the system (6,7), a trick that we recall below because it works in general (i.e. when the system has some intrinsic evolution while the monitoring is performed). Note also that W t is not simply the conditional expectation of B t knowing H t . The interested reader should consult the appendix for a detailed discussion.

Density matrix evolution

Let us now give a brief description of the off-diagonal elements of the density matrix kernel ρ t (x, y) = x|ρ t |y whose evolution is governed by eq.( 2). These simple results will be useful in the following. As is well known [START_REF] Belavkin | tsov, Semiclassical asymptotics of quantum stochastic equations[END_REF], the solution of this equation are obtained by changing variables and defining a un-normalized density matrix kernel ρt (x, y), solution of the linear stochastic differential equation

dρ t (x, y) = - γ 2 2 (x -y) 2 ρt (x, y)dt + γ(x + y)ρ t (x, y) dS t , (8)
driven by the output signal S t . The normalized density kernel ρ t (x, y) is then reconstructed by setting

ρ t (x, y) = ρt (x, y) Z t ,
with the normalization factor

Z t := dx ρt (x, x) satisfying dZ t = 2γ X t Z t dS t . Since dS 2 t = dt, solution of eq.(8) reads ρt (x, y) = ρ 0 (x, y) e -γ 2 t (x 2 +y 2 )+γ (x+y) St . Z t = dµ 0 (x) e -2γ 2 t x 2 +2γ x St .
For the diagonal elements one recovers the solution (5) described above. The mean position is then 2γ

X t = (∂ S log Z)(S t ) so that dS t = (∂ S log Z)(S t ) dt + dW t .
The analysis of the previous subsection implies that the signal S t can also be decomposed as S t = 2γ x t + B t , with B t another standard Brownian motion and x an independent random variable sampled with measure dµ 0 (x). As a consequence, ρt (x, y) = ρ 0 (x, y) e 2γ 2 tx 2 e -γ 2 t (x-x) 2 +(y-x) 2 +O(γ √ t) , and dµ t (x) is a Gaussian distribution 2 centred at the random position x with width of order ∼ 1/ γ 2 t. Alternatively, this tells us that under QND monitoring of the position the system state has approximatively 'collapses' on Gaussian state of width of order after a time of order 1/ 2 γ 2 .

The diagonal components of the density matrix kernel have a well-defined large γ limit, in the sense that dµ t (x) → δ(x -x)dx in this limit, but the off-diagonal components don't. This is illustrated by the behaviour of the momentum distribution, e iaP t = e -1 2 γ 2 t a 2 , which implies that P 2k t → ∞ and f (P ) t → 0 for any function f with bounded Fourier transform, at large γ as expected from Heisenberg principle.

Monitoring continuous spectrum observables with dissipative dynamics

The aim of this section is to understand what dynamical processes emerge when strongly monitoring an observable with a continuous spectrum for a quantum system subject to a dissipative dynamics.

The simplest way to present the discussion is to consider a quantum particle on the real line, with quantum Hilbert space H = L 2 (R), and a monitoring of the position observable X. The (stochastic) dynamical equation for the density matrix is then

dρ t = L(ρ t ) dt - γ 2 2 [X, [X, ρ t ]]dt + γ Xρ t + ρ t X -2 X t ρ t dW t , (9)
with W t a normalized Brownian motion and L some Lindblad operator. Alternatively, the evolution of the density kernel ρ t (x, y) = x|ρ t |y reads

dρ t (x, y) = L(ρ t )(x, y) dt - γ 2 2 (x -y) 2 ρ t (x, y)dt + γ (x + y -2 X t )ρ t (x, y) dW t .
The aim of this section is to understand the limit of large γ.

If one want to get a precise statement, one should not look directly at the dynamics of the limiting density kernel but at the dynamics of the measures associated to system observables induced by the density matrix. Let us recall that, given a system density matrix ρ, to any

system observable O = O † is associated a measure µ O on Spec(O), the spectrum of O, via Tr ρ ϕ(O) = dµ O [o] ϕ(o),
for any (bounded) function ϕ. Now, a time evolution ρ → ρ t , as specified by eq.( 9), induces a time evolution of the measure

µ O → µ O t via dµ O t [o] ϕ(o) = Tr ρ t ϕ(O) . Since ρ t is random, so is µ O t .
The statements about the large γ limit are going to be formulated in terms of the limiting behavior of those measures for appropriately chosen observables.

The measure we should look at is that associated to the position, the observable that is monitored, defined by

dµ X t (x) ϕ(x) = Tr ρ t ϕ(X) ,
for any function ϕ. To simplify notation, we drop the upper index X on µ X and let dµ t (x) := dµ X t (x). Alternatively, dµ t (x) = ρ t (x, x) dx.

A simple example: Monitoring quantum diffusion.

Let us first look at the simplest model for which the Lindblad operator L is the so-called quantum Laplacian

L(ρ) = - D 2 P, P, ρ , with D some diffusion constant, so that L(ρ t )(x, y) = D 2 (∂ x + ∂ y ) 2 ρ t (x, y).
With this choice, the SDE for the density kernel reads

dρ t (x, y) = D 2 (∂ x +∂ y ) 2 ρ t (x, y) dt- γ 2 2 (x-y) 2 ρ t (x, y)dt+γ (x+y-2 X t )ρ t (x, y) dW t . ( 10
)
Interestingly enough, this equation is closed on the diagonal elements so that the measure

dµ t (x) := ρ t (x, x) dx satisfies d (dµ t (x)) = D 2 ∂ 2 x dµ t (x) dt + 2γ (x -X t ) dµ t (x) dW t , with X t = dµ t (x) x =: µ t [x]
. Alternatively, introducing a test function ϕ and denoting its "moment" by µ t [ϕ] = dµ t (x) ϕ(x), the stochastic evolution reads:

d µ t [ϕ] = D 2 µ t [∆ϕ] dt + 2γ µ t [x • ϕ] c dW t , (11)
with ∆ = ∂ 2 x the Laplacian, and 11) defines a process on measures on the real line. It is specified by its transition kernel which is going to be generated by a second order differential operator. We have to spell out the class of functions (of the measure µ t ) used to test the process on which this operator or kernel is acting. By construction, the functions we are going to consider are polynomials in the moments of the measure µ t and their appropriate completions (that is, say, convergent series of the moments of the measure µ t , with a finite radius of convergence).

µ t [x•ϕ] c := µ t [xϕ]-µ t [x]µ t [ϕ] the "connected [x•ϕ]-moment". Eq.(
Let ϕ j , j = 1, • • • , n, be test functions and let µ t [ϕ j ] be the corresponding moments. Let f be a function of n variables,

f ( µ) = f (µ 1 , • • • , µ n ),
defined by its series expansion (with a finite radius of convergence). The class of functions (of the measure µ t ) we consider are defined by

F ϕ f (µ t ) := f (µ t [ϕ 1 ], • • • , µ t [ϕ n ]). We set f ϕ = f • ϕ, that is f ϕ (x) = f (ϕ 1 (x), • • • , ϕ n (x)).
We can then state: Proposition: Let µ 0 be the initial condition µ 0 = µ t=0 . At γ → ∞ the limiting process for the measure µ t is that of measures concentrated on a Brownian motion started at an initial position chosen randomly with distribution µ 0 . That is 3 :

lim γ→∞ dµ t = δ Yt , Y t := B D 2 t with Y t=0 µ 0 -distributed, (12)
with B t a normalized Brownian motion. The limit is weak in the sense that it holds for all moments F ϕ f (µ t ). Namely

lim γ→∞ E F ϕ f (µ t ) = E µ 0 [f ϕ (Y t )] = µ 0 [e t D 2 ∆ • f ϕ ]. ( 13
)
with E µ 0 refers to the measure on Brownian motion with initial conditions distributed according to µ 0 . Proof: The proof goes in three steps: (i) first identify the differential operators (acting on F ϕ f ) generating the process; (ii) then identify the differential operator echoing the monitoring and look at its limiting action at large γ; and (ii) finally use a perturbation theory to deal with the effect of Lindblad dynamics on top of the monitoring.

(i) Let F ϕ f (µ t ) be some moments of the random measure µ t . Using eq.( 11) and Itô calculus, it is easy to check (as for general stochastic process generated by a stochastic differential equation) that these moments satisfies a stochastic differential equation of the form

dF ϕ f (µ t ) = D • F ϕ f (µ t ) dt + noise
, with D a second order differential operator. Equivalently, their expectation reads

E µ 0 F ϕ f (µ t ) = [e t D ] • F ϕ f (µ 0
). The form of eq.( 11) ensures that the second order differential operator D decomposes as

D = D 0 +4γ 2 D 2 .
Here D 0 is a first order differential operator whose action on linear functions is such that

D 0 • µ[ϕ] := µ[ D 2
∆ϕ], by definition. It is extended to any functions F ϕ f (µ) via Leibniz's rules:

D 0 • F ϕ f (µ) = n k=1 µ[ D 2 ∆ϕ k ] (∇ k f )(µ[ϕ 1 ], • • • , µ[ϕ n ]).
The operator D 2 is a second order differential operator whose action on first and second moments is

D 2 • µ[ϕ] = 0 and D 2 • µ[ϕ 1 ] µ[ϕ 2 ] = µ[x • ϕ 1 ] c µ[x • ϕ 2 ] c
, by definition. It is extended to any functions F ϕ f (µ) as a second order differential operator:

(D 2 • F ϕ f (µ) = 1 2 n k,l=1 µ[x • ϕ k ] c µ[x • ϕ l ] c (∇ k ∇ l f )(µ[ϕ 1 ], • • • , µ[ϕ n ]).
Both operators D 2 and D = D 0 + 4γ 2 D 2 are non-positive, because they are both associated to well defined stochastic differential equations.

(ii) The operator 4γ 2 D 2 is the one associated to QND X-measurement (without extra evolution). From the analysis of the previous, we know that in a pure QND monitoring of the position the measure µ t behave at large γ as

dµ t (x)| qnd = 1 Z t e -2γ 2 t (x-x) 2 +O(γ -1 ) dµ 0 (x) ,
with x µ 0 -distributed and Z t μ0 (x) π/2γ 2 t (with dµ 0 (x) = μ0 (x)dx). Thus we have lim

γ→∞ dµ t (x)| qnd = δ x,
with x µ 0 -distributed. Alternatively, this implies that

E qnd µ t [ϕ 1 ] • • • µ t [ϕ n ] → µ 0 [ϕ 1 • • • ϕ n ], as γ → ∞, which yields that lim γ→∞ E qnd µ 0 F ϕ f (µ t ) = µ 0 [f ϕ ],
for any function f . Since [e 4γ 2 t D 2 ] is the transition kernel for QND measurements, this is equivalent to lim

γ→∞ [e 4γ 2 t D 2 ] • F ϕ f (µ 0 ) = µ 0 [f ϕ ].
From this we learn that:

-The kernel of D 2 is made of linear functions: KerD 2 = {µ[ϕ], ϕ test function}.

-Let Π be the projector on KerD 2 defined by Π := lim a→∞ e a D 2 . Let F ϕ f | be the projection of F ϕ f on KerD 2 , defined by

F ϕ f | = Π • F ϕ f . Then F ϕ f | (µ) = µ[f ϕ ]. (iii)
We now study the (original) process whose generator is D = D 0 + 4γ 2 D 2 . By construction, we have

E µ 0 F ϕ f (µ t ) = [e t (D 0 +4γ 2 D 2 ) ] • F ϕ f (µ 0 ). It is clear that if F ϕ f | = 0 (that is F ϕ f | has no linear component), then lim γ→∞ E µ 0 F ϕ f (µ t ) = 0.
That is: only expectations of functions in KerD 2 have a non trivial limit as γ → ∞, and the limiting process is reduced to that space of zero modes (i.e. to that kernel).

We now use an algebraic strategy to prove convergence. Since D 0 : KerD 2 → KerD 2 , perturbation theory tells us (as recalled below) that the dynamics on KerD 2 reduces to

lim γ→∞ E µ 0 F ϕ f (µ t ) = [e t D 0 • Π] • F ϕ f (µ 0 ), or equivalently, lim γ→∞ E µ 0 F ϕ f (µ t ) = [e t D 0 ] • µ 0 [f ϕ ]. ( 14
)
This last equation is equivalent to the claim [START_REF] Bassi | Collapse models: analysis of the free particle dynamics[END_REF] because

[e t D 0 ] • µ 0 [f ϕ ] = µ 0 [e t D 2 ∆ • f ϕ ].
It is thus remains to argue for eq.( 14). Let

K γ t := [e t (D 0 +4γ 2 D 2 ) ] and Π γ t := [e t 4γ 2 D 2 ]. We have ∂ t K γ t = K γ t (D 0 + 4γ 2 D 2 ) and lim γ→∞ Π γ t = Π with Π the project on linear function (i.e. on KerD 2 ). Let F ∈ KerD 2 . Then ∂ t (K γ t • F ) = K γ t D 0 • F (because D 2 • F = 0). Equivalently ∂ t (K γ t Π) = K γ t (D 0 Π). Now because D 0 maps KerD 2 onto KerD 2 we have D 0 Π = Π D 0 Π, and thus ∂ t (K γ t Π) = (K γ t Π) (Π D 0 Π).
Integrating and taking the large γ limit yields eq.( 14).

Let us end this sub-section by a remark. It is easy to verify that the dynamical equation (10) admits a separation of variables so that its general solutions are density kernels ρ t (x, y) of the following form

ρ t (x, y) = σ 0 ( x -y 2 ) • e -γ 2 2 (x-y) 2 t • μt ( x + y 2 ),
with σ 0 an arbitrary function (normalized to σ 0 (0) = 1) and μt the density (with respect o the Lebesgue measure) of the measure dµ t , i.e. dµ t (x) = μt (x) dx. This gives the complete solution of the position monitoring a simple quantum diffusion. It is clear that, except for the observable position (the observable X), the measures associated to any other system observables have no well defined large γ limit. This in particular holds true for the momentum observable P , as expected.

Generalization : General stochastic diffusion

We can reverse the logic and ask ourselves whether it is possible to obtain any stochastic differential equation, of the form dY t = U (Y t )dt + V (Y t )dB t , as the strong monitoring limit of a quantum dynamical systems. That is: we ask whether, given two real functions U (y) and V (y) (sufficiently well-behaved), we can choose a Lindbladian L such that the large γ limit of the quantum trajectories

dρ t = L(ρ t ) dt - γ 2 2 [X, [X, ρ t ]]dt + γ Xρ t + ρ t X -2 X t ρ t dW t , (15)
leads to solutions of the stochastic differential equation

dY t = U (Y t )dt + V (Y t )dB t .
Proposition: Let L = L U +L V be the sum of two Lindblad operators such that their duals L * U and L * V act on any observable φ on H = L 2 (R), as follows (recall that V (X) * = V (X) and U (X) * = U (X))

L * V ( φ) = V (X)P φ P V (X) - 1 2 φ V (X)P 2 V (X) + V (X)P 2 V (X) φ , L * U ( φ) = i 2 [U (X)P + P U (X), φ],
with X the position observable and P the momentum observable (such that [X, P ] = i). Let µ t be the measure on the real line induced by ρ t via dµ t (x)ϕ(x) = Tr ρ t ϕ(X) for any function ϕ with ρ t solution of eq.( 15). Then, in the large γ limit, µ t concentrates on solutions of the stochastic differential equation

dY t = U (Y t )dt + V (Y t )dB t , in the sense that lim γ→∞ dµ t = δ Yt with dY t = U (Y t )dt + V (Y t )dB t .
Proof: Recall the definition µ t [ϕ] := dµ t (x)ϕ(x) = Tr ρ t ϕ(X) . By duality, if ρ t evolves according to eq.( 15), then because Tr L(ρ t ) ϕ(X) = Tr ρ t L * (ϕ(X)) by definition the dual Lindbladian L * . The operator D exists and is well-defined because, as we shall see, our choice of L ensures that L * (ϕ(X)) is again a function of the observable X. To prove the claim we need to check that L * is such that

d µ t [ϕ] = µ t [ D • ϕ] dt + 2γ µ t [x • ϕ] c dW t , with µ t [x • ϕ] c = µ t [x • ϕ] -µ t [x]µ t [ϕ]
L * (ϕ(X)) = 1 2 V 2 (X)∂ 2 x ϕ(X) + U (X)∂ x ϕ(X) =: (D st • ϕ)(X),
because the differential operator associated to the SDE

dY t = U (Y t )dt + V (Y t )dB t is D st = 1 2 V 2 (x)∂ 2 x + U (x)∂ x . Now, if φ = ϕ(X)
, so that it commutes with V (X) and U (X), we have

L * V (ϕ(X)) = - 1 2 V (X) [P, [P, ϕ(X)]] V (X) = + 1 2 V (X) 2 ∂ 2 x ϕ(X), L * U (ϕ(X)) = i 2 U (X)[P, ϕ(X)] + [P, ϕ(X)]U (X) = U (X)∂ x ϕ(X), so that (L * V + L * U )ϕ(X)) = (D st • ϕ)(X)
as required. The rest of the proof is as before. We look at the functions F ϕ f (µ t ). By identical arguments (with D st replacing the Laplacian ∆), we then have

lim γ→∞ E µ 0 F ϕ f (µ t ) = µ 0 [e t Dst • f ϕ ]. (16)
In parallel, let dµ ∞

t := δ Yt with Y t solution of dY t = U (Y t )dt + V (Y t )dB t with initial condition Y t=0 µ 0 -distributed. Then, µ ∞ t [ϕ] = ϕ(Y t ) and F ϕ f (µ ∞ t ) = f ϕ (Y t
), and we have

E F ϕ f (µ ∞ t ) = E µ 0 [f ϕ (X t )] = µ 0 [e t Dst • f ϕ ]. ( 17
)
Comparing eq.( 16) and eq.( 17) proves the claim. Let us end this sub-section by a remark. This construction generalizes to higher dimensional systems. Indeed considered a system of stochastic differential equations, dY j t = U j (Y t )dt + V j a (Y t )dB a t , (with implicit summation over repeated indices) on M variables Y j driven by N motions B a t with quadratic variations dB a t dB d t = κ ab dt. We may then ask under which conditions a quantum system concentrates along trajectories solutions of these SDEs. Of course, the system has to be in dimension M with Hilbert space H = L 2 (R M ). Let us consider the evolution equation [START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF] generalized in dimension M (with monitoring of the M observables X j ) with Lindblad operator L = L U + L V with (with implicit summation on repeated indices)

L * V ( φ) = κ ab V j a (X)P j φ P k V k b (X) - 1 2 φ V j a (X)P j P k V k b (X) + V j a (X)P j P k V k b (X) φ , L * U ( φ) = i 2 [U j (X)P j + P j U j (X), φ],
with P j the momentum operator conjugated to the position observable X j (i.e. [X j , P k ] = iδ j k ). It is then easy to check that the measure on R M associated to X j and induced by the density matrix evolving according to the M -dimensional generalization of eq.( 15) concentrates in the large γ limit along the trajectories solutions of dY j t = U j (Y t )dt + V j a (Y t )dB a t .

It remains an open question to decipher what are the stochastic processes describing the strong monitoring limit for Lindladians not quadratic in the impulsion operators.

Monitoring continuous spectrum observable with Hamiltonian dynamics

The aim of this section is to analyze similarly the large monitoring limit for a system undergoing a Hamiltonian non dissipative dynamics. We consider a particle on the real line with Hilbert space H = L 2 (R) and monitoring its positions. The density matrix dynamical equation is (we put back the Planck constant for later convenience)

dρ t = - i [H, ρ t ] dt - γ 2 2 [X, [X, ρ t ]]dt + γ Xρ t + ρ t X -2 X t ρ t dW t , (18)
for some Hamiltonian H. As is well known, this equation preserves pure states (by construction, because monitoring preserves the state purity), so that we can equivalently write it on wave functions ψ t (x) [START_REF] Caves | Quantum-mechanical model for continuous position measurements[END_REF]:

dψ t (x) = - i (Hψ t )(x) dt - γ 2 2 (x -X t ) 2 ψ t (x) dt + γ (x -X t ) ψ t (x) dW t , (19)
with

X t = dx x|ψ t (x)| 2 and (Hψ t )(x) = - 2 2m ∂ 2 x ψ t (x) + V (x)ψ t (x)
for a (non-relativistic) particle of mass m in a potential V .

As recalled in the introduction, equation [START_REF] Kallenberg | Foundations of Modern Probability[END_REF] encodes for three different regimes [START_REF] Belavkin | tsov, Semiclassical asymptotics of quantum stochastic equations[END_REF][START_REF] Kolokol | tsov, Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate[END_REF][START_REF] Bassi | Collapse models: analysis of the free particle dynamics[END_REF][START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF]: (a) a collapse regime, (b) a classical regime, and c) a diffusive regime. The aim of this section is to show that we may define a scaling limit which describes the cross-over from the classical regime to the diffusive regime.

A simple case: Particle in a harmonic potential

Let us start with this simple case which includes a free particle. It will allow us to decipher what strong monitoring limit we may expect and which features may be valid in a more general setting. We closely follow methods of analysis used refs. [START_REF] Bassi | Collapse models: analysis of the free particle dynamics[END_REF][START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF][START_REF] Bhattacharya | Continuous quantum measurement and the classical to quantum transition[END_REF].

Let V (x) = 1 2 mΩ 2 x 2 be the potential. As is well known, eq.( 19) is better solved by representing the wave function as ψ t (x) = φ t (x)/ √ Z t with the normalization Z t and φ t (x) solution of the linear equation

dφ t (x) = i -1 2 2m ∂ 2 x φ t (x) -V (x)φ t (x) dt - γ 2 2 x 2 φ t (x) dt + γ x φ t (x) dS t , (20)
where S t is the monitoring signal (with dS 2 t = dt), solution of dS t = 2γ X t dt + dW t . The normalization factor Z t is such that dZ t = 2γ X t Z t dS t = 2γ dxx|φ t (x)| 2 dS t . Besides the frequency Ω associated to the harmonic potential, there is another frequency scale ω and a length scale , both arising from the position monitoring, with

γ 2 m .
Eq.( 20) is a Schrödinger equation in a complex harmonic potential and can be exactly solved via superposition of Gaussian wave packets. Thus, as in [START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF] we take a Gaussian ansatz for the un-normalized wave function written as

φ t (x) = φ 0 exp -a t (x -xt ) 2 + i kt x + α t , (21)
where all the time-indexed quantities have to be thought as stochastic variables. For a single Gaussian packet ansatz -the case we shall consider -, xt and kt are the mean position and mean wave vector. This single Gaussian packet is then solution of eq.( 19) if [START_REF] Kolokol | tsov, Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate[END_REF][START_REF] Bassi | Collapse models: analysis of the free particle dynamics[END_REF][START_REF] Bassi | On the long time behavior of stochastic Schroedinger evolutions[END_REF]]

da t = -2 1 -i2 4 a 2 t + i Ω 2 2ω 2 ωdt dx t = vt dt + √ ω 2 a R t dW t dv t = -Ω 2 xt dt -ω 3 2 a I t a R t dW t
with vt is the mean velocity. Here a R t (a I t ) denotes the real (imaginary) part of a t respectively (i.e. a t = a R t + ia I t ). From these equations, it is clear that τ c = 1/ω is the typical time for the wave function to collapse. After a typical time of order τ c , the Gaussian packet reaches its minimum size with a t a ∞ for t τ c with

a ∞ = 1 2i 4 1 + i Ω 2 ω 2 1/2
Taking ω → ∞ while keeping Ω fixed allows us to simplify :

a ∞ = e -iπ/4 √ 2 -2 . ( 22
)
In other words, monitoring stabilizes the wave function in a Gaussian wave packet with constant (minimal) width . In this collapsed wave packet, the position and velocity dispersions are

σ x = 2 -1/4 , σ v = 2 -3/4 ω .
After this transient collapsing period, for t τ c , the mean position and velocity evolve according to

dx t = vt dt + √ 2ω 2 dW t , ( 23
)
dv t = -Ω 2 xt dt + ω 3 2 dW t . ( 24
)
We now may wonder if there is a well defined strong monitoring limit (i.e. a limit γ → ∞). On physical ground, this limit should be such that the time to collapse vanishes, that is τ c → 0 or equivalently ω → ∞. It is then clear from eqs.(23,24) above that should simultaneously vanish for this limit to make sense, so that the strong monitoring limit is the double scaling limit ω → ∞, → 0. A closer inspection of eqs. (23,24) shows that we should take this double limit with ε := ω 3 2 fixed (so that √ 2 ω → 0). Note that, in this limit, the wave packet is localized both in space and in velocity, σ x → 0 and σ v → 0, so it is actually a classical limit (i.e. γ → ∞ and → 0 with γ/m fixed).

We can summarize this discussion: Proposition: In the double limit ω → ∞ and → 0 at ε := ω 3 2 = 2 γ 2 /m 2 fixed, solution of the quantum trajectory equation [START_REF] Kallenberg | Foundations of Modern Probability[END_REF] in an harmonic potential V (x) = 1 2 mΩx 2 localizes in the sense that the probability density |ψ t (x)| 2 = δ xt with xt solution of the stochastic equations

dx t = vt dt, ( 25
)
dv t = -Ω 2 xt dt + √ ε dW t . ( 26
)
This behavior describes the cross-over from a semi-classical behavior, which occurs just after the transient collapsing period, to the diffusion behavior due to monitoring back action. As is well known, eqs.(25, 26) can be solved exactly with solution:

xt = x 0 cos(Ωt) + ε Ω 2 t 0 dW s sin(Ω(t -s)),
where we chose for simplicity the initial conditions x(t = 0) = x 0 , v(t = 0) = 0. It reflects the cross-over behavior from the classical solution xt x 0 cos(Ωt) at small time to the diffusion behavior xt ε Ω 2 t 0 dW s sin(Ω(t -s)) at large time. The fuzziness of the trajectory can be testified by computing the variance of the position. We have (∆x t ) 2 = Ω 2 ( 2Ωt-sin(2Ωt)

4Ω

), so that (∆x t ) 2 t/2Ω 2 for Ωt 1 which is typical of a diffusive behavior and (∆x t ) 2 t 3

3 for Ωt 1 which can be interpreted as a state localized with accuracy t 3 for small times. The two behaviors are showed in fig. 1.

Generalization: A particle in a smooth potential

We can now borrow the previous observation to state what the strong monitoring limit could be for a particle in an arbitrary potential. As suggested sometime ago [START_REF] Habib | Decoherence, Chaos, and the Correspondence Principle[END_REF][START_REF] Kolokol | tsov, Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate[END_REF][START_REF] Bassi | Collapse models: analysis of the free particle dynamics[END_REF], after a transient time of order τ c = 1/ω with ω 2 := γ 2 /m, continuous monitoring of the position leads to a collapse of the wave function on a Gaussian state with a minimal width of order with 4 := /mγ 2 . In view of the previous analysis we are lead to suggest that Conjecture: In the double limit γ → ∞ and /m → 0 at ε := 2 γ 2 /m 2 fixed, solution of the quantum trajectory equation [START_REF] Kallenberg | Foundations of Modern Probability[END_REF] in a potential V (x) localizes at a position xt solution of the Langevin equation

dx t = vt dt (27)
dv t = - 1 m ∂ x V (x t ) dt + √ ε dW t (28)
Let us give a few arguments in favour of this claim. After the transient collapse time, we can take the Gaussian wave packet (21) for the (unnormalized) wave function. Taylor expanding the potential around the mean position xt and keeping only the terms up to second order leads to the stochastic equations for the width, the mean position and the mean velocity :

da t = γ 2 -2i m a 2 t + i 2 ∂ 2 x V (x t ) dt dx t = vt dt + γ 2a R t dW t dv t = - 1 m ∂ x V (x t ) dt - γ a I t ma R t dW t
with a t = a R t + ia I t and where we supposed that

(x -xt ) 3 ∂ 3 x V (x t ) (x -xt ) 2 ∂ 2 x V (x t ), (29)
for every x, xt . As for the harmonic potential, at time t τ c the width reaches its stationary values a ∞ = (

1 2i 4 (1 + i 2 γ 2 ∂ 2 x V (x t ))) 1/2 .
Demanding that this width be independent of dynamical aspects, that is, independent of the position, imposes |∂ 2

x V (x t )| γ 2 . This is indeed satisfied in the double scaling limit (if the potential is smooth enough) and it coincides with the condition of ref. [START_REF] Bhattacharya | Continuous quantum measurement and the classical to quantum transition[END_REF] forcing a localisation of the wave packet. We then simply have

a ∞ = e -i π 4 √ 2
-2 and, as in the harmonic case, the position and speed dispersions are given by

σ x = 2 -1/4 , σ v ≡ 2 -3/4 ω , in this limit.
Plugging these asymptotic values in the above equations and taking the scaling limit at ε := 2 γ 2 /m 2 fixed, leads to the Langevin equations (27,28).

Remark that we can make an a posteriori self-consistent check for the approximation (29). The typical length on which the wave function admits non zero value in the ω → ∞, → 0 limit at ε fixed scales like so that condition (29) amounts to

∂ 3 x V (x t ) ∂ 2 x V (x t )
. This is indeed valid for → 0 if the potential is smooth enough (with finite second and third spatial derivative) everywhere in space.

Acknowledgements: This work was in part supported by the ANR project "StoQ", contract number ANR-14-CE25-0003. D.B. thanks C. Pellegrini for discussions and for his interest in this work.

A Exchangeable processes and QND measurements

We here give the details of the arguments needed to prove the main proposition of the Section 2. Before discussing the continuous time limit, it is useful to have a look at the case of discrete time, which is more elementary. Our point is to show two equivalent descriptions of the monitoring during a non-demolition measurement process.

A.1 Discrete time monitoring

The evolution equation for the diagonal elements of the density matrix (this is a measure in general, a subtlety that becomes unavoidable for an observable with continuous spectrum) in the pointer states basis in repeated quantum non-demolition measurements in discrete time reads, The natural situation of the experimenter is to have only access to the observation process, and possibly to the initial condition. From this viewpoint, the fate of the observation process S n and of dµ n at large n is not so easy to decipher.

dµ n+1 (α) = dµ n (α)p(i|α) dµ n (β)p(i|β
However the evolution equations are easily "solved" to yield the joint law of

ε 1 , • • • , ε n which is Prob(ε 1 , • • • , ε n ) = dµ 0 (α) i p(i|α) Sn(i) ,
and the value of dµ n which is

dµ n (α) = dµ 0 (α) i p(i|α) Sn(i) dµ 0 (β) i p(i|β) Sn(i)
This law, which involves only the random variables S n = n m=1 ε m is obviously invariant under permutation of ε 1 , • • • , ε n , which expresses the property that the sequence ε 1 , • • • , ε n is exchangeable. De Finetti's theorem expresses that if this holds for each n = 1, 2, • • • then the random variables ε 1 , ε 2 , • • • are conditionally independent and identically distributed. This is also apparent from the explicit formula for the law: conditionally on a choice of α (sampled with the law dµ 0 (α)) the joint law of

ε 1 , • • • , ε n is i p(i|α) Sn(i) = n m=1 im p(i m |α) εm(im) ,
which says that the m th measurement yields observation of i m with probability p(i m |α) independently of the other observations. From this viewpoint, it is clear that the sequence of frequencies Sn(i) n converges almost surely to p(i|α) so that if the conditional probability distributions p(•|β) are distinct for different βs the asymptotics of the observation process allow to recover the value α sampled initially.

To summarize, the natural situation of an experimenter if to have access at time n to ε 1 , • • • , ε n , but the law of the observation process is exactly the same as if, before the experiment begins, a "cheater" samples the state of the system with dµ 0 so that he has access not only to ε 1 , • • • , ε n but also to α. The cheater knows in advance the asymptotics (i.e. α) while the experimenter discovers it only progressively as time goes by. To make the point again: the same process has two different interpretations depending on the information you have at your disposal. De Finetti's theorem is also closely related to the notion of reverse martingales. Fix 0 < l < m. Due to exchangeability, it is easy to see that knowing Sn n = ( Sn(i) n ) i for n ≥ m, the best estimate (i.e. the conditional expectation) for S l l is Sm m . In words, the best estimate for a value in the past knowing the future is the present. This is the notion of reverse martingales, to be contrasted with (usual) martingales for whom the best estimate for a value in the future knowing the past is the present. It turns out that the notion of reverse martingales is even more rigid than that of martingales: without any conditions (apart from the existence of conditional expectations implied by their very definition), reverse martingales converge at large times, almost surely and in L 1 . Of course, in the case at hand, we can rely on the explicit formula for the law of S n and the strong law of large numbers to be sure that Sn n has a limit at large n but this is deceptive: if K n , n ≥ 0, is the sequence of partial sums of independent identically distributed integrable random variables, then Kn n is an example of a reverse martingale, so the reverse martingale convergence theorem immediately implies the strong law of large numbers and yields a conceptual proof of it (see e.g. page 484 in [START_REF] Fristedt | A modern Approach to Probability Theory[END_REF]).

A.2 Continuous time monitoring

Not only do the frequencies Sn(i) n converge at large n. In fact, a stronger property, a central limit theorem holds: if the limiting pointer state is α, Sn(i)-np(i|α) n 1/2 converges to a Gaussian. Note that i S n (i) = n so there is one degree of freedom less than the number of possible measurement outcomes. To take a continuous time limit, the situation resembles that of random walks: one has to replace p(i|α) by p δ (i|α) where δ 0 is the time increment, with p δ (i|α) = p 0 (i) + O(δ 1/2 ) so that each observation has only a small effect (on the correct order of magnitude) on dµ. Assuming for simplicity that the pointer state basis is indexed by a real number A and that i takes only two values (so that there is only one degree of freedom) one is naturally led to an observation process described up to normalization by S t = B t + At, t ∈ [0, +∞[ where B t is a standard Brownian motion and A is sampled from an initial distribution dµ 0 . This is the description from the perspective of the "cheater", whose knowledge at time t is α and S u , u ∈ [0, t], or α and B u , u ∈ [0, t]. In more mathematical terms, the "cheater" observes the process via the filtration G t := σ{A and S u , u ∈ [0, t]} = σ{A and B u , u ∈ [0, t]}. Let us note that a general theorem (see e.g. page 322 in [START_REF] Kallenberg | Foundations of Modern Probability[END_REF]) based on a natural extension of the notion of exchangeability ensures that CB t + At with C, A random and B t t ∈ [0, +∞[ an independent Brownian motion is the most general continuous exchangeable process on [0, +∞[. A random conditional variance such as C in the above formula plays no role in the forthcoming discussion.

Our goal is to get the description of the same process for the experimenter, who knows only S u , u ∈ [0, t] at time t, i.e. uses the filtration H t := σ{S u , u ∈ [0, t]}. It is also useful to introduce the filtration F t := σ{B u , u ∈ [0, t]}. The relations between the filtrations F t , G t , H t are the clue to solve our problem. It is crucial that F t is independent of A and that F t , H t ⊂ G t , and we use these properties freely in conditional expectations in what follows.

We let E denote the global (i.e. over both A and the Brownian motion) expectation symbol.

A.3 Change of filtration

The crucial computation is an identity for the joint law of the random variable A and the process S t .

Proposition: Let f be a nice (measurable and such that the following expectations make sense, non-negative or bounded would certainly do) function from R k+1 to R, and

0 = t 0 < t 1 • • • < t k = t. Then E [f (A, S t 1 , • • • , S t k )] = E f (A, B t 1 , • • • , B t k )e ABt-A 2 t/2 .
The general tool to understand such a formula is Girsanov's theorem, but in the case at hand, an easy (if tedious) explicit computation does the job. The idea is to write

E [f (A, S t 1 , • • • , S t k )] = E [f (A, B t 1 + At 1 , • • • , B t k + At k )] = dµ 0 (α)E [f (α, B t 1 + αt 1 , • • • , B t k + αt k )] ,
where the first equality is the definition the observation process and the second makes use of the fact that the Brownian motion is independent of A. Thus we are left to prove the identity

E [f (α, B t 1 + αt 1 , • • • , B t k + αt k )] = E f (α, B t 1 , • • • , B t k )e αBt-α 2 t/2
for every α ∈ R. This is done by writing the left-hand side using the explicit expression of the finite dimensional distributions of Brownian motion in terms of the Gaussian kernel and translating the integration variables x l associated to the positions at time t l , l = 1, • • • , k, by αt l . Setting x l + αt l = y l (with the convention x 0 = y 0 = 0) one gets

- (x l -x l-1 ) 2 2(t l -t l-1 ) = - (y l -y l-1 ) 2 2(t l -t l-1 ) + α(y l -y l-1 ) -α 2 (t l -t l-1 )/2
which leads to a telescopic sum k l=1 α(y l -y l-1 ) -α 2 (t l -t l-1 )/2 = αy k -α 2 t k /2 yielding an expression which is recognized as the right-hand side.

We use this identity to understand the conditional distribution of A when the measurement has been observed up to time t, i.e. to have an explicit representation of H t := E[h(A)|H t ] for an arbitrary measurable function h such that h(A) is integrable. Note that by construction H t is a closed H t -martingale. Note also that, at least if h(A) is square integrable, a conditional expectation is a best mean square approximation so that H t is the best estimate of h(A) (known exactly to the cheater) for someone whose knowledge is limited to the observations. To get a hold on this conditional expectation we introduce a bounded measurable function g

(S t 1 , • • • , S t k ) where 0 = t 0 < t 1 • • • < t k = t and use the general formula to get E [h(A)g(S t 1 , • • • , S t k )] = dµ 0 (α)h(α)E g(B t 1 , • • • , B t k )e αBt-α 2 t/2 . (30)
and for each

β ∈ R E e βSt-β 2 t/2 dµ 0 (γ)e γSt-γ 2 t/2 g(S t 1 , • • • , S t k ) = dµ 0 (α)E e βBt-β 2 t/2 dµ 0 (γ)e γBt-γ 2 t/2 g(B t 1 , • • • , B t k )e αBt-α 2 t/2 which simplifies to E e βSt-β 2 t/2 dµ 0 (γ)e γSt-γ 2 t/2 g(S t 1 , • • • , S t k ) = E e βBt-β 2 t/2 g(B t 1 , • • • , B t k ) .
Integrating this identity against dµ 0 (β)h(β) and comparing with eq.(30) we get E dµ 0 (β)h(β)e βSt-β 2 t/2 dµ 0 (γ)e γSt-γ 2 t/2 g(B

t 1 , • • • , B t k ) = E [h(A)g(S t 1 , • • • , S t k )] .
As dµ 0 (β)h(β)e βS t -β 2 t/2 dµ 0 (β)e βS t -β 2 t/2 is H t -measurable and g is arbitrary, we have obtained our major result, an explicit representation for the closed H t -martingale H t :

H t = E[h(A)|H t ] =
dµ 0 (β)h(β)e βSt-β 2 t/2 dµ 0 (β)e βSt-β 2 t/2 .

This can be rephrased by saying that the measure dµ 0 conditional on H t is the measure (in fact a measure-valued H t -martingale) dµ t (α) := dµ 0 (α) e αSt-α 2 t/2 dµ 0 (β)e βSt-β 2 t/2 .

As emphasized in the main text, the same expression for dµ t holds for the diagonal of the density matrix at time t as a functional of the diagonal of the density matrix at time 0 when the results of the measurements have been taken into account. That this must be the case is strongly supported by the discrete time counterpart recalled above.

What remains to be deciphered is how S t can be analyzed from the point of view of stochastic processes under the filtration H t . The general formula for the joint distribution of the random variable A and the process S t leads easily to E[S t -S s |H s ] = (t -s)E[A|H s ] for 0 ≤ s ≤ t, so that the process W t := S t -t 0 E(A|H s ) ds is an H t -martingale. This can also be checked as follows. As H t ⊂ G t and B t is a G t -martingale, the process

S t -E[A|H t ] t = E[S t -At|H t ] = E[B t |H t ]
is an H t -martingale, and we are left to check that the process E[A|H t ] t -t 0 E[A|H s ] ds is an H t -martingale. This is easy either by formal manipulations of conditional expectations or by integration by parts:

E[A|H t ] t - t 0 E[A|H s ] ds = t 0 s dE[A|H s ]
which is a martingale as the stochastic integral of an adapted (in fact deterministic) integrand s against the martingale integrator dE[A|H s ].

The quadratic variation of S t (which has continuous trajectories) is dS 2 t = dB 2 t = dt and t 0 E[A|H s ] ds = t 0 dµ 0 (β)h(β)e βSs-β 2 s/2 dµ 0 (β)e βSs-β 2 s/2 ds is a finite variation process with continuous trajectories. Hence W t := S t -t 0 E[A|H s ] ds is an H t -martingale with continuous trajectories and quadratic variation dW 2 t = dS 2 t = dt. By Lévy's characterization theorem, W t is an H t Brownian motion. Thus, from the point of view of the observer, the signal S t decomposes as an H t -semimartingale

S t = W t + t 0 E[A|H s ] ds.
With some tedious manipulations of the general formula allowing to go back and forth between expectations of the observation process S t and the Brownian motion B t we could do without Lévy's characterization theorem, i.e. get an explicit formula for the finite dimensional distributions of W t and recognize those of a Brownian motion. It is worth noticing that E[B t |H t ] which is an H t -martingale with continuous trajectories is not a Brownian motion.

Let us note that setting Z(x, t) := dµ 0 (β)e βx-β 2 t/2 , so that Z t := Z(S t , t) is the normalization factor for µ t , we obtain E[A|H t ] = (∂ x log Z)(S t , t), which leads to the following form for the stochastic differential equation for S t as an H t -semi-martingale dS t = dW t + (∂ x log Z)(S t , t)dt = dW t + dt dµ t (α)α .

The first expression of the drift term is typical for a so-called h-transform and points to a systematic (but less direct and less elementary) derivation of the above results via Girsanov's theorem. The second expresses the instantaneous drift term as the average of the observable A at time t (i.e. in a state described by a density matrix whose diagonal in the pointer states basis is the measure dµ t ).

One

  as before and D a linear operator on function ϕ such that µ t [ D • ϕ] = Tr L(ρ t ) ϕ(X) = Tr ρ t L * (ϕ(X)) ,

  (a) Small time behavior (b) Long time behavior

Figure 1 :

 1 Figure 1: Typical behaviors of a particle trapped in an harmonic oscillator in the strong monitoring regime. The plots correspond to the evolution of the position of the particle viewed at different time scales and evolving according to (25,26) with Ω = 1, = 1, x0 = 2, v0 = 0. We clearly have a transition from an oscillatory behavior at time of order 1/Ω to a diffusive regime for t 1/Ω.

  ) with probability dµ n (β)p(i|β), where Greek letters α, β, ... index pointer states and Latin letters i, j, • • • are the outcomes of the indirect measurements. The observation process is defined by S n (i) := #{m ≤ n, m th measurement has given outcome i} =: n m=1 ε m (i) so that ε n (i) = 1 if the outcome of the n th measurement is i (probability dµ n-1 (β)p(i|β)) and 0 otherwise.

  checks easily that E[h(A)|H t ], which is automatically an H t -martingale, satisfiesdE[h(A)|H t ] = (E[Ah(A)|H t ] -E[h(A)|H t ]E[A|H t ]) dW t .The combinatorics ensuring the absence of dt terms is embodied in the relation∂ t + 1 2 ∂ 2x e βx-β 2 t/2 = 0 valid for every β. This leads todE[B t |H t ] = 1 -t(E[A 2 |H t ] -E[A|H t ) 2 ] dW twhere the conditional variance on the right-hand side can be rewritten asE[A 2 |H t ]-E[A|H t ] 2 = dµ 0 (α)dµ 0 (β) (α -β) 2 e αSt-α 2 t/2 e βSt-β 2 t/2 2 dµ 0 (α)dµ 0 (β) e αSt-α 2 t/2 e βSt-β 2 t/2 = dµ t (α)dµ t (β) (α -β) 2 2 .At large times this conditional variance vanishes and dE[B t |H t ] approaches the Brownian motion increment dW t .

Don't get confused between the notation d associated to the integration of the variable α or x and the notation d related to differentiation with respect to time variable t.

But the kernel ρt(x, y) are not a two dimensional normalized Gaussian.

δy denotes de Dirac measure centered at y: δy := δ(x -y)dx.

:= mγ 2 , ω 2 :=


Fanny Morin

Hadrien Courtecuisse

Ingerid Reinertsen

Florian Le Lann

Olivier Palombi

Yohan Payan

Matthieu Chabanas

Florian Le Lann

Resection-induced brain-shift compensation using vessel-based methods

Keywords: Resection-induced brain-shift, Vessel-based registration method, Intraoperative ultrasound

come

INTRODUCTION

For brain tumor procedures, planning and image-guided navigation are based on preoperative Magnetic Resonance (pMR) images. However, these images do not account for the intraoperative deformation of soft tissues, called brain-shift, affecting the accuracy of the procedure. While causes of this shift are numerous, for example gravity or cerebrospinal fluid loss, [START_REF] Gerard | Brain shift in neuronavigation of brain tumors: A review[END_REF] tumor resection is one of the most important. A solution is to reimage the brain during surgery, to acquire data reflecting the current shape of the tissue. An image-to-patient registration can then map high-resolution pMR images to these intraoperative data, providing updated images to the surgeon.

In the literature, craniotomy-induced brain-shift occuring before or just after dura opening is a widely studied topic. [START_REF] Reinertsen | Intra-operative correction of brain-shift[END_REF][START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF][START_REF] Garlapati | More accurate neuronavigation data provided by biomechanical modeling instead of rigid registration[END_REF][START_REF] Miga | Clinical evaluation of a model-updated image-guidance approach to brain shift compensation: experience in 16 cases[END_REF][START_REF] Fuerst | Automatic ultrasoundMRI registration for neurosurgery using the 2d and 3d LC2 Metric[END_REF][START_REF] Rivaz | Automatic Deformable MR-Ultrasound Registration for Image-Guided Neurosurgery[END_REF] However, fewer works deal with the image-to-patient registration during the whole procedure, to take into account the deformation induced by tissue retraction and tumor resection itself. Some authors have simply used their craniotomy-induced compensation methods, while others have proposed to take account for the cavity formed by the resection. The following paragraphs, organized by the modality of intraoperative images, summarize a state of the art.

Different groups have proposed to register pMR with intraoperative Magnetic Resonance (iMR) images, acquired once or even several times during tumor resection. While of reduced quality with respect to pMR, For further information, send correspondence to matthieu.chabanas@univ-grenoble-alpes.fr this imaging system provides dense volumetric information. Images can be mapped using fiducials [START_REF] Nimsky | Intraoperative Compensation for Brain Shift[END_REF] or imagebased registration methods. [START_REF] Nabavi | Serial Intraoperative Magnetic Resonance Imaging of Brain Shift[END_REF][START_REF] Archip | Non-rigid alignment of pre-operative MRI, fMRI, and DT-MRI with intraoperative MRI for enhanced visualization and navigation in image-guided neurosurgery[END_REF][START_REF] Fan | Image Updating for Brain Shift Compensation During Resection[END_REF] Other authors use a biomechanical model for the registration, [START_REF] Clatz | Robust Nonrigid Registration to Capture Brain Shift From Intraoperative MRI[END_REF] sometimes with the removal of elements (or anniliation of their mechanical effects) localized inside the resection cavity. [START_REF] Ferrant | Serial registration of intraoperative MR images of the brain[END_REF][START_REF] Vigneron | Serial FEM/XFEM-Based Update of Preoperative Brain Images Using Intraoperative MRI[END_REF] Nonetheless, and regardless of their accuracy, these methods cannot be widely spread in clinical practice due to the cumbersomeness and cost of intraoperative MR devices. Another option is to digitize the brain cortical surface during resection, through the craniotomy, using stereovision image pairs or laser range systems. Their coupling with a mechanical model accounting for the retraction and/or resection yielded to significant results. [START_REF] Fan | Image Updating for Brain Shift Compensation During Resection[END_REF][START_REF] Miga | Modeling of Retraction and Resection for Intraoperative Updating of Images[END_REF] A recent retrospective study also compared outcomes from simulated surface data with actual iMR imaging. [START_REF] Luo | Retrospective study comparing model-based deformation correction to intraoperative magnetic resonance imaging for image-guided neurosurgery[END_REF] Finally, another intraoperative framework is Ultrasound (iUS) imaging. In addition to providing sub-surface data, this system is in widespread use, portable, cost-effective and do not require major changes in the surgical procedure. On one hand, image-based methods can be used to register pMR and iUS acquired during or after resection, using generated peudo-US images, [START_REF] Mercier | Registering pre-and postresection 3-dimensional ultrasound for improved visualization of residual brain tumor[END_REF] LC metrics 6 or patch-based correlation ratio, [START_REF] Rivaz | Deformable registration of preoperative MR, pre-resection ultrasound, and post-resection ultrasound images of neurosurgery[END_REF] with promising or less conclusive results. [START_REF] Riva | 3d intra-operative ultrasound and MR image guidance: pursuing an ultrasoundbased management of brainshift to enhance neuronavigation[END_REF] On the other hand, iUS were used with a model-based method, [START_REF] Bucki | Doppler Ultrasound Driven Biomechanical Model of the Brain for Intraoperative Brain-Shift Compensation: A Proof of Concept in Clinical Conditions[END_REF] with the neutralization of mechanical elements inside the resection cavity. While encouraging, a difficulty is to precisely define which elements have to be removed, especially when tissue debulking occur.

In this paper, we consider Power Doppler iUS to assess the value of blood vessels for brain-shift compensation during resection. Two methods initially developed for craniotomy-induced brain-shift compensation are evaluated, with no further treatments.

METHODS

Vessel-based methods

This section provides a short description of the evaluated vessel-based methods. Both rely on the acquisition of pMR and pMR Angiography (pMRA) exams before surgery, then intraoperative navigated Doppler and/or B-mode iUS images. A 3D iUS volume is reconstructed out of the navigated 2D slices. Ultrasound acquisitions are performed directly in contact with the brain and, during resection, the cavity formed by the removed tissue is filled with saline.

Modified ICP method [START_REF] Reinertsen | Intra-operative correction of brain-shift[END_REF] Blood vessels are segmented then skeletonized from the pMRA and Doppler iUS images. A modified Iterative Closest Point (ICP) algorithm is then used to register the vascular trees with the particularity to only consider the n% pairings having the smallest distances. The computed rigid transformation is finally applied to the iUS images. [START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF] Preoperatively, brain soft tissues and blood vessels are respectively extracted from pMR and pMRA. A tetrahedral Finite Element (FE) mesh, only accounting for the hemisphere affected by the tumor, is then mechanically coupled with the vascular tree skeleton. Intraoperatively, Power Doppler and B-mode information are recorded simultaneously, respectively allowing to extract the blood vessels and probe footprint. A biomechanical simulation, relying on a linear constitutive law solved with a co-rotational approach, is used to compensate for brain-shift, accounting for boundary conditions (contacts with the dura mater, the falx cerebri and the tentorium cerebelli) and loads (applied to register blood vessels and maintain brain tissue under the US probe footprint) expressed as Lagrangian Multipliers constraints. After resolution, pMR images are updated with the FE model deformations.

Constraint-based biomechanical method

Experiments

Four patients suffering from low grade tumor are included in this study. In the first three cases, the tumor was located in the brain periphery. For the fourth one, is was located deeper into the brain. All patients underwent surgery at Saint Olav Hospital, Trondheim, Norway, and data were collected by SINTEF Medical Technology. During the procedure, the navigation system Sonowand 21 was used to acquire navigated B-mode and Doppler iUS images. These acquisitions' characteristics are presented in Table 1. For each patient, an iUS acquisition was performed in the middle or near the end of the resection procedure.

Table 1. Description of the experimental data. t0 refers to the dura opening time. The number of landmarks set on blood vessels and the anatomical structures delineated for the validation are specified for each patient, as well as the tumor location in depth. For the evaluation, five to nine paired landmarks were manually identified on vessels in the pMRA and Doppler iUS images. Note that there is a bias since vessels are also used to drive the registration, although not specifically these landmarks. In addition, anatomical structures such as sulci were delineated by a clinician in the pMR and B-mode iUS acquisitions. This evaluation process is similar to the one used to validate our vessel-based methods at the opening of the dura. [START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF]

Case Acquisition time Vessel landmarks

RESULTS

Quantitative results on blood vessels landmarks and delineated anatomical structures are presented in Table 2.

Table 2. Mean distances±sd between paired landmarks identified on blood vessels, then mean closest-point Euclidian distances±sd between the delineated anatomical structures. Distances are given before compensation, after compensation using the modified ICP, 2 then after the constraint-based biomechanical method. [START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF] Maximal distances are provided in parenthesis. All values are specified in mm. The percentage of associated points is shown in square brackets.

Case Before compensation Modified ICP method 2 Constraint-based biomech. method [START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF] Evaluation on vessels landmarks

Surface tumor, cases 1-3

All distances (mean, max and sd) between blood vessels landmarks are significantly reduced using both brain-shift compensation techniques, proving an accurate registration in the vessel areas. The constraint-based biomechanical method provides better registration results with an average compensation score of 63% against 56% for the rigid modified ICP. In addition, the registration of the anatomical surfaces is also globally improved (distances are lower and the number of associated points is increased).

The order of magnitude of the remaining shit is similar to the ones obtained just after dura opening. [START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF][START_REF] Miga | Clinical evaluation of a model-updated image-guidance approach to brain shift compensation: experience in 16 cases[END_REF][START_REF] Rivaz | Deformable registration of preoperative MR, pre-resection ultrasound, and post-resection ultrasound images of neurosurgery[END_REF] During resection, these results can therefore be considered very satisfying. This can be confirmed qualitatively, as shown in Figures 1 and2.

Deep tumor, case 4

Case 4 is clearly more complex. With regard to the distance between vessels landmarks, the remaining error after both compensation methods is much higher, in mean, max as well as standard-deviation. While an improvement is shown, from 9 to 6.5 mm approximately, all these factors indicate a non-sufficient compensation of the brainshift during resection. Surprisingly, the distances based on the Sylvian sulcus does not confirm this, as they are similar to previous cases. An explanation can be given by the non-discriminant geometry of the delineated sulcus near the tumor, alike a 3D "U shape". As shown in Figure 3, the shift is in fact well corrected in depth, but an important sliding movement can still be observed in the lateral direction. In addition to deformations larger than in cases 1-3, the vessels are here located on one side of the tumor only which can explain the poor lateral accuracy.

CONCLUSION

This paper addresses the topic of resection-induced brain-shift compensation. We have shown the interest of accounting for the vascular tree to correct these deformation, using existing vessel-based methods without additional treatment. [START_REF] Reinertsen | Intra-operative correction of brain-shift[END_REF][START_REF] Morin | Brainshift compensation using intraoperative ultrasound and constraint-based biomechanical simulation[END_REF] While more cases must be obviously be considered, first results show a significant improvement of the pre-to-intraoperative registration in the case of surface tumors. In a more complex case, the model-based methods in their current state are not successful.

Two conclusions can be established. First, while deformation and topological changes occur locally within the tissues, surrounding vessels still provide a reliable mean to track the brain and tumor location. However, unless in simple cases, methods designed for craniotomy-induced brain-shift are too limited and must be improved to correctly match the deformation during retraction and resection.

Future works include the development of such a method, plus the combine use of Doppler and B-mode iUS images to drive the registration with both vessels and anatomical structures, like the sulci themselves.

Figure 1 .

 1 Figure 1. Brain-shift compensation for patient 1 using the constraint-based biomechanical method. Slices of the pMR (left), B-mode iUS images (middle) and updated pMR (right) are shown. A pointer first shows the borders of the exposed cortical surface (top row) then a deep sulcus bifurcation point (bottom row). The black area in the iUS image is the current resection cavity.

Figure 2 .

 2 Figure 2. Intraoperative iUS image during tumor resection of case 3 (left). The resected cavity is segmented, then overlaid on the preoperative pMR (middle) and the updated pMR images (right). Nb: for rendering clarity, the iUS image scale is different.

Figure 3 .

 3 Figure3. Brains-shift compensation for case 4, resection of a deep tumor. An anatomical landmark of the Sylvian sulcus is pointed on the pMR, iUS and updated pMR images. While the correction is adequate in depth, it is not satisfactory in the lateral direction. Qualitatively, the cavity (in this case a retraction then a resection area) cannot be well surperposed on the updated pMR images.

Anatomical structure Tumor location

	1	t 0 + 107 min	5	Prefrontal sulcus	Surface
	2	t 0 + 94 min	8	Sylvian fissure	Surface
	3	t 0 + 97 min	5	Central sulcus	Surface
	4	t 0 + 192 min	9	Sylvian fissure	Deep

Acknowledgment

This work was partly supported by the French National Research Agency (ANR) through the frameworks Investissements d'Avenir Labex CAMI (ANR-11-LABX-0004) and Infrastructure d'Avenir en Biologie et Santé (ANR-11-INBS-0006), and by a France-Norway partnership (PHC Aurora 2015/Research Council of Norway).


Georges Bossis

A Ciffreo

Yann Grasselli

Olga Volkova

Discontinuous

Discontinuous shear thickening in magnetorheological suspensions

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

At high enough volume fraction some suspensions of microparticles show a discontinuous shear thickening (DST) which happens when the applied stress is large enough to overcome the repulsive barrier preventing the aggregation between particles [START_REF] Bergström | Shear thinning and shear thickening of concentrated ceramic suspensions[END_REF][START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions as reentrant jamming transition[END_REF] such as a ionic layer or a coating polymer. If the suspension is flowing under an applied stress then, above a critical one, the flow can stop suddenly which is quite undesirable in some industrial processes like the pumping of concrete. On the contrary, if this phenomenon can be controlled, it gives birth to an efficient two-state material for on-off applications in force or torque transmission. We recently demonstate [START_REF] Bossis | Tunable discontinuous shear thickening with magnetorheological suspensions[END_REF][START_REF] Bossis | Outstanding magnetorheological effect based on discontinuous shear thickening in the presence of a superplastifier molecule[END_REF] that such a fluid can be made using carbonyl iron particles and some superplastifier molecules used in cement industry [START_REF] Mosquet | diphosphonates as efficient dispersing polymers for aqueous suspensions[END_REF] . In this presentation we shall first compare the usual behavior of a MRS with a typical volume fraction Φ =45% with the one of a very concentrated one at Φ=64%. This very high volume fraction is obtained thanks to the use of a superplastifier molecule called polyphosphonate polyoxyethlene(PPP). This molecule has a head made of two phosphonate groups (PO 3 H -) and an hydrophilic tail made of a polyoxyethylene chain. In a second part we shall present experiments made in microgravity during parabolic flights in order to check the effect of the sedimentation on the transition.

HIGH VERSUS MIDDLE VOLUME FRACTION AT IMPOSED STRESS

The experiments were performed with suspensions made of commercial carbonyl iron particles (CIP grade HQ from BASF) dispersed in ethylene glycol with the addition of the PPP molecule at a mass fraction of 0.2% relatively to the mass of the particles. The rheological curves are obtained in plate-plate geometry on a MCR502 from Anton Paar with serrated plates and a serrated plateau in order to prevent the slipping of the suspension on the walls. The stress was increased at a rate of 20Pa/min for which we do not observe a thixotropic behavior. The magnetic field is perpendicular to the plates and is applied with the help of a coil surrounding the sample. It is worth noting that in this configuration, the magnetic field inside the

Discontinuous shear thickening in magnetorheological suspensions

with for H=0 a plastic viscosity η p =0.080 Pa.s and a yield stress τ y =8.9Pa. Then at higher field, the curve is better represented by a Herschel-Buckley law:

n y K τ = τ + γ ɺ .......( 2
) With for instance for the largest field: H=43kA/m τ y =114.1Pa , K=38.7, n=0.31. One can see that this behavior differs from the standard Bingham model where all the curves have the same plastic viscosity, meaning that the average stress induced by the field does not depend on the state of the flow, or in other words that the relative trajectories of the particles remain the same when we are above the yield stress. In order to emphasize this difference we can compare the differential viscosities,which is the slope of the curve, η d , around a shear rate of 500s -1 . We have η d = 0.08Pa.s for H=0 but η d =0.19Pa.s for H=43kA/m., so a factor of 2.37 between these two fields. The first major difference is that now we have a discontinuous transition with a strong decrease of the shear rate. At H=0 this transition happens for a shear rate of 45 s -1 and a critical stress of 246Pa. When the field is increased the critical shear rate strongly decreases; for instance it is equal to 3.5 s -1 for a field H= 43kA/m and the critical stress is now σ c =440Pa. We must emphasize that this increase of the critical stress with the field is not expected, because this is the stress (and not the shear rate) which controls the onset of the jamming transition. The explanation is that, above a critical stress, the repulsive force between two particles ,here due to the molecule PPP, no longer resists to the applied stress and the particles come into frictionnal contact, which triggers the transition. In our situation the applied stress is splitted in a magnetic and an hydrodynamic stress, but the total critical stress should remain the same. Nevertheless there is a different kind of action, since the magnetic force between two particles is mainly a radial force whereas the hydrodynamic force has both a radial and a tangential component. The radial component due to the magnetic force prevails on the hydrodynamic one when the field increases and the consequence is a more important entanglement of the polymer layers in the space between two particles especially at high volume fraction where the average gap between the particles becomes smaller. It appears as the increase of the differential viscosities with the magnetic field. For example we have η d = 5.86 Pa.s for H=0 and 41.5 Pa.s for H=43kA/m, that is to say an increase by a factor of 7.08. For the same fields the increase of η d was only by a factor of 2.37 at Φ=0.45. If it is the tangential hydrodynamic stress which must reach a critical value to sweep the polymer out of the gap between the particles, then the yield stress which characterizes the effect of the field does not intervene and the value of the hydrodynamic stress (as given by η d times the critical shear rate) appears to be approximately constant around 200Pa.

Until now the experiments were performed at imposed stress, we are now going to see what happens at imposed shear rate. In order to reduce the role of the sedimentation which could greatly modifies the jamming transition, we have made some experiments in microgravity during parabolic flights.

JAMMING TRANSITION AT IMPOSED VELOCITY IN MICROGRAVITY

When, in the imposed stress mode, the shear rate decreases during a ramp of stress, it means that, if we are in the imposed velocity mode, we shall have a jump of stress when we reach the critical shear rate.

That is actually what we observe and the jump of stress is so strong that it overcomes the maximum torque of the rheometer of 0.3N.m. On the other hand in the imposed stress mode, if we increase the stress well above the critical one, the particles are then expelled out of the suspending fluid [START_REF] Cates | Dilatancy, jamming and the physics of granulation[END_REF] with, as a consequence, an artificial decrease of the viscosity. In order to avoid these two drawbacks, we have designed a homemade rheometer based on the high torque rheometer (10N.m) developped by CAD company for the rheology of concrete. The cell, represented in fig. 3 is closed by a lip seal in order to avoid the leakage of the suspension at high stress. The geometries we used are either a double helicoidal ribbon [START_REF] Aït-Kadi | Quantitative analysis of mixer-type rheometers using the Couette analogy[END_REF] or a vane geometry in order to prevent the slippage of the suspension on the wall. Furthermore the outer cylinder was serrated for the same reason. As sedimentation can also change the conditions of the jamming transition, we have used the opportunity of parabolic flights offered by the CNES to realize experiments in microgravity to see in which extent the pressure due to gravity can play a role on the jamming transition [START_REF] Fall | Yield stress and shear banding in granular suspensions[END_REF] . A comparison of two experiments made one after the other in an interval of approximately 1mn is represented in Fig. 4. In this experiment a ramp of shear rate between 0 and 1.6 s -1 was applied during 2s then the shear rate was kept constant. The ramp was applied just at the beginning of the microgravity period which is of 22s. A field of 19.2kA/m was applied at the same time. The resulting curve is the one with the solid line which grows up to 170kPa and then decreases slowly with time. The second curve with crosses was recorded in the same conditions immediately after but in normal gravity between two parabolas. We see that the rising of the stress is the same but, once the maximum the stress is reached, it continues to rise and the experiment stops because the maximum torque was reached. It appears that in microgravity the final stress is lower than in normal one. This effect was confirmed in several experiments. An other demonstration of the effect of the gravity on the triggering of the jamming transition is illustrated in Fig. 5. In this experiment we have used the helicoidal geometry of diameter 24mm and the shear rate was kept constant at a value of 24 s -1- (corresponding to a rotational velocity of 20rpm for a gap of 1mm) after a rising ramp of 8s, a field of 17.6 kA/m (220Oe) was applied just at the beginning of the microgravity period and kept constant. We see a peak of stress at a value of 60kPa when the field is turned on but followed by a quick decrease towards a plateau of 10kPa , then an abrupt increase which overcomes the maximum torque of the motor and triggers its shutdown. This sudden increase occurs during the period of hypergravity (1.8g) and this is not a coincidence because it was observed several times. It means that, despite the rotation of the geometry at 20rpm which prevents the sedimentation, the larger pressure due to the extra weight of a column of particles of height h: ∆P=138 Pa (∆P=ρ ∆g h with h=1mm, ρ=7.8g/cm3, ∆g=1.8x9.81) is enough to trigger the transition.

CONCLUSION

The possibility to modulate the jamming transition in a supension of iron microparticles by the application of a magnetic field is an opportunity to get a better understanding of this phenomenon. The increase of the differential viscosity with the applied magnetic field before the jamming transition indicates that the layers of polymer adsorbed on the particles interpenetrate each other more and more. The hypothesis that it is the tangential hydrodynamic stress which is responsible for the jamming transition and not the total stress, can explain the increase of the critical stress with the magnetic field even if additional experiments are needed to confirm this point. Finally, experiments made in microgravity emphasize the sensitivity of the jamming transition to the gravity force.

G. Bossis 1 ,

 1 A.Ciffreo 1 , Y.Grasselli 2 and O.Volkova 1 sample is lower than the external one: H i =H e /µ(H i ) because of the demagnetizing field. In Fig.(1) we have presented a set of curves showing the shear stress versus the shear rate for five different amplitudes of the magnetic field.

Figure 1 .

 1 Figure 1. Volume fraction Φ=0.45; the magnetic field H is in kA/m

  Now we are going to see what is the rheology at high volume fraction, for Φ=0.64 still with the same proportion of 0.2% of the PPP molecule. The result is shown in Fig.2for five different fields.

Figure 2 .

 2 Figure 2. Volume fraction Φ=0.64; the magnetic field H is in kA/m

Figure 3 .

 3 Figure 3. Sketch of the cell installed on the high torque rheometer

Figure 4 .

 4 Figure 4. Vane geometry. Shear stress versus time during a ramp of shear rate. The upper curve is in microgravity, the lower one in 1g. H=19.2kA/m (240Oe)

Fig. 5

 5 Fig.5 Lower curve: stress versus time, Upper curve (right scale): shear rate versus time. The dotted lines separate the different values of gravity during a parabolic flight. Field amplitude: H=17.6kA/m turned on at the start of the 0g period.

University of Nice Sophia-Antipolis, Institut de Physique de Nice,UMR7010,Parc Valrose,06108 Nice Cedex

[START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF] Skema Bachelors,60 rue Dostoievski,BP085,06902 Sophia-Antipolis,France

ACKNOWLEDGMENTS

The authors want to thank the Centre National d'Etudes Spatiales (CNES, the French Space Agency) for having supported this research