Polina Feldmann

Carsten Klempt

Augusto Smerzi

Luis Santos

Manuel Gessner

Interferometric Order Parameter for Excited-State Quantum Phase Transitions in Bose-Einstein Condensates

Quantum phase transitions (QPTs) are sudden changes in the ground-state properties of a system. The ground-state energy and wave function behave nonanalytically and the gap between the ground state and the first excited state closes when, at zero temperature, a control parameter is adiabatically varied across a critical value [START_REF] Sachdev | Quantum Phase Transitions[END_REF]. The idea of QPTs has been extended in recent years to out-of-equilibrium quantum many-body systems [START_REF] Diehl | Dynamical phase transitions and instabilities in open atomic many-body systems[END_REF][START_REF] Heyl | Dynamical quantum phase transitions: a review[END_REF]. For example, a sudden shift of a parameter (quantum quench), can lead to dynamical QPTs, which are characterized by a non-analyticity of physical quantities as a function of time [START_REF] Heyl | Dynamical quantum phase transitions: a review[END_REF]. A direct generalization of QPTs beyond the ground state is given by excitedstate quantum phase transitions (ESQPTs) [START_REF] Cejnar | Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei[END_REF][START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF][START_REF] Cejnar | Excited-state quantum phase transitions[END_REF]. Their distinguishing signature is a closing gap at nonzero energies: excited states cluster at a critical energy, which leads to a singularity in the density of states (DOS). Typically, the critical energy is a continuous function of a control parameter. Thus, in contrast to ground-state QPTs, ESQPTs can be crossed both by varying a control parameter at constant energy and by varying the energy at fixed parameters.

ESQPTs have been theoretically studied in a large variety of many-body quantum systems [START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF][START_REF] Stránský | Excited-state quantum phase transitions in systems with two degrees of freedom: Level density, level dynamics, thermal properties[END_REF][START_REF] Stránský | Excited-state quantum phase transitions in systems with two degrees of freedom: II. finite-size effects[END_REF], including the Lipkin-Meshkov-Glick (LMG) model [START_REF] Leyvraz | Large-n scaling behavior of the lipkin-meshkov-glick model[END_REF], Dicke and Jaynes-Cummings models [START_REF] Pérez-Fernández | Quantum quench influenced by an excited-state phase transition[END_REF][START_REF] Brandes | Excited-state quantum phase transitions in dicke superradiance models[END_REF], interacting boson models [START_REF] Cejnar | Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei[END_REF][START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF][START_REF] Macek | Excited-state quantum phase transitions in systems with two degrees of freedom. III. interacting boson systems[END_REF], molecular bending transitions [START_REF] Pérez-Bernal | Algebraic approach to two-dimensional systems: Shape phase transitions, monodromy, and thermodynamic quantities[END_REF][START_REF] Larese | A study of quantum phase transitions and quantum monodromy in the bending motion of non-rigid molecules[END_REF], and the quasi-energy spectrum of driven systems [START_REF] Bastidas | Quantum criticality and dynamical instability in the kicked-top model[END_REF]. Experimentally, ESQPTs have been confirmed in microwave Dirac billiards [START_REF] Dietz | Lifshitz and excited-state quantum phase transitions in microwave dirac billiards[END_REF] and in molecular spectroscopy [START_REF] Zobov | Monodromy in the water molecule[END_REF][START_REF] Winnewisser | Experimental confirmation of quantum monodromy: The millimeter wave spectrum of cyanogen isothiocyanate ncncs[END_REF]. Signatures of ESQPTs have been predicted in the many-body dynamics after a quench [START_REF] Pérez-Fernández | Quantum quench influenced by an excited-state phase transition[END_REF][START_REF] Santos | Structure of eigenstates and quench dynamics at an excited-state quantum phase transition[END_REF][START_REF] Kloc | Quantum quench dynamics in dicke superradiance models[END_REF] and in time-averaged expectation values [START_REF] Engelhardt | Excited-state quantum phase transitions and periodic dynamics[END_REF]. However, identifying order parameters that distinguish neighboring excited-state quantum phases from each other remains a challenge [START_REF] Cejnar | Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei[END_REF][START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF].

Spinor Bose-Einstein condensates (BECs) attract since several years a major interest as an exceptional tool for the study of many-body quantum dynamics [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Stamper-Kurn | Spinor bose gases: Symmetries, magnetism, and quantum dynamics[END_REF], including coherent spinor dynamics [START_REF] Chang | Coherent spinor dynamics in a spin-1 bose condensate[END_REF], classical bifurcations [START_REF] Zibold | Classical bifurcation at the transition from rabi to josephson dynamics[END_REF], and the generation of highly-entangled many-body states [START_REF] Lücke | Twin matter waves for interferometry beyond the classical limit[END_REF][START_REF] Gross | Atomic homodyne detection of continuous-variable entangled twin-atom states[END_REF][START_REF] Luo | Deterministic entanglement generation from driving through quantum phase transitions[END_REF][START_REF] Pezzè | Heralded generation of macroscopic superposition states in a spinor Bose-Einstein condensate[END_REF]. So far systematic investigations of critical behavior in such systems have focused on the ground-state QPTs [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Luo | Deterministic entanglement generation from driving through quantum phase transitions[END_REF][START_REF] Liu | Quantum phase transitions and continuous observation of spinor dynamics in an antiferromagnetic condensate[END_REF][START_REF] Bookjans | Quantum phase transition in an antiferromagnetic spinor bose-einstein condensate[END_REF][START_REF] Zhang | Generation of massive entanglement through an adiabatic quantum phase transition in a spinor condensate[END_REF], though observations of diverging oscillation periods can be interpreted as signatures of ESQPTs [START_REF] Zhang | Coherent spin mixing dynamics in a spin-1 atomic condensate[END_REF][START_REF] Zhao | Dynamics in spinor condensates tuned by a microwave dressing field[END_REF]. Very recently, a study of the quench dynamics of a spinor BEC revealed a dynamical QPT, which has been related to a phase transition in the highest-energy level [START_REF] Tian | Observation of dynamical quantum phase transitions with correspondence in an excited state phase diagram[END_REF].

In this Letter, we propose spinor BECs as a platform to explore ESQPTs in a paradigmatic class of models. We identify ESQPTs in a ferromagnetic spin-1 BEC, and show that the different excited-state quantum phases can be distinguished by the topology of classical phase-space trajectories. We use this to introduce an order parameter that is related to the dynamics of coherent states. This order parameter can be accessed by interferometry in existing experimental setups. Our work is, hence, an important step towards the characterization of excitedstate quantum phases and towards the systematic exploration of ESQPTs with controllable many-body quantum systems.

Ground-state quantum phases.-We consider a ferromagnetic spin-1 BEC of N atoms with three spin states m = ±1, 0. We assume a tight enough external trapping of the BEC such that, to a good approximation, all spin states share a common spatial mode (single-mode approximation). The spin degrees of freedom are then well described by the Hamiltonian density [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF]

ĥ = q 2N N -2 N0 (1) + c N 2 â † 1 â † -1 â2 0 + â †2 0 â1 â-1 + N0 N -N0 + 1 2 + D2 2 ,
where â † m and âm are the bosonic creation and annihilation operators for state m, Nm ≡ â † m âm with m Nm = N , and D ≡ N1 -N-1 is the magnetization. The interaction strength c depends on the spatial wave function and on the mass and scattering lengths of the atoms. A ferromagnetic BEC is characterized by c < 0 [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF]. The effective quadratic Zeeman shift q incorporates microwave dressing and thus may be both positive and negative [START_REF] Zhao | Dynamics in spinor condensates tuned by a microwave dressing field[END_REF]. The linear Zeeman effect has been eliminated by moving to a rotating frame. The Hamiltonian density (1) conserves D and the parity Î = (-1) N0 . In the eigenspace of D with eigenvalue D = 0, ĥ features three groundstate phases [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Luo | Deterministic entanglement generation from driving through quantum phase transitions[END_REF] depending on the ratio ξ ≡ q 2|c| : the Twin-Fock (TF) phase for ξ < -1, the Polar (P) phase for ξ > 1, and the Broken-Axisymmetry (BA) phase for |ξ| < 1.

Excited-state quantum phases.-To reveal the excitedstate phases, we study the mean-field limit [START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF][START_REF] Duffield | Mean-field dynamical semigroups on C -algebras[END_REF][START_REF] Duffield | Classical Hamiltonian dynamics for quantum Hamiltonian mean-field limits[END_REF][39] N → ∞ of Model (1) for the case of zero magnetization. We introduce the coherent states |α, N ≡

1 √ N ! ( m α m â † m ) N |0 , where α ≡ (α 1 , α 0 , α -1 ), α m ≡ √ n m e iφm , n m ≥ 0, φ m ∈ [0, 2π), and m n m = 1. The coherent states with D = 0, i. e., |α 1 | 2 = |α -1 | 2 , yield the classical Hamiltonian [39] h mf (α) |c| = 1 |c| lim N →∞ α, N | ĥ|α, N = ξ(1 -2n 0 ) -2n 0 (1 -n 0 ) cos 2 φ, (2)
where φ ≡ φ 0 -(φ 1 + φ -1 )/2. Note that parity conservation results in h mf (φ + π) = h mf (φ). The mean-field dynamics is governed by the equations of motion [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Zhang | Coherent spin mixing dynamics in a spin-1 atomic condensate[END_REF]39]

d dτ n 0 = ∂ ∂φ h mf |c| , d dτ φ = - ∂ ∂n 0 h mf |c| , and
d dτ (φ 1 -φ -1 ) = 0 (3)
with τ ≡ |c|t/ . The mean-field limit of the DOS ν 0 (η) in the D = 0 subspace can be computed according to [39] lim

N →∞ |c| ν 0 (η) N = Dαδ(n 1 -n -1 ) δ h mf (α) |c| -η , (4)
where Dα ≡ 1 Extending the ground-state phase diagram to the entire energy spectrum, we identify three excited-state phases in the ξ-η-plane: the TF phase for η > -|ξ| and ξ < 0, the P phase for η > -|ξ| and ξ > 0, and the BA phase for η < -|ξ|. The phases are indicated in Fig. 1a, where we have subtracted η 0 (ξ) = -1 2 (ξ 2 + 1), which corresponds to the ground-state energy in the mean-field limit, from η. The excited-state phases are separated by ESQPTs at η * = -|ξ| with 0 < |ξ| < 1. In the limit |ξ| → 0, η * hits the maximum of h mf /|c|. As |ξ| approaches 1, the ESQPTs evolve into the known groundstate QPTs.

Signatures of ESQPTs.-As expected for ESQPTs [START_REF] Cejnar | Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei[END_REF][START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF], the DOS (4) diverges at η * (ξ). Fig. 1a displays the mean-field DOS as a function of ξ and η -η 0 . Furthermore, it shows that in a finite-size system the ESQPTs reveal themselves by a sequence of avoided crossings in the energy spectrum [START_REF] Cejnar | Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei[END_REF]. The divergence of the DOS is due to stationary points of h mf . At a stationary point, ∂ ∂φ h mf = ∂ ∂n0 h mf = 0 causes the integrand in Eq. ( 4) to become singular. There are three stationary points at each 0 < |ξ| < 1: a saddle point at η * and two minima at η 0 . The saddle point is located at n 0 = 0 for ξ < 0 or at n 0 = 1 for ξ > 0, and the minima are at n 0 = (ξ + 1)/2 and cos 2 (φ) = 1, see Fig. 1b. Note that these stationary points do not depend on the restriction to coherent states with D = 0. However, the unrestricted DOS [39] remains finite at η * .

The phase-space trajectories [39] of h mf provide further signatures of the ESQPTs. The classical phase space is a sphere with z-axis n 0 and azimuthal angle φ. Figure 1b shows exemplary trajectories for ξ = 0.5. The trajectories reflect the symmetry h mf (φ + π) = h mf (φ). Since h mf (ξ, n 0 , φ) = h mf (-ξ, 1 -n 0 , φ), for ξ < 0 the phase space would appear upside down. As in the LMG model [START_REF] Ribeiro | Exact spectrum of the lipkin-meshkov-glick model in the thermodynamic limit and finite-size corrections[END_REF], the sets of trajectories at fixed ξ and η (the energy hypersurfaces) change topology at η * (ξ)-at the critical energy hypersurfaces called separatrices. For η > η * , i. e., in the TF and P phases, there is only one trajectory per ξ and η. By contrast, for η < η * , i. e., in the BA phase, the evolution can follow one of two disconnected trajectories. Each of these trajectories breaks the classical symmetry h mf (φ + π) = h mf (φ). Note, however, that the corresponding quantum symmetry I cannot be broken in the D = 0 subspace, where all states belong to a single eigenspace of I.

Order parameter.-The solutions n 0 (t) and φ(t) of the classical equations of motion, Eq. ( 3), are periodic [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Zhang | Coherent spin mixing dynamics in a spin-1 atomic condensate[END_REF]39]. In the TF and P phases, the phasespace trajectories encircle the n 0 -axis (green curves in Fig. 1b)-clockwise in the TF phase and counterclockwise in the P phase. By contrast, the trajectories in the BA phase do not enclose the n 0 -axis (yellow curves). We define our order parameter w as the winding number of the classical trajectories with respect to the n 0 -axis, such that w = -1 in the TF , w = 1 in the P , and w = 0 in the BA phase. We observe that w can be expressed in a particularly simple form. Let us denote the period of n 0 (t) at fixed ξ and η by T . In the BA phase, the periods of φ(t) and n 0 (t) coincide and, thus, φ(t + T ) = φ(t).

In the TF and P phases, however, φ(t

+ T ) = φ(t) ± π. Hence, w = 1 π [φ(T ) -φ(0)]. (5)
In contrast to most observables that have been studied in the context of ESQPTs [START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF][START_REF] Pérez-Fernández | Quantum quench influenced by an excited-state phase transition[END_REF][START_REF] Brandes | Excited-state quantum phase transitions in dicke superradiance models[END_REF][START_REF] Bastidas | Quantum criticality and dynamical instability in the kicked-top model[END_REF][START_REF] Santos | Structure of eigenstates and quench dynamics at an excited-state quantum phase transition[END_REF][START_REF] Kloc | Quantum quench dynamics in dicke superradiance models[END_REF][START_REF] Ribeiro | Exact spectrum of the lipkin-meshkov-glick model in the thermodynamic limit and finite-size corrections[END_REF][START_REF] Heiss | The large-N behaviour of the Lipkin model and exceptional points[END_REF], w is not merely singular at the phase transitions. It qualitatively distinguishes the entire excited-state phases by the dynamics of coherent states.

In the following, we present an interferometric scheme that extracts p ≡ cos(πw) [START_REF] Cejnar | Excited-state quantum phase transitions[END_REF] and therefore distinguishes neighboring excited-state phases from each other: in the BA phase p = 1, while in the TF and P phases p = -1. To measure p, first, an initial point (n 0 (0), φ(0)) on a trajectory at the ξ and η of interest is selected. Then the corresponding coherent state with φ 1 = φ -1 , |ψ(0) , is prepared at q = 2|c|ξ. The state freely evolves for the time T . Next, the spin states m = 0 and m = ±1 are coupled by the internal-state beamsplitter exp(-i π 2 Ŝθ0 ) with Ŝθ ≡1 2 (e -iθ â † 0 ĝ + e iθ ĝ † â0 ), ĝ ≡ (â 1 + â-1 )/ √ 2, and θ 0 ≡ π/2 -φ(0). Finally, the expectation value of N0 /N is measured. In the mean-field limit, this yields [39] lim

N →∞ 1 N ψ(T )| e i π 2 Ŝθ 0 N0 e -i π 2 Ŝθ 0 |ψ(T ) = 1 -V p 2 , (7)
where we have introduced the visibility V = 2 1 -n 0 (0) n 0 (0). As long as n 0 (0) = 0, 1, this unambiguously determines p.

Experimental realization.-We detail the measurement of p for 87 Rb atoms in their hyperfine ground state [START_REF] Lücke | Twin matter waves for interferometry beyond the classical limit[END_REF][START_REF] Luo | Deterministic entanglement generation from driving through quantum phase transitions[END_REF]. However, most of our discussion applies to any ferromagnetic spin-1 BEC. We assume that, initially, the condensate is in the state â †N 0 |0 / √ N !. Then a coherent state characterized by n 0 (0), φ(0), and φ 1 = φ -1 can be obtained by applying exp(-iχ Ŝθ0 ) with cos 2 (χ/2) = n 0 (0). Thus, both the state preparation and the beamsplitter are generated by Ŝθ0 and can be implemented by a sequence of a phase shift exp(i(θ 0 -θ rf ) N0 ), a radiofrequency pulse exp(-iζ Ŝθ rf ) 1 with ζ = χ or ζ = π/2, respectively, and another phase shift exp(-i(θ 0 -θ rf ) N0 ). Since we aim at the expectation value in Eq. ( 7), the first step of the state preparation and the last one of the beamsplitter can be omitted. N 0 can be measured, e. g., by a magnetic-field gradient that spatially separates the different spin states and subsequent absorptive imaging.

Reliably distinguishing p = ±1 requires a large visibility V , which can be maximized by choosing n 0 (0) as close to 1/2 as possible. The optimal n 0 (0), n opt , is [39]:

n opt =      1 2 (1 + ξ -ξ |ξ| 1 + 2η + ξ 2 ) for η < -1 2 1 2 for -1 2 ≤ η ≤ 0 1 2 (1 -η ξ ) for 0 < η (8) A corresponding φ(0), φ opt , is obtained from cos 2 φ opt = ξ(1 -2n opt ) -η 2n opt (1 -n opt ) . ( 9
)
Figure 2a shows that the optimized visibility is large throughout the vast majority of the phase diagram. The coherence time in typical BEC experiments is limited to few seconds. This constrains the accessible periods T . It is known [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Zhang | Coherent spin mixing dynamics in a spin-1 atomic condensate[END_REF]39] that where

|c| T = √ y -1 K(x/y) for η < η * √ x -1 K(y/x) for η > η * , (10)
K(k 2 ) = π/2 0 dγ 1 -k 2 sin 2 γ -1
is the complete elliptic integral of the first kind, x = |ξ| 1 + ξ 2 + 2η, and y = (x -ξ 2 -η)/2. T diverges at the ESQPTs. Figure 2b displays T for the typical interaction strength |c|/ = 2π × 4 Hz. Fortunately, T exceeds a moderate value of, e. g., 0.3 s only in the immediate vicinity of the ESQPTs.

So far we have considered only the mean-field limit, N → ∞. To study the impact of a finite system size, we simulate a measurement of p for N = 100 bosons by exact diagonalization of the Hamiltonian density (1), see Fig. 3. The jump discontinuities signaling the ESQPTs in the mean-field limit are, as expected, smoothed at finite N . However, the BA phase can still be clearly distinguished from the TF and P phases. In typical experiments, N is of the order of 10 4 and, thus, a much better convergence to the mean-field limit can be expected.

Conclusions.-Ferromagnetic spin-1 BECs exhibit ES-QPTs, which, in the mean-field limit, show up as a diverging DOS and a change in the topology of phase-space trajectories. We characterize the mean-field dynamics by a winding number w that distinguishes the excitedstate quantum phases from each other and, thus, is an order parameter. Adjacent phases differ in |w| and can be told apart by interferometrically monitoring the coherent many-body dynamics in present-day experiments. Note that the local order parameter N0 /N that characterizes the ground-state QPTs [START_REF] Luo | Deterministic entanglement generation from driving through quantum phase transitions[END_REF][START_REF] Zhang | Generation of massive entanglement through an adiabatic quantum phase transition in a spinor condensate[END_REF] cannot be directly generalized to excited states. The topological order parameter w, instead, is defined for all energies apart from the very ground state, where the trajectories reduce to single points. Our results show that ESQPTs can be studied in well-controlled atomic quantum many-body systems, and that these studies are not limited to properties of the transition itself. We propose a feasible experiment for characterizing excited-state quantum phases. This rep-resents an important step towards employing ESQPTs in quantum state engineering.

Finally, we remark that our findings apply to any of the numerous quantum systems with the same meanfield limit, including bosonic two-level pairing models at zero generalized angular momentum [START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF]. Our theoretical treatment of ESQPTs complements previous studies for opposite interaction sign [START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF]. Bosonic two-level pairing models comprise, e. g., the LMG model, the vibron model for molecules, and the interacting boson model for nuclei. We thank Dmytro Bondarenko, Pavel Cejnar, Ignacio Cirac, and Reinhard Werner for valuable discussions. We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the SFB 1227 "DQ-mat", project A02, and under Germany's Excellence Strategy -EXC-2123 QuantumFrontiers -390837967, and by the LabEx ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.

Appendix A: Mean-field limit of bosonic systems We consider a system of N ∈ N (pseudo-)spin-j bosons with j ∈ N 0 /2 or, equivalently, a system of N bosons distributed among 2j + 1 ∈ N modes. Such systems can be treated in terms of creation and annihilation operators â † m and âm , where m ∈ {j, j -1, . . . , -j} denotes the spin projection quantum number. Then Nm ≡ â † m âm counts the number of particles in mode m. Our Hilbert space is restricted to eigenstates of N ≡ m Nm with eigenvalue N .

We will focus on operators  with the following properties: For example, the c k (N ) may be independent of N or include O(N -1 ) corrections. Note that  is simultaneously defined for all N and [ Â, N ] = 0. We call the set of all such operators A, and the subset of Hermitian operators E ⊂ A. Let Ĥ be the system's Hamiltonian and ĥ ≡ Ĥ/N the Hamiltonian density. We require that ĥ ∈ E.

1. Â is a polynomial in â † m âl /N 2
Let us discuss some further properties of A. For any Â, B ∈ A:

4. Obviously, Â B ∈ A.

N [ Â, B] ∈ A.

To confirm № 3 of the defining properties, one may iteratively apply

[ D Ê, F ] = [ D, F ] Ê + D[ Ê, F ] and [ D, Ê F ] = [ D, Ê] F + Ê[ D, F ], (A1)
which holds for any operators D, Ê, and F . This yields a finite number of terms, each of which contains a single elementary commutator, [â m / √ N , â † m /

√ N ] = 1/N . For the coefficients of these terms, property № 3 follows immediately.

6. Let • N denote the spectral norm in the N -particle Hilbert space. Then there is a c ∈ R such that  N ≤ c ∀N .

Note that • N is sub-additive and sub-multiplicative, and that â( †) m N = √ N . Hence, c can be chosen to be the (finite) sum of the d k defined in property № 3.

We employ the projective coherent states |α, N ≡ 1 k! = e |zc| is finite. Note that this argument can be immediately generalized to arbitrary N -independent complex analytic functions on A, the set of which we denote by C(A):

√ N ! ( m α m â † m ) N |0 , where α ∈ C 2j+1 comprises the α m , α m ≡ √ n m e iφm , n m ≥ 0, φ m ∈ [0,
lim N →∞ α, N |f ( Â)|α, N = f (A mf (α)) ∀f ∈ C(A), Â ∈ A. (A4)
Similarly, employing the Baker-Campbell-Hausdorff formula, we obtain lim

N →∞ α, N | e zN Â B e -zN Â |α, N = k z k k! K (k) mf (α) with K(0) ≡ B, K(k) ≡ N [ Â, K(k-1) ]. (A5)
The key step in proving Eq. ( A5) is to demonstrate that we can find a suitable N -independent bound on K(k) .

For  N and B N , we first construct upper bounds c A and c B as suggested in the proof of property № 6. Then, iteratively applying the relation (A1), we find that K(k (A6) Note that, in general, e zN  B e -zN  / ∈ A and thus Eq. (A6) is not implied by Eq. (A2). Instead, it can be derived from Eq. (A5) by using Eqs. (A1) and (A2) to observe that

) ≤ 2 r B c B (2 r A c A ) k ,
K (n) mf [ B Ĉ] = n l=0 n l K (l) mf [ B]K (n-l) mf [ Ĉ], (A7)
where the argument of K (k) mf specifies the operator K(0) , and K(k) ensues inductively as defined in Eq. (A5). In the following two sections, we use Eqs. (A3), (A5), and (A6) to derive the mean-field limits of the density of states and of the equations of motion. In Section A 3 we summarize, for completeness, some of the mathematical theorems we use.

Density of states

We denote the energy per particle by and define the density of states (DOS) ν( ) by way of its Fourier transform:

F[ν](ζ) ≡ d e iζ ν( ) ≡ Tr e iζ ĥ, ζ ∈ R, (A8)
where the trace is taken over the respective N -particle Hilbert space. To obtain the DOS in the mean-field limit, we argue that lim

N →∞ 1 N 2j
Tr e iζ ĥ = lim

N →∞ C N N 2j Dα α, N | e iζ ĥ |α, N = Dα e iζh mf (α) = d e iζ Dα δ(h mf (α) -) (A9)
and conclude that lim

N →∞ ν( ) N 2j = Dα δ(h mf (α) -). (A10)
In the following we comment on some details of this derivation. First of all, note that ν( ) is well defined by Eq. (A8). For any N , the inverse Fourier transform of Tr e iζ ĥ is a unique tempered distribution, ν ∈ S (R).

Next, we discuss each step of Eq. (A9). The first equality follows from the resolution of the identity in terms of coherent states:

Tr D = Tr C N Dα |α, N α, N | • D = C N Dα α, N | D|α, N (A11)
for any operator D. The second equality in Eq. (A9) comprises several steps. First, we note that lim N →∞ C N /N 2j = 1. Second, we apply Eq. (A3) to the integrand:

lim N →∞ α, N | e iζ ĥ |α, N = e iζh mf (α) (A12)
Third and last, we argue by Lebesgue's dominated convergence theorem that we can interchange the operation of taking the N → ∞ limit with the integration. To check the assumptions of the theorem it is helpful to note that the domain of integration is compact, α, N | e iζ ĥ |α, N is a continuous function of α, and that | α, N | e iζ ĥ |α, N | ≤ e iζ ĥ N ≤ 1 ∀N . The last step of Eq. (A9) is, essentially, a change of variables. Some caution is needed at values of α where the gradient of h mf (α) vanishes. For measurable sets of α with h mf (α) = c the equality can be proven directly. Measure-zero sets with ∇h mf (α) = 0, e. g., isolated stationary points of h mf , can be excluded from the integration.

Finally, to arrive at Eq. (A10) we demonstrate that, for any sequence of tempered distributions f N ∈ S (R), lim

N →∞ f N = f ⇔ lim N →∞ F[f N ] = F[f ]. (A13)
Since the f N are distributions, we can demand convergence only in the following weak sense: lim

N →∞ f N = f ⇔ lim N →∞ dx f N (x)t(x) = dx f (x)t(x) (A14)
for all test functions t ∈ S(R). Similarly, the Fourier transform of any g ∈ S (R) is defined by

dx F[g](x)t(x) ≡ dx g(x)F[t](x) ∀t ∈ S(R). (A15)
Since the Fourier transformation is an automorphism on S(R), we can replace the arbitrary test function t(x) in Eq. (A14) by its Fourier transform F[t](x). This connects Eq. (A14) with Eq. (A15) and yields Eq. (A13).

Equations of motion

We consider the Heisenberg representation ÂH (t) ≡ e i Ĥt/  e -i Ĥt/ of an operator  ∈ E. The Heisenberg equation of motion for α,

N | ÂH (t)|α, N reads d dt α, N | ÂH (t)|α, N = i α, N |[ Ĥ, ÂH (t)]|α, N . (A16)
This section contains two results. We demonstrate that lim

N →∞ α, N | ÂH (t)|α, N = A mf (α t ), (A17)
where α 0 ≡ α and α t consists of α m (t) ≡ n m (t) e iφm(t) with n m (t) ≥ 0, φ m (t) ∈ [0, 2π), and m n m (t) = 1. Furthermore, we prove that the dynamics of A mf (α t ) is governed by

d dt A mf (α t ) = i K mf (α t ) with K ≡ [ Ĥ, Â]. (A18)
To derive Eq. (A17), we recall that  is a polynomial in â † m âl /N with coefficients c k (N ). Let us define λ ml (t)

≡ lim N →∞ 1 N α, N |[â †
m âl ] H (t)|α, N . Then, according to Eq. (A6), lim N →∞ α, N | ÂH (t)|α, N is a polynomial in λ ml (t) with the respective coefficients lim N →∞ c k (N ). Next, we argue that λ ml (t) can be parametrized, without loss of generality, by n m (t)n l (t) e -i(φm(t)-φ l (t)) with n m (t) and φ m (t) as introduced above. At t = 0 this parametrization is obviously correct. For m = l, λ mm (t) = n m (t) is a valid parametrization because λ mm (t) ≥ 0 and m λ mm (t) = lim N →∞ 1 N α, N | NH (t)|α, N = 1. Employing, again, Eq. (A6), we find

λ ml λ lm = lim N →∞ 1 N 2 α, N |[â † m âl â † l âm ] H (t)|α, N = lim N →∞ 1 N 2 α, N |[â † m âm â † l âl + â † m âm ] H (t)|α, N = n m (t)n l (t). ( A19
)
Recall that q ∈ R, c < 0, D = N1 -N-1 is the magnetization, and [ ĥ, D] = 0. We are particularly interested in the case of zero magnetization. This has several reasons. First, the present work is motivated by the utility of ground-state quantum phase transitions (QPTs) in the magnetization-free subspace [START_REF] Pezzè | Heralded generation of macroscopic superposition states in a spinor Bose-Einstein condensate[END_REF][START_REF] Feldmann | Interferometric sensitivity and entanglement by scanning through quantum phase transitions in spinor bose-einstein condensates[END_REF]. Second, previous results on excited-state QPTs (ESQPTs) suggest [START_REF] Caprio | Excited state quantum phase transitions in many-body systems[END_REF] that the signatures should be most pronounced at zero magnetization. Third, the restriction to zero magnetization eases the computations. As discussed in Section A, we base our mean-field study on spin-1 projective coherent states. Most of these states are no eigenstates of D. It is therefore not obvious how to restrict the N -particle Hilbert space to the eigenspace of D with eigenvalue D = 0. In Section B 1 we demonstrate that the mean-field limit of the restricted density of states can be still expressed in terms of spin-1 coherent states.

For the mean-field dynamics of expectation values, we simply confine ourselves to coherent states |α, N with α, N | D|α, N = 0 or, equivalently, with |α 1 | 2 = |α -1 | 2 . Importantly, such states can be readily realized experimentally. To identify the mean-field Hamiltonian h mf (α) = lim N →∞ α, N | ĥ|α, N , we substitute the â( †) m by

√ N α ( * ) m
with α m ≡ √ n m e iφm and take the limit N → ∞, as explained in the introduction to Section A. We introduce

φ ≡ φ 0 -(φ 1 + φ -1
)/2 and ξ ≡ q 2|c| and, for n 1 = n -1 , obtain

h mf |c| = ξ(1 -2n 0 ) -2n 0 (1 -n 0 ) cos 2 φ, (B2)
cf. Eq. ( 2) from the main text. Applying Eq. (A18) to N0 /N , (â andâ † -1 â1 /N yields the equations of motion in Eq. ( 3) from the main text:

† 1 â † -1 â2 0 + â †2 0 â1 â-1 )/N 2 , â † 1 â-1 /N ,
d dτ n 0 = ∂ ∂φ h mf |c| , d dτ φ = - ∂ ∂n 0 h mf |c| , and d dτ (φ 1 -φ -1 ) = 0 (B3)
The first two equations of motion are Hamilton's equations for the Hamiltonian h mf and the canonical coordinates n 0 and φ. The corresponding phase space is a sphere with z-axis 0 ≤ n 0 ≤ 1 and azimuthal angle φ ∈ [0, 2π). Below, Section B 2 provides the classical phase-space trajectories. In Section B 3 we review the dynamics of n 0 .

In the main text, we have introduced an order parameter for the ESQPTs in ferromagnetic spin-1 BECs with zero magnetization and have proposed to reveal this order parameter by interferometry. In Section B 4 we supplement some mathematical details regarding our measurement prescription. Tr P0 e iζ ĥ = Dαδ(n 1 -n -1 ) e iζh mf (α) . (B6)

In the same way as in Section A 1, this implies lim

N →∞ ν 0 ( ) N = Dαδ(n 1 -n -1 ) δ(h mf (α) -) (B7)
for the restricted DOS in the mean-field limit. Expressing ν 0 as a function of η = /|c| yields Eq. ( 4) from the main text.

In the following, we prove that lim

N →∞ 1 N Tr P0 f ( Â) = Dαδ(n 1 -n -1 ) f (A mf (α)) ∀f ∈ C(A), Â ∈ A, (B8)
where, as before, Dα ≡ 1 (2π) 3 m dn m dφ m δ( m n m -1) and C(A) denotes the complex analytic functions on A. Equation (B6) immediately follows as a special case. First, we assume that lim

N →∞ 1 N Tr P0 f ( Â) = m dn m dφ m µ(α) f (A mf (α)) (B9)
and determine µ(α). We observe that P0 is invariant under phase shifts: e -iθ Nm P0 e iθ Nm = P0 ⇒ Tr P0 f (e iθ Nm  e -iθ Nm ) = Tr P0 f ( Â) ∀θ, m (B10)

The mean-field limit of e iθ Nm  e -iθ Nm is A mf (α ) with φ m = φ m -θ and, apart from that, α coinciding with α.

Hence, µ(α) cannot depend on any of the φ m . For an arbitrary function f ∈ C(A) of ( N1 + N0 + N-1 )/N , lim

N →∞ 1 N Tr P0 f N1 + N0 + N-1 N = lim N →∞ 1 N N/2 k=0 f (1) = 1 2 f (1). (B11)
Similarly, for functions of D/N , lim

N →∞ 1 N Tr P0 f D N = 1 2 f (0). (B12)
Hence, µ(α) = μ(n 0 )δ( m n m -1)δ(n 1 -n -1 ). To determine μ(n 0 ), we consider functions of N0 /N : lim

N →∞ 1 N Tr P0 f N0 N = lim N →∞ 1 N N/2 k=0 f N -2k N = 1 2 dn 0 f (n 0 ). (B13)
This, finally, yields µ(α) = 1 (2π) 3 δ( m n m -1)δ(n 1 -n -1 ) as required for Eq. (B8). The proof of our assumption, Eq. (B9), relies on results from Ref. [START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF]. Recall that all  ∈ A and f ( Â) with f ∈ C(A) are endomorphisms on N -particle Hilbert spaces with arbitrary N . Considered as sequences in N , they belong to the set of approximately symmetric sequences Ỹ defined in Ref. [START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF]. On each N -particle Hilbert space, we introduce the state ρN ≡ 1 N/2 +1 P0 and the corresponding linear functional T N : Ỹ → C, T N ( Ŷ ) ≡ Tr ρN ŶN , where ŶN denotes the sequence elements of Ŷ . The T N constitute a sequence T N in a compact space [START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF]. The compactness has two consequences. First, T N has a convergent subsequence. Second, if all convergent subsequences of T N converge to the same T ∞ , so does the entire T N . According to Propositions III.3 and IV.5 in Ref. [START_REF] Raggio | Quantum statistical mechanics of general mean field systems[END_REF], the limit of any convergent subsequence of T N assumes the form

T ∞ ( Ŷ ) = m dn m dφ m µ(α) lim N →∞ α, N | Ŷ |α, N . (B14)
The arguments from the previous paragraph immediately yield µ

(α) = 2 (2π) 3 δ( m n m -1)δ(n 1 -n -1 )
, where the additional factor of 2 reflects the different normalization of ρN and 1 N P0 . Hence, T ∞ does not depend on the convergent subsequence under consideration and T N converges to T ∞ . Finally, we observe that lim

N →∞ 1 N Tr P0 f ( Â) = 1 2 T ∞ (f ( Â)). ( B15
)
Recalling that lim

N →∞ α, N |f ( Â)|α, N = f (A mf (α)
), see Eq. (A4), completes the proof.

Phase-space trajectories

An energy hypersurface at ξ and η consists of all phase-space points (n 0 , φ) which fulfill

η = h mf |c| = ξ(1 -2n 0 ) -2n 0 (1 -n 0 ) cos 2 φ. ( B16
)
For 0 < |ξ| < 1, Eq. (B16) can be rewritten as

n 0 (φ) =            m ± (φ) ∀ cos 2 φ ≥ η 2 -ξ 2 -η for η < η * m + (φ) ∀φ for ξ < 0 m -(φ) ∀φ for ξ > 0 for η > η * 1 2 1 -ξ |ξ| + ξ cos 2 φ ∀ cos 2 φ ≥ |ξ| for η = η * (B17) with m ± (φ) = 1 2 cos 2 φ (cos 2 φ + ξ ± √ ∆) and ∆ = cos 4 φ + 2η cos 2 φ + ξ 2 .
Recall that η 0 = -1 2 (ξ2 + 1), η 0 ≤ η ≤ |ξ|, and η * = -|ξ|.

A phase-space trajectory is the set of all points (n 0 , φ) which are connected by the Hamiltonian dynamics. Particularly, any closed line of constant η which does not pass through a stationary point of h mf is a trajectory. For each η > η * , the energy hypersurface is a closed line by itself, while for η 0 < η < η * each energy hypersurface comprises two disconnected closed lines. Since the stationary points of h mf are at η 0 and η * , we conclude that the phase-space trajectories for η / ∈ {η 0 , η * } are the connected components of the energy hypersurfaces (B17). This result can be extended to η 0 , where the energy hypersurface consists of two stationary points, each of which is its own trajectory. At η * , the energy hypersurface has the shape of an eight with the stationary point located at the intersection. There are, hence, three trajectories: the two wings of the eight excluding the stationary point, and the stationary point itself.

Phase-space trajectories are commonly assigned the direction in which they are traced by the evolution forward in time. This direction is determined by the equations of motion, see Eq. (B3).

Dynamics

The dynamics of n 0 is governed by

d dτ n 0 = ∂ ∂φ h mf |c| = 4n 0 (1 -n 0 ) cos φ sin φ. (B18)
We square Eq. (B18) and, exploiting the conservation of η, cf. Eq. (B16), obtain for ξ = 0

d dτ n 0 2 = 16ξ(n 0 -z 0 )(n 0 -z + )(n 0 -z -) (B19) with z 0 = 1 2 (1 -η/ξ) and z ± = 1 2 (1 + ξ ± 1 + ξ 2 + 2η).
Recall that ESQPTs at η * = -|ξ| and 0 < |ξ| < 1 divide the ξ-η-plane into three excited-state quantum phases: the TF phase for η > η * and ξ < 0, the P phase for η > η * and ξ > 0, and the BA phase for η < η * . In the TF and P phases z -≤ z 0 ≤ z + , while in the BA phase z 0 ≤ z -≤ z + for ξ < 0 and z -≤ z + ≤ z 0 for ξ > 0.

Let us introduce x 1 = 1 2 (1 + |ξ| -1 + ξ 2 + 2η) and Note that x 1 ≤ x 2 ≤ x 3 and that for ξ > 0 the x i coincide with the appropriately ordered zeroes z 0 and z ± . According to Refs. [START_REF] Kawaguchi | Spinor Bose-Einstein condensates[END_REF][START_REF] Zhang | Coherent spin mixing dynamics in a spin-1 atomic condensate[END_REF],

x 2 =
n 0 (τ ) = x 2 -(x 2 -x 1 ) cn 2 2 |ξ|(x 3 -x 1 )τ + u 0 , x 2 -x 1 x 3 -x 1 ≡ ñ0 (τ ) for ξ > 0, ( B21
)
where cn(u; k 2 ) is the Jacobi elliptic cosine and u 0 accounts for the initial conditions 2 . It can be easily verified that Eq. (B21) solves Eq. (B19). The dynamics for ξ < 0 and ξ > 0 are related by h mf (ξ, n 0 , φ) = h mf (-ξ, 1 -n 0 , φ).

Combining this with the equations of motion in Eq. (B3) and using the time-reversal symmetry of ñ0 (τ ) one can show that n 0 (τ ) = 1 -ñ0 (τ ) for ξ < 0. (B22)

The evolution n 0 (τ ) is periodic with period

T = 1 |ξ|(x 3 -x 1 ) K x 2 -x 1 x 3 -x 1 , (B23)
where

K(k 2 ) = π/2 0 dγ 1 -k 2 sin 2 γ -1
is the complete elliptic integral of the first kind. Plugging in the respective expressions for the x i yields T = √ y -1 K(x/y) for η < η * √ x -1 K(y/x) for η > η * (B24) with x = |ξ| 1 + ξ 2 + 2η and y = (x -ξ 2 -η)/2. To obtain Eq. ( 10) from the main text, recall that τ = |c|t/ . The periodicity diverges at the ESQPTs, as can be derived from lim k 2 1 K(k 2 ) = ∞.

Measuring the order parameter w

In the main text, we have introduced the order parameter w, which distinguishes between the TF (w = -1), the P (w = 1), and the BA (w = 0) phase. Our measurement prescription for p = cos(πw) is summarized in Eq. ( 7) from the main text. It relies on the mean-field dynamics at a given ξ and η. To evaluate the corresponding mean-field limit, we have employed Eq. (A17).

The visibility V = 2 1 -n 0 (0) n 0 (0), which depends on the initial condition n 0 (0), quantifies how well one can experimentally tell |w| = 1 from |w| = 0 and, thus, neighboring excited-state quantum phases from each other. To optimize V , n 0 (0) has to be chosen as close to 1/2 as is compatible with the periodic dynamics n 0 (τ ) at the ξ and η under consideration. We denote the optimal value of n 0 (0) by n opt .

According to Section B 3, n 0 (τ ) oscillates in the TF phase between the minimum value z 0 and the maximum value z + , in the P phase between the minimum z -and the maximum z 0 , and in the BA phase between z -and z + . We observe that z 0 ≤ 1/2 ⇔ η/ξ ≥ 0 and z 0 ≥ 1/2 ⇔ η/ξ ≤ 0. Furthermore, for ξ < 0 it is obvious that z -< 1/2, and for ξ > 0 that z + > 1/2. Finally, one can show for ξ > 0 that z -≤ 1/2 ⇔ η ≥ -1/2 and for ξ < 0 that z + ≥ 1/2 ⇔ η ≥ -1/2. Combining these findings yields Eq. ( 8) from the main text:

n opt =     

FIG. 1 .

 1 FIG. 1. Excited state quantum phases of a ferromagnetic spin-1 BEC with zero magnetization. (a) DOS in the mean-field limit as a function of ξ and η -η0 with η0(ξ) = -1 2 (ξ 2 + 1). The ESQPTs at η * = -|ξ| (black) divide the ξ-η-plane into three phases: the TF phase, the P phase, and the BA phase. The DOS diverges at the ESQPTs. The inset shows the DOS along lines of constant ξ = -0.2 (red, dashed) and η -η0 = 0.2 (orange, solid). The spectrum of a BEC of N = 100 atoms (gray, every third eigenvalue) exhibits avoided crossings at the ESQPTs. (b) Classical phase space and trajectories for ξ = 0.5. The separatrix (black) separates trajectories in the P phase (η > η * , green) with winding number w = 1 from trajectories in the BA phase (η < η * , yellow) with w = 0. Stationary points of h mf are marked in red.

(2π) 3 m

 3 dn m dφ m δ( m n m -1) and η denotes the energy divided by N |c|. Below we employ Eqs. (3) and (4) to study the signatures of ESQPTs.

FIG. 2 .

 2 FIG. 2. Measuring p = cos(πw) to distinguish adjacent excited-state quantum phases requires a large optimized visibility Vopt and a short periodicity T . (a) Vopt is large throughout the vast majority of the phase diagram. (b) T for |c|/ = 2π × 4 Hz. A moderate value of 0.3 s (gray) is surpassed only at the immediate vicinity of the ESQPTs. (a, b) Black lines mark the ESQPTs. The insets show Vopt and T along lines of constant ξ = -0.2 (red, dashed) and η -η0 = 0.2 (orange, solid).

FIG. 3 .

 3 FIG. 3. Simulated measurement of p for N = 100 atoms. The finite-size results closely resemble the mean-field limit, where p = 1 in the BA phase and p = -1 in the TF and P phases. Black lines mark the ESQPTs. The inset shows p along lines of constant ξ = -0.2 (red, dashed) and η -η0 = 0.2 (orange, solid). The shaded regions indicate the standard deviation.

  . the polynomial coefficients c k (N ) are time-independent 3. the N -dependence of the c k (N ) is such that for any k there are d k ∈ R and e k ∈ C with |c k (N )| ≤ d k ∀N and lim N →∞ c k (N ) = e k

1 (2π) 3 m

 13 2π), and m n m = 1. These states are separable and fulfill âm |α, N = √ N α m |α, N -1 and C N Dα |α, N α, N | = 1 N with Dα ≡ dn m dφ m δ( m n m -1) and C N = (N +2j)! N ! . 1 N denotes the identity operator on the N -particle Hilbert space. Let us now turn towards the mean-field limit N → ∞. To start with, we consider the coherent-state expectation value of  ∈ A. Let :  : denote the normal ordering of Â. Using that âm |α, N = √ N α m |α, N -1 and, for any finite k ∈ N 0 , lim N →∞ N -k N = 1, we obtain lim N →∞ α, N | :  : |α, N from :  : by substituting the âlimit N → ∞. Note that, since the α m commute, it does not matter whether we substitute the â( †) m in :  : or in Â. We denote the result by A mf (α). From the scaling of [â m / √ N , â † m / √ N ] = 1/N with N we can conclude that lim N →∞ α, N | Â-:  : |α, N = 0. Hence, lim N →∞ α, N | Â|α, N = A mf (α). For Â, B ∈ A, obviously, lim N →∞ α, N |  B|α, N = A mf (α)B mf (α). (A2) This is a central observation, which we can further generalize by means of Tannery's theorem, which we state below. Let us first show that lim N →∞ α, N | e z  |α, N = e zA mf (α) ∀z ∈ C. (A3) We already know that lim N →∞ α, N | Âk |α, N = A k mf (α) ∀k ∈ N 0 . Tannery's theorem ensures that we can pull the N → ∞ limit into the exponential series. Its assumptions are fulfilled since, by property № 6, there is some c ∈ R such that | α, N | Âk |α, N | ≤ c k ∀N , and since k |zc| k

  where r A and r B are the polynomial degrees of  and B, respectively. Finally, including a Ĉ ∈ A, we can show that lim N →∞ α, N | e zN  B Ĉ e -zN  |α, N = lim N →∞ α, N | e zN  B e -zN  |α, N lim N →∞ α, N | e zN  Ĉ e -zN  |α, N .

1 .

 1 Restricting the density of states The Fock basis of the N -particle Hilbert space consists of the joint eigenstates |N 1 , N 0 , N -1 of the Nm with eigenvalues N m and m N m = N . In this basis, the projection onto the eigenspace of D with eigenvalue D = 0 reads P0 = N/2 k=0 |k, N -2k, k k, N -2k, k|. (B4) We define the density of states (DOS) in the D = 0 subspace by F[ν 0 ](ζ) = d e iζ ν 0 ( ) ≡ Tr P0 e iζ ĥ, ζ ∈ R.

1 2 ( 1 + 1 2 ( 1 - 1 2 1 2 ( 1 +

 1111111 |ξ| + 1 + ξ 2 + 2η) for η ≤ η * η/|ξ|) for η > η * , x 3 = (1 -η/|ξ|) for η ≤ η * |ξ| + 1 + ξ 2 + 2η) for η > η * . (B20)

1 2 ( 1 +

 11 ξ -ξ |ξ| 1 + 2η + ξ 2 ) for η < -1

The phase θ rf is fundamentally fixed by the interaction of the atoms with the radio-frequency pulse. While we do not state its value, it may be both derived and measured.

Note that at (|ξ| = 1, η = -1) the denominator x

-x 1 vanishes. Hence, ñ0 (τ ) has to be computed by taking an appropriate limit.

Together with λ ml (t) = λ * lm (t) this entails |λ ml (t)| = |λ lm (t)| = n m (t)n l (t). The phases of all λ ml (t) with m = l can be deduced from the phases of λ 1l (t) by using the relations λ ml (t) = λ * lm (t) and λ ml (t)λ lk (t)λ km (t) = n m (t)n l (t)n k (t) ∈ R. The parametrization λ ml (t) = n m (t)n l (t) e -i(φm(t)-φ l (t)) reflects these relations without constraining the λ ml any further.

To obtain the mean-field equation of motion (A18), we take the N → ∞ limit of Eq. (A16). In any time interval [t 1 , t 2 ], Theorem 3 permits to interchange the limit with the time derivative because the right-hand side (RHS) of Eq. (A16) is continuous in t and uniformly converges for N → ∞. Let us prove the uniform convergence. We know from Eq. (A5) that the pointwise limit of

It is sufficient to show that the RHS of

uniformly converges to zero as N → ∞. Similarly to the proof of Eq. (A5), we can find some c, c ∈ R such that

Hence, we can apply Tannery's theorem, which yields that the RHS of Eq. (A22) converges to zero pointwise. The RHS of Eq. (A22) is a strictly increasing function of |t|. Let us assume, without loss of generality, that

Then, for any N , the RHS of Eq. ( A22) is absolutely bounded by its value at t 2 and the pointwise convergence to zero in t 2 implies uniform convergence.

Mathematical supplement

For completeness, we state here some well-known theorems which we have used above:

Theorem 1 (Tannery [START_REF] Loya | Amazing and aesthetic aspects of analysis[END_REF]). Consider the sequence a k (n) ∈ C with k ∈ N 0 , n ∈ N and assume that for any k there are

Theorem 2 (Lebesgue [START_REF] Königsberger | Analysis[END_REF]). Let f n : U ⊂ R d → C, n ∈ N be Lebesgue integrable functions which, for n → ∞, converge pointwise to a function f and are dominated by some Lebesgue integrable function g, i. e., |f n (x)| ≤ g(x) ∀n ∈ N, x ∈ U . Then f is integrable and In this section we focus on a ferromagnetic spin-1 Bose-Einstein condensate (BEC), which we model by the Hamiltonian density in Eq. ( 1) from the main text, ĥ = q N


Amel Ben Othmane

Andrea Tettamanzi
email: andrea.tettamanzi@unice.fr

Serena Villata
email: villata@i3s.unice.fr

Nhan Le Thanh

Michel Buffa

A Multi-context Framework for Modeling an Agent-based Recommender System

Keywords: multi-agent systems, BDI architecture, multi-context systems, possibility theory, ontology, recommender systems

In this paper, we propose a multi-agent recommender system based on the Belief-Desire-Intention (BDI) model applied to multi-context systems. First, we extend the BDI model with additional contexts to deal with sociality and information uncertainty. Second, we propose an ontological representation of planning and intention contexts in order to reason about plans and intentions. Moreover, we show a simple real-world scenario in healthcare in order to illustrate the overall reasoning process of our model.

INTRODUCTION

Human activities take place in particular locations at specific times. The increasing use of wearable devices enables the collection of information about these activities from a diverse population varying in physical, cultural, and socioeconomic characteristics. Generally, the places you have been and spent regularly or occasionally time on, reflect your lifestyle, which is strongly associated to your socioeconomic features. This amount of information about people, their relations, and their activities are valuable elements to personalize healthcare being sensitive to medical, social, and personal characteristics of individuals. Besides, the decision-making process in human beings uses not only logical elements, but also emotional components that are typically extra-logical. As a result, behavior can also be explained by other approaches, which additionally consider emotions, intentions, beliefs, motives, cultural and social constraints, impulsive actions, and even the simple willingness to try. Hence, building recommender systems that take user behavior into account requires a step toward personalization.

To the best of our knowledge, there are no recommender systems that combine all these features at the same time. The following is a motivating example that had driven this research. Bob, a 40 year-old adult, wants to get back to a regular physical activity (pa).

Bob believes that a regular physical activity reduces the risk of developing a non-insulin dependant diabetes mellitus (rd). Mechanisms that are responsible for this are weight reduction (wr), increased insulin sensitivity, and improved glucose metabolism. Due to his busy schedule (bs), Bob is available only on weekends (av). Hence, he would be happy if he can do his exercises only on weekends (w). Bob prefers also not to change his eating habits (eh). Besides all the aforementioned preferences, Bob should take into account his medical concerns (c) and certainly refer to a healthcare provider for monitoring. This scenario exposes the following problem: how can we help Bob to select the best plan to achieve his goal based on his current preferences and restrictions? This problem raises differents challenges. First, the proposed solution should take into account Bob's preferences and restrictions (e.g. medical and physical concerns) in the recommendation process. Second, information about the environment in which Bob acts and people that might be in relationship with him may have impact in his decision-making process. Third, the system should be able to keep a trace of Bob's activities in order to adapt the recommendation according to his progress. Finally, the information or data about Bob's activities is distributed geographically and temporarily.

In order to address these challenges, multi-agent systems stand as a promising way to understand, manage and use distributed, large-scale, dynamic, and heterogeneous information. The idea is to develop recommender systems to help users confronted with situations in which they have too many options to choose from with the aim of assisting them to explore and to filter out their preferences from a number of different possibilities. Based on this real-world application scenario, we propose in this paper a multiagent-based recommender system where agents are described using the BDI model as a multi-context system. The system's goal is to recommend a list of activities according to user preferences. We propose also an extension of the BDI model to deal with sociality and uncertainty in dynamic environments.

The originality of what we are proposing with respect to existing works is the combination of an extended possibilistic BDI approach with multi-context systems. The resulting framework is then used as a healthcare recommender system. There are several advantages of such combination. First, the use of a multi-context architecture allows us to have different syntaxes, e.g. ontology to represent and reason about plans and intentions. Besides, we believe that extending the classical BDI model with goals and social contexts better reflects humans behavior. The proposed approach deals with goal-belief consistency and proposes also a a belief revision process. The idea of extending the BDI model with social contexts is not novel. Different works explored trust or reputation [START_REF] Koster | Opening the black box of trust: reasoning about trust models in a bdi agent[END_REF][START_REF] Pinyol | Reputation-based decisions for logic-based cognitive agents[END_REF] while in our approach we consider trust measures between two agents only if they are similar. The rest of this paper is organized as follows. Section 2 includes a literature overview on the related work. In Section 3 we summarize the main concepts on which is based this work. We introduce after, in Section 4, the multi-context BDI agent framework. In order to give a view of how the model works, we show in Section 5 a real-world scenario in healthcare domain. Conclusions end the paper.

RELATED WORK

Recommender systems (RS) are information-filtering systems that help users to deal with the problem of information overload by recommending only relevant items. [START_REF] Bobadilla | Recommender systems survey[END_REF] undertook a literature review and classification of recommender systems. They came up with the conclusion that approaches reviewed still require further improvements to make recommendation methods more effective in a broader range of applications. For example, in order to exploit information coming from various sensors and devices on the Internet of things and the acquisition and integration of trends related to the habits, consumption and tastes of individual users in the recommendation process. The main recommendation algorithms can be divided into four categories: contentbased (CB), collaborative filtering (CF), Knowledgebased (KB) and hybrid recommendation (HR). The CB method recommends objects that are similar to the ones the user showed to prefer in the past. However, this method has a tendency to produce recommendations with a limited degree of novelty (serendipity). CF has been the most successful recommendation system technology. In CF, we make recommendations according to the assumption that users who share similar preferences choose similar items. However, the performance of CF is significantly limited by data sparsity. Knowledge-based recommender approaches [START_REF] Trewin | Knowledge-based recommender systems[END_REF] appear to be more promising to tackle those challenges by exploiting explicit user requirements and specific domain knowledge. There are two approaches to knowledge-based recommendation: case-based [START_REF] Bridge | Case-based recommender systems[END_REF] and constraint-based recommendation [START_REF] Felfernig | Constraint-based recommender systems[END_REF]. Case-based recommenders determine recommendations on the basis of similarity metrics while constraint-based recommenders exploit a predifined knowledge base that contains explicit rules to rely user requirements with item features. Finally, HR is currently the most popular approach. As its name suggests, it combines at least two recommendation algorithms to determine a recommendation. New trends of recommender system appeared with the abundance of smart devices like smartphones. This trend of recommender systems is called context-aware RS. Context-aware recommender systems [START_REF] Adomavicius | Context-aware recommender systems[END_REF] focus on additional contextual information, such as time, location, and wireless sensor networks [START_REF] Gavalas | A web-based pervasive recommendation system for mobile tourist guides[END_REF]. However, those traditionnal recommendation approaches are not well-suited for the recommendation of complex products and services. Despite all these advances, the current generation of recommender systems still requires further improvements to make recommendation methods more effective and applicable to broader range of real-life applications, including recommending holidays plans (flight, places, and accommodation), certain types of financial services to investors, and workout plans for healthcare purpose. These improvements include better methods for representing user behavior and the information about the items to be recommended, more advanced recommendation modeling methods, incorporation of contextual information into the recommendation process, utilization of multi-criteria rat-ings, development of less intrusive and more flexible recommendation methods that also reason beyond user preferences. A solution to this research problem can be provided by the use of recommender agents, which are agentbased recommender systems that take into account users preferences to generate relevant recommendations. Agents are also well suited for autonomous applications operating in a dynamic environment. Building systems based on agents gives a more natural way to simulate complex real-world situations [START_REF] Jennings | On agent-based software engineering[END_REF]. A complete taxonomy of this kind of recommendation systems can be found in [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions[END_REF]. They are widely used in the tourism domain as [START_REF] Casali | A graded bdi agent model to represent and reason about preferences[END_REF][START_REF] Batet | Turist@: Agent-based personalised recommendation of tourist activities[END_REF][START_REF] Gavalas | Mobile recommender systems in tourism[END_REF], healthcare applications [START_REF] Amir | Collaborative health care plan support[END_REF], and traffic ones aiming at improving the travel efficiency and mobility [START_REF] Chen | A review of the applications of agent technology in traffic and transportation systems[END_REF].

BACKGROUND

In this section, we summarize the main insights and notions, which the present contribution is based on. An agent in a BDI architecture is defined by its beliefs, desires and intentions. Beliefs encode the agent's understanding of the environment, desires are those states of affairs that an agent would like to accomplish and intentions those desires that the agent has chosen to act upon. Many approaches tried to formalize such mental attitudes (e.g. [START_REF] Cohen | Intention is choice with commitment[END_REF], [START_REF] Rao | Bdi agents: From theory to practice[END_REF], [START_REF] Wooldridge | The gaia methodology for agent-oriented analysis and design[END_REF] and [START_REF] Singh | Semantical considerations on intention dynamics for bdi agents[END_REF]). However, all these works concentrated on the human decision-making process as a single approach without considering social influences. They did not take the gradual nature of beliefs, desires, and intentions into account. Incorporating uncertainty and different degrees of attitudes will help the agent in the decision-making process. In order to represent and reason about uncertainty and graded notions of beliefs, desires, and intentions, we follow the approach proposed by (da Costa Pereira and Tettamanzi, 2010) where uncertainty reasoning is dealt with by possibility theory. Possibility theory is an uncertainty theory dedicated to handle incomplete information. It was introduced by [START_REF] Negoita | Fuzzy sets as a basis for a theory of possibility[END_REF] as an extension to fuzzy sets which are sets that have degrees of membership in [0, 1]. Possibility theory differs from probability theory by the use of dual set functions (possibility and necessity measures) instead of only one. Possibility distribution assigns to each element ω in a set Ω of interpretations a degree of possibility π(ω) ∈ [0, 1] of being the right description of a state of affairs. It represents a flexible restriction on what is the actual state with the following conventions:

• π(ω) = 0 means that state ω is rejected as impossible;

• π(ω) = 1 means that state ω is totally possible (plausible).

While we chose to adopt a possibilistic BDI model to include gradual mental attitudes, unlike (da Costa Pereira and Tettamanzi, 2010), to represent our BDI agents we use multi-context systems (MCS) [START_REF] Parsons | Agent specification using multi-context systems[END_REF]. According to this approach, a BDI model is defined as a group of interconnected units {C i }, i ∈ I, ∆ br , where:

• For each i ∈ I, C i = L i , A i , ∆
i is an axiomatic formal system where L i , A i and ∆ i are the language, axioms, and inference rules respectively. They define the logic for context C i whose basic behavior is constrained by the axioms.

• ∆ br is a set of bridge rules; i.e. rules of inference, which relate formulas in different units.

The way we use these components to model BDI agents is to have separate units for belief B, desires D and intentions I, each with their own logic. The theories in each unit encode the beliefs, desires, and intentions of specific agents and the bridge rules (∆ br ) encode the relationships between beliefs, desires and intentions. We also have two functional units C and P, which handle communication among agents and allow to choose plans that satisfy users desires. To summarize, using the multi-context approach, a BDI model is defined as follows:

Ag = ({BC,

THE MULTI-CONTEXT BDI FRAMEWORK

The BDI agent architecture we are proposing in this paper extends Rao and Georgeffs well-known BDI architecture [START_REF] Rao | Bdi agents: From theory to practice[END_REF]. We define a BDI agent as a multi-context system being inspired by the work of [START_REF] Parsons | Agent specification using multi-context systems[END_REF]. Following this approach, our BDI agent model visualized in Figure 1 is defined as follows:

Ag = ({BC, DC, GC, SC, PC, IC,CC}, ∆ br )
where GC and SC represent respectively the Goal Context and the Social Context.

In order to reason about beliefs, desires, goals and social contexts we follow the approach developed by (da Costa Pereira and Tettamanzi, 2010; da Costa Pereira and Tettamanzi, 2014) where they adopt a classical propositional language for representation and possibility theory to deal with uncertainty.

Let A be a finite set of atomic propositions and L be the propositional language such that A ∪ { , ⊥} ⊆ L and ∀φ, ψ ∈ L,¬φ ∈ L,φ ∨ ψ ∈ L,φ ∧ ψ ∈ L. These propositions can contain temporal elements that are left as future work. As in (da Costa Pereira and Tettamanzi, 2010), L is extended and we will denote with Ω = {0, 1} A the set of all interpretations on A. An interpretation ω ∈ Ω is a function ω : A → {0, 1} assigning a truth value p ω to every atomic proposition p ∈ A and, by extension, a truth value φ ω to all formula φ ∈ L.

[φ] denotes the set of all interpretations satisfying φ. (i.e., [φ] = ω ∈ Ω : ω φ).

In the planning and intentions contexts, we propose an ontological representation for plans and intentions in order to offer to agents a computer-interpretable description of the services they offer and the information they have access to (workout plans in our case). In the following subsections, we will outline the different theories defined for each context in order to complete the specification of our multi-context agent model.

Belief Context

The BC language and semantics

In order to represent beliefs, we use the classical propositional language with additional connectives, following (da Costa Pereira and Tettamanzi, 2010). We introduce also a fuzzy operator B over this logic to represent agents beliefs. The belief of an agent is then represented as a possibility distribution π. A possibility distribution π can represent a complete preorder on the set of possible interpretations ω ∈ Ω. This is the reason why, intuitively, at a semantic level, a possibility distribution can represent the available knowledge (or beliefs) of an agent. When representing knowledge, π(ω) acts as a restriction on possible interpretations and represents the degree of compatibility of interpretation ω with the available knowledge about the real world. As in (da Costa Pereira and Tettamanzi, 2010), a graded belief is regarded as a necessity degree induced by a normalized possibility distribution π on the possible worlds ω. The degree to which an agent believes that a formula Φ is true is given by:

B(φ) = N([φ]) = 1 -max ω φ {π(ω)} (1)
(ω) =    π(ω) Π([φ]) if ω φ and B(¬φ) < 1; 1 if ω φ and B(¬φ) = 1; min{π(ω), (1 -τ)} if ω φ.
(2) where τ is the trust degree toward a source about an incoming information φ.

BC Axioms and Rules

Belief context axioms include all axioms from classical propositional logics with weight 1 as in [START_REF] Dubois | Possibility theory and its applications: a retrospective and prospective view[END_REF]. Since a belief is defined as a necessity measure, all the properties of necessity measures are applicable in this context. Hence, the belief modality in our approach is taken to satisfy these properties that can be regarded as axioms. The following axiom is then added to the belief unit:

BC : B(φ) > 0 → B(¬φ) = 0
This axiom is a straightforward consequence of the properties of possibility and necessity measures. It means that if an agent believes φ to a degree then it cannot believe ¬φ at all. Other consequences are:

B(φ ∧ ψ) ≡ min{B(φ), B(ψ)} B(φ ∨ ψ) ≥ max{B(φ), B(ψ)} The inference rules are: • B(¬p ∨ q) ≥ α, B(p) ≥ β B(q) ≥ min(α, β) (modus ponens) • β ≤ α, B(p) ≥ α B(p) ≥ β(weight weakening)
where denote the syntactic inference of possibilistic logic.

Desire Context

Desires represent a BDI agent's motivational state regardless its perception of the environment. Desires may not always be consistent. For example, an agent may desire to be healthy, but also to smoke; the two desires may lead to a contradiction. Furthermore, an agent may have unrealizable desires; that is, desires that conflict with what it believes possible.

The DC language and semantics

In this context, we make a difference between desires and goals. Desires are used to generate a list of coherent goals regardless to the agent's perception of the environment and its beliefs. Inspired from (da Costa Pereira and Tettamanzi, 2014), the language of DC (L DC ) is defined as an extension of a classical propositional language. We define a fuzzy operator D + , which is associated with a satisfaction degree (D + (φ) means that the agent positively desires φ) in contrast with a negative desire, which reflects what is rejected as unsatisfactory. For sake of simplicity, we will only consider the positive side of desires in this work and the introduction of negative desire is left as future work. In this theory, (da Costa Pereira and Tettamanzi, 2010) use possibility measures to express the degree of positive desires. Let u(ω) be a possibility distribution called also qualitative utility (e.g. u(ω) = 1 , means that ω is fully satisfactory). Given a qualitative utility assignment u (formally a possibility distribution), the degree to which the agent desires φ ∈ L DC is given by:

D(φ) = ∆([φ]) = min ω φ {u(ω)} (3)
where ∆ is a guaranteed possibility measure that, given a possibility distribution π, is defined as follows, for all set S ⊆ Ω:

∆(S) = min ω∈S {π(ω)}. (4)

DC Axioms and Rules

The axioms consist of all properties of possibility measures such as D(φ ∨ ψ) ≡ min{D(φ), D(ψ)}. The basic inference rules, in the propositional case, associated with ∆ are:

• [D(¬p ∧ q) ≥ α], [D(p ∧ r) ≥ β] [D(q ∧ r) ≥ min(α, β)](resolution rule) • if p entails q classically, [D(p) ≥ α] [D(q) ≥ α](formula weakening) • for β ≤ α, [D(p) ≥ α] [D(p) ≥ β] (weight weak- ening) • [D(p) ≥ α]; [D(p) ≥ β] [D(p) ≥ max(α, β)]
(weight fusion).

Goal Context

Goals are sets of desires that, besides being logically "consistent", are also maximally desirable, i.e., maximally justified. Even though an agent may choose some of its goals among its desires, nonetheless there may be desires that are not necessarily goals. The desires that are also goals represent those states of the world that the agent might be expected to bring about precisely because they reflect what the agent wishes to achieve. In this case, the agent's selection of goals among its desires is constrained by three conditions. First, since goals must be consistent and desires may be inconsistent, only the subsets of consistent desires can be the potential candidates for being promoted to goal-status, and also the selected subsets of consistent desires must be consistent with each other. Second, since desires may be unrealizable whereas goals must be consistent with beliefs (justified desires), only a set of feasible (and consistent) desires can be potentially transformed into goals. Third, desires that might be potential candidates to be goals should be desired at least to a degree α. Then only the most desirable, consistent, and possible desires can be considered as goals.

Example: Let us consider one agent representing Alice. Alice believes that her usual road to work is congested and that there are other alternative routes that she probably did not know. She would like to be at her office at time without leaving earlier. She also prefers a route without stops. Some of Alice desires can not be moved to goal status such as desiring a route without stops because Alice's agent does not believe that a route without stops is possible.

The GC language and semantics

The language L GC to represent the Goal context is defined over the propositional language L extended by a fuzzy operator G having the same syntactic restrictions as D + . G(φ) means that the agent has goal φ. As explained above, goals are a subset of consistent and possible desires. Desires are adopted as goals because they are justified and achievable. A desire is justified because the world is in a particular state that warrants its adoption. For example, one might desire to go for a walk because he believes it is a sunny day and may drop that desire if it starts raining. A desire is achievable, on the other hand, if the agent has a plan that allows it to achieve that desire.

GC Axioms and Rules

Unlike desires, goals should be consistent, meaning that they can be expressed by the D G axiom (D from the KD45 axioms [START_REF] Rao | Bdi agents: From theory to practice[END_REF]) as follows:

D G GC : G(φ) > 0 → G(¬φ) = 0
Furthermore, since goals are a set of desires ,we use the same axioms and deduction rules as in DC.

Goals-beliefs and goals-desires consistency will be expressed with bridge rules as we will discuss later on the paper.

Social Context

One of the benefits of the BDI model is to consider the mental attitude in the decision-making process, which makes it a more realistic than a purely logical model. However, this architecture overlooks an important factor that influences this attitude, namely the sociality of an agent. There are a number of ways in which agents can influence one another's mental states such as authority where an agent may be influenced by another to adopt a mental attitude whenever the latter has the power to guide the behavior of the former, trust where an agent may be influenced by another to adopt a mental attitude merely on the strength of its confidence in the latter or persuasion where an agent may be influenced to adopt another agents mental state via a process of argumentation or negotiation.

In this work we will only consider trust as a way by which agents can influence each others.

The SC language and semantics

In our model, we consider a multi-agent system MAS consisting of a set of N agents {a 1 , .., a i , ..a N }. The idea is that those agents are connected in a social network such as agents with the same goal. Each agent has links to a number of other agents (neighbors) that change over time. In this paper, we do not consider dynamic changes in the social network, but we assume to deal with the network in a specific time instant. Between neighbors, we consider a trust relationship. The trustworthiness of an agent a i toward an agent a j about an information φ is interpreted as a necessity measure τ ∈ [0, 1] as in [START_REF] Paglieri | Trusting the messenger because of the message: feedback dynamics from information quality to source evaluation[END_REF] and is expressed by the following equation:

T a i ,a j (φ) = τ (5)
where a i , a j ∈ MAS = {a 1 , .., a i , .., a N }. Trust is transitive in our model, which means that, trust is not only considered between agents having a direct link to each others but, as showed in Figure 2, indirect links are also considered. Namely if agent a i trusts agent a k to a degree τ 1 which trusts agent a j with a trust degree τ 2 then a i can infer its trust to agent a j and T a i ,a j (φ) = min{τ 1 , τ 2 }. We only consider first and second order neighbors in our work e.g. agent a i can be influenced by agent a k and agent a j .

SC Axioms and Rules

As sociality is expressed as a trust measure, which is interpreted as a necessity measure, SC axioms include properties of necessity measures as in BC (e.g.

N(φ ∧ ψ) ≡ min{N(φ), N(ψ)}).
When an agent is socially influenced to change its mental attitude, by adopting a set of beliefs and/or desires, the latter should maintain a degree of consistency. Those rules will be expressed with bridge rules that link the social context to the belief and the desire contexts.

Planning and Intention Contexts

The aim of this functional context is to extend the BDI architecture in order to represent plans available to agents and provide a way to reason over them. In this context, we were inspired from [START_REF] Batet | Turist@: Agent-based personalised recommendation of tourist activities[END_REF] to represent and reason about plans and intentions. Plans are described using ontologies. [START_REF] Gruber | What is an ontology[END_REF] defines an ontology as 'the specification of conceptualizations, used to help programs and humans share knowledge'. According to the World Wide Web Consortium 1 (W3C), ontologies or vocabularies define the concepts and relationships used to describe and represent an area of concern. We use the 5W 2 (Who, What, Where, When, Why) vocabulary which is relevant for describing different concepts and constraints in our scenario. The main concepts and relationships of this ontology are illustrated by Figure 3.

The main task of this context is to select plans that satisfy maximally the agents goals. To go from the abstract notions of desires and beliefs to the more concrete concepts of goals and plans, as illustrated by Figure 4, the following steps are considered: (1) new information arrives and updates beliefs or/and desires which trigger goals update; (2) these goal changes invoke the Plan Library.

The selection process is exressed by Algorithm 1

1 http://www.w3.org/standards/semanticweb/ontology 2 http://ns.inria.fr/huto/5w/ which looks in a knowledge base (KB) for all plans that satisfy maximally these goals; CB and/or CF techniques can be used in the selection process but will be investigated more thoroughly in further work. The algorithm complexity is significantly reduced since we discard from the beginning goals without plans.

(3) one or more of these plans are then chosen and moved to the intention structure; finally (4) a task (intention) is selected for execution and once ex-

Data: G Result: S //S is a list of plans G * = {φ 1 , φ 2 , .., φ n } m ← 0; S ← / 0; G ← / 0; for each φ i in G * do //Search in the KB for a plan satisfying φ i S φ i ← SearchInKB(φ i ); if S φ i <> / 0 then //Discard goals without plans Append(G , S φ i ); end end for i in 1..Lenght(G ) do //Combination of i elements in G' S ← Combination(G , i); for j in 1..Length(S ) do if S [ j] <> / 0 then //Compute the satisfaction degree of S α i = G(S [ j]); //Select the maximum α i if α i > m then m ← α i ; Initialize(S); Append(S, S'); else if α i = m then Append(S,S'); end end end end end Return S;
Algorithm 1: RequestForPlan Function ecuted or failed this lead to the update of the agents beliefs (5).

Example (continued): Suppose that Alice's agent accepts to change its belief regarding 'the route without stops'. Then this desire will become a goal. So Alice's agent will look in its KB for alternatives routes without stops. Suppose that Algorithm 1 returns two plans (routes) A and B. If Alice chooses Route A then the first action a 1 ="Take Route Dolines" becomes the agent intention. Once done, Alice's agent updates its beliefs with information that a 1 is completed successfully.

Bridge Rules

There are a number of relationships between contexts that are captured by so-called bridge rules. A bridge rule is of the form: u1 : φ, u2 : ψ → u3 : θ and it can be read as: if the formula φ can be deduced in context u1 and ψ in u2 then the formula θ is to be added to the theory of context u3. A bridge rule allows to relate formulae in one context to those in another one. In this section we present the most relevant rules illustrated by numbers in Figure 1.∀a i ∈ MAS, the first rule relating goals to beliefs can be expressed as follows:

(1) GC : G(a i , φ) > 0 → BC : B(a i , ¬φ) = 0 which means that if agent a i adopt a goal φ with a satisfaction degree equal to β φ then φ is believed possible to a degree β φ by a i . Concerning rule (2) relating goal context to desire context, if φ is adopted as goal then it is positively desired with the same satisfaction degree.

(

) GC : G(a i , φ) = δ φ → DC : D + (a i , φ) = δ φ 2
An agent may be influenced to adopt new beliefs or desires. Beliefs coming from other agents are not necessarily consistent with agent's individual beliefs. This can be expressed by the following rule:

(3) BC : B(a j , φ) = β φ , SC :

T a i ,a j (φ) = t → BC : B(a i , φ) = β φ
where β φ is calculated using Equation 1 with τ = min{β φ ,t} to compute the possibility distribution and Equation 1 to deduce the Belief degree.

Similarly to beliefs, desires coming from other agents need not to be consistent with agent's individual desires. For example, an agent may be influenced by another agent to adopt the desire to smoke, and at the same time having the desire to be healthy as shown by the following rule:

(4) DC : D + (a j , ψ) = δ ψ , SC :

T a i ,a j (ψ) = τ → DC : D + (a i , ψ) = δ ψ
where δ ψ = min{δ ψ , τ}. Desire-generation rules can be expressed by the following rule:

(5) BC : min{B(φ 1 ) ∧ ... ∧ B(φ n )} = β, DC :

min{D + (ψ 1 ) ∧ ... ∧ D + (ψ n ))} = δ → DC : D + (Ψ) ≥ min{β, δ}
Namely, if an agent has the beliefs B(φ 1 ) ∧ ... ∧ B(φ n ) with a degree β and positively desires D + (ψ 1 ) ∧ ... ∧ D + (ψ n ) to a degree δ, then it positively desires Ψ to a degree greater or equal to min{β, δ}.

According to (da Costa Pereira and Tettamanzi, 2014), goals are a set of desires that, besides being logically 'consistent", are also maximally desirable, i.e., maximally justified and possible. This is expressed by the following bridge rule: where χ(φ, ψ) = ElectGoal(φ, ψ), as specified in Algorithm 2, is a function that allows to elect the most desirable and possible desires as goals. If ElectGoal returns / 0 then G( / 0) = 0, i.e. no goal is elected. As expressed by the bridge rule above, once goals are generated, our agent will look for plans satisfying goal φ by applying RequestForPlan function and intend to do the first action of the recommended plan. (7) GC : G(a i , φ) = δ, PC : RequestForPlan(φ) → IC : I(act i , PostConditon(act i )) where RequestForPlan is a function that looks for plans satisfying goal φ in the plan library as specified in Algorithm 1. Rule (8) means that if an agent has the intention of doing an action act i with PostCondition(act i ) then it passes this information to the communication unit and via it to other agents and to the user.

(8) IC : I(act i , PostConditon(act i )) → CC : C(does(act i , PostConditon(act i )))

If the communication unit obtains some information that some action has been completed then the agent adds it to its beliefs set using rule (3) with B(PostConditon(act i )) = 1.

A Running Example

To illustrate the reasoning process of our BDI architecture, we use the example introduced in Introduction and illustrated by Figure 5. The belief set is represented by formulae describing the world (e.g. B(ψ 1 ) = 1, means that ψ 1 is necessary and totally possible). Desires are all possible states that the agent wishes to achieve. Notice that they can be conflicting like D + (wr) and D + (¬eh) or unachievable like D + (wr ∧¬eh). D + (wr) = 0.8, means that wr is desired to a degree equal to 0.8. Desire-generation rules from bridge rule (5) can be described as follows:

R Then the desire base of Bob, derived from desiregeneration rules will be as follows: D = {(pa, 0.8), (wr, 0.8), (w, 0.75), (¬eh, 0.9)}

We may now apply rule (6) to elect Bob's goals, given his belief base and his desire base. This rule will apply the function electGoal() which will choose from the desire base the most desirable and possible desires. Then, Img(B) = {0.75, 0.8, 0.9, 0.4} and Img(D) = {0.75, 0.8, 0.9}. We begin by calling Algorithm 2 with γ = 0; δ is set to maxImg(D) = 0.9 and the corresponding desire in D is D δ = {¬eh}. Now if we verify B(¬(¬eh)) = 0.4 > γ we move to the next less desired value which sets δ to Img(D) = 0.8 < δ = 0.9. δ = 0.8 > 0, then we go back to Step 2. In this case D δ = {(pa, wr}. Now B(¬pa) = B(pa) = 0 because we ignore yet whether pa is possible or nor. Similarly, B(¬wr) = 0 and Algorithm 2 will terminate with G * = G γ = {pa, wr}, i.e. Bob's recommender agent will elect as goal 'get back to a regular physical activity and reduce weight'. Given these goals, Bob's agent (a 1 ) will look in the plan library for a plan satisfying them. As explained in rule ( 7), the agent will invoke function Request-ForPlan, which will look for a plan satisfying pa and wr. Applying Algoritm 1, we have G = {pa, wr} and S = [pa, wr, {pa, wr}] with the same satisfaction degree α 1 = α 2 = α 3 = 0.8. Suppose that it returns three plans p 1 , p 2 and p 3 satisfying respectively goals pa, wr and {pa, wr}. Bob's recommender agent will propose plan p 3 to the user because it meets more Bob's requirements with the same satisfaction degree. We suppose that Bob chooses Plan p 3 . Therefore, the first action (activity) in Plan p 3 will become the agent's intention. The intended action will be proposed to the user via the communication unit by applying rule (8). Finally, if Bob starts executing the activity, information such as speed, distance or heart rate are collected via sensors (i.e. smart watch) and transmitted to the communication unit in order to update the agent's beliefs. The revision mechanism of beliefs is the same as in (da Costa Pereira and Tettamanzi, 2010) defined by Equation 2. Once the activity is completed, rule(3) is triggered in order to update the belief set of Bob's agent with B(postCondition(action1) = 1) which will permit to move to the next action in Plan α.

In order to illustrate the social influence between agents, we suppose that Bob's Doctor uses our application with the same goal as Bob i.e. reduce his diabete risk. Then, there is a direct link between agents a 1 , a 2 representing respectively Bob and Bob's doctor with T a 1 ,a 2 (φ) = 0.9 where φ represents any message coming from Bob's doctor (see [START_REF] Paglieri | Trusting the messenger because of the message: feedback dynamics from information quality to source evaluation[END_REF] for more details). Now that Bob is executing his plan in order to get back to a physical activity, his recommender agent recieves the following information from a 2 : B(¬pa) = 1 which means that Bob's doctor believes that physical activity is not possible (not recommended). This information will trigger bridge rule (3). Knowing the belief degree of a 2 about pa and given the trust degree of a 1 toward a 2 about information pa (T a 1 ,a 2 (pa)), a 1 decides to update its mental state according to Equation 2, and sets the new belief to B (pa) = 0 according to Equation 1. This will trigger the goal generation process, which updates the elected goals. pa will be removed because B(¬pa) = 1. Hence, a new plan is proposed to Bob.

CONCLUSIONS

We have presented in this paper a multi-context formalisation of the BDI architecture. We use a possibilistic approach, based on (da Costa Pereira and Tettamanzi, 2010), to deal with graded beliefs and desires which are used to determine agent's goals as suggested by [START_REF] Casali | A graded bdi agent model to represent and reason about preferences[END_REF] in their future works. We also take into account the social aspect of agents as a similarity-trust measure. The proposed model is conceived as a multi-context system where we define a mental context containing the beliefs, desires and intentions, a social context representing the social influence among agents in an implicit relationship, two functional contexts allowing to select a feasible plan among a list of precompiled plans, and a communication context that enables to communicate with other agents and with users. Short-term objectives of our research concern the realisation of a simulation of the MAS presented using Netlogo [START_REF] Sakellariou | Enhancing netlogo to simulate bdi communicating agents[END_REF]. This simulation will help to have a testing of initial design ideas and choices and also to understand how the system will behave when it will be implemented. For the implementation of the proofof-concept MCS framework, we have investigated the approaches of [START_REF] Casali | A tourism recommender agent: from theory to practice[END_REF] and [START_REF] Besold | Towards an implementation of a multi-context system framework[END_REF]. We will explore also approaches such as [START_REF] Costabello | Contextaware access control for rdf graph stores[END_REF] for multi-context access in RDF graph for our planning module. We consider that extending this model in order to handle temporal reasoning in dynamic environments (e.g. when executing recommended plan) will be more representative of real world applications. This includes a revision mechanism of the mental attitudes (beliefs, desires and intentions) and taking into account the evolution of the social relationship over time. Extending the social context in order to get into a communication process with agent via argumentation [START_REF] Mazzotta | Portia: A user-adapted persuasion system in the healthyeating domain[END_REF] is also part of our future work.

Figure 1 :

 1 Figure 1: Extended Multi-context BDI Agent Model

  An agent's belief can change over time because new information arrives from the environment or from other agents. A belief change operator is proposed in (da Costa Pereira and Tettamanzi, 2010), which allows to update the possibility distribution π according to new trusted information. This possibility distribution π which induces the new belief set B after receiving information φ is computed from the possibility distribution π with respect to the previous belief set B (B = B * τ Φ , π = π * τ Φ ) as follows: for all interpretation ω, π

Figure 2 :

 2 Figure 2: An example of a social multi-agent trust network

Figure 3 :

 3 Figure 3: Main Concepts and relationships in the 5W ontology

Figure 4 :

 4 Figure 4: Planning and Intention Contexts

( 6 )

 6 BC : B(a i , φ) = β φ , DC : D + (a i , ψ) = δ ψ → GC : G(χ(φ, ψ)) = δ Data: B,D Result: G * 1: γ ← 0; 2: Compute G γ by Algorithm 3; if G γ = ø then terminate with γ * = 1γ, G * = G γ ; else //Move to the next more believed value in B γ ← min{α ∈ Img(B)} i f α > γ 1 i f α end if γ <1 then go back to Step 2; end terminate with G * = ø;//No goal can be elected Algorithm 2: Goal Election Function Data: B,D Result: G γ //Img(D) is the degree of Desire 1: δ ← maxImg(D); //Verify if ψ is believed possible 2: if min ψ∈D δ B(¬ψ) ≤ γ then terminate with G γ = D δ ; else //move to the next less desired value of D δ← max{α ∈ Img(D)} i f α < δ 0 i f α end if δ > 0 thengo back to Step 2; end terminate with G γ = ø;Algorithm 3: Computation of G γ

Figure 5 :

 5 Figure 5: An Illustrating ExampleTo implement such a scenario using the BDI formalism, a recommender agent has a knowledge base (KB) like that shown in Table1initially specified by Bob.

  5 1 : B(pa → rd), D + (rd) → D + (pa), R 5 2 : B(wr → rd), D + (rd) → D + (wr), R 5 3 : B(bs), D + (pa) → D + (w), R 5 4 : B(pa → wr), D + (wr) → D + (¬eh).

Table 1 :

 1 Initial Knowledge Base of Bob's Recommender Agent

ACKNOWLEDGEMENTS

The authors would like to thank the French Agency of the Environment and the Energy Management (ADEME) and the Provence Alpes Cote d'Azur (PACA) region for the scholarship that is funding this research and the reviewers for their insightful comments.