N Pernet

P St-Jean

D D Solnyshkov

G Malpuech

N Carlon Zambon

Q Fontaine

B Real

O Jamadi

A Lemaître

M Morassi

L Le Gratiet

T Baptiste

A Harouri

I Sagnes

A Amo

S Ravets

J Bloch

Gap solitons in a 1D driven-dissipative topological lattice

Lasers Atomes et Molécules (PhLAM), 59000 Lille, France (Dated: November 29, 2021) Nonlinear topological photonics is an emerging field aiming at extending the fascinating properties of topological states to the realm where interactions between the system constituents cannot be neglected. Interactions can trigger topological phase transitions, induce symmetry protection and robustness properties for the many-body system. Here, we report on the nonlinear response of a polariton lattice implementing a driven-dissipative version of the Su-Schrieffer-Heeger model. We first demonstrate the formation of topological solitons bifurcating from a linear topological edge state.

We then focus on the formation of dissipative gap solitons in the bulk of the lattice and show that they exhibit robust nonlinear properties against defects, thanks to the underlying sublattice symmetry. Leveraging on the driven-dissipative nature of the system, we discover a new class of bulk gap solitons with high sublattice polarization.

We show that these solitons provide an all-optical way to create a non-trivial interface for Bogoliubov excitations. Our results show that coherent driving can be exploited to stabilize novel many-body phases and establish dissipatively stabilized solitons as a powerful resource for nonlinear topological photonics.

The topology of band structures in periodic systems is related to the existence of a nonzero Berry phase, and gives rise to fascinating phenomena like anomalous velocity, chiral edge states that are robust to disorder or topological pumps [1,[START_REF] Qi | Topological insulators and superconductors[END_REF]. First discovered in solid-state systems, topological physics can also be emulated in artificial lattices, including atomic [START_REF] Cooper | Topological bands for ultracold atoms[END_REF], photonic [START_REF] Lu | Topological photonics[END_REF][START_REF] Ozawa | Topological photonics[END_REF], mechanical [START_REF] Huber | Topological mechanics[END_REF], optomechanical [START_REF] Peano | Topological phases of sound and light[END_REF], and polaritonic [START_REF] Klembt | Exciton-polariton topological insulator[END_REF][START_REF] Gianfrate | Measurement of the quantum geometric tensor and of the anomalous Hall drift[END_REF] systems. These platforms have allowed engineering topological phases hardly achievable in condensed matter, involving synthetic dimensions [START_REF] Ozawa | Topological quantum matter in synthetic dimensions[END_REF], disorder effects [START_REF] Meier | Observation of the topological Anderson insulator in disordered atomic wires[END_REF][START_REF] Stützer | Photonic topological Anderson insulators[END_REF], quasi-crystalline structures [START_REF] Kraus | Topological states and adiabatic pumping in quasicrystals[END_REF] or higher-order multipoles [START_REF] Serra-Garcia | Observation of a phononic quadrupole topological insulator[END_REF][START_REF] Peterson | A quantized microwave quadrupole insulator with topologically protected corner states[END_REF]. The physics becomes even richer when inter-particle interactions are considered [START_REF] Smirnova | Nonlinear topological photonics[END_REF]. In the weakly interacting regime, nonlinearities can induce topological phase transitions [START_REF] Bleu | Interacting quantum fluid in a polariton Chern insulator[END_REF][START_REF] Hadad | Self-induced topological protection in nonlinear circuit arrays[END_REF][START_REF] Maczewsky | Nonlinearity-induced photonic topological insulator[END_REF], wave-mixing among topological modes [START_REF] Mittal | A topological source of quantum light[END_REF][START_REF] Kruk | Nonlinear light generation in topological nanostructures[END_REF], enable the formation of solitons in a topologically non trivial gap [START_REF] Lumer | Self-localized states in photonic topological insulators[END_REF][START_REF] Solnyshkov | Chirality of Topological Gap Solitons in Bosonic Dimer Chains[END_REF][START_REF] Bisianov | Stability of topologically protected edge states in nonlinear fiber loops[END_REF][START_REF] Mukherjee | Observation of Floquet solitons in a topological bandgap[END_REF][START_REF] Guo | Weakly nonlinear topological gap solitons in su-schrieffer-heeger photonic lattices[END_REF] or the realization of Thouless pumping with solitons [START_REF] Jürgensen | Quantized nonlinear thouless pumping[END_REF][START_REF] Jrgensen | The chern number governs soliton motion in nonlinear thouless pumps[END_REF][START_REF] Mostaan | Quantized transport of solitons in nonlinear thouless pumps: From wannier drags to topological polarons[END_REF]. In the strongly interacting regime, novel symmetry protected phases may appear [START_REF] De Léséleuc | Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms[END_REF], and fractional quantum Hall physics can be emulated with the possible stabilization of multi-particle Laughlin states [START_REF] Clark | Observation of Laughlin states made of light[END_REF].

Recently, photonic platforms have allowed pushing this exploration beyond the realm of conservative Hamiltonians, mainly through the engineering of gain and loss. Non-Hermitian topological systems [START_REF] Bergholtz | Exceptional topology of non-hermitian systems[END_REF] have led, for example, to the development of topological lasing on 0D [START_REF] Solnyshkov | Kibble-zurek mechanism in topologically nontrivial zigzag chains of polariton micropillars[END_REF][START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF][START_REF] Zhao | Topological hybrid silicon microlasers[END_REF][START_REF] Parto | Complex Edge-State Phase Transitions in 1D Topological Laser Arrays[END_REF] or 1D [START_REF] Bahari | Nonreciprocal lasing in topological cavities of arbitrary geometries[END_REF][START_REF] Bandres | Topological insulator laser: Experiments[END_REF] edge states, and PT-symmetric phases [START_REF] Weimann | Topologically protected bound states in photonic paritytime-symmetric crystals[END_REF][START_REF] Xia | Nonlinear tuning of pt symmetry and non-hermitian topological states[END_REF]. Most of these works on non-Hermitian topology have focused on probing, stabilizing or amplifying the linear response of the system. These recent advances now offer the possibility to experimentally explore topological photonics in a regime where non-Hermiticity and nonlinearity are combined [START_REF] Xia | Nonlinear tuning of pt symmetry and non-hermitian topological states[END_REF][START_REF] Konotop | Nonlinear waves in PT -symmetric systems[END_REF][START_REF] Jeon | Non-Hermitian C T -Symmetric Spectral Protection of Nonlinear Defect Modes[END_REF][START_REF] Liu | Gain-induced topological response via tailored long-range interactions[END_REF]. Driven-dissipative photonic platforms are particularly suitable to probe this physics because in addition to the non-Hermiticity of the Hamiltonian, the external drive can be used to stabilize novel nonlinear solutions going beyond what is possible in systems subject solely to gain and loss, and to modify the underlying topology [START_REF] Bardyn | Chiral Bogoliubov excitations in nonlinear bosonic systems[END_REF].

In this article, we investigate the physics of gap solitons in a 1D driven-dissipative topological lattice. We emulate a nonlinear version of the well-known Su-Schrieffer-Heeger (SSH) model: a 1D bipartite topological lattice with staggered hopping energies forming a chain of coupled dimers. [START_REF] Su | Solitons in Polyacetylene[END_REF]. Cavity polaritons are well suited for exploring this nonlinear topological physics [START_REF] Klembt | Exciton-polariton topological insulator[END_REF][START_REF] Gianfrate | Measurement of the quantum geometric tensor and of the anomalous Hall drift[END_REF][START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF][START_REF] Tanese | Polariton condensation in solitonic gap states in a one-dimensional periodic potential[END_REF][START_REF] Cerda-Méndez | Exciton-polariton gap solitons in two-dimensional lattices[END_REF][START_REF] Whittaker | Effect of photonic spin-orbit coupling on the topological edge modes of a Su-Schrieffer-Heeger chain[END_REF]. Indeed, their excitonic fraction provides repulsive interactions, resulting in a Kerr-type nonlinearity, while their photonic component makes the system intrinsically non-Hermitian via loss and possibly gain. Moreover the system can be resonantly driven by an external source and reach a steady-state determined by the balance between drive, dissipation and nonlinearity [START_REF] Carusotto | Quantum fluids of light[END_REF].

We demonstrate the formation of dissipative solitons that bifurcate either from a topological interface state or from dimers in the lattice bulk. When excited within the topological gap, the solitons present tails that are highly sublattice polarized. This endows them with robustness properties that we probe using an optically controlled non-Hermitian defect (i.e. a local perturbation to the real and imaginary parts of the potential landscape). We demonstrate that these solitons are robust to defects located on one sublattice, a property inherited from the chiral symmetry of the underlying model. The crucial asset brought by the drivendissipative character of the system appears when engineering the phase of the driving field.

By imposing a phase frustration between the phase pattern of the driving field and that of the Bloch eigenstates, we demonstrate the generation of novel gap solitons that have no counterpart in conservative systems and show high sublattice polarization in the core. Crucially, we show how the modification of the potential landscape induced by these asymmetric solitons can effectively split the chain and form a non-trivial interface. This latter aspect is evidenced through the emergence of a topological edge state in the calculated Bogoliubov excitation spectrum.

THE NONLINEAR DRIVEN-DISSIPATIVE SSH MODEL

The physics of the polariton nonlinear SSH model can be captured by a discretized Gross-Pitaevskii equation in presence of drive and dissipation:

i d dt   a n b n   = E 0 -i γ 2   a n b n   + g   |a n | 2 a n |b n | 2 b n   -J   b n a n   -J   b n-1 a n+1   + i   F a,n F b,n   e iωt (1)
where ψ n = [a n ; b n ] T is a spinor describing the wavefunction of the A and B sites in the n th unit cell; E 0 is the on-site energy,γ the polariton linewidth, g the interaction energy, J (J )

is the intracell (intercell) coupling energy, and ω/(2π) the driving field frequency which sets the frequency of the polariton field. The driving field F n = [F a,n ; F b,n ] T can be engineered with a specific amplitude and phase on each site, F α,n = |F α,n |e iϕα,n .

The topological nature of the SSH model is related to chiral (or sublattice) symmetry, which imposes identical on-site energies and restricts the coupling terms to sites belonging to distinct sublattices, e.g. no next-nearest neighbour coupling. Zero-energy states which emerge in lattices ending with weak links are protected by this symmetry. They are localised on a single sublattice, thus presenting well defined sublattice pseudo-spin:

S = n |a n | 2 -|b n | 2 n |a n | 2 + |b n | 2 = ±1, (2)
The sign of the spin reflects the sublattice localization, either on A (+1) or B (-1). The existence of topological edge states can also be related to the value of the winding number W computed over the first Brillouin zone [START_REF] Delplace | Zak phase and the existence of edge states in graphene[END_REF] that can take the values 0 (strong link termination)

or 1 (weak link termination). Hereafter, we will show that the specific properties of gap solitons generated in the nonlinear driven-dissipative SSH model are strongly linked to the pseudospin evaluated in their core and also in their exponential tails.

To emulate this system with cavity polaritons, we design arrays of coupled micro-pillars We first evidence a topological soliton that bifurcates from a topological interface state by focusing the pump on a micropillar defining an interface defect. We tune the laser energy to 180 µeV (approximatively 3γ) above the linear topological edge state and measure the transmitted signal when scanning the excitation power up and down. As shown in Fig. 1 (e), an hysteresis cycle is observed in the transmitted intensity, as expected in presence of Kerr nonlinearities. In the high intensity branch of the cycle, we generate a gap soliton that has the same spatial profile as the linear topological interface state (Fig. 1 (g)). This topological soliton shows the expected profile: a core self-localized at the defect position surrounded by two sublattice-polarized exponentially decaying tails.

BULK GAP SOLITONS

In the following, we address the nonlinear response in the bulk of the SSH lattice, see Fig. 1 (a). We implement a quasi-resonant excitation scheme using an excitation laser that is spectrally detuned with respect to the energy E 0 of the center of the topological gap by ∆E = ω -E 0 (see Fig. 2 (a)). We focus the pump laser (3.5 µm FWHM) ten unit cells away from the lattice edges to avoid finite size effects, and measure the transmitted intensity.

We first consider an excitation scheme where the pump frequency is tuned to the center of the topological gap (∆E = 0) and is spatially aligned with the center of a dimer. This corresponds, in Eq. ( 1), to a driving field localized on a single dimer with equal amplitude and phase on both A and B sites. As we ramp up the driving power, we observe the formation of gap solitons that are symmetric with respect to the pump and localized on a discrete number of dimers (see Fig. 2 (b)). When the incident power reaches a first threshold denoted P 1 , we observe the formation of a bulk soliton with a core extending over one-dimer. Above threshold P 2 , the soliton core extends to three dimers. Looking at the total transmitted intensity when scanning the power up and down, we evidence bistability and hysteretic behavior at each nonlinear threshold (see Supplementary Section S3). Each threshold occurs when the polariton field locally enters the nonlinear regime, i.e. when the nonlinear blueshift induced by the polariton interaction energy within a dimer overcomes the spectral detuning between the pump and the top of the lower band. These solitons are composed of a self-localized core region (grey-shaded areas in Fig. 2) and of exponentially decaying tails on both sides (white areas). Due to the lattice inversion symmetry, the soliton pseudospin computed over the entire profile vanishes [START_REF] Smirnova | Topological Edge States and Gap Solitons in the Nonlinear Dirac Model[END_REF]: Stot = -0.10 ± 0.06 after P 1 and Stot = 0.09 ± 0.04 after P 2 . Note that variations of the threshold powers are observed when pumping different dimers. They are due to disorder-induced fluctuations of the on-site energy. These fluctuations, on the order of 15 µeV, are smaller than the polariton linewidth (γ ≈ 70 µeV) and do not influence the physics discussed here.

It is insightful to probe how the pseudospin locally varies over the profile. Inside the soliton core, each dimer shows similar amplitude on both sublattices, leading to a locally vanishing pseudospin [START_REF] Smirnova | Topological Edge States and Gap Solitons in the Nonlinear Dirac Model[END_REF] Score = -0.03 ± 0.09 after P 1 and Score = 0.01 ± 0.06 after P 2 , see Fig. 2 (c-d). In contrast, the evanescent tails are strongly localized on a single sublattice:

the A sublattice on the left, and the B on the right. Consequently, the pseudo-spins SL and SR integrated over the left and right soliton tails do not vanish and present opposite signs: SL > 0 and SR < 0, see Fig. 2 (c-d).

The spin properties of the bulk soliton tails can be further explored when varying the laser energy. In Fig. 2 To accurately reproduce the experimental data, we use an effective 1D continuous model, which allows taking into account the finite size of the lattice pillars and the mixing of s-p bands. We look for steady-state solutions of the following Gross-Pitaevskii equation:

i dψ(x, t) dt = - 2 2m ∇ 2 -i γ 2 + V (x) + g|ψ(x, t)| 2 ψ(x, t) + iF (x)e -iωpt (3)
where m is the polariton mass. The strong and weak links are respectively represented by barriers of small and large amplitudes in the potential V (x) (see Supplementary Section S5).

Numerical results shown in Fig. 2.(c-d) (and also Fig. S6) reproduce the measured soliton profiles with their characteristic tails of opposite spin polarization [54]. Note that since the energy splitting between s and p orbitals is an order of magnitude bigger than the effective coupling between pillars, the deviation from perfect chiral symmetry induced by the s-p coupling (which is taken into account in the 1D continuous model) does not significantly alter the sublattice spin properties of bulk gap solitons.

ROBUSTNESS OF BULK GAP SOLITONS IN THE PRESENCE OF A NON-HERMITIAN DEFECT

It was theoretically predicted for conservative SSH lattices [START_REF] Solnyshkov | Chirality of Topological Gap Solitons in Bosonic Dimer Chains[END_REF], that these spin-polarized tails play a crucial role in the robustness properties of solitons in the topological gap, and in their interaction with a local defect. Hereafter, we extend this idea to the driven-dissipative framework, and show that the nonlinear threshold for the lateral growth of the soliton core is immune to the presence of a defect on one sublattice but not on the other one.

In order to probe the effect of a perturbation, we consider a defect on a pillar belonging to a dimer neighboring the soliton core. We then monitor how the spatial expansion of the soliton core is affected by this perturbation. The defect is optically generated by nonresonantly pumping a pillar belonging to the A or B sublattice with a second laser (see Fig. 3 a). This creates a local reservoir of excitons whose effect is twofold: 1-it locally blueshifts the pillar on-site energy thus acting as a perturbation on the real part of the potential; and 2-it induces a local gain through stimulated relaxation of excitons toward polariton modes, thus acting as a perturbation of the imaginary part of the potential. This defect is therefore intrinsically non-Hermitian.

To calibrate the real part of this perturbation, we measure the spectral shift of the ground-state emission from a single micro-pillar presenting the same characteristics as the ones forming the chain (see Fig. 3 (b)). For the imaginary part, we evaluate the lasing threshold in the same pillar (P th ∼ 0.6 mW), which enables estimating the gain induced by the non-resonant pump creating the defect (see Supplementary Section S7).

The evolution of the spatial profile of solitons as we ramp up the resonant pump power is displayed in Fig. 3 (d)-(g) for different amplitudes and positions of the defect. For comparison, the case with no defect is shown in Panel (c). We first monitor the soliton core expansion in presence of a defect located on a B sublattice site, where the soliton tail intensity presents local maxima (see panels (d)-(f) and Supplementary Section S7 for the calculated profiles).

As the defect breaks the system spatial symmetry with respect to the excitation spot, and locally reduces the laser detuning, the soliton core expansion becomes asymmetric. It is first favored toward the defect at power P * 2 . At a higher power, close to power P 2 measured without defect, the soliton eventually recovers a symmetric profile. We observe in Fig. 3 (h) that P * 2 strongly varies with the defect amplitude. For instance, a defect as small as 20 µeV (less than a tenth of the detuning between the pump laser and the top of the lower s-band) is sufficient to reduce P * 2 by a factor of two with respect to P 2 . In the simulated curve shown in Fig. 3 (h), this strong sensitivity is well accounted for by including both calibrated values of the real and imaginary parts of the perturbation (see Supplementary Section S7 and Fig. S8). This high sensitivity to defects located on the B sublattice contrasts with what we measure when the defect is localized on the A sublattice, where the amplitude of the tails vanishes. In that case, the second nonlinear expansion simultaneously occurs toward the right and left dimer, regardless of the defect amplitude (see Fig. 3 (g)-(h)) and for a power close to the one measured in absence of defect. This difference in sensitivity to defects located on A or B sublattice is a direct consequence of the spin polarization of the soliton tails. These results demonstrate that the lattice chiral symmetry provides solitons in the topological gap with high robustness against non-Hermitian defects located on one sublattice. This would also apply to the topological solitons discussed earlier, which bifurcate from a topological edge state. Moreover, as discussed in Ref. [START_REF] St-Jean | Lasing in topological edge states of a one-dimensional lattice[END_REF], linear topological edge states at zero energy are immune to disorder in the SSH lattice coupling constants, as long as the topological gap remains open. These properties still hold for the gap solitons discussed here.

GENERATING SPIN POLARIZED GAP SOLITONS USING PHASE-ENGINEERED

DRIVE

The bulk solitons we have considered so far present a globally vanishing pseudospin, like in conservative systems [START_REF] Smirnova | Topological Edge States and Gap Solitons in the Nonlinear Dirac Model[END_REF]. Hereafter, we show how we can depart from this family of unpolarized solutions thanks to the driven-dissipative nature of polaritons. To do this, we turn to an excitation scheme where two pillars of a dimer are driven with two beams with same amplitude and tunable phase difference ∆ϕ = ϕ A -ϕ B (see sketch above Fig. 4 (a)).

In Eq. (1), this corresponds to a driving field still localized on a single dimer with equal amplitude on both sublattices, but different phases. We select a total excitation power P > P 2 in order to obtain a three-dimer soliton when ∆ϕ = 0. Fig. 4 (a) presents the evolution of the intensity distribution along the lattice upon increasing ∆ϕ. A sequence of abrupt switchings is revealed between distinct regimes where the soliton core changes size. Close to ∆ϕ = π, the driving field becomes orthogonal to the Bloch modes, strongly reducing light injection into the lattice and explaining why the system switches to the linear regime.

Interestingly, for non-zero values of ∆ϕ, the soliton exhibits a non-zero total sublattice polarization, which reaches a maximum close to | STot | = 0.5 in the vicinity of ∆ϕ = π (see Fig. 4 (b)). Remarkably, the soliton core is then localized on a single site of the driven dimer, while the other driven site experiences a destructive interference induced by the nonlinearity. We show, in Fig. 4 (c), experimental and theoretical spatial profiles of this soliton for ∆ϕ = 1.13 π. As can be derived analytically (see Supplemental Section 9), when assuming perfect destructive interference in one of the two sites of the driven dimer, the resulting profile is found to be the same as that of a SSH topological edge state. We emphasize that the novel family of spin-polarized gap solitons we report here does not exist in conservative or undriven systems, as a phase-patterned driving field is required to stabilize them (see Supplementary Section 8).

Highly spin-polarized dissipative solitons have striking consequences on the physics of the underlying excitations. These consequences are related to the nonlinear onsite energy blueshifts locally induced by the solitons. For ∆ϕ = 1.13 π, the soliton core is localized on a single site so that the energy of this particular site is strongly shifted and lies above the upper SSH band. This creates a local potential barrier strongly localized on a single site and effectively splits the lattice in two semi-infinite SSH chains, one of them showing a weak link termination (W = 1). This suggests the optical creation of a non-trivial interface. For a more quantitative analysis of this effect, we compute the Bogoliubov excitation spectrum of the system (see Methods), in presence of a spin-polarized soliton. We consider a situation similar to the experimental one, where the soliton is generated at energy ∆E = 0 using two pumps localized on sites 0 and 1 and ∆ϕ = 1.13 π. The spatial profile of the soliton (obtained for the excitation corresponding to the optimal destructive interference within the driven dimer), is shown in the top part of Fig. 4(d), and the spatial profile of the Bogoliubov modes is shown in the bottom. The Bogoliubov spectrum clearly shows a high energy mode strongly localized at the soliton core position on site 1 (mode highlighted with a white dashed ellipse), which is a signature of the large blueshift experienced by this site. The vanishing amplitude of the Bloch band states at this position hints towards the fact that the lattice has effectively been split into two parts. Remarkably, we observe the emergence of a Bogoliubov mode within the topological gap (see arrow), together with its particle-hole symmetric with respect to the pump energy E = 0 [START_REF] Castin | Bose-einstein condensates in atomic gases: Simple theoretical results[END_REF]. This mode is localized on site 0 (weak link termination with W = 1 next to the soliton core), and presents a strong spin polarization of S = 0.67, thus pointing towards the non-trivial character of the optically-created interface. Since the soliton does not create an infinite potential barrier, the emerging topological edge state in the excitation spectrum is not fully spin-polarized as one would expect otherwise if the optical potential barrier were infinite. These simulations appeal for further explorations (beyond the scope of the present paper) to experimentally demonstrate the emergence of an edge state induced by spin-polarized dissipative solitons in the topological gap of the SSH lattice.

OUTLOOK

This work provides new perspectives to nonlinear topological photonics. Indeed, we have shown that the driven-dissipative nature of polaritonic systems provides a general way to stabilize novel nonlinear solutions that find no equivalent in conservative systems, to engineer non-Hermitian defects, as well as to reshape the potential landscape probed by Bogoliubov excitations and to modify its topology [START_REF] Diehl | Topology by dissipation in atomic quantum wires[END_REF][START_REF] Bardyn | Topology by dissipation[END_REF]. The control over the imaginary part of the potential could enable the exploration of PT-symmetric phases in nonlinear drivendissipative systems [START_REF] Konotop | Nonlinear waves in PT -symmetric systems[END_REF]. It will also be of the highest interest to extend these ideas in more complex non-Hermitian landscapes with chiral symmetry like 2D polariton lattices [START_REF] Bleu | Interacting quantum fluid in a polariton Chern insulator[END_REF][START_REF] Bardyn | Chiral Bogoliubov excitations in nonlinear bosonic systems[END_REF][START_REF] Kartashov | Modulational instability and solitary waves in polariton topological insulators[END_REF][START_REF] Gulevich | Exploring nonlinear topological states of matter with exciton-polaritons: Edge solitons in kagome lattice[END_REF]] or higher-order topological insulators [START_REF] Mittal | Photonic quadrupole topological phases[END_REF][START_REF] El Hassan | Corner states of light in photonic waveguides[END_REF][START_REF] Banerjee | Coupling between exciton-polariton corner modes through edge states[END_REF].

METHODS

Sample description

The periodic structure used in this work is etched out of a planar semiconductor microcavity with nominal quality factor of Q ≈ 75, 000 grown by molecular beam epitaxy. The microcavity is composed of a λ GaAs layer embedded between two Ga 0.9 Al 0.1 As/Ga 0.05 Al 0.95 As distributed Bragg reflectors with 28 (top) and 32 (bottom) pairs. A single 15 nm In 0.05 Ga 0.95 As quantum well is inserted at the center of the cavity, resulting in strong exciton-photon coupling, with an associated 3.2 meV Rabi splitting (measured by probing the polariton dispersion via angularly resolved photoluminescence as explained in the Supplementary Section 1). After the epitaxy, the sample is processed with electron beam lithography and dry etching into arrays of coupled pillars arranged in a SSH lattice. The exciton-photon detuning, defined as the energy difference between the uncoupled cavity mode at the bottom of the s-band and the exciton resonance, is of the order of -4.9 meV for all experiments (see Supplementary Section 1).

Experimental techniques

The sample is held at T = 4 K in a closed-cycle cryostat. Non-resonant photoluminescence measurements are realized with a single-mode continuous-wave laser at 780 nm. The exci-tation spot is elongated (FWHM ∼ 30 µm) by focusing the beam with a cylindrical lens at the back-focal plane of the excitation objective. The emission is collected through a lens with NA 0.55 and imaged on the entrance slit of a spectrometer coupled to a charge-coupled device camera with ∼ 30 µeV spectral resolution. Real-and momentum-space photoluminescence images are realized by imaging the sample surface and the Fourier plane of the objective, respectively. A polarizer is used to select emission polarized along the long axis of the lattice. Experiments with quasi-resonant excitation are realized in transmission geometry, with the excitation (detection) on the epitaxial (substrate) side of the sample and a spot of 3.5 µm FWHM. The optical defect is created by focusing onto a 3 µm FWHM spot a 825 nm cw laser on the epitaxial side. For the resonant two-spot experiment (each with 3.5 µm FWHM), the phase difference is controlled with a delay line with one of the mirrors mounted on a piezoelectric actuator. Error bars in Fig. 3 (h) correspond to the standard deviation on the measurement of P 2 obtained by performing a repeatibility study on the resonant pump alignment. Error bars in Fig. 4 (b) are calculated by evaluating the impact of modifications in the area chosen for intensity integration on the pseudo spin calculation.

Numerical methods

We use the 3-rd order Adams-Bashforth method for numerical integration of the nonlinear Gross-Pitaevskii Eq. (3) with a time step of 10 -3 ps. CPU-based parallel computing is used to evaluate the kinetic energy term via Fast Fourier Transform. The numerical grid of 2 9 = 512 points allows to describe a lattice of 21 dimers with a step of 0.25 µm. Increasing the resolution further or considering a longer chain does not change the results. The rampup and ramp-down times are 20 ns, which is a compromise between the required adiabaticity and simulation time. We have checked that increasing the ramp-up time further does not change the results. The parameters were as follows: m = 3 × 10 -5 m 0 (m 0 is the free electron mass), γ = 70 µeV , g = 5 µeV • µm 2 pumping spot FWHM ≈ 2.8 µm, and the pumping frequency detuning with respect to the band edge is ≈ 0.2 meV. Other relevant parameters, such as the potential profile, are provided and discussed in the Supplementary materials. Bogoliubov spectra were calculated by linearizing the discretized Gross-Pitaevskii equation around the steady state solution ψ s using the ansatz ψ n = (ψ s n + δψ n ) exp(iωt) with δψ n = u n exp(-i t/ ) + v * n exp(i * t/ ), where is the complex-valued energy [START_REF] Sarchi | Coherent dynamics and parametric instabilities of microcavity polaritons in double-well systems[END_REF] and u and v are two components vectors describing the amplitude of the wavefunction of the A and B sites in the n th unit cell. The spectrum is symmetric with respect to the pump energy E = 0 meV, as imposed by particle-hole symmetry. In the graph Fig. 4(d), we display the energy versus position distribution of the component |u n | 2 that is associated to the creation operator. The squared amplitudes corresponding to the A and B sites of the n th unit cell are associated to site numbers 2n and 2n + 1, respectively.

( 3

 3 Fig.S2bin Supplementary materials Section 2).

FIG. 1 .

 1 FIG. 1. Implementation of the SSH lattice. a-c, Scanning electron microscope images of SSH polariton lattices. We highlight some of the micro-pillars with yellow circles and an interface defect in (c) with a green circle. Inset of (a): Schematic representation of a micro-pillar, composed of a Fabry-Perot cavity defined by two distributed Bragg reflectors (DBR) and a spacer (orange layer) embedding a single quantum well (QW). Right: Typical shape of the s and p polariton modes. b-d, Spectrally resolved photoluminescence intensity measured as a function of momentum k under non-resonant excitation using an elongated spot centered in the bulk of lattice (a) (as in panel (b)) and around the interface defect in lattice (c) (as in panel (d)). e, Transmitted intensity measured while ramping the excitation power up (blue circles) or down (red circles) under quasi-resonant excitation of the interface defect. f, 2D real space image of the transmitted intensity measured on topological soliton. The excitation pump is localized on the interface state, as sketched on top of the panel. g, Intensity profiles (integrated along the transverse direction) measured for (grey line) the topological interface state in the linear regime, and (black line) a topological gap soliton bifurcating from the interface state.

FIG. 2 .

 2 FIG. 2. Generation of topological gap solitons. a, Spectral density of states (DOS) of the SSH lattice. The drive (purple arrow) is detuned from the center of the topological gap by ∆E. b, Spatially resolved intensity profiles (integrated along the direction perpendicular to the lattice and normalized to their maximum value), measured as we ramp up the driving power. Dotted horizontal red lines mark the P 1 and P 2 thresholds. c-d, Experimental (solid line) and simulated (dashed line) spatial profiles of the soliton above (c) P 2 and (d) P 1 thresholds (corresponding excitation powers are indicated with dotted black lines in (b)). e, Pseudo-spins SL (red full circles) and -SR (blue open circles) measured as a function of ∆E. The amount of optical power available in the experiment enables exploring positive ∆E values up to 0.1 meV. The grey solid line shows the values of SL obtained using a tight-binding calculation, where the intensity on each site is convoluted with a Gaussian profile of FWHM 2.7 µm (around the FHWM of the pillar s-mode). f,. Soliton profile measured when the laser energy is tuned to the upper trivial gap (∆E = 1.0 meV). In all panels (b,c,d,f), location of the drive is indicated with a purple arrow in the sketch above. In panels (c,d,f), red (blue) dashed lines indicate A (B) sites with peaked intensity, while gray areas represent the extent of the soliton cores.

  (e), we report the measured spin polarization SL and SR of the left and right evanescent tails when varying ∆E within the topological gap. When the laser energy departs from the center of the topological gap, the tails remain spin-polarized as long as the laser energy stays within the gap (see Supplemental Section S4 for measured profiles). It vanishes when the excitation reaches the edges of the topological gap. The spin polarization of the tails observed in solitons generated in the topological gap is thus directly linked to the chiral symmetry of the SSH model. To highlight this specific property, we tune the laser energy to the upper gap separating the s and p bands [53]. The soliton profile obtained when exciting this trivial gap is presented in Fig. 2(f). Both evanescent tails present high intensities on all pillars, and thus no spin polarization.

FIG. 3 .

 3 FIG.3. Robustness of gap solitons against a defect. a, Scheme of the two-beam experiment to generate both a soliton (quasi-resonant pump, purple arrow) and an optically induced defect (non-resonant pump, red arrow). b, Calibration of the optical defect: blueshift ∆ as a function of the non-resonant pump power measured on an isolated micro-pillar. c-g, Spatially resolved intensity profiles (integrated along the direction perpendicular to the lattice and normalized to their maximum value), measured as we ramp up the driving power for (c) no defect, (d-f ) a defect on the B sublattice generated with various non-resonant powers, and (g) a defect on the A sublattice generated with same power as in (f ). For each panel, the driven dimer (defect) is shown on top with purple (red) circles. h, Threshold power P * 2 measured as a function of the blueshift (deduced from (b)) induced by a defect localized on the A (red symbols) or B (blue symbols)sublattice.The red (blue) line corresponds to numerical simulations.

FIG. 4 .

 4 FIG. 4. Spin-polarized solitons. a, Spatially resolved intensity profiles (integrated along the direction perpendicular to the lattice and normalized to their maximum value), measured as a function of ∆ϕ for a total excitation power P > P 2 (P = 80 mW). The driven dimer is depicted on top (purple circles). The white vertical dashed line indicates the center of the driven dimer. b, Measured (symbols) and calculated (dashed line) soliton global pseudospin as a function of ∆ϕ c, Intensity profile of the soliton measured (solid line) and simulated (dashed line) for a phase difference ∆ϕ = 1.13 π (indicated by the horizontal dashed line in Panel a). Blue dashed lines mark B pillars with peaked intensity. d, Calculated Bogoliubov excitation spectrum (bottom panel) when driving a dimer with ∆E = 0, ∆ϕ = 1.13 and an excitation power corresponding to the maximum spin polarization of the soliton core. The white dashed ellipse highlights the blueshifted mode, and the arrow points towards the mode emerging in the gap (see the mode profile in inset). Above the Bogoliubov spectrum, we show the intensity profile of the bulk soliton (red bars), as well as an illustrative schematics depicting the chain: the red disc shows the site which eigenmode has effectively been decoupled from the bands, while the white discs show the remaining semi-infinite chains together with their respective winding numbers related to the coupling strength at the termination.

Acknowledgements

We thank one of the anonymous reviewers for their insightful comments, that led to major improvements of the article. This work was supported by the Paris Ile-de-France Région in the framework of DIM SIRTEQ, the Marie Sklodowska-Curie individual fellowship ToPol, the H2020-FETFLAG project PhoQus (820392), the QUANTERA project Interpol (ANR-QUAN-0003-05), the French National Research Agency project Quantum Fluids of Light (ANR-16-CE30-0021), European Research Council via projects EmergenTopo (865151) and ARQADIA (949730), the French RENATECH network, the French government through the Programme Investissement dAvenir (I-SITE ULNE / ANR-16-IDEX-0004 ULNE) and IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25), managed by the Agence Nationale de la Recherche, the Labex CEMPI (ANR-11-LABX-0007).


Célia Martinie

Philippe Palanque

Task models based gameful design as a mean to increase engagement with automation

Keywords: Interactive systems and tools, automation, gameful design, task modelling Engaging with Automation, CHI'22

Gameful design, including gamification, has proved helpful in increasing user engagement while using interactive systems in various contexts and domains including professional, leisure, healthcare…. The use of game-related elements correspond to the addition of new goals and tasks for the user to perform. In the context of work, users' goals and tasks strongly relate to the users' missions and objectives in the work organization. As part of the tasks may be automated at design time, allocation of tasks between the user and the interactive systems supporting these users' tasks has to be carefully analyzed, elicited and designed, so that the user can reach the objectives and goals assigned with adequate performance and acceptable error rate. This paper argues that the gameful design of interactive systems in the workplace requires precisely identifying and describing automation in work tasks, as well as users' tasks with the game elements. For that purpose, the paper presents a task models based approach that enables to analyze and design automation while applying gameful design. The paper presents an illustration of the approach using the example of an operator performing a monitoring task.

Introduction

In the context of work, the users have missions and objectives defined by the organization employing them, and automation is a mean to support these users in accomplishing their missions and reaching their assigned objectives. However, as demonstrated by Yerkes and Dodson [START_REF] Wilson | Gamification challenges and a case study in online learning Internet Learning[END_REF] more automation might reduce operator vigilance (complacency and boredom) and prevent operators from reacting promptly to adverse events (e.g. due to loss of situation awareness). Design of automation results then in the identification of pros and cons for each, and every user task.

Gameful design is defined in [START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF] by the "End of affording ludic qualities or gamefulness (the experiential qualities characteristic for gameplay) in nongame contexts". In that context, gameful design (which includes gamification), increases user engagement while interacting with products, services and systems, in various contexts (e.g. professional, industrial, leisure, health…). Using a gamified interactive system to perform their missions implies that additional goals and tasks are added to the work of users and may be intertwined with these work tasks. This means that, at design time, there is a need to identify both work automation and game-related tasks, in order to ensure that the automation is aligned and not conflicting with the game-related tasks. In addition, it is important to ensure that the allocation of tasks and functions between the user and the gamified interactive system matches the objectives of the organization in term of key performance indicators such as performance or safety.

This paper discusses the importance of being able to precisely describe and carefully analyze the allocation of tasks and functions between a user and a gamified interactive system at work, and proposes a model-based approach to support this analysis. The paper is structured as follows. Section 2 highlights the benefits of using gameful design to increase user engagement. Section 3 discusses the impact of the use of a gamified interactive system on user tasks. Section 4 presents the task-models-based approach for the design of automation and game elements. Section 5 illustrates the proposed approach with a simple example based on the Mackworth clock experiment. We highlight the fact that this task is representative of monitoring tasks in industrial contexts. Section 6 concludes the paper and presents perspectives for future work.

Gameful design and engagement

Although gameful design and gamification techniques have not been studied evenly across application domains (most of the contributions come from the application domains of education and health), Seaborn and Fels [START_REF] Roto | Engaging Automation at Work -A Literature Review[END_REF] surveyed the literature and established that the results of gamification are mostly positive. They highlight that gameful design, including gamification, raise engagement and user performance. Indeed, a study on the gamification of the tutorial of the AutoCAD 3D objects modelling application [START_REF] Roto | Engaging Automation at Work -A Literature Review[END_REF] reported higher subjective engagement levels with the gamified application, and that users performed a set of testing tasks from 20% to 76% faster after using the tutorial with the gamified components. Gameful design can also encourage participation and collaboration with other users as demonstrated in the context of online learning [START_REF] Tondello | Elements of Gameful Design Emerging from User Preferences[END_REF]. These examples (and many other ones) show that the range of potential benefits of gameful design is quite wide if design choices are carefully elicited.

Gameful design requires fine-tuning of the game mechanics that are integrated in the interactive system, and game elements have to match what the user is able to perform in the context of use. Wilson et al. [START_REF] Tondello | Elements of Gameful Design Emerging from User Preferences[END_REF] argue that some design choices may be counterproductive and that engagement and motivation may vary a lot depending on the type of game element. Adding artificial challenge to a system supporting functional needs engenders frustration [START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF]. Moreover, Korn et al [START_REF] Korn | Towards a gamification of industrial production: a comparative study in sheltered work environments[END_REF] showed that in an industrial environment, some gamification elements may improve production speed but may also increase the error rate.

Impact of gameful design on user tasks

As stated above, adding game elements and challenges to motivate the user leads to add additional objectives and tasks to perform with the interactive system. These additional tasks interleave with work tasks, and have thus to integrate in a consistent manner with them in order to avoid frustration and errors (especially capture and interference ones). Moreover, the design of game elements also requires identifying automation opportunities (for these game elements), with the objective of increasing the overall usability of the system but also to make sure that proposed challenges are important and motivating experiences for the user [START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF].

Gameful design for interactive systems at work thus requires being able to describes exhaustively and systematically:

• User work tasks; • User tasks while interacting with work automation; • User tasks to reach game elements objectives; Analyzing all these elements together is also critical in order to detect conflicting elements.

Task models for the design of automation in both gamified and work tasks

Existing gamification and gameful design methods rely on the identification of user tasks [START_REF] Aparicio | Analysis and application of gamification[END_REF][START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF][START_REF] Tondello | Elements of Gameful Design Emerging from User Preferences[END_REF], but there is very little information about how to describe those tasks beyond text-based sequential lists of action.

We propose a task-models-based systematic approach for the identification of both tasks coming from game elements and the ones coming from work-related activity. Together with these task models we make explicit automation aspects (especially migration of tasks from the user activity to the system). Task models consist in a graphical, hierarchical and temporally ordered representation of the tasks the users have to perform with an interactive application or system in order to reach their goals. Task models is one of the very few means for describing explicitly and exhaustively user tasks [START_REF] Martinie | Formal tasks and systems models as a tool for specifying and assessing automation designs[END_REF]. Task models support several different stages of interactive systems design and development (e.g. user roles identification, system functions identification, user interface design, testing, training program design…) which requires the involvement of various stakeholders (e.g. human factor experts, system engineers, software engineers). The expressive power of the task modelling notation is critical for performing analysis on the models produced. Indeed, what is not represented cannot be analyzed. In this paper we use the HAMSTERS|XL notation [START_REF] Martinie | Formal tasks and systems models as a tool for specifying and assessing automation designs[END_REF] which is able to describe many types of task such as user task (cognitive, perceptive, and motor), abstract tasks, interactive tasks (input, output) and system tasks. Distinction between user and system task is critical for the current paper as this is how automation, using migration of tasks in functions can be represented. In addition to tasks, HAMSTERS|XL makes explicit in models the representation of information and knowledge required for performing each task. HAMSTERS|XL task models also enable the explicit and exhaustive description of allocation of tasks and functions between users and interactive systems [START_REF] Bouzekri | Should I Add Recommendations to My Warning System? The RCRAFT Framework Can Answer This and Other Questions About Supporting the Assessment of Automation Designs[END_REF][START_REF] Martinie | Formal tasks and systems models as a tool for specifying and assessing automation designs[END_REF]. This makes them a perfect candidate for describing the user tasks at work and the user tasks to achieve objectives of the game elements, as well as the user tasks to interact with work automation and the user tasks to interact with game elements.

Illustrative example: The Mackworth clock

The Mackworth clock is an interactive system designed and developed to study the performance of operators monitoring information on airborne radars [START_REF] Mackworth | The breakdown of vigilance during prolonged visual search[END_REF]. The system was developed to build experiments to assess vigilance capabilities of human being while monitoring autonomous systems. The Mackworth autonomous system (presented in Figure 1 a)) includes a green point, which moves in steps every second (like the second hand of a clock). In the experiment, at irregular time intervals, the green point moves the double of the usual distance (jumps one step). The operator has to detect this unexpected movement (representing a problem or a failure) and to press a button to prove that the malfunction has been detected. The hypothesis behind this experiment was that human performance in detecting malfunctions would decrease over time. Indeed, the study of performance of operators monitoring this autonomous system confirmed the fact that attention and vigilance decrease over time. The main outcome of this study was to propose identify the best compromise between the duration of the monitoring period (the watch-time of the operator) and the errors. In the following sections, we discuss the opportunity to alter this experiment adding game elements to the monitoring task to assess the possible increase of attention and vigilance for longer monitoring tasks.

5.1.

Task model based gameful design of a new version of the Mackworth Clock

We selected the gameful design method called "The lens intrinsic skill atoms" [START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF] because this method provides a practical and detailed set of steps. Furthermore, this method integrates several major concepts identified as requirements for gameful design, amongst which designing for basic need satisfaction (specifically competence) and designing around inherent skill-based challenges. Figure 2 presents the main steps of this gameful design method. It is composed of five main steps respectively named Strategy, Research, Synthesis, Ideation and Prototyping. Each step is decomposed in several activities, and for our illustrative example, we focus on the activities that benefits from and can be described using of task models. These activities are 2a) Translate user activities into behavior chains and and 3a) Formulate Activity-Challenge-Motivation triplet systematically, which we refine using task model support. We also refine step four of the method and propose the activity task model based ideation.

Translate user activities into task models (beyond behavior chains)

In the method "The lens intrinsic skill atoms", the behavior chains are to be used when activity is complex, because they are useful to "deconstructs complex activities into chains of behaviors by different actors" [START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF]. It is important to note that our proposed approach goes further that the concept of behavior chains. Task modelling enables to go beyond the decomposition in a set of event sequences, as it enables to breakdown all possible activities, as well as their hierarchy and all of their possible temporal interleaving (sequence, interleaving, interruption…).

Figure 3 presents the detailed description of the tasks the operator has to perform to monitor the clock. The main goal, represented at the top of the task model, is "Monitor clock". Under this main goal, the temporal ordering operator "|||" named concurrency has two branches that describe the tasks performing in parallel. On the left branch under the main goal, the system updates the clock every second (abstract system task "Update numeric watch" with an incoming arrow from the calendar event "Every second"). This task decomposes in a sequence (temporal ordering operator ">>") of a system task (a choice, indicated with the choice temporal ordering operator "[]", between the tasks "Update to next" and "Update (malfunction)") and the interactive output task "Display green point" which uses the output device "screen". The system tasks update the value of the software object "position of green point", which is required to perform the system output task "Display green point". In the right branch under the main goal "Monitor clock", the abstract iterative task "Monitor green point" decomposes in a sequence of user tasks. First, the user performs the perceptive task "Look at green point" using the output device "screen", which produces the information "current position of green point'. Then the user performs the cognitive task "Recall previous position of green point" using the information "Memorized position of green point" and producing the information "previous position of green point". Then the user performs the cognitive analysis task "Compare previous and current position" using both information "current position of green point" and "previous position of green point". This task produces the information "size of shift". Then, the user performs the cognitive analysis task "Analyze if shift is standard or not" using the information "size of shift" and the declarative knowledge "Standard shift is +1". Then from the result of this analysis task, the user makes a choice (described using the choice temporal ordering operator "[]" combined with the test arcs on the value of the information "shift type"). If the shift type is standard, the user decides not to do anything. If the shift type is different from standard, then the user decides to press the response key, and then presses the response key. At last, the user performs the cognitive task "Memorize position of green point" which produces the information "Memorized position of green point".

Formulate Activity-Challenge-Motivation triplet systematically

The term "activity" refers to the tasks the operator will have to perform. This part of the method thus enables to identify systematically the possible motivations of the operator for the tasks, as well as inherent skill-based challenges of the tasks. Table 1 presents the list of activity (task) -challengemotivation triplets produced using the task model. We produced this table by systematically going through the user tasks in the task model, and associating to each of them a relevant challenge and a relevant motivation.

Task model based ideation

The method "The lens intrinsic skill atoms" guides the designers using questions when brainstorming design options. In particular, the questions "What challenges can be removed through automation or improving usability?" and "What challenges remain that the user can learn to get better at?" highlight that automation is also a tool when gamifying an application. On one hand, the main goal of the user task is to monitor the clock and the challenge of being attentive and staring at the clock should remain (framed row in Table 1). On the other hand, the "Recall previous position of green point" and "Memorize position of green point" may be difficult and error-prone, we thus propose to migrate them from the operator to the system. The automation of these work tasks decrease the number of tasks to perform for the operator. Figure 4 presents two screenshots to illustrate this proposal. Figure 5 presents the task model modified for this automation proposal. In the left main branch of the model, we added two system tasks: an output system task named "display previous position of green point", as well as a storing system task. We also added a new software object "stored position of green point" to describe that the system will be storing the value of the previous position instead of the user doing it. In the abstract iterative task "Monitor green point", we replaced the cognitive task "recall previous position of green point" with the perceptive task "Look at previous position of green point", and removed the cognitive task "Memorize position of green point", as well as the associated memorized information "Memorized position of green point".

Figure 5. Task model modified for the work automation

This proposal of work automation should decrease cognitive load but may lead operator to be less attentive. We thus propose to integrate a game element, focused on a continuous input from the operator, to increase immersion. The game element the continuous tracking of the green point using a mouse pointer. The operator has one hand on the mouse device and has to move the mouse pointer to the green point each time the green point shifts. Figure 6 presents three screenshots that illustrate this design proposal. At the bottom of the screens, a panel indicates the total time on the green point and the total time outside of the green point. 7 presents the task model modified to include the tasks related to the game element "mouse pointer tracking using a mouse device", as well as the automation of work (explained previously). In the left main branch, we added the system output task "Display mouse pointer" which requires the software object "Mouse pointer position". In the right part of the model, we added a new branch which main task is "Monitor mouse pointer" and is iterative. This task is part of the abstract iterative task "Monitor green point and mouse pointer". The temporal ordering operator "|=|" (order independent) means that the operator can perform the abstract task "Monitor green point" first and then the abstract interactive task "Monitor mouse pointer", or that the operator can perform the abstract task "Monitor mouse pointer" and then the abstract task "Monitor green point". It decomposes using the same pattern as the green point monitoring task, but for the mouse pointer. The user first performs the perceptive task "Look at mouse pointer", which produces the information "mouse pointer position". The user then performs the cognitive analysis task "Compare mouse pointer position with green point position", using both the information "current position of green point" and "mouse pointer position". This task produces the information "mouse pointer position relative to green point". Depending on the value of this information, mouse pointer is on green point or mouse pointer is not on green point, either the user will perform the cognitive decision task "Decide not to do anything" or the cognitive decision task "Decide to move the mouse pointer" followed by the interactive input task "Move mouse pointer".

Figure 7. Task model modified to integrate both automation and game element

This task model helps to figure out the impact of the game element on the operator's tasks. We see that the new set of tasks that we introduced represents as many tasks as the set of tasks to monitor the green point and that this set of tasks interleaves with the work task, and share a common information to process for the tasks (Information "current position of green point" at the bottom in Figure 7). This confirms that the design proposal should increase the operator's workload while helping the operator to focus on the main goal.

By supporting the precise comparison of the original work tasks with the tasks altered by adding work automation and game elements, we argue that a task models based approach enables to identify relevant automation and gamification opportunities. The level of precision of task descriptions enables to filter out tasks that should be migrated to the system and tasks for which gamification will benefit to user performance and engagement. In that way, it supports reaching an optimum level of workload as exemplified in Figure 8.

Conclusion

This paper has presented the potential benefits of using game elements interleaved with operators' work in order to increase user experience and engagement of operators. We argue that describing how these game elements transform the operators' work is critical in order to be able to assess the impact (positive and negative) of the activities added by the game elements on operators' work.

We have revisited the Mackworth clock experiment which is centered on the monitoring (by an operator) of an autonomous system. We took the position that degradation of the monitoring performance of the operators was related to a loss of engagement, and an increase of boredom and was due to the lack of active participation of the operator on the system.

To improve performance we decided to integrate in the work tasks additional tasks related to game elements added to the system. We used the HAMSTERS|XL notation to describe the original and the altered task models and used these models to demonstrate the higher engagement of operators when game elements are added.

This early work is a first step toward a more generic approach to improve operators' performance when, for instance in the context of automation, user activity is reduced to monitoring and supervision of autonomous systems. This approach proposes to add non-work activities to keep the user active and aware of the situation. We represented the expected results in a schematic way on Figure 8 where Macworth Clock is positioned in the underload region while the clock with the game elements is positioned in the optimum region.

There are several perspectives for future work. First, the modelling technique, beyond supporting the analysis of work automation, could also support the analysis of automation in game elements and the impact on user tasks. This would enable to differentiate the impact of work automation from the impact of game elements automation on user tasks. Then, the proposed task-model based approach could be applied to the whole method "The lens intrinsic skill atoms", this by including steps that require users in the loop (e.g. interviews and evaluations). The main expected benefits of such complete integration of task modelling within the gameful design process is to ensure systematically precise identification of the most appropriate candidate tasks for gamification as well as fine tuning of the design solutions.

Figure 1 :

 1 Figure 1: Original Mackworth Clock from[START_REF] Aparicio | Analysis and application of gamification[END_REF] and modified Mackworth Clock from[START_REF] Martel | EEG predictors of covert vigilant attention[END_REF]

Figure 1 b

 1 Figure 1 b) presents a modified and digitalized version of the Mackworth clock. We chose to present it because it makes more explicit (as a symbolic representation) how the Mackworth autonomous system was functioning.The hypothesis behind this experiment was that human performance in detecting malfunctions would decrease over time. Indeed, the study of performance of operators monitoring this autonomous system confirmed the fact that attention and vigilance decrease over time. The main outcome of this study was to propose identify the best compromise between the duration of the monitoring period (the watch-time of the operator) and the errors. In the following sections, we discuss the opportunity to alter this experiment adding game elements to the monitoring task to assess the possible increase of attention and vigilance for longer monitoring tasks.

Figure 2 .

 2 Figure 2.Five steps in gameful design from[START_REF] Deterding | The Lens of Intrinsic Skill Atoms: A Method for Gameful Design[END_REF]

Figure 3 .

 3 Figure 3. Task model describing the tasks to "Monitor clock"

Figure 4 .

 4 Figure 4. Illustration of the design proposal for automating operator's work

Figure 6 .

 6 Figure 6. Illustration of the game element added to the modified Mackworth Clock

Figure

  Figure7presents the task model modified to include the tasks related to the game element "mouse pointer tracking using a mouse device", as well as the automation of work (explained previously). In the left main branch, we added the system output task "Display mouse pointer" which requires the software object "Mouse pointer position". In the right part of the model, we added a new branch which main task is "Monitor mouse pointer" and is iterative. This task is part of the abstract iterative task "Monitor green point and mouse pointer". The temporal ordering operator "|=|" (order independent) means that the operator can perform the abstract task "Monitor green point" first and then the abstract interactive task "Monitor mouse pointer", or that the operator can perform the abstract task "Monitor mouse pointer" and then the abstract task "Monitor green point". It decomposes using the same pattern as the green point monitoring task, but for the mouse pointer. The user first performs the perceptive task "Look at mouse pointer", which produces the information "mouse pointer position". The user then performs the cognitive analysis task "Compare mouse pointer position with green point position", using both the information "current position of green point" and "mouse pointer position". This task produces the information "mouse pointer position relative to green point". Depending on the value of this information, mouse pointer is on green point or mouse pointer is not on green point, either the user will perform the cognitive decision task "Decide not to do anything" or the cognitive decision task "Decide to move the mouse pointer" followed by the interactive input task "Move mouse pointer".

Figure 8 .

 8 Figure 8. Using Yerkes-Dodson curve as a mean to represent evolution of user engagement and performance when game elements are added to work tasks

Table 1

 1 List of Activity-Challenge-Motivation triplets produced using the task model

	Task (activity or	Challenge	Motivation (would require
	behavior)		interviewing the operators, we
			cite examples from [3] and [12])
	Monitor the clock	Detect all of the malfunctions	Example of motivation is
			incentive
	Look at green point	Be attentive, always stare at the	Example of motivation is
	(visual perceptive task)	clock	immersion
	Recall previous position	Recall the green point last	Example of motivation is to use
	of green point	position, do not confuse with	special abilities
	(cognitive task)	other previous position
	Compare previous and	Evaluate the right distance	Example of motivation is self-
	current position	between the positions	achievement
	(cognitive analysis task)
	Analyze if shift is	Identify that the distance	Example of motivation is
	standard or not	matches a standard shift	acquiring skills
	(cognitive analysis task)
	Decide not to do	Identify that the distance	Example of motivation is fear of
	anything (cognitive	matches a double shift	not being at the same level of
	decision task)		competence as others
	Decide to press	Take appropriate action plan	Example of motivation is to feel
	response key (cognitive		engaged in protecting the
	decision task)		organization
	Press response key	Correctly press the response	Example of motivation is fear of
	(interactive input task)	key, do not slip	being laughed at
	Memorize position of	Memorize the position of the	Example of motivation is to feel
	green point (cognitive	green point	in full possession of one's
	task)		abilities