Franc ¸ois Copie

Pierre Suret

Stéphane Randoux

Space-time observation of the dynamics of soliton collisions in a recirculating optical fiber loop

Keywords: solitons, recirculating fiber loop, inverse scattering transform method, integrable nonlinear wave systems

We present experiments performed in a recirculating fiber loop in which we realize the single-shot observation of the space and time interaction of two and three bright solitons. The space-time evolutions observed in experiments provide clear evidence of a nearly-integrable nonlinear wave dynamics that can be easily interpreted within the framework of the inverse scattering transform (IST) method. In particular collisions between solitons are found to be almost perfectly elastic in the sense that they occur without velocity change and with only a position (time) shift quantitatively well described by numerical simulations of the integrable nonlinear Schrödinger equation. Additionally our experiments provide the evidence that the position (time) shifts arising from the interaction among three solitons are determined by elementary pairwise interactions, as it is well known in the IST theory.

Introduction

The conditions under which an intense electromagnetic beam can produce its own dielectric waveguide and propagate without spreading were examined by Chiao et al. as early as 1964 in Ref. [START_REF] Chiao | Self-Trapping of Optical Beams[END_REF]. In 1973, Hasegawa and Tappert pointed out that nonlinearity of the index of refraction could indeed compensate the pulse broadening effect of chromatic dispersion in low-loss optical fibers leading to undistorted propagation of short pulses. In particular, they theoretically and numerically investigated propagation of light pulses using the one-dimensional nonlinear Schrödinger equation (1D-NLSE) both in normal and anomalous dispersion regimes [START_REF] Hasegawa | Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion[END_REF][START_REF] Hasegawa | Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion[END_REF]. Following this, Mollenaueur et al. experimentally reported for the first time in 1980 the propagation of solitons in a single-mode optical fiber [START_REF] Mollenauer | Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers[END_REF].

The discovery of the optical soliton immediately led to many attempts at applications in ultra-high-speed fiber communications, see Ref. [START_REF] Hasegawa | Optical soliton: Review of its discovery and applications in ultra-high-speed communications[END_REF] for a recent historical review on this subject. Many works have been realised to construct and to study all-optical transmission systems in which fiber losses are compensated by optical amplification, see e.g. Ref. [START_REF] Hasegawa | Solitons in optical communications[END_REF][START_REF] Taylor | Optical solitons: theory and experiment, no. 10 in Cambridge studies in modern optics[END_REF][START_REF] Ablowitz | Optical solitons: Perspectives and applications[END_REF][START_REF]Fiber-optic communication systems[END_REF][START_REF] Haus | Solitons in optical communications[END_REF] for relevant monographs and reviews that are available on this subject. In particular Mollenauer and Smith carried out pioneering experiments on long distance optical communications using solitons [START_REF] Mollenauer | Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain[END_REF]. They developed a fiber loop system into which a bit stream is loaded and recirculates, thereby simulating propagation over very large distances using a few hundred kilometers of fiber. Recirculating fiber loops are systems where the concept of all optical soliton transmission by periodic amplifications has been investigated in details [START_REF] Mollenauer | Soliton propagation in long fibers with periodically compensated loss[END_REF] and where soliton propagation has been achieved over distances as large as one million kilometers [START_REF] Nakazawa | 10 Gbit/s soliton data transmission over one million kilometres[END_REF].

From a physical point of view, solitons can be described as nonlinear wavepackets that propagate without changing shape, thanks to a balance between nonlinearity and dispersion of the wavepacket. From a mathematical point of view, solitons represent specific solutions of some nonlinear dispersive partial differential equations (PDEs), such as the 1D-NLSE, which are integrable and can be solved using the inverse scattering transform (IST) method. In the IST theory of the 1D-NLSE, a fundamental soliton is parameterised by its amplitude and velocity, each of these parameters being encoded into a single complex discrete eigenvalue λ 1 of the associated Zakharov-Shabat scattering problem [START_REF] Zakharov | Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media[END_REF][START_REF] Novikov | Theory of solitons: the inverse scattering method[END_REF][START_REF] Yang | Nonlinear waves in integrable and nonintegrable systems[END_REF][START_REF] Taylor | Optical solitons: theory and experiment, no. 10 in Cambridge studies in modern optics[END_REF]. Due to the integrabillity of the 1D-NLSE, the discrete eigenvalue λ 1 is a constant of the evolutionary motion, which means that it is preserved with the propagation distance in an optical fiber experiment. A fundamental soliton in the 1D-NLSE is also characterized by its position in space and by its phase. In the IST theory, these phase and position parameters, which generally change with the evolution variable, are encoded in a complex coefficient C 1 termed norming constant [START_REF] Zakharov | Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media[END_REF][START_REF] Gelash | Strongly interacting soliton gas and formation of rogue waves[END_REF]. Noteworthy, the concepts of the IST theory are now used in optical fiber communication where the nonlinear Fourier transform (NFT), originally introduced as eigenvalue communication [START_REF] Hasegawa | Eigenvalue communication[END_REF], is now used for optical data processing and fiber transmission, see. e.g. Ref. [START_REF] Turitsyn | Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives[END_REF].

As originally noted in the pioneering work by Zabusky and Kruskhal, solitons in nonlinear wave systems described by integrable PDEs exhibit the remarkable property that they retain their shape, amplitude and velocity upon interactions with other solitons [START_REF] Zabusky | Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States[END_REF]. The process of elastic collision between two solitons occurs without energy exchange between them. It has been studied experimentally in great detail in many physical systems [START_REF] Ikezi | Formation and Interaction of Ion-Acoustic Solitions[END_REF][START_REF] Nguyen | Collisions of matter-wave solitons[END_REF][START_REF] Aossey | Properties of solitonsoliton collisions[END_REF][START_REF] Slunyaev | Laboratory and numerical study of intense envelope solitons of water waves: Generation, reflection from a wall, and collisions[END_REF][START_REF] Mitschke | Experimental observation of interaction forces between solitons in optical fibers[END_REF][START_REF] Andrekson | Observation of multiple wavelength soliton collisions in optical systems with fiber amplifiers[END_REF][START_REF] Aitchison | Experimental observation of spatial soliton interactions[END_REF][START_REF] Shalaby | Experimental observation of spatial soliton interactions with a $\pi$/2 relative phase difference[END_REF][START_REF] Parker | Collisions of bright solitary matter waves[END_REF][START_REF] Stellmer | Collisions of Dark Solitons in Elongated Bose-Einstein Condensates[END_REF][START_REF] Xin | Evidence of Chaotic Dynamics in Three-Soliton Collisions[END_REF][START_REF] Xin | Intense Wave Formation from Multiple Soliton Fusion and the Role of Extra Dimensions[END_REF]. In particular, the elastic collision between two individual solitons parametrised by discrete eigenvalues λ 1 and λ 2 is accompanied by phase/position shifts that can be measured far outside the interaction region for each soliton. As it is well known in the IST theory, a soliton with the spectral parameter λ 1 that interacts with another soliton with the spectral parameter λ 2 experiences a position shift ∆ 1↔2 that only depends on λ 1 and λ 2 : ∆ 1↔2 = ∆(λ 1 , λ 2 ), and not on the position and phase parameters that are encoded in the norming constants C 1 and C 2 associated with each soliton [START_REF] Novikov | Theory of solitons: the inverse scattering method[END_REF][START_REF] Yang | Nonlinear waves in integrable and nonintegrable systems[END_REF][START_REF] Taylor | Optical solitons: theory and experiment, no. 10 in Cambridge studies in modern optics[END_REF]. In an interaction process involving three solitons with spectral (IST) parameters λ j ( j = 1, 2, 3), the IST theory predicts that the total position shift ∆ 1↔(2,3) experienced by the soliton with the spectral parameter λ 1 is equal to the algebraic sum of the shifts accumulated during the paired collisions: ∆ 1↔(2,3) = ∆ 1↔2 + ∆ 1↔3 [START_REF] Zakharov | Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]. This fundamental property that solitons exhibit particle-like properties with elastic, pairwise interactions arises from the integrable nature of the PDE describing their evolution in space and time.

Beyond regular solitons that are localised structures sitting on zero background, the 1D-NLSE supports a broad class of solutions referred to as breathers or solitons on finite background which includes the so-called Akhmediev, Peregrine and Kuznetsov-Ma breathers [START_REF] Dudley | Instabilities, breathers and rogue waves in optics[END_REF]. The interaction of such breather waves features an even richer dynamics that has been investigated experimentally in several domains including fiber optics and hydrodynamics systems [START_REF] Kibler | Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity[END_REF][START_REF] Xu | Ghost Interaction of Breathers[END_REF][START_REF] Gelash | Management of breather interactions[END_REF].

In this paper, we present experiments performed in a recirculating fiber loop in which we observe the interaction in space and time of some small ensembles composed of two or three bright solitons. Despite the presence of weak dissipation occurring over propagation, the observed dynamics is found to be nearly integrable, which means that soliton collisions are found to be nearly elastic without significant changes of their velocities due to the interactions with other solitons. Considering the collision between two bright solitons, we observe that their relative phase determines the shape of the field in the interaction region where their overlap is significant. On the other hand, the position (time) shift measured far outside the interaction region only depends on the amplitudes and velocities of the two solitons without being influenced by the phase-dependent dynamics of the field in the interaction region. Considering the interaction between three bright solitons, we observe that the position shift depends on the spectral IST parameters (amplitude and velocity) of the solitons without depending on their position and phase parameters. The space-time behaviours observed in the experiments are reproduced quantitatively well by numerical simulations of the 1D-NLSE. The space-time evolutions recorded in the experiments provide the clear evidence of almost perfectly integrable nonlinear wave dynamics that can be easily interpreted within the framework of the IST theory.

Experimental setup

The principle of operation of the experimental setup is schematically depicted in Fig. 1(a). It is essentially similar to the one reported in Ref. [START_REF] Kraych | Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics[END_REF][START_REF] Kraych | Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics[END_REF][START_REF] Copie | Spatiotemporal observation of higherorder modulation instability in a recirculating fiber loop[END_REF] and used for experimental investigations of the nonlinear stage of modulation instability of a plane wave in various configurations. A continuous wave field from a narrow-band laser at 1555 nm is temporally shaped using a combination of 20 GHz bandwidth electro-optic (intensity) modulator (I-EOM) and phase modulator (ϕ-EOM). A 12.5 GHz arbitrary waveform generator (AWG) drives the I-EOM to generate a train of short pulses and synchronously drives the ϕ-EOM to imprint a linear phase across each pulse with an alternating sign of the slope. Using this synchronous phase modulation scheme, the group velocity of each pulse can be controlled in an accurate way.

In the experiments reported in the present paper, the phase modulation is designed in such a way that adjacent pulses have opposite velocities in order to trigger their collision at the desired propagation distance. After the modulation stage, the wave field is amplified to reach Watt-level peak power and chopped down by an acousto-optic modulator (not shown for clarity) to create a 1 µs-long signal burst at a 20 Hz repetition rate which is used as input of the recirculating optical fiber loop.

The fiber loop itself is mostly composed of ∼ 5 km of single mode fiber (SMF) closed on itself by a 90/10 fiber coupler. The coupler is arranged in such a way that 10 % of the signal is initially injected in the loop and 90 % of the intra-loop power is recirculating. The optical signal circulates in the clockwise direction and at each roundtrip, 10 % of the circulating power is extracted and directed towards a photodetector (PD) coupled to a fast sampling oscilloscope leading to an overall 32 GHz detection bandwidth. Experimental data are recorded with the oscilloscope at a sampling rate of 160 GSa/s. They consist in a succession of sequences (one per roundtrip) that are subsequently processed numerically to construct single shot space-time diagrams showing the wavefield dynamics. Note that in addition to the short pulses, the input signal contains flat-top pulses of durations ranging from 500 ps to 10 ns (not illustrated in Fig. 1) that are used as reference markers for post-processing synchronisation and power calibration. Importantly, the losses accumulated over one circulation in the fiber loop are partially compensated using a counter-propagating Raman pump coupled in and out of the loop via wavelength division multiplexers (WDMs). As illustrated in Fig. 1(b) which shows the normalised evolution of the power in a typical experimental run, this enables reduction of the effective exponential power decay rate of the circulating field to α eff ∼ 1.92 × 10 -4 km -1 or equivalently ∼ 0.000 84 dB/km. This value can be normalised with respect to the typical nonlinear length of the experiments presented in this work which is L NL = 1/(γP 0 ) = 14.7 km (see caption of Fig. 2 for the values of the parameters). Accordingly, the losses expressed in unit of nonlinear length is

α eff L NL = 2.8 • 10 -3 .
As already mentioned in Sec. 1, recirculating fiber loops represent useful platforms for the research and development of long-haul transmission systems. In configurations typically implemented in the context of optical communications, the fiber loop system is usually first loaded with a bit stream which subsequently circulates over a given number of roundtrips before the detection/diagnostic of the optical field is made. The architecture of such loop systems is based on the use of several optical switches that must be appropriately activated at the loading, circulating and detection stages, see e.g. Ref. [START_REF] Bergano | Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers[END_REF]. Such a fiber loop device has been recently used for the space-time observation of Fermi-Pasta-Ulam-Tsingou recurrence in a longhaul optical fiber transmission system [START_REF] Goossens | Experimental realization of Fermi-Pasta-Ulam-Tsingou recurrence in a long-haul optical fiber transmission system[END_REF]. In contrast with these loops incorporating optical switches, our system is based on a fiber coupler which is used both to inject and to extract light, thereby using an architecture similar to that proposed in Ref. [START_REF] Desurvire | Raman amplification of recirculating pulses in a reentrant fiber loop[END_REF]. The recirculating fiber loop shown in Fig. 1(a) has the strong advantage of allowing real-time periodic recording of the evolution of the temporal dynamics of a signal at each roundtrip. In other words, our experimental setup permits the reconstruction of the spatiotemporal dynamics of a signal in single shot, which enables the observation of the dynamics of non-repetitive events.

Collision of two solitons

Using the experimental system described previously, we control with a satisfying accuracy the process of collision between individual solitons and observe the associated spatiotemporal dynamics in single shot. Firstly, we designed the pulses launched in the fiber loop to have a typical duration of 35.7 ± 0.9 ps FWHM and a peak power of 101 ± 4 mW in such a way that their initial shape is compatible with fundamental solitons. However, the soliton generation process is not perfect and our initial pulses unavoidably contain some radiative (nonsolitonic) content. Importantly, owing to the slight variations of the pulses parameters from shot to shot, the phase difference between adjacent pulses is not controlled in the experiment. The synchronous phase modulation imprints a linear phase slope to each pulse which results in a relative velocity difference between adjacent pulses that can be translated in relative linear drifts ∆V ∼ 1.2 ± 0.04 ps/km or equivalently a frequency detuning ∆ f = ∆ω/(2π) = ∆V/(2π |β 2 |) ∼ 8.62 ± 0.3 GHz. This is calculated a posteriori by measuring and fitting the trajectories of the solitons in the reconstructed spatiotemporal diagrams. A single experimental recording features several tens of soliton collisions from which the fluctuations given above are estimated. The input signal is engineered such that collisions occur after a propagation distance of ∼ 3000 km which allows the initial radiative content of the pulses to be cleared out (see the collision window in Fig. 1(b)). That way, we ensure that the collisions involve almost purely solitonic pulses with a limited contribution from dispersive waves. The characteristics of the temporal shape of the pulses are calculated via fitting with sech 2 functions (see caption of Figure 2(e-h) shows the spatiotemporal dynamics of four different collisions recorded experimentally within a single experimental run including a total of 50 collision events. Each collision event is presented over a propagation distance range of 500 km with the collision occurring after a propagation distance that has been artificially shifted for simplicity around 250 km (the actual collision occur after a propagation distance of ∼ 3000 km inside the loop). It is noteworthy that this propagation distance corresponds to 34 nonlinear lengths (L NL = 14.7 km). The space-time diagrams in Fig. 2 are presented in a reference temporal frame moving at a group velocity corresponding to the mean group velocity of the two pulses, which results in collisions that appear nearly symmetric with respect to the vertical axis (t = 0). The relative phase between colliding solitons not being controlled in the generation process, the interaction region takes a different shape depending on this phase that is randomly distributed between 0 and 2π.

We show in Fig. 2(a-d) numerical simulations of the 1D-NLSE expressed with physical variables

i ∂Ψ ∂z = β 2 2 ∂ 2 Ψ ∂t 2 -γ|Ψ| 2 Ψ, (1)
where Ψ is the complex envelope of the wavefield, β 2 and γ the group velocity dispersion and Kerr coefficients respectively, z the longitudinal variable describing propagation distance, and t the time in the reference frame introduced previously. We take as initial conditions two well separated solitons

Ψ(t, z = 0) = P 0 sech[(t -t 0 )/T 0 ]e -i(∆ω/2)t + P 0 sech[(t + t 0 )/T 0 ]e +i[(∆ω/2)t+∆ϕ] , (2)
with peak power P 0 and frequency detuning ∆ω fixed to the average value experimentally estimated before collision, their typical duration T 0 is set to the exact value leading to fundamental solitons (see parameters' values in the caption of Fig. 2). Spatiotemporal evolution for four particular values of ∆Φ are illustrated: in-phase (∆Φ = 0) collision (a) is translated into a strong peak of the optical intensity whereas the pattern resulting from out-of-phase (∆Φ = π) collision (c) creates the false impression that solitons exactly bounce (or repel) each other while they still actually pass through one another. Intermediate phase differences ∆Φ = π/2 and 3π/2 respectively lead to similar evolutions where a peak optical intensity appears but shifted towards the left (right) with respect to the collision center.

Experimental observations presented in Fig. 2(e-h) have been respectively selected for their striking quantitative agreement with the simulations illustrated above. Each of the experimental realisations, not shown here for brevity, matches with numerical simulation with appropriate phase difference. This confirms that we indeed realise the collision of solitonic pulses, the coherence of which is well preserved all along their interaction. Note that numerical simulations do not take account of dissipation that is present in experiment but we stress that the counterpropagating Raman amplification limits the power loss over the 500 km shown in Fig. 2(e-h) to less than 10 % which barely affects the dynamics with respect to the conservative case.

As exposed in the introduction, a remarkable feature of solitons is that they emerge unaltered from their pairwise interaction although their relative phase difference drastically impacts the exact dynamics of the collision as confirmed by the experiments presented Fig. 2. However, the collision is accompanied by a time shift that is solely parametrised by the IST spectral parameters of the individual solitons (amplitude and velocity). Our experiments enable clear observation of the collisioninduced temporal shift as presented in Fig. 3. We first take the case of the collision between balanced solitons (i.e. solitonic pulses with comparable amplitudes) similar to those illustrated in Fig. 2 and focus on the temporal shift experienced by one of the soliton. To do so, it is convenient to operate a change of temporal reference frame to one that travels at the group velocity of the soliton under consideration. That way, the latter appears as following a straight vertical trajectory when not interacting with other solitons. This operation is done numerically a posteriori for ease of visualisation and does not modify the physics of the system. Figure 3(b) shows the space-time dynamics of a collision in the same conditions as in Fig. 2 but plotted in the reference frame of the right (steady) soliton. The vertical dashed white line emphasises the trajectory of the soliton before collision. After collision with the left (moving) soliton, it has clearly experienced a temporal shift towards negative times of the or-der of the soliton's width while retaining almost perfectly its velocity close to zero after the interaction, thus demonstrating that the observed collision is of elastic nature.

The temporal trace recorded ∼ 250 km after the collision is shown in Fig. 3(a) in which the collision-induced shift is marked by the arrows. The two pulses are well fitted by sech 2 functions (dashed lines) whose parameters still confirm their solitonic nature. Note that the signal exhibit significant noise due to the rather small optical power detected in single shot (we recall that less than 10 % of the circulating power is detected so the ∼ 50 mW peak power indicated in Fig. 3(a) correspond effectively to pulses of less than ∼ 5 mW peak power incident on the photodiode). In this configuration, the moving soliton also experiences a shift of almost the same magnitude but opposite sign (as visible in the inset showing the same collision in the reference frame that travels at the mean group velocity of the two pulses).

As previously mentioned, we have recorded 50 similar realisations of this scenario of collision which allows us to evaluate an average shift of the solitons with a certain error interval. To do so, we extracted from the spatiotemporal diagrams the linear trajectories of the two solitons before and after interaction for each realisation. The collision-induced shift is then calculated from the parameters of the linear trajectories. Note that parameters of each collisions (amplitude, duration of the solitons and their relative velocities) are slightly different such that the expected temporal shift also varies from shot to shot. Also, care must be taken since small changes of solitons' velocity are detected between before and after collision which is a consequence of weak non-integrability of our system. This prevents straightforward estimation of the temporal shift which is formally defined at asymptotically long propagation distance from the interaction [START_REF] Zakharov | Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media[END_REF][START_REF] Novikov | Theory of solitons: the inverse scattering method[END_REF]. To limit the impact of this velocity change, we estimated the shift 75 km after the point of maximum interaction (marked by the horizontal dotted white line) which is sufficiently large for the solitons to separate but not too much for the change of velocity to dramatically alter the measurement of the shift. Using this methodology, the average temporal shift measured is 29.1 ± 4.2 ps. For comparison, numerical simulations of the 1D-NLSE taking account of the realistic variation of all the relevant parameters give a shift of 30.7 ± 0.9 ps. Eventhough our experimental estimation is associated to a larger uncertainty, it remarkably agrees with realistic prediction from the integrable 1D-NLSE model.

To complement the result shown in Fig. 3(a-b), we present an example of a so-called imbalanced collision, i.e. a collision between solitons of significantly different peak power. The spatiotemporal evolution of the pulses plotted in the reference frame moving at the velocity of the weakest soliton is shown in Fig. 3(d) and exhibit a similar dynamics as the one described previously. For this particular realisation, the weak soliton clearly experiences a large temporal shift and keeps its velocity, demonstrating once again that the observed imbalanced collision is of elastic nature. The inset in Fig. 3(d) additionally illustrates that the weak soliton emerges from the collision with a larger shift than the strong one.

Three solitons interaction

In this last section, we report experiments on the interaction process between three solitons. In this configuration, the IST theory predicts that the total shift experienced by one of the soliton is the algebraic sum of the shifts acquired over the paired collisions. The result of the interaction between three solitons is thus fully determined by the three constitutive pairwise collisions. Importantly, the relative positions and phase of each soliton before the interaction do not influence the time shift that are measured well after the interaction among the three solitons.

To illustrate this, we designed a set of experiments where three solitons of nearly the same amplitude (comparable to that shown in Fig. 2) interact and we show the results in Fig. 4. The velocity and initial position of the solitons are precisely controlled to generate three distinct interaction scenarios. In all cases, frequency detuning (relative velocity differences) between adjacent pulses is constant and equal to ∆ f ∼ 4.24 GHz (∆V ∼ 0.59 ps/km). Also, because the total interaction between the three solitons takes place over a larger span of distance we engineered the input signal so that it occurs at shorter propagation distance compared to the previous experiments (centered around 1000 km of propagation instead of 3000 km) which provides better control over the solitons parameters. This means that a greater dispersive content is present in the interaction area with respect to previous experiments. The dynamics of the solitons is therefore visibly impacted by this interplay.

In the first case illustrated in Fig. 4(a), the solitons are initially equally spaced leading to an interaction that is very localised in space and time. In fact, individual pairwise collisions are not distinguishable and the exact collision dynamics depends on the relative phase differences between the three solitons. This results in a greater complexity of the interaction region as compared to the dynamics associated with a two soliton interaction pictured in Fig. 2. Noticeably, the central soliton emerges almost unshifted from the interaction since in this configuration, the shifts induced from the two indivual collisions is in principle of the same magnitude but of opposite sign. The two other solitons experience a total shift larger than those observed in Fig. 3 because they both cumulate two shifts of same sign. Also, it is clear that the velocity of the central soliton has slightly changed during the interaction. The weak but non null effective losses and the periodic evolution of the power imposed by the recirculating loop configuration break the integrability of the system to some extent and are involved in the slightly inelastic features observed in the experiments.

If the in-between soliton is now initially slightly offsetted, the interaction window expands and pairwise separable collisions start to become apparent though individual trajectories still cannot be identified as shown in Fig. 4(c). Again, the dynamics of the interaction is highly sensitive to the relative phases between the solitons. However, the central soliton comes out of the interaction quasi unperturbed in terms of position and velocity. Finally, for a large offset of the in-between soliton the three individual pairwise collisions are perfectly identified (see Fig. 4(e)) and each exhibit a dynamics comparable to those reported in Fig. 2 and3. Paying specific attention to the in-between soli-ton, the latter experiences a clear positive shift from the first collision that is almost perfectly cancelled after the second collision. This is overall remarkably translated by its broken but straight trajectory. The experimental observations presented in Fig. 4 illustrate qualitatively yet with a great level of details the full complexity of the particle-like behaviour of solitons' pairwise interaction.

Conclusion

In conclusion, a recirculating fiber loop has been used to realize the single-shot observation of the interaction in space and time of two and three bright solitons. To the best of our knowledge, it is the first time in nonlinear fiber optics that the fundamental process of soliton collision is observed both in singleshot (without averaging) and in space and time. In our experiments, collisions between two solitons have been found to be almost perfectly elastic, without significant change in the velocity of the solitons after their interaction. The position (time) shift measured in experiments is quantitatively well described by numerical simulations of the 1D-NLSE. Our experiments have also provided the evidence that the position (time) shifts arising from the interaction among three solitons are determined by the elementary pairwise interactions, as it is well known in the IST theory.

The recirculating fiber loop is a weakly dissipative experimental platform that is well adapted to the space-time observation of the evolution of solitons and solitary waves. We plan to use it in the near future to investigate theoretical questions behind the spectral theory of soliton gases [START_REF] El | Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation[END_REF][START_REF] Suret | Soliton refraction through an optical soliton gasPublisher[END_REF] and generalized hydrodynamics, the hydrodynamic theory of many-body quantum and classical integrable systems [START_REF] Doyon | Lecture notes on Generalised Hydrodynamics[END_REF].

Figure 1 :

 1 Figure 1: (a) Principle of operation of the experimental setup. (b) Normalised evolution of the circulating optical power computed from the experimental recording from which data presented in Fig. 2 are extracted. Light blue dots are experimental values calculated at each roundtrip and the black dashed line is the exponential fit giving α eff ∼ 1.92 × 10 -4 km -1 . cw: Continuous Wave, I-EOM: Intensity-Electro-Optic Modulator, ϕ-EOM: Phase-Electro-Optic Modulator, EDFA: Erbium Doped Fiber Amplifier, WDM: Wavelength Division Multiplexer, PD: Photodetector, OSC: Oscilloscope.

Fig. 2

 2 for the expression) 250 km before collision and are as follows: P 0 = 55.3 ± 8 mW, T 0 = 19.9 ± 3 ps which gives N = γP 0 T 2 0 /|β 2 | = 1.11 ± 0.1 [43].

Figure 2 :

 2 Figure 2: Spatiotemporal dynamics of the collision of two identical solitons, impact of the phase difference. (a-d) Numerical simulations of the 1D-NLSE for 4 remarkable relative phase differences. (e-h) Corresponding experimental observations selected from a batch of 50 collisions. Vertical axis is shifted for direct comparison since collisions actually occur after ∼ 3000 km of propagation. The color scale is common to all the presented results. Initial condition for the simulations is of the form Ψ(t, z = 0) = √ P 0 sech[(tt 0 )/T 0 ]e -i(∆ω/2)t + √ P 0 sech[(t + t 0 )/T 0 ]e +i((∆ω/2)t+∆ϕ) , with P 0 = 55.3 mW, β 2 = -22 ps 2 /km, γ = 1.23 W -1 km -1 , T 0 = |β 2 |/(γP 0 ) ∼ 18 ps, ∆ω/(2π) = 8.62 GHz, t 0 = 168 ps.

Figure 3 :

 3 Figure 3: Experimental observation of collision-induced soliton shift in (a-b) balanced and (c-d) imbalanced configurations. The space-time diagrams are plotted in the reference frame of the right soliton for clarity. (a, c) Temporal traces extracted from the top of the space-time diagrams with sech 2 fits superimposed. Vertical dashed white lines in (b) and (d) highlight the linear trajectories of the soliton before collision. Insets in (b) and (d) show the same collisions in the temporal reference frame that travels at the mean group velocity of the two pulses.

Figure 4 :

 4 Figure 4: Spatiotemporal dynamics of three solitons interaction for various initial positions of the solitons (from evenly (a) to largely unequally (c) spaced). Dashed white lines highlight the linear trajectories of the solitons before interaction. (b, d, f) Temporal traces at the bottom of the corresponding space-time diagrams (i.e. before interaction).

Acknowledgment

The authors are very grateful to H. Damart, and G. Dekyndt for the technical support and to A. Amo for providing access to the fast waveform generator. This work has been partially supported by the Agence Nationale de la Recherche through the StormWave (ANR-21-CE30-0009) and SOGOOD (ANR-21-CE30-0061) projects, the LABEX CEMPI project (ANR-11-LABX-0007), the Ministry of Higher Education and Research, Hauts de France council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Région (CPER Photonics for Society P4S). The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves" when part of the work on this paper was undertaken.


T Cavalié

J Lunine

O Mousis

A subsolar oxygen abundance or a radiative region deep in Jupiter revealed by thermochemical modelling

Jupiter's deep abundances help to constrain the formation history of the planet and the environment of the protoplanetary nebula. Juno recently measured Jupiter's deep oxygen abundance near the equator to be 2.2 +3.9 -2.1 times the protosolar value (2σ uncertainties). Even if the nominal value is supersolar, subsolar abundances cannot be ruled out. Here we use a stateof-the-art one-dimensional thermochemical and diffusion model with updated chemistry to constrain the deep oxygen abundance with upper tropospheric CO observations. We find a value of 0.3 +0.5 -0.2 times the protosolar value. This result suggests that Jupiter could have a carbon-rich envelope that accreted in a region where the protosolar nebula was depleted in water. However, our model can also reproduce a solar/supersolar water abundance if vertical mixing is reduced in a radiative layer where the deep oxygen abundance is obtained. More precise measurements of the deep water abundance are needed to discriminate between these two scenarios and understand Jupiter's internal structure and evolution.

Introduction

Giant planets are the true architects of planetary systems given their relatively short formation timescales compared with terrestrial planets, their gravity and migration. Their deep composition holds part of the key to understanding the formation of giant planets in planetary systems, along with other measurements such as their gravity and magnetic field. It can also help to constrain the processes that led to the condensation or trapping of the primordial ices in the protosolar nebula [START_REF] Helled | [END_REF]Bar-Nun et al. 1988).

The Galileo probe plunged in the atmosphere of Jupiter in 1995, reaching the 22 bar level, and measured its elemental and isotopic composition. The main result is the relatively uniform enrichment in volatiles by a factor of two to four with respect to the protosolar value (Wong et al. 2004). The only element that did not follow this trend, and for which there was no internal process to explain its depletion, is oxygen. The accepted hypothesis is that Galileo entered a dry area where water, the main carrier of oxygen at depths, was still not uniformly mixed at 22 bar. The oxygen measurement from Galileo has thus been mostly accepted as a lower limit ever since.

One of the reasons the Juno mission was designed was to fill this gap left by Galileo regarding the deep oxygen abundance in Jupiter. The idea was to observe the microwave spectrum of Jupiter at various emission angles to retrieve the global water abundance (Janssen et al. 2005). The main difficulty then resides in the dependency of the spectrum on the temperature profile and on absorbers other than water, such as ammonia. While Juno confirmed previous observations that ammonia was depleted below its condensation level (de Pater 1986), it surprisingly showed that this depletion was latitude dependent and that it extended as deep as ∼50 bar (Li et al. 2017), except near the equator. Li et al. (2020) used the microwave spectra seen by Juno at this latitude to retrieve the vertical water profile of Jupiter at its equator and derive its deep oxygen abundance. They found that the oxygen was nominally enriched by a factor of 2.7 +2.4 -1.7 times the protosolar value but the 1σ lower bound was weakly determined, and they state that subsolar values are possible. Helled et al. (2022) emphasized that these error bars are sufficiently large that the depletion seen by Galileo may not be a local anomaly, but could instead reflect a global depletion. In what follows, we use protosolar abundances determined by Lodders (2021). The Jovian oxygen abundance of Li et al. (2020) then translates into 2.2 +2.0 -1.4 times the protosolar value. Long before Galileo and Juno, the reported (Beer 1975) detection of CO in the troposphere of Jupiter with an abundance higher than thermochemical equilibrium predictions by orders of magnitude triggered the development of numerous models to constrain the deep oxygen abundance by solving the balance between thermochemistry and vertical mixing (for example, Lunine & Stevenson 1987;Fegley & Prinn 1988;Yung et al. 1988). Below the cloud level, in the deep troposphere, CO and H 2 O are in thermochemical equilibrium resulting from the equilibrium reaction CO + 3H 2 = H 2 O + CH 4 . As the temperature decreases with height, the thermochemical equilibrium is shifted in favour of H 2 O. The detection of CO in the upper troposphere then demonstrates that thermochemical equilibrium is quenched by vertical mixing when the mixing timescale becomes shorter than the chemical conversion timescale. Although the initial studies cited above focused on identifying the rate-limiting reaction, more recent work incorporated comprehensive chemical schemes (Visscher et al. 2010;Wang et al. 2016). By probing the abundance of CO in the upper troposphere and modelling thermochemistry and diffusion, one can then reconstruct the vertical profile of the species and tie it back to the deep water (and thus to the deep oxygen) abundance in the planet.

We used a 1D thermochemical and diffusion model (Cavalié et al. 2017) to fit a CO upper tropospheric mole fraction of 0.9 ± 0.3 ppb, as measured by previous studies (Bézard et al. 2002;Bjoraker et al. 2018). Although the chemical scheme used by Cavalié et al. (2017) was validated globally by the combustion industry over a wide range of temperatures and pressures, Wang et al. (2016) showed that the model failed to agree with most competing chemical schemes regarding the quench chemistry of CO in giant planets. Moses (2014) identified the main source of the disagreement as the kinetics of the conversion reaction of methanol into the methyl radical and water measured by Hidaka et al. (1989). This led Venot et al. (2020) to fully update their CH 3 OH submechanism by adopting the work of Burke et al. (2016), which noticeably provided an explicit logarithmic dependence on pressure of some reaction rates, and removing of the controversial reaction of Hidaka et al. (1989) that produced the methyl radical from methanol (and thus from CO). This updated scheme21 then destroys less CO than that of Venot et al. (2012), in agreement with other studies (Wang et al. 2015(Wang et al. , 2016;;Moses 2014) (Extended Data Fig. 1). As a consequence, a given CO abundance requires a lower deep oxygen abundance with the new scheme (Methods).

Results

With our nominal parameter set (Methods), we found that a deep oxygen abundance of 0.3 times the protosolar value is required in Jupiter's deep troposphere. The corresponding vertical profiles are displayed in Fig. 1. In our simulations, we allowed three parameters to vary to fit the tropospheric CO and estimate the uncertainty of the deep oxygen abundance with respect to the methane mole fraction y top CH 4 at the top of the troposphere and the vertical eddy diffusion coefficient K zz . We allowed these two parameters to vary within their respective uncertainty ranges; that is, y top CH 4 = 0.00204 ± 0.0050 (ref.

3) and K zz = 10 8 cm 2 •s -1 within a factor of two (Wang et al. 2016;Grassi et al. 2020). When y top CH 4 was allowed to vary within its uncertainty range and K zz was fixed to 10 8 cm 2 •s -1 , we obtained our nominal fits to the CO mole fraction with a deep oxygen abundance of 0.3 +0.3 -0.2 times the protosolar value (Fig. 2). Conversely, when we varied K zz and fixed y top CH 4 to 0.00204, we obtained a deep oxygen abundance of 0.3 +0.3 -0.2 times the protosolar value (Fig. 3). The deep carbon abundance was 3.2 ± 0.8 times the protosolar value, depending on the adopted value of y top CH 4 . Finally, if we accounted for the uncertainty ranges of K zz and y top CH 4 , then the result for the deep oxygen was 0.3 +0.5 -0.2 times the protosolar value. According to our model, oxygen is subsolar and the C/O ratio is 6 +10 -5 , suggesting that Jupiter could have a surprisingly carbon-rich envelope.

The deep oxygen abundance in giant planets has long been debated as it is one of the key elements pertaining to the formation of solids and trapping of more volatile species in the protosolar nebula, which are later released to the growing envelopes of the giant planets (Owen et al. 1999;Gautier et al. 2001). Remote sensing observations and in situ measurements provide values ranging from 0.25 to 4.2 times the protosolar value. The Galileo measurement, when translated into an O/H abundance using the protosolar composition of Lodders ( 2021), is 0.37 ± 0.12 (Wong et al. 2004). This value has often been considered a lower limit because the water abundance was still increasing in the measurements when the signal from the Galileo probe was lost, and the probe entered a dry region of Jupiter's atmosphere (a 5 µm hotspot). Bjoraker et al. (2018) found that their Great Red Spot spectrum around 5 µm was best fitted by fixing the water cloud base at 5 ± 1 bar, which translates nominally into a near-protosolar deep oxygen abundance, but the 1σ range encompasses 0.3 to 3 times the protosolar value. Finally, a deep water abundance of 2.2 +2.0 -1.4 times the protosolar value was obtained (Li et al. 2020) with the Juno Microwave Spectrometer (MWR) in the sole equatorial region where vertical mixing and meteorology seem to maintain a well-mixed atmosphere throughout the probed gas column. Whether this value is representative of the whole planet remains to be verified, especially given the unexpected results for the meridional distribution of ammonia and its depletion at pressures lower than 30 bar (Li et al. 2017) and the role water seems to play in this depletion (Guillot et al. 2020b). The fact that Li et al. (2017) found about half the deep ammonia measured by Galileo could indicate that this oxygen measurement is also not representative of the global value. In any case, their oxygen abundance retrieval presents an error bar in which the lower end is more weakly determined than the higher end. The lower 2σ limit lies at 0.1 times the protosolar value. 3 times the protosolar value). The dashed lines correspond to alternative Jupiter abundance and T profiles obtained with a more sluggish mixing (K zz is reduced to 2.5 × 10 6 cm 2 •s -1 ) to match the nominal Juno O/H value (2.2 times the protosolar value). This lower K zz is then indicative of a radiative region located around the quench level of CO (that is, at pressure p ∼ 0.4-0.5 kbar and T ∼ 1000 K). The dash-dotted lines correspond to a final model in which a deep radiative layer with reduced K zz is inserted to match solar oxygen and produce 0.9 ppb CO in Jupiter's upper troposphere: K zz is set to 1 cm 2 •s -1 between T = 1400 K and 2200 K and transitions linearly with T towards our nominal value of 10 8 cm 2 •s -1 at 970 K to ensure that PH 3 and GeH 4 are quenched at ∼800 K. This is a non-unique solution.

Discussion

We nominally found a subsolar deep oxygen abundance in Jupiter. Our results, compatible with the range obtained from similar modelling in Visscher et al. (2010), indicate marginal agreement with the Juno MWR analysis (Li et al. 2020) as discussed above. Cloud models often require one times solar oxygen or higher (Iñurrigarro et al. 2022), but can also accommodate subsolar oxygen (Hueso & Sánchez-Lavega 2001). More problematic are the results from lightning data. While overall modelling of lightning frequency (Aglyamov et al. 2021) permits a subsolar value, the Galileo detection of a potentially deeper lightning flash would imply a water enrichment exceeding solar values (Dyudina et al. 2002). Decreasing K zz from the nominal value of 10 8 cm 2 •s -1 to 2.5 × 10 6 cm 2 •s -1 would raise the deep oxygen abundance from 0.3 to 2.2 times solar (Fig. 1 and Extended Data Fig. 2). However, such a low level of vertical mixing must be confined to the altitudes below which temperatures reach 1000-1100 K, where CO quenching starts, as demonstrated in Fig. 1; at 800 K it must be at or close to our nominal value to fit the data on other disequilibrium species such as PH 3 and GeH 4 (Wang et al. 2016;Grassi et al. 2020), as described in Methods. Our revised chemical network, taken from Venot et al. (2020), probably still suffers from some uncertainties in the reaction rates. However, any change in the rates to produce a solar or supersolar water abundance would require the deep oxygen abundance derived in the ice giants (Venot et al. 2020) to increase as well, raising the problem of the consistency between the deep oxygen abundance and deep D/H ratio when compared with that found in Oort cloud comets (Ali-Dib et al. 2014). Our results therefore require either (1) a carbon-rich envelope in Jupiter or (2) a deep layer of reduced vertical mixing.

Option (1) was already proposed by Mousis et al. (2012), assuming that the Galileo O determination corresponds to the bulk abundance. A high C/O ratio in Jupiter's envelope such as that of 6 +10 -5 found here resulting from a relatively low oxygen abundance was also proposed by previous studies (Lodders 2004;Mousis et al. 2019). In one model (Mousis et al. 2019), the radial drift of pebbles through the amorphous-to-crystalline-ice transition front in the protoplanetary disk releases carbon-rich supervolatiles into the gas while stranding water in the ice. Other scenarios include the agglomeration of building blocks condensed in the vicinity of the condensation lines of C-rich volatiles by the growing Jupiter. The C/O ratio in icy solids formed in those regions is expected to rise steeply, as found by Mousis et al. (2021a) to explain the water-poor composition of Comet C/2016 R2 (PanSTARRS). Another study (Mousis et al. 2021b) suggested that a wide range of protosolar nebula compositions can match Jupiter's metallicity, including several types of icy phase (clathrates and pure condensates). It should be noted that current Jupiter formation and structure models predict a low-metallicity envelope (Helled et al. 2022;Schneider & Bitsch 2021). Option (2) implies a dramatic decrease in vertical mixing at temperatures higher than 1000-1100 K, corresponding to a pressure level of roughly 0.6 kbar. Various mechanisms may lead to stable regions throughout Jupiter's deep atmosphere, and one mechanism that corresponds roughly to the p-T region here is a radiative zone extending from 1200-2000 K (Guillot et al. 1994). Such a zone is the result of low opacity, obtained only in the case of a depletion in the alkali metals [START_REF] Guillot | Jupiter. The Planet, Satellites and Magnetosphere[END_REF]), and Juno MWR data hint at such a depletion [START_REF] Bhattacharya | AAS/Division for Planetary Sciences Meeting Abstracts[END_REF]. It is therefore plausible that the vertical mixing, represented by K zz in our model, is very low in the region just below the CO quench level, begins to increase at that level and reaches its full convective value by 900 K (300 bar) where PH 3 and GeH 4 quenching determine our nominal value for K zz . A model in which K zz was set to the very low value of 1 cm 2 •s -1 (it could in principle be as low as the molecular diffusivity) at temperatures higher than 1400 K and to 10 8 cm 2 •s -1 at temperatures lower than 970 K and then interpolated logarithmically between the two levels produced satisfactory results with solar oxygen (Fig. 1 and Extended Data Fig. 2).

Either of the two possibilities -depleted oxygen or a deep radiative zone -would be important for understanding the nature of Jupiter's interior below its visible atmosphere. Further analysis of Juno MWR data during the extended mission, particularly at the longest wavelength channel, will help to distinguish between them. The results for Jupiter also provide important context for the elemental composition of giant planets in our Solar System, and beyond. In situ probes, despite their inherent limitation of a single entry-point, would provide invaluable compositional data for Saturn and the ice giants, especially regarding noble gases, as presented other works (Mousis et al. 2014(Mousis et al. , 2018)). Cavalié et al. (2020) have shown how the use of thermochemical simulations can increase the science return of in situ probe measurements.

Methods

Thermochemical model

We used the 1D thermochemical and diffusion model initially developed in Venot et al. (2012) for warm exoplanets, and adapted in Cavalié et al. (2014Cavalié et al. ( , 2017) ) to study the deep oxygen abundance in Uranus and Neptune. The model solves the continuity equation as a function of time at each altitude, for 111 carbon, oxygen and nitrogen species through 1912 reactions.

Our model required boundary conditions regarding the composition of the upper troposphere. We took the upper tropospheric mole fraction of He and CH 4 from von Zahn et al. (1998) and Wong et al. (2004), respectively y top He = 0.1359 ± 0.0027 and y top CH 4 = 0.00204 ± 0.0050, both resulting from the Galileo probe measurements. For CO, we adopted an upper tropospheric mole fraction of 0.9 ± 0.3 ppb, following measurements from Bézard et al. (2002) and Bjoraker et al. (2018).

To constrain the deep oxygen abundance of Jupiter from upper tropospheric CO observations, we also needed to make assumptions on the vertical mixing and on the temperature profile. Following Wang et al. (2015) and recent Juno results of Grassi et al. (2020) on tropospheric abundances of disequilibrium species, we nominally adopted a vertical eddy mixing coefficient of 10 8 cm 2 •s -1 with a factor of two uncertainty. Even though Juno observations of NH 3 (Li et al. 2017) and models to explain downward ammonia transport (Guillot et al. 2020b,a) show that chemical transport does not obey a pure diffusion equation between 0.1 and tens of bars, we assumed that this is the case at several hundred bars in the more homogeneously mixed deeper troposphere where the CO thermochemistry is quenched. Our temperature profile follows the Galileo profile (Seiff et al. 1998) within 1 K down to 22 bar and we extrapolated temperatures using a wet adiabat from the diffusion-dominated upper levels down to the deep levels where thermochemical equilibrium prevails. The deep oxygen abundance measured with Juno for Jupiter (Li et al. 2020) is not high enough to produce a radiative layer, resulting from a mean molecular weight gradient at the water condensation level in which the temperature would sharply increase, as opposed to the case of the ice giants (Cavalié et al. 2017). We stopped our temperature extrapolations at 1700 K, much deeper than the levels where thermochemistry is quenched by vertical mixing. We ensured that our results reached steady state with an integration time of 10 10 s.

We fixed the deep nitrogen abundance to ∼4 times the protosolar value to reproduce the Galileo measurement (Wong et al. 2004). The abundance of N 2 was then ∼10 -5 in the upper troposphere. Our model results regarding oxygen were, however, insensitive to this value, because the nitrogen and oxygen chemistries were mostly uncoupled. We adopted the protosolar abundances reported in Lodders (2021) throughout this Article. We used them to express our model results and to convert previous results to a common scale.

Chemical scheme

The chemical network in Venot et al. (2012) is a C/H/O/N mechanism initially validated for the combustion industry to help to understand the combustion of fuels in car engines and thus limit their environmental impact. It is based on a C 0 -C 2 mechanism to which a nitrogen reaction base was added. It comprises 105 species and 1926 reactions. Although it was validated against experiments for pressures ranging from 0.01 bar to several hundred bars and for temperatures ranging from 300 to 2500 K, Wang et al. (2016) showed that the conversion of H 2 O into CO was less efficient in Jupiter with the network shown in Venot et al. (2012) compared with several others (see their fig. 17), resulting in CO abundance 10 times lower than in simulations involving competing networks. A similar issue with CH 4 ?CO chemistry had been found in applications to hot Jupiters by Moses (2014), even though Venot et al. (2020) found even more compelling differences in cooler planets. Moses (2014) further narrowed down the main difference in the networks to the kinetics of methanol through the H + CH 3 OH = CH 3 + H 2 O reaction. The chemical rate of this reaction had been set in Venot et al. (2012) to that estimated by Hidaka et al. (1989), but was found to be over-estimated by Visscher et al. (2010) in their work on Jupiter's thermochemistry. This led Venot et al. (2020) to fully revise the CH 3 OH sub-network of their chemical scheme. They adopted experimental data (Burke et al. 2016) that are remarkable in several aspects. First, the reaction of Hidaka et al. (1989) is no longer explicitly present in the network. This does not prevent CH 3 OH from being destroyed and producing CH 3 and H 2 O, but this is achieved through other destruction pathways. Second, the kinetic rates of several reactions of this network (more specifically those involving methoxide and the methyl radical) have an explicit logarithmic dependence with pressure defined for up to five pressure decades, which thus increase the accuracy and robustness of the kinetics over this wide range of pressure conditions. This new scheme was validated (Venot et al. 2020) over a wide range of temperature and pressure conditions and showed improved agreement with experimental data. We have produced a CO profile for Jupiter in the same conditions as in fig. 17 of Wang et al. (2016). It is shown in Extended Data Fig. 1 and it fully agrees with the profiles presented in fig. 17 of Wang et al. (2016). When applied to the ice giants (Venot et al. 2020) this scheme resulted in the production of the observed CO in ice giants with substantially lower oxygen enrichments.

As stated by Moses (2014), "the exact mechanism involved with CH 4 -CO quenching in reducing environments has not been strictly identified". This is caused by the high nonlinearity and high coupling between the various chemical reactions of the scheme. It is thus not possible, as found in initial studies that assumed a rate-limiting reaction in the CO destruction mechanism (Fegley & Prinn 1988;Yung et al. 1988;Bézard et al. 2002), to easily identify a single reaction and quantify uncertainties on the results from the uncertainty on the rate of this reaction. A methodology of uncertainty propagation and global sensitivity analysis has been developed (Dobrijevic et al. 2010), but it required running several hundred simulations following a Monte Carlo scheme. Applying this methodology is beyond the scope of this study.

Deep radiative region in Jupiter?

It has been shown (Wang et al. 2016) that PH 3 and GeH 4 are quenched at ∼700-800 K (p ≈ 0.1 kbar; see their figs. 6 and 11) and the abundances observed with Juno JIRAM (Grassi et al. 2020) in the upper troposphere imply that K zz ≈ 10 7 -10 9 cm 2 •s -1 from PH 3 and ∼10 8 cm 2 •s -1 from GeH 4 at this level. For uniform K zz > 10 7 cm 2 •s -1 our model predicts a subsolar oxygen to fit the observed abundance of CO (Extended Data Table 1).

The first hint that Jupiter's troposphere could harbour a radiative region in the vicinity of the layers where CO is quenched (T ≈ 1000-1100 K; Fig. 1) was then obtained when the deep oxygen abundance was raised to the Juno MWR nominal value of 2.2 times protosolar. This required us to decrease the vertical mixing K zz from its nominal value of 10 8 cm 2 •s -1 to 2.5 × 10 6 cm 2 •s -1 (Fig. 1). Fitting the whole range of 1σ uncertainties of the Juno measurement led to the K zz value reported in Extended Data Table 1. We thus found that convection needs to be less efficient in the CO quench region, with K zz lowered by a factor 10 to 100, to obtain solar-to-supersolar oxygen.

A decrease in the Rosseland opacity due to hydrogen and helium opacity between 1200 and 4000 K in Jupiter can result in a radiative region, as initially pointed out by a previous study (Guillot et al. 1994). It was subsequently confirmed [START_REF] Guillot | Jupiter. The Planet, Satellites and Magnetosphere[END_REF]) that this region may exist between p ≈ 1.5 kbar (T ≈ 1400 K) and p ≈ 8.0 kbar (T ≈ 2200 K) provided that the Jovian atmosphere is also depleted in alkali metals. This depletion seems to be confirmed by recent Juno MWR observations [START_REF] Bhattacharya | AAS/Division for Planetary Sciences Meeting Abstracts[END_REF]). If such a radiative region exists, vertical heat transport and chemical mixing would strongly be inhibited, and our assumption of a vertically uniform K zz would not hold in this region. A previous work (Cavalié et al. 2017) investigated the effect of an insulation layer produced in the ice giants by the rapid change in mean molecular weight gradient where water condenses. Despite the presence of this insulation layer in the altitudes where CO is quenched and vertical transport prevails, they found very limited impact on their results, mostly because the radiative layer was very thin. The radiative layer in Jupiter, which is of a different origin than that in the ice giants, may extend from 1.5 to 8 kbar [START_REF] Guillot | Jupiter. The Planet, Satellites and Magnetosphere[END_REF]). Even if the radiative layer itself has a limited impact on the CO O/H (× the protosolar value) Required K zz in the CO quench region 0.3 1×10 8 cm 2 •s -1 0.8 2.5×10 7 cm 2 •s -1 2.2 2.5×10 6 cm 2 •s -1 4.2 4×10 5 cm 2 •s -1

Table 1: Relationship between Jupiter's deep O/H and the K zz required in the quench region of CO to fit the observed upper tropospheric CO mole fraction. This essentially shows that the higher the deep oxygen abundance, the more inhibited the mixing to produce the right abundance of CO.

profile, because it is located in the region where thermochemical equilibrium between CO and H 2 O prevails over vertical transport, we needed to assess the effect of such a layer and how it connects to upper layers in our simulations.

We found that our model could reconcile solar oxygen with the observed tropospheric CO by adding the presence of a deep radiative layer in which K zz could be as low as the molecular diffusivity. We set K zz to 1 cm 2 •s -1 for layers with T > 1400 K and interpolated K zz between this value at 1400 K and our nominal value of 10 8 cm 2 •s -1 at 970 K, such that PH 3 and GeH 4 are quenched in the ∼800 K region as expected from models and observations. This ensures that K zz is low enough where CO is quenched. The resulting CO profile is shown in Fig. 1.

The K zz profiles used in this work that correspond to the results presented in K zz [cm 2 .s -1 ]

Figure 5: K zz profiles used in this work. The black profile is our nominal model (where K zz =10 8 cm 2 •s -1 , constant with altitude) which results in an oxygen abundance of 0.3 times the protosolar value. The blue profile (K zz =2.5 × 10 6 cm 2 •s -1 , constant with altitude) results constrains oxygen to 2.2 times the protosolar value, i.e., the Juno MWR nominal measurement of ( 7). An intermediate constant value of 2.5 × 10 7 cm 2 •s -1 (purple line) will produce the observed CO with nearly solar oxygen. The red profile (variable with altitude) indicates the presence of a stable radiative layer at depth with a transition region such that K zz reaches our nominal value at the levels where PH 3 and GeH 4 are quenched.

Figure 1 :

 1 Figure1: Abundances and temperature profiles for Jupiter. Vertical temperature (T), CO, H 2 O and CH 4 profiles from our nominal model with K zz = 10 8 cm 2 •s -1 and y top CH 4 = 0.00204 in which oxygen is subsolar (O/H = 0.3 times the protosolar value). The dashed lines correspond to alternative Jupiter abundance and T profiles obtained with a more sluggish mixing (K zz is reduced to 2.5 × 10 6 cm 2 •s -1 ) to match the nominal Juno O/H value (2.2 times the protosolar value). This lower K zz is then indicative of a radiative region located around the quench level of CO (that is, at pressure p ∼ 0.4-0.5 kbar and T ∼ 1000 K). The dash-dotted lines correspond to a final model in which a deep radiative layer with reduced K zz is inserted to match solar oxygen and produce 0.9 ppb CO in Jupiter's upper troposphere: K zz is set to 1 cm 2 •s -1 between T = 1400 K and 2200 K and transitions linearly with T towards our nominal value of 10 8 cm 2 •s -1 at 970 K to ensure that PH 3 and GeH 4 are quenched at ∼800 K. This is a non-unique solution.

Figure 2 :

 2 Figure 2: K zz and oxygen dependence of Jupiter's upper tropospheric CO mole fraction. The CO mole fraction (colour scale) as a function of tropospheric mixing and the deep oxygen abundance relative to solar. The solid line shows the computations that result in 0.9 ppb CO. The grey dashed lines represent the range that results in 0.6 to 1.2 ppb CO (full uncertainty range). The range of possible K zz values, constrained from laboratory experiments and matching observed values of PH 3 and GeH 4 , is shown by the dashed blue lines. The nominal value of our model is shown by the black dot.

Figure 3 :

 3 Figure 3: Carbon and oxygen dependence of Jupiter's upper tropospheric CO mole fraction. The CO mole fraction (colour scale) as a function of the carbon abundance (represented by y top CH 4 ) and oxygen water abundance relative to protosolar according to the thermochemical model results when assuming a constant K zz of 10 8 cm 2 •s -1 . The layout is similar to figFig2.

Figure 4 :

 4 Figure4: CO vertical profile in Jupiter computed in the same conditions as in (15) with our chemical scheme, i.e., that of (21) with revised methanol chemistry kinetics. The profile is obtained for K zz = 10 9 cm 2 •s -1 and seven times solar oxygen. It is in full agreement with those obtained with other chemical schemes and shown in Figure17of (15), which are indicated by the grey area.

Fig

Acknowledgements

T.C. acknowledges funding from CNES and the Programme National de Planétologie (PNP) of CNRS/INSU. J.L. acknowledges support from the Juno mission through a subcontract from the Southwest Research Institute.