Planned: examples from the literature

The gallery shows what the solvers do. It does not yet show what they are for, which is the experiments the LKB quantum optics group runs on paraxial fluids of light. This is the plan for closing that, one example per result, each one a page in the gallery with the figure the script drew.

Nothing here is implemented, and none of it should be written from scratch. There are already scripts for several of these outside the repository, and the parameters are in the papers; both are better starting points than anything reconstructed from the physics. What went wrong on the first attempt, at the end, is what reconstructing it costs.

The dimensionless form of the equation — which is how these problems are posed in the group, and the natural way to write an example that is about a regime rather than about one vapour cell — is set out in the Jones-Roberts soliton paper below.

The framework these all sit in is the review, Paraxial fluids of light (Glorieux, Piekarski, Schibler, Aladjidi, Baker-Rasooli), which sets out the mapping from the paraxial NLSE to a 2D+1 Gross-Pitaevskii equation.

The budget every one of these has to fit

A gallery example is executed when the documentation builds, so it has to be worth seconds rather than minutes. Two relations make that predictable rather than a matter of trying:

• Steps ≈ 10 × \(L/z_{nl}\). The nonlinear length \(z_{nl} = 1/(k\Delta n)\) is by definition one radian of nonlinear phase, and it is the same quantity as \(\xi/c_s\), since \(\xi = 1/(kc_s)\) and \(c_s^2 = \Delta n\). The solver aims at 0.1 rad per step, so a run of 20 \(z_{nl}\) is about 200 steps whatever the medium — the two scales are the same number, which is what makes the cost of one of these examples answerable before it is written.

• The window has to hold the wake. These are periodic grids. A wake that wraps round and re-enters the obstacle is not the physics being modelled, and it is what turned a first attempt into noise.

1. Swimming against the flow — CNLSE

Swimming against a superfluid flow: self-propulsion via vortex-antivortex shedding in a quantum fluid of light — Baker-Rasooli, Aladjidi, Ferreira, Bramati, Albert, Larré, Glorieux.

A mobile impurity in a flowing 2D superfluid of light sheds vortex-antivortex pairs once it is above the critical velocity, and the recoil from them drives it upstream.

The impurity is the second component, not a potential. That distinction is the whole result: a static V cannot recoil, so it cannot swim. So

• component 1 is the fluid: broad, defocusing, carrying a transverse velocity from a phase ramp;

• component 2 is the defect: a small, self-focusing beam, which holds itself together and therefore has a finite mass;

• n12 couples them, which is what makes the defect an obstacle at all.

What the figure should show: the vortex pairs downstream, and the defect’s position against \(z\) going the other way.

2. An analogue horizon — DDGPE

Polariton fluids as quantum field theory simulators on tailored curved spacetimes — Falque, Delhom, Glorieux, Giacobino, Bramati, Jacquet, Phys. Rev. Lett. 135, 023401 (2025).

A polariton fluid crossing from subsonic to supersonic flow carries an acoustic horizon, and shaping the drive shapes the spacetime it stands in. DDGPE is the solver for this: it is driven and dissipative by construction.

What the figure should show: \(v(x)\) and \(c_s(x)\) crossing, with the horizon marked where they meet.

One caveat belongs with this one. The coupled solvers still apply their lossy real-space step frozen rather than solved, so they are first order in a lossy medium — see Open in the optimization log. DDGPE is where that matters most, and an example that leans on it is also a reason to fix it.

3. Spin and density modes — CNLSE

Spin and density modes in a binary fluid of light — Piekarski, Cherroret, Aladjidi, Glorieux, Phys. Rev. Lett. 134, 223403 (2025).

A two-component fluid has two Bogoliubov branches: the components moving together, and against each other, at different speeds set by \(g \pm g_{12}\).

Perturb in phase and out of phase, and measure how fast each spreads. This is the most quantitative of the four — the two speeds have closed forms, so it belongs in tests/physics/ as well as in the gallery, next to the checks that are already there.

4. Rayleigh-Taylor fingers — CNLSE — done

Two immiscible components — \(g_{12}^2 > g_{11}g_{22}\) — with the heavier pushed into the lighter. The interface is unstable and breaks into fingers.

Built as examples/rayleigh_taylor.py, though not in the geometry planned here. A flat interface driven by an index ramp cannot be made to finger in an affordable window: keeping the density positive caps the drive at \(g<\mu/X\), and a free beam dilutes as it expands, which together leave under two e-foldings however large the beam. The example instead lets a harmonic trap supply the gravity — the dense component in an outer shell falling inward through a light core — which makes the acceleration \(Cr_0\) and hands the mode number to the circumference. Seven fingers against a predicted 8.2, growing at 11 m⁻¹ against 16.

5. Jones-Roberts solitons — NLSE

Observation of Jones-Roberts solitons in a paraxial quantum fluid of light — Baker-Rasooli, Aladjidi, Krause, Bradley, Glorieux, Phys. Rev. Lett. 134, 233401 (2025).

Single component, and the easiest to start with: the initial condition is a known ansatz and the result is that it propagates without changing shape, which is a sharp thing for a figure to show and a sharp thing to assert. Like §3, it could be a physics test as much as an example.

What went wrong on the first attempt

Recorded because it is the cost of guessing at a regime, and two of the three mistakes are not specific to that example.

1. A static Gaussian potential was used as the obstacle. It cannot recoil, so the physics being reproduced was absent by construction.

2. The phase ramp did not close on the periodic grid. exp(i k_x x) with \(k_x\) not a multiple of \(2\pi/W\) leaves a phase discontinuity at the wrap, which seeds vortices along the whole seam: 85 000 phase singularities on a 512² grid, none of them physical. \(k_x\) has to be \(2\pi m / W\).

3. The propagation was ~30× too long, from sizing \(z\) against the window rather than against \(z_{nl}\). See the budget above.

With 1 and 2 fixed the obstacle behaves — the density at its centre falls to 0.006 of the mean at Mach 0.3 and fills back to 0.52 at Mach 2.0 — but no clean shedding appeared: too short and nothing sheds, longer and the whole box goes turbulent. That is where a dedicated session starts: from the existing scripts and the published parameters, in the dimensionless variables, rather than from reverse-engineered ones.