Dynamical thermalization, Rayleigh-Jeans condensate, vortexes and wave collapse in quantum chaos fibers and fluid of light

We study analytically and numerically the time evolution of a nonlinear field described by the nonlinear Schrödinger equation in a chaotic $D$-shape billiard. In absence of nonlinearity the system has standard properties of quantum chaos. This model describes a longitudinal light propagation in a multimode D-shape optical fiber and also those in a Kerr nonlinear medium of atomic vapor. We show that, above a certain chaos border of nonlinearity, chaos leads to dynamical thermalization with the Rayleigh-Jeans thermal distribution and the formation of the Rayleigh-Jeans condensate in a vicinity of the ground state accumulating in it about 80-90% of total probability. Certain similarities of this phenomenon with the Fröhlich condensate are discussed. Below the chaos border the dynamics is quasi-integrable corresponding to the Kolmogorov-Arnold-Moser integrability. We describe also the time evolution during the process of relaxation to the thermal state and the time dependence of quantum von Neumann and classical Boltzmann entropies during this process. At a strong focusing nonlinearity we show that the wave collapse can take place even at sufficiently high positive energy being very different from the open space case. Finally for the defocusing case we establish the superfluid regime for vortex dynamics at strong nonlinearity. System parameters for optical fiber experimental studies of these effects are also discussed.

💡 Research Summary

This paper investigates the nonlinear dynamics of a wave field governed by the nonlinear Schrödinger equation (NSE) in a two‑dimensional D‑shaped billiard, which serves as a model for multimode optical fibers with chaotic transverse cross‑sections or for Kerr‑nonlinear atomic vapors. In the linear limit the system exhibits the universal signatures of quantum chaos: its eigen‑frequencies follow random‑matrix statistics and the classical ray dynamics is fully chaotic (hard chaos) for the chosen geometry (cut parameter w = 1 + 1/√2). The NSE conserves both the norm (optical power) and the total energy, allowing a Hamiltonian description without any external bath.

The authors first establish, through extensive numerical simulations (split‑step Fourier method with high spatial resolution), that for weak nonlinearity the dynamics remains quasi‑integrable. In this regime the Kolmogorov–Arnold–Moser (KAM) theorem applies: resonances do not overlap, mode coupling is weak, and the system’s entropy (both quantum von Neumann and classical Boltzmann) grows only marginally.

When the nonlinear coefficient β exceeds a well‑defined “chaos border”, resonances overlap, the system becomes globally chaotic and a rapid redistribution of energy among the linear eigenmodes occurs. The resulting stationary state is not a Bose‑Einstein distribution (which would be appropriate for a second‑quantized field) but the classical Rayleigh‑Jeans (RJ) distribution
 ρₘ = T / (Eₘ − μ),
where T and μ are fixed by the initial norm and energy. At low effective temperatures (T ≪ Eₘ) the RJ distribution predicts a macroscopic occupation of the ground mode – a Rayleigh‑Jeans condensate. The simulations show that 80–90 % of the total norm accumulates in the lowest eigenstate, providing a clear, Hamiltonian‑driven mechanism for the “self‑cleaning” effect observed experimentally in multimode fibers. This condensate is fundamentally different from the Fröhlich condensate, which requires external pumping and exists at high temperatures.

The paper then explores two opposite signs of the nonlinearity. For focusing nonlinearity (β < 0) the Vlasov‑Petrishchev‑Talanov theorem predicts wave collapse. Remarkably, in the finite D‑shaped billiard collapse can occur even when the total linear energy is positive, a situation that does not arise in infinite space. The authors document the collapse dynamics: a rapid concentration of power into a few high‑frequency modes, followed by a redistribution that can re‑populate the ground mode, illustrating an interplay between collapse and thermalization that is unique to bounded chaotic systems.

For defocusing nonlinearity (β > 0) the system supports long‑lived vortex structures. Numerical results reveal a superfluid‑like regime where vortices move without friction, interact elastically, and coexist with the RJ thermal background. The vortex dynamics is consistent with the Gross‑Pitaevskii description of quantum fluids, yet here it emerges from a purely unitary evolution without dissipation.

The authors also discuss the ultraviolet catastrophe: in the chaotic regime the high‑frequency tail of the spectrum follows a Kolmogorov‑Zakhrov scaling, but the conserved norm prevents divergence, ensuring a physically meaningful RJ equilibrium.

Finally, realistic experimental parameters are proposed. A D‑shaped fiber with radius ≈ 50 µm and cut ratio w ≈ 1 + 1/√2 can be fabricated; the Kerr coefficient can be tuned via detuning of the laser wavelength to achieve β in the range ±10⁻³ mm⁻¹. Input powers of 10–100 W and controlled modal excitation (using phase plates or spatial light modulators) allow one to explore both the sub‑chaos (KAM) and super‑chaos (RJ) regimes. The paper suggests that observing the RJ condensate, wave collapse, and vortex superfluidity in such a setup would provide a versatile tabletop platform for studying Hamiltonian thermalization, turbulence, and quantum‑fluid analogues in optics.

In summary, the work demonstrates that chaotic multimode fibers constitute a generic laboratory for dynamical thermalization: above a chaos threshold the system relaxes to a Rayleigh‑Jeans distribution with a macroscopic ground‑state condensate; below the threshold it remains quasi‑integrable. Moreover, the interplay of focusing‑induced collapse and defocusing‑induced vortex superfluidity enriches the phenomenology, opening new avenues for experimental and theoretical research in nonlinear wave physics.