Rayleigh-Jeans and Boltzmann distributions emerge in quantum light

Introduction

A recent collaboration between Humboldt‑Universität zu Berlin and Saint Louis University has demonstrated that quantum light—single‑ and two‑photon states—can thermalize when co‑propagated with classical light in a multimode nonlinear optical platform. By exploiting weak Kerr‑type interactions, the experiment reveals the emergence of both Rayleigh‑Jeans and Boltzmann statistical distributions in the quantum subsystem, providing a practical route to explore quantum many‑body dynamics using ordinary laboratory optics.

Classical chaos as a driver of quantum thermalization

The key ingredient is a multimode waveguide (or lattice) with a modest Kerr nonlinearity. A bright classical beam, injected with random phase across many spatial modes, evolves into a chaotic intensity pattern. This chaotic field modulates the refractive index, creating a time‑varying, spatially disordered potential that the quantum photons experience. Because the classical and quantum fields are kept distinguishable—by orthogonal polarizations or frequency offsets—their interaction remains linear for the quantum component while still allowing energy exchange through the nonlinear index change.

When the classical field reaches a statistical steady state, its mode occupations follow the Rayleigh‑Jeans distribution, the optical analogue of classical equipartition. Remarkably, a single‑photon input inherits this distribution almost exactly: the photon’s probability of occupying a given supermode matches the classical occupation probability. This demonstrates a direct imprint of classical chaos onto quantum dynamics, despite the underlying Schrödinger equation being linear.

Distinct thermal signatures for single‑ and two‑photon states

The situation changes when the input is a two‑photon state. Quantum indistinguishability and bosonic correlations modify the effective interaction with the chaotic background. The resulting steady‑state occupation follows a Boltzmann‑like distribution rather than Rayleigh‑Jeans, and the associated optical temperature differs from that of the classical field. Experiments in various geometries—one‑dimensional lattices, two‑dimensional waveguide arrays, and coupled‑cavity networks—show temperature values that depend sensitively on the mode spectrum and propagation distance, confirming that the quantum statistics of the input state shape the thermal outcome.

Why weak nonlinearity matters

Strong Kerr effects would generate solitons, concentrating energy into a few modes and suppressing the mode‑mixing needed for thermalization. Conversely, an excessively weak nonlinearity would require impractically long propagation lengths for the system to reach equilibrium. The reported work identifies an intermediate regime where the nonlinearity is sufficient to induce chaotic mode mixing without forming localized structures. This balance enables observable thermalization on laboratory‑scale waveguides (centimeter‑scale propagation distances), making the approach experimentally viable.

Implications and future directions

• Quantum simulators with classical hardware – By mapping quantum many‑body Hamiltonians onto the mode structure of a multimode waveguide, researchers can study phenomena such as quantum phase transitions, thermalization dynamics, and entropy production without needing strong photon‑photon interactions.
• Temperature engineering – The ability to assign distinct optical temperatures to classical and quantum subsystems opens avenues for controlled nonequilibrium studies, where energy flow between the two can be monitored in real time.
• Scalable platforms – Multimode fibers, photonic lattices, and synthetic time‑lattice platforms already exist; integrating weak Kerr media into these systems could provide a versatile testbed for a broad class of quantum statistical problems.

Practical take‑away

The experiment shows that carefully designed classical optical environments can act as effective thermal baths for quantum light. By tuning the strength of the Kerr nonlinearity and the geometry of the multimode structure, researchers can select whether a quantum state relaxes to a Rayleigh‑Jeans or Boltzmann distribution, and even set its effective temperature. This bridges a gap between classical thermodynamics and quantum statistical mechanics, offering a low‑cost, high‑control platform for probing the foundations of quantum many‑body physics.

--- The findings illustrate how classical chaos, when coupled through weak nonlinearities, can steer quantum photons toward well‑defined thermal states, paving the way for new experimental explorations of quantum thermodynamics using everyday optical components.