Why high modal dimensionality is necessary but not sufficient, and the conditions a substrate must meet simultaneously.
A passive optical cavity filled with linear media implements a linear input–output map, however elaborate its engineered disorder. Such a map can be of arbitrarily high rank and still computes no nonlinear function of its input history. The conditions under which a driven, open, disordered cavity becomes a genuine computational substrate are therefore conditions on three coupled physical observables — dimensionality, memory, and a nonlinearity internal to the field evolution — that all enter through the same complex spectrum. This note states those conditions and reduces the computation question to a measurable criterion on the Volterra kernels of the input–output map.
Consider a resonator with a spatially structured permittivity \(\varepsilon(\mathbf r)\) — a plano-hemispherical cavity seeded with subwavelength dielectric scatterers. In the frequency domain the passive field obeys
Because the cavity is open — radiative leakage through the curved wall — the relevant operator is non-Hermitian, and its natural modes are quasinormal modes with complex eigenfrequencies
The real part fixes the spectral position; the imaginary part fixes the linewidth and the photon lifetime. The scatterers break integrability and mix these modes.
For linear media, Eq. (1) is linear in \(\mathbf E\). The map from drive to field is therefore a linear operator \(A(\omega)\),
Equivalently, expanding the field in quasinormal modes, \(\mathbf E(\mathbf r,t)=\sum_m a_m(t)\,\boldsymbol\psi_m(\mathbf r)\), the modal amplitudes obey a coupled-mode equation
where \(\kappa_{mn}\) is the disorder-induced linear mode coupling, \(s_m\) the projected drive, and \(N_m\) the coupling generated by any field-dependence of \(\varepsilon\). In the passive linear limit \(N_m=0\), and the solution is a convolution,
This is the constraint in one line: the internal state is a linear functional of the drive history. No matter how many modes participate or how complex \(A(\omega)\) is, the state never depends on products of its own past. A high-rank linear map is a random projection, not a dynamical system that computes.
Dimensionality. Let \(M\) be the state matrix sampled by the readout, with singular values \(\sigma_i\). The effective dimensionality is the participation ratio of its singular spectrum,
set by the eigenfunction statistics (participation ratio, IPR) of the mixed mode field — not by the detector count.
Memory. The fading-memory timescale is the decay-rate spectrum measured against the drive period: \(\tau_m=1/\gamma_m\), with the cavity scale \(\tau\simeq \mathcal F L/\pi c\) for finesse \(\mathcal F\). Modal individuality is governed by the modal-overlap (Thouless) parameter
with \(\Delta\omega\) the mean level spacing. Resolved modes require \(\delta\lesssim1\); for \(\delta\gg1\) resonances overlap and individuality is lost.
Nonlinearity. A genuine activation enters only through a field-dependent permittivity,
for example saturable gain \(\chi_{\text{eff}}=\chi_0\,(1+I/I_{\text{sat}})^{-1}\) or a Kerr shift \(\Delta n=n_2 I\). What matters is not only its magnitude but its response time relative to \(\tau_m\): a nonlinearity slow compared with the photon lifetime acts as a quasi-static background, not a per-symbol activation.
The computation question, free of any reservoir-computing vocabulary, is a statement about the Volterra expansion of the input–output map:
A linear system has only a first-order kernel — \(h_1=G\) and \(h_{k\ge2}\equiv0\). Nonlinearity in the state evolution appears as a nonvanishing higher-order kernel, and the physically meaningful requirement is that it be sourced by the intracavity field-dependence rather than by a nonlinear detector, with temporal support set by the photon lifetime:
This collapses three fuzzy properties into one measurable quantity: drive the system with impulses and impulse pairs and read off whether \(h_2\) survives over the memory window.
A nonvanishing \(h_2\) is a necessary criterion, not a sufficient one. A substrate can be nonlinear and still compute poorly if its memory capacity or readout conditioning is weak; the sufficient figure of merit is task-level — memory capacity, or accuracy on a benchmark that demands nonlinear memory (NARMA, parity-\(N\)) measured against a linear-readout control.
By coexistence we mean the simultaneous satisfaction of the conditions below — not phase or modal coexistence, nor bistability. The difficulty is that the observables are not independent — disorder, loss, and the nonlinearity all enter through the same complex spectrum \(\{\omega_m-i\gamma_m\}\) and the same susceptibility. Disorder raises \(r_{\text{eff}}\) but adds radiative loss, which shortens memory and drives \(\delta\) past unity; a nonlinearity strong enough to make \(h_2\neq0\) — gain saturation near threshold, say — tends to induce mode competition that collapses the field onto a few modes and sends \(r_{\text{eff}}\to1\). The same non-Hermiticity exerts one further pull: the cavity's modes are non-orthogonal, and that non-orthogonality amplifies quantum-limited noise by the Petermann factor \(K\), which diverges near the exceptional points that strong mixing drives toward — so the high-\(r_{\text{eff}}\) regime is intrinsically the high-noise regime. The open question is whether a single operating point satisfies all of the following at once:
The linear-disorder half — the eigenfunction statistics and decay spectrum — is well-trodden open-wave-chaos physics. The unsettled half is the multimode nonlinear regime: whether a field-dependence strong enough to register in \(h_2\) on the photon-lifetime timescale can coexist with high modal participation, or whether it generically destroys it. Whether the favorable region is nonempty is the whole problem.