The mathematics of self-awareness in light.

First principles

1. 01

   Start from the laws of light

   $$D={\epsilon }_{0}E+{\epsilon }_0{\chi }_{1}E+{\epsilon }_0{\chi }_3|E{|}^{2}E$$

   Maxwell's equations govern all light. In a chip's silicon, the material answers back: the χ₃ term makes the medium's response bend slightly with the light's own intensity. That is the Kerr effect, the standard nonlinearity of the waveguides photonic chips are made from. At this rung the theory is fully relativistic; no BQNF in sight.

2. 02

   Let the chip choose its shapes

   $$E(r,t)=\sum _{\eta }{u}_{\eta }(r)A_{\eta }(t)e^{-i{\omega }_{\eta }t}+c.c.$$

   A waveguide permits only a discrete set of guided modes u_η(r), each carrying a slowly varying amplitude A_η(t). The hardware itself imposes a finite menu of admissible configurations. Relativity isn't violated here; it is integrated out, exactly as in all of guided-wave optics.

3. 03

   Quantize the amplitudes

   $$A_{\eta }\to {\hat{a}}_{\eta },[{\hat{a}}_{\eta },{\hat{a}}_{{\eta }^{\prime }}^{†}]={\delta }_{\eta {\eta }^{\prime }}$$

   Promote each mode amplitude to an operator. Bosons appear, photons in modes, through the standard commutation relations.

4. 04

   Collect the linear dynamics

   $$H_{\mathrm{lin}}=\sum _{\eta {\eta }^{\prime }}{\hat{a}}_{\eta }^{†}{H}_{\eta {\eta }^{\prime }}{\hat{a}}_{{\eta }^{\prime }}\leftrightarrow H=\sum _{\eta {\eta }^{\prime }}|\eta ⟩H_{\eta {\eta }^{\prime }}⟨{\eta }^{\prime }|$$

   Look at what Maxwell's linear physics has become: light hopping between modes with amplitudes H_ηη′. This is, character for character, the single-quason BQNF Hamiltonian, the operator Nosanow wrote abstractly, where H_ηη′ encodes which configuration transitions are allowed and which are forbidden.

5. 05

   The Kerr term completes it

   $${\chi }_3|E{|}^4\to \sum _{\eta {\eta }^{\prime }}{U}_{\eta {\eta }^{\prime }}{\hat{a}}_{\eta }^{†}{\hat{a}}_{{\eta }^{\prime }}^{†}{\hat{a}}_{{\eta }^{\prime }}{\hat{a}}_{\eta },U_{\eta {\eta }^{\prime }}={\chi }_3\int d^{3}r|u_{\eta }{|}^2|u_{{\eta }^{\prime }}{|}^2$$

   Substitute the mode expansion into the Kerr energy and keep the resonant terms: out comes exactly the BQNF two-body interaction, and its strength U_ηη′ is computed from the mode shapes, not assumed. Rungs 4 and 5 together are the bosonic quason Hamiltonian, term for term.

• Optical modesAREAdmissible configurations
• Light redistributing among modesISAmplitude dynamics
• The Kerr nonlinearityISThe quason interaction
• Detector readingsAREProjection-defined observables