Resonant quartet dynamics

Four distinct field modes

In frequency units with canonical equation $i\dot a_j=\partial H/\partial a_j^*$, choose

$$H=\sum_{j=1}^4\omega_j|a_j|^2+ \kappa a_1^*a_2^*a_3a_4+kappa^*a_1a_2a_3^*a_4^*.$$

For example $i\dot a_1=\omega_1a_1+\kappa a_2^*a_3a_4$ and $i\dot a_3=\omega_3a_3+\kappa^*a_1a_2a_4^*$; exchange $1\leftrightarrow2$ and $3\leftrightarrow4$ for the other two equations. The positions of the complex conjugates are essential to reciprocity.

Let $I_j=|a_j|^2$ and $Z=\kappa a_1^*a_2^*a_3a_4$. Direct differentiation gives

$$\dot I_1=\dot I_2=2\operatorname{Im}Z, \qquad \dot I_3=\dot I_4=-2\operatorname{Im}Z.$$

Every charge $Q_c=\sum c_jI_j$ with $c_1+c_2=c_3+c_4$ is conserved. These include total norm, $I_1-I_2$, and $I_3-I_4$. The Hamiltonian itself is conserved and total norm bounds every amplitude, giving global solutions for finite initial data. Mode amplitudes add no physical spatial dimensions.

Frequency matching

Writing $a_j=b_je^{-i\omega_jt}$ makes the first interaction proportional to $e^{i\Delta t}$ with

$$\Delta=\omega_1+\omega_2-\omega_3-\omega_4.$$

A label equality $n_1+n_2=n_3+n_4$ implies $\Delta=0$ when all four frequencies follow the same affine law $\omega_j=An_j+B$. It need not do so under a nonlinear count law or unequal sector scales. For example $1+4=2+3$, while $S(1,2)+S(4,2)=11$ and $S(2,2)+S(3,2)=9$. In a first-order calculation that freezes the other three rotating amplitudes, the forced change is proportional to $\int_0^t e^{i\Delta s}ds$ and is bounded by $2/|\Delta|$ times that fixed forcing strength. This bound belongs to that approximation.

Exact exchange with undepleted pumps

For independently maintained pumps $b_2,b_4$, the $1,3$ subsystem obeys

$$i\dot b_1=g e^{i\Delta t}b_3,\qquad i\dot b_3=g^*e^{-i\Delta t}b_1,\qquad g=\kappa b_2^*b_4.$$

Set $b_1=c_1e^{i\Delta t/2}$ and $b_3=c_3e^{-i\Delta t/2}$. Then

$$i\frac d{dt}\begin{pmatrix}c_1\\c_3\end{pmatrix} =\begin{pmatrix}\Delta/2&g\\g^*&-\Delta/2\end{pmatrix} \begin{pmatrix}c_1\\c_3\end{pmatrix}.$$

The matrix square is $(|g|^2+\Delta^2/4)I$. Starting with $(c_1,c_3)=(0,1)$,

$$|c_1(t)|^2=\frac{4|g|^2}{\Delta^2+4|g|^2} \sin^2\left(\frac{\sqrt{\Delta^2+4|g|^2}}2t\right).$$

This derives resonance-enhanced transfer and its finite detuning width. The full four-mode dynamics includes pump depletion; neither model alone selects the persistent particle spectrum.

• Counting mode tools — separates available frequency, excitation, and occupation.

• Continuum mode memory — supplies an exact route to response with a continuum.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.