Polarized response spectrum

Summary: A normalized two-sided mode coupling gives an exact finite-time polarization response matrix whose dark inputs depend on both spatial overlap and the target spectrum.

Both preparations and their mode spaces

Choose normalized perturbation modes of two objects in the same field $\Psi_\infty$. Let $p\in\mathbb C^m$ and $q\in\mathbb C^n$ be their amplitudes, with spatial mode inclusions $I_a,I_b$. The mode counts $m,n$ are not physical spatial dimensions. Both families describe polarized spatial patterns on their actual nested submanifolds.

Assume for this benchmark that the retained perturbation equation has a self-adjoint canonical generator

$$i\frac{d}{dt}\begin{pmatrix}p\\q\end{pmatrix} =\begin{pmatrix}H_a&B^\dagger\\B&H_b\end{pmatrix} \begin{pmatrix}p\\q\end{pmatrix}. \tag{1}$$

The block $B$ is obtained from the spatial variational operator, not chosen solely from the two mode counts. Nonorthogonal families require the full kinetic Gram normalization. Equation (1) preserves $\|p\|^2+\|q\|^2$ and includes reciprocal response of both objects. These are modes of one field, with no mediator particles.

This self-adjoint normal-mode form is an assumption. A general nonlinear Dirac linearization has the ordinary and conjugate blocks derived in Dirac source linearization. It must be analyzed in that full space or reduced to (1) by a justified stable-mode construction; simply discarding its conjugate block is not valid.

Coupling between two definite polarizations

For normalized source and target directions $c\in\mathbb C^m$ and $f\in\mathbb C^n$, the specified channel amplitude is

$$\mathcal A(f,c)=f^\dagger Bc. \tag{2}$$

Both polarizations occur explicitly. Reversing these matched channels uses $c^\dagger B^\dagger f=\overline{\mathcal A(f,c)}$. A zero for one $f$ does not imply $Bc=0$: another target response may be available.

Under unitary changes of either mode basis, $B$ transforms as $B\mapsto U_b^\dagger BU_a$. Its singular values are invariant and are the same for $B$ and $B^\dagger$. Thus a reciprocal linear vertex has the same nonzero channel strengths in either direction, even though the actual source may be prepared in the higher sector.

For uniform unit-vector preparations in the two mode spaces,

$$\mathbb E_c\|Bc\|^2=\frac{\operatorname{tr}(B^\dagger B)}m, \qquad \mathbb E_f\|B^\dagger f\|^2 =\frac{\operatorname{tr}(B^\dagger B)}n. \tag{3}$$

These different averages describe different preparations. Replacing $m,n$ by spatial depth or by an oscillator count would require a derived identification and occupation rule.

Resolve the target's temporal response

For a weak prescribed input $p(t)=ce^{-i\omega t}$ and $q(0)=0$, the target equation in (1) has the exact forced solution

$$q(T)=-i\int_0^T e^{-iH_b(T-t)}Bc\,e^{-i\omega t}dt. \tag{4}$$

This is exact for the driven row. Treating $p$ as unchanged is the leading approximation to the reciprocal system when its depletion is small. It does not remove the $B^\dagger q$ term from the underlying action.

Let $H_bu_\lambda=\Omega_\lambda u_\lambda$ in an orthonormal target basis and write $b_\lambda=u_\lambda^\dagger B$. Integration gives

$$|u_\lambda^\dagger q(T)|^2 =F_T(\Omega_\lambda-\omega)|b_\lambda c|^2, \qquad F_T(\Delta)=T^2\operatorname{sinc}^2\left(\frac{\Delta T}{2}\right), \tag{5}$$

where $\operatorname{sinc}(x)=\sin(x)/x$ and $\operatorname{sinc}(0)=1$. Summing distinguishable orthogonal output modes yields

$$\boxed{\|q(T)\|^2=c^\dagger W_T(\omega)c, \qquad W_T(\omega)=B^\dagger F_T(H_b-\omega)B\geq0.} \tag{6}$$

The matrix $W_T$ combines the spatial compatibility in $B$ with the target's allowed motions and their frequencies. Its eigenvectors give the input polarizations with greatest and least response during the specified exposure. For one photon plane $m=2$, its two eigenvalues need not be equal, and neither is set by the number two.

Source zeros, spectral zeros, and robust dark inputs

Because every term in (6) is nonnegative,

$$c\in\ker W_T(\omega) \quad\Longleftrightarrow\quad b_\lambda c=0\ \text{for every }\lambda \text{ with }F_T(\Omega_\lambda-\omega)>0. \tag{7}$$

A finite-time zero can also occur at a zero of the sinc factor. It is not protected against changing the exposure time or frequency. In contrast $Bc=0$ gives zero driven response for every $T$ and $\omega$. Conversely, since $q(T)=-iTBc+O(T^2)$, zero response for every sufficiently small $T$ implies $Bc=0$.

If $W$ is positive definite and an omitted perturbation changes it by a Hermitian $\delta W$, then

$$\lambda_{\min}(W+\delta W) \geq\lambda_{\min}(W)-\|\delta W\|. \tag{8}$$

A positive response bounded above the uncertainty cannot become exactly dark under that perturbation. For a candidate dark unit vector $c$ with $Bc=0$ and a corrected coupling $B+\delta B$,

$$\|q(T)\|^2\leq T^2\|\delta B\|^2 \tag{9}$$

in the forced model, because $F_T\leq T^2$. This gives a quantitative way to distinguish an exact selection rule from a small response caused by an explicitly bounded correction.

Coherence and incomplete output spaces

For a coherent detector direction $f$, the measured amplitude is $f^\dagger q(T)$, so contributions from several target modes can interfere. It is not generally the sum of their separate intensities. If the target already has an initial polarization amplitude $q_0$, its solution includes $e^{-iH_bT}q_0$ and interference with the driven term. The initial phases of both objects therefore remain relevant.

If only a target subspace $P$ is retained, $PBc=0$ does not establish $Bc=0$ when $(I-P)Bc\ne0$. Moreover, $P$ must be invariant under $H_b$ to use $PH_bP$ as a closed generator without a complement response. The projected-current calculation gives a concrete example where omitted modes carry the response.

For a finite discrete spectrum, (5) describes reversible excitation and interference; it does not supply an irreversible long-time rate. A continuum with a sufficiently regular spectral density permits the distributional limit $F_T(\Delta)/T\to2\pi\delta(\Delta)$. Applying that limit requires the continuum and weak-depletion assumptions, as discussed in continuum mode memory. Interpreting the resulting norm or flux as an observed cross section still requires the physical incident and detector normalization.

Verification

polarization_coupling_checks.py checks the displayed identities with independent spatial quadrature, energy variations, or full finite-system evolution as appropriate. These checks verify the declared mathematics and benchmarks, not empirical couplings.

Related

• Reciprocal scattering of spatial modes — includes the wave's backreaction and derives conserved reflection and transmission from the same spatial vertex.

• Bilateral polarization coupling — distinguishes a background source from the mixed response of two polarized objects.

• Dirac source linearization — supplies explicit ordinary and conjugate spatial matrix elements from the written nonlinear term.

• Canonical coupling normalization — fixes the mode metrics before interpreting singular values.

• Continuum mode memory — states when finite-mode response can be approximated by an irreversible continuum law.

• Projected Dirac current — shows why a truncated dark subspace need not imply zero full response.