An ideal flat two-path distribution: exact coherent and which-path bin laws

Adopt the declared two-path preparation and detection model The declared two-path preparation model and its ideal detection bins in its spin-zero reading $r=1$, and the coherent and which-path detection laws Coherent and which-path detection laws of the declared two-path model. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), as required by the preparation and detection-law suppliers. Fix lengths $0<d<L<\mathrm{\infty }$, measured in metres, and declare the two intervals $I_1:=[-\frac{d}{2}-\frac{L}{2},-\frac{d}{2}+\frac{L}{2}],I_2:=[\frac{d}{2}-\frac{L}{2},\frac{d}{2}+\frac{L}{2}]$ on the first spatial coordinate, measured in metres, whose overlap region $O:=I_1\cap I_2=[\frac{d}{2}-\frac{L}{2},-\frac{d}{2}+\frac{L}{2}]$ has length $L-d>0$; let $g\in L^2({\mathbb{R}}^2;\mathbb{C})$ satisfy $\mathrm{\parallel }g{\mathrm{\parallel }}_2=1$ and choose a Borel representative, which exists by completion of Borel Lebesgue measure ($\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets). Its amplitude unit is ${\mathrm{m}}^{-1}$ in the two-dimensional position measure; declare the flat path amplitudes $f_1:=L^{-1\mathrm{/}2}{\mathbf{1}}_{I_1},f_2:=L^{-1\mathrm{/}2}{\mathbf{1}}_{I_2}\mathrm{.}$ Define the declared path components by ${\psi }_j(x_1,x_2,x_3):=f_j(x_1)g(x_2,x_3)(j=1,2)\mathrm{.}$ In these SI conventions each $f_j$ has unit ${\mathrm{m}}^{-1\mathrm{/}2}$ and each ${\psi }_j$ has unit ${\mathrm{m}}^{-3\mathrm{/}2}$, as required for a normalized position wave function on ${\mathbb{R}}^3$; the probabilities below are dimensionless. A slab bin is a set $B_1\times {\mathbb{R}}^2$ with $B_1\subseteq \mathbb{R}$ Borel. Then:

(a) ${\psi }_1,{\psi }_2\in H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ with $\mathrm{\parallel }{\psi }_1\mathrm{\parallel }=\mathrm{\parallel }{\psi }_2\mathrm{\parallel }=1$, so the declared data are admissible, and the model is admissible at every phase: on $I_1\setminus I_2$ the combination $f_1+e^{i\phi }{f}_2$ equals $L^{-1\mathrm{/}2}$ independently of $\phi$, and that region has length $d>0$. Since $\mathrm{\parallel }g\mathrm{\parallel }=1$, $\mathrm{\parallel }{\psi }_1+e^{i\phi }{\psi }_2{\mathrm{\parallel }}^2=\mathrm{\parallel }{f}_1+e^{i\phi }{f}_2{\mathrm{\parallel }}^2\ge d\mathrm{/}L>0$.

(b) For every slab bin, with $N_{\phi }:=\mathrm{\parallel }{f}_1+e^{i\phi }{f}_2{\mathrm{\parallel }}^2=2+\frac{2(L-d)}{L}\cos \phi$, ${\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G(B_1\times {\mathbb{R}}^2)=\frac{1}{N_{\phi }}{\int }_{B_1}\mathrm{\mid }{f}_1+e^{i\phi }{f}_2{\mathrm{\mid }}^{2}dx,{\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w)}^G(B_1\times {\mathbb{R}}^2)=\frac{w\mathrm{\mid }{B}_1\cap I_1\mathrm{\mid }+(1-w)\mathrm{\mid }{B}_1\cap I_2\mathrm{\mid }}{L}\mathrm{.}$

(c) The coherent density $\mathrm{\mid }{f}_1+e^{i\phi }{f}_2{\mathrm{\mid }}^2$ equals $\frac{2(1+\cos \phi )}{L}$ on $O$, equals $\frac{1}{L}$ on the symmetric difference $I_1\mathrm{△}{I}_2$, and vanishes outside $I_1\cup I_2$; consequently ${\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G(O\times {\mathbb{R}}^2)=\frac{(L-d)(1+\cos \phi )}{L+(L-d)\cos \phi },{\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G((I_j\setminus I_{3-j})\times {\mathbb{R}}^2)=\frac{d}{2L+2(L-d)\cos \phi }$ for $j=1,2$, and for the balanced which-path preparation $w=\frac{1}{2}$ ${\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(1\mathrm{/}2)}^G(O\times {\mathbb{R}}^2)=\frac{L-d}{L},{\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(1\mathrm{/}2)}^G((I_j\setminus I_{3-j})\times {\mathbb{R}}^2)=\frac{d}{2L}\mathrm{.}$

(d) At $\phi =\pi$ the coherent law assigns probability zero to the whole overlap region $O$ of positive length, while the balanced which-path law assigns it probability $(L-d)\mathrm{/}L>0$; every slab bin contained in $O$ has coherent probability zero, every slab bin meeting $I_1\setminus I_2$ or $I_2\setminus I_1$ in a set of positive length has positive coherent probability, and every slab bin meeting $O$ in a set of positive length has positive which-path probability.

The computation is exact in the declared model; it asserts no detector response, no sampling model, no measured count and no laboratory realization.

• The dark region is a feature of the declared flat model. The vanishing of the coherent density on all of $O$ at $\phi =\pi$ uses the flat amplitudes and the exact cancellation $f_1+e^{i\phi }{f}_2=0$ there; a model with unequal weights or non-flat amplitudes need not vanish on a region. At $\phi =\pi$, a slab bin $B_1\times {\mathbb{R}}^2$ has positive coherent probability exactly when $\mathrm{\mid }{B}_1\cap (I_1\mathrm{△}{I}_2)\mathrm{\mid }>0$; it has zero coherent probability exactly when this intersection has length zero. Adding points in $O$ or outside $I_1\cup I_2$ does not change that probability. These are ideal-bin probabilities; recorded-event probabilities require a declared response model.
• No data are used. The example computes probabilities in the declared model; it does not report counts, and the flat amplitudes are a declared idealization, not a measured profile.