= Photon-number measurement backaction
{title2=$\rho_{nm}\mapsto\rho_{nm}\cos((n-m)\chi/2)$}
An explicit original ideal meter model explains how <photon number> survives while field <quantum coherence> changes. Let $\hat N$ be the <number operator> and prepare an atomic <qubit> in $|+\rangle=(|e\rangle+|g\rangle)/\sqrt2$. The dispersive <unitary operator>
$$
U=\exp(-i\chi\hat N\otimes\sigma_z/2)
$$
commutes with $\hat N\otimes I$. For a field <quantum superposition> $\sum_n c_n|n\rangle$, it produces $\sum_n c_n|n\rangle|a_n\rangle$, where $|a_n\rangle=(e^{-in\chi/2}|e\rangle+e^{in\chi/2}|g\rangle)/\sqrt2$. The atomic meter states encode $n$ through relative <quantum phase> and can become <entangled> with the field. The second <Ramsey interferometry> pulse and detector turn those correlations into the outcome <probabilities> above when $C=1$.
Tracing out the atom multiplies a field <density matrix> entry $\rho_{nm}$ by $\langle a_m|a_n\rangle=\cos((n-m)\chi/2)$. Hence the <nonselective quantum measurement> preserves every <photon number> population $\rho_{nn}$ while changing off-diagonal <quantum coherence>. For $\chi=\pi/2$, coherence between $|0\rangle$ and $|1\rangle$ is multiplied by $1/\sqrt2$ per fresh meter, even with no photons absorbed. Field <quantum phase> information can therefore be disturbed by an energy-preserving probe. No universal self-adjoint phase operator or naive canonical number–phase commutator is needed for this conclusion.
One possible finite-contrast <quantum instrument> makes the distinction sharper. Define diagonal <Kraus operators> $K_e=\cos((\phi+\chi\hat N)/2)$ and $K_g=\sin((\phi+\chi\hat N)/2)$. After their ideal readout, flip the reported label with <probability> $r=(1-C)/2$. The unnormalized selective maps are
$$
\mathcal J_e(\rho)=(1-r)K_e\rho K_e^\dagger+rK_g\rho K_g^\dagger,\qquad
\mathcal J_g(\rho)=rK_e\rho K_e^\dagger+(1-r)K_g\rho K_g^\dagger.
$$
Their <traces> reproduce $L_e,L_g$ on a <Fock state>, and their sum reproduces the displayed dephasing <quantum channel>, independently of $r$. At $C=0$ the recorded labels carry no information although the physical meter interaction can still cause <measurement backaction>. This is an illustrative instrument, not a reconstruction of all laboratory imperfections; equal <likelihoods> alone do not determine the disturbance.
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