Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

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Take a single electric-dipole transition ∣g⟩↔∣e⟩|g\rangle \leftrightarrow |e\rangle∣g⟩↔∣e⟩ of frequency ω0=2πc/λ\omega_0=2\pi c/\lambdaω0=2πc/λ, dipole moment d\mathbf dd, driven by an optical field (laser or broadband), with detuning Δ=ω−ω0\Delta=\omega-\omega_0Δ=ω−ω0.

Interaction (electric-dipole)

Hint=− d⋅E(t).H_{\rm int} = -\,\mathbf d\cdot \mathbf E(t).Hint=−d⋅E(t).
Master equation (Born–Markov, RWA)

ρ˙=−iℏ[H,ρ]+Γ↓ D[σ−]ρ+Γ↑ D[σ+]ρ+γϕ D[12σz]ρ,\dot\rho
= -\frac{i}{\hbar}[H,\rho]
+ \Gamma_\downarrow \,\mathcal D[\sigma_-]\rho
+ \Gamma_\uparrow \,\mathcal D[\sigma_+]\rho
+ \gamma_\phi \,\mathcal D[\tfrac{1}{2}\sigma_z]\rho,ρ˙=−ℏi[H,ρ]+Γ↓D[σ−]ρ+Γ↑D[σ+]ρ+γϕD[21σz]ρ,
where D[L]ρ=LρL†−12{L†L,ρ}\mathcal D[L]\rho=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}D[L]ρ=LρL†−21{L†L,ρ}, σ−=∣g⟩ ⁣⟨e∣\sigma_- = |g\rangle\!\langle e|σ−=∣g⟩⟨e∣, σ+=∣e⟩ ⁣⟨g∣\sigma_+=|e\rangle\!\langle g|σ+=∣e⟩⟨g∣.
* Spontaneous emission (geometry of a dipole radiating over 4π4\pi4π):

 Γ≡Γsp=ω03 ∣d∣23π ε0 ℏ c3 .\boxed{\ \Gamma \equiv \Gamma_{\rm sp} = \frac{\omega_0^3\,|\mathbf d|^2}{3\pi\,\varepsilon_0\,\hbar\,c^3}\ }. Γ≡Γsp=3πε0ℏc3ω03∣d∣2 .
* Stimulated absorption/emission from a radiation field of energy density u(ω0)u(\omega_0)u(ω0):

Γ↑=B u(ω0),Γ↓=Γ+B u(ω0),\Gamma_\uparrow = B\,u(\omega_0),\qquad
\Gamma_\downarrow = \Gamma + B\,u(\omega_0),Γ↑=Bu(ω0),Γ↓=Γ+Bu(ω0),
with Einstein relation A≡Γ=8πhν03c3BA\equiv\Gamma=\frac{8\pi h \nu_0^3}{c^3}BA≡Γ=c38πhν03B (so B=Γ c3/(8πhν03)B = \Gamma\,c^3/(8\pi h\nu_0^3)B=Γc3/(8πhν03)). Using intensity I=c uI=c\,uI=cu,

Γ↑=Bc I,Γ↓=Γ+Bc I.\Gamma_\uparrow = \frac{B}{c}\,I,\quad \Gamma_\downarrow = \Gamma + \frac{B}{c}\,I.Γ↑=cBI,Γ↓=Γ+cBI.
Coherence (off-diagonal) decay
For ρeg\rho_{eg}ρeg one finds

ρ˙eg=−(iΔ+Γ2)ρeg, Γ2=12(Γ↓+Γ↑)+γϕ=Γ2+BcI+γϕ .\dot\rho_{eg} = -\Big(i\Delta + \Gamma_2\Big)\rho_{eg},\qquad
\boxed{\ \Gamma_2
= \tfrac{1}{2}(\Gamma_\downarrow+\Gamma_\uparrow) + \gamma_\phi
= \frac{\Gamma}{2} + \frac{B}{c}I + \gamma_\phi\ }.ρ˙eg=−(iΔ+Γ2)ρeg, Γ2=21(Γ↓+Γ↑)+γϕ=2Γ+cBI+γϕ .
This says: even without population change, elastic light scattering and stimulated processes add a decoherence rate proportional to intensity III. That’s your “every change is accompanied by photon interaction.”

==== It’s often handier to express things with the saturation parameter ====

s≡IIsat, Isat=πhc Γ3 λ3 s \equiv \frac{I}{I_{\rm sat}},\qquad
\boxed{\ I_{\rm sat} = \frac{\pi h c\,\Gamma}{3\,\lambda^3}\ }s≡IsatI, Isat=3λ3πhcΓ 
(for a closed cycling dipole transition; other transitions differ by order-1 factors).

Define the elastic scattering rate (events per atom per second; absorption + re-emission) as

 Rsc(I,Δ)=Γ2 s1+s+(2Δ/Γ)2 .\boxed{\ R_{\rm sc}(I,\Delta)
= \frac{\Gamma}{2}\,
\frac{s}{1+s+(2\Delta/\Gamma)^2}\ }. Rsc(I,Δ)=2Γ1+s+(2Δ/Γ)2s .
Then a robust, experiment-ready decoherence rate is

 Γ2(I,Δ)≃Γ2  +  Rsc(I,Δ)  +  γenv .\boxed{\ \Gamma_2(I,\Delta)
\simeq \frac{\Gamma}{2} \;+\; R_{\rm sc}(I,\Delta) \;+\; \gamma_{\rm env}\ }. Γ2(I,Δ)≃2Γ+Rsc(I,Δ)+γenv .
* Γ/2\Gamma/2Γ/2 is the irreducible quantum limit from spontaneous emission (geometry of dipole radiation over 4π4\pi4π).
* RscR_{\rm sc}Rsc is how fast the light field “touches” the system (absorption + elastic scattering), i.e. QAT’s photon-event cadence.
* γenv\gamma_{\rm env}γenv accounts for other dephasing (collisions, thermal phonons, technical noise).

This already encodes your idea: no macroscopic change without photon exchange → increase III or resonant overlap (small ∣Δ∣|\Delta|∣Δ∣) and the “unfolding” accelerates (larger Γ2\Gamma_2Γ2); turn the light down or detune (or cool to reduce background photons) and the world “slows” (smaller Γ2\Gamma_2Γ2). In the T→0T\to0T→0 (dark) limit, I→0I\to0I→0, stimulated terms vanish, matching your Third-Law/“field lines lock” picture.

==== - Resonant cross-section (spherical/dipole geometry): ====

 σ0=3λ22π (peak on resonance).\boxed{\ \sigma_0 = \frac{3\lambda^2}{2\pi}\ }\quad\text{(peak on resonance)}. σ0=2π3λ2 (peak on resonance).
Event rate = photon flux Φ=I/(ℏω)\Phi = I/(\hbar\omega)Φ=I/(ℏω) times σ\sigmaσ:
R≈Φ σ(ω)R \approx \Phi\,\sigma(\omega)R≈Φσ(ω). The λ2\lambda^2λ2 area and the 3/2π3/2\pi3/2π factor come from integrating a dipole pattern over 4π4\pi4π—just the spherical geometry you emphasize.
* Einstein A (spontaneous rate) has the 13π\tfrac{1}{3\pi}3π1 from angular averaging of the dipole over the sphere and the ω3/c3\omega^3/c^3ω3/c3 from photon mode density in kkk-space (again a geometric count of directions).
* Elastic (Rayleigh) scattering far off resonance Using the Rabi frequency Ω=∣d⋅E∣/ℏ\Omega = |\mathbf d\cdot\mathbf E|/\hbarΩ=∣d⋅E∣/ℏ and I=12cε0E2I=\tfrac12 c\varepsilon_0 E^2I=21cε0E2:

 Rsc≈ΓΩ24Δ2=ΓIIsat11+(2Δ/Γ)2 .\boxed{\ R_{\rm sc} \approx \Gamma \frac{\Omega^2}{4\Delta^2}
= \Gamma \frac{I}{I_{\rm sat}} \frac{1}{1+(2\Delta/\Gamma)^2}\ }. Rsc≈Γ4Δ2Ω2=ΓIsatI1+(2Δ/Γ)21 .
So even pure scattering (no net energy change) produces dephasing proportional to intensity and to the dipole geometry.

==== Take a typical strong optical line (e.g., an alkali D line; λ∼589 nm\lambda\sim 589\,\mathrm{nm}λ∼589nm, Γ/2π∼10 MHz\Gamma/2\pi\sim 10\,\mathrm{MHz}Γ/2π∼10MHz, hence Γ≈6.3×107 s−1\Gamma\approx 6.3\times10^7\ \mathrm{s^{-1}}Γ≈6.3×107 s−1; Isat∼6 mW/cm2I_{\rm sat}\sim 6\ \mathrm{mW/cm^2}Isat∼6 mW/cm2). ====
* On resonance (Δ=0\Delta=0Δ=0) - I=0.1 IsatI=0.1\,I_{\rm sat}I=0.1Isat: Rsc≈0.045 Γ⇒Γ2≈3.36×107 s−1⇒T2∼30 nsR_{\rm sc}\approx 0.045\,\Gamma\Rightarrow \Gamma_2\approx 3.36\times10^7\ \mathrm{s^{-1}}\Rightarrow T_2\sim 30\ \mathrm{ns}Rsc≈0.045Γ⇒Γ2≈3.36×107 s−1⇒T2∼30 ns. - I=1 IsatI=1\,I_{\rm sat}I=1Isat: Rsc≈0.25 Γ⇒Γ2≈4.6×107 s−1⇒T2∼22 nsR_{\rm sc}\approx 0.25\,\Gamma\Rightarrow \Gamma_2\approx 4.6\times10^7\ \mathrm{s^{-1}}\Rightarrow T_2\sim 22\ \mathrm{ns}Rsc≈0.25Γ⇒Γ2≈4.6×107 s−1⇒T2∼22 ns. - I=10 IsatI=10\,I_{\rm sat}I=10Isat: Rsc≈0.455 Γ⇒Γ2≈5.9×107 s−1⇒T2∼17 nsR_{\rm sc}\approx 0.455\,\Gamma\Rightarrow \Gamma_2\approx 5.9\times10^7\ \mathrm{s^{-1}}\Rightarrow T_2\sim 17\ \mathrm{ns}Rsc≈0.455Γ⇒Γ2≈5.9×107 s−1⇒T2∼17 ns.
* Far detuned (∣Δ∣=5Γ|\Delta|=5\Gamma∣Δ∣=5Γ) - I=1 IsatI=1\,I_{\rm sat}I=1Isat: Rsc≈0.0049 Γ⇒Γ2≈3.1×107 s−1⇒T2∼32 nsR_{\rm sc}\approx 0.0049\,\Gamma\Rightarrow \Gamma_2\approx 3.1\times10^7\ \mathrm{s^{-1}}\Rightarrow T_2\sim 32\ \mathrm{ns}Rsc≈0.0049Γ⇒Γ2≈3.1×107 s−1⇒T2∼32 ns.

Takeaway: turning the photon field up (or bringing it on resonance) accelerates decoherence. In your language: the rate of “world update” rises with photon event rate.

==== - The only knobs that appear are geometric/field ones: λ\lambdaλ (sets the λ2\lambda^2λ2 cross-section area), the dipole pattern over 4π4\pi4π (sets the constants), and the photon number flux I/(ℏω)I/(\hbar\omega)I/(ℏω). No ad-hoc time operator is introduced; time’s arrow emerges from the irreversible flow of EM quanta into (and out of) matter plus the environment (decoherence). ====
* In the T→0T\to0T→0 / dark limit, background I→0I\to 0I→0: stimulated terms vanish and you’re left with the smallest possible Γ2=Γ/2+γenv\Gamma_2=\Gamma/2+\gamma_{\rm env}Γ2=Γ/2+γenv. This is your “as the field lines lock, the process slows” statement, aligned with the Third Law.

==== 1. Decoherence vs intensity (Ramsey fringes on a two-level atom/ion): measure T2T_2T2 as a function of III and Δ\DeltaΔ; fit to the boxed formulas. QAT wins if the “rate of change” of the system’s classicality tracks the photon event rate. ====
# Scattering-only decoherence (far detuned probe): no energy level changes, yet coherence still decays at Rsc∝I/Δ2R_{\rm sc}\propto I/\Delta^2Rsc∝I/Δ2. That directly implements your “even light scattering is geometry” claim.
# Cross-section geometry: verify σ0=3λ2/2π\sigma_0=3\lambda^2/2\piσ0=3λ2/2π (standard), then use it to predict event rates from photon flux. This is your “spherical boundary → area → rates” pathway.

If you want, I can now:
* plug in a specific line (e.g., hydrogen Hα at 656.3 nm) with its Γ\GammaΓ and compute Isat,Rsc,Γ2I_{\rm sat}, R_{\rm sc}, \Gamma_2Isat,Rsc,Γ2 vs I,ΔI,\DeltaI,Δ;
* or derive the same scaling starting from the boundary action we built (surface current jaAaj^a A_ajaAa), showing explicitly how the Poynting flux through a 4πr24\pi r^24πr2 surface produces the same RscR_{\rm sc}Rsc and Γ2\Gamma_2Γ2.