06 / TECHNICAL DEEP DIVE
The assumptions
behind the demonstration.
One consistent convention connects the master equation, coherence decay, Ramsey contrast, and conditional bounds.
The timing explorer uses a phenomenological Milburn-type model as a candidate test of temporal granularity. It is an additional dynamical assumption; the philosophical idea of continual creation does not uniquely imply this master equation. The coefficient must ultimately be derived or independently justified within a complete theory.
ρ is the density operator; H is the Hamiltonian; ε ≥ 0 is the proposed time-scale parameter. The second term damps coherences between distinct energies while preserving their populations in this idealized model.
With this definition of Γ, the same master equation can be written with −(Γ/4)[σz,[σz,ρ]]. The off-diagonal coherence and normalized Ramsey contrast decay at the same rate Γ.
The demo sets C₀ = 1 and assumes independent exponential background decay. Real noise may be non-exponential or correlated, so this convenient model must be checked against actual data.
Derivation: random unitary steps and the small-step limit
A useful mathematical realization takes Hamiltonian evolution steps of duration ε occurring at Poisson rate 1/ε. Its generator is [exp(−iεH/ℏ) ρ exp(iεH/ℏ) − ρ]/ε. Expanding in ε produces the usual commutator and the double-commutator term above. For a two-level system the exact decay rate is [1 − cos(ωε)]/ε and approaches εω²/2 when |ωε| ≪ 1. There is also a small frequency correction in the exact generator.
This is a stochastic elapsed-evolution model. A single global phase multiplying the entire state cancels out of the density operator and cannot cause this decoherence. The relevant randomness must change relative phases between energy components.
All parameter settings offered in the timing laboratory are in the small-|ωε| regime. That mathematical consistency does not establish experimental viability or acceptance of the model.
Units, numerical example, and normalization
ε has units of seconds and ω has units of inverse seconds, so Γ has units of inverse seconds. For ν = 429 THz and ε = 5.39 × 10⁻⁴⁴ s, Γ is approximately 1.96 × 10⁻¹³ s⁻¹. At T = 10⁴ s, the intrinsic contrast loss is approximately 1.96 × 10⁻⁹, before background effects. This is a mathematical estimate, not a demonstrated sensitivity.
If Γmax = 10⁻¹³ s⁻¹ at the same frequency, the conditional bound is ε ≤ 2.75 × 10⁻⁴⁴ s. There is no square root in this bound: a square root would give the wrong units for the stated rate law. Small losses are calculated with expm1 to avoid subtractive numerical cancellation.
Why no universal Allan-deviation curve is shown
Ramsey contrast alone does not determine clock stability. The mapping requires the interrogation sequence, atom number, measurement noise, dead time, feedback system, and noise spectrum. A credible stability forecast must explicitly state those assumptions. A future implementation could simulate a complete clock servo and compare Allan deviation or related statistics against experimentally sampled data.
What would be distinctive about a DTT interpretation?
A familiar decoherence equation may be compatible with several physical stories. To identify a distinctive DTT prediction, researchers must derive its parameters and correlations from the framework and show how they differ from technical noise and competing intrinsic-decoherence models. Compatibility with one observed decay curve would not establish spatial emergence or the full SMM/DTT framework.