Lab

This page defines the research programme for the lab. You will simulate clock networks of increasing complexity, benchmark them within both frequentist and Bayesian frameworks, and interpret your results through the causal-geometry lens. Each lab block builds on the previous; record everything in your lab notes as you go.

1.1 Research Question

Given an oscillator with a specified noise model, can you fully characterise its stability using both frequentist and Bayesian methods, and do the two approaches agree?

1.2 Tasks

Frequentist Characterisation

• Simulate a single oscillator with a composite noise model: white frequency + flicker frequency + random-walk frequency. Choose physically motivated amplitudes (document your choices).
• Compute the Allan deviation σ_y(τ) over at least four decades of τ. Identify the three noise regimes from the slope.
• Compute the modified Allan deviation. Confirm that it resolves white phase from flicker phase (add white phase noise to test this).
• Estimate confidence intervals on σ_y(τ) using chi-squared statistics (see Riley, NIST SP 1065, §5).

Bayesian Characterisation

• Parametrise the noise model as h = (h₋₂, h₋₁, h₀). Define priors for each amplitude.
• Run MCMC to estimate the posterior p(h | data). Report posterior medians and 95% credible intervals.
• Perform posterior predictive checks: generate synthetic Allan deviation curves from the posterior and compare to the observed curve.
• Test model selection: compare a two-component model (white + flicker) against the full three-component model using the Bayes factor.

Cross-Validation

• Do the frequentist confidence intervals and Bayesian credible intervals overlap? Where do they diverge? Explain.
• Record all results, code, and reasoning in your lab notes.

2.1 Research Question

For two clocks separated by a comparison link with its own noise, does the predicted η_opt from Exercise D.4.2 match the simulated optimum?

2.2 Tasks

Simulation

• Simulate two independent clocks with identical noise models from Block 1.
• Model the comparison link: add white phase noise with amplitude h_2,link representing propagation and detection noise.
• Compute the comparison time series y_AB(t) = y_A(t) − y_B(t) + y_link(t).
• Compute σ_AB(τ) and identify τ_opt — the minimum of the total comparison instability.

The η Landscape

• Define a range of comparison distances L_comparison (e.g. 1 m to 10⁸ m).
• For each L, compute η_opt = L/(cτ_opt).
• Plot η_opt vs L_comparison and compare to the analytic prediction.
• Introduce asymmetric noise (different clocks) and repeat. How does η_opt shift?

Bayesian Extension

• From the comparison time series, perform Bayesian inference on the link noise amplitude h_2,link treating the clock noise as known (from Block 1 posterior).
• Compute the posterior predictive σ_AB(τ) with uncertainty bands.

3.1 Research Question

How does network topology (line, triangle, star, complete graph) affect the achievable comparison accuracy, and can triangular closure detect hidden systematics?

3.2 Tasks

Triangle Network

• Simulate three clocks with different noise profiles: one “good” (low h₀), one “mediocre”, one “poor”.
• Compute all three pairwise comparisons. Verify closure: y_AB + y_BC + y_CA ≈ 0.
• Inject a systematic frequency offset on one clock. Show that closure breaks and the residual localises the faulty clock.
• Inject a systematic on the link rather than the clock. Show how closure residuals now point to the link rather than a clock.

Scaling to Larger Networks

• Using NetworkX, construct networks of N = 4, 6, 8 clocks in different topologies (line, ring, star, complete).
• For each topology, compute the number of independent closure constraints and the resulting constraint-to-parameter ratio.
• Simulate and compute the network-averaged stability. Which topology yields the best average comparison accuracy per unit of comparison resource?

Bayesian Network Analysis

• Treat all clock frequencies and noise amplitudes as unknowns. Use the full set of pairwise comparisons as data. Run a hierarchical Bayesian model.
• Compare the posterior on each clock’s frequency to the true (simulated) value. Does the network information improve estimation compared to isolated pairwise analysis?

4.1 Research Question

Can a single comparison-network framework meaningfully rank clocks of fundamentally different kinds? Where does the framework succeed, and where does it break down?

4.2 Tasks

Clock Zoo

Simulate clocks inspired by real physical systems. For each, assign noise parameters from literature values (document sources):

Unified η Map

• Place all clock types in a single network (complete graph). Compute η_opt for every pair.
• Create a comprehensive η_opt map: x-axis = L_comparison, y-axis = τ_opt, colour = clock-pair type.
• Overlay the L = cτ boundary. Discuss which comparisons approach the causal limit.

Limits of the Framework

• For pulsar timing: the noise is non-stationary and the comparison interval is intrinsically set by orbital mechanics. Discuss how η behaves when τ is not freely choosable.
• For optical lattice clocks: L_source is quantum-mechanically determined and cannot be decoupled from L_apparatus. Discuss what this implies for the three-length separation.
• State clearly where the framework works, where it requires modification, and where it breaks.

Findings Session Preparation

Before the findings session (Step 3), prepare a ∼30-min presentation covering:

1. Key results with uncertainties from each block.
2. Comparison to expectations: where did simulation match theory? Where did it diverge?
3. η_opt map as the central figure.
4. Unresolved questions and their implications for the framework.

Short Report

The short report documents your analysis pathway. It is due 7 calendar days before your seminar. See Rulebook: Seminar Experiment for format requirements and the short report checklist.

Seminar Presentation

The seminar develops your short report into a coherent scientific narrative (60 min total). See Rulebook: Seminar Presentations Checklist.

Return to Framework or proceed to the Rulebook →