Tutorial

This tutorial bridges the Framework and the Lab. Part I develops derivations by hand; Part II translates them into working code. Both parts are prerequisites for the entrance session.

D.1 From Oscillator to Phase Variable

Start with a damped harmonic oscillator driven by a restoring potential. The general solution is

where A(t) captures amplitude noise and φ(t) captures phase noise.

Exercise D.1.1

Show that for a high-Q oscillator (Q >> 1), amplitude fluctuations decay on a timescale τ_A = 2Q/ω₀, while phase fluctuations accumulate without bound. Argue why clock metrology focuses on φ(t) rather than A(t).

Exercise D.1.2

Define the fractional frequency deviation y(t) = (1/2πν₀) dφ/dt. Show that the time-averaged fractional frequency over an interval [t, t+τ] is

D.2 Allan Variance from First Principles

Exercise D.2.1

Starting from the definition of ¯y_k, derive the two-sample (Allan) variance:

Show that this is equivalent to

Exercise D.2.2

Evaluate the integral above for S_y(f) = h₀ (white frequency noise). Show that σ_y(τ) ∝ τ^−1/2 and determine the proportionality constant in terms of h₀.

Exercise D.2.3

Repeat for S_y(f) = h₋₁f⁻¹ (flicker frequency noise). Show that σ_y(τ) becomes independent of τ (flat on the log-log plot). Explain physically why averaging does not improve stability in this regime.

D.3 Bayesian Parameter Estimation

Exercise D.3.1

Consider N independent measurements of fractional frequency, y_i, drawn from a Gaussian with unknown mean μ and known variance σ². Write down the likelihood, choose a conjugate Gaussian prior for μ, and derive the posterior analytically. Show that the posterior mean is a precision-weighted average of the prior mean and the sample mean.

Exercise D.3.2

Now suppose both μ and σ² are unknown. Write down the joint likelihood. Argue why a Jeffreys prior p(σ) ∝ 1/σ is appropriate for the scale parameter. Show that the marginalised posterior for μ is a Student-t distribution. Discuss: what does the heavier tail of the Student-t imply for outlier sensitivity compared to the Gaussian case?

D.4 From Pairwise Comparison to the η Parameter

Exercise D.4.1

Two clocks are separated by distance L. A phase comparison requires a signal round trip (or one-way with synchronisation). Show that the minimum comparison interval is T_min = L/c (one-way) or T_min = 2L/c (round trip). Define η(τ) = L/(cτ) and argue that η ≤ 1 is a necessary condition for a meaningful comparison.

Exercise D.4.2

For a clock with white frequency noise (S_y = h₀), the Allan deviation improves as τ^−1/2 with averaging. But as τ increases, so does the accumulation of propagation noise on the comparison link. Model the link noise as white phase noise with amplitude h_2,link. Derive the total comparison uncertainty σ_total(τ) = σ_clock(τ) ⊕ σ_link(τ) and find the τ_opt that minimises it. Express η_opt = L/(cτ_opt) in terms of h₀ and h_2,link.

Exercise D.4.3

Interpret η_opt physically. Sketch the two competing contributions and their sum on a log-log plot. Explain: why is η_opt the falsifiable prediction of the causal-geometry framework?

C.1 Noise Generation

Exercise C.1.1 — White Frequency Noise

Generate N = 10⁶ samples of white frequency noise y[n] with amplitude h₀. Compute the Allan deviation using the overlap estimator and verify the τ^−1/2 slope.

import numpy as np

def white_freq_noise(N, h0, dt=1.0):
    """Generate white frequency noise samples."""
    sigma_y = np.sqrt(h0 / (2 * dt))
    return np.random.normal(0, sigma_y, N)

# Your code: compute phase by cumulative sum, then Allan deviation

Exercise C.1.2 — Power-Law Noise via Spectral Shaping

Generate flicker frequency noise (α = −1) by filtering white noise in the frequency domain. Compare the resulting Allan deviation slope to your analytic prediction from D.2.3.

def power_law_noise(N, alpha, h_alpha, dt=1.0):
    """Generate power-law noise S_y(f) = h_alpha * f^alpha
       via spectral shaping of white noise."""
    freqs = np.fft.rfftfreq(N, d=dt)
    freqs[0] = freqs[1]  # avoid division by zero
    # Shape the spectrum
    white = np.fft.rfft(np.random.normal(0, 1, N))
    shaped = white * np.sqrt(h_alpha * np.abs(freqs)**alpha * dt)
    return np.fft.irfft(shaped, n=N)

Exercise C.1.3 — Mixed Noise

Generate a time series with both white frequency and flicker frequency noise. Plot the Allan deviation and identify the crossover τ where the dominant noise type changes. Compare to the analytic prediction.

C.2 Allan Deviation from Scratch

Exercise C.2.1

Implement the non-overlapping Allan variance estimator directly from the phase time series φ[n]:

def allan_variance(phase, taus, dt=1.0):
    """Compute non-overlapping Allan variance from phase data.
    
    Parameters
    ----------
    phase : array  - cumulative phase samples
    taus  : array  - averaging times to evaluate
    dt    : float  - sampling interval
    
    Returns
    -------
    adev : array - Allan deviation for each tau
    """
    adev = np.zeros(len(taus))
    for i, tau in enumerate(taus):
        n = int(tau / dt)
        # Second differences of phase
        diffs = phase[2*n::n] - 2*phase[n:-n:n] + phase[:-2*n:n]
        adev[i] = np.sqrt(np.mean(diffs**2) / (2 * tau**2))
    return adev

Test against allantools.adev() for each noise type. Quantify agreement.

C.3 Bayesian Fitting

Exercise C.3.1 — Frequency Offset Estimation

Generate 100 samples of fractional frequency from a clock with known offset μ = 1×10⁻¹³ and white frequency noise. Use MCMC to recover μ with a Gaussian prior. Compare the posterior width to the frequentist standard error.

import emcee

def log_posterior(theta, data, prior_mu, prior_sigma, sigma_known):
    mu = theta[0]
    # Log-likelihood (Gaussian)
    ll = -0.5 * np.sum((data - mu)**2 / sigma_known**2)
    # Log-prior (Gaussian)
    lp = -0.5 * ((mu - prior_mu) / prior_sigma)**2
    return ll + lp

# Your code: set up sampler, run chains, check convergence

Exercise C.3.2 — Model Selection

Generate a dataset from white + flicker frequency noise. Fit two models: (1) pure white frequency, (2) white + flicker. Estimate the Bayes factor using the Savage-Dickey density ratio or thermodynamic integration. Which model is preferred, and by how much?

Exercise C.3.3 — Posterior Predictive Checks

From the posterior of C.3.2, generate 200 synthetic datasets. For each, compute the Allan deviation at τ = 1, 10, 100, 1000 s. Plot the distribution of synthetic Allan deviations against the observed values. Does the posterior predictive distribution contain the observations?

C.4 Comparison Network

Exercise C.4.1 — Two-Clock Comparison

Simulate two clocks with different noise profiles. Add white phase noise on the comparison link. Compute the comparison Allan deviation σ_AB(τ) and identify τ_opt. Calculate η_opt and compare to your derivation in D.4.2.

Exercise C.4.2 — Triangular Closure

Extend to three clocks. Compute all three pairwise comparisons and verify closure: y_AB + y_BC + y_CA → 0. Inject a systematic offset on one link and show that the closure residual reveals it.

Exercise C.4.3 — The η Landscape

For a three-clock network with varying separations, compute η_opt for each pair. Plot a map of η_opt vs L_comparison. Overlay the analytic prediction from D.4.2. Discuss where the prediction breaks down and why.

Submission

Bring your completed derivations (handwritten or typeset) and working Jupyter notebooks to the entrance session. Push notebooks to your fork of the repository before the session.

Continue to the Lab →