Q020 · Phase accumulation and spectroscopy

Ramsey Frequency Measurement

Can a longer wait reveal a smaller frequency difference? Yes. During a longer dark interval the same frequency difference builds up more phase, so the fringes narrow and sensitivity rises; doubling the wait halves the fringe spacing. The cost is a smaller range you can read without ambiguity, and decoherence fades the fringe contrast.

Record Ramsey spectra and phase counts, then test a clock correction at several dark times and effective densities, with the cavity phase in both directions. Infer the zero-density frequency from the recorded atoms, and see how aliases, weak contrast, a bad density calibration and a nonlinear collision law can mislead it.

Model positive probability $\text{—}$

Positive / negative / rejected $\text{—}$

Current-setting trials $\text{—}$

Acquired positive fraction $\text{—}$

95% Wilson interval $\text{—}$

Output-state purity $\text{—}$

Model fringe spacing $\text{—}$

Pulses are instantaneous and the cavity phases are set in advance; the Lab does not simulate the cavity field in space or the fountain. The original replay uses milliseconds, and the separate clock protocol uses seconds. The correction from counts alone assumes a stated prior range for the frequency, a phase that stays fixed across dark times and calibrated effective densities. Linewidth, counting error, systematic bias and long-term stability are separate quantities.

No acquired record yet.

Inference from acquired counts

Acquire a scan to estimate the signal.

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Density matrix and count conventions

\rho=\frac{I+\boldsymbol r\cdot\boldsymbol\sigma}{2}

Latest raw events · full record in CSV

Event	Setting	Outcome

Scientific basis: Feynman III.6 · MIT 8.05 · NIST Ramsey

Q020 · Count records and systematic shifts

A narrow fringe can still be biased.

Take independent phase scans over several dark times, effective densities and both settings of the reversible phase. The fit uses only the counts, the scan coordinates and the stated calibration.

This protocol is separate and runs on a scale of seconds. The state-space replay above still shows the original millisecond experiment. Cavity geometry, the fountain trajectory and the clock servo are not simulated.

Generator and independent calibration settings

Generator settings change future simulated records. Calibration and search range are assumptions of the analysis. Editing any setting starts a new sample. The middle effective density is halfway between the low and high settings; atom counts do not calibrate this ratio.

Unshifted offset · generator

\delta f_0

0.18\,\mathrm{Hz}

Linear collision coefficient · generator

k

0.3\,\mathrm{Hz}

Density curvature · counterexample

q

0\,\mathrm{Hz}

Even cavity phase · generator

\phi_{\rm e}

0.25\,\mathrm{rad}

Reversible cavity phase · generator

\phi_{\rm o}

0.4\,\mathrm{rad}

Coherence time · generator

T_2

4\,\mathrm{s}

Actual effective density ratio · generator

r_{\rm true}

0.3

Calibrated effective density ratio

r

0.3

Density ratio standard uncertainty

u_r

0.01

Frequency search half-range · declared prior

B

1.5\,\mathrm{Hz}

Atoms per phase setting

N

600

Acquisition seed

s

741

Dark-time design

No protocol record yet.

Zero-density offset · counts only

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Count standard uncertainty

\text{—}

Separate density-calibration uncertainty

\text{—}

Measured density slope

\text{—}

Even cavity phase · joint branch

\text{—}

Reversible phase · modulo a half turn

\text{—}

The correction needs complete counts that pin down every fitted parameter.

Latest completed phase scan: excited fraction and 95% Wilson intervals. Lines join acquired values only.

The frequency comes from how the phase changes with dark time, then is extrapolated in density. Error bars are standard uncertainties from the counts. The zero-density point is an extrapolation.

Phase record and retained atoms

Density	Time / s	Reversal	Phase estimate / rad	Contrast

Latest settings: every atom, excited or ground, stays in the denominator for its setting. The CSV has every event in order.

Cell	Excited	Ground	Trials

Separate generator reference and model limits

\delta f_0=0.18\,\mathrm{Hz}

The even phase is fixed across dark times; the odd phase changes sign exactly. These are prescribed reduced-model assumptions. A real distributed cavity phase shift depends on fields, trajectories and detection weighting; simple reversal does not remove every component. Density ratios require independent preparation calibration with a common collision coefficient. A wrong ratio can bias a perfectly straight fit.

The search interval limits the total atomic detuning, including the density shift. It is an outside assumption, not something the data prove. The reversible phase is half the wrapped phase difference, modulo a half turn. The even phase is reported modulo a full turn, given that principal branch of the odd phase. Adding a half turn to both phases leaves both reversal responses unchanged, and this joint ambiguity is not resolved. No true-phase label is used to pick the branch. Count uncertainties use a large-sample delta approximation and assume independent atoms. Calibration uncertainty is treated separately, to first order. Oscillator noise, finite pulse sensitivity, drift and long-term Allan stability are omitted.

Equations and primary sources

P_e=\tfrac12[1+C(T)\cos(\Phi-\varphi)]

\Phi=2\pi(\delta f_0+kn+qn^2)T+\phi_{\rm e}+s\phi_{\rm o}

C(T)=0.9e^{-T/T_2},\quad s\in\{-1,+1\}

\nu_0=\frac{\nu_{\rm L}-r\nu_{\rm H}}{1-r}

The last identity describes the two-density linear limit; this workbench fits three densities and tests a residual. Linewidth, count error, systematic bias and long-term stability are separate quantities.

Weyers et al. 2011 · Hoth et al. 2022 / NIST · Pereira Dos Santos et al. 2002

Start with the essentials

Focus question: Does doubling the dark time improve every frequency measurement?
One-sentence intuition: A longer dark time packs the fringes closer and makes the phase more sensitive, but it narrows the range of frequencies you can identify without ambiguity and lowers the contrast.

Core mathematical model

Phase built up during the dark time

\Phi=2\pi\,\delta f\,T

The phase grows in proportion to both the frequency offset and the dark time.

Ramsey fringe with ideal half-flip pulses

P_e=\tfrac12[1+e^{-T/T_2}\cos(\Phi-\varphi_2)]

With instantaneous half-flip pulses and pure dephasing, the excited population oscillates with the accumulated phase, and the fringe contrast fades as the dark time grows.

One phase fits many frequencies

\delta f=\frac{\Phi+2\pi k}{2\pi T},\quad k\in\mathbb Z

A measured phase fixes the frequency offset only up to whole multiples of the inverse dark time.

Common difficulties

Linewidth is not accuracy

Typical misconceptionA narrow fringe in the simulation proves the atomic clock is accurate.

Better mental modelThe original phase scan leaves out systematic shifts. The separate clock protocol includes a set cavity phase and a density shift, but it leaves out oscillator noise and long-term drift. A correction from counts that holds only under these assumptions does not certify accuracy or stability.

Long waits create aliases

Typical misconceptionThe estimate in the principal interval is the only possible frequency.

Better mental modelAdding any whole multiple of the inverse dark time gives the same phase. A second, independent interrogation time or a prior frequency range is needed to pick the right one.

Run the experiment

01
Acquire both scans
Choose Sensitive slope and acquire a scan. Compare the spectrum with the phase scan.
What to observe: The spectrum is plotted against the known frequency settings. The frequency estimate uses only the first harmonic of the phase counts.
02
Lengthen the dark time
Double the dark interval and acquire again.
What to observe: The fringe spacing halves. The frequency is now known only up to multiples of the new inverse dark time.
03
Test the estimate with weak contrast
Choose Lost contrast and acquire a scan. Then increase the shots or shorten the wait.
What to observe: When the contrast cannot be resolved, no frequency is reported. More shots reduce projection noise, but they cannot bring back lost coherence.
04
Move the working point
Change the second-pulse phase and the pulse-area error, then replay the sequence.
What to observe: The second pulse now reads out a different quadrature. Pulse errors change the contrast and the baseline of the simulated evolution.
05
Separate phase from frequency
In the seconds-scale clock panel, choose Cavity bias and acquire the complete protocol.
What to observe: The phase intercept changes when the cavity phase is reversed. The frequency comes from the slope of the count-derived phase across three dark times, and the estimate never sees the offset used to generate the data.
06
Test the density correction
Compare the Wrong calibration, Curved density law and One dark time presets.
What to observe: A wrong density ratio gives a fit that still looks straight but is biased. Curvature, unresolved aliases or too few dark times stop the Lab from reporting a zero-density correction.

Ramsey Frequency Measurement

Make a prediction, then collect the evidence.

Inference from acquired counts

A narrow fringe can still be biased.

Trade waiting time against coherence

Core mathematical model

Phase built up during the dark time

Ramsey fringe with ideal half-flip pulses

One phase fits many frequencies

Common difficulties

Linewidth is not accuracy

Long waits create aliases

Run the experiment

Acquire both scans

Lengthen the dark time

Test the estimate with weak contrast

Move the working point

Separate phase from frequency

Test the density correction

Related experiments and articles

Clocks, traps and single-quantum control