---
title: "Ramsey Frequency Measurement Interactive Lab | Sandbox Physics"
description: "Can a longer wait reveal a smaller frequency difference? Yes."
canonical_url: "https://sandboxphysics.com/labs/ramsey-frequency/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/ramsey-frequency/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Simulated acquisition

## Make a prediction, then collect the evidence.

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](https://www.feynmanlectures.caltech.edu/III_06.html) · [MIT 8.05](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/lecture-notes-7/) · [NIST Ramsey](https://www.nist.gov/publications/ramsey-spectroscopy-displaced-frequency-jumps)

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

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Separate density-calibration uncertainty

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Measured density slope

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Even cavity phase · joint branch

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Reversible phase · modulo a half turn

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

> Plot of retained counts and their inference; definitions and raw-event export remain available in text.

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

> Plot of retained counts and their inference; definitions and raw-event export remain available in text.

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](https://arxiv.org/pdf/1110.2590) · [Hoth et al. 2022 / NIST](https://tf.nist.gov/general/pdf/3156.pdf) · [Pereira Dos Santos et al. 2002](https://arxiv.org/pdf/physics/0209010)

Physics tutorial

## Trade waiting time against coherence

**Background** The first pulse creates transverse coherence. A phase builds up during the dark time, and the second pulse turns that phase into population.

**Why it matters** Ramsey's method underlies precision spectroscopy. Fringe contrast, aliases and systematic shifts limit what a finite set of measurements can show.

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 misconception** A narrow fringe in the simulation proves the atomic clock is accurate.

**Better mental model** The 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 misconception** The estimate in the principal interval is the only possible frequency.

**Better mental model** Adding 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

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

Physical conclusion

For the original scan, report the frequency with its alias spacing and contrast limits. For the clock protocol, also report the search prior, the density calibration, the residual checks and the count uncertainty on its own.

## Related experiments and articles

Quantum evidence · Reading path

### Clocks, traps and single-quantum control

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How Ramsey and ion traps gave atoms more time**](https://sandboxphysics.com/stories/quantum-evidence/ramsey-ion-traps-measurement/)
2. [Lab **Trapped-Ion Sideband Spectroscopy** Compare the motional sidebands of a trapped ion](https://sandboxphysics.com/labs/trapped-ion-sideband-spectroscopy/)
3. [Article **Reading a cavity phase error in an atomic fountain**](https://sandboxphysics.com/stories/quantum-evidence/ramsey-cavity-phase-frequency-bias/)
4. [Article **Counting the quantum noise left by a perfect detector**](https://sandboxphysics.com/stories/quantum-evidence/quantum-projection-noise-population-counts/)
5. [Lab **Quantum State & Born Measurement** Measure one qubit along more than one direction](https://sandboxphysics.com/labs/qubit-state/)
6. [Article **Following a fountain clock’s extrapolation to zero density**](https://sandboxphysics.com/stories/quantum-evidence/fountain-clock-collision-density-extrapolation/)
7. [Lab **Rabi Pulse Calibration** Calibrate a pulse that moves atoms between levels](https://sandboxphysics.com/labs/rabi-pulse-calibration/)
8. [Article **How one trapped ion switches its fluorescence off and on**](https://sandboxphysics.com/stories/quantum-evidence/electron-shelving-single-ion-jumps/)
9. [Lab **Fluorescence & Quantum Jumps** Count simulated fluorescence clicks from one atom](https://sandboxphysics.com/labs/fluorescence-quantum-jumps/)
10. [Article **Let a second ion read out a narrow transition**](https://sandboxphysics.com/stories/quantum-evidence/quantum-logic-spectroscopy-shared-motion/)
11. [Lab **Two-Ion Quantum Logic Spectroscopy** Read one ion through the simulated fluorescence of its partner](https://sandboxphysics.com/labs/two-ion-quantum-logic-spectroscopy/)
12. [Lab **Quantum State & Process Tomography** Find out which measurements test coherence](https://sandboxphysics.com/labs/quantum-state-process-tomography/)
13. [Article **Counting stored photons one atom at a time**](https://sandboxphysics.com/stories/quantum-evidence/qnd-photon-counting-atomic-records/)
14. [Lab **Cavity Photon Counting & Feedback** Accumulate probe-atom outcomes and infer cavity photon number](https://sandboxphysics.com/labs/cavity-photon-qnd-feedback/)
15. [Lab **Decoherence Channels & Information Loss** Distinguish information gain from coherence preservation](https://sandboxphysics.com/labs/decoherence-channels/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)