---
title: "Rabi Pulse Calibration Interactive Lab | Sandbox Physics"
description: "How long must a pulse last to flip a state? On resonance, half a Rabi period; this is a pi pulse."
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last_modified: "2026-10-27"
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Q019 · Coherent drive and calibration

# Rabi Pulse Calibration

**How long must a pulse last to flip a state?** On resonance, half a Rabi period; this is a pi pulse. Take the Rabi frequency from a measured scan of pulse length, not from the nominal drive setting. Detuning makes the oscillation faster but lowers its peak, so a detuned pulse reaches its maximum sooner yet never flips the state completely.

Set the drive amplitude, phase and detuning, then scan the pulse length. Estimate the real drive strength from a finite number of counts and test the pulse you calibrated. An amplitude error shows how the nominal setting and the measured calibration can disagree.

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

The model is an isolated two-level system in a rotating frame, driven by square pulses under the rotating-wave approximation. Frequencies are in cycles per millisecond. The fit assumes the detuning is known from an independent measurement. Off resonance, the first maximum is not a full flip, and population alone does not measure gate fidelity.

Simulated acquisition

## Make a prediction, then collect the evidence.

No acquired record yet.

### Inference from acquired counts

Acquire a scan to estimate the signal.

$\text{—}$

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)

Physics tutorial

## Calibrate a pulse from counts

**Background** A coherent square pulse rotates a two-level state around an effective axis. Detuning tilts this axis.

**Why it matters** When the control electronics have a calibration error, the amplitude you set is not the amplitude the atom feels. Scanning the pulse length measures the population and gives a calibration from data.

Start with the essentials

- **Focus question**: Does the first peak of the oscillation always flip the state completely?
- **One-sentence intuition**: Detuning speeds up the oscillation but lowers its peak, so a faster oscillation can come with a worse flip.

### Core mathematical model

#### Square-pulse Hamiltonian in the rotating frame

$$
H_{\rm rot}=\frac{\hbar}{2}(\Omega\cos\varphi\,\sigma_x+\Omega\sin\varphi\,\sigma_y+\Delta\sigma_z)
$$

The drive strength and phase set a rotation axis in the equatorial plane, and detuning adds a component along Z. All frequencies here are angular frequencies.

#### Excited-state population after a pulse

$$
P_e(\tau)=\frac{\Omega^2}{\Omega^2+\Delta^2}\sin^2\left(\frac{\sqrt{\Omega^2+\Delta^2}\,\tau}{2}\right)
$$

This two-level result is exact under the rotating-wave approximation. Detuning lowers the peak and makes the oscillation faster.

#### Resonant flip and half-flip durations

$$
\tau_\pi=\frac{1}{2f_R},\quad\tau_{\pi/2}=\frac{1}{4f_R}\quad(\Delta=0)
$$

On resonance, a full flip takes half a Rabi period and a half flip takes a quarter of one.

### Common difficulties

#### Population is not gate fidelity

**Typical misconception** If the excited population is high, the gate must work well for every input.

**Better mental model** This experiment tests one starting state with one readout. Measuring process fidelity takes more input states and measurements that are sensitive to phase.

#### The fit depends on its model

**Typical misconception** The drive estimate still holds when the detuning is unknown, or in a real atom with many levels.

**Better mental model** The fit assumes a known detuning, the chosen scan range and an isolated two-level system under the rotating-wave approximation. It ignores other levels and any leakage into them.

### Run the experiment

1. 01

   #### Find the first peak

   Choose Full flip and acquire a duration scan. Read the inferred drive, then set the duration to the fitted peak.

   **What to observe:** The drive estimate comes from a binomial likelihood fit to the counts, with the detuning taken from an independent measurement.

2. 02

   #### Prepare a half flip

   Choose Half flip and compare the final Bloch vector with the measured population.

   **What to observe:** Here the equal populations come from a coherent transverse state, not from an unpolarized mixture.

3. 03

   #### Find a calibration error

   Choose Calibration error and acquire a scan. Compare the fitted drive with the nominal drive.

   **What to observe:** A 20% amplitude error moves the best pulse length. The setting and the fitted value are shown separately.

4. 04

   #### Tilt the rotation axis

   Choose Off resonance, acquire a scan and rotate the drive phase.

   **What to observe:** The first peak never reaches a full flip. The drive phase changes the transverse part of the state, but the excited population reached from the ground state does not depend on it.

Physical conclusion

Calibrate the pulse from a finite number of counts, then test the fitted duration under the same model assumptions.

## 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 **Following a fountain clock’s extrapolation to zero density**](https://sandboxphysics.com/stories/quantum-evidence/fountain-clock-collision-density-extrapolation/)
2. [Lab **Ramsey Frequency Measurement** Extrapolate the clock frequency to zero atom density](https://sandboxphysics.com/labs/ramsey-frequency/)

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)