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.

Model positive probability $\text{—}$

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

Current-setting trials $\text{—}$

Acquired positive fraction $\text{—}$

95% Wilson interval $\text{—}$

Output-state purity $\text{—}$

Actual drive · model truth $\text{—}$

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.

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

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 misconceptionIf the excited population is high, the gate must work well for every input.

Better mental modelThis 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 misconceptionThe drive estimate still holds when the detuning is unknown, or in a real atom with many levels.

Better mental modelThe 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

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.
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.
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.
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.

Rabi Pulse Calibration

Make a prediction, then collect the evidence.

Inference from acquired counts

Calibrate a pulse from counts

Core mathematical model

Square-pulse Hamiltonian in the rotating frame

Excited-state population after a pulse

Resonant flip and half-flip durations

Common difficulties

Population is not gate fidelity

The fit depends on its model

Run the experiment

Find the first peak

Prepare a half flip

Find a calibration error

Tilt the rotation axis

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