Q021 · Refocusing / noise correlations

Spin Echo & Noise Spectroscopy

Watch phase arrows spread, flip and come together. Compare the same steady detunings with and without a pulse, predict the result, then test it with simulated counts. Open the full experiment to explore fluctuating noise and spectrum inference.

Your sequence · alignment $100\%$

No pulse · alignment $100\%$

The prepared state is ready. Move time forward to see the change.

Clock hands are illustrative transverse components in a rotating analysis frame; they are not measured spins. Bars use the ensemble model, not the six displayed examples.

Endpoint model and acquisition readouts

Model refocused coherence $\text{—}$

Ideal-pulse reference $\text{—}$

Numerical probability error $\text{—}$

Acquired positive / negative $\text{—}$

95% Wilson fraction interval $\text{—}$

ONE CHANGE · WATCH WHAT HAPPENS

Can the phases meet again?

Both rows begin aligned. Steady detunings make the arrows turn at different speeds. Compare their alignment with and without a pulse.

1 · Choose and predict

At the end, will your sequence keep more alignment than no pulse?

2 · Play or scrub the time

Observation time

0\,\mathrm{ms}

Make a prediction, then play to the end to check it.

3 · Test with simulated counts

The animation is a model preview. These actions acquire fresh preparations at the selected endpoint; raw counts and uncertainty appear below.

Go further · parameters and state sphere

The simple comparison above resets to its stated preparation. Here you can change the full experiment. The state sphere is a mathematical representation, not a spatial spin direction.

Total free evolution

T

2\,\mathrm{ms}

RMS detuning noise

\sigma_f

0.35\,\mathrm{kHz}

Noise: static / correlatedOU correlation time

\tau_c

1.5\,\mathrm{ms}

Inversions: zero is Ramsey

n

1

Pulse offset in each cell

\eta

0.5

Inversion area error

\epsilon_\pi

0\,\mathrm{\%}

Markov pure-dephasing rate

\Gamma_\varphi

0\,\mathrm{ms^{-1}}

Shots per basis and setting

N

400

Acquisition seed

s

271

Drag the orange pulse marker to shift the inversion times. Each shot starts with a fresh preparation. Replay never adds counts.

Steady detunings stay fixed within each preparation. Correlation time acts only for fluctuating OU noise. Ideal pulses use an exact Gaussian filter; pulse errors use 1,024 trajectories.

Rotating-frame two-level model with classical Gaussian detuning noise. Pulses are instantaneous with declared area errors; their finite width and energy relaxation are omitted. Ideal pulses have an exact filter reference. Imperfect pulses use a finite trajectory ensemble with a separate numerical error. A finite decay scan cannot uniquely reconstruct an arbitrary noise spectrum.

Explore the evidence · counts, fits and raw data

Predict whether the selected inversions beat Ramsey. Acquire both time records, then inspect the conditional noise fit. Dashed curves are model references; colored points and bars are acquired fractions and 95% Wilson intervals.

No acquired record yet.

Inference from acquired counts

\text{—}

Current counts and 95% intervals

Basis	Positive / total	95% Wilson interval

Model density matrix · not reconstructed data

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

Latest raw events · full record in CSV

Event / run	Protocol / basis	Duration (ms)	Outcome

Scientific basis: Hahn 1950 · Álvarez & Suter 2011

Start with the essentials

Focus question: When do more inversion pulses preserve less coherence?
One-sentence intuition: Compare the selected sequence and an independently acquired Ramsey baseline. Slow detuning is refocused; fast changes, irreversible dephasing and pulse area error remain.

Core mathematical model

Gaussian detuning model

C_f(u)=\sigma_f^2e^{-|u|/\tau_c},\quad H/\hbar=\pi\delta f(t)\sigma_z

The OU process is stationary with zero mean. Detuning is in kHz and time in ms; a separate static mode holds one Gaussian detuning fixed during each preparation.

Ideal-pulse coherence

W(T)=e^{-\Gamma_\varphi T-\frac{(2\pi)^2}{2}\int_0^T\!\int_0^T y(t)y(s)C_f(t-s)\,dt\,ds}

The switching function changes sign at each instantaneous inversion. This exact Gaussian expression applies only to ideal pulse areas.

Quasi-static Hahn limit

W(T)=e^{-\Gamma_\varphi T-2\pi^2\sigma_f^2T^2(2\eta-1)^2}

A centered single inversion refocuses the static distribution exactly; moving it away from the midpoint leaves a residual signed time.

Declared OU spectrum

S_f(\omega)=\frac{2\sigma_f^2\tau_c}{1+\omega^2\tau_c^2},\quad \operatorname{Var}\Phi=(2\pi)^2\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}S_f(\omega)|Y(\omega)|^2

The lower plot shows model PSD and filter weight, each normalized for display. It is a diagnostic, not a spectrum reconstructed from measured counts.

Common difficulties

Coherence is not energy

Typical misconceptionEvery lost transverse vector means energy was lost.

Better mental modelThis experiment contains dephasing and rotations, with no energy-relaxation bath. An imperfect rotation can change population without thermal relaxation.

A finite filter is not a narrow spectral point

Typical misconceptionOne decay curve identifies the whole noise spectrum.

Better mental modelThe filter has finite width and side lobes. The two-parameter OU fit is conditional on the chosen family, scan window and independently known Markov rate.

Trajectory spread is not counted data

Typical misconceptionThe small dots are simultaneously measured individual spin directions.

Better mental modelThey are latent simulation trajectories, displayed in a toggling analysis frame. Individual acquired records contain only binary outcomes; the density matrix averages unobserved noise.

Run the experiment

01
Recover a static ensemble
Choose Exact static refocusing. Replay the protocol, then acquire a time scan.
What to observe: The selected Hahn counts remain close to certainty while the independent Ramsey record decays. The short vector is an ensemble density matrix.
02
Break the timing
Drag the orange inversion marker away from the midpoint and reacquire.
What to observe: A residual phase spread returns. Keyboard sliders set the same timing; camera movement changes neither physics nor counts.
03
Probe finite memory
Choose Hahn echo, acquire the two decay records, then choose Four-pulse train and repeat.
What to observe: The count-only likelihood fit assumes an OU family and a known Markov rate. Broad or boundary-touching parameter ranges report limited identifiability.
04
Expose a failed inference
Choose Imperfect inversions, compare the selected counts with the dashed ideal reference, then choose Fast noise.
What to observe: The ideal-pulse spectrum fit is disabled when area error is present. A separate trajectory integration error never becomes a shot confidence interval.

Spin Echo & Noise Spectroscopy

Follow the evidence through the noise.