Q021 · Echoes and the noise behind dephasing

Spin Echo & Noise Spectroscopy

Which lost coherence can a pulse bring back? Only dephasing from frequency offsets that stay steady through the sequence. A flip halfway through reverses the phase each spin has gathered, so the spins line up again at the end. Fast fluctuations, irreversible dephasing and errors in the pulse area are not undone.

Watch the phase arrows fan out, flip and come back together. Compare the same fixed detunings with and without a pulse, predict the result, then test it with simulated counts. Open the full experiment to explore fluctuating noise and how its spectrum can be inferred.

Your sequence · alignment $100\%$

No pulse · alignment $100\%$

The starting state is ready. Move time forward to see what changes.

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 $\text{—}$

Let the phases meet again

The arrows spread as they turn at different speeds. Flip them halfway through and watch them regroup.

Total free evolution

T

2\,\mathrm{ms}

Frequency spread

\sigma_f

0.35\,\mathrm{kHz}

Pulse position

\eta

0.5

Playback position

40\%

Elapsed model time $0.8\,\mathrm{ms}$

Drag the orange pulse marker to change when the flip happens.

Make a prediction, then test it

At the endpoint, will this sequence keep the arrows better aligned than no pulse?

Choose a prediction and drag time to the end.

Noise, pulse errors and state sphere

Noise: static / correlatedOU correlation time

\tau_c

1.5\,\mathrm{ms}

Inversions: zero is Ramsey

n

1

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 move the inversion pulses in time. Each shot prepares the state from scratch, and replaying the protocol adds no counts.

A static detuning keeps one value for the whole of each preparation. The correlation time matters only for the fluctuating OU noise. With ideal pulses the coherence is computed exactly from a Gaussian filter; with pulse errors it is averaged over 1,024 trajectories.

The model is a two-level system in the rotating frame, with classical Gaussian noise in the detuning. Pulses act instantly and can carry a set area error; finite pulse width and energy relaxation are left out. Ideal pulses have an exact filter-function reference. Imperfect pulses are averaged over a finite set of noise trajectories, and the numerical error of that average is reported separately. 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: Compared with an independently measured Ramsey baseline, the chosen pulse sequence refocuses slow detuning. Fast changes, irreversible dephasing and pulse-area errors 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 and has zero mean. Detuning is in kHz and time in ms. A separate static mode keeps one Gaussian detuning fixed throughout each preparation.

Coherence with ideal pulses

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 holds only when every pulse has the ideal area.

Quasi-static Hahn limit

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

A single inversion at the midpoint refocuses a static spread exactly. Moving it off the midpoint leaves a net signed free-evolution time that no longer cancels.

The OU noise 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 the model's power spectral density and the filter weight, each normalized for display. It is a diagnostic, not a spectrum reconstructed from measured counts.

Common difficulties

Coherence is not energy

Typical misconceptionWhenever the transverse vector shrinks, energy has been lost.

Better mental modelThis experiment has dephasing and rotations but no bath for energy relaxation. An imperfect rotation can change the populations without any thermal relaxation.

A filter covers a band, not a single frequency

Typical misconceptionOne decay curve identifies the whole noise spectrum.

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

The dots are simulation, not data

Typical misconceptionThe small dots are the directions of individual spins, all measured at the same time.

Better mental modelThey are hidden simulation trajectories, drawn in a toggling frame used for analysis. Each acquired record holds only a two-valued outcome, and the density matrix averages over noise that is never observed.

Run the experiment

01
Refocus a static spread
Choose Exact static refocusing. Replay the protocol, then acquire a time scan.
What to observe: The Hahn-echo counts stay close to certainty, while the independent Ramsey record decays. A shortened vector stands for the density matrix of the whole ensemble.
02
Break the timing
Drag the orange inversion marker away from the midpoint and reacquire.
What to observe: Some phase spread comes back. The keyboard sliders set the same timing, and moving the camera changes neither the physics nor the counts.
03
Probe noise with a finite memory
Choose Hahn echo, acquire the two decay records, then choose Four-pulse train and repeat.
What to observe: The likelihood fit uses only the counts and assumes OU noise with a known Markov rate. A parameter range that comes out broad, or touches the search boundary, means the data pin that parameter down only weakly.
04
Watch an inference fail
Choose Imperfect inversions, compare the selected counts with the dashed ideal reference, then choose Fast noise.
What to observe: When a pulse-area error is present, the spectrum fit that assumes ideal pulses is switched off. The numerical error of the trajectory integration is reported separately and never becomes part of a shot confidence interval.

Spin Echo & Noise Spectroscopy

Use the records to identify the noise.