Decoherence Timescale Estimation

This notebook guide specifies a reproducible calculation for estimating decoherence times from pure-dephasing models. The goal is to compute coherence envelopes for chosen noise spectra, vary spectral parameters, compare Ramsey and echo-style filter functions, and estimate $T_2$, $T_2^*$, and $T_\phi$ with explicit conventions.

As of this review, no executable notebook under notebooks/density-open-systems/decoherence-timescale-estimation/ is promoted as a reproduced artifact. This page is the admission contract for that notebook: it states the model, spectra, numerical quadrature, fitting rules, validation tests, and outputs required before estimated timescales should be cited.

Purpose

The notebook should demonstrate how to:

• compute pure-dephasing coherence envelopes from noise spectra;
• compare Ramsey, echo, and simple dynamical-decoupling filters;
• vary white, Lorentzian, and low-frequency noise models;
• estimate $T_2^*$, echo $T_2$, and pure-dephasing time $T_\phi$;
• distinguish fitted timescales from microscopic mechanisms;
• check numerical quadrature and cutoff sensitivity;
• record conventions for one-sided versus two-sided spectra.

The first version should stay focused on a single qubit with classical Gaussian frequency noise. Relaxation, non-Gaussian baths, and full quantum noise can be added only after this baseline is reproducible.

Directory Plan

Use a dedicated directory:

notebooks/density-open-systems/decoherence-timescale-estimation/
  decoherence-timescale-estimation.ipynb
  README.md

The opening notebook cell or README.md should state:

• Python and package versions;
• frequency units and angular-frequency convention;
• whether spectra are one-sided or two-sided;
• filter-function convention;
• quadrature grid and cutoffs;
• sequence definitions;
• fit model and fit window;
• random seed if stochastic time traces are generated;
• date and commit identifier when the notebook is promoted.

Baseline Dephasing Model

Use a qubit Hamiltonian with a stochastic frequency shift,

$$H(t) = \frac{\hbar}{2} \left[ \omega_0+\xi(t) \right] \sigma_z .$$

In an interaction picture with respect to $\omega_0$, the coherence accumulates random phase

$$\phi(T) = \int_0^T y(t)\xi(t)\,dt,$$

where $y(t)$ is the modulation function set by the pulse sequence. For Ramsey, $y(t)=1$. For a simple spin echo,

$$y(t) = \begin{cases} 1,&0\le t\lt T/2,\\ -1,&T/2\le t\le T. \end{cases}$$

For zero-mean Gaussian noise, the coherence envelope is

$$W(T) = \left\langle e^{-i\phi(T)}\right\rangle = e^{-\chi(T)}.$$

The notebook should estimate $\chi(T)$ from a spectral-density integral and verify selected cases against analytic expressions.

Filter-Function Convention

Define

$$Y(\omega,T) = \int_0^T y(t)e^{i\omega t}\,dt.$$

For a two-sided classical frequency-noise spectrum $S_{\xi\xi}(\omega)$ with

$$\langle \xi(t)\xi(0)\rangle = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, e^{-i\omega t} S_{\xi\xi}(\omega),$$

use

$$\chi(T) = \frac12 \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, S_{\xi\xi}(\omega) |Y(\omega,T)|^2.$$

This convention assumes $\xi(t)$ has units of angular frequency. If the notebook uses one-sided spectra or ordinary frequency $f$, it must convert before comparing numbers.

For Ramsey,

$$Y_R(\omega,T) = \frac{e^{i\omega T}-1}{i\omega}.$$

The notebook may compute $Y$ by numerical integration for arbitrary sequences, but it should compare the Ramsey result with this analytic expression.

Noise Spectra

Implement at least three spectra.

White noise

Use

$$S_{\xi\xi}(\omega)=S_0.$$

With the convention above, Ramsey coherence decays exponentially:

$$\chi(T) = \frac{S_0T}{2}.$$

This is the primary analytic validation case.

Lorentzian noise

Use an Ornstein-Uhlenbeck-style spectrum,

$$S_{\xi\xi}(\omega) = \frac{2\sigma^2\tau_c}{ 1+\omega^2\tau_c^2 }.$$

Here $\sigma^2$ is the variance of $\xi$ and $\tau_c$ is the correlation time. This model should interpolate between slow quasi-static noise and faster Markov-like noise as $\tau_c$ changes.

Low-frequency noise with cutoffs

Use a regularized low-frequency model such as

$$S_{\xi\xi}(\omega) = \frac{A^2}{ |\omega|+\omega_{\mathrm{ir}} } \,e^{-|\omega|/\omega_{\mathrm{uv}}}.$$

The infrared cutoff $\omega_{\mathrm{ir}}$ and ultraviolet cutoff $\omega_{\mathrm{uv}}$ are part of the model. The notebook should show how estimated times can change when the cutoffs are moved.

Timescale Definitions

The notebook should define each reported time explicitly.

For a computed envelope $|W(T)|$, define a threshold time by

$$|W(T_*)| = e^{-1}.$$

Use labels tied to the sequence:

• Ramsey threshold time: $T_2^*$ in common qubit language;
• echo threshold time: echo $T_2$ or $T_{2,\mathrm{echo}}$;
• decoupled threshold time: state the sequence name and pulse count.

If an exponential fit is used,

$$|W(T)| \approx e^{-T/T_2},$$

then the notebook should state the fit window and residuals. If a Gaussian fit is used,

$$|W(T)| \approx e^{-(T/T_G)^2},$$

then $T_G$ is not the same parameter as an exponential $T_2$ unless the convention is explicitly defined.

T2 and Tφ Bookkeeping

If a separate population-relaxation time $T_1$ is supplied, estimate pure dephasing only through the standard weak-coupling Markovian relation

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}.$$

Thus

$$\frac{1}{T_\phi} = \frac{1}{T_2} - \frac{1}{2T_1}.$$

The notebook should refuse or warn if this expression gives a negative rate, if the fitted $T_2$ is not an exponential transverse decay time, or if the data correspond to a decoupling sequence rather than free transverse decay.

Numerical Workflow

The notebook should follow this workflow:

1. Define all spectral conventions and units.
2. Implement Ramsey and echo modulation functions.
3. Compute $Y(\omega,T)$ analytically for Ramsey and numerically for general $y(t)$.
4. Implement white, Lorentzian, and low-frequency spectra.
5. Integrate $\chi(T)$ over a symmetric frequency grid.
6. Validate the white-noise Ramsey result.
7. Sweep spectral parameters and pulse sequences.
8. Extract threshold times and optional fit times.
9. Store all times with their sequence, spectrum, cutoffs, and fit window.

For low-frequency spectra, the notebook should not hide cutoff dependence. If a quoted time changes materially with $\omega_{\mathrm{ir}}$, that is a result, not a nuisance to erase.

Validation Tests

The accepted notebook should include automated checks:

Check	Required behavior
Ramsey filter	numerical $Y_R$ agrees with analytic $Y_R$
white noise	$\chi(T)$ agrees with $S_0T/2$
grid convergence	estimated times stabilize as grid is refined
cutoff sensitivity	low-frequency results report cutoff changes
sequence labels	Ramsey, echo, and decoupled times are not merged
fit residuals	exponential or Gaussian fits report residuals
Tφ bookkeeping	$T_\phi$ estimate warns on negative inferred rate

The validation file should contain enough metadata that a reader can reproduce every reported time without guessing hidden cutoffs.

Accepted Outputs

Save accepted outputs under:

notebooks/density-open-systems/decoherence-timescale-estimation/outputs/
  coherence-envelopes.svg
  filter-functions.svg
  spectrum-sweep-timescales.svg
  timescale-estimates.csv
  decoherence-timescale-validation.json

The CSV should include:

• spectrum name;
• spectrum parameters;
• sequence;
• threshold definition;
• estimated time;
• fit model, if any;
• fit window;
• quadrature cutoffs;
• grid size;
• validation status.

Common Mistakes

• Reporting “the $T_2$” without saying Ramsey, echo, or decoupled sequence.
• Mixing one-sided and two-sided spectra without converting factors of $2$.
• Mixing angular frequency $\omega$ and ordinary frequency $f$.
• Fitting a Gaussian envelope and calling the parameter an exponential $T_2$.
• Inferring $T_\phi$ from $T_2$ and $T_1$ outside the weak-coupling Markovian bookkeeping model.
• Treating low-frequency cutoff choices as harmless when they control the result.
• Using a filter function without checking its normalization against a known case.

Exercises

White-Noise Ramsey Envelope

Using the convention in this page, white noise gives $\chi(T)=S_0T/2$. What is the threshold time $T_*$ defined by $|W(T_*)|=e^{-1}$?

Solution

The coherence envelope is

$$|W(T)| = e^{-\chi(T)} = e^{-S_0T/2}.$$

Set this equal to $e^{-1}$:

$$\frac{S_0T_*}{2} = 1.$$

Therefore

$$T_* = \frac{2}{S_0}.$$

Quasi-Static Gaussian Noise

Suppose $\xi$ is constant during each Ramsey shot but varies between shots with Gaussian variance $\sigma^2$. Show that the Ramsey envelope is $e^{-\sigma^2T^2/2}$.

Solution

For one shot,

$$\phi(T)=\xi T.$$

The ensemble coherence is the characteristic function of a zero-mean Gaussian:

$$W(T) = \left\langle e^{-i\xi T}\right\rangle = e^{-\sigma^2T^2/2}.$$

This is a Gaussian decay envelope, not an exponential Markovian decay.

Tφ from T1 and T2

Assume the Markovian relation

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}.$$

If $T_1=40\,\mu\mathrm{s}$ and $T_2=20\,\mu\mathrm{s}$, what is $T_\phi$?

Solution

Compute

$$\frac{1}{T_\phi} = \frac{1}{T_2} - \frac{1}{2T_1} = \frac{1}{20\,\mu\mathrm{s}} - \frac{1}{80\,\mu\mathrm{s}}.$$

Thus

$$\frac{1}{T_\phi} = \frac{3}{80\,\mu\mathrm{s}},$$

so

$$T_\phi = \frac{80}{3}\,\mu\mathrm{s} \approx 26.7\,\mu\mathrm{s}.$$

Cutoff Dependence

Why should a low-frequency noise calculation report the infrared cutoff?

Solution

Low-frequency spectra such as regularized $1/f$ models can place substantial weight near zero frequency. Ramsey dephasing is especially sensitive to slow fluctuations, so changing the infrared cutoff can change the computed coherence envelope and the inferred $T_2^*$. The cutoff may represent finite measurement time, drift subtraction, feedback, or a physical low-frequency rolloff. It is therefore part of the model and must be reported.

Cross-Links

• Pure Dephasing Master Equation
• Noise Spectra
• Common Noise Spectra
• Dynamical Decoupling
• Dephasing vs Dissipation
• Quantum Sensing
• Spin Qubits
• NV Centers and Solid-State Defects
• Bloch Vector Noise Models

References

• L. Cywiński, R. M. Lutchyn, C. P. Nave, and S. Das Sarma, “How to enhance dephasing time in superconducting qubits,” Physical Review B 77, 174509 (2008).
• G. Ithier et al., “Decoherence in a superconducting quantum bit circuit,” Physical Review B 72, 134519 (2005).
• R. de Sousa, “Electron spin as a spectrometer of nuclear-spin noise and other fluctuations,” Topics in Applied Physics 115, 183-220 (2009).
• C. Álvarez and D. Suter, “Measuring the spectrum of colored noise by dynamical decoupling,” Physical Review Letters 107, 230501 (2011).
• C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Reviews of Modern Physics 89, 035002 (2017).