One-Over-F Noise

One-over-f noise is low-frequency noise whose spectrum is approximately proportional to the inverse of frequency over a finite band. It is common in solid-state devices, materials, electronics, and precision measurements, and it is one of the standard reasons that Ramsey coherence decays are nonexponential and protocol-dependent.

The phrase usually means a spectrum like

$$S(f) \propto \frac{1}{f^\alpha}, \qquad \alpha\approx1,$$

over the measured frequency window. In angular-frequency notation one writes similarly

$$S_{\xi\xi}(\omega) \propto \frac{1}{|\omega|^\alpha}.$$

The qualification “over a finite band” is essential. A literal $1/f$ spectrum from zero frequency to infinite frequency is not a stationary noise process with finite variance.

Why It Matters

One-over-f noise is dangerous because it is concentrated where many coherent protocols are most vulnerable: near zero frequency. It produces slow detuning drift, shot-to-shot variation, calibration wander, inhomogeneous broadening, and dephasing that can be strongly improved by echo or dynamical-decoupling sequences.

Common examples include:

• charge noise in quantum dots, superconducting circuits, and nanoscale devices;
• flux noise in superconducting loops;
• critical-current and resistance fluctuations;
• magnetic noise from surface spins or defects;
• fluctuating two-level defects in materials;
• slow laser, microwave, voltage, or bias drift in apparatus.

Hardware-specific microscopic detail belongs in quantum-information hardware and materials pages. The open-system role of this page is the noise-spectrum and dephasing language.

Spectral Convention

Use a two-sided angular-frequency convention:

$$C_{\xi\xi}(t) = \langle \xi(t)\xi(0)\rangle, \qquad S_{\xi\xi}(\omega) = \int_{-\infty}^{\infty} dt\,e^{i\omega t}C_{\xi\xi}(t).$$

A simple one-over-f model is

$$S_{\xi\xi}(\omega) = \frac{A^2}{|\omega|}, \qquad \omega_{\mathrm{ir}} \le |\omega| \le \omega_{\mathrm{uv}},$$

with cutoffs outside that band. The parameter $A$ has units set by $\xi$ and the spectral convention.

The variance in this model is

$$\langle \xi^2\rangle = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} S_{\xi\xi}(\omega) = \frac{A^2}{\pi} \ln \left( \frac{\omega_{\mathrm{uv}}} {\omega_{\mathrm{ir}}} \right).$$

The logarithm is the warning. The variance depends on the measurement bandwidth and observation time. A one-over-f model without cutoffs is not a complete stochastic process.

Cutoffs and Observation Time

The infrared cutoff often comes from the longest timescale in the experiment:

$$\omega_{\mathrm{ir}} \sim \frac{1}{T_{\mathrm{obs}}},$$

or from a physical saturation of the slow fluctuator distribution. The ultraviolet cutoff can come from a microscopic switching-rate limit, a filter, a bandwidth, or the point where the spectrum crosses over to another shape.

Changing the experiment can therefore change the apparent noise variance. A slow drift that looks static during a single shot can look noisy over many hours. Conversely, a high-frequency tail may matter for a fast pulse sequence even if it is invisible in a slow drift measurement.

This is why one-over-f noise is often reported with an amplitude at a reference frequency and a measured exponent:

$$S(f) = \frac{A_f^2}{(f/1\,\mathrm{Hz})^\alpha} \quad \text{in a stated band}.$$

The units and one-sided versus two-sided convention must be stated.

Dephasing from Frequency Noise

For a qubit with frequency noise,

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

the Ramsey coherence is

$$W(T) = \left\langle \exp \left[ -i\int_0^T dt\,\xi(t) \right] \right\rangle.$$

For Gaussian stationary noise,

$$W(T)=e^{-\chi(T)},$$

where, in the convention used here,

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

For Ramsey evolution,

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

Since this filter has strong low-frequency weight, Ramsey experiments are highly sensitive to one-over-f noise. The resulting envelope is often closer to a Gaussian or stretched exponential than to a single Markovian exponential.

Ramsey Versus Echo

A Hahn echo changes the modulation function so that the time integral of static noise vanishes. Its filter is

$$Y_{\mathrm{echo}}(\omega,T) = \frac{ 1-2e^{i\omega T/2}+e^{i\omega T} } {i\omega}.$$

Near $\omega=0$, the Ramsey filter satisfies

$$|Y_{\mathrm R}(\omega,T)|^2 \approx T^2,$$

while the echo filter vanishes at zero frequency. Echo therefore suppresses the slowest part of one-over-f noise. CPMG and other dynamical-decoupling sequences extend this idea by moving the filter weight to frequencies set by the pulse spacing.

For the control side of the formalism, see Dynamical Decoupling. For threshold-time conventions, see Decoherence Timescales.

Fluctuator Picture

A common microscopic cartoon is a collection of independent two-state fluctuators. One fluctuator that switches between two values gives random telegraph noise. If its coupling amplitude is $v$ and its switching rate is $\gamma$, a common symmetric-switching spectrum has the Lorentzian form

$$S_{\mathrm{RTN}}(\omega) = \frac{4v^2\gamma} {\omega^2+4\gamma^2},$$

up to switching-rate conventions.

A broad distribution of switching rates can produce one-over-f behavior. If

$$P(\gamma) \propto \frac{1}{\gamma}$$

over many decades, then summing Lorentzians over $\gamma$ gives an approximate $1/|\omega|$ spectrum over the corresponding frequency band.

This picture is useful but not universal. Real devices can have interacting fluctuators, non-Gaussian switching, nonstationary drift, spatial correlations, temperature-dependent defects, and device-specific microscopic mechanisms.

Classical or Quantum?

Low-frequency one-over-f noise is often modeled as classical stochastic noise because the relevant fluctuators are slow compared with the system frequency and may be effectively measured by the device environment. This can be a good approximation for dephasing estimates.

It is not automatically an equilibrium thermal bath. Technical one-over-f noise generally need not satisfy the Fluctuation–Dissipation Relation or detailed balance. When a noise source can exchange energy near a transition frequency, or when quantum backaction matters, a purely classical dephasing model may be incomplete.

The practical diagnostic is:

Does the model need ordered quantum spectra, or only slow classical parameter fluctuations?

For many Ramsey and echo dephasing analyses, the classical model is the right first approximation. For relaxation, quantum-limited detection, or low-temperature absorption-emission asymmetry, it may not be.

Noise Spectroscopy

Dynamical-decoupling experiments can be used as filters to estimate noise spectra. A pulse sequence with filter $Y(\omega,T)$ samples the weighted integral

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

By changing pulse number and spacing, one changes which band contributes most to $\chi(T)$. This is powerful, but it is an inverse problem:

• the result depends on filter calibration;
• pulse errors and finite pulse widths matter;
• non-Gaussian noise is not fully characterized by $S(\omega)$;
• finite-time data cannot determine arbitrarily low frequencies;
• high-frequency tails can be aliased or filtered by the control hardware.

Noise spectroscopy should therefore report the assumed spectral model and bandwidth, not only a fitted one-over-f amplitude.

Relation to Markovian Dephasing

Markovian pure dephasing often uses a rate proportional to $S_{\xi\xi}(0)$. For one-over-f noise, this expression is not meaningful without an infrared cutoff; $S(0)$ is divergent in the ideal model.

Instead of a constant rate, one usually computes a protocol-specific envelope:

$$W(T) = e^{-\chi(T)}.$$

Only over a limited time window might this envelope be fitted by

$$W(T) \approx e^{-T/T_{\mathrm{fit}}}.$$

The fitted $T_{\mathrm{fit}}$ is then a useful experimental summary, not a fundamental Markovian rate.

Common Mistakes

• Writing $S(\omega)=A^2/|\omega|$ without infrared and ultraviolet cutoffs.
• Calling one-over-f noise white or Markovian.
• Using $S(0)$ to define a pure-dephasing rate.
• Comparing amplitudes without checking one-sided versus two-sided spectra.
• Mixing ordinary frequency $f$ and angular frequency $\omega$.
• Treating a Gaussian spectrum model as complete when rare telegraph jumps dominate.
• Assuming echo improvement proves all dephasing is reversible.
• Extrapolating a measured one-over-f law outside the measured band.
• Applying equilibrium fluctuation–dissipation relations to technical drift.
• Treating hardware-specific microscopic mechanisms as universal.

Cross-Links

• Quantum Noise for the broader noise taxonomy.
• Noise Spectra for spectral conventions and filter functions.
• Universal Conductance Fluctuations distinguishes a static reproducible magnetofingerprint from temporal conductance noise and explains why moving defects can strongly change the fingerprint.
• Single-Electron Devices shows how charge-offset drift, telegraph switching, and detector noise enter practical electrometry and pump error budgets.
• Pure Dephasing Model for exact coherence envelopes from phase noise.
• Dynamical Decoupling for echo, CPMG, and filter-function control.
• Decoherence Timescales for Ramsey, echo, and threshold-time conventions.
• Fluctuation–Dissipation Relation for why equilibrium thermal noise is more constrained than technical drift.
• Approximation Checklist for Markov and classical-noise checks.

References

• P. Dutta and P. M. Horn, “Low-frequency fluctuations in solids: 1/f noise,” Reviews of Modern Physics 53, 497–516 (1981).
• M. B. Weissman, “1/f noise and other slow, nonexponential kinetics in condensed matter,” Reviews of Modern Physics 60, 537–571 (1988).
• G. Ithier et al., “Decoherence in a superconducting quantum bit circuit,” Physical Review B 72, 134519 (2005).
• 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).
• E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, “1/f noise: Implications for solid-state quantum information,” Reviews of Modern Physics 86, 361–418 (2014).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).

Exercises

Variance with cutoffs

For

$$S_{\xi\xi}(\omega) = \frac{A^2}{|\omega|}, \qquad \omega_{\mathrm{ir}} \le |\omega| \le \omega_{\mathrm{uv}},$$

compute $\langle\xi^2\rangle$ in the two-sided convention.

Solution

Use

$$\langle\xi^2\rangle = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} S_{\xi\xi}(\omega).$$

The spectrum is even, so

$$\langle\xi^2\rangle = \frac{1}{\pi} \int_{\omega_{\mathrm{ir}}}^{\omega_{\mathrm{uv}}} d\omega\, \frac{A^2}{\omega} = \frac{A^2}{\pi} \ln \left( \frac{\omega_{\mathrm{uv}}} {\omega_{\mathrm{ir}}} \right).$$

Why S of zero fails

Why is the Markovian estimate $\Gamma_\phi\propto S_{\xi\xi}(0)$ inappropriate for ideal one-over-f noise?

Solution

The ideal model has $S_{\xi\xi}(\omega)=A^2/|\omega|$, which diverges as $\omega\to0$. The zero-frequency value is therefore not finite. A physical estimate must include an infrared cutoff and the actual experimental filter function. The result is a protocol-dependent coherence envelope rather than a universal constant Markovian rate.

Telegraph-noise mixture

Explain qualitatively why a broad distribution $P(\gamma)\propto1/\gamma$ of random-telegraph switching rates can produce a $1/|\omega|$ spectrum.

Solution

Each fluctuator contributes a Lorentzian spectrum centered at zero frequency with width set by its switching rate. Fast fluctuators contribute at higher frequencies; slow fluctuators contribute at lower frequencies. If the number of fluctuators per logarithmic interval of $\gamma$ is roughly constant, meaning $P(\gamma)\propto1/\gamma$, then each frequency decade receives comparable weight from fluctuators with rates near that frequency. The summed spectrum is then approximately proportional to $1/|\omega|$ over the range of available switching rates.

Echo rejects static noise

Use the Hahn-echo toggling function to show that static detuning noise does not contribute to the accumulated phase.

Solution

For static noise $\xi(t)=\xi_0$, the accumulated phase is

$$\phi(T) = \xi_0 \int_0^T dt\,y(t).$$

For Hahn echo,

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

Thus

$$\int_0^T dt\,y(t) = \frac{T}{2} - \frac{T}{2} = 0.$$

The static contribution cancels.