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.

Slow Fluctuations and Qubit Coherence

One-over-f noise places substantial weight near zero frequency, where an unrefocused qubit accumulates phase from slow fluctuations. 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.

For concrete charge-noise examples, see Single-Electron Devices. Here we connect the measured spectrum to coherence and echo experiments.

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 the same units as $\xi$. When $\xi$ is an angular-frequency fluctuation, as below, both have units $\mathrm{s}^{-1}$.

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 model variance depends on its physical cutoffs; the variance inferred from a finite record also depends on measurement bandwidth and processing. A one-over-f model without cutoffs is not a complete stochastic process.

Cutoffs and Observation Time

A finite observation of duration $T_{\mathrm{obs}}$ usually cannot resolve frequencies much below an effective analysis scale of order

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

where the numerical factor depends on the window and whether ordinary or angular frequency is used. This resolution scale is not necessarily a physical cutoff of the bath. Slower modes can act as unknown offsets during one record and fluctuate between records; calibration and detrending determine how much of that variation is retained. A physical infrared cutoff instead describes saturation of the noise itself. Similarly, distinguish a physical ultraviolet crossover from the bandwidth imposed by a detector or control filter.

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

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

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 finite-band $A^2/|\omega|$ model in the range $\omega_{\mathrm{ir}}T\ll1\ll\omega_{\mathrm{uv}}T$, the leading Gaussian-noise estimates are

$$\begin{aligned} \chi_{\mathrm R}(T)&=\frac{A^2T^2}{2\pi} \ln\!\frac{1}{\omega_{\mathrm{ir}}T}+O(A^2T^2),\\ \chi_{\mathrm{echo}}(T)&\simeq\frac{A^2T^2\ln2}{2\pi}. \end{aligned}$$

Echo removes the leading infrared logarithm, not all noise. These coefficients use the two-sided angular-frequency spectrum above; comparing them with a one-sided spectrum requires converting conventions. Cywiński et al. (2008) develop the filter-function approach for both Gaussian noise and individual fluctuators.

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 it takes values $\pm v$ and switches from either value to the other at rate $\gamma$, its stationary correlation is $v^2e^{-2\gamma|t|}$ and its spectrum is

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

Here the correlation decay rate is $2\gamma$, twice the rate of leaving either state.

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.

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

Let independent fluctuators have equal amplitude $v$ and density of rates $P(\gamma)=C/\gamma$ between $\gamma_{\min}$ and $\gamma_{\max}$. Integrate their spectra and identify the range with one-over-f behavior.

Solution

The sum becomes

$$\begin{aligned} S(\omega)&=4Cv^2\int_{\gamma_{\min}}^{\gamma_{\max}} \frac{d\gamma}{\omega^2+4\gamma^2}\\ &=\frac{2Cv^2}{|\omega|}\left[ \arctan\frac{2\gamma_{\max}}{|\omega|} -\arctan\frac{2\gamma_{\min}}{|\omega|}\right]. \end{aligned}$$

For $2\gamma_{\min}\ll|\omega|\ll2\gamma_{\max}$, the bracket approaches $\pi/2$, giving $S\simeq\pi Cv^2/|\omega|$. Below this band it becomes approximately constant; above it it falls as $\omega^{-2}$. Thus the fluctuator model supplies physical cutoffs as well as the intermediate power law.

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.

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). doi:10.1103/PhysRevB.77.174509; arXiv:0712.2225.
• P. Dutta and P. M. Horn, “Low-frequency fluctuations in solids: 1/f noise,” Reviews of Modern Physics 53, 497–516 (1981).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).
• G. Ithier et al., “Decoherence in a superconducting quantum bit circuit,” Physical Review B 72, 134519 (2005).
• 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). doi:10.1103/RevModPhys.86.361; arXiv:1304.7925.
• M. B. Weissman, “1/f noise and other slow, nonexponential kinetics in condensed matter,” Reviews of Modern Physics 60, 537–571 (1988).