Common Noise Spectra

This page is a compact lookup guide for spectra commonly used in open quantum systems, measurement theory, and noise modeling. It is not a substitute for specifying the correlation convention, units, bandwidth, and whether the spectrum is classical, symmetrized quantum, or ordered quantum.

For the canonical convention discussion, see Noise Spectra. For equilibrium constraints, see Fluctuation–Dissipation Relation.

Convention Used Here

For a stationary bath operator or classical noise process $B(t)$, this page uses the two-sided convention

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

The inverse is

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

Classical spectra for real noise are symmetric. Quantum ordered spectra generally are not symmetric; positive and negative frequencies encode different energy exchange processes.

Fast Map

Spectrum	Schematic form	Correlation picture	First caution
white noise	$S(\omega)=S_0$ over a band	delta-correlated idealization	needs ultraviolet and infrared cutoffs
Ohmic noise	$J(\omega)\propto\omega$ at low frequency	frictional oscillator bath	spectral-density convention matters
Lorentzian	$S(\omega)\propto[1+(\omega-\omega_c)^2\tau_c^2]^{-1}$	exponentially decaying memory or damped mode	can be non-Markovian when narrow
1/f noise	$S(\omega)\propto1/	\omega	$ between cutoffs
thermal Bose spectrum	weighted by $n_B(\omega)$ and $n_B(\omega)+1$	equilibrium absorption and emission	ordered spectrum is asymmetric
vacuum noise	zero-temperature ordered spectrum	bath can absorb energy but not thermally excite	symmetrized spectrum still has zero-point noise

White Noise

White noise has a frequency-independent spectrum over the frequencies of interest:

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

The corresponding ideal two-sided correlation is proportional to a delta function:

$$\langle \xi(t)\xi(0)\rangle = S_0\delta(t)$$

up to Fourier-convention factors.

Use white noise when the bath correlation time is much shorter than the system timescales and only a finite frequency window matters. It is the natural idealization behind many Markovian Langevin equations and master-equation rates.

Main caution: perfectly white noise at all frequencies has infinite total power. A physical model always has cutoffs or a finite bandwidth. See Markov Approximation.

Ohmic Noise

An Ohmic oscillator bath has spectral density linear in frequency at low frequency:

$$J(\omega) = \eta\omega f_c(\omega), \qquad \omega>0,$$

where $f_c(\omega)$ is a high-frequency cutoff function. In a common Drude-like convention,

$$f_c(\omega) = \frac{\omega_c^2}{\omega^2+\omega_c^2}.$$

Use Ohmic noise for coordinate damping, friction, resistive electromagnetic environments, and the standard Caldeira–Leggett Brownian model.

Main caution: $J(\omega)$ is not always the same object as an ordered noise spectrum $S(\omega)$. In equilibrium, the fluctuation–dissipation relation supplies the thermal occupation factors that turn dissipative response into noise. See Caldeira–Leggett Model.

Lorentzian Spectrum

A Lorentzian spectrum centered at $\omega_c$ has the schematic form

$$S(\omega) = \frac{A\tau_c} {1+(\omega-\omega_c)^2\tau_c^2}.$$

For a zero-centered classical Lorentzian, the time-domain correlation is exponential:

$$C(t) \propto e^{-|t|/\tau_c}.$$

Use Lorentzian spectra for damped modes, pseudomodes, resonator-filtered noise, random telegraph noise in a simple limit, and environments with a single dominant correlation time.

Main caution: a narrow Lorentzian has long memory. If $\tau_c$ is comparable to system timescales, replacing it by white noise can erase non-Markovian physics. See Pseudomode Methods and Memory Kernels.

1/f Noise

One-over-f noise has the schematic form

$$S(\omega) = \frac{A}{|\omega|}$$

over a finite range

$$\omega_{\mathrm{ir}} \le |\omega| \le \omega_{\mathrm{uv}}.$$

It is common in solid-state qubits, charge noise, flux noise, materials defects, and slow control drift. It is often modeled as a broad distribution of fluctuators with many switching timescales.

Main caution: $1/|\omega|$ cannot extend to zero frequency or infinite frequency. Without cutoffs the variance diverges. Low-frequency noise may be quasistatic over one experiment and drifting over longer averaging times, so the distinction between ensemble dephasing and single-shot noise matters.

Thermal Bose Spectrum

For a bosonic equilibrium bath, occupation at positive frequency is

$$n_B(\omega) = \frac{1} {e^{\beta\hbar\omega}-1}, \qquad \omega>0.$$

With the ordered-spectrum convention used here, a common schematic form is

$$S(+\omega) \propto J(\omega) \left[ n_B(\omega)+1 \right], \qquad S(-\omega) \propto J(\omega)n_B(\omega),$$

for $\omega>0$. Thus

$$S(-\omega) = e^{-\beta\hbar\omega} S(+\omega).$$

Use this structure for weak-coupling transition rates, detailed balance, thermal master equations, and equilibrium oscillator baths.

Main caution: the $+1$ term is spontaneous emission or vacuum contribution in this convention. It is not a high-temperature classical fluctuation. See Thermal Master Equations and Detailed Balance.

Vacuum Noise

Vacuum noise is the zero-temperature limit of a quantum bath. In the ordered convention above, for $\omega>0$,

$$S(+\omega) \propto J(\omega), \qquad S(-\omega)=0.$$

Physically, the vacuum can absorb energy from an excited system, but it cannot thermally excite the system. This is why spontaneous emission can occur at zero temperature while thermal absorption is absent.

The symmetrized vacuum spectrum is different: it contains zero-point fluctuations. That is useful in linear-response and detector-noise contexts, but it should not be confused with an ordered excitation rate.

Main caution: when a formula uses a symmetrized spectrum, vacuum noise appears at positive and negative frequencies. When a transition-rate formula uses an ordered spectrum, the negative-frequency thermal excitation side vanishes at zero temperature.

Common Unit and Convention Checks

Before comparing two spectra, check:

• angular frequency $\omega$ versus ordinary frequency $f$;
• one-sided versus two-sided normalization;
• classical, symmetrized quantum, or ordered quantum definition;
• whether $S(\omega)$ or a bath spectral density $J(\omega)$ is being plotted;
• whether a cutoff is implicit;
• whether positive frequency means bath absorption or system absorption in the author’s convention.

These checks are not bookkeeping trivia. They change rates, detailed-balance factors, and inferred temperatures.

Self-Checks

1. A spectrum is written as $S(\omega)=S_0$ for every real $\omega$. What physical qualification is missing?

Solution

It needs a finite bandwidth or cutoff. Perfectly white noise over all frequencies has infinite integrated power and is only an idealization over a relevant frequency window.

2. For a zero-temperature bath, should an ordered spectrum produce upward thermal excitation of a positive-frequency transition?

Solution

No. In the convention used here, $S(-\omega)=0$ for $\omega>0$ at zero temperature. The bath can absorb energy from the system, giving spontaneous emission, but it cannot thermally excite the system.

3. Why is a narrow Lorentzian spectrum dangerous to replace by white noise?

Solution

A narrow Lorentzian corresponds to a long correlation time. If that memory time is comparable to the system dynamics, a Markov approximation can miss memory effects, coherent exchange with a damped mode, or nonexponential decay.

References

• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 2004.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• U. Weiss, Quantum Dissipative Systems, World Scientific, 2012.
• A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155, 2010.
• 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, 2014.