Noise Spectra

A noise spectrum is the frequency-domain version of a correlation function. It tells which fluctuation frequencies are available to dephase a system, drive transitions, broaden a line, or pass through a control sequence.

In open quantum systems, spectra are not just plotting tools. They enter weak-coupling master equations, detailed-balance relations, dynamical-decoupling estimates, quantum-limited measurement, and experimental noise budgets.

For the conceptual distinction between classical stochastic noise, quantum bath noise, vacuum noise, thermal noise, and technical noise, see Quantum Noise. For the modeling distinction between a bath, reservoir, noise source, and record, see Baths, Reservoirs, and Environments.

The main warning is simple:

classical spectra are usually symmetric;
quantum unsymmetrized spectra generally are not.

That asymmetry is what lets a thermal bath distinguish absorption from emission.

Stationary Correlations

Let $B(t)$ be a centered Hermitian bath operator, $\langle B\rangle=0$, in the bath Heisenberg picture, and let the bath state be stationary. A nonzero mean should be treated separately as a Hamiltonian shift; otherwise it contributes $2\pi\langle B\rangle^2\delta(\omega)$ to the spectrum. The two-time correlation depends only on the time difference:

$$C_{BB}(t) = \langle B(t)B(0)\rangle.$$

With the Fourier convention used on this page, the two-sided unsymmetrized spectrum is

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

The inverse convention is

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

If $B$ has units of energy, then $S_{BB}$ has units of energy squared times time. If $B$ is a frequency fluctuation, then its spectrum has units of angular frequency.

Fourier conventions vary. Some authors place $2\pi$ factors in the forward transform, and some use $e^{-i\omega t}$ instead of $e^{i\omega t}$. Before comparing rates, check the definition.

Symmetrized Spectrum

For a quantum operator, the order of $B(t)$ and $B(0)$ matters. The symmetrized spectrum is

$$S_{BB}^{\mathrm{sym}}(\omega) = \frac12 \int_{-\infty}^{\infty} dt\, e^{i\omega t} \langle \{B(t),B(0)\}\rangle.$$

Symmetrized noise is useful for quadrature variances and many calibrated linear-detector measurements. It is not the universal spectrum reported by a detector: photon counting, a two-level spectrometer, and a linear amplifier can sample different operator orderings and include their own added noise.

The symmetrized spectrum does not by itself determine upward and downward quantum transition rates. Those rates use ordered, unsymmetrized correlations because the bath must either absorb or supply energy.

Classical Noise

For a real classical stationary random process $\xi(t)$,

$$C_{\xi\xi}(t) = \mathbb E[\xi(t)\xi(0)], \qquad S_{\xi\xi}(\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t} C_{\xi\xi}(t).$$

Since $C_{\xi\xi}(t)=C_{\xi\xi}(-t)$ for a real stationary process with ordinary multiplication,

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

This symmetry is not generally true for a quantum unsymmetrized spectrum. It is a common source of wrong temperature factors in relaxation formulas.

One-Sided and Two-Sided Spectra

Physics derivations usually use two-sided spectra over positive and negative frequencies. Many experimental plots use one-sided spectra over $\omega\ge0$ or $f\ge0$.

A common classical convention is

$$S_{\xi\xi}^{(1)}(\omega) = 2S_{\xi\xi}(\omega), \qquad \omega>0,$$

with the same total variance:

$$\langle \xi^2\rangle = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} S_{\xi\xi}(\omega) = \int_{0}^{\infty} \frac{d\omega}{2\pi} S_{\xi\xi}^{(1)}(\omega).$$

For ordinary frequency define $S_f(f)=\int dt\,e^{2\pi ift}C(t)$. Then $S_f(f)=S_\omega(2\pi f)$ and $\langle\xi^2\rangle=\int df\,S_f(f)$: there is no extra factor of $2\pi$ in that conversion because the angular-frequency inverse transform already contains $1/(2\pi)$. An angular-frequency density defined instead by $\langle\xi^2\rangle=\int d\omega\,P_\omega(\omega)$ is $P_\omega=S_\omega/(2\pi)$. Specify the integration measure as well as the frequency label.

Quantum Positive and Negative Frequencies

In the convention

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

positive and negative frequencies have different physical meanings. For a thermal bath and a system transition of angular frequency $\omega_0>0$:

• $S_{BB}(+\omega_0)$ controls processes in which the bath absorbs energy from the system;
• $S_{BB}(-\omega_0)$ controls processes in which the bath supplies energy to the system.

In equilibrium the two are related by detailed balance:

$$S_{BB}(-\omega) = e^{-\beta\hbar\omega} S_{BB}(+\omega), \qquad \omega>0,$$

for a single Hermitian bath operator with this convention. At low temperature, the negative-frequency thermal excitation spectrum is suppressed.

This is the spectral form of Detailed Balance. The same thermal structure underlies the general Fluctuation–Dissipation Theorem, while the open-system Fluctuation–Dissipation Relation develops bath noise and damping applications.

Spectra in Master Equations

For a system–bath interaction

$$H_I = \sum_\alpha A_\alpha\otimes B_\alpha,$$

take dimensionless $A_\alpha$ and energy-valued Hermitian $B_\alpha$. Weak-coupling Markovian master equations involve the rate matrix

$$\Gamma_{\alpha\beta}(\omega) = \frac{1}{\hbar^2} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \langle B_\alpha(t)B_\beta(0)\rangle.$$

The system coupling operators are decomposed into Bohr-frequency components:

$$[H_S,A_\alpha(\omega)] = -\hbar\omega A_\alpha(\omega).$$

After a secular approximation, the dissipator has the schematic form

$$\sum_{\omega,\alpha,\beta} \Gamma_{\alpha\beta}(\omega) \left( A_\beta(\omega)\rho A_\alpha^\dagger(\omega) - \frac12 \{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho\} \right),$$

plus a Lamb-shift Hamiltonian from the imaginary principal-value part of the same bath response.

This formula shows why spectra must be defined with their ordering and sign conventions. Changing the definition of $A(\omega)$ or the Fourier transform changes which side of the spectrum is called positive frequency.

Relaxation Rates

For a two-level system with transition frequency $\omega_0$, transverse bath noise drives energy exchange. Schematically,

$$\Gamma_\downarrow \propto S_\perp(+\omega_0), \qquad \Gamma_\uparrow \propto S_\perp(-\omega_0).$$

The longitudinal relaxation time satisfies

$$\frac{1}{T_1} = \Gamma_\downarrow+\Gamma_\uparrow.$$

At zero temperature, $\Gamma_\uparrow$ vanishes for an ordinary equilibrium bath, but $\Gamma_\downarrow$ need not vanish because the bath can absorb energy. This is the spectral language behind spontaneous emission and amplitude damping.

Pure Dephasing Rates

Longitudinal frequency noise samples the spectrum near zero frequency. For a qubit with

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

the off-diagonal density-matrix element accumulates the random phase

$$\varphi(t) = \int_0^t ds\,\xi(s).$$

For zero-mean Gaussian stationary noise,

$$\langle e^{-i\varphi(t)}\rangle = \exp\!\left[ - \frac12 \langle\varphi(t)^2\rangle \right].$$

If the noise is effectively white on the timescale of the experiment, then

$$\frac{1}{T_\phi} = \frac12 S_{\xi\xi}(0),$$

with the two-sided convention above. Other Hamiltonian normalizations move factors of $2$ between $\xi$ and $S_{\xi\xi}$.

The key physical point is robust: pure dephasing is sensitive to low-frequency or elastic fluctuations, while relaxation samples noise at transition frequencies.

Filter Functions

Control sequences reshape how a system samples noise. For a toggling function $y(t)=\pm1$ that changes sign under ideal refocusing pulses, the accumulated phase from classical dephasing noise is

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

Define the filter amplitude

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

For Gaussian dephasing noise,

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

and the coherence factor is $e^{-\chi(T)}$.

This formula explains why spin echo suppresses very slow noise: the sign-changing control makes $Y(\omega,T)$ small near $\omega=0$. The canonical pulse-sequence discussion is Dynamical Decoupling.

Common Spectral Shapes

Several idealized spectra recur across open-system modeling.

Spectrum	Time-domain source	Typical use
white	short correlation time	Markovian dephasing or diffusion
Lorentzian	exponential correlation	finite-bandwidth noise, random telegraph approximations
Ohmic	oscillator bath with $J(\omega)\propto\omega$ at low frequency	damping, Brownian motion, circuit environments
sub-Ohmic	enhanced low-frequency bath weight	slow environments, strong memory effects
super-Ohmic	suppressed low-frequency bath weight	phonon-like environments in some dimensions
$1/f$	broad distribution of slow fluctuators	solid-state dephasing and drift

For an exponential classical correlation

$$C(t)=\sigma^2e^{-|t|/\tau_c},$$

the two-sided spectrum is Lorentzian:

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

The correlation time $\tau_c$ sets the bandwidth. In the limit $\tau_c$ becomes very short while the area remains fixed, the noise approaches a white-noise approximation.

Spectral Density Versus Noise Spectrum

The phrase spectral density is used in two related but distinct ways. The mode-density convention is treated directly in Spectral Densities.

For an oscillator bath, one often defines a mode spectral density such as

$$J(\omega) = \sum_k |g_k|^2 \delta(\omega-\omega_k), \qquad \omega>0.$$

If $B=\sum_k(g_kb_k+g_k^*b_k^\dagger)$ and $g_k$ has units of energy, $J$ has units of energy squared times time. A thermal bath then gives the exact conversion

$$S_{BB}(+\omega) = 2\pi J(\omega)\,[n(\omega)+1], \qquad S_{BB}(-\omega) = 2\pi J(\omega)\,n(\omega),$$

for $\omega>0$. If the Hamiltonian instead writes $\hbar g_k$ with frequency-valued $g_k$, the corresponding energy-noise spectrum carries an additional $\hbar^2$.

Thus $J(\omega)$ and $S_{BB}(\omega)$ are not always interchangeable. The spectrum relevant for rates includes both coupling density and bath state.

For the canonical coordinate-coupled oscillator-bath model where Ohmic and Drude spectral densities become damping kernels, see Caldeira–Leggett Model.

For the canonical two-level-system version, where Ohmic, sub-Ohmic, and super-Ohmic spectra control dissipative tunneling and dephasing, see Spin-Boson Model.

Estimating a Correlation Time

A Markov approximation needs the bath correlation to decay on a timescale short compared with system evolution. Spectrally, a short correlation time usually means a broad, smooth spectrum over the system’s relevant transition frequencies.

Useful checks:

• Is $S(\omega)$ smooth near each Bohr frequency?
• Does the spectrum have sharp resonances, gaps, or band edges?
• Is there strong low-frequency weight that produces slow drift?
• Are there finite-size recurrences in the bath?
• Is the system being driven so that sidebands sample additional frequencies?

If the spectrum has narrow structure on the scale of the system linewidth, a memoryless Lindblad model may miss nonexponential decay, coherent exchange with a mode, or non-Markovian information return. Lorentzian structured spectra are a common entry point for Pseudomode Methods.

For Gaussian baths, another common numerical strategy is to expand the bath correlation function into exponentials and evolve the corresponding Hierarchical Equations of Motion.

Exercises

Variance from a Two-Sided Spectrum

Show that, with the convention on this page,

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

Solution

Set $t=0$ in the inverse transform:

$$C_{\xi\xi}(0) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} S_{\xi\xi}(\omega).$$

But

$$C_{\xi\xi}(0)=\mathbb E[\xi(0)^2]=\langle\xi^2\rangle.$$

Lorentzian Spectrum

Compute the spectrum of

$$C(t)=\sigma^2e^{-|t|/\tau_c}.$$

Solution

Use the evenness of $C(t)$:

$$S(\omega) = 2\sigma^2 \int_0^\infty dt\, e^{-t/\tau_c} \cos(\omega t).$$

The integral is

$$\int_0^\infty dt\, e^{-t/\tau_c} \cos(\omega t) = \frac{\tau_c}{1+\omega^2\tau_c^2}.$$

Therefore

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

Detailed Balance Ratio

For a thermal bath with

$$S(-\omega) = e^{-\beta\hbar\omega}S(+\omega), \qquad \omega>0,$$

what is the ratio $\Gamma_\uparrow/\Gamma_\downarrow$ for a two-level system of transition frequency $\omega_0$?

Solution

With the convention used here,

$$\Gamma_\downarrow\propto S(+\omega_0), \qquad \Gamma_\uparrow\propto S(-\omega_0).$$

Thus

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = \frac{S(-\omega_0)}{S(+\omega_0)} = e^{-\beta\hbar\omega_0}.$$

Upward transitions are Boltzmann suppressed in thermal equilibrium.

Dephasing from White Noise

For

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

assume Gaussian white frequency noise with two-sided spectrum $S_{\xi\xi}(\omega)=S_0$. Show that the coherence decays with rate $S_0/2$.

Solution

The random phase is

$$\varphi(t) = \int_0^t ds\,\xi(s).$$

For white noise,

$$\langle \varphi(t)^2\rangle = \int_0^t ds \int_0^t ds'\, \langle \xi(s)\xi(s')\rangle = S_0t.$$

Gaussian averaging gives

$$\langle e^{-i\varphi(t)}\rangle = \exp\!\left[ - \frac12S_0t \right].$$

Therefore

$$\frac{1}{T_\phi} = \frac{S_0}{2}.$$

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer (1999).
• 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, 1155–1208 (2010), doi:10.1103/RevModPhys.82.1155.
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer (2004).
• R. Kubo, “The fluctuation-dissipation theorem,” Reports on Progress in Physics 29, 255–284 (1966).
• U. Weiss, Quantum Dissipative Systems, World Scientific (2012).