Quantum Noise, Dissipation, and Baths

Quantum noise describes fluctuations of degrees of freedom that influence a system. Their correlations determine which transitions occur, how phases fluctuate, and how the environment responds. Quantum operator ordering matters: the noise available to absorb energy from a system generally differs from the noise available to excite it. This chapter connects spectra and response to concrete bath models.

Required background. Baths, Reservoirs, and Environments, System–Bath Hamiltonians, and Spectral Densities specify the microscopic model. The Markov Approximation and Lindblad–GKSL Equation explain when correlations lead to simple rates.

Ordered spectra distinguish absorption from emission

For a stationary bath and a Hermitian, zero-mean bath operator $B$, use

$$\begin{gathered} C(t)=\langle B(t)B(0)\rangle,\\ S(\omega)=\int_{-\infty}^{\infty}e^{i\omega t}C(t)\,dt. \end{gathered}$$

This is an ordered, two-sided spectrum in angular frequency. For coupling $H_I=A\otimes B$, with dimensionless system operator $A$ and $B$ in energy units, weak-coupling transitions of a qubit with energy gap $\hbar\omega_0>0$ have rates proportional to

$$\begin{gathered} \Gamma_\downarrow =\frac{|A_{ge}|^2}{\hbar^2}S(+\omega_0),\\ \Gamma_\uparrow =\frac{|A_{ge}|^2}{\hbar^2}S(-\omega_0), \end{gathered}$$

for a Hermitian coupling and the usual stationary-bath golden-rule limit. Positive-frequency noise in this convention accepts energy released by the system. Reversing the Fourier sign reverses those labels.

At thermal equilibrium,

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

The upward rate vanishes in the zero-temperature limit when the downward spectrum stays finite. Vacuum fluctuations have not vanished; their frequency asymmetry matters. Quantum Noise, Correlation Functions, and Noise Spectra develop these definitions and their limits.

Symmetrized noise and response answer different questions

A real stationary classical scalar noise process has a symmetric two-sided autocorrelation spectrum. A quantum bath generally does not. Symmetrizing its spectrum keeps the average of the two frequency directions and removes the asymmetry needed to distinguish excitation from relaxation.

The Fluctuation–Dissipation Relation connects equilibrium fluctuations with dissipative linear response after operator and Fourier conventions are fixed. It is an equilibrium relation, not a universal formula for an arbitrarily driven environment. Thermal and Vacuum Noise explains the classical high-temperature limit and what remains at zero temperature.

From fluctuations to equations of motion

Quantum Langevin Equations describes system operators driven by input fluctuations and damped by environmental response. The noise term is essential: omitting it can destroy the system commutators even when the mean damping equation looks reasonable.

Input–Output Theory relates incoming fields, system motion, and outgoing fields available to a detector. The outgoing field is not interchangeable with an unobserved bath once a measured record is used for conditioning. Port normalization and direct reflected fields also matter when calculating detector signals.

Choose a bath model by the coupling

Model or regime	What to investigate
Caldeira–Leggett Model and Quantum Brownian Motion	A coordinate coupled to oscillator modes, including friction, diffusion, counterterms, and cutoff dependence.
Spin–Boson Model	A two-level system coupled to bosonic modes; longitudinal and transverse components can generate different dephasing and relaxation processes.
Pure Dephasing Model	Coupling that preserves system-energy populations, allowing coherence decay to be studied separately from energy relaxation.
One-Over-F Noise	Slow fluctuations over a specified frequency band, with infrared and ultraviolet cutoffs and protocol-dependent coherence.

For Brownian motion, check the full friction convention before copying a diffusion coefficient. The term $-i\gamma[q,\{p,\rho\}]/(2\hbar)$ gives momentum damping rate $\gamma$ and high-temperature $D_{pp}=M\gamma k_BT$ in the convention $-D_{pp}[q,[q,\rho]]/\hbar^2$. The simplified high-temperature equation is not automatically completely positive at arbitrarily short times; its regime and any additional diffusion terms must be examined.

“Ohmic,” “Gaussian,” and “Markovian” name different properties. Low-frequency scaling, higher correlation functions, and memory times must be assessed separately. For slowly decaying correlations or a narrow environmental resonance, continue to Non-Markovian Dynamics.

Exercises

Temperature from asymmetry. A weakly coupled qubit has $\Gamma_\uparrow/\Gamma_\downarrow=1/4$ in an equilibrium bath. Find $\beta\hbar\omega_0$ and its stationary excited population.

Solution

Detailed balance gives $\beta\hbar\omega_0=\ln4$. The stationary ratio is $p_e/p_g=1/4$, hence $p_e=1/5$. A symmetric spectrum would predict equal rates and populations for this transition, missing the finite-temperature asymmetry.

A finite correlation time. For classical noise with $C(t)=\sigma^2e^{-|t|/\tau}$, find the two-sided spectrum and its low-frequency limit.

Solution

Evenness gives $S(\omega)=2\sigma^2\int_0^\infty e^{-t/\tau}\cos(\omega t)\,dt=2\sigma^2\tau/(1+\omega^2\tau^2)$. It is approximately flat only for $|\omega|\tau\ll1$. A white-noise approximation must preserve the relevant area $2\sigma^2\tau$ and be restricted to the frequency range used by the system.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Correlations and weak-coupling rates.
• Clerk, A. A., 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. Spectral asymmetry and measurement noise.
• Gardiner, C. W., and P. Zoller. Quantum Noise. 3rd edition, Springer, 2004. Langevin equations and input–output fields.
• Weiss, U. Quantum Dissipative Systems. 4th edition, World Scientific, 2012. Oscillator and spin–boson environments.