Spectral Densities

A spectral density summarizes how strongly a system couples to environmental modes as a function of frequency. It is the object that replaces a long discrete sum over bath modes by a continuum function.

The most common schematic definition is

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

This should be read with a convention label attached. Different communities include factors of $\pi$, $2$, $\hbar$, mode masses, or powers of $\omega_k$ in the definition. The physics is not in the symbol alone; it is in the pair consisting of the definition of $J(\omega)$ and the formula in which it is used.

This page is about mode spectral densities. Ordered quantum noise spectra, one-sided experimental spectra, symmetrized spectra, and filter-function conventions are treated in Noise Spectra.

From Mode Sums to Continua

Consider a bosonic bath with

$$H_B = \sum_k \hbar\omega_k b_k^\dagger b_k,$$

and an interaction of the schematic form

$$H_I = S\otimes \sum_k \left( g_k b_k + g_k^* b_k^\dagger \right).$$

In weak-coupling formulas, sums such as

$$\sum_k |g_k|^2 f(\omega_k)$$

often appear. The spectral density is defined so that

$$\sum_k |g_k|^2 f(\omega_k) \longrightarrow \int_0^\infty d\omega\, J(\omega)f(\omega).$$

If the mode density is $D(\omega)$ and the coupling is smooth enough to write $g_k\to g(\omega)$, then

$$J(\omega) = D(\omega)|g(\omega)|^2$$

in this convention.

Thus $J(\omega)$ is not merely a density of states. It is a coupling-weighted density of states.

Coordinate-Bath Convention

For the Caldeira–Leggett coordinate bath, a common Hamiltonian uses oscillator coordinates $x_j$ and couplings $c_j$:

$$H_I = - q \sum_j c_j x_j.$$

A standard convention is

$$J_{\mathrm{CL}}(\omega) = \frac{\pi}{2} \sum_j \frac{c_j^2}{m_j\omega_j} \delta(\omega-\omega_j), \qquad \omega>0.$$

This differs from the simple $|g_k|^2$ definition because the oscillator coordinate normalization contributes factors of $m_j$ and $\omega_j$. Both definitions are legitimate if used consistently.

The coordinate-bath convention is natural when $J(\omega)$ controls damping kernels, force-noise kernels, and counterterm renormalization. See Caldeira–Leggett Model.

What J Does and Does Not Include

A mode spectral density tells how strongly the system couples to bath modes at frequency $\omega$. By itself it usually does not include:

• the bath temperature;
• thermal occupation factors;
• whether the bath is absorbing or supplying energy;
• operator ordering;
• one-sided versus two-sided measurement conventions;
• detector filtering or experimental bandwidth.

For a thermal bosonic bath, ordered spectra often contain combinations of the form

$$S_{BB}(+\omega) \propto J(\omega)\,[n_B(\omega)+1],$$

and

$$S_{BB}(-\omega) \propto J(\omega)\,n_B(\omega),$$

for $\omega>0$, up to Fourier and sign conventions. Here

$$n_B(\omega) = \frac{1}{e^{\beta\hbar\omega}-1}$$

is the Bose occupation.

At zero temperature, $n_B(\omega)=0$. The bath can still absorb energy from the system, but it cannot thermally supply energy at positive frequency in the same way. This absorption-emission asymmetry belongs to ordered quantum spectra and detailed balance, not to $J(\omega)$ alone.

Ohmic Classifications

A common family is

$$J_s(\omega) = \eta\, \omega_c^{1-s} \omega^s F(\omega/\omega_c), \qquad \omega>0,$$

where:

• $s$ controls the low-frequency behavior;
• $\omega_c$ is a cutoff scale;
• $F(x)$ suppresses high frequencies;
• $\eta$ sets the coupling scale in the chosen convention.

The low-frequency classification is:

Regime	Condition	Low-frequency behavior	Typical implication
sub-Ohmic	$0\lt s\lt1$	enhanced low-frequency weight	slow memory, strong dephasing, nonperturbative effects
Ohmic	$s=1$	linear in $\omega$	friction-like damping in coordinate baths
super-Ohmic	$s>1$	suppressed low-frequency weight	weaker low-frequency dephasing in many models

The classification describes the limit $\omega\to0$. A bath can be Ohmic at low frequency and still have strong finite-frequency structure.

Cutoffs Are Part of the Model

A power law cannot usually extend to arbitrarily high frequency. Common cutoff functions include:

$$F_{\mathrm{exp}}(x)=e^{-x},$$

and the Drude–Lorentz shape

$$F_{\mathrm{D}}(x) = \frac{1}{1+x^2}.$$

For an Ohmic coordinate bath, one often writes

$$J(\omega) = M\gamma\omega \frac{\Omega_c^2}{\omega^2+\Omega_c^2},$$

or an equivalent convention with factors of $\pi$ and $2$.

The cutoff affects:

• short-time behavior;
• memory kernels;
• Lamb shifts and counterterms;
• convergence of reorganization-energy integrals;
• whether a Markov approximation is plausible.

Saying “Ohmic bath” without specifying a cutoff is usually incomplete.

Structured Spectral Densities

Many environments are not smooth continua. A structured spectral density may have:

• a Lorentzian peak;
• a band edge;
• a spectral gap;
• underdamped vibrational modes;
• a sharp cavity resonance;
• finite propagation delay;
• several separated mode families.

A common single-peak model is

$$J_{\mathrm{L}}(\omega) = \frac{1}{2\pi} \frac{\lambda^2\kappa} {(\omega-\omega_c)^2+(\kappa/2)^2},$$

with peak frequency $\omega_c$, width $\kappa$, and coupling scale $\lambda$ in one common convention.

If $\kappa$ is large compared with the relevant system rates, the peak may act like a short-memory reservoir. If $\kappa$ is small, the system can exchange excitation coherently with the structured mode before that mode damps away. Then a Markovian rate built only from $J(\omega_0)$ may miss revivals or nonexponential decay.

Structured spectral densities are the natural entry point for Pseudomode Methods and Reaction-Coordinate Mapping.

Correlation Functions

The bath correlation function is the time-domain object that appears in memory kernels and weak-coupling derivations. For a bosonic thermal bath, a typical convention gives

\begin{aligned} C(t) ={}& \int_0^\infty d\omega\, J(\omega) \left[ (n_B(\omega)+1)e^{-i\omega t} \right. \\ &\left. \hspace{6rem} + n_B(\omega)e^{+i\omega t} \right]. \end{aligned}

The exact prefactor depends on the definition of $J(\omega)$ and the bath operator $B$. The structural point is stable:

smooth broad J(ω)       -> short correlation time
narrow structured J(ω)  -> long-lived correlation
low-frequency weight    -> slow dephasing and memory

This is why spectral densities are not just fitting curves. They encode memory times, transition rates, thermal noise, and non-Markovian structure.

Reorganization Energy

In molecular and condensed-phase applications, a spectral density often determines a reorganization energy. A common convention is

$$\lambda_{\mathrm{reorg}} = \frac{1}{\pi} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega}.$$

This quantity measures the bath-induced energetic shift associated with changing the system configuration. It is especially common in electron transfer, exciton dynamics, and vibronic models.

The same warning applies: the numerical value of $\lambda_{\mathrm{reorg}}$ depends on the convention used for $J(\omega)$.

Spectral Density and Markov Rates

In weak-coupling Markovian theory, a transition with Bohr frequency $\omega_0$ often samples the bath near $\omega_0$. Schematically,

$$\Gamma_{\downarrow} \propto J(\omega_0)[n_B(\omega_0)+1],$$

and

$$\Gamma_{\uparrow} \propto J(\omega_0)n_B(\omega_0).$$

This local sampling picture assumes the spectral density and occupation factors are smooth across the linewidth and memory scale relevant to the system. It can fail near band edges, narrow resonances, strong coupling, low-frequency singularities, or rapidly varying density of states.

For the approximation step, see Markov Approximation. For the full correlation-based language, see Memory Kernels.

Examples

Acoustic phonons

Acoustic phonon baths often have power-law low-frequency behavior. The exponent depends on spatial dimension, phonon dispersion, and the coupling mechanism. Super-Ohmic behavior is common in some deformation-potential models.

Resistive electromagnetic environment

A resistor or transmission line can produce an Ohmic electromagnetic environment over a finite bandwidth. The physical bandwidth and impedance determine the useful cutoff.

Cavity or waveguide reservoir

A cavity resonance, photonic-crystal band edge, or waveguide with delay produces a structured spectral density. A single Markovian decay rate may be inadequate if the spectral feature is narrow or the propagation time is comparable to system dynamics.

Molecular vibrational environment

Molecular spectral densities often contain broad solvent background plus discrete or underdamped vibrational peaks. The peaks can be promoted to explicit modes when weak-coupling rates are not sufficient.

Common Mistakes

Treating J as the full noise spectrum

$J(\omega)$ describes coupling-weighted mode density. Thermal factors and operator ordering are needed for ordered quantum noise spectra.

Comparing formulas with different conventions

Two papers may both write $J(\omega)$ but differ by $\pi$, $2$, $\hbar$, or coordinate-normalization factors.

Forgetting the cutoff

Power-law spectral densities usually need high-frequency cutoffs. Without them, short-time behavior and renormalization integrals can become unphysical.

Assuming Ohmic means Markovian

Ohmic describes low-frequency scaling. Markovianity depends on correlation times, bandwidth, coupling strength, and system timescales.

Ignoring low-frequency weight

Low-frequency spectral weight can dominate pure dephasing, tunneling renormalization, and long-memory behavior.

Fitting a curve without a model

A fitted spectral density should be checked against a plausible bath, temperature, coupling operator, and approximation regime.

Exercises

1. Mode density and coupling. Suppose bath modes have density $D(\omega)=A\omega^2$ and coupling $|g(\omega)|^2=g_0^2/\omega$ over a frequency range. What is the spectral density in the convention

$$\sum_k |g_k|^2 f(\omega_k) \to \int d\omega\,J(\omega)f(\omega)?$$

Is it sub-Ohmic, Ohmic, or super-Ohmic at low frequency?

Solution

Using $J(\omega)=D(\omega)|g(\omega)|^2$,

$$J(\omega) = A\omega^2 \frac{g_0^2}{\omega} = Ag_0^2\omega.$$

At low frequency, $J(\omega)\propto\omega$, so it is Ohmic in this convention.

2. Cutoff dependence. For an Ohmic form $J(\omega)=\eta\omega e^{-\omega/\omega_c}$, evaluate the convention-dependent reorganization-energy integral

$$\lambda_{\mathrm{reorg}} = \frac{1}{\pi} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega}.$$

Solution

Substituting the spectral density gives

$$\lambda_{\mathrm{reorg}} = \frac{\eta}{\pi} \int_0^\infty d\omega\, e^{-\omega/\omega_c}.$$

The integral is $\omega_c$, so

$$\lambda_{\mathrm{reorg}} = \frac{\eta\omega_c}{\pi}.$$

The answer depends explicitly on the cutoff. Without a cutoff, the integral would diverge.

3. Thermal factors. At zero temperature, $n_B(\omega)=0$ for $\omega>0$. What happens to the schematic upward and downward rates

$$\Gamma_{\downarrow}\propto J(\omega_0)[n_B(\omega_0)+1], \qquad \Gamma_{\uparrow}\propto J(\omega_0)n_B(\omega_0)?$$

Solution

At zero temperature,

$$\Gamma_{\downarrow} \propto J(\omega_0), \qquad \Gamma_{\uparrow}=0.$$

The bath can absorb energy emitted by the system, but it has no thermal occupation available to excite the system upward. This asymmetry is not contained in $J(\omega)$ alone; it enters through the ordered quantum spectrum and the bath state.

4. Structured reservoir warning. A two-level system has linewidth scale $\Gamma$ and couples to a Lorentzian spectral peak of width $\kappa$. Explain why $\kappa\gg\Gamma$ is more favorable for a Markovian rate model than $\kappa\lesssim\Gamma$.

Solution

A broad peak has a short correlation time of order $\kappa^{-1}$, so the environment forgets quickly compared with the system decay time $\Gamma^{-1}$. Then a rate that samples $J(\omega_0)$ can be a useful approximation.

If $\kappa\lesssim\Gamma$, the reservoir correlation persists over the timescale on which the system changes. The system can exchange excitation coherently with the structured mode, leading to memory, nonexponential decay, or revivals. A Markovian rate model is then suspect.

5. Convention check. Two sources define spectral densities by

$$J_1(\omega) = \sum_k |g_k|^2\delta(\omega-\omega_k),$$

and

$$J_2(\omega) = \pi \sum_k |g_k|^2\delta(\omega-\omega_k).$$

If a rate formula in source 1 is $\Gamma=2\pi J_1(\omega_0)$, what is the corresponding formula using $J_2$?

Solution

The definitions imply

$$J_2(\omega)=\pi J_1(\omega).$$

Therefore

$$J_1(\omega_0) = \frac{J_2(\omega_0)}{\pi}.$$

Substituting into $\Gamma=2\pi J_1(\omega_0)$ gives

$$\Gamma = 2J_2(\omega_0).$$

The physics has not changed; only the convention for $J$ changed.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• U. Weiss, Quantum Dissipative Systems, World Scientific, 4th ed. (2012).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587-616 (1983).
• A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Reviews of Modern Physics 59, 1-85 (1987).
• Y. Tanimura, “Numerically exact approach to open quantum dynamics: The hierarchical equations of motion,” Journal of Chemical Physics 153, 020901 (2020).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).