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.

Helpful background. System–bath Hamiltonians define the coupling, and noise spectra explain ordered correlation functions and Fourier conventions.

From Mode Sums to Continua

Consider a bosonic bath with

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

and a Hermitian, dimensionless system operator $S$ coupled through

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

Here $g_k$ has energy units, $\omega$ is angular frequency, and $J$ has units energy squared times time. With the discrete delta-function definition, sums such as

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

often appear. The spectral density is defined so that

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

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.

The sum-to-integral identity is exact for the delta-function distribution. Replacing that distribution by a smooth function is the continuum approximation. Thus $J(\omega)$ is a coupling-weighted density of states, not just a count of modes.

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

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

Quantizing $x_j=\sqrt{\hbar/(2m_j\omega_j)}(b_j+b_j^\dagger)$ makes the force coefficients $g_j^{(F)}=c_j\sqrt{\hbar/(2m_j\omega_j)}$. Therefore the force-weighted density is

$$J_F(\omega)=\sum_j|g_j^{(F)}|^2\delta(\omega-\omega_j) =\frac\hbar\pi J_{\mathrm{CL}}(\omega).$$

Because $q$ has length units, these coefficients are forces rather than the energy-valued coefficients used with dimensionless $S$ above. Both the factor and the units must be converted.

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 the energy-valued bath operator above, a thermal state and the convention $S_{BB}(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega t}\langle B(t)B(0)\rangle$ give exactly

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

and

$$S_{BB}(-\omega) =2\pi J(\omega)\,n_B(\omega),$$

for $\omega>0$. 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.

Assume $F(0)=1$ and a nonsingular cutoff. The low-frequency classification is:

Regime	Exponent	Low-frequency comparison at fixed coupling and cutoff
Sub-Ohmic	$0<s<1$	More weight than the Ohmic form as $\omega/\omega_c\to0$
Ohmic	$s=1$	Linear in frequency; related to friction in coordinate baths
Super-Ohmic	$s>1$	Less low-frequency weight; the effect still depends on temperature and coupling operator

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

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

In the coordinate convention $J_{\mathrm{CL}}$, an Ohmic Drude bath can be defined by

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

where $\gamma$ is the momentum damping rate in the broad-band limit of the associated Langevin equation. This definition is not the energy-valued $J$ used above.

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

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

Here $\omega_c>0$ is the center, $\kappa>0$ is the full width at half maximum, and $\lambda$ has energy units in the first convention. Near a high-frequency resonance with $\omega_c\gg\kappa$, at zero temperature, one often extends the frequency integral to the entire real line. That approximation gives

$$C(t)\simeq\lambda^2e^{-i\omega_ct-\kappa|t|/2}.$$

The correlation-envelope time is $2/\kappa$. The full-line extension is not an exact positive-frequency bath spectrum: the lower-frequency edge changes long-time tails. Moreover, an unmodified Lorentzian has nonzero $J(0)$ and can give an infrared-divergent thermal correlation through $n_B\sim1/\omega$; a finite-temperature model must specify physically appropriate low-frequency behavior.

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 the first, energy-valued convention and a thermal state of independent modes, the ordered correlation is

$$C(t) ={} \int_0^\infty d\omega\, J(\omega) \left[ (n_B(\omega)+1)e^{-i\omega t} \qquad + n_B(\omega)e^{+i\omega t} \right].$$

There is no additional prefactor in this convention. The complete product of spectral density and occupation factors controls memory; a broad $J$ alone is insufficient. For example, at zero temperature the exponential-cutoff Ohmic form gives

$$C(t)=\eta\int_0^\infty d\omega\,\omega e^{-\omega/\omega_c-i\omega t} =\frac{\eta\omega_c^2}{(1+i\omega_ct)^2}.$$

Its magnitude has an algebraic tail, $|C(t)|=\eta\omega_c^2/[1+(\omega_ct)^2]$, rather than an exponential envelope. Whether replacing this memory by a local rate is accurate still depends on the system timescale and the observable.

Reorganization energy and displacement units

For a dimensionless system coordinate taking the values zero and one, a linear coupling $B=\sum_k(g_kb_k+g_k^*b_k^\dagger)$ displaces each oscillator when the coordinate changes. Completing the square gives the relaxation energy

$$E_{\mathrm r}=\sum_k\frac{|g_k|^2}{\hbar\omega_k} =\int_0^\infty\frac{J(\omega)}{\hbar\omega}\,d\omega.$$

This has energy units with the first convention. For a physical displacement $\Delta q$ in the coordinate-bath convention, the corresponding energy is

$$E_{\mathrm r}=\frac{(\Delta q)^2}{\pi} \int_0^\infty\frac{J_{\mathrm{CL}}(\omega)}{\omega}\,d\omega.$$

The coordinate separation and the definition of $J$ are essential. Omitting them can turn an alleged energy into a spring constant or an energy squared times time. Some molecular conventions absorb these factors into the spectral density itself.

Spectral Density and Markov Rates

In weak-coupling Markovian theory, a transition with Bohr frequency $\omega_0$ often samples the bath near $\omega_0$. For a transition matrix element $S_{ge}$ of the dimensionless Hermitian coupling, the convention above gives

$$\Gamma_{\downarrow} =\frac{2\pi}{\hbar^2}|S_{ge}|^2 J(\omega_0)[n_B(\omega_0)+1],$$

and

$$\Gamma_{\uparrow} =\frac{2\pi}{\hbar^2}|S_{ge}|^2 J(\omega_0)n_B(\omega_0).$$

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.

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 the energy-valued coupling convention and $J(\omega)=\eta\omega e^{-\omega/\omega_c}$, evaluate the reorganization energy

$$E_{\mathrm r} = \frac{1}{\hbar} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega}.$$

Solution

Substituting the spectral density gives

$$E_{\mathrm r} = \frac{\eta}{\hbar} \int_0^\infty d\omega\, e^{-\omega/\omega_c}.$$

The integral is $\omega_c$, so

$$E_{\mathrm r} = \frac{\eta\omega_c}{\hbar}.$$

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

For the narrow-resonance full-line approximation, the correlation envelope decays on $2/\kappa$. A broad peak therefore 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 $\Gamma=2\pi J_1(\omega_0)/\hbar^2$, 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 $\Gamma=2\pi J_1(\omega_0)/\hbar^2$ gives

$$\Gamma = 2J_2(\omega_0)/\hbar^2.$$

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). Publisher.
• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983). DOI.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004). Publisher.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.
• U. Weiss, Quantum Dissipative Systems, 4th ed., World Scientific (2012).