noise - What information is contained in the quantum spectral density?

Consider a harmonic oscillator system with Hamiltonian

$$\hat{H} = \frac{1}{2} A \hat{u}^2 + \frac{1}{2} B \hat{v}^2 \qquad [\hat{u}, \hat{v}]=i \gamma$$

where $A$, $B$, and $\gamma$ are all real. This system has resonance frequency $\omega_0 = \gamma \sqrt{A B}$. Suppose we are at nonzero temperature with $\beta \equiv 1/k_b T$. Denote the zero point fluctuations in $\hat{u}$ as

$$u_{\text{zpf}}^2 \equiv \langle 0 | \hat{u}^2 | 0 \rangle = \frac{\gamma}{2} \left( \frac{B}{A} \right)^{1/2}\,.$$

Using the usual Heisenberg time dependence $a(t) = a(0)e^{-i \omega_0 t}$ and the Planck distribution

$$\langle \hat{n} \rangle = \frac{1}{\exp \left( \beta \hbar \omega_0 \right) - 1}$$

one computes the correlation function for $\hat{u}$ as

$$\langle \hat{u}(t) \hat{u}(0)\rangle = u_{\text{zpf}}^2 \left[ \coth(\hbar \omega_0 \beta / 2)\cos(\omega_0 t) - i \sin(\omega_0 t) \right] \, .$$

One can then define a spectral density as the Fourier transform of the time correlation function

$$\begin{align} S_{uu}(\omega) &\equiv \int dt \langle \hat{u}(t)\hat{u}(0)\rangle e^{i \omega t} \\ &= \frac{u_{\text{zpf}}^2}{2} \left\{ \delta(\omega + \omega_0) \left[ \coth(\hbar \omega_0 \beta / 2) - 1 \right] - \delta(\omega - \omega_0) \left[ \coth(-\hbar \omega_0 \beta / 2) - 1 \right] \right\} \, . \end{align}$$

What does $S_{uu}(\omega)$ mean? In other words, what information about the oscillator does $S_{uu}(\omega)$ tell me?

Some notes:

I understand the spectral density of a random process in classical physics. In the simplest sense it is the amount of power per unit frequency range in the process. However, the quantum version is somewhat different. Unlike the usual classical spectral density it is different at positive and negative frequencies (because the quantum correlation function is complex). I have read that the quantum spectral density is related to emission and absorption rates into and out of a thermal bath. In particular, the negative frequency part of the spectral density supposedly corresponds to emission of a quantum of energy, whereas the positive frequency part corresponds to absorption. However, I have never seen a proof of this idea or an example problem in which it can be seen that those precesses are described by the spectral density. A good answer to this question could focus on that relationship, possibly showing the connection between the two delta functions in the $S_{uu}(\omega)$ computed in this example and their corresponding emission and absorption processes.

References:

Michel Devoret's Les Houches notes on quantum fluctuations in electrical systems

Paper by J. Martinis which partially rehashes the Devoret notes and uses the spectral density to compute decay rates, but doesn't explain why that works

Fairly comprehensive notes by Ingold

Similar question with answer unsatisfactory for my question

will add more as they are found and/or suggested

Answer

The spectral density, or spectral function, describes the coupling between a small quantum system that is coupled to a larger environment. In many cases, this environment can be modelled effectively as a system of free bosonic or fermionic modes, with Hamiltonian (working in units with $\hbar = 1$)

$$H_B = \sum_k \omega_k b_k^{\dagger}b_k.$$ The mode operators satisfy $[b_k,b_l^{\dagger}] = \delta_{k,l}$ or $\{b_k,b_l^{\dagger}\} = \delta_{k,l}$ for bosonic and fermionic modes respectively.

The small quantum system (hereafter referred to simply as the system) is described by an autonomous Hamiltonian $H_A$, which we leave unspecified. In many cases, the system-environment coupling takes the form $H_{AB} = AB,$ where the operator $A$ acts on the system only, while the environment noise operator is linear:

$$B = \sum_k \left( g_k b_k + g_k^{\ast} b_k^{\dagger}\right ).$$ This type of Hamiltonian (or its close relatives) can successfully model atoms coupled to the radiation field, electron-phonon coupling in solids, the coupling between a mesoscopic quantum conductor and macroscopic electrical leads or a superconducting qubit coupled to its electromagnetic environment, to name just a handful of examples.

In this setting, one finds in practice that all the effects of the environment are encapsulated by a single quantity, the spectral density, defined as

$$J(\omega) = 2\pi\sum_k \lvert g_k\rvert^2 \delta(\omega-\omega_k).$$ This is the coupling strength weighted by the density of states of the environment. It describes how easy it is to exchange a quantum of energy $\omega$ with the environment.

In the simplest cases, the dissipation can be modelled by a Markovian master equation (Lindblad equation). I am not going to give tedious details of the derivation here, for more information see Breuer & Petruccione. In the master equation description, the effect of the environment is to cause incoherent transitions between energy eigenstates of $H_A$. The transition rate for the process that increases the energy of the system by an amount $\epsilon$ (which may be positive or negative) is found to be

$$\begin{align*} \gamma(\epsilon) & = \int_{-\infty}^{\infty}\mathrm{d}t\; e^{-\mathrm{i}\epsilon t} \langle B(t) B(0)\rangle, \\ & = \int_0^{\infty}\mathrm{d}\omega\; J(\omega)\left [ n(\omega) \delta(\omega-\epsilon) + (1\pm n(\omega)) \delta(\omega+\epsilon) \right ], \end{align*}$$ where, in the factor $(1\pm n(\omega)),$ the plus sign is for bosons and the minus sign for fermions. Here, $\langle \bullet\rangle$ indicates a thermal average over the environment variables, $B(t)$ indicates the Heisenberg picture evolution under $H_B$, while $n(\omega)$ indicates the thermal occupation number of a mode with energy $\omega$. Of course, one finds that $n(\omega) = (e^{\beta (\omega-\mu)} \mp 1)^{-1}$ is the Bose-Einstein (Fermi-Dirac) distribution at temperature $T = 1/k_B\beta$ and chemical potential $\mu$.

Now the similarity with the OP's expression should be clear (one can rewrite the $\coth$ terms into the form I have given). The dissipation is determined completely by the power spectrum of the quantum noise operator $B$. In evaluating this expression, one finds two terms, both proportional to the spectral density. The first term describes the probability of absorption of energy from the environment, which is only possible if a mode of energy $\epsilon$ is occupied. The second term describes emission of energy into the environment, which comes with an additional $+1$ due to quantum non-commutativity. This $+1$ allows spontaneous emission even when the environment mode is empty (this may be colourfully attributed to quantum zero-point fluctuations). When the environment mode is occupied, we have either enhanced (stimulated) emission due to bosonic bunching, or suppressed emission due to the Pauli exclusion principle.