# Notation for spectral density in experimental community

In an open quantum system setup where the system is coupled to infinite harmonic oscillators as bath (as in the Caldeira leggett model, for example), through the Hamiltonian,

$$H = \frac{\hat{p}^2}{2m} + V(\hat{x}) + \sum_n \frac{\hat{p}^2_n}{2m_n}+ \frac{1}{2}m_n \omega_n ^2 \hat{x}^2 - \hat{x} \sum_n g_n \hat{x}_n$$

Here, $$\hat{p}, \hat{x}, \hat{p_n}, \hat{x_n}$$ are, respectively, system momentum and position operator and nth bath oscillator momentum and position operator and $$g_n$$ is the coupling strength between the system and the nth bath oscillator.

One then defines the spectral density as following,

$$j(w)=\frac{1}{m}\sum_i \frac{g_i ^2}{\omega_i ^2}\delta(\omega-\omega_i)$$

The dimension of $$j(\omega)$$ can be calculated to be $$\frac{mass}{time}$$.

But in the same paper as linked above (in the supplementary info), the spectral density is later expressed in terms of $${time}^{-1}$$. (please refer to the following image). What happened to the mass dimension? Are they now calling $$\frac{j(\omega)}{m}$$ spectral density? On the other hand, the following paper expresses the weighted spectral density $$\rho(\omega)\omega^2$$ as $$cm^{-1}$$. (please refer to the following image).It appears that in the notation of this paper, $$\rho(\omega)$$ is the spectral density. If so, then the correct dimension of weighted spectral density should be $$\frac{mass}{{time}^3}$$. How is it $$cm^{-1}$$ instead? I know that sometimes, in experimental literature, frequency is expressed as wavenumber and one needs to multiply the term with speed of light expressed in cms ($$c=3 \times 10^{10}$$)in order to get the frequency. But in the second paper I have linked, one would need to multiply the expression with $$c^3$$ instead, because the original expression is in terms of $$\frac{mass}{{time}^3}$$.

• I believe the main problem is the pigeon hole effect with this terminology. Too many people use the term "spectral density" and it is a different version for each in each paper. It is a loosely defined term in my opinion. Nov 17, 2022 at 2:41
• You will literally have to follow each equation in a given paper and try to recreate their figure if possible. Nov 17, 2022 at 2:43