Bosonic Spectral Density $\rho (\omega )$

All the dependence on the coupling constants $\gamma_Q$ is encapsulated within the spectral density $\rho (\omega )$. The latter is often parametrised as

$$\fbox{$ \begin{array}{rcl} \displaystyle \rho(\omega) = {2}\alpha \omega_c^{1-s} \omega^{s}e^{-{\omega}/{\omega_c}}, \end{array}$\ }$$ (54)

where $\alpha$ is the dimensionless coupling parameter and $\omega_c$ is the cutoff frequency. Note that $\rho (\omega )$ has the dimension $[\omega]$ which is the reason for the pre-factor $\omega_c^{1-s}$. The parameter $s$ determines the low-frequency behaviour of $\rho (\omega )$, and one calls couplings with

$$s<1 :$$    sub-ohmic

$$s=1 :$$    ohmic

$$s>1 : .$$ (55)

This classification has its origin in the analysis of the dissipative two-level (spin-boson) system which we will discuss below.

The case $s=1,\omega_c\to \infty$

$$\rho(\omega) = {2}\alpha \omega$$ (56)

is called scaling limit of the ohmic bath and has the special property of homogeneity $\rho(k \omega) = k \rho(\omega)$.