# How to physically interprut coupling to a heat bath?

In studying quantum optics I commonly see and use a model environment represented as a sum of harmonic modes. I am familiar with the idea of quantizing Maxwell's wave description of light so it doesn't seem entirely abstract to me but it feels like I'm lacking some intuition.

The interaction Hamiltonian's I see usually have the form,

$H_{int}=i\hbar\int_{\omega_{0}-\theta}^{\omega_{0}+\theta}d\omega \sqrt{\frac{\gamma}{2\pi}}(b^{\dagger}(\omega)e^{i(\omega-\omega_{0})t}c-c^{\dagger}b(\omega)e^{-i(\omega-\omega_{0})t})$

where $b$ and $c$ are the bath and system ladder operators respectively.

• When coupling to a continuum of modes like this, can I think of an annihilation event in the system (at a single frequency) corresponding to an excitation of several of the bath modes?
• This Hamiltonian assumes smooth coupling to each of the bath modes (I believe known as the Markovian approximation). The range of coupled modes is set by $\theta$, what is a good physical bound on this value? Recently I have seen an author set $\gamma \ll \theta \ll \omega_{0}$, where the first inequality is stated to amount to the Born approximation (the system and environment dynamical scales are large compared to the induced coupling dynamics).
• Could this model be used to describe general dissipation to other environments, not just to an electromagnetic field?
• • "can I think of an annihilation event...?": Yes, in the sense that annihilating a system quantum generates some environment state that is expressed as a superposition of single-quantum bath harmonic states. – udrv May 1 '17 at 20:35
• • "known as the Markovian approximation" Actually the Markov approximation means the system-bath dynamics at any given time does not depend on the "memory" of the dynamics at any previous time. "what is a good physical bound on this value?" In principle the limits go from 0 to $\infty$ (or even $-\infty$ to $+\infty$ if mathematically convenient) and the actual "range" is set by the environment density of states $g(\omega)$. So your interaction Hamiltonian is a model that simplifies the density of states to get that particular form. – udrv May 1 '17 at 20:35
• •" Could this model be used ...?" Absolutely, provided you adjust the density of states accordingly, and also the nature of the spectrum, because in a condensed state environment you may want to distinguish between acoustic and optical branches. – udrv May 1 '17 at 20:35