In open quantum system dynamics it is often stated that a continuum of bath modes is required to obtain irreversible dynamics. Why is this the case? Is there a general theorem or precise statement about it?
Here are the concrete examples:
- Caldeira-Leggett model of quantum Brownian motion: When deriving the Master equation one starts out with a bath of harmonic oscillators. The Master equation contains two main terms, one for dissipation, one for noise. Both can be expressed in terms of the so called spectral density, which is only non-zero at the frequencies of the individual bath oscillators. In order to obtain irreversible dynamics one can go to a continuum of oscillators and phenomenologically introduce an Ohmic spectral density with a Lorentz-Drude cutoff function. References: CaldeiraLeggett1983, BreuerPetruccione2002.
- Medium Absorption: When looking at the optical response of a quantum mechanical level system for a classical harmonic driving field one will find that absorption only occurs if at least one of the levels couples to a continuum of other levels.
- Spontaneous Emission: (I might be wrong on this one, correct me if I am) One can see this e.g. in the Wigner-Weisskopf approximation. The derivation seems to crucially rely on an integral over the frequency and the use of the Sokhotski-Plemelj theorem. Reference: e.g. this script.