# Why is a continuum of bath modes required for irreversible dynamics?

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.

The way I think of it is by analogy of two coupled pendulums with similar frequencies. We set one swinging and it gradually transfers its energy to the second that then starts swinging. After a while, the second pendulum starts to transfer its energy back to the first. There is no permananet loss of energy (i.e no true friction or dissipation) whith only two pedulums. With the first pendulum coupled to two other pendulms, the same thing happens: an initial enegy loss but gradual return. The return usually takes longer though. The more pendulums the longer it takes for the other pendulms to swing in phase and return energy tothe first. If you want the energy never to return --- true dissipation --- you need infinitely many other pendulums. This is an oscillator "bath" as in Caldeira-Leggett.

A similiar thing happens with tunneling out of a potential well. The particle needs an infinite volume to escape into (and hence a continuum of unbound energy states) or it will originally return to the well.

• thanks for your answer! i wasnt really looking for an intuitive answer unfortunately, since i already understand that. also i think your example isnt very good. are you saying that with infinitely many pendulums you can get irreversible dynamics?? that's still discrete... May 4, 2017 at 15:35
• @Wolpertinger If you don't like this answer, but you already understand the Caldeira-Leggett (CL) model, then what do you want? The CL model is a rigorous derivation, so it's not clear what else you're looking for. Please clarify. May 4, 2017 at 16:01
• @Wolpertinger Yes you should get irreversible dynamics with infinitely many discrete (with incommensurate periods) pendula. The continuum in Caldera Leggett is really just for convenience: it provides a simple formula for a spectral fuction that gives linear friction. May 4, 2017 at 18:01
• @mikestone I didn't know that. If that is true it would make the question I posed indeed pointless. If you could provide an explanation of that particular point or a reference that would change my view of your answer. I am a bit doubtful of this though, since I have seen calculation for the absorption example where this is certainly not the case. But maybe I was wrong on CL. May 4, 2017 at 20:04
• @Wolpertinger It's not usually satisfactory as an answer to appeal to Authority --- but sometimes informative. Tony Leggett sits in the next office but one to me, so I asked him your question about Caldeira-Leggett dynamics. He broadly agreed with my "intuitive" ideas: 1) We need infinitely many incomensurate-frequency oscillators to avoid Poincare recurrence, but once this is done permanent enegy loss ensues; 2) A continuum is probably needed to get a monotonic decay of energy, but he knows no analytic proof. May 5, 2017 at 13:56

The situation is somewhat similar to a spreading wave packet: if it is spreading in a region bounded by two potential walls, then at some point the reflected waves come back to the point where the packet was localized at $$t=0$$. We may even have a situations where the original shape of the packet is restored. In order to have really irreversible spread, we need to consider infinite space or limit our observation to the times shorter than the time that it takes for the tails of the packet to come in contact with the walls.

Dynamics of a system coupled to an oscillator bath functions in a similar way - we treat it as Hamiltonian/coherent dynamics and, sooner or later, this shows itself in a phenomenon known as Quantum collapse and revival. Thus, in order to make the dynamics irreversible one needs an infinite number of oscillators or to make it effectively infinite: e.g., by considering sufficiently short time scales. One could also see it as taking a thermodynamic limit, which may sound abstract, but makes sense in many practical applications. Considering coupling to a continuous spectrum is yet another way to slip in the infinite number of modes - a trick innocently slipped into the basic derivations of the Fermi Golden rule or scattering cross-sections.