Quantising a Damped Mass on a Spring

Question

Background: this question discusses Lagrangian/Hamiltonian formulation of a dissipative problem. However, I'm not clear if this can be made quantum and would like a more explicit roadmap if possible.

I'm interested if there are quantum systems whose classical limits are not Hamiltonian, and how one would describe such a system if they do exist. I have in mind something like the following:

1) There is a Hilbert space of states.

2) Time evolution is completely positive. This may have to not be unitary/Hamiltonian, but this is acceptable since I have in mind some effective theory of a subsystem.

3) The classical limit has time evolution given by the dissipative EoMs:

$$m\ddot{x} +\gamma \dot{x} +kx = 0$$

How do I define such a system, what makes it quantum, and how do I achieve the quantisation such that I get the correct classical behaviour?

Answer 1

In order to have dissipation you need somewhere for the energy to go, and this somewhere has to be included in the quantization process. In addition to absorbing the energy, the extra system causes quantum decoherence and so makes the problem quite tricky.

The damped harmonic oscillator is one of the problems that were attacked and solved by Caldeira and Leggett (A.Caldiera, A.J.Leggett, Phys. Rev. Lett. vol 46 (1981) p 211). Calderia and Leggett showed that you can almost always model the "somewhere" as a bath of infinitely many harmonic oscillators. In this they derive that the physics is given by the effective (Euclidean) action:

$$ S_{\text{eff}} = \int_0^\tau dt \left( \frac{1}{2}m\dot{x}^2+V(x)+\frac{\gamma}{4\pi}\int_{-\infty}^\infty dt' \left( \frac{x(t)-x(t')}{t-t'} \right)^2 \right)$$