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As it pertains to state preparation, but also quite generally, why should the method of waiting until decay to the ground state work at all?

Ballentine writes "It is possible to prepare the lowest energy state of a system simply by waiting for the system to decay to its ground state." What conditions on the system are necessary for this to be true? Does one require $T \to 0$? Are there other conditions for which this method works? Or is this just a "general rule of thumb" as long as we shield our isolated quantum system?

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  • $\begingroup$ Yep, the finite temperature of any system with $T>0$ prevents 100% decay $\endgroup$
    – Quantum Mechanic
    Commented Aug 8, 2023 at 14:04
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    $\begingroup$ the only condition will be that the system is not in a bound state. If it is excited it will decay to the ground state with appropriate time constants for the material . relevant researchgate.net/figure/… , so that the remaining excited states are very small in number $\endgroup$
    – anna v
    Commented Aug 8, 2023 at 14:06
  • $\begingroup$ @annav I am confused, the ground state itself is often a bound state, no? $\endgroup$
    – EE18
    Commented Aug 8, 2023 at 14:21
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    $\begingroup$ I've just checked in the book and Ballentine give justifications for his statement e.g. that the radiated photon heads off to infinity never to return, that the EM field around the system is in its ground state and that the temperature is low. This seems a pretty complete list to me. Can you say what more you think might be required? $\endgroup$
    – John Rennie
    Commented Aug 8, 2023 at 15:02
  • $\begingroup$ @JohnRennie Ballentine gives specific instances/models, whereas I think I am looking for a more general framework for deciding whether a given system will decay to the ground state. Or is the best we can do is to say “if the system is such that it decays to the ground state…” and then analyze whether that premise holds on a case-by-case basis? $\endgroup$
    – EE18
    Commented Aug 8, 2023 at 16:13

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In any excited quantum system there is always a non-zero probability for the transition to the ground state. This probability can be small if the transition is prohibited, but if we wait long enough the system will always visit the ground state. The only question is whether the system will stay there.

Ballentine is considering a radiative transition in an isolated system like an atom, and he points out that when the system decays by emission of a photon the photon heads off to infinity making the transition to the ground state irreversible. He also comments that there must be no EM field capable of re-exciting the atom and the temperature must be low enough not to cause thermal excitation.

Given the huge range of quantum systems I think it's hard to make general statements about what restrictions are necessary to prevent the ground state from being excited by its environment.

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  • $\begingroup$ It is important for that to consider an open system...If the system is in any excited eigenstate of the Hamiltonian, a closed system will remain there forever. You implicitly assumed that, but I think it is worth to spell out. $\endgroup$
    – Tobias Fünke
    Commented Aug 8, 2023 at 17:09
  • $\begingroup$ When you say “prohibited”, what do you mean? Obviously it’s not truly prohibited since, as you later say, decay eventually occurred? Does prohibited mean prohibited to some order in a perturbation theory argument? $\endgroup$
    – EE18
    Commented Aug 8, 2023 at 17:16
  • $\begingroup$ I know the term "prohibited" is used when the spin is not conserved. What is meant, is that the "prohibited transition" is far less likely to take place, since a spin flip has to occur. Look up inter-system-crossing for example. So I think in general "prohibited" means less probability due to breaking of some (non-strict) conservation law. $\endgroup$
    – Martin
    Commented Aug 9, 2023 at 8:19
  • $\begingroup$ @EE18 This is really a separate question, but we normally consider a transition prohibited when there is a conservation law that (in principle) prevents it. However in the real world there is almost always some mechanism for evading the conservation law so in practice the transition is possible though it may be very slow. For example take the hydrogen line transition. This is forbidden because there is no electric dipole to couple to the electric field of a photon. However it can occur through magnetic dipole coupling but it takes 10⁷ years! $\endgroup$
    – John Rennie
    Commented Aug 9, 2023 at 9:27

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