# Particle creation through a time dependent Hamiltonian

We know that a time dependent Hamiltonian can create particles. We know this for instance from field theory in curved spacetime, where for instance in an expanding or contracting universe creation and annihilation of particles can take place.

My question is, is it possible for a particle to be created in a scenario where the Hamiltonian is a periodic function of time (i.e are there particles at the end of the period)? So this would be asking, if the universe expanded and then contracted back to its original size, following say a sine curve, could particles content have changed at the end of it? In more quantum mechanical terms, suppose I have a Hamiltonian $H_0 f(t)$ where $f(t)$ is periodic in time and the initial state is the ground state of $H_0$. Would I have an excited state at the end of one period? I know for instance for Rabbi oscillation I would be back at the same state. But is it always true and if so, is there a general proof or argument for a periodic (or time-symmetric) Hamiltonian?

• Note that Rabi oscillations do not occur at the frequency of the driving field! The Rabi frequency is a (linear) function of the amplitude of the driving field. Given that your Hamiltonian is just $H_0 f(t)$, your chosen initial state is an eigenstate of the time evolution operator. Now, do you think there will be an excited state after one period? (PS. In general time-dependent Hamiltonians do put energy into a system and create particles, but you need at least two terms which don't commute.) – Mark Mitchison Feb 25 '16 at 0:55
• Hm, I guess I am wrong about Rabi oscillations. You are right that generally time-dependent Hamiltonians put energy into a system and create particles. But for a periodic Hamiltonian one would think as much energy as has been pumped in is also taken out, so one is left with no particles. Or is that interpretation only true in the absence of non-commuting terms? – Nirmalya Kajuri Feb 25 '16 at 3:16
• Yes, that idea only works when the oscillating part of the Hamiltonian commutes with everything else. Otherwise you generically expect the system to become excited over time. This is exactly what also happens in classical physics. If you push a child on a swing with a periodically varying force, the swing goes higher and higher. – Mark Mitchison Feb 25 '16 at 10:44
• That sounds counter-intuitive....classically there is no commutation relation..shouldn't there be some form of energy conservation at the end of a cycle? – Nirmalya Kajuri Feb 25 '16 at 13:19
• In the classical case you just replace commutators with Poisson brackets. The appearance of a commutator/Poisson bracket here has nothing to do with Heisenberg uncertainty, it is about different contributions to the energy affecting each other (e.g. potential and kinetic energy being interchanged). And no, systems with time-dependent Hamiltonians do not have energy conservation, periodic or not. – Mark Mitchison Feb 25 '16 at 13:21