Stimulated emission in harmonic oscillator

[kelly0303]

Hello! Is stimulated emission possible for a harmonic oscillator (HO) i.e. you send a quanta of light at the right energy, and you end up with 2 quantas and the HO one energy level lower (as you would have in a 2 level system, like an atom)?

 

[WernerQH]

Yes, of course. Cyclotron emission (from electrons spiralling in a magnetic field) is very much like that. Electrons jump from one Landau level to the next and emit photons of the cyclotron frequency with each step. And this can be self-amplifying due to stimulated emission. It is called electromagnetic cyclotron instability and is well known in plasma physics. It happens when the electron velocity distribution becomes unstable, i.e. there is a "population inversion" and in a region of velocity space faster electrons are more numerous than slow ones.

It is wrong to think of stimulated emission as a quantum effect. Already Einstein explained that it has a classical analog. An oscillating charge in an electromagnetic wave can gain energy or lose energy (giving it to the wave) depending on the phase relation between the oscillator and the wave.

 

[kelly0303]

Yes, of course. Cyclotron emission (from electrons spiralling in a magnetic field) is very much like that. Electrons jump from one Landau level to the next and emit photons of the cyclotron frequency with each step. And this can be self-amplifying due to stimulated emission. It is called electromagnetic cyclotron instability and is well known in plasma physics. It happens when the electron velocity distribution becomes unstable, i.e. there is a "population inversion" and in a region of velocity space faster electrons are more numerous than slow ones.

It is wrong to think of stimulated emission as a quantum effect. Already Einstein explained that it has a classical analog. An oscillating charge in an electromagnetic wave can gain energy or lose energy (giving it to the wave) depending on the phase relation between the oscillator and the wave.

Thanks a lot for your reply. I got a bit confused as in my QM book, they presented HO as if when you add more quanta you go higher and higher in energy. But it seems like you will end up having Rabi oscillations between 2 levels (ignoring the lifetime of excited states). I haven't really seen any mention of Rabi oscillations in the context of the HO.

 

[f95toli]

Thanks a lot for your reply. I got a bit confused as in my QM book, they presented HO as if when you add more quanta you go higher and higher in energy. But it seems like you will end up having Rabi oscillations between 2 levels (ignoring the lifetime of excited states). I haven't really seen any mention of Rabi oscillations in the context of the HO.

I think there might be difference between theory and experiment here. A perfect HO where the levels are exactly equidistant won't exhibit Rabi oscillations since the system will just continue to climb up the "ladder" as it absorbs more photons. To get oscillations you need some form of anhamonicity so that the energy difference between the states differ.
Now, I don't know much about cyclotron physics; but I suspect the there is some anharmonicity between the different Landau levels; if not you could just keep "pumping" the system to higher and higher levels forever.

Note that the dynamics will also be different for a multi-level system where the lifetime of the levels differ

 

[WernerQH]

in my QM book, they presented HO as if when you add more quanta you go higher and higher in energy. But it seems like you will end up having Rabi oscillations between 2 levels (ignoring the lifetime of excited states). I haven't really seen any mention of Rabi oscillations in the context of the HO.

It is correct that in a harmonic oscillator you can go up to higher and higher energies. Rabi oscillations are specific to two-level systems.