Excited system in a harmonic potential

[Malamala]

Hello! Assume we have a simple harmonic oscillator potential, in 3D (say created by some electric fields, such as a Paul trap) and inside it we have a 2 level system in the excited state (say an ion in which we care only about 2 levels, for example the lowest 2). The translational energy of the ion can change only if the difference is given by the frequency difference between the levels of the harmonic oscillator, $\omega_{osc}$. However, when we have a decay from the excited state, the atom will achieve a given kinetic energy, in order to conserve the momentum (this is the principle behind the Doppler cooling of ions/atoms). Can the transition frequency $\omega_{ion}$ be such that the ion will be forever stuck in the excited state, just because it can't achieve a recoil energy that is a multiple of $\omega_{osc}$?

 

[jambaugh]

Be careful with your thinking. In a harmonic well, momentum is not well defined nor conserved. Consider a classical SHO at a given energy. What is it's momentum? Well that depends on what part of the cycle you observe it. It is continuously exchanging momentum with the well. So rethink your query without the "...in order to conserve momentum..." step. (Also, along the same lines, don't equate kinetic energy with total energy [which is what defines the energy levels of the QHO].)

Also note that decay of such an excited system depends on it's interaction with an external system. Excited electrons in an atom decay by e-m interaction releasing a photon. Absent the e-m interaction or any other they would never decay. Note that these energy eigen-states are eigen-vectors of the non-interacting Hamiltonian which is itself the generator of time evolution. They are, thus, time translation symmetries i.e. stationary states i.e. not decaying absent that interaction. Given an interaction it will depend on the nature of that interaction and the additional qualities of the system. E.g. one-photon interactions may be "forbidden" due to spin/angular momentum conservation. But then higher order e.g. 2-photon interactions, which are generally much less likely may still be allowed.

 

[vanhees71]

Momentum is well defined in an oscillator potential. In the position representation it's given by the usual operator $\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla}$. Of course an energy eigenstate of the harmonic oscillator is no momentum eigenstate.

 

[Malamala]

Be careful with your thinking. In a harmonic well, momentum is not well defined nor conserved. Consider a classical SHO at a given energy. What is it's momentum? Well that depends on what part of the cycle you observe it. It is continuously exchanging momentum with the well. So rethink your query without the "...in order to conserve momentum..." step. (Also, along the same lines, don't equate kinetic energy with total energy [which is what defines the energy levels of the QHO].)

Also note that decay of such an excited system depends on it's interaction with an external system. Excited electrons in an atom decay by e-m interaction releasing a photon. Absent the e-m interaction or any other they would never decay. Note that these energy eigen-states are eigen-vectors of the non-interacting Hamiltonian which is itself the generator of time evolution. They are, thus, time translation symmetries i.e. stationary states i.e. not decaying absent that interaction. Given an interaction it will depend on the nature of that interaction and the additional qualities of the system. E.g. one-photon interactions may be "forbidden" due to spin/angular momentum conservation. But then higher order e.g. 2-photon interactions, which are generally much less likely may still be allowed.

Thank you for your reply! Using momentum was probably not the best phrasing, but I am still confused. In the CM frame of the atom, after the electron decays, we have the atom moving, so it acquires a kinetic energy in the CM frame, which can then be moved to the lab frame. So the atom gained a given kinetic energy $E_{atom}$ and this is something real in the lab, as it is used for laser cooling. Also you don't need an external field for this, if we have spontaneous decay. So if we have an excited atom in free space and it decays, it will change its energy, and there is no constraint on this energy change, as we don't have quantized energy levels. But if the decay happens in a quantized level, doesn't the change of energy need to be equal to the difference between some of these levels?

 

[jambaugh]

You mention spontaneous decay but remember that this still involves emission of photons. It seems to me by your query that you're ignoring those photons. It does require an external field. It is just interacts with that field by emitting rather than absorbing. But also note that if the transition is the same as is used for cooling this effect will be swamped by stimulated emissions which favor the effective trapping potential. Note that the total momentum, angular momentum, and energy is conserved for the system of atom plus field. The COM of the excited atom will equal the COM of the spontaneously de-excited atom plus emitted photon.

Remember the emitted photon is (rest) massless it still has mass-energy and momentum.

Also note that this spontaneous decay will not leave the atom with an arbitrary change in KE. The momentum of the emitted photon is tied to its energy/frequency which is tied to the transition energy of the atom.

In particular if that transition is the one being used in the trap then stimulated emission will decay the atom much more quickly and impart on it exactly that direction of its change of momentum to favor the effective trapping potential and cooling effect. That is, in the method of laser trapping/cooling that I've read about where they tune and direct the lasers so that the Doppler effect for moving atoms favors such a momentum exchange via excitation and stimulated de-excitation of transitions between the particular ground and excited states.

If you like I could attempt to describe this process in more detail to the limit of my understanding.