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I am dealing with a problem of a 2 level system (an ion in my case) placed in a Penning trap. Basically the ion is moving inside the trap under the influence of the magnetic and electric field and I need to study its inner 2 level system (basically the lowest 2 energy states) while it is moving. For simplicity assume that we look only at the axial motion, so that the ion oscillates up and down. If I treat the ion motion classically, assuming it moves like $z = z_0 cos(\omega t)$ and the electric field in the z direction is (I care about the electric field in my case, as I want to mix the 2 levels of opposite parity): $E = E_0z$, the field that the ion feels in its intrinsic frame is $E=E_0z_0cos(\omega t)$. From here I just treat the 2 level system under the influence of an oscillatory electric field, which is doable. However, now I need to solve the same problem assuming the ion motion is quantized. I can't write its position as $z = z_0 cos(\omega t)$ anymore, as its position is described by a wavefunction now. But now I am not sure what does the ion see in its own reference frame. I am not sure how to move from the ion motion in the lab frame (and by this I mean the wavefunction squared distribution) to the ion frame, such that I can extract the field that the ion sees and then proceed with calculating the effect on the 2 level system. Can someone advise me about this (or point me towards any readings)? Thank you!

JohnDoe122
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  • Why do you care what does ion see in its reference frame? Such frame is hard to define and non-inertial. Why not describe everything in the lab frame? – Ján Lalinský Mar 03 '21 at 15:50
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    I am not totally sure how to do that either. The 2 level system (which in this case is basically given by the lowest 2 levels of the electron in the ion) is defined in the ion frame i.e. when solving SE for an atom you assume the center of coordinates is the nucleus. But in my problem I have a lab frame where the electric field is easily definable and an intrinsic frame, where the 2 level system is easily definable. I am not sure which one is easier to work with. Could you elaborate a bit on your suggestion? – JohnDoe122 Mar 04 '21 at 16:06
  • Ok. I would start by writing down the Schr. equation for the ion in the external potential of the trap. The ion moves so nuclear coordinates have to be accounted for, the psi function depends on three nuclear coordinates and 3n electronic coordinates where $n$ is number of electrons in the ion. This is complicated equation so then I would try to apply some simplifying assumptions about the psi function dependence on nuclear coordinates, similar to the Born-Oppenheimer approximation, to get simpler equations. – Ján Lalinský Mar 05 '21 at 00:15
  • Maybe before that, I would first try the simplest approach, to assume the nucleus is a point that just oscillates sinusoidally and EM field in its comoving frame is just the Lorentz transformed field of the trap. This is easier than the general approach but it neglects the uncertainty of position/momentum of nucleus. Whether it is good enough I don't know. Maybe be a good preparation for the harder calculation using the full Schr. equation. – Ján Lalinský Mar 05 '21 at 00:18
  • In any case, effect of acceleration due to oscillation on the field of the nucleus and electrons probably can't be captured correctly either way, one would have to go into relativistic theory. – Ján Lalinský Mar 05 '21 at 00:21
  • @JánLalinský I was wondering, can't I just assume the (point-like) ion moves in a harmonic potential, so I can describe its motion using the HO wavefunctions. Then, I calculate the expectation value of the electric field (E_0z) in that state and use that as the field that the 2 level system inside the ion sees. As you said, this implies the use of BO approximation, but beside that it should be no further assumptions. What do you think? – JohnDoe122 Mar 05 '21 at 17:57
  • Maybe. I don't know what your ultimate goal with this is. There are many ways to approximate the Schroedinger's equation for such many particle system using various approximations and semi-classical ideas (classical electric field). Sometimes these are enough to explain some measurement (spectra), sometimes they are not (spontaneous emission, optical activity). The first problem I see with your idea is that electrons are not located at nucleus, so the EM field they experience isn't the same as expectation value based on psi function of the nucleus. – Ján Lalinský Mar 05 '21 at 20:02
  • The electrons experience also EM field of the nucleus which is much stronger than the field of the trap. This field of the nucleus is different from the Coulomb field because the nucleus oscillates. Maybe this is important and changes the resulting 2-level toy model parameters and behaviour, maybe not so much. You have to estimate these effects and try to find the simplest reasonable approach. – Ján Lalinský Mar 05 '21 at 20:04
  • One general advice I can give, try to analyze this system using just classical theoretic methods first, in the simplest way possible. E.g. physical electric dipole (heavy positive particle, light negative particle) bound together by harmonic force that keeps the particle in some equilibrium non-zero distance, and this whole thing placed in Hooke-like external field. When you get familiar with behaviour of that model, it may be of help finding the simplest quantum theoretical model to analyze and compare. – Ján Lalinský Mar 05 '21 at 20:09
  • related: https://physics.stackexchange.com/q/630781/226902 – Quillo Feb 16 '22 at 11:28

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