Different reference frames in QM

[BillKet]

Hello! 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!

 

[Twigg]

I don't think you need to transform reference frames. You can find the electric field in the lab frame. The axial motion will be harmonic oscillator motion, so you know what those eigenstates are. If you have an operator expression for the force, all you have to do is divide by the ion charge and you have the electric field operator.

To get you started:

Given that $E=E_0 z$, you know that you have a simple harmonic oscillator where the spring constant is $k = qE_0$. You can look up the eigenstates in any quantum textbook. Then you know that the force operator in this system is just $\hat{F} = k\hat{x}$, and you know that $q\hat{E} = \hat{F}$. You can calculate the expectation value of $\hat{E}$ versus time by integration, or by rewriting the operator in terms of ladder operators. Then you can use that expected E-field versus time in your two-level calculations.

 

[BillKet]

I don't think you need to transform reference frames. You can find the electric field in the lab frame. The axial motion will be harmonic oscillator motion, so you know what those eigenstates are. If you have an operator expression for the force, all you have to do is divide by the ion charge and you have the electric field operator.

To get you started:

Given that $E=E_0 z$, you know that you have a simple harmonic oscillator where the spring constant is $k = qE_0$. You can look up the eigenstates in any quantum textbook. Then you know that the force operator in this system is just $\hat{F} = k\hat{x}$, and you know that $q\hat{E} = \hat{F}$. You can calculate the expectation value of $\hat{E}$ versus time by integration, or by rewriting the operator in terms of ladder operators. Then you can use that expected E-field versus time in your two-level calculations.

Thanks a lot for this! I am a bit confused. The ion moves in a harmonic potential and from there I can get its eigenstate as you said. But if I calculate the expectation value of $\hat{E}$, that is basically a constant times the expectation value of $\hat{z}$, right (as $E = E_0 z$)? And once I calculate that, I get a fixed number, for the given eigenstate in which the ion is. I am not sure I understand where does the time dependence appear. (Actually isn't the expectation value of $\hat{z}$ zero, as that is the center of the oscillation?)

 

[Twigg]

You're right about the expectation value of $\hat{E}$ being 0 for any of the eigenstates.

Sorry, when I was talking about a time-dependent expectation value, I was thinking of non-stationary states because I was thinking of nearly classical trajectories. It sounds like you're actually more interested in the eigenstates. My bad.

The only thing I didn't mention before is that if your two-level Hamiltonian has any terms that scale with $E^2$, then you'll also need to evaluate $\langle\hat{E}^2\rangle$, and that will not be zero even for an eigenstate.

 

[BillKet]

You're right about the expectation value of $\hat{E}$ being 0 for any of the eigenstates.

Sorry, when I was talking about a time-dependent expectation value, I was thinking of non-stationary states because I was thinking of nearly classical trajectories. It sounds like you're actually more interested in the eigenstates. My bad.

The only thing I didn't mention before is that if your two-level Hamiltonian has any terms that scale with $E^2$, then you'll also need to evaluate $\langle\hat{E}^2\rangle$, and that will not be zero even for an eigenstate.

I don't think I have a term that scales like $E^2$ (why would that appear from $Ez$?). So assuming there is no $E^2$ term, does it mean that the 2 level system inside the ion feels no electric field, so it is not perturbed? How can that be? Shouldn't the expectation value of any term in QM reproduce the classical result? And given that classically the 2 level system feel the electric field, shouldn't the QM case be such that the 2 level system also feels some electric field?

 

[Twigg]

(why would that appear from $Ez$?)

For all I know you may have been interested in a 2nd order Stark shift. Not trying to be snarky, just saying it's hard to tell what your hamiltonian looks like over the interwebs. I was just trying to cover all my bases.

As far as why the eigenstates don't get perturbed, it's because the eigenstates don't behave like classical trajectories. The states that behave like classical trajectories are the coherent states. Try taking the expectation value of $\hat{z}$ on a coherent state $|\alpha(t)\rangle$, and you'll wind up with a result like $\langle z \rangle \propto Re[\alpha(t)]$. Plug in the usual time evolution of a coherent state: $\alpha(t) = \alpha_0 e^{-i\omega t}$, and you wind up with $\langle z \rangle \propto cos(\omega t)$

 

[BillKet]

For all I know you may have been interested in a 2nd order Stark shift. Not trying to be snarky, just saying it's hard to tell what your hamiltonian looks like over the interwebs. I was just trying to cover all my bases.

As far as why the eigenstates don't get perturbed, it's because the eigenstates don't behave like classical trajectories. The states that behave like classical trajectories are the coherent states. Try taking the expectation value of $\hat{z}$ on a coherent state $|\alpha(t)\rangle$, and you'll wind up with a result like $\langle z \rangle \propto Re[\alpha(t)]$. Plug in the usual time evolution of a coherent state: $\alpha(t) = \alpha_0 e^{-i\omega t}$, and you wind up with $\langle z \rangle \propto cos(\omega t)$

Thanks a lot for this! Actually yes, in theory we can have second order Stark shift, but for this experiment we only care about off-diagonal terms, so we would need odd powers of E.

I understand what you mean now. Actually I am not sure if I am interested in eigenstates or coherent states. What do you actually see in practice, in a real experiment, if you place a particle in a harmonic potential i.e. can you place a particle in a coherent state?

 

[Twigg]

No problem

In every experiment I've worked on, the ion temperature was much much higher than the trapping frequency ($kT >> \hbar \omega$), so we always just assumed classical trajectories. If you have an ion that is cooled to below the trapping frequency ($kT < \hbar \omega$), then it's safe to assume that your ion is in the ground eigenstate. The colder the ion is, the more likely the ion is in the ground eigenstate. The probability of being in an excited state goes with the Boltzmann factors $e^{-(E-E_0)/kT} = e^{-n\hbar\omega/kT}$. Of course, the ion could be in a superposition of the ground and excited states, so you don't really know what's going on. The only case that you really know the quantum state is when that ion is cooled well below the trapping frequency.

 

[BillKet]

No problem

In every experiment I've worked on, the ion temperature was much much higher than the trapping frequency ($kT >> \hbar \omega$), so we always just assumed classical trajectories. If you have an ion that is cooled to below the trapping frequency ($kT < \hbar \omega$), then it's safe to assume that your ion is in the ground eigenstate. The colder the ion is, the more likely the ion is in the ground eigenstate. The probability of being in an excited state goes with the Boltzmann factors $e^{-(E-E_0)/kT} = e^{-n\hbar\omega/kT}$. Of course, the ion could be in a superposition of the ground and excited states, so you don't really know what's going on. The only case that you really know the quantum state is when that ion is cooled well below the trapping frequency.

So as you said, in an eigenstate, the expectation value of the electric field in my case would be zero, so the 2 levels would not be perturbed. So I guess that what I need in order to make the 2 levels feel the electric field is to place them in a superposition of eigenstates, and as you said, a coherent state, would be ideal as it reproduces the classical trajectories. But is it possible to control the mixing of different eigenstates that the ion is in? Or is it possible, in practice, to place the ion in a coherent state? (I am really new to this field so I am sorry if some of these questions are silly). Thank you!

 

[Twigg]

Sorry, I don't know. On paper I feel like you could "kick" the trapped ion with a pulsed potential to excite it from the ground state into higher motional states, but I don't have any experience with that kind of thing in practice.

Edit: the "kick" procedure assumes it starts in the ground state, so it's in a known starting point

 

[Twigg]

I remembered something. If, by slim chance, your ion happens to be a polar molecule, you could consider applying a rotating electric field in the transverse plane like they do in the HfF+ eEDM search. In that case, you really do need to think carefully about the ion frame vs the lab frame, as the rotating field can cause levels to mix in weird ways (you'd have to get comfy with your Wigner D-matrices ). Again, only valid for polar ionic molecules and I imagine it wouldn't be easy to implement in a penning trap. Just food for thought.