Transition rules for a trapped ion

[jamie.j1989]

Hi, I am studying the dynamics of a trapped ion in a laser and am trying to simulate the dynamics via a quantum jump, in order to do this I need to know the allowed transitions that the ion external and internal degrees of freedom can take, being the number state (n) of the approximated harmonic potential of the laser and the ground and excited states g and e respectively.

I figured that you can determine the allowed transitions from assuming the ion is in an initial state and then just acting the hamiltonian on the state. The RWA (Rotating wave approximation) hamiltonian in natural units is

$$H= \frac{1}{4}\left[2\Omega\left(a^+\sigma^++a\sigma^-\right)-\delta\left(2a^+a-\sigma^z\right)\right]$$

Where we have, $\Omega$ is the Rabi frequency, $a$ and $a^+$ are the lowering and raising operators, $\sigma^-=|g><e|$, $\sigma^+=|e><g|$, $\sigma^z=|e><e|-|g><g|$ and $\delta$ is the detuning.

So if we start our system in the state $|en>$ (in the nth external state and excited internal state), acting on $H$,

$$H|en>=\frac{\Omega}{2}\sqrt{n}\left|g,n-1\right>+\frac{\delta}{2}\left(\frac{1}{2}-n\right)\left|en\right>$$

And if we start our system in the state $|gn>$ (in the nth external state and ground internal state), acting on $H$,

$$H|gn>=\frac{\Omega}{2}\sqrt{n+1}\left|e,n+1\right>-\frac{\delta}{2}\left(n-\frac{1}{2}\right)\left|gn\right>$$

My interpretation of this is that if in the |en> state the ion can transition to the |g,n-1> state, and if in the |gn> state it can transition to the |e,n+1> state, this doesn't seem right to me as the ion would just surely transition between two states endlessly?

 

[DrClaude]

If the Hamiltonian is as you wrote it, then the conclusion is correct. But I agree with you that the conclusion is strange, so my hunch is that the Hamiltonian is not correct. You should probably have terms of the type $a \sigma^+$ and $a^+ \sigma^-$ in there.

 

[jamie.j1989]

The Hamiltonian is the anti Jaynes-Cummings Hamiltonian I got it from this paper http://arxiv.org/pdf/1412.1863v2.pdf setting j=1 and J = 0, as I'm only looking at 1 ion. If we're only looking at a laser exciting the two level ion at resonance then when a photon has been absorbed it can't absorb any more energy so can only emit a photon, so the repeating process might actually make sense?

 

[naima]

Did you read
"Decoherence of quantum superpositions through coupling to engineered reservoirs" by wineland?

You have the details of the two lasers with Raman transitions.

 

[DrClaude]

The Hamiltonian is the anti Jaynes-Cummings Hamiltonian I got it from this paper http://arxiv.org/pdf/1412.1863v2.pdf setting j=1 and J = 0, as I'm only looking at 1 ion. If we're only looking at a laser exciting the two level ion at resonance then when a photon has been absorbed it can't absorb any more energy so can only emit a photon, so the repeating process might actually make sense?

Indeed, in that model only two levels can be coupled at a time by the laser: |n,g> and |n+1,e>. Note however that there is a decay term involving only σ^-, so that there is a channel |n+1,e> → |n+1,g>, which can then lead to |n+1,g> → |n+2,e>, etc.

 

[jamie.j1989]

Would we be neglecting recoil if that transition (|n+1,e> $\rightarrow$ |n+1,g> ) were possible, surely if a decay by spontaneous emission occurs which is described by the $\gamma D[\sigma^-](\rho)$ term in the master equation, heating or cooling would take place from the recoil which would amount to a change in the n state? And if this transition is valid why doesn't the Hamiltonian describe it?