Hi, Im 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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# I Transition rules for a trapped ion

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