Mostly a collection of notes to myself. I'm reading through a review article called Quantum Dynamics of single trapped ions, (Leibfreid, et al, Rev. Mod. Phys. 75, 281). It seems to be a pretty good description of the basic physics involved in ion traps, or in Paul traps at any rate. What follows is some highlights that I thought were interesting.
- Sec. II starts with the classical solution of an ion in an oscillating rf potential with some additional static potential. We start with a potential of the form
and then try to write down the equations of motion. If you write down the classical EOM, and make the following substitutions
then the equation of motion can be transformed into the Mathieu equation, which is (sort of) nicely solvable.
- The stable solutions of the Mathieu equation are periodic and have the form
In general one obtains continued fractions for the coefficients and $\beta_x$. I won't go into the details, but I need to get the notation started. One important result: a trapped particle is stable in all three dimensions if for all .
- It's easiest to understand the dynamics in the lowest order case, where . We get:
- In the above: we see motion at the secular frequency of (denoted by the authors), modulated by micromotion at the rf frequency.
- The quantum solution isn't really much different (which is not exactly surprising). The authors follow a fully quantum mechanical treatment due to Glauber. You basically use the eigenstates of a static potential oscillator of frequency to build a basis for the time-dependent problem. These states are not actually energy eigenstates of the full time-dependent potential, because their energy is periodically altered through micromotion.
- We start with the usual (and instructive) two-level atom approximation with the states and .
- With some re-definitions, we can use the following Hamiltonian to account for dipole, quadrupole, and Raman transitions:
- Then you go into an interaction picture to get rid of the motion and electronic terms in the Hamiltonian, and apply the rotating wave approximation. We get:
is a periodic function that basically accounts for driven motion, and is discussed in detail in Sec. II. is the Lamb-Dicke parameter.
- Plugging in the exact solution for , you can see that there are resonant terms at multiples of the secular and micromotion frequencies and their combinations. If you look only at the secular sidebands, and make the usual assumptions of and , then the interaction Hamiltonian becomes
is a rescaled interaction strength.
- The Lamb-Dicke regime is where the ion's wave function is much smaller than . (i.e. the light wave does not vary much over the extent of the ion's wavefunction.) In this regime, the interaction contains three resonant terms: the carrier resonance, and sidebands at the secular frequency. The Rabi frequencies are different:
- Higher order sidebands are suppressed by successive powers of the Lamb-Dicke parameter.