Laser cooling of ions stored in a Penning trap: A phase-space picture

We present a phase-space picture of laser cooling of the radial motion of ions in a Penning trap. This picture enables a particularly simple derivation of the condition for simultaneous cooling of all degrees of freedom of a single ion to be obtained. It also allows a physically intuitive approach to be taken to the cooling process. Using this approach, we discuss different aspects of the cooling of a single ion in a Penning trap, including the formation of ``trapped'' states where the steady-state motion amplitude is nonzero for either the magnetron or modified cyclotron motion. We use an analytical approach where the approximation of small-amplitude motions can be made, and we use numerical calculations for larger-amplitude motions. The best procedure to use for effective laser cooling is derived. A similar approach is then used to treat the laser cooling of two ions in the Penning trap. We show that the use of the phase-space approach allows new insights into many aspects of the laser-cooling process to be gained, and we indicate how these insights may be applied in experimental investigations.