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 i ons in a Penning trap. This picture enables a particularly simple derivatio n 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 t o be taken to the cooling process. Using this approach, we discuss differen t aspects of the cooling of a single ion in a Penning trap, including the f ormation of ''trapped'' states where the steady-state motion amplitude is n onzero for either the magnetron or modified cyclotron motion. We use an ana lytical approach where the approximation of small-amplitude motions can be made, and we use numerical calculations for larger-amplitude motions. The h st procedure to use for effective laser cooling is derived. A similar appro ach 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 ho w these insights may be applied in experimental investigations. [S1050-2947 (99)02406-3].