If damping is applied to N ions in the Mathieu regime, the loss of kinetic energy induces a loss of potential energy, which in turn decreases ion displacements from the trap center, bringing the ions closer to each other and rendering
Coulomb interactions non-negligible. Heating is observed in Paul-trapped ions when the root-mean squared radius, 
, where 
is the 
ion's displacement from the trap center, of the ion cloud is small enough that the nonlinearities of the Coulomb interaction significantly affect the ions' motions. In this heating regime, the ions' trajectories are characterized by
deterministic chaos
rather than by the quasiperiodicity of the Mathieu regime's virtually uncoupled trajectories. The nonlinear Coulomb interaction in the presence of
an oscillating potential leads to deterministic chaos and renders possible the absorption of kinetic
energy from the oscillating electric field.[24,36] Blümel
demonstrates the origin of ion heating by numerical calculation of the work
done by an oscillating field acting on Coulomb point charges.[4]
Figure: Blümel's four dynamical regimes for a collection of Paul-trapped
particles--- ( 1): in the high-energy Mathieu regime, interparticle
spacing is large, trajectories are virtually uncorrelated, and no heating occurs; ( 2a): in region C
of the chaotic regime, non-negligible Coulomb interactions give rise to heating, and higher-density configurations (smaller 
) correspond to higher heating rates; ( 2b): in region C
of the chaotic regime, higher-density configurations correspond to slower heating rates; ( 3): the quasiperiodic regime is characterized by quasiperiodic trajectories and the absence of heating; ( 4): the crystalline phase.
Figure: Running averages of energy and 
for a melted 15-ion
crystal which evolves through the heating regime to the Mathieu regime.
Note the gain in kinetic energy and increase in 
for the
heating phase. When the Mathieu regime is reached, 
and <KE>
oscillate about stable average values. source file: 15melt.trp.
The chaotic heating regime has been delineated by Blümel into two distinct regions (Fig. 
). Let 
refer to the root-mean square cloud radius at the
peak of the heating curve and 
to the root-mean square ion cloud radius
which marks the transition between chaos and the Mathieu regime. The two domains of the chaotic regime are defined by
: (1) region 
, in which 
; (2) region 
, bounded by 
. In the region 
, the slope of the heating curve is negative, indicating a faster rate of energy gain for
smaller 
. If one removes energy from an ion cloud in region 
at a rate of 
, the ions move closer to the center of the trap in accordance with lower potential energy. Assuming that the rate of energy
removal is less than the rate of energy gain at 
, the ion cloud condenses toward the center of the trap--- and therefore toward a region of faster energy gain--- until the rate of energy gain via rf heating equals the rate of energy removal via damping.
In the absence of an absolutely perturbation-free system, the positive slope of
the heating curve in region 
negates the possibility of attaining a steady-state 
within the region 
. Suppose the root-mean square radius of the ion cloud, 
, corresponds to a damping of 
. The slightest perturbation, 
, in a
compensating damping rate of 
, will either decrease the ion radius and put the cloud in a region of slower heating or increase the radius and put the cloud in a region of faster heating. In the event that 
is decreased, the slower rate of heating will not be sufficient to compensate the damping of 
and the ion cloud will continue to condense until Coulomb repulsion forces it into a crystalline state. If 
is increased, the damping of 
will be unable to offset the faster rate of heat gain, and the root-mean square radius will continue to increase
until a stable equilibrium is established in region 
or in the nonheating Mathieu regime.
File cloud9A.trp depicts nine damped Mg
ions in the chaotic heating regime. Observe
the lack of definition in Fourier transforms of axial and radial positions.
The system is no longer adequately described by the linear Mathieu equation, which would yield motions composed of discrete frequencies. Also observe the diffusive gain of energy and increase in 
which occurs when damping is turned off. These phenomena are
enabled by the nonlinearities of Coulomb interactions, which are no longer negligible.
Figure: Fourier transforms (2048 data points) of x-position
( left) and
( right) axial position of one of 5 Paul-trapped ions in a non-heating cloud configuration (Blümel's ``quasiperiodic'' regime) for region A of the Mathieu stability.
m. source file:
cloud5A.trp.