Molecular Dynamics Simulation The method of molecular dynamics simulations (MD) is one of the routine
tools in the theoretical study of a manybody system. This computational
method gives direct numerical solutions of the equations of motion of particles
and calculates their time dependent behaviors, which allow us to check the
models, offer insights to the experimentalists, assist in the interpretation
of new results, predict the properties of the dynamics and other complicated
phenomena that cannot be found out in other ways. At the same time, they
reveal hidden details behind the experimental measurements. It acts as a
bridge between microscopic length and time scales and the macroscopic world
of the laboratory, and between theory and experiment. Equations of motion Molecular dynamics simulations are based
on solving the classical equations of motion, which for trapped ions in a
linear Paul trap may be written as 



where m_{i} and r_{i} are
mass and position vector of the ith
particle, respectively. 



is the total
force acting on the ith particle, which may
depend on the positions and velocities of all other particles, and the time t.
There are several contributions to this force. 

. 

The first term F_{i}^{trap} is
the force from the trapping potential and can be written as 

. 

For the rf
electric field it can be weell approximated as a
time averaged harmonic pseudopotential, The motion of the particle is
separated into slow and fast parts (called macromotion and micromotion,
respectively). The
second term F_{i}^{Coulomb} is
due to the Coulomb repulsive force between each pair of particle in the trap. The stochastic force F_{i}^{stochastic}
results from the random interactions of particles with surroundings, such as
collisions with residual gas, the scattering light, electric field noise, and
so forth. In simulation all the factors are modelled in a
simple way by velocity “kicks” applied to each particle k of species i and at each integration step. The last term F_{i}^{laser} acts
only on laser cooled particles. This force can be simplified to be a constant
light pressure force independent of the ion velocity plus a linear viscous
damping force F_{L}=bdx/dt with friction coefficients in the
range b=(1.28)*10^{22} kg/s.
Ion number determination: In the simulation, the ion number can be
adjusted arbitrarily, and the parameters of external electric fields can be
modified near the experimental values conveniently, so we see different shape
and structure of the laser cooled ion ensemble in realtime, and select the
best set to fit the CCD image of experiment. Since the ion number of laser
cooled ions is determined, the counting coefficient of a channeltron
ion detector in the experiment is calibrated. Now the ion number of
sympathetically cooled complex molecular ions can also be determined. Fig. 1 is an example: Through the
comparison of simulation and experiment we know there are 27 barium ions and
3 isotopes. While it is easy to count the number of barium ions in the case,
it is impossible to count them in Coulomb crystal consisting of several
hundred or thousand ions, there you rely on the simulations.. 

experiment simulation Fig. 1.
Compare the simulation to the experiment to determine the number of ions, 

Temperature determination: Another important feature of the
simulations is the determination of the final temperatures of different ion
ensembles. Therefore, we simulate the pure laser cooled ion ensemble with
different temperatures, and compare them to the CCD image to know which
temperature the experiment is close to. Fig. 2 shows a simulated pure
barium ion ensemble (300 ions) with different temperatures. Second, we keep
the parameters set in the first step such as cooling and heating rate of the
barium ions. Then we add the sympathetically cooled complex molecular ions,
which are hotter and will heat the barium ion ensemble. By adjusting the heating
rate of the complex molecular ions we can get the barium ion ensemble images
with different final temperatures. Finally, we compare the simulated images
with the CCD image of barium ion ensemble (with the complex molecular ions in
the trap simultaneously) and get the right heating rate for the complex
molecular ions. Since we know all the parameters, the final temperature of
the complex molecular ion ensemble is determined. Fig. 3 is a good example. 

4mk 

8mk 

20mk 

50mk 

100mk Fig. 2 Simulated ion ensemble with
different temperatures 


CCD image of a pure barium ion crystal,
through this we find the heating rate of barium ions 


CCD image of barium ion ensemble with AF350
ions simultaneously, through this we find the heating rate for the complex
molecular ions. 

Fig. 3. Simulation of a threespecies Coulomb
crystal containing 830 lasercooled ^{138}Ba^{+} ions (blue)
at a temperature of 25 mK, 420 sympathetically cooled
ions of barium isotopes (red) and 200 protonated
AlexaFluor350 molecules (AF350^{+}, green) at 120 mK 


