Ph.D. Thesis - Physics

where Q is the charge of the trapped particle (equal to ec for the atomic ions used in this
thesis) and m is its mass. We will use this formula in all our calculations of ion trap
potentials, and find that it agrees quite well with our results. Dissecting this formula a bit,
we see that Ψ is proportional to ecQ/r 2 0 ; the 1/r2 0
dependence comes from the gradient of the
potential squared. This parameter depends on the trap geometry, and in most of our traps
is computed numerically. For ion traps that admit approximate analytical solutions, r0 has
a clear physical significance; for example, in the case of the ring trap, it is the distance from
the trap center to the nearest point on the rf ring electrode.
For the quadrupole potential above, close to the its minimum, the pseudopotential has
the form of a three-dimensional harmonic oscillator:
where the ωi are the secular frequencies.
Ψ(x, y, z) = 1
m
2
i
ω 2 i u 2 i , (4.9)
The classical solutions for a particle in a harmonic oscillator will not be repeated here.
We will briefly present the quantum mechanical solution, since this is essential to the theory
of quantum simulation with ion traps. The Hamiltonian of the oscillator in a given direction
with frequency ω is
H = ω a † a + 1
, (4.10)
2
where [a, a † ] = 1 (see Ref. [Sak85] for further details). The operators a and a † are called the
annihilation and creation operators, respectively. Their effect on the state |n〉 is to remove
or add one quantum of vibrational energy to the state |n〉:
a |n〉 = √ n |n − 1〉 ; a † |n〉 = √ n + 1 |n + 1〉 . (4.11)
The eigenbasis is the set of states {|n〉} that satisfy, in addition to the above,
H |n〉 = ω n + 1
|n〉 . (4.12)
2
The ground state wave function, for which a |0〉 = 0, is written in coordinate space as
where x0 =
2mω
〈0|x〉 =
x0
π e−
“ ” 2
x
2x0 , (4.13)
is the width of the ground state wavefunction.
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