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