Ph.D. Thesis - Physics

to address an ion in state |↑〉 ˆα but not in state |↓〉 ˆα ? One way is to set the polarization
of this laser such that only one state in a ground state hyperfine or Zeeman manifold is
permitted by electric dipole selection rules to couple to a given excited state. Another is
to make the laser much closer to resonance with one state than the other, for instance by
using an optical qubit. Note that if the laser pushes on both |↑〉 ˆα and |↓〉 ˆα , but in opposite
directions, the derivation will change but the same basic physics will be implemented.
The first step is to write down the Hamiltonian. The full Hamiltonian has three parts:
Hv, the vibrational Hamiltonian, Hf, the interaction term due to the state-dependent force,
and Hm, a term due to an externally-applied magnetic field that is a key part of the
transverse Ising model, and is important for the observation of quantum phase transitions.
Hv is written, including the Coulomb repulsion between the ions, as a set of harmonic
oscillators at the normal mode frequencies labeled by n:
Hv =
ωna † nan. (4.29)
Here the magnitudes of the ωn are determined by the Coulomb interaction.
form
n
The state-dependent force term, which is assumed to only act upon state |↑〉 ˆα , has the
Hf = −2
ˆα,i
Fˆαqˆα,i| ↑〉〈↑ |ˆα,i, (4.30)
where qˆα,i is the position operator for the i th qubit along direction ˆα, and Fˆα is the mag-
nitude of the state-dependent force along ˆα. An intuitive picture of the effect of the state-
dependent force is depicted in Fig. 4-4.
Finally, the magnetic term is written as
Hm =
µˆαBˆασˆα,i , (4.31)
ˆα,i
where Bˆα and µˆα are the magnetic field and atomic magnetic moment, respectively, along
the direction ˆα. The full Hamiltonian H is given by H = Hv + Hf + Hm.
Porras and Cirac derive the spin-spin interaction by means of a canonical transformation:
H ′ = e −S He S , where
and
S =
α,i,n
ηˆα,i,n
a †
ˆα,i − aˆα,i (1 + σˆα,i) (4.32)
Mˆα,i,n
ηˆα,i,n = Fˆα
ωˆα,n
. (4.33)
The Mˆα,i,n are the elements of the matrix that diagonalizes the vibrational Hamiltonian,
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