Ph.D. Thesis - Physics

Figure 4-4: Schematic of the action of a state-dependent force on a pair of trapped ions.
The force is created by an optical standing wave, and we assume that the frequency and
polarization of this laser is such that an ion in one state will be pushed in one direction,
and an ion in the opposite state in the opposite direction. On the left, the ions in the same
state are pushed in the same direction by the optical force, and thus the system energy is
unchanged by the force. On the right, the ions in opposite states are pushed toward one
another. This results, through the Coulomb interaction, in an overall increase in the energy
of the system. This is equivalent to a ferromagnetic interaction: it is energetically favorable
for the ions to be in the same state.
and are defined by the equation
qˆα,i =
n
Mˆα,i,n
a
2mωˆα,n/
†
ˆα,n + aˆα,n . (4.34)
It is clear that the nontransformed Hamiltonian already contains the coupling of the
internal to the vibrational modes, given the facts that Hf contains qˆα,i and that the aˆα,n
and a †
ˆα,n depend on these coordinates. The purpose of this transformation is merely to
rewrite the Hamiltonian in a form where these interactions are evident.
The transformed Hamiltonian H ′ is given by
H ′ =
ˆα,n
ωˆα,na †
ˆα,na† 1
ˆα,n +
2
Jˆα,i,jσˆα,iσˆα,j +
α,i,j
where the effective J-coupling rate is given by
− Jˆα,i,j =
n
F 2 ˆα
mω 2 ˆα,n
α,i
µˆαB ′ ˆα σˆα,i + Hr
(4.35)
Mˆα,i,nMˆα,j,n = 2
ηˆα,i,nηˆα,j,nωˆα,n , (4.36)
and an additional effective magnetic field is given by µˆαB ′ ˆα = µˆαBˆα +
n F 2 ˆα /(mω2 ˆα,n ). In
the case of the transverse Ising model, this induced field will be parallel to the direction of
J coupling and perpendicular to the applied external magnetic field.
The term Hr represents a residual coupling between the effective spins and the vibra-
tional modes. Fortunately, this can be neglected in the case of anisotropic traps, or when
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n