Ph.D. Thesis - Physics

4.2.1 The Ising and Heisenberg models
We review here the basic physics and of the Ising and Heisenberg models in preparation
for seeing how they could be simulated with trapped ions. These models are Hamiltonians
that describe the physics of interacting spins. Although there is a classical Ising model, we
are concerned here only with the quantum-mechanical version.
The Ising model has the following general form:
HIsing = −
i=j
Jijσiσj
(4.26)
where the summation is, as noted, over all spins i and j for which i = j, the Jij are the
coupling energies between spins, and σi and σj are the Pauli matrices for spins i and j.
Most often, we shall choose ˆz to be the quantization axis, and focus on nearest-neighbor
interactions, in which case the Ising Hamiltonian takes the form
HIsing = −
i,j=i+1
Ji,jZiZj. (4.27)
The preceding Hamiltonian is one-dimensional, in that it models spins that are arranged
along a line segment. Other configurations are possible, and Eq. 4.27 can readily be gener-
alized to two and three dimensions.
The Heisenberg model is like the Ising model, but involving spin-spin interactions, gen-
erally, along more than one direction. For spins arranged in one dimension, the Heisenberg
model has this form:
HHeis = −
i,j
(JˆxXiXj + JˆyYiYj + JˆzZiZj). (4.28)
4.2.2 Porras and Cirac’s proposal for simulating quantum spin models
We begin with showing how a chain of ions in a linear Paul trap can be used to simulate
spin models. In this section we will follow the paper of Porras and Cirac [PC04b] quite
closely. Although a linear ion trap was used as a model in their work, the scheme extends
quite easily to 2-D, a fact that motivates the rest of our work in this part of the thesis.
We consider a chain of N trapped ions aligned along the ˆz direction. We define the
index ˆα to indicate spatial direction; ˆα = 1, 2, and 3 represents ˆx, ˆy, and ˆz, respectively
1 . Accordingly, the Pauli operators are written as σˆα and the corresponding eigenstates as
|↑〉 ˆα and |↓〉 ˆα . We assume that lasers can be applied to the ions that couple the internal
state to the motional state only if the ion is in a specific internal state. This state-dependent
force is the key ingredient of implementing the simulation. How can one cause a laser force
1 Here, as throughout the thesis, we specify the directionality of a given quantity with a subscripted unit
vector, even if only the magnitude (a scalar) is represented by the quantity. We hope that this will clarify
symbols with multiple subscripts.
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