Ph.D. Thesis - Physics

increase the dephasing time, using linear superpositions of qubit states that are insensitive
to magnetic field fluctuations [HSKH + 05].
Equally important is the uncontrolled heating of the motional state of the trapped ions,
leading to motional state decoherence. In ion trap quantum computing, the motional states
are important since they actually encode quantum information; in fact, the classic way of
doing a quantum logic gate is to map the state of one ion to the shared motional mode,
then perform an operation on the second ion that changes its internal state only if a given
motional mode is excited. This is the basic principle of the Cirac-Zoller gate [CZ95], which
has been used with great success by the Innsbruck group (e.g. in Ref. [GRL + 03]). Clearly,
unwanted excitations of the motional state can ruin this process.
Studies have shown that the heating rate in an ion trap scales roughly as h −4 , where
h is the distance from the ion to the nearest trap electrode, and is strongly dependent on
temperature. The process is often called anomalous heating, since it scales more strongly
with h than Johnson noise (which would be 1/h 2 ) and is of uncertain origin [TKK + 99]. The
leading hypothesis for the cause of this heating is fluctuating patch potentials on the trap
electrodes. However, as stated before, it is now known that this heating can be reduced by
orders of magnitude by cryogenic cooling of the trap electrodes [LGA + 08]. We will make
use of this approach in Ch. 7. Further work from this experiment has narrowed down the
possible causes of this noise by studying its temperature dependence [LGL + 08]. However,
this phenomenon is still not perfectly explained.
Although there are some schemes for quantum simulation that do not require ground
state cooling (e.g. the Porras/ Cirac scheme below), it is nevertheless important that the
heating rate is minimized. Extra quanta of motion incoherently gained during the simulation
decrease the accuracy, and excessive heating could even push the ions out of the Lamb-Dicke
regime or cause the ions to transition from a crystal to a cloud state.
4.2 Quantum simulation of quantum spin models
Many proposals have been published for performing quantum simulation with trapped ions.
Milburn proposed the simulation of nonlinear spin models using trapped ions [Mil99]. A
paper from Porras and Cirac explains how to simulate Bosonic physics such as Bose-Einstein
condensation using the vibrational degrees of freedom in a crystal of trapped ions [PC04a].
The group of Wineland, in 2002, proposed and implemented the simulation of a nonlinear
interferometer using a single trapped ion [LDM + 02]. New and exotic ideas such as the
simulation of quantum fields in an expanding universe [ADM05] have also emerged. Al-
though many of these proposals are quite compelling, in this thesis we choose to focus on
the physics of quantum spin model simulation.
Spin physics in 2-D is most interesting because of the phenomenon of spin frustration.
On certain types of lattices, spins that have an antiferromagnetic interaction (meaning that
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