Ph.D. Thesis - Physics

selection rule ∆m = ±1 is applied to the ions until, with high probability, they are in the
desired state. This optical pumping technique was first demonstrated by Kastler (although
not in an ion trap) [Kas50], and won him a Nobel prize.
Measurement is one of the strongest advantages of ion traps, in that measurement
fidelities well over 90% have been demonstrated [MSW + 08]. The basic principle is quite
simple: the same laser used for Doppler cooling is also used for state measurement. If the
ion is in state S, then many photons will be scattered; the rate, as argued above, is Γp↓.
This is millions of photons per second, and even given a finite light collection angle and
inefficient detectors, the scattering rate tremendous: to take one example, from a single ion
in the Innsbruck experiment (Part III), it was not uncommon to observe 30,000 photons
per second in our photomultiplier tube. By contrast, if the ion is in state D, no photons
will be scattered.
In the proposals for quantum simulation studied in this thesis, it is not necessary to
measure the motional state of the ion, nor even to cool it to the ground state, but this
could in principle be done in the following way. First assume the ion is in internal state |↑〉.
Suppose you want to determine if the motional state in a given direction is |0〉 or |1〉. A
pulse on the red sideband will take the state |↑ 1〉 to |↓ 0〉 and subsequent measurement of
the internal state will reveal the state is indeed |↓〉. However, if the state is in |↑ 0〉, there
literally is no red sideband transition. The ion will remain in |↑ 0〉, and the fluorescence
signal will determine that the motional state was |0〉.
To summarize, we have assembled all the necessary ingredients for quantum simulation
with trapped ions, with the exception of two-qubit operations. Following a brief discussion
of decoherence in ion traps, we present, in Sec. 4.2, an interesting method for producing
two-body interactions between trapped ions.
4.1.5 Decoherence
Decoherence afflicts all quantum simulators, including a set of trapped ions being used
as such. It includes amplitude damping and dephasing of both the internal and motional
states of the trapped ions. The choice of qubit states is important. Qubits that consist
of a ground state and an excited state separated by a quadrupole transition have as their
fundamental limitation the spontaneous emission rate of the excited state, which is on
the order of a few Hz. However, classical control errors such as fluctuations in the laser
intensity and frequency currently limit the amplitude damping rate in many experiments.
By contrast, hyperfine qubits can exhibit coherence times on the order of seconds. They
are subject, however, to errors due to spontaneous scattering from the excited electronic
state used to implement Raman transitions between the levels. Dephasing due to static
or slowly-varying sources, such as stray magnetic fields, can be compensated for by using
spin-echo techniques, but there always remains some dephasing that cannot be removed. A
technique known as the decoherence-free subspace can be used, however, to dramatically
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