Ph.D. Thesis - Physics

coupling to the motional states by setting η = 0. The Hamiltonian that acts on the internal
state of the ion is
and the unitary transformation done on the ion-qubit is
Hint = Ω σ + e −iφ + σ − e iφ
, (4.17)
U(t) = exp(−iHintt/) . (4.18)
The rotations Rˆx(θ) and Rˆy(θ) may be performed by turning on this Hamiltonian for
an appropriate period of time and setting the phase φ correctly. A rotation along ˆz may be
composed of ˆx and ˆy rotations (similar to a method discussed in Sec. 2.1).
We have thus shown that the interaction of a single ion with a laser may be used
to perform single-qubit rotations; however, two-qubit operations are also required for the
quantum simulations of interest to us. A number of references ([CZ95, MS98, LDM + 03])
detail methods for performing two-qubit gates that are sufficient for universal quantum
computation. We do not go into detail on these now, since the quantum simulation schemes
we study in Sec. 4.2 do not directly make use of these methods.
4.1.3 Control of the ion’s external state
We now turn to laser manipulation of the external, i.e. motional states of trapped ions.
The interaction Hamiltonian (Eq. 4.16) already contains this physics. It can be shown that
by detuning the laser above or below the transition by an amount equal to the motional
frequency ωˆz, a quantum of vibrational energy may be added to or subtracted from the ion.
That is, for δl = ωˆz, ∆n = 1, and for δl = −ωˆz, ∆n = −1. These transitions are called the
blue sideband (∆n = 1) and red sideband (∆n = −1). Repeated cycles of the latter process
result in sideband cooling, which has been demonstrated by several ion trap groups, e.g. in
Ref. [DBIW89].
Sideband cooling requires that the linewidth of the laser Γl is much less than ωˆz, for
the somewhat that for Γl > ωˆz the ∆n = 1 and ∆n = −1 transitions are simultaneously
addressed. The other case, Γl > ωˆz, can still be very useful for laser cooling, as we will
show below. This process is called Doppler cooling. It was first observed in 1978 [WDW78],
and formed the basis of the Nobel prize-winning research in laser cooling of neutral atomic
ensembles. Because of the prevalence of Doppler cooling over sideband cooling in this thesis,
we will focus on that from this point on. However, it is important to know that ground
state cooling is possible, because many quantum information protocols rely on it to initialize
the system.
We will treat the problem of a single two-level atom interacting with a monochromatic
radiation field (laser) that is detuned by δl from the atomic transition frequency ω0, as be-
fore. The ion is also characterized by “natural” linewidth Γ, which is the rate of spontaneous
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