Ph.D. Thesis - Physics

angular momentum along the quantization axis, which is determined by the orientation of
the external magnetic field.
We now describe the interaction of the trapped ion with laser radiation, which is essential
for understanding how quantum operations may be performed on trapped ions, as well as
other techniques such as Doppler cooling and state measurement. We focus on electric
dipole transitions, which occur between atomic levels that differ by one unit of orbital
angular momentum. For simplicity, we will limit the discussion to the coupling of two
discrete levels. The interaction is that of the dipole moment of the electron interacting with
the oscillating laser field. The interaction is written as
Hdip = − d · E = −
d ·ǫE0 cos −ωlt +
k · z + φ , (4.14)
where d = ecz is the dipole operator, z is the displacement vector of the ion from its
equilibrium position, ǫ is the polarization of the laser, E0 is the amplitude of the laser’s
electric field, ωl is the frequency of the laser radiation, k is the wavevector of the radiation
field, and φ is a phase that may be chosen by the experimenter. We will assume that the
wavevector of the light is in the ˆz direction.
We note that the dipole operator is proportional to σ + + σ − , where σ + = |↑〉 〈↓| and
σ − = |↓〉 〈↑| are the raising and lowering operators for the atomic state, respectively. Making
use of the relation of the creation and annihilation operators to the operators x and p, and
making a rotating wave approximation, the above Eq. 4.14 takes the form
HI = Ω e iη(a+a†
) + −i(ωlt+φ)
σ e + h.c. , (4.15)
where h.c. denotes Hermitian conjugate. The Rabi frequency Ω is the rate at which the
atomic state |↑〉 is flipped to |↓〉 and vice-versa. For an electric dipole transition, Ω =
|ecE0 〈↓|z |↑〉 |, that is, it depends on the applied electric field and on the expectation value
of the dipole operator ecz. The Lamb-Dicke parameter η is equal to kz0, where k is the
norm of the wavenumber of the radiation field and z0 is the width of the ground state wave
function along ˆz: z0 = . A small η (η ≪ 1) means that the ion in its ground state
2mωˆz
is well-localized in space to within the wavelength of the laser light, which implies that the
light field will couple well to the harmonic oscillator mode in that direction. This condition
is called the Lamb-Dicke limit, and when it is satisfied, the interaction Hamiltonian looks
like
HI = Ωe iφ
σ +
−i(ωl−ω0)t
e 1 + iη ae −iωˆzt † iωˆzt
+ a e
+ h.c., (4.16)
where ωˆz is the ion’s secular frequency in the ˆz direction.
Eq. 4.16 actually reveals all the physics of how manipulations may be made on both the
internal and external states of a single trapped ion using laser radiation. We focus first on
the case in which the laser detuning δl, defined as δl ≡ ωl − ωˆz, is zero. We also neglect
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