Ph.D. Thesis - Physics

α = 2k 2 4Iδ
I0Γ 1 + (2δ/Γ) 2.
(4.23)
Although it may at first seem that cooling to zero velocity is possible, it is important
to remember that the absorption events are random and themselves add some amount of
entropy to the atom’s state. The atom thus executes a random walk in momentum space.
The rate of heating due to this process is
˙Eheat
= ˙ 〈p〉 2
2m = 22k2Γp↓ m
= D/m, (4.24)
where ˙
〈p〉 is the rate of change of the average value of the ion’s momentum, m is the mass
of the ion, and
the constant
D,
defined
in Eq. 4.24, is the momentum diffusion constant.
Equating ˙Eheat with − ˙Ecool , and noting that the temperature of the ion is related
to the mean kinetic energy by m v 2 /2 = kBT/2, the ultimate temperature attainable by
Doppler cooling is
TDopp = Γ
, (4.25)
2kB
which is known as the Doppler cooling limit. A simple numerical estimate for a 88 Sr + ion
in a trap of frequency 1 MHz shows that this limit is about 10 motional quanta. At this
point, sideband cooling can take the ion to the motional ground state, if desired.
In practice, the heating rate due to momentum diffusion is often not the limiting factor
in ion trap Doppler cooling. Heating of the ions due to electric potentials plays a strong
role. In the cloud state, the micromotion of the ions couples into the secular motion, and
the rf voltages can directly cause heating of the ions. In a crystal state, this does not occur;
the micromotion and secular motion are largely decoupled. Even though micromotion
does not directly create heating if the ions are in a crystalline state, the line broadening
due to it can raise the ultimate Doppler temperature attainable. This has been treated in
Ref. [CGB + 94]. However, even in a compensated trap, heating due to fluctuating potentials
on the trap electrodes can exceed that due to the spontaneous emission events. This heating
process is discussed in Sec. 4.1.5.
4.1.4 State preparation and measurement
Methods for preparing the qubits in some initial state and measuring their state are essential
to every quantum information protocol, and are briefly presented here.
The internal state of the ion is prepared not only in the electronic ground state, but
also typically in a specific magnetic sublevel of it. A controlled bias usually field breaks the
degeneracy of the ground state; this is done to avoid pumping to “dark states,” superposi-
tions of S and D states that do not fluoresce. A beam of circularly polarized radiation with
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