Controlling the motion of an atom in an optical cavity

24 3.1. Model
operators vanish. This can be seen as follows: The reservoir consists of modes which can
be described by a harmonical oscillator model. In the rotating wave approximation, the
noise operators are a linear combination of annihilation operators of the reservoir’s field
modes (MS99). At room temperature, the modes associated with visible frequencies are
in their ground state. As the annihilation operators yield zero when they are applied to
a ground state, the noise operators also yield zero. The temperature of the reservoir is
assumed to be zero throughout this thesis, which is a very good approximation in the
optical regime.
Finding an exact solution of these equationsisbynomeanseasy. Therefore,asan
approximation, the Hilbert space of each two-level atom is replaced by the Hilbert space
of a harmonic oscillator. In the following, it will be investigated in which cases this
approximation yields correct results.
First, the approximation will certainly fail if the population of the higher excited states
of the atomic harmonic oscillator is not small. In this case, as those states do not exist in
reality, there would be a large difference between the quantum state of the approximate
system and the one of the real system. For a small enough pump parameter η, however,
this will not be the case. Therefore, the approximation is valid only for small saturation
of the atoms, which can be achieved by a small enough pump strength η.
Second, even for weak excitation, where only leading order terms in η are important,
the coupled-oscillator model can introduce significant errors when calculating higher-order
expectation values. For example, it is clear from our ansatz that the exact result σ − j σ − j =0
will not be reproduced in the simplified theory. Also, one cannot hope to find correct
solutions for the second-order photon correlation function, which involves the operator
product a † a † aa. The reason is that here the leading order in η comes from the population
of the second excited state of the cavity field. In the harmonic oscillator model, this
two-photon state couples indirectly to the second excited state of the atomic harmonic
oscillator, see Fig. 3.2. The strength of this coupling is given by the gij and thus does
not vanish with vanishing pump strength η. As the second excited state of the atom
does not exist in the exact theory, the amplitude of the two-photon state will also deviate
from the correct result, even in the limit of small saturation. However, only expectation
values where the leading-order term in η depends on the amplitudes of the first excited
states of the atomic and cavity harmonic oscillators, respectively, will be calculated in
this chapter. These expectation values will be correct in leading order of η. Therefore,
the exact equation of motion (3.4) can be approximated by
˙σ − j = i∆a,jσ − Nc
j − i gij
i=1
∗ ai − γj σ − j + Fatom,j. (3.5)
To check in which regime this approximation is good, the steady state of the system was
calculated numerically for a single atom inside the cavity, including the saturation effects.
These calculations indicate that the harmonic-oscillator approximation leads to results
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