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2.1.1 The dressed state model
Chapter 2: Theoretical Background
To describe the situation of an atom in a light field correctly, one needs to
treat the atom, the light field, and their interaction quantum mechanically.
This leads to the so-called dressed states model [Dal85], which will be used
later (section 2.1.4) to explain how atoms can be trapped with light. In the
following, that derivation will be outlined.
If the light is monochromatic, in the sense that the width of the frequency
distribution of the light is small compared to the energy difference between
atomic levels, the atomic polarisability can be calculated from an ansatz that
treats the atom as a two-level quantum system. The ground state is denoted
by |g〉, the only excited state by |e〉. The Hamiltonian of this atomic system
is then
HA = ¯hω0|e〉〈e| (2.2)
where the energy of the ground state has been set to zero and the energy
difference between the levels is ¯hω0. If the light has frequency ωL = ω0 + ∆, it
can be described by the Hamiltonian
HL = ¯hωL(a † a + 1
) (2.3)
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where a † and a are the creation and annihilation operators, and ˆn = a † a is the
number operator with eigenvalue n, the number of photons in the field. It is
then said that the light is detuned by ∆ with respect to the transition.
To include the interaction of atom and light, we need a third operator
VAL = − � d · � E(�r) (2.4)
Here � d is the operator of the induced electric dipole moment of the atom and
�E(�r) is the operator of the electric field strength at position �r.
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