Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.1. Optical speckle potential
In continuous systems, however, there is no spacing, and the actual correlation
length is important, so similar correlation lengths in all directions
are desirable. This could be achieved by the superposition of several speckle
fields, where at least a second speckle field for the third dimension is needed.
This would achieve similar correlation lengths in all directions, but with an
anisotropic correlation function. Ideally, the speckle pattern should be obtained
in a closed cavity [23, 24], which restores the isotropy. In this case,
the complex degree of coherence reads γ(k) = 2π 2 σ 2 δ(|k|−kL), which results
in a k-space correlator C3(kσ) = (8πkσ) −1 Θ(2 − |kσ|) with a divergence at
k = 0.
Lacking a simple experimentally realized model, we prefer to follow Pilati
et al. [118] and define the three-dimensional speckle disorder from a more
abstract point of view. Independently of a possible experimental realization,
we declare Eq. (3.7) the definition of the speckle field also in dimension three.
This preliminary isotropic three-dimensional speckle field grasps the important
features of laser speckles: the asymmetric intensity distribution (3.10)
and the finite support of the correlator in k-space (3.12), see below. Like the
two-dimensional speckle it might have to be adjusted to the experimental
details of future experimental setups, in particular to anisotropies.
3.1.3. Intensity and potential correlations
In typical experimental setups, the laser frequency ωL is chosen to be close to
an internal resonance ω0 of the atom such that ωL/ω0 ≈ 1, but far-detuned
in the sense that the detuning ∆ = ωL −ω0 is larger than the line width Γ of
the transition or the inverse lifetime of the excited state. Then, the energy
levels of the atoms are shifted due to the interaction of their induced dipole
moment with the electric laser field [39]. This shift is proportional to the
laser intensity I(r) = |E(r)| 2 , where the sign and the magnitude depend on
the detuning
∆E(r) = 3πc2 L
2ω 3 0
Γ
∆ 2ɛ0cL I(r). (3.8)
In this formula, ɛ0 is the vacuum permittivity and cL is the speed of light.
The origin of energy is shifted such that the potential has zero mean
V (r) = V0(I(r) − I0)/I0. (3.9)
Here, the signed disorder strength is defined as V0 = 3πc 3 L ɛ0ΓI0/(ω 3 0∆).
The magnitude of the parameter V0 is the rms value of V (r), V 2
0 = V (r) 2 .
Its sign is determined by the detuning of the laser frequency and states
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