Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
The quadratic terms have been absorbed as phase factors in
˜E(x) = e −i k L
2L x 2
e −ikLL E(r) and ˜η(x ′ ) = e i k L
2L x ′2
η(x ′ ). (3.5)
The mixed term in (3.4) connects ˜ E(x) and ˜η(x ′ ) as a Fourier transform.
Up to a complex phase, the field (3.1) is identified as the Fourier transform
of the random field ˜η(x ′ ), evaluated at k = kLx/L. Accordingly, the Fourier
components of the field ˜ Ek are given by the random field ˜η(x ′ ) evaluated at
x ′ = −Lk/kL. The Fourier components inherit all statistical properties
from the surface of the diffusor:
˜E ∗ k ˜ E k ′ ∝ ˜η ∗ (−Lk/kL)˜η(−Lk ′ /kL) ∝ δ(k − k ′ )Θ(σ −1 − |k|), (3.6)
with the disorder correlation length σ = L/(RkL). As only the intensity
of the electric fields is relevant for the light-shift potential [see below], we
identify E with ˜ E in the following. In terms of the average intensity in the
plane of observation I0 = |E(x)| 2 , the field correlator reads
E ∗ k E k ′ = (2πσ)d
Vol(d) I0 Θ(1 − kσ) (2π) d δ(k − k ′ ) =: γ(k)(2π) d δ(k − k ′ ), (3.7)
which defines the so-called complex degree of coherence γ(k) [23]. Here,
Vol(d) is the volume of the d-dimensional unit sphere. All modes with
k < 1/σ = RkL/L are statistically independent with a common Gaussian
probability distribution.
3.1.2. Generalization to 3D
In the preceding derivation of the speckle correlations, a two-dimensional or
one-dimensional diffusor plate and a corresponding plane of observation was
considered (figure 3.2). The third dimension was needed for the distance L
between both. How can this setup be generalized to three dimensions? Also
in the third dimension, the speckle field has a certain grain size, but this
axial correlation length is typically an order of magnitude longer [66, 67].
Such a 3D speckle field with an anisotropic correlation function has been
used e.g. in an experiment on the 3D Bose-Hubbard model [117]. There, the
transverse correlation length was shorter than the lattice spacing, and the
axial correlation length was of the order of the lattice spacing. In a lattice,
variations on a scale shorter than the lattice spacing do not matter, so the
disorder was effectively uncorrelated in all three dimensions, fulfilling the
demands of the disordered Bose-Hubbard problem.
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