Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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1.6. Standing <strong>of</strong> this work<br />
are restricted to a particular parameter range. There are several works<br />
dealing with <strong>Bogoliubov</strong> excitations and disorder, using methods that are<br />
restricted to one dimension, like the transfer-matrix approach [79] or the<br />
phase formalism [33, 79]. Often, the disorder is approximated by an uncorrelated<br />
white-noise disorder. In present-day experiments, however, the<br />
situation is usually different. For speckle potentials (section 3.1), the finite<br />
correlation length can in general not be neglected.<br />
One <strong>of</strong> the central quantities <strong>of</strong> interest is the speed <strong>of</strong> sound in disordered<br />
<strong>Bose</strong> gases. I.e. the dispersion relation at low energies, entering the Landau<br />
criterion <strong>of</strong> superfluidity. The question how the speed <strong>of</strong> sound is influenced<br />
by disorder has been investigated in different parameter regimes and<br />
dimensions with different methods leading to different predictions. Using<br />
perturbation theory, Giorgini et al. [78] find a positive correction for uncorrelated<br />
disorder in three dimensions, which has been reproduced by Lopatin<br />
et al. [75] and Falco et al. [80]. Within a self-consistent non-perturbative<br />
approach, Yukalov and Graham [81, 82] report a decrease <strong>of</strong> the sound velocity<br />
in three dimensions, even in the case <strong>of</strong> δ-correlated disorder, which is<br />
in clear contradiction to [78]. For disordered hard-core bosons on a lattice,<br />
Zhang [83] finds a decrease <strong>of</strong> c to fourth order in disorder strength, without<br />
information on the second-order effect.<br />
Thus, the knowledge <strong>of</strong> the speed <strong>of</strong> sound in disordered systems is far<br />
from comprehensive. A major goal <strong>of</strong> this work is to provide a formalism<br />
for describing the excitations <strong>of</strong> disordered BEC, that covers a range <strong>of</strong><br />
parameters as wide as possible. In particular, different dimensions and<br />
arbitrary types <strong>of</strong> disorder should be covered.<br />
The disordered <strong>Bogoliubov</strong> problem is not expected to be simple. Concerning<br />
the spectrum <strong>of</strong> the non-uniform <strong>Bose</strong> gas, Nozières and Pines write<br />
in their book Theory <strong>of</strong> Quantum Liquids [84, chapter 10]:<br />
In practice, one faces enormous mathematical difficulties, except<br />
in the case <strong>of</strong> the ground state, for which Φ(r) is constant<br />
. . . The coupled equations (. . . ) [equation (2.65) in this work],<br />
though certainly complex in character, are rich in physical content.<br />
It may be expected that detailed study <strong>of</strong> these and similar<br />
equations will yield much new information concerning the nonuniform<br />
superfluid <strong>Bose</strong> liquid.<br />
In this work, this very problem is tackled, in the case where the condensate<br />
is non-uniform due to a disorder potential.<br />
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