Background: For a BEC close to its ground state (at temperature T = 0 K), its excitations are well described by small perturbations around the stationary state of the BEC. The energy of an excitation is then given by the Bogoliubov dispersion relation (derived previously in Mapleprimes "Quantum Mechanics using computer algebra II").
where G is the atom-atom interaction constant, n is the density of particles, m is the mass of the condensed particles, k is the wave-vector of the excitations and
their pulsation (
time the frequency). Typically, there are two possible types of excitations, depending on the wave-vector
:
Problem: An impurity of mass
moves with velocity
within such a condensate and creates an excitation with wave-vector
. After the interaction process, the impurity is scattered with velocity
.
a) Departing from Bogoliubov's dispersion relation, plus energy and momentum conservation, show that, in order to create an excitation, the impurity must move with an initial velocity
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When , no excitation can be created and the impurity moves through the medium without dissipation, as if the viscosity is 0, characterizing a superfluid. This is the Landau criterion for superfluidity.
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b) Show that when the atom-atom interaction constant
(repulsive interactions), this value
is equal to the group velocity of the excitation (speed of sound in a condensate).