The Landau criterion for Superfluidity Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 (1) Laboratoire PhLAM, UMR CNRS 8523, Universit\303\251 Lille 1, F-59655, France (2) Maplesoft, Canada A Bose-Einstein Condensate (BEC) is a medium constituted by identical bosonic particles at very low temperature that all share the same quantum wave function. Let's consider an impurity of mass M, moving inside a BEC, its interaction with the condensate being weak. At some point the impurity might create an excitation of energy 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 and momentum LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EpJmhzbGFzaDtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRi4vJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== 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. We assume that this excitation is well described by Bogoliubov's equations for small perturbations LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEwJmRlbHRhOyZ2YXJwaGk7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn around the stationary solutions LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEnJiM5ODE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==JSFHof the field equations for the system. In that case, the Landau criterion for superfluidity states that if the impurity velocityLUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZORj9GPy8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRlc= is lower than a critical velocity LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJjRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= (equal to the BEC sound velocity), no excitation can be created (or destroyed) by the impurity. Otherwise, it would violate conservation of energy and momentum. So that, if LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZORj9GPy8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRlc= < LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJjRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= the impurity will move within the condensate without dissipation or momentum exchange, the condensate is superfluid (Phys. Rev. Lett. 85, 483 (2000)). Note: low temperature liquid 4He is a well known example of superfluid that can, for instance, flow through narrow capillaries with no dissipation. However, for superfluid helium, the critical velocity is lower than the sound velocity. This is explained by the fact that liquid 4He is a strongly interacting medium. We are here rather considering the case of weakly interacting cold atomic gases.

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"). 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 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 LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiw2JVEia0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= their pulsation (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGOEYvRi8= time the frequency). Typically, there are two possible types of excitations, depending on the wave-vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: In the limit: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLSZSaWdodEFycm93O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMEYnRjlGOQ==, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJjRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGQg== with 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, this relation is linear in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and is typical of a massless quasi-particle, i.e. a phonon excitation. In the limit: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLSZSaWdodEFycm93O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RKCZpbmZpbjtGJ0Y5RjtGPi9GQUY9RkJGREZGRkgvRktRJjAuMGVtRicvRk5GVEY5, 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 which is the dispersion relation of a free particle of mass LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGOQ==i.e. one single atom of the BEC. Problem: An impurity of mass LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= moves with velocity 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 within such a condensate and creates an excitation with wave-vector 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. After the interaction process, the impurity is scattered with velocity LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JJW1yb3dHRiQ2Ji1JI21pR0YkNiVRIndGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GNlEnbm9ybWFsRictSSNtb0dGJDYtUScmcmFycjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlRGUA==. 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 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 , 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. b) Show that when the atom-atom interaction constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5 (repulsive interactions), this value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJjRidGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn is equal to the group velocity of the excitation (speed of sound in a condensate).

The version of Physics used below is from April/10 (or later), available at the Maplesoft Physics Research & Development webpage a) The system consists of a condensate in its ground state and an impurity. Originally there is no excitation so the energy of the system can be approximated to the kinetic energy of the impurity. When an excitation with energy 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 is created, the total energy will be the kinetic energy of the impurity after being scattered (final velocity is LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JJW1yb3dHRiQ2Ji1JI21pR0YkNiVRIndGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GNlEnbm9ybWFsRictSSNtb0dGJDYtUScmcmFycjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlRGUA==) plus the energy 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 of the excitation. Conservation of energy then implies on 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

Likewise, conservation of linear momentum implies on 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

LyomSSJNRzYiIiIiSSN2X0dGJUYmLCYqJkknJmhiYXI7R0YlRiZJI2tfR0YlRiZGJiomRiRGJkkjd19HRiVGJkYm

where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtb3ZlckdGJDYlLUYsNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZOLyUrZXhlY3V0YWJsZUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGPw== is the wave vector of the excitation with respect to the condensate (a referential with its origin at the condensates's center of mass). Before proceeding further, indicate the real variables of this problem 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 The key observation now is that, for an excitation in a state that is a small perturbation around a stationary state, the wave vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtb3ZlckdGJDYlLUYsNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZOLyUrZXhlY3V0YWJsZUdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGPw== of the excitation is related to its energy by Bogoliubov's dispersion relation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEoJiM4NDYzO0YnLyUnaXRhbGljR0Y0Ri9GKy1JJW1zdWJHRiQ2JS1GRzYlUScmIzk2OTtGJ0ZKRi8tRkc2JVEia0YnL0ZLUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRiw2LVEpJmVxdWFscztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGam4tSSZtc3FydEdGJDYjLUYjNiYtSSZtZnJhY0dGJDYoLUYjNiYtSSVtc3VwR0YkNiVGUi1JI21uR0YkNiRRIjRGJ0YvLyUxc3VwZXJzY3JpcHRzaGlmdEdGZW4tRiw2LVExJkludmlzaWJsZVRpbWVzO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtRmdvNiUtRkc2JVEnJmhiYXI7RidGVUZXRmlvRl1wRi8tRiM2JkZpb0YrLUZnbzYlLUZHNiVRIm1GJ0ZVRlctRmpvNiRRIjJGJ0YvRl1wRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZxLyUpYmV2ZWxsZWRHRjQtRiw2LVEnJnBsdXM7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRl9yLUZibzYoLUYjNitGX3AtRmdvNiVGUkZecUZdcEZfcC1GZ282JUZkcEZecUZdcEZfcC1GRzYlUSJHRidGVUZXRl9wLUZHNiVRIm5GJ0ZVRldGL0ZbcUZhcUZkcUZncUZpcUYvLUZHNiNRIUYnLyUrZXhlY3V0YWJsZUdGNEYv

LyomSScmaGJhcjtHNiIiIiImSSZvbWVnYUdGJTYjSSJrR0YlRiYsJComI0YmIiIjRiYpLCYqKClGKiIiJUYmKUYkRjNGJilJIm1HRiVGLiEiIkYmKi5GM0YmKUYqRi5GJilGJEYuRiZJIkdHRiVGJkkibkdGJUYmRjZGN0YmRi1GJkYm

where 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. So to obtain the minimum (critical) value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiQtSSZtb3ZlckdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZyYXJyO0YnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURlBGQUZBLyUlb3BlbkdRJiZwYXI7RicvJSZjbG9zZUdGWS1GNDYjUSFGJ0ZB such that an excitation exists (is created) we need to combine these three equations , and . Start removing 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 from using 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

L0kjd19HNiIsJComLCYqJkknJmhiYXI7R0YkIiIiSSNrX0dGJEYqRioqJkkiTUdGJEYqSSN2X0dGJEYqISIiRipGLUYvRi8=

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

Eliminate now LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiw2JVEia0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= using . On the way, rewrite 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 as 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the angle between the two vectors 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

Isolate now the norm of LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21pR0YkNidRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1JI21vR0YkNi9RJyZyYXJyO0YnRjJGNC9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVEZBRlA= 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

Simplify this result taking into account that 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

Since 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, in the case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5, the minimum possible value for 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which must be kept positiveLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=, i.e. 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) is obtained for the lowest LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= values, i.e. in the phonon regime for the excitations, as discussed in the Background: 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

Combine the square roots LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbWlHRiQ2JVEoY29tYmluZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRKXNpbXBsaWZ5RidGL0YyLUY2NiQtRiM2Jy1GLDYjUSFGJy1JKG1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUWZebC8tSSVOb3JtRzYlJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkoVmVjdG9yc0c2JSUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIvSTFoaWRlbW9kdWxlcHJlZml4RzYiSSV0cnVlRyUqcHJvdGVjdGVkRy9JMWhpZGVtb2R1bGVwcmVmaXhHNiJJJXRydWVHJSpwcm90ZWN0ZWRHNiNJI3ZfRzYiKigpKihJIkdHNiIiIiJJIm1HNiIiIiJJIm5HNiIiIiIjIiIiIiIjIiIiSSJtRzYiISIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnRoZXRhRyUqcHJvdGVjdGVkRyEiIkYnLyUrYWN0aW9udHlwZUdRPG1hcGxlc29mdC5jb206bGFiZWwoTDE3MjE4KUYnLyUldmlld0dRJmxhYmVsRidGQS8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZURlFGVEZULUkjbW9HRiQ2LVEifkYnRlQvJSZmZW5jZUdGUy8lKnNlcGFyYXRvckdGUy8lKXN0cmV0Y2h5R0ZTLyUqc3ltbWV0cmljR0ZTLyUobGFyZ2VvcEdGUy8lLm1vdmFibGVsaW1pdHNHRlMvJSdhY2NlbnRHRlMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zkby1GVzYtUSlhc3N1bWluZ0YnRlRGWkZmbkZobkZqbkZcb0Zeb0Zgb0Zib0Zlb0ZWLUYsNiVRIm1GJ0YvRjJGVi1GVzYtUSI+RidGVEZaRmZuRmhuRmpuRlxvRl5vRmBvL0Zjb1EsMC4yNzc3Nzc4ZW1GJy9GZm9GYXBGVi1JI21uR0YkNiRRIjBGJ0ZULUZXNi1RIixGJ0ZURlovRmduRjFGaG5Gam5GXG9GXm9GYG9GYm8vRmZvUSwwLjMzMzMzMzNlbUYnRlYtRiw2JVEibkYnRi9GMkZdcEZWRmNwRmdwRlYtRiw2JVEiR0YnRi9GMkZWLUZXNi1RLyZHcmVhdGVyRXF1YWw7RidGVEZaRmZuRmhuRmpuRlxvRl5vRmBvRmBwRmJwRmNwRlFGVA==

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

Finally, the lowest possible value for LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZORj9GPy8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRlc=, the critical velocity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJjRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRkI=is obtained minimizing with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, that is, for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjItSSNtb0dGJDYtUSI9RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGMkYy: 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

LyZJInZHNiI2I0kiY0dGJSokKSooSSJHR0YlIiIiSSJtR0YlISIiSSJuR0YlRiwjRiwiIiNGLA==

In summary: the three equations , and together, that is conservation of energy, momentum and Bogoliubov's dispersion relation for an excitation of the condensate, imply that, for an excitation to be scattered by the impurity, LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZORj9GPy8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRlc=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEvJkdyZWF0ZXJFcXVhbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGQ0Yv 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 . Below LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJjRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGQg==there is no solution for LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUScmcmFycjtGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUkZORj9GPy8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRlc=, so no interaction, in which case the motion is superfluid. One could also notice that if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZXL0ZORlhGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJjRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnL0Y2USdub3JtYWxGJw===0, that is, the superfluidity behavior is closely related to the existence of an atom-atom interaction term, represented by the non-linear term of the Gross-Pitaevskii equation (derived previously in Mapleprimes "Quantum Mechanics using Computer Algebra"). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Note: Bogoliubov spectrum has two branches. In , we have chosen the '+' one, but the conclusions are the same for the other one.\313\231 ______________________________________________________________________ b) The group velocity is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImdGJ0YyRjUvJStleGVjdXRhYmxlR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y9Rj9GQg==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, so departing from Bogoliubov's dispersion relation we have 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

LyZJJm9tZWdhRzYiNiNJImtHRiUsJCooIyIiIiIiI0YrKSwmKigpRiciIiVGKylJJyZoYmFyO0dGJUYxRispSSJtR0YlRiwhIiJGKyouRjFGKylGJ0YsRispRjNGLEYrSSJHR0YlRitJIm5HRiVGK0Y1RjZGK0YqRitGM0Y2Ris=

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

LyZJInZHNiI2I0kiZ0dGJSwkKiojIiIiIiIlRispLCYqKClJImtHRiVGLEYrKUknJmhiYXI7R0YlRixGKylJIm1HRiUiIiMhIiJGKyouRixGKylGMUY2RispRjNGNkYrSSJHR0YlRitJIm5HRiVGK0Y1RjdGKyNGK0Y2RjdGM0Y3LCYqKkYsRispRjEiIiRGK0YyRitGNEY3RisqLiIiKUYrRjFGK0Y6RitGO0YrRjxGK0Y1RjdGK0YrRis=

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

LyZJInZHNiI2I0kiZ0dGJSooLCYqJilJJyZoYmFyO0dGJSIiIyIiIilJImtHRiVGLUYuRi4qKkYtRi5JIkdHRiVGLkkibUdGJUYuSSJuR0YlRi5GLkYuRjMhIiIpLCZGKkYuKioiIiVGLkYyRi5GM0YuRjRGLkYuI0YuRi1GNQ==

For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5, and in the limit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= (phonon branch), considered in the previous section (see ) we have 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

LyZJInZHNiI2I0kiZ0dGJSooSSJHR0YlIiIiSSJuR0YlRiopKihGKUYqSSJtR0YlRipGK0YqI0YqIiIjISIi

Combining the square roots 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

LyZJInZHNiI2I0kiZ0dGJSokKSooSSJHR0YlIiIiSSJtR0YlISIiSSJuR0YlRiwjRiwiIiNGLA==

In summary, the group velocity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHZfX2dgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn (speed of sound in a condensate) is equal to the critical velocity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHZfX2NgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn that an impurity needs to have in order to move within the condensate creating an excitation and dissipation.

References LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= [1] Suppression and enhancement of impurity scattering in a Bose-Einstein condensate [2] Superfluidity versus Bose-Einstein condensation [3] Bose\342\200\223Einstein condensate (wiki) [4] Dispersion relations (wiki)