The attached presentation is the last one of a sequence of three on Quantum Mechanics using Computer Algebra, covering the field equation for a quantum system of identical particles, its stationary solutions and the equations for small perturbations around them and, in this third presentation, the conditions for superfluidity of such a system of identical particles at low temperature. The novelty is again in how to tackle these problems in a computer algebra worksheet.

The Landau criterion for Superfluidity

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université 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 `&hbar;`*omega[k] and momentum `&hbar;` `#mover(mi("k"),mo("&rarr;"))`. We assume that this excitation is well described by Bogoliubov's equations for small perturbations `&delta;&varphi;` around the stationary solutions `&varphi;```of the field equations for the system. In that case, the Landau criterion for superfluidity states that if the impurity velocityLinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) is lower than a critical velocity v[c] (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 LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) < v[c] 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.

Landau criterion for superfluidity



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").


epsilon[k] = `&hbar;`*omega[k] and `&hbar;`*omega[k] = `&+-`(sqrt(k^4*`&hbar;`^4/(4*m^2)+k^2*`&hbar;`^2*G*n/m))


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 omega[k] their pulsation (2*Pi time the frequency). Typically, there are two possible types of excitations, depending on the wave-vector k:


In the limit: proc (k) options operator, arrow; 0 end proc, "epsilon[k]&sim;`&hbar;`*k*"v[c] with v[c] = sqrt(G*n/m), this relation is linear in k and is typical of a massless quasi-particle, i.e. a phonon excitation.


In the limit: proc (k) options operator, arrow; infinity end proc, `&sim;`(epsilon[k], `&hbar;`^2*k^2/(2*m)) which is the dispersion relation of a free particle of mass "m,"i.e. one single atom of the BEC.


Problem: An impurity of mass M moves with velocity `#mover(mi("v"),mo("&rarr;"))` within such a condensate and creates an excitation with wave-vector `#mover(mi("k"),mo("&rarr;"))`. After the interaction process, the impurity is scattered with velocity `#mover(mi("w"),mo("&rarr;"))`.


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


LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) >= v[c] and v[c] = sqrt(G*n/m)



When LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) < v[c] , 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 G >= 0 (repulsive interactions), this value v[c] is equal to the group velocity of the excitation (speed of sound in a condensate).






[1] Suppression and enhancement of impurity scattering in a Bose-Einstein condensate

[2] Superfluidity versus Bose-Einstein condensation
[3] Bose–Einstein condensate (wiki)

[4] Dispersion relations (wiki)


Download   QuantumMechanics3.pdf

Edgardo S. Cheb-Terrab
Physics, Maplesoft

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