383. Puzzling kinetics of Bose-Enstein condensation

Posted by Dmitry

May 1, 2009

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Just finished reading a really review “Magnon BEC and spin superfluidity” by Yu. Bunkov and G. Volovik, which left me with quite a bit of material to think about… Probably the thing that stroke me most after digesting the review is how poorly I (or in truth – it’s better to say “we”) actually understand the kinetics of Bose-Einstein condensation. But before I’ll try to explain why I think so, let me briefly describe the setup discussed by the authors.

1. Bose-Einstein condensation of magnons

They talk about Bose-Einstein condensation (BEC) in ${}^3He$ , but their BEC is not the one that leads to superfluidity of ${}^3He$ . The story is actually much funnier: ${}^3He$ are fermions carrying spin 1/2. As a result of spin-spin and spin-orbital interactions, spin waves can be excited in ${}^3He$ . The ground state of the system (at T=0) corresponds to all spins of all nuclei oriented in the same direction. An excitation above the ground state corresponds to one or several spins oriented downwards. There are of course quasi-particles associated with these excitations called magnons, that carry spin 1 (since the difference between S_z of nuclei with spins oriented upwards and downwards is 1). Since these quasi-particles are bosons, it is natural to expect that Bose-Einstein condensation is possible for them.

Magnon BEC corresponds to a situation when spins of all nuclei in the superfluid precess coherently. Such state was indeed observed in ${}^3He$ experimentally not so long ago (you can find references to original papers in the bibliography section of the review). More over, it was even possible experimentally to observe vortices – topological excitations in the corresponding Bose-Einstein condensate.

What surprised me the most is the time scale at which Bose-Einstein magnon condensate gets established after the system’s cooling.

Formation of magnon BEC. From Bunkov, Volovik, 09.

First of all, note that magnons are quasi-particles. The true ground state does not contain any spin excitations – in other words, all magnons decay before the ground state is reached. As seen from the plot on the left, the characteristic time scale of their decay is rather short – about 0.5-1 sec. On the other hand, as follows from the plot on the right, true BEC (that is, off-diagonal long range order) is formed at time scales 10 times shorter – at 0.05 sec after cooling. So, why am I excited so much about it?

2. How kinetics of BEC formation is understood at present time

The main reason is that according to our present understanding of BEC kinetics, formation of the condensate peak in a weakly interacting Boise gas requires infinite time (in the system with infinite time, see Nozieres, Levich-Yakhot), although formation of Bose distribution with chemical potential close to zero is of the order of mean free time for particles forming the condensate. By the condensate peak I mean as usual a delta-function contribution into the distribution function of elementary excitations at k=0 .

Condensation proceeds in several steps. First of all, particles start to condense in the energy interval

$\epsilon < n_c U$ ,

where n_c is equilibrium density of condensate, is effective interaction vertex for elementary excitations. It is said that precondensate or short range order is formed. The state of the liquid is then described by the non-linear Schrodinger (or better say Gross-Pitaevskii) equation

$i\frac{\partial \Psi}{\partial t}=-\frac{\nabla^2}{2m}\Psi + U|\Psi|^2\Psi$ ,

where $\Psi$ is the wavefunction of the precondensate.
The modulus and the phase of this wave function still fluctuate strongly after formation of precondensate.

Short range order is formed at time scales

$\tau_{\rm sr}\sim\frac{1}{n_c U}$ ,

a relatively short time scale.

The order is called short since the correlation length exists in the system that is of the order

$r_c\sim(mn_cU)^{-1/2}$ –

inside domains of the size r_c the phase of the wave function is correlated, and it remains uncorrelated between different domains.

But what we ultimately want to see is not a precondensate in the sense above but the formation of the delta-function in the distribution function n_k at k=0 . The latter does not happen even at $t\gg\tau_{\rm sr}$ .

The reason is that the phase is rapid variable, and while the absolute value of the wave function achieves its equilibrium value at time scale of the order $\tau_{\rm sr}$ , the phase of the wavefunction continues to fluctuate strongly. In a sense, the system of magnons can be described in terms of vortices (topological excitations associated to fluctuations of the phase) and “phonons” – spin waves.

In order for off-diagonal long range order to appear, both vortex field and spin waves should relax.

Characteristic time scale for the appearance of the topological LR order (in other words, the time scale for decay of vortices) depends on the size of the system (this is a kind of apparent, since long vortices may be present with characteristic size of the same order of magnitude as ) can be estimated as follows:

$\tau_{\rm top}\sim\frac{1}{\Gamma\kappa}\frac{L^2}{{\rm log}L/r_c}$ ,

where $\kappa \sim \left(\frac{n_cU}{T_c}\right)^{3/2}$ and $\Gamma$ is the quantum of circulation. As you see, this is rather long time scale that depends on the size of the system.

Finally, for the long range order spin waves should also relax, and their relaxation time is the longest one in the system and is given by

$\tau_{\rm ODLRO}\sim\frac{L}{aT}$

where is the scattering length for spin waves (related to the parameter above in the known way). This time scale is also proportional to the size of the system…

So, why the time scale of magnon BEC formation is so short? Probably, something goes wrong with considerations above, but I am not sure what exactly…

Condensed Matter, Fellow Craft, Journal club

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383. Puzzling kinetics of Bose-Enstein condensation

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