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In 2006, Demokritov et al have reported that they have achieved the observation of quasi-equilibrium magnon Bose-Einstein condensation (BEC) in YIG at finite (room) temperature by using the method called "microwave pumping" http://www.nature.com/nature/journal/v443/n7110/abs/nature05117.html.

I know there are various (sometimes controversial) discussion on their experiments and the understanding of the microscopic mechanism; about spontaneous coherence (spontaneous symmetry breaking), spontaneous condensation[A] or driven condensation, thermodynamics properties, the novel dispersion relation of YIG, and thermalization effects.

From the view point of theoretical issues (to the best of my knowledge), Prof. Bunkov and Volovik http://arxiv.org/abs/1003.4889 has generalized (extended) the idea of magnon condensation and have proposed a non-equilibrium magnon BEC theory[B], in which they have viewed the magnon annihilation operator associating with (you call) the off-diagonal long-range order, $a_0=\sqrt{N_0} \exp(i\mu t-i\alpha)$, as the macroscopic condensate order paramater.

Taking the experimental result by Demokritov et al into account, we can easily expect that the magnitude of the condensate order parameter decreases due to thermalization effects[C] (i.e. The dissipation into environments, phonons) and it takes the following form; $a_0= \sqrt{N_0} \exp(i\mu t-i\alpha -t/\tau)$, [D] where the variable $\tau$ reflects the dissipation.

Thus although we can understand an aspect of the above experiment by Demokritov et al, I wonder whether we can regard it as Quasi-equilibrium or non-equilibrium magnon BEC (until $t<\tau$) also in the case where the magnitude of the condensate order paramater is time-dependent; $a_0= \sqrt{N_0} \exp(i\mu t-i\alpha-t/\tau)$ ?

[A] In the standard or traditional equilibrium magnon BEC theory treating only equilibrium situations, the idea of "the spontaneous $U(1)$-symmetry breaking" plays the key role.

[B] Magnon BEC corresponds to coherent precessions in the language of original spins.

[C] By tracing out the enviromental degrees of freedoms, the time-development of the system we focus on becomes a non-unitary form. As an eaxample, please see the famous Caldeira-Leggett model or quantum master equation.

[D] When we neglect the thermalization effects or the dissipation, we can easily show and see that the magnitude of the macroscopic condensation order parameter takes a time-independent constant value after switching off microwaves because the $U(1)$-symmetry of the system is recovered. I guess although this theoretically means the stable magnon BEC, it does not reflect the experimental result by Demokritov et al.

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I should apologize that a part of the last document is not correct and hence, I have modified it. –  Kouki Nakata Jun 26 '13 at 22:38
    
I think this is a very nice high level question that would deserve the research-level tag to get the right attention, +1. Unfortunately there can not be more than 5 tags attached, but maybe you would like to trade one of the 5 you have against research-level? Just an idea to help, which can savely be ignored if it is not helpful ... –  Dilaton Jun 26 '13 at 22:46
    
Taking the comment from Dilaton into account, I have tagged "research-level". Thank you for your contributions. –  Kouki Nakata Jun 27 '13 at 5:25
    
I think that the number of pumped magnons (by ferromagnetic resonance) is different from that of magnons which belong to BEC through thermalization processes. I wonder there are such studies which have closely studied the dynamics microscopically (i.e. microscopic description of thermalization processes or the explicit evaluation of the number of magnons in BEC after thermalization processes or pumping)? –  Kouki Nakata Jul 10 '13 at 15:09
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