BEC of magnons in Superfluid ³He-B and symmetry breaking fields

States with a coherent precession of magnetization like a homogeneously precessing domain (HPD) or a persistently precessing domain, created in the superfluid ³He-B represent the macroscopic examples of the Bose-Einstein condensates of magnons. Once the magnons form one of these states, this many-magnon coherent quantum state is described by a "single magnon wave function" (or an order parameter). A suitable external perturbation may cause the condensate to oscillate around the state of coherent precession, which demonstrates a collective rigidity of the condensate against scattering a single magnon out of it. The states corresponding to a free coherent precession of magnetization are degenerate in the phase of precession, so there exist oscillations around such state with a gapless dispersion relation, known as the Goldstone modes. Here we present both experimental and theoretical results of the study of the spin density oscillations superposed on a homogeneously precessing domain in superfluid ³He-B in the presence of a high-frequency excitation field B_rf. We show that the presence of this field lifts the degeneracy of the precessing state with respect to the phase of precession, that is it violates the symmetry of the magnon condensate and former Goldstone modes become non-Goldstone ones, as they acquire the energy gap in their spectrum.

A schematic visualization of various kinds of the normal modes of an oscillating HPD-SD structure: (a) the HPD-SD stable state, (b) the torsion oscillation mode, (c) the planar mode and (d) the the first axial surface mode.

Left figure shows the measured resonance frequencies of the axial surface mode as a function of the field gradient measured at the temperature of 0.49T_C and pressure of 11 bar. Black dashed lines correspond to the theoretical calculations according to Fomin's theory for the first and the second axial modes, while the red line corresponds to the theoretical model presented below. The right figure shows the resonance frequency of the torsion mode as a function of the HPD length scaled by the cell length. The line is the fit to experimental data using below mentioned equation. Inset shows the resonance characteristics measured for three various HPD lengths at the constant amplitude of B_rf. The general equation describing collective oscillation modes of the BEC of magnons in form of HPD-SD structure in superfluid ³He-B was found in form

As one can see from the above equation, the spectrum (dispersion relation) acquires an energy gap due to presence of the high frequency excitation field B_rf as this field violates U(1) symmetry of the magnon condensate.

M. Kupka, P. Skyba:
BEC of Magnons in Superfluid ³He-B and Symmetry Breaking Fields,
Phys. Rev. B 85, 184529 (2012).