  We study the thermodynamic properties of dimerized spin-gapped quantum
magnets with and without exchange anisotropy (EA) and Dzyaloshinsky and Moriya
(DM) anisotropies within the mean-field approximation (MFA). For this purpose
we obtain the thermodynamic potential $\Omega$ of a triplon gas taking into
account the strength of DM interaction up to second order. The minimization of
$\Omega$ with respect to self-energies $\Sigma_n$ and $\Sigma_{an}$ yields the
equation for $X_{1,2}=\Sigma_n\pm\Sigma_{an}-\mu$, which define the dispersion
of quasiparticles $E_k=\sqrt{\epsilon_k+X_1}\sqrt{\epsilon_k+X_2}$ where
$\epsilon_k$ is the bare dispersion of triplons. The minimization of $\Omega$
with respect to the magnitude $\rho_0$ and the phase $\Theta$ of triplon
condensate leads to coupled equations for $\rho_0$ and $\Theta$. We discuss the
restrictions on $\rho_0$ and $\Theta$ imposed by these equations for systems
with and without anisotropy. The requirement of dynamical stability conditions
$(X_1>0, X_2>0)$ in equilibrium, as well as the Hugenholtz-Pines theorem,
particularly for isotropic Bose condensate, impose certain conditions to the
physical solutions of these equations. It is shown that the phase angle of a
purely homogenous Bose-Einstein condensate (BEC) without any anisotropy may
only take values $\Theta=\pi n $ (n=0,$\pm 1,\pm 2$...) while that of BEC with
even a tiny DM interaction results in $\Theta=\pi/2+2\pi n$. In contrast to the
widely used Hartree-Fock-Popov approximation, which allows arbitrary phase
angle, our approach predicts that the phase angle may have only discrete
values, while the phase of the wave function of the whole system remains
arbitrary as expected. The consequences of this phase locking for interference
of two Bose condensates and to their possible Josephson junction is studied.
