Singular sound in a trapped quantum gas driven by two phase-conjugated optical vortices

The trapped atomic cloud irradiated by two counter-propagating $\delta \omega$ frequency detuned Laguerre-Gaussian optical vortices with opposite angular momenta $\pm \ell \hbar$ is considered.\footnote{F. Dalfovo et al. Rev.Mod.Phys., \textbf{71}, 463 (1999).}$^,$\footnote{A.Yu.Okulov. J.Low Temp.Phys., in press (2013).} When period of spatial modulation $\lambda/2$ and LG carrier frequency detuning $\delta \omega$ \footnote{A.Yu.Okulov. J. Phys. B , \textbf{41}, 101001 (2008).} are in resonance with dispersion curve $\epsilon(p)$ \footnote{A.Yu.Okulov. Phys.Lett.A, \textbf{376}, 650-655 (2012).} the acoustical vortices carrying orbital angular momentum\footnote{J.L.Thomas and R.Marchiano. Phys.Rev.Lett., \textbf{91},244302 (2003).} are expected to occur.\footnote{A.Yu.Okulov. JETP.Lett., \textbf{88}, 487 (2008).}$^,$\footnote{Fam Le Kien,V.I.Balykin,K.Hakuta.Phys.Rev.A, \textbf{73},053823 (2006).} The experimentally accessible range of $\lambda$ and $\delta \omega$ \footnote{A.Yu.Okulov. J. Opt. Soc. Am. B, \textbf{29}, 714-718 (2012).} is analyzed from the point view of direct measurements of the excitation spectrum $\epsilon(p)$\footnote{R.P.Feynman. {\itshape ``Statistical mechanics''}, Ch.11(1972).}$^,$\footnote{M. R. Andrews et al. Phys.Rev.Lett.}