The aim of this work is to improve the study of a phonon laser (saser) proposed by us several years ago[1]. This is a device capable to generate an intense coherent beam of acoustical phonons. Our acoustic laser consists in a double barrier heterostructure tailored such the energy difference between the ground and the first excited state in the well is close to the energy of the LO phonon. The electrons are directly injected into the excited level. Therefore they decay producing a high rate of LO phonons. These phonons are confined inside the well and decay into a pair of phonons[2]: LO -> <img src="http:/img/fbpe/bjp/v29n4/lotil.gif">+ TA. The TA phonons escape the well in the [111] direction constituting an intense coherent beam. Recently were studied (and sometimes realized experimentally) several kinds of phonon lasers. Up to our knowledge our saser is the only that has a very short wavelength (smaller than 25 Å) and a very long range (greater than 1000 mum). Because of that, such beam could have applications to acoustic nanoscopy, acoustic nanolithography and phonoelectronics. In early articles[1, 3, 4, 5, 6] we get the kinetic equations for the averaged electron and phonon populations. Quantum fluctuations were not taken into account. The system Hamiltonian is H = He + Hph + He<FONT FACE=Symbol>-</FONT>ph + Hph<FONT FACE=Symbol>-</FONT>ph + He<FONT FACE=Symbol>-</FONT>e. To solve this Hamiltonian we expand their eigenfunctions in the basis of the eigenstates¦jn1n2n3<FONT FACE=Symbol>ñ</FONT> of the single particle part of it. We obtain a set of coupled equations for the expansion coefficients that can be solved with some approximations. The results are qualitatively similar to those obtained previously.
