The damping of collective excitations in two-component Bose-Einstein condensates (BECs) is studied. The damping includes Landau and Baliaev mechanisms. Elementary excitation in BECs is the basic subject of statistical physics and condensed matter physics. With the development of Feshbach resonance technology in ultra-cold atomic gases and the use of highly controllable ultra-cold quantum systems, significant progress has been made in related research. The attenuation of collective excitation amplitude is called damping, which is generated by the interaction between particles. Damping is an important feature of low-energy collective excitation in the BEC experiment. Accurate calculation of damping is very important for understanding the essence of quantum multi-body physics. Since the damping of collective excitation is one of the long-term and important topics in the study of quantum multi-body physics, and the two-component BEC system has rich physical properties, the application of Hartree-Fock-Bogoliubov (HFB) mean-field theory in this paper is extended, so as to provide ideas for the smooth development of related research work.

HFB mean-field theory is used, and the theoretical framework of the two-component system is constructed based on the original work of the one-component system theory. The collective excitation damping formula is derived strictly according to the original work method, including the Bogoliubov-de Gennes equations of non-condensed quasi-particles obtained by diagonalizing the giant canonical Hamiltonian. The three-mode coupling matrix element describing the interaction between particles is obtained by commutation calculation and Fourier transform of normal and abnormal quasi-particle distribution function motion equations, and the relation for the perturbed eigenfrequency and the damping rate of collective excitation are obtained by Fourier transform of collective excitation motion equations. Landau damping of collective excitation in a continuous two-component BEC homogeneous system with an ideal energy level is taken as an example, and a semi-classical approximate calculation is carried out to show the details of particle interaction. In addition, the physical significance and application method of the theory are explained.

The main differences between the two-component system and the one-component system in the construction of the theoretical framework are analyzed. The two-component system includes three cases: two kinds of atoms of the same kind with different fine structures, two kinds of atoms of different isotopes of the same element, and two kinds of atoms of different kinds. The first two cases are not exactly the same. In the construction of the mean-field theory of two-component BECs, two different quasi-particle generation and annihilation operators are used respectively, and the Bogoliubov transform is applied to the non-condensed part operators of the two components, respectively. Although the construction of the one-component theoretical framework is complex, that of the two-component theoretical framework is more complicated. Since two different quasi-particle generation and annihilation operators are used respectively, the cross terms of two different quasi-particle generation and annihilation operators are generated after the Bogoliubov transform of the non-condensed part operators of the two components, and more approximations than the construction of the one-component theoretical framework are needed to introduce and thus diagonalize the giant canonical Hamiltonian. In the semi-classical approximate calculation process of Landau damping of collective excitation in a two-component BEC homogeneous system, the dependence of Landau damping on temperature is analyzed, and the quasi-particle transitions that contribute to damping are analyzed, including various cases between the same type of quasi-particles and between different types of quasi-particles, which are expressed by dimensionless damping functions. The contribution of quasi-particle transitions in different energy ranges to damping is analyzed and expressed by error function.

In this paper, HFB mean-field theory for studying collective excitation damping in BEC is successfully extended from a one-component system to a two-component system, and the application of the original mean-field theory is extended. The observation of collective excitation damping is the evidence for the realization of BECs in early magnetic trap experiments, and the theory of related one-component systems has been used to carry out more in-depth research on the coupling interaction between excitations. Similarly, further experimental research on collective excitation damping of two-component systems is also important for understanding the essence of quantum multi-body physics. Because of the complexity of the axisymmetric system in the magnetic trap, the homogeneous system in the box trap is a simple and ideal example that is easy to be calculated in the early theoretical study of collective excitation damping in one-component BECs. At present, the damping of collective excitation in one-component BECs in the box trap has been experimentally studied. With the further development of cold atom technology, the experimental study of collective excitation damping in two-component BECs in the box trap will also be conducted. It is hoped that the theoretical research in this paper can play a certain role in facilitating experimental work on two-component condensates.