Abstract
In these two lectures (March 18 and 25, 2022), we describe a powerful method for treating the case of an interacting Bose gas, the Bogoliubov approach. This approach makes it possible to describe the gas's ground state as well as its low-energy excitation spectrum, subject to a number of approximations that we describe in detail. The method is based on a binary particle interaction potential, and assumes that the action of this potential does not significantly alter the fundamental state of the fluid compared to the case of the perfect gas.
Bogoliubov's method, although a commonly used tool, has certain subtleties stemming from the fact that it is difficult to use it with the actual interatomic potential. For all the atomic species used in the laboratory, this potential - , which describes the van der Waals interaction - contains many states linked to two particles. The true ground state of the system is therefore very different from the Bose-Einstein condensate formed from the monoatomic gas found in the non-interacting case, and also far removed from the fluid prepared, in a metastable state, in cold atom experiments.
Bogoliubov's method is frequently used with a contact potential, i.e. one of zero range. The coupling is then defined on the basis of the diffusion length a of the physical problem. However, we know (cf. lecture 2020-2021) that such a potential leads to divergences as early as 2 in the Born series. A fortiori, it does not provide a simple description of the interaction between N particles. Some expressions, such as the speed of sound or quantum depletion, can be calculated without difficulty, while others, such as ground-state energy, diverge.
The approach we are exploring initially uses a regular potential, of non-zero range, whose Fourier transform is also regular. We assume that the action of this two-body potential can be described in the Born approximation. In an interaction-free Bose gas, the ground state is obtained by placing the N particles in the zero impulse state k = 0. We therefore begin by using the fact that the potential can be treated as a weak perturbation to perform a systematic expansion of the N body Hamiltonian, assuming that the average population in the zero impulse state remains in the majority. This allows us to obtain an approximate expression of the Hamiltonian containing only quadratic terms in the creation and destruction operators of a particle in a non-zero k-impulse state. More precisely, the structure of the Hamiltonian reveals a sum of independent terms, each relating to a (+k, -k) pair. We begin by focusing on a given pair to detail Bogoliubov's method, which uses a canonical transformation to diagonalize this Hamiltonian of pairs. We illustrate this diagonalization method on the case of a spiner gas in the single-space mode approximation. Finally, we describe the return to an infinite number of pairs and to a contact potential, with the convergence problems that may then arise. We calculate the ground-state energy of the system, known as the Lee-Huang-Yang energy, and discuss the validity of the development underlying this method. Finally, we describe a number of recent experiments that have enabled us to accurately measure the value of various physical quantities predicted by this approach.
