CANONICAL TRANSFORMATION AND THE PROPERTIES OF A MIXTURE OF FERMIONS AND BOSONS AT LOW TEMPERATURE

Abstract

Properties of free and interacting systems of bosons and fermions have been studied theoretically using canonical transformation. In the last few years (1995 to be exact), condensates have been obtained practically in the laboratories for a mixture of bosons and fermions, where optical lattice system is created to trap the mixture and study its properties at very low temperature. Theoretical studies on the properties of such a mixture were undertaken in this work, where a canonical transformation was developed in terms of canonical operators for bosons and fermions. The Hamiltonian H, for the mixture of interacting bosons and fermions was constituted assuming the possible interactions that can exist in the mixture such as boson-boson and boson-fermion interactions. However, the other possible interaction between fermion-fermion is disallowed by the Pauli’s principle. The scattering lengths between boson-boson (𝛼𝐵𝐵) and between boson-fermion (𝛼𝐵𝐹) give a measure of interaction strength between the interacting boson-boson and boson-fermion respectively. The values of 𝛼𝐵𝐵 and 𝛼𝐵𝐹 vary with the types of bosons and fermions in the mixture and can also be changed by the method of Feshbach resonance. In this thesis, the values of quasi-particle energy of the kth state (𝐸𝑘) were calculated for different values of 𝛼𝐵𝐵 and 𝛼𝐵𝐹 to determine their values for different mixtures. For 𝑅3787𝑏+𝐾1940 mixture with 𝛼𝐵𝐹 of 150×10−8 𝑐𝑚, 162×10−8 𝑐𝑚, 300×10−8 𝑐𝑚, and −209×10−8 𝑐𝑚, the corresponding calculated values of 𝐸𝑘 were 1.237×10−12 𝑒𝑟𝑔𝑠, 1.337×10−12 𝑒𝑟𝑔𝑠, 2.475×10−12 𝑒𝑟𝑔𝑠, and −1.103×10−12 𝑒𝑟𝑔𝑠. Similarly, for 𝐿𝑖+𝐿𝑖3736 mixture with 𝛼𝐵𝐹 of 0.2158×10−8 𝑐𝑚 had 𝐸𝑘 of 5.227×10−17 𝑒𝑟𝑔𝑠, while for 𝑁𝑎+𝐿𝑖361123 mixture with 𝛼𝐵𝐹 of −1.45×10−8 𝑐𝑚 had 𝐸𝑘 of −1.22×10−21 𝑒𝑟𝑔𝑠. The negative values of 𝐸𝑘 mean that the interaction is attractive, where interacting species overlap, whereas positive values of 𝐸𝑘