Quantum size effect and two-step condensation of ideal Bose atoms in an optical hard-wall nanotube

We consider a quasi-one-dimensional (quasi-1D) system that is an optical hard-wall nanotube with axial harmonic potential. This quasi-1D system possesses a quasi-1D harmonic potential that consists of a 1D harmonic trap along the z axis and a two-dimensional box trap in the x-y plane. Within the framework of quantum statistical mechanics, we find that in a quasi-1D harmonic potential, the generalized Bose-Einstein condensation (GBEC) can be classified into the two types. The first type of GBEC (GBEC_{1}) corresponds to T_{m}r_{c}, then the mediate condensate fraction N_{m}/N=0; (2) if r>r_{c}, then the standard transition temperature T_{c}=0; (3) if r>r_{c}, then T_{m}>T_{c} and this is the state of GBEC_{2}. The results of numerical calculation of the analytical solution predict many new experimental phenomena in the GBEC of ideal atoms in the quasi-1D harmonic potential. In the thermodynamic limit, the analytical expressions of the three critical temperatures and the three condensate fractions are derived.