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 $x\text{−}y$ 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 $\left({\mathrm{GBEC}}_1\right)$ corresponds to $T_m<T_c$. The second type of GBEC $\left({\mathrm{GBEC}}_2\right)$ corresponds to $T_c<T_m$. Here, $T_c$ is the critical temperature of the first-step condensation and $T_m$ is the critical temperature of the second-step condensation. There is a competitive mechanism between normal Bose-Einstein condensation (NBEC) and GBEC. We shall show that the standard transition temperature $T_c$ is much larger than the transition temperature $T_n$ of NBEC. Therefore, the state of NBEC does not occur in a quasi-1D harmonic potential. We have proposed an analytical solution to the problem of GBEC of ideal atoms in the quasi-1D harmonic potential. The number of noncondensed atoms in the first-step condensation is characterized by a series of elliptic theta functions and the number of noncondensed atoms in the second-step condensation is characterized by a single $q$-digamma function. If $r$ denotes the tube radius, then this quasi-1D system can show the quantum size effect. There is a critical tube radius $r_c$. (1) If $r>r_c$, 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 ${\mathrm{GBEC}}_2$. 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.

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• Cold atoms & matter waves
• Bose-Einstein condensates