1D system: local density approximation

If the discretization of levels in the longitudinal direction can be neglected, i.e. if $E/N-\hbar\omega_\perp\gg\hbar\omega_z$, the 1D system can be described within the local density approximation (LDA). In this case, the chemical potential of the system is calculated through the local equilibrium equation (1.126) which we write separating in an explicit way the dominant contribution of the transverse confinement $\hbar\omega_\perp$:

$$\mu=\hbar\omega_\perp + \mu_{hom}(n_{1D}(z))+\frac{m}{2}\omega_z^2z^2,$$ (3.4)

Here $\mu_{hom}(n_{1D})$ is the chemical potential corresponding to a homogeneous 1D system of density $n_{1D}$. If the ratio $a_{3D}/a_{\perp}\ll 1$, the local chemical potential can be obtained from the Lieb-Liniger (LL) equation (5.1) of state with the effective 1D coupling constant $g_{1D}=g_{3D}/(2\pi a_{\perp}^2)$ (1.124).One finds: $\mu_{hom}=\partial[n_{1D}\epsilon_{LL} (n_{1D})]/\partial n_{1D}$, where $\epsilon_{LL}$ is the LL energy per particle. The LDA problem in one-dimension was already studied in Sec. 1.6.2. The chemical potential $\mu$ as a function of $N$ and trap parameters is given by formula (1.140). The ground-state energy of the system with a given number of particles can then be calculated through direct integration of $\mu(N)$.

If $n_{1D}a_\perp^2/a_{3D}\gg 1$, the system is weakly interacting and the LL equation of state coincides with the mean-field prediction: $\epsilon_{LL}=g_{1D}n_{1D}/2$. In the notation of Sec. 1.6.2 it corresponds to $C_1 = 2, \gamma_1 =1,C_2=0$. From formula (1.140) one finds the following results for the energy per particle

$$\frac{E}{N}-\hbar\omega_\perp=\frac{3}{10}\left(3N\lambda\frac{a_{3D}}{a_\perp}\right)^{2/3}\hbar\omega_\perp \;,$$ (3.5)

and exploiting formula (1.138) one obtains an expression for the mean square radius of the cloud in the longitudinal direction

$$\sqrt{\langle z^2\rangle}=\left(3N\lambda\frac{a_{3D}}{a_\perp}\right)^{1/3}\frac{a_\perp}{\sqrt{5}\lambda} \;.$$ (3.6)

In the opposite limit, $n_{1D}a_\perp^2/a\ll 1$, the system enters the Tonks-Girardeau regime and the LL equation of state has the Fermi-like behavior (1.102) $\epsilon_{LL}=\pi^2\hbar^2n_{1D}^2/6m$. The energy per particle and the mean square radius of the trapped system are easily extracted from the results for a purely one-dimensional system (1.151),(1.153)

$$\frac{E}{N}-\hbar\omega_\perp=\frac{N\lambda}{2}\hbar\omega_... ...\sqrt{\langle z^2\rangle}=\sqrt{\frac{N}{2\lambda}}a_\perp \;.$$ (3.7)

In terms of the parameters of the system, the two regimes can be identified by comparing the corresponding energies. The mean-field energy becomes favorable if $N\lambda a_\perp^2/a^2_{3D}\gg 1$, whereas the Tonks-Girardeau gas is preferred if the condition $N\lambda a_\perp^2/a^2_{3D}\ll 1$ is satisfied^3.2.