Trapped system

Now let us consider effects of the external trap. We consider the trapping potential to be a harmonic oscillator. The effective one-dimensional Hamiltonian is then given by

$$\hat H^{trap}_{LL} = -\frac{\hbar^2}{2m}\sum\limits_{i=1}^N\frac{... ...omega_z^2}{2}\sum\limits_{i=1}^N z_i^2 +g_{1D}\sum\limits_{i<j}\delta(z_i-z_j),$$ (5.16)

where the effective coupling constant depends both on the value of the 3D s-wave scattering length and the oscillator length of the transverse confinement $a_\perp=\sqrt{\hbar/m\omega_\perp}$ through relation $g_{1D} = 2\hbar^2a/ma^2_\perp$ (1.124). The relevant parameters are: the ratio $a_{3D}/a_\perp$, the anisotropy parameter $\lambda=\omega_z/\omega_\perp$ and the number of particles $N$.

In the construction of the trial wave function used in our DMC calculation we introduce one-body Jastrow term $f_1(z_i)$ in addition to the two-body terms $f_2(z_{ij})$ already contained in homogeneous trial wave function (2.55). Taking into account the harmonic nature of the external potential we choose the one-body term in the Gaussian form $f_1(z) = \exp(-\alpha_z z^2)$ with $\alpha _z$ being the variational parameter. The correlations at distances much larger than the longitudinal oscillator length $a_z=\sqrt{\hbar/m\omega_z}$ are dominated by the oscillator confinement and two-body correlations become irrelevant.

We consider the following configurations: $a_{3D}/a_\perp =0.2$, $\lambda = 10^{-3}$ and number of particles $N = 5, 20, 100$. In Sec. 3 we have proven that in these conditions the ground-state energy and structure of the cloud is correctly described by the Lieb-Liniger equation of state in local density approximation.

In Fig. 5.10 we plot the pair distribution function (2.146) for 5, 20 and 100 particles. The short-range dependence is dominated by two-body interactions. We do not find oscillations which means that strong shell structure is absent. At large distances the external trapping suppresses density.

Figure 5.10: Pair distribution of a trapped system for 5, 20, 100 particles and $a_{3D}/a_\perp = 0.2, \lambda = 10^{-3}$.

We refer to general definition of the static structure factor in terms of the momentum distribution $n_k$ (2.133):

$$S(k) = \frac{1}{N}(\langle n_{-k} n_k\rangle - \vert\langle n_k\rangle\vert^2)$$ (5.17)

On the contrary uniform case, the last term is no longer vanishing for $k\ne 0$in a trap. In Fig. 5.11 we present the static structure factor obtained for the same set of parameters. We are interested in evidences of the linear behavior characteristic for the phonon propagation. We discover that Feynmann formula (5.9) with the speed of sound taken at the center of the trap provides relatively good description also for the trapped systems. Of course, the very low momenta part is different due to the finite size effects.

For the smallest number of particles considered ($N=5$), the density is always small $n_{1D}a_{1D}<0.18$ and we can derive an explicit expression for the $S(k)$ exploiting knowledge of the static structure factor in the limit of small density (5.5) as explained in Sec. 1.6.4. The resulting expression is given by formula (1.154). an be calculated. We find that thr linear behavior at small $\k$ matches the asymptotic constant in a smoother way than it happens in a homogeneous system (see TG static structure factor in Fig. 5.4).

Figure 5.11: Static structure factor of a trapped system for $a_{3D}/a_\perp = 0.2, \lambda = 10^{-3}$ and 5, 20, 100 particles (solid lines from up to down). The dashed lines show linear behavior residual of the phononic part of $S(k)$ in a homogeneous system and given by formula (5.9). We use the density in the center of the trap to estimate the sound velocity. The dash-dotted line for $5$ particles is obtained within the local density approximation for the TG-equation of state and is given by Eq. 1.154.

In Fig. 5.12 we show the results for the momentum distribution $n(k)$. On the contrary to homogeneous case, $n(k)$ in a finite system always remain finite due to natural limitations on the minimal possible value of wave vector $k_{min} \simeq 1/R_z$, where $R_z$ is the size of the cloud in the axial direction. We are looking for traces of the divergent behavior at small momenta similar to (5.8). In the case of $N=5$ and $N=20$ the rounding off of $n(k)$ at $k\sim k_{min}$ washes out completely the divergent behavior. For the largest system with $N=100$ we find some evidence of the infrared behavior (see inset in Fig. 5.12) in the region of wave vectors $1/R_z < k < 1/\xi$. The healing length is estimated by the density in the center of the trap $n_0\vert a_{1D}\vert \simeq 1.1$. We also plot a power law function with the exponent $\alpha \simeq 0.19$ which corresponds to the value in a homogeneous system with same density and the coefficient of proportionality obtained by best fit. In order to see a cleaner signature of the infrared behavior one should consider much larger systems.

Figure 5.12: Momentum distribution of a trapped system. Inset: momentum distribution for $N=100$ (solid line) on a log-log scale. The dashed line is a fit to $1/k^{1-\alpha }$ with $\alpha =0.19$. The momentum distribution is in units of $a_z=\sqrt {\hbar /(m\omega _z)}$.