Boson pairs in a one-dimensional split trap

We describe the properties of a pair of ultracold bosonic atoms in a one-dimensional harmonic trapping potential with a tunable zero-ranged barrier at the trap center. The full characterization of the ground state is done by calculating the reduced single-particle density, the momentum distribution, and the two-particle entanglement. We derive several analytical expressions in the limit of infinite repulsion (Tonks-Girardeau limit) and extend the treatment to finite interparticle interactions by numerical solution. As pair interactions in double wells form a fundamental building block for many-body systems in periodic potentials, our results have implications for a wide range of problems.

Figure 1

(Color online) Ground-state wave function for a boson pair in a harmonic trap with a $\delta$ barrier along $x_1=x_2=0$ of strength (a) $\kappa =0$, (b) $\kappa =1$, (c) $\kappa =2$, and (d) $\kappa =\infty$. The corresponding scaled ground-state energies are $E_0=2.0$, 2.4, 2.6, and 3.0, respectively. In each plot the horizontal and vertical axes run from $-6$ to $+6$, in scaled units.

Figure 2

(Color online) Reduced single-particle density matrix for the Tonks molecule for barrier strength (a) $\kappa =0$, (b) $\kappa =1$, (c) $\kappa =2$, and (d) $\kappa =\infty$. Each plot spans the range $-6<x$, $x^{\prime }<6$.

Figure 3

(Color online) Momentum distribution $n\left(k\right)$ for $\kappa =0$, 1, 5 and 10, with the corresponding normalized ground-state energies $E_0=2$, 2.4, 2.8, and 2.9. The momentum distributions broaden for increased barrier strength due to the localization of the particles in separate halves of the trap. Also shown is the momentum distribution for the $\kappa =\infty$ case, given by Eq. (16), for which $E_0=3.0$.

Figure 4

(Color online) von neumann entropy $S$ for the Tonks molecule as a function of the barrier strength $\kappa$. When $\kappa =0$, then $S\approx 0.985$. As the barrier is strengthened the entropy increases to a maximum at $\kappa \approx 3.4$. In the limit of $\kappa \to \infty$, then $S\to 1$, corresponding to a nonentangled state. The bar charts show the values of the Schmidt numbers at the point where ($\kappa =1.33$, $S=1$) and for the limit ($\kappa =\infty$, $S=1$).

Figure 5

(Color online) RSPD $\rho (x,x^{\prime })$ as a function of interaction strength $g_{1D}=0$, 1, 5, and 500 and barrier strength $\kappa =0$, 1, 2, and 10. Each individual plot spans the range $-6.4<x$, $x^{\prime }<6.4$.

Figure 6

(Color online) Quantity $Q\equiv \int \rho (x-x)dx$, as a function of barrier strength $\kappa$. Three intermediate values of $g_{1D}$ are considered: 1 (solid line, black), 2 (dashed line, red), and 5 (dotted line, green). For sufficiently high $g_{1D}$ and $\kappa$ one is considering the insulator regime for which $Q$ will effectively vanish.

Figure 7

(Color online) Momentum distributions for varying interparticle interaction strength $g_{1D}=0$ (a), 1 (b), 5 (c), and 500 (d). Within each plot the distribution is considered for different values of the barrier strength: $\kappa =0$ (solid line, black), 1 (dashed line, red), 5 (dash-dotted line, green), and 10 (dotted line, blue). These calculations have been carried out by means of DVR discretization of the spatial coordinates $x_1$ and $x_2$, with $N=61$ DVR mesh points in each dimension and a scale factor of $\Delta x=0.16$.

Figure 8

(Color online) Effect of varying interaction strength $\left(g_{1D}\right)$ on the von Neumann entropy $\left(S\right)$ of the ground state. The strength of the $\delta$ barrier is taken to be $\kappa =0$ (thick solid line), 1 (dashed line), 2 (dash-dotted line), 5 (dash-double-dotted line), and 10 (dotted line). It is seen that for increased strength of the central barrier, the entropy shows an increased sensitivity to the interaction parameter about the value $g_{1D}=0$.

Figure 9

(Color online) Effect of varying barrier strength on the von Neumann entropy $S$ of the ground state. The strength of the interparticle interaction is set to $g_{1D}=1$ (solid line), 2 (dashed line), 5 (dash-dotted line), and 500 (dotted line). The initial value of the entropy (i.e., in the absence of any barrier) is dictated by the strength of the interparticle interaction, with larger interaction leading to increased entropy. One observes that in all cases, in the limit of a strong barrier, the von Neumann entropy saturates at a value $S=1$.