Excitations of few-boson systems in one-dimensional harmonic and double wells

We examine the lowest excitations of one-dimensional few-boson systems trapped in double wells of variable barrier height. Based on a numerically exact multiconfigurational method, we follow the whole pathway from the noninteracting to the fermionization limit. It is shown how, in a purely harmonic trap, the initially equidistant, degenerate levels are split up due to interactions, but merge again for strong enough coupling. In a double well, the low-lying spectrum is largely rearranged in the course of fermionization, exhibiting level adhesion and (anti)crossings. The evolution of the underlying states is explained in analogy to the ground-state behavior. Our discussion is complemented by illuminating the crossover from a single to a double well.

Figure 1

Lowest energies $E_m$ in a harmonic trap $(h=0)$ for $N=3$, 4, and 5 bosons. (The lines connect the data points to guide the eye. Please note the different abscissa scale for $N=5$, where convergence for $g_{1\mathrm{D}}>15$ becomes overly expensive.)

Figure 2

(Color online) Single-particle spectrum $\left\{{ϵ}_a\right\}$ in a double well with barrier height $h=5$.

Figure 3

Lowest energies $E_m$ in a double well $(h=5)$ for $N=3$, 4, and 5 bosons. Inset $(N=4)$: Level adhesion for the states $m=3$ and 4 (counted from below at $g=0$).

Figure 4

Density profiles of $N=4$ bosons in a harmonic trap $(h=0)$ for the excited states $m=1$, 2, and 3 (from top to bottom).

Figure 5

Two-particle density ${\rho }_2(x_1,x_2)$ for $N=4$ bosons in a harmonic trap. Top row: Excited state $m=1,\dots$, bottom row: $m=3$, shown are the interaction strengths $g=0.2$, 2.2, and 15 from left to right.

Figure 6

Density profiles of $N=4$ bosons in a double well $(h=5)$ for the lowest excited states $m=1,\dots ,4$ (from top to bottom).

Figure 7

Two-particle density ${\rho }_2(x_1,x_2)$ for $N=4$ bosons in a double-well trap $(h=5)$. Left: Excited state $m=3$; right: $m=4$; shown are the interaction strengths $g=0.2$, 1.3, and 25 from top to bottom. The densities for both states are practically indistinguishable for mediate $g\sim 1$.

Figure 8

Natural populations $n_a\left(g\right)$ ($a=0,\dots ,3$ from top) for the excited states $m=3$ (—) and $m=4$ $(\cdots )$ of $N=4$ bosons in a double well with barrier height $h=5$. Both are practically equal in energy and densities for mediate $g$.

Figure 9

Evolution of the natural orbitals ${\varphi }_a$ as $h\to \infty$ for the case $N=3$ $(g=0.2)$. Top: The first symmetric orbital ${\varphi }_0$ is notched at $x=0$. Bottom: The antisymmetric one $\left({\varphi }_1\right)$ is barely altered.

Figure 10

Crossover of the lowest energies $E_m\left(h\right)$ with varying barrier strength $h$ for $N=3$ bosons at interaction strengths (a) $g=0.2$ and (b) $g=15$. The line styles are assigned so as to distinguish the different level groups at $h=0$.