Correlation versus commensurability effects for finite bosonic systems in one-dimensional lattices

We investigate few-boson systems in finite one-dimensional multiwell traps covering the full interaction crossover from uncorrelated to fermionized particles. Our treatment of the ground-state properties is based on the numerically exact multiconfigurational time-dependent Hartree method. For commensurate filling, we trace the fingerprints of localization as the interaction strength increases, in several observables like reduced-density matrices, fluctuations, and momentum distribution. For a filling factor larger than 1 we observe on-site repulsion effects in the densities and fragmentation of particles beyond the validity of the Bose-Hubbard model upon approaching the Tonks-Girardeau limit. The presence of an incommensurate fraction of particles induces incomplete localization and spatial modulations of the density profiles, taking into account the finite size of the system.

Figure 1

Sketch of the finite three-well lattice and the corresponding single particle states for the first two bands.

Figure 2

(a) One-body density $\rho (x)$ for six wells and six particles. Shown are four different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.2,0.6$), and Mott insulator-fermionization limit ($g=10.0$). (b) Particle number fluctuations as $g$ increases: red, green, and blue lines correspond to the left three wells of the potential $s=1,2,3$ counting from the outermost one. (inset) On-site population as $g$ increases.

Figure 3

Two-body density ${\rho }_2(x_1,x_2)$ for six wells and six particles. Shown are four values of the interaction strength (a) $g=0.0$, (b) $g=0.2$, (c) $g=0.6$, and (d) $g=10.0$.

Figure 4

Off-diagonal one-body density matrix ${\rho }_1(x,x^{\prime })$ for six wells and six particles. Shown are four values of the interaction strength (a) $g=0.0$, (b) $g=0.2$, (c) $g=0.6$, and (d) $g=10.0$.

Figure 5

(a) Momentum distribution for six wells and six particles. Shown are five different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.05,0.2,0.6$), and Mott insulator-fermionization limit ($g=10.0$). (b) Population of the natural orbitals as a function of the interaction strength. (c) Profile of lowest natural orbital for several values of the interaction.

Figure 6

(a) $\rho (x)$ for three wells and six particles. Shown are five different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.02$, $g=0.2$), and on-site fermionization crossover ($g=5.0$, $g=20.0$). (b) Particle number fluctuations as $g$ increases for the left ($s=1$) and the middle ($s=2$) well. (inset) On-site populations.

Figure 7

${\rho }_2(x_1,x_2)$ for three wells and six particles for (a) $g=0.0$, (b) $g=0.2$, (c) $g=5.0$, and (d) $g=20.0$.

Figure 8

(a) ${\rho }_1(x,x^{\prime })$ for three wells and six particles in the fermionization limit $g=20.0$. (b) Population of the natural orbitals as a function of the interaction strength. (c) Momentum distribution: shown are four different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.02,0.2,0.5$), and on-site fermionization crossover ($g=5.0,20.0$). (d) Profile of the lowest natural orbital for several values of the interaction.

Figure 9

(a) Ground-state energies per particle as the interaction increases for commensurate filling $\nu =1$ for the cases of four particles in four wells and six particles in six wells. (b) Two first eigenstates of the spectrum of three particles in three wells as the interaction strength increases. (inset) weak interactions.

Figure 10

(a) $\rho (x)$ for five particles and seven wells. Shown are four different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.5,g=1.0$), and fermionized limit ($g=3.0$). (b) Particle number fluctuations as $g$ increases for the sites $s=1,2,3,4$. (inset) On-site population.

Figure 11

(a) ${\rho }_1(x,x^{\prime })$ for seven wells and five particles in the fermionization limit. (b) Population of the natural orbitals as a function of $g$. (c) Momentum distribution for five different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.1$), and fermionization limit ($g=3.0,20.0$).

Figure 12

(a) $\rho (x)$ for five particles and four wells. Shown are four different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.05,1.0,5.0$), and fermionization limit ($g=20.0$). (b) Particle number fluctuations as $g$ increases for the sites $s=1,2$. (inset) On-site populations.

Figure 13

(a) One-particle density matrix $\rho (x,x^{\prime })$ for five particles in four wells in the noninteracting $g=0.0$ and (b) in the fermionization limit $g=20.0$. (c) Momentum distribution. Shown are five different values of $g$: ($g=0.0,0.05,1.0,5.0,20.0$). (d) Population of the natural orbitals as a function of the interaction strength.

Figure 14

(a) One body-density for six particles and four wells. Shown are six different values of $g$: noninteracting ($g=0.0$), weakly interacting ($g=0.1,1.0$), strongly interacting ($g=10.0,30.0$), and fermionization limit ($g=10.0$). (inset) The case of a superimposed harmonic trap with strong interactions $g=30.0$ for five particles. (b) Particle number fluctuations as a function of $g$ for $s=1,2$. (inset) On-site populations. (c) Same plot for the weak interaction regime. (d) One-body density for five particles in three wells for $g=0.0,0.05,5.0,20.0$.

Figure 15

Two body-density ${\rho }_2(x_1,x_2)$ for six particles and four wells for (a) $g=0.1$, (b) $g=10.0$, (c) $g=30.0$, and (d) $g=100.0$.

Figure 16

One-body density matrix $\rho (x,x^{\prime })$ for six particles in four wells (a) $g=30.0$ and (b) $g=100$ and (c) two particles in four wells fermionization limit $g=20.0$. (d) Momentum distribution for six particles in four wells. Shown are six different values of $g$: noninteracting ($g=0.0$), weak interacting ($g=0.005,0.1,2.0$), strong interactions ($g=10.0,30.0$), and fermionization limit ($g=100.0$) which is compared with the fermionization limit of two particles in four wells.