Quantum Monte Carlo simulations of confined bosonic atoms in optical lattices

We study properties of ultracold bosonic atoms in one-, two-, and three-dimensional optical lattices by large scale quantum Monte Carlo simulations of the Bose-Hubbard model in parabolic confining potentials. Our results indicate that local properties of the atoms can be accessed by probing the system’s response to local potential perturbations. Furthermore, we show how the formation of Mott insulating regions is reflected in the momentum distribution of the atoms, amenable to experimental detection. We disprove previous claims concerning the relevance of fine structure in the momentum distribution function. Furthermore, we discuss limitations of local density approximations for confined systems, and demonstrate the absence of quantum criticality due to the inhomogenous potential. Instead, we show that quantum critical behavior can be observed in flat confining potentials. Our results indicate that the experimental detection of the Mott transition in moderately sized optical lattices would be significantly eased in flat confinement potentials.

Figure 1

Spatial dependence of the local density for two-dimensional confined atomic boson systems: (a) in the superfluid phase; (b) at stronger lattice depths a central Mott insulating region is formed. The simulations were performed for a parabolic trap with curvature $V∕U=0.002$, at $\mu ∕U=0.37$, for $U∕t=6.7$ (a) and 25 (b).

Figure 2

Spatial dependence of the local compressibility ${\kappa }^{\text{local}}$ of bosons in a two-dimensional parabolic trap with curvature $V∕U=0.002$, for $\mu ∕U=0.37$ and $U∕t=25$. A superfluid ring surrounding the $n=1$ central Mott plateau is clearly resolved.

Figure 3

Radial dependence of the density $n$ and local compressibility ${\kappa }^{\text{local}}$ of bosons in a two-dimensional parabolic trap with curvature $V∕U=0.002$, for $\mu ∕U=0.37$ and $U∕t=25$, with a superfluid ring surrounding the $n=1$ central Mott plateau. The inset shows the radial dependence of the local density fluctuations $\Delta$ and the on-site compressibility ${\kappa }^{\text{on}\text{‐}\text{site}}$ which remain finite within the Mott plateau region.

Figure 4

Spatial dependence of the correlation function $g(i,j)=⟨b_i^{†}{b}_j⟩$ between bosons at site ${\mathbf{r}}_j=(12.3,0)$ inside the superfluid shell, and all other sites $i$ within a two-dimensional parabolic trap with curvature $V∕U=0.002$, at $\mu ∕U=0.37$ and $U∕t=25$.

Figure 5

Dependence of the correlation function $g\left(d\right)=⟨b_i^{†}{b}_j⟩$ on the distance $d$ between sites $i$ and $j$ along the superfluid ring for sites within $r∕a=12.3\pm 1.2$ lattice units from the center of a two-dimensional parabolic trap with curvature $V∕U=0.002$, at $\mu ∕U=0.37$ and $U∕t=25$. For $d∕a>5$ the data fit well $({\chi }^2=1.6)$ to an exponential decay, with a finite value of $c=0.033\pm 0.003$, and correlation length $\xi ∕a=21.6\pm 0.4$. The right inset shows the correlation at the largest distance, $g\left(\pi r\right)$, as a function of the distance $r$ from the trap center. The left inset illustrates the distance $d$ between two sites $i,j$ along a ring of radius $r$ employed here.

Figure 6

Local density $n$ and local compressibility ${\kappa }^{\text{local}}$ of bosons in a two-dimensional parabolic trap with curvature $V∕U=0.002$, at $\mu ∕U=0.37$ and $U∕t=25$, as functions of ${\mu }^{\mathrm{eff}}$. The results for ${\kappa }^{\text{local}}$ from the density fluctuations, Eq. (3) (circles) agree with the numerical derivative $\partial n∕\partial {\mu }^{\mathrm{eff}}$ (triangles).

Figure 7

Density $n$ and compressibility $\kappa$ of the Bose-Hubbard model on the 2D square lattice for $U∕t=25$, as a function of $\mu$. The data are taken from simulations on a $(24\times 24)$-site square lattice.

Figure 8

Different lattice topologies and expected superfluid regions (denoted in gray) for a quadratic confining potential for such topologies: (a) square lattice, (b) polar lattice, (c) ladder lattice.

Figure 9

Local density $n$ and local compressibility ${\kappa }^{\text{local}}$ for a two-dimensional parabolic trap on a square lattice and for a ladder, as functions of ${\mu }^{\mathrm{eff}}$. The parameters for the trapping potential are $V∕U=0.002$, at $\mu ∕U=0.37$, and the parameters of the ladder model are chosen to cover the whole superfluid region (see text). The density profiles within the trap and the ladder coincide almost perfectly. Small differences are exhibited by the two different compressibility curves, which nevertheless share the same overall shape.

Figure 10

Local density $n$ for bosons confined in one-dimensional parabolic confinement potentials of different curvatures $V$, as a function of ${\mu }^{\mathrm{eff}}$. For comparison, the behavior of the density as a function of $\mu$ in the uniform 1D system is also shown. The insets focus on the regions close to $n=0$ and $n=1$, where differences between the trapped case and the uniform case are most pronounced.

Figure 11

Local density $n$ and local compressibility ${\kappa }^{\text{local}}$ for the Bose-Hubbard model on ladders with different lengths as a function of ${\mu }^{\mathrm{eff}}$. The other parameters of the ladder model are chosen as in Fig. 9. The inset shows the compressibility for the uniform 1D case as a function of $\mu$ for a chain with 64 sites, for the same value of $U∕t$ as used in the ladder model.

Figure 12

Momentum distribution function of bosons in a three-dimensional parabolic trap with curvature $V∕U=0.008$, along the line $(0,0,0)\text{‐}(\pi ∕a,0,0)$ in momentum space, for $\mu ∕U=-0.2$ and $U∕t=24$, values taken from Ref. 6. The lower inset exhibits the presence of satellite peaks in this system, without a Mott plateau, as seen from the radial density distribution shown in the upper inset.

Figure 13

Momentum distribution function of bosons in a three-dimensional parabolic trap with curvature $V∕U=0.0122$, along the line $(0,0,0)\text{‐}(\pi ∕a,0,0)$ in momentum space, for $\mu ∕U=-0.3125$ and $U∕t=80$, values taken from Ref. 6. The corresponding radial density distribution is shown in the inset.

Figure 14

Density $n$ as a function of $U∕t$ for bosons along a constant $\mu ∕U=0.37$ scan, for both periodic (PBC) and open (OBC) boundary conditions on a $(24\times 24)$-site square lattice, the OBCs corresponding to an infinitely sharp trapping potential. The Mott transition in the uniform case is indicated by the dashed line. The inset locates this constant-$\mu$ scan within the phase diagram of the uniform system.

Figure 15

Scaling plot of the stiffness ${\rho }_s$ for the Bose-Hubbard model on the square lattice for periodic boundary conditions, while tuning along a constant $\mu ∕U=0.37\mathrm{scan}$. The inset shows the stiffness ${\rho }_s$ for a $(28\times 28)$-site system as a function of $U∕t$.

Figure 16

Momentum distribution function of bosons on a $(24\times 24)$-site square lattice with periodic boundary conditions (PBC) for the commensurate momenta along the line $(0,0)\text{‐}(\pi ∕a,0)$ in momentum space, for different values of $U∕t$, while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 14. The loss of coherence due to the Mott transition at $U∕t=16.7$ is reflected by the reduced coherence peak height $n(0,0)$.

Figure 17

Evolution of the coherence fraction, i.e., the height of the coherence peak, $n(0,0)$, as a function of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 14, for bosons on a $(24\times 24)$ site square lattice for both periodic (PBC) and open (OBC) boundary conditions. For OBCs the full width at half maximum (FWHM) of the coherence peak is also shown, clearly signaling the onset of the Mott phase. The Mott transition point is indicated by the vertical dashed line.

Figure 18

Momentum distribution function of bosons on a $(24\times 24)$-site square lattice with open boundary conditions, corresponding to an infinitely sharp trapping potential, along the line $(0,0)\text{‐}(\pi ∕a,0)$ in momentum space, for different values of $U∕t$, while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 14. Within the superfluid regime satellite peaks appear, which diminish upon emergence of the Mott phase.

Figure 19

Variation of ${\mu }^{\mathrm{eff}}$ in a two-dimensional parabolic trap with curvature $V∕U=0.002$ for three different values of $U∕t$ while tuning $t$ along a constant $\mu ∕U=0.37$ scan, shown within the phase diagram of the Bose-Hubbard model on the square lattice. Along the dash-dotted line the density has a constant value of $n=1$.

Figure 20

Top: Radial density distribution of bosons within a two-dimensional parabolic trap with curvature $V∕U=0.002$ for different values of $U∕t$ for $\mu ∕U=0.37$. Bottom: Evolution of the density in the trap center while tuning $t$ along a constant $\mu ∕U=0.37$ scan. For comparison the density of the uniform system along the same constant-$\mu$ scan is shown. The arrow indicates the threshold for Mott plateau formation within the trap, which deviates by about 6% from the position of the Mott transition in the uniform case (vertical dashed line).

Figure 21

Momentum distribution function of bosons in a two-dimensional parabolic trap with curvature $V∕U=0.002$ along the line $(0,0)\text{‐}(\pi ∕a,0)$ in momentum space, for different values of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 20. Satellite peaks near $(k_x,k_y)=(0.1\pi ∕a,0)$ appear unrelated to the phase structure within the trap.

Figure 22

Evolution of the coherence fraction, i.e., the height $n(0,0)$, and of the full width at half maximum (FWHM) of the coherence peak as a function of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 20 for bosons in a two-dimensional parabolic trap with curvature $V∕U=0.002$. The threshold for Mott plateau formation is indicated by the dashed line.

Figure 23

Top: Radial density distribution of bosons within a three-dimensional parabolic trap with curvature $V∕U=0.0125$ for different values of $U∕t$ for $\mu ∕U=0.25$. Bottom: Evolution of the density in the trap center while tuning $t$ along a constant $\mu ∕U=0.25$ scan. The arrow indicates the threshold for Mott plateau formation within the trap, which deviates by about 30% from the position of the Mott transition in the uniform case within mean field theory (vertical dashed line).

Figure 24

Momentum distribution function of bosons in a three-dimensional parabolic trap with curvature $V∕U=0.0125$ along the line $(0,0,0)\text{‐}(\pi ∕a,0,0)$ in momentum space, for different values of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.25$ scan of Fig. 23. Satellite peaks near $\mathbf{k}=(0.3\pi ∕a,0,0)$ appear unrelated to the phase structure within the trap.

Figure 25

Evolution of the coherence fraction, i.e., the height $n(0,0,0)$, and of the full width at half maximum (FWHM) of the coherence peak as a function of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.25$ scan of Fig. 23 for bosons in a three-dimensional parabolic trap with curvature $V∕U=0.0125$. The threshold for Mott plateau formation is indicated by the dashed line.

Figure 26

Top: spatial density distribution of bosons within a one-dimensional parabolic trap with curvature $V∕U=0.0004$ for different values of $U∕t$, for $\mu ∕U=0.37$. Bottom: evolution of the density in the trap center while tuning $t$ along a constant $\mu ∕U=0.37$ scan. The threshold for Mott plateaus formation within the trap, and merging of the two plateaus is indicated by vertical arrows.

Figure 27

Momentum distribution function of bosons in a one-dimensional parabolic trap with curvature $V∕U=0.0004$ along the line $0\text{‐}\pi ∕a$ in momentum space, for different values of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 26.

Figure 28

Evolution of the coherence fraction, i.e., the height $n\left(0\right)$, and of the full width at half maximum (FWHM) of the coherence peak as a function of $U∕t$ while tuning $t$ along the constant $\mu ∕U=0.37$ scan of Fig. 26 for bosons in a one-dimensional parabolic trap with curvature $V∕U=0.0004$. The thresholds for Mott plateaus formation within the trap and for the collapse of the two plateaus into a single plateau are indicated by dashed lines.