Dynamics of the superfluid to Mott-insulator transition in one dimension

We numerically study the superfluid to Mott insulator transition for bosonic atoms in a one-dimensional lattice by exploiting a recently developed simulation method for strongly correlated systems. We demonstrate this method's accuracy and applicability to Bose-Hubbard model calculations by comparison with exact results for small systems. By utilizing the efficient scaling of this algorithm we then concentrate on systems of comparable size to those studied in experiments and in the presence of a magnetic trap. We investigate spatial correlations and fluctuations of the ground state as well as the nature and speed at which the superfluid component is built up when dynamically melting a Mott insulating state by ramping down the lattice potential. This is performed for slow ramping, where we find that the superfluid builds up on a time scale consistent with single-atom hopping and for rapid ramping where the buildup is much faster than can be explained by this simple mechanism. Our calculations are in remarkable agreement with the experimental results obtained by Greiner et al. [Nature (London) 415, 39 (2002)].

Figure 1

(a) The sequence of contiguous partitions of the system in which the SD are computed. The coefficients and states from these SD are then used to form the $\Gamma$ and $\lambda$ tensors. (b) A depiction of the $\Gamma$ tensors associated to lattice sites and $\lambda$ tensors associated to bonds between those sites.

Figure 2

Comparisons of the numerical (엯) and exact (∗) calculations with $U∕2J=2$ for spatial correlations $\mid {\rho }_{4,4+d}\mid$ with the central site $m=4$ obtained for (a) $\chi =3$ and (b) $\chi =5$, the standard deviation of the site occupation $\sigma \left({\rho }_{m,m}\right)$ obtained for (c) $\chi =3$ and (d) $\chi =5$, and the spectrum $e_{\gamma }$ of the one-particle density matrix obtained for (e) $\chi =3$ and (f) $\chi =5$. The dashed and dotted curves shown are to guide the eye.

Figure 3

Comparisons of the numerical ($엯$) and exact (∗) calculations, where the numerics were all performed with $\chi =3$, for spatial correlations $\mid {\rho }_{4,4+d}\mid$ with the central site $m=4$ obtained with (a) $U∕2J=6$ and (b) $U∕2J=20$, the standard deviation of the site occupation $\sigma \left({\rho }_{m,m}\right)$ obtained with (c) $U∕2J=6$ and (d) $U∕2J=20$, and the spectrum $e_{\gamma }$ of the one-particle density matrix obtained with (e) $U∕2J=6$ and (f) $U∕2J=20$. Note the differing scales and that the dashed and dotted curves are shown only to guide the eye.

Figure 4

The absolute value $\mid {\rho }_{m,n}\mid$ of the one-particle density matrix as a function of site indices $m,n$ for (a) $U∕2J=2$, (b) $U∕2J=6$, and (c) $U∕2J=20$.

Figure 5

Specific plots for the three regimes. For $U∕2J=2$ there is (a) the site occupancy $\mid {\rho }_{m,m}\mid$, (b) the standard deviation of the site occupancy $\sigma \left({\rho }_{m,m}\right)$, both with the MF calculation shown as the dashed curve, (c) the spatial correlations from the central site $m=25$, and (d) the spectrum $e_{\gamma }$ of the one-particle density matrix, showing only the 40 nonzero eigenvalues. The same plots are presented for $U∕2J=6$ in (e)–(h) and for $U∕2J=20$ in (i)–(l).

Figure 6

Slow dynamics of the small system: (a) the resulting ramping profile of the parameter $U∕2J$ with time, where the time parameters for the $V_0\left(t\right)$ profile are $t_c=60\mathrm{ms}$, $t_w=24\mathrm{ms}$, and $t_s=18\mathrm{ms}$, (b) the spectrum of the one-particle density matrix $e_{\gamma }$ with time obtained from the numerical calculation, (c) the standard deviation of the site occupancy $\sigma \left({\rho }_{m,m}\right)$ with time obtained from the numerical calculation, (d) the maximum deviation between the numerical and exact spectrum $\Delta e$ with time, and (e) the infidelity $1-F$ of the numerical many-body state compared to the exact state with time.

Figure 7

The spectrum of the one-particle density matrix $e_{\gamma }$ for the slow dynamics of the larger inhomogeneous system, showing only the largest ten eigenvalues. The dashed lines denote the two times (i) $t=t_c∕2$ and (ii) $t=t_c$ for which the absolute value $\mid {\rho }_{m,n}\mid$ of the one-particle density matrix is plotted.

Figure 8

Slow dynamics of the larger inhomogeneous system with the variation in the correlation cutoff ${\xi }_c$, measured in lattice sites, for (a) the complete slow ramping profile starting from the SF ground state and for (b) the MI ground state beginning at the bottom of the ramping. The momentum distribution width $f_w$, also measured in lattice sites, for the same situations is shown in (c) and (d), respectively. The dashed curves present in all plots are the fitted smooth “box” or “well” functions to the data.

Figure 9

A comparison between the speeds at which the correlation cutoff length ${\xi }_c$ changes in time for (i) the full slow ramping profile starting from the SF ground state and (ii) the last half of the ramping profile starting from the MI ground state at the bottom of the ramping. The time profile for the single-atom hopping speed $v_{\text{tunnel}}\left(t\right)$ is also shown. All speeds are expressed in lattice sites per ms.

Figure 10

Results for rapid dynamics, (a) the final correlation cutoff length ${\xi }_c$ obtained for different linear ramping times $t_{\text{ramp}}$, focusing on $t_{\text{ramp}}=8\mathrm{ms}$ we have (b) the spectrum of the one-particle density matrix $e_{\gamma }$ showing only the largest ten eigenvalues, and (c) the variation in ${\xi }_c$ over the simulation run, along with a fitted smooth box function shown as the dashed curve.

Figure 11

Rapid ramping of an optical lattice from the MI $\left(U∕2J=20\right)$ to the SF $\left(U∕2J=2\right)$ regime for $N=40$ atoms in $M=49$ lattice sites superimposed by a magnetic trapping potential. The width of the central interference fringe $f_w$ as a function of the ramping time $t_{\text{ramp}}$ is shown in (a). The solid curve is a fit using a double-exponential decay (${\tau }_1=0.22\mathrm{ms}$, ${\tau }_2=2.14\mathrm{ms}$) (cf. [1]). In (b) the rate of change of the correlation cutoff length ${\xi }_c$ is shown for the ramping performed with $t_{\text{ramp}}=8\mathrm{ms}$, along with the profile for the characteristic tunneling speed $v_{\text{tunnel}}\left(t\right)$ for the ramping. Both are plotted in units of lattice sites per $\mathrm{ms}$.