Self-trapping dynamics in a two-dimensional optical lattice

We describe theoretical models for the recent experimental observation of macroscopic quantum self-trapping (MQST) in the transverse dynamics of an ultracold bosonic gas in a two-dimensional lattice. The pure mean-field model based on the solution of coupled nonlinear equations fails to reproduce the experimental observations. It greatly overestimates the initial expansion rates at short times and predicts a slower expansion rate of the cloud at longer times. It also predicts the formation of a hole surrounded by a steep square fortlike barrier which was not observed in the experiment. An improved theoretical description based on a simplified truncated Wigner approximation (TWA), which adds phase and number fluctuations in the initial conditions, pushes the theoretical results closer to the experimental observations but fails to quantitatively reproduce them. An explanation of the delayed expansion as a consequence of a type of self-trapping mechanism, where quantum correlations suppress tunneling even when there are no density gradients, is discussed and supported by numerical time-dependent density matrix renormalization group (t-DMRG) calculations performed in a simplified two coupled tubes setup.

Figure 1

Schematic of the experimental setup. A BEC of ${}^{87}$Rb atoms was initially prepared in the crossed-dipole-harmonic trap. A 2D optical lattice along the $x$-$y$ direction was then adiabatically ramped on to create an array of quasi-1D tubes. The crossed dipole trap was then turned off and the expansion dynamics in the presence of the 2D periodic potential were investigated by direct imaging. The line of sight was at 45 degrees from the lattice directions (45 degrees from $x$ and $y$). The 2D atom density distribution was recorded after various expansion times.

Figure 2

The acceleration of $x_{\mathrm{rms}}$ (${\ddot{x}}_{\mathrm{rms}}$) obtained from the mean-field calculations for all lattice depths used in the experiment: (a) 7.25$E_R$, (b) 9.25$E_R$, (c) 11$E_R$, and (d) 13$E_R$. From the sign of ${\ddot{x}}_{\mathrm{rms}}$, the evolution can be separated into three different regimes indicated by a solid green line and a dashed blue line. The three regimes are discussed in detail in the text. MQST is signaled in the $x_{\mathrm{rms}}$ as a negative acceleration. It starts at the solid green line and stops at $\sim t_c$ (indicated by the dashed blue line) when the acceleration curves cross 0 from below. The values of $U\rho \left(t_c\right)/J{a}^3$ for the four depths in consideration at $t_c$ are $\{0.4,3.0,3.0,3.0\}$. Those correspond to $\rho \left(t_c\right)/J=\{60,350,320,300\}$ $\mu {\mathrm{m}}^{-3}$$E_R^{-1}$, respectively.

Figure 3

Time evolutions of the kinetic energy along the lattice direction ($E_{kx}$, dashed orange line), the kinetic energy along $z$ ($E_{kz}$, dotted black line), and the interaction energy ($E_{\mathrm{int}}$, solid red line). Here energy is in units of $E_R$. This plot is computed using the mean-field model for a $7.25E_R$ lattice. The vertical lines are at the same positions as in Fig. 2a.

Figure 4

Expansion dynamics predicted by the mean-field calculations for the lowest lattice depth (7.25$E_R$): (a) Displays the evolution of the density profiles at $z=0$, after integrating along one transverse direction; (b) displays the same density profiles, but after integrating along a direction that is 45 degrees between the $x$ and $y$ axes (as done in the experiment); (c) shows transverse density profiles at $z=0$, during the MQST regime at $t=8.7\text{ms}$. The insets in (a) and (b) are intersecting profiles at $t=8.7\text{ms}$ marked by white lines in the 3D density plots and have the same units as in the 3D plots.

Figure 5

(a) A schematic picture of the tunneling in 2D optical lattices. In 2D, no site is self-trapped in all directions so atoms are not fully frozen. The arrows show the directions that atoms may tunnel while the crosses mean that tunneling along those directions is forbidden by MQST. (b) A density ($\rho$, at $z=0$) contour plot of a square fortlike barrier developed during the MQST regime.

Figure 6

The evolution of (a) ${\sigma }_{n^2}$ and (b) $R_z$ for $7.25E_R$ from mean-field calculations and the experiment. Each black point is the average of 10 experimental measurements. (a) The error bars represent random uncertainty, but an overall systematic uncertainty of $0.5\mu$m associated with the imaging resolution is not included. To illustrate the dependence of the mean-field dynamics on the initial width, we show the evolution of ${\sigma }_n^2$ for different initial conditions for the $7.25E_R$ lattice. Each line corresponds to a particular initial condition, subject to the constraint of matching axial expansion rates at long times. (b) Error bars are not shown, since we are mainly interested in the long-time dynamics when $R_z$ is very large and the error bars are negligible. Even with different initial conditions, the expansion rates along $z$ are almost the same, therefore in (b) all lines are almost on top of each other.

Figure 7

Time evolution of (a) ${\sigma }_{n^2}$ and (b) $R_z$ for $13E_R$ from mean-field calculations and the experiment. As in Fig. 6, the solid and dashed lines are simulation results and each corresponds to a particular initial condition. The long-time dynamics of ${\sigma }_{n^2}$ is sensitive to the initial width of the cloud also at $13E_R$. The black points are experimental data.

Figure 8

Dimensionless coupling strength at $t=0$ as a function of tube location along the $y$ axis, if the tubes are approximated to be independent. Here $V_0=13E_R$. The approximation is never fully warranted in the regime of this work, where the tubes are manifestly not independent. Still, we suspect that this $\gamma$ gives a sense of the initial axial correlations in the relatively deep lattices.

Figure 9

Transverse density profiles at $z=0$ during the MQST regime at $t=8.7$ ms for the $13E_R$ lattice obtained from the aTWA results.

Figure 10

Mean-field, aTWA, and experimental results for the evolution of ${\sigma }_n^2$ for all lattice depths used in the experiment. The lattice depths are (a) 7.25$E_R$, (b) 9.25$E_R$, (c) 11$E_R$, and (d) 13$E_R$. At low lattice depth, a small $\eta$ seems to account for the observed behavior at short times. The dashed blue line is the result of a mean-field calculation of the dynamics with no random phase between the tubes ($\eta =0$). The dotted purple line is for $\eta =0.2$, the dashed-dotted green lines are for (a) $\eta =0.4$ and (b) $\eta =0.5$, and the solid orange line is for $\eta =1$. The dashed vertical red lines indicate the transition from MQST to BE predicted by the aTWA. The yellow shadow regions indicate the $t_c$ inferred from the experimental data.

Figure 11

Evolution of $R_z$ for all lattice depths used in the experiment. The dots show experimental data and the solid and dashed lines show the predictions obtained from the aTWA for different $\eta$. The lattice depths are (a) 7.25$E_R$, (b) 9.25$E_R$, (c) 11$E_R$, and (d) 13$E_R$. $R_z$ is obtained by fitting the density along $z$ to a Thomas-Fermi profile. The theoretical curves use the same parameters as the ones shown in Fig. 10.

Figure 12

${\ddot{x}}_{\mathrm{rms}}$ calculated using the aTWA for different lattice depths and for the corresponding optimal $\eta$, (a) $7.25E_R,\eta =0.4$ and (b) $13E_R,\eta =1$. The boundaries between MQST and BE are indicated by the dashed blue lines. In (a), the boundary between EXP and MQST is indicated by the solid vertical green line. In (b), because ${\ddot{x}}_{\mathrm{rms}}$ is almost 0 at all times, it can be interpreted as the self-trapping starts at $t=0$ ms.

Figure 13

Transverse density profiles. The solid and dashed lines are obtained from the aTWA results for the $V=7.25E_R$ lattice and for $\eta =0.4$. The profiles are taken at $z=0$ and viewed transversely at an angle ${45}^{\circ }$ from the lattice axis and normalized to 1. The solid blue lines indicate the theoretical profiles before convolution and the dashed black lines after convolution. The red points are averaged experimental data of 10 measurements.

Figure 14

Transverse density profiles. The solid and dashed lines are obtained from the aTWA results for the $V=13E_R$ lattice and for $\eta =1$. The profiles are taken at $z=0$ and viewed transversely at an angle ${45}^{\circ }$ from the lattice axis and normalized to 1. The solid blue lines indicate the theoretical profiles before convolution and the dashed black lines after convolution. The red points are averaged experimental data of 10 measurements.

Figure 15

Time evolution of ${\sigma }_{n^2}$ computed for a $7.25E_R$ lattice and for $\eta =0$. The dashed red line shows the result when number fluctuations are neglected, and the solid blue line shows the result when number fluctuations ($\approx 1/\sqrt{N_{mn}}$) are included. The plot clearly shows that number fluctuations suppress the expansion, but only by a small amount.

Figure 16

aTWA results for the time evolution of ${\sigma }_{n^2}$ for a 7.25$E_R$ lattice and for $\eta =0.35$. The dashed red line shows the result when only phase fluctuations are included, and the solid blue line shows the result when both number fluctuations ($\approx 1/\sqrt{N_{mn}}$) and phase fluctuations are included. The plot shows that number fluctuations slightly suppress the initial expansion rate.

Figure 17

Schematic configuration of the two-tube system.

Figure 18

$\frac{{\ddot{N}}_1}{N_{\mathrm{total}}}{|}_{t=0}$ (in units of $J_{\parallel }^2$) as a function of $\frac{N_1(t=0)}{N_{\mathrm{total}}},J=0.05,{UN}_{\mathrm{total}}=30$. The solid black line shows the GPE limit and the dash-dotted gray line the hard-core boson limit [see Eq. (12)]. The red squares, magenta up-triangles, blue diamonds, and orange disks show the initial curvature $\frac{{\ddot{N}}_1}{N_{\mathrm{total}}}{|}_{t=0}$ extracted from the t-DMRG results at $U=1$, $U=2$, $U=3$, and $U=5$, respectively. The dashed black lines show the aTWA solutions using $\eta$ as a fitting parameter. The values of $\eta$ are listed in Table . This plot is computed using the “case 2” parameters shown in Table .

Figure 19

Evolutions of ${UN}_1$ (in units of $J_{\parallel }$) and $N_1$ as functions of time (in units of $1/J_{\parallel }$) calculated using the DMRG and GPE methods. In panel (a) and (b), the values of $U$ are 1 and 5, respectively, while ${UN}_{\mathrm{total}}$ is fixed to 30. In panel (c) and (d), the values of $U$ are 2 and 5, respectively, while $N_{\mathrm{total}}$ is fixed to 30. In each panel, the solid line is the t-DMRG solution; the dashed black line is the mean-field calculation with no random phase (GPE). The dotted gray line is for (a) $\eta =0.3$, (b) $\eta =0.7$, (c) $\eta =0.4$, and (d) $\eta =0.53$.

Figure 20

$g_l^{\left(2\right)}$ as function of time in different regimes. The correlations are extracted from the t-DMRG results for the parameters describing case 2. Panel (a) shows $g_1^{\left(2\right)}$, $\frac{L}{N_1^2}{\sum }_i\langle {\widehat{n}}_{1,i}^2\rangle -\langle n_{1,i}\rangle$ and panel (b) shows $g_2^{\left(2\right)}$, $\frac{L}{N_2^2}{\sum }_i\langle {\widehat{n}}_{2,i}^2\rangle -\langle n_{2,i}\rangle$. In each panel, from top to bottom, the lines correspond to $U=1$, $U=2$, $U=3$, and $U=5$, respectively.