Phase-induced transport in atomic gases: From superfluid to Mott insulator

Recent experimental realizations of artificial gauge fields for cold atoms are promising for generating steady states carrying a mass current in strongly correlated systems, such as the Bose-Hubbard model. Moreover, a homogeneous condensate confined by hard-wall potentials from laser sheets has been demonstrated, which provides opportunities for probing the intrinsic transport properties of isolated quantum systems. Using the time-dependent density matrix renormalization group, we analyze the effect of the lattice and interaction strength on the current generated by a quench in the artificial vector potential when the density varies from low values (continuum limit) up to integer filling in the Mott-insulator regime. There is no observable mass current deep in the Mott-insulator state as one may expect. Other observable quantities used to characterize the quasisteady state in the bulk of the system are the Drude weight and entanglement entropy production rate. The latter in particular provides a striking signature of the superfluid–Mott-insulator transition. Furthermore, an interesting property of the superfluid state is the formation of shock and rarefaction waves at the boundaries due to the hard-wall confining potentials. We provide results for the height and the speed of the shock front that propagates from the boundary toward the center of the lattice. Our results should be verifiable with current experimental capabilities.

Figure 1

(a) Schematic representation of the Bose-Hubbard Hamiltonian with a complex time-dependent hopping term $J{e}^{i\varphi \left(t\right)}{\widehat{b}}_i^{†}{\widehat{b}}_{i+1}+\mathrm{H}.\mathrm{c}.$ and interaction term $U{\widehat{n}}_i({\widehat{n}}_i-1)/2$. The system is confined by hard-wall (h.w.) boundaries at the edges of the lattice [43, 44, 45]. (b) Density profiles observed after the quench of the hopping phase $\varphi (t<0)=0\to \varphi (t\ge 0)=0.05$ for $U/J=0.4$. The light-gray profile corresponds to $t=0$ ($t_0=\hslash /J$), the gray to $t=200t_0$, and the black to $t=400t_0$. The bulk current $j=\overline{n}v$ is driven to the left of the system by the phase quench ($\overline{n}$ is the bulk density in the initial state and $v$ is the particle velocity). A shock wave with height $\Delta n_{\mathrm{step}}$ and front speed $v_{\mathrm{shock}}$ forms at the left boundary, while a rarefaction wave with the same height forms at the right boundary.

Figure 2

(a)–(c) Current $j\left(t\right)$ in the middle of the chain [Eq. (4)] as a function of time $t$ for the fillings (a) $n=0.1$, (b) $n=0.5$, and (c) $n=1$, and selected values of the Hubbard interaction $U/J$ after quenching the hopping phase to the value ${\varphi }_0=0.05$ (small phase quench). The blue and red dotted lines are the results for free bosons (f.b., $U=0$) and hardcore bosons, equivalent to free fermions (f.f., $U=+\infty$), respectively. (d)–(f) Same as the above panels after a quench of the hopping phase to the value ${\varphi }_0=0.5$ (large phase quench). The fillings are the same as in the panels immediately above. Note the nonstationary character of the current induced by the quench in the case of free bosons.

Figure 3

The quantity $D/\left(J\overline{n}\right)$ as a function of the density-rescaled Hubbard interaction $\gamma =U/\left(2J\overline{n}\right)$ (or Lieb-Liniger parameter; see Sec. 3). The different symbols correspond to different fillings. The black lines are TDMRG results and the gray lines refer to the time-dependent Gutzwiller ansatz [58]. The two vertical dotted lines are the exact ($U_{c,\mathrm{exact}}/J=3.4$) [68] and mean-field ($U_{c,\mathrm{mean}\mathrm{field}}/J=11.7$) [31] values of the critical interaction strength. The horizontal dashed lines are the asymptotic limits in the corresponding hard-core case ($U/J=+\infty$). These data have been extracted from the results for $j\left(t\right)$ shown in Fig. 2 according to Eq. (7). The value of the velocity in Eq. (7) is taken at time $t^{*}=10t_0$ if the data are available. Otherwise, $t^{*}$ is the maximum time reached in each simulation. The upper panel refers to the small phase quench ${\varphi }_0=0.05$, while the lower panel refers to the large phase quench ${\varphi }_0=0.5$.

Figure 4

The symbols shown are the Drude weight extracted from our TDMRG simulations of a phase quench in the XXZ model (2) at half filling for values of the anisotropy parameter in the range $-1<\Delta <1$. Equation (7) is used to relate the current obtained from the simulations to the Drude weight. The blue curve is the result of the exact Bethe ansatz solution of the XXZ model [69] [Eq. (8)]. The upper panel refers to a phase quench with ${\varphi }_0=0.025$ while the lower panel refers to ${\varphi }_0=0.05$. Different symbols refer to the time $t^{*}$ at which the Drude weight is extracted. Note that for $\Delta >0$, the measured value of $D$ depends both on $t^{*}$ and ${\varphi }_0$, while this is not the case for $\Delta <0$ where a well-defined value is obtained which coincides with the Bethe ansatz result.

Figure 5

In the upper panels, we show the entropy as a function of time $S_{L/2}\left(t\right)$ relative to the bipartition of the system in two halves. The plots from top to bottom correspond to increasing interaction strength $[S_{L/2}\left(0\right)$ decreases with increasing $U/J]$. On the left, results for small phase quench ${\varphi }_0=0.05$ are shown, and, on the right, for the large phase quench ${\varphi }_0=0.5$. In the lower plots, the corresponding values of the entropy production rate $d{S}_{L/2}\left(t\right)/dt$ (in units of $t_0^{-1}$) at time $t=10t_0$ (left) and $t=1.5t_0$ (right) are reported as a function of the interaction strength. The filled circles correspond to the values of $U/J$ shown in the upper plots. The vertical dotted lines indicate the critical value of the interaction strength, $U_{c,\mathrm{exact}}/J=3.4$. All of the data refer to filling $n=1$.

Figure 6

(a)–(c) Density step height $\Delta n_{\mathrm{step}}$ as a function of the interaction strength at fillings (a) $n=0.1$, (b) $0.5$, and (c) 1 extracted from the time-dependent Gutzwiller ansatz (gray dots) and TDMRG (black dots) simulations in the case of a small phase quench ${\varphi }_0=0.05$. The step height for increasing interaction tends asymptotically to the value for hard-core bosons (horizontal dashed line). In Fig. 5, the two vertical dotted lines are the exact ($U_{c,\mathrm{exact}}/J=3.4$) [68] and mean-field ($U_{c,\mathrm{mean}\mathrm{field}}/J=11.7$) [31] values of the critical interaction strength. Insets: A snapshot of the density profile $n_i=\langle {\widehat{n}}_i\rangle$ obtained from TDMRG at $t=0$ (dashed line) and at a later time [solid line, (a) $t=266t_0$, (b) $40t_0$, (c) $13t_0]$ for the selected values of the interaction strength. The dynamics in a Mott insulator is substantially different without well-defined shock and rarefaction waves [compare inset of panel (c) for $U/J=4.0$ and $n=1$ to the other insets and Fig. 1].

Figure 7

The shock wave propagation speed $v_{\mathrm{shock}}=\overline{n}v/\Delta n_{\mathrm{step}}$ (triangles) for different initial fillings ($n=0.1,0.25,0.5,0.75$, from bottom to top) compared to the sound speed in a weakly interacting Bose gas $v_{\mathrm{sound}}{t}_0=2\overline{n}\sqrt{\gamma }$ (circles) as a function of the density-rescaled interaction strength $\gamma =U/\left(2J\overline{n}\right)$.